Weakly coupled map lattice models for multicellular patterning and collective normalization of abnormal single-cell states

We present a weakly coupled map lattice model for patterning that explores the effects exerted by weakening the local dynamic rules on model biological and artificial networks composed of two-state building blocks (cells). To this end, we use two cellular automata models based on (i) a smooth majority rule (model I) and (ii) a set of rules similar to those of Conway's Game of Life (model II). The normal and abnormal cell states evolve according to local rules that are modulated by a parameter $\kappa$. This parameter quantifies the effective weakening of the prescribed rules due to the limited coupling of each cell to its neighborhood and can be experimentally controlled by appropriate external agents. The emergent spatiotemporal maps of single-cell states should be of significance for positional information processes as well as for intercellular communication in tumorigenesis, where the collective normalization of abnormal single-cell states by a predominantly normal neighborhood may be crucial.

Figure 1

The cell state $u_{t+1}^{i,j}\in [0,1]$ vs the neighborhood sum $s_t^{i,j}$ for the values of $\kappa$ indicated beside the curves. Panels are separated to better show the effect of varying $\kappa$ in Eq. (8). Note that values of $s_t^{i,j}>9$ are considered only to show the mathematical trends of Eq. (8).

Figure 2

Spatiotemporal evolution of the cell states $u_t^{i,j}$ taking values between 0 and 1 (right bar) for model I obtained by iterating Eq. (8) in a multicellular ensemble of $159\times 159=25281$ cells at different times $t$ for three $\kappa$ values. The initial ($t=0$) state with cells randomly distributed in the 0 and 1 states is the same for the three cases.

Figure 3

Bifurcation diagram calculated from the asymptotic behavior of Eq. (20). The stationary cell state $u_{\infty }$ obtained after $3\times {10}^4$ time steps is shown as a function of the parameter $\kappa$.

Figure 4

Fraction of initial conditions in the unit interval that are attracted as $t\to \infty$ to the stable state 0 ($x_0$ curve) and to state 1 ($x_1$ curve) as a function of $\kappa$ and to a third stable state, $u_{\infty ,2}$, found at intermediate $\kappa$ values ($1-x_0-x_1$ dashed curve).

Figure 5

Spatiotemporal evolution of the cell states $u_t^{i,j}$ taking values between 0 and 1 (right bar) for model II obtained by iterating Eq. (17) in a multicellular ensemble of $159\times 159=25281$ cells for six $\kappa$ values. The initial ($t=0$) state with cells randomly distributed in the 0 and 1 states is the same for all cases.

Figure 6

Spatiotemporal evolution of the cell states $u_t^{i,j}$ taking values between 0 and 1 (right bar) for model II obtained by iterating Eq. (17) in a multicellular ensemble of $159\times 159=25281$ cells for six $\kappa$ values. The initial ($t=0$) state is the same for all cases and consists of a random distribution of 0 and 1 values in a central square region of the lattice of $25\times 25$ size, the rest of the lattice being in state 0.

Figure 7

$M_t$ versus time $t$ obtained from Eqs. (17) and (21) for the same initial condition as in Fig. 6 and the values of $\kappa$ indicated on the curves. $M_t$ constitutes a measure of the ensemble abnormality.

Figure 8

Bifurcation diagram calculated from the asymptotic behavior of Eq. (22). Black curves correspond to stationary states $u_{\infty }$ obtained at long times. The red points (light in grayscale) indicate the period-doubling bifurcation cascades into chaos. Note the correspondence of this figure with the results in Figs. 5 and 6.

Figure 9

Value of the Lyapunov exponent vs $\kappa$ for two initial conditions $u_0$.