Statement of Research Interests

My general research interests lie in the statistical inference of stochastic processes superimposed on stochastically evolving networks. Stochastic processes are broadly defined to include Poisson processes, Markov chains, and non-linear stochastic differential equations. Stochastically evolving networks are meant to provide the biologically realistic contexts within which our stochastic processes arise. For example, such contexts are provided by phylogenetic X-trees in molecular evolution, structured ancestral recombination graphs in population genetics, contact networks of susceptible and infected nodes in epidemiology, or gene interaction networks in cell biology. I am interested in statistically sound, biologically realistic, computationally efficient, and numerically rigorous approaches to statistical inference in the biological sciences. During the next decade, I plan to conduct research and teach courses at the integrative interface of biological, computational, mathematical, and statistical sciences. My recent and current research can be broadly classified into two areas; (1) statistical genetics and (2) computational statistics. Statistical Genetics Microsatellites: Microsatellites are simple sequence repeats in DNA, for example the motif AT repeated twentyfive times in a row. Microsatellites mutate by changing the number of their repeats. They are used in the construction of genome-wide maps, in the search for disease-causing genes, and in identification for both forensics applications and paternity tests. For a recent mathematical treatment of microsatellites see Calabrese and Sainudiin (2004). My M.S. thesis was an effort to place several continuous time Markov chain models of microsatellite evolution in the literature into a cohesive framework that allows for statistical inference (Sainudiin et al., 2004). Biologically realistic models of complicated random walks on finitely many states in continuous time were shown to provide better descriptions of microsatellite evolution through their likelihood ratios using real data. Subtle differences in the nature of microsatellite evolution between humans and chimpanzees were captured in that study through the maximum penalized likelihoods of time non-homogeneous Markov chain models superimposed on species trees. Robustness in population genetic inference with microsatellites to model mis-specification, raised in Sainudiin et al. (2004), is currently being studied via simulations. More realistic molecular sequence data contains both fast-mutating repetitive DNA (microsatellite) and slow-mutating non-repetitive DNA. The first phase of my post-doctoral research at Oxford’s Mathematical Genetics group is to develop novel sequential importance samplers that can obtain the likelihood under coalescent models from such DNA molecules which simultaneously evolve over two scales of time that can differ by up to five orders of magnitude. Comparative Phylogenetics: We provided a simple generalization of a fairly popular codon substitution model used to detect positive Darwinian selection in rapidly evolving proteins (Nielsen and Yang, 1998), such as the human leukocyte antigen (HLA) of the immune system. Multiple independent projections of the stochastic dynamics of such molecules, modeled as simple mixtures of Markov chains, as they evolve down a phylogenetic tree, can shed light on positive physicochemical selection at the individual amino-acid level in the HLA system as well as the plant self-incompatibility (SI) system of crucifers (Sainudiin et al., 2005). First order empirical mutagenetic studies in the SI system have recently confirmed our predictions ( Nasrallah, personal communication). Finer site-specific mutagenetic studies on the SI system are expected to provide empirical feed-backs that can improve the models employed in that study. We have extended these models to test the extent and nature of site-specific physicochemical pressures acting on the evolving proteins through likelihood ratio tests between more general mixtures of Markov chain models of codon substitution superimposed on phylogenetic trees (Wong et al., 2005). Exact Bayesian Inference from Summaries: When full likelihood is computationally infeasible for realistic models of sequence evolution in a demographically complex population with rich sub-population structure, one may infer a population’s history based on summary statistics of the data in a simulation-based framework. Such methods are known as approximate Bayesian computations (ABC) in population genetics (Marjoram et al., 2003).