Special Reviews

From Signals to Patterns: Space, Time, and Mathematics in Developmental Biology

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Science  17 Oct 2008:
Vol. 322, Issue 5900, pp. 399-403
DOI: 10.1126/science.1166154


We now have a wealth of information about the molecular signals that act on cells in embryos, but how do the control systems based on these signals generate pattern and govern the timing of developmental events? Here, I discuss four examples to show how mathematical modeling and quantitative experimentation can give some useful answers. The examples concern the Bicoid gradient in the early Drosophila embryo, the dorsoventral patterning of a frog embryo by bone morphogenetic protein signals, the auxin-mediated patterning of plant meristems, and the Notch-dependent somite segmentation clock.

Developmental biologists are preoccupied with patterns in space and patterns in time. What causes cells in one part of an embryo regularly to adopt one character, and those elsewhere to adopt another? What controls the timing of these patterning events as the organism develops from the egg? What halts growth when a structure has reached its proper size?

Early discussions of developmental patterning considered, in abstract terms and in total ignorance of the molecules involved, how signals exchanged between cells might drive formation of the patterns we observe. Although they lacked a concrete molecular basis, several of these speculative theories turn out to have been remarkably prescient. Moreover, right or wrong, they drive home a general lesson. To explain how embryos generate their spatial patterns and their temporal programs, we need more than lists of the molecules involved and more than simple cartoon diagrams of the qualitative control relationships between them. To make sense of the control circuitry, qualitative data and unaided intuition are not enough. Even for the simplest cases, we need mathematics and detailed measurements, just as we need mathematics and measurements to explain the orbits of the planets or the swing of a pendulum.

The necessity arises especially because of the fundamental part that feedback plays in developmental control processes, as discussed elsewhere in this issue of Science by Brandman and Meyer (1). Positive feedback can give a system a flip/flop choice between alternative steady states; it can endow a system with enduring memory of its exposure to past signals; it can generate inhomogeneity in a system that starts out spatially uniform. Negative feedback can smooth out irregularities; it can enable a system to respond to a signal more rapidly; operating with a delay, it can give rise to temporal oscillations. These behaviors are not immediately obvious intuitively, but mathematics allows us to predict and compute them.

Molecular biology has been triumphant in identifying the key signaling molecules that govern development. Developmental biologists are now beginning to confront in earnest the problem of how these molecules work together as control systems to generate patterns in space and time. I discuss here a few examples to show how the interplay of quantitative experimentation and mathematical modeling leads to new insights and new ways of thinking about how molecular signals are used to pattern the embryo.

How a Morphogen Gradient Is Made and Used: The Bicoid Gradient in the Drosophila Embryo

Conceptually, at least, the simplest way to set up a spatial pattern is by means of a morphogen gradient. A signal substance, the morphogen, is synthesized at one end of a developing structure and diffuses away from this source, suffering degradation as it goes and thereby creating a concentration gradient. Cells at varying distances from the source experience different morphogen concentrations and somehow “read” those concentrations, adopting different characters as a result. We now have evidence for mechanisms of this sort in innumerable systems (2); a particularly clear example is that of the Bicoid gradient in the early Drosophila embryo. Recent quantitative analysis has provided a uniquely detailed and thought-provoking portrait of how this particular morphogen gradient works.

The early Drosophila embryo (Fig. 1) is a syncytium, in which repeated cycles of mitosis without cell division generate a steadily increasing number of nuclei. These nuclei are distributed uniformly in a cortical layer along the 500-μm length of the cigar-shaped embryo. Molecules of maternal bicoid mRNA are strictly localized at the anterior pole of the embryo and begin to be translated at the time of fertilization. Their product, the Bicoid protein, is a transcription regulator: It acts on the nuclei in the cortex, controlling their expression of genes such as hunchback (coding for another transcription regulator) that specify differences between regions of the embryo. The same molecules and the same mechanism operate in other related species of fly, some with embryos as little as 300 μm long, others as big as 1500 μm long. Curiously, the shape of the Bicoid gradient and the proportions of the resulting pattern are practically the same in every case, scaling with the size of the embryo (3).

