GENETIC DRIFT IN CLINES WHICH ARE MAINTAINED BY MIGRATION AND NATURAL SELECTION

Genetics ◽  
1975 ◽  
Vol 81 (1) ◽  
pp. 191-207 ◽  
Author(s):  
Joseph Felsenstein

ABSTRACT Genetic drift will cause a migration-selection cline to wobble about its expected position. A rough linear approximation is developed, valid when local populations are large. This is used to calculate effects of genetic drift on clines in a stepping-stone model with abrupt and with gradual changes of selection coefficients at a single haploid locus. Among the quantities calculated are measures of slope, standardized variation of gene frequencies around their expected values, and correlation among neighboring populations with respect to deviations from the expected gene frequencies. These quantities appear to be primarily functions of Ns and Nm for a given pattern of selection. Computer simulation gives rough confirmation of these results. Standardized variances of gene frequencies and correlation of neighbors differ along the cline in the case of smooth changes in selection. In no case is pathological behavior of gene frequency deviations found near the boundaries of selective regions. Local behavior of gene frequencies of nearby colonies is approximately predicted by a simple adaptation of the stepping-stone theory of Kimura and Weiss. Approximate measures of the lateral variation of the midpoint of a cline and the probability of non-monotonicity are also calculated and discussed.

1970 ◽  
Vol 2 (02) ◽  
pp. 229-258 ◽  
Author(s):  
Takeo Maruyama

The stepping stone model of population structure, of finite length, is analysed with special reference to the variance, and correlation coefficients of gene frequencies. Explicit formulas for these quantities are obtained. The model is also analysed for the genetic variability maintained in the population. In order to check the validity of the analytical results, several numerical computations were carried out using two different methods: iterations and Monte Carlo experiments. The values obtained by these numerical methods agree well with the theoretical values obtained by formulas derived analytically.


1986 ◽  
Vol 28 (6) ◽  
pp. 1048-1050
Author(s):  
Curtis Strobeck

The expected values of the probabilities of identity by descent are derived for the circular stepping-stone model. The results are more easily interpreted than those derived previously.Key words: model, population, ecology.


1997 ◽  
Vol 05 (04) ◽  
pp. 509-520
Author(s):  
Michel Gillois

The two 1944 and 1945 Malécot's papers are each a face of the same coin. The first paper is the study of gene frequency evolution in a small population as a Markoff chain. The second paper is the study of consanguinity coefficient evolution in a small population. Coefficient of consanguinity is defined between two homologous genes and have narrow relation with second moment of gene frequencies following the relation [Formula: see text] with [n>0,1<k<2N] and qn the frequence of an allelic form at the generation n, and [Formula: see text] the probability of identity for k homologous gens slamped at random in the population at the generation n to belong to j identity subsets [8]. In this paper we use the concept of "generalized genotype" in order to synthetize these two independent Malécot's results. Four roots control dynamics: [Formula: see text] and form the kernel of all genetic dynamics. All other roots introduced by mutation and/or zygotic selection derive by duplication from one of these four primitive roots.


1970 ◽  
Vol 2 (2) ◽  
pp. 229-258 ◽  
Author(s):  
Takeo Maruyama

The stepping stone model of population structure, of finite length, is analysed with special reference to the variance, and correlation coefficients of gene frequencies. Explicit formulas for these quantities are obtained. The model is also analysed for the genetic variability maintained in the population. In order to check the validity of the analytical results, several numerical computations were carried out using two different methods: iterations and Monte Carlo experiments. The values obtained by these numerical methods agree well with the theoretical values obtained by formulas derived analytically.


