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A survey of unequal crossover systems and their mathematical properties

Bulletin of Mathematical Biology volume 64, Article number: 703 (2002) Cite this article

An Erratum to this article was published on 01 November 2002

Abstract

We present a model of gene duplication by means of unequal crossover (UCO) where the probability of any given pairing between homologous sequences scales as a penalty factor p z ≤ 1, with z the number of mismatches due to asymmetric sequence alignment. From this general representation, we derive several limiting case models of UCO, some of which have been treated elsewhere in the literature. One limiting case is random unequal crossover (RUCO), obtained by setting p = 1 (corresponding to equiprobable pairings at each site). Another limiting case scenario (the ‘Krueger-Vogel’ model) proposes an optimal ‘endpoint’ alignment which strongly penalizes both overhang and deviations from endpoint matching positions. For both of these scenarios, we make use of the symmetry properties of the transition operator (together with the more general UCO properties of copy number conservation and equal parent-offspring mean copy number) to derive the stationary distribution of gene copy number generated by UCO. For RUCO, the stationary distribution of genotypes is shown to be a negative binomial, or alternatively, a convolution of geometric distributions on ‘haplotype’ frequencies. A different type of model derived from the general representation only allows recombination without overhang (internal UCO or IntUCO). This process has the special property of converging to a single copy length or a distribution on a pair of copy lengths in the absence of any other evolutionary forces. For UCO systems in general, we also show that selection can readily act on gene copy number in all of the UCO systems we investigate due to the perfect heritability (h 2 = 1) imposed by conservation of copy number. Finally, some preliminary work is presented which suggests that the more general models based on misalignment probabilities seem to also converge to stationary distributions, which are most likely functions of parameter value p.

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References

  • Axelrod, D. E., K. A. Baggerly and M. Kimmel (1994). Gene amplification by unequal sister chromatid exchange: probabilistic modeling and analysis of drug resistance data. J. Theor. Biol. 168, 151–159.

    Article  Google Scholar 

  • Bailey, W. J., J. Kim, G. P. Wagner and F. H. Ruddle (1997). Phylogenetic reconstruction of vertebrate Hox cluster duplications. Mol. Biol. Evol. 14, 843–853.

    Google Scholar 

  • Barker, W. C., L. C. Ketchum and M. O. Dayhoff (1978). Duplications in protein sequences, in Atlas of Protein Sequences and Structure, Vol. 5, Supplement 3, Silver Springs, MD: National Biomedical Research Foundation, pp. 359–362.

    Google Scholar 

  • Biebricher, C. K., M. Eigen and T. S. McCaskill (1993). Template directed and template free RNA synthesis. J. Mol. Biol. 231, 175–179.

    Article  Google Scholar 

  • Biebricher, C. K. and K. Luce (1992). In vitro recombination and terminal elongation of RNA. EMBO J. 11, 5129–5135.

    Google Scholar 

  • Bowler, M. G. (1982). Lectures on Statistical Mechanics, New York: Pergamon.

    Google Scholar 

  • Buongiorno-Nardelli, M., F. Amaldi and P. A. Lava-Sanchez (1972). Amplification as a rectification mechanism for redundant rRNA genes. Nat. New Biol. 238, 134.

    Google Scholar 

  • Charlesworth, B., P. Sniegowski and W. Stephan (1994). The evolutionary dynamics of repetitive DNA in eukaryotes. Nature 371, 215–220.

    Article  Google Scholar 

  • Dayhoff, M. O. (1978). Atlas of Protein Sequences and Structure, Vol. 5, Supplement 3, Silver Springs, MD: National Biomedical Research Foundation.

    Google Scholar 

  • Du Pasquier, L. (1992). Origin and evolution of the vertebrate immune system. Apmis 100, 383–392.

    Google Scholar 

  • Durrett, R. and S. Kruglyak (1998). A new stochastic model of microsatellite evolution. J. Appl. Probability 36, 621–631.