Fig. 1.

The Bicoid gradient in a fly embryo. The early embryo is a syncytium, with its many nuclei distributed in a layer just beneath the cell cortex. bicoid mRNA is localized at the anterior end of this giant cell and is translated to produce Bicoid protein, which diffuses along the anteroposterior axis, creating an exponential gradient. During interphase, the protein is concentrated inside the nuclei, where it controls position-specific expression of other genes. Closely related fly species vary enormously in the sizes of their embryos but have closely similar numbers of nuclei at corresponding stages. The Bicoid gradient, remarkably, scales in proportion, giving a scaled body pattern.

Gregor et al. (4, 5) used immunofluorescence staining and Bicoid protein tagged with green fluorescent protein (GFP) to see how the Bicoid concentration gradient in Drosophila is set up and how it controls hunchback expression. They measured the precise form of the concentration gradient, the time taken to establish it (about 1 hour), the rise and fall of Bicoid concentration in the nuclei with each mitotic cycle, and the surprisingly slow rate of diffusion of Bicoid protein through the cytoplasm. Putting all these measurements together, they showed how creation of the Bicoid gradient is related to the dynamics of release, reuptake, and degradation of Bicoid by the nuclei and argued that it must involve some sort of facilitated transit from nucleus to nucleus: The gradient is not explicable in terms of simple diffusion of Bicoid through the cytoplasm. The central role of the nuclei in localizing and degrading Bicoid hints, in a still-tantalizing way, at an explanation of how the pattern scales with size in embryos of different fly species: For if distance along the embryo is measured in units of the spacing between nuclei, instead of μm, all these embryos are alike.

Gregor et al. (4) examined how the Bicoid gradient controls expression of genes such as hunchback with such remarkable precision and sharpness that each nucleus behaves as though it knows its position along the 500-μm Drosophila embryo axis to an accuracy of better than 10 μm. They found that at the threshold for hunchback activation, about halfway along the embryo, there are on average 690 molecules of Bicoid protein per nucleus; through some statistical physics calculations, they showed that this means that the observed precision of patterning is close to the limit set by molecular noise (thermal fluctuations). They went on to measure the amazingly steep relationship between levels of Bicoid and Hunchback proteins in the critical neighborhood, and from this they were able to estimate that about five molecules of Bicoid must bind cooperatively to the hunchback regulatory locus to switch on hunchback expression.

These and many other ingenious deductions by the authors show how careful measurements and quantitative reasoning can highlight new questions and lead to deeper understanding, linking the simple concept of the morphogen gradient to other levels of description.

How a Morphogen Gradient Is Scaled to the Size of the Organism: BMP Signaling and the Patterning of the Dorsoventral Axis of the Frog

The Bicoid gradient in Drosophila is intracellular: It is set up inside a single giant cell that only later becomes partitioned into many separate cells. Most of the morphogen gradients that pattern embryos are extracellular or at least based on signals that pass from one cell to another. An archetypal example is the gradient that originates in Spemann's organizer, the group of cells on the dorsal side of an early amphibian embryo that orchestrates the movements of gastrulation and governs creation of the main axes of the body (6). When cells from the organizer region are grafted to a new position in the embryo, opposite to their original location, they induce their new neighbors to join with them in forming a new body axis. Cells furthest from the organizer form ventral structures, and those closest to it form dorsal structures: Some sort of signal from the organizer apparently tells cells their position along the dorsoventral axis. The patterning mechanism has, however, a remarkable property, encountered also in many other systems (as we have just seen for fly embryos of different species): It scales with embryo size. Thus when an embryo is cut in half, the half that retains organizer tissue develops a miniature but perfectly proportioned body pattern (7). This would not be expected if the organizer were dictating the position-dependent characters of cells in the embryo by acting straight-forwardly as the source of a simple morphogen gradient based on diffusion and degradation. The scaling mechanism in this multicellular system cannot be the same as in the syncytial fly embryo. Recent studies have revealed how it works (Fig. 2).

