Biophysical studies of morphogen gradient formation in Drosophila

Biophysical studies of morphogen gradient formation in Drosophila

Biophysical studies of
morphogen gradient formation
in Drosophila melanogaster
Jeffrey Alfred Drocco
A THESIS
PRESENTED TO THE FACULTY
OF PRINCETON UNIVERSITY
IN CANDIDACY FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
RECOMMENDED FOR ACCEPTANCE
BY THE DEPARTMENT OF PHYSICS
Advisor: David W. Tank
September 2011
c Copyright by Jeffrey Alfred Drocco, 2011.
All rights reserved.
Abstract
Gradients of chemical substances called morphogens underlie the differentiation of naive
cells into the various tissue types of a mature organism. By measuring the concentration
of a given morphogen in their local area, cells obtain information about their position relative to some axis of the developing embryo, and control the expression of distinct genetic
elements accordingly.
An important question is how organisms establish these gradients early in life, with
sufficient precision and reproducibility to accurately execute the developmental program.
It has long been thought that morphogens were produced at various extrema of the embryo during development, and that long-range gradients were formed by diffusion of the
morphogen away from the source. On this model, gradient length is determined by competition between the rates of diffusion and degradation of the morphogen. However, this idea
has recently come into controversy, with some suggesting that the diffusion-degradation
model does not explain gradient shape, and that gradients must be largely preformed prior
to fertilization.
In this thesis, we study the formation of the gradient of Bicoid, a morphogen that controls anterior-posterior axis specification in the fruit fly Drosophila melanogaster. We use
a fusion of Bicoid and the photoswitchable fluorescent protein Dronpa to measure and optically modulate the degradation rate of the morphogen. We find that decreasing the lifetime
of the morphogen shortens the spatial extent of the gradient, providing support for the
diffusion-degradation model of gradient formation.
i
Additionally, we present experimental results suggesting that previous measurements
underestimated the Bicoid diffusion coefficient, as a result of incorrect assumptions of
isotropic diffusion in the embryo cortex. We extend our measurements to examine Bicoid gradient formation in unfertilized eggs, finding that the diffusion-degradation model
works equally well in this system, and that the only parameter needing modification is the
morphogen degradation rate.
ii
Acknowledgments
One of the benefits of completing a dissertation in biophysics is the opportunity to work
with scientists from a wide variety of backgrounds, and I have had many such opportunities
during my time in Princeton. For this I have to thank most of all my advisor David Tank.
It has been a privilege to learn from him what doing science is all about and I am grateful
for his continued support. He let a young and very green graduate student join his lab,
and pointed me in the direction of Drosophila morphogenesis, which would eventually
become the topic of my thesis. To him and the members past and present of the Tank
lab–Peter, Anton, Zhao, Amina, Dave, Tom, Guy, Kelly, Andrew, Dmitriy, Chris, Emily,
Ryan, Cristina, Forrest, Emre, Dan, Jeff, and Ben–I owe all of the skills I have learned as
an experimental physicist as well as many fun and enjoyable experiences both in and out
of the lab. I am indebted to everyone but I would like to thank especially Peter Rickgauer,
who helped edit a large part of this thesis.
Eric Wieschaus has truly been a second advisor to me, and it was my good fortune that
he was willing to risk letting another physicist cause trouble in his fly lab and provide humor
for his group meetings. I have benefited much from his biological intuition and scientific
insight, not to mention all of the personal lessons in fly genetics. In Eric’s lab I also
met my closest collaborator during graduate school, Oliver Grimm, whose knowledge of
developmental biology has been the antidote to my ignorance in too many conversations to
count, and whose genetic handiwork makes appearances throughout this thesis. I would like
to thank Xuemin Lu for the opportunity to contribute to an interesting project on collective
iii
behavior in syncytial nuclei. Shawn Little and Stefano di Talia have also been helpful
points of reference and sources of advice.
When I first began work in biophysics, it was Thomas Gregor who introduced me to the
field as well as to all the nitty gritty aspects of doing experiments, and he has continued to
be a mentor to me even now as a professor. Bill Bialek deserves credit for challenging us
all to come to a deeper understanding of the physical basis of morphogenesis, and I thank
him for his inspiration as well as a number of helpful discussions. I will miss chalk talks
and problem solving sessions with my fellow graduate students in biophysics, especially
Sidhartha Goyal, Julien Dubuis, Dima Krotov, and Anand Murugan.
My years at Princeton have been memorable thanks to the friendship of many others,
among them: Adam Hincks, Matt Buican, Ryan Fisher, Scott McIsaac, Emily Snow, and
John Doherty. Special thanks to Scott McIsaac and Xin Wang for their hospitality during
the last few weeks of my dissertation. Jitendra Kanodia, Denise Pauler, Oliver Grimm,
and Julien Dubuis provided useful comments on drafts of this work. Cynthia and Charles
Reichhardt have been both occasional collaborators and constant mentors throughout my
graduate school career, and I look forward to working with them again in the near future.
I was supported financially during my early years in graduate school by the Department
of Energy Computational Science Graduate Fellowship and the excellent staff of the Krell
Institute. Finally, for all their love and support, I thank my parents Carol and Dan and my
brother Brian.
iv
Contents
1
Introduction
1
2
Measurement and perturbation of morphogen lifetime
11
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3
2.2.1
Measurement of Bcd lifetime by Dronpa-Bcd photoconversion . . . 15
2.2.2
Correction of Bcd lifetime measurement for core-to-cortex flux . . 18
2.2.3
The Bcd gradient does not reach steady state prior to interpretation . 21
2.2.4
Optically mimicked degradation shortens the Bcd gradient . . . . . 23
2.2.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1
Table of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2
Synthesis, characterization, and preparation of samples . . . . . . . 29
2.3.3
Description of Dronpa photophysics . . . . . . . . . . . . . . . . . 30
2.3.4
Description of optical methods . . . . . . . . . . . . . . . . . . . . 30
2.3.5
Measurement of spatially integrated bright-state fluorescence . . . . 31
2.3.6
Determination of irreversible photoconversion rate . . . . . . . . . 36
2.3.7
Isolation of photoconverted state . . . . . . . . . . . . . . . . . . . 38
2.3.8
Uniformity of photoconversion . . . . . . . . . . . . . . . . . . . . 40
2.3.9
Relation of total Bcd quantities to measured fluorescence values . . 42
v
2.3.10 Description of degradation measurement . . . . . . . . . . . . . . 44
2.3.11 Correction for Dronpa-Bcd flux . . . . . . . . . . . . . . . . . . . 45
2.3.12 Estimation of Dronpa-Bcd flux correction . . . . . . . . . . . . . . 47
2.3.13 Establishment of upper bound on Dronpa-Bcd flux . . . . . . . . . 49
2.3.14 Removal of cell-cycle-periodic oscillation . . . . . . . . . . . . . . 50
2.3.15 Measurement of total Bcd protein by Western blot . . . . . . . . . 51
2.3.16 Measurement of gradient shift under optically mimicked degradation 52
2.3.17 Calculation of Bcd synthesis rate from Western analysis . . . . . . 55
2.3.18 Simulation of gradient shape . . . . . . . . . . . . . . . . . . . . . 57
2.3.19 Measurement of newly maturing Dronpa-Bcd . . . . . . . . . . . . 58
2.3.20 Measurement of Bcd lifetime is insensitive to position along the
anterior-posterior axis . . . . . . . . . . . . . . . . . . . . . . . . 61
3
In vivo measurements of diffusion during embryogenesis
63
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1
Theoretical considerations relating to FRAP in presence of binding
sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.2
Bcd FRAP recovery times are anomalous relative to bleaching spot
width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.3
FLIP demonstrates that exchange of Bcd between neighboring nuclei is minimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.4
Effect of boundary conditions on FRAP recovery times . . . . . . . 74
3.2.5
FRAP recovery is faster at locations more basal in the embryo . . . 78
3.2.6
Can recent FCS measurements of Bcd diffusivity be reconciled
with gradient shape? . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
vi
3.3
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.1
Synthesis and preparation of samples . . . . . . . . . . . . . . . . 82
3.3.2
Description of optical methods . . . . . . . . . . . . . . . . . . . . 83
3.3.3
Nuclear identification and motion correction of image series . . . . 84
3.3.4
Measurement of corona effect in FRAP experiments . . . . . . . . 84
3.3.5
Determination of FLIP bleaching spot width . . . . . . . . . . . . . 86
3.3.6
Diffusion is substantially faster within nucleus-associated compartments than between compartments . . . . . . . . . . . . . . . . . . 88
3.3.7
4
Determination of confocal bleaching spot depth . . . . . . . . . . . 88
Bicoid gradient formation in fertilized and unfertilized eggs
91
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.1
Bcd lifetime in unfertilized eggs is not down-regulated within several hours of oviposition . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.2
High-contrast imaging confirms greater spatial extent of unfertilized Bcd gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.3
Two-photon imaging shows time evolution of unfertilized Bcd gradient
4.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.4
Simulation of unfertilized gradient dynamics . . . . . . . . . . . . 98
4.2.5
Measurement of total Bcd quantity in fertilized and unfertilized eggs 101
4.2.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3.1
Preparation of fixed samples . . . . . . . . . . . . . . . . . . . . . 106
4.3.2
Optical lock-in detection imaging of Dronpa-Bcd . . . . . . . . . . 106
4.3.3
Computational identification of nuclear mask . . . . . . . . . . . . 110
4.3.4
Preparation of samples for live imaging . . . . . . . . . . . . . . . 111
vii
4.3.5
Two-photon laser scanning microscopy . . . . . . . . . . . . . . . 111
4.3.6
Two-photon image analysis and gradient computation . . . . . . . . 112
4.3.7
Calibration of two-photon fluorescence intensity to absolute eGFP
concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A Choice of embryo parametrization for Bcd gradient quantification
117
B Diffusion in driven, athermal systems
122
B.1 Method and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.2 Nonthermal Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.3 Application of the Fluctuation Theorem . . . . . . . . . . . . . . . . . . . 127
B.4 Nonequilibrium Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 132
B.5 Large Pinning Trap Regimes and Negative Events . . . . . . . . . . . . . . 133
B.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
viii
List of Figures
1.1
French flag model of tissue specification . . . . . . . . . . . . . . . . . .
3
1.2
The Bicoid gradient in Drosophila melanogaster . . . . . . . . . . . . .
5
1.3
Downstream targets of bicoid . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1
Description of Dronpa-Bcd construct and degradation measurement . . 16
2.2
Measurement of Bcd degradation and correction for flux . . . . . . . . 19
2.3
Measurement of total Bcd quantity . . . . . . . . . . . . . . . . . . . . . 22
2.4
Optical augmentation of Dronpa-Bcd degradation . . . . . . . . . . . . 24
2.5
Determination of rates of photoconversion . . . . . . . . . . . . . . . . 32
2.6
Determination of PMT offset value . . . . . . . . . . . . . . . . . . . . . 33
2.7
Example of cortical mask selection for degradation rate measurement . 35
2.8
Description of photoconversion quantity calculation . . . . . . . . . . . 37
2.9
Quantification of irreversible photobleaching and dark state isolation . 39
2.10 Transparency of embryo to 496 nm and 405 nm photoconversion . . . . 41
2.11 Illustration of Bcd flux quantification from embryonic cross-sections . . 48
2.12 Establishment of upper bound on kdeg . . . . . . . . . . . . . . . . . . . 50
2.13 Calibration of blot intensity to molecule count . . . . . . . . . . . . . . 52
2.14 Quantification of mimicked Dronpa-Bcd degradation induced by photoconversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.15 Identification of nuclear mask for Bcd gradient computation . . . . . . 54
2.16 Calculation of Bcd synthesis rate . . . . . . . . . . . . . . . . . . . . . . 56
ix
2.17 Measurement of newly matured Dronpa-Bcd and agreement with simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.18 Measurement of kdeg dependence on A-P axis position . . . . . . . . . . 62
3.1
Bcd FRAP recovery curves for bleaching spots of variable width . . . . 71
3.2
Intensity profile of adjacent nuclei and cytoplasm during FLIP . . . . . 73
3.3
FRAP recovery times for various geometries . . . . . . . . . . . . . . . 77
3.4
Measurement of Bcd FRAP recovery at various depths . . . . . . . . . 79
3.5
Determination of FRAP bleaching spot widths . . . . . . . . . . . . . . 85
3.6
Determination of FLIP bleaching spot width . . . . . . . . . . . . . . . 87
3.7
Time evolution of fluorescence intensity profile during FLIP . . . . . . 89
3.8
Determination of axial bleaching spot depth . . . . . . . . . . . . . . . . 90
4.1
Measurement of Bcd lifetime in unfertilized eggs . . . . . . . . . . . . . 94
4.2
Measurement of Bcd gradient in fertilized and unfertilized eggs by optical lock-in imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3
Evolution of unfertilized and fertilized Bcd gradients in time . . . . . . 99
4.4
Comparison of fertilized and unfertilized Bcd gradients with simulation 102
4.5
Calculation of total number of Bcd molecules in embryos and unfertilized eggs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6
Detection of Dronpa-Bcd by correlation image . . . . . . . . . . . . . . 108
4.7
Determination of two-photon point spread function . . . . . . . . . . . 112
4.8
Verification of uniform radial distribution of Bcd in unfertilized eggs
and illustration of axes for gradient computation . . . . . . . . . . . . . 114
4.9
Imaging of purified eGFP under two-photon illumination . . . . . . . . 116
A.1 Illustration of different choices of metric along the Drosophila A-P axis 119
A.2 Test of different metrics applied to the Bcd gradient . . . . . . . . . . . 121
x
B.1 Determination of noise properties . . . . . . . . . . . . . . . . . . . . . 126
B.2 Illustration of particle trajectories . . . . . . . . . . . . . . . . . . . . . 127
B.3 Demonstration of FT in a nonthermal system driven over quenched
disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.4 Limits of regime in which FT is verified . . . . . . . . . . . . . . . . . . 130
B.5 Nonequilibrium temperature and scaling with applied drive . . . . . . 132
B.6 Scaling of time required to reach an asymptotic value of βτ with trap size133
xi
Chapter 1
Introduction
All complex organisms develop from a single cell, the egg, which after fertilization divides
into a group of cells, each carrying the same genetic information. Eventually the fates of
the various cells diverge, in a process called cell differentiation. Importantly, however, cells
do not choose randomly from among a variety of possible fates; rather, they are patterned
into distinct tissue types by reproducible mechanisms. This process, in which cells and
tissues are organized in a way that gives the organism its characteristic shape and function,
is called morphogenesis [1, 2]. A physicist might prefer to classify it in the somewhat
broader category of pattern formation, a phenomenon which occurs in both the biotic and
abiotic spheres of the natural world [3].
Pattern formation in biology encompasses a diverse range of phenomena which cannot
easily be grouped together. In some cases, the particular features of a pattern need not
be reproducible at all; such is the case with the ridges and whorls that form a fingerprint,
for example. In other instances, however, the precise location of a feature is crucial to
the function of an organism, such as the positioning of a head or a wing in relation to the
body plan of an animal. In these cases, the reproducibility and precision of a patterning
mechanism are of paramount importance during development.
The model system we consider in this thesis falls within this last class of patterning
1
modalities. To adopt the correct fate, a cell must first acquire positional information which
specifies its location relative to the major axes of the organism. It does this by sensing the
presence or absence of chemical factors that control the cell differentiation process. This
idea has its root in the experiments of Spemann and Mangold [4], who in 1924 demonstrated that transplantation of embryonic sections in the salamander was able to induce an
altered fate in the cells surrounding the area to which the transplant was grafted. That the
“organizing principle” responsible for this effect was of a chemical nature was shown later
by Waddington and Needham [5, 6]. This era also marks the nascence of the concept of a
morphogen, or form-giving substance [7].
The refutation of vitalistic as opposed to biochemical explanations of the fate specification process was a major leap in scientific understanding, but many questions remained
about how organizer molecules were able to break the symmetry of homogeneous cells. As
the twentieth century progressed, a number of theoretical models were advanced to explain
the physical mechanisms underlying this process. These can be divided into roughly two
groups. The first, credited largely to Turing [8] and later developed further by Gierer and
Meinhardt [9], sees cell differentiation essentially as a process of self-organization. In this
class of models, competition between local self-enhancement and long-range inhibition
amplifies local inhomogeneities to generate a stable pattern [10]. In Turing’s model, for
example, a system of two interacting chemical substances with different rates of diffusion
is shown to induce an inhomogeneous spatial structure in a previously unpatterned space.
In the other group of models, the symmetry breaking inherent in the pattern formation
process is predetermined by some prior, asymmetrically localized factor. In one sense,
this type of model could be entirely trivial: a factor which takes a binary distribution in
space might induce naive cells to choose between two fates according to the same pattern,
or its complement. If, however, the factor takes a graded distribution, then the positional
information it encodes is more than binary, and a single factor could potentially induce
a variety of cell fates at different thresholds. This is the idea proposed by Wolpert in
2
concentration
threshold
blue
threshold
white
position
Figure 1.1: French flag model of tissue specification Position along an embryonic axis is
indicated on the horizontal, and concentration of the relevant morphogen is on the vertical
axis. The theory’s name, attributed to Lewis Wolpert [11], derives from the idea that a set
of morphogen concentration thresholds are responsible for patterning undifferentiated cells
into a multiplicity of fates, represented schematically by the red, white, and blue colors of
the French flag.
his French flag model (Fig. 1.1) [11]. On this model, undifferentiated cells measure the
concentration of a chemical substance that exists in a gradient along some embryonic axis.
Different concentration thresholds induce different cell fates. This particular activity has
come to define the concept of a morphogen in the modern sense of the term [2].
For Wolpert’s model to work in practice, two issues must be addressed. First, cells must
have some means of making precise measurements of morphogen concentration in some
local region. If the morphogen is a transcription factor, for example, then this measurement
might effectively be reflected in the expression of some downstream gene. Fundamentally,
there are physical limits to the precision of any read-out mechanism that are determined
by the absolute number of molecules present and the integration time available [12]. In
any case, however, the positional information obtained will be limited by the positional
information encoded in the original gradient itself. Hence the precision and reproducibility
of the process by which morphogen gradients are formed directly impacts the robustness
of the patterning system as a whole.
With regard to the problem of establishing a precise gradient, there again exists an
essentially trivial solution. If the source of the morphogen exists in some precisely graded
3
distribution, and the role of inherently stochastic processes such as diffusion is limited,
then, in principle, an infinite amount of positional information can be encoded in a gradient.
However, in this case the central problem simply regresses to the question of how a precise
source distribution can be formed. In 1970, Crick proposed a far simpler and more elegant
model, suggesting that morphogen gradients are produced by a source-sink mechanism,
mediated by diffusion [13]. On Crick’s model, the source of a morphogen is localized at
one end of a field of cells and a sink at the other, and the morphogen is allowed to freely
diffuse between them. In steady state, the intervening gradient has a constant slope, the
source-free solution of Laplace’s equation. With some rough estimates of the size of a
typical embryo and the diffusion coefficient within it, he concludes that this is a reasonable
mechanism for gradient formation on the time scales available for development.
Up to this point, however, the morphogen concept remained a purely theoretical one,
with continued doubts about its realization in living systems. The first gene to be identified displaying the function of a morphogen was the maternal effect gene bicoid in the
fruit fly Drosophila melanogaster. bicoid drives the development of anterior pattern in the
Drosophila embryo, as shown by Frohnhofer and Nusslein-Volhard [15], who found that injections of bicoid-containing cytoplasm were capable of inducing the formation of mouth
structures at various injection points in the embryos of mothers lacking the bicoid gene.
However, the authors also noted that the strength of the anteriorization response seemed to
vary continuously, in relation to the quantity of bicoid cytoplasm transplanted, rather than
in a binary fashion. It was later shown by Driever and Nusslein-Volhard that the product
of bicoid, a transcription factor1 , takes a graded distribution along the anterior-posterior
axis of the embryo [16] (Fig. 1.2), and that it drives the expression of downstream genes
in a concentration dependent manner [17], thereby satisfying the criteria of a morphogen.
The gradient they observe takes roughly exponential shape; however, this distribution is
1
In this thesis, the lower-case italicized bicoid (bcd) refers to the gene or the messenger RNA, while the
capitalized Bicoid (Bcd) refers to the gene product, a protein and transcription factor.
4
Figure 1.2: The Bicoid gradient in Drosophila melanogaster a) Bright field image of
a formaldehye-fixed mid-cycle-14 Drosophila embryo. Anterior is at left and posterior,
distinguished by pole cells, is at right. Nuclei occupy a syncytial layer at the embryo surface. b) Fluorescence image of nuclei in the same embryo, imaged by yellow excitation of
fluorescently-tagged Histone, which marks DNA. After 13 mitotic division cycles, approximately 6000 nuclei appear at the surface of the embryo [14]. c) Gradient of the Bicoid
protein in the same embryo, imaged by green excitation of fluorescently-tagged Bicoid.
Anterior nuclei show much higher concentration of the transcription factor, and the protein
takes a graded distribution along the anterior-posterior axis of the embryo.
5
the steady-state solution to a slightly modified version of Crick’s model, which includes
ubiquitous rather than localized degradation.
In the roughly 20 years since the identification of bicoid, a number of other morphogens
in a variety of species have been found [18, 19, 20]. However, bicoid continues to generate
interest as perhaps the paradigmatic example of morphogen system [21, 22]. bicoid is essential to dipteran development, and many downstream targets have been categorized (Fig.
1.3) [23]. Also, many factors both old and new combine to make Drosophila an appealing species in which to study development. Prior to gastrulation, the embryo undergoes
thirteen nuclear cleavage divisions, resulting in an excess of 6000 largely undifferentiated
nuclei in a layer near the surface of the embryo [14]. Combined with other recent developments, such as the invention of genetically-encoded fluorescent probes [24], this makes
the Drosophila embryo an ideal system for visualizing the fate specification process as it
occurs [25].
Recent work, in the bicoid system in particular, has also given rise to a number of
controversies that challenge traditional views regarding cell differentiation and morphogen
gradient formation. Gregor et al. measured absolute concentrations of Bicoid molecules
at putative regulatory thresholds and found them to be near the lower limit of that which
could produce the observed sharp boundaries of downstream target gene expression [26].
Additionally, several recent papers have called into question the mechanism long assumed
to be responsible for producing the characteristic shape and length of the Bicoid gradient. The traditional hypothesis, named the synthesis-diffusion-degradation (SDD) model
by Gregor et. al. in 2007 [27], but described in essence by Driever and Nusslein-Volhard
in 1988 [16], is the slightly modified version of the Crick model mentioned above [13]. On
the SDD model, bicoid mRNA is localized to a small region at the anterior of the embryo
[28, 29, 30] and translation of the protein occurs at a constant rate following egg deposition. Bicoid then diffuses throughout the embryo with a diffusion coefficient D and is
degraded ubiquitously at a rate k, leading to an exponential profile in steady state with
6
1
[Bcd] (norm.)
a
otd
0.5
btd
gt
hb Kr
0
0
0.2
0.4
0.6
0.8
1
x/L
2
log[η(6xbcd)/η(2xbcd)]
b
1
0
CG31054
tsh Antp
Kr
su(w[a])
CR40457
ems
btd
knrl
kni
h tll kn bcd
cad
eve hb
gt
opa
CG4476
run
−1
CG31813
CG13159
odd
salm
Ubi−p5E
ftz
−2
−2
−1
0
1
2
log[η(2xbcd)/η(1xbcd)]
Figure 1.3: Downstream targets of bicoid a) Several representative proximate targets of
bcd (out of 17 putative direct targets, see Ref. [23]), with the locations of the posterior
expression boundaries along the anterior-posterior axis shown: orthodenticle (28% egg
length), buttonhead (33%), giant (39%), hunchback (50%), Kruppel (56%). Each boundary is set by the minimum concentration of Bcd necessary to drive zygotic expression.
b) Global effect of bcd, as determined by progressively increasing a flattened Bcd profile
(1x, 2x, and 6x, with flat bcd expression denoted by bcd) and measuring RNA expression
changes by microarray experiment. η indicates hybridization intensity for each gene. Distance from zero indicates the amplitude of response to Bcd, with genes in the upper right
quadrant showing an overall positive response and genes in the lower left quadrant showing
an overall negative response. Source: Ochoa-Espinosa et al., Ref. [23].
7
length constant λ =
p
D/k.
In their original paper on the Bicoid protein gradient [16], Driever and Nusslein-Volhard
make several observations in support of the view that it is established by the mechanism we
refer to here as the SDD model. First, they note that the distributions of bcd mRNA and Bcd
protein are strikingly different, with the former being localized at the anterior 20% of the
embryo and the latter extending well past the midpoint of the anterior-posterior axis. Second, they observe that the syncytial Drosophila embryo lacks cell boundaries which would
inhibit diffusion of the protein. Citing estimates of cytoplasmic protein diffusion coefficients [31, 32], they note that they are nearly in the range necessary to explain the spread of
the protein during the allowed developmental time, and further suggest that the contractions
of the cytoplasm described by Foe and Alberts in Ref. [14] might serve to augment this
process. Third, they remark that the Bcd protein contains several PEST (proline-glutamic
acid-serine-threonine) sequences, which are thought to be present in short-lived proteins
[33, 34], a necessary condition if the gradient is to reach its equilibrium shape in the available time. Finally, they plot the concentration of protein on semi-log axes and note that it
appears linear along the majority of the anterior-posterior length of the cell, consistent with
the prediction of the equilibrium SDD model.
Over 20 years later, little new evidence that the Bicoid gradient is established by an
SDD mechanism has been produced, while nearly all of the evidence adduced by Driever
and Nusslein-Volhard has been called into question. Gregor et al. [27] measured the Bicoid
diffusion coefficient and found it too small to explain the observed extent of the gradient
regardless of the protein lifetime. While the Bicoid lifetime has not been measured, Grimm
and Wieschaus [35] deleted a PEST sequence from bicoid, expecting this to increase both
the protein lifetime and the characteristic gradient length, and found no observable effect
on the gradient. Bergmann et al. [36] note that pre-steady-state gradients can appear almost
indistinguishable from exponential gradients, and suggest that there is nothing implausible
about the idea that the gradient would be read out prior to steady state to begin with. Finally,
8
recent work by Spirov et al. [37] disputes the claim that the distributions of bcd mRNA and
protein differ, which renders the protein gradient formation process trivial. They advance
an alternative model, called the ARTS (active RNA transport and synthesis) model, in
which the shape of the Bicoid gradient is due primarily to delocalization of bicoid mRNA,
and which has been said to have explanatory superiority compared to the SDD model [38].
In this thesis, we deal with each of these controversies and present new evidence in
support of a non-equilibrium, or pre-steady-state, SDD model of Bicoid gradient formation.
In Chapter 2, we demonstrate a new method for measuring protein lifetime in vivo using
timed photoconversions of a photoconvertible fluorescent protein. We apply this method
to a fusion of Bicoid and the fluorophore Dronpa to make the first direct measurement of
the Bicoid lifetime. In partial vindication of Bergmann et al., we find that the lifetime is
longer than estimated by Driever and Nusslein-Volhard, and that the Bicoid gradient does
not reach steady state prior to interpretation. Additionally, we use an optical technique
with this same fusion protein to determine that amplification of the protein degradation rate
shortens the characteristic length of the gradient, refuting the model of identical protein
and mRNA distributions advanced by Spirov et al.
In Chapter 3, we use transgenic embryos expressing a uniform concentration of Bicoid
fused to the fluorescent protein Venus to make new measurements of the diffusivity of Bicoid, using fluorescence recovery after photobleaching (FRAP) and fluorescence loss in
photobleaching (FLIP). These results suggest that existing estimates of the Bicoid diffusion coefficient are limited by an incomplete understanding of the geometry at the embryo
cortex, and invite a rethinking of the idea that the syncytial structure of the Drosophila
blastoderm provides an ideal environment for fast morphogen diffusion.
Finally, in Chapter 4, we explore the boundaries of in vivo experimentation and make
precise measurements of the Bicoid gradient in unfertilized as well as fertilized eggs. We
compare the gradients as well as measurements of the protein lifetime and examine whether
the SDD model can offer a consistent physical picture of Bicoid gradient formation in both
9
cases.
In addition, set apart somewhat from the question of morphogen gradient formation, at
several points in the thesis we make observations which are of relevance to the question of
determining absolute concentrations of Bicoid in the embryo during the interval when the
gradient is read out.
Together, these measurements advance the effort to place our understanding of morphogenesis on rigorously quantitative footing, supported by physical principles as well as those
of traditional genetics. Furthermore, our work emphasizes the importance of perturbative
experimental techniques, as opposed to those purely descriptive, which we expect to play
an increasingly important role in biophysics in the immediate future.
10
Chapter 2
Measurement and perturbation of
morphogen lifetime
Protein lifetime is of critical importance for most biological processes and plays a central
role in cell signaling and embryonic development, where it impacts the absolute concentration of signaling molecules and, potentially, the shape of morphogen gradients [39, 40, 41].
Early conceptual and mathematical models of gradient formation proposed that steady-state
gradients are established by an equilibration between the lifetime of a morphogen and its
rates of synthesis and diffusion [16], though whether gradients in fact reach steady state
before being read out is a matter of controversy [42, 36, 43, 44]. Regardless, this class of
models predicts that protein lifetime is a key determinant of both the time to steady state
and the spatial extent of a gradient. Using a novel method which employs repeated photoswitching of a fusion of the morphogen Bicoid (Bcd) and the fluorescent protein Dronpa,
we measure and modify the lifetime of Dronpa-Bcd in living Drosophila embryos. We
find that the lifetime of Bicoid is dynamic, changing from 50 minutes prior to mitotic cycle 14 to 15 minutes during cellularization. Moreover, by measuring total quantities of
The material in this chapter is developed upon a manuscript co-authored with Oliver Grimm, David
Tank, and Eric Wieschaus.
11
Bicoid over time, we find that the gradient does not reach steady state. Finally, using a
nearly continuous low-level conversion to the dark state of Dronpa-Bcd to mimic the effect
of increased degradation, we demonstrate that perturbation of protein lifetime changes the
characteristic length of the gradient, providing direct support for a mechanism based on
synthesis, diffusion, and degradation [13].
2.1
Introduction
With the development of photoactivable and photoconvertible fluorophores [45], a broad
spectrum of new techniques has become available for use in in vivo biophysical experimentation. Bi-stable probes such as PA-GFP [46], PS-CFP [47], Kaede [48], Dronpa
[49, 50], and Dendra [51] allow for imaging and photolabeling of proteins over a variety of excitation regimes. Recent applications include cell tracking [52], observation of
organelle fusion [53], measurement of nucleocytoplasmic shuttling of proteins [49], and
measurement of degradation of autophagosomes [54].
To date, measurements of protein degradation have generally been carried out with
other labeling methods, such as incorporation of radioactive amino acids [55]. Such methods typically reveal degradation lifetimes ranging from 30 minutes to several hours for
regulatory proteins [34]. Alternatively, degradation rates can be measured by observing
total quantities of a target molecule while inhibiting synthesis. This method has been used
extensively in large-scale studies of mRNA decay rates [56], though results for proteins
are significantly noisier [57, 58]. In fly embryos, blocking translation has the additional
drawback of arresting the nuclear cleavage division cycles occurring prior to blastoderm
formation. Recently, Eden et al. devised a means of measuring protein degradation via
photobleaching of fluorescent tags [59], and apply it to a large library of proteins in cancer
cells, though the particulars of the method make it impractical in large cells such as the fly
embryo.
12
The synthesis-diffusion-degradation (SDD) model of morphogen gradient formation
[27] is one biophysical model in which protein lifetime plays a key role. Let c(x, t) represent morphogen concentration on some embryonic axis 0 < x < L. Its evolution is
described by
∂c(x, t)
= s(x, t) + D∇2 c(x, t) − kdeg c(x, t)
∂t
(2.1)
where s is the source function, D is the diffusion coefficient, and kdeg the morphogen degradation rate. No-flux boundary conditions are generally imposed at the ends,
∂c
|
∂x x=0,L
= 0.
