Abstract
Mathematical approaches from dynamical systems theory are used in a range of fields. This includes biology where they are used to describe processes such as protein-protein interaction and gene regulatory networks. As such networks increase in size and complexity, detailed dynamical models become cumbersome, making them difficult to explore and decipher. This necessitates the application of simplifying and coarse graining techniques to derive explanatory insight. Here we demonstrate that Zwanzig-Mori projection methods can be used to arbitrarily reduce the dimensionality of dynamical networks while retaining their dynamical properties. We show that a systematic expansion around the quasi-steady-state approximation allows an explicit solution for memory functions without prior knowledge of the dynamics. The approach not only preserves the same steady states but also replicates the transients of the original system. The method correctly predicts the dynamics of multistable systems as well as networks producing sustained and damped oscillations. Applying the approach to a gene regulatory network from the vertebrate neural tube, a well-characterized developmental transcriptional network, identifies features of the regulatory network responsible for its characteristic transient behavior. Taken together, our analysis shows that this method is broadly applicable to multistable dynamical systems and offers a powerful and efficient approach for understanding their behavior.
Highlights
In complex dynamical systems, comprising multiple interacting components, it can be difficult to identify causal mechanisms and to dissect the function of parts of a system
Another option is to expand the dynamical equations around a fixed point and derive memory functions from this approximation [21,22], but for multistable or oscillatory systems the memory functions obtained in this way do not capture all the qualitative behaviors
We develop a method, based on the formalism of Ref. [11], that allows the calculation of memory functions for generic dynamical systems without prior knowledge of the dynamics
Summary
In complex dynamical systems, comprising multiple interacting components, it can be difficult to identify causal mechanisms and to dissect the function of parts of a system. These functions describe how the current subnetwork state feeds back, through the activity of molecular species in the bulk, to affect the subnetwork at a later time This approach, its nonlinear version, was originally developed for the dynamics of physical systems [10] but later generalized by Chorin et al [11,12], with related uses in optimal coarse graining [13]. An alternative is to map the nonlinear system to a physical system consistent with the original; while this can be effective it does not necessarily simplify the problem [19,20] Another option is to expand the dynamical equations around a fixed point and derive memory functions from this approximation [21,22], but for multistable or oscillatory systems the memory functions obtained in this way do not capture all the qualitative behaviors. The analysis introduces a broadly applicable method for the investigation and analysis of complex dynamical systems
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