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T Nagylaki

Publications and source records attributed to T Nagylaki.

At least 19 recordsLinked to original sources
PubMed1991

Error bounds for the fundamental and secondary theorems of natural selection.

Error bounds are derived for the approximation delta Z approximately C/W, where delta Z, C, and W denote the rate of change of the mean of a character under selection, the genic (or additive genetic) covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character), and the mean fitness, respectively. Generations are discrete and nonoverlapping; the monoecious population mates at random. The character is determined by arbitrarily many multiallelic loci without epistasis; the linkage map is also arbitrary. The genotypic values of the character are constant. Rounds on the absolute error in the above approximation are deduced for an arbitrary character, and these are converted to bounds on the relative error when the character is fitness itself. In that case, C is the genic variance in fitness, and the relative error cannot exceed one half the greatest genotypic selection coefficient.

Alleles↗
PubMed1990

Gene conversion, linkage, and the evolution of repeated genes dispersed among multiple chromosomes.

The evolution of the probabilities of genetic identity within and between the loci of a multigene family dispersed among multiple chromosomes is investigated. Unbiased gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. The linkage map is arbitrary, but the same for every chromosome; the dependence of the probabilities of identity on the location on each chromosome is formulated exactly. The greatest of the rates of gene conversion, random drift, and mutation is epsilon much less than 1. Under the assumption of loose linkage (i.e., all the crossover rates greatly exceed epsilon, though they may still be much less than 1/2), explicit approximations are obtained for the equilibrium values of the probabilities of identity and of the linkage of disequilibria. The probabilities of identity are of order one [i.e., O(1)] and do not depend on location; the linkage disequilibria are of O(epsilon) and, within each chromosome, depend on location through the crossover rates. It is demonstrated also that the ultimate rate and pattern of convergence to equilibrium are close to that of a much simpler, location-independent model. If intrachromosomal conversion is absent, the above results hold even without the assumption of loose linkage. In all cases, the relative errors are of O(epsilon). Even if the conversion rate between genes on nonhomologous chromosomes is considerably less than between genes on the same chromosome or homologous chromosomes, the probabilities of identity between the former genes are still almost as high as those between the latter, and the rate of convergence is still not much less than with equal conversion rates. If the crossover rates are much less than 1/2, then most of the linkage disequilibrium is due to intrachromosomal conversion. If linkage is loose, the reduction of the linkage disequilibria to O(epsilon) requires only O(-ln epsilon) generations.

Alleles↗
PubMed1989

Diffusion approximations of the two-locus Wright-Fisher model.

Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is incomplete. Under certain conditions, explicit formulas are obtained for the stationary distributions of the two diffusions with loose linkage.

Genetic Linkage↗
PubMed1989

The maintenance of genetic variability in two-locus models of stabilizing selection.

The maintenance of genetic variability at two diallelic loci under stabilizing selection is investigated. Generations are discrete and nonoverlapping; mating is random; mutation and random genetic drift are absent; selection operates only through viability differences. The determination of the genotypic values is purely additive. The fitness function has its optimum at the value of the double heterozygote and decreases monotonically and symmetrically from its optimum, but is otherwise arbitrary. The resulting fitness scheme is identical to the symmetric viability model. Linkage disequilibrium is neglected, but the results are otherwise exact. Explicit formulas are found for all the equilibria, and explicit conditions are derived fro their existence and stability. A complete classification of the six possible global convergence patterns is presented. In addition to the symmetric equilibrium (with gene frequency 1/2 at both loci), a pair of unsymmetric equilibria may exist; the latter are usually, but not always, unstable. If the ratio of the effect of the major locus to that of the minor one exceeds a critical value, both loci will be stably polymorphic. If selection is weak at the minor locus, the more rapidly the fitness function decreases near the optimum, the lower is this critical value; for rapidly decreasing fitness functions, the critical value is close to one. If the fitness function is smooth at the optimum, then a stable polymorphism exists at both loci only if selection is strong at the major locus.

Chromosome Mapping↗
PubMed1989

Gustave Malécot and the transition from classical to modern population genetics.