The form of the steady-state gradient C(x) is determined by an equilibration between the
rates of diffusion and degradation [13, 41]. The gradient relaxes to this equilibrium shape
after a time determined by the morphogen lifetime and rate of change, if any, in morphogen
synthesis [60]. In the idealized case, where synthesis occurs at a constant rate strictly at
the anterior tip, such that
∂c
|
∂x x=0
=
−p
D
where p is some constant production rate, and
the concentration at the posterior boundary is negligible, the equilibrium solution is well
p
known to be a decaying exponential C(x) ∝ e−x/λ with characteristic length λ = D/kdeg
[16, 21, 27, 41]. That the source distribution is strongly localized, implying that the exponential solution well approximates the actual steady-state gradient, is considered an important feature of the SDD model. Formally, this criterion can be expressed as
R Tdev R L
xs(x, t)dxdt
λ
R0 Tdev R0 L
s(x, t)dxdt
0
0
(2.2)
where Tdev is the total developmental time.
Although this model may be theoretically attractive, it is not known whether any gradient forms in such a simple manner in a complex cellular medium. Yu et al. [61] study
the fibroblast growth factor (Fgf8) morphogen system in zebrafish, creating a local source
distribution by injection of mRNA at one end of the embryo and taking degradation to be
represented by receptor-mediated endocytosis. They demonstrate that mutants which are
known to have impaired or augmented rates of endocytosis show longer or shorter gra13
dients, respectively, providing evidence for a source-sink gradient formation mechanism
such as previously described. However, while irreversible internalization is mathematically
equivalent to degradation [62], quantitative estimates of the expected shift in the internalization rates of the various mutants are not known. Kicheva et al. use fluorescence recovery
after photobleaching (FRAP) to perform a comprehensive study the Decapentaplegic (Dpp)
and Wingless (Wg) systems in the Drosophila wing imaginal disc, and find that the gradient lengths of different morphogens are best explained by different degradation rates [63].
However, their analysis extracts morphogen lifetime τ from optical measurements of the
√
diffusion constant D and gradient length λ, combined with the assumption that λ = Dτ ,
as predicted by the equilibrium SDD model. In the present work, our goal is to test this
basic hypothesis. We measure the lifetime of a morphogen in vivo and modulate this lifetime downwards in precisely measurable steps. The observed shifts in the length of the
gradient provide the first quantitative test of the central claim of the SDD model, that the
characteristic length of a morphogen gradient scales approximately as the square root of the
lifetime, with remaining deviation explained by pre-steady-state effects and the specifics of
the source distribution.
To investigate these questions, we study the formation of the Bicoid (Bcd) morphogen
gradient in the Drosophila melanogaster embryo. This gradient has been studied extensively by quantitative and non-quantitative techniques [16, 17, 41, 64], but whether the
SDD model accurately describes the formation of this gradient remains unclear [21]. It
is known that transcripts of maternal bcd mRNA at the anterior of the embryo act as the
source of Bcd protein [29, 65]. According to the SDD model, the Bcd protein lifetime then
determines when levels of morphogen in any region of the embryo become stable [60], as
well as the characteristic length λ of the gradient. These assumptions are controversial.
Bergmann et al., for example, suggest that equilibrium is neither necessary nor achieved
[36]. More recently, Spirov et al. claim that the distribution of Bcd protein is in fact nearly
identical to that of bcd mRNA [37]. This model, which they name the ARTS (active RNA
14
transport and synthesis) model, is formally described by Eq. 2.1, but the shape of the protein gradient is at all times virtually identical to that of the mRNA distribution itself [37]:
c(x, t) u A(t)s(x, t)
(2.3)
2
∂ s
This is an approximate solution to Eq. 2.1 if | 1s ∂s
| kdeg and | 1s ∂x
2| ∂t
kdeg
.
D
Provided
these criteria are satisfied, as the ARTS model claims, the rates of protein diffusion and
degradation are otherwise irrelevant to gradient shape.
2.2
Results and Discussion
To distinguish between these competing hypotheses and quantitatively evaluate the importance of morphogen degradation to gradient formation, we generated a fusion between
Bcd and the photoswitchable protein Dronpa (Dronpa-Bcd) [49, 50]. The photoconversion
properties of Dronpa-Bcd provide a means of measuring the Bcd lifetime, as conversion to
the Dronpa dark state effectively allows tagging a subpopulation of the protein and following its evolution in time. Moreover, Dronpa-Bcd allows for the experimental application of
an optical effect mimicking an augmented Bcd degradation rate, as photoconversion to the
dark state has a quantifiable effect on the rate at which bright-state Dronpa-Bcd disappears,
and the subsequent effects on gradient shape can be examined.
2.2.1
Measurement of Bcd lifetime by Dronpa-Bcd photoconversion
The Dronpa-Bcd fusion we created is flanked by the endogenous 5’ and 3’ regulatory regions (see section 2.3.2), ensuring that mRNA localization occurs in the wild-type fashion
[66]. Transgenes carrying this construct rescue the sterility of females homozygous for a
null mutant for bcd, and form gradients identical to those previously described for EGFPBcd fusion proteins (Fig. 2.1d) [27]. Dronpa-Bcd has bistable bright and dark states,
15
Fluorescence Intensity
a
496 nm
405 nm
b
1
c
time
0.5
496 nm
0
0
1
CGFP
C (norm.)
1
ee
1
0.5
0.5
x/L
0.5
0
0
0.5
C
1
Dronpa
d
0
0
0.5
1
Qdf (t0+T)
Qdf (t0)
recovered
focal
(arb.)
13 s
converted
x/L
496
nm
S
T = 4, 8, 15 min.
f
405 nm 8.2
8.1
0.1
time
Figure 2.1: Description of Dronpa-Bcd construct and degradation measurement Confocal images of in vivo Dronpa-Bcd expressing embryos with Dronpa-Bcd predominantly
in the bright (a) or dark (b) state. Switching from bright-to-dark and dark-to-bright occurs via 496 nm or 405 nm illumination, respectively. c) Dronpa-Bcd gradients plotted as
fractional anterior-posterior (A-P) axis position, shown for a sequence of 496 nm images.
Imaging effects simultaneous conversion to the Dronpa dark state. d) Mean A-P gradients
in early cycle 14 of 19 Dronpa-Bcd embryos (blue) and 18 EGFP-Bcd embryos (green),
from fluorescence images obtained by confocal microscopy and processed as described in
Ref. [27], with background subtraction using Oregon-R under identical conditions. Inset:
Scatter plot of Dronpa-Bcd (CDronpa ) vs. EGFP-Bcd (CGF P ) intensities. e) Confocal image of a fixed Dronpa-Bcd embryo with Dronpa predominantly converted to the dark state
except an inscription produced by a targeted 405 nm reactivation pulse. Species are stable
for multiple days. f) Schematic of degradation measurement, beginning with a Dronpa-Bcd
population converted to the dark state, Qdf (t0 ), followed by an interval T and subsequent
measurement of the surviving population Qdf (t0 + T ). Actual values of Qdf are determined
by a fitting procedure described in section 2.3.5. Sf ocal indicates bright-state Dronpa-Bcd
integrated over the region within 14 mm of the embryo surface.
16
conversions between which are rapid and inducible in living and fixed material [49]. Illumination with 496 nm light allows visualization of Dronpa-Bcd, but also converts it to
the dark state (Fig. 2.1a, c). Reactivation from the dark to a stable bright state is achieved
by exposure to 405 nm light (Fig. 2.1b, e), and, importantly, under the conditions we use
this reactivation pulse is sufficient to reactivate virtually all (98%) dark-state protein to
the bright state (see Fig. 2.5c and section 2.3.10). Photoconversion is uniform over the
entire egg, and maturation of newly-synthesized Dronpa-Bcd occurs exclusively into the
bright state (see section 2.3.7 and 2.3.8). Together, these properties facilitate a strategy in
which the Dronpa-Bcd degradation rate can be calculated from time-lapse measurements
of a population of dark-state Dronpa-Bcd.
The conceptual idea in our strategy to measure Bcd lifetime is illustrated in Fig. 2.1f.
A sequence of 8 bright-state fluorescence images is produced using 496 nm excitation in
a confocal microscope. The entire sequence occurs in a time (13 s) that is short compared
with changes in morphology and Bcd synthesis. With each image, there is a progressive decline in bright-state fluorescence produced by photoconversion of a fraction of the
Dronpa-Bcd molecules to the dark state. The total amount of Bcd converted from bright to
dark by this sequence can be measured by determining the difference in intensity between
the last image in this sequence and the first (see section 2.3.10 and Fig. 2.8 for more details). This represents the dark-state-tagged Bcd population. Then, after a fixed delay T
that can be varied in different experiments, the bright-state fluorescence is measured again
and excitation at 405 nm is used to reconvert the dark-state-tagged Dronpa-Bcd that has
not degraded back into the bright state; this recovered Dronpa-Bcd is again measured as a
difference in bright-state intensity, this time comparing the intensities before and after 405
nm conversion. The loss between the amplitude of the initially converted and that recovered after a delay represents the amount of dark-state-tagged Bcd that has degraded during
time T . Note that because intensity differences closely spaced in time are used to measure
both converted and recovered dark-state quantities, the method is insensitive to changes
17
in bright-state Dronpa-Bcd concentration (due to a combination of fluorophore maturation
and degradation) that also occur during the experiment.
If we let Qdf (t0 ) represent the amount of Dronpa-Bcd converted to the dark state and
Qdf (t0 + T ) represent the amount recovered from the dark state after a time T , it follows
that the degradation rate, kdeg , can be determined according to the equation
Qdf (t0 + T ) = e−kdeg T Qdf (t0 )
(2.4)
We verify that this relation holds in Fig. 2.2c, showing that log (Qdf (t0 +T )/Qdf (t0 )) is well
fit by a linear function of T in early cycle 14, consistent with first-order degradation.
The degradation rate kdeg at this developmental time point, indicated by the slope, is
0.028 min.−1 , which corresponds to a Bcd lifetime, τBcd , of 36 minutes. However, performing this measurement at different time points after fertilization demonstrates that the
degradation rate is developmentally regulated (Fig. 2.2a). At the onset of cycle 14 kdeg is
0.020 min.−1 , corresponding to a lifetime of 50 minutes. By the time the embryo begins
gastrulation, kdeg has increased and the lifetime of the protein has fallen to 15 minutes.
2.2.2
Correction of Bcd lifetime measurement for core-to-cortex flux
The Bcd gradient forms prior to cycle 14 and therefore it is important to measure its lifetime
during these earlier stages. We can first measure τBcd using our photoconversion method
when nuclei reach the surface of the embryo in cycle 11, ∼100 minutes after fertilization.
Prior to this time, the signal-to-noise ratio for bright-state Dronpa-Bcd concentration measurements is too low. In Fig. 2.2b the fraction of surviving Bcd, Qdf (t0 +T )/Qdf (t0 ), which can
be used to provide an estimate of τBcd through Eq. 2.4, is plotted against developmental
time since fertilization. From cycle 11 until the beginning of cycle 14, this quantity shows
cyclic fluctuations that appear correlated with the cell cycle, including periods where the
ratio exceeds unity, implying that more is recovered than converted.
18
1.5
Qdf(t0+T)/Qdf(t0)
a
0.06
0.04
k
deg
(min.−1)
0.08
0.02
160
t (min.)
170
180
0.5
12
100
0.2
0.1
0
0
13
120
d
c
4 8 12 16
T (min.)
1
kdeg (min.−1)
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
0
150
Q(r)/ΣQ
df 0
df 0
log(Q (t +T)/Q (t ))
0
140
b
0.5
r/R
1
14
140
t (min.)
0.06
160
180
e
0.04
0.02
12 13 14
0
100 120 140 160 180
t (min.)
Figure 2.2: Measurement of Bcd degradation and correction for flux a) kdeg (x’s) in
cycle 14 displayed for all degradation experiments with duration T = 8 min., comprising
60 embryos. t indicates the midpoint of the 8 minute interval, in minutes since fertilization.
b) Qdf (t0 +T )/Qdf (t0 ) calculated for all T = 8 min. experiments. Labels indicate mitotic cycle
number. Solid line represents smoothing spline fit to all data smoothed by a Savitzky-Golay
filter [67] of span 6. c) Scaling of surviving fraction according to waiting interval duration
T , measured early in interphase 14 (squares). Error bars indicate standard error. Least
squares fit (dashed line) has slope -0.028 min−1 , corresponding to the degradation rate
shortly after the onset of interphase 14. Each data point bins all experiments with a given
T whose midpoint falls between the dashed lines in (a). d) Measurement of fractional
distribution of Dronpa-Bcd in cross-sections of embryos in cycle 11 and prior (blue), in
cycles 12 and 13 (green), and in cycle 14 (red). Q(r)/ΣQ indicates the fraction of total
Dronpa-Bcd in an annulus of fixed width at distance r from the longitudinal axis of the
embryo. Dashed line indicates the uniform distribution. A larger percentage of Dronpa-Bcd
is located at the surface region of integration, indicated by yellow shading, in later stages. e)
Degradation rate found by removing cell-cycle-periodic oscillations but with no correction
for core-to-cortex flux (black). Estimated kdeg obtained by assuming redistribution of Bcd
as observed in cross-sections (green). Estimated kdeg obtained by assuming Dronpa-Bcd
redistribution identical to that of H2A-RFP (red). Shading indicates standard error of this
curve found by binning a moving window of 10 adjacent values.
19
A possible explanation for this effect is Bcd flux into and out of the volume of cytoplasm
from which the fluorescence measurements are made. By cycle 14, nuclear cleavage has
ceased and compartmentalization of the cytoplasm has begun, suggesting that the surface
layer in the embryo approximates a closed volume [68]. Prior to cycle 14, however, two
types of spatial flux have been described that could impact our measurements. The first
is the movement of Bcd into nuclei during interphase and its accretion in deep cytoplasm
during mitosis (Fig. 3d in Ref. [27]). Due to scattering, single-photon fluorescence imaging
does not capture emission from the deep cytoplasm, and we expect experiments that begin
in interphase and end in mitosis to show reduced recovery and high apparent degradation
rates, since a significant fraction of the initial dark converted Bcd has moved out of the
surface layer when reactivation is induced. Conversely, experiments that begin in mitosis
and end in interphase will report artificially small degradation rates. In fact, as shown in
Fig. 2.2b, we observe precisely this sort of oscillation in the surviving fraction of DronpaBcd. If the cyclic flux associated with mitosis conserves total Bcd levels, the most accurate
estimate of lifetime will be obtained by replacing the oscillatory components with their
mean using Fourier transform methods (see section 2.3.14). The result of this correction
is shown in Fig. 2.2e (black trace), where kdeg is plotted versus developmental time for all
time points from cycle 11 through the end of cycle 14.
Recovery of dark Dronpa-Bcd might also be influenced by a second type of flux, namely
the slow redistribution of Bcd from the core of the embryo towards the cortex (Fig. 6
in Ref. [27]). To estimate its magnitude, we measured the cross-sectional distribution
of Dronpa-Bcd in fixed embryos. At cycle 12, we find that Bcd in the surface volume
we measure represents about 47% of the total signal in the cross-section. Thirty minutes
later, in early cycle 14, levels in the surface layer had increased to 54% of the total signal
(Fig. 2.2d). A similar estimate for general cytoplasmic flux between cycle 12 and 14 was
obtained using live imaging to follow the redistribution of the Histone-RFP (see section
2.3.12). Correcting for this flux has a minor effect on the degradation rate as a function of
20
developmental time (Fig. 2.2e; red trace) but yields a degradation rate prior to cycle 14 of
0.020 ± .006 min−1 , and a lifetime estimate of 50 minutes, identical to that measured at the
beginning of cycle 14. We conclude that, from cycle 11 to the beginning of cycle 14, the
degradation rate of Bcd protein is approximately constant.
2.2.3
The Bcd gradient does not reach steady state prior to interpretation
An important question that follows is whether the Bcd gradient is in equilibrium. Given
a constant degradation rate, an examination of the total Bcd quantity versus time can provide information about the constancy of Bcd synthesis. In particular, if the synthesis rate
of Bcd is constant and commences at fertilization, the total amount of Bcd should relax
exponentially to its equilibrium value with a time constant equal to its lifetime. To test this
hypothesis, we used Western blotting to measure the accumulation rate of Bcd protein during stages prior to cycle 11. We extracted total protein from embryos staged in 30 minute
time windows between fertilization and gastrulation. A typical Western blot is shown in
Fig. 2.3a; in Fig. 2.3b, the quantification of 5 different experiments that represent a total of
400 embryos per time point is plotted. We calibrate the blot to known quantities of GFP to
determine the absolute number of Bcd molecules in the embryo. The number of molecules
continues to increase until cycle 14, at which time point we find a total of 1.5 ± 0.2 × 108
Bcd molecules in the embryo (see section 2.3.15).
Until cycle 11, 100 minutes after fertilization, there is an almost linear increase in the
total amount of Bcd. This linearity is inconsistent with the expected exponential relaxation
to an equilibrium quantity given τBcd = 50 minutes, but is consistent with increasing Bcd
synthesis after fertilization (see section 2.3.17). An independent estimate of Bcd synthesis
can be made by analyzing the changes in bright-state fluorescence during the delay time T
in our degradation experiments. In section 2.3.19, we show that it gives a result consistent
21
20
0m
in.
in.
0m
17
0m
in.
in.
14
mi
80
50
11
0m
n.
n.
mi
n.
mi
20
R
Or
e
a
Bcd
α-Tub
7
15
b
10
5
in.
0m
c
20
15
0m
in.
in.
200
0m
in.
13
0m
mi
n.
100
t (min.)
90
60
mi
n.
n.
mi
20
no
RT
0
0
11
QBcd (molec.)
x 10
bcd
mRNA
Figure 2.3: Measurement of total Bcd quantity a) Time course of Bcd quantity measured
by Western blotting, 10 embryos per lane. Time t indicates the mean time of the embryo
collection, reported in minutes post fertilization, which is assumed to occur 5 minutes prior
to oviposition. b) Quantification of an ensemble of 5 Bcd Western blots such as shown in
(a). Error bars indicate standard error. Solid line represents smoothing spline fit to data.
Peak in QBcd occurs between cycles 12 and 14. QBcd is expressed as the total number of
Bcd molecules in the embryo at each time point. c) Measurement of bcd mRNA levels by
RT-PCR. (Measurements in (a) and (c) by O. Grimm)
22
with the Western blot analysis. An increasing rate of Bcd synthesis is also compatible with
reported dynamics of the polyA tail of bcd [69, 70]. Hence our results support the idea that
the Bcd gradient is not in equilibrium when it is read out [36].
2.2.4
Optically mimicked degradation shortens the Bcd gradient
While our results suggest the gradient is not in equilibrium, we can nevertheless test whether
the spatial extent of the Bcd gradient scales with the morphogen’s lifetime. In the equilib√
rium version of the SDD model, the gradient has a characteristic length λ = Dτ , where
D and τ are the diffusion coefficient and lifetime of the morphogen, respectively. If the distribution of bcd mRNA were identical to the protein gradient, as reported in Ref. [37], then
one would not expect any change in gradient shape corresponding to a shortened protein
lifetime. On the contrary, if the Bcd gradient arises by diffusion from a localized anterior
source, protein lifetime will play a crucial role in determining the final spatial distribution
of the morphogen. Shorter lived versions of the protein will give rise to shorter, steeper
gradients because the protein has less time to diffuse before it is degraded. This scaling
relationship holds even in the absence of a true equilibrium, as shown by numerical simulation (see section 2.3.18 and Fig. 2.4b, inset, red curve).
To test this idea, we developed an optical strategy that mimics enhanced degradation.
While photoconversion of Dronpa-Bcd to the dark state does not affect the biological properties of Dronpa-Bcd, it creates an effect that is indistinguishable from an enhanced endogenous Bcd degradation from the standpoint of bright-state fluorescence measurements.
Thus, we can augment the endogenous degradation of Bcd while simultaneously imaging
Dronpa-Bcd. We take a continuous series of images of Dronpa-Bcd expressing embryos
beginning prior to nuclei appearing at the surface and continuing for 70 minutes until the
early phase of cycle 14. Each image produces an increment of conversion to the dark state;
because the loss is proportional to concentration it mimics a first-order degradation. The
added degradation rate, kadd , is specified by modulating the laser intensity and scan fre23
0
10
b
3
0.6
2
1
0
0.4
k
0.02
0.04
(min.−1)
0
20
−2
10
a
0.2
0.4
x/L
0.6
0.8
60
40
add
0.2
0
−1
10
Δ L2 (µm)
0.8
C/Cmax
1
Qmat/Qapp
C Dronpa-bright (norm.)
1.2
0
−1
0 k
(min. ) 0.05
add
0.2
1
0.4
x/L
0.6
0.8
Figure 2.4: Optical augmentation of Dronpa-Bcd degradation a) Mean mid-cycle-14
Dronpa-Bcd gradients, averaged over an ensemble of N embryos after background subtraction determined by identical measurement of H2A-RFP embryos, after 70 minutes of equilibration at the additional artificial degradation rate kadd = 0.043 min−1 (blue, N = 13),
kadd = 0.022 min−1 (red, N = 8), and the identical embryos immediately after reactivation
of all Dronpa-Bcd in the embryo (black, N = 21). Error bars indicate standard deviation.
Inset: Fit of equilibrium model for Qmat /Qapp for a range of kadd = 0, 0.008, 0.011, 0.016,
0.019, 0.022, 0.026, and 0.043 min−1 . Qmat indicates the total quantity of mature DronpaBcd protein. Error bars indicate standard deviation. Least squares fit (red line) predicts
kdeg = 0.027 min−1 . b) Gradients normalized to Dronpa-Bcd concentration near the anterior pole, on semilog axes, for kadd = 0.043 min−1 (blue, N = 13), and the identical
embryos immediately after reactivation of all Dronpa-Bcd in the embryo (black), showing
a change in gradient length. Inset: Measurement of gradient contraction for various values
of kadd . L2 is the point at which the mean concentration decreases to ∼ 13% of the maximal
value: L2 = inf {x|CDronpa−bright (x) < e−2 CDronpa−bright (30µm)}. ∆L2 is the difference
in this quantity between the pre-activation gradient, which is the product of augmented
degradation, and the post-activation gradient, which results from endogenous degradation
alone: ∆L2 = Lpost
− Lpre
2
2 . Error bars indicate standard error. Red curve indicates value
of ∆L2 predicted by simulation, assuming the Bcd degradation rate shown in Fig. 2.2e, a
Dronpa maturation lifetime of 60 minutes, and a diffusion coefficient D = 4 µm2/s, chosen
to match the characteristic length of the unperturbed gradient. Blue curve indicates the
same, except for the assumption of instantaneous maturation, showing that the effect is less
pronounced partly due to the effect of fluorophore maturation. Green curve indicates the
∆L2 shift predicted for a shallow source distribution as measured in Ref. [37], using a
diffusion coefficient D = 1.2 µm2/s, similarly chosen to fit the length of the unperturbed
gradient.
24
quency (see section 2.3.16). After the bright-state Bcd gradient equilibrates to the optically
augmented degradation rate, ktotal = kdeg + kadd , the full Bcd gradient, reflecting solely
endogenous degradation, can be measured after a complete reactivation series with 405 nm
excitation.
In Fig. 2.4a, we show gradients obtained without augmentation and for two distinct
values of kadd . Both changes in amplitude and in shape are observed and can be quantified.
As ktotal is increased, the total apparent quantity of Dronpa-Bcd, Qapp , decreases (see inset
plot of Qmat/Qapp ). In the equilibrium model, Qapp will depend on the sum of endogenous
and induced degradation rates according to the equation
Qapp =
m
ktotal
=
m
kdeg + kadd
(2.5)
where we assume a constant rate of newly maturing Dronpa-Bcd m. Though our results
suggest that the rate of Bcd synthesis, and hence the rate of newly maturing Dronpa-Bcd,
is not constant, Eq. 2.5 nevertheless holds if the mean value of m(t) is approximately
equivalent for the various time intervals of length (kdeg + kadd )−1 . The endogenous degradation rate kdeg , obtained by estimating the point where kadd produces Qapp =
Qmat/2,
is
0.027 min−1 (τBcd = 37 minutes) and is within 25% of the value obtained in the direct
experiments in Fig. 2.2. Furthermore, gradients obtained under optically mimicked degradation have shapes that are distinguishable from each other as well as the full gradient
after Dronpa reactivation (Fig. 2.4b). As the apparent degradation rate is increased, the
characteristic length of the gradient decreases and the gradient becomes steeper (see inset). This provides an in vivo demonstration that morphogen gradient shape is altered by
changes in protein lifetime. Using numerical simulations of the SDD model (Fig. 2.4b,
inset, red curve), we find that the magnitude of the shifts as well as the characteristic length
of the full gradient is well explained by our measured and decremented Bcd lifetimes,
D = 4 µm2/s, of the same order of magnitude as recently reported in Ref. [71], and a chro25
mophore maturation lifetime of 60 minutes, consistent with our measurements of newly
maturing Dronpa-Bcd (see section 2.3.19). It is notable that the maturation time attenuates
the observed gradient length shift; this is because the maturation time effectively smooths
the source distribution of mature fluorophore in space. The slightly larger shifts that would
be expected in the case of instantaneous maturation are given by the blue curve in Fig.
2.4, inset. Finally, we simulate the gradient shift expected based on the extended mRNA
distribution reported in Ref. [37]; this is given by the green curve, which does not fit the
data.
2.2.5
Conclusion
In conclusion, we have developed a novel morphogen fusion and optical method that can
be used to measure morphogen lifetime and explore the effects of increased protein degradation. Applying this method, we find that Bcd has a lifetime of 50 minutes prior to mitotic
cycle 14, becoming shorter at later times. We confirm this result by optically modulating
the Bcd lifetime, which additionally refutes the claim that the bcd mRNA distribution mirrors the shape of the protein gradient and provides direct support for a synthesis-diffusiondegradation model of gradient formation. Additionally, the demonstration that protein lifetime is a potential lever with which an organism can change the extent of its morphogen
gradients is of significance for the presently unsolved mechanism by which morphogen
gradients scale between variably sized organisms [72].
One remaining question which might be investigated to high precision using this technique is whether regulated protein degradation in complex cellular environments follows
first-order kinetics. Our results are consistent with this hypothesis, as we have shown by
relating the surviving fraction to the waiting time interval (Fig. 2.2c) and by measuring
approximately equal lifetimes at different concentrations of Bcd (see section 2.3.20). However, we are limited in our ability to measure deviations from first-order behavior primarily
by the developmental time available. Application of this technique to systems in which
26
the experimental time is much longer than the measured lifetime might provide further
mechanistic insights into the nature of protein degradation in general.
2.3
Methods
The work in this chapter uses photoconversion of fluorescently-tagged proteins as a means
of measuring protein lifetime. In particular, we prepare a Dronpa-Bcd fusion and subsequently use the dark state of Dronpa as a means of tagging a subpopulation of protein
to measure the remaining quantity at a later time. For this method to work, the DronpaBcd dark state must be stable, meaning that spontaneous reactivation of dark-state protein
is negligible, and isolated, meaning that newly-synthesized protein enters the bright state
alone. Additionally, the photoconversion process must be reversible, meaning that there
is a negligible amount of irreversibly bleached fluorophore during each photoconversion
series, and uniform, meaning that it affects all fluorophore throughout the embryo equally.
We verify all these facts in this section. Moreover, corrections must be made for flux of
Dronpa-Bcd between the region of integration of fluorescence intensity, which for technical
reasons is limited to the embryo surface, and the remainder of the embryo. This flux can
be decomposed into a sinusoidal cell-cycle-periodic component, which we remove, and a
component that varies steadily in time, which we estimate by two different methods. In
addition to estimating the degradation rate, kdeg , we describe methods used for estimating
the rate of synthesis of Bcd.
2.3.1
Table of symbols
Symbol
Description
Itrans
excitation beam intensity measured by trans-side PMT
27
pointwise fluorescence intensity measured by green channel
If luor
epi-side PMT
η
calibrated offset for trans-side PMT
power of transmitted excitation beam, corrected for PMT offset,
P
P = Itrans − η, used to normalized If luor
kof f /kon
measured rates of Dronpa dark-conversion/reactivation
fraction of dark-state Dronpa-Bcd reactivated by a 405 nm
ξ
photoactivation series, ξ = 98%
total quantity of Bcd protein in the embryo, equivalent to total
QBcd
quantity of Dronpa-Bcd in bcdE1 embryos
total quantity of bright-state Dronpa-Bcd under conditions of
Qapp
optically augmented degradation
Qmat /Qimm
Qf ocal /Qobscure
quantity of mature/immature Dronpa-Bcd in the embryo
quantity of mature Dronpa-Bcd inside/outside the visible region of
integration at the embryo surface, Qmat = Qf ocal + Qobscure
quantity of bright-state/dark-state Dronpa-Bcd in the region of
Qbf /Qdf
integration at the embryo surface, Qf ocal = Qbf + Qdf
fraction of mature Dronpa-Bcd in the dark state at all points and
G(~r, t)
times in the embryo
fraction of mature Dronpa-Bcd located in the region of integration
γ
at the embryo surface, γ = Qf ocal/Qmat
p(t)
Dronpa-Bcd synthesis function
kmat
rate of maturation of Dronpa chromophore
m(t)
newly maturing Dronpa-Bcd function, m(t) = kmat Qimm
Sf ocal
integration of If luor over embryo surface region
28
contribution to If luor representing specific signal, the concentration
CDronpa−bright
of bright-state Dronpa-Bcd
A
contribution to If luor representing autofluorescence
flux of mature (dark-state) Dronpa-Bcd into the region of
Φ (Φdark )
integration, Φ = Φbright + Φdark
fraction of newly matured Dronpa-Bcd that appears initially in the
U
dark state
2.3.2
δ/δa
duration of major/minor dark conversion series, δ = 13 s, δa = 7 s
kinc
rate of incidental or spontaneous reactivation of Dronpa
kdeg
rate of Dronpa-Bcd degradation
τBcd
Bcd protein lifetime, τBcd = 1/kdeg
ksimp
rate of Dronpa-Bcd degradation assuming Φ = 0
Synthesis, characterization, and preparation of samples
The GFP of the GFP-bicoid rescuing construct in Refs. [66] and [27] was replaced with
Dronpa [49] by Oliver Grimm, using standard cloning techniques, and verified by sequencing. The resulting fusion, under control of its normal 5’ and 3’ regulatory regions, rescues
the sterility of females that are homozygous mutant for bicoid.
dronpa-bcd embryos are collected over a period of 90 minutes on a plate of agar and
apple juice mixture. Embryos are dechorionated in bleach, glued to a glass slide and immersed in halocarbon oil. No coverslip is used. Where fixed specimens are used, for power
calibration, embryos are dechorionated in bleach, fixed in a solution of 20% paraformaldehyde in 1X PBS buffer for 45 minutes, devitellinized by hand with a sharp needle, and
mounted on a glass slide in Aqua PolyMount (PolySciences).
29
2.3.3
Description of Dronpa photophysics
Reversibility of photoconversion and near-bistability of fluorescent and non-fluorescent
states make Dronpa an especially suitable fluorophore for measuring degradation[50]. As
shown in Fig. 5 of Ref. [50], it has a deprotonated fluorescent state (B), which absorbs
photons at 503 nm and reemits at 518 nm with a high fluorescence quantum yield of 0.85.
We refer to B as the Dronpa “bright” or “on” state. With a relatively small cross-section,
molecules excited at 496 nm are photoswitched to a protonated state (A2 ), which has a
minor absorbance peak at 388 nm, but is for our purposes nonfluorescent. We refer to
A2 as the Dronpa “dark” or “off” state. Fluorophores in the dark state are returned to the
bright state with a comparatively large cross-section by illumination with 405 nm light.
Thermal conversion to the bright state in the absence of light is also reported, though reported metastable lifetimes vary from 840 min. [73] to several days (Ref. [50] and Fig.
2.1e). There is an additional nonfluorescent protonated state (A1 ) whose occupancy is
determined by the solution pH. At pH 7.4, approximating physiological values, 0.4% of
Dronpa molecules are in state A1 [74]. As such, A1 is a negligibly small reservoir. Finally,
there is a triplet dark state (D) which causes intermittent blinking of the fluorophores. This
state is cycled back to the bright state with lifetime 15 s−1 and does not significantly distort
measurements of the quantity of fluorophores transferred to the dark state over conversion
series of the duration used in the experiment.
2.3.4
Description of optical methods
Imaging is performed on a Leica TCS SP5 laser scanning confocal microscope with a 0.7
NA multi-immersion objective (Leica 20x HC PL APO). For imaging bright-state fluorophore and conversion to the Dronpa dark state, we use the 496 nm line of an argon laser
with average power 350 mW at the objective. For reconversion from the dark state to the
Dronpa bright state, we use a 405 nm diode laser with average power 130 mW at the objec30
tive. For imaging Histone-RFP fluorescence, we use a 561 nm He-Ne laser at 40 mW. In all
images the laser is focused at the mid-sagittal plane of the egg and scanned over the entire
cross-section, with a single 576 x 1536 pixel frame acquisition time of 0.8 s.
