Embryonic lateral inhibition as optical modes: An analytical framework for mesoscopic pattern formation

Cellular checkerboard patterns are observed at many stages of embryonic development. We study an analytically tractable model for lateral inhibition and show that the steady states are analogous to optical phonons at the $\mathrm{\Gamma }$ point, which have the wave number $k=0$. We study the cases of cells arranged in linear and hexagonal lattices. To determine how the final pattern is selected it is necessary to take into account the granularity of the pattern and, analogously to solid-state physics, to redefine the basis and lattice sites in terms of a periodic crystal. The sites and basis are determined by looking at the symmetries of inhibitory interactions between cells. The redefined basis for cells placed in a linear lattice is composed by two cells which are embedded in another linear lattice, while for cells placed in a hexagonal lattice the redefined basis consists of three cells embedded in another hexagonal lattice. The pattern in hexagonal lattices can be driven into three different states: two of those states are periodic checkerboards and a third in which both periodic states coexist. These observations provides new predictions for experiments.

Figure 1

Periodic fine-grained patterns. (a) Example from the auditory epithelium with a green fluorescent marker for actin adapted from Ref. [9], where $\lambda$ depicts the characteristic wavelength of the pattern. (b) Depiction of the notation used for Eq. (1). (c) Pattern in a hexagonal lattice, where at each lattice point (blue points) there is a basis structure with three cells, and (d) the possible basis structures for a periodic pattern.

Figure 2

Linear stability analysis of Eq. (2), where $k$ corresponds to the wave number and $\alpha$ is the mode growth rate.

Figure 3

Impulse response for a linear chain of cells. We set ${\mathrm{\Omega }}_n={\mathrm{\Omega }}_o{\delta }_{n0}$, where (a) corresponds to $\epsilon <0$ and (b) to $\epsilon >0$ for a finite chain of 20 cells. Response to random input ${\mathrm{\Omega }}_n$, where (c) shows the next-neighbor correlations $\langle u_n{u}_{n\pm 1}\rangle$ as a function of $\epsilon$, symbols correspond to numerical simulations, and solid curve was obtained analytically [Eq. (6)], and (d) examples of noisy anticorrelated patterns as a function of $\epsilon$.

Figure 4

Potential well for a linear chain of cells. (a) Redefinition of the crystal basis from a single-component basis $u$ to a two-component basis $u,v$ (diatomic). (b) An example of a potential $\mathrm{\Phi }(v,w)$ with $\epsilon <1/2$. The potential has minima at $(+,-)$ (positive $v_s$ and negative $w_s$) and $(-,+)$ (negative $v_s$ and positive $w_s$). (c) Time evolution of the continuum approximation of $v(x,t)$ and $w(x,t)$ defined by Eqs. (7) and (8) and (d) time evolution of the chain of $u_n$ given by Eq. (1).

Figure 5

Increase in cell anticorrelation with increasing coupling strength. Stochastic patterns given by $f_n=2\theta \left(u_n\right)-1$ generated with noisy input ${\mathrm{\Omega }}_n$, where black corresponds to $f_n=-1$ and red to $f_n=1$. (a) $\epsilon =0$, (b) $\epsilon =-0.1$, (c) $\epsilon =-0.2$, and (d) $\epsilon =-0.3333$.

Figure 6

Redefining the basis and metastable and stable points of $\mathrm{\Phi }(v,w,z)$. (a) Redefinition of the crystal basis from a single-component $\left(u\right)$ to a three-component basis $(v,w,z)$ in hexagonal lattices. [(b) and (c)] Stable states (black symbols) and metastable states (white symbols) of the potential $\mathrm{\Phi }(v,w,z)$ given by Eq. (18) projected onto the $v-z$ axis for the parameters $\gamma =0.1$ and $\epsilon =-2$.

Figure 7

Conceptual double-well potential and corresponding pattern selection. (a) Double-well potential at ${\mathrm{\Omega }}_o=0$, where the minima represents the stability of the basis $B_{u,d,d}$ and $B_{d,u,u}$, and [(b) and (c)] simulations initiated with uniform initial conditions $u_{n,n^{\prime }}=0$ except for three cells set to (b) $B_{u,d,d}$ and (c) $B_{d,u,u}$. In this regime the final pattern supports a periodic repetition of either basis. For ${\mathrm{\Omega }}_o>{\mathrm{\Omega }}_c$, (d) the single-well potential shows the basis $B_{d,u,u}$ is stable. Simulations initiated with (e) $B_{u,d,d}$ and (f) $B_{d,u,u}$, show that the final pattern supports only the basis $B_{d,u,u}$. For ${\mathrm{\Omega }}_o<{\mathrm{\Omega }}_c$, (g) the single-well potential shows the basis $B_{u,d,d}$ is stable. Simulations initiated with (h) $B_{u,d,d}$ and (i) $B_{d,u,u}$ show that the final pattern supports only the basis $B_{u,d,d}$.

Figure 8

Coexistence of both base at ${\mathrm{\Omega }}_o=0$. A simulation is initiated with uniform initial conditions and three cells forming the basis $B_{u,d,d}$ and three cells forming $B_{d,u,u}$. At the end $(t=4)$ the stable pattern contains clusters of both base which meet at interfaces.

Figure 9

Hysteresis curve between patterns. The system is initiated with a pattern filled with the basis $B_{u,d,d}$ and the parameter ${\mathrm{\Omega }}_o$ is swept back and forth between $-60$ and 60. The lower branch corresponds to a pattern filled with the basis $B_{u,d,d}$ and the upper branch to $B_{d,u,u}$.