Two-locus Case Research Articles

Finding the boundary between evolutionary basins of attraction, and implications forWright’s fitness landscape analogy

In 1932 Wright introduced the notion of the fitness landscape. By analogy with a physicallandscape, whose gradient predicts a rolling marble’s spatial trajectory, the contours of thefitness landscape are meant to predict an evolving population’s genetic trajectory. Wright’schief interest was in the possibility that mutational interactions might frustrate naturalselection, giving rise to multiple maxima on the fitness landscape. Here we study adynamical system over the state space defined by allele frequencies and linkagedisequilibria between alleles. We first analytically locate the saddle between basins ofattraction in infinite-sized populations evolving under the influence of selectionand recombination for the simplest two-locus case. We further show numericallythat the boundary between basins is approximately linear with respect to linkagedisequilibrium, though not allele frequency. We also employ this framework to developnovel perspectives on two venerable results for single-peaked fitness landscapes.Finally we sought the potential function whose contours would predict evolutionarytrajectories through this state space. Importantly not every dynamical system canbe described by a potential function, and the present problem is provably onesuch case. Thus in the parlance of Wright’s analogy, in locating the floor of thefitness valley we have lost the landscape, and this conclusion is not limited to ourchoice of parameterization, nor of problem. This result motivates us to carefullyreview the formal implications and requirements of this widely used analogy.

TESTING FOR HOMOGENEITY OF GAMETIC DISEQUILIBRIUM AMONG POPULATIONS.

Testing for homogeneity of gametic disequilibria among populations can be informative in discriminating among the evolutionary agents generating them in natural populations. The standard statistical procedure that tests for homogeneity of disequilibrium for the same locus pair in different populations is based on the Fisher's test of homogeneity among correlation coefficients (Fisher 1925; Steel and Torrie 1980; Sokal and Rohlf 1995). This statistical approach makes it necessary to measure the magnitude of disequilibrium between loci by the coefficient of correlation of gene frequencies. This coefficient for the two-allele, two-locus case is defined as r = D/l[p(l p)q(l q)]?12 (Hill and Robertson 1968). In this formula, D is the covariance of allelic frequencies defined as D = f(A1B1) pq, where f(A1B1) is the frequency of the haplotype AIB1, andp and q denote the allele frequencies at the loci (Lewontin and Kojima 1960). When samples of haplotypes for a given pair of loci are available from k populations, each correlation coefficient of allele frequencies is transformed to a variable normal z, and a weighted sum of squares of the corresponding z-values has a x2 distribution with k 1 degrees of freedom, which is used for testing the null hypothesis of no heterogeneity. This statistical approach is recommended by Weir (1996) in his influential book Genetic Data Analysis (pp. 136-138), and it has been used in testing for homogeneity of disequilibrium for either allozyme loci or DNA polymorphisms (see Laurie-Ahlberg and Weir 1979; Miyashita et al. 1993). The test for homogeneity of r-values is a valid test for r-values but is not suitable to detect disequilibrium heterogeneity among populations. This is because this statistical approach uses the strength of disequilibrium as the correlation coefficient of allele frequencies whose range is strongly frequency dependent (Sved 1971; Hedrick et al. 1978; Hedrick 1988; Lewontin 1988). Thus, the r coefficient ranges from 1.0 to 1.0 only if allele frequencies at the two loci are 0.5. The range of r will be smaller to the extent that allele frequencies at loci are more different. This property makes it inappropriate to compare disequilibrium values among populations that have different allelic frequencies. In general, any test of homogeneity based on a disequilibrium measure whose range is frequency dependent, such as r or D coefficients, will suffer from the same problem. Alternatively, tests for homogeneity of disequilibrium based on a measure that have the same range for all allelic frequencies are desirable. Although no measure of disequilibrium is completely independent of allele frequencies (Lewontin 1988), the D' disequilibrium coefficient suggested by Lewontin (1964) has a range that is frequency independent, and, therefore, D' is very useful for comparisons of disequilibrium for pairs of loci with different allelic frequencies (Hedrick 1987, 1988; Lewontin 1988). The D' coefficient is the ratio of the D coefficient to its theoretical maximum value, Dmax, given the gene frequencies and the sign of D. The Dmax value is min[p(l q), (1 p)q], when D > 0, or min [pq, (1 p)(l q)], when D < 0. Because D' is a measure of disequilibrium standardized by the Dmax value, its range varies from 1.0 to 1.0 for all combinations of allele frequencies at two loci. Of course, the r disequilibrium coefficient can

Variational principles in evolution

For a one-locus selection model, Svirezhev introduced an integral variational principle by defining a Lagrangian which remained stationary on the trajectory followed by the population undergoing selection. It is shown here (i) that this principle can be extended to multiple loci in some simple cases and (ii) that the Lagrangian is defined by a straightforward generalization of the one-locus case, but (iii) that in two-locus or more general models there is no straightforward extension of this principle if linkage and epistasis are present. The population trajectories can be constructed as trajectories of steepest ascent in a Riemannian metric space. A general method is formulated to find the metric tensor and the surface in the metric space on which the trajectories, which characterize the variations in the gene structure of the population, lie. The local optimality principle holds good in such a space. In the special case when all possible linkage disequilibria are zero, the phase point of the n-locus genetic system moves on the surface of the product space of n higher dimensional unit spheres in a certain Riemannian metric space of gene frequencies so that the rate of change of mean fitness is maximum along the trajectory. In the two-locus case the corresponding surface is a hyper-torus.

