Analysis of Dynamics and Stability of Hybrid System Models of Gene Regulatory Networks

We present hybrid system based gene regulatory network models for lambda, HK022 and Mu bacteriophages and analysis of dynamics and possible stable behaviours of the modelled networks. Lambda phage model LPH2 is the result of further development of an earlier LPH1 model taking into account more recent biological assumptions about the underlying biological gene regulatory mechanism. HK022 and Mu phage models are new. All three models provide accurate representations of lytic and lysogenic behavioural cycles, and, importantly, allow to conclude that lysis and lysogeny are the only stable behaviours that can occur in the modelled networks. Along with these models we describe also some new analysis techniques for hybrid system model state spaces. The models also allow to derive switching conditions that irrevocably lead to one of these two stable behaviours (these are consistent with proposed biological models) and also constraints on binding site affinities that are required for biologically feasible lysis and lysogeny processes. One of the derived constraints in LPH2 model is required for lambda lysis cycle feasibility and places conditions on cro protein binding site affinities. This is consistent with the constraint obtained previously for LPH1 model, although parts of state spaces that describe lysis in these models are different. Another constraint on protein cI binding affinities that is required for biologically feasible lysogeny cycle is new (and likely has been overlooked earlier). At the same time dynamics of HK022 model (which, notably, lacks N antitermination protein) turns out to be independent of both these constraints, although the involved genes and binding their sites are very similar. The used HSM system framework also allows to reproduce biologically different lysis-lysogeny switching mechanisms that are used by Mu phage. In general the results show that HSM hybrid system framework can be successfully applied to modelling small gene regulatory networks (with up to ∼ 20 genes) and for comprehensive analysis of model state space stability regions.