Group-theoretic models of the inversion process in bacterial genomes

The variation in genome arrangements among bacterial taxa is largely due to the process of inversion. Recent studies indicate that not all inversions are equally probable, suggesting, for instance, that shorter inversions are more frequent than longer, and those that move the terminus of replication are less probable than those that do not. Current methods for establishing the inversion distance between two bacterial genomes are unable to incorporate such information. In this paper we suggest a group-theoretic framework that in principle can take these constraints into account. In particular, we show that by lifting the problem from circular permutations to the affine symmetric group, the inversion distance can be found in polynomial time for a model in which inversions are restricted to acting on two regions. This requires the proof of new results in group theory, and suggests a vein of new combinatorial problems concerning permutation groups on which group theorists will be needed to collaborate with biologists. We apply the new method to inferring distances and phylogenies for published Yersinia pestis data.Comment: 19 pages, 7 figures, in Press, Journal of Mathematical Biolog

Weighted Minimum-Length Rearrangement Scenarios

We present the first known model of genome rearrangement with an arbitrary real-valued weight function on the rearrangements. It is based on the dominant model for the mathematical and algorithmic study of genome rearrangement, Double Cut and Join (DCJ). Our objective function is the sum or product of the weights of the DCJs in an evolutionary scenario, and the function can be minimized or maximized. If the likelihood of observing an independent DCJ was estimated based on biological conditions, for example, then this objective function could be the likelihood of observing the independent DCJs together in a scenario. We present an O(n^4)-time dynamic programming algorithm solving the Minimum Cost Parsimonious Scenario (MCPS) problem for co-tailed genomes with n genes (or syntenic blocks). Combining this with our previous work on MCPS yields a polynomial-time algorithm for general genomes. The key theoretical contribution is a novel link between the parsimonious DCJ (or 2-break) scenarios and quadrangulations of a regular polygon. To demonstrate that our algorithm is fast enough to treat biological data, we run it on syntenic blocks constructed for Human paired with Chimpanzee, Gibbon, Mouse, and Chicken. We argue that the Human and Gibbon pair is a particularly interesting model for the study of weighted genome rearrangements

Quantifying evolutionary constraints on B cell affinity maturation

The antibody repertoire of each individual is continuously updated by the evolutionary process of B cell receptor mutation and selection. It has recently become possible to gain detailed information concerning this process through high-throughput sequencing. Here, we develop modern statistical molecular evolution methods for the analysis of B cell sequence data, and then apply them to a very deep short-read data set of B cell receptors. We find that the substitution process is conserved across individuals but varies significantly across gene segments. We investigate selection on B cell receptors using a novel method that side-steps the difficulties encountered by previous work in differentiating between selection and motif-driven mutation; this is done through stochastic mapping and empirical Bayes estimators that compare the evolution of in-frame and out-of-frame rearrangements. We use this new method to derive a per-residue map of selection, which provides a more nuanced view of the constraints on framework and variable regions.Comment: Previously entitled "Substitution and site-specific selection driving B cell affinity maturation is consistent across individuals