Remote access

How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax

A Mutation-Selection Model with Recombination for General Genotypes

About this Title

Steven N. Evans, Department of Statistics, 367 Evans Hall, University of California, Berkeley, California 94720-3860, David Steinsaltz, Department of Statistics, 1 South Parks Road, Oxford, OX1 3TG, United Kingdom and Kenneth W. Wachter, Department of Demography, 2232 Piedmont Avenue, University of California, Berkeley, CA 94720-2120

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 222, Number 1044
ISBNs: 978-0-8218-7569-8 (print); 978-0-8218-9511-5 (online)
Published electronically: September 5, 2012
Keywords:Measure-valued, dynamical system, population genetics, quasi-linkage equilibrium, Poisson random measure, Wasserstein metric, Palm measure, shadowing, stability, attractivity
MSC: Primary 60G57, 92D15; Secondary 37N25, 60G55, 92D10

View full volume PDF

View other years and numbers:

Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Definition, existence, and uniqueness of the dynamical system
  • Chapter 3. Equilibria
  • Chapter 4. Mutation, selection, and recombination in discrete time
  • Chapter 5. Shattering and the formulation of the convergence result
  • Chapter 6. Convergence with complete Poissonization
  • Chapter 7. Supporting lemmas for the main convergence result
  • Chapter 8. Convergence of the discrete generation system
  • Appendix A. Results cited in the text


We investigate a continuous time, probability measure-valued dynamical system that describes the process of mutation-selection balance in a context where the population is infinite, there may be infinitely many loci, and there are weak assumptions on selective costs. Our model arises when we incorporate very general recombination mechanisms into an earlier model of mutation and selection presented by Steinsaltz, Evans and Wachter in 2005 and take the relative strength of mutation and selection to be sufficiently small. The resulting dynamical system is dynamical system a flow of measures on the space of loci. Each such measure is the intensity measure of a Poisson random measure on the space of loci: intensity measure the points of a realization of the random measure record the set of loci at which the genotype of a uniformly chosen individual differs from a reference wild type due to an accumulation of ancestral mutations. Our motivation for working in such a general setting is to provide a basis for understanding mutation-driven changes in age-specific demographic schedules that arise from the complex interaction of many genes, and hence to develop a framework for understanding the evolution of aging.

We establish the existence and uniqueness of the dynamical system, provide conditions for the existence and stability of equilibrium states, and prove that our continuous-time dynamical system is the limit of a sequence of discrete-time infinite population mutation-selection-recombination models in the standard asymptotic regime where selection and mutation are weak relative to recombination and both scale at the same infinitesimal rate in the limit.

References [Enhancements On Off] (What's this?)

