Oscillators in Animal Segmentation

SCHUMACHER COLLEGE
An International Centre for Ecological Studies

From In Silico Biology 2, 0010 (2002)

Oscillators in Animal Segmentation
Johannes Jaeger and Brian C. Goodwin

Edited by E. Wingender; received September 29, 2001; revised and accepted
December 8, 2001; published January 24, 2002

Abstract

Kinetic modeling of developmental dynamics requires detailed knowledge about genetic and metabolic networks that underlie developmental processes. However, such knowledge is not available for a vast majority of developmental processes. Here, we present an coarse-grained, phenomenological model of periodic pattern formation in multicellular organisms based on cellular oscillators (CO) that can be applied to systems for which little or no molecular data is available. An oscillatory process within cells serves as a developmental clock whose period is tightly regulated by cell-autonomous and non-autonomous mechanisms. A spatial pattern is generated as a result of an initial temporal ordering of the cell oscillators freezing into spatial order as the clocks slow down and stop at different times or phases in their cycles. When applied to vertebrate somitogenesis, the CO model can reproduce the dynamics of periodic gene expression patterns observed in the presomitic mesoderm. Different somite lengths can be generated by altering the period of the oscillation. There is evidence that a CO-type mechanism might also underlie segment formation in certain invertebrates, such as annelids and short germ insects. This suggests that the dynamical principles of sequential segmentation might be equivalent throughout the animal kingdom although most of the genes involved in segment determination differ between distant phyla.

Keywords: dagstuhl01261, segmentation, somitogenesis, short-germ band insects, cellular oscillators, phenomenological modeling, coarse-grained modeling, evolution of development, developmental dynamics.

Introduction

Many biologists today agree that to understand evolution we need to understand the evolution of development [Hall, 1992; Raff, 1996]. Over the last 25 years, various evolutionarily conserved genes have been discovered that play central roles in development. Examples of such genes include the Hox genes [Krumlauf, 1994], the segmentation gene engrailed [Patel, 1994] or Pax6, a gene which plays a central role in the development of photosensitive tissues in species ranging from jellyfish to mammals [Callaerts et al., 1997]. However, despite all this progress, we still lack explanations for the nonlinearity of evolutionary dynamics which is reflected in the fossil record [Eldredge and Gould, 1972] and we lack understanding of how development can constrain the evolution of morphologies [Webster and Goodwin, 1996].

There is growing evidence that development is not solely based on the action of single key genes but involves whole networks of genes and metabolites interacting in the context of growing tissues (see for example Kauffman, 1993; Goodwin, 1994; Webster and Goodwin, 1996; von Dassow et al., 2000; Sole and Goodwin, 2000; >Salazar-Ciudad et al., 2000; Salazar-Ciudad et al., 2001a,Salazar-Ciudad et al., 2001b). Developing organisms are dynamical systems that can be described using two different conceptual frameworks: a compositional and a dynamical one. The composition of a dynamical system is defined by the materials it is made of, by its chemical building blocks. For a genetic network, these would be the particular genes and their products which constitute the network. Up to now, evolutionary developmental biology, in its search for conserved molecular factors, has mainly focussed on this compositional aspect. A mathematician, on the other hand, would define a dynamical system by describing the interactions that take place between its parts (see for example Hirsch and Smale, 1974). This is the dynamical aspect of the system, which constrains and defines its behavior and which is largely independent of the materials the system is made of.

This implies that a developmental mechanisms in different organisms can be equivalent in two different ways. First, there can be compositional equivalence, in which the genes involved in some developmental process have similar or identical sequences in different species. This means that their products also have similar molecular structures. On the other hand, we find developmental mechanisms that are dynamically equivalent, but need not involve similar genes in different species. For example, the same type of feedback loop can be implemented by entirely different types of transcription factors but still lead to the same kind of dynamical behavior of the system. Such dynamical equivalence has been shown to be very useful for the study of the evolution of developmental mechanisms and the constraints that act upon it (see for example Wagner, 1994; Webster and Goodwin, 1996; Sole and Goodwin, 2000; Salazar-Ciudad et al., 2000; Salazar-Ciudad et al. 2001b).

