Scaling and Regeneration of Self-Organized Patterns

Biological patterns generated during development and regeneration often scale with organism size. Some organisms, e.g., flatworms, can regenerate a rescaled body plan from tissue fragments of varying sizes. Inspired by these examples, we introduce a generalization of Turing patterns that is self-organized and self-scaling. A feedback loop involving diffusing expander molecules regulates the reaction rates of a Turing system, thereby adjusting pattern length scales proportional to system size. Our model captures essential features of body plan regeneration in flatworms as observed in experiments.

Figure 1

Classical Turing patterns show more periodic repeats in larger systems as a result of fixed intrinsic length scales (a), instead of being a scaled-up version of the patterns in small systems (b).

Figure 2

Classical Turing patterning implies that in larger systems higher-order patterns form. (a) Steady-state patterns of Eq. (1) are classified by two pattern numbers $(m,n)$: $m$ is the total number of contiguous source regions, while $n$ is the number of source regions touching the system boundaries. Typical profiles of the activator concentration $A_{(m,n)}(x)$ for the $(m,n)$-pattern are shown in red. Size ranges are shown, where the $(m,n)$-pattern is linearly stable (black), or exists, but is not stable (gray). In the blue region, the (1,1)-pattern is the only stable pattern. (b) Basins of attraction: final pattern type at steady state as a function of system size on the horizontal axis and initial conditions on the vertical axis. Initial conditions linearly interpolate between the (1,1)- and (1,0)-pattern, i.e., $A(x,t=0)=(1-q)A_{(1,1)}(x)+q{A}_{(1,0)}(x)$, and analogously for $B(x,t=0)$. Parameters: $D_B/D_A=30$, ${\alpha }_B/{\alpha }_A=4$, ${\beta }_B/{\beta }_A=2$, $h\to \infty$ (a), $h=5$ (b).

Figure 3

Scalable pattern formation in a Turing system with expander feedback. (a) The Turing system and the expander mutually control their degradation rates, resulting in a stable feedback loop. (b) Scaling corresponds to morphogen profiles that collapse as a function of relative position $x/L$ (normalized by respective concentrations $A_0$, $B_0$, $E_0$ at $x=0$). (c) The feedback self-consistently adjusts the length scales ${\lambda }_A$ and ${\lambda }_B$ of the morphogen profiles and thus the source size $\ell$ with system size (symbols: numerical results; lines: analytical solution of Eqs. (1) and (4) at steady state for homogeneous expander concentration and $h\to \infty$). Here, ${\mathcal{E}}_0=({\alpha }_A/{\kappa }_A{)}^{1/2}$ and ${\lambda }_0=[D_A/({\mathcal{E}}_0{\kappa }_A){]}^{1/2}$ denote the characteristic concentration and length scales of the system. (d) Example trajectories, mimicking amputation experiments (labeled i, ii), and uniform, one-time injection of the expander (labeled iii, iv); all converge to the same stable fixed point, an appropriately scaled (1,1)-pattern. (e) Parameter regions for stable, self-scaling pattern formation (green), and regions of expander divergence (orange, purple). Parameters of panels (a)–(d) indicated by cross. (f)–(g) For adiabatically slow expander dynamics, the system relaxes along the nullclines of the Turing system $f_{(m,n)}$ (shown for $h\to \infty$, ${\lambda }_E\gg L$). As each nullcline intersects the steady-state condition of Eq. (6) twice, the system possesses two fixed points $(n,m{)}^{+}$ and $(n,m{)}^{-}$ for each pair $(n,m)$. In the blue region, the (1,1)-pattern is the only stable steady state of the Turing system, compare to Fig. 2, implying that all trajectories must converge to this fixed point. Parameters: $D_B/D_A=30$, $D_E/D_A=10$, ${\alpha }_B/{\alpha }_A=4$, ${\alpha }_E/{\alpha }_A=0.4$, ${\kappa }_B/{\kappa }_A=2$, ${\kappa }_E/{\kappa }_A=2$, $h=5$, $L/{\lambda }_0=10$, unless indicated otherwise.