Memory functions reveal structural properties of gene regulatory networks

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, but the size and complexity of these models 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. These memory functions provide probes of near-steady state dynamics, revealing information not easily accessible otherwise. We illustrate the method on a simple cross-repressive transcriptional motif to show that memory functions not only simplify the analysis of the subnetwork but also have a natural interpretation. 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. The memory functions reveal the function of specific links within the neural tube network and identify features of the regulatory structure that specifically 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.Author summary: Gene regulatory networks are essential for cell fate specification and function. But the recursive links that comprise these networks often make determining their properties and behaviour complicated. Computational models of these networks can also be difficult to decipher. To reduce the complexity of such models we employ a Zwanzig-Mori projection approach. This allows a system of ordinary differential equations, representing a network, to be reduced to an arbitrary subnetwork consisting of part of the initial network, with the rest of the network (bulk) captured by memory functions. These memory functions account for the bulk by describing signals that return to the subnetwork after some time, having passed through the bulk. We show how this approach can be used to simplify analysis and to probe the behaviour of a gene regulatory network. Applying the method to a transcriptional network in the vertebrate neural tube reveals previously unappreciated properties of the network. By taking advantage of the structure of the memory functions we identify interactions within the network that are unnecessary for sustaining correct patterning. Upon further investigation we find that these interactions are important for conferring robustness to variation in initial conditions. Taken together we demonstrate the validity and applicability of the Zwanzig-Mori projection approach to gene regulatory networks.