Lecture 1:  Physics problems in early embryonic development

(last updated 17 March 2008)

 

 

One of the most beautiful phenomena in nature is the emergence of a fully formed, highly structured organism from a single undifferentiated cell, the fertilized egg. Biologists have shown that in many cases the “blueprint” for the body is laid out with surprising speed and is readable as variations in the expression levels of particular genes. As we try to understand how these molecules interact to form the patterns that we recognize as characteristic of the mature organism, we face a number of physics problems:

 

How can spatial patterns in the concentration of these molecules scale with the size of the egg, so that organisms of different sizes have similar proportions?

 

What insures that the spatial patterns are reproducible from one embryo to the next?

 

Since the concentrations of all the relevant molecules are small, does the random behavior of individual molecules set a limit to the precision with which patterns can be constructed?

 

Although the phenomena of life are beautiful, one might worry that these systems are just too complicated and messy to yield to the physicists' desire for explanation in terms of powerful general principles. For the past several years, a small group of us have been struggling with these problems in the context of the fruit fly embryo. To our delight, we have been able to banish much of the messiness, and to reveal some remarkably precise and reproducible phenomena. In particular, the first crucial step in the construction of the blueprint really does involve the detection of concentration differences so small that they are close to the physical limits set by the random arrival of individual molecules at their targets. This problem may be so serious that the whole system for constructing the blueprint has to be tuned to maximize how much signal can be transmitted against the inevitable background of noise, and this idea of optimization can be turned into a theoretical principle from which we can actually predict some aspects of how the system works (which carries us into the next lecture).

 

While I am very excited about the particular things my colleagues and I are doing on this set of problems, I also hope that this lecture will give us a chance to talk about the more general question of how one builds bridges between general theoretical principles and the details of specific biological systems.  This theme will continue throughout the course.

 


 

In retrospect, the lecture seemed to have a logical outline.  I emphasize that what is written here is quite rough, and should be viewed more as a guide to what was said than as a reference text!

 

 

Getting started

 

Some background, and some interesting side issues

 

Back to the embryo

 

Formulating the problems

 

Some progress

 

An aside about collaboration

 

 

Although meant to be informal, I certainly would appreciate feedback on these notes.  Please tell me what’s helpful, what is unclear, and where you could use more help with background etc..  Needless to say, if there is something you find especially interesting, I’d be happy to chat about that too!

 


 

Getting started

 

The goal for this first half of the lecture is to explore what we know about embryonic development in fruit flies, and to pull out of this knowledge some physics problems (the three questions listed above) that we think might be of general interest.  We’ll start by going through what the biologists have taught us, which really is wonderful, and then see how if we take seriously what they tell us then we have some things which we clearly don’t understand.  One way to say this is that the biologists have given us a qualitative description of what is going on, but as physicists we would like a quantitative theory.  As we try to make qualitative ideas quantitative, two things can happen.  The first possibility is that everything works, and what we are doing is taking the cartoons that one sees in biology books and putting “numbers on the arrows.”  This would be a good thing.  But one would have to admit that if all we do is to make quantitative versions of the biologists’ qualitative models, then the biologists might be right to complain that we aren’t doing anything new and we should find ways of translating our mathematical analyses back into qualitative terms.  The other possibility, of course, is that when we try to map qualitative ideas into quantitative ones, something seems to be missing.  What’s missing might be a detail, or it might be a clue about some general principles, and it’s hard to know which is which when we start.  I think it’s clear, though, that we should be on the lookout for these cases where something goes wrong, where we can’t make all the numbers fit together in a clear physical picture, because these are the places where we have a chance to discover something really new about the physics of life.  I hope you’ll agree that the fly embryo offers us some of these opportunities.

 

We recall that even big, complicated animals like us start out as a single cell, the fertilized egg.  As the embryo develops, this one cell becomes many cells, but importantly these cells become differentiated from one another.  Cells seem to fall into discrete types, and these types are in different locations, so this differentiation is also a problem of spatial pattern formation.

 

Every cell in your body has the same DNA (assuming nothing has gone wrong!).  What makes the different cells different is that they “express” different genes.  The genes code for proteins, but not all of the proteins are made in all cells; the reading of the code to make the proteins is called the “expression” of the genes.  Importantly, the regulation of gene expression is not just the flipping of a switch sometime in development, but rather something that all cells (from neurons in our brain down to bacteria) are doing all the time.

