Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities - PubMed (original) (raw)

Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities

Janko Gravner et al. J Theor Biol. 2007.

Abstract

We study how correlations in the random fitness assignment may affect the structure of fitness landscapes, in three classes of fitness models. The first is a phenotype space in which individuals are characterized by a large number n of continuously varying traits. In a simple model of random fitness assignment, viable phenotypes are likely to form a giant connected cluster percolating throughout the phenotype space provided the viability probability is larger than 1/2(n). The second model explicitly describes genotype-to-phenotype and phenotype-to-fitness maps, allows for neutrality at both phenotype and fitness levels, and results in a fitness landscape with tunable correlation length. Here, phenotypic neutrality and correlation between fitnesses can reduce the percolation threshold, and correlations at the point of phase transition between local and global are most conducive to the formation of the giant cluster. In the third class of models, particular combinations of alleles or values of phenotypic characters are "incompatible" in the sense that the resulting genotypes or phenotypes have zero fitness. This setting can be viewed as a generalization of the canonical Bateson-Dobzhansky-Muller model of speciation and is related to K-SAT problems, prominent in computer science. We analyze the conditions for the existence of viable genotypes, their number, as well as the structure and the number of connected clusters of viable genotypes. We show that analysis based on expected values can easily lead to wrong conclusions, especially when fitness correlations are strong. We focus on pairwise incompatibilities between diallelic loci, but we also address multiple alleles, complex incompatibilities, and continuous phenotype spaces. In the case of diallelic loci, the number of clusters is stochastically bounded and each cluster contains a very large sub-cube. Finally, we demonstrate that the discrete NK model shares some signature properties of models with high correlations.

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Figures

Figure 1

Figure 1

A four-dimensional example. (a) Start with the four-dimensional hypercube. (b) On the first step, create conformist clusters by randomly eliminating each edge with probability 1 − q. In the example, there are 7 conformist clusters left. (c) On the second step, remove each conformist cluster with probability 1 − p; the vertices to be removed are black. In the example, there are 3 viable conformist clusters left. (d) The remaining viable genotypes have nearest neighbors connected by edges. In the example, all remaining viable genotypes are connected in a single component.

Figure 2

Figure 2

Simulated lower bounds λm (dashed curves) and upper bounds λM (solid curves), and ζ (thick curve) plotted against μ, for n = 10,…, 20. Lower bounds increase with n, and upper bounds decrease, for this range of n.

Figure 3

Figure 3

Simulated number of clusters, vs. c for n = 20, with the proportion (out of 1000) of trials with exactly one (squares), exactly two (triangles), and at least three (diamonds) clusters. The solid curve is (1−c)ec.

Figure 4

Figure 4

Simulated lower bounds λm (dashed curves) and upper bounds λM (solid curves), and ζ (thick curve) plotted against μ, for n = 10,…, 20 in the model from Appendix B. Lower bounds increase with n, and upper bounds decrease, for this range of n. Compare to Figure 2.

Figure 5

Figure 5

The upper bound ru(c) for the number of fixed loci (non-*’s) in the implicant of shortest length included in every cluster of viable genotypes, plotted against c. As explained in the text, 1 − ru(c) is a lower bound on the dimension of the subcube specified by the implicant.

References

    1. Achlioptas D, Kirousis LM, Kranakis E, Krizanc D. Rigorous results for (2 + p)-SAT. Theoretical Computer Science. 2001;265:109–129.
    1. Achlioptas D, Moore C. Random k-SAT: two moments suffice to cross a sharp threshold. SIAM Journal on Computing. 2004;17:947–973.
    1. Achlioptas D, Peres Y. The threshold for random k-SAT is 2k log 2 − o(k) Journal of the American Mathematical Society. 2004;17:947–973.
    1. Athreya K, Ney P. Branching processes. Springer-Verlag; 1971. reprinted by Dover 2004.
    1. Barbour AD, Holst L, Janson S. Poisson Approximation. Oxford University Press; 1992.

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