Motoo Kimura

A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population

A new model of mutational production of alleles was proposed which may be appropriate to estimate the number of electrophoretically detectable alleles maintained in a finite population. The model assumes that the entire allelic states are expressed by integers (…, A −1 , A 0 , A 1 , …) and that if an allele changes state by mutation the change occurs in such a way that it moves either one step in the positive direction or one step in the negative direction (see also Fig. 1). It was shown that for this model the ‘effective’ number of selectively neutral alleles maintained in a population of the effective size N e under mutation rate υ per generation is given by When 4 N e υ is small, this differs little from the conventional formula by Kimura & Crow, i.e. n e = 1 + 4 N e υ, but it gives a much smaller estimate than this when 4 N e υ is large.

Evolution in Sexual and Asexual Populations

In an asexual population two favorable mutants can be incorporated into the population only if one occurs in a descendant of the individual in which the other occurred. In a sexual population both mutants can be incorporated through recombination. A mathematical formulation is given of the relative rates of incorporation of the new mutations with and without recombination. Recombination is of the greatest advantage when the double mutant is more advantageous than either single mutant, when the mutant effects are small, when mutations occur with high frequency, and when the population is large. On the other hand, for the incorporation of individually deleterious but collectively beneficial mutations, recombination can be disadvantageous. Close linkage has effects similar to those of asexual reproduction. Experimental data on DDT resistance in Drosophila and chloramphenicol resistance in bacteria are cited showing greater development of coadaptation in an asexual system. The evolution of diploidy from haploidy confers an immediate reduction in the mutation load by concealment of deleterious recessives, but this advantage is lost once a new equilibrium is reached. Thus the development of diploidy may be because of an immediate advantage rather than because of any permanent benefit. On the other hand, there are other possible advantages of diploidy, such as heterosis and protection from somatic mutations.

THE MEASUREMENT OF EFFECTIVE POPULATION NUMBER

Abstract : The effective population number can be defined either in terms of the amount of increase in homozygosity (inbreeding effective number) or the amount of gene frequency drift (variance effective number). Under many circumstances these are the same, but not in general. The effective number is considered in terms of an idealized population in which each individual parent has an equal expectation of progeny. The effective number of an actual population is defined as the size of an idealized population with the same amount of inbreeding or random gene frequency drift as the population under consideration. Formulae are given for determining both kinds of effective numbers when the population is monoecious (including self fertilization) and when there are separate sexes. The formulae are summarized along with special cases of interest. (Author)

The role of compensatory neutral mutations in molecular evolution

A pair of mutations at different loci (or sites) which are singly deleterious but restore normal fitness in combination may be called compensatory neutral mutations. Population dynamics concerning evolutionary substitutions of such mutants was developed by making use of the diffusion equation method. Based on this theory and, also, by the help of Monte Carlo simulation experiments, a remarkable phenomenon was disclosed that the double mutants can easily become fixed in the population by random drift under continued mutation pressure if the loci arc tightly linked, even when the single mutants are definitely deleterious. More specifically, I consider two loci with allelesA andA’ in the first locus, and allelesB andB’in the second locus, and assign relative fitnesses 1, 1-s’, 1-s’ and 1 respectively to the four gene combinationsAB, A’B, AB’ andA’B’, wheres’ is the selection coefficient against the single mutants (s’ > 0). Letv be the mutation rate per locus per generation and assume that mutation occurs irreversibly fromA toA’ at the first locus, and fromB toB’ at the second locus, whereA andB are wild type genes, andA’ andB’ are their mutant alleles. In a diploid population of effective size Ne (or a haploid population of 2Ne breeding individuals), it was shown that the average time (T) until joint fixation of the double mutant (A’B’) starting from the state in which the population consists exclusively of the wild type genes (AB) is not excessively long even for large 4Nes’ values. In fact, assuming2Nev = 1 we have -T = 54Ne for 4Nes’ = 400, and -T = 128Ne for 4Nes’ = 1000. These values are not unrealistically long as compared with -T~ 5Ne obtained for 4Nes’ = 0. The approximate analytical treatment has also been extended to estimate the effect of low rate crossing over in retarding fixation. The bearing of these findings on molecular evolution is discussed with special reference to coupled substitutions at interacting amino acid (or nucleotide) sites within a folded protein (orrna) molecule. It is concluded that compensatory neutral mutants may play an important role in molecular evolution.

On the stochastic model for estimation of mutational distance between homologous proteins.

SummaryA set of simple equations is derived which gives the relationship between the observed amino acid differences per 100 codons and the evolutionary distance per 100 codons using Holmquists stochastic model of molecular evolution.