We determine the rates of photoconversion between the bright and dark states in our
setup by plotting the mean fluorescence intensity If luor in a region of nuclei at the anterior
of the embryo during the interphase of mitotic cycle 14. In this region nuclei strongly fluoresce with Dronpa-Bcd and autofluorescence is small relative to specific signal (A/If luor ∼
3%). As shown in Fig. 2.5a, Dronpa-Bcd can be repeatedly switched between bright and
dark states in vivo. The rates of conversion are determined in Figs. 2.5b and 2.5c by fitting
to a single exponential. We also observe that the rate of conversion to the dark state koff
scales linearly with the excitation power (Fig. 2.5d).
We normalize all measured fluorescence intensities by the excitation power to correct
for small fluctuations in laser intensity. To measure the excitation power, we acquire a
bright field image simultaneous with each fluorescence image and measure the intensity of
transmitted light Itrans in a region of the bright field image far from the embryo. We assume
that the actual excitation power P = Itrans − η, where η is a constant offset particular to the
PMT used. We calibrate η by varying Itrans in a sawtooth pattern and choosing the value of
η that minimizes the fractional variation in the corrected fluorescence intensities (Fig. 2.6).
2.3.5
Measurement of spatially integrated bright-state fluorescence
The measured quantity we use to calculate estimates of degradation and synthesis is Sf ocal ,
which is obtained by integrating the intensity of fluorescence at the embryo surface over the
volume it represents. For this purpose we parametrize the embryo in cylindrical coordinates
31
3
7
a
log(−∆I/∆n)
Ifluor (arb.)
2.5
2
1.5
1
5
4
3
0.5
0
2
4
t (min.)
6
20
n (scans)
40
200
P (µW)
400
0.08
d
(scans−1)
c
1
off
0.5
k
I−I0/Imax−I0
koff =0.08
2
0
8
1.5
kon =1.23
0
0
b
6
2
4
n (scans)
6
0.06
0.04
0.02
0
0
Figure 2.5: Determination of rates of photoconversion a) Mean fluorescence intensity
If luor at 496 nm excitation. 50 images with exclusively 496 nm excitation are followed by
a sequence of 10 images for which a single scan at 405 nm is interposed between each.
This is repeated six times. t is in minutes. Decrease in peak intensity is consistent with
endogenous degradation rate in late cycle 14, as determined in Fig. 2.2. b) Differentiated
mean If luor from the dark conversion series in (a) plotted on a log scale. n represents the
scan number. The rate of photoconversion to the dark state koff is 0.08 in units of inverse
scans, as shown by the linear fit. c) Fractional recovery of If luor from the reactivation series
in (a). Imax is the mean of the final five images in each activation series. Fit to 1 − e−kn
gives k = koff + kon = 1.31 scans−1 . kon is 1.23 scans−1 . d) Scaling of koff with excitation
power, measured in fixed tissue. koff is determined as in (b) for a range of excitation power
P at 496 nm, reported in mW.
32
−3
7
5450
6
σI
Itrans
5400
x 10
5350
5
5300
a
5250
0
5
n
b
10
4
−6000 −4000 −2000
η
0
2000
Figure 2.6: Determination of PMT offset value a) Variation of excitation power as observed by intensity of transmitted light Itrans in a bright field image. Separate curves indicate multiple realizations. n indicates scan number. b) Fractional variation
in corrected
2
I
I
I
Dronpa-Bcd fluorescence intensity σI = Σ ( /P − < /P >) / < /P > . σI is minimized
by η = −1600.
and sum as follows:
Z
Sf ocal =
If luor (~r)d~r
ZL Rmax
Z (l)
= 2π
If luor (r, l)rdrdl
(2.6)
(2.7)
0 Rmin (l)
L is the length of the embryo from anterior tip to posterior tip. Rmin to Rmax represents
the volume within 14 mm of the embryo surface, which is equivalent to approximately 33%
of the total embryo volume and encompasses all syncytial nuclei. We use a phase symmetry calculation function (“phasesym.m”) from a MATLAB computer vision library [75] to
score the composite fluorescence image of the embryo at the mid-sagittal plane (Fig. 2.7a).
The phase symmetry image is then thresholded to identify the vitelline membrane. The
resulting mask is filled to determine the embryo mask, and this mask is then eroded with
a disk structuring element of diameter corresponding to 14 mm. This gives a total of three
masks identifying the embryo, the vitelline membrane, and the yolk, respectively. To calculate Sf ocal , we create a cortical mask, consisting of the region that belongs to the embryo
mask but not to the vitelline mask or to the yolk mask (Fig. 2.7b). We integrate over the
33
cortex alone rather than the entire embryo as including fluorescence from the yolk only reduces the signal-to-noise ratio of the measurement. As Sf ocal is intended to be a measure of
quantity of fluorescent molecules rather than concentration, it remains to multiply fluorescence intensity measurements in the cortical mask, which are proportional to fluorophore
concentration, by the volume they represent. Practically, this means that cortical intensity
measurements must be weighted by their distance r from the anterior-posterior (A-P) axis,
since we assume that the embryo has azimuthal symmetry. To determine the A-P axis, we
find the maximal line segment that can be inscribed on the previously calculated embryo
mask. We then subdivide the cortical mask into 400 boxes of approximately equal size using the prolate spheroidal coordinate system (see Appendix A), with the dorsal and ventral
sides containing 200 boxes each. Rather than performing a explicit pointwise integration
between Rmin (l) and Rmax (l), we instead find the mean radius R̄(l) for A-P axis position l
of each box by calculating the distance between the centroids of corresponding dorsal and
ventral boxes. Similarly, we find the mean fluorescence intensity I¯f luor (r, l) in each box
of the cortical mask. The integration of Sf ocal over r is then performed by trapezoidal rule
and the integration over l by Riemann sum.
Sf ocal includes signal from both Dronpa-Bcd and extraneous autofluorescence in the
embryo.
Z
Sf ocal = α
CDronpa−bright (~r)d~r + A
(2.8)
CDronpa−bright (~r) is the concentration of bright-state Dronpa-Bcd at all points in the embryo. A represents the total autofluorescence and may vary with time. There is an unknown
factor a that converts absolute fluorophore concentration to fluorescence intensity, which
for the purposes of the degradation measurement we set to unity.
To obtain values for the absolute quantity of mature Dronpa-Bcd in this region, Qf ocal ,
we acquire a sequence of images at 496 nm and differentiate Sf ocal with respect to scan
34
a
b
Figure 2.7: Example of cortical mask selection for degradation rate measurement a)
Image of green channel fluorescence intensity (If luor ) of a Dronpa-Bcd expressing embryo
in mid-cycle-14. b) Cortical mask selected by eroding an embryo mask created after identifying the vitelline membrane by computer vision methods (see section 2.3.5). This mask
is rotated around the A-P axis to create a volume of revolution which represents the region
of integration for Sf ocal , or the volume within approximately 14 mm of the embryo surface.
35
index. The initial quantity of fluorophore in the bright state, Qbf (0), can be fit using the
dark conversion rate koff determined in Fig. 2.5b.
Z
Sf ocal (n) =
CDronpa−bright (~r, n)d~r + A(n)
Z
=
CDronpa−bright (~r, 0)e−kof f n d~r + A(n)
= e−kof f n Qbf (0) + A(n)
(2.9)
(2.10)
(2.11)
The mean time series of autofluorescent signal A is measured by imaging wild-type (OregonR) embryos under the same experimental conditions as Dronpa-Bcd embryos, and is subtracted from Sf ocal . We then use a weighted least squares fitting method to extrapolate the
original amount of bright-state Dronpa-Bcd Qbf (0) and the final amount after the conversion series Qbf (t = δ). We find that variance in the degradation rate measurement is minimized by setting the weights of all time points but equal to zero; this approach is optimal
if error is dominated by variation in koff . However, we find that the result is substantially
identical if all weights are chosen to be equal.
An example of this fitting procedure is shown in Fig. 2.8b. The value of koff used in the
fit is adjusted for laser power fluctuations based on the linearity established in Fig. 2.5d.
Experimentally, we use a series of 16 496 nm scans intensity averaged into 8 images for this
measurement. If the conversion series immediately follows a complete activation sequence
with 405 nm light, then we assume Qf ocal (0) = Qbf (0). If the activation sequence is not
complete, then a correction for the residual dark state amount is added, as described in
section 2.3.10.
2.3.6
Determination of irreversible photoconversion rate
The dark state conversion rate is composed of both reversible and irreversible photoconversion: koff = koff-rev + koff-irrev . As reported in Ref. [50], we observe excellent reversibility of
36
Sfocal (arb.)
10
a
9
T
8
7
6
0
0.1
0.2
7.9
t (min.)
8.0
8.1
8.2
Sfocal − A (arb.)
4
3
Qbf(t0−δ)
Q (t +T+δ )
bf 0
2
b
a
Q (t +T)
bf 0
Q (t )
bf 0
1
0
0
0.1
0.2
7.9
t (min.)
8.0
8.1
8.2
Figure 2.8: Description of photoconversion quantity calculation a) Integrated fluorescence intensity Sf ocal at the embryo surface, for a Dronpa-Bcd expressing embryo (blue)
and an Oregon-R wild-type embryo (black, also denoted as autofluorescence level A). Each
square represents integrated fluorescence for a single image, which is produced by averaging two individual scans. T indicates waiting interval duration, which is 8 minutes in this
example. Arrow indicates the time of 405 nm reactivation pulse. b) Integrated fluorescence intensity after background subtraction, Sf ocal − A, indicated by black squares. Actual endpoints of each conversion series (red asterisks) indicate the quantity of bright-state
Dronpa-Bcd Qbf at various times and are extrapolated assuming a single conversion rate
koff , determined as in Fig. 2.5b. Differences between these endpoints are used to calculate
amount of Dronpa-Bcd transferred to the dark state, Qdf (t0 ), and amount of Dronpa-Bcd
recovered from the dark state, Qdf (t0 + T ), after a minor correction for residual dark-state
Dronpa from incomplete Dronpa reactivation (Eq. 2.26).
37
conversion between the bright and dark states of Dronpa with our dronpa-bcd transgene.
In Fig. 2.9a, we show the values of Qf ocal measured for two embryos imaged on the same
slide but with one at twice the repetition rate of the other. Normalizing each to the quantity observed at the beginning of cycle 14, the difference accumulating due to irreversible
bleaching arising from the imaging process is small despite a relative excess of 600 scans
at 496 nm and 75 scans at 405 nm. Hence we neglect the irreversible component and take
koff u koff-rev .
2.3.7
Isolation of photoconverted state
The concept behind using Dronpa-Bcd to measure degradation rests on the assumption that
a population of molecules can be transferred to a state isolated from synthesis, allowing the
decay rate of the transferred population to be observed. With Dronpa-Bcd we use the dark
state to achieve this isolation. Though it is not possible to measure its occupancy directly
we can measure it indirectly via the differential in bright-state fluorophore after a 405 nm
pulse.
Separating the observable fluorophore population into bright-state and dark-state components, we can write the full equations for the time evolution of these quantities:
∂Qdf
= U m(t)γ(t) − kdeg Qdf − kinc Qdf + Φdark (t)
∂t
∂Qbf
= (1 − U )m(t)γ(t) − kdeg Qbf + kinc Qdf + Φbright (t)
∂t
(2.12)
(2.13)
where m(t) represents newly maturing Dronpa-Bcd, γ(t) is the fraction of Dronpa-Bcd
in the surface region of integration, and Φdark (t)/Φbright (t) represent the flux of dark and
bright Dronpa-Bcd into the surface region, respectively (see section 2.3.9 for details). The
incidental relaxation rate of Dronpa-Bcd from the nonfluorescent dark state back to the
bright state is given by kinc , and includes effects due to thermal relaxation as well as photoconversion from ambient light. We let U denote the fraction of newly matured fluorophore
38
a
Qfocal/Q0
1.6
1.4
1.2
1
0.8
0
20
p(∆log(Qn+δ/Qn))
0.4
b
0.3
0.2
0.1
0
−0.1
40
t
8
c
1.15
Sfocal
0.1
x 10
1.1
d
0.2
0.15
1.05
1
0.95
0
p(∆Sfocal/Sfocal)
1.2
0
∆log(Qn+δ/Qn)
0.1
0.05
20
40
0
−0.02
t
0
∆Sfocal/Sfocal
0.02
Figure 2.9: Quantification of irreversible photobleaching and dark state isolation a)
Qf ocal normalized to the value at onset of interphase 14 in two embryos. t is in minutes.
Bullets: Embryo imaged once per 45 s. Squares: Embryo imaged once per 90 s. Each time
point represent a dark conversion and reactivation series of 24 scans at 496 nm and 3 scans
at 405 nm. b) Histogram of difference in changes in Qf ocal between the embryos in (a).
δ = 2 for the more frequently imaged embryo and δ = 1 for the less frequently imaged em-
emb1
emb2
bryo. The quantity plotted is ∆log(Qn+δ /Qn ) = log Qemb1
− log Qemb2
n+2 /Qn
n+1 /Qn
for all n. The mean value is −0.005±.006, where .006 is the standard error. c) Sf ocal shown
for three embryos (bullets) extracted from individual images. Arrows indicate 405 nm photoactivation pulses, occurring between consecutive images. ∆Sf ocal = Sffocal − Sfi ocal ,
where Sfi ocal is taken immediately preceding the 405 nm pulse, and Sffocal is taken immediately after. A positive value of ∆Sf ocal would indicate maturation of new Dronpa-Bcd into
the dark state. t is in minutes. d) Distribution of ∆Sf ocal /Sfi ocal , over all embryos.
39
that appears initially in the dark state.
In Fig. 2.9c, we verify that U is negligibly small in this experiment. We collect a set of
20 embryos at various stages prior to gastrulation and acquire two single images at 496 nm,
performing a single scan at 405 nm in between. Following a time lapse of 8 minutes this
procedure is repeated. The fractional change in Sf ocal between the 496 nm images (Fig.
2.9d) is distributed around zero, demonstrating that no fluorophore entered the dark state
during the time interval. Additionally, we know that kinc is negligibly small based on our
experiment described in Fig. 2.1e, and as reported in Ref. [50].
2.3.8
Uniformity of photoconversion
Ideally Dronpa-Bcd will be photoconverted uniformly throughout the embryo, to avoid
equilibration between variably converted regions masking the effect of degradation. Lipid
droplets in the embryo [76] inhibit imaging beyond the surface layer of the embryo on
the side facing the objective, though they may only impede emission collection rather than
absorption.
Let G(~r, t) be the fraction of mature fluorophore in the dark state at all points in the
embryo: G(~r, t)=CDronpa−dark (~r, t)/CDronpa (~r, t). This quantity changes in time as new
synthesis affects the bright and dark state populations asymmetrically. We consider simply
G0 (~r), the profile of the conversion percentage G obtained after a single conversion series
of 16 scans at 496 nm, assuming all fluorophore begins in the bright state.
To determine the spatial dependence in G0 (~r), we acquire a fluorescence image on the
obverse face of an embryo (Fig. 2.10a), followed by a 496 nm photoconversion series at the
mid-sagittal plane, followed by a second image at the original location (Fig. 2.10b). The
slide is then flipped over and the reverse side of the embryo is imaged at the same distance
from the mid-sagittal plane (Fig. 2.10c). Intensity gradients extracted for an ensemble of
embryos pre- and post-conversion are shown in Fig. 2.10d, indicating that the obverse and
reverse face gradients are indistinguishable. We assume that Dronpa-Bcd in the yolk of the
40
60
a
I (arb.)
b
d
50
40
30
20
c
10
0
0.2
0.4
x/L
0.6
60
e
I (arb.)
f
h
50
40
30
20
g
10
0
0.2
0.4
x/L
0.6
Figure 2.10: Transparency of embryo to 496 nm and 405 nm photoconversion a) Obverse face of embryo 60 mm above mid-sagittal plane prior to photoconversion b) Same as
(a) post-photoconversion c) Reverse face of embryo 60 mm above mid-sagittal plane postphotoconversion d) Mean intensity gradients acquired from images in (a-c) for a set of 5
embryos. (a) diamonds, (b) squares, (c) circles. Error bars represent standard error. e)
Obverse face of embryo 60 mm above mid-sagittal plane prior to photoactivation f) Same
as (e) post-photoactivation g) Reverse face of embryo 60 mm above mid-sagittal plane postphotoactivation h) Mean intensity gradients acquired from images in (e-g) for a set of 3
embryos. (e) diamonds, (f) squares, (g) circles.
embryo is photoconverted in the same way, and take spatial inhomogeneities in G0 to be
negligible. If degradation and synthesis occur in equal proportion throughout the embryo
as well, then G(~r, t) = G(t) in general.
Additionally we require that 405 nm photoactivation be applied uniformly to the entire
embryo. Although the delay between 496 nm images pre- and post-activation is too short
for diffusion to be a concern, we want to be able to do multiple degradation experiments
in a single embryo while avoiding accumulation of dark-state fluorophore. To measure
transparency we first apply a 496 nm photoconversion series at the mid-sagittal plane to
41
populate the dark state. This is followed by a 496 nm image at the obverse (Fig. 2.10e),
a 405 nm photoactivation pulse at the mid-sagittal plane, a second 496 nm image at the
original location (Fig. 2.10f), and finally a 496 nm image on the reverse side at the same
distance from the mid-sagittal plane (Fig. 2.10g). Fig. 2.10h shows that the intensity
gradients extracted from the obverse and reverse sides are indistinguishable following the
405 nm pulse.
2.3.9
Relation of total Bcd quantities to measured fluorescence values
We assume that the concentration of bright-state Dronpa-Bcd in the focal volume is linearly proportional to fluorescence intensity when corrected for excitation power variation.
However, to obtain measurements of Bcd synthesis and degradation, the true quantity of
interest is the amount of Dronpa-Bcd in a closed volume rather than concentrations of fluorophore. While the entire egg is guaranteed to be a closed volume, for technical reasons
we are unable to image the egg in its entirely. Rather, we use a region of integration at the
embryo surface as the closed volume, and attempt to quantify the amount of Bcd flux that
violates this assumption.
The total quantity of Bcd in the egg, QBcd , has a mature and immature component
QBcd = Qmat + Qimm . Assuming first-order kinetics, these quantities evolve according to
the following equations:
∂QBcd
= p(t) − kdeg QBcd
∂t
∂Qimm
= p(t) − kmat Qimm − kdeg Qimm
∂t
∂Qmat
= kmat Qimm − kdeg Qmat
∂t
(2.14)
(2.15)
(2.16)
p(t) represents the rate of synthesis of new protein, and kmat and kdeg are the rates of maturation and degradation, respectively. Measurement of the maturation lifetime of Dronpa
42
in vivo is beyond the scope of this work, and Qimm cannot be measured; therefore, we
combine these quantities into m(t) = kmat Qimm to simplify the equations:
∂Qmat
= m(t) − kdeg Qmat
∂t
(2.17)
Only a fraction of the total mature fluorophore will be located such that its emission will
be observed in the focal region of a confocal microscope. This is distinguished from the
question of absorption of incident light, because while scattered photons from the excitation
beam may be absorbed far from the focus, their emission signal will be excluded by the
confocal pinhole. As mentioned previously, the Drosophila embryo is known to contain a
distribution of yolk granules as well as lipid droplets which are rearranged in a controlled
fashion during the mitotic division cycles prior to gastrulation [76]. These droplets scatter
light in visible wavelengths and affect the size of the region of the embryo observable under
fluorescence microscopy. The total quantity of mature Dronpa-Bcd thus breaks down into
that which is observable and located in our region of integration (Qf ocal ) and that which is
not (Qobscure ), such that Qmat = Qf ocal + Qobscure :
∂Qf ocal
= m(t)γimm (t) − kdeg Qf ocal + Φ(t)
∂t
∂Qobscure
= m(t)(1 − γimm (t)) − kdeg Qobscure − Φ(t)
∂t
(2.18)
(2.19)
We let Φ(t) denote the flux of mature fluorophore into the region of integration, comprising the effect due to both the actual transport of fluorophores and the resizing of the
observable region. Φ is a potentially significant quantity as a nonzero value invalidates
the assumption that the region of integration is a closed volume, introducing an error into
our attempt to measure the degradation. γimm (t) represents the fraction of total immature
fluorophore located in the focal region. For simplicity, we assume that both immature and
mature fluorophore is distributed between the observable and obscure regions in the same
43
way, making the substitution γimm (t) = γmat (t) = γ(t). Hence Qf ocal = γ(t)Qmat and
Qobscure = (1 − γ(t))Qmat . Subsequently:
∂Qf ocal
= γ(t) [m(t) − kdeg Qmat ] + Φ(t)
∂t
∂Qmat
= γ(t)
+ Φ(t)
∂t
∂Qobscure
= (1 − γ(t)) [m(t) − kdeg Qmat ] − Φ(t)
∂t
∂Qmat
− Φ(t)
= (1 − γ(t))
∂t
(2.20)
(2.21)
(2.22)
(2.23)
However, by chain rule we also obtain Q̇f ocal = γ Q̇mat + γ̇Qmat and so we can simply
replace Φ(t) with Φ(t) = γ̇(t)Qmat . This quantity will be used in section 2.3.12 to correct
for flux after obtaining estimates of γ(t).
2.3.10
Description of degradation measurement
As outlined previously, the degradation of a population of Dronpa-Bcd is measured by the
time evolution of a quantity of dark-state fluorophore. We separate the total quantity of
mature Dronpa-Bcd in the surface region of integration into bright-state (Qbf ) and darkstate (Qdf ) components, such that Qf ocal = Qbf + Qdf . An initial quantity of bright-state
Dronpa-Bcd Qbf (t0 − δ) is determined according to the method described in section 2.3.5.
The quantity transferred into the dark state in the same imaging series is given by:
Qdf (t0 ) − Qdf (t0 − δ) = Qbf (t0 − δ) 1 − e−kof f ∗16
(2.24)
Using koff as determined in Fig. 2.5b, this is equivalent to 72% of the total mature fluorophore being photoconverted within the series acquisition time, δ, which is 13 s. Following a waiting interval of duration T another set of images is taken at t0 + T . The quantity
of bright-state fluorophore Qbf (t0 + T ) is measured first with a series of 8 496 nm scans
averaged into 4 images by the same quantification method. The duration of this conversion
44
series, δa , is 7 s. Immediately following is a 405 nm photoactivation pulse of 3 scans, resulting in ξ = 1 − e−kon ∗3 , or 98%, of the dark-state population being returned to the bright
state. A final conversion series of 16 496 nm scans is then performed and used to determine
the quantity Qbf (t0 + T + δa ), as well as providing an initial dark-state population for a
subsequent repetition of the experiment.
The amount of dark-state fluorophore at the end of the waiting interval is given by:
Qdf (t0 + T ) = ξ −1 (Qbf (t0 + T + δa ) − Qbf (t0 + T ))
(2.25)
The initial concentration of dark-state fluorophore Qdf (t0 − δ), preceding even the 496
nm photoconversion series, is determined from the previous photoactivation pulse:
Qdf (t0 − δ) =
1−ξ
(Qbf (t0 − δ) − Qbf (t0 − δ − δa ))
ξ
(2.26)
In a series of sequential conversions and reactivations, this correction will become small if
there is a slowly varying total Dronpa-Bcd population.
Degradation rates were measured from a batch of 150 embryos on six slides, repeating
the degradation measurement from 10 to 30 times on each as they age. All embryos used
in the analysis reach gastrulation. Also, for data quality, a minimum Dronpa-Bcd intensity
measured at the peak of interphase 14 is chosen and embryos not meeting this threshold
are eliminated. Degradation rates are extracted from the remaining embryos and synchronized according to stage, by inspection, at a temporal resolution of approximately 1 minute,
significantly more precise than grouping by mitotic division cycle alone.
2.3.11
Correction for Dronpa-Bcd flux
In the simplest interpretation of our data, the rate of Dronpa-Bcd degradation kdeg can be
directly inferred from the quantities of dark-state fluorophore measured at times t0 and
45
t0 + T . Let ksimp be the degradation rate measured according to the assumption that Φ = 0.
ksimp =
log (Qdf (t0 )) − log (Qdf (t0 + T ))
T
(2.27)
Allowing nonzero flux of dark-state fluorophore into the focal region Φ implies that kdeg 6=
ksimp . However, with an estimate for Φ(t) we can solve for kdeg .
Removing terms denoting thermal reactivation of Dronpa and maturation into the dark
state, both of which are negligible, Eq. 2.12 becomes
∂Qdf
= −kdeg Qdf + Φdark (t)
∂t
(2.28)
which is solved by
Zt
Qdf (t) =
0
e−kdeg (t−t ) Φdark (t0 )dt0 + Qdf (t0 )e−kdeg (t−t0 )
(2.29)
t0
We then note that this can be further simplified:
 t

Z
0
Qdf (t)
Φdark (t ) 0
0
= e−kdeg (t−t0 )  ekdeg (t −t0 )
dt + 1
Qdf (t0 )
Qdf (t0 )
t

 0t
Z
0
γ̇(t ) 0
= e−kdeg (t−t0 ) 
dt + 1
γ(t0 )
(2.30)
(2.31)
t0
0
where in the integral we have made use of the fact that Qdark (t0 ) = Qdark (t0 )e−kdeg (t −t0 ) .
46
We can solve explicitly for the degradation rate, now corrected for flux:
 t +T

Z0
0
γ̇(t ) 0
Qdf (t0 + T )
1
1
+ log 
dt + 1
= − log
T
Qdf (t0 )
T
γ(t0 )
t0
 t +T

Z0
γ̇(t0 ) 0
1

= ksimp + log
dt + 1
T
γ(t0 )
kdeg
(2.32)
(2.33)
t0
Eq. 2.33 is used in Fig. 2.2e to obtain curves of kdeg after correction for each estimate
of monotonic flux.
2.3.12
Estimation of Dronpa-Bcd flux correction
As shown in section 2.3.9, Φ(t) is given by γ̇(t)Qmat , and can be determined exactly if the
fraction of fluorophore in the observable region γ is known at all times. We can estimate
γ indirectly by measuring the distribution of Dronpa-Bcd between cortex and core in fixed
embryonic cross-sections. Additionally, though we cannot obtain an absolute value for γ,
we can estimate the relative change in γ, or γ̇/γ , by measuring the time evolution of Qf ocal
for fluorescently-tagged histone, H2A-RFP. The relative change γ̇/γ is all that is necessary
to compute the correction to the degradation rate measurement, as shown in Eq. 2.33.
We fix 100 Dronpa-Bcd expressing embryos of various ages less than 3 hours postoviposition and slice them in cross-sections perpendicular to the A-P axis. We acquire images of specific Dronpa-Bcd fluorescence by using 405 and 496 nm illumination to switch
between Dronpa bright and dark states, and then subtract consecutive images. These images are computationally segmented into 14 annuli of maximum radius r, and the fraction of Dronpa-Bcd in each annulus, Q(r)/ΣQ, is calculated. A uniform distribution of
Dronpa-Bcd will give Q(r)/ΣQ ∝ r. We then stage the slices according to mitotic cycle
by inspection, and estimate γ(t) by summing the values in the outermost annuli, corresponding to the surface region of integration. The result of this experiment is illustrated in
47
Q (r)/ΣQ
0.3
0.2
0.1
0
0
cycle: <12
12/13
14
r
a
b
0.5
r/R
1
Figure 2.11: Illustration of Bcd flux quantification from embryonic cross-sections a)
As in Fig. 2.2d, bars indicate a measurement of the fractional distribution of Dronpa-Bcd in
cross-sections of embryos in cycle 11 and prior (blue), in cycles 12 and 13 (green), and in
cycle 14 (red). Q(r)/ΣQ represents the fraction of total Dronpa-Bcd in an annulus of fixed
width at distance r from the longitudinal axis of the embryo. The dashed line indicates the
uniform distribution. b) Sample image of an anterior cross-section from a Dronpa-Bcdexpressing embryo fixed in mitotic cycle 13. Intensity is computed by subtraction of a
image with Dronpa-Bcd predominantly in the dark state from an image with Dronpa-Bcd
in the bright state. Bright ovals near the embryo surface are nuclei. Diffuse fluorescence
extending inwards is taken to represent Dronpa-Bcd, consistent with the high specificity of
Dronpa photoconversion (see section 4.3.2).
Fig. 2.11.
Though it is believed that all histones necessary for the pre-gastrula are present at fertilization and stored in lipid droplets until they become needed after several cycles of DNA
replication [77], an increase in the observed intensity of RFP-tagged histone is nevertheless
observed when imaging embryos in the mitotic cycles prior to gastrulation under confocal
microscopy. Many of the same factors that effect an increase in intensity of fluorescently
tagged histone, such as clearing of the cortical cytoplasm, movement of nuclei to the cortex,
and changes in pH, also impact the intensity of fluorescently tagged Bcd. Combined with
the supposition that the quantity of histone molecules is constant with no new synthesis
from translation of maternal RNA, it seems reasonable to consider H2A-RFP intensity as a
proxy for determining γ̇/γ . We use embryos for which part of the maternally deposited H2A
is conjugated to RFP and measure surface intensity Sf ocal , here using the red rather than
the green channel, by the method of section 2.3.5. The background value A is determined
48
by an identical measurement of Oregon-R embryos. As there is no significant photoconversion of H2A-RFP, we assume Qf ocal ∝ Sf ocal − A for a series of time points spaced
over the mitotic cycles prior to gastrulation, and assume that all change in Qf ocal is due to
redistribution of fluorophore or changes in the opacity of the surface region. The relative
change in γ, γ̇/γ , can be directly extracted given these assumptions.
2.3.13
Establishment of upper bound on Dronpa-Bcd flux
In addition to the estimates of reasonable values of Φ(t), as shown in section 2.3.12, it
is possible to establish an upper bound on Φ(t) based on measurements of Qf ocal over
the course of the mitotic cycles prior to gastrulation. Let us first assume that Qmat (0) =
Qimm (0) = 0 and that ṗ(t|t[0, T ]) ≥ 0. It follows that Q̇mat (t|t[0, T ]) ≥ 0. As shown in
section 2.3.9, Φ(t) = Q̇f ocal − γ Q̇mat , and since γ as well as Q̇mat is positive definite, we
see that Φ(t|t[0, T ]) ≤ Q̇f ocal . Note that this result obtains only as long as the synthesis
rate p(t) is a monotone increasing function. We place no lower bound on Φ(t), limiting us
to an upper bound on kdeg .
Analogous to previous notation, Φdark (t) = D(t)Φ(t) is the flux of dark-state fluorophore into the focal region. Assuming that dark-state fluorophore is transported identically as other fluorophore, we replace γdark with γ and see Φdark (t) = γ̇(t)Qdark =
D(t)γ̇(t)Qmat . As D(t) ≤ D0 , then Φdark (t) = D(t)Φ(t) ≤ D0 Φ+ (t), where Φ+ (t) denotes the positive part of Φ(t). This gives a bound Φdark (t) ≤ D0 Q̇+
f ocal (t), which we can
use in Eq. 2.28.
We can find the upper bound on kdeg , kdeg ≤ kdegmax by substituting the bound on Φdark
and solving Eq. 2.30 implicitly. This is possible because ∂Qdf/∂kdeg is negative definite, and
when substituting a bound on flux, Φ0 (t0 ) ≥ Φdark (t0 ) ∀ t0 [0, t], then Q0df (t) − Qdf (t) =
Rt
0
e−kdeg t 0 ekdeg t (Φ0 (t0 ) − Φdark (t0 )) dt0 ≥ 0. This guarantees that kdegmax which we obtain
from substituting the bound is greater than kdeg . Practically, we find kdegmax by using
49
k deg (min -1)
0.06
0.04
0.02
0
100
12
13
120
14
140 160
t (min.)
180
Figure 2.12: Establishment of upper bound on kdeg Best estimates of kdeg as shown in
Fig. 2.2e, where the black curve shows the degradation rate after removal of cell-cycleperiodic oscillations (see section 2.3.14) and the green and red curves represent kdeg after
correction for core-to-cortex flux Φ(t) as estimated by cross-section measurements and
Histone-RFP measurements, respectively (see section 2.3.12). The blue curve represents
kdegmax , calculated as described in section 2.3.13 using the assumption that Bcd synthesis
is monotonically increasing prior to cycle 14. While kdegmax provides a firm upper bound
on kdeg , it is assumed that the green and red curves provide better estimates of the actual
flux, and, by extension, the value of kdeg .
