Parameter Invariance Analysis of Moment Equations Using Dulmage-Mendelsohn Decomposition

Living organisms maintain stable functioning amid environmental fluctuations through homeostasis, a property that preserves a system’s behavior despite changes in environmental conditions. To elucidate homeostasis in stochastic biochemical reactions, theoretical tools for assessing population-level invariance under parameter perturbations are crucial. In this paper, we propose a systematic method for identifying the stationary moments that remain invariant under parameter perturbations by leveraging the structural properties of the stationary moment equations. A key step in this development is addressing the underdetermined nature of moment equations, which has traditionally made it difficult to characterize how stationary moments depend on system parameters. To overcome this, we utilize the Dulmage-Mendelsohn (DM) decomposition of the coefficient matrix to extract welldetermined subequations and reveal their hierarchical structure. Leveraging this structure, we identify stationary moments whose partial derivatives with respect to parameters are structurally zero, facilitating the exploration of fundamental constraints that govern homeostatic behavior in stochastic biochemical systems.