Stochastic Networks
Our research in the category of theory and tool development for stochastic networks is motivated by the fact that modeling and analysis of such complex networks presents a number of mathematical challenges for which existing tools are inadequate.
In many applications, the state-space can be modeled by a jump Markov process with a very large and often infinite number of states. The probability density function of these states evolves according to the Forward-Kolmogorov equation. Due to the large size of the state-space, computational techniques have commonly relied on Monte-Carlo simulations to get samples of the density.
Our approach has been to develop computational methods that exploit the fact that the density functions are often supported on much smaller subsets thereby allowing one to devise projections on finite subsets that give an extimate of the actual density with a guaranteed error bound.
Although our approach was motivated by problems in stochastic chemical kinetics in biology, its domain of application is much broader, and includes any system that can be modeled by a Markov chain.
Computational tools:
Motivated by our study of the pap pili stochastic switch, we have developed a new method to compute the probability density function arising in stochastic chemical kinetics, called the Finite State Projection (FSP) method. In stochastic chemical kinetics, the configuration space defines a very large or infinite continuous-time discrete-state Markov chain. The FSP approach relies on a projection that preserves an important subset of the state space (e.g. that supporting the bulk of the probability density) while projecting the remaining states onto a single 'absorbing' state.
Probabilities for the resulting finite state Markov chain can be computed exactly, and can be shown to give an error bound for the corresponding probability for the original full system. These guaranteed error bounds make it possible to study rare events that often play a key role in biology, e.g. low switch rates of genetic switches. The FSP method provides a simple and powerful approach to the sensitivity and robustness analyses--with respect to parameter and initial condition variations--of stochastic systems.
Selected Publications:
- Brian Munsky and Mustafa Khammash, "Finite state projection approaches for stochastic analyses of gene regulatory networks”, IEEE Transactions on Automatic Control and IEEE Transactions on Circuits and Systems I, Joint Special Issue on Systems Biology, in press.
- Brian Munsky and Mustafa Khammash, “A multiple time interval finite state projection alorithm for the solution of the chemical master equation,” Journal of Computational Physics, in press.
- Slaven Peles, Brian Munsky and Mustafa Khammash, "Reduction and Solution of the Chemical Master Equation Using Time-Scale Separation and Finite State Projection," Journal of Chemical Physics 125 (22), 503646 (December 2006).
- Brian Munsky and Mustafa Khammash, "The Finite State Projection Algorithm for the Solution of the Chemical Master Equation," Journal of Chemical Physics124, 044104 2006.