Arguably, the best way to study the dynamics of developmental processes is by data-driven mathematical modeling, i.e. modeling combined with some sort of numerical optimization to fit the model to real biological data [Reinitz and Sharp, 1996]. However, there is a regrettable lack of data for such an approach. Most biological data produced today is compositional data (such as sequence data). Moreover, techniques to obtain dynamical information still have severe limitations. DNA microarrays, for instance, are only useful in a very limited way for the study of pattern formation, since the tissue which is examined needs to be homogenized before it is applied to the array. DNA microarrays have been successfully used for identifying candidate genes involved in developmental processes [White et al., 1999; Furlong et al., 2001 Altmann et al., 2001]. However, the spatial aspects of the gene network involved still had to be studied using traditional low-throughput techniques such as whole mount in-situ hybridization [Furlong et al., 2001] or tissue dissection [Altmann et al., 2001]. Therefore, in spite of some efforts to make in-situ hybridization and antibody staining techniques suitable for large-scale data gathering (see for example Kosman et al., 1998), it is quite probable that developmental modeling will remain a data-poor discipline in the near future.

Currently, the only developmental system for which a relatively detailed topology of its underlying molecular network is available is the early embryo of the fruit fly Drosophila melanogaster (reviewed in Gilbert, 2000). A data-driven approach to modeling segment determination in the Drosophila embryo has been successfully applied [Reinitz et al. 1995; Reinitz and Sharp, 1995; Reinitz et al., 1998]. To our knowledge, there is no other developmental system that has reached a sufficient level of available molecular data for data-driven approaches to dynamical modeling of pattern formation. In view of this lack of data, we argue that presently there is no alternative to phenomenological models of developmental processes. We argue that such coarse-grained, phenomenological models can be a valuable tool for gaining insight into developmental mechanisms, as long as their assumptions are kept as realistic and as simple as possible (see also Monk, 2000). Moreover, such models should be formulated in a biologically plausible and straightforward way, in view of the growing importance of interdisciplinary communication between mathematical modelers and experimental biologists

In the work presented here, we have used a very simple phenomenological model to investigate the dynamics of sequential segmentation in animals (see also Jaeger and Goodwin, 2001). Sequential segmentation is a very common developmental mechanism that occurs in vertebrates, annelids and arthropods [Sander, 1976; Shankland and Savage, 1997; Gilbert, 1997; Gossler and Hrabe de Angelis, 1998]. This mechanism of segmentation is quite different from what we find in Drosophila, where all segments are formed simultaneously and no tissue growth is involved in the process [Ingham, 1988]. Sequential segmentation in invertebrates occurs as follows: at the beginning of the segmentation process, the embryo shows very few segments or no segmental structure at all. As the embryo grows at its posterior end, new segments are added, usually one by one, until the final number of segments is reached [Sander, 1976; Shankland and Savage, 1997; Gilbert, 1997]. In an analogy to computer terminology, we could call this type of segmentation the serial type, whereas Drosophila shows a parallel type of segmentation (Fig. 1).

Figure 1: Parallel vs. serial/sequential segmentation. The two species of insects shown here show two extreme cases of parallel (fruit fly, Drosophila) and serial (locust, Schistocerca gregaria ) segmentation. Most insects show a combination of these two mechanisms. In Drosophila, an initial morphogen gradient is translated into 14 stripes through intermediate striped steps. The colored patterns show typical expression patterns of different classes of segmentation genes (brown: gap genes, green: pair-rule genes, gray: segment-polarity genes). Note that the embryo is a syncytium at all stages except the last one shown and no tissue growth is involved in the segmentation process (after Ingham, 1988). In Schistocerca, the embryo starts unsegmented. The first three morphological segments to appear are thoracic segments. Abdominal segments are then added in a sequential manner. The gray stripes represent the expression domain of the segment-polarity gene engrailed (after Sander, 1976; Patel, 1994).

A serial type of segmentation can also be observed in vertebrate somitogenesis. Somites are segmental patches of cells that form along the developing neural tube and go on to develop into vertebrae, axial muscles, blood vessels and dermal tissue (reviewed in Gossler and Hrabe de Angelis, 1998). They form from a tissue called the pre-somitic mesoderm (PSM). At the posterior end of the PSM, there is a structure called Hensen’s Node, which recruits cells into the PSM as it moves posteriorly along the embryo’s axis. During this process, new somites are formed in the PSM at a fixed distance from Hensen’s node (Fig 2a, Gossler and Hrabe de Angelis, 1998).

Figure 2: Vertebrate somitogenesis: (A) Schematic representation of somite formation in chicken. Cells become incorporated into the pre-somitic mesoderm (PSA) as Hensen’s node (represented by a triangle) moves posteriorly through the embryo. Somites (S1, ... , S12) form each 90 mins along the developing spinal cord (dark gray) at a fixed distance from Hensen’s node. The black spot represents a cell at its fixed position in the PSM, developing into a somite cell 18 hrs after being incorporated into the PSM (redrawn, with permission, from Palmeirim et al., 1997). (B) Schematic representation of oscillatory gene expression patterns observed during the formation of one somite. The expression starts in a broad posterior domain and narrows down as it advances anteriorly until it becomes fixed into one half of the newly formed somite. A anterior, P posterior, S somites (redrawn, with permission from Palmeirim et al., 1997).

