A stochastic framework for the design of transient and steady state behavior of biochemical reaction networks

Stochasticity plays an essential role in the dynamics of biochemical systems. Stochastic behaviors of bimodality, excitability, and fluctuations are present in biochemical reaction networks at low molecular numbers. These stochastic dynamics can be captured by modeling the system using a forward Kolmogorov equation known in the biochemical literature as the chemical master equation. The chemical master equation describes the time evolution of probability distributions of molecule species in the system. We develop a stochastic framework for the design of these time evolving probability distributions. Our design specifications include their uni-/multi-modality, the locations of their modes, and their rate of convergence to the stationary distribution. We formulate these specifications as constraints in an optimization program that determines the desired reaction rate values. We apply our design framework to examples of biochemical reaction networks to illustrate its strengths and limitations.