The cell cycle model of Stern and collaborators 〚19, 20〛 links the cell cycle with somite segmentation. This hypothesis is supported by the following observations: (1) the existence of discrete regions of cell synchrony in the PSM, (2) a cell cycle duration of 9 hours, which corresponds to the development of six to seven somites in the chick embryo, and (3) periodic anomalies caused by heat shock experiments are mimicked by drugs inhibiting the progression of the cell cycle.

The formation of a somite is explained by assuming that there are two successive time points, P₁ and P₂, a time interval of 1/7 of the cell cycle time apart. Somite cells recognise the P₁–P₂ time window. According to the model, cells destined to form somites leave Hensen's node strictly in the order in which they were derived from founder cells in the node, and they remain in that order. Hence there is some degree of cell cycle synchrony of somite cells at the same level, and these cells will be arranged along the embryonic axis in the same order as they are positioned in their cell cycles; older cells further anterior than younger ones 〚35〛. We assume that a fraction of the cohort of cells destined to segment together reach P₂ before others and produce a signal to which cells situated between P₁ and P₂ would respond by later increasing their adhesion to each other shortly before segmentation, regardless of their position within the PSM. The signal produces a somitic factor, which according to our hypothesis is the precursor of a CAM. It has been demonstrated that CAMs can direct cell differentiation, regulate gene expression and tissue formation 〚36, 37〛. Thus, the resulting group of cells would actually undergo segmentation one cell cycle after the P₁–P₂ time window. As they move out of the P₁–P₂ window, cells become refractory to the signal and/or unable to signal (see Fig. 1).

The model proposes that heat shock temporarily blocks the cell cycle, so altering the number of cells that segment together. Such an alteration would occur once in each cell cycle in the segmental plate posterior to the P₁–P₂ time window, accounting for the repetitive anomalies resulting from heat shock.

We have recently formulated these ideas mathematically 〚see for details Appendix A〛 as a coupled system of nonlinear partial differential equations and have shown that the resultant mathematical model can account for the periodic abnormalities observed after a single heat shock, as well as the cell autonomous character of somite formation in the PSM explants and the transplantation experiments reversing the AP axis 〚4〛. The model postulates the existence of two chemicals: a somitic factor and a signalling factor, with concentrations denoted, respectively, by u and v. In words, the signalling part of the model 〚equations (A.1) and (A.2)〛 takes the form:

It is assumed that the production of u is of an autocatalytic nature enhanced by v but saturating for large u. The signal v is emitted at P₂ and is assumed to diffuse rapidly. Its production rate is assumed to be inhibited by the somitic factor u, so that cells which have already been specified as somitic (and hence have a high u concentration) cannot signal. We assume that both u and v degrade linearly. The particular form of kinetics are chosen to exhibit a switch-like behaviour – the emission of v activates the production of u, which then quickly reaches its saturating level (specifying a block of tissue as 'somitic'). In turn, this concentration of u inhibits further production of v so that only a spatially confined block of tissue is specified as somitic.

The above model, for somite determination, operates one cell cycle before a somite actually forms. If we wish to model the cell movements during somitogenesis, we have to add to this model, an equation for the response of cell density to the signalling 〚equation (A.3) in Appendix A〛. The cells are assumed to diffuse randomly and to respond haptotactically to gradients in v. Furthermore, we assume that movement depends on the concentration level of v. In words, we assume that a block of cells, specified as somitic, clump together in response to adhesive gradients (determined by v) in regions of sufficiently high v concentration 〚equations (A.4) and (A.5) in Appendix A〛.

In our numerical simulations, the equations are solved on a closed bounded interval of $ℝ$, which we denote by $\Omega$; typically $\Omega =\left[-10,10\right]$. This domain is large enough to ensure that conditions at the boundary do not affect the patterning process. As an approximation to these infinite domain boundary conditions we impose zero flux boundary conditions at x=-10 and x=10 (diffusion of v out of the domain is not considered).

2.1 The signalling behaviour of the model

A typical numerical solution to (A.1)–(A.2) is shown in Fig. 2. The results show successive pulses of v, occurring about 200 time units apart. A pulse in the signal v is emitted at x≈0 leading to a jump in the somitic factor u between points x≈0 and x≈1. This jump in the somitic factor u corresponds to a collection of cells that will later form a somite. The first pulse at x≈0 is followed by a second at x≈1 and a third at x≈2, causing a series of jumps in the somitic factor u along the x-axis.

In Fig. 3, we present the results of a heat shock on the pulse of signalling molecule u. In this figure, a schematic representation of the first ten somites shows a sequence of one normal, two abnormal, four normal, two abnormal and one normal somite. This shows that the model can exhibit periodic anomalies due to heat shock.

