Robustness of spatial patterns in buffered reaction-diffusion systems and its reciprocity with phase plasticity

The robustness of spatial patterns against perturbations is an indispensable property of developmental processes for organisms, which need to adapt to changing environments. Although specific mechanisms for this robustness have been extensively investigated, little is known about a general mechanism for achieving robustness in reaction-diffusion systems. Here, we propose a buffered reaction-diffusion system, in which active states of chemicals mediated by buffer molecules contribute to reactions, and demonstrate that robustness of the pattern wavelength is achieved by the dynamics of the buffer molecule. This robustness is analytically explained as a result of the scaling properties of the buffered system, which also lead to a reciprocal relationship between the wavelength's robustness and the plasticity of the spatial phase upon external perturbations. Finally, we explore the relevance of this reciprocity to biological systems.

Figure 1

Schematic of a buffered reaction-diffusion system. $x_i^i$ and $x_i^a$ are inactive and active $i\mathrm{th}$ molecules, respectively. $x_i^i$ turns into $x_i^a$ by binding with a buffer molecule $w$. Reactions occur only among active molecules. Solid, flat-headed, and dotted arrows represent activation, inhibition, and catalysis, respectively.

Figure 2

Wavelength robustness and spatial-phase plasticity of the buffered Brusselator with $A=2, B=4, D_u=5$, and $D_v=30$. (a) Environmental dependency of the wavelength under the Dirichlet boundary condition. Different line colors indicate spatial patterns at different $\beta$, while $E_{g1}$ is set at 0 (up) and 1.0 (down). Both $E_f$ and $E_{g1}$ are fixed at 1.0. ESFs are given as the Arrhenius function as $\mathcal{F}\left(\beta \right)=e^{E_f}exp(-\beta E_f), {\mathcal{G}}_1\left(\beta \right)=3.0e^{E_{g1}}exp(-\beta E_{g1})$, and ${\mathcal{G}}_2\left(\beta \right)=e^{E_{g2}}exp(-\beta E_{g2})$. (b) Spatial-phase shift after a stimulus, given as a transient change in $\beta$ from the spatially uniform $\beta ={\beta }_1$ to a gradually distributed $\beta$ between ${\beta }_1=1.0$ and ${\beta }_2=0.8$, shown as the gray dotted line, with a duration time of 100.0. The red line is the spatial pattern before the stimulus, and the others indicate spatial patterns after the stimulus, for varied $E_{g1}$. A periodic boundary condition is adopted to compute the phase shift of the pattern. (c) Spatial-phase response curve (SPRC) of the buffered Brusselator against the stimulus. The horizontal axis shows the phase differences between the spatial pattern generated by the buffered Brusselator and given by the $\beta$ gradient, as defined by the difference between the peaks of the pattern. The applied $\beta$ gradient is normalized by the pattern wavelength. Different colors indicate SPRCs for different $E_{g1}$ values.

Figure 3

Reciprocity between wavelength robustness and spatial-phase plasticity of the buffered Brusselator. Red circles represent the difference in the wavelength at ${\beta }_1=1.0$ and at ${\beta }_2=0.8$ under the periodic boundary condition, normalized by the wavelength at ${\beta }_1$. Green squares represent the difference between the maximal and minimal values of the SPRC, as shown in Fig. 2.

Figure 4

Reciprocity between the robustness of period and phase plasticity in the temporal rhythm of a buffered Brusselator. $A$ is set to 1.0 and other parameters are the same as the spatial pattern. $\mathrm{\Delta }T/T$ is the difference in periods at ${\beta }_1=1.0$ and ${\beta }_2=0.8$ normalized by the period at ${\beta }_1$. The difference between the maximal and minimal values of the phase response curve is $\mathrm{\Delta }\varphi$ [3, 22].