Newton’s method to locate the root of the following function:

f (kdegmax ) =
Qdf (t)
− e−kdegmax (t−t0 ) 
Qdf (t0 )
Zt
Q̇+ (t0 )
kdegmax (t0 −t0 ) f ocal
e
Qf ocal (t0 )

dt0 + 1
t0
where in the integral we have made use of the fact that Qdf (t0 ) = D0 Qf ocal (t0 ). The result
is shown in Fig. 2.12, where the blue curve represents the upper bound on kdeg obtained
from consideration of the time evolution of Qf ocal alone.
2.3.14
Removal of cell-cycle-periodic oscillation
To arrive at a value of kdeg corrected for cell-cycle-periodic oscillations but not for any
monotonic flux, we assume that γ(t) = 1 + Csin(ωt + φ), where ω is the frequency of
mitotic divisions. This will describe a general sinusoidal flux of a single frequency. For
50
single-exponential decay, this will give
Qdf (t + T )
−T kdeg (t) 1 + Csin (ω(t + T ) + φ)
=e
Qdf (t)
1 + Csin (ωt + φ)
The bracketed term oscillates with period
2π/ω .
(2.34)
We eliminate the periodic oscillation
as follows: a linear least squares fit is performed on the curves f (t) =
Qdf (t+T )/Q (t)
df
for
waiting intervals T = 4, 8, and 15 minutes. The curves are then de-trended, by removing any
linear trend as determined by least-squares fit, and decomposed into Fourier vectors. We
remove Fourier components with period greater than 12 minutes, which is the approximate
length of mitotic cycles 12 and 13. The Fourier transform is then inverted and the linear
trend reintroduced to give an adjusted fe(t) with cell-cycle-periodic oscillations removed,
from which the corrected degradation rate can be directly extracted.
2.3.15
Measurement of total Bcd protein by Western blot
We extracted total protein from embryos that develop within consecutive 30 minute time
windows following fertilization. More than 95% of embryos collected in this way are
within the predicted 30 minute time window as determined by scoring the onset of cephalic
furrow formation. Anti-GFP antibody is a mixture of two mouse monoclonal antibodies
purchased from Roche (Cat# 11 814 460 001) used 1:100 in 1xPBS, 0.1% Tween 20, and
10% non-fat dry milk for ~3 hours at room temperature. The relative amounts of Bcd are
then established by detecting a chemi-luminescent signal with a CCD camera [78]. Shown
in Fig. 2.3a is a typical western blot and, in Fig. 2.3b, the quantification of 5 different experiments that represent a total of 400 embryos per lane. By measuring the integrated CCD
intensity of known quantities of rGFP, we can calibrate the Bcd western blot to determine
the absolute number of Bcd molecules in the embryo at each stage. Recombinant GFP
(rGFP) was purchased from Clontech (Cat# 63 23 73). In Fig. 2.13 is plotted the intensity
obtained for a variety of masses M of rGFP, which are well fit to a line that passes through
51
I (arb.)
15
10
5
0
0
200
400
M (pg)
600
Figure 2.13: Calibration of blot intensity to molecule count Calibration of Bcd Western
blots to absolute number of Bcd molecules. Circles indicate integrated CCD intensity of
known masses M of rGFP, measured in picograms. Dashed line is a least squares fit of these
points anchored to the origin. Blue line indicates integrated CCD intensity of GFP-Bcd at
time point t = 135 min., which intersects the best-fit line at M = 75 pg of rGFP.
the origin. This establishes linearity of protein mass with blot intensity in the regime tested,
which includes the intensity corresponding to GFP-Bcd from early cycle 14 embryos.
2.3.16
Measurement of gradient shift under optically mimicked degradation
We determine the rate of apparent Dronpa-Bcd degradation induced by imaging at a given
496 nm laser power in units of 1/scans, as shown in Fig. 2.14. In contrast to the lifetime
experiments, rapidity of photoconversion is not of primary importance; hence we use a
lower excitation power and the conversion rate is not identical to that found in Fig. 2.5.
Having determined the rate of degradation added by photoconversion, we then mount
embryos expressing Dronpa-Bcd as well as Histone-RFP (H2A-RFP) on a slide along with
embryos expressing only Histone-RFP, and image both groups in a looping sequence for 70
minutes. Fluorescence images in green (520-570 nm) and red (590-660 nm) channels are
collected. A distribution of optically augmented degradation rates is obtained by choosing
a different total number of embryos over which to loop (N = 12, 16, 20, 28, and 36) for each
52
1.2
log(I)
1
0.8
0.6
0.4
0.2
0
kadd = 0.009 scans−1
20
40
n (scans)
60
Figure 2.14: Quantification of mimicked Dronpa-Bcd degradation induced by photoconversion I indicates fluorescence intensity of a region containing highly Dronpa-Bcd
expressing nuclei at the anterior of the embryo, after subtraction of Histone-RFP background. n indicates scan number. Blue line is a linear least-squares fit. In units of scan
frequency, each scan augments fluorophore degradation by kadd = 0.009 scans−1 . Each
70 scan experiment takes less than 50 s, rendering the effect of endogenous degradation
negligible.
slide, which effects a change in the scan frequency and, by extension, the rate of optically
augmented degradation.
After 70 minutes of images have been collected at the augmented optical degradation
rate, a single loop of 405 nm scans is performed to reactivate all dark-state Dronpa-Bcd in
the embryo. Subsequently two additional imaging loops at 496 nm are performed. These
two images are averaged for each embryo to determine the full Bcd gradient after Dronpa
reactivation, and the two images of each embryo immediately prior to 405 nm reactivation are averaged to determine the gradient after equilibration to the optically augmented
degradation rate.
The nuclear concentration gradient for each embryo is determined by creating a nuclear
mask in MATLAB using the red channel fluorescence image, which indicates the HistoneRFP signal (Fig. 2.15a). The fluorescence intensity in the red channel is thresholded to
create an initial mask and the final nuclear mask is given by the union between the initial mask and the cortical mask calculated by the method described in section 2.3.5. An
53
a
b
Figure 2.15: Identification of nuclear mask for Bcd gradient computation a) Image
of red channel fluorescence intensity of a Dronpa-Bcd/H2A-RFP expressing embryo in
mid-cycle-14. Red channel shows Histone-RFP fluorescence and negligible Dronpa-Bcd
fluorescence. b) Nuclear mask selected by thresholding the image in (a) and taking the
union with the cortical mask as calculated by the method described in section 2.3.5 and
Fig. 2.7.
example of the resulting nuclear mask is shown in Fig. 2.15b.
Absolute Dronpa-Bcd intensity is determined in the following way. We divide the nuclear mask of each image into 200 boxes projected onto the A-P axis, and calculate the
mean intensity in both green and red channels in each box. The mean green intensity in
each box is then divided by the corresponding mean red intensity to effectively normalize
the green signal by the DNA content. Smoothed green channel gradients from the HistoneRFP expressing embryos are then subtracted from the Dronpa-Bcd/Histone-RFP gradients
for which they are matched controls, as determined by identical scan frequency and pre/post- activation classification.
54
In the one-dimensional synthesis-diffusion-degradation model with idealized source
conditions (constant synthesis strictly located at the anterior tip), the equilibrium morphogen distribution takes the form of an exponentially decaying profile combined with
a reflection due to the imposition of a no-flux boundary condition at the posterior end:
c = B e− /λ + e
x
x−2L/λ
(2.35)
where c is the morphogen concentration, B is a constant, and λ is the characteristic length
√
Dτ . We fit the observed gradients to this equation using a least-squares method to quantify the change in characteristic length effected by the enhanced degradation of the morphogen. We find a result that is qualitatively similar to that obtained from fitting to a more
accurate model, incorporating a realistic source distribution, as described in section 2.3.18
and Fig. 2.4b, inset.
2.3.17
Calculation of Bcd synthesis rate from Western analysis
Models that account for the dynamics of the Bcd gradient generally assume constant production of Bcd beginning at fertilization [21]; however, existing evidence for this assumption is equivocal. On one hand, the mRNA encoding Bcd appears constant until cycle 14
when it decays (Fig. 2.3c) [79, 21]. On the other, the dynamics of the polyA tail length
suggest that synthesis of Bcd might not be constant [69, 70]. We perform two independent
measurements that each indirectly reflect the synthesis rate of Bcd. One is the total accumulation of Bcd in the embryo over time, extracted from Western blotting, from which we
can extract the production rate based on our measurement of Bcd degradation. The other
is based on the appearance of new Dronpa-Bcd fluorescence after prior photo-conversion,
reflecting the rate of synthesis plus a delay dependent on the maturation time of the fluorophore.
55
7
6
x 10
b
10
5
0
0
x 10
a
p (molec./min.)
QBcd (molec.)
15
6
100
t (min.)
200
4
2
0
0
100
t (min.)
200
Figure 2.16: Calculation of Bcd synthesis rate a) Quantification of an ensemble of 5
Bcd Western blots (Fig. 2.3b). b) Production rate p(t) of new Bcd protein implied by
Bcd quantity in (a) and degradation rate time series in Fig. 2.2e, extended based on the
assumption that kdeg = 0.020 min−1 prior to cycle 11 and smoothed by moving average to
match time resolution of western blot. Production rate increases until cycle 14 and drops
off to zero once gastrulation begins. Dashed lines indicate error as propagated from QBcd
and kdeg . Units indicate the number of Bcd molecules translated per minute.
If the overall amount of Bcd in the embryo changes according to
dQBcd
= p(t) − kdeg QBcd
dt
(2.36)
where p(t) is the time-dependent synthesis of Bcd, kdeg the degradation rate, and QBcd
the total quantity of Bcd in the embryo, then, assuming that kdeg = 0.020 min−1 prior
to cycle 14, and that QBcd is as determined from Western blotting, we obtain the Bcd
synthesis function, p(t), shown in Fig. 2.16b. Though error bars are large, as this number
is calculated from two independent quantities, we see a definite increase up to cycle 14. If
we allow that the dynamics of the polyA tail of bcd mRNA could effect an increasing rate
of protein synthesis even with a constant mRNA level, it is consistent with measurement of
bcd mRNA by RT-PCR (Fig. 2.3c), which indicates relatively constant mRNA levels until
cycle 14, followed by complete extinction in the gastrula.
56
2.3.18
Simulation of gradient shape
The predicted Bcd gradient and expected gradient shifts shown in Fig. 2.4b, inset, are
calculated by a computational simulation of Bcd gradient formation. We use the onedimensional diffusion equation with uniform degradation
∂c(x, t)
= s(x, t) + D∇2 c(x, t) − kdeg (t)c(x, t)
∂t
(2.37)
This equation is integrated using a forward-time centered-space finite difference scheme,
imposing the Neumann condition ∂c/∂x = 0 at the anterior and posterior boundaries of the
embryo, x = 0 and x = L = 500 µm, respectively. We use a mesh size of 5 µm and a
time step of 2 s. Neither the source function, s(x, t), nor the degradation rate, kdeg (t), is
assumed to be constant. kdeg is given by the red curve in Fig. 2.2e, extended backwards in
time on the assumption that kdeg (0 < t < 105 min.) = 0.020 min−1 . We further separate
the source function into time-varying and space-varying components.
s(x, t) = p(t)s̄(x)
(2.38)
where p(t) is the production function calculated in section 2.3.17 and shown in Fig. 2.16b,
RL
and s̄(x) is the normalized source distribution, such that 0 s̄(x)dx = 1. The spatial
distribution of s̄(x) is given by the early (pre-cycle-11) mRNA distribution as reported in
Ref. [65], or, in the case of the shallow source, the cycle 10 mRNA distribution given
in Fig. 3E of Ref. [37]. We neglect small variations in the mRNA distribution over time,
reported in both Ref. [65] and Ref. [37], as the result in Fig. 2.4 is qualitatively unchanged.
Where maturation is not taken to be instantaneous, the concentration c(x, t) is decomposed into mature and immature components: c(x, t) = cimm (x, t) + cmat (x, t). These
57
quantities evolve separately according to
∂cimm (x, t)
= s(x, t) + D∇2 cimm (x, t) − (kmat + kdeg (t)) cimm (x, t)
∂t
∂cmat (x, t)
= kmat cimm (x, t) + D∇2 c(x, t) − kdeg (t)c(x, t)
∂t
(2.39)
(2.40)
In general, maturation is taken to be a single-step process with time constant 60 minutes
(kmat = 0.0167 min−1 ).
2.3.19
Measurement of newly maturing Dronpa-Bcd
Although we can infer the rate of synthesis of Bcd, p(t), from our measurement of degradation and the quantification of total Bcd in the embryo (see section 2.3.17), we can also determine the amount of newly maturing Dronpa-Bcd from our experiments. The amount of
newly matured Dronpa-Bcd is not equivalent to the amount of newly synthesized DronpaBcd, as it is filtered through the Dronpa maturation process. However, we can verify that
this measurement is consistent with a simple kinetic model incorporating our previously reported synthesis and degradation rates and a reasonable assumption for the in vivo Dronpa
maturation lifetime.
Since our measurements give complete information about the evolution of dark-state
Dronpa-Bcd, we can use this to determine the fraction of the change in bright-state fluorophore that is attributable to newly matured protein. As described previously, all newly
matured Dronpa-Bcd appears in the bright state. No assumptions need to be made to correct
for the possibility of fluorophore redistribution.
The mature fluorophore in the observable region is distributed into the bright and dark
58
states of Dronpa:
∂Qdf
= −kdeg Qdf + Φdark (t)
∂t
∂Qbf
= m(t)γ(t) − kdeg Qbf + Φbright (t)
∂t
(2.41)
(2.42)
With algebra we can rewrite the last two terms in the equation for bright fluorophore evolution. Recall that Qdf = G(t)Qf ocal and correspondingly Qbf = (1 − G(t))Qf ocal .
∂Qbf
= mγ − kdeg (1 − G)Qf ocal + (1 − G)Φ
∂t
1−G
= mγ +
(−kdeg GQf ocal + GΦ)
G
Qbf ∂Qdf
= mγ +
Qdf ∂t
(2.43)
(2.44)
(2.45)
Integration allows us to solve for the newly matured protein in the observable region,
mγ
Zt
t0
m(t0 )γ(t0 )dt0 =
Zt
t0
∂Qbf 0
dt −
∂t0
Zt
Qbf ∂Qdf 0
dt
Qdf ∂t0
(2.46)
t0
Qbf (t) + Qbf (t0 )
∼
(Qdf (t) − Qdf (t0 ))
= Qbf (t) − Qbf (t0 ) −
Qdf (t) + Qdf (t0 )
(2.47)
We make the assumption that the change in fraction of dark-state fluorophore G is small
and use a linear approximation for 1−G/G. An estimation of the true value of m/Qmat can be
obtained by dividing out Qf ocal = γQmat .
In Fig. 2.17a, we show the fraction of mature Dronpa-Bcd fluorophore newly matured
within the past minute, m/Qmat , as calculated by this method. We fit this data to a line,
which shows a negative trend in time, starting with a mean value of 0.033 min−1 at the
midpoint of interphase 11 and decreasing to 0.027 min−1 at the end of interphase 14.
To verify that these values agree with the previously reported degradation and synthe-
59
0.08
0.06
a
mat
b
0.04
m/Q
m/Qmat
0.06
0.02
0.04
0.02
0
12
−0.02
100
13
120
14
140 160
t (min.)
12
180
0
100
13
120
14
140 160
t (min.)
180
Figure 2.17: Measurement of newly matured Dronpa-Bcd and agreement with simulation a) Fraction of mature Dronpa-Bcd fluorophore newly matured within the past minute,
m/Qmat , in units of inverse minutes, for all T = 8 and 15 minute experiments. 4 minute experiments are too short to obtain precise measurements of new maturation. t is in minutes
since fertilization, and the onset of each mitotic cycle is shown. Least squares fit indicated
by solid line. b) m/Qmat from (a) (circles) input to moving average filter of span 13 minutes,
with standard deviation indicated by dashed lines. Predicted value of m/Qmat obtained by
simulation of our estimate of kdeg (Fig. 2.2e) and production time series p(t) (Fig. 2.16b)
extracted from Western blots, assuming that maturation and degradation are Poisson processes with maturation lifetime 10 minutes (green), 30 minutes (red), 100 minutes (blue),
and 300 minutes (magenta).
sis rates, we find a predicted value of m/Qmat assuming the egg is activated with zero Bcd
protein and that maturation and degradation subsequently occur as single-step Poisson processes. For the degradation rate we assume that kdeg = 0.02 min−1 prior to cycle 14 and
increases within cycle 14 as indicated in Fig. 2.2e. We do not assume a constant production
rate; we use the time-dependent function that we extract from our estimate of kdeg and QBcd
as determined by Western blot. This function, p(t), is shown in Fig. 2.16b. Assuming a
Dronpa maturation rate of 100 minutes or longer, we find that m/Qmat takes a time series
that very closely approximates the measured value, as indicated in Fig. 2.17b. Shorter
maturation lifetimes are less consistent with our data, but only those of 10 minutes or less
yield values that fall outside one standard deviation of the measurement. This justifies our
assumption of a 60 minute maturation lifetime used for simulations of gradient shape in
section 2.3.18 and Fig. 2.4b.
60
2.3.20
Measurement of Bcd lifetime is insensitive to position along the
anterior-posterior axis
As described in section 2.3.9, the use of Dronpa photoconversion to measure protein degradation relies on the assumption that the region over which fluorophore concentration is integrated is a closed volume. In the earlier part of this chapter, the region of integration is
taken to be the entire cortex or surface region of the embryo, extending the length of the
A-P axis and penetrating 14 mm basally from the vitelline membrane. The core-to-cortex
flux measured and corrected for in sections 2.2.2 and 2.3.11 represents a deviation from
this assumption. However, while there is reason to believe that flux of Bcd from core-tocortex is significant, it is at least plausible that flux between surface regions along the A-P
axis is negligible in later syncytial blastoderm stages, if hypotheses about compartmentalization of cortical cytoplasm [68] and small diffusion constants [27] in the cortical region
are assumed (see section 3.2.3).
With the assumption that exchange of Dronpa-Bcd between neighboring cortical A-P
sections is negligible, it becomes possible to measure the Bcd degradation rate kdeg as a
function of A-P axis position, kdeg (x). Spatial inhomogeneities in kdeg could simply reflect
differences in absolute Bcd concentration along the A-P axis, which would suggest a nonfirst-order degradation process. This has been hypothesized as a mechanism for creating
gradients of shape other than exponential [80]. Alternatively, the degradation mechanism
might itself be variable with position, which is itself a conceivable mechanism for creating morphogen concentration gradients, and which has been observed in aspects of the C.
elegans patterning system [81].
In Fig. 2.18b is shown the degradation rate, uncorrected for flux, calculated for various
distinct regions of integration along the A-P axis (Fig. 2.18a). Error bars are of the same
magnitude as in Fig. 2.2e and are not shown. However, error is dominated by embryo-toembryo variation, suggesting that differences in degradation between regions, for the same
61
a
0.08
kdeg (min−1)
0.06
0.04
0.02
0
−0.02
100
b
120
140 160
t (min.)
180
Figure 2.18: Measurement of kdeg dependence on A-P axis position a) Sample midcycle-14 embryo with three distinct regions of integration highlighted, at the anterior tip
(blue), at the eventual cephalic furrow position (green), and immediately posterior of the
embryo midline (red). As in Fig. 2.7, each mask is rotated around the A-P axis to create
a volume of revolution which represents the region of integration for Sf ocal . b) ksimp (x)
calculated for each of the three regions in (a) as a function of time since fertilization. Measurements prior to cycle 14 are uncorrected for flux, either cell-cycle-periodic or monotonic. As in Fig. 2.2a, each solid line represents smoothing spline fit to all data smoothed
by a Savitzky-Golay filter [67] of span 6.
set of embryos, are much more likely to be significant. Regardless, no significant difference
is seen between the degradation rate calculated for the mid-anterior and posterior regions
despite a difference in absolute concentration exceeding a factor of two, casting doubt on
the hypothesis of second-order degradation [80]. A slightly less pronounced cell-cycleperiodic oscillation is observed in the uncorrected degradation rate from the anterior-most
region, which is potentially explained by the smaller surface area connecting this region to
the basal cytoplasm and yolk as compared to the more posterior regions.
62
Chapter 3
In vivo measurements of diffusion
during embryogenesis
3.1
Introduction
Techniques for measuring the diffusivity of fluorescent probes, such as fluorescence recovery after photobleaching (FRAP), fluorescence correlation spectroscopy (FCS), and fluorescence loss in photobleaching (FLIP) have a long history [82]; however, the comparatively recent discovery of non-invasive, genetically-encoded fluorescent markers has led to
a large increase in studies examining protein mobility in vivo [24, 83]. While no family
of proteins has been untouched by this development, it has been remarked that studies of
morphogen diffusion are particularly challenging and consequently rare, given their relatively slow dynamics compared to other signaling molecules [7]. The work of Kicheva et
al. in the developing Drosophila wing disc, for example, involved measuring FRAP recovery curves of multiple hours, requiring them to account for complicating factors, such as
production, degradation, and tissue growth, typically ignored in simpler experiments [63].
As the first morphogen identified, Bicoid has attracted the interest of a number of studies of gradient formation, and nearly all make estimates of the diffusion coefficient of Bcd
63
protein [16, 27, 37, 72, 84]. As mentioned in Chapter 1, it has been hypothesized that the
lack of cell boundaries in the developing Drosophila embryo provides an ideal environment
for the diffusion of proteins. One of the first measurements to directly examine diffusivity in this setting was made by Gregor et al. [72]. The authors inject fluorescent dextran
molecules of sizes in the neighborhood of that of Bcd and estimate their diffusion coefficients by fitting the observed spread of fluorescence intensity in the embryo. As expected,
larger molecules diffuse more slowly, and they find that the result is well fit to a modified
Stokes-Einstein relation of the form D =
kB T/6πηrs
+ b, where the first term is the classic
Stokes-Einstein relation and b is a positive constant which they attribute to cytoplasmic
stirring. Substituting the presumed hydrodynamic radius of Bcd based on the assumption
that it is a globular protein with molecular mass 55 kDa, their results predict a Bcd diffusion coefficient of ∼ 16 µm2/s, four times larger than that implied by the measured protein
lifetime and gradient length as described in Chapter 2.
Despite the prediction of the Stokes-Einstein relation, it is not surprising that Bcd does
not diffuse at the same rate as fluorescently-labeled dextran molecules of the same molecular weight. Proteins in general are known to form multiple-body complexes and to bind
to stationary structures in the cell [85]. As a transcription factor, Bcd is known to bind to
a number of regulatory regions in the genome, and, combined with nonspecific binding,
is similarly expected to spend a significant fraction of time in the bound state. However,
binding reactions alone do not significantly complicate the analysis of Bicoid diffusion1 . In
fact, if the concentration of bound Bcd is at all points directly proportional to the concentration of free Bcd, then diffusion is governed by the same equations but with an effective
diffusion coefficient Deff , which is the free diffusion coefficient Df simply rescaled by a
1
While binding reactions in general can lead to anomalous subdiffusion, or the state in which the meansquare displacement of a diffusing particle scales as a power of time less than one (< x2 >∝ tα , 0 <
α < 1), Saxton [86] has shown that particles in thermal equilibrium with traps diffuse normally at all times.
Anomalous subdiffusion due to obstacles, however, can occur even in thermal equilibrium [87]. A more
recent treatment of subdiffusion and its implications for the gradient formation process is given by Yuste et
al. [88].
64
constant representing the fraction of total Bcd in the free state [62]. This is true regardless
of the geometry or dimensionality of the system. The effective diffusion coefficient Deff
is the quantity measured by FRAP experiments in which the recovery time is sufficiently
long for the free particles to reach equilibrium with the traps (see section 3.2.1).
The first direct perturbative measurement of Bcd diffusion was a FRAP experiment
reported by Gregor et al. in Ref. [27]. Puzzlingly, they report a diffusion coefficient of
0.30 ± 0.09 µm2/s in the cortical cytoplasm immediately prior to mitotic cycle 14. This
value is more than an order of magnitude smaller than the value of 4 µm2/s expected given
a Bcd lifetime of 50 minutes, and is too small to explain the extent of the gradient even
if there is no Bcd degradation whatsoever. They perform the same experiment on a set of
unfertilized eggs also expressing eGFP-Bcd and arrive at only a slightly larger value, which
they use to argue that diffusion at the embryo core is not significantly different from that at
the cortex, since the unfertilized egg appears uniform throughout (see Chapter 4).
A variety of explanations have been advanced to explain this paradoxical measurement.
Some have suggested that a local measurement of diffusivity over a single minute may
simply not be relevant to Bcd gradient formation across the entire embryo, over the course
of several hours [22]. For example, some sort of intermittent stirring or advective transport
might complement Bcd diffusivity in a way not observable during the FRAP experiment
[27, 89]. Others have used the measurement as motivation for a more radical rethinking of
Bcd gradient formation, suggesting that very little diffusion of Bcd from the translation site
is necessary at all [37].
In contrast to these interpretations, which accept the measurement of low Bcd diffusivity at face value, another camp views the reported 0.30 µm2/s diffusion coefficient as the
result of a mistaken interpretation of FRAP data [22, 71]. They point out that there are
many sources of error which can cause FRAP experiments to produce erroneous diffusion
coefficients, and that these tend towards systematic underestimation of true diffusion coefficients. The sources of error can be separated into three categories: anomalous recovery due
65
to binding kinetics, fitting errors due to the finite duration of the bleaching pulse, and fitting errors due to boundaries and incorrect geometry. It is these concerns which we treat in
this chapter. The first two have received somewhat more attention to date, as they are well
known complications which are of general interest to nearly all experimenters using FRAP
techniques [90, 91]. The third, relating to geometry, is considerably more system-specific
and perhaps for this reason is less frequently mentioned; however, we present results which
suggest that the magnitude of this effect is significant and may explain the majority of the
discrepancy in Bcd diffusion coefficients reported to date.
3.2
3.2.1
Results and Discussion
Theoretical considerations relating to FRAP in presence of binding sites
Following Sprague et al. [90] and the notation of Pando et al. [92], let Pf represent the
concentration of free fluorescent molecules and Pb represent the concentration of fluorescent molecules in bound complexes. We assume a substrate of vacant binding sites of
concentration S. The three quantities take equilibrium values Pf −eq , Pb−eq , and Seq , and
are related by a set of reaction-diffusion equations
∂Pf
= Df ∇2 Pf − kon SPf + kof f Pb
∂t
∂S
= Ds ∇2 S − kon SPf + kof f Pb
∂t
∂Pb
= Db ∇2 Pb + kon SPf − kof f Pb
∂t
(3.1)
(3.2)
(3.3)
where Df , Ds , and Db are the diffusion coefficients for free molecules, traps, and bound
complexes, respectively. The on- and off- rates are given by kon and kof f , and it is standard
∗
to substitute a pseudo-on rate constant kon
= kon Seq . Additionally we assume that diffusion
66
of traps and bound complexes is negligible: Ds = Db = 0. This simplifies the above
equations (omitting the behavior of unoccupied traps) to:
∂Pf
∗
= Df ∇2 Pf − kon
Pf + kof f Pb
∂t
∂Pb
∗
Pf − kof f Pb
= kon
∂t
(3.4)
(3.5)
In a FRAP experiment, a bleaching pulse inactivates the fluorophores of some fraction
of the molecules, separating the free particles and bound complexes into “tagged” and
“untagged” populations: Pf = Pft + Pfu and Pb = Pbt + Pbu . It is assumed that the medium
is infinite and that production and degradation of the molecules are negligible over the time
of the FRAP experiment; the subsequent fluorescence intensity time series then tracks the
recovery of the tagged population to its original equilibrium value, as tagged molecules
diffuse back into the bleached space. The measured quantity is
Z
Pft (~r, t) + Pbt (~r, t)d~r
IF RAP (t) =
(3.6)
bleached−region
and the most general solution for an infinite cylinder, the idealized geometry typically used
to model FRAP in confocal microscopy, is given in Ref. [90]. Additionally, the measured
curve is typically renormalized such that the recovery ranges from 0 to 1 [93].
There are three principal regimes in which approximate solutions exist. In the first,
∗
binding is negligible (kon
/kof f 1) and the classic FRAP analysis applies, based on free
diffusion alone [93]. The solution to this case for uniform, instantaneous bleaching of an
infinite cylinder is given by [94]:
−τcyl/2t
IF RAP (t) = e
τ i
h τ cyl
cyl
+ I1
I0
2t
2t
67
(3.7)
where I0 and I1 are modified Bessel functions and the characteristic recovery time τcyl is
τcyl =
w2
Df
(3.8)
where w is the radius of the bleach spot. It is notable that τcyl is the only quantity by means
of which the recovery is affected by the bleach spot size.
In the second approximate solution, there is non-negligible binding of the fluorophore,
but the binding reaction equilibrates rapidly compared to the time scale of free diffusion
∗ w2
within the bleach spot (kon
/Df 1). In this case, the recovery is of the functional form
given in Eq. 3.7 with the substitution τcyl = w2/Def f , where
Def f =
Df
1+
∗
kon
kof f
(3.9)
This is termed the effective diffusion regime [90], and, though here we only apply it to
the infinite cylindrical bleaching spot, an analogous approximation exists for any FRAP
experiment in which the binding reaction rapidly achieves local equilibrium regardless of
the particular geometry [62].
Finally, there is a third approximate solution which is the opposite of the effective
diffusion solution, called the reaction dominant regime. In this case free diffusion within
∗ w2
the bleach spot is much faster than the relaxation time of the binding reaction, kon
/Df 1
∗
(and the equilibrium concentration of bound complexes is non-negligible, kon
/kof f & 1). In
the reaction dominant regime, the FRAP recovery curve is not affected by the geometry of
the bleach spot at all, and the experiment only gives information about the binding reaction,
such that
IF RAP (t) = 1 − Ceq e−kof f t
The full derivation of this solution can be found in Ref. [90].
68
(3.10)
These considerations underscore the importance of determining that FRAP recovery
times scale with bleaching spot size prior to the extraction of a molecular diffusion coefficient. In the case of Bcd, whose function is intimately related to a binding reaction, it seems
likely that concentration in the bound state is not negligible, and it has been conjectured
that the ratio of bound to free Bcd protein may be as high as 50 [95]. There are theoretical
grounds to suppose that transcription factors should want to minimize time in the bound
state [96], but the specific values of the on- and off-rate constants for Bcd binding are not
known.
Additionally, in treating mobility of Bcd, it is important to consider not only Bcdspecific binding sites but all sites which bind the protein. In the real system, there is most
likely a hierarchy of traps which bind Bcd with different affinities. Sprague et al. give
some consideration to the case where there are two or more species of binding sites, and
find that the effective diffusion and reaction dominant regimes continue to exist, though
their share of parameter space is diminished when there is a multiplicity of binding site
type with different rate constants [90].
3.2.2
Bcd FRAP recovery times are anomalous relative to bleaching
spot width
Results of FRAP experiments at the cortex of the egg suggest that the necessary assumptions of the free or effective diffusion regimes described in section 3.2.1 are not satisfied
in the case of Bcd FRAP using bleaching spots of order 10 mm width. We use unfertilized
eggs expressing eGFP-bcd of the stock used in Ref. [27], and perform FRAP experiments
approximately 4 hours after oviposition, on bleaching spots of radius wsmall = 5 mm and
wlarge = 14 mm. In Fig. 3.1a, we show the raw intensities for the recovery of both size
bleach spots as well as the photobleaching due to imaging of an unbleached control spot.
The normalization of each FRAP curve is shown in Fig. 3.1b, where the larger spot shows
69
a similarly rapid recovery as the smaller spot. However, in both the free and effective diffusion regimes, the characteristic recovery time scales as the square of the linear dimension
of the bleaching spot: τcyl−small = τcyl−large (wsmall/wlarge )2 . In this case,
wlarge/w
small
> 2,
and no scaling of the recovery time is observed. Two possible explanations exist. First, it
may be that the equilibration time of the binding reaction is longer than the characteristic
recovery time of the FRAP experiment, in which case only the full or reaction dominant
solutions may be used [90]. A second possibility is that the infinite cylinder geometry typically assumed in confocal FRAP experiments is incorrect. On this model, the predominant
diffusion of unbleached fluorophore is in the z-direction, or along the apical-basal axis.
