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T Nagylaki

Publications and source records attributed to T Nagylaki.

16 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↗
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↗
PubMed1975

Polymorphisms in cyclically-varying environments.

We analysed both continuous and discrete two-allele models of cyclically-varying environments with an arbitrary degree of dominance. In continuous models, the gene frequency fluctuates with the period of the environmental oscillation. For the discrete case, the calculations were carried out to second order in selection. In contrast to the continuous models, and depending on the amount of dominance and the intitial gene frequency, fixation is possible as well as polymorphism.

Environment↗
PubMed1975

The deterministic behavior of self-incompatibility alleles.

For a system of n self-incompatibility alleles, neglecting mutation and random drift, it is shown that the completely symmetric equilibrium is locally stable, and any allelic frequency less than q equals 1 + a minus the square root of 1 + a-2, where a equals [2(n minus 1)]- minus 1, will increase. For all n, q greater than (2n)- minus 1, but if n greater than 1, q is approximately equal to (2n)- minus 1.

Alleles↗
PubMed1975

Conditions for the existence of clines.

A very general partial differential equation in space and time satisfied by the gene frequency in a monoecious population distributed continuously over an arbitrary habitat is derived. The treatment is restricted to a single diallelic locus in the absence of mutation and random drift, and it is supposed that time is continuous, births and deaths occur at random, and migration is independent of genotype. With the further assumptions that migration is isotropic and homogeneous, the population density is constant and uniform (as permitted by the population regulation mechanism included in the formulation), and Hardy-Weinberg proportions obtain locally, this partial differential equation reduces to the simplest multidimensional generalization of the classical Fisher-Haldane cline model. The efficacy of migration and selection in maintaining genetic variability at equilibrium in this model is investigated by deducing conditions for the existence of clines under various circumstances. The effects of the degree of dominance, a neutral belt between the regions where a particular allele is advantageous and deleterious, finiteness of the habitat, and habitat dimensionality are evaluated. Provided at least one of the alleles is favored only in a finite region, excluding the special case in which its total effective selective coefficient is zero, if conditions for supporting a cline are too unfavorable because migration is too strong, selection is too weak, or both, a cline cannot exist at all. Thus, unless there is overdominance, the population must be monomorphic. It is possible for a cline which can barely exist under the prevailing ecological circumstances to show a large amount of variation in gene frequency.

Alleles↗