Interestingly, several genes have been described that show oscillating expression patterns during somitogenesis in zebrafish, chicken and mouse [Palmeirim et. al, 1997; McGrew et al. 1998; Forsberg et al. 1998; Leimeister et al., 2000; Jouve et al., 2000; Sawada et al., 2000; Holley et al., 2000]. The oscillation periods of these patterns generally coincide with the time it takes to form one somite. Expression starts in a broad domain at the posterior end of the PSM, then narrows as it moves anteriorly through the tissue until the expression becomes stationary, usually in either the anterior or posterior half of the newly forming somite (Fig. 2b). These oscillations are not due to cell movements and are thought to be an effect of an unknown cell-intrinsic oscillation in PSM cells [Dale and Pourquie, 2000].

It has been proposed some time ago that cell-intrinsic oscillations can be used to generate spatial patterns in developing organisms [Goodwin and Cohen, 1969; Goodwin, 1976] and several oscillation-based models of somitogenesis have been proposed [Cooke and Zeeman, 1976; Keynes and Stern, 1988; Cooke, 1998; Kerszberg and Wolpert, 2000; Collier et al., 2000]. However, all the above models fail to reproduce the oscillatory gene expression patterns observed in the PSM. Since we consider the dynamics of the developmental mechanism as crucial for the understanding of pattern formation, we will describe a model that can not only successfully reproduce the sequential formation of segments, but also produces the oscillatory gene expression patterns mentioned above.

The model we use here is based on three basic assumptions about cells in growing tissues (see Methods for details):

1. The cells show an intrinsic physiological oscillation. We call such oscillating cells cellular oscillators (CO). The oscillating property of the cell will be called ‘cell state’ for simplicity.

2. The cellular oscillation serves as a molecular clock, which can be tightly regulated by cell-autonomous and non-autonomous mechanisms. The period of the clock needs to be different from the cell-division cycle.

3. There is a localized progress zone in the tissue, where tissue growth occurs (see also Kerszberg and Wolpert, 2000). Tissue growth can be due to cell division or recruitment of neighboring cells into the tissue.

Periodic patterns will be formed in any tissue for which the three above assumptions apply (see also >Newman, 1990). Note that a very similar model has been proposed based on fluid-distributed oscillators (FDO; Kaern and Menzinger, 1999; Kaern et al., 2000a,b). In fact, the CO mechanism described above corresponds to a simplified subset of FDO mechanisms, which covers all the FDO meachanims relevant to developmental processes [Jaeger and Goodwin, 2001].

Below, we discuss simulations that show the various kinds of pattern that can be generated using a CO-type mechanism in a growing row of cells. Then, we will compare these simulations to experimental evidence from various animal systems. Lastly, we will briefly discuss the possible relevance of such a coarse-grained mechanism for the study of evolutionary developmental dynamics.

Methods

Simulations were implemented in C using the CodeWarrior integrated programming environment on a Macintosh PowerBook G3. The source code is available from JJ. Results were plotted using Mathematica 4.0.

The following discrete algorithm was used for simulations of the CO mechanism (see also Jaeger and Goodwin, 2001): The growing tissue is represented by a line of cellular oscillators, each with its own phase () and its oscillation period (T). At one end of the tissue is a progress zone. Cells in this progress zone are thought to be undifferentiated and oscillate with a period that can vary through time in any biologically plausible way (see Results). The progress zone is represented as a single cell, since all its cells are thought to oscillate in phase. As cells in the progress zone divide, new cells become incorporated into the tissue inheriting initial phase ((0)) and period (T(0)) of their oscillation from the progress zone. The rate at which cells become incorporated into the tissue is called the growth rate of the tissue.

At every step t, the following three actions are performed for each cell:

1. The cell’s physiological age is increased by , which is the increase in its oscillation’s phase

We have deliberately chosen this physiological measure of time, since we think it is more realistic to assume an cell-intrinsic measure of time than a mechanism by which the cell could sense ‘objective’ or ‘outside’ time (t).

2. The cell’s period is evaluated according to:

The period of the oscillation increases exponentially starting at the initial period T(0) at the time when the cell leaves the progress zone. This amounts to an exponential slowing down of the oscillation after a certain constant time delay (defined by B). This time delay is constant for all cells in the tissue and depends on the cell-intrinsic molecular clock. Parameter A determines how abruptly the cell will slow its oscillation.

3. The cell’s phase is evaluated according to:

The above three steps are repeated to generate a two-dimensional array of phases for each cell at each time step (t). The cell state (z) of each cell is then evaluated as:

Note that we have chosen a simple harmonic cosine function for simplicity. Nonlinear periodic functions, such as saw tooth or square waves would probably be closer to the behavior of a real biological oscillator, but would not fundamentally alter the results of our simulations.

