Epistasis and Entropy

Epistasis is a key concept in the theory of adaptation. Indicators of epistasis are of interest for large system where systematic fitness measurements may not be possible. Some recent approaches depend on information theory. We show that considering shared entropy for pairs of loci can be misleading. The reason is that shared entropy does not imply epistasis for the pair. This observation holds true also in the absence of higher order epistasis. We discuss a refined approach for identifying pairwise interactions using entropy

GenEpi: gene-based epistasis discovery using machine learning.

BackgroundGenome-wide association studies (GWAS) provide a powerful means to identify associations between genetic variants and phenotypes. However, GWAS techniques for detecting epistasis, the interactions between genetic variants associated with phenotypes, are still limited. We believe that developing an efficient and effective GWAS method to detect epistasis will be a key for discovering sophisticated pathogenesis, which is especially important for complex diseases such as Alzheimer's disease (AD).ResultsIn this regard, this study presents GenEpi, a computational package to uncover epistasis associated with phenotypes by the proposed machine learning approach. GenEpi identifies both within-gene and cross-gene epistasis through a two-stage modeling workflow. In both stages, GenEpi adopts two-element combinatorial encoding when producing features and constructs the prediction models by L1-regularized regression with stability selection. The simulated data showed that GenEpi outperforms other widely-used methods on detecting the ground-truth epistasis. As real data is concerned, this study uses AD as an example to reveal the capability of GenEpi in finding disease-related variants and variant interactions that show both biological meanings and predictive power.ConclusionsThe results on simulation data and AD demonstrated that GenEpi has the ability to detect the epistasis associated with phenotypes effectively and efficiently. The released package can be generalized to largely facilitate the studies of many complex diseases in the near future

Historical contingency and entrenchment in protein evolution under purifying selection

The fitness contribution of an allele at one genetic site may depend on alleles at other sites, a phenomenon known as epistasis. Epistasis can profoundly influence the process of evolution in populations under selection, and can shape the course of protein evolution across divergent species. Whereas epistasis between adaptive substitutions has been the subject of extensive study, relatively little is known about epistasis under purifying selection. Here we use mechanistic models of thermodynamic stability in a ligand-binding protein to explore the structure of epistatic interactions between substitutions that fix in protein sequences under purifying selection. We find that the selection coefficients of mutations that are nearly-neutral when they fix are highly contingent on the presence of preceding mutations. Conversely, mutations that are nearly-neutral when they fix are subsequently entrenched due to epistasis with later substitutions. Our evolutionary model includes insertions and deletions, as well as point mutations, and so it allows us to quantify epistasis within each of these classes of mutations, and also to study the evolution of protein length. We find that protein length remains largely constant over time, because indels are more deleterious than point mutations. Our results imply that, even under purifying selection, protein sequence evolution is highly contingent on history and so it cannot be predicted by the phenotypic effects of mutations assayed in the wild-type sequence.Comment: 42 pages, 13 figure

Epistasis not needed to explain low dN/dS

An important question in molecular evolution is whether an amino acid that occurs at a given position makes an independent contribution to fitness, or whether its effect depends on the state of other loci in the organism's genome, a phenomenon known as epistasis. In a recent letter to Nature, Breen et al. (2012) argued that epistasis must be "pervasive throughout protein evolution" because the observed ratio between the per-site rates of non-synonymous and synonymous substitutions (dN/dS) is much lower than would be expected in the absence of epistasis. However, when calculating the expected dN/dS ratio in the absence of epistasis, Breen et al. assumed that all amino acids observed in a protein alignment at any particular position have equal fitness. Here, we relax this unrealistic assumption and show that any dN/dS value can in principle be achieved at a site, without epistasis. Furthermore, for all nuclear and chloroplast genes in the Breen et al. dataset, we show that the observed dN/dS values and the observed patterns of amino acid diversity at each site are jointly consistent with a non-epistatic model of protein evolution.Comment: This manuscript is in response to "Epistasis as the primary factor in molecular evolution" by Breen et al. Nature 490, 535-538 (2012

Strong Selection Significantly Increases Epistatic Interactions in the Long-Term Evolution of a Protein

Epistatic interactions between residues determine a protein's adaptability and shape its evolutionary trajectory. When a protein experiences a changed environment, it is under strong selection to find a peak in the new fitness landscape. It has been shown that strong selection increases epistatic interactions as well as the ruggedness of the fitness landscape, but little is known about how the epistatic interactions change under selection in the long-term evolution of a protein. Here we analyze the evolution of epistasis in the protease of the human immunodeficiency virus type 1 (HIV-1) using protease sequences collected for almost a decade from both treated and untreated patients, to understand how epistasis changes and how those changes impact the long-term evolvability of a protein. We use an information-theoretic proxy for epistasis that quantifies the co-variation between sites, and show that positive information is a necessary (but not sufficient) condition that detects epistasis in most cases. We analyze the "fossils" of the evolutionary trajectories of the protein contained in the sequence data, and show that epistasis continues to enrich under strong selection, but not for proteins whose environment is unchanged. The increase in epistasis compensates for the information loss due to sequence variability brought about by treatment, and facilitates adaptation in the increasingly rugged fitness landscape of treatment. While epistasis is thought to enhance evolvability via valley-crossing early-on in adaptation, it can hinder adaptation later when the landscape has turned rugged. However, we find no evidence that the HIV-1 protease has reached its potential for evolution after 9 years of adapting to a drug environment that itself is constantly changing.Comment: 25 pages, 9 figures, plus Supplementary Material including Supplementary Text S1-S7, Supplementary Tables S1-S2, and Supplementary Figures S1-2. Version that appears in PLoS Genetic