Pleiotropy and the Preservation of Perfection
10.1126/science.279.5354.1210
Crossref journal-article
American Association for the Advancement of Science (AAAS)
Science (221)
Abstract

A mathematical model is presented in which a single mutation can affect multiple phenotypic characters, each of which is subject to stabilizing selection. A wide range of mutations is allowed, including ones that produce extremely small phenotypic changes. The analysis shows that, when three or more characters are affected by each mutation, a single optimal genetic sequence may become common. This result provides a hypothesis to explain the low levels of variation and low rates of substitution that are observed at some loci.

Bibliography

Waxman, D., & Peck, J. R. (1998). Pleiotropy and the Preservation of Perfection. Science, 279(5354), 1210–1213.

Authors 2
  1. David Waxman (first)
  2. Joel R. Peck (additional)
References 40 Referenced 113
  1. R. A. Fisher The Genetical Theory of Natural Selection (Clarendon Oxford 1930). (10.5962/bhl.title.27468)
  2. Foley P., Evolution 46, 763 (1992). (10.2307/2409644) / Evolution by Foley P. (1992)
  3. Kimura M., Proc. Natl. Acad. Sci. U.S.A. 78, 5773 (1981). (10.1073/pnas.78.9.5773) / Proc. Natl. Acad. Sci. U.S.A. by Kimura M. (1981)
  4. M. Nei Philos. Trans. R. Soc. London B 319 615 (1988). (10.1098/rstb.1988.0069)
  5. S. Wright Evolution and the Genetics of Populations vol. 3 (Univ. of Chicago Press Chicago 1977).
  6. Mather K., Heredity 58, 267 (1987). (10.1038/hdy.1987.41) / Heredity by Mather K. (1987)
  7. G. Bell Selection: The Mechanism of Evolution (Chapman & Hall New York 1997).
  8. J. F. Crow and M. Kimura in Proceedings of the XI International Congress of Genetics (Pergamon Oxford 1964) vol. 2 pp. 405–505.
  9. Foley P., Proc. Natl. Acad. Sci. U.S.A. 84, 7996 (1987). (10.1073/pnas.84.22.7996) / Proc. Natl. Acad. Sci. U.S.A. by Foley P. (1987)
  10. Kimura M., ibid. 54, 731 (1965). / ibid. by Kimura M. (1965)
  11. ___ The Neutral Theory of Molecular Evolution (Cambridge Univ. Press Cambridge UK 1983).
  12. J. F. Crow and M. Kimura An Introduction to Population Genetics Theory (Burgess Minneapolis MN 1970).
  13. Turelli M., Theor. Popul. Biol. 25, 138 (1984). (10.1016/0040-5809(84)90017-0) / Theor. Popul. Biol. by Turelli M. (1984)
  14. ___, Genetics 111, 165 (1985). (10.1093/genetics/111.1.165) / Genetics by ___ (1985)
  15. Gavrilets S., Dejong G., ibid. 134, 609 (1993). / ibid. by Gavrilets S. (1993)
  16. Bulmer M. G., Genet. Res. 19, 17 (1972). (10.1017/S0016672300014221) / Genet. Res. by Bulmer M. G. (1972)
  17. Lande R., Am. Nat. 116, 463 (1980). (10.1086/283642) / Am. Nat. by Lande R. (1980)
  18. Wagner G. P., Genetics 122, 223 (1989). (10.1093/genetics/122.1.223) / Genetics by Wagner G. P. (1989)
  19. Kondrashov A. S., Turelli M., ibid. 132, 603 (1992). / ibid. by Kondrashov A. S. (1992)
  20. Keightley P. D., Hill W. G., Proc. R. Soc. London B 242, 95 (1990). (10.1098/rspb.1990.0110) / Proc. R. Soc. London B by Keightley P. D. (1990)
  21. Barton N. H., Genetics 124, 773 (1990). (10.1093/genetics/124.3.773) / Genetics by Barton N. H. (1990)
  22. Wagner G. P., ibid. 143, 617 (1996). / ibid. by Wagner G. P. (1996)
  23. A. Caballero and P. D. Keightley ibid. 138 883 (1994). (10.1093/genetics/138.3.883)
  24. 10.1146/annurev.ge.23.120189.002005
  25. Caspari E., Evolution 6, 1 (1952). (10.2307/2405500) / Evolution by Caspari E. (1952)
  26. Bürger R., Hofbauer J., J. Math. Biol. 32, 193 (1994). (10.1007/BF00163878) / J. Math. Biol. by Bürger R. (1994)
  27. Bürger R., ibid. 24, 341 (1986). / ibid. by Bürger R. (1986)
  28. ___, Math. Biosci. 91, 67 (1988). (10.1016/0025-5564(88)90024-7) / Math. Biosci. by ___ (1988)
