Comparing discrete, continuous and hybrid modelling approaches of gene regulatory networks
Mathematical modelling of biological networks can help us understand the complex mechanisms that are behind cell proliferation, differentiation or other cellular processes. From these models, we are able to replicate and predict system behaviour that can help in the design of experiments in the systems biology context. Multiple formalisms capture the evolution or dynamics of a system as implied by the network. Ordinary differential equation (ODE) models provide a precise representation of the system, where the concentrations of network components evolve based on chemical kinetics, e.g. mass action kinetics. The kinetic parameters required to generate the dynamics accurately, however, are often lacking, which has led to the development of more qualitative or discrete modelling methods. Discrete formalisms, like the well known Thomas formalism, provide a very coarse representation of the systems dynamics, whilst still highlighting fundamental features of the network structure. When modelling a given system, it could occur that the different approaches yield contrary dynamics. From a modelling perspective, this is highly impractical as we expect the system to behave uniquely irrespective of the modelling approach used. By mathematically relating different formalisms, we can analyse the dynamics of the formalisms and determine conditions for which the dynamics of each formalism are common or contrary between formalisms. Hybrid modelling approaches, that is formalisms that combine discrete and continuous methods, help in relating the purely discrete Thomas formalism with the purely continuous ODE formalism. Approximating the ODEs, we obtain piecewise affine differential equations (PADEs), which have well defined dynamics that can be discretised to reflect features of the Thomas formalism. Incorporating the hybrid formalism of PADEs into our analysis, we can break up the otherwise rough transformation between ODE and Thomas formalisms in order to specify the conditions for contrary dynamics to occur between formalisms. Our main result compares the qualitative approach of PADEs with the Thomas formalism. In particular, we show that even though the qualitative parameter information of the PADEs is inherent in the Thomas formalism and vice versa, the dynamics in both models still yield contrary dynamics. However, with the well-defined correspondences of the transition systems implied by the two approaches, we can provide proofs of paths and terminal strongly connected components that are common between both formalisms. With our analysis, we bridge the gap between discrete and continuous modelling methods. More specifically, we establish the dynamics that is common regardless of the choice of formalism and the dynamics that can be seen as artefacts of the formalism. From this analysis, therefore, we achieve a more rigorous modelling framework that allows us to model and predict biological systems with greater accuracy.