Appendix to Paper by Wall and Herbeck

A method is described for determining the number of preferred codons in taxa in which G+C levels differ. If the hypothesis of random codon usage is not rejected, there are no preferred codons. If that hypothesis is rejected, then a model with one or two preferred codons is fitted to the data and a likelihood ratio test is used to determine whether there are one or two preferred codons. A C++ program is freely available to perform the calculations.     

Hitchhiking and associative overdominance at a microsatellite locus

The possible effects of a selected locus on a closely linked microsatellite locus are discussed and analyzed in terms of coalescent theory and models of the mutation process. Background selection caused by recurrent deleterious mutations will reduce the variance of allele size at a microsatellite locus. The occasional substitution of advantageous alleles (genetic hitchhiking) will also reduce the variance, but a high mutation rate at a microsatellite locus can restore the variance relatively rapidly. Overdominance at the selected locus will increase the variance at the microsatellite locus and create partitioning of the variation in allele size among gametes carrying one or the other of the overdominant alleles. These results suggest that neutral microsatellite loci can provide indicators of selective processes at closely linked loci.      

Overdominant alleles in a population of variable size.

An approximate method is developed to predict the number of strongly overdominant alleles in a population of which the size varies with time. The approximation relies on the strong-selection weak-mutation (SSWM) method introduced by J. H. Gillespie and leads to a Markov chain model that describes the number of common alleles in the population. The parameters of the transition matrix of the Markov chain depend in a simple way on the population size. For a population of constant size, the Markov chain leads to results that are nearly the same as those of N. Takahata. The Markov chain allows the prediction of the numbers of common alleles during and after a population bottleneck and the numbers of alleles surviving from before a bottleneck. This method is also adapted to modeling the case in which there are two classes of alleles, with one class causing a reduction in fitness relative to the other class. Very slight selection against one class can strongly affect the relative frequencies of the two classes and the relative ages of alleles in each class.     

Gene Flow and Selection in a Cline

A model of the effect of gene flow and natural selection in a continuously distributed, infinite population is developed. Different patterns of spatial variation in selective pressures are considered, including a step change in the environment, a "pocket" in the environment and a periodically varying environment. Also, the problem of the effect of a geographic barrier to dispersal is analyzed. The results are: (1) there is a characteristic length scale of variation of gene frequencies, (see PDF). The population cannot respond to changes in environmental conditions which occur over a distance less than the characteristic length. The result does not depend either on the pattern of variation in selective pressures or on the exact shape of the dispersal function. (2) The reduction in the fitness of the heterozygote causes a cline in gene frequencies to become steeper. (3) A geographic barrier to dispersal causes a drastic change in the gene frequencies at the barrier only when almost all of the individuals trying to cross the barrier are stopped.     

Exchangeable models of complex inherited diseases

A model of unlinked diallelic loci affecting the risk of a complex inherited disease is explored. The loci are equivalent in their effect on disease risk and are in Hardy-Weinberg and linkage equilibrium. The goal is to determine what assumptions about dependence of disease risk on genotype are consistent with data for diseases such as schizophrenia, bipolar disorder, autism, and multiple sclerosis that are relatively common (0.1-2% prevalence) and that have high concordance rates for monozygotic twins (30-50%) and high risks to first-degree relatives of affected individuals (risk ratios exceeding 4). These observations are consistent with a variety of models, including generalized additive, multiplicative, and threshold models, provided that disease risk increases rapidly for a narrow range of numbers of causative alleles. If causative alleles are in relatively high frequency, then the combined effects of numerous causative loci are necessary to substantially elevate disease risk.     

LIKELIHOOD-BASED INFERENCE IN ISOLATION-BY-DISTANCE MODELS USING THE SPATIAL DISTRIBUTION OF LOW-FREQUENCY ALLELES

Estimating dispersal distances from population genetic data provides an important alternative to logistically taxing methods for directly observing dispersal. Although methods for estimating dispersal rates between a modest number of discrete demes are well developed, methods of inference applicable to "isolation-by-distance" models are much less established. Here, we present a method for estimating ρσ², the product of population density (ρ) and the variance of the dispersal displacement distribution (σ²). The method is based on the assumption that low-frequency alleles are identical by descent. Hence, the extent of geographic clustering of such alleles, relative to their frequency in the population, provides information about ρσ². We show that a novel likelihood-based method can infer this composite parameter with a modest bias in a lattice model of isolation-by-distance. For calculating the likelihood, we use an importance sampling approach to average over the unobserved intraallelic genealogies, where the intraallelic genealogies are modeled as a pure birth process. The approach also leads to a likelihood-ratio test of isotropy of dispersal, that is, whether dispersal distances on two axes are different. We test the performance of our methods using simulations of new mutations in a lattice model and illustrate its use with a dataset from Arabidopsis thaliana .     

Likelihood analysis of ongoing gene flow and historical association

Abstract.— We develop a Monte Carlo-based likelihood method for estimating migration rates and population divergence times from data at unlinked loci at which mutation rates are sufficiently low that, in the recent past, the effects of mutation can be ignored. The method is applicable to restriction fragment length polymorphisms (RFLPs) and single nucleotide polymorphisms (SNPs) sampled from a subdivided population. The method produces joint maximum-likelihood estimates of the migration rate and the time of population divergence, both scaled by population size, and provides a framework in which to test either for no ongoing gene flow or for population divergence in the distant past. We show the method performs well and provides reasonably accurate estimates of parameters even when the assumptions under which those estimates are obtained are not completely satisfied. Furthermore, we show that, provided that the number of polymorphic loci is sufficiently large, there is some power to distinguish between ongoing gene flow and historical association as causes of genetic similarity between pairs of populations.     

RARE ALLELES AS INDICATORS OF GENE FLOW

A. method for estimating the average level of gene flow in a subdivided population, as measured by the average number of migrants exchanged between local populations, Nm, is presented. The results from a computer simulation model show that the logarithm of Nm is approximately linearly related to the logarithm of the average frequency of private alleles, p?(1), in a sample of alleles from the population. It is shown that this result is relatively insensitive to changes in parameters of the model other than Nm and the number of individuals sampled per population. The dependence of the value of p?(1) on the numbers of individuals sampled provides a rough way to correct for differences in sample size. This method was applied to data from 16 species, showing that estimated values of Nm range from much greater than 1 to less than 0.1. These results confirm the qualitative analysis of Slatkin (1981). This method was also applied to subsamples from a population to show how to measure the extent of isolation of local populations.