数理解析研究所講究録
1193 巻 2001 年 54-63

54Compensatory Neutral Mutation Model with Dominance
Masaru Iizuka1 (飯塚勝) and Tomoyuki Kado2 (角友之)
1 Divisionof Mathematics, Kyushu Dental College
Kokurakita-ku, Kitakyushu 803-8580, Japan
Manazuru,
2-6-1
2 Departmentof Biology, Graduate School of Science, Kyushu University
4-2-1 Ropponmatsu, Chuo-ku, Fukuoka 810-8560, Japan
Abstract
We consider two loci (or DNA sites) in which mutation occurs from the wild-type allele) in the first (resp. second) locus. Gene
(resp.
(resp.
) to a mutant allele, are deleterious. The
orcombinations or haplotypes with a single mutation,
double mutant, however, does not have the harmful effect of the single mutations. Thismodel is a two-locus two-allele model with compensatory fitness interaction. For this,model with finite population size, the time until fixation of the double mutant,, is investistarting from the state in which the population consists exclusively ofgated. The average time until fixation and other aspects of the time until fixation arecompared. The effect of dominance in diploid population makes the average time until
fixation much shorter than the case of no dominance.
$A_{1}$$A_{2}$$B_{1}$$B_{2}$$A_{1}B_{2}$$A_{2}B_{1}$$A_{2}B_{2}$$A_{1}B_{1}$1. Introduction
A mutation may cause a structural change in the gene product, and is deleterious when thechange is harmful. A second mutation, however, may bring a change by which the structureof the gene product is almost the same as that of the original molecule, and the protein may
be functional (Chelvanayagam et al. (1997), Lei et al. (2000), Stephan and Kirby (1993),
Wilson et al. (1992)
The evolutionary model discussed here is one in which the first mutation is deleterious
and the second mutation compensates for the deleterious effect of the first mutation. Under
continued mutation pressure, the fixation of the double mutant is certain. If it takes muchtime until fixation of the double mutant, however, molecular evolution by compensatory
fitness interaction would not be common. In other words, the time until fixation is not long
if this kind of evolution is an important one.
proposed a population genetics model called the compensatory neutral
Kimuramutation model. This is a two-locus two-allele model in which a single mutation at either
one of the two loci is deleterious but mutations at both loci confer the same fitness as thewild type. This model incorporates compensatory fitness interaction. Under the stochastic
effect of random genetic drift, the time until fixation is a random variable. The main result ofis that the average time until fixation of the double mutant starting from
Kimuraa population consisting exclusively of the wild type increases quickly as the deleterious effectof single mutants increases when the linkage of two loci is weak. In other words, molecular
evolution by compensatory fitness interaction is difficult when two loci are loosely linked. To
have this conclusion, a haploid model or a diploid model with no dominance is assumed (seealso Iizuka and Takefu (1996) and Stephan (1996) .
$)$$(1985\mathrm{a}, \mathrm{b})$$(1985\mathrm{a}, \mathrm{b})$

$)$

55
If the distribution of the time until fixation ofhas a tail toward the infinity, theaverage time until fixation may not be an appropriate quantity to analyze this model. In otherwords, even if the average time until fixation is large, molecular evolution by compensatory
fitness interaction may be possible, for example, when 10 percent point of the distributionfunction of the time until fixation is not large. For this reason, it is interesting to see whether
the conclusion on the possibility of molecular evolution by compensatory fitness interactionis altered by using other aspects of fixation time than the average time until fixation or not.
For this reason, we will introduce new quantities associated to the distribution of time until
fixation in this paper.
Further, the deleterious effects of single mutant may be weaken by dominance effects of
selection for diploid organisms. Then the effects of dominance for diploid organisms seemto enlarge the possibility of molecular evolution by compensatory fitness interaction. In thispaper, we consider a diploid model with dominance and show that the conclusion obtained by
Kimurathat molecular evolution by compensatory fitness interaction is difficult
when two loci are loosely linked is not applicable to the diploid model with dominance.
$A_{2}B_{2}$$(1985\mathrm{a}, \mathrm{b})$2. ModelWe consider the following Wright-Fisher model with two loci and two alleles for a randomly
mating, diploid population with the effective size $N$ . For the first (resp. second) locus, we(resp.
) the wild-type allele that mutates to allele $A_{2}(B_{2})$ . We use locus
denote byeven for DNA site in a gene in this paper.
haplotype $(i, j=1,2)$ in generation
Let $X_{ij}(n)$ be the relative frequency of$(n=0,1,2,\ldots)$ . The genetic composition of the population $X(n)=(X12(n), x_{2}1(n),$ $X22(n))$
changes deterministically by mutation, recombination and selection. Further, $X(n)$ changes
stochastically by random genetic drift.
.
$Y_{ij}=Y_{ij}(n)$ofafter mutation is given by
The relative frequency$A_{1}$$B_{1}$$n$$A_{i}B_{j}$1$A_{i}A_{j}$$Y_{11}=(1-2u)X11(n)$$Y_{12}=(1-u)X_{12}(n)+ux_{11}(n)$$Y_{21}=(1-u)X_{21}(n)+ux_{11}(n)$$Y_{22}=X_{22}(n)+ux_{12}(n)+uX_{21}(n)$ ,where is the mutation rate per generation.
By recombination, $Y_{ij}(n)$ changes to $Z_{ij}=Z_{ij}(n)$ as$u$$Z_{ij}=Y_{ij}-(-1)^{i}+jDr$ ,where$D=\mathrm{Y}_{11}Y22-\mathrm{Y}12\mathrm{Y}21$

