Hybrid reaction-diffusion and clock-and-wavefront model for the arrest of oscillations in the somitogenesis segmentation clock
Authors: Pantoja-Hernández Jesús , Breña-Medina Víctor F. , Santillán Moisés 
Published: September 2020

AbstractThe clock and wavefront paradigm is arguably the most widely accepted model for explaining the embryonic process of somitogenesis. According to this model, somitogenesis is based upon the interaction between a genetic oscillator, known as segmentation clock, and a differentiation wavefront, which provides the positional information indicating where each pair of somites is formed. Shortly after the clock and wavefront paradigm was introduced, Meinhardt presented a conceptually different mathematical model for morphogenesis in general, and somitogenesis in particular. Recently, Cotterellet al. rediscovered an equivalent model by systematically enumerating and studying small networks performing segmentation. Cotterellet al. called it a progressive oscillatory reaction-diffusion (PORD) model. In the Meinhardt-PORD model, somitogenesis is driven by short-range interactions and the posterior movement of the front is a local, emergent phenomenon, which is not controlled by global positional information. With this model, it is possible to explain some experimental observations that are incompatible with the clock and wavefront model. However the Meinhardt-PORD model has some important disadvantages of its own. Namely, it is quite sensitive to fluctuations and depends on very specific initial conditions (which are not biologically realistic). In this work, we propose an equivalent Meinhardt-PORD model, and then amend it to couple it with a wavefront consisting of a receding morphogen gradient. By doing so, we get a hybrid model between the Meinhardt-PORD and the clock-and-wavefront ones, which overcomes most of the deficiencies of the two originating models.Somitogenesis, the process by which somites are formed, is an essential developmental stage in many vertebrates. This process occurs with a strikingly regular periodicity, that is preserved among embryos of a single species. The clock and wavefront paradigm is arguably the most widely accepted model for explaining somitogenesis. However, it is incapable of explaining some experimental facts, like the appearance of somites in the absence of an external wavefront (i.e. a receding morphogen gradient). Shortly after the clock and wavefront paradigm was introduced, Meinhardt presented a conceptually different mathematical model for morphogenesis in general, and somitogenesis in particular. Recently, Cotterellet al. rediscovered an equivalent model by systematically enumerating and studying small networks performing segmentation, and called it a progressive oscillatory reaction-diffusion (PORD) model. The Meinhardt-PORD model tackles some of the deficiencies of the clock and wave-front models, but it has some serious issues of its own. In the present work, we introduce an equivalent Meinhardt-PORD model, and then amend it to couple it with a receding morphogen gradient. By doing so, we get a hybrid model that incorporates characteristics of the Meinhardt-PORD and clock-and-wavefront models. We show that this hybrid model under-goes a bifurcation, from a stable to an unstable limit cycle, as the value of the parameter accounting for a background regulatory input (associated to the receding morphogen gradient) decreases. This bifurcation allows the model to explain why somites can form in the absence of an external wavefront, reassesses the role of the receding morphogen gradient as a conductor for somitogenesis, and makes the model behavior robust to random fluctuations, as well as independent from specific initial conditions (the latter, are two of the weak points of the Meinhardt-PORD model). We argue that this findings provide convincing evidence that reaction-diffusion and positional information (receding morphogen gradient) mechanisms could work together in somitogenesis.