Impacts of statistical feedback on the flexibility-accuracy trade-off in biological systems

Biological systems possess the ability to adapt quickly and adequately to both environmental and internal changes. This vital ability cannot be explained in terms of conventional stochastic processes because such processes are characterized by a trade-off between flexibility and accuracy, that is, they either show short transition times (large Kramers escape rates) to broad steady-state distributions or long transition times to sharply peaked distributions. To develop a stochastic theory for systems exhibiting both flexibility and accuracy, we study systems under the impact of white noise multiplied with an accordant statistical measure, here the probability density. This results in negative feedback and circular causality: the more probable a stable state the less it will be affected by noise and, conversely, the less a stable state is affected by noise the more probable it is. Using nonlinear Fokker-Planck equations, steady states are computed via transformations of solutions of the corresponding linear Fokker-Planck equations. Transients reveal rapidly evolving and sharply peaked probability densities and thus mimic systems characterized by both flexibility and accuracy.