andis the recombination fraction between the loci per generation.
By selection, $Z_{ij}(n)$ changes to $Q_{ij}=Q_{ij}(n)$ as follows. Let$r$$p_{ij}^{kl}=Zijzk\iota$

56
be the relative frequency of$(A_{i}B_{j,\mathrm{t}}A_{k}B)$

individuals. Note that
$p_{ij}^{klij}=p_{kl}$and$i,j,k, \iota\sum_{=1}^{2}pij1k\iota_{=}$

be the relative fitness ofetuals. By definition,$\tilde{\mathrm{L}}$$w_{ij}^{kl}$.

individuals relative to $(A_{1}B_{1}, A_{1}B_{1})$ individ-$(A_{i}B_{j}, A_{k}B_{l})$$w_{11}^{11}=1$$w_{ij}^{k\iota}=wk\iota ij,$and by compensatory neutral fitness interaction, we assume that$0<w_{i}k\mathrm{t}j\leq 1,$$w_{22}^{22}=1$

We will consider four cases of assigning the value ofchanges to.

$w_{ij}^{kl}$later in this section. By selection,$p_{ij}^{kl}$$q_{ij}^{kl}= \frac{w_{ijij}^{k\iota_{p}}kl}{\overline{W}}$

,

where
$\overline{W}=\sum_{ki,,j,,\iota=1}w_{ij}^{kl}p_{i}^{k\mathrm{t}}2j$

is the mean of relative fitness of the population and note that$q_{ij\iota}^{k\mathrm{t}ij}=qk$

Then the relative frequency of$A_{i}B_{j}$.

haplotype after selection is$Q_{ij}= \sum_{=k,,\iota 1}^{2}q^{kl}ij$.

As concrete examples of compensatory fitness interaction, we consider the following four
selection schemes.
Selection Scheme [I]: No dominance and tight coupling
individual produces two kinds of molecules
In the case of tight coupling, each
andwith the same proportion. The relative fitness ofandcoded byindividual$1-s_{kl}$oftherelative
fitness$1-s_{ij}$
Then
the
,
respectively.
and
are
is
,
$(A_{i}B_{j}, A_{k}B\downarrow)$$A_{i}B_{j}$$A_{i}B_{j}$$A_{k}B_{l}$$(A_{i}B_{j}, AkB\downarrow)$$w_{ij}^{k\iota_{=\frac{1-s_{ij}}{2}+\frac{1-s_{kl}}{2}}}$where$s_{11}=s_{22}=0,$$s_{12}=s21=s$ .

$A_{k}B_{l}$57Selection Scheme [II]: No dominance and free coupling
In the case of free coupling, each $(A_{i}B_{j,k}AB\iota)$ individual produces four kinds of molecules$AB$ andcoded by $A_{i}B_{j,kj}A_{i}Bl,$,
with the same proportion. The relative fitness of$1-s_{il},$$1-s_{kj}$ and $1-skl,$
andare $1-s_{ij},$. Then thefitness of theindividual is$A_{k}B_{l}$$A_{i}B_{l},$$A_{k}B_{j}$$A_{i}B_{j}$resp,e$A_{k}B_{l}$r.elative$\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{V}.\mathrm{e}\mathrm{l}\mathrm{y}$$(A_{i}B_{j}, A_{k}B_{l})$,$w_{ij}^{kl}= \frac{1-s_{ij}}{4}+\frac{1-s_{il}}{4}+\frac{1-s_{kj}}{4}+\frac{1-s_{kl}}{4}$where$s_{11}=s_{22}=0$,$s_{12}=s_{21}=S$.