 

As physicists, when we look at the phenomena of life we need to think about what is general and what is particular.  In this lecture, we are going to look closely at the very first stages of development and pattern formation in the embryo of a fruit fly.  Much of what happens surely is specific to this system.  But I hope you will see some general questions emerging from these details.  As noted above, at least one of these questions (the scaling problem) is about pattern formation, and indeed it is about a striking contrast between pattern formation in biological vs. non-biological systems.  The other two major questions (about reproducibility and precision) are really questions about how accurately cells can regulate the expression of their genes, and I hope to convince you that these questions are not just general in biological terms but also connected to some reasonably fundamental physical problems about noise in counting molecules.

 

The fly embryo is an interesting model system for many reasons.  One is that there is a well developed genetics for fruit flies (the species Drosophila melanogaster), made possible not least by their rapid growth and reproduction.  Embryonic development itself is rapid as well, leading from a fertilized egg to the hatching of a fully functional maggot (the larvae of flies, like caterpillars for butterflies) within 24 hours.  All of this happens inside an egg shell, so there is no growth—pattern formation occurs at constant volume.  The egg is ~1/2 mm long, so one starts with one rather large cell, which has one nucleus.  In the maggot there are ~50,000 cells.  For the first three hours of development, which will concern us most, something special happens:  the nuclei multiply without building walls to make separated cells.  Thus, for about three hours, the fly embryo is close to the physicists’ idealization of a box in which chemical reactions are occurring, with the different molecules free to move from one end of the box to the other (perhaps even by diffusion, although this is a more subtle question). 

 

The duplication of the nuclei is more or less synchronous for the first 13 divisions, which is visually quite striking.  At cycle 9 or so (I don’t remember exactly, sorry), almost all of the nuclei move to the surface of the egg, where they form a fairly regular two dimensional lattice.  With each subsequent cycle, this lattice dissolves and reforms.  You can get a sense of all this by watching a movie here.  I recommend that you open this in a new window, since it’s a big file and you might want to flip back and forth.  This movie was made by Thomas Gregor when he was still at Princeton (see my remarks about collaboration, below).  The fly you are looking at has been genetically engineered so that the histone proteins in the nucleus (proteins around which the DNA is wrapped in the chromosomes) are fluorescent, so what you see are the chromosomes in each nucleus.  As each cycle proceeds, the chromosomes bunch together, duplicate and separate, and then one nucleus suddenly becomes two.  The movie isn’t perfect, so there is some artefactual “flashing” in the early cycles.  Also, this is a mutant fly, so the stuff at the end is weird; more about that later.  Still, I hope you get the idea.

 

With cycle 14, the synchronous duplication of nuclei stops, and there is a pause while embryo builds walls between the nuclei to make separate cells.  If you stop the action at this point and take an electron micrograph of the embryo, this is what you see (this and the subsequent images courtesy of Eric Wieschaus).  Again, the embryo is roughly ½ mm long.

 

 

 

 

 

If you count, you’ll find that there are ~6000 cells on the surface.  This is smaller than 213, but that’s because not all of the nuclei make it to the surface; some stay in the interior of the embryo, probably not by accident since these become cells with special functions.  Notice that all the cells look pretty much alike.

 

If instead of stopping at this point, we wait just 15 minutes more, we see something very different:

 

 

 

Notice that there is a vertical cleft, about one-third of the way from the left edge of the embryo.  This is the “cephalic furrow,” and defines which part of the body will become the head.  There is also a furrow along the bottom of the embryo, which is where the one layer of cells on the surface starts to fold in on itself so that you can have two “outside” surfaces (think about the inside and outside of your cheek, both of which are outside of the body from the topological point of view … we are not simply connected!), a process called “gastrulation.”  To get more of a flavor for this, you can watch a movie of the formation of the cephalic furrow and some perspective on gastrulation.  Again I recommend that you open this in another window.  In this fly the fluorescent proteins are along the cell membranes, so it is the boundaries between the cells that are visible.  As the cells move around, these boundaries are of course deformed in three dimensions, and this is what you see in the movie.

 

It’s not just that the embryo breaks into a head and a non-head.  In fact there are many different pieces to the body, usually called segments.  Our bodies also have segments, such as the vertebrae along our spine.  But in insects, the segments are obvious even from the outside.  Here’s an image of the fruit fly maggot (at right), and you can see the lines on the skin, which should also be familiar if you’ve looked closely at caterpillars (such as this lovely tiger moth caterpillar, at left).

