Stochastic Modeling of Protein Field with a Delayed Feedback

Abstract

Abstract Protein fields synthesized by genes play a principal role in the functioning of living systems. The processes of gene regulation determine the properties of these fields. Since the number of nucleotides usually is not large, a deterministic description of the field dynamics is insufficient. In this work, we consider a special kind of protein field, the dynamic behavior of which is described by the non-Markov process. Generally, the dynamics of complex organic compounds is time-dependent and spatially extended, and it may depend on all the previous evolution of the system. We consider a time-delayed repressilator as a model system. We study this system numerically using a modified Gillespie algorithm. New dynamic phenomena, which are visible only within a stochastic description, are reported. We show that synchronization in a gene expression occurs much faster due to the non-linear interaction of noise and delay. It results in almost regular oscillations even below the neutral curve derived within the deterministic analysis. We apply a hybrid approach to study the spatial dynamics of the repressilator proteins. This approach includes a deterministic calculation of the diffusion fluxes between cells and the stochastic simulation of gene regulation processes. We found that the combined action of time-delay, noise, and spatial signaling can lead to pattern formation even when the deterministic description predicts the absolute stability of the system.