Copyright 0 1991 by the Genetics Society of America A Study on a Nearly Neutral Mutation Model in Finite Populations Hidenori Tachida National Institute of Genetics, Mishima, Shizuoka-ken 41 1,Japan Manuscript received August 16, 1990 Accepted for publication January 19, 199 1 ABSTRACT As a nearly neutral mutation model, the house-of-cards model is studied in finite populations using computer simulations. The distribution of the mutant effect is assumed to be normal. The behavior is mainly determined by the product of the population size, N, and the standard deviation, u, of the distribution of the mutant effect. If 4Nu is large compared to one, a few advantageous mutants are quickly fixed in early generations. Then most mutation becomes deleterious and very slow increase of the average selection coefficient follows. It takesvery long for the population to reach the equilibrium state. Substitutions of alleles occur very infrequently in the later stage. If 4Na is the order of one or less, the behavior is qualitatively similar to that of the strict neutral case. Gradual increase of the average selection coefficient occurs and in generations of several times the inverse of the mutation rate the population almost reaches the equilibrium state. Both advantageous and neutral (including slightly deleterious) mutations are fixed. Except in the early stage, an increase of the standard deviation of the distribution of the mutant effect decreases the average heterozygosity. The substitution rate is reduced as 4Nu is increased. Three tests of neutrality, one using the relationship between the average and the variance of heterozygosity, another using the relationship between the average heterozygosity and the average number of substitutions and Watterson’s homozygosity test are applied to the consequences of the present model. It is found that deviation from the neutral expectation becomes apparent only when 4Na is more than two. Also a simple approximation for the model is developed which works well when the mutation rate is very small. he mechanismof protein evolution has been one severe competition used in the experiment. Another T of the most debated issues in the study of evo- observation is that of AQUADRO,LADO and NOON lution. KIMURA (1968) emphasized the effectof ran- (1988). They estimated DNA sequence variation ofa dom genetic drift and the neutral theorywhich states region including rosy locus in both Drosophilamela- that the main cause of evolutionary change at the nogaster and Drosophilasimulans. In contrast to the molecular level is random fixation of selectively neu- resultfor protein polymorphism which shows that tral or very nearly neutral mutants rather than posi- both species have almost the same amountof protein tive Darwinian selection is gaining support from most variation(CHOUDHARY and SINCH, 1987),they ob- molecular genetic data rapidly accumulating these ten served that D. simulans has several times more DNA years (see KIMURA 1983, 1987). Although some dis- variation than D. melanogaster has. They hypothesize crepancies between the prediction of the strict neutral that D. simulans has a larger population size and that theory with the assumption of a constant mutation this causes the increase of complete neutral variation rateand observations may exist (GILLESPIE 1987; (DNA variation) in this species while slightly deleteri- TAKAHATA1987), we can say at this point that the ous variation (protein variation) does not increase due effect of random genetic drift is very important in tomore effectiveoperation of selection in larger molecularevolution and should be considered in populations. Finally, GOJOBORI(1 982) observed that models of protein evolution. someof the substrate-specificenzymes involved in On the other hand there are observations which mainpathways or singlepathways have lower vari- indicatethe importance of very weakselection in ancesof heterozygosity than those expected from proteinevolution. DEAN, DYKHIZEN and HARTL complete neutrality. Although statistical significances (1988) have shown that new mutations which caused were not examined and another study which consid- replacements of amino acids in @-galactosidasein Esch- ered statistical significances did not find any discrep- erichia coli have small fitness effects. Effects of many ancy fromthat expected from complete neutrality of them are not detectableby their method, but some (FUERST,CHAKRABORTY andNEI 1977), Gojoborisus- caused detectable increase or decrease of fitness. Al- pects that slightly deleterious mutations (OHTA1973) though the detected selection coefficients are in the have occurred in these enzymes. Thus,is worthwhile it order of one percent, they speculate that the selection to investigate models which incorporate