Selection Scheme [III]: Dominance and tight coupling
In this case, each $(A_{i}B_{j,kl}AB)$ individual produces the molecules coded by
andwith the same proportion. By dominance effect, the relative fitness of an individual with at
leastor
is 1. Then the relative fitness of $(A_{i}B_{j,k}AB\iota)$ individual is$A_{i}B_{j}$$A_{1}B_{1}$$A_{k}B_{l}$$A_{2}B_{2}$$w_{11}^{kl}=w_{22}=w_{i}k\mathrm{t}1j1=wi22j=1,$$w_{ij}^{kl}=1-s$,otherwise.

Selection Scheme [IV]: Dominance and free coupling
In this case, each $(A_{i}B_{j}, A_{kl}B)$ individual produces the molecules coded by $A_{i}B_{j,\iota,kj}A_{i}BAB$
andwith the same proportion. By dominance effect, the relative fitness of an individualwith $i=j$ or $i=l$ or $k=j$ or $k=l$ is 1. Then the relative fitness of the relative fitness of
individual is$w_{12}^{12}=w^{2}2^{1}1=1-s,$,$A_{k}B_{l}$$(A_{i}B_{j}, A_{k}B_{\mathrm{t}})$$w_{ij}^{k}=1l$otherwise.
For the case that two loci is closely linked, tight coupling is realistic. On the other hand, freecoupling may be realistic when these loci code two subunits of a molecule that combine toform the total molecule. Note that selection scheme [I] is the compensatory neutral mutation
if $2N$ in our paper
model of haploid population that was investigated by Kimura
is replaced by $N$ .
By random sampling drift, $Q_{ij}(n)$ changes to $X_{ij}(n+1)$ stochastically. The transition
probability law of $X(n+1)$ conditional on $(Q_{12}(n), Q_{21}(n),$ $Q_{2}2(n))$ is given by$(1985\mathrm{a}, \mathrm{b})$$P(X_{12}(n+1)= \frac{m_{12}}{2N}, X_{21}(n+1)=\frac{m_{21}}{2N},$ $X_{22}(n+1)= \frac{m_{22}}{2N}|Q12(n),$ $Q21(n),$ $Q22(n))$$= \frac{m_{11}!m_{12}!m_{2}1!m22!}{(2N)!}Q_{11}^{m_{11}}Q_{12}m12Q_{2}m121Q_{22}^{m_{22}}$where$m_{11}+m_{1}2+m_{21}+m22=2N$

,

58
and
$Q_{1}1=1-Q12-Q21-Q_{2}2$ .

3. Diffusion Model
Let $x(t)=(x_{12}(t), x21(t),$ $x22(t)),$ $0\leq t<\infty$ be the diffusion model corresponding to the
The continuous time stochastic process $x(t),$ $t\geq 0$
Wright-Fisher model $X(n),$ $n=0,1,2,$
is a three dimensional diffusion process on $\{(x_{12}, X21, X_{22})\in R^{3}|x_{ij}\geq 0, x_{12}+x_{21}+x_{22}\leq 1\}$
with the infinitesimal generator$..$.

$L= \frac{1}{2}\sum_{)(i,j),(k,l)=(1,2),(2,1),(2,2}\frac{x_{ij}(\delta ik\delta_{j}\downarrow-X_{kl})}{2N}\frac{\partial^{2}}{\partial x_{ijl}\partial_{X_{k}}}$$+ \sum_{)(i,j)=(1,2),(2,1),(2,2}\{fij(x_{12}, X21, x_{2}2)$$-(-1)^{i+j}r(X11^{X-}22x12X_{21})+x_{ij(} \sum_{k,\iota=}21\tilde{w}^{k}ijx_{k}\iota\downarrow-\tilde{W})\}\frac{\partial}{\partial x_{ij}}$

