Morphogen fluctuations in space and time during development
The precision of reaction-diffusion mechanisms for the specification of positional information during development is limited by fluctuations. A stochastic reaction-diffusion equation that incorporates spatio-temporal white noise has an infinite standard deviation, and modifications that introduce microscopic correlations are inconvenient in physical contexts of wide interest. We instead estimate the magnitude of fluctuations by coarse-graining solutions of the Van Kampen equation at a relevant mesoscopic scale. The ensuing theory yields fluctuations of finite magnitude. Our approach is demonstrated for a specific biophysical model: the encoding of positional information by bicoid signaling in Drosophila. The analysis and numerical methods developed here can be applied in physical problems to predict the magnitude of fluctuations. This general approach can also be extended to other classes of dynamical systems that are described by partial differential equations.
A morphogen produced at the left boundary at concentration a0 diffuses along an embryo, which extends between positions 0 and L. The morphogen’s steady-state concentration a(x), which is set by a degradation reaction, decreases monotonically toward the impenetrable right boundary. The instantaneous concentration of the morphogen a(t,x), however, is subject to spontaneous fluctuations; this poses a problem for cells whose fates are specified by the values.
