Abstract:
The focus of this thesis is the inclusion of stochasticity into mathematical models of assembly with particular interest to the in vitro polymerisation of intermediate filaments, one of three components of the cytoskeleton. From the chemical master equation (CME), two additional models (the reaction rate equations or RREs and the two-moment approximation equations or 2MA equations) are derived. As analysis of the CME is generally intractable, we present the stochastic simulation algorithm (SSA) as a means of reproducing the most probable state of the CME at a given time. The results from the SSA are compared to simulations of both the RREs and the 2MA equations and we find that the three models are in good agreement. Further, the numerical results are compared to mean lengths and length distributions of experimental data which all models are shown to mimic. Mathematical analyses of the RREs demonstrate the conservation of mass in the system, and the unique positive equilibrium is proven to be globally asymptotically stable. Further, the 2MA equations are also shown to have conservation of mass and to possess an analogous equilibrium to the one found in the case of the RREs. In general, this study illustrates how randomness can be incorporated in polymerisation models and highlights the advantages and disadvantages of the different approaches.
