Nonlinear memory functions for subnetwork dynamics in gene regulatory networks

Gene regulatory networks (GRNs) are complex biological systems that can generate a multitude of dynamical behaviours such as temporal oscillations, transients of expression or multistability. Such networks can be modelled using classical methods for dynamical systems, but the models often become so complex that it can be difficult to understand the qualitative properties of the system. To address this issue we study how the dynamics can be projected onto that of biologically relevant subnetworks. The interactions with the remainder of the network, the bulk, then lead to memory effects in the subnetwork dynamics. Previous approaches to determining the corresponding memory functions using Zwanzig-Mori projections have provided new insights into GRN, e.g. on the role of individual interactions between subnetwork and bulk [1], but they were restricted to studying the dynamics close to a specific steady state and thus unable to capture multistability effects. In the present work we utilise nonlinear projection methods to calculate full nonlinear memory functions that can be used across even in GRNs with multiple steady states. Our method is based on a systematic expansion around the limit where the bulk network is always in quasi-steady state with the subnetwork and is applicable to a broad class of dynamical systems. We show using a number of examples that with the memory structure we calculate, the projected subnetwork dynamics gives an accurate account of the full system dynamics, capturing key qualitative behaviours like multistability and sustained oscillations. We also consider the interpretation of the nonlinear memory functions, allowing us to gain novel insights into the functioning of GRNs that would be difficult to obtain using e.g. numerical simulation. [1] Herrera-Delgado, E., Perez-Carrasco, R., Briscoe, J., & Sollich, P. (2018). Memory functions reveal structural properties of gene regulatory networks. PLoS Computational Biology, 14(2), 1–25. http://doi.org/10.1371/journal.pcbi.1006003