The contributions of Gustave Malécot to theoretical population genetics are described, discussed, and put into perspective relative to earlier and later work. In this context, certain aspects of the theory of inbreeding, the correlation between relatives, the evolution of finite panmictic populations, and (in more depth) spatial variation are reviewed. A brief biographical sketch of Malécot is also presented.

France↗
PubMed1988

Regular systems of inbreeding.

Regular systems of inbreeding with discrete, nonoverlapping generations and the same number of individuals and mating pattern in every generation are studied. The matrix Q that specifies the recursion relations satisfied by the probabilities of identity is expressed in terms of the matrix M that describes the mating system. Necessary and sufficient conditions for convergence to genetic uniformity are given, and it is determined which probabilities of identity approach one. If the mating system has certain symmetries and these are imposed initially, then a matrix R, of lower dimension than Q, specifies the recursion relations. For such a mating system, for generic initial conditions, the condensed matrix R suffices for determining whether convergence to uniformity occurs and which probabilities of identity approach one. If Q is irreducible, the maximal eigenvalue of R is the same as that of Q. If Q is also aperiodic, this implies that the asymptotic rate of convergence to homogeneity of the condensed system is the same as that of the complete one. The above results apply to autosomal loci in monoecious (with or without selfing) and dioecious populations and to X-linked loci. As an example, all the eigenvalues and right and left eigenvectors of Q for circular mating are found.

Animals↗
PubMed1988

Gene conversion, linkage, and the evolution of multigene families.

The evolution of the probabilities of genetic identity within and between the loci of a multigene family is investigated. Unbiased gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. The linkage map is arbitrary, and the location dependence of the probabilities of identity is formulated exactly. The greatest of the rates of gene conversion, random drift, and mutation is epsilon much less than 1. For interchromosomal conversion, the equilibrium probabilities of identity are within order epsilon [i.e., O(epsilon)] of those in a simple model that has no location dependence and, at equilibrium, no linkage disequilibrium. At equilibrium, the linkage disequilibria are of O(epsilon); they are evaluated explicitly with an error of O(epsilon 2); they may be negative if symmetric heteroduplexes occur. The ultimate rate and pattern of convergence to equilibrium are within O(epsilon 2) and O(epsilon), respectively, of that of the same simple model. If linkage is loose (i.e., all the crossover rates greatly exceed epsilon, though they may still be much less than 1/2), the linkage disequilibria are reduced to O(epsilon) in a time of O(-ln epsilon). If intrachromosomal conversion is incorporated, the same results hold for loose linkage, except that, if the crossover rates are much less than 1/2, then the linkage disequilibria generally exceed those for pure interchromosomal conversion.

Biological Evolution↗
PubMed1979

Selection in dioecious populations.

Weak selection at a single mutiallelic locus in a dioecious population is analysed under the assumptions of panmixia and discrete non-overlapping generations. The results hold for both autosomal and X-linked loci after several generations have elapsed. With an error of the order of s (i.e. O(s)), where s is the selection intensity, the population evolves as if it were monoecious. The equivalent monoecious fitnesses must be calculated by weighting each sex by the number of genes carried by an individual at the locus under consideration. Provided the explicit time dependence (if any) of the genotypic fitnesses in each sex is O(s2), the rate of change of the male--female allelic frequency differences is O(s2). If the change per generation of the genotypic fitnesses is smaller than second order in s (i.e. o(s2)), then to O(s2) the rate of change of the unweighted average of the male and female mean fitnesses is equal to the genic variance. Hence, as long as there is significant gene frequency change, this measure of the mean fitness of the population will increase.

Alleles↗
PubMed1978

The correlation between relatives with assortative mating.

The equilibrium correlation between various close relatives is calculated for phenotypic assortative mating for a character determined by additive loci without dominance and an uncorrelated environment. It is supposed that the phenotypes of spouses have a bivariate normal distribution and environment and heredity are normally distributed. Heredity is Gaussian if either there are many alleles with approximately normally distributed effects at each of an arbitrary number of loci or the trait is controlled by many loci, each of which makes only a small contribution. It is also assumed that the regression of the phenotype or genotype of an individual on the phenotype or genotype of any one of his relatives is linear. The results show that with the above assumptions Fisher's formulae for the correlation between relatives hold with no restrictions on the linkage map.