Fig. 2.

Dorsoventral patterning in the Xenopus embryo: a current view. A self-organizing gradient of BMP signaling gives rise to a pattern that scales according to the size of the embryo. The gradient is set up by two signaling centers. The ventral center secretes BMP. The dorsal center (Spemann's organizer) secretes both a BMP-like molecule called ADMP and an inhibitor, chordin, which can bind to BMP and ADMP and block their action. High levels of BMP pathway activation stimulate expression of BMP and inhibit expression of ADMP and chordin; low levels do the opposite. Thus, local positive feedback maintains expression of BMP at the ventral center and maintains expression of chordin and ADMP at the dorsal center. At the same time, chordin, by binding to ADMP, speeds ADMP diffusion, thereby shuttling ADMP from the dorsal center toward the ventral, where the ADMP helps to stimulate BMP expression. This interaction between the two signaling centers coordinates their formation and ensures that the BMP gradient scales with the size of the embryo.

The critical condition governing the dorsoventral character of the cells is the strength of activation of the BMP (bone morphogenetic protein) signaling pathway: Strong activation causes cells to adopt a ventral character; weak or zero activation causes them to adopt a dorsal character. BMP itself is produced by cells at the ventral side of the embryo. Meanwhile, the organizer secretes at least eight different signal molecules, at least four of which are antagonists of BMP signaling, preventing activation of the pathway on the dorsal side of the embryo. Chordin, for example, is secreted by the organizer and binds to BMP, preventing it from activating the BMP receptor. Thus, it seems that a gradient of BMP inhibition, spreading out from the organizer, underlies dorsoventral patterning. Curiously, however, the organizer also secretes a BMP-related protein, called ADMP (antidorsalizing morphogenetic protein), that behaves, like BMP, as an activator of the BMP pathway and is also blocked in this action when bound to chordin. Reversade and De Robertis (8) analyzed the regulatory relationships between these components and showed that they all play essential parts in producing a dorsoventral pattern that scales with the size of the embryo. Their paper suggests in a qualitative way how they may do so, through feedback regulation of the expression of the various BMP agonists and antagonists, leading to a BMP “seesaw” between the ventral and dorsal poles of the embryo, with high levels of production of chordin and ADMP by the organizer at the dorsal pole somehow forcing creation of an opposite type of signaling center at the ventral pole. The paper ends with this remark: “Despite the complexity... there is hope that, with the rise of systems biology, one day this intricate extracellular machinery will be understood as an integrated molecular circuit” (8).

Ben-Zvi et al. (9) have picked up the challenge. The first task is to discover whether the identified classes of molecules, interacting according to the rules proposed, really are sufficient in principle to give the observed scalable patterning. Unaided intuition is a feeble and often misleading guide to the behavior of systems with feedback, and all the more so for multiple cross-regulatory molecules diffusing in space. To answer the question of principle, therefore, one must turn to mathematics. It is easy enough to write down equations that represent the proposed rules of synthesis, diffusion, degradation, binding, and cross-regulation for each representative type of molecule—BMP, ADMP, and chordin—as revealed by the experiments. It is not too hard to program a computer to solve these equations. The difficulty is that different choices of the rate constants, diffusion coefficients, and other parameters in the mathematical model lead to wildly different outcomes. To make even qualitative predictions, we need quantitative data. In contrast with the case of the Bicoid gradient in Drosophila, the relevant quantitative data for BMP signaling in the frog embryo are largely lacking: We don't yet have properly developed tools to make the measurements in the context of the living organism.