Results

Using the CO algorithm described in Methods, we have simulated rows of cells that grow at their posterior end (rows of cells in Fig. 3). The posterior progress zone is represented by a single cell (discs in Fig. 3), since all cells in it are assumed to oscillate in phase. The progress zone releases a certain amount of new cells into the tissue (squares in Fig. 3), depending on its growth rate. As cells enter the tissue, they start differentiating, i.e. they undergo a few more oscillations before the oscillation slows down or stops completely (Fig. 3).

Figure 3: The CO model of sequential segmentation: Each row in the figure represents a row of cells at a specific moment in time. Anterior (A) is to the left, posterior (P) to the right. The progress zone is represented by discs with red frames, tissue cells are represented as squares. Different cell states are shown in different shades of grey. Growth occurs only at the posterior end of the tissue. Cells enter the tissue at a given rate (one cell per time step in this figure) and inherit the initial phase and period of their oscillation from the progress zone. In this figure, cells will only undergo two more oscillation cycles in the tissue before the oscillation stops abruptly.

If the period of the oscillation in the progress zone is kept constant, a regular striped pattern is formed in the tissue. Spatial stripes form at a fixed distance from the progress zone (Fig. 4a). The width of the stripes is directly proportional to the period of the oscillator and the growth rate of the tissue. If the period of the oscillation is very long compared to the time it takes for the tissue to grow, a gradient is formed instead of a periodic pattern (Fig. 4b).

Figure 4: CO computer simulations: Patterns created using a constant period of the progress zone oscillation. (A) shows a typical striped pattern created using a period which is short compared to the time the tissue takes to grow. Oscillating waves of identical cell states that travel through the tissue can be observed at the transition between homogeneous oscillations (horizontal stripes) and spatial stripes (vertical stripes). See text for details. (B) Same as (A) using an oscillation period which is larger than the amount of time required for the tissue to grow. In this case, a gradient is formed instead of periodic stripes. Rows of the graph represent rows of cells at a specific moment in time. Anterior is to the left, posterior to the right. The progress zone is located at the posterior end of the tissue. The gray zone at the bottom of the graph represents space in which no cells have grown yet.

The dynamics by which the periodic stripes in Fig. 4a are formed are strongly reminiscent of the waves of gene expression patterns observed during vertebrate somitogenesis (see Introduction). Initially, large, almost stationary domains of similar cell states appear at the posterior end of the tissue (horizontal stripes in Fig. 4a). As times moves on, these expression domains start to move anteriorly in the tissue. Meanwhile, they become narrower and their movement slows down until the stripes appear stationary for all practical purposes (see also Jaeger and Goodwin, 2001).

More complex patterns can be generated, if the period of the progress zone oscillation or the tissue’s growth rate are allowed to change over time. In both cases, stripes of variable width are formed. If we assume an exponentially slowing tissue growth rate, we observe few and large stripes at the anterior end of the tissue, whereas numerous narrow stripes form more posteriorly (Fig. 5a). This corresponds to the segmentation pattern observed in mammals, where there are few large lumbar, but many small tail vertebrae. Similarly, we can produce stripes of alternating width, by alternating large and small periods of the progress zone oscillation (Fig. 5b). Such alternating segmental patterns can be seen in certain centipede species [Barnes et al., 1993].

Figure 5: CO computer simulations: Patterns created using a variable growth rate or period of the progress zone oscillation. (A) shows stripes of variable width, created by slowing down the growth rate of the tissue over time. Few large anterior stripes and a large number of narrow posterior stripes can be seen. (B) Stripes of alternating width can be created by switching the period of the progress zone oscillation between a large and a small period. Figure layout same as Fig. 4. Anterior is to the left, posterior to the right.

Apart from being able to reproduce the dynamics of gene expression during somitogenesis as well as a wide range of patterns seen in different organisms, a CO-type mechanism of segmentation is also consistent with embryological experiments. Cooke (1975) has observed that embryos of the clawed frog Xenopus laevis are capable of size regulation. This means, that embryos reduced in size will develop the typical number of somites, which are accordingly reduced in size. We can explain this in terms of a CO-type mechanism, assuming that such a reduced embryo will have a reduced progress zone. Therefore, the growth rate of the tissue will be proportionally smaller, which will produce more, but narrower stripes.

