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Average Fitness Differences on NK Landscapes

Abstract

The average fitness difference between adjacent sites in a fitness landscape is an important descriptor that impacts in particular the dynamics of selection/mutation processes on the landscape. Of particular interest is its connection to the error threshold phenomenon. We show here that this parameter is intimately tied to the ruggedness through the landscape’s amplitude spectrum. For the NK model, a surprisingly simple analytical estimate explains simulation data with high precision.

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Acknowledgements

WH thanks the Institute for Advanced Study, Amsterdam, for financial support through a fellowship. Discussions with Bärbel M. R. Stadler are gratefully acknowledged.

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Correspondence to Wim Hordijk.

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In memoriam Manfred Eigen $$^*$$∗9 May 1927–$$^\dagger $$† 6 Feb 2019.

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Appendices

An example of an NK landscape

We include here a brief example of an NK landscape, taken from (Kauffman 1993). The fitness contributions \(f_i(b)\) for the individual positions are tabulated for each value of the bit \(b_i\) itself and \(K=2\) additional relevant bits. The final fitness value for each bit string is the average of the position-wise contributions. Together with the adjacency relation of the Boolean hypercube, this defines the landscape (Fig. 3).

Fig. 3
figure 3

A simple example of an instance of the NK model for \(N=3\) and \(K=2\). Top: The fitness contributions for the three bits for each of the \(2^{K+1}=8\) possible neighborhood configurations are assigned at random. The fitness of the entire string is the average of the individual fitness contributions. Bottom: The boolean hypercube representing the fitness landscape defined in the table above

Independence of Parity and Neighborhood Structure

Figure 4 shows a scatter plot of \((\varDelta f)(x)/N\) against f for landscapes with different parity and different neighborhood structures (random and adjacent). We have chosen the different values of K such that the data sets are distinguishable. The predicted slopes for \(N=50\) and \(K=5\), 10, 15, and 20 are \(s=0.12\), 0.22, 0.32, and 0.42, while the empirical values from the data displayed here are \(\hat{s}=0.127\), 0.230, 0.328, and 0.412, respectively. The empirical and theoretical values are in excellent agreement.

Fig. 4
figure 4

Scatter plot of \((\varDelta f)(x)/N\) against f, illustrating that the slopes do not depend on parity, i.e., even vs. odd values of K, or the choice of adjacency, i.e., random (’rnd’) vs. adjacent (’adj’)

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Hordijk, W., Kauffman, S.A. & Stadler, P.F. Average Fitness Differences on NK Landscapes. Theory Biosci. 139, 1–7 (2020). https://doi.org/10.1007/s12064-019-00296-0

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Keywords

  • NK model
  • Fitness landscapes
  • Elementary landscape
  • Graph Laplacian
  • Error threshold