Female↗
PubMed1977

Decay of genetic variability in geographically structured populations.

The ultimate rate and pattern of approach to equilibrium of a diploid, monoecious population subdivided into a finite number of equal, large, panmictic colonies are calculated. The analysis is restricted to a single locus in the absence of selection, and every mutant is assumed to be new to the population. It is supposed that either the time-independent backward migration pattern is symmetric in the sense that the probability that an individual at position x migrated from y equals the probability that one at y migrated from x, or it depends only on displacements and not on initial and final positions. Generations are discrete and nonoverlapping. Asymptotically, the rate of convergence is approximately (I-u)2t[I-(2NT)-1]t, where u, NT, and t denote the mutation rate, total population size, and time in generations, respectively; the transient part of the probability that two homologous genes are the same allele is approximately independent of their spatial separation. Thus, in this respect the population behaves as if it were panmictic.

Genetic Variation↗
PubMed1977

The evolution of one- and two-locus systems. II.

Weak selection in a monoecious population is studied in two multiallelic panmictic models. In the first, a single locus is considered with continuous time and age-independent fertilities and mortalities. If the fertilities of the various matings and the genotypic mortalities may be expressed with an error at most of the second order in s (i.e., O(s2)), where s is the intensity of selection, as sums of terms corresponding to the different genotypes and alleles, respectively, then after several generations the deviations from Hardy-Weinberg proportions are of O(s2). In the second model, two loci are treated with discrete nonover-lapping generations. It is shown that if the epistatic parameters are of O(s2), then after several generations the linkage disequilibria are reduced to O(s2). Assuming only weak selection, it is proved that in both models, after several generations, the total change is mean fitness is generally positive. It is likely that the exclusion of the initial period is usually unnecessary in natural populations. Exceptions are discussed.

Alleles↗
PubMed1977

Selection and mutation at an X-linked locus.

The most general mutation-selection model with discrete non-overlapping generations is formulated for a multiallelic X-linked locus. The difference equations for the gene frequencies, though dependent on the female genotypic proportions, are shown to have a simple form. If mating is random and the rates at which various unions produce females may be expressed as products of factors depending on the male and female parental genotypes, then the female zygotic frequencies are proved to be generalized Hardy-Weinberg proportions with respect to the male and female gametic frequencies. As the main application of the formalism, assuming only that mutation is much weaker than selection, the total frequencies of mutant alleles in males and females at equilibrium are related to the mutation rates and mean selection coefficients in the two sexes.

Alleles↗
PubMed1976

Clines with variable migration.

The consequences of a discontinuity in the migration rate and of a geographical barrier in the habitat are studied in a diffusion model of migration and selection. The treatment is restricted to a single diallelic locus in a monoecious population in the absence of mutation and random drift. It is supposed further that migration is independent of genotype, the population density remains constant and uniform, and Hardy-Weinberg proportions obtain locally. It is shown that a discontinuity in the migration rate leads to a jump in the slope of the gene frequency, but not in the gene frequency itself, while a localized geographical barrier has precisely the opposite effect. These features of the gene frequency behavior are quantitatively related to the migration rate. The influence of the above inhomogeneities in migration on the maintenance of an allele in an environmental pocket is examined. The extent to which the critical condition for polymorphism is made less stringent by decreased migration outside the pocket and by a geographical barrier between the pocket and the rest of the habitat is evaluated.

Alleles↗
PubMed1976

The evolution of one- and two-locus systems.

Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O(s2), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s2), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s2). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s2)), then to O(s2) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O(s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behaviour of one and two loci, deviations from random combination appear to have very little evolutionary significance.

Biological Evolution↗
PubMed1975

A continuous selective model for an X-linked locus.

Neglecting age-structure, but taking into account matings with differential fertility in Mendelian reproduction, a continuous selective model is formulated for a single X-linked locus with an arbitrary number of alleles. Without restricting the mating system, differential equations are derived for the genotypic and allelic frequencies. Assuming random mating, no selection, and constant fertilities and mortalities, these differential equations are solved explicitly. For this case, in contrast to the corresponding phenomenon in the usual model with discrete, non-overlapping generation, the difference between the frequencies of any allele in males and females approaches zero without oscillation.

Age Factors↗