The snails Cepaea nemoralis and C. hortensis are remarkable for an extensive and stable polymorphism involving the colour and banding of the shell. It was formerly thought that the variation in frequency of the different morphs between populations was random. Cain & Sheppard, for nemoralis and Clarke, for C. hortensis , have shown that in many English colonies visual selection by thrushes, and no doubt other predators, strongly influences the frequencies of the morphs, the more conspicuous on a given background being more heavily predated. In consequence populations tend to match their backgrounds, but remain polymorphic. In some districts of high chalk downland, this correspondence with background does not occur. The predominance of a few morphs irrespective of habitat and background characterizes areas vastly larger than that of a panmictic population. Such a constancy of morph frequencies over a large and diverse area in spite of visual selection we call an area effect . The principal district we have studied is the Marlborough Downs, where in an area of several square kilometres there are no five-banded C. nemoralis although in a contiguous area they predominate. Part of the non-five-banded area has a vast excess of browns, and another part of yellows. The form spread-banded and the cross-product ratio of pink and yellow to unbanded and banded also show such effects. In some places the morph frequencies change with extraordinary abruptness over 100 to 300 m. The area effects are not due to differential incidence of visual predation, nor, since they bear no relation to variation of habitat, to differences in its direction. In only two subareas do we think that visual selection is affecting morph frequencies. The observed frequency distributions cannot be accounted for by sampling drift (‘genetic drift’) at the present day since the numbers involved are far too large and the frequencies too constant over large areas. In the few populations that have been observed for up to 10 years, no major changes of frequency have been found. The probability of a reduction to a few isolated populations because of ploughing up or drought in the last 200 years and subsequent drift and expansion is shown by the known agricultural history of the district to be slight. Restriction by spread of C. hortensis is also unlikely. A few colonies with restricted variation which might seem to show the action of drift or the founder effect are only extreme examples of local tendencies. Moreover, subfossil material from just off the south-western corner of the district strongly suggests that the area effects seen there have been in existence since Neolithic times. A survey of another district of high downland (Lambourn Downs) has shown a similar state of affairs to that on the Marlborough Downs, with a large area characterized by excess of yellow and mid-banded, and adjacent to it localities in which visual selection is effective. Observations from various other places on and off the Chalk also indicate that area effects are frequent on the Chalk, but that away from it visual selection is the principal agent determining local variation in gene frequencies. There is good evidence that the pigmentation of the body, which is apparently multifactorially controlled, also shows area effects; and part of the correspondence between body colour and background shade reported by Cain & Sheppard may be due to them. The evidence available for C. hortensis suggests that this species also shows area effects in shell characters. Yellow, pink or brown may predominate in C. nemoralis , but area effects in banding seem due mostly to the excess or defect of the modifier M 3 which reduces the five-banded phenotype to the form with only the middle band. It seems clear that the area effects are caused by some form of selection, but the topography, geology and vegetation of the Marlborough Downs gives no clue to what this could be for banding. Brown is known to be common only in the northern half of the range of nemoralis, and hortensis to extend much farther north than does nemoralis . A study of the distributions of the two species and of the brown morph on the Marlborough Downs suggests that local features of topography of open downland may produce localized climatic conditions influencing the relative distributions of the species and the abundance of brown. The abruptness of change of gene frequencies in both colour and banding might be caused by the change-over from one balanced gene complex to another requiring very different frequencies. Examination of Fisher’s equation for stability of a polymorphism maintained by heterosis (the most likely condition in these species) shows that in the districts where visual selection is effective in altering gene frequencies in nemoralis, the heterozygote advantage can only be of the order of a few percent, and that local differences of a few percent in the selective disadvantages of the homozygotes concerned could well produce area effects as marked as those we have observed. For biological purposes it is essential to recognize the difference between changes in gene frequency caused by selection and those produced by the effects of sampling error. ‘Genetic drift’ has been generally used to refer to the latter, but Sewall Wright uses it for all apparently random changes, whatever their cause, and perhaps for all changes in gene frequency; we therefore use sampling drift for the effects of sampling error. Surveys based only on the observations of frequencies and population size in widely scattered populations do not allow one to distinguish between the effects of selection that varies in direction and intensity from place to place (although more or less constant in time) and those of sampling drift. In general it is exceedingly difficult to identify the result of sampling drift in the wild except in certain situations. Casual collecting over such a district as the Marlborough Downs might well give the impression that sampling drift was effective there, but a more intensive survey shows the contrary.


1969 ◽  
Vol 13 (2) ◽  
pp. 117-126 ◽  
Author(s):  
Derek J. Pike

Robertson (1960) used probability transition matrices to estimate changes in gene frequency when sampling and selection are applied to a finite population. Curnow & Baker (1968) used Kojima's (1961) approximate formulae for the mean and variance of the change in gene frequency from a single cycle of selection applied to a finite population to develop an iterative procedure for studying the effects of repeated cycles of selection and regeneration. To do this they assumed a beta distribution for the unfixed gene frequencies at each generation.These two methods are discussed and a result used in Kojima's paper is proved. A number of sets of calculations are carried out using both methods and the results are compared to assess the accuracy of Curnow & Baker's method in relation to Robertson's approach.It is found that the one real fault in the Curnow-Baker method is its tendency to fix too high a proportion of the genes, particularly when the initial gene frequency is near to a fixation point. This fault is largely overcome when more individuals are selected. For selection of eight or more individuals the Curnow-Baker method is very accurate and appreciably faster than the transition matrix method.


Genetics ◽  
1994 ◽  
Vol 136 (1) ◽  
pp. 361-381
Author(s):  
T Nagylaki

Abstract A model for the evolution of the local averages of a quantitative character under migration, selection, and random genetic drift in a subdivided population is formulated and investigated. Generations are discrete and nonoverlapping; the monoecious, diploid population mates at random in each deme. All three evolutionary forces are weak, but the migration pattern and the local population numbers are otherwise arbitrary. The character is determined by purely additive gene action and a stochastically independent environment; its distribution is Gaussian with a constant variance; and it is under Gaussian stabilizing selection with the same parameters in every deme. Linkage disequilibrium is neglected. Most of the results concern the covariances of the local averages. For a finite number of demes, explicit formulas are derived for (i) the asymptotic rate and pattern of convergence to equilibrium, (ii) the variance of a suitably weighted average of the local averages, and (iii) the equilibrium covariances when selection and random drift are much weaker than migration. Essentially complete analyses of equilibrium and convergence are presented for random outbreeding and site homing, the Levene and island models, the circular habitat and the unbounded linear stepping-stone model in the diffusion approximation, and the exact unbounded stepping-stone model in one and two dimensions.


1975 ◽  
Vol 25 (2) ◽  
pp. 89-94 ◽  
Author(s):  
Edward Pollak ◽  
Barry C. Arnold

SUMMARYThe distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of bij's, where bij is the mean number of visits to frequency j/2N, given initial frequency i/2N. The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.


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