It is crucial for a CO-type mechanism, that there be an oscillating progress zone. The progress zone in our simulations corresponds to Hensen’s node in case of the chick embryo. Dale and Pourquie (2000) have recently presented preliminary evidence for periodic expression of mRNAs in Hensen’s node. Moreover, a CO-type mechanism is consistent with evidence that the molecular clock acting during somitogenesis is linked to, but different from the cell cycle. Heat shocks applied to chick embryos result in malformations that coincide with the period of cell division in the developing PSM [Primmett et al., 1989]. Such defects can be explained by a lasting damage to a specific phase of the cell cycle, which also affects the oscillation that underlies somite formation. Coupling of somite clock and cell cycle is further supported by the fact that the period of the clock seems to correspond roughly to about 20% of the cell cycle period in zebrafish [Stickney et al., 2000], chick [Gossler and Hrabe de Angelis, 1998] and mouse [Forsberg et al., 1998.

In the case of invertebrate segmentation, there is less experimental evidence for or against a CO-type mechanism. No oscillating gene expression patterns have been detected in insects yet. However, the existence of a posterior progress zone has been shown by fragmentation experiments in embryos of the African plague locust Schistocerca gregaria [Mee, 1986], an insect which is evolutionarily much more primitive than Drosophila. Furthermore, heat shocks applied to developing Schistocerca embryos cause deletions of segments at a fixed distance from where the progress zone was at the time of the heat shock [Mee and French, 1986a;Mee and French, 1986b]. Mee and French (1986a) mention that heat shocks temporarily inhibit cell division. If we interrupt cell division in our simulations for a brief period, but allow the cellular oscillator to continue during this interruption, we observe deletions in the final stripe pattern, which are only detectable some time after the interruption actually occurred (Fig. 6).

Figure 6: CO computer simulations: The effect of heat shock experiments on Schistocerca segmentation. During the heat shock, cell division ceases, but the cellular oscillation continues, causing a deletion in the segmentation pattern. The duration of the heat shock is indicated to the right of the graph. Figure layout same as Fig. 4. Anterior is to the left, posterior to the right.

A recent paper suggests that abdominal segments of Schistocerca form with a two segment period which then becomes refined into the final segmental pattern [Davis et al., 2001]. This is good evidence that there might be more than one oscillatory process involved in invertebrate sequential segmentation. In the CO model, this dynamical behavior can be observed, if we assume two oscillations with two-segment periods which are phase-shifted by half a period, i.e one segment’s length. However, it is also a strong indication that the actual mechanism at work in the insect abdomen is much more complex than our very simple model suggests.

Discussion

Although the molecular nature of the oscillatory mechanism in vertebrate somitogenesis remains unknown [Dale and Pourquie, 2000], there is strong molecular and morphological evidence for a CO-type mechanism (see also Kaern et al, 2000b). It has been argued that due to their astonishing simplicity, CO-type mechanisms are bound to be a very widespread phenomenon in growing cellular tissues throughout evolution [Newman, 1993; Kaern et al. 2000b; Jaeger and Goodwin, 2001]. However, the mechanism of invertebrate sequential segmentation remains elusive, in spite of some experimental evidence in favor of a CO-type mechanism from dissection and heat shock experiments in insects. Therefore, far more data will be required to be able to conclusively prove that a CO-type sequential segmentation mechanism is at work in vertebrate somitogenesis as well as in invertebrate sequential segmentation.

CO-type mechanisms can be easily detected by the typical growth-rate dependent oscillatory patterns they produce in the growing tissue. We believe that the discovery of such physiological oscillations in the invertebrate abdomen during segment formation is very likely in the near future. As mentioned earlier, these oscillations need not appear as oscillatory gene expression patterns, but might also be implemented as waves of protein modification or even oscillations in the concentration of metabolites or ions [Jaeger and Goodwin, 2001].

Of course, the coarse-grained model presented here will have to be adapted and refined to suit specific systems. This is why we have formulated the CO model as an open framework, that allows for easy inclusion of additional complexity, such as cell-cell signaling, interactions with other pattern forming processes or specific molecular oscillatory mechanisms. An example of such an application of a CO-type mechanism to the evolution of a specific developmental process is reported in Salazar-Ciudad et al. (2001b). These authors have used simulated evolution of simple genetic networks to show that both parallel and sequential segmentation can be implemented by the same gene network acting either in a syncytium or the cellular context of a growing tissue. This study therefore provides an explanation of why both types of segmentation occur in most insect orders and can vary even between closely related species [Sander, 1976].

We have tried to show in this paper that coarse-grained phenomenological models of developmental processes can be a very useful tool to study the evolution of development in silico. However, we think that the usefulness of such models heavily depends on their being based on probable and experimentally testable biological mechanisms. This is true for both CO and FDO mechanisms [Kaern et al., 2000b], which makes them preferable over classical reaction-diffusion (RD) models for the study of pattern formation in cellular tissues.

