Precision and dissipation of a stochastic Turing pattern

Spontaneous pattern formation is a fundamental scientific problem that has received much attention since the seminal theoretical work of Turing on reaction-diffusion systems. In molecular biophysics, this phenomenon often takes place under the influence of large fluctuations. It is then natural to inquire about the precision of such pattern. In particular, spontaneous pattern formation is a nonequilibrium phenomenon, and the relation between the precision of a pattern and the thermodynamic cost associated with it remains largely unexplored. Here, we analyze this relation with a paradigmatic stochastic reaction-diffusion model, i.e., the Brusselator in one spatial dimension. We find that the precision of the pattern is maximized for an intermediate thermodynamic cost, i.e., increasing the thermodynamic cost beyond this value makes the pattern less precise. Even though fluctuations get less pronounced with an increase in thermodynamic cost, we argue that larger fluctuations can also have a positive effect on the precision of the pattern.

Figure 1

Spatial patterns. (a) Homogeneous profile $\mathrm{\Delta }\mu =10.5<\mathrm{\Delta }{\mu }_c$. (b) Formation of pattern for $\mathrm{\Delta }\mu =11.8>\mathrm{\Delta }{\mu }_c$.

Figure 2

Average densities as functions of the thermodynamic force $\mathrm{\Delta }\mu$. The values for the homogeneous fixed point solution in Eq. (7) are represented by the full magenta line.

Figure 3

Relative standard deviations of the densities ${\mathrm{\Sigma }}_X$ and ${\mathrm{\Sigma }}_Y$ as functions of $\mathrm{\Delta }\mu$. The crosses represent the values obtained from numerical simulations and the line is a guide to the eye.

Figure 4

Average rate of entropy production $\sigma$ as a function of $\mathrm{\Delta }\mu$. The crosses represent the values obtained from numerical simulations and the line is a guide to the eye.

Figure 5

Spatial profile of the stochastic rate of entropy production ${\sigma }_i$ for $\mathrm{\Delta }\mu =11.8$.

Figure 6

Illustration of the variable $w$, which is the distance between two peaks. For a given profile, the variable $w$ is the sum of the distance between neighboring peaks divided by the total number of peaks. These distances are illustrated as $w_1, w_2$, and $w_3$ in the figure.

Figure 7

Relation between precision of a pattern and thermodynamic cost. The relative standard deviation ${\mathrm{\Sigma }}_w$ as a function $\mathrm{\Delta }\mu$. The lines are guides to the eye.

Figure 8

Spatial correlation $C_i$ for (a) $\mathrm{\Delta }\mu =11$, (b) $\mathrm{\Delta }\mu =11.3$, and (c) $\mathrm{\Delta }\mu =12$.