3.2.3
FLIP demonstrates that exchange of Bcd between neighboring
nuclei is minimal
To examine the relative contributions of diffusion in the plane of the vitelline membrane
as compared to diffusion along the apical-basal axis to Bcd FRAP recovery, we perform a
fluorescence loss in photobleaching (FLIP) experiment. FLIP is an alternative technique
for studying dynamics of fluorophore-conjugated proteins in living cells [82]. In contrast to
FRAP, which requires only a fluorescence intensity time series at a single point, FLIP takes
into account spatial intensity profiles to observe distributed fluorescence loss resulting from
photobleaching of a point or other isolated region. Though the time necessary for FLIP
limits its usefulness during mitosis, we perform FLIP during interphase 14 of fertilized
embryos and look at the fluorescence loss in an arrangement of neighboring nuclei.
The usefulness of FLIP as a technique for studying Bcd dynamics has to date also been
limited by the fact that Bcd forms a protein gradient. Hence, it is not uniformly expressed
and inhomogeneities in fluorescence intensity cannot be entirely attributed to photobleaching. We eliminate this difficulty by using a transgene in which bcd is expressed under control of the spaghetti squash (squ) promoter and 3’ UTR (see section 3.3.1) , and is therefore
70
4
x 10
1
a
3
(t)
0.8
0.6
0.4
FRAP
I(t) (a.u.)
3.5
I
4
2.5
2
0.2
b
1.5
0
50
t (s)
100
0
0
50
t (s)
100
Figure 3.1: Bcd FRAP recovery curves for bleaching spots of variable width a) Intensity
time series for FRAP bleaching regions of radii 5 mm (blue curve) and 14 mm (red curve) in
the cortex of unfertilized eGFP-Bcd-expressing eggs. Green curve represents mean intensity of a region of 10 mm radius between the bleached regions and with an equivalent initial
intensity, to quantify residual photobleaching due to imaging. Black curves (dashed lines)
represent mean intensity of a region of 10 mm radius posterior to the bleached areas but with
comparable initial intensity to the immediate post-bleach intensity of the bleached regions,
with a constant added such that the intensities at t=0 are precisely equivalent for each
size bleaching spot; this represents the photobleaching due to imaging of the remaining
unbleached eGFP-Bcd within the bleaching spot. Separate curves are necessary because
the bleach depths are different for differently sized regions. b) FRAP recovery curves for
bleaching regions of radii 5 mm (blue curve) and 14 mm (red curve), normalized on 0 to 1 using the double normalization method outlined in Ref. [93]. To account for photobleaching
from imaging, the blue and red curves in (a) are plotted as the fractional distance between
their respective black curve, representing background, and the green curve, representing
full recovery.
71
ubiquitously and uniformly expressed in the embryo. Additionally, the squ-bcd construct
is fused to Venus, a GFP-family yellow fluorescent protein [97], for in vivo fluorescence
imaging. The a priori knowledge that all intrinsic fluorescence intensity variation between
nuclei is stochastic renders this construct superior to eGFP-Bcd for studying diffusion by
perturbative methods.
Fig. 3.2a shows a yellow channel fluorescence image of a mid-cycle-14 embryo, before formation of cell walls, expressing Venus-Bcd uniformly, with a photobleaching pulse
applied every 8 seconds to a single nucleus for 5 minutes. In between bleaching pulses,
images of the nucleus and surrounding cluster are acquired. From these images, we use
morphological filtering to make nuclear and cytoplasmic masks (Fig. 3.2b), as described
in section 4.3.3. The approximate radius of the bleaching pulse is 2 mm, and therefore does
not touch adjacent nuclei (see section 3.3.5).
If there is appreciable exchange of Bcd between neighboring nuclei, one would expect
to see the fluorescence intensity in adjacent nuclei decline according to their distance from
the bleached nucleus, with a delay determined by the Bcd diffusion constant. Surprisingly,
as shown in Fig. 3.2c, only a minimal change in Venus-Bcd fluorescence is observed even
in nearest neighbor nuclei. The effect is small enough to demonstrate that fluorescence
recovery in Bcd FRAP experiments that measure nuclear refilling, typically of duration ~1
minute [35], does not arise from contributions by neighboring nuclei.
The extraction of a Bcd diffusion constant from this experiment is complicated by two
factors. First, as previously mentioned, the fluorescence loss experienced by adjacent nuclei is small enough to be nearly insignificant. Second, it can be objected that nuclei are
enveloped in a semi-permeable membrane which controls import and export of protein,
such as the Venus-Bcd imaged in this experiment. Hence, fluorescence loss might be limited by the rates of nuclear import and export rather than the Bcd diffusion constant; this
would occur if the nuclear residence time is long compared to the duration of FLIP. However, nuclear residence time of Bcd was estimated to be 74 ± 22 seconds by Grimm et
72
r
5 µm
1
1
0.8
0.8
I(x,tf)/I(x,t0)
I(x,tf)/I(x,t0)
a
0.6
0.4
0.2
0
0
10
15
r (µm)
20
0.6
3 µm
0.4
0.2
c
5
bb
25
0
0
d
5
r (µm)
10
15
Figure 3.2: Intensity profile of adjacent nuclei and cytoplasm during FLIP a) Mean
squ-Venus-Bcd fluorescence image showing mid-cycle-14 nuclei during FLIP. Photobleaching pulse is applied to dim nucleus at 8 s intervals. b) Extraction of nuclear (blue)
and cytoplasmic (red) masks from fluorescence image in (a) (see section 4.3.3). Nuclear
mask is dilated by 4 pixels to create a small buffer (green) before inverting to find the cytoplasmic mask. Distance r of each nucleus from the bleach point is measured in microns.
Scale bar indicates 5 mm. c) Mean nuclear fluorescence intensities in 5 squ-Venus-Bcd
expressing embryos in mid-cycle-14 (blue dots), plotted as distance from the bleach point,
after tf = 5 minutes of nearly continuous photobleaching. Intensities of bleached nuclei
(r = 0) are determined by finding the mean intensity in a manually selected mask. Red
curve indicates mean cytoplasmic fluorescence intensity in a set of concentric annuli of
width 1 mm, centered at the bleach point. Error bars indicate standard error. Contamination
of the cytoplasmic mask by the bleached nucleus is eliminated for r & 3 µm. d) Nuclear
intensities from (c) (blue dots) fit to a Gaussian fluorescence profile centered at the bleach
point (magenta) by least squares. Fit profile has σ = 3 µm.
73
al. [35], which is significantly shorter than the 5 minute duration of the FLIP experiment,
and we verify by FRAP that this residence time is not substantially changed when Bcd
is expressed uniformly (data not shown). Moreover, if the nuclear export rate were preventing the loss of fluorescence from neighboring nuclei, the cytoplasmic intensity profile
should show a broader decline in fluorescence intensity corresponding to the cytoplasmic
diffusion constant. In fact, as shown in Fig. 3.2c (red curve), the cytoplasmic fluorescence
decay profile is not broader than the nuclear fluorescence profile. This suggests that nuclear
protein concentrations are in equilibrium with the surrounding cytoplasm by the time the
final profile is measured. Hence we fit the nuclear intensity profile to a Gaussian distribution to obtain the width of the final bleached region. Assuming anisotropic diffusion [62]
with Dx (= Dy ) representing the cortical diffusion coefficient in the plane of the nuclear
layer and Dz representing the diffusion coefficient along the apical-basal axis, the width
of this distribution allows us to obtain an upper bound on Dx . Fitting to the solution of
the diffusion equation for a continuous circular source in the plane [62], we calculate that
Dx < 0.01 µm2/s.
This result is broadly consistent with observations that the Drosophila plasma membrane forms functional compartments around individual nuclei even prior to cellularization,
which does not occur until late in mitotic cycle 14 [68, 98] (see also section 3.3.6). Efforts
to incorporate this effect into modeling of Bicoid gradient formation have also been made
[99]. However, it is unclear whether the features of this effect which constitute an impediment to diffusion in the x-y plane is limited to interphase or if it persists through mitosis,
and perhaps even exists in analogous form in unfertilized eggs, offering an explanation for
the anomalous results shown in section 3.2.2.
3.2.4
Effect of boundary conditions on FRAP recovery times
As described in section 3.2.1, Eq. 3.7 is the recovery curve predicted for free or effective
diffusion recovery of a bleached region that can be approximated as an infinite cylinder.
74
This geometry is justified for FRAP experiments using confocal microscopes to bleach relatively transparent media (see Fig. 3.8a), with objectives of sufficiently small numerical
aperture. The assumption is that all recovery originates from unbleached molecules diffusing in from the r = w surface of the cylinder, and none from the ends. This leads to a
characteristic recovery time which scales with the square of the radius of the cylinder and
inversely as the diffusion coefficient (Eq. 3.8).
It is not immediately obvious that this geometry is unsuitable for analysis of FRAP
experiments on Bcd in the cortex of the Drosophila embryo. The vitelline membrane constitutes a boundary at one end of the cylindrical bleached region, which does not conflict
with the infinite cylinder geometry. However, as shown in section 3.3.7 and Fig. 3.8b-d,
photobleaching in the visible spectrum does not penetrate through to the reverse side of the
embryo. If the photobleaching pulse stops at the boundary between cytoplasm and yolk, as
seems reasonable from the scattering properties of the lipid droplets as discussed in section
2.3.9, then recovery from the basal z-surface of the cylinder may not be negligible.
Based on our observations in sections 3.2.2 and 3.2.3, we consider the opposite extreme
case, in which only a negligible contribution to the FRAP recovery arises from the r =
w surface of the cylinder and the recovery is instead dominated by flux of unbleached
fluorophores from the ends. In this case, the solution to the diffusion equation becomes
one-dimensional. As is standard for deriving FRAP recovery curves, we consider the export
of bleached molecules from the region of interest, which, assuming a constant equilibrium
concentration, is complementary to the import of unbleached fluorophores. The solution to
the diffusion equation with the following initial conditions:
∂ 2 c(z, t)
∂c(z, t)
=D
∂t
∂z 2
(3.11)
c(z, 0) = C 0
c(z, 0) = 0
75
−l < z < l
(3.12)
|z| > l
(3.13)
is given by [62]:
C0
c(z, t) =
2
l+z
l−z
√
√
+ erf
erf
2 Dt
2 Dt
(3.14)
To determine the FRAP recovery curve, we integrate the quantity remaining over time
on −l < z < l:
Zl
Qrod (t) =
c(z, t)dz = 2lC0 erf
−l
l
√
Dz t
√
√
2C0 Dz t 2C0 Dz t
√
−
+ l2 √
π
e /Dz t π
(3.15)
and make the substitution
τrod =
l2
Dz
(3.16)
which gives a FRAP recovery curve, normalized on 0 to 1, described by
Qrod (t)
IF RAP (t) = 1 −
= 1 − erf
2lC0
r
τrod
t
r
+
t
−τ
1 − e rod/t
πτrod
(3.17)
As in the case of the infinite cylinder, the characteristic recovery time τrod depends
only on the square of the linear dimension l and the diffusion coefficient. As mentioned
previously, the vitelline membrane constitutes a barrier to diffusion at the apical end of a
FRAP region at the surface of an embryo, so it is also informative to compare the recovery
time when diffusion occurs through only one end of the rod. Using the method of images,
the solution for this case is easily obtained by choosing a constant initial concentration over
the region −2l < z < 2l (in Eq. 3.12) and integrating the concentration remaining over
time on 0 < z < 2l. The characteristic FRAP recovery time, τrod =
(2l)2/Dz ,
is precisely 4
times longer.
We want to compare this recovery time to the case where all surfaces of the bleached
76
1
1
10
0
10
τ1 (a.u.)
IFRAP (norm.)
0.8
0.6
0.4
0.2
−2
10
a
0
0
−1
10
5
b
−1
10
10
t (a.u.)
0
10
w/l
Figure 3.3: FRAP recovery times for various geometries a) FRAP recovery curve IF RAP
for a cylindrical bleaching region with radius equal to its length,w = l = 1. Blue curve is
identical to IF RAP −cyl (Eq. 3.7), which assumes the r = w surface is permeable and the z =
0 and z = l surfaces are insulating. Green and magenta curves show FRAP recovery when
the r = w surfaces are insulating and either one or both of the z = 0, z = l surfaces are
permeable, respectively. Red curve shows FRAP recovery when all surfaces are permeable.
Crossings of the dashed line indicate τ1 , which is defined such that IF RAP (τ1 ) = 1 − e−1 .
b) Recovery times τ1 for various half-width-to-length ratios w/l, choosing the length l to
be constant at unity. Color scheme is identical to (a). Red curve, indicating the recovery
time when all surfaces are permeable, is bounded from above by each of the other curves.
region are permeable, which is the geometry typically assumed in FRAP experiments using
multiphoton microscopy. The solution to the diffusion equation for the infinite cylinder
with r = w, initialized with constant concentration C0 inside the cylinder, is given in Ref.
[100]:
Z∞
c(r, t) = wC0
2
J1 (uw)J0 (ur)e−Dx tu du
(3.18)
0
Because the initial conditions are separable, the solution for the combined case, in
which recovery arises from all surfaces, is simply the product of the solutions for the rod
and infinite cylinder, Eqs. 3.14 and 3.18 [101]. The FRAP recovery curves for each type
of boundary condition, for the cylinder w = l = 1 with constant initial concentration, are
shown in Fig. 3.3a. Notably, the curves are dramatically different and the time to recover
to 1 − e−1 varies by more than an order of magnitude.
77
In Ref. [27], Gregor et al. perform a FRAP experiment by bleaching a 16 x 16 x 7 mm
rectangular volume in the cortex of the embryo during mitosis. As the bleaching volume
is produced by two-photon illumination, it cannot have effectively infinite extent in the zdimension, and its aspect ratio is approximately equal to one in the units of Fig. 3.3b. In
the cylindrical geometry, the recovery in the case that all surfaces are permeable is 13 times
faster than the recovery in the case that only the z = l surface is permeable. Applying this
rescaling factor to the measurement of Gregor, the estimate of the diffusion coefficient Dz
instead becomes 3.8 ± 1.1 µm2/s, within 10% of the predicted value given in Chapter 2.
3.2.5
FRAP recovery is faster at locations more basal in the embryo
It might be objected that, even if Dz is larger than previously believed, the diffusion coefficient along the apical-basal axis is not the relevant quantity determining gradient formation
along the anterior-posterior axis. A possible resolution is that the value of Dz in the cortex
is identical to the isotropic diffusion coefficient in the bulk of the embryo, which would be
the case if the small Dx observed in section 3.2.3 is due to barriers that are present only
at the embryo cortex. We test this hypothesis by extending the photobleaching experiment
of Ref. [27] to greater depths. This is limited by available laser power, which in our case
was 180 mW at the sample, permitting effective photobleaching of a ∼ 103 mm3 bleaching
region at a maximum of 30 mm below the vitelline membrane. We image a 11 x 13 x 7
mm3 region approximately one-third of the egg length from the anterior end of the embryo,
and photobleach by increasing the laser intensity for 20 seconds. Interestingly, the aspect
ratio of our photobleaching region is slightly smaller than that of Gregor et al., and likewise our measurement of 0.27 ± .07 µm2/s near the embryo surface is just slightly smaller
than their reported 0.30 ± .09 µm2/s, as would be predicted if not all surfaces of the fitting
geometry are permeable as suggested in section 3.2.4. There is indeed an increase in the
apparent diffusivity as depth, though we do not recover the value of 4 µm2/s expected in the
core. Our maximum observed fit diffusion coefficient was 0.65 µm2/s , which could be due
78
0.8
0.7
D (µm2/s)
0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
15
z (µm)
20
25
30
Figure 3.4: Measurement of Bcd FRAP recovery at various depths Apparent diffusion
constants at various depths below the vitelline membrane of a fertilized embryo in mitosis
13, measured by photobleaching a 103 mm3 region for 20 s and observing the recovery
curve. Nuclei are localized within approximately 10 mm of the surface and the cortical
cytoplasm extends to 25 mm. A least-squares fit of the data is shown with dashed lines
representing 20% variation.
to dissolution of boundaries in the x-y plane, but also, as it is less than a factor of 4 greater
than the cortical value, could be similarly due to diffusion from the apical surface of the
bleaching region. While only an increase greater than a factor of 4 would provide conclusive evidence that lateral boundaries dissolve approaching the embryo core, it remains a
plausible hypothesis as even the deepest FRAP experiments presented here extend to less
than one-third of the embryo radius.
3.2.6
Can recent FCS measurements of Bcd diffusivity be reconciled
with gradient shape?
The preceding sections of this chapter have dealt with potential sources of error in measuring diffusion coefficients by FRAP methods. Recently, however, Abu-Arish et al. [71]
reported a value of the Bcd diffusion coefficient as measured by fluorescence correlation
spectroscopy (FCS), a method which tracks the fluctuations in time of fluorescence in-
79
tensity observed in a small focal volume. Conceptually, it is substantially different from
FRAP and not subject to the same limitations. Additionally, unlike published FRAP measurements of the Bcd diffusion coefficient, the diffusion coefficient they report is significantly greater than that predicted by our measurement of the Bcd lifetime and gradient
shape (section 2.2.4); according to Ref. [71], eGFP-Bcd in the cortical cytoplasm of living
embryos between mitotic cycles 12 and 14 has a diffusion coefficient of 7.4 ± 0.4 µm2/s.
The authors correctly note that this value is more consistent with the observed extent of the
gradient than that reported in Ref. [27]. However, combined with the Bcd lifetime reported
in Chapter 2 and recent measurements of the bcd mRNA distribution [65], it predicts a
gradient length significantly in excess of the observed value.
There are, though, theoretical reasons to believe that measurements of diffusivity by
FRAP and FCS will produce different effective diffusion coefficients in the presence of a
substrate to which the fluorescently-tagged species can bind.
As shown by Pando et al., different effective diffusion coefficients describe the behavior of a diffusing species in equilibrium with a substrate of traps to which the molecules
may bind [92]. The first, Dt (Dtagged ), describes the spread of a bolus of tagged particles
introduced into a medium in which untagged particles are already in equilibrium with the
binding substrate. The second, Du (Duntagged ), describes the spread of a bolus of untagged
particles introduced into the same medium. The difference is due to the fact that the untagged particles are indistinguishable from those which are already bound. In other words,
Dt describes the effective diffusion coefficient of a single particle, while Du describes the
propagation of density fluctuations in the medium.
It is shown in Ref. [92] that Dt = Deff (Eq. 3.9), and is the same quantity measured
by FRAP experiments in the effective diffusion regime described in section 3.2.1. In Ref.
[102], Sigaut et al. demonstrate that the quantity measured by FCS is instead Du , which is
80
given by:
Du =
1+
Df
∗
Seq
kon
kof f S+Pb
(3.19)
Dt ≤ Du in general (provided the diffusion coefficient of the traps is less than that of
the free particles), and may be arbitrarily smaller depending on the fraction of traps which
are occupied. Dt remains equivalent to the single particle effective diffusion coefficient,
and is the relevant number which determines the gradient length. Hence it is not surprising
that a FCS experiment would measure a diffusion coefficient which is significantly larger
than the effective diffusion coefficient necessary for gradient formation.
Notably, as in mentioned in section 3.2.1, traps in the real system should not be considered Bcd specific binding sites alone, but rather all sites which bind the protein. Determining the specific numbers in Eq. 3.19 is complicated by the fact that the theory has not
been extended to the multiple binding state case. Including only Bcd specific binding sites
in the competitive binding case leads to a region-dependent effective diffusion coefficient,
which we do not consider in detail.
3.2.7
Conclusion
In conclusion, we have performed in vivo fluorescence photobleaching experiments which
demonstrate that the lateral exchange of Bcd between neighboring nuclear compartments
in the late syncytial blastoderm is much smaller than previously believed. This has consequences for the interpretation of FRAP experiments, which have in the past assumed
isotropic or infinite cylindrical recovery geometries [27, 71]. We also notice similarly
anomalous FRAP recovery behavior in the case of unfertilized eggs, which we treat in
further detail in Chapter 4.
One of the most puzzling consequences of the Bcd FLIP experiment is that it makes it
difficult to understand the mechanism by which neighboring nuclei might collectively av81
erage their individual measurements of Bcd concentration when the gradient is being read
out. In Ref. [26], Gregor et al. report that noise in the hb output response shows a correlation length of 5 ± 1 nuclear spacings. Our results suggest that exchange of Bcd between
nuclei is limited to a much shorter distance over the relevant developmental time. It remains
possible that correlation in output noise is due to a nuclear lineage effect, as daughter nuclei
remain in close proximity after mitosis. Alternatively, if diffusion occurs primarily along
the apical-basal axis, correlation in output noise could reflect spatial inhomogeneities in
Bcd concentration at the embryo core.
A recent computational study has also considered the possibility that lateral transport
between nucleus-associated compartments in the syncytial blastoderm might be minimal,
as we have shown [99]. The study found this hypothesis compatible only with a distributed
mRNA model such as advanced in Ref. [37], which we dismiss in Chapter 2. However,
their modeling approach assumes that the yolk is an impermeable region from which Bcd
is excluded. The synthesis of these observations suggests an important role for the embryo
core with regard to Bcd diffusion. A direct measurement of the Bcd diffusion coefficient in
this region would be of substantial value to our understanding of gradient dynamics.
3.3
3.3.1
Methods
Synthesis and preparation of samples
The eGFP-bcd expressing embryos used in sections 3.2.2 and 3.2.5 were obtained from
Thomas Gregor and are of the same genotype as used in Ref. [27]. Eggs were collected
from plates of agar and apple juice mixture, dechorionated in bleach, and glued to a glass
slide. In section 3.2.2, we use eggs fertilized only by sterile (XO) males. After mounting,
the eggs are immersed in halocarbon oil and covered with a glass coverslip. In section
3.2.5, we chose pre-gastrula embryos with visible eGFP-bcd expression and perform a
82
FRAP experiment on a 11 x 13 x 7 mm3 region approximately one-third of the egg length
from the anterior end of the embryo. Embryos are immersed in halocarbon oil with no
coverslip, and the vitelline membrane height is determined as described in Chapter 4.
The uniform Venus-bcd expressing embryos used in sections 3.2.3 and 3.3.6 were provided by Oliver Grimm. To obtain a spatially uniform distribution of Bcd, he substituted the
3’ UTR of bcd, which normally controls anterior localization, with the 3’ UTR of spaghetti
squash (squ), which is not localized. The construct was inserted into random sites in the
Drosophila genome by P-element mediated transformation, and the lines with the highest
level of Venus-Bcd expression were selected. After collection, embryos are glued to a glass
slide and immersed in halocarbon oil. No coverslip is used for imaging.
3.3.2
Description of optical methods
Imaging in sections 3.2.2, 3.2.3, and 3.3.6 is performed on a Leica TCS SP5 laser scanning
confocal microscope with a 0.7 NA multi-immersion objective (Leica 20x HC PL APO).
The 488 nm line of an argon laser is used to excite eGFP and the 514 nm line is used for
excitation of Venus. Photobleaching is performed by simultaneously zooming in on the
bleached region and ramping up the laser intensity by increasing the transmittance of the
acousto-optic tunable filter (AOTF). This is done in software using the Live Data Mode of
the Leica LAF software suite.
In section 3.2.5, a sequence of images is obtained at low laser power to verify that no
bleaching occurs, followed by a 20 s bleaching pulse and 60 s recovery curve measurement.
Beam intensity is modulated via Pockels cell. Given the irregularly shaped bleaching region, the apparent diffusion constant is determined by fitting the observed recovery curve
to a numerical integration of the 3D diffusion equation, using a Nelder-Mead simplex minimization.
83
3.3.3
Nuclear identification and motion correction of image series
Living Drosophila embryos display significant motion of nuclei in the cycles leading up to
gastrulation. For FRAP and FLIP experiments which involve measuring nuclear and cytoplasmic fluorescence intensities, this motion must be removed prior to evaluating intensity
profiles and time series. To accomplish this, we create a nuclear mask using the method
described in section 4.3.3, though a uniform Venus-Bcd fluorescence image is used in place
of a Histone image. The motion vector between images is determined by translating the nuclear mask pixelwise up to 2 mm in both x and y directions and determining the translation
which maximizes the mean intensity in the nuclear mask. This vector is then subtracted
at each time point, and borders trimmed where necessary, to create a motion corrected image series. Movies of the corrected series appear stationary by inspection, and lack large
jumps which would suggest a need for a higher-order motion correction algorithm [103].
Rotational motion is negligible and not corrected for. Similarly, jitter from the microscope
apparatus itself does not contribute significantly to motion artifacts and images of fixed
specimens are therefore not motion corrected.
3.3.4
Measurement of corona effect in FRAP experiments
A concern in FRAP experiments is that bleaching pulses often cannot be approximated as
instantaneous [91]. In such cases, bleached molecules have time to diffuse away from the
bleaching region, creating a “corona” of bleached particles. This corona effect delays the
fluorescence recovery in the bleached region and can cause systematic underestimation of
diffusion coefficients by up to a factor of 4 for realistic FRAP scenarios using a confocal
microscope [91]. This is a valid complicating factor in FRAP experiments which have been
used to infer the diffusion coefficient of Bcd [22]; however, in section 3.2.2 our primary
aim is understanding the dependence of FRAP recovery on bleaching spot width rather than
the extraction of a Bcd diffusion coefficient.
84
10000
Ipre−Ipost (a.u.)
a
b
5000
0
10 µm
0
20
40
60
x (µm)
Figure 3.5: Determination of FRAP bleaching spot widths a) Mean intensity of three
images immediately following (t = 1 s) bleaching pulse subtracted from mean intensity of
three images immediately preceding (t = −7 s) bleaching pulse, showing two bleaching
spots of radii 7 mm and 14 mm. The image is smoothed using a Gaussian kernel of σ =
.4 µm, and deblurred using Lucy-Richardson deconvolution [104]. Scale bar represents 10
mm. b) Anterior-posterior oriented cross-section of unfiltered image in (a) over the larger
bleaching spot, showing the effective bleaching width (black curve). Red curve indicates
region to which bleaching pulse was applied in software. Minor intensity depression at
distance & 10 µm from the bleached region represents primarily residual photobleaching
from imaging.
85
Fig. 3.5a shows the effective size of the small and large bleached regions from section 3.2.2, and Fig. 3.5b shows a one-dimensional cross-section of the large region (black
curve) against the width of the bleaching pulse (red curve). These indicate a corona of approximately 4 mm for both the large and small bleached regions. We find that the width of
the corona appears to reach an asymptotic value by the time of the first post-bleach image,
suggesting a compartmentalization effect similar to that observed in section 3.3.6. In fact,
the size of the coronas are comparable in both cases, even though the FRAP experiment
takes place in an unfertilized egg whereas the FLIP experiment is in interphase, though the
mitosis corona is slightly larger (4 mm vs. 2 mm). We dilate the regions of integration in
section 3.2.2 by 4 mm to take account of the larger effective bleach spot size; however, we
find that the recovery time does not scale with the bleach spot width regardless of whether
the adjustment is made.
3.3.5
Determination of FLIP bleaching spot width
While it is in principle possible to analytically determine the size of a photobleached region
in an idealized apparatus using the numerical aperture of the objective and the wavelength
of the beam, we instead use fixed specimens under identical imaging conditions to determine the width of the photobleaching pulse, as shown in Fig. 3.6. We find that it is
significantly smaller than the width of a single nucleus.
Motion of nuclei in living specimens does not appear to contribute significantly to spot
width, as motion artifacts occur with equal probability over the duration of an experiment,
while the width of the in vivo spot does not show any increase over time (Fig. 3.7b).
86
a
b
5 µm
5 µm
d
0.6
I(0,t)/I(0,t0)
I(x,t=3 min.)/I(x,t0)
1
0.5
0.4
0.3
0.2
c
−10
0
x (µm)
0.8
0.6
0.4
0.2
0
10
2
4
6
t (min.)
Figure 3.6: Determination of FLIP bleaching spot width a) Mean intensity of original
image relative to final bleached image over 5 fixed samples of mid-cycle-14 squ-Venus-Bcd
syncytial nuclei. Each image is centered on the bleaching spot, smoothed using a Gaussian
kernel of σ = 1 µm, and the mean resulting image is deblurred using Lucy-Richardson
deconvolution [104] with the same kernel. Scale bar represents 5 mm. b) Same procedure as
in (a) applied to 5 in vivo samples, showing a slight but not dramatic increase in dispersion
of fluorescence loss. c) Cross-section showing fluorescence loss, calculated by finding the
fluorescence intensity along a line segment passing through the center of the bleached area
and taking the mean after rotation of the line segment through 180 degrees. Live sample
(blue) shows greater dispersion of fluorescence loss than the fixed sample (red). d) Decay of
fluorescence intensity during photobleaching at the center of the bleached region, showing
greater decay in fixed (red) as compared to live (blue) tissue.
87
3.3.6
Diffusion is substantially faster within nucleus-associated compartments than between compartments
Because the width of the bleach point as determined by photobleaching fixed embryos (see
section 3.3.5) is smaller than both the width of an individual nucleus and that of the region
in which fluorescence is lost, it should in principle be possible to extract a diffusion constant
from the data by examining the spread of the fluorescence loss over time. In Fig. 3.7a, we
show the evolution of the mean fluorescence loss profile over time, as photobleaching is
occurring at x = 0. Because the curves show the ratio of initial fluorescence intensity
to final fluorescence intensity, the heightening of the peak is consistent with continued
photobleaching at the central point (compare to Fig. 3.6d). Similarly there is an overall loss
of fluorescence, even far from the bleach point, which represents photobleaching caused by
the imaging alone. However, the width of the region in which fluorescence is depleted does
not increase in time, as would be expected for diffusive behavior (Fig. 3.7b). In fact, the
full width is attained prior to the acquisition of the first bleached image, at 15 seconds. This
allows us to estimate an intracompartmental diffusion coefficient D & 1 µm2/s, significantly
greater than the effective diffusion coefficient between compartments.
3.3.7
Determination of confocal bleaching spot depth
It is assumed that a confocal microscope using an objective with sufficiently small numerical aperture will produce a photobleached region that approximates an infinite cylinder. We
test this assumption with the microscope and objective used in this experiment. In Fig. 3.8a,
we show the bleaching pattern in an x-z cross-section of a slab of fluorescein-infused agar,
bleached in a single focal plane. The bleached pattern extends through 100 mm in depth,
suggesting that the infinite cylinder approximation is valid for transparent media such as
agar. In Fig. 3.8b-d, we perform a similar experiment to determine whether the bleaching
pattern likewise penetrates through the yolk of a syncytial blastoderm embryo. We find
88
4
8
a
3
b
5
t (min.)
W(t) (µm)
I(x,t0)/I(x,t)
0
2
6
4
2
1
−10
0
x (µm)
10
0
2
4
6
t (min.)
Figure 3.7: Time evolution of fluorescence intensity profile during FLIP a) Mean profile of bleached region in 5 in vivo mid-cycle-14 squ-Venus-Bcd embryos, computed by
smoothing and deconvolution by the method described in section 3.3.5. Color indicates
time t since initiation of photobleaching (t0 ). b) Full width at half maximum W (t) calculated from each of the curves displayed in (a) (blue curve). Half-max is calculated relative
to the deviation between the peak and the point farthest from the bleach spot. Red curve
shows the identical calculation applied to measurements from 5 fixed embryos. Error bars
indicate standard deviation.
that the bleaching pattern does not extend to nuclei on the reverse side of the embryo, at a
distance of ∼ 180 mm in z, suggesting that the infinite cylinder approximation is not valid
for living embryos. Interestingly, scattering in the yolk appears to inhibit photobleaching
of Venus-Bcd but not photoconversion of Dronpa-Bcd (see section 2.3.8), possibly because
the latter is understood to be a single-photon absorption process [73] while photobleaching
is a higher-order process.
89
100
b
z (µm)
80
60
c
40
20
a
0
0
d
100 200 300 400 500
x (µm)
Figure 3.8: Determination of axial bleaching spot depth a) Intensity z-stack for a slab
of agar mixed uniformly with fluorescein dye, bleached in a square pattern with the focal
plane at z = 0, using the confocal microscope and 0.7 NA objective as described in section
3.3.2. Images are then acquired at 50 planes deeper in the agar slab and displayed in
(a); increasing z indicates increasing depth in agar. Bleaching pattern extends with minor
blurring through 100 mm. b-d) Mid-cycle-14 embryo uniformly expressing Venus-Bcd (b),
after bleaching in a circular pattern at the obverse surface layer of nuclei (c), and the reverse
side of the same embryo <1 minute later (d), showing no evidence of the bleaching pattern.