  29. The proofs we present here are in accordance with the rather general mathematical results of Bürger and collaborators (26-28 39). The population is described in equilibrium by the distribution Ψ( x 1 x 2  …  x k ). Thus the proportion of the population with genotypic values in the infinitesimal volume d k x ≡ dx 1 dx 2  …  dx k centered at ( x 1 x 2  …  x k ) is Ψ ( x 1 x 2  …  x k ) d k x. Every mutation affects Ω characters and we deal with the sets of characters ( x 1 x 2  …  x Ω ) ( x Ω+1 x Ω+2  …  x 2Ω )  … . It is possible to show that in equilibrium the probability density factorizes into the form Ψ ( x 1 x 2  …  x k ) = Φ( x 1 x 2  …  x Ω )Φ( x Ω+1 x Ω+2  …  x 2Ω ) … . From the life cycle that we have specified it follows that with x ≡ def ( x 1 x 2  …  x Ω ) Φ( x ) obeys Φ(x) =(1−Θ)w1(x)Φ(x)+Θ∫f1(x−y)w1(y)Φ(y)dΩy∫w1(y)Φ(y)dΩywhere w1(x)=∏i=1Ω exp −xi22Vs f1(x)=∏i=1Ω12πm2 exp −xi22m2and all integrals cover the full range of the integration variables. Equation 3 can be written as [w¯1−(1−Θ)w1(x)]Φ(x) =Θ∫f1(x−y)w1(y)Φ(y)dΩywhere w 1 ≡ def ∫ w 1 ( y )Φ( y ) d Ω y and this quantity is determined by the condition that Φ( x ) is normalized to unity namely ∫Φ(x)dΩx=1General features of Φ( x ) follow from f 1 ( x − y ) w 1 ( x ) and Φ( x ) being ≥0. In particular from Eq. 5 it follows that [ w 1 − (1 − Θ) w 1 ( x )] ≥ 0. The smallest value of [ w 1 − (1 − Θ) w 1 ( x )] occurs at x = (0 0  …  0) ≡ 0 where w 1 ( 0 ) = 1; hence generally w 1 ≥ 1 − Θ [closely related results have been derived by Bürger and his collaborators (26 28 39)]. The inequality w 1 > 1 − Θ and the equality w 1 = 1 − Θ lead to qualitatively different forms for Φ( x ) and we discuss these separately. Case i: w 1 > 1 − Θ: This case yields from Eq. 5 Φ(x)=Θ∫f1(x−y)w1(y)Φ(y)dΩyw¯1−(1−Θ)w1(x)Φ( x ) is a peaked but nonsingular function of x because for x = 0 the right-hand side is finite. The constant w 1 is determined by the condition of normalization (Eq. 6). The application of the normalization condition Eq. 6 to Eq. 7 leads to the x integral: ∫ f 1 ( x − y )/[ w 1 − (1 − Θ) w 1 ( x )] d Ω x and this as a function of w 1 is unbounded from above when Ω = 1 and Ω = 2. As a consequence irrespective of how small Θ is w 1 can be chosen so that Φ( x ) in Eq. 7 is normalized to unity. Therefore the case w 1 > 1 − Θ applies for Ω = 1 and Ω = 2. If α << 1 this case cannot apply for Ω ≥ 3. If we do not assume α << 1 then w 1 > 1 − Θ may apply for larger values of Ω and hence yield nonsingular distributions for these values of Ω. As an example if V s /m 2 = 100 then we numerically find that when α < 0.67 w 1 > 1 − Θ will only apply for Ω = 1 and Ω = 2 but if 1.67 > α > 0.67 w 1 > 1 − Θ will apply for Ω = 1 Ω = 2 and additionally Ω = 3. Case ii: w 1 = 1 − Θ: For this case we cannot simply solve Eq. 5 to obtain the result of Eq. 7 because [ w 1 − (1 − Θ) w 1 ( x )] vanishes as x = 0 and the solution to Eq. 5 must include the singular function δ( x ) ≡ ∏ i =1 Ω δ( x i ) where δ( x ) is a Dirac delta function of argument x. Derivatives of Dirac delta functions cannot be present in the solution because they correspond to distribution functions that are negative for some x. Thus when w 1 = 1 − Θ Eq. 5 is equivalent to Φ(x)=Aδ(x) +Θ1−Θ∫f1(x−y)w1(y)Φ(y)dΩy1−w1(x) where A (≥0) is determined by normalization (Eq. 6). When Ω = 1 and Ω = 2 the x integral that results from the normalization condition ∫ f 1 ( x − y )/[1 − w 1 ( x )] d Ω x diverges and hence definitely rules out these Ω values. For Ω ≥ 3 the same integral is finite and when α << 1 the delta function term must be present (that is A ≠ 0) in order that Φ( x ) is normalized to unity. Thus Φ( x ) contains a singular delta function part for Ω ≥ 3 when α << 1. If for a given value of Ω the mutation rate Θ (and hence α) is large enough that the condition of normalization yields A < 0 then we can infer that the case w 1 = 1 − Θ does not apply to this value of Ω. For example if V s /m 2 = 100 then when 1.67 > α > 0.67 case ii does not apply to Ω = 3 although it does apply for Ω ≥ 4. Distributions: We determine approximate forms for the distribution of a single character say x 1 and we denote the single character distribution by Φ 1 ( x 1 ). We use the house-of-cards approximation (13 31) which entails replacing ∫ f 1 ( x − y ) w 1 ( y )Φ( y ) d Ω y in Eqs. 7 and 8 by f 1 ( x ) w 1 ( 0 )∫Φ( y ) d Ω y = f 1 ( x ). This approximation can be shown to be highly accurate when α << 1. Assuming m 2 / V s << 1 which is apparently reasonable (13) and furthermore that Ω << V s /m 2 we can replace the Gaussian w 1 ( x ) by 1 − Σ i =1 Ω x i 2 /(2 V s ) without any substantial loss of accuracy. When Ω = 1 Φ 1 ( x 1 ) ≡ Φ( x 1 ) and we obtain Φ1(x1)≈α2πm2exp−x122m2x122m2+πα2To obtain the single character distribution Φ 1 ( x 1 ) in a pleiotropic model we integrate Φ( x ) over x 2 x 3  …  x Ω . When Ω = 2 we have Φ1(x1)≈α exp (c2)2πm2Γ12 x122m2+c2x122m2+c2where c 2 = exp (−γ − α −1 ). For Ω ≥ 3 we have Φ1(x1)≈1−2αΩ−2δ(x1) +Γ−(Ω−3)2 x122m2 α2πm2x122m2(Ω−3)/2In the above γ = 0.5772 … is Euler's constant and Γ( a b ) ≡ def ∫ b ∞ u a −1 e −u du is the incomplete gamma function. Origin and explanation of the results: The fundamental origin of the results we have produced arises from the suppression of beneficial mutations when pleiotropic mutations are present. To see this consider a single mutation that affects the genotypic value x in one of the sets of Ω characters. The probability that the mutation will change this genotype to a genotype with associated fitness lying in the range w 1 to w 1 + dw 1 is when w 1 ≈ 1 approximately porportional to f1(x)(1−w1)(Ω−2)/2dw1When Ω ≥ 3 this probability is much smaller than that for Ω = 1 or Ω = 2. When α = Θ V s /m 2 << 1 it is the suppression of beneficial mutations to w 1 ≈ 1 that results in singular distributions for Ω ≥ 3. Larger values of α may push the delta function singularity to occur at a larger value of Ω. Inspection of cases i and ii considered above indicates the mathematical reason why delta functions in Φ( x ) are not possible when Ω = 1 or Ω = 2 yet are possible when Ω ≥ 3. The reason is that the integral with respect to x 1 x 2 …  x Ω of 1/( x 1 2 + x 2 2 + …  x Ω 2 ) over a region near (and including) the origin x = 0 is divergent when Ω = 1 or Ω = 2 but finite when Ω ≥ 3. The extension of the results given in this work to more general fitness functions is straightforward and the convergence of analogous integrals is the key to the presence of delta functions in Φ( x ).
  30. Lande R., Genet. Res. 26, 221 (1975). (10.1017/S0016672300016037) / Genet. Res. by Lande R. (1975)
  31. J. F. C. Kingman J. Appl. Probab. 15 1 (1978). (10.1017/S0021900200105534)
  32. 10.1038/336435a0
  33. 10.1017/S0016672300032365
  34. Sharp P. M., J. Mol. Evol. 33, 23 (1991). (10.1007/BF02100192) / J. Mol. Evol. by Sharp P. M. (1991)
  35. R. C. Lewontin The Genetic Basis of Evolutionary Change (Columbia Univ. Press New York 1974).
  36. W. H. Li Molecular Evolution (Sinauer Sunderland MA 1997).
  37. Schaeffer S. W., Miller E. L., Genetics 132, 471 (1992). (10.1093/genetics/132.2.471) / Genetics by Schaeffer S. W. (1992)
  38. Selander R. K., et al., Infect. Immun. 58, 1891 (1990). (10.1128/iai.58.6.1891-1901.1990) / Infect. Immun. by Selander R. K. (1990)
  39. R. Bürger and I. M. Bomze Adv. Appl. Probab. 28 227 (1996). (10.1017/S0001867800027348)
  40. We thank N. H. Barton R. Bürger B. Charlesworth A. Eyre-Walker W. G. Hill C. H. Langley J. Maynard Smith N. H. Smith and M. Turelli for their kind advice and assistance.
Dates
Type When
Created 23 years ago (July 27, 2002, 5:42 a.m.)
Deposited 1 year, 7 months ago (Jan. 12, 2024, 11:57 p.m.)
Indexed 2 weeks, 6 days ago (Aug. 6, 2025, 8:18 a.m.)
Issued 27 years, 6 months ago (Feb. 20, 1998)
Published 27 years, 6 months ago (Feb. 20, 1998)
Published Print 27 years, 6 months ago (Feb. 20, 1998)
Funders 0

None

@article{Waxman_1998, title={Pleiotropy and the Preservation of Perfection}, volume={279}, ISSN={1095-9203}, url={http://dx.doi.org/10.1126/science.279.5354.1210}, DOI={10.1126/science.279.5354.1210}, number={5354}, journal={Science}, publisher={American Association for the Advancement of Science (AAAS)}, author={Waxman, David and Peck, Joel R.}, year={1998}, month=feb, pages={1210–1213} }