Evolutionary constraints in fitness landscapes.

In the last years, several genotypic fitness landscapes-combinations of a small number of mutations-have been experimentally resolved. To learn about the general properties of "real" fitness landscapes, it is key to characterize these experimental landscapes via simple measures of their structure, related to evolutionary features. Some of the most relevant measures are based on the selectively acessible paths and their properties. In this paper, we present some measures of evolutionary constraints based on (i) the similarity between accessible paths and (ii) the abundance and characteristics of "chains" of obligatory mutations, that are paths going through genotypes with a single fitter neighbor. These measures have a clear evolutionary interpretation. Furthermore, we show that chains are only weakly correlated to classical measures of epistasis. In fact, some of these measures of constraint are non-monotonic in the amount of epistatic interactions, but have instead a maximum for intermediate values. Finally, we show how these measures shed light on evolutionary constraints and predictability in experimentally resolved landscapes.

The amount of DNA polymorphism when population size changes linearly.

Population size is one of the main factors that determine the amount of DNA polymorphism. We examined a model under which the population size changed linearly. Because of the simplicity of this model, we could analytically obtain the expectation of nucleotide diversity, E(π), and the expectation of the amount of DNA polymorphism, E(θ), based on the number of segregating sites. The results suggest that E(π) is larger than E(θ) when the population size decreased and that E(π) is smaller than E(θ) when the population size increased. The expected time to the most recent common ancestor could also be obtained under this model.

Measuring epistasis in fitness landscapes: The correlation of fitness effects of mutations.

Genotypic fitness landscapes are constructed by assessing the fitness of all possible combinations of a given number of mutations. In the last years, several experimental fitness landscapes have been completely resolved. As fitness landscapes are high-dimensional, simple measures of their structure are used as statistics in empirical applications. Epistasis is one of the most relevant features of fitness landscapes. Here we propose a new natural measure of the amount of epistasis based on the correlation of fitness effects of mutations. This measure has a natural interpretation, captures well the interaction between mutations and can be obtained analytically for most landscape models. We discuss how this measure is related to previous measures of epistasis (number of peaks, roughness/slope, fraction of sign epistasis, Fourier-Walsh spectrum) and how it can be easily extended to landscapes with missing data or with fitness ranks only. Furthermore, the dependence of the correlation of fitness effects on mutational distance contains interesting information about the patterns of epistasis. This dependence can be used to uncover the amount and nature of epistatic interactions in a landscape or to discriminate between different landscape models.

New weighting methods for phylogenetic tree reconstruction using multiple loci.

Efficient determination of evolutionary distances is important for the correct reconstruction of phylogenetic trees. The performance of the pooled distance required for reconstructing a phylogenetic tree can be improved by applying large weights to appropriate distances for reconstructing phylogenetic trees and small weights to inappropriate distances. We developed two weighting methods, the modified Tajima-Takezaki method and the modified least-squares method, for reconstructing phylogenetic trees from multiple loci. By computer simulations, we found that both of the new methods were more efficient in reconstructing correct topologies than the no-weight method. Hence, we reconstructed hominoid phylogenetic trees from mitochondrial DNA using our new methods, and found that the levels of bootstrap support were significantly increased by the modified Tajima-Takezaki and by the modified least-squares method.

Simple method for analyzing the pattern of DNA polymorphism and its application to SNP data of human.

In order to analyze the pattern of DNA polymorphism in detail, we have developed a simple method using a new statistic theta(i) which estimates 4Nmu from the number of segregating sites whose allelic nucleotide frequency is i/n among n DNA sequences, where N is the effective population size and mu is the mutation rate per generation per nucleotide site. Under the assumption that mutations are selectively neutral and a population size is constant, the expectation of theta(i) is equal to that of theta, which estimates 4Nmu from the number of segregating sites, so that the distribution of theta(i) is flat. Therefore, the departure of the distribution of theta(i) from the horizontal line, which represents the value of theta, reflects change in population size and natural selection. Results of the coalescent simulation show that the distributions of theta(i) in the populations which experienced expansion and reduction are U-shaped and upside-down U-shaped, respectively. And the distributions of theta(i) in some populations that experienced bottleneck are W-shaped. Furthermore, we have applied this method to the SNP data in the International HapMap Project. Results of data analyses show that the distributions of theta(i) in the CEU (European), CHB and JPT (Asian) populations are different from that in the YRI population (African). From these results of data analyses in nuclear DNA and the pattern of polymorphism in human mitochondrial DNA already known, we infer that the CEU, CHB and JPT populations experienced the bottleneck.

