Error Thresholds in a Mutation–selection Model with Hopfield-type Fitness

  • Tini GarskeEmail author
Original Article


The deterministic limit of a Hopfield-type mutation–selection model in the sequence space approach is investigated. Genotypes are identified with two-letter sequences. Mutation is modelled as a Markov process, fitness functions are of Hopfield type, where the fitness of a sequence is determined by the Hamming distances to a number of predefined patterns. Using a maximum principle for the population mean fitness in equilibrium, the error threshold phenomenon is studied for quadratic Hopfield-type fitness functions with small numbers of patterns. Different from previous investigations of the Hopfield model, the system shows error threshold behaviour not for all fitness functions, but only for certain parameter values.


Mutation–selection model Hopfield model Error threshold Maximum principle 


  1. Amit, D.J., Gutfreund, H., Sompolinsky, H., 1985a. Spin-glass models of neural networks. Phys. Rev. A 32(2), 1007–1018.MathSciNetCrossRefADSGoogle Scholar
  2. Amit, D.J., Gutfreund, H., Sompolinsky, H., 1985b. Storing infinite numbers of patterns in a spin-glass model of neural networks. Phys. Rev. Lett. 55 (14), 1530–1533.CrossRefADSGoogle Scholar
  3. Baake, E., Baake, M., Bovier, A., Klein, M., 2005. An asymptotic maximum principle for essentially linear evolution models. J. Math. Biol. 50(1), 83–114.PubMedzbMATHMathSciNetCrossRefGoogle Scholar
  4. Baake, E., Baake, M., Wagner, H., 1997. Ising quantum chain is equivalent to a model of biological evolution. Phys. Rev. Lett. 78(3), 559–562, erratum, Phys. Rev. Lett. 79(1997), 1782.Google Scholar
  5. Baake, E., Gabriel, W., 2000. Biological evolution through mutation, selection, and drift: An introductory review. In: Stauffer, D., (Ed.), Annual Reviews of Computational Physics VII. World Scientific, Singapore, pp. 203–264.Google Scholar
  6. Baake, E., Wagner, H., 2001. Mutation–selection models solved exactly with methods of statistical mechanics. Genet. Res. 78, 93–117.PubMedCrossRefGoogle Scholar
  7. Boerlijst, M.C., Bonhoeffer, S., Nowak, M.A., 1996. Viral quasi-species and recombination. P. Roy. Soc. Lond., Series B 263(1376), 1577–1584.ADSGoogle Scholar
  8. Bonhoeffer, S., Stadler, P.F., 1993. Error thresholds on correlated fitness landscapes. J. Theo. Biol. 164(3), 359–372.CrossRefGoogle Scholar
  9. Bürger, R., 2000. The Mathematical Theory of Selection, Recombination, and Mutation. Wiley, Chichester.zbMATHGoogle Scholar
  10. Campos, P.R.A., Adami, C., Wilke, C.O., 2002. Optimal adaptive performance and delocalization in NK fitness landscapes. Physica A 304(3–4), 495–506.zbMATHCrossRefADSGoogle Scholar
  11. Crotty, S., Cameron, C.E., Andino, R., 2001. RNA virus error catastrophe: Direct molecular test by using ribavirin. P. Natl. Acad. Sci. USA 98(12), 6895–6900.CrossRefADSGoogle Scholar
  12. Crow, J.F., Kimura, M., 1970. An Introduction to Population Genetics Theory. Harper & Row, New York.zbMATHGoogle Scholar
  13. Domingo, E., Escarmis, C., Sevilla, N., Moya, A., Elena, S.F., Quer, J., Novella, I.S., Holland, J.J., 1996. Basic concepts in RNA virus evolution. FASEB J. 10(8), 859–864.PubMedGoogle Scholar
  14. Domingo, E., Holland, J.J., 1988. High error rates, population equilibrium, and evolution of RNA replication systems. In: Domingo, E. (Ed.), RNA Genetics. vol. 3. CRC Press, Boca Raton, p. 3.Google Scholar
  15. Domingo, E., Holland, J.J., 1997. RNA virus mutations and fitness for survival. Annu. Rev. Microbiol. 51, 151–178.PubMedCrossRefGoogle Scholar
  16. Eigen, M., 1971. Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58(10), 465–523.PubMedCrossRefADSGoogle Scholar
  17. Eigen, M., 1993. Viral quasispecies. Sci. Am. 269(1), 42–49.PubMedCrossRefGoogle Scholar
  18. Eigen, M., Biebricher, C.K., 1988. Sequence space and quasispecies evolution. In: Domingo, E. (Ed.), RNA Genetics. vol. 3. CRC Press, Boca Raton, pp. 211–245.Google Scholar
  19. Eigen, M., McCaskill, J., Schuster, P., 1989. The molecular quasi-species. Adv. Chem. Phys. 75, 149–263.Google Scholar
  20. Ewens, W.J., 2004. Mathematical Population Genetics, 2nd edition. Springer, New York.zbMATHGoogle Scholar
  21. Franz, S., Peliti, L., 1997. Error threshold in simple landscapes. J.Phys. A 30 (13), 4481–4487.zbMATHMathSciNetCrossRefADSGoogle Scholar
  22. Franz, S., Peliti, L., Sellitto, M., 1993. An evolutionary version of the random energy model. J. Phys. A 26 (23), L1195–L1199.CrossRefADSGoogle Scholar
  23. Garske, T., 2005. Mutation–Selection Models of Sequence Evolution in Population Genetics. PhD thesis, The Open University, Milton Keynes, UK.Google Scholar
