Noise-induced Mixing and Multimodality in Reaction Networks

We analyze a class of chemical reaction networks under mass-action kinetics
and involving multiple time-scales, whose deterministic and stochastic models
display qualitative differences. The networks are inspired by gene-regulatory
networks, and consist of a slow-subnetwork, describing conversions among the
different gene states, and fast-subnetworks, describing biochemical
interactions involving the gene products. We show that the long-term dynamics
of such networks can consist of a unique attractor at the deterministic level
(unistability), while the long-term probability distribution at the stochastic
level may display multiple maxima (multimodality). The dynamical differences
stem from a novel phenomenon we call noise-induced mixing, whereby the
probability distribution of the gene products is a linear combination of the
probability distributions of the fast-subnetworks which are `mixed' by the
slow-subnetworks. The results are applied in the context of systems biology,
where noise-induced mixing is shown to play a biochemically important role,
producing phenomena such as stochastic multimodality and oscillations.