Abstract
Dominance divides the fixation of favourable mutations in diploids, and additive ones come near half the haploid value (Paper 982). Different numbers of males and females (the sex ratio) were not included there. For a diploid population of 40 males and females in total, this paper solves exactly a transition matrix counting mutant copies in males and in females separately, and compares how the fixation probability differs for a mutation arising in a male or a female with the effective-size formula. No new theorem or law is claimed.
First, under neutrality, a mutation in the rarer sex fixes more readily: a mutation in a male agreed with 1/(4 times the number of males) and in a female with 1/(4 times the number of females) within $10^{-10}$; with 2 males and 38 females these were $\mathbf{0.125000}$ and $0.006579$ (Section 2).
Second, the neutral average does not depend on the sex ratio: the average weighted by the numbers of males and females in which mutations arise was 1/(2N) = $0.012500$ for all four sex ratios (Section 2).
Third, favourable mutations arising in males also fix far more readily: with advantage 0.05 and 2 males to 38 females, a mutation in a male fixed with $0.212137$ and in a female with $0.012209$, a ratio of $\mathbf{17.375044}$ (19 under neutrality) (Section 3).
Fourth, the average for favourable mutations is further below the effective-size formula: the average over the diffusion formula with effective size was $0.970541$ at 20:20 and $\mathbf{0.921808}$ at 2:38, smaller with fewer males; relative to the equal sex ratio, 2:38 gave $0.240347$ exactly against $0.253053$ by the formula (Section 3).
Fifth, the separator is whether one looks at the sex in which a mutation arose or at the population average: a single copy weighs more in the rarer sex, and under neutrality that weight was exactly the fixation probability. Looking only at the average, the sex ratio vanished under neutrality and departed from the effective-size formula for favourable mutations (Section 4).
Keywords: sex ratio, effective population size, fixation probability, diploid, transition matrix, population genetics.
Scope (what is not claimed)
No new theorem or law is claimed: that unequal numbers of males and females give an effective size of 4 times males times females over their sum, that a neutral mutation’s fixation probability equals the share of copies it represents, and that each sex contributes half the genes of the next generation, are known in population genetics. No real values are claimed: 40 individuals in total, male to female numbers 20:20, 10:30, 5:35 and 2:38, and a per-copy advantage of 0.05 are model values and correspond to no animal. The model is not extended: one autosomal locus, with the same advantage applied to the copy frequency in each sex and offspring frequency the mean of the two sexes. Sex chromosomes, sex determination, sex-specific advantages and overlapping generations are not included. Mechanisms are not separated: the departure of the average from the effective-size formula for favourable mutations is not split into the error of the diffusion approximation and the effect of the sex ratio. Relation to earlier papers: Paper 982, in its scope note, did not extend the model, taking randomly mating diploids with non-overlapping generations and leaving out inbreeding, sex ratio, overlapping generations, linkage between genes and the mechanism of dominance; this paper sits in the seat of sex ratio (no earlier occupant; the inbreeding seat was taken by Paper 984). Paper 982 showed that dominance divides the fixation of favourable mutations in diploids, and that the diffusion formula exceeds the exact transition-matrix solution. Here, again by transition matrix, different numbers of males and females are added. Paper 984 showed that inbreeding lifts recessive fixation most. Here not inbreeding but imbalance in the numbers of the sexes is counted. What is added: finding exactly the fixation probabilities of mutations in males and in females for four sex ratios, confirming agreement with 1/(4 times the number of that sex) under neutrality, confirming that the weighted average does not depend on sex ratio under neutrality, comparing the average for favourable mutations with the effective-size formula, and placing the separator on “which sex, or the average”.
1. Notation and the Question
Model: at one autosomal locus of a diploid population with $N_m$ males and $N_f$ females ($N_m+N_f=40$), let $x$ of the $2N_m$ male copies and $y$ of the $2N_f$ female copies be mutant. Selection with per-copy advantage $s$ takes each sex’s frequency $p$ to $p(1+s)/(1+ps)$, the offspring frequency is the mean $\bar p$ of the two sexes, and the next generation’s $2N_m$ male and $2N_f$ female copies are drawn independently from binomials with $\bar p$. The transition matrix over states $(x,y)$ (row sums off by at most $5.1\times10^{-14}$) was solved for the probability that all copies become mutant, starting with one mutant copy in a male or in a female.
Formula compared: with effective size $N_e=4N_mN_f/(N_m+N_f)$ and initial frequency $p_0=1/(2N)$,
$$u=\frac{1-e^{-4N_e s p_0}}{1-e^{-4N_e s}}$$and $u=p_0$ under neutrality. The exact solutions were averaged with weights given by the numbers of males and females in which mutations arise ($N_m$ and $N_f$).
Question: with unequal numbers of males and females, how does the fixation probability differ by the sex in which a mutation arises, and how does the population average agree with the effective-size formula?
