Simultation Models and Zeno Path Avoidance in a Class of Piecewise Linear Biochemical Process

Abstract

Hybrid automata represent a natural modelling abstraction for biochemical interactions. Within discrete states, the local dynamics is usually described by linear differential equations, and generically, the continuous variables are protein concentrations. Transitions between discrete states occur, with possible reset variables, when specific threshold values are reached, thus producing abrupt changes in the right hand side of the differential system. This paper firstly analyses a simple hybrid automaton modelling a repressor mechanism in molecular biology, which presents a hysterezis behaviour. Alternate modelling frameworks, limitations of the model and simulation challenges are discussed. Secondly, a second-order piecewise-linear model of a generic biochemical interaction process is abstracted to a Zeno hybrid automaton and a Zeno path avoidance scheme, based on fixed-step time discretization is proposed.