RD models are based on chemical reactions involving reactants with different diffusion rates [Turing, 1952] and have been applied to various problems of biological pattern formation (see for example Meinhardt, 1982, Meinhardt, 1999). Muratov (1997) has shown that both RD and oscillation-based mechanisms can be treated within the same mathematical framework. However, an RD mechanism is very unlikely to apply to pattern formation in cellularized tissues, since diffusion cannot readily occur across cell membranes (see for example Entchev et al., 2000; Pfeiffer et al., 2000). We argue that although mathematically equivalent to RD models, CO or FDO models are therefore more powerful in yielding useful predictions and interesting insights into the underlying developmental processes during pattern formation.

We are well aware of the limitations of CO-type models. For instance, they cannot tell us anything about the specific molecular mechanism underlying the oscillations in different developmental processes. In the future, they will also need to be extended to include cell-cell interactions or interacting oscillations with periods on different time scales. Nevertheless, we hope that coarse-grained but biologically plausible CO or FDO models might provide a conceptual framework for studying the evolution of periodic patterns formation in various organisms and might also stimulate new experimental research, aimed at obtaining more and better data on the dynamical aspect of such oscillations in development and evolution.

Acknowledgments

We would like to thank Hilde Janssens, Nick Monk and John Reinitz for critical comments and discussion of this work. We thank David Bilton for pointing out to us the alternating segmentation pattern of tropical centipedes.

This research (JJ) was funded by a Roche Research Foundation fellowship.

References

Altmann, C. R., Bell, E., Sczyrba, A., Pun, J., Bekiranov, S., Gaasterland, T. and Brivanlou, A. H. (2001). Microarray-based analysis of early development in Xenopus laevis. Dev. Biol. 236, 64-75.

Barnes, R. S. K., Calow, P. and Olive, P. J. W. (1993). The Invertebrates: A New Synthesis. Blackwell Science, Oxford.

Callaerts, P., Halder, G. and Gehring, W. J. (1997). PAX-6 in development and evolution. Ann. Rev. Neurosci. 20, 483-532.

Collier, J. R., McInerney, D., Schnell, S., Maini, P. K., Gavaghan, D. J., Houston, P. and Stern, C. D. (2000). A cell cycle model for somitogenesis: mathematical formulation and numerical simulation. J. Theor. Biol. 207, 305-316.

Cooke, J. (1975). Control of somite number during development in a vertebrate, Xenopus laevis. Nature 254, 196-199.

Cooke, J. (1998). A gene that resuscitates a theory – somitogenesis and a molecular oscillator. Trends Gen. 14, 85-88.

Cooke, J. and Zeeman, E. C. (1976). A clock and wavefront model for control of the number of repeated structures during animal morphogenesis.
J. Theor. Biol. 58, 455-476.

Dale, K. J. and Pourquie, O. (2000). A clock-work somite. BioEssays 22, 72-83.

Davis, G. K., Jaramillo, C. A. and Patel, N. H. (2001). Pax group III genes and the evolution of insect pair-rule patterning. Development 128, 3445-3458.

Eldredge, N. and Gould, S. J. (1972). Punctuated equilibria: an alternative to phyletic gradualism. In: Models in Paleobiology. Schopf, T.J.M. (ed.), Freeman Cooper, San Francisco.

Entchev, E. V., Schwabedissen, A. and Gonzalez-Gaitan, M. (2000). Gradient formation of the TGF-ß homolog Dpp. Cell 103, 981-991.

Forsberg, H., Crozet, F. and Brown, N. A. (1998). Waves of mouse lunatic fringe expression, in four-hour cycles at two-hour intervals, precede somite boundary formation. Curr. Biol. 8, 1027-1030.

Furlong, E. E. M., Andersen, E. C., Null, B., White, K. P. and Scott, M. P. (2001). Patterns of Gene Expression During Drosophila Mesoderm Development. Science 293, 1629-1633.

Gilbert, S. F. (1997). Arthropods: the Crustaceans, Spiders and Myriapods. In: Embryology: Constructing the Organism, Gilbert, S. F. and Raunio, A. M. (eds.), Sinauer Associates, Sunderland, MA, pp. 237-258.

Gilbert, S. F. (2000). Developmental Biology, 6th Edition. Sinauer Associates, Sunderland, MA.

Goodwin, B. C. (1976). Analytical Physiology of Cells and Developing Organisms.
Academic Press, London.

Goodwin, B. C. (1994). How the Leopard Changed Its Spots. Charles Scribner’s Sons, New York.

Goodwin, B. C. and Cohen, M. H. (1969). A phase-shift model for the spatial and temporal organisation of developing systems. J. Theor. Biol. 25, 49-147.

