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Algorithms and monte carlo methods in computational biology

We study three different aspects of computational biology: interacting particle systems, phylogenetic distance methods, and Markov Chain Monte Carlo. 
Our study of interacting particle systems has two components: First, a Windows-based simulator has been developed. It serves as a platform for many spatial biology studies; for example, plasmid transfer in biofilms. Second, a specific ecological model of multi-species cross-feeding has been studied using the simulator, where we have observed spatial self-organization and pattern formation (particularly, spiral waves). A mathematical derivation, which is based on analysis of partial differential equations, explains the phase-transitions observed in simulations. 
Distance methods have been widely used to infer phylogenies (evolution trees among different species) because they are fast and reasonably accurate. In this study, we have proved that a popular distance method named neighbor-joining is in fact a special case of least squares methods. The proof relates neighbor-joining to least squares statistics. Hence it provides an explanation for the robustness and accuracy of neighbor-joining. 
In an empirical study of MCMC, we investigate, through extensive simulations., how a mixture of local and heavy-tailed proposals (a.k.a. "small world proposals") can dramatically increase the convergence rates of Markov chains that admit stationary measures that are multi-modal. In the theoretical part of the study, we have proved, using techniques of state decomposition and isoperimetric inequalities for log-concave distributions, that the ''small world proposals" turn a "slowly mixing" chain into a "rapidly mixing" chain.