both random coefficients in nature are much smaller because of the genetic drift andweak selection. Genetics 128: 183-192 (May, 1991) 184 H. Tachida OHTA and TACHIDA(1990) proposed a model of MODEL protein evolution in which effects of random genetic Consider a random matingpopulation of N diploid drift and very weak selection are incorporated.In this individuals. The standard Wright-Fisher model in nearly neutral mutation model,the distribution ofthe population genetics is assumed (see, forexample, effect of mutant allele on selection coefficient is fixed. More precisely, the fitness effect of any mutation is a CROWand KIMURA1970). A mutation occurs with a random number sampled from a fixed distribution rate u per generation per gene. If mutation occurs, regardless of the original state of the allele. This the selection coefficient of the mutatedgene is a model is differentfrom previous models of nearly random numbers drawn from afixed distributionf(s) neutral mutation in which the distribution of mutant regardless of the original state of thegene. This effect is shifted so that the difference between the mutation model is the house-of-cards model (KING- original and the mutant alleles has a constant distri- MAN 1977). The fitness of a genotype AiAj is 1 + si + bution (see OHTA 1977;KIMURA 1979). A motivation sJ where si is the selection coefficient of the allele i. In for the model of OHTA and TACHIDA(1 990) is our the presentstudy, we assume that f(s) is normally biological intuition that there must be a limit in the distributed with mean zero and variance a*. Because improvement of a protein and that after major im- we are interested in mutations with very small effects, provements there would be some fine tuning of the the magnitude of u is assumed to be 0(1/N) through- function. In this model, the proportion of advanta- out the paper. The assumption of mean zero is not geous mutations decreases as the population accumu- restrictive since changing the mean by m corresponds lates advantageous mutations and in consequence has to changing the initial selection coefficients by -m in higher average fitness. Such behavior may explain the the zero-mean case. pattern of amino acids substitution in the globin gene Since it is difficult to obtain analytical results for family where adaptations leading to responses to new this model,I used mainly computer simulation to chemical stimuli have evolved by only a few amino investigate theproperties of the model. However, acid substitutions in key positions (PERUTZ 1983).The before going into simulation studies, I try to obtain fixed mutation model is the same as the "house-of- some results using approximations to guide the simu- cards" model of KINCMAN(1978). The model is also lation studies. First 4Nu << 1 is assumed. Then, the adopted in the studies of evolution of quantitative population is mostly monomorphic experiencing in- characters and selection limits (COCKERHAMand frequent transitions among monomorphic states. Let TACHIDA1987; ZENG, TACHIDAand COCKERHAM p(s, t)be the density function of the population being 1989). fixed with an allele whose selection coefficient is s at The behavior of the house-of-cards model in finite time t. From (A2) in APPENDIX, the equilibrium dis- populations with finite allelic states was studied in the tribution p(s) = p(s, ") is equilibrium state by ZENG, TACHIDAand COCKERHAM (1989). However, we are also interested in the tran- sient state since adaptation of protein occurs in this stage. Moreover, their concern was directed toward Puttingf(s) = (&a)" exp(-s2/2a2) into (l), we ob- selection limits in quantitative characters. Thus, quan- tain tities of interest in molecular evolution such as substi- tutionrates and heterozygosities were notstudied p(s) = (&a)"exp[-(s - (2) there. Some aspect of the transient state was studied At equilibrium the effect, s, of the allele fixed in the in OHTAand TACHIDA(1990) in conjunction with population is normally distributed with mean (4Na)a molecular evolution. However, their model incorpo- and variance u2. The distribution does not dependon rates population structures assuming spatial fluctua- tion of selection coefficient and thus systematic study the mutation rate. In other words, the distribution of was not performed due to the computational limita- the allelic effect is shifted upward by 4Na times the tion. In the present study, I investigate the house-of- standard deviation fromthat of theneutral case. cards model in finite panmictic populations. I first Thus, selection is very effective in bringing the pop- examine how long does it take for the population to ulation fitness to a high value if 4Na is, say, greater reach the equilibrium state in this model. Then var- than four. Next we consider the