The imprint of natural selection on protein coding genes is often difficult to identify because selection is frequently transient or episodic, i.e. it affects only a subset of lineages. Existing computational techniques, which are designed to identify sites subject to pervasive selection, may fail to recognize sites where selection is episodic: a large proportion of positively selected sites. We present a mixed effects model of evolution (MEME) that is capable of identifying instances of both episodic and pervasive positive selection at the level of an individual site. Using empirical and simulated data, we demonstrate the superior performance of MEME over older models under a broad range of scenarios. We find that episodic selection is widespread and conclude that the number of sites experiencing positive selection may have been vastly underestimated.

Model-based analyses of natural selection often categorize sites into a relatively small number of site classes. Forcing each site to belong to one of these classes places unrealistic constraints on the distribution of selection parameters, which can result in misleading inference due to model misspecification. We present an approximate hierarchical Bayesian method using a Markov chain Monte Carlo (MCMC) routine that ensures robustness against model misspecification by averaging over a large number of predefined site classes. This leaves the distribution of selection parameters essentially unconstrained, and also allows sites experiencing positive and purifying selection to be identified orders of magnitude faster than by existing methods. We demonstrate that popular random effects likelihood methods can produce misleading results when sites assigned to the same site class experience different levels of positive or purifying selection--an unavoidable scenario when using a small number of site classes. Our Fast Unconstrained Bayesian AppRoximation (FUBAR) is unaffected by this problem, while achieving higher power than existing unconstrained (fixed effects likelihood) methods. The speed advantage of FUBAR allows us to analyze larger data sets than other methods: We illustrate this on a large influenza hemagglutinin data set (3,142 sequences). FUBAR is available as a batch file within the latest HyPhy distribution (http://www.hyphy.org), as well as on the Datamonkey web server (http://www.datamonkey.org/).

This paper continues the study of self-dual axioms on forms, i.e. faithful amnestic functors, motivated by properties of subobject forms in non-abelian algebra (which in many cases are Grothendieck bifibrations), and in particular, by properties of forms of substructures of group-like structures. In this paper we explore axiomatic origins of this kind, of a hierarchy of contexts introduced by M. Grandis for his projective approach to non-abelian homological algebra. This reveals new links between those contexts and the theory of factorization systems. Among other things, we show that a Grandis exact category is the same as an Isbell bicategory whose form (fibration) of projections is isomorphic to the form (opfibration) of injections.

We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn's Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are hybrid large scale normal, and characterize those locally compact abelian groups which (as hybrid large scale spaces) are hybrid large scale normal. Finally, we look at some properties of the Higson compactifications and coronas of hybrid large scale normal spaces.