 

LEAD Technologies Inc. V1.01

tiger moth caterpillar: http://www.hsu.edu/content.aspx?id=7435

fruit fly larva:  http://commons.wikimedia.org/wiki/Image:Fruit_fly_larva_01.jpg

 

The obvious question is how the cells at different points in the embryo “know” to become parts of different segments.  The answer is quite striking, and one of the great triumphs of modern biology.  Long before cells start moving around and making the three dimensional structures that one sees in the fully developed organism, there is a “blueprint” that can be made visible by asking about the expression levels of particular genes.  We know which genes to look at because of pioneering experiments first by EB Lewis and then by EF Wieschaus and C Nüsslein–Vollhard.  Lewis identified a series of puzzling mutant flies where a mutation in a single gene could generate flies that were missing segments, or had extra segments.  It is as if the “program” of embryonic development has sub–routines (!).  Wieschaus and Nüsslein–Vollhard decided to search for all the genes such that mutations in those genes would perturb the formation of spatial structure in the embryo, and they found that there are surprisingly few such genes, on the order of 100 out the roughly 25,000 genes in the whole fly genome.  Most of these genes code for proteins that control the expression of other genes, which means that these are “transcription factors.” 

 

Suppose we stop that action in the embryo at cycle 14 and measure the concentration of two of these key proteins.  One way to do this is to make antibodies against the protein we are interested in, and then make antibodies against the antibodies, but before using the secondary antibodies we attach to them a fluorescent dye molecule.  So if we expose the embryo first to one antibody (which should stick to the protein we are interested in, and not anywhere else, if we’re lucky) and to the other, we should have the effect of attaching fluorescent dyes to the protein we are looking for, and hence if we look under a microscope the brightness of the fluorescence should indicate the concentration of the protein (not obvious if this relationship is quantitative; hold that question).  One such experiment is shown here.

 

 

 

Evidently the concentration of the proteins varies with position, and this variation corresponds to a striped pattern.  The stripes should remind you of the segments in the fully develop animal, and this is actually quite precise.  Mutations that move the stripes around, or delete particular stripes, have the expected correlates in the pattern of segmentation.  To illustrate this point, we can blow up corresponding pieces of this image and the electron micrograph above, showing the cephalic furrow:

 

 

Hopefully you can see how the furrow occurs at a place defined by the locations of the green and orange stripes.  At the moment the names of these molecules don’t really matter.  What is important is to realize that the macroscopic structure of the fully developed organism follows a blueprint laid out within about three hours after fertilization, and that this blueprint is “written” as variations in the expression level of different genes.  Furthermore, we know which genes are the important ones, and there aren’t too many of them.

 


 

Some background, and some interesting side issues

 

At this point it would be useful if you had a little knowledge of basic molecular biology.  You need to know that DNA sequences encode proteins, and the process of getting from DNA to protein goes in two steps:  DNA is “transcribed” into messenger RNA (mRNA), and then the mRNA is “translated” into protein.  Generating the mRNA strand is like copying the DNA strand itself, since the mRNA is also a polymer complementary in structure to the DNA.  This polymerization is catalyzed by a molecule called RNA polymerase (RNAP), although this is an oversimplification; in eukaryotes (cells with nuclei, including us, fruit flies and yeast), there is a complex of roughly twenty proteins involved.  Nonetheless, transcription is a polymerization reaction, catalyzed by enzymes that “walk” along the DNA strand and leave the mRNA strand in their wake.  Translation occurs at the ribosome, which importantly is (again, in eukaryotes) outside the nucleus, so the mRNA needs to be exported from the nucleus in order to function.

 

One of the basic mechanisms of regulation is to control access of the RNA polymerase to the point at the start of the gene.  In the simplest view (which probably isn’t too bad in bacteria, although a bit more metaphorical for eukaryotes), we can imagine controlling the expression of a gene with other proteins that either block the RNA polymerase (and thus repress transcription) or provide an extra place for the RNA polymerase to bind (and thus activate transcription).  These proteins are called transcription factors, and they bind to specific sequences of DNA near the start site for transcription; these regions of DNA that serve as control elements are called “promoters” or “enhancers.”  A schematic of all this is shown here (thanks to G Tkacik for the figure), where we add also the ribosomes moving along the mRNA and making protein. 

 

 

 

 

 

 

I should admit that learning more about all of this, without learning everything, isn’t so easy. Indeed, it has become more difficult as the classic texts have become more like encyclopedias.  There isn’t really a brief introduction aimed specifically at students with a strong physics background, although many people are struggling with this problem in their lectures.  If you can get your hands on an early edition of Watson’s Molecular Biology of the Gene, by all means read it; it is a lovely book, with real style and personality. 

