Qualitative Control Strategies for Synchronization of Bistable Gene Regulatory Networks

Abstract

In this paper we investigate the emergent dynamics in a network of N coupled cells, each expressing a similar genetic bistable switch. The bistable switch is modeled as a piecewise affine system and the cells are diffusively coupled. We show that both the coupling topology and the strength of the diffusion parameter may introduce new steady state patterns in the network. We study the synchronization properties of the coupled network and, using a control set of only three possible values (u min , u max , or 1), propose different control strategies which stabilize the system into a chosen synchronization pattern, both in the weak and strong coupling regimes. The results are illustrated by several numerical examples.

Weak synchronization and convergence in coupled genetic regulatory networks: Applications to damped oscillators and multistable circuits

“…This setting is slightly different from the case (X j,q , X j,k ) = c j (X j,q − X j,k ), which appears more naturally when studying diffusion processes. 12,15,22,23 Indeed, in our case, two coupled cells having a given protein concentration less than 𝜃 cannot cooperate in order to increase concentrations, so that they evolve independently. In the same way, two coupled cells having a given protein concentration larger than 𝜃 cannot cooperate.…”

“…Approximate Hill's functions by their limit value when , we have In other words, the diffusion mechanism activates with discrete values, and the cells decide to exchange only when one of the two cells has a large enough concentration of a given protein, while the other one has a low concentration of it. This setting is slightly different from the case , which appears more naturally when studying diffusion processes 12,15,22,23 . Indeed, in our case, two coupled cells having a given protein concentration less than cannot cooperate in order to increase concentrations, so that they evolve independently.…”

“…In our recent work, 12 we studied the diffusive coupling of bistable genetic regulatory networks and showed that the system may exhibit emergent behaviors leading to diverse synchronization patterns, similar to those described in Reference 13. In our recent work, 12 as well as in References 14,15, coupled genetic networks are studied using piecewise affine (PWA) models, which approximate continuous sigmoidal‐like activity functions by step functions and facilitate an algorithmic analysis of the dynamics. This is an interesting framework since various tools are available for analysis of PWA systems 16 and a practical method for finding Lyapunov functions for PWA systems was also recently proposed in Reference 17.…”

“…For instance, a network of globally asymptotically stable subsystems connected by diffusive coupling may exhibit oscillatory behaviors 11 . In our recent work, 12 we studied the diffusive coupling of bistable genetic regulatory networks and showed that the system may exhibit emergent behaviors leading to diverse synchronization patterns, similar to those described in Reference 13. In our recent work, 12 as well as in References 14,15, coupled genetic networks are studied using piecewise affine (PWA) models, which approximate continuous sigmoidal‐like activity functions by step functions and facilitate an algorithmic analysis of the dynamics.…”

Time‐optimal control of piecewise affine bistable gene‐regulatory networks

“…This allows genetic toggle switches to act as biological memory units capable of storing 1 bit of information, by sustaining one of the two possible states through time, 14 which offers a biosynthetic alternative to the classical electronic flip-flop. Since its creation, understanding how to regulate bistable systems in a reliable manner (e.g., by suppressing undesirable oscillations 15 or achieving transitions between states [16][17][18] ) has become highly relevant for their vast implications in biotechnology and biocomputing.…”