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Mathematical approaches from dynamical systems theory are used in a range offields. This includes biology where they are used to describe processes such asprotein-protein interaction and gene regulatory networks. As such networksincrease in size and complexity, detailed dynamical models become cumbersome,making them difficult to explore and decipher. This necessitates theapplication of simplifying and coarse graining techniques in order to deriveexplanatory insight. Here we demonstrate that Zwanzig-Mori projection methodscan be used to arbitrarily reduce the dimensionality of dynamical networkswhile retaining their dynamical properties. We show that a systematic expansionaround the quasi-steady state approximation allows an explicit solution formemory functions without prior knowledge of the dynamics. The approach not onlypreserves the same steady states but also replicates the transients of theoriginal system. The method also correctly predicts the dynamics of multistablesystems as well as networks producing sustained and damped oscillations.Applying the approach to a gene regulatory network from the vertebrate neuraltube, a well characterised developmental transcriptional network, identifiesfeatures of the regulatory network responsible dfor its characteristictransient behaviour. Taken together, our analysis shows that this method isbroadly applicable to multistable dynamical systems and offers a powerful andefficient approach for understanding their behaviour.

Tractable nonlinear memory functions as a tool to capture and explain dynamical behaviours