 

Another text of great relevance is Ptashne’s book(s) about the regulation of gene expression.  In the first instance, he focused on one model system, explored in detail, and then briefly considered what were the general lessons to come out of this system.  In later versions, he too tried to be more encyclopedic, but maybe less so than others.

 

A Genetic Switch:  Gene Control and Phageλ. M Ptashne (Cell Press, Cambridge, 1986).

A Genetic Switch:  Phageλand Higher Organisms (2nd edition). M Ptashne (Cell Press, Cambridge, 1992).

A Genetic Switch:  PhageλRevisited (3rd edition). M Ptashne (Cold Spring Harbor Press, New York, 2004).

 

Genes and Signals.   M Ptashne & A Gann (Cold Spring Harbor Press, New York, 2002).

 

There are many beautiful biophysics problems lurking in the processes of transcription and translation, even before we consider regulation.  Perhaps the most famous of the problems is the tremendous accuracy of these reactions.  The issue of how accurately genetic information is transmitted goes back (at least) to Schrodinger’s What is Life? —another book very much worth reading! 

 

Of course the great lesson of the Watson-Crick structure for DNA is that it provides a basis for transmitting information by complementarity:  A fits with T, C fits with G.  For better or worse, this can’t be the whole story.  The difference in binding energy between the “correct” and “incorrect” pairings corresponds to a Boltzmann factor of ~104, when in fact the observed accuracy is more like 1 error in every 108 base pairs, or even less in higher organisms.  In the 1970s, Hopfield and Ninio independently realized that this problem was much more general, and comes up in many biological processes.  In effect, when cells need to link molecules together, they seem to sort out the correct components with a reliability better than expected from equilibrium thermodynamics, which means that they function as Maxwell demons.  This is possible, of course, only if there is some appropriate expenditure of energy.  Looking carefully at what was known about the underlying biochemistry, Hopfield and Ninio proposed that there were some general rules which linked energy expenditure to synthesis, and that these served to power the Maxwell demon.  In outline, all of this has proven to be correct, although the details still sometimes cause problems.  I think one of the most important things (which is clearest in Hopfield’s paper) is that many different biological processes face a common physics problem, and thus share mechanistic features which provide the solution to this probem but otherwise look arbitrary.  An important lesson!

 

Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity.  JJ Hopfield, Proc Nat’l Acad Sci (USA) 71, 4135-4139 (1974).

 

In the intervening 30+ years, the experimentalists have made considerable progress.  The emergence of methods for observing single molecules in action has given us a new window into the processes of transcription and translation, even to the point of being able to “see” the proofreading steps as they happen.  Some recent work along these lines, very much at the forefront of a whole field of single molecule experiments, is in the following references:

 

Direct observation of base-pair stepping by RNA polymerase.  EA Abbondanzieri, WJ Greenleaf, JW Shaevitz, R Landick & SM Block, Nature 438, 460-465 (2005).

 

Following translation by single ribosomes one codon at a time.  J-D Wen, L Lancaster, C Hodges, A-C Zeri, SH Yoshimura, HF Noller, C Bustamante & I Tinoco, Jr, Nature in press (2008).

 

Once we start to think about regulation, new problems emerge.  If we want transcription factors to regulate particular genes, then they need to recognize something about the sequence of the DNA near those genes, in the “promoter” regions.  In one major stream of work, one uses examples of real binding sites to try and learn what it is that these proteins are “looking for.”  One can view this as a purely statistical problem:  make up a rule that sorts the possible short DNA sequences into two groups, one of which corresponds to binding sites and one of which does not.  But this misses the fact that binding sites don’t have p-values, they have binding energies (!).  There is a difference between saying that a particular sequence should have a small binding energy, and that you are not sure if it’s a binding site. 

 

On the experimental side, several groups have developed methods that test a transcription factor protein against, for example, all of the promoters for the ~6000 different genes in yeast.  These methods are a bit messy, and there is much discussion of how to do the processing to clean things up and get some definite, convincing answers.  Recently, some of my colleagues at Princeton have taken a new look at this problem, taking seriously the physical picture that there is a binding energy characterizing the interaction of the protein with each possible sequence.  They also make use of an old idea that this binding energy is composed of a sum of terms, one for each base pair contact between the protein and the DNA.  They also take a very interesting view of the uncertainties in the data:  they assume that we don’t really know how the physical binding energy relates to the measured fluorescence signal in the experiments, and so they average over all possible functions characterizing this relationship (more or less).  Remarkably, this agnostic approach works beautifully, even clearing up many mysteries that were left in more “fine tuned” statistical approaches that have less physical meaning. 

