  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.