2.2 The inclusion of cell density

Fig. 4 depicts a series of periodic cell aggregations of equal height. These clusters of somitic cells of equal size are formed at regular time intervals. Note that changes in cell density are highly localised, leading to deep troughs in cell density. Cell aggregation occurs in much of the spatio-temporal domain where v is high (see previous simulation in Fig. 2). The compaction process begins at the anterior end of the embryo and moves towards the posterior as a wave. This wave is static in the trunk of the embryo, with cells moving up the PSM as development proceeds.

In this paper we have presented and extended a mathematical formulation of the cell cycle model for somitogenesis by adding a third equation to account for cell density. Our mathematical model is based on two key assumptions: (i) the signalling process is decoupled from the compaction process, and (ii) somites are formed as clusters of cells by changes in the concentration of a signalling molecule.

The first assumption is supported by the experimental evidence of Stern and collaborators 〚19, 20〛 linking the cell cycle with somite segmentation. Cell aggregation and compaction during somitogenesis is thought to be due to the interaction of CAMs in the PSM. The mechanisms that regulate the spatio-temporal activation and inhibition of CAMs are unknown. We postulate that the formation of clusters of somitic cells is driven by an adhesive gradient related to the signalling molecule.

In the full mathematical formulation (A.1)–(A.3) of our model, the somitic potential factor u is produced in response to the signalling molecule v. Cells at positions on the axis where u≈1 are termed somitic, while those where u≈0 are non-somitic. The signalling molecule v acts as a chemical morphogen providing a spatio-temporal pre-pattern of somitic and non-somitic cells, while u is providing the positional information for the cells in the PSM, which move depending on the concentration of u 〚42〛. This sub-model may be thought of as providing a chemical (or molecular) pre-pattern to which cells respond via movement 〚equation (A.3)〛. To our knowledge, this is the first model to combine two of the major theories for spatial patterning.

Our results indicate that the cell cycle mechanism can indeed give rise to the periodic pattern of somites observed in normal embryos and also to the abnormal patterns observed after heat shock. It is worth noting that one possible scenario is that two separate clocks linked to the cell cycle regulate somitogenesis. Supporting this suggestion is the finding that another member of the hairy-enhancer of split family, her-1, has been characterised in zebrafish as having an expression pattern in every other somite 〚43〛, unlike any gene thus far described in other vertebrates 〚4〛.

If one clock exists, then a direct link between the cell cycle and the c-hairy-1/l-fng cycling is not obvious. To date, there is no molecular mechanism relating the cell cycle and the c-hairy-1 and the l-fng oscillations. Nevertheless, there is evidence 〚44, 45〛 suggesting that the cell cycle and the c-hairy-1 oscillations are regulated by post-translational modifications such as phosphorylation, and therefore could be part of the same clock. Recently, Kaern and collaborators 〚34〛 have proposed, in agreement with Meinhardt 〚31〛, that segmentation is controlled by one segmentation clock (which has c-hairy-1 and l-fng as components or products) and by the AP subdivision of somites.

In our model, we formulated mathematically a mechanism which regulates segmentation independently of the c-hairy-1/l-fng oscillations. This model also assumes that segmentation occurs independently from AP subdivision. One important point to make is that several mouse mutants reveal that although AP subdivision plays a role in determining somite boundaries, this is not inextricably linked to the initial subdivision of the PSM into discrete units. For example, in most of the Notch pathway mutants, some sort of somites start forming despite the failure of the normal AP subdivision mechanism, but by the time that they should split into AP halves they fall apart and regular segmentation disappears 〚46〛. This suggests that the hairy or lunatic oscillations (these genes are part of the Notch signalling system) are more likely to be part of the mechanisms for the AP subdivision of somites and affect boundary formation indirectly (or later).

In a previous paper, we have addressed the AP subdivision mechanism by developing our own model, which we call the clock-and-induction model 〚33〛. This theoretical description introduces a counting mechanism for cells in the PSM produced by a stationary phase gradient of l-fng. Cell maturation is the result of motion along this gradient. We suggest that l-fng mRNA is expressed periodically and its stable product – a protein – accumulates in a cellular membrane. The formation of a new intersomite boundary could be triggered by the Notch-Delta pathway when a certain threshold is reached; the timing of this response will correspond to a certain number of completed cycles of l-fng expression. Interestingly, several independent experimental groups have recently shown that l-fng mRNA is translated into a stable protein that accumulates in the endoplasmic reticulum and which forms a complex with the Notch protein 〚47–50〛. Then, once a somite is formed the formation of the AP pattern would be triggered by a signalling pathway. In the clock and induction model, we proposed that an induction mechanism by the Notch and Delta signalling pathway would be responsible for the AP pattern. However, in the light of recent experimental evidence 〚51〛, the fibroblast growth factor (FGF) signalling pathway could play an important role in this process. Members of the FGF family are important signalling molecules in several inductive and patterning processes during early embryogenesis. During the formation of the vertebrate nervous system 〚52–54〛 and somitogenesis 〚55〛, FGF8 acts as a signal to determine the AP patterning.