    MathSciNet  Google Scholar 

  • Ewens, W. J. (1979). Mathematical Population Genetics, New York: Springer.

    MATH  Google Scholar 

  • Fuchs, E. and C. Byrne (1994). The epidermis: rising to the surface. Curr. Opin. Gen. Dev. 4, 725–736.

    Article  Google Scholar 

  • Go, M. (1981). Correlation of DNA exonic regions with protein structural units in hemoglobin. Nature 291, 90–92.

    Article  Google Scholar 

  • Grenier, J. K., T. L. Garber, R. Warren, P. M. Whittington and S. Carroll (1997). Evolution of the entire arthropod Hox gene set predated the origin and radiation of the onychophoran/arthropod clade. Curr. Biol. 7, 547–553.

    Article  Google Scholar 

  • Haken, H. (1977). Synergetics, New York: Springer.

    Google Scholar 

  • Houle, D. (1992). Comparing evolvability and variability of quantitative traits. Genetics 130, 195–204.

    Google Scholar 

  • Huynen, M. and E. van Nimwegen (1998). The frequency distribution of gene family sizes in complete genomes. Mol. Biol. Evol. 15, 583–598.

    Google Scholar 

  • Krueger, J. and F. Vogel (1975). Population genetics of unequal crossing over. J. Mol. Evol. 4, 201–247.

    Article  Google Scholar 

  • Kruglyak, S., R. T. Durrett, M. D. Schug and C. F. Aquadro (1998). Equilibrium distributions of microsatellite repeat length resulting from a balance between slippage events and point mutations. PNAS 95, 10774–10778.

    Google Scholar 

  • Lande, R. (1976). Natural selection and random drift in phenotypic evolution. Evolution 30, 314–334.

    Article  Google Scholar 

  • Lande, R. and S. Arnold (1983). The measurement of selection on correlated characters. Evolution 37, 1210–1226.

    Article  Google Scholar 

  • Li, W.-H. (1983). Evolution of duplicate genes and pseudogenes, in Evolution of Genes and Proteins, M. Nei and R. K. Koehn (Eds), Sunderland, MA: Sinauer Assoc., pp. 14–37.

    Google Scholar 

  • Li, W.-H. and D. Grauer (1994). Fundamentals of Molecular Biology, Sunderland, MA: Sinauer Assoc.

    Google Scholar 

  • Lynch, M. and J. B. Walsh (1996). Genetics and Analysis of Quantitative Traits, Sunderland, MA: Sinauer Assoc.

    Google Scholar 

  • Maeda, N. and O. Smithies (1986). The evolution of multigene families: human haptoglobin genes. Ann. Rev. Genet. 20, 81–108.

    Article  Google Scholar 

  • Nagylaki, T. (1984a). The evolution of multigene families under intrachromosomal gene conversion. Genetics 106, 529–548.

    Google Scholar 

  • Nagylaki, T. (1984b). Evolution of multigene families under intrachromosomal gene conversion. PNAS 81, 3796–3800.

    Article  Google Scholar 

  • Nowak, M. A., M. C. Boerlist, J. Cooke and J. Maynard Smith (1997). Evolution of genetic redundancy. Nature 388, 167–171.

    Article  Google Scholar 

  • Ohno, S. (1970). Evolution by Gene Duplication, Berlin: Springer.

    Google Scholar 

  • Ohta, T. (1980). Functional Variation in Multigene Families, Berlin: Springer-Verlag.

    Google Scholar 

  • Ohta, T. (1983). On the evolution of multigene families. Theor. Popul. Biol. 23, 216–240.

    MATH  Article  Google Scholar 

  • Ohta, T. (1987). Simulating evolution by gene duplication. Genetics 115, 207–213.

    Google Scholar 

  • Ohta, T. and G. Dover (1980). Population genetics of multigene families that are dispersed into two or more chromosomes. PNAS 80, 4079–4083.

    Article  Google Scholar 

  • Pendleton, J. W., B. K. Nagai, M. T. Murtha and F. H. Ruddle (1993). Expansion of the Hox gene family and the evolution of chordates. PNAS 90, 6300–6304.