Ben-Zvi et al. (9), therefore, guided by their experience with a closely related dorsoventral patterning system in Drosophila (10), tried out different possibilities for the numerical parameters, systematically varying each one and exploring the behavior of the mathematical model for 26,000 different combinations of values. In this way, they showed that the model can indeed give the observed behavior, creating a pattern that scales in proportion to the size of the embryo. However, only a very small subset of parameter values, 21 out of the 26,000 combinations tested, will yield this outcome. These “successful” combinations share a striking feature: They correspond to conditions where BMP and ADMP molecules are only able to diffuse rapidly when they have their inhibitor, chordin, bound to them. In the process, the chordin itself undergoes degradation, creating a concentration gradient. The chordin, in these model cases, serves as a shuttle for movement of BMP and ADMP molecules across the embryo. Ben-Zvi et al. (9) therefore turned from mathematical modeling back to experimentation, asking whether such shuttling occurs in the real system: Does diffusion of BMP and ADMP through the tissue actually depend on binding to chordin? The answer is yes.

This work illustrates again how quantitative mathematical reasoning can sharpen understanding and suggest new lines of experimentation. At the same time, it highlights the central frustration of mathematical modeling in developmental biology: For most of the signaling systems that pattern the embryo, with their intricate feedback loops, we simply do not have the quantitative data that are needed to go beyond proof of plausibility to the type of solidly based quantitative theory that is commonplace in the physical sciences. But that is the challenge.

How Signaling Centers Are Created and Positioned: Auxin Transport in the Plant Meristem

Given a signaling center, one can easily imagine how it can organize the pattern of cell differentiation in its neighborhood. But how does the signaling center itself arise ? If we start with a more or less homogeneous field of cells, what internal mechanism can make one region different from another and break the symmetry? Symmetry-breaking typically depends on positive feedback mechanisms that amplify small fluctuations into full-blown inhomogeneities. Turing (11) long ago showed with a mathematical model how this could happen in a developing tissue: Symmetry can be broken if the cells secrete two kinds of molecules that diffuse at different rates and cross-regulate one another's synthesis in an appropriate way; spatial variations in their synthesis can then become self-amplifying. Gierer and Meinhardt (12, 13) developed this abstract mathematical idea further, showing how the combination of a highly diffusible inhibitor molecule and a less diffusible activator can give rise to a self-organizing pattern in which tightly defined clusters of cells become signaling centers, secreting both types of molecules and thereby sustaining their own character while forcing different behavior in their neighbors. The organizer in the frog embryo, as we have just seen, is a signaling center that fits this description, and it probably arises more or less in this way.

Such self-organizing pattern formation is nicely exemplified in plant apical meristems, the tiny (∼100 μm) growing tips of shoots in which the microscopic primordia of new leaves or floral organs are successively generated. As a meristem grows, the preexisting primordia act as signaling centers to control the positioning of new primordia, which go on to become signaling centers in their own right. But the molecular mechanism that creates these centers and governs their spacing turns out to be rather different from that imagined by Turing or Gierer and Meinhardt.

The crucial signal in this system is the plant hormone auxin (14, 15). A dab of auxin placed at the side of the meristem will induce formation of an ectopic primordium. Converse effects result from mutation of the PIN1 gene, coding for a protein that transports auxin out of cells into the extracellular space: The mutant meristem continues to grow but fails to generate any lateral organs, forming instead a single long thin unadorned stalk—a “pin.” The dynamics of the normal auxin-dependent patterning process can be watched and measured in Arabidopsis by live imaging of the meristem using fluorescent reporters (16, 17): A transgenic construct consisting of an auxin response element linked to a GFP coding sequence lights up cells in which auxin levels are high; a transgene coding for GFP-tagged PIN1 reveals the distribution of PIN1.

The patterning thus displayed involves movements of auxin in the epidermis, the outer surface layer of the meristem. Within this layer, auxin becomes concentrated at the site of each new primordium as it starts to form. The changing distribution of auxin goes hand in hand with a shifting epidermal distribution of PIN1 protein. This is most strongly expressed in the regions where levels of auxin are highest, that is, in the neighborhood of nascent primordia. Moreover, in each cell in these regions the PIN1 is localized predominantly on the side of the cell that faces toward the center of the nascent primordium. These and other findings suggest that the pattern of primordia is generated through a feedback loop in which auxin controls the synthesis and localization of its own transporter protein, PIN1: Where a local peak of auxin concentration starts to develop, the surrounding auxin gradient somehow orients the distribution of PIN1 transporters so as to drive still more auxin uphill toward the peak. This positive feedback based on active transport not only builds up the auxin concentration at the site of each nascent primordium, it also depletes auxin in the surrounding neighborhood, inhibiting formation of other primordia in that vicinity. Mathematical modeling shows that a mechanism of this type can explain the genesis of primordia in detail, including the precise geometry of primordium spacing and the quantitative data that come from live imaging (15, 1820).