Effect of population structure on the amount of polymorphism and the fixation probability under overdominant selection.

Under overdominant selection, mutants substantially contribute to increase the amount of polymorphism. It is also known that under neutrality as the migration rates among demes decrease in a subdivided population, the amount of polymorphism increases along with the increase of the effective population size, N(e). In this study, under overdominant selection the effect of population subdivision on the amount of polymorphism was investigated using the diffusion approximation and the low migration approximation. It was shown that if selection is medium or strong (e.g., N(T)s > 1, where N(T) is the population size and s is the selective advantage of heterozygotes), the nucleotide diversity, pi, decreases along with the decrease of Nm against the increase of N(e), where N is the size of demes and m is the migration rate per deme. In addition, the ratio of the nucleotide diversity to the evolutionary rate also decreases along with the decrease of Nm. In some cases the ratio becomes smaller than that expected under neutrality as Nm decreases.

Evolution of coadaptation in a two-locus epistatic system.

Although recent advances in genome biology have dramatically increased our understanding of the contribution of gene interactions to the development of complex phenotypes, we still lack general agreement on the process and mechanisms responsible for the evolution of epistatic systems. Even if genes in a species are indeed integrated into coadapted complexes of interacting components, simple additive evolution may eventually result in epistatic differentiation of populations. Consequently, the prevalence of epistatic gene action does not tell us anything about the role of epistatic selection in the history of population divergence. To elucidate the contribution of epistatic selection in the evolution of coadaptation, we investigate the fixation process of two mutations that interact synergistically to enhance fitness. We show by diffusion analysis and simulations that epistatic selection on cosegregating variants does not by itself promote the evolution of epistatic systems; rather, accumulation of neutral mutations may play a crucial role, creating an appropriate genetic milieu for adaptive evolution in the future generations.

Effect of dominance on heterozygosity and the fixation probability in a subdivided population.

Under the assumptions of a subdivided population and the presence of dominance for fitness, the expected sum of heterozygosity in the total population during the lifetime of mutant was investigated. It was shown analytically and by computer simulations that in the island model the effect of dominance on the expected sum of heterozygosity decreases as the migration rate decreases and is lost almost completely when the migration rate is very low. In addition to the expected sum of heterozygosity, the fixation probability of mutant was also investigated. The effect of dominance on the fixation probability also decreases as the migration rate decreases but is not completely lost when the migration rate is very low.

A statistical test for the difference in the amounts of DNA variation between two populations.

A statistical test for the difference in the amounts of DNA variation between two populations is developed. The test statistic involves the covariance of the amount of variation between two populations, which is given by a function of their divergence time, T0. Accordingly, the power (rejection probability) of the test depends on T0. In this article, T0 is treated as unknown because it is very difficult to estimate. The test is most conservative when T0 = infinity is assumed because the covariance is zero. If T0 = 0 is assumed, the largest value of the rejection probability is obtained. Thus, the test provides a range of rejection probability unless we have a reliable estimate of T0. The test is applied to the PgiC region in three mustard species: Leavenworthia stylosa, L. crassa and L. uniflora. The results of our test show that the level of variation in L. stylosa is significantly higher than those in the other species.

The effect of migration during the divergence.

The mean and variance of the number of nucleotide differences were obtained when the ancestral population diverged with migration. The number of nucleotide differences obtained indicates that not only the migration rate but also the period of migration has influence on a population structure. According to the migration rate and the period of migration, populations behave approximately as a single unit, diverged and isolated populations, two populations under equilibrium, or none of them. When sigma m(t) is about one, the variance of the number of nucleotide differences becomes large, where sigma m(t) is the sum of the migration rate for the period of migration. The distribution of the estimated divergence time was also obtained using computer simulations. It was found that the divergence time can be explained by sigma m(t). That is, the divergence time is mostly estimated as the time when sigma m(t) is less than 1.