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The precision of reaction-diffusion models for mesoscopic physical systems islimited by fluctuations. To account for this uncertainty, Van Kampen derived astochastic Langevin-like reaction-diffusion equation that incorporatesspatio-temporal white noise. The resulting solutions, however, have infinitestandard deviation. Ad hoc modifications that address this issue by introducingmicroscopic correlations are inconvenient in many physical contexts of wideinterest. We instead estimate the magnitude of fluctuations by coarse-grainingsolutions of the Van Kampen equation at a relevant mesoscopic scale. Theensuing theory yields fluctuations of finite magnitude. Our approach isdemonstrated for a specific biophysical model--the encoding of positionalinformation. We discuss the properties of the fluctuations and the role playedby the macroscopic parameters of the underlying reaction-diffusion model. Theanalysis and numerical methods developed here can be applied in physicalproblems to predict the magnitude of fluctuations. This general approach canalso be extended to other classes of dynamical systems that are described bypartial differential equations.

Fluctuation theory in space and time: white noise in reaction-diffusion models of morphogenesis