Stochastic modelling for control and interconnections of biological systems
Biochemical reactions often occur in small volumes within a cell, restricting molecule numbers to the hundreds or even tens. At this scale, reactions are inherently discrete and stochastic. Traditional models based on differential equations and mass-action kinetics fail to capture the behaviour of such systems. In contrast, stochastic models can suffer the curse of dimensionality, complicating system analysis and design in this low-copy number setting. The first part of this dissertation explores a novel decision-making mechanism in small cell compartments by introducing a novel self-regulating signalling motif: the Size-Regulated Switch (SRS). The SRS allows reliable switching at small system sizes and transitions to a stable behaviour as the size increases. Since many cellular compartments grow in size, the system size can act as a feedback signal to self-regulate the switching behaviour. These results are then generalised to different network topologies with two or three species, allowing both inhibition and excitation, finding that inhibitory connections robustly acquire size-dependent multimodality, while excitatory connections tend to suppress this behaviour. These results provide a potential solution to the contradictory findings surrounding the CaMKII/protein phosphatase-1 pathway, showing that bistability can be sensitive to the absolute quantity of reactants present. The second part of this dissertation develops a novel input/output framework for ap- proximating the stationary relationship between dependent species within a feedforward biochemical reaction network. These networks can consist of many species, including nonlinear interactions and combining series and parallel interconnections, making simu- lation methods computationally expensive and approximation methods inaccurate in the low copy number regime. The Conditional Poisson Approximation CPA method allows the approximation of the stationary behaviour of these stochastic systems, accounting for the inherent discreteness and stochasticity of the molecular species, as well as the nonlinear interactions such as Hill-type functions. Overall, the CPA method provides a scalable and accurate approximation of nonlinear feedforward biochemical networks, including both series and parallel interconnection. Furthermore, its potential extension to feedback systems is illustrated through a simple example.
Biological systems, Engineering, Master Equation, Stochastic processes, Conditional Poisson Approximation, Size-Regulated Switch
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