Gossler, A. and Hrabe de Angelis, M. (1998). Somitogenesis. Curr. Top. Dev. Biol. 38, 225-287.

Hall, B. K. (1992). Evolutionary Developmental Biology. Chapman and Hall, London.

Hirsch M. W. and Smale, S. (1974). Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, Orlando, FL.

Holley, S. A., Geisler, R. and Nusslein-Volhard, C. (2000). Control of her1 expression during zebrafish somitogenesis by a Delta-dependent oscillator and an independent wave-front activity. Genes Dev. 14, 1678-1690.

Ingham, P. W. (1988). The molecular genetics of embryonic pattern formation in Drosphila. Nature 335, 25-34.

Jaeger, J. and Goodwin, B. C. (2001). A Cellular Oscillator Model for Periodic Pattern Formation. J. Theor. Biol. 213, 171-181.

Jouve, C., Palmeirim, I., Henrique, D., Beckers, J., Gossler, A., Ish-Horowicz, D. and Pourquie, O. (2000). Notch signalling is required for cyclic expression of the hairy-like gene hes1 in the presomitic mesoderm. Development 127, 1421-1429.

Kaern, M. and Menzinger, M. (1999). Flow-distributed oscillations: stationary chemical waves in a reacting flow. Phys. Rev. E 60, R3471-R3474.

Kaern, M., Menzinger, M. and Hunding, A. (2000a). A chemical flow system mimics waves of gene expression during segmentation. Biophys. Chem. 87, 121-126.

Kaern, M., Menzinger, M. and Hunding, A. (2000b). Segmentation and somitogenesis derived from phase dynamics in growing oscillatory media.
J. Theor. Biol. 207, 473-493.

Kauffman, S. (1993). The Origins of Order. Oxford University Press, Oxford.

Kerszberg, M. and Wolpert, L. (2000). A clock and trail model for somite formation, specialization and polarization. J. Theor. Biol. 205, 505-510.

Keynes, R. J. and Stern, C. D. (1988). Mechanisms of vertebrate segmentation. Development 103, 413-429.

Kosman, D., Reinitz, J. B. and Sharp, D. H. (1998). Automated assay of gene expression at cellular resolution. Pac. Symp. Biocomput., 6-17.

Krumlauf, R. (1994). Hox genes in vertebrate development. Cell 78, 191-201.

Leimeister, C., Dale, K. J., Fischer, A., Klamt, B., Hrabe de Angelis, M., Radtke, F. McGrew, M. J., Pourquie, O. and Gessler, M. (2000). Oscillating expression of c-hey2 in the presomitic mesoderm suggests that the segmentation clock may use combinatorial signaling through multiple interacting bHLH factors. Dev.
Biol. 227, 91-103.

McGrew, M. J., Dale, J. K., Fraboulet, S. and Pourquie, O. (1998). The lunatic fringe gene is a target of the molecular clock linked to somite segmentation in avian embryos. Curr. Biol. 8, 979-982.

Mee, J. E. (1986). Pattern formation in fragmented eggs of the short germ insect Schistocerca gregaria. Roux Arch.

Mee, J. E. and French, V. (1986a). Disruption of segmentation in a short germ insect embryo: I. the location of abnormalities induced by heat shock. J. Embryol. Exp. Morph. 96, 245-266.

Mee, J. E. and French, V. (1986b). Disruption of segmentation in a short germ insect embryo: II. the structure of segmental abnormalities induced by heat shock. J. Embryol. Exp. Morph. 96, 267-294.

Meinhardt, H. (1982). Models of Biological Pattern Formation. Academic Press, London.

Meinhardt, H. (1999). On pattern and growth. In: On Growth and Form: Spatio-temporal Pattern Formation in Biology, Chaplain, M.A.J., Singh, G.D. and McLachlan, J.C. (eds.), John Wiley & Sons, London, pp. 129-148.

Monk, N. A. M. (2000). Elegant hypothesis and inelegant fact in developmental biology. Endeavour 24, 170-173.

Muratov, C. B. (1997). Synchronization, chaos, and the breakdown of collective domain oscillations in reaction-diffusion systems. Phys. Rev. E 55, 1463-1477.

Newman, S. A. (1990). Generic physical mechanisms of morphogenesis and pattern formation as determinants in the evolution of multicellular organization. In: The Principles of Organization in Organisms, Mittenthal, J.E. and Baskin, A.B. (eds.), Addison-Wesley, Reading, MA.

Newman, S. A. (1993). Is segmentation generic? BioEssays 15, 277-283.

Palmeirim, I., Domingos, H., Ish-Horowicz, D. and Pourquie, O. (1997). Avian hairy gene expression identifies a molecular clock linked to vertebrate segmentation and somitogenesis. Cell 91, 639-648.