90
Chapter 4
Bicoid gradient formation in fertilized
and unfertilized eggs
4.1
Introduction
Previous chapters have dealt with experiments almost exclusively on fertilized eggs, which
undergo nuclear cleavage divisions for multiple hours post-fertilization. In this chapter,
we treat the subject of gradient formation in unfertilized eggs, which undergo no nuclear
cleavage divisions. bcd mRNA is nevertheless localized and protein is translated from
oviposition; the event of fertilization is known to not be necessary for bcd translation to
occur [16].
Only limited mention of the shape of the unfertilized gradient exists in the published
literature. Imaging of unfertilized eggs, fluorescence imaging in particular, presents special challenges due the the lack of a region of transparent cortical cytoplasm which forms
with the syncytial blastoderm in fertilized eggs. Two studies examine the unfertilized case
in some detail. Driever and Nusslein-Volhard quantified immunostaining intensities from
unfertilized eggs of various ages, reporting that the development of the gradient proceeds
normally until 2-4 hours after egg deposition, at which point levels of total Bcd protein
91
in unfertilized eggs begin to exceed those in fertilized eggs [16]. Gregor et al. measured
the ratio of eGFP-Bcd fluorescence intensity at the anterior as compared to the posterior of
fertilized and unfertilized eggs and found a smaller value in the unfertilized case, reflecting
a putatively shallower gradient in the unfertilized egg [27].
Nevertheless, traditional views on the Bcd gradient formation process make somewhat
vague and inconsistent predictions about the properties of the unfertilized Bcd distribution.
On the one hand, one might expect the unfertilized gradient to be longer and shallower,
reflecting the presumed greater stability of Bcd protein. On the other, one might expect a
shorter, steeper gradient, reflecting the absence of cytoplasmic contractions coincident with
nuclear cleavage divisions, which are only present in fertilized eggs and which have been
conjectured to augment Bcd diffusivity [16].
In this chapter, we apply the optical technique for measuring protein lifetime described
in Chapter 2 to unfertilized eggs expressing Dronpa-Bcd, providing a direct test of the assumption that the protein is more stable in unfertilized eggs. Next, we use a technique
similar to optical lock-in detection imaging, a method devised to obtain high-contrast images of photoconvertible fluorophores in environments where background autofluorescence
is substantial, to quantitatively compare the Bcd gradient in fixed populations of fertilized
and unfertilized eggs. Finally, we obtain time course data on the unfertilized gradient
formation process using two-photon laser scanning microscopy, which is notable for its
capacity to acquire fluorescence images deeper in living tissue than confocal microscopy
[105]. We note several significant differences from the fertilized case, such as monotonically increasing Bcd levels and the presence of measurable Bcd concentration extending to
the posterior end of the egg, and discuss whether these can be understood in the context of
the SDD model and current knowledge of the Bcd lifetime and diffusion coefficient.
92
4.2
4.2.1
Results and Discussion
Bcd lifetime in unfertilized eggs is not down-regulated within
several hours of oviposition
The technique of measuring Bcd lifetime by means of timed photoconversions of DronpaBcd as described in Chapter 2 is extensible to unfertilized eggs with relatively minor modifications. Two principal challenges exist. First, the signal-to-background ratio of the green
channel fluorescence intensity of Dronpa-Bcd is significantly lower in unfertilized eggs,
due both to the contamination of the cortical cytoplasm with yolk granules and vesicles
and to the lack of accretion of Bcd at the embryo surface. This challenge is substantially
similar to that encountered in fertilized eggs prior to the clearing of the cortical cytoplasm
in mitotic cycle 14. However, a second complication exists in the unfertilized case. In the
fertilized case, nuclei are visible at the embryo surface beginning at cycle 10, and can be
counted to make a precise determination of embryo age. This method was used to establish
the existence of and average out cyclic Bcd flux as shown in Fig. 2.2b. In the unfertilized
case, no morphological cues give evidence of the developmental age of the egg, and the
time since oviposition or since the eggs were mounted on the slide is used instead.
To obtain a sufficiently strong Dronpa-Bcd fluorescence signal in the unfertilized eggs,
we wait a longer time post-oviposition (∼ 4 hrs.) before imaging and performing the
photoconversion experiment on the eggs. As a result, the distribution of egg ages is larger
and not directly comparable to the fertilized case; however, if Bcd lifetime has decreased
within several hours of oviposition we expect this to be evident in our data. In Fig. 4.1a
is plotted the fraction of Dronpa-Bcd recovered from the dark state following an 8 minute
delay interval; in Fig. 4.1b is shown the corresponding unfertilized Bcd degradation rate
u
u
kdeg
(black curve). After a transient, kdeg
takes an approximately constant value of 0.009 ±
0.006 min.−1 , which corresponds to a lifetime range of 70 to 300 minutes, confirming the
93
1.2
0.06
(min.−1)
b
deg
1
0.8
0.6
20
k
Qdf(t0+T)/Qdf(t0)
a
40
60
t (min.)
80
0.04
0.02
0
20
40
60
t (min.)
80
Figure 4.1: Measurement of Bcd lifetime in unfertilized eggs a) Fraction of Dronpa-Bcd
recovered from the dark state Qdf (t0 +T )/Qdf (t0 ) in a population of 8 unfertilized eggs for a
series of photoconversion experiments with delay T = 8 min. (see section 2.3.10). Solid line
represents smoothing spline fit to all data smoothed by a Savitzky-Golay filter [67] of span
6. Time is expressed in minutes since the start of imaging, which occurs at approximately
4 ± 1 hours post-oviposition for the unfertilized eggs. Initial lower recovery appears to
be a transient principally attributable to build-up of endogenous autofluorescence in the
unfertilized eggs, as it is correlated much more strongly with total image count than would
be reasonable if it were a function of developmental age alone. b) Red curve indicates
f
kdeg
, the degradation rate time series in fertilized eggs, as shown in Fig. 2.2e. Black
u
, the degradation rate time series in unfertilized eggs. Shading indicates
curve indicates kdeg
standard error. Time is expressed in minutes since the start of imaging, which is taken to
be at a mean of 90 minutes post-oviposition in the fertilized case.
94
conjecture of Driever and Nusslein-Volhard that Bcd protein is more stable in unfertilized
eggs [16].
4.2.2
High-contrast imaging confirms greater spatial extent of unfertilized Bcd gradient
If the lifetime of the protein is longer in unfertilized eggs, then our result in Chapter 2
predicts that the length of the gradient will similarly be longer in this case. To test this
hypothesis, our objective is to measure and compare the Bcd distribution as it appears in
fertilized versus unfertilized eggs. However, as mentioned previously, measurements of
the Bcd gradient in unfertilized eggs present special challenges, due to the scattering and
autofluorescence properties of the egg yolk. In vivo confocal microscopy is poorly-suited
to this task, as the depth penetrance in tissue is limited. Measurements of the unfertilized
eGFP-Bcd gradient have been made in fixed tissue [95], but precision remains limited by
high background levels.
We use the Dronpa-Bcd/Histone-RFP transgene described in Chapter 2, combined with
an adaption of the high-contrast optical lock-in detection imaging method developed in Ref.
[106] (see section 4.3.2), to make measurements of the Bcd gradient in unfertilized eggs
by photoactivation. The gradients in both fertilized (blue curve) and unfertilized (green
curve) eggs are shown in Fig. 4.2a and b, respectively. The background intensity value
for each case is given by the red curve, representing fertilized (Fig. 4.2a) and unfertilized (Fig. 4.2b) Histone-RFP expressing eggs. Because the specific signal is identified
by a decorrelation procedure using the fertilized Histone-RFP as a control, the fertilized
background intensity has a zero mean value. While the intensity of the fertilized gradient
decreases nearly to background at the egg posterior, the deviation from zero is significant.
The ratio of Dronpa-Bcd concentration at 0.9L to that at 0.1L is .036 ± .017, which is
consistent with the in vivo measurements in sections 2.2.4 and 4.2.3, as well as theoretical
95
and computational studies of the gradient, which suggest that it should extend to the posterior of the embryo [107]. As shown in Fig. 4.2c, the unfertilized gradient is shallower,
and, correspondingly, shows a larger posterior-to-anterior ratio than the fertilized gradient:
u
CDronpa−Bcd
(3h,0.9L)/C u
Dronpa−Bcd (3h,0.1L)
= 0.14 ± 0.07. We conclude that the observation
that unfertilized Bcd gradients are longer and shallower than their fertilized counterparts is
verified with higher precision imaging techniques.
4.2.3
Two-photon imaging shows time evolution of unfertilized Bcd
gradient
While the gradients we measure with lock-in imaging are more precise than previous measurements of the unfertilized Bcd gradient, the essential observation that the unfertilized
gradient is longer and shallower than the fertilized gradient, at some time several hours after oviposition, is not new [16, 27]. Of more interest to the study of the gradient formation
process is the evolution of the gradient in time. An in vivo measurement in the unfertilized
case, to the extent it can be called this, does not exist in the published literature.
The time evolution of the unfertilized gradient is of particular interest because it lacks
the interference of nuclei, which, as we show in section 2.3.12 and as reported elsewhere
[27], amplify the observed intensity of the gradient by shifting the total distribution of Bcd
towards the cortex of the embryo. Nuclei have also been conjectured on theoretical grounds
to modify the length of the gradient [27, 84, 99, 108], though experimental evidence suggests otherwise [35].
In Fig. 4.3, we show the mean eGFP-Bcd concentration gradients observed in sets of
unfertilized (Fig. 4.3a,c) and fertilized (Fig. 4.3b,d) eggs under two-photon microscopy.
As mentioned previously, multiphoton microscopy is noted for its ability to penetrate more
deeply in tissue than single-photon fluorescence microscopy, and we in fact observe greatly
improved signal-to-noise and signal-to-background ratios compared to in vivo measure96
1
0.8
0.6
0.4
0.2
0
0
0.5
1
Figure 4.2: Measurement of Bcd gradient in fertilized and unfertilized eggs by optical
lock-in imaging a) B represents decorrelated Dronpa-Bcd signal, after prior division by
red channel intensity to normalize for Histone-RFP signal (see section 4.3.2). Blue curve
indicates the mean signal extracted from 27 fixed, fertilized Dronpa-Bcd/H2A-RFP expressing embryos, of age 3±0.3 hours at the beginning of fixation. Shaded region indicates
standard deviation. Red curve is the decorrelation transform applied to 16 fixed, fertilized
H2A-RFP expressing embryos of the same age b) Green curve indicates the mean decorrelated Dronpa-Bcd signal from 19 fixed, unfertilized Dronpa-Bcd/H2A-RFP expressing
eggs of the same age as in (a), with shading indicating standard deviation. Red curve is the
same calculation applied to 5 fixed, unfertilized H2A-RFP expressing eggs. c) Blue and
green curves indicate the A-P axis Dronpa-Bcd concentration gradients in fixed fertilized
and unfertilized eggs, as in (a) and (b), here plotted on semi-log axes. Shading indicates
standard error. Unfertilized gradient has a lower maximum intensity, reflecting primarily
the lack of Bcd accretion at the embryo cortex (see Figs. 4.8 and 2.11), and a longer decay
length, reflecting lower Bcd degradation rates (see section 4.2.1).
97
ments in the confocal system (data not shown). Several differences between the fertilized
and unfertilized cases become apparent with this data set.
First, as noted in section 4.2.2, the fertilized gradient is noticeably steeper and shorter
than the unfertilized gradient. This is most obvious when considering the region near the
posterior pole of the egg, at approximately 0.8L. In the fertilized case the increase in spef
f
is the stan, where σ0.8
cific fluorescence over the observed 100 minutes is small, 3.9σ0.8
dard deviation of the wild-type (OreR) fertilized egg intensity at x = 0.8L over the same
time period. In the unfertilized case, the increase in specific fluorescence is much larger,
u
u
is the standard deviation of the wild-type unfertilized egg intensity at
, where σ0.8
18.6σ0.8
x = 0.8L. Second, the unfertilized gradient rises steadily throughout the entire period of
observation, rising monotonically in each segment of the egg. The fertilized gradient, by
contrast, plateaus at approximately 2.5 hours after oviposition, equivalent to 15 minutes
after the beginning of interphase 14, and subsequently regresses slightly at the anterior end
of the embryo. This is consistent with our measurement of up-regulated Bcd degradation
in fertilized eggs (section 2.2.1), and the absence of the same in unfertilized eggs (section
4.2.1).
4.2.4
Simulation of unfertilized gradient dynamics
To determine the extent to which the unfertilized gradient time series is quantitatively consistent with our measurements of lifetime and the predictions of the SDD model, we perform a coarse-grained, one-dimensional simulation of Bcd gradient evolution. We use the
method described in section 2.3.18, and incorporate a realistic source distribution derived
from recent measurements of bcd mRNA [65]. We assume a fluorophore maturation lifetime of 60 minutes, which is supported by experimental evidence in the case of eGFP
[109].
In Fig. 4.4a we show the experimentally observed time series of eGFP-Bcd unfertilized gradients, and in Fig. 4.4b, the corresponding simulated time series using a diffusion
98
87
87
a
b
111
t (min.)
t (min.)
111
135
135
159
159
183
0.1
183
0.1
0.3
0.5
x/L
0.7
0.9
c
20
15
time
10
5
0
0.1
0.5
x/L
0.7
0.9
50
[eGFP−Bcd] (nM)
[eGFP−Bcd] (nM)
25
0.3
0.3
0.5
x/L
0.7
30
20
10
0
0.1
0.9
d
40
0.3
0.5
x/L
0.7
0.9
Figure 4.3: Evolution of unfertilized and fertilized Bcd gradients in time a) Mean eGFPBcd concentration gradient observed in eGFP-bcd unfertilized eggs, with concentration
indicated as a heat map spanning 0 to 50 nM. Mean is taken over a population of 12 eggs.
Time in minutes since egg deposition is indicated on the vertical axis. Horizontal axis
indicates normalized position along the A-P axis. b) Identical to (a), for a population of 6
fertilized embryos. c) Mean unfertilized gradients from (a) displayed with concentration on
the y-axis, after background subtraction and normalization to absolute eGFP concentrations
(see sections 4.3.6 and 4.3.7). Time is indicated by color, from cyan to magenta, and
the increasing intensity at the egg posterior is clearly seen. Error bars overlap, obscuring
consecutive gradients, and are omitted. d) Identical to (c), calculated from 6 fertilized
eGFP-bcd embryos.
99
coefficient of 4
µm2/s
(as in Chapter 2) and a constant protein degradation rate of 0.009
min.−1 . Only the mature component of the simulated Bcd distribution is displayed. We
convert to absolute concentration values by setting the anterior value (at 0.1L) of the final
simulated profile equal to that of its experimentally observed counterpart. By this method,
the simulation predicts well the final concentration at the far posterior (0.9L) of the unfertilized egg three hours after egg deposition, predicting a value of 2.4 nM, as compared to
an observed value of 2.3 ± 0.9 nM1 . The evolution in time of the concentration at the anterior end (0.1L) is captured fairly well by the simulation, predicting a mean concentration
increase of 0.13 nM · min−1 , as compared to an observed value of 0.16 ± .02 nM · min−1 .
The same procedure applied to the fertilized gradient time series (Fig. 4.4c) does not
reproduce the observed data. This is because the one-dimensional simulation predicts only
a single concentration over the entire radial cross-section at each A-P axis position. This
assumption appears to be satisfied in the unfertilized case, as shown in Ref. [27] and in section 4.3.6. In the fertilized case, however, the distribution of Bcd between core and cortex
is highly inhomogeneous and, importantly, changes in time (see section 2.3.12). To correct
for this we float an additional parameter, Γ(t), to convert simulated Bcd concentrations to
absolute values. We define Γ by the following:
sim−cort
sim
CBcd
= Γ(t)CBcd
(x, t) =
obs
CBcd
(0.1L, t) sim
C (x, t)
sim
CBcd (0.1L, t) Bcd
(4.1)
sim
obs
where CBcd
and CBcd
are simulated and observed Bcd concentrations, respectively, and
sim−cort
CBcd
is the simulation value rescaled to predict cortical concentration in the embryo.
Γ effectively converts observed Bcd concentration at the embryo cortex to the mean consim−cort
centration over the entire cross-section. We plot CBcd
in Fig. 4.4d. The predicted
mean value at the posterior (0.9L) over the entire experiment time is 1.2 nM, as compared
1
The reported uncertainty here reflects error only from the distribution in observed intensity values and
background subtraction; it does not include uncertainty from the absolute eGFP concentration calibration.
The true quantities of interest in this section are the relative concentration ratios.
100
to an observed value of 0.8 ± 0.8 nM. That the observed concentration increase is also
significantly less was discussed in section 4.2.3.
Though we take the predicted behavior of posterior intensity levels as evidence that
the unfertilized gradient length in a broad sense can be explained by the SDD model, the
full shape of the gradient along the A-P axis is not well captured by the above simulation
in either the unfertilized or the fertilized case. The actual gradients are steeper and more
concave that the simulated gradients; a similar effect was noticed comparing the simulated
and observed gradients in Chapter 2. We emphasize that this is a coarse-grained simulation, intended principally to capture the scaling of gradient length with protein lifetime
according to the SDD model, incorporating a realistic source distribution and maturation
lifetime. It ignores potentially important effects such a nuclear trapping, nuclear scattering,
cytoplasmic stirring, and hydrodynamic effects. Other work has simulated the system in
greater detail [89, 99, 108]. However, we do note that the discrepancies between our simplified simulation and the actual gradient are qualitatively similar in both the fertilized and
unfertilized cases. This suggests that the incompleteness of the simplified model does not
lie predominantly in its failure to account for nuclei; it is instead more likely that nuclei are
irrelevant to gradient formation, as argued in Ref. [35], or that the microscopic structure
of the unfertilized egg is less different from that of the syncytial embryo than it appears, as
suggested by our data in section 3.2.2.
4.2.5
Measurement of total Bcd quantity in fertilized and unfertilized
eggs
Knowledge of absolute concentrations of signaling molecules is of special biophysical importance because it allows for the determination of physical limits on the precision of the
signaling process [26, 40]. In Ref. [26], the authors find absolute eGFP-Bcd concentrations in syncytial Drosophila nuclei by bathing an embryo in a solution of known eGFP
101
25
[eGFP−Bcd] (nM)
[eGFP−Bcd] (nM)
25
a
20
15
time
10
5
0
0.1
0.3
0.5
x/L
0.7
10
5
0.3
0.5
x/L
0.7
0.9
50
[eGFP−Bcd] (nM)
[eGFP−Bcd] (nM)
50
c
40
30
20
10
0
0.1
15
0
0.1
0.9
b
20
0.3
0.5
x/L
0.7
0.9
d
40
30
20
10
0
0.1
0.3
0.5
x/L
0.7
0.9
Figure 4.4: Comparison of fertilized and unfertilized Bcd gradients with simulation a)
Mean eGFP-Bcd gradients in unfertilized eggs from 87 to 183 minutes after egg deposition,
as shown in Fig. 4.3c, with time indicated by color proceeding from blue to red. 12
minutes elapse between successive gradients b) eGFP-Bcd gradients obtained from a onedimensional finite-difference simulation of gradient dynamics (section 2.3.18) assuming a
realistic source distribution [65], a 60 minute eGFP maturation lifetime [109], and a 110
minute Bcd lifetime (section 4.2.1). The absolute concentration is obtained by normalizing
the anterior value of simulated gradient to the observed value at the last time point only. c)
Identical to (a), showing the observed fertilized eGFP-Bcd gradients d) Simulated gradients
in the fertilized case, obtained as in (b) but assuming a protein lifetime as determined in
Chapter 2, and normalizing the anterior (0.1L) value to the observed concentration at each
time point.
102
concentration and comparing the nuclear eGFP-Bcd fluorescence intensities to that of the
reference solution. We use a similar procedure to calibrate the intensities obtained from our
two-photon imaging data to absolute eGFP concentrations, as described in section 4.3.7.
Unlike the measurements of Gregor et al., the concentrations we report are not exclusively
nuclear concentrations; rather they are averages over larger regions containing both nuclei and cytoplasm in the fertilized case and cytoplasm alone in the unfertilized case. For
this reason, our measurement of approximately 40 nM peak eGFP-Bcd concentration at
the anterior of the embryo is not necessarily inconsistent with the measurement of 55 nM
eGFP-Bcd concentration in anterior nuclei as reported by Gregor [26]. Measurements of
nuclear concentrations are not equal to cortical concentrations in general unless the nucleocytoplasmic ratio is 1, which is not the case [35]. However, because Bcd is believed to be
distributed uniformly between cortex and core in unfertilized eggs [27], the measurement
of cortical unfertilized eGFP-Bcd concentrations offers an especially good opportunity to
determine the absolute number of Bcd molecules in the entire egg.
One major limitation of calibrating absolute intensities by using a reference solution
is the problem of fluorophore maturation. GFP-like proteins expressed in vivo show some
delay, called a maturation time, prior to becoming fluorescently excitable [24]. Hence our
direct measurements by the method described above are limited to mature fluorophore only.
In Fig. 4.5a, we show the total quantity of mature eGFP-Bcd in fertilized and unfertilized
eggs, obtained by the summation procedure of section 4.3.6. The unfertilized quantity
increases steadily, as observed previously [16, 95], and the fertilized quantity peaks shortly
after the onset of cycle 14, consistent with the up-regulation of degradation and decline of
synthesis described in Chapter 2. The fertilized and unfertilized quantities are not identical
prior to this time; this is perhaps due to the fact that we fix Γ to unity at the beginning of
data acquisition. This is equivalent to the assumption that the core eGFP-Bcd concentration
is equal to the cortical concentration at t = 87 minutes. If we relax this assumption, then it
is possible that unfertilized and fertilized concentrations follow equivalent trajectories prior
103
7
7
x 10
(molec.)
8
3
eGFP−Bcd
2
1
Q
Qmat (molec.)
4
x 10
6
4
2
a
0
100
150
t (min.)
b
200
0
100
150
t (min.)
200
Figure 4.5: Calculation of total number of Bcd molecules in embryos and unfertilized
eggs a) Total number of mature eGFP-Bcd molecules in fertilized (red) and unfertilized
(blue) eggs, computed from the gradients in Fig. 4.3. Time indicates minutes since egg
deposition. b) Total number of Bcd molecules, in eGFP-bcd/bcd- embryos, calculated by
determining the fraction of mature fluorophore to total fluorophore at each point and time
in the fertilized (red) and unfertilized (blue) egg by simulation (section 4.2.4).
to cycle 14.
To compensate for the omission of immature fluorophore, we find the ratio of mature
eGFP-Bcd to total eGFP-Bcd, both mature and immature, at all points and times in our
simulation from section 4.2.4, which assumes an eGFP maturation time of 60 minutes. We
use this matrix to rescale the measured concentrations of mature fluorophore to total fluorophore, and the integrated result is shown in Fig. 4.5b. The shapes of each curve are
qualitatively similar, though we obtain a peak total eGFP-Bcd quantity more than double
that of the mature component alone. This value is 6.6 ± 0.8 × 107 molecules of Bcd at time
t = 124 min., which corresponds to mid-cycle-13, and is broadly consistent with the time
of peak Bcd expression reported by Little et al. [65]. The error reported here and in the figure consists of uncertainty introduced by the fluorescence measurement and inter-embryo
expression differences alone; it does not include error introduced from the calibration process, which is substantial. Including uncertainty from all sources, the measurement of peak
eGFP-Bcd quantity is 6.6 ± 1.8 × 107 molecules.
104
Despite the relatively large (~25%) error bars, this number is still inconsistent with the
peak value of 1.5 ± 0.2 × 108 Bcd molecules which we report in section 2.2.3. Clearly
there is some source of systematic error which remains unaccounted for in one or both of
these measurements. Potential candidates for this source of error with regard to the in vivo
measurement are pH imbalances, opacity of the vitelline membrane, scattering of visible
light within the embryo, and mistaken assumptions regarding core-to-cortex concentration
ratios. Further quantitative controls are needed in both cases to arrive at better estimates of
absolute protein concentration values.
4.2.6
Conclusion
In this chapter, we presented evidence that the formation of the Bcd gradient in unfertilized
eggs is well explained by the SDD model. Extending our technique of measuring protein
lifetime from Chapter 2, we find that the Bcd lifetime is longer in unfertilized eggs, and
we additionally show that direct measurements of the unfertilized gradient show a correspondingly longer length, in way that is qualitatively and quantitatively consistent with the
predictions of the SDD model. However, the full features of gradient shape are not well
predicted by the simplest version of the SDD model. In fact, as we argue, the measurement of the unfertilized gradient represents an important constraint for biophysical models
of gradient formation in Drosophila, as it requires modelers to provide an explanation for
the observed steepness of the gradient without resorting to hypothetical effects caused by
nuclei.
105
4.3
4.3.1
Methods
Preparation of fixed samples
We use the combined dronpa-bcd/h2a-rfp transgene described in section 2.3.16 for comparison of the fertilized and unfertilized gradients in fixed tissue. The Dronpa-Bcd expression
allows for imaging by the high-contrast method described in section 4.3.2, and the H2ARFP expression allows for staging and identification of nuclei in the fertilized embryos.
Flies were kept at 22◦ C and fed baker’s yeast. Eggs were collected on plates of apple
juice and agar mixture for 60 minutes, and plates collected at this time were aged for
two hours. Eggs were dechorionated for 90 seconds in bleach and fixed in a solution of
20% paraformaldehyde and 1X PBS for 45 minutes. For imaging endogenous DronpaBcd/Histone-RFP fluorescence, methanol immersion for removal of the vitelline membrane
was avoided; rather, fixed eggs were stuck to a piece of double-sided tape and devitellinized
by hand with a sharp needle. Eggs were then mounted on a glass slide in Aqua PolyMount
(PolySciences), and the preparation was hardened by overnight incubation.
4.3.2
Optical lock-in detection imaging of Dronpa-Bcd
Pioneered by Marriott et al., optical lock-in detection imaging (OLID) uses the photoconversion properties of switchable fluorophores such as Dronpa to produce high-contrast
images of specific fluorescent probes in systems with large and dynamic non-specific background signals [106]. It accomplishes this by using some periodic optical drive to push the
fluorophore of interest between its two emission states. It is related to and in a sense a development upon fluorescence lifetime imaging (FLIM) [110]. In OLID, a cross-correlation
analysis is performed between the intensity signal obtained over time from each pixel in
an image as compared to the reference intensity signal Iref from some region in which the
fluorophore of interest is expressed purely. The pixelwise correlation coefficient ρ is given
106
by [106]:
ρ(x, y) =
X (I(x, y, t) − µI ) (Iref (t) − µR )
σI σR
t
(4.2)
Pixels in the image containing large quantities of fluorophore relative to background
will show large values of ρ, and, if the background noise shows little cross-correlation
with the reference signal, pixels in which the amount of fluorophore is small relative to
background will show small values of ρ. In this way, OLID not simply removes but also
uses the autofluorescent background to discriminate between intensity levels of specific
fluorophore. As we find, this aspect of OLID can also be a drawback in systems in which
the signal-to-noise ratio is already high.
In our experiment, we fix embryos expressing Dronpa-Bcd/Histone-RFP according to
the method in section 4.3.1. Each embryo is then imaged in a repetitive sequence with
6-8 images at 496 nm excitation followed by a single image at 405 nm excitation. These
choices of wavelength correspond to the conversion and activation wavelengths of Dronpa
(see section 2.3.3), and serve to push the fluorophore population between bright and dark
states. This procedure is repeated through 10-30 cycles. Additionally, during each cycle,
we acquire a single image at 561 nm to measure Histone-RFP fluorescence and to mark
nuclei.
In Ref. [106], the authors compared the results computed by obtaining a reference signal from a pure sample of the fluorophore of interest; in particular, microspheres containing
a high concentration of the fluorophore Dronpa. They find that the correlation image obtained by this method is substantially identical to the correlation image produced by using
a region of the specimen expressing Dronpa at a high-level. Following this technique, we
use as a reference signal the integrated fluorescence intensity of a nucleus at the anterior of
the embryo.
In our system, we find that the correlation image obtained by the OLID method of Mar107
−5
4
2
x 10
12
b
10
ρ/IH2A (a.u.)
ref
(a.u.)
a
I
x 10
1.5
1
IH2A
8
6
4
0.5
0
20
40
n
60
80
2
0
0.5
x/L
1
Figure 4.6: Detection of Dronpa-Bcd by correlation image a) Reference drive Iref , obtained by computing the green channel intensity over time for a single anterior nucleus
of a fixed mid-cycle-14 embryo expressing Dronpa-Bcd at a high level. b) A-P axis nuclear intensity profile in the correlation image produced by by the method of Marriott et
al. [106]. Dots represent the correlation coefficient ρ (Eq. 4.2) between the intensity time
series of a single nucleus and the reference drive Iref , divided by the red channel H2A-RFP
intensity in the same nucleus. Gradient in the correlation image principally reflects minor
inhomogeneities in beam intensity, as indicated similarly by the gradient in IH2A .
riott et al. [106] produces correlation coefficients in excess of 0.8 across the entire A-P axis
of the embryo, and gradients which are largely flat after normalization for inhomogeneities
in the excitation beam intensity (Fig. 4.6b). This is because the correlation coefficient (Eq.
4.2) is insensitive to changes in the magnitude of the specific signal if noise is negligible
relative to signal, as is often the case with fluorescence images of fixed specimens, which
can be averaged for arbitrarily long times.
Instead, we use a modified implementation which is better suited for the low-noise
regime which characterizes our system. The measured intensity signal y(t) can be written
as:
y(t) =
Igreen (t)
= As1 (t) + Bs2 (t) + ξ(t) + σn(t)
< Ired >t
(4.3)
where A and B are constants, s1 and s2 are zero-mean unit-norm processes, ξ(t) is a
general function of time, σ is the noise level, and n(t) is a Gaussian random variable. We
108
normalize by the mean red channel intensity at each point to correct for small inhomogeneities due to variance in DNA content as represented by Histone-RFP, such as differing
numbers of nuclei in a given slice of the image. ξ(t) represents the general loss of fluorescence from photobleaching; we define it by filtering the measured signal by moving
mean:
1
ξ(t) =
2T
ZT
y(t)dt
(4.4)
−T
where T is the period of the conversion-reactivation cycle. The detrended intensity
signal y(t) − ξ(t) then takes the form of the two-user synchronous channel defined in Ref.
[111]. Our problem is effectively to determine the received amplitude of the the second
“user”, representing Dronpa-Bcd concentration. As all GFP-like fluorophores have some
photoactivation cross-section [112], there is also some residual oscillation due to H2A-RFP.
We let s1 represent the signature waveform of H2A-RFP emission2 ; s1 is found by imaging
embryos containing H2A without Dronpa-Bcd and choosing B = 0. The characteristic
waveform of Dronpa-Bcd emission s2 is found by subtracting the posterior intensity signal
y(0.9L, t) from the anterior signal y(0.1L, t). Provided beam intensity is uniform across
the sample, the difference between the signals is due to Dronpa-Bcd alone.
The maximum likelihood estimator for B is given by the decorrelating detector c2 [113,
111]:
c2 =
s2 − ρ12 s1
1 − ρ12
(4.5)
where ρ12 is the cross-correlation between the Dronpa-Bcd and H2A-RFP waveforms:
Z
mT
ρ12 =
s1 s2 dt
(4.6)
0
2
s1 additionally includes any conversion/reactivation oscillations produced by endogenous autofluorescent molecules.
109
and m is the total number of cycles. B is determined at each point by the inner product
R mT
B = 0 c2 (t) (y(t) − ξ(t)) dt. To obtain the gradients in section 4.2.2, we subdivide the
A-P axis of the embryo into 200 boxes and find the mean signal y(t) in each box, using this
time series as the raw input to the detector, after detrending to remove ξ(t).
4.3.3
Computational identification of nuclear mask
Because the surface of the syncytial Drosophila embryo at cycle 14 contains in excess of
6000 nuclei [14, 114], manual identification of nuclear boundaries is not practical. We use
a computational feature extraction routine to identify nuclei locations from an image of
fluorescently-tagged Histone, which we expect to be expressed with approximately equal
intensity in each syncytial nucleus, as the surface nuclei are uniformly diploid [14]. To
segment the image into a mask consisting of individual nuclei, we use the morphological
top-hat filter of F. Meyer [115], originally developed for identifying nuclei in cytological
specimens. More specifically, we estimate the mean nuclear radius a from the image, in
pixels, and filter the Histone fluorescence image using the MATLAB function “imtophat”
with a disk structuring element of radius a. The output from the top-hat filter is then
sequentially thresholded beginning with the maximum intensity in the filtered image and
stepping downwards. At each step, regions in the thresholded binary image of area less
than ∼ 5a2 , chosen to avoid identifying clusters, are counted as nuclei and the intensity of
each putative nucleus is zeroed out in the filtered image. This process is repeated iteratively,
proceeding to lower thresholding intensities until the nuclear mask becomes contaminated
with artifacts. The minimum threshold is set by inspection and varied based on the signalto-noise ratio of the data set.
