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The structure of genotype-phenotype maps makes fitness landscapes navigable

Nature Ecology & Evolution volume 6, pages 1742–1752 (2022)Cite this article

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Abstract

Fitness landscapes are often described in terms of ‘peaks’ and ‘valleys’, indicating an intuitive low-dimensional landscape of the kind encountered in everyday experience. The space of genotypes, however, is extremely high dimensional, which results in counter-intuitive structural properties of genotype-phenotype maps. Here we show that these properties, such as the presence of pervasive neutral networks, make fitness landscapes navigable. For three biologically realistic genotype-phenotype map models—RNA secondary structure, protein tertiary structure and protein complexes—we find that, even under random fitness assignment, fitness maxima can be reached from almost any other phenotype without passing through fitness valleys. This in turn indicates that true fitness valleys are very rare. By considering evolutionary simulations between pairs of real examples of functional RNA sequences, we show that accessible paths are also likely to be used under evolutionary dynamics. Our findings have broad implications for the prediction of natural evolutionary outcomes and for directed evolution.

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Fig. 1: High-dimensional bypasses facilitate landscape navigability.
Fig. 2: The relationship of navigability to the GP map properties of redundancy, deleterious frequency and mean genotypic robustness.
Fig. 3: The relationship of navigability to neutral correlations, dimensionality and ruggedness with the associations visualized for an RNA phenotype network.
Fig. 4: Example of an accessible path for a specific L = 30 fRNA source–target pair.
Fig. 5: Navigability under evolutionary dynamics with source and target phenotypes sampled from the fRNAdb with random and Hamming fitness assignments under monomorphic (left) and polymorphic (right) evolutionary dynamics.

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Data availability

The dataset containing fRNA (fRNAdb) used in this paper is available at: https://doi.org/10.18908/lsdba.nbdc00452-001. The GP maps analysed are available in the Code availability section.

Code availability

The ViennaRNA package (v.1.8.5), RNAshape package https://anaconda.org/bioconda/rnashapes and custom C++ and Python source code was used to construct GP maps and perform computational simulations. The source code is available at: https://github.com/sgreenbury/gp-maps-nav.

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Acknowledgements

S.E.A. was supported by the Royal Society and the Gatsby Foundation. S.F.G. was supported by the Engineering and Physical Sciences Research Council. We thank M. Weiß for helpful discussions and insights.

Author information

Authors and Affiliations

  1. Theory of Condensed Matter Group, Cavendish Laboratory, University of Cambridge, Cambridge, UK

    Sam F. Greenbury

  2. The Alan Turing Institute, British Library, London, UK

    Sam F. Greenbury & Sebastian E. Ahnert

  3. Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, UK

    Ard A. Louis

  4. Department of Chemical Engineering and Biotechnology, University of Cambridge, Cambridge, UK

    Sebastian E. Ahnert

Authors
  1. Sam F. Greenbury
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  2. Ard A. Louis
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  3. Sebastian E. Ahnert
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Contributions

S.F.G., A.A.L. and S.E.A. conceived and designed the experiments. S.F.G. performed the experiments. S.F.G., A.A.L. and S.E.A. analysed the data. S.E.A. supervised the work. S.F.G., A.A.L. and S.E.A. wrote the paper.

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Correspondence to Sam F. Greenbury, Ard A. Louis or Sebastian E. Ahnert.

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Nature Ecology & Evolution thanks Jacobo Aguirre and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Depiction of the different biological systems, specific GP maps considered and example genotype, phenotype and encoding of phenotype.

Each row is a specific GP map included in this work and is situated within one of the four categories of system: RNA, Polyomino, HP (compact), and HP (non-compact). RNA and HP genotypes are depicted with distinct colours for their constituent bases. Polyomino genotypes are shown as numerical sequences that map to the edges of distinctly coloured tiles with arrows used to indicate the tile orientation. The corresponding phenotype (the structure that is formed following the self-assembly process on the example genotype) is shown with the colours and arrows used in the genotype depiction highlighting the mechanism by which bonds are formed. The encoding of the example phenotypes are shown in the final column: dot-bracket and shape notation for RNA, grid coordinates for tile placements of polyominoes, and the lattice directions for the HP lattice fold.

Extended Data Table 1 GP map properties and navigability estimates. All GP maps studied with their properties (base K, sequence length L, number of phenotypes NP, proportion of genotypes with the deleterious phenotype fdel, average redundancy \({\log }_{10}R\), mean genotypic robustness \(\left\langle {\rho }_{g}\right\rangle\)) and estimate with standard error of navigability \(\left\langle \psi \right\rangle \pm SE(\left\langle \psi \right\rangle )\). RNA, Polyomino and compact HP GP maps all have navigable fitness landscapes (\(\left\langle \psi \right\rangle > 0.6\)) under random fitness assignment. By contrast, non-compact HP models have very low navigability (\(\left\langle \psi \right\rangle \le 0.013\))
Full size table
Extended Data Table 2 Terminology. A summary of terms and their representations used in the paper. The first column (left) provides the term used and its description, while the second column (right) has the corresponding mathematical symbol and equation where relevant
Full size table

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Greenbury, S.F., Louis, A.A. & Ahnert, S.E. The structure of genotype-phenotype maps makes fitness landscapes navigable. Nat Ecol Evol 6, 1742–1752 (2022). https://doi.org/10.1038/s41559-022-01867-z

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