Vertebrae in different regions of the vertebral column are morphologically different, suggesting that somites have defined positional information identities. After somites have been formed a subsequent differentiation process gives rise to unique anatomical structures depending on position along the AP axis. Experiments in chick embryos demonstrate that the positional specification of somites occurs early during somitogenesis 〚56–62〛. When cervical somites are replaced with somites from the trunk region, rib-like structures develop in the cervical vertebral column of the embryo. When thoracic somites are replaced by cervical somites, embryos do not develop ribs 〚56〛.

To our knowledge, the only model that addresses the regional differences of somites is Meinhardt's reaction diffusion type model 〚31, 63〛 for segmentation. This model proposes that the specification of anterior (a) and posterior (p) half-somites occurs by wave-like processes initiated at the posterior end of the PSM, moving anteriorly until it comes to rest at a given distance from the last formed half-somite. The two states, a and p, locally exclude each other, but stimulate each other over a long range. Cells switch from one state to the other until finally reaching a stable state. This can lead to a pattern of stable ... apapap ... stripes forming from anterior to posterior. If the transition from, say p to a, allows a change of segmental specification then each ap pair (or segment) will have a more posterior specification than its predecessor. Thus a segmental pattern can be generated in which segments have different regional characteristics. To set up this pattern, Meinhardt proposed an extracellular diffusive morphogen gradient in which threshold concentrations of the morphogen are required for successive p to a transitions.

However, Meinhardt's model cannot easily explain the results of the experiments demonstrating the positional specification of somites. If an extracellular morphogen gradient set up the somitic pattern, the model would predict that somites would differentiate according to their new position, but in reality specification occurs according to their original location. One would have to assume that rostral-caudal determination occurs very early and is fixed before isolation or rotation of the PSM. This possible explanation requires more detailed investigation.

The cell cycle model proposed originally by Stern and collaborators 〚19, 20〛 does not account for the regionalisation of individual somites leading to specific vertebrae. However, it is very important to notice (as mentioned in Section 1) that vertebrae, at least in the chick embryo, are regionalised in multiples of 6 to 7 somites, which correspond to the number of somites formed per cell cycle. This correlation implicates the cell cycle in the temporal regulation of regional identity specification but with specific reference to the role of Hox genes.

We proposed a mechanism of control which invokes, as in the clock-and-induction model 〚33〛, a cytoplasmatic gradient in the concentration of a factor regulating regional identity specification. Cell division would distribute the cytoplasmic determinant changing its concentration between the mother and daughter cells and thereby confer different fates on the mother and daughter cells activating the family of the Hox genes.

There is now a large body of experimental work showing that positional specification of the PSM requires members of the Hox gene family 〚25〛. Hox gene activation during development correlates with gene position in the Hox complex, a property referred to as collinearity. The spatial and temporal collinearity in the expression of these genes results in unique combinations of Hox genes in defined groups of somites and their derivatives along the AP axis 〚64, 65〛. This led to the suggestion that a Hox code specifies the identity of somites 〚26, 27〛. The role of Hox genes in positional specification has been analysed by interfering with or altering the expression of single Hox genes or by simultaneously perturbing the expression with retinoic acid, which is implicated in the specification of the axes during development 〚66〛. It has also been proposed that the FGF signalling pathway is involved in the caudalisation of somites by coordinating either the segmentation process or the mechanism involved in somite boundary formation and the spatio-temporal Hox gene activation 〚51, 55, 67〛.

The mechanism we proposed for the activation of the Hox genes is the same as that proposed during limb development 〚68〛. The cytoplasmatic determinants are molecules which bind with high affinity to the Hox complex blocking access to the transcription machinery. The transcriptional antagonist would become diluted by successive cell divisions allowing access to the genes of the Hox complex. As dilution progresses many more genes of the Hox complex are expressed allowing somitic regionalisation.

A similar sort of behaviour of cell cycle kinetics has been modelled for cell growth and maturation during carcinogenesis (see, for example, 〚69〛). In the literature, there has been a predominant trend to model this process by (i) a branching process approach and (ii) a partial differential equations approach 〚70〛. A typical partial differential equation formulation is the Lotka-von Foster equation, where chronological time, cell age, population density and growth rate terms are described in a transport equation with additional boundary conditions 〚71〛.

This idea can be also applied to lineage intrinsic specification, in which an asymmetric cell division distributes some sort of cytoplasmic determinant unequally between daughter cells and thereby confers different fates on distinct daughter cells and their descendants. In certain organisms, such as the nematode Caenorhabditis elegans and other protostomes, the invariant patterns of asymmetric cell division and differentiation arise from mechanisms operating through lineage cell ancestry and cell-cell adhesion interactions.

Experimental research is presently being carried out to study the key components of the molecular clock and how the molecular oscillations are modulated by the cellular aspects of somitogenesis. A challenging future problem for mathematical modelling will involve linking the pattern formation mechanisms at the cellular level with the molecular control of cell properties in other developmental processes. So far, somitogenesis has proven to be an excellent potential example for a detailed study of the marriage between cellular and molecular biology.

Appendices