  24. Garske, T., Grimm, U., 2004a. Maximum principle and mutation thresholds for four-letter sequence evolution. Journal of Statistical Mechanics: Theory and Experiment P07007, (Preprint q-bio.PE/0406041).Google Scholar
  25. Garske, T., Grimm, U., 2004b. A maximum principle for the mutation–selection equilibrium of nucleotide sequences. B. Math. Biol. 66(3), 397–421, (Preprint physics/0303053).Google Scholar
  26. Hamming, R.W., 1950. Error detecting and error correcting codes. Bell Syst. Tech. J. 26(2), 147–160.MathSciNetGoogle Scholar
  27. Hermisson, J., Redner, O., Wagner, H., Baake, E., 2002. Mutation selection balance: Ancestry, load, and maximum principle. Theor. Popul. Biol. 62, 9–46.PubMedzbMATHCrossRefGoogle Scholar
  28. Hermisson, J., Wagner, H., Baake, M., 2001. Four-state quantum chain as a model of sequence evolution. J. Stat. Phys. 102(1/2), 315–343.zbMATHMathSciNetCrossRefGoogle Scholar
  29. Higgs, P., 1994. Error thresholds and stationary mutant distributions in multilocus diploid genetics models. Genet. Res. Cambridge 63(1), 63–78.Google Scholar
  30. Holland, J.J., Domingo, E., de la Torre, J.C., Steinhauer, D.A., 1990. Mutation frequencies at defined single codon sites in vesicular stromatitis-virus and poliovirus can be increased only slightly by chemical mutagenesis. J. Vir. 64(8), 3960–3962.Google Scholar
  31. Hopfield, J. J., 1982. Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. USA 79 (8), 2554–2558.Google Scholar
  32. Huynen, M. A., Stadler, P. F., Fontana, W., 1996. Smoothness within ruggedness: the role of neutrality in adaptation. Proc. Nat. Acad. Sci. USA 93 (1), 397–401.Google Scholar
  33. Karlin, S., 1966. A First Course in Stochastic Processes. Academic Press, New York.zbMATHGoogle Scholar
  34. Kauffman, S., Levin, S., 1987. Towards a general theory of adaptive walks on rugged landscapes. J. Theo. Biol. 128, 11–45.MathSciNetGoogle Scholar
  35. Kemeny, J.G., Snell, J.L., 1960. Finite Markov Chains. Van Nostrand Reinhold Company, New York.zbMATHGoogle Scholar
  36. Leuthäusser, I., 1987. Statistical mechanics of Eigen's evolution model. J. Stat. Phys. 48(1/2), 343–360.CrossRefADSGoogle Scholar
  37. Loeb, L.A., Essigmann, J.M., Kazazi, F., Zhang, J., Rose, K.D., Mullins, J.I., 1999. Lethal mutagenesis of HIV with mutagenic nucleoside analogs. Proc. Nat. Acad. Sci. USA 96, 1492–1497.Google Scholar
  38. Nowak, M., Schuster, P., 1989. Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller's ratchet. J. Theo. Biol. 137(4), 375–395.Google Scholar
  39. Ohta, T., Kimura, M., 1973. A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population. Genet. Res. 22, 201–204.MathSciNetCrossRefGoogle Scholar
  40. Peliti, L., 2002. Quasispecies evolution in general mean-field landscapes. Europhys. Lett. 57(5), 745–751.CrossRefADSGoogle Scholar
  41. Reidys, C., Forst, C.V., Schuster, P., 2001. Replication and mutation on neutral networks. B. Math. Biol. 63(1), 57–94.CrossRefGoogle Scholar
  42. Reidys, C.M., Stadler, P.F., 2002. Combinatorial landscapes. SIAM Rev. 44(1), 3–54.zbMATHMathSciNetCrossRefGoogle Scholar
  43. Rumschitzky, D.S., 1987. Spectral properties of Eigen's evolution matrices. J. Math. Biol. 24, 667–680.MathSciNetGoogle Scholar
  44. Sierra, S., Dávila, M., Lowenstein, P.R., Domingo, E., 2000. Response of foot-and-mouth disease virus to increased mutagenesis: Influence of viral load and fitness in loss of infectivity. J. Virol. 74(18), 8316–8323.PubMedCrossRefGoogle Scholar
  45. Talagrand, M., 2003. Spin Glasses: A Challenge for Mathematicians. Springer, Berlin.Google Scholar
  46. Tarazona, P., 1992. Error thresholds for molecular quasispecies as phase transitions: From simple landscapes to spin-glass models. Phys. Rev. A 45(8), 6038–6050.PubMedCrossRefADSGoogle Scholar
  47. Thompson, C.J., McBride, J.L., 1974. On Eigen's theory of the self-organization of matter and the evolution of biological macromolecules. Math. Biosci. 21(1–2), 127–142.zbMATHMathSciNetGoogle Scholar
  48. van Lint, J.H., 1982. Introduction to Coding Theory. Springer, Berlin.zbMATHGoogle Scholar
  49. Whittle, P., 1976. Probability. Wiley, London.zbMATHGoogle Scholar
  50. Wiehe, T., 1997. Model dependency of error thresholds: The role of fitness functions and contrasts between finite and infinite sites models. Genet. Res. Cambridge 69, 127–136.Google Scholar
  51. Wiehe, T., Baake, E., Schuster, P., 1995. Error propagation in reproduction of diploid organisms. A case study on single peaked landscapes. J. Theor. Biol. 177(1), 1–15.PubMedCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2006

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Applied Maths Department, Faculty of Mathematics and ComputingThe Open UniversityMilton KeynesUK

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