110
4.3.4
Preparation of samples for live imaging
The D. melanogaster transgene expressing eGFP-bcd in a bcd− background, used in Ref.
[27], was obtained from Thomas Gregor. Sterile XO males were generated by crossing
males from a stock with a compound XY chromosome (C(X,Y),yB/0) with Oregon-R wildtype females. The XO males were crossed to females from the eGFP-bcd stock to produce
the unfertilized eggs, which do not undergo mitotic division cycles and contain only a single
female pronucleus.
Eggs were collected on plates of apple juice and agar mixture. Plates were changed at
60-80 minute intervals during experimental collections, though we notice that the agar was
generally free of eggs up to 40 minutes following a plate change, and thus estimate mean
egg age at collection to be 15 minutes after oviposition. Eggs were dechorionated for 1
minute in bleach, washed in dH2 O, arranged on a fresh piece of agar, and glued to a glass
slide with adhesive derived from scotch tape and heptane. Eggs were allowed to dry briefly
and then immersed in halocarbon oil (Sigma-Aldrich). No coverslip was used.
4.3.5
Two-photon laser scanning microscopy
The microscope apparatus used is similar to that described in Ref. [116]. A Coherent
Verdi V10 diode-pumped laser is fed into a mode-locked Coherent Mira 900 Ti-sapphire
oscillator, generating 80 fs pulses at 918 nm with a repetition rate of 80 MHz. We use
a 918 nm excitation beam as it is close to the two-photon action cross-section maximum
of eGFP as reported in Ref. [117]. Intensity is modulated with a Pockels cell (Conoptics,
model 350-50). The beam passes through a shutter (UniBlitz) and is scanned via a pair of
galvo driven mirrors. We use a 20x infinity-corrected 0.75 NA multi-immersion objective
(Nikon, CFI Plan Fluor 20x MI). Light emitted from the specimen is reflected by a dichroic
beamsplitter into a two-channel photomultiplier system, with a HQ610/75 two-photon 18◦
emission filter (Chroma) in the first bay and a BG39 filter in the second. PMT output is
111
1
I (a.u.)
0.8
0.6
0.4
ωz
ωxy
0.2
0
−2
0
2
s (µm)
4
6
Figure 4.7: Determination of two-photon point spread function Fluorescence intensity
of a single 100 nm fluorescent microsphere, imaged in a z-stack with 200 nm stepping and
averaged over four realizations. Red curve is a Gaussian distribution fit to the lateral (x-y)
intensity profile. Blue curve is a Gaussian fit to the axial (z) intensity profile. Lateral and
axial 1/e widths are 410 nm and 2750 nm, respectively. This compares to theoretical optima
of 290 nm and 1600 nm [119].
fed into a set of pre-amplifiers (Stanford Research Systems, model SR570) and a DAQ
(National Instruments, model BNC-2090). High resolution 1024x192 images are acquired
using MATLAB and the ScanImage software suite [118]. Laser power was measured to be
85 mW at the sample. The point spread function of this apparatus is determined by imaging
a set of four 100 nm diameter fluorescent microspheres (Polysciences). As shown in Fig.
4.7, we recover an approximately Gaussian relationship.
4.3.6
Two-photon image analysis and gradient computation
Spatial inhomogeneities in beam intensity across the field of view are mitigated by normalizing all images by an image of uniform concentration of 100 mM fluorescein. Autofluorescence is then minimized and specific eGFP signal maximized by creating a compound
image whose intensity at each pixel is a linear combination of measured intensity in both
PMT channels:
IeGF P = IBG39 − AI610 + B
112
(4.7)
Coefficients are determined by acquiring images in wild-type embryos lacking fluorescent
markers. The method is based on the observation that the autofluorescent signal in the red
channel is strongly correlated with that in the green channel, and that eGFP bleed through
into the red channel is a small effect which can be corrected for (∼ 4%, see section 4.3.7).
2
2
The coefficient A is chosen to minimize the variance < IeGF
P > − < IeGF P > inside the
egg, and B is chosen such that < IeGF P >= 0. The ensemble average is taken over the
length of the egg and the developmental time range of interest (1-3 hours after oviposition).
Gradient computation in unfertilized eggs requires a technique different from that used
in fertilized eggs, as late syncytial embryos have a layer of cortical cytoplasm 25 mm thick
[14] which is transparent to visible light. In unfertilized eggs, by contrast, the cortex and
core are equally opaque, and autofluorescent yolk granules extend nearly to the surface of
the egg. To obtain an accurate reading of the Bcd gradient it is necessary to compute the
fluorescence intensity at equal optical depth along the length of the egg. A stack of 30
images, each averaged over two realizations, with stepping 4 mm is acquired beginning at
the mid-sagittal plane of the embryo and proceeding upward. The anterior-posterior (A-P)
axis is chosen manually and spanned with 200 equally-spaced points. A MATLAB routine
calculates the mean intensity in a 8 mm x 8 mm box centered at each AP axis point, and for
each of the 30 z-stacked images. The vitelline membrane height Svit (x) at each point is
located by plotting I610 (x, z), applying a smoothing spline, and finding the first zero of the
first derivative approaching the egg from above. Statistical outliers in the vitelline membrane location, due to occasional debris adhering to the membrane surface, are replaced by
a moving mean. The Bcd gradient GBcd is computed by summing
dmax /∆z
GBcd (x, t) =
X
i=1
IeGF P (x, z = Svit (x) − i∆z, t)
dmax/∆z
(4.8)
This assumes that observed intensities at various depths, after background subtraction, are
linearly proportional to each other with a single value along the entire A-P axis of the
113
Figure 4.8: Verification of uniform radial distribution of Bcd in unfertilized eggs and
illustration of axes for gradient computation (a) Image from Ref. [27], Fig. 6, showing
longitudinal cross-section of wild-type anti-Bcd stained unfertilized egg cut near the AP
axis midpoint. (b) Fluorescence intensity in (a) plotted as r, indicating no radial dependence of Bcd concentration in unfertilized eggs. Error bars represent standard deviation.
(c) Sample eGF P − Bcd expressing unfertilized egg 2.5 hr after oviposition, IeGF P (see
section 4.3.6) shown. Concentration of Bcd protein (orange) is visible from the anterior
pole. Gradient is computed at 200 points along three parallel axes (red), with the center
being the A-P axis.
embryo. We choose the maximum depth dmax to be 40 mm and the z stepping ∆z to be 5
mm. To reduce noise, this calculation is repeated, beginning with the vitelline membrane
computation, for two different axes parallel to the A-P axis and separated by 16 mm on each
side. Fig. 4.8c shows the three axes (red) overlayed on the mid-sagittal plane image of a
sample egg. The z-stack extends out of the page.
obs
GBcd is converted to an absolute concentration of eGFP-Bcd (CBcd
) using the scale fac-
tor obtained in section 4.3.7 and adjusting for the loss of specific eGFP intensity removed
during the background subtraction process due to bleed-through.
Total quantity of mature eGFP-Bcd Qmat is determined by the integration
114
ZL
Qmat (t) =
obs
Γ(t)CBcd
(x, t)πR2 (x)dx
(4.9)
0
p
where R = b 1 − x2/a2 , which assumes the embryo is a prolate spheroid with major
axis a = 250 µm and minor axis b = 90 µm. The integral is computed by Riemann sum
using 200 equipartitioned boxes along the A-P axis. Lacking data at the poles of the embryo
obs
(x < 0.1L, x > 0.9L), we use the nearest value of CBcd
at these locations; however, due to
the elliptical geometry these make only a small contribution to the embryo volume. The
scale factor Γ(t) is set to unity in the unfertilized case. In the fertilized case, Γ is fixed at
unity at the first time point, t = 87 min. post-oviposition, and subsequently computed by
fitting to simulation according to Eq. 4.1.
4.3.7
Calibration of two-photon fluorescence intensity to absolute eGFP
concentration
Following the method of Gregor et al. [27], we convert fluorescence intensities to absolute eGFP concentrations by imaging a sample of purified eGFP. We obtained a set of
four samples of purified eGFP from Lorraine Schepis, and measure the absorption in a
spectrophotometer (Biorad SmartSpec Plus). The molar extinction coefficient of eGFP
is 56 × 103 M−1 cm−1 [24], and using this number we determine that the samples are of
concentration 2.8 µM, 3.8 µM, 4.4 µM, and 7.5 µM. Dilution of each sample 100-fold produces a fluorescence intensity under two-photon microscopy within the dynamic range of
the settings used for embryo imaging. We fill a square glass capillary of width 100 mm with
the diluted eGFP solution and acquire a z-stack of fluorescence images through the sample
(Fig. 4.9). Importantly, an intensity plateau is reached, indicating that the axial point spread
function is contained entirely within the solution at the maximum intensity value. As the
PMT attached to the BG39 filter has a negative offset, we use the intensity in the 610/75
115
300
I (a.u.)
BG39
610/75
200
100
0
0
50
100
150
z (µm)
Figure 4.9: Imaging of purified eGFP under two-photon illumination Mean fluorescence intensity in a 250 x 600 mm raster-scanned image of purified eGFP in a glass capillary of depth 100 mm, in green (BG39) and red (610/75) emission channels. A z-stack of
images is acquired with 10 mm spacing. Blue dots indicate intensity in the green (BG39)
channel and red dots indicate intensity in the red channel (610/75). An intensity plateau
is reached, indicating that the entire point spread function is contained within the eGFP
solution at these points. The red channel intensity is used to determine the half-maximum
of the intensity plateau, as well as the eGFP emission bleed-through into the 610/75 filter.
channel to determine the location of the half-maximum of the plateau, and find the total
fluorescence intensity in the green channel by doubling the difference between the plateau
and the half-maximum. Applying this calculation to all four samples, we find that green
channel intensity converts to eGFP concentration with the coefficient 0.092 ± .023 nM/i.u.,
where i.u. indicates green channel intensity units for the excitation power, emission filter,
and PMT settings used in this chapter. Additionally, the plateau in the red channel allows
for the calculation of the bleed-through of eGFP fluorescence into the 610/75 filter, relative
to the BG39, which is 4% in the units of the settings in this chapter.
116
Appendix A
Choice of embryo parametrization for
Bcd gradient quantification
While the Bicoid gradient is typically plotted and modeled as a one-dimensional gradient along the anterior-posterior (A-P) axis, the Drosophila embryo is not a quasi-onedimensional system. A reasonably close approximation is to model the embryo as a prolate
spheroid with an aspect ratio of slightly less than 3 [84], but real embryos deviate from
this shape [26] because the dorsal and ventral surfaces are not identical. More significantly
for the case of Bcd, as detailed in Fig. 2.11, the distribution along the apical-basal axis is
not uniform and changes with developmental stage. However, most published studies of the
Bcd gradient report it as an array of protein concentration values, measured at the surface of
the embryo for imaging reasons, and plotted as the projection of the measurement position
along the anterior-posterior axis [16, 27, 65]. Furthermore, this coordinate is often plotted
as fractional position along the A-P axis to obscure small variations in embryo length.
Genes downstream from bcd, most especially pair-rule genes such as eve [120], display significant curvature along the dorso-ventral axis, and as such are poorly represented
when expression intensity values are collapsed onto the A-P axis [121]. This curvature,
also called splay, has generated considerable interest in “stripe-straightening” algorithms,
117
which, though numerically non-trivial, allow splayed gene expression patterns to be plotted along a single axis with good reproducibility between embryos [121, 122]. It has been
argued [122] that, since A-P axis development is under control of bcd, the Bcd protein gradient itself should show evidence of splay1 . More complicated, splay-friendly coordinate
systems have also been used in modeling work related to the Bcd gradient [84].
We use the high-contrast optical lock-in imaging technique described in section 4.3.2
to reexamine the question of the optimal choice of coordinates to use in describing the Bcd
protein gradient. We define the optimal parametrization to be that which minimizes the
variance of Dronpa-Bcd fluorescence intensity at each point along the axis in the respective
coordinate systems. We consider three possible metrics. The first is the aforementioned
projection onto the A-P axis, normalized to the total length of the A-P axis.
ξ1 =
(~x − ~rant ) · (~rpos − ~rant )
|~rant − ~rpos |2
(A.1)
Here ~rant and ~rpos specify the anterior and posterior poles, respectively. The contours
of this coordinate are shown in Fig. A.1a. The second metric represents absolute distance
from the anterior pole, again normalized to the length of the A-P axis.
ξ2 =
|~x − ~rant |
|~rant − ~rpos |
(A.2)
The contours of this coordinate are shown in Fig. A.1b. Finally, we consider a variant
of the prolate spheroidal coordinates used by Coppey et al. [84] and described in Ref.
[124], defined such that ξ3 ∈ [0, 1] for purposes of comparison to the previous metrics.
ξ3 =
|~x − ~rant |
|~x − ~rant | + |~x − ~rpos |
(A.3)
1
More recent work by the same authors reports that later segmentation gene expression patterns show
substantial independence from maternal gene products such as Bcd [123].
118
a
b
c
Figure A.1: Illustration of different choices of metric along the Drosophila A-P axis
a) Contours of coordinate ξ1 as defined in Eq. A.1, for a single mid-sagittal cross-section,
representing the projection along the anterior-posterior axis. Color from blue to red indicates scale from 0 to 1. b) Same as in (a), illustrating the contours of coordinate ξ2 , distance
from the anterior pole. c) Same as in (a) and (b), illustrating the contours of coordinate ξ3 ,
derived from the prolate spheroidal coordinate system [124].
119
The contours of this coordinate are shown in Fig. A.1c. While it is not identical to
the stripe-straightening method described by Spirov and Holloway [121], which requires a
fitting routine applied to an image of striped gene expression, it will nevertheless accommodate some splay should it be present in Bcd expression.
In Fig. A.2a is shown the quantification of the Bcd gradient in a fixed mid-cycle14 embryo expressing Dronpa-Bcd and Histone-RFP, using the optical lock-in method of
section 4.3.2, for each of the three coordinate choices described as above. The shape of the
gradient is qualitatively identical in each case. In Fig. A.2b is plotted the relative deviation
of the nuclear intensity values at each point from 0 to 1 along the three coordinate axes.
The 10% precision level described in Ref. [26] is recovered in each case, though the A-P
axis projection method, coordinate ξ1 , is superior in the anterior-most part of the gradient,
where the source is located.
The data suggest that the precision of the Bcd gradient is not especially sensitive to
the specific choice of parametrization. As the known 10% relative deviation in Bcd was
previously attributed predominantly to intrinsic variance rather than measurement error
[26], we further conclude that the embryo does not make use of dorso-ventral variation in
Bcd to establish the splay pattern of the pair-rule genes, which occurs by other means. In
this thesis, we use the A-P axis projection coordinate ξ1 , except in Chapter 2, where ξ3 is
used as it is suited to the creation of equal area boxes along the surface of the embryo.
120
0.3
0.5
0.4
0.2
δBnuc/Bnuc(ξ n)
Bnuc(ξ n) (a.u.)
a
0.1
0
0
0.5
ξn
1
b
0.3
0.2
0.1
0
0
0.5
ξn
1
Figure A.2: Test of different metrics applied to the Bcd gradient a) Dronpa-Bcd nuclear
intensities Bnuc (see section 4.3.2) plotted as coordinate ξ1 (magenta), ξ2 (green), and ξ3
(blue). b) Relative standard deviation ∂Bnuc/Bnuc , as used in Ref. [26], shown for each case,
with coloring as in (a). Error bars are determined by randomly shifting the locations of the
anterior and posterior poles up to 50 mm.
121
Appendix B
Diffusion in driven, athermal systems
In Chapter 3, we reviewed work studying diffusion of proteins in biological media. Interestingly, in many cases, the measured diffusion coefficients of various proteins in cytoplasm
show only a weak dependence on protein size [31]. In Ref. [72], the authors studied the
diffusivity not only of proteins but also of inert fluorescent markers of various sizes in living cells. They found that the diffusion coefficients of the inert markers scaled according
to size and were well predicted by the Stokes-Einstein relation plus some positive offset,
which they attribute to stirring.
Phenomena such as these motivate further work to understand the extent to which diffusive behavior mimicking Brownian motion can arise from fluctuations that are non-thermal
in nature. Here we study this general phenomenon in a highly idealized computational
simulation of interacting spherical particles moving over random quenched disorder, which
obviously cannot be considered to model the behavior of proteins in cells. However, examples of real physical systems that can be modeled in this way include colloidal particles
moving over random substrates [125], vortices in strongly pinned type-II superconductors
[126], and charge transport through metallic dot arrays [127]. When a directional drive
The material in this appendix is developed upon a manuscript co-authored with Cynthia OlsonReichhardt and Charles Reichhardt.
122
is applied, these systems exhibit a transition from a pinned phase to a disordered moving
phase, where the particles flow plastically and tear past their neighbors [128, 129]. The
plastic flow is associated with non-Gaussian velocity distributions [129, 130, 131], 1/f velocity noise [132, 133], and rapidly changing flow channels [134]. Additionally, particle
motion in the plastic flow regime can have diffusive properties that are anisotropic due to
the symmetry breaking imposed by the applied drive [135].
Since the plastic flow regimes show nonequilibrium fluctuations, a possible new method
for characterizing the plastic flow is with the recently developed fluctuation theorems [136,
137, 138, 139] which have been applied to various systems out of equilibrium [140, 141,
142]. Previous work has already shown that generalized fluctuation-dissipation theorems
can be applied to driven vortex matter in order to extract an effective temperature of the
moving vortices as a function of drive [143]. This suggests that a similar application of
fluctuation theorems could also provide a useful way to characterize the dynamics of these
plastic systems.
In this work we show how the fluctuation theorem (FT) of Ref. [136], which relates the
frequency of entropy-destroying or “second-law-violating” trajectories to entropy-creating
trajectories, can be used as a method to characterize plastic depinning. The GallavottiCohen FT was demonstrated to hold analytically in the long time limit for a class of timereversible dynamical systems, and has been verified numerically in many others [144, 145].
However, to our knowledge the FT has not previously been applied to systems exhibiting
plastic depinning.
It is known to apply to the power dissipated by a thermal bath for
particles obeying Langevin dynamics [146, 147], but it is not guaranteed to apply in the zero
temperature limit [148]. Here, we consider strictly non-thermal systems; the noise which
produces fluctuations in our system is athermal and arises as a result of driving the system.
In a sense, the time window over which the FT appears to be valid can provide insight into
how the flow dynamics change with parameters such as drive or disorder strength.
123
B.1
Method and Simulation
We specifically use the form of the FT given in Ref. [145]. One of the main predictions of
the FT is that the probability density function (p.d.f.) of the injected power p(Jτ ) obeys the
following relation:
p(Jτ )
= eJτ τ /βτ ,
p(−Jτ )
(B.1)
where Jτ is the injected power, τ is the duration of the trajectory, and βτ approaches a
constant value β∞ as τ → ∞. The value β∞ is referred to as a “nonequilibrium temperature” [144, 141, 142, 149]. It is not related to an ambient temperature, which is zero for
our system. Wang et al. [140] experimentally measured the quantity in Eq. 1 from the
trajectories of a colloid driven through a thermal system. In the system we consider, there
is no thermal bath and T = 0; instead, the particles experience only an external drive, a
random quenched background, and interactions with other particles. Here, the fluctuations
are generated via the plastic motion of the particles.
We consider colloidal spheres confined to two dimensions and driven with an electric
field in the presence of randomly distributed pinning sites. This particular model system
has been shown to exhibit the same general dynamical features, including plastic flow and
moving crystalline phases [129], observed in other collectively interacting particle systems
driven over random disorder such as vortices in type-II superconductors [128, 130, 150];
thus, we believe the behavior we observe will be generic to other systems of this type.
Additionally, experimental realizations of this system permit the direct measurement of the
particle trajectories [151].
We simulate a system of Nc = 792 colloids at density ρ = 0.5 with periodic boundary conditions in the x and y directions, and employ overdamped dynamics such that the
equation of motion for a single colloid i is
η
dri
= fYi + fpi + fd .
dt
124
(B.2)
All quantities are rescaled to dimensionless units, and the damping constant η is set to
unity. There is no thermalization. The colloid interaction force fYi is given by the following
P c
1
4
−4rij
r̂ij . Here Ac is an adjustable
screened Coulomb repulsion: fYi = N
j6=i Ac ( rij + r2 )e
ij
coefficient, ri(j) is the position of vortex i(j), rij = |ri − rj | and r̂ij = (ri − rj )/rij . The
quenched disorder introduces a force fpi which is modeled by Np nonoverlapping randomly
placed attractive parabolic pinning sites of strength Ap = 0.5 and radius rp = 0.45, fpi =
PNp
k=1 (−Ap rik /rp )Θ(rp −rik )r̂ik , where Θ is the Heaviside step function. Np varies slightly
based on the number of sites which can be placed given the random seed used in a particular
realization, but is approximately normally distributed with µN = 1000 and σN = 17. The
driving force fd = fd x̂ is a constant unidirectional force applied equally to all colloids. We
initialize the system using simulated annealing in order to eliminate undesirable transient
effects due to relaxation, and apply a driving force. The equations of motion are then
integrated by the velocity Verlet method for 105 − 107 simulation time steps, depending on
Nc . The time step dt = 0.002.
We compute the longitudinal and transverse diffusivities Dα with α = x, y by fitting
h[(ri (t + ∆t) − ri (t)) · α̂]2 i = 2Dα ∆t.
(B.3)
The injected power computed for a single colloid i over a time period of length τ is given
by:
Z
t+τ
fd · vi (s)ds
Jτ =
(B.4)
t
where vi represents the instantaneous velocity of colloid i. A particle which moves in
a retrograde fashion, or opposite to the direction of the driving force, makes a negative
contribution to the injected power. We measure Jτ for a series of individual particles in a
single run and combine this data to obtain p(Jτ ). We identify Jτ for a variety of τ ranging
from a minimum of 10 simulation time steps to roughly one tenth the duration of the entire
simulation.
125
−2
10
0.8
a
0.6
−6
S(ν)
v(t)
b
−4
10
0.4
0.2
10
−8
10
−10
10
0
0
−12
0.5
1
t
1.5
2
4
10
−4
10
−2
0
10
10
2
10
ν
x 10
Figure B.1: Determination of noise properties a) Velocity time series v(t) for a single
colloid in a system with fd = 0.345, showing strong intermittency. Occasional motion
against the driving force, indicated by negative values, appears. b) Noise power spectrum
S(ν) for the v(t) shown in (a) averaged over an ensemble of 100 colloids. For ν > 0.1, we
find S(ν) ∝ ν −2 .
B.2
Nonthermal Fluctuations
We first demonstrate that our colloidal system exhibits strong nonthermal fluctuations under T = 0 dynamics, as is characteristic of the class of systems that exhibit crackling noise.
This system has previously been shown to exhibit disordered plastic flow in both simulations and experiments [125, 151]. We focus on the plastic flow phase in which a portion
of the particles are temporarily trapped for a period of time while the remaining particles
are mobile. The distribution of particle velocities in the plastic flow regime is bimodal.
The noise induced in collectively interacting systems with quenched disorder differs significantly from that found in simulations of thermal noise. In Fig. B.1a, we plot a velocity
time series v(t) for a single colloid at fd = 0.345, near the depinning transition. Frequent,
aperiodic starts and stops are observed, as well as intermittent motion in the direction opposite to the driving force (indicated when the signal drops below zero). The velocity noise is
not Gaussian, unlike the noise found for colloids in a thermal bath. In addition, the power
126
y
y
(a)
x
(b)
x
Figure B.2: Illustration of particle trajectories Colloid trajectories (lines) during 4 × 105
simulation time steps in a T = 0 system with quenched disorder at (a) fd = 0.27; (b)
fd = 0.34.
spectrum of the velocity noise is not white, as shown by the plot of
Z
2
−i2πνv(t)
S(ν) = e
dt
(B.5)
in Fig. B.1b. We find a 1/f 2 scaling of the noise power at high frequencies, and a 1/f
noise signature at lower frequencies. Power spectra of this form are characteristic of nonthermal systems that exhibit crackling noise, such as dislocation dynamics and plastically
deforming superconducting vortex matter.
B.3
Application of the Fluctuation Theorem
We inspect the distribution of power dissipated by individual colloids to test whether Eq. B.1
is satisfied in the interacting colloid system. For fd < 0.33 the colloids are pinned and there
is no nontransient motion, as illustrated in Fig. B.2a. Just above the depinning transition at
fd = 0.34, the colloid motion persists with time and the trajectories are highly disordered
as shown in Fig. B.2b. Approximately one third of the colloids are pinned at any given
time; however, all of the colloids take part in the motion. In Fig. B.3a we plot the strongly
127
non-Gaussian p(Jτ ) that appear in the absence of thermalization for τ = 0.02, 4.02, 8.02,
12.02, and 16.02 at fd = 0.345. The τ = 0.02 curve, most closely representative of
the instantaneous distribution, peaks at Jτ = 0 and is skewed in the positive direction by
the applied drive. The power p.d.f. at a slightly larger drive of fd = 0.375, shown in
Fig. B.3b, shares this feature, though the distribution of longer trajectories resolves to a
peak at Jτ > 0. Near the depinning transition, however, each p.d.f. peaks at Jτ = 0,
where it has a cusp-like discontinuous derivative (Fig. B.3c) due to the motionless particles
trapped in pinning sites, so there is an effective jump discontinuity at the sampled resolution of Jτ = 0, violating the implicit assumption of continuity in Eq. B.1. We expect that
this will be a very general feature of systems with pinned states. While there is a positive
correlation between log (p(Jτ )/p(−Jτ )) and Jτ , the fluctuation theorem is not satisfied for
small τ as these curves do not pass through the origin. Taking τ larger effectively smooths
the p.d.f. and lessens the discontinuity, making a fit to Eq. B.1 possible at large τ , as shown
in Fig. B.3d.
The quality of the fit to Eq. B.1 depends on fd . Earlier studies showed that systems
with depinning transitions can exhibit different dynamical regimes as a function of external
drive, including a completely pinned phase where there is no motion, a stable filamentary
channel phase at depinning where a small number of particles move in periodic orbits
[128], chaotic flow at higher drives when the filaments change rapidly with time [128, 130],
and a dynamically recrystallized phase at even higher drives where the particle paths are
mostly ordered [130]. To quantify the quality of the fits to Eq. 1 we calculate the Pearson
product-moment correlation coefficient r [152], which is a measure of the linear correlation
between two variables, and the coefficient of determination R2 , which measures the fraction
of variance explained by the fitted model [153]. In Fig. B.4a we plot the mean dissipation
ηhvi/fd versus fd for the system in Fig. B.2b, along with the corresponding longitudinal
and transverse diffusivities Dx and Dy .
128
−1
b
10
−2
10
p(Jτ)
p(Jτ)
−2
a
10
−3
−3
10
−4
10
10
0
0.05
0.1
J
0.15
0.2
0
0.05
τ
0.1
J
0.15
0.2
τ
0.5
−1
τ−1log[p(Jτ)/p(−Jτ)]
p(Jτ)
10
−2
10
−3
10
−4
0.3
0.2
0.1
c
−4
10
0.4
−2
0
Jτ
2
d
4
0
0
−3
x 10
2
4
Jτ
6
8
−3
x 10
Figure B.3: Demonstration of FT in a nonthermal system driven over quenched disorder a) p(Jτ ) for all observed trajectories with fd = 0.345 at τ = 0.02, 4.02, 8.02, 12.02,
and 16.02 (from upper right to lower right). b) Identical to (a) with fd = 0.375. c) A blowup of panel (a) highlighting the discontinuous behavior near Jτ = 0. d) Fit to Eq. B.1 of
the driven system with fd = 0.345 for all 10.02 ≤ τ ≤ 30.02, with darker color indicating
larger tau. The slope of the fit gives the inverse of the nonequilibrium temperature β∞ .
129
0.9
0.1
0.75
0.08
0.6
0.06
0.45
0.04
0.3
a
0.02
0
0.3
0.32
0.34
0.36
0.38
fd
0.4
0.42
0.44
η<
v>/f
η<v>/f
D
0.12
0.15
0.46
0
50
40
0
0.5
1
τ
30
20
b
10
0.3
0.32
0.34
0.36
0.38
fd
0.4
0.42
0.44
0.46
Figure B.4: Limits of regime in which FT is verified a) Solid curve: mean dissipation
ηhvi/fd vs fd , relating colloid displacements to applied drive. Upper crosses: longitudinal
diffusivity Dx vs fd . Lower crosses: transverse diffusivity Dy vs fd . b) Coefficient of
determination R2 of the fit log(p(Jτ )/p(−Jτ )) = mJτ as a function of τ and fd . Values
closer to 1 indicate better agreement with the FT. The FT holds over the largest range of
τ in the fluctuating plastic flow regime near fd ≈ 0.34 illustrated in Fig. B.2b; however,
curves for τ . 10 are poor fits in this region given the discontinuous p.d.f.
130
In Fig. B.4b we show the value of R2 for varied fd and for all τ < τc (fd ), where
τc (fd ) = sup {t | r(fd , τ ) ≥ 0.5 ∀ τ < t} .
(B.6)
Agreement with the FT, indicated by R2 ≈ 1, holds over the largest range of τ at fd ≈ 0.34,
in the plastic flow region coinciding with peaks in both Dx and Dy . Here, the colloids flow
in plastic fluctuating channels, as shown in Fig. B.2b, which are rapidly changing over
time. Previous studies of vortex plastic flow have shown that motion just above threshold
occurs in the form of a small number of flowing channels that are generally static in time
and that produce a periodic time signal as the vortices repeat the same path over and over
again [135]. We observe such a static channel plastic flow regime for fd < 0.335. In this
case, the chaotic hypothesis does not apply even though the system is undergoing plastic
flow, and we find that the fits to the FT are poor. At higher drives where more flow channels
open, it was shown previously that the velocity fluctuations become much more random,
the velocity power spectrum is no longer periodic but has a 1/f noise characteristic, and
that the flow becomes chaotic [135, 154, 134]. In the regime 0.335 < fd < 0.355, R2
comes the closest to reaching R2 = 1 for all the ranges of τ we analyze, indicating that
this regime gives the best agreement with the FT. For higher drives fd & 0.4, Dx and Dy
drop and we find an onset of dynamical ordering where the particles form an anisotropic
crystal structure that is similar to the dynamical reordering observed in vortex systems at
sufficiently high drive [126, 155, 130]. As fd increases, the FT continues to hold for small
τ , with the maximum value of τ for which R2 > 0.5 decreasing with increasing fd . This is a
result of the fact that on short time scales, the particles experience a “shaking temperature”
Ts as they move over the pinning sites which decreases as Ts ∝ 1/fd [126, 143]. At longer
times, however, the formation of a partially ordered crystal structure limits the size of the
fluctuations the particles can experience and eliminates retrograde trajectories, causing the
agreement with the FT to fail at long times.
131
0.025
0.04
a
b
0.02
0.03
βτ
β20
0.015
0.02
0.01
0.01
0.005
0
0
10
20
τ
30
40
0
0.32
0.36
0.4
fd
0.44
0.48
Figure B.5: Nonequilibrium temperature and scaling with applied drive a) Nonequilibrium temperature βτ in a nonthermal system with quenched disorder at fd = 0.335, 0.340,
0.345, and 0.350 (from top to bottom). In all cases βτ relaxes to the intrinsic noise level
within τ . 10. b) β20 = hβ15<τ <25 i vs fd for ρ = 0.1 (x), 0.2 (+), 0.3 (♦), 0.5 (), and 0.6
().
B.4
Nonequilibrium Temperature
As described previously, the FT allows the definition of a “nonequilibrium temperature”
βτ →∞ when sufficient retrograde trajectories of duration exceeding the microscopic time
scales of the system can be sampled. This necessarily involves a balance of time scales
since the second law of thermodynamics guarantees that p(Jτ < 0) = 0 as τ → ∞.
