  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 phenomena 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
unexplored. Here, we analyze this relation with a paradigmatic stochastic
reaction-diffusion model, 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.
