Applied Mathematics Seminars: A constrained approach to the simulation and analysis of stochastic multiscale chemical kinetics, and application to multiscale approximations of stiff ODEs
3 November 2017 14:00 in CM219
In many applications in cell biology, the inherent underlying stochasticity and discrete nature of individual reactions can play a very important part in the dynamics. The stochastic simulation algorithm has been around since the 1970s, which allows us to simulate trajectories from these systems, by simulating in turn each reaction, giving us a Markov jump process. However, in multiscale systems, where there are some reactions which are occurring many times on a timescale for which others are unlikely to happen at all, this approach can be computationally intractable. Several approaches exist for the efficient approximation of the dynamics of the “slow” reactions, the majority of which rely on the “quasi-steady state assumption” (QSSA). In this talk, we will present the Constrained Multiscale Algorithm. This method requires us to find the stationary distribution of stochastic reaction network with non-mass action kinetics in order to estimate the effective slow rates in the system. We will present a new result stating the analytical form of these distributions, which allow us to use these highly accurate multiscale approximations without the need for costly stochastic simulations or approximations. We will then present some results comparing the resulting approximations of the effective dynamics with those arising from the QSSA. If time allows, we will then consider how the ideas arising from these results can allow us to find highly accurate approximations of the slow manifold for stiff ODE systems, with potential for the development of accurate and stable explicit numerical solvers.
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