The Annals of Applied Probability

A time-dependent Poisson random field model for polymorphism within and between two related biological species

Amei Amei and Stanley Sawyer

Full-text: Open access

Abstract

We derive a Poisson random field model for population site polymorphisms differences within and between two species that share a relatively recent common ancestor. The model can be either equilibrium or time inhomogeneous. We first consider a random field of Markov chains that describes the fate of a set of individual mutations. This field is approximated by a Poisson random field from which we can make inferences about the amounts of mutation and selection that have occurred in the history of observed aligned DNA sequences.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 5 (2010), 1663-1696.

Dates
First available in Project Euclid: 25 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1282747397

Digital Object Identifier
doi:10.1214/09-AAP668

Mathematical Reviews number (MathSciNet)
MR2724417

Zentralblatt MATH identifier
1210.92009

Subjects
Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D10: Genetics {For genetic algebras, see 17D92} 92D20: Protein sequences, DNA sequences
Secondary: 92D15: Problems related to evolution 60F99: None of the above, but in this section

Keywords
Poisson random field DNA sequences diffusion approximation population genetics

Citation

Amei, Amei; Sawyer, Stanley. A time-dependent Poisson random field model for polymorphism within and between two related biological species. Ann. Appl. Probab. 20 (2010), no. 5, 1663--1696. doi:10.1214/09-AAP668. https://projecteuclid.org/euclid.aoap/1282747397


