Global density equations for a population of actively switching particles

Abstract

There are many processes in cell biology that can be modelled in terms of an actively switching particle. The continuous degrees of freedom of the particle evolve according to a hybrid stochastic differential equation (hSDE) whose drift term depends on a discrete internal or environmental state that switches according to a continuous time Markov chain. Examples include Brownian motion in a randomly switching environment, membrane voltage fluctuations in neurons, protein synthesis in gene networks, bacterial run-and-tumble motion, and motor-driven intracellular transport. In this paper we derive generalized Dean-Kawasaki (DK) equations for a population of actively switching particles, either independently switching or subject to a common randomly switching environment. In the case of a random environment, we show that the global particle density evolves according to a hybrid DK equation. Averaging with respect to the Gaussian noise processes in the absence of particle interactions yields a hybrid partial differential equation for the one-particle density. We use this to show how a randomly switching environment induces statistical correlations between the particles. We also discuss methods for handling the moment closure problem for interacting particles, including dynamical density functional theory and mean field theory. We then develop the analogous constructions for independently switching particles. In order to derive a DK equation, we introduce a discrete set of global densities that are indexed by the single-particle internal states, and take expectations with respect to the switching process. However, the resulting DK equation is no longer closed when particle interactions are included. We conclude by deriving Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) path integrals for the global density equations in the absence of interactions, and relate this to recent field theoretic studies of Brownian gases and run-and-tumble particles (RTPs).