 

Precise physical models of protein-DNA interaction from high throughput data.  JB Kinney, G Tkacik & CG Callan, Jr, Proc Nat’l Acad Sci (USA) 104, 501-506 (2007).

 

Although off to the side of what we want to discuss in this lecture, all of these topics have lots of currently open problems, and I am happy to chat with you about them (or at least point you toward people who know more).

 


 

Back to the embryo

 

We have seen that the blueprint for the organism is laid out as spatial patterns of gene expression.  But how do these arise?  You could imagine, as Turing did, that these patterns reflect a spontaneous breaking of symmetries in the egg.  This, for better or worse, is not how it works.  When the mother makes the egg, she places the mRNA for a handful of proteins at cardinal points. For example, there is a protein called Bicoid for which the mRNA is placed at the end that will become the head, as shown in this figure.

 

 

Here the embryo is stained by making DNA strands that are complementary to the mRNA we are looking for, and attaching a label to these synthesized DNA.   Importantly, the mRNA is attached to the end of the egg, not free to move. Once the egg is laid, translation of this mRNA begins, and the resulting Bicoid protein is free to move through the embryo.  If we use the same trick as above and stop the action, staining with fluorescent antibodies against the protein, we see images like this:

 

 

Evidently there is a rather smooth gradient in the concentration of Bicoid protein, high at one end and low at the other.  A cell sitting at some point in the embryo thus can “know” where it is along this long (anterior-posterior) axis by measuring the Bicoid concentration.

 

In fact Bicoid is a transcription factor, so it regulates the expression of other genes.  One such gene is hunchback.  If we use the antibody staining methods to look at the concentration of the Hunchback protein, what we see is this:

 

 

 

It seems as if the transformation from Bicoid to Hunchback is almost the flipping of a switch, so that if the Bicoid concentration is above some threshold, then Hunchback is “one,” otherwise it’s “off.”  In fact things are more gradual, but this isn’t a bad cartoon.  There are several genes at the same level as hunchback, and again they all code for transcription factors.  Bicoid turns all of these genes on with different thresholds, and they interact with each other, mostly turning each other off, so that the end result is a collection of genes which go on and off in different combinations along the length of the embryo.  These genes in turn (along with Bicoid itself) provide input to a second layer of genes that form the beautiful striped patterns shown above.  Hunchback and its partners are called “gap genes,” because deletion of one of these genes causes a gap of several segments in the structure of the organism; the genes in the next layer of the systems are called “pair rule” genes.

 

I think you can imagine (even without looking into all the details) how a smooth gradient, processed by a set of threshold elements, provides at each location a sort of combinatorial code.  This code is then “read” by the next layer of genes to produce more complex patterns.  Our task, as mentioned at the outset, is to see what happens as we try to take this qualitative scenario and turn it into something quantitative.

 


 

Formulating the problems

 

Imagine that we take everything which has been said in the previous paragraphs and formalize it in equations.  In each nucleus there will be chemical reactions corresponding to the transcription of the relevant genes, and the rates of these reactions will be determined by the concentrations of the appropriate transcription factors.  More equations will be needed to describe translation (although maybe one can simplify, if, for example, mRNA molecules degrade quickly and proteins live longer).  Different points in space are coupled, presumably through diffusion of all these molecules, although we should worry about whether diffusion is the correct description.  Even if you’re not sure about the details,  you can see the form of the equations.

 

The first problem, then, is that we have seen equations like these before in the study of non-biological pattern forming systems such as Rayleigh-Bernard convection, directional solidification, … .   For a (very long) review, see

 

Pattern formation outside of equilibrium. MC Cross & PC Hohenberg, Revs Mod Phys 65, 851-1112 (1993).

 

In these problems, one can observe the formation of periodic spatial patterns which remind us of the segments in the insect and the patterns of pair rule gene expression. The scale of these patterns, however, is set by combinations of parameters in the equations.  For example, we can combine a diffusion constant with a reaction rate to get a length.  What happens if you put these equations in a larger box?  Well, from Rayleigh-Bernard convection, we know the answer.  Recall that in this system (a fluid layer heated from below) we see a collection of convective rolls, sometimes in stripes and sometimes in 2D cellular patterns.   Again, the length scale of the stripes is determined by the parameters of the equation(s).  If you put the whole system in a bigger, you get more stripes, not wider stripes. 