Genetic variability maintained in a finite population due to mutational production of neutral and nearly neutral isoalleles

1. The average and the effective numbers of alleles maintained in a finite population due to mutational production of neutral isoalleles were studied by mathematical analysis and computer simulation. 2. The exact formula was derived for the effective number ( n e ) of alleles maintained in a population of effective size N e , assuming that there are K possible allelic states and mutation occurs with equal frequency in all directions. If the number of allelic states is so large that every mutation is to a new, not pre-existing, allele, we have n e = 4 N e u +1 − 2 N e u 2 , where u is the mutation rate. Thus, the approximation formula, n e = 4 N e u +1, given by Kimura & Crow (1964) is valid as long as 2 N e u 2 ≪ 1. 3. The formula for the average number of alleles ( n a ) maintained in a population of actual size N and effective size N e was derived by using the method of diffusion approximation. If every mutation is to a new, not pre-existing, allele, we obtain where M = 4 N e u . The average number of alleles as a function of M and N is listed in Table 1. 4. In order to check the validity of the diffusion approximations, Monte Carlo experiments were carried out using the computer IBM 7090. The experiments showed that the approximations are satisfactory for practical purposes. 5. It is estimated that among the mutations produced by DNA base substitutions, synonymous mutations, that is, those which cause no alterations of amino acids, amount roughly to 0·2–0·3 in vertebrates. Incompletely synonymous mutations , that is, those which lead to substitution of chemically similar amino acids at a different position of the polypeptide chain from the active site and therefore produce almost no phenotypic effects, must be very common. Together with synonymous mutations, they might constitute at least some 40% of all mutations. These considerations suggest that neutral and nearly neutral mutations must be more common than previously considered.

On the constancy of the evolutionary rate of cistrons.

SummaryThe variations of evolutionary rates in hemoglobins and cytochrome c among various lines of vertebrates are analysed by estimating the variance. The observed variances appear to be larger than expected purely by chance.If the amino acid substitutions in evolution are the result of random fixation of selectively neutral or nearly neutral mutations, the evolutionary rate of cistrons can be represented by the integral of the product of mutation rate and fixation probability in terms of selective values around the neutral point. This integral is called the effective neutral mutation rate.The influence of effective population number and generation time on the effective neutral mutation rate is discussed. It is concluded that the uniformity of the rate of amino acid substitutions over diverse lines is compatible with random fixation of neutral or very slightly deleterious mutations which have some chance of being selected against during the course of substitution. On the other hand, definitely advantageous mutations will introduce significant variation in the substitution rate among lines. Approximately 10% of the amino acid substitutions of average cistrons might be adaptive and create slight but significant variations in evolutionary rate among vertebrate lines, although the uniformity of evolutionary rate is still valid as a first approximation.

On the evolutionary adjustment of spontaneous mutation rates

Evolutionary factors which tend to decrease the mutation rate through natural selection and those which tend to increase the mutation rate are discussed from the standpoint of population genetics. The authors theory of optimum mutation rate based on the principle of minimum genetic load is re-examined, assuming that mutation rate is adjusted in the course of evolution in such a way that the sum of mutational and substitutional load is minimized. Another hypothesis is also examined that only selection toward lowering the mutation rate is effective and the present mutation rate in each organism represents the physical or physiological limit that may be attained by natural selection. The possibility cannot be excluded that the spontaneous mutation rate is near the minimum that may be attained under the present mode of organization of the genetic material, and at the same time is not very far from the optimum in the sense of minimizing the genetic load.

On the maximum avoidance of inbreeding

Mating systems in which the least related individuals are mated have been designated by Wright as having maximum avoidance of inbreeding. For such systems the initial rate of decrease in heterozygosity is minimum. However, some other systems have a lower rate of decrease in later generations. Circular mating, in which each individual is mated with the one to his right and to his left, leads to an asymptotic rate of decrease in heterozygosity of 1– λ ˜ π 2 /(2 N + 4) 2 compared with 1/4 N for maximum avoidance systems. Circular pair mating, in which for example each male progeny is moved one cage to the right, leads to 1– λ ~ π 2 /( N + 12) 2 . Other similar systems are discussed. For minimum gene frequency drift, a mating system should have a constant number of progeny per parent and the population should be broken up as rapidly as possible into the maximum number of lines. The gene frequency variance at generation T within a line is where N is the number in the line and H t is the proportion of heterozygotes in generation t . Although the three mating systems, circular, circular pair, and maximum avoidance (and many others) have the same amount of random drift ultimately, at any generation circular mating has the smallest drift variance, V T , and circular pair next smallest.

Linkage disequilibrium due to random genetic drift

The behaviour of linkage disequilibrium between two segregating loci in finite populations has been studied as a continuous stochastic process for different intensity of linkage, assuming no selection. By the method of the Kolmogorov backward equation, the expected values of the square of linkage disequilibrium z 2 , and other two quantities, xy (1 − x ) (1 − y ) and z (1 − 2 x ) (1 − 2 y ), were obtained in terms of T , the time measured in N e as unit, and R , the product of recombination fraction ( c ) and effective population number ( N e ). The rate of decrease of the simultaneous heterozygosity at two loci and also the asymptotic rate of decrease of the probability for the coexistence of four gamete types within a population were determined. The eigenvalues λ 1 , λ 2 and λ 3 related to the stochastic process are tabulated for various values of R = N e c .