Export citation

References

  • [1] Abel, H. J. (2009). The role of positive selection in molecular evolution: Alternative models for within-locus selective effects. Ph.D. thesis, Washington Univ. in St. Louis.
  • [2] Akashi, H. (1999). Inferring the fitness effects of DNA mutations from polymorphism and divergence data: Statistical power to detect directional selection under stationarity and free recombination. Genetics 151 221–238.
  • [3] Baines, J. F., Sawyer, S. A., Hartl, D. L. and Parsch, J. (2008). Effects of X-linkage and sex-biased gene expression on the rate of adaptive protein evolution in Drosophila. Mol. Biol. Evol. 25 1639–1650.
  • [4] Bierne, N. and Eyre-Walker, A. (2004). The genomic rate of adaptive amino acid substitution in Drosophila. Mol. Biol. Evol. 21 1350–1360.
  • [5] Boyko, A. R., Williamson, S. H., Indap, A. R., Degenhardt, J. D., Hernandez, R. D. et al. (2008). Assessing the evolutionary impact of amino acid mutations in the human genome. PLoS Genetics 4 e1000083.
  • [6] Bustamante, C. D., Wakeley, J., Sawyer, S. A. and Hartl, D. L. (2001). Directional selection and the site-frequency spectrum. Genetics 159 1779–1788.
  • [7] Bustamante, C. D., Nielsen, R., Sawyer, S. A., Purugganan, M. D., Olsen, K. M. and Hartl, D. L. (2002). The cost of inbreeding: Fixation of deleterious genes in Arabidopsis. Nature 416 531–534.
  • [8] Bustamante, C. D., Nielsen, R. and Hartl, D. L. (2003). Maximum likelihood and Bayesian methods for estimating the distribution of selective effects among classes of mutations using DNA polymorphism data. Theory Popul. Biol. 63 91–103.
  • [9] Caicedo, A. L., Williamson, S. H., Hernandez, R. D., Boyko, A., Fledel-Alon, A. et al. (2007). Genome-wide patterns of nucleotide polymorphism in domesticated rice. PLoS Genetics 3 e163.
  • [10] Dunford, C. and Schwartz, J. (1958). Linear Operators. Part I: General Theory. Interscience, New York.
  • [11] Dynkin, E. B. (2006). Theory of Markov Processes. Dover, Mineola, NY.
  • [12] Durrett, R. (2002). Probability Models for DNA Sequence Evolution. Springer, New York.
  • [13] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [14] Ewens, W. J. (2004). Mathematical Population Genetics, 2nd ed. Springer, New York.
  • [15] Eyre-Walker, A. and Keightley, P. D. (2007). The distribution of fitness effects of new mutations. Nat. Rev. Genet. 8 610–618.
  • [16] Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2) 55 468–519.
  • [17] Feller, W. (1955). On second order differential operators. Ann. of Math. (2) 61 90–105.
  • [18] Hartl, D. L. (2000). A Primer of Population Genetics, 3rd ed. Sinauer, Sunderland, MA.
  • [19] Hartl, D. L. and Clark, A. (2007). Principles of Population Genetics, 4th ed. Sinauer, Sunderland, MA.
  • [20] Hartl, D. L., Moriyama, E. N. and Sawyer, S. A. (1994). Selection intensity for codon bias. Genetics 138 227–234.
  • [21] Huerta-Sanchez, E., Durrett, R. and Bustamante, C. D. (2008). Population genetics of polymorphism and divergence under fluctuating selection. Genetics 178 325–337.
  • [22] Itô, K. and McKean, H. P. Jr. (1965). Diffusion Processes and Their Sample Paths. Academic Press, New York.
  • [23] Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.
  • [24] Keightley, P. D. (1994). The distribution of mutation effects on viability in Drosophila melanogaster. Genetics 138 1315–1322.
  • [25] Keightley, P. D. and Eyre-Walker, A. (2007). Joint inference of the distribution of fitness effects of deleterious mutations and population demography based on nucleotide polymorphism frequencies. Genetics 177 2251–2261.
  • [26] Kemeny, J. G., Snell, J. L. and Knapp, A. W. (1966). Denumerable Markov Chains. Van Nostrand, New York.
  • [27] Kimura, M. (1955). Solution of a process of random genetic drift with a continuous model. Proc. Natl. Acad. Sci. USA 41 144–150.
  • [28] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. Oxford Univ. Press, New York.
  • [29] Lewontin, R. C. (1974). The Genetic Basis of Evolutionary Change. Columbia Univ. Press, New York.
  • [30] Li, W. H. (1997). Molecular Evolution. Sinauer, Sunderland, MA.
  • [31] McDonald, J. H. and Kreitman, M. (1991). Adaptive protein evolution at the Adh locus in Drosophila. Nature 351 652–654.
  • [32] Moran, P. A. P. (1959). The survival of a mutant gene under selection. J. Aust. Math. Soc. 1 121–126.
  • [33] Nielsen, R., Bustamante, C., Clark, A. G., Glanowski, S., Sackton, T. B. et al. (2005). A scan for positively selected genes in the genomes of humans and chimpanzees. PLoS Biology 3 e170.
  • [34] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2007). Numerical Recipes: The Art of Scientific Computing, 3rd ed. Cambridge Univ. Press, Cambridge.
  • [35] Protter, M. H. and Weinberger, H. F. (1967). Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs, NJ.
  • [36] Riesz, F. and Sz. Nagy, B. (1955). Functional Analysis. Frederick Ungar Publishing, New York.
  • [37] Sawyer, S. (1974). A Fatou theorem for the general one-dimensional parabolic equation. Indiana Univ. Math. J. 24 451–498.
  • [38] Sawyer, S. A. and Hartl, D. L. (1992). Population genetics of polymorphism and divergence. Genetics 132 1161–1176.
  • [39] Sawyer, S. A. (1994). Inferring selection and mutation from DNA sequences: The McDonald–Kreitman test revisited. In Non-Neutral Evolution: Theories and Molecular Data (G. B. Golding, ed.) 77–87. Chapman and Hall, New York.
  • [40] Sawyer, S. A., Kulathinal, R. J., Bustamante, C. D. and Hartl, D. L. (2003). Bayesian analysis suggests that most amino acid replacements in Drosophila are driven by positive selection. J. Mol. Evol. 57 S154–S164.
  • [41] Sawyer, S. A., Parsch, J., Zhang, Z. and Hartl, D. L. (2007). Prevalence of positive selection among nearly neutral amino acid replacements in Drosophila. Proc. Natl. Acad. Sci. USA 104 6504–6510.
  • [42] Smith, N. G. C. and Eyre-Walker, A. (2002). Adaptive protein evolution in Drosophila. Nature 415 1022–1024.
  • [43] Templeton, A. R. (1996). Contingency tests of neutrality using intra/interspecific gene trees: The rejection of neutrality for the evolution of the mitochondrial cytochrome oxidase II gene in the hominoid primates. Genetics 144 1263–1270.
  • [44] Trotter, H. F. (1958). Approximation of semi-groups of operators. Pacific J. Math. 8 887–919.
  • [45] Wakeley, J. (2003). Polymorphism and divergence for island-model species. Genetics 163 411–420.
  • [46] Williamson, S., Alon, A. F. and Bustamante, C. D. (2004). Population genetics of polymorphism and divergence for diploid selection models with arbitrary dominance. Genetics 168 463–475.
  • [47] Williamson, S., Hernandez, R., Alon, A. F., Zhu, L., Nielsen, R. and Bustamante, C. D. (2005). Simultaneous inference of selection and population growth from patterns of variation in the human genome. Proc. Natl. Acad. Sci. USA 102 7882–7887.
  • [48] Wright, S. (1938). The distribution of gene frequencies under irreversible mutation. Proc. Natl. Acad. Sci. USA 24 253–259.
  • [49] Zhu, L. and Bustamante, C. D. (2005). A composite-likelihood approach for detecting directional selection from DNA sequence data. Genetics 170 1411–1421.