Topological-numerical analysis of global dynamics in the discrete-time two-gene Andrecut-Kauffman model

TL;DR

A topological-numerical analysis of global dynamics in a discrete-time two-gene Andrecut-Kauffman model that describes gene expression regulation through nonlinear interactions and introduces new symbols to convey the information provided by the Conley index in an easy to understand schematic way.

Abstract

We conduct a topological-numerical analysis of global dynamics in a discrete-time two-gene Andrecut-Kauffman model. This model describes gene expression regulation through nonlinear interactions. We use rigorous numerical methods to construct Morse decomposition of the system across a wide range of parameters. We obtain qualitative results by effectively computing the Conley indices of the constructed isolating neighborhoods that form the Morse decomposition. We introduce new symbols to convey the information provided by the Conley index in an easy to understand schematic way. We additionally conduct numerical simulations aimed at confirming the presence of complex dynamical phenomena, including multistability and the existence of chaotic attractors. The results demonstrate the usefulness of topological methods in understanding the global structure of dynamics in a gene regulatory model and highlight the richness of dynamics that can be observed in such a system when parameter values change.

Figures (23)

• Figure 1: Phase space diagram for the Andrecut--Kauffman model \ref{['eq:model']} with $\alpha_1=\alpha_2=25$, $\beta_1=0.18$, $\beta_2=0.42$, $\varepsilon=0.1$, and $n=3$. Isolating neighborhoods for all stable and unstable bounded invariant sets are shown in different colors. The sets in this figure have been drawn slightly thicker than their actual size for readability. An approximation of the attractor (consisting of two loops) obtained in numerical simulations is shown inside the neighborhood in the shape of two rings. Arrows represent the stable and unstable directions of each set.
• Figure 2: Example of a very complicated isolated invariant set with the Conley index of a hyperbolic fixed point. An isolating neighborhood is shown in green, together with the corresponding exit set drawn in a lighter tone. The trajectories are shown for a flow; consider the time-$1$ map for a discrete-time dynamical system.
• Figure 3: We represent different types of the Conley index by pictograms. The arrows indicate the stability type (source, sink, saddle). The "$-$" symbol indicates a flip in the unstable direction. The dots inside indicate the number of connected components of the isolating neighborhood.
• Figure 4: Continuation diagram for the Andrecut--Kauffman model \ref{['eq:model']} with $(\alpha_1,\alpha_2) \in [0,80] \times [0,80]$, $\beta=0.2$, $\varepsilon=0.8$, and $n=3$.
• Figure 5: The largest Lyapunov exponent calculated for $\alpha_1=\alpha_2$ at 1024 equally spaced points. The same randomly generated, uniformly distributed set of 128 initial conditions from $\left\{(x, y)\in[0.01, 50]\times[0.01, 50]\colon x\geq y\right\}$ was used for each parameter value. Plotted values are estimated from 1000 iterations of the system, following 5000 iterations of burn-in. Parameter values and colors correspond to those in Fig. \ref{['fig:contDiag']}.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3