### Abstract

The genotypic variance within, sigma 2w, and between, sigma 2b, random mating populations and rates and times for convergence to equilibrium values from different founder populations are formulated for an additive genetic model with an arbitrary number of alleles k, number of loci m, population size N, and mutation rate u, with unequal mutation rates for alleles. As a base of reference, the additive variance sigma 2a in an infinite equilibrium population is used. sigma 2a increases as k increases and decreases with variation in the mutation rates. Both transitional and equilibrium values of the variance within populations could be expressed as sigma 2w = (1 - theta)sigma 2a, where theta is the coancestry with mutations of individuals within populations. Thus, rates of convergence and evolutionary times are a function of those for theta, which involves both N and u. When the founder population is fixed, very long times are required to obtain a perceptible increase in sigma 2w and equilibrium values of sigma 2w are very small when 4Nu less than or equal to 10(-1). The variance between populations can be expressed as sigma 2b = 2 theta sigma 2a when the founder population is an infinite equilibrium population, and as sigma 2b = 2(theta - alpha)sigma 2a when the founder population is fixed, where alpha is a function only of u. In this latter case, rates of divergence, while affected by both N and u, are dominated by u and asymptotically a function of u only. With u = 10(-5), very long times (10(3) generations) are required for any perceptible divergence, even for N = 1-10. At equilibrium, most of the variance is between small populations and within very large populations. Migration increases the variance within populations and decreases the variance between populations.

Original language | English |
---|---|

Pages (from-to) | 6205-6209 |

Number of pages | 5 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 84 |

Issue number | 17 |

DOIs | |

Publication status | Published - Sep 1987 |

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### All Science Journal Classification (ASJC) codes

- General

### Cite this

*Proceedings of the National Academy of Sciences of the United States of America*,

*84*(17), 6205-6209. https://doi.org/10.1073/pnas.84.17.6205

**Evolution and maintenance of quantitative genetic variation by mutations.** / Cockerham, C. C.; Tachida, H.

Research output: Contribution to journal › Article

*Proceedings of the National Academy of Sciences of the United States of America*, vol. 84, no. 17, pp. 6205-6209. https://doi.org/10.1073/pnas.84.17.6205

}

TY - JOUR

T1 - Evolution and maintenance of quantitative genetic variation by mutations.

AU - Cockerham, C. C.

AU - Tachida, H.

PY - 1987/9

Y1 - 1987/9

N2 - The genotypic variance within, sigma 2w, and between, sigma 2b, random mating populations and rates and times for convergence to equilibrium values from different founder populations are formulated for an additive genetic model with an arbitrary number of alleles k, number of loci m, population size N, and mutation rate u, with unequal mutation rates for alleles. As a base of reference, the additive variance sigma 2a in an infinite equilibrium population is used. sigma 2a increases as k increases and decreases with variation in the mutation rates. Both transitional and equilibrium values of the variance within populations could be expressed as sigma 2w = (1 - theta)sigma 2a, where theta is the coancestry with mutations of individuals within populations. Thus, rates of convergence and evolutionary times are a function of those for theta, which involves both N and u. When the founder population is fixed, very long times are required to obtain a perceptible increase in sigma 2w and equilibrium values of sigma 2w are very small when 4Nu less than or equal to 10(-1). The variance between populations can be expressed as sigma 2b = 2 theta sigma 2a when the founder population is an infinite equilibrium population, and as sigma 2b = 2(theta - alpha)sigma 2a when the founder population is fixed, where alpha is a function only of u. In this latter case, rates of divergence, while affected by both N and u, are dominated by u and asymptotically a function of u only. With u = 10(-5), very long times (10(3) generations) are required for any perceptible divergence, even for N = 1-10. At equilibrium, most of the variance is between small populations and within very large populations. Migration increases the variance within populations and decreases the variance between populations.

AB - The genotypic variance within, sigma 2w, and between, sigma 2b, random mating populations and rates and times for convergence to equilibrium values from different founder populations are formulated for an additive genetic model with an arbitrary number of alleles k, number of loci m, population size N, and mutation rate u, with unequal mutation rates for alleles. As a base of reference, the additive variance sigma 2a in an infinite equilibrium population is used. sigma 2a increases as k increases and decreases with variation in the mutation rates. Both transitional and equilibrium values of the variance within populations could be expressed as sigma 2w = (1 - theta)sigma 2a, where theta is the coancestry with mutations of individuals within populations. Thus, rates of convergence and evolutionary times are a function of those for theta, which involves both N and u. When the founder population is fixed, very long times are required to obtain a perceptible increase in sigma 2w and equilibrium values of sigma 2w are very small when 4Nu less than or equal to 10(-1). The variance between populations can be expressed as sigma 2b = 2 theta sigma 2a when the founder population is an infinite equilibrium population, and as sigma 2b = 2(theta - alpha)sigma 2a when the founder population is fixed, where alpha is a function only of u. In this latter case, rates of divergence, while affected by both N and u, are dominated by u and asymptotically a function of u only. With u = 10(-5), very long times (10(3) generations) are required for any perceptible divergence, even for N = 1-10. At equilibrium, most of the variance is between small populations and within very large populations. Migration increases the variance within populations and decreases the variance between populations.

UR - http://www.scopus.com/inward/record.url?scp=0023414712&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0023414712&partnerID=8YFLogxK

U2 - 10.1073/pnas.84.17.6205

DO - 10.1073/pnas.84.17.6205

M3 - Article

C2 - 3476941

AN - SCOPUS:0023414712

VL - 84

SP - 6205

EP - 6209

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 17

ER -