In Fig. B.5a, we plot βτ versus τ showing the existence of an asymptotic nonequilibrium
temperature β∞ in a nonthermal system with quenched disorder. We find that the time scale
of relaxation to the asymptotic value is unchanged when varying both the applied drive
and particle density (data not shown). We note that other approaches have been used to
establish effective temperatures for systems of vortices driven over random disorder using
generalized fluctuation-dissipation relationships [131].
Having established that we are able to measure a nonequilibrium temperature β∞ within
the time scale explored by the simulation, we observe how this quantity changes as we vary
fd and colloid density ρ by calculating β20 = hβ15<τ <25 i. As predicted, we see in Fig. B.5b
that β20 decreases with increasing fd and with decreasing ρ; however, the variance with ρ
132
1.4
1.2
βτ/β100
1
0.8
rp = .45
0.6
r = .55
p
rp = .65
0.4
rp = .75
rp = .85
0.2
r = .95
p
0
0
10
20
30
τ
40
50
60
70
Figure B.6: Scaling of time required to reach an asymptotic value of βτ with trap size
Nonequilibrium temperature βτ /β100 for a system with fd = 0.345 and varied pinning
radius rp = 0.45, 0.55, 0.65, 0.75, 0.85, and 0.95.
appears secondary as the value of fd required for depinning is anticorrelated with ρ. We
cannot measure a broader range of fd as we can only define a nonequilibrium temperature
for those values of fd where the FT holds over a wide range of τ , limiting us to drives near
the depinning threshold where plastic flow occurs.
B.5
Large Pinning Trap Regimes and Negative Events
We next examine βτ for a system with the same parameters but for varying trap radii up
to rp = 0.95 as shown in Fig. B.6. The time required to reach an asymptotic value of βτ
increases with increasing rp . This occurs when a portion of the colloids are able to undergo
significant rotational motion within the traps while still remaining confined, producing an
excess of retrograde trajectories. We also observe an interesting overshoot effect at intermediate times for the intermediate trap sizes. The overshoot arises because each colloid
spends only a finite amount of time trapped in a well. For the smaller trap radii, the colloids
are trapped only for a very short time which is less than τ , while for larger trap radii, some
133
colloids remain within a particular trap for the entire duration of the simulation. At the
intermediate trap radii of 0.45 < rp < 0.85 for 10 − 20τ , the colloids each spend a portion
of the time in the trap and than hop. For rp > 0.75, the colloids located within the traps
almost never leave during the duration of the simulation and the dynamics are dominated
by the flow of unpinned colloids. This result shows that the time evolution of βτ can be an
effective way to probe the dynamics of the system.
B.6
Summary
We have shown that the fluctuation theorem can be used to characterize the class of far
from equilibrium nonthermal systems of interacting particles driven over quenched disorder where there is a transition from a stationary to a moving state. The FT holds for the
regime near the onset of motion where the transport is nonlinear, the particle trajectories are
disordered with 1/f noise characteristics, and the particle displacements exhibit diffusive
behavior, in agreement with the expectation that regimes that agree best with the FT should
be chaotic. As a function of the trajectory length τ considered and the magnitude of the
external drive, we use the goodness of fit to the FT to construct a dynamic phase diagram
showing that in the plastic flow regime the FT fits well out to long trajectory lengths. In the
dynamical reordering regime, we find that the FT fits well only for short trajectory lengths
due to the short time effective temperature experienced by the particles as a result of being
driven over the quenched disorder, but that for longer trajectory lengths, the partial ordering
of the particle lattice cuts off the fluctuations and gives poor fits to the FT when the system
drops out of the chaotic regime. For drives right at the onset of motion, where the flow
occurs in a small number of static channels through the system, the FT does not hold since
the particle orbits are periodic and the chaotic behavior is lost. The FT cannot be applied
to the pinned phase since there are no fluctuations. We demonstrate that it is possible to
extract an effective temperature based on the FT in the plastic flow regime, and that this
134
temperature complements the effective shaking temperature defined in earlier work which
cannot be applied to the plastic flow regime. Systems in which these results could be tested
with local transport measurements include magnetic domain walls, vortices in type-II superconductors, sliding Wigner crystals, driven stripe systems, and earthquake models. It
would also be interesting to apply this approach to analyze other non-thermal systems that
exhibit similar crackling noise, such as dislocation dynamics.
135
Bibliography
[1] Gilbert, S. Developmental Biology (Sinauer Associates, Sunderland, MA, 2003), 7th
edn.
[2] Slack, J. Essential developmental biology (Blackwell Publishing, 2001).
[3] Cross, M. C. & Hohenberg, P. C. Pattern-formation outside of equilibrium. Reviews
of Modern Physics 65, 851–1112 (1993).
[4] Spemann, H. & Mangold, H. The induction of embryonic predispositions by implantation of organizers foreign to the species. Archiv Fur Mikroskopische Anatomie Und
Entwicklungsmechanik 100, 599–638 (1924).
[5] Waddington, C. H., Needham, J., Nowinski, W. W. & Lemberg, R. Studies on the
nature of the amphibian organisation centre I - Chemical properties of the evocator.
Proceedings of the Royal Society of London Series B-Biological Sciences 117, 289–
310 (1935).
[6] Waddington, C. H. & Needham, D. M. Studies on the nature of the amphibian
organization centre II - Induction by synthetic polycyclic hydrocarbons. Proceedings
of the Royal Society of London Series B-Biological Sciences 117, 310–317 (1935).
[7] The, I. & Perrimon, N. Morphogen diffusion: the case of the Wingless protein.
Nature Cell Biology 2, E79–E82 (2000).
136
[8] Turing, A. M. The chemical basis of morphogenesis. Philosophical Transactions of
the Royal Society of London Series B-Biological Sciences 237, 37–72 (1952).
[9] Gierer, A. & Meinhardt, H. Theory of biological pattern formation. Kybernetik 12,
30–39 (1972).
[10] Koch, A. J. & Meinhardt, H. Biological pattern-formation - from basic mechanisms
to complex structures. Reviews of Modern Physics 66, 1481–1507 (1994).
[11] Wolpert, L. Positional information and spatial pattern of cellular differentiation.
Journal of Theoretical Biology 25 (1969).
[12] Berg, H. C. & Purcell, E. M. Physics of chemoreception. Biophysical Journal 20,
193–219 (1977).
[13] Crick, F. Diffusion in embryogenesis. Nature 225, 420 (1970).
[14] Foe, V. E. & Alberts, B. M. Studies of nuclear and cytoplasmic behavior during the
5 mitotic-cycles that precede gastrulation in Drosophila embryogenesis. Journal of
Cell Science 61, 31–70 (1983).
[15] Frohnhofer, H. G. & Nusslein-Volhard, C. Organization of anterior pattern in the
Drosophila embryo by the maternal gene bicoid. Nature 324, 120–125 (1986).
[16] Driever, W. & Nusslein-Volhard, C. A gradient of bicoid protein in drosophila embryos. Cell 54, 83–93 (1988).
[17] Driever, W. & Nusslein-Volhard, C. The bicoid protein determines position in the
drosophila embryo in a concentration-dependent manner. Cell 54, 95–104 (1988).
[18] Nellen, D., Burke, R., Struhl, G. & Basler, K. Direct and long-range action of a Dpp
morphogen gradient. Cell 85, 357–368 (1996).
137
[19] Hogan, B. L. M. Bone morphogenetic proteins: Multifunctional regulators of vertebrate development. Genes & Development 10, 1580–1594 (1996).
[20] Jessell, T. M. Neuronal specification in the spinal cord: Inductive signals and transcriptional codes. Nature Reviews Genetics 1, 20–29 (2000).
[21] Grimm, O., Coppey, M. & Wieschaus, E. Modelling the Bicoid gradient. Development 137, 2253–2264 (2010).
[22] Porcher, A. & Dostatni, N. The bicoid morphogen system. Current Biology 20,
R249–R254 (2010).
[23] Ochoa-Espinosa, A., Yu, D. Y., Tsirigos, A., Struffi, P. & Small, S. Anteriorposterior positional information in the absence of a strong Bicoid gradient. Proceedings of the National Academy of Sciences of the United States of America 106,
3823–3828 (2009).
[24] Tsien, R. Y. The green fluorescent protein. Annual Review of Biochemistry 67,
509–544 (1998).
[25] Mavrakis, M., Rikhy, R., Lilly, M. & Lippincott-Schwartz, J. Fluorescence imaging
techniques for studying Drosophila embryo development. Curr Protoc Cell Biol
Chapter 4, Unit 4.18 (2008).
[26] Gregor, T., Tank, D. W., Wieschaus, E. F. & Bialek, W. Probing the limits to positional information. Cell 130, 153–164 (2007).
[27] Gregor, T., Wieschaus, E. F., McGregor, A. P., Bialek, W. & Tank, D. W. Stability
and nuclear dynamics of the bicoid morphogen gradient. Cell 130, 141–152 (2007).
[28] Berleth, T. et al. The role of localization of bicoid RNA in organizing the anterior
pattern of the Drosophila embryo. Embo Journal 7, 1749–1756 (1988).
138
[29] St. Johnston, D., Driever, W., Berleth, T., Richstein, S. & Nusslein-Volhard, C. Multiple steps in the localization of bicoid RNA to the anterior pole of the Drosophila
oocyte. Development 107, 13–19 (1989).
[30] Weil, T. T., Forrest, K. M. & Gavis, E. R. Localization of bicoid mRNA in late
oocytes is maintained by continual active transport. Developmental Cell 11, 251–
262 (2006).
[31] Mastro, A. M., Babich, M. A., Taylor, W. D. & Keith, A. D. Diffusion of a
small molecule in the cytoplasm of mammalian-cells. Proceedings of the National
Academy of Sciences of the United States of America-Biological Sciences 81, 3414–
3418 (1984).
[32] Wojcieszyn, J. W., Schlegel, R. A., Wu, E. S. & Jacobson, K. A. Diffusion of
injected macromolecules within the cytoplasm of living cells. Proceedings of the
National Academy of Sciences of the United States of America-Biological Sciences
78, 4407–4410 (1981).
[33] Rechsteiner, M. & Rogers, S. W. PEST sequences and regulation by proteolysis.
Trends in Biochemical Sciences 21, 267–271 (1996).
[34] Rogers, S., Wells, R. & Rechsteiner, M. Amino-acid-sequences common to rapidly
degraded proteins - the PEST hypothesis. Science 234, 364–368 (1986).
[35] Grimm, O. & Wieschaus, E. The Bicoid gradient is shaped independently of nuclei.
Development 137, 2857–2862 (2010).
[36] Bergmann, S. et al. Pre-steady-state decoding of the bicoid morphogen gradient.
Plos Biology 5, 232–242 (2007).
[37] Spirov, A. et al. Formation of the bicoid morphogen gradient: an mRNA gradient
dictates the protein gradient. Development 136, 605–614 (2009).
139
[38] Lipshitz, H. D. Follow the mRNA: a new model for bicoid gradient formation.
Nature Reviews Molecular Cell Biology 10, 509–512 (2009).
[39] Hershko, A. & Ciechanover, A. The ubiquitin system. Annual Review of Biochemistry 67, 425–479 (1998).
[40] Bialek, W. & Setayeshgar, S. Physical limits to biochemical signaling. Proceedings
of the National Academy of Sciences of the United States of America 102, 10040–
10045 (2005).
[41] Houchmandzadeh, B., Wieschaus, E. & Leibler, S. Establishment of developmental
precision and proportions in the early Drosophila embryo. Nature 415, 798–802
(2002).
[42] Bergmann, S., Tamari, Z., Schejter, E., Shilo, B. Z. & Barkai, N. Re-examining the
stability of the bicoid morphogen gradient. Cell 132, 15–17 (2008).
[43] Bialek, W., Gregor, T., Tank, D. W. & Wieschaus, E. F. Can we fit all of the data?
response. Cell 132, 17–18 (2008).
[44] Saunders, T. & Howard, M. When it pays to rush: interpreting morphogen gradients
prior to steady-state. Physical Biology 6, 046020 (2009).
[45] Lukyanov, K. A., Chudakov, D. M., Lukyanov, S. & Verkhusha, V. V. Photoactivatable fluorescent proteins. Nature Reviews Molecular Cell Biology 6, 885–891
(2005).
[46] Patterson, G. H. & Lippincott-Schwartz, J. A photoactivatable GFP for selective
photolabeling of proteins and cells. Science 297, 1873–1877 (2002).
[47] Chudakov, D. M. et al. Photoswitchable cyan fluorescent protein for protein tracking. Nature Biotechnology 22, 1435–1439 (2004).
140
[48] Ando, R., Hama, H., Yamamoto-Hino, M., Mizuno, H. & Miyawaki, A. An optical marker based on the UV-induced green-to-red photoconversion of a fluorescent
protein. Proceedings of the National Academy of Sciences of the United States of
America 99, 12651–12656 (2002).
[49] Ando, R., Mizuno, H. & Miyawaki, A. Regulated fast nucleocytoplasmic shuttling
observed by reversible protein highlighting. Science 306, 1370–1373 (2004).
[50] Habuchi, S. et al. Reversible single-molecule photoswitching in the gfp-like fluorescent protein dronpa. Proceedings of the National Academy of Sciences of the United
States of America 102, 9511–9516 (2005).
[51] Gurskaya, N. G. et al. Engineering of a monomeric green-to-red photoactivatable
fluorescent protein induced by blue light. Nature Biotechnology 24, 461–465 (2006).
[52] Hatta, K., Tsujii, H. & Omura, T. Cell tracking using a photoconvertible fluorescent
protein. Nature Protocols 1, 960–967 (2006).
[53] Karbowski, M. et al. Quantitation of mitochondrial dynamics by photolabeling of
individual organelles shows that mitochondrial fusion is blocked during the Bax activation phase of apoptosis. Journal of Cell Biology 164, 493–499 (2004).
[54] Hailey, D. W. & Lippincott-Schwartz, J. Using photoactivable proteins to monitor autophagosome lifetime, vol. 452 of Methods in Enzymology, 25–45 (Elsevier
Academic Press Inc, San Diego, 2009).
[55] Jamieson, J. D. & Palade, G. E. Intracellular transport of secretory proteins in pancreatic exocrine cell: I. Role of peripheral elements of Golgi complex. Journal of
Cell Biology 34, 577 (1967).
141
[56] Wang, Y. L. et al. Precision and functional specificity in mRNA decay. Proceedings
of the National Academy of Sciences of the United States of America 99, 5860–5865
(2002).
[57] Belle, A., Tanay, A., Bitincka, L., Shamir, R. & O’Shea, E. K. Quantification of protein half-lives in the budding yeast proteome. Proceedings of the National Academy
of Sciences of the United States of America 103, 13004–13009 (2006).
[58] Plotkin, J. B. The lives of proteins. Science 331, 683–684 (2011).
[59] Eden, E. et al. Proteome half-life dynamics in living human cells. Science 331,
764–768 (2011).
[60] Berezhkovskii, A. M., Sample, C. & Shvartsman, S. Y. How long does it take to
establish a morphogen gradient? Biophys J 99, L59–61 (2010).
[61] Yu, S. R. et al. Fgf8 morphogen gradient forms by a source-sink mechanism with
freely diffusing molecules. Nature 461, 533–U100 (2009).
[62] Crank, J. The Mathematics of Diffusion (Oxford University Press, 1975).
[63] Kicheva, A. et al. Kinetics of morphogen gradient formation. Science 315, 521–525
(2007).
[64] Struhl, G., Struhl, K. & Macdonald, P. M. The gradient morphogen bicoid is a
concentration-dependent transcriptional activator. Cell 57, 1259–1273 (1989).
[65] Little, S., Tkacik, G., T.B., K., Wieschaus, E. & Gregor, T. The formation of
the bicoid morphogen gradient requires protein movement from anteriorly localized
mRNA. PLoS Biology 9, e1000596 (2011).
142
[66] Hazelrigg, T., Liu, N., Hong, Y. & Wang, S. X. GFP expression in Drosophila
tissues: Time requirements for formation of a fluorescent product. Developmental
Biology 199, 245–249 (1998).
[67] Savitzky, A. & Golay, M. J. E. Smoothing & differentiation of data by simplified
least squares procedures. Analytical Chemistry 36, 1627 (1964).
[68] Mavrakis, M., Rikhy, R. & Lippincott-Schwartz, J. Plasma membrane polarity and
compartmentalization are established before cellularization in the fly embryo. Developmental Cell 16, 93–104 (2009).
[69] Salles, F. J., Lieberfarb, M. E., Wreden, C., Gergen, J. P. & Strickland, S. Coordinate initiation of Drosophila development by regulated polyadenylation of maternal
messenger-RNAs. Science 266, 1996–1999 (1994).
[70] Lieberfarb, M. E. et al. Mutations that perturb poly(A)-dependent maternal mRNA
activation block the initiation of development. Development 122, 579–588 (1996).
[71] Abu-Arish, A., Porcher, A., Czerwonka, A., Dostatni, N. & Fradin, C. High mobility
of Bicoid captured by fluorescence correlation spectroscopy: Implication for the
rapid establishment of its gradient. Biophysical Journal 99, L33–L35 (2010).
[72] Gregor, T., Bialek, W., van Steveninck, R. R. R., Tank, D. W. & Wieschaus, E. F.
Diffusion and scaling during early embryonic pattern formation. Proceedings of the
National Academy of Sciences of the United States of America 102, 18403–18407
(2005).
[73] Stiel, A. C. et al. 1.8 angstrom bright-state structure of the reversibly switchable
fluorescent protein Dronpa guides the generation of fast switching variants. Biochemical Journal 402, 35–42 (2007).
143
[74] Habuchi, S. et al. Photo-induced protonation/deprotonation in the GFP-like fluorescent protein Dronpa: mechanism responsible for the reversible photoswitching.
Photochemical & Photobiological Sciences 5, 567–576 (2006).
[75] Kovesi, P. D. MATLAB and Octave functions for computer vision and image processing. <http://www.csse.uwa.edu.au/˜pk/research/matlabfns/>. School of Computer Science & Software Engineering, The University of Western Australia.
[76] Welte, M. A. Fat on the move: intracellular motion of lipid droplets. Biochemical
Society Transactions 37, 991–996 (2009).
[77] Cermelli, S., Guo, Y., Gross, S. P. & Welte, M. A. The lipid-droplet proteome reveals
that droplets are a protein-storage depot. Current Biology 16, 1783–1795 (2006).
[78] Ghaemmaghami, S. et al. Global analysis of protein expression in yeast. Nature
425, 737–741 (2003).
[79] Surdej, P. & Jacobs-Lorena, M. Developmental regulation of bicoid mRNA stability
is mediated by the first 43 nucleotides of the 3’ untranslated region. Molecular and
Cellular Biology 18, 2892–2900 (1998).
[80] Wartlick, O., Kicheva, A. & Gonzalez-Gaitan, M. Morphogen gradient formation.
Cold Spring Harbor Perspectives in Biology 1, a001255 (2009).
[81] DeRenzo, C. & Seydoux, G. A clean start: degradation of maternal proteins at the
oocyte-to-embryo transition. Trends in Cell Biology 14, 420–426 (2004).
[82] Lippincott-Schwartz, J., Snapp, E. & Kenworthy, A. Studying protein dynamics in
living cells. Nature Reviews Molecular Cell Biology 2, 444–456 (2001).
[83] Reits, E. A. J. & Neefjes, J. J. From fixed to FRAP: measuring protein mobility and
activity in living cells. Nature Cell Biology 3, E145–E147 (2001).
144
[84] Coppey, M., Berezhkovskii, A. M., Kim, Y., Boettiger, A. N. & Shvartsman, S. Y.
Modeling the bicoid gradient: Diffusion and reversible nuclear trapping of a stable
protein. Developmental Biology 312, 623–630 (2007).
[85] Phair, R. D. & Misteli, T. Kinetic modelling approaches to in vivo imaging. Nature
Reviews Molecular Cell Biology 2, 898–907 (2001).
[86] Saxton, M. J. Anomalous diffusion due to binding: A Monte Carlo study. Biophysical Journal 70, 1250–1262 (1996).
[87] Saxton, M. J. Anomalous diffusion due to obstacles: A Monte Carlo study. Biophysical Journal 66, 394–401 (1994).
[88] Yuste, S. B., Abad, E. & Lindenberg, K. Reaction-subdiffusion model of morphogen
gradient formation. Physical Review E 82, 090602 (2010).
[89] Hecht, I., Rappel, W. J. & Levine, H. Determining the scale of the Bicoid morphogen
gradient. Proceedings of the National Academy of Sciences of the United States of
America 106, 1710–1715 (2009).
[90] Sprague, B. L., Pego, R. L., Stavreva, D. A. & McNally, J. G. Analysis of binding
reactions by fluorescence recovery after photobleaching. Biophysical Journal 86,
3473–3495 (2004).
[91] Weiss, M. Challenges and artifacts in quantitative photobleaching experiments. Traffic 5, 662–671 (2004).
[92] Pando, B., Dawson, S. P., Mak, D. O. D. & Pearson, J. E. Messages diffuse faster
than messengers. Proceedings of the National Academy of Sciences of the United
States of America 103, 5338–5342 (2006).
145
[93] Axelrod, D., Koppel, D. E., Schlessinger, J., Elson, E. & Webb, W. W. Mobility measurement by analysis of fluorescence photobleaching recovery kinetics. Biophysical
Journal 16, 1055–1069 (1976).
[94] Soumpasis, D. M. Theoretical analysis of fluorescence photobleaching recovery
experiments. Biophysical Journal 41, 95–97 (1983).
[95] Gregor, T. Biophysics Problems in Early Embryonic Development: Precision and
Dynamics in the Bicoid Morphogen Gradient. Ph.D. thesis, Princeton University
(2005).
[96] Endres, R. G. & Wingreen, N. S. Maximum likelihood and the single receptor.
Physical Review Letters 103, 4 (2009).
[97] Nagai, T. et al. A variant of yellow fluorescent protein with fast and efficient maturation for cell-biological applications. Nature Biotechnology 20, 87–90 (2002).
[98] Frescas, D., Mavrakis, M., Lorenz, H., DeLotto, R. & Lippincott-Schwartz, J. The
secretory membrane system in the Drosophila syncytial blastoderm embryo exists
as functionally compartmentalized units around individual nuclei. Journal of Cell
Biology 173, 219–230 (2006).
[99] Kavousanakis, M. E., Kanodia, J. S., Kim, Y., Kevrekidis, I. G. & Shvartsman, S. Y.
A compartmental model for the bicoid gradient. Developmental Biology 345, 12–17
(2010).
[100] Rideal, E. & Tadayon, J. On overturning and anchoring of monolayers: 2. Surface
diffusion. Proceedings of the Royal Society of London Series a-Mathematical and
Physical Sciences 225, 357–361 (1954).
[101] Carslaw, H. S. & Jaeger, J. C. Conduction of Heat in Solids (Oxford University
Press, 1959).
146
[102] Sigaut, L., Ponce, M. L., Colman-Lerner, A. & Dawson, S. P. Optical techniques
provide information on various effective diffusion coefficients in the presence of
traps. Physical Review E 82, 051912 (2010).
[103] Dombeck, D. A., Khabbaz, A. N., Collman, F., Adelman, T. L. & Tank, D. W.
Imaging large-scale neural activity with cellular resolution in awake, mobile mice.
Neuron 56, 43–57 (2007).
[104] Lucy, L. B. Iterative technique for rectification of observed distributions. Astronomical Journal 79, 745–754 (1974).
[105] Svoboda, K. & Yasuda, R. Principles of two-photon excitation microscopy and its
applications to neuroscience. Neuron 50, 823–839 (2006).
[106] Marriott, G. et al.
Optical lock-in detection imaging microscopy for contrast-
enhanced imaging in living cells. Proceedings of the National Academy of Sciences
of the United States of America 105, 17789–17794 (2008).
[107] Wu, Y. F., Myasnikova, E. & Reinitz, J. Master equation simulation analysis of
immunostained Bicoid morphogen gradient. Bmc Systems Biology 1, 52 (2007).
[108] Sample, C. & Shvartsman, S. Y. Multiscale modeling of diffusion in the early
Drosophila embryo. Proceedings of the National Academy of Sciences of the United
States of America 107, 10092–10096 (2010).
[109] Evdokimov, A. G. et al. Structural basis for the fast maturation of arthropoda green
fluorescent protein. Embo Reports 7, 1006–1012 (2006).
[110] Trugnan, G., Fontanges, P., Delautier, D. & Ait-Slimane, T. FRAP, FLIP, FRET,
BRET, FLIM, PRIM... new techniques for a colourful life. M S-Medecine Sciences
20, 1027–1034 (2004).
147
[111] Verdu, S. Multiuser Detection (Cambridge University Press, 1998).
[112] Elowitz, M. B., Surette, M. G., Wolf, P. E., Stock, J. & Leibler, S. Photoactivation
turns green fluorescent protein red. Current Biology 7, 809–812 (1997).
[113] Lupas, R. & Verdu, S. Linear multiuser detectors for synchronous code-division
multiple-access channels. Ieee Transactions on Information Theory 35, 123–136
(1989).
[114] Foe, V. E., Odell, G. M. & Edgar, B. A. Mitosis and morphogenesis in the Drosophila
embryo: Point and counterpoint. The development of Drosophila melanogaster,
Vols. 1 and 2 (Cold Spring Harbor Laboratory Press a, 1993).
[115] Meyer, F. Iterative image transformations for an automatic screening of cervical
smears. Journal of Histochemistry & Cytochemistry 27, 128–135 (1979).
[116] Svoboda, K., Denk, W., Kleinfeld, D. & Tank, D. W. In vivo dendritic calcium
dynamics in neocortical pyramidal neurons. Nature 385, 161–165 (1997).
[117] Blab, G. A., Lommerse, P. H. M., Cognet, L., Harms, G. S. & Schmidt, T. Twophoton excitation action cross-sections of the autofluorescent proteins. Chemical
Physics Letters 350, 71–77 (2001).
[118] Pologruto, T. A., Sabatini, B. L. & Svoboda, K. ScanImage: flexible software for
operating laser scanning microscopes. Biomed Eng Online 2, 13 (2003).
[119] Zipfel, W. R., Williams, R. M. & Webb, W. W. Nonlinear magic: multiphoton
microscopy in the biosciences. Nature Biotechnology 21, 1368–1376 (2003).
[120] Nusslein-Volhard, C. & Wieschaus, E. Mutations affecting segment number and
polarity in Drosophila. Nature 287, 795–801 (1980).
148
[121] Spirov, A. & Holloway, D. M. Evolutionary techniques for image processing a large
dataset of early Drosophila gene expression. Eurasip Journal on Applied Signal
Processing 2003, 824–833 (2003).
[122] Spirov, A. V., Kazansky, A. B., Timakin, D. L. & Reinitz, J. Reconstruction of the
dynamics of Drosophila genes expression from sets of images sharing a common
pattern. Real-Time Imaging 8, 507–518 (2002).
[123] Holloway, D. M., Harrison, L. G., Kosman, D., Vanario-Alonso, C. E. & Spirov,
A. V. Analysis of pattern precision shows that drosophila segmentation develops
substantial independence from gradients of maternal gene products. Developmental
Dynamics 235, 2949–2960 (2006).
[124] Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).
[125] Pertsinidis, A. & Ling, X. S. Statics and dynamics of 2D colloidal crystals in a
random pinning potential. Physical Review Letters 100, 028303 (2008).
[126] Koshelev, A. E. & Vinokur, V. M. Dynamic melting of the vortex lattice. Physical
Review Letters 73, 3580–3583 (1994).
[127] Middleton, A. A. & Wingreen, N. S. Collective transport in arrays of small metallic
dots. Physical Review Letters 71, 3198–3201 (1993).
[128] Jensen, H. J., Brass, A. & Berlinsky, A. J. Lattice deformations and plastic-flow
through bottlenecks in a two-dimensional model for flux pinning in type-II superconductors. Physical Review Letters 60, 1676–1679 (1988).
[129] Olson, C. J., Reichhardt, C. & Nori, F. Nonequilibrium dynamic phase diagram for
vortex lattices. Physical Review Letters 81, 3757–3760 (1998).
149
[130] Faleski, M. C., Marchetti, M. C. & Middleton, A. A. Vortex dynamics and defects
in simulated flux flow. Physical Review B 54, 12427–12436 (1996).
[131] Kolton, A. B., Dominguez, D. & Gronbech-Jensen, N. Hall noise and transverse
freezing in driven vortex lattices. Physical Review Letters 83, 3061–3064 (1999).
[132] Kokubo, N. et al. Dynamic ordering of driven vortex matter in the peak effect regime
of amorphous MoGe films and 2H-NbSe2 crystals. Physical Review B 75, 184512
(2007).
[133] Okuma, S., Inoue, J. & Kokubo, N. Suppression of broadband noise at mode locking
in driven vortex matter. Physical Review B 76, 172503 (2007).
[134] Olive, E. & Soret, J. C. Chaos and plasticity in superconductor vortices: Lowdimensional dynamics. Physical Review B 77, 144514 (2008).
[135] Gronbech-Jensen, N., Bishop, A. R. & Dominguez, D. Metastable filamentary
vortex flow in thin film superconductors. Physical Review Letters 76, 2985–2988
(1996).
[136] Gallavotti, G. & Cohen, E. G. D. Dynamical ensembles in nonequilibrium statisticalmechanics. Physical Review Letters 74, 2694–2697 (1995).
[137] Gallavotti, G. Nonequilibrium in statistical and fluid mechanics. ensembles and
their equivalence. entropy driven intermittency. Journal of Mathematical Physics
41, 4061–4081 (2000).
[138] Evans, D. J., Cohen, E. G. D. & Morriss, G. P. Probability of 2nd law violations in
shearing steady-states. Physical Review Letters 71, 2401–2404 (1993).
[139] Evans, D. J. & Searles, D. J. The fluctuation theorem. Advances in Physics 51,
1529–1585 (2002).
150
[140] Wang, G. M., Sevick, E. M., Mittag, E., Searles, D. J. & Evans, D. J. Experimental
demonstration of violations of the second law of thermodynamics for small systems
and short time scales. Physical Review Letters 89, 050601 (2002).
[141] Feitosa, K. & Menon, N. Fluidized granular medium as an instance of the fluctuation
theorem. Physical Review Letters 92, 164301 (2004).
[142] Majumdar, S. & Sood, A. K. Nonequilibrium fluctuation relation for sheared micellar gel in a jammed state. Physical Review Letters 101, 078301 (2008).
[143] Kolton, A. B., Exartier, R., Cugliandolo, L. F., Dominguez, D. & Gronbech-Jensen,
N. Effective temperature in driven vortex lattices with random pinning. Physical
Review Letters 89, 227001 (2002).
[144] Aumaitre, S., Fauve, S., McNamara, S. & Poggi, P. Power injected in dissipative
systems and the fluctuation theorem. European Physical Journal B 19, 449–460
(2001).
[145] Schmick, M. & Markus, M. Fluctuation theorem for a deterministic one-particle
system. Physical Review E 70, 065101 (2004).
[146] Kurchan, J. Fluctuation theorem for stochastic dynamics. Journal of Physics aMathematical and General 31, 3719–3729 (1998).
[147] Lebowitz, J. L. & Spohn, H. A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. Journal of Statistical Physics 95, 333–365
(1999).
[148] Kurchan, J. Gallavotti-Cohen theorem, chaotic hypothesis and the zero-noise limit.
Journal of Statistical Physics 128, 1307–1320 (2007).
[149] Gallavotti, G. & Cohen, E. G. D. Nonequilibrium stationary states and entropy.
Physical Review E 69, 035104 (2004).
151
[150] Bassler, K. E., Paczuski, M. & Altshuler, E. Simple model for plastic dynamics of a
disordered flux-line lattice. Physical Review B 64, 224517 (2001).
[151] Reichhardt, C. & Olson, C. J. Colloidal dynamics on disordered substrates. Physical
Review Letters 89, 078301 (2002).
[152] Rodgers, J. L. & Nicewander, W. A. 13 ways to look at the correlation-coefficient.
American Statistician 42, 59–66 (1988).
[153] Willett, J. B. & Singer, J. D. Another cautionary note about R2 - its use in weighted
least-squares regression-analysis. American Statistician 42, 236–238 (1988).
[154] Olson, C. J., Reichhardt, C. & Nori, F. Fractal networks, braiding channels, and
voltage noise in intermittently flowing rivers of quantized magnetic flux. Physical
Review Letters 80, 2197–2200 (1998).
[155] Moon, K., Scalettar, R. T. & Zimanyi, G. T. Dynamical phases of driven vortex
systems. Physical Review Letters 77, 2778–2781 (1996).
152