Spatiotemporal pattern in somitogenesis: A non-Turing scenario with wave propagation

Living organisms maintain their lives under far-from-equilibrium conditions by creating a rich variety of spatiotemporal structures in a self-organized manner, such as temporal rhythms, switching phenomena, and development of the body. In this paper, we focus on the dynamical process of morphogens in somitogenesis in mice where propagation of the gene expression level plays an essential role in creating the spatially periodic patterns of the vertebral columns. We present a simple discrete reaction-diffusion model which includes neighboring interaction through an activator, but not diffusion of an inhibitor. We can produce stationary periodic patterns by introducing the effect of spatial discreteness to the field. Based on the present model, we discuss the underlying physical principles that are independent of the details of biomolecular reactions. We also discuss the framework of spatial discreteness based on the reaction-diffusion model in relation to a cellular array, by comparison with an actual experimental observation.

Figure 1

Illustration of a reaction-diffusion system (activator-inhibitor type). Note that we exclude the diffusion of the inhibitor $v$, i.e., Turing instability can never be satisfied in our model.

Figure 2

Geometry of nullclines corresponding to (a) oscillatory state in a big $\gamma$ case and (b) bistable state when $\gamma$ is small enough.

Figure 3

Spatiotemporal diagrams of $u\left(t,x\right)$ illustrating the manner of wave propagation, depending on the geometrical distribution of $\gamma$. The grayscale from black to white indicates the level of $u$ from 0 to 1. (a) Parameter $\gamma$ is given as an invariant spatial gradient. A single pulse triggered at the left boundary propagates to the right, and finally generates a stationary band at a specific position. (b) If we consider the growth of the embryo (the posterior end of PSM grows from $x=0.55$ to $x=1$), the kinetic parameter $\gamma$ is a function of both space and time. In this case, a pulse train leads to a static, periodic structure. (c) Parameter $\gamma$ is set to decrease as a wave passes, while we ignore the growth of the embryo. The pulse train can also cause a static periodic structure. In contrast to (b), there is large diversity in the band width of the final pattern.

Figure 4

Snapshots at different time points. Left and right columns show the time evolutions of $u$ and $\gamma$, respectively. (a), (b), and (c) correspond to three cases as shown in Fig. 3.

Figure 5

Two length scales showing the differences between continuous space and a discontinuous cellular array. (a) Characteristic scale $l^{\ast }$ represents the diffusion length of activator $u$ in its lifetime. By using the wave velocity $c$, we can obtain the representation $l^{\ast }\sim \sqrt{D{\tau }_2}=c\sqrt{{\tau }_1{\tau }_2}$ which consists of observable quantities. (b) $\eta \sim \sqrt{D{\tau }_1}$ is the edge width of the traveling wave front in the continuum. By comparing this length scale with the distance $\Delta x$ between two cells, we can evaluate the discreteness of the field.