    Article  Google Scholar 

  • Perelson, A. S. and G. Bell (1977). Mathematical models for the evolution of multigene families by unequal crossing over. Nature 265, 304–310.

    Article  Google Scholar 

  • Rabani, Y., Y. Rabinovich and A. Sinclair (1995). A computational view of population genetics, Proceedings of the 27th ACM Symposium on the Theory of Computing, Las Vegas, NV: pp. 83–92.

  • Rabinovich, Y., A. Sinclair and A. Wigderson (1992). Quadratic dynamical systems, Proceedings of the 33rd IEEE Symposium on the Foundations of Computer Science, pp. 304–313.

  • Roughgarden, J. (1979). Theory of Population Genetics and Evolutionary Ecology: An Introduction, New York: Prentice Hall.

    Google Scholar 

  • Schluter, S. F., E. Schroeder, E. Wang and J. J. Marschalonis (1994). Recognition molecules and immunoglobin domains in invertebrates. Ann. N. Y. Acad. Sci. 712, 74–81.

    Google Scholar 

  • Shpak, M. and G. P. Wagner (2000). Asymmetry of configuration spaces induced by unequal crossover. Artif. Life 6, 25–43.

    Article  Google Scholar 

  • Smith, G. P. (1973). Unequal crossover and the evolution of multigene families. Cold Springs Harb. Symp. Quant. Biol. 38, 507–513.

    Google Scholar 

  • Smith, K. A., P. A. Gorman, M. B. Stark, R. P. Groves and G. P. Stark (1990). Distinctive chromosome structures are formed very early in the amplification of CAD genes in Syrian hamster cells. Cell 63, 1219–1227.

    Article  Google Scholar 

  • Smith, K. A., M. B. Stark, P. A. Gorman and G. R. Stark (1992). Fusions near telomeres occur very early in the amplification of CAD genes in Syrian hamster cells. PNAS USA 89, 5427–5431.

    Article  Google Scholar 

  • Smithies, O. (1964). Chromosomal rearrangements and protein structure. Cold Springs Harb. Symp. Quant. Biol. 29, 309–323.

    Google Scholar 

  • Stadler, B., P. F. Stadler, M. Shpak and G. P. Wagner (2002). Recombination spaces, metrics, and pretopologies. Z. Phys. Chemie 216, 217–234.

    Google Scholar 

  • Takahata, N. (1981). A mathematical study on the distribution of the number of repeated genes per chromosome. Genet. Res. 38, 97–102.

    Article  Google Scholar 

  • Walsh, J. B. (1987). Persistence of tandem arrays: implications for satellite and simplesequence DNA’s. Genetics 115, 553–567.

    Google Scholar 

  • Zimmer, E. A., S. L. Martin, S. M. Beverley, Y. W. Kan and A. C. Wilson (1986). Rapid duplication and loss of genes coding for the α chains of hemoglobin. PNAS 77, 2158–2162.

    Article  Google Scholar 

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Author notes

  1. Kevin Atteson

    Present address: Doubleprime Inc., 250 W 89th Street, #5E, New York, NY, 10024, USA

Affiliations

  1. Department of Ecology and Evolutionary Biology, Yale University, P.O. Box 208106, New Haven, CT, 06520-8106, USA

    Max Shpak & Kevin Atteson

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  1. Max Shpak
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  2. Kevin Atteson
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Correspondence to Max Shpak.

Additional information

An erratum to this article is available at http://dx.doi.org/10.1006/bulm.2002.0312.

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Shpak, M., Atteson, K. A survey of unequal crossover systems and their mathematical properties. Bull. Math. Biol. 64, 703 (2002). https://doi.org/10.1006/bulm.2001.0299

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  • DOI: https://doi.org/10.1006/bulm.2001.0299

Keywords

  • Stationary Distribution
  • Gene Copy Number
  • Global Attractor
  • Detailed Balance
  • Negative Binomial