2. Neutral Mutations
Computation 2.1: fixation probability under neutrality ($s=0$).
| males : females | effective size | mutation in a male | in a female | average |
|---|---|---|---|---|
| 20 : 20 | $40.0000$ | $0.012500$ | $0.012500$ | $0.012500$ |
| 10 : 30 | $30.0000$ | $0.025000$ | $0.008333$ | $0.012500$ |
| 5 : 35 | $17.5000$ | $0.050000$ | $0.007143$ | $0.012500$ |
| 2 : 38 | $7.6000$ | $\mathbf{0.125000}$ | $0.006579$ | $0.012500$ |
For all four sex ratios, a mutation in a male agreed with $1/(4N_m)$, in a female with $1/(4N_f)$, and the average with $1/(2N)$, within $10^{-10}$.
3. Favourable Mutations
Computation 3.1: fixation probability with advantage $s=0.05$.
| males : females | in a male | in a female | male over female | average | average over effective-size formula |
|---|---|---|---|---|---|
| 20 : 20 | $0.092390$ | $0.092390$ | $1.000000$ | $0.092390$ | $0.970541$ |
| 10 : 30 | $0.135656$ | $0.047478$ | $2.857251$ | $0.069522$ | $0.959773$ |
| 5 : 35 | $0.159990$ | $0.024608$ | $6.501456$ | $0.041531$ | $0.940902$ |
| 2 : 38 | $0.212137$ | $0.012209$ | $\mathbf{17.375044}$ | $0.022206$ | $\mathbf{0.921808}$ |
The effective-size formula gave $0.095195$, $0.072436$, $0.044140$ and $0.024089$ from 20:20 to 2:38. Relative to the equal sex ratio, the average for 10:30, 5:35 and 2:38 was $0.752485$, $0.449519$ and $0.240347$ exactly and $0.760927$, $0.463679$ and $0.253053$ by the formula (exact over formula $0.988906$, $0.969461$ and $0.949788$).
Observation 3.2 for favourable mutations, the effect of sex ratio was slightly stronger than the effective-size formula. Even at 20:20 the average was $0.970541$ of the formula, which includes the error of the diffusion approximation independent of sex ratio: relative to the equal sex ratio, the exact value at 2:38 was $0.949788$ of the formula. Male over female was $2.857251$, $6.501456$ and $17.375044$ with advantage 0.05, against 3, 7 and 19 under neutrality.
4. The Separator
Observation 4.1 the separator is whether one looks at the sex in which a mutation arose or at the population average. Each sex contributes half the next generation’s copies, so a single copy in the rarer sex carries more weight than one in the commoner sex: under neutrality those weights, $1/(4N_m)$ and $1/(4N_f)$, were exactly the fixation probabilities, and averaging by where mutations arise returned $1/(2N)$. For favourable mutations the average departed from the effective-size formula more with fewer males, but that departure was not split into approximation error and the effect of sex ratio.
Observation 4.2 what is not said. Sex chromosomes, sex determination, sex-specific advantages, overlapping generations, the sex ratios of real animals and the shape for larger populations are not treated. What is shown is how, in a model of 40 individuals, fixation probabilities divide by the sex in which a mutation arises and by the sex ratio.
5. Roll Call and Closing
- a transition matrix counting male and female copies separately (Section 1): exact solution.
- neutral mutation in one of 2 males: $\mathbf{0.125000}$ (Computation 2.1): 1/(4 times the number of males).
- the neutral average stays 0.012500 (Computation 2.1): independent of sex ratio.
- male over female for favourable mutations $\mathbf{17.375044}$ (Computation 3.1): fixes more readily in males.
- average over effective-size formula $\mathbf{0.921808}$ (Computation 3.1): departs more with fewer males.
- which sex, or the average (Observation 4.1): the separator; the scope note of Paper 982.
Closing: With few males, mutations arising in males fix more readily, yet the neutral average ignores the sex ratio. Solving by transition matrix with male and female copies counted separately, under neutrality a mutation in a male fixed with 1/(4 times the number of males) and in a female with 1/(4 times the number of females), $0.125000$ and $0.006579$ with 2 males and 38 females, while the average was $0.012500$ for all four sex ratios: with advantage 0.05, male over female was $17.375044$ and the average was $0.921808$ of the effective-size formula. The separator is whether one looks at the sex in which a mutation arose or at the population average. Placed among the earlier papers: to the diploids Paper 982 counted without distinguishing sexes, the sex ratio Paper 982 did not include was added. To be honest: fixation probabilities by sex were set out for one model of 40 individuals. How the sex ratio divides fixation, and no further.
References
- Hartl, D. L., & Clark, A. G. (2007). Principles of Population Genetics (4th ed.). Sinauer.
- Kimura, M. (1962). On the probability of fixation of mutant genes in a population. Genetics.
- Yamagishi, Y. (2026). Dominance Divides the Fixation of Favourable Mutations in Diploids, and Additive Ones Come Near Half the Haploid Value [Paper 982]. YOI: https://www.yamagishiyuuki.com/yoi/982
- Yamagishi, Y. (2026). Inbreeding Lifts Recessive Fixation Most, and at Inbreeding Coefficient 1 Dominance Stops Mattering [Paper 984]. YOI: https://www.yamagishiyuuki.com/yoi/984
Reference
Yamagishi, Y. (2026). With Few Males, Mutations Arising in Males Fix More Readily, Yet the Neutral Average Ignores the Sex Ratio [Paper 994]. YOI: https://www.yamagishiyuuki.com/yoi/994
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 994. This paper is a preprint and has not been peer reviewed.