Memory effects in gene regulatory networks reveal key properties of dynamical spatial patterning

Gene regulatory networks (GRNs) control cellular function and decision making during tissue development and homeostasis. Mathematical tools based on dynamical systems theory are often used to model these networks. The size and complexity of these models, however, mean that their behaviour is not always intuitive and the underlying mechanisms can be difficult to decipher. For this reason, methods that simplify and aid exploration of complex networks are necessary. To this end we develop a broadly applicable form of the Zwanzig-Mori projection. By first converting a thermodynamic state ensemble model of gene regulation into mass action reactions we derive a general method that produces a set of time evolution equations for a subset of components of a network. The influence of the rest of the network, the bulk, is captured by memory functions that describe how the subnetwork reacts to its own past state via components in the bulk. We demonstrate that the method captures the full dynamics of a simple cross-repressive transcriptional motif while simplifying analysis of the subnetwork and providing a natural interpretation of the memory functions. We then apply the approach to a GRN from the vertebrate neural tube, a well characterised developmental transcriptional network composed of four interacting transcription factors. This reveals the function of specific links within the neural tube network and identifies features of the regulatory structure that increase the robustness of the network to initial conditions. Taken together, the study provides evidence that Zwanzig-Mori projections offer powerful and effective tools for simplifying and exploring the behaviour of GRNs.