How Cells Use Time Signals to Trace Out a Spatial Pattern: The Somite Segmentation Clock

Cells in embryos, like sailors at sea, use timers as well as spatial signals to tell them where they are and how they should behave. Such is the case for the cells that form the rudiments of the segments of the main vertebrate body axis, with their vertebrae, ribs, and axial muscles. These repetitive structures originate from somites, blocks of mesodermal cells that form sequentially, in head-to-tail succession and in two neat rows on either side of the midline of the embryo. Each somite is made out of cells that emerge from a region of apparently undifferentiated proliferative tissue at the tail end of the embryo, called the presomitic mesoderm, or PSM (Fig. 3). The PSM cells are kept in their special state by a combination of Wnt and FGF (fibroblast growth factor) proteins; the genes for these signal molecules are transcribed in the cells at the tail end of the PSM itself, creating a morphogen gradient that defines the PSM's extent (21, 22). As the PSM cells grow and divide, the body axis elongates, but the morphogen gradient retreats in step with the tail bud, leaving behind a trail of cells that differentiate to form somites as the concentrations of FGF and Wnt to which they are exposed fall below a critical level.

Fig. 3.

The somite segmentation clock and its coordination. (A) Side view of a 14-somite zebrafish embryo. [Adapted from (38).] (B) Diagram of the gene control circuitry proposed to underlie the oscillations of gene expression in the cells of the PSM of the zebrafish. For simplicity, just two neighboring cells are shown. Oscillations arise from a very simple intracellular negative feedback loop: The protein product of a gene of the her family (or of a set of such genes) feeds back with a delay to shut off its own expression. Each cell is capable of oscillating independently, but neighboring cells are normally entrained to the same rhythm by Delta-Notch signaling. The oscillating levels of Her protein drive oscillating expression of a Notch ligand of the Delta family, which thus serves as a periodic time signal that each cell exchanges with its neighbors. This keeps them synchronized, provided that the delays in the signaling pathway are appropriate.

The eye-catching feature of this process is its periodicity, the regular alternating pattern of the somites, each separated from the next by an intersomitic cleft. Cooke and Zeeman (23) speculated that this spatial periodicity could arise from a temporal oscillation in the PSM: a “clock” interacting with a signal, a “wavefront,” that swept back along the embryo, marking the boundary between differentiating somite cells and PSM cells. As the wavefront swept over them, cells would become consigned to different fates according to their clock phase at the critical moment, creating the periodic spatial pattern. Experiments by Palmeirim et al. (24) revealed that there is indeed a clock in the PSM, manifest in the oscillatory behavior of certain genes, which switch their expression on and off cyclically in this region. The period of the oscillation is equal to the time taken to form a somite: 2 hours in a mouse, 90 min in a chick, and 30 min in a zebrafish. The discovery of this segmentation clock has triggered an explosion of interest in somitogenesis and the mechanism of the clock (25).

Chick, mouse, and zebrafish all show the same phenomenon, and in all of them the Notch signaling pathway is critical for clock function. The first of the genes discovered to oscillate, in each of these species, were genes in the Notch pathway, especially members of the Hes/her family (called Hes genes in mouse and chick and her genes in zebrafish). These are all direct targets of regulation by Notch signaling, and all code for transcription regulators that bind to DNA to inhibit gene expression. Moreover, mutation of almost any component of the Notch pathway disrupts the organized oscillations and the regular pattern of somites (26). An obvious suggestion, therefore, is that Notch pathway components somehow generate the oscillations.

