# Population evolution in a single peak fitness landscape how high are the clouds?

## Abstract

A theory for evolution of molecular sequences must take into account that a population consists of a finite number of individuals with related sequences. Such a population will not behave in the deterministic way expected for an infinite population, nor will it behave as in adaptive walk models, where the whole of the population is represented by a single sequence. Here we study a model for evolution of population in a fitness landscape with a single fitness peak. This landscape is simple enough for finite size population effects to be studied in detail. Each of the N individuals in the population is represented by a sequence of L genes which may either be advantageous or disadvantageous. The fitness of an individual with k disadvantageous genes is w_{k}=(1−s)^{k}, where s determines the strength of selection. In the limit L tends to infinity the model reduces to the problem of Muller's Ratchet: the population moves away from the fitness peak at a constant rate due to the accumulation of disadvantageous mutations. For finite length sequences, a population placed initially at the fitness peak will evolve away from the peak until a balance is reached between mutation and selection. From then on the population will wander through a spherical shell in sequence space at a constant mean Hamming distance <k> from the optimum sequence. This has been likened to the idea of a cloud layer hanging below the mountain peak. We give an approximate theory for the way <k> depends on N, L, s, and the mutation rate u. This is found to agree well with numerical simulation.

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## References

- Bonhoeffer, S. and Stadler, P.F. (1993) Error thresholds on correlated fitness landscapes.
*J. Theor. Biol.***164**, 359–72.Google Scholar - Derrida, B. and Peliti, L. (1991) Evolution in a flat fitness landscape.
*Bull. Math. Biol.***53**, 355–382.Google Scholar - Eigen, M., McCaskill, J. and Schuster, P. (1989) The molecular quasispecies.
*Adv. Chem. Phys.***75**, 149–263.Google Scholar - Flyvbjerg, H. and Lautrup, B. (1992) Evolution in a rugged fitness landscape.
*Phys. Rev. A.***46**, 6714–18.Google Scholar - Fontana, W., Stadler, P.F., Bornberg-Bauer, E.G., Griesmacher, T., Hofacker, I.L., Tacker, M., Tarazona, P., Weinberger, E.D., and Schuster, P. (1993) RNA folding and combinatory landscapes.
*Phys. Rev. E***47**, 2083–99.Google Scholar - Gabriel, W., Lynch, M. and Bürger, R. (1993) Muller's Ratchet and Mutational Meltdowns.
*Evolution***47**, 1744–57.Google Scholar - Haigh, J. (1978) The accumulation of deleterious genes in a population — Muller's ratchet.
*Theor. Pop. Biol.***14**, 251–267.Google Scholar - Higgs, P.G. (1994) Error thresholds and stationary mutant distributions in multi-locus diploid genetics models.
*Genet. Res. (Camb)***63**, 63–78.Google Scholar - Higgs, P.G. (1995) Frequency Distributions in Population Genetics Parallel those in Statistical Physics.
*Phys. Rev. E.***51**, 95–101.Google Scholar - Higgs, P.G. and Derrida, B. (1991) Stochastic models for species formation in evolving populations.
*J. Phys. A (Math. & Gen.)***24**, L985–L991.Google Scholar - Higgs, P.G. and Derrida, B. (1992) Genetic distance and species formation in evolving populations.
*J. Mol. Evol.***35**, 454–465.Google Scholar - Higgs, P.G. and Woodcock, G. (1995) The accumulation of mutations in asexual populations and the structure of genealogical trees in the presence of selection.
*J. Math. Biol.*(in press).Google Scholar - Kauffman, S.A. (1993)
*The Origins of Order*. Oxford University Press.Google Scholar - Kauffman, S.A. and Levin, S. (1987) Towards a general theory of adaptive walks on rugged landscapes.
*J. Theor. Biol.***128**, 11–45.Google Scholar - Lynch, M., Bürger, R., Butcher, D. and Gabriel, W. (1993) The mutational meltdown in asexual populations.
*J. Hered.***84**, 339–344.Google Scholar - Macken, C.A., Hagan, P.S. and Perelson, A.S. (1991) Evolutionary walks on rugged landscapes.
*SIAM J. Appl. Math.***51**, 799–827.Google Scholar - Stephan, W., Chao, L. and Smale, J.G. (1993) The advance of Muller's ratchet in a haploid asexual population: approximate solutions based on diffusion theory.
*Genet. Res. (Camb)***61**, 225–231.Google Scholar - Tarazona, P. (1992) Error thresholds for molecular quasispecies as phase transitions: From simple landscapes to spin-glass models.
*Phys. Rev. A.***45**, 6038–50.Google Scholar - Woodcock, G. and Higgs, P.G. (1995) Population evolution on a single peak fitness landscape (in press).Google Scholar