,

where
$f_{12}(x_{11}, X_{12}, X_{21})=u(_{X_{1}-}1X_{12})$$f21(x_{11}, X12, X21)=u(x_{11}-x_{21})$$f_{22}(x_{1}1, X_{12}, X_{21})=u(X12+x_{21})$$kl$$\tilde{w}_{ij}^{k\iota_{=}}s\mathrm{l}\mathrm{i}\mathrm{m}Sarrow 0\frac{w_{ij}}{s}$$\tilde{W}=s\lim_{sarrow 0}\frac{\overline{W}}{s}$and $x_{11}=1-x_{12}-x_{21}-X_{22}$ . Note that
selection scheme
in Section 4, however,$\tilde{w}_{ij}^{kl}=w_{ij}^{kl}$$\mathrm{I}\sim \mathrm{I}\mathrm{V}$for selection scheme [I]. Let$\sim[\mathrm{I}\mathrm{V}]$. For$\tilde{w}_{ij}^{kl}\neq w_{ij}^{kl}$$T(x_{12,21,22}xX)= \inf\{t|X_{22}(t)=1, (x_{12}(\mathrm{o}), X21(0), X22(0))=(x_{12}, X21, X_{22})\}$

be the time until fixation of
The average time until fixation$A_{2}B_{2}$conditional on $(x_{12}(0), x_{21}(0),$ $X_{22}(\mathrm{o}))=(X_{1221,22}, XX)$ .

$\overline{T}(x_{12,2}x1, X_{22})=E[T(x12, x21, X22)]$

satisfies$L\overline{T}(x_{12}, x21, X22)+1=0$,(1)

with $\overline{T}(0,0,1)=0$ andbeing finite.
Kimurainvestigated the case of selection scheme [I] by computer simulationof the Wright-Fisher model and by an approximate method for the diffusion model. Stephan(1996) analyzed some extension of selection scheme [I] by an approximate method proposed. Iizuka and Takefu (1996) considered some extension of selection
by Kimurascheme [I] by solving the partial differential equation (1) numerically.
$\overline{T}(0,0,0)$$(1985\mathrm{a}, \mathrm{b})$$(1985\mathrm{a}, \mathrm{b})$59$\bullet$

4. Results and Discussion
It is difficult to solve the partial differential equation (1) analytically. Further, we are interested in not only the average time until fixation but also the distribution of the time until
fixation as mentioned in Introduction. It is also difficult to obtain the properties of distribution function by analytical method. For these reasons, we will investigate the compensatory
neutral mutation model with dominance by computer simulations. Computer simulations are
performed as follows. For the initial condition that $(X_{12}(0), x_{2}1(0),$, thegenetic composition of the population is changed deterministically by mutation, recombination and selection using formulas in Section 2. Then random sampling drift is performed using
the rejection method (Press et al. (1992)), by which we obtain $(X12(1), x_{2}1(1),$ $X22(1))$ . We
iterate this operation until $X_{22}(\tau)=1$ . For this sample path, the time until fixationis . We repeat independently this procedure $m$ times. We denote by the time until fixation in the i-th simulation. We can obtain the distribution of. andby
the average time of fixation can be obtained by$\overline{T}$$x_{2}2(\mathrm{o}))=(0,0, \mathrm{o})$$T(\mathrm{O}, 0,0)$$\tau$$\tau_{i}$$T(\mathrm{O}, 0,0)$$\{\tau_{i}\}_{i=1},2,\ldots,m$$\overline{T}=\overline{T}(0,0,0)=\frac{1}{m}\sum_{i=1}^{m}\mathcal{T}_{i}$The selection schemes described in Section 2 are refered to as additive model. In our
simulations, we use the alternative model of multiplicative model defined in the following.

Selection Scheme I: No dominance and tight coupling
The relative fitness of the $(A_{i}B_{jkl}, AB)$ individual is$w_{ij}^{kl}=(1-S_{ij})(1-S_{kl})$

,

where
$s_{11}=s_{22}=0,$$s_{12}=s_{21}=s$.

Selection Scheme II: No dominance and free coupling
The relative fitness of the $(A_{i}B_{j,l}A_{k}B)$ individual is$w_{ij}^{k\iota_{=\sqrt{(1-s_{ij})(1-Si\downarrow)(1-skj)(1-S_{k\downarrow)}}}}$

,

where
$s_{11}=s_{22}=0,$$s_{12}=s_{21}=s$.

Selection Scheme III: Dominance and tight coupling
The relative fitness of$(A_{i}B_{j}, AkB_{\iota})$individual is$w_{11}^{k\downarrow}=w_{2}^{k}2\iota 11=wij=wij=122,$$w_{ij}^{k\iota_{=}2}(1-S)$,

60
otherwise.