 

You maybe have noticed that people come in a variety of sizes, but (for example) the fraction of the body which is head varies much less.  Almost everyone you know has five fingers, despite variations in the size of their hands.  These sound like jokes, but they are hints about a serious problem, which can be made more quantitative for insects, and especially for fruit flies.  There are closely related species of flies with embryos that vary in length by a factor of five, and these embryos use molecules that are almost identical to those in the fruit fly (you can compare the DNA sequences of the genes).   Certainly it is not the case that embryos that are five times longer produce insects with five times as many segments!  Indeed, what happens, to a very good approximation, is that the number of segments stays fixed and the size of the segments scale to the size of the egg.  You can see this scaling not just in the macroscopic patterns of the developed organisms, but also in the patterns of gene expression. These examples are from Thomas Gregor.

 

 

 

So, scaling is our first problem.  How can a system of chemical reactions with diffusion produce patterns that scale with the size of the container in which they occur?  Do we have to imagine that evolution tweaks the parameters of the system slightly as we move across flies of different sizes?  Even if this works, how does a single species deal with the variations in length of eggs laid by the same mother?

 

The next problem is precision.  Two neighboring nuclei along the long axis of the embryo can certainly adopt different fates in the fully developed organism, and one can also see that they have different patterns of expression for the pair rule genes (although this hasn’t been checked as carefully as one might like!).  One way to summarize this is that a close look at the stripes shows that borders are only one cell wide, as in the image below.  One doesn’t see the sort of “salt and pepper” patterns that one might if individual cells were unsure about what to do.

 

 

What are the signals that drive these differences between neighbors?  Well, for Bicoid we pretty much know the answer.  The concentration profile is a pretty good fit to an exponential decay along the length of embryo, which is also what you expect if the profile reflects a (steady state) balance between diffusion of molecules from their source at one end of the embryo and simple first-order degradation reactions.  The length constant of the decay is about 80 microns, and the distance between neighboring nuclei is about 8 microns, so differences of Bicoid concentration are in the range of ~10%.

 

Of course we don’t know that any single step in the readout of the Bicoid gradient is accurate to 10%, but I think this is a really interesting possibility.  Obviously if there are billions of Bicoid molecules, noticing a 10% change is easy.  But how many molecules are there?  Bicoid is a transcription factor, and it acts by binding to specific sites in the promoter region of its target genes.  Presumably the “threshold” for turning the genes on is when about half of the relevant sites are occupied, so the concentration scale is set by the binding energy or binding constant of the protein to its specific site along the DNA.  Protein-DNA binding has been studied in many systems, and the usual result (including some experiments with Bicoid) is that this binding constant is in the range of nanoMolar, which corresponds to ~0.6 molecules per cubic micron.  The nucleus is a few microns across, so we expect that there are hundreds of molecules of Bicoid in the nucleus at the point where the “decision” to express Hunchback (for example) is being made.  Suddenly being sensitive to ~10% changes seems more interesting.

 

In fact, the number of molecules in the nucleus is not the relevant question, but rather the rate at which molecules find their way to their target sites along the DNA.  Long ago, in the context of bacterial chemotaxis, Berg and Purcell tried to estimate the limits to the precision with which a system can respond to small changes in concentration, assuming that the molecules have to arrive via diffusion at a fixed target.  Their argument (indeed, the whole paper) is a classic of physicists’ back-of-the-envelope reasoning, also with some idiosyncrasies.  You should read the original:

 

Physics of chemoreception.  HC Berg & EM Purcell, Biohys J  20, 193-219 (1977).

 

I won’t repeat the arguments here (although I sketched them in the lecture); a summary in the context of the fly embryo is given by Gregor et al (2007b).  When the dust settles, given what we know about the Bicoid concentration, typical diffusion constants for proteins in the cytoplasm, and the size of the target, it would take many hours of averaging for the embryo to achieve ~10% accuracy in reading out the Bicoid signal.  This is implausible, since the whole pattern forms within this time.  There seems to be a paradox here.

 

The problem of precision really is several problems.  First, there is the purely theoretical question of whether the Berg-Purcell argument is sufficiently rigorous and general to be useful in this context.  Second, the Berg-Purcell limit on precision depends on quantities such as the absolute concentration of Bicoid, which we only estimated indirectly from the literature; it would be good to have a direct measurement.  Third, and perhaps most crucially, we need to look carefully at a single step in the process (e.g., the transformation from Bicoid to Hunchback) and see if this one step really reaches the 10% level of precision; this is a question about noise in the control of gene expression, which has been an active area recently, but mostly in engineered model systems.  Fourth, if the system is reading out the Bicoid concentration with a precision of ~10%, does this mean that the mother must control the absolute concentration with similar precision in order to make sure that all embryos develop reproducible patterns?  Finally, depending on the answers to the first three questions, we might still have a problem to solve, and it would be nice to know the solution!

