Multiscale Monte Carlo methods to cope with separation of scales in stochastic simulation of biological networks

The emerging paradigm of systems biology aims at gaining a quantitative understanding of the organization, dynamics and control of biological phenomena, via an iterative process of experimentation and computation. Building a close link between the system-level physiology and the underlying molecular machinery has been made possible only by the recent advances in genomic and proteomic technologies. The idea of mathematical modeling is not new to biology; however, the formal introduction of computational biology as a scientific discipline was driven by the need for an efficient and systematic way of organizing and analyzing the vast amount of information generated by high throughput experimental platforms. Currently, the field of computational biology is in its infancy and is plagued by numerous challenges.; One of the key challenges facing computational biology is building and simulating hierarchical models that span multiple length and time scales. Biological systems are inherently multiscale; not only in terms of time and length scales of intracellular processes, but also in the terms of the populations of species participating in these processes. Separation of scales reduces the efficiency and speed of most dynamic and spatial simulation techniques. In this thesis, we develop a multiscale approach to circumvent the problem of numerical stiffness in stochastic simulation of well-mixed reaction networks. The focus on a stochastic framework was motivated by two factors---firstly, the presence and role of stochasticity in biological systems is a well-established experimental fact, and secondly, accelerated stochastic algorithms to deal with numerical stiffness are currently unavailable.; In this work, we develop a multiscale Monte Carlo (MSMC) method to efficiently deal with computational challenges stemming from the disparity of time scales in well-mixed stochastic networks. Broadly speaking, the developed multiscale framework extends the deterministic quasi-equilibrium (QE) approximation, computational singular perturbation (CSP), and low-dimensional manifold (LDM) concepts to stochastic simulation. We address various issues to enable a seamless and probabilistically accurate algorithm. Specifically, dynamic network partitioning, on-the-fly relaxation of the fast network and numerical approximation and generation of the QE probability distribution function are some key issues addressed. Finally, incorporating the hybrid solvers enables us to deal with systems having mixed population scales in an efficient way. The modified method, called the hybrid multiscale Monte Carlo (HyMSMC) method, represents a significant improvement over the MSMC method. Especially, for stiff systems involving large populations the HyMSMC method is significantly faster than the MSMC method, as demonstrated with several examples.