Domain nucleation and confinement in agent-controlled bistable systems

We report a mechanism of pattern formation in growing bistable systems coupled indirectly. A modified Fujita et al. model is studied as an example of a reaction-diffusion system of nondiffusive activator and inhibitor molecules immersed in the medium of a fast diffusive agent. Here we show that, as the system grows, a new domain nucleates spontaneously in the area where the local level of the agent becomes critical. Newly nucleated domains are stable and the pattern formation is different from Turing's mechanism in monostable systems. Domains are spatially confined by the agent even if the activator and inhibitor molecules diffuse. With the spatial extension of the system, a larger domain may undergo a wave number instability, and the concentrations of active molecules within the neighboring elements of a domain can become sharply different. The mechanism reported in this work could be generic for pattern formation systems involving multistability, growth, and indirect coupling.

Figure 1

Nullclines of Eq. (5). Solid line, the nullcline of $X$. Dashed lines, nullclines of $Y$. At $H_S=0$, there is a single stable steady solution shown by the open circle. At $H_S=1$, bistable solutions are possible (filled circles). The parameters are $E=1,B=2,C=0.25,D=1,A_d=1,A_s=1.9,u_{\max }=10,n=20,H_{S0}=0$, and $S_Y=1$.

Figure 2

Domain nucleation in a growing system described by Eqs. (2) and (3). Here we assume that the number of cells is fixed but the distances between the cells are increasing at a constant rate. (a) Space-time dynamics of $H$. Higher values of $H$ are shown in red (white), and lower values in dark blue (black). (b) Space-time dynamics of $X_i$'s. Higher values (near $X_{02})$ are shown in red (white), and lower values (near $X_{01})$ in dark blue (black). The parameters are the same as in Fig. 1, except $D_H=100,\epsilon =0.01,S_Y=0.5,\Delta X_{02}=1.8$, and $L_{S_0}=12$.

Figure 3

Snapshots of $H$ and $X$ from Fig. 2 at different time moments. (a) Distributions of $H$. (b) Distributions of $X_i$'s.

Figure 4

Domain nucleation at different values of the size of initial domain. (a) The size of the newly nucleated domain $\Delta X_{02}^{\mathrm{new}}$ vs the size of the initial domain $\Delta X_{02}$. (b) The distance from the center to the new domain $L_{\mathrm{nuc}}$ vs the size of the initial domain $\Delta X_{02}$. The other parameters are the same as in Fig. 2. The size of the system is fixed at $L_S=64$.

Figure 5

Domain confinement. Left: Space-time plot of the $X_i$'s. Right: Snapshots of the $X_i$'s, $Y_i$'s, and $H$ at time $t=200$. The parameters are the same as in Fig. 2, except $D_X=0.1,D_Y=0.1$, and the system size is fixed at $S_L=12$.

Figure 6

A domain instability leading to the coexistence of large- and small-domain solutions. Left: Nondiffusive $X$ and $Y$. Right: $D_X=0.035$ and $D_Y=0.1$. The parameters are the same as in Fig. 4, except $S_Y=0.14$ and $B=2.5$.

Figure 7

Wiring diagrams of minimal models of SAM regulation. (a) A diagram of a minimal model of SAM regulation by Nikolaev et al. [13]. (b) A modified version of the diagram. By the intense lines for WUS, we express a self-enhancement mechanism of WUS. The dashed line shows activation of $Y$ by WUS. The last two features, the characteristics of activator-inhibitor interactions, are adopted from the Fujita et al. model [9].

Figure 8

Simulations of Eqs. (A1) and (A2). Left: $D_W=1$. Right: $D_W=0$. The other parameters are $h_Y=-0.1,h_W=h_C=-88.9379,d_h=0.4,T_{WH}=2500,T_{WC}=-2900,T_{HW}=300$, and $T_{CH}=250$.

Figure 9

A cusp bifurcation in Eq. (5) of the main text. The parameters are $A_s=1.8,A_d=1,B=2,C=0.25,D=1,E=0.1,u_{\max }=10$, and $H_S=0.8$.

Figure 10

The potential in Eq. (B2) at three different values of $A_0$. When $A_0<A_{0c}=2.05$, the front moves in the direction of the expansion of the $X_{02}$ solution. When $A_0>A_{0c}=2.05$, the front moves in the direction of the expansion of the $X_{01}$ solution. At $A_0=A_{0c}$, the front is stationary. The other parameters are the same as in Fig. 9.

Figure 11

Front velocity $v$ vs $A_0$. The dashed line is obtained via bifurcation analysis of Eq. (B3), as the heteroclinic connections of the steady-state solutions. The symbols are obtained from simulations of Eq. (B1).

Figure 12

The potential in Eq. (B2) in the case of global coupling. The parameters are the same as in Fig. 10. When $\kappa \overline{X}<A_{0c}$, the front moves in the direction of the expansion of $X_{02}$. Therefore, $\overline{X_{02}}$ increases with time, until the front reaches the point $\kappa \overline{X}=A_{0c}$, where the front is motionless.

Figure 13

Domain confinement in the global coupling model Eq. (B4). In the case for $\kappa \overline{X(x,t)}=\text{const}$, the front propagates with velocity $v_0$. However, as $\kappa \overline{X(x,t)}$ increases, the front velocity decreases and eventually becomes zero at $\kappa \overline{X(x,t)}=A_{0c}$. No-flux boundary conditions were used for the simulations of Eq. (B4) with the parameters in Fig. 12, except $\kappa =0.16$.

Figure 14

The solid lines show domain confinement in Eq. (B5) when $A_0$ is replaced by $S_Y\overline{X(x,t)}$ and $D_Y=0$. The dashed lines show simulations of Eq. (B6). The other parameters are the same as in Fig. 5 of the main text.

Figure 15

Linear stability spectra of the uniform states near saddle-node points. The parameters are, for $H_{c1},H_{S0}=0.442$ and $S_Y=0.055$; and for $H_{c2},H_{S0}=1$ and $S_Y=0.0455$. The other parameters are the same as in Fig. 6 of the main text and $D_X=0.1,D_Y=0.1$, and ${\kappa }^{\prime }=0.05$.

Figure 16

Projection of patterns on the $(Y,X)$ plane. Black symbols: Turing patterns near the saddle-node point $Y_{c1}$ in Fig. 1. The parameters are $E=0.3,D_H=0.36$, and $\delta x=0.05$. Blue symbols (open circles): patterns in the bistable region. The parameters are $E=0.1,D_H=100$, and $\delta x=0.5$. The other parameters are $B=2,C=0.25,D=1,A_d=1,A_s=1.8,u_{\max }=10,D_X=0.05,D_Y=0.1,H_{S0}=0,S_Y=0.25,N=256,\epsilon =1$, and $n=20$.

Figure 17

Space-time profiles of the stationary Turing patterns shown by black symbols in Fig. 16. The formation of the pattern does not involve the mechanism described in Fig. 2.

Figure 18

Space-time profiles of the stationary patterns shown by blue symbols (open circles) in Fig. 16. The formation of the pattern does involve the mechanism described in Fig. 2.

Figure 19

$X$ patterns in space dimension 2 for simulations with no-flux boundary conditions, from the same initial conditions. The parameters are $E=1,B=2.5,C=0.25,D=1,A_d=1,A_s=1.9,u_{\max }=10,D_X=0.3,D_Y=0.3,H_{S0}=0,S_Y=0.5,D_H=100$, and $n=20$. Left: $\epsilon =0$, Turing-like patterns. Right: $\epsilon =0.01$, irregular patterns.