 

I should say that there is an overwhelming prejudice against biological systems functioning in a highly precise fashion.  Depending on how things turn out, it might be that the fly embryo is providing an example of extreme precision, down to the physical limits sets by the statistics of counting molecules. 

 

 


 

Some progress

 

As noted at the outset, a small group of us have been working on these problems for a few years, bringing together theory and experiment, as well as crossing the physics/biology border.  The first (maybe 1.5) generation of papers has now come out, so I will just provide pointers, with very brief summaries.

 

Our first foray into the scaling problem was

 

Diffusion and scaling during early embryonic pattern formation.  T Gregor, W Bialek, RR de Ruyter van Steveninck, DW Tank & EF Wieschaus, Proc Nat’l Acad Sci (USA) 102, 18403-18407 (2005).

 

This work showed that, at least for inert molecules, the diffusion equation really does describe molecular motion on the relevant length and time scales, and that diffusion constants (perhaps not surprisingly) do not vary significantly across embryos of very different sizes. On the other hand, the patterns of gene expression do scale with embryo size across species, and this scaling can be traced back to the initial anterior-posterior gradient of Bicoid.  Although there is scaling of the spatial patterns, large and small embryos in fact develop on the same temporal schedules.  I should say that this work was done before we had control over the issues of reproducibility in the Bicoid profile (see below), and it would be good to go back and do a cleaner job.  The short summary, though, is that scaling is not an emergent property of the whole network of gene interactions, but rather a property of the Bicoid gradient itself.

 

Almost everything we know about morphogen gradients has been learned from snapshots of embryos that are fixed and stained.  An alternative is to genetically engineer flies that make a “fusion” of the protein you are interested in (e.g., Bicoid) and the green fluorescent protein, and this is the technical core of the next paper,

 

Stability and nuclear dynamics of the Bicoid morphogen gradient.  T Gregor, EF Wieschaus, AP McGregor, W Bialek & DW Tank, Cell  130, 141-152 (2007).

 

These fusion proteins are widely used in modern biology, but for our purposes one needs to know that the fusion quantitatively replaces the normal protein; along the way to doing this we accumulated some of the best evidence for the linearity of the traditional antibody staining methods, which will be important below.  Armed the green flies, one can observe (live, and at least in one color) the dynamics of the Bicoid gradient in live embryos as they develop.  An example of the resulting movies is here; as before you probably want to open the movie in a separate window.  What you are seeing are three optical sections through the embryo,  showing the complex dynamics of Bicoid as it is repeatedly trapped and released from nuclei during successive cell cycles.    This dynamics coexists with a constancy of the concentration inside the nuclei, so that as the nuclei form again after one round of division the Bicoid concentration is restored to its value at the start of the previous cycle.  Photobleaching experiments demonstrate that this constancy is the result of a rapid equilibration between the interior of the nucleus and the surrounding cytoplasm, and the slight deviations from constancy can be beautifully explained as an effect of changing nuclear size during the cell cycle, balancing the dilution of molecules as the volume increases against the increased flux as the surface area increases.  Many of these results would be understandable if the system were in an effective steady state, but the diffusion constant of Bicoid in the cytoplasm surrounding the nuclei (which can be estimated consistently from several different experiments) is too small to allow establishment of this steady state in the available time.    Models in which the observed nuclear trapping is crucial to the dynamics (rather than just sampling the surrounding cytoplasm) may also relate to the scaling problem, because the number of nuclei is the same in embryos of different size—and hence the local density of nuclei provides a measure of size that can be used to drive scaling behavior. 

 

One can also use the fusion idea more aggressively, extracting the sequences of Bicoid from flies of different sizes and re-inserting green versions of these different Bicoids into the Drosophila genome.

 

Shape and function of the Bicoid morphogen gradient in dipteran species with different sized embryos.  T Gregor, AP McGergor & EF Wieschaus, Dev Biol in press (2008).

 

The striking result is the resulting spatial profiles are those appropriate to the host embryo, not the source of the Bicoid.  Thus, the scaling problem is solved at the level of Bicoid itself, but not by evolutionary tinkering!  Rather there is something about the environment or geometry of the embryo itself (perhaps, as suggested above, the spacing of the nuclei) that couples the global changes in the size of the embryo to the local dynamics. Our papers discuss models along these lines, but I don’t think we have everything fitting together yet.  I do think the theoretical problem has been sharpened, though.

 

For me, interest in the fly embryo as a model system began with the attempt to understand theoretically what defines the real limits to the precision of signaling in biological systems.  As noted above, we have an intuitive answer from Berg and Purcell, but some work needed to be done to make this rigorous.