  • [AGS05] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. MR 2129498
  • [B000] R. Bürger, The mathematical theory of selection, recombination, and mutation, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000. MR 1885085
  • [Baa01] Ellen Baake, Mutation and recombination with tight linkage, J. Math. Biol. 42 (2001), no. 5, 455–488. MR 1842838, 10.1007/s002850000077
  • [Baa05] Michael Baake, Recombination semigroups on measure spaces, Monatsh. Math. 146 (2005), no. 4, 267–278. MR 2191729, 10.1007/s00605-005-0326-z
  • [Baa07] Michael Baake, Addendum to: “Recombination semigroups on measure spaces” [Monatsh. Math. 146 (2005), no. 4, 267–278; MR2191729], Monatsh. Math. 150 (2007), no. 1, 83–84. MR 2297255, 10.1007/s00605-006-0431-7
  • [Bau05] Annette Baudisch, Hamilton's indicators of the force of selection, Proceedings of the National Academy of Sciences 102 (2005), no. 23, 8263-8268.
  • [Bau08] -, Inevitable aging?: Contributions to evolutionary-demographic theory, Demographic Research Monographs, Springer Verlag, 2008.
  • [BB03] Michael Baake and Ellen Baake, An exactly solved model for mutation, recombination and selection, Canad. J. Math. 55 (2003), no. 1, 3–41. MR 1952324, 10.4153/CJM-2003-001-0
  • [BT91] N. H. Barton and M. Turelli, Natural and sexual selection on many loci, Genetics 127 (1991), 229-55.
  • [Bür98] Reinhard Bürger, Mathematical properties of mutation selection models, Genetica 102/103 (1998), 279-298.
  • [BW01] Ellen Baake and Holger Wagner, Mutation-selection models solved exactly with methods of statistical mechanics, Genet. Res., Camb. 78 (2001), 93-117.
  • [CE09] Aubrey Clayton and Steven N. Evans, Mutation-selection balance with recombination: convergence to equilibrium for polynomial selection costs, SIAM J. Appl. Math. 69 (2009), no. 6, 1772–1792. MR 2496717, 10.1137/070702783
  • [Cha94] Brian Charlesworth, Evolution in age-structured populations, 2nd ed., Cambridge Studies in Mathematical Biology, vol. 13, Cambridge University Press, Cambridge, 1994. MR 1294137
  • [Cha01] -, Patterns of age-specific means and genetic variances of mortality rates predicted by the mutation-accumulation theory of ageing, J. Theor. Bio. 210 (2001), no. 1, 47-65.
  • [Çın11] Erhan Çınlar, Probability and stochastics, Graduate Texts in Mathematics, vol. 261, Springer, New York, 2011. MR 2767184
  • [CKP95] Brian A. Coomes, Hüseyin Koçak, and Kenneth J. Palmer, A shadowing theorem for ordinary differential equations, Z. Angew. Math. Phys. 46 (1995), no. 1, 85–106. MR 1315738, 10.1007/BF00952258
  • [CT03] James R. Carey and Shripad Tuljapurkar (eds.), Life span: Evolutionary, ecological, and demographic perspectives, Population Council, 2003, Pop. Dev. Rev. vol. 29 suppl.
  • [Daw99] Kevin J. Dawson, The dynamics of infinitesimally rare alleles, applied to the evolution of mutation rates and the expression of deleterious mutations, Theor. Popul. Biol. 55 (1999), 1-22.
  • [DS88] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
  • [DU77] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
  • [DVJ07] D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer Series in Statistics, Springer-Verlag, New York, 1988. MR 950166
  • [EK86] Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
  • [Hor03] Shiro Horiuchi, Interspecies differences in the lifespan distribution: Humans versus invertebrates, Life Span: Evolutionary, Ecological, and demographic perspectives (James R. Carey and Shripad Tuljapurkar, eds.), Population and Development Review, vol. 29 (Supplement), The Population Council, New York, 2003, pp. 127-151.
  • [Ken74] D. G. Kendall, Foundations of a theory of random sets, Stochastic geometry (a tribute to the memory of Rollo Davidson), Wiley, London, 1974, pp. 322–376. MR 0423465
  • [Kin93] J. F. C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207584
  • [KJB02] Mark Kirkpatrick, Toby Johnson, and Nick Barton, General models of multilocus evolution, Genetics 161 (2002), 1727-50.
  • [KM66] Motoo Kimura and Takeo Maruyama, The mutational load with epistatic gene interactions in fitness, Genetics 54 (1966), 1337-1351.
  • [Kon82] A.S. Kondrashov, Selection against harmful mutations in large sexual and asexual populations, Genet. Res. 40 (1982), 325-332.
  • [LC60] Lucien Le Cam, An approximation theorem for the Poisson binomial distribution, Pacific J. Math. 10 (1960), 1181–1197. MR 0142174
  • [Med52] Peter Medawar, An unsolved problem in biology: An inaugural lecture delivered at University College, London, 6 December, 1951, H. K. Lewis and Co., London, 1952.
  • [Nev75] J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed; North-Holland Mathematical Library, Vol. 10. MR 0402915
  • [PC98] Scott D. Pletcher and James W. Curtsinger, Mortality plateaus and the evolution of senescence: Why are old-age mortality rates so low?, Evolution 52 (1998), no. 2, 454-64.
  • [Rac91] Svetlozar T. Rachev, Probability metrics and the stability of stochastic models, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1991. MR 1105086
  • [RR98] Svetlozar T. Rachev and Ludger Rüschendorf, Mass transportation problems. Vol. II, Probability and its Applications (New York), Springer-Verlag, New York, 1998. Applications. MR 1619171
  • [SEW05] David Steinsaltz, Steven N. Evans, and Kenneth W. Wachter, A generalized model of mutation-selection balance with applications to aging, Adv. in Appl. Math. 35 (2005), no. 1, 16–33. MR 2141503, 10.1016/j.aam.2004.09.003
  • [SH92] Stanley A. Sawyer and Daniel L. Hartl, Population genetics of polymorphism and divergence, Genetics 132 (1992), 1161-1176.
  • [Vil03] Cédric Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483
  • [Vil09] Cédric Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR 2459454
  • [Wac03] Kenneth W. Wachter, Hazard curves and life span prospects, Life Span: Evolutionary, Ecological, and Demographic Perspectives (James R. Carey and Shripad Tuljapurkar, eds.), Population and Development Review, vol. 29 (Supplement), The Population Council, New York, 2003, pp. 270-291.
  • [WBG98] Holger Wagner, Ellen Baake, and Thomas Gerisch, Ising quantum chain and sequence evolution, J. Statist. Phys. 92 (1998), no. 5-6, 1017–1052. MR 1657856, 10.1023/A:1023048711599
  • [WES08] Kenneth W. Wachter, Steven N. Evans, and David R. Steinsaltz, The age-specific force of natural selection and walls of death, Tech. Report 757, Department of Statistics, University of California at Berkeley, 2008, Available at
  • [WF97] Kenneth W. Wachter and Caleb E. Finch (eds.), Between Zeus and the salmon: The biodemography of longevity, National Academy Press, Washington, D.C., 1997.
  • [Wil57] George C. Williams, Pleiotropy, natural selection, and the evolution of senescence, Evolution 11 (1957), 398-411.
  • [WSE09] Kenneth W. Wachter, David R. Steinsaltz, and Steven N. Evans, Vital rates and the action of mutation accumulation, Journal of Population Ageing 2 (2009), no. 1-2, 5-22.