Selection S.cheme$\mathrm{I}\mathrm{V}$: Dominance and free coupling

The relative fitness of the relative fitness of$(A_{i}B_{j}, A_{k\iota}B)$$w_{12}^{12}=w^{21}21=(1-s)^{2},$individual is$w_{ij}^{k}=1l$,otherwise.
The additive model and the multiplicative model for each selection scheme are almost thesame if the selection parameter $s(0<s<1)$ is small and the parameter in the additivemodel is replaced by in the multiplicative model.
We use $N=50,2Nu=1$ and $m=10000$ in our simulations. The distribution functionof the time until fixation $T=T(\mathrm{O}, 0,0)$ for election scheme I is presented in Fig.1. In thisfigure, we use the window of width 50 such as with $50n+1\leq\tau_{i}<50(n+1)$ being plottedon the point $50n+25(n=0,1,2, \ldots)$ . The scaled recombination haction is $2Nr=50$ (loose
linkage). Four cases of the scaled selection parameter ( $2Ns=0,5,10$ and 15) are given inthis figure. Fig.1 shows that the distribution has the tail toward the infinity.
$s$$\frac{s}{2}$$\dot{\mathrm{s}}$$\tau_{i}$Figure 1
Freauencv$\mathrm{X}10000$$\mathrm{T}$Distribution of$\mathrm{T}$for scheme 1$(2\mathrm{N}\mathrm{u}=1,2\mathrm{N}\mathrm{r}=50)$

Mean (the average time until fixation), median, 10 percent point and 90 percent pointof the distribution are shown in Fig.2 for the case of selection scheme I with $2Ns=10$ and$2Nr=50$ . We use again the window of width 50 in this figure. In this case, $T=134$ for
10 percent point, $T=317$ for median, $T=$ 405.8 for mean and $T=792$ for 90 percent
point. The ratioof 10 percent point to mean is 0.330. The ratio is decreases as $2Ns$
increases. For example, $R=$ 0.103 for $2Ns=24$ . The absolute value of 10 percent point,$R$61
however, increases as $2Ns$ increases. Therefore, using 10 percent point of the distribution
may not change the conclusion on the possibility of molecular evolution by compensatory
fitness interaction obtained by using the average time until fixation for large $2Ns$ .

Frequency$\mathrm{x}$Fioure 2

10000$\mathrm{T}$Mean, median, 10 % and 90 % of the distributions

for scheme 1$(2\mathrm{N}\mathrm{u}=1,2\mathrm{N}\mathrm{r}=50,2\mathrm{N}\mathrm{S}=10)$

The average time until fixation
given in Fig.3 for selection scheme$\overline{T}$as a function of the scaled selection parameter $2Ns$ iswith $2Nr=0$ and $2Nr=50$ .

$\mathrm{I}\sim \mathrm{I}\mathrm{V}$$\log_{\mathrm{l}0}\mathrm{T}-$Fioure 3$2\mathrm{N}\mathrm{s}$Effects of Selection Scheme

62
increases quickly as $2Ns$ increases in the case of loose linkage
For selection scheme I,(see also Iizuka and$(2Nr=50)$ . This result is consistent with that of Kimura
Takefu (1996) and Stephan (1996) . For selection scheme II with loose linkage $(2Ns=50)$ ,increase quickly as $2Ns$ increases even in thethis property is more enhanced. Note thatcase of no recombination $(2Nr=0)$ for selection scheme II. The reason of the enhancement
may be that the effect of free coupling is similar to that of recombination in this case.
The effects of dominance are remarkable both for tight coupling and free coupling. For
free coupling (selection scheme IV), is slowly increasing function of $2Ns$ for $2Nr\cdot=0$ and$2Nr=50$ contrast to the case of hee coupling without dominance (selection scheme II). For
tight coupling with dominance (selection scheme III), is much shorter than the case of tightfor the case of loose linkage
coupling without dominance. This tendency is much$\overline{T}$$(1985\mathrm{a}, \mathrm{b})$$)$$\overline{T}$$\overline{T}$$\overline{T}$remarkabl.e$(2Nr=50)$ .

By these results, molecular evolution by compensatory fitness interaction is difficult when
two genes (DNA sites) are far distant (loosely linked) if there is no dominance. By introducing
dominance effects in selection, molecular evolution by compensatory fitness interaction may
be possible even if two genes (DNA sites) are far distant (loosely linked) and the conclusion
does not hold for the compensatory neutral mutation modelobtained by Kimurawith dominance.
Finally, we note that the minimum of is attained by some positive $2Ns$ . This phenomof this phenomena is givenThe
ena seems to be inconsistent with our.
by Iizuka and Takefu (1996).
$(1985\mathrm{a}, \mathrm{b})$$\overline{T}$intuitio.n.

$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}-$The authors are grateful to M. Ichinose and H. Tachida for useful discussion. M. I. is partially
supported by a grant-in-aid from the Ministry of Education, Science and Culture of Japan.

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