 

Physical limits to biochemical signaling. W Bialek & S Setayeshgar, Proc Nat’l Acad Sci (USA) 102, 10040-10045 (2005); physics/0301001.

 

Cooperativity, sensitivity and noise in biochemical signaling.  W Bialek & S Setayeshgar, q–bio.MN/0601001 (2006).

 

Diffusion, dimensionality and noise in transcriptional regulation. G Tkacik & W Bialek, arXiv:0712.1852 [q–bio.MN] (2007).

 

The result, perhaps surprisingly, is that Berg and Purcell really did identify a lower bound on the effective noise level of a signaling system, and this bound depends only on basic physical parameters.  The (often unknown) details of the biochemistry can add noise, but can’t lower the noise floor.  This was a fun bit of statistical physics.

 

With more confidence in our understanding of the physical limits, we went back to the embryo to see how well it does:

 

Probing the limits to positional information.  T Gregor, DW Tank, EF Wieschaus & W Bialek, Cell 130, 153-164 (2007). 

 

We used the fluorescent Bicoid to make a measurement of the absolute concentration, confirming that it is in the nanoMolar range.  We then combined classical antibody staining methods with image processing to map the input/output relation between Bicoid and Hunchback, nucleus by nucleus, characterizing the noise in the system (and struggling with noise in the measurements!). The noise indeed is below 10%.  To resolve the conflict with the theoretical limits, one possibility is that nuclei, even at this early stage of development, communicate to “agree” on the concentration of Bicoid.  If this is correct, then the noise in the Hunchback concentration should be correlated among nearby nuclei, and we found that this is true, with exactly the correlation length needed to solve the noise problem.  Finally, by lining up many embryos under the microscope we could compare the absolute Bicoid concentrations at corresponding positions, and this reproducibility is also at the ~10% level. All of this suggests that the reliability of a developmental decision is set by fundamental physical limits, which is a very attractive idea.

 

In the same way that the initial theoretical work provided motivation for the experiments, the experimental results have sharpened the theoretical questions.  We’ll do more in the next lecture, but here let me point to the problem of connecting theory and experiment more closely in the analysis of noise:

 

The role of input noise in transcriptional regulation. G Tkacik, T Gregor & W Bialek, q–bio.MN/0701002 (2007).

 

This work showed that not just the overall magnitude, but also the pattern of noise in the Bcd/Hb transformation is consistent with the idea that the dominant source of noise is the irreducible physical randomness in the diffusion of Bcd to its targets along the Hb promoter.

 


 

An aside about collaboration

 

I hope that I’ve been able to convey some of my excitement about this work without overstating what we have accomplished.  I think we are at the beginning rather than an end, but perhaps with some paths now clearer.  One of the crucial things that theory can do at the interface of physics and biology is to point out which experiments are critical in deciding among alternative directions.  But to deliver on this promise requires strong cooperative interactions among theorists and experimentalists—not just experimental biology, but also experimental physics.  I’ve been incredibly fortunate to have some wonderful collaborators, and to have the whole collaboration run in a friendly and easy way, with little if any formal structure or negotiation.

 

There is a lot of discussion at the moment of how to foster interactions between physics and biology, and one should admit that there is much frustration on both sides.  I don’t know if there are general “rules” that one can lay down for successful collaboration.  Maybe the whole point is that, at its best, collaboration is a very intimate relationship among individuals.  In particular, if two people (e.g., a physicist and a biologist) want to accomplish more together than the sum of what they could do separately, then they have to discuss with one another the things that they don’t understand.  This is hard, since there are many forces encouraging us to hide, rather than to advertise, our ignorance.  Neither physicists nor biologists are going to learn everything about their collaborators’ field as a prerequisite to collaboration (it’s just like peace talks; if everybody had agreed on terms before sitting down there probably wouldn’t have been a war). Realistically, we may never become masters of the other field, or more precisely we may never fully assimilate its culture, and maybe that’s a good thing, because the novelty comes from the clash of cultures—looking at the same phenomena, physicists and biologists ask different questions.  I suspect that these thoughts on the importance of our ignorance and the preservation of physics vs biology culture run a bit counter to the conventional wisdom about the path to a more homogenized quantitative biology, and I’d be happy to discuss these issues more informally if you’re interested.

 

As for our particular collaboration looking at the fly embryo, it attracted some attention, so the University newspaper ran a story about it.  If you can forgive some of the pubic relations aspects, it does give a good sense of the different personalities involved.