Abstract

Co-authors: Radek Erban (University of Oxford), Ioannis Kevrekidis (Princeton), Konstantinos Zygalakis (University of Southampton)

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 Gillespie 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, some of which rely on the “quasi-steady state assumption” (QSSA). In this talk, we will present the Constrained Multiscale Algorithm, a method based on the equation free approach, which was first used to construct diffusion approximations of the slowly changing quantities in the system. We will compare this method with other methods which rely on the QSSA to compute the effective drift and diffusion of the approximating SDE . We will then show how this method can be used, back in the discrete setting, to approximate an effective Markov jump generator for the slow variables in the system, and quantify the errors in that approximation. If time permits, we will show how these generators can then be used to sample approximate paths conditioned on the values of their endpoints.