Global analysis of regulatory network dynamics: equilibria and saddle-node bifurcations

TL;DR

It is demonstrated that adapting and combining classical techniques with recently developed combinatorial methods provides a richer picture of the global dynamics despite the high parameter dimension.

Abstract

In this paper we describe a combined combinatorial/numerical approach to studying equilibria and bifurcations in network models arising in Systems Biology. ODE models of the dynamics suffer from high dimensional parameters which presents a significant obstruction to studying the global dynamics via numerical methods. The main point of this paper is to demonstrate that adapting and combining classical techniques with recently developed combinatorial methods provides a richer picture of the global dynamics despite the high parameter dimension. Given a network topology describing state variables which regulate one another via monotone and bounded functions, we first use the {\em Dynamic Signatures Generated by Regulatory Networks} (DSGRN) software to obtain a combinatorial summary of the dynamics. This summary is coarse but global and we use this information as a first pass to identify ``interesting'' subsets of parameters in which to focus. We construct an associated ODE model with high parameter dimension using our {\em Network Dynamics Modeling and Analysis} (NDMA) Python library. We introduce algorithms for efficiently investigating the dynamics in these ODE models restricted to these parameter subsets. Finally, we perform a statistical validation of the method and several interesting dynamical applications including finding saddle-node bifurcations in a $54$ parameter model.

Figures (11)

• Figure 1: The network structure for the EMT model.
• Figure 2: (a) Toggle Switch. (b) Parameter graph for Toggle Switch. (c) Morse graph at node 5 of parameter graph. (d) Morse graph at nodes 1, 2, and 4 of parameter graph. Note that the element $a$ of these Morse graph can be directly identified with the element $a$ of the Morse graph at node 5. (e) Morse graph at nodes 6, 8, and 9 of parameter graph. Note that the element $b$ of these Morse graph can be direcctly identified with the element $b$ of the Morse graph at node 5. (f) Morse graph at nodes 3 and 7.
• Figure 3: Schematic description of the strategy of this paper. $R(4)$, $R(5)$, and $R(6)$ are regions of parameter space $\Xi$ identified by DSGRN for Toggle Switch. As indicated in Figure \ref{['fig:TS']} DSGRN identifies bistability for $R(5)$ and monostability for $R(4)$ and $R(6)$. We are interested in understanding the equilibrium structure for \ref{['eq:Toggle_ODEs']}. Set $d=d_{1,2}=d_{2,1}$. By duncan:gedeon:kokubu:mischaikow:oka and as indicated by the gray regions the equilibria identified by DSGRN persist for $d$ sufficiently large. For $d=1$\ref{['eq:Toggle_ODEs']} has a unique equilibrium. There therefore must be an interface between the monostability region and the bistability region. In this paper, we use paths of the form of ${\bf r}$ to justify that the main mechanism to pass from monostability to bistability is through a saddle node bifurcation and we investigate the "shape" of the boundary between stability regions.
• Figure 4: Coherency rate as a function of the Hill coefficient depending on the DSGRN prediction. In blue all parameters, in green only parameter predicted to be monostable, in orange parameters predicted to be bistable. As we can see, for the bistable region a choice of a large enough Hill coefficient is paramount for the detection of the bistability, while this does not hold true in the monostable region, where monostability is reliably detected independently of the Hill coefficient.
• Figure 5: Toggle Switch nullclines for a parameter in $R^*(5)$ with Hill exponent varying through a saddle-node bifurcation. As both Hill functions become steeper, the nullclines tend to be closer to one another until the saddle-node bifurcation, at which point the nullclines intersect tangentially. Further increasing $d$ increases the area bounded between the nullclines and the three equilibria appear to move further apart from each other.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Proposition 1
• proof
• Theorem 2
• Remark 2
• Definition 5