Maximum mutational robustness in genotype–phenotype maps follows a self-similar blancmange-like curve
Creators
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1.
University of Oxford
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2.
Massachusetts Institute of Technology
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3.
Harvard University
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4.
University of Cambridge
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The Alan Turing Institute
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Wellcome/MRC Cambridge Stem Cell Institute
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Gulf University for Science & Technology
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8.
California Institute of Technology
Abstract
Phenotype robustness, defined as the average mutational robustness of all the genotypes that map to a given phenotype, plays a key role in facilitating neutral exploration of novel phenotypic variation by an evolving population. By applying results from coding theory, we prove that the maximum phenotype robustness occurs when genotypes are organized as bricklayer's graphs, so-called because they resemble the way in which a bricklayer would fill in a Hamming graph. The value of the maximal robustness is given by a fractal continuous everywhere but differentiable nowhere sums-of-digits function from number theory. Interestingly, genotype–phenotype maps for RNA secondary structure and the hydrophobic-polar (HP) model for protein folding can exhibit phenotype robustness that exactly attains this upper bound. By exploiting properties of the sums-of-digits function, we prove a lower bound on the deviation of the maximum robustness of phenotypes with multiple neutral components from the bricklayer's graph bound, and show that RNA secondary structure phenotypes obey this bound. Finally, we show how robustness changes when phenotypes are coarse-grained and derive a formula and associated bounds for the transition probabilities between such phenotypes.
Copyright and License
© 2023 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Acknowledgement
The authors thank Nora Martin and Akshay Jaggi for helpful discussions.
Funding
V.M. was supported by a Marshall Scholarship and by award nos. T32GM007753 and T32GM144273 from the National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of General Medical Sciences, National Institutes of Health or the Marshall Aid Commemoration Commission. S.N. was supported by an NSF Graduate Fellowship, a Simons Investigator award, the NSF TRIPODS program and a Google Fellowship. S.F.G. was supported by the Engineering and Physical Sciences Research Council. S.E.A. was supported by the Royal Society and the Gatsby Foundation.
Contributions
V.M.: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, validation, visualization, writing—original draft, writing—review and editing; S.F.G.: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, resources, validation, writing—review and editing; T.S.: data curation, investigation, validation, writing—review and editing; S.N.: formal analysis, funding acquisition, investigation, writing—review and editing; K.D.: writing—review and editing; S.E.A.: conceptualization, funding acquisition, investigation, methodology, resources, supervision, validation, writing—review and editing; A.A.L.: conceptualization, formal analysis, investigation, methodology, project administration, resources, supervision, validation, writing—original draft, writing—review and editing.
All authors gave final approval for publication and agreed to be held accountable for the work performed therein.
Data Availability
We have introduced the web tool RoBound Calculator, a Google Colaboratory notebook which can generate, for specified ℓ and k, a continuous interpolation of the maximum robustness curve, tight upper and lower bounds on the maximum robustness curve, the exact robustnesses of bricklayer's graphs comprising 1 to kℓ genotypes, the random null expectation of robustness and the minimum robustness curve for a single neutral component. The RoBound Calculator is available free of charge, with open-source code at the GitHub link in [52]. The data are available from the Dryad Digital Repository: https://datadryad.org/stash/dataset/doi:10.5061/dryad.sj3tx969f.
Conflict of Interest
We declare we have no competing interests.
Files
rsif.2023.0169.pdf
Additional details
Identifiers
- PMCID
- PMC10369032
Related works
- Is variant form of
- https://authors.library.caltech.edu/records/erdht-rbw55 (URL)
Funding
- National Institutes of Health
- NIH Predoctoral Fellowship T32GM007753
- National Institutes of Health
- NIH Predoctoral Fellowship PMC10369032
- National Science Foundation
- NSF Graduate Research Fellowship
- Simons Foundation
- Google PhD Fellowship
