On Stabilizing the Rate of Isonymy Divergence

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Abstract

A theoretical analysis of the surname state of the population (the vector of namesake concentrations in the male component of the population) and its dynamics as a result of random surname drift is presented. An approximation of such a process by the Wright-Fisher model of a population with non-overlapping generations without selection pressure is used, i.e., an approximation by a sequence of nested random samples with the replacement from fathers’ surnames in the population. The sample size is N/2 according to the size of the male component in the population of size N. In the same population, processes of random drift of both surnames and genes simultaneously occur. Their cardinal difference is that the sample size of surnames is four times smaller than the sample size of autosomal locus alleles. The analysis of random drift is simplified when moving from concentration coordinates to the square roots of them. As generations change, the state receives a sample deviation, measured by angular distance, and its mean square gives the rate of divergence, stabilizing in the new coordinates. An adaptation (in relation to the analysis of surname drift) of a known in population genetics result about the nature of divergence at a stage of a relatively small number of generations compared to the size of the population is given. The divergence of surnames occurs four times faster than the divergence of allele concentrations.

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About the authors

V. P. Passekov

Federal Research Center “Computer Science and Control”, Russian Academy of Sciences

Author for correspondence.
Email: pass40@mail.ru
Russian Federation, Moscow 119991

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2. Fig. 1. The space of population states in different coordinate systems. a – shaded angle θ between the radius vector y₀ and the abscissa axis; b – shaded part of the plane as the space of population states in terms of group concentrations; c – shaded part of the sphere as the space of population states in terms of square roots of concentrations. See text for explanations.

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3. Fig. 2. Between-group, within-group, and complete covariance matrices of population state concentrations: □ – population designation; y(0) – state of the parent population; y(1) – random states of its descendants, first-generation populations, the spread of possible states y(1) is characterized by the covariance matrix V(y(1)|y(0)), which serves as the between-group covariance matrix for the next-generation populations, where y(0) is fixed; y(2) – random states of populations, second-generation descendants of the parent population. They form a metapopulation consisting of groups originating from individual first-generation populations and with within-group covariance matrices V(y(2)|y(1)). Here y(1) varies randomly between groups; The next line refers to the second generation without dividing its populations into groups, the spread of which is characterized by the full covariance matrix Vs(y(2)). The arrows pointing from top to bottom connect the parent population with the offspring population.

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