Variance bounds for a class of biochemical birth/death like processes via a discrete expansion and spectral properties of the Master equation

We consider a class of birth/death like process corresponding to coupled biochemical reactions and consider the problem of quantifying the variance of the molecular species in terms of the rates of the reactions. In particular, we address this problem in a configuration where a species is formed with a rate that depends nonlinearly on another spontaneously formed species. By making use of an appropriately formulated expansion based on the Newton series, in conjunction with spectral properties of the master equation, we derive an analytical expression that provides a hard bound for the variance. We show that this bound is exact when the propensities are linear, with numerical simulations demonstrating that this bound is also very close to the actual variance. An analytical expression for the covariance of the species is also derived.