Patel, N. H. (1994). Developmental evolution: insights from studies of insect segmentation. Science 266, 581-590.

Pfeiffer, S., Alexandre, C., Calleja, M. and Vincent, J. P. (2000). The progeny of wingless-expressing cells deliver the signal at a distance in Drosophila embryos. Curr. Biol. 10, 321-324.

Primmett, D. R. N., Norris, W. E., Carlson, G. J., Keynes, R. J. and Stern, C. D. (1989). Periodic segmental anomalies induced by heat shock in the chick embryo are associated with the cell cycle. Development 105, 119-130.

Raff, R. A. (1996). The Shape of Life. University of Chicago Press, Chicago.

Reinitz, J. B., Mjolsness, E. and Sharp, D. H. (1995). Cooperative control of positional information in Drosophila by bicoid and maternal hunchback. J. Exp. Zool. 271, 47-56.

Reinitz, J. B. and Sharp, D. H. (1995). Mechanism of eve strip formation. Mech Dev. 49, 133-158.

Reinitz, J. B. and Sharp, D. H. (1996). Gene circuits and their uses. In: Integrative Approaches to Molecular Biology, Collado, J., Magasanik, B. and Smith, T. (eds.), MIT Press, Cambridge, MA.

Reinitz, J. B., Kosman, D., Vanario-Alonso, C. E. and Sharp, D. H. (1998). Stripe forming architecture of the gap gene system. Dev. Genet. 23, 11-27.

Salazar-Ciudad, I., Garcia-Fernandez, J. and Sole, R. V. (2000). Gene networks capable of pattern formation: from induction to reaction-diffusion. J. Theor. Biol. 205, 587-603.

Salazar-Ciudad, I., Newman, S. A. and Sole, R. V. (2001a). Phenotypic and dynamical transitions in model genetic networks I. emergence of patterns and genotype-phenotype relationships. Evol. Dev. 3, 84-94.

Salazar-Ciudad, I., Sole, R. V. and Newman, S. A. (2001b). Phenotypic and dynamical transitions in model genetic networks II. application to the evolution of segmentation mechanisms. Evol. Dev. 3, 95-103.

Sander, K. (1976). Specification of the basic body pasttern in insect embryogenesis. Adv. Insect Physiol. 12, 125-238.

Sawada, A., Fritz, A., Jiang, Y., Yamamoto, A., Yamasu, K., Kuroiwa, A., Saga, Y. and Takeda, H. (2000). Zebrafish Mesp family genes, mesp-a and mesp-b]are segmentally expressed in the presomitic mesoderm, and mesp-b confers the anterior identity to the developing somites. Development 127, 1691-1702.

Shankland, M. and Savage, R. M. (1997). Annelids, the Segmented Worms. In: Embryology: Constructing the Organism, Gilbert, S. F. and Raunio, A. M. (eds.), Sinauer Associates, Sunderland, MA, pp. 219-236.

Sole, R. V. and Goodwin, B. C. (2000). Signs of Life. Basic Books, New York.

Stern, C. D. and Vasiliauskas, D. (1998). Clocked gene expression in somite formation. BioEssays 20, 528-531.

Stickney, H. L., Barresi, M. J. F. and Devoto, S. H. (2000). Somite development in zebrafish. Dev. Dyn. 219, 287-303.

Turing, A. (1952). The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. London B 237, 37-72.

Von Dassow, G., Meir, E., Munro, E. M. and Odell, G. M. (2000). The segment polarity network is a robust developmental module. Nature 406, 188-192.

Wagner, A. (1994). Evolution of gene networks by gene duplications: a mathematical model and its implications on genome organization. Proc. Natl. Acad. Sci. USA 91, 4387-4391.

Webster, G. and Goodwin, B. C. (1996). Form and Transformation. Cambridge University Press, Cambridge.

White, K. P., Rifkin, S. A., Hurban, P. and Hogness, D. S. (1999). Microarray analysis of Drosophila development during metamorphosis. Science 286, 2179-2184.

For more information about Schumacher College and its courses, please contact :

The Administrator, Schumacher College,
The Old Postern, Dartington, Totnes, Devon, TQ9 6EA.
Tel : +44 (0)1803 865934 Fax : +44 (0)1803 866899
Email : admin@schumachercollege.org.uk
Website : www.schumachercollege.org.uk

Schumacher College is part of the Dartington Hall Trust, a company limited by guarantee, registered in England and as a charity (company no. 1485560, charity no. 279756). Registered office: The Elmhirst Centre, Dartington Hall, Totnes, Devon TQ9 6EL, UK.