Notch signaling, unlike the other signaling pathways mentioned above, depends on direct cell-cell contact: A transmembrane protein, typically Delta, on the surface of one cell binds to a transmembrane receptor, Notch, on the surface of its neighbor. This triggers cleavage of Notch, releasing an intracellular fragment, NotchICD. NotchICD translocates to the cell nucleus and acts there as a transcription regulator, stimulating expression of the Hes/her genes. Notch signaling is famous for its role in lateral inhibition, a process that drives neighboring cells to become different as the result of a feedback loop based, in most cases, on negative regulation of Delta by Hes/Her proteins. But this is clearly not what is happening in the PSM, where neighboring cells remain similar and oscillate in synchrony. Indeed, experiments in the zebrafish suggested exactly the opposite: Notch signaling appeared to be needed to keep neighbors in synchrony. In mutants where signal transmission failed, it seemed as though the individual cells in the PSM continued to oscillate but gradually drifted out of synchrony over the course of half-a-dozen clock cycles, after having started out in synchrony (27).

Oscillations typically arise through delayed negative feedback. It is thus an attractive idea that the oscillations of the individual cells might be produced in a very simple way, through autoinhibition of Hes/her genes by their own products (2831). Active transcription of a Hes/her gene would lead to synthesis of Hes/Her protein, which would act back on the promoter of the Hes/her gene, shutting off transcription; this would allow the protein levels to fall, thereby enabling transcription to start up again; and so on, cyclically.

Mathematical modeling of this simple feedback loop reveals properties that one might not otherwise have guessed. Sustained oscillations can indeed be generated, but only if certain conditions are satisfied. In particular, the delays involved in transcription and translation are critical: delays, that is, from the time when a fresh transcript or protein molecule begins to be synthesized to the time when synthesis of that molecule is completed and the functional molecule is delivered to its site of action. Without such delays, the system will not oscillate. If it is to oscillate, the lifetimes of the mRNA and protein molecules must be short compared with the sum of the delays. And if the system does oscillate, the predicted period is given by a simple formula: It is, to a good approximation, just twice the sum of the delays plus the lifetimes (31).

But is the model just a mathematical toy, or does it correspond to reality? In the zebrafish, it has been possible to estimate experimentally the transcriptional delays for the relevant her genes as well as their mRNA lifetimes. The estimates, although crude, support the model (32). The logic of the autoregulatory gene circuitry can also be checked experimentally and also fits the model. Moreover, the model can be extended to show how Notch signaling, through oscillating expression of a Delta gene, can serve to couple the oscillations in adjacent cells and keep them synchronized in the face of random fluctuations (31). Here too, delays are critical, and measurements indicate that the actual delays in Delta production have values consistent with the theory (32).

In the zebrafish, several independent lines of experimentation confirm that Notch signaling serves to keep the clocks of individual PSM cells synchronized. Cells grafted between embryos in different phases of the clock fall into step, and cells that misexpress Notch pathway components disturb the phase of their neighbors (33). When the strength of the coupling mediated by Notch is gradually reduced by increasing doses of an inhibitor that blocks Notch activation, synchrony is lost in just the manner predicted by the mathematical theory of systems of weakly coupled noisy oscillators and is regained when the block is removed (34). And when Notch signaling is blocked acutely with large doses of the same inhibitor at different stages in development, somitogenesis is disrupted only after a long delay, in a manner that implies that clock synchronization is the only function of Notch signaling in this system (35).

Despite all this evidence, we still lack proof that the proposed simple feedback circuitry is truly the pacemaker of the segmentation clock. Moreover, what is true for the zebrafish may not be true for mouse and chick. In the mouse and the chick, although not in the zebrafish, PSM cells are known to also show oscillating expression of various genes in the Fgf and Wnt signaling pathways (21, 36, 37). Which of these oscillators is the master and which the slave? Or do they operate in parallel, with loose coupling to keep them synchronized? Could there be some other as-yet undiscovered master oscillator at the root of it all, driving the oscillations in multiple signaling pathways? Firm answers, as for a host of other open questions about the systems of temporal and spatial signals that govern development, will require a combination of experiments, measurements, and mathematics.

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