Variational principles in evolution.

For a one-locus selection model, Svirezhev introduced an integral variational principle by defining a Lagrangian which remained stationary on the trajectory followed by the population undergoing selection. It is shown here (i) that this principle can be extended to multiple loci in some simple cases and (ii) that the Lagrangian is defined by a straightforward generalization of the one-locus case, but (iii) that in two-locus or more general models there is no straightforward extension of this principle if linkage and epistasis are present. The population trajectories can be constructed as trajectories of steepest ascent in a Riemannian metric space. A general method is formulated to find the metric tensor and the surface in the metric space on which the trajectories, which characterize the variations in the gene structure of the population, lie. The local optimality principle holds good in such a space. In the special case when all possible linkage disequilibria are zero, the phase point of the n-locus genetic system moves on the surface of the product space of n higher dimensional unit spheres in a certain Riemannian metric space of gene frequencies so that the rate of change of mean fitness is maximum along the trajectory. In the two-locus case the corresponding surface is a hyper-torus.

Secondary Theorem of Natural Selection in biocultural populations

The “Secondary Theorem of Natural Selection,” an extension of Fisher's fundamental theorem, states that the rate of change in the mean of an arbitrary character in response to selection is proportional to the additive genetic covariance between the character and fitness. Here I derive an expression for the change in the mean value of a trait subject to both genetic and cultural transmission. I start with the one-locus case under generalized mating and cultural transmission from parents to offspring, then proceed to the two-locus case. My results support previous work on the effects of nongenetic inheritance by showing that (i) cultural transmission introduces a timelag in the population response to selection; (ii) with cultural transmission the effects of selection persist even after selection is relaxed; and (iii) cultural transmission can either enhance or retard phenotypic evolution relative to that obtained under purely genetic transmission.

The Evolution of Social Interactions by Family Selection

A population genetics model for the evolution of social behaviors by selection between families is presented. The effects of dominance and epistasis on both the conditions for an increase in gene frequency and the rate of gene frequency change are described for diploid and haplo-diploid organisms. 1. The conditions for the spread of genes determining social behaviors are identical to those of Hamilton's kinship theory for diploid organisms, namely, rb > c. 2. These conditions are independent of the degree of dominance (one- and two-locus case) or epistasis (two-locus case) for the linear fitness model investigated here. 3. Haplo-diploidy alone confers no advantage in terms of the evolution of social behaviors.

Two-locus inbreeding measures for recurrent selection

For a population undergoing recurrent selection, a method is presented for determining the average inbreeding coefficients at the end of each breeding cycle. The coefficients are derived in terms of probability measures that genes are identical by descent. For the one-locus case, two digametic measures are defined and employed in the derivation of a recurrence formula for the inbreeding coefficient. Two further classes of measures, trigametic and quadrigametic, are required for transition from one cycle to the previous one to allow the calculation of the inbreeding function for the two-locus case. Numerical values of the average probability of double identity by descent for populations with various imposed assumptions are listed to illustrate the effects of linkage and population size on the accrual of inbreeding and hence of homozygosity.

THE PROBABILITY OF FIXATION OF A MUTANT: THE TWO-LOCUS CASE.

The quantitative investigation of problems in population genetics has mainly been concerned, until recently, with the analysis of gene frequencies at a single locus. Such an approach is natural as a first step, but necessarily requires assumptions about the interaction effects of other loci to be meaningful. Despite the latter restriction a number of important results have been obtained from this approach, to a large extent by the researches of R. A. Fisher and Sewall Wright. Latterly interest has moved to multilocus behavior, and as a first step in this direction the two-locus case has received increasing attention (see, for example, Kimura (1956), Lewontin and Kojima (1960), Bodmer and Parsons (1962), Moran (1964)). Although such an extension from the single locus case still requires assumptions about the interactions of the loci considered with the remaining loci, it is at least possible to treat, for example, the effect of linkage between loci, which would be impossible in a single locus analysis (see in particular Kimura (1956) and Bodmer and Parsons (1962)). In another direction, Moran (1964) has shown that the analogues of several single locus theorems (including Fisher's fundamental theorem of natural selection) are no longer always true in the two locus case. All the investigations referred to above consider deterministic models, a stochastic treatment not having been attempted so far. A notable feature of the deterministic analyses is that close linkage between the loci appears, in all cases considered, to be desirable. An example of this, considered

The Probability of Fixation of a Mutant: The Two-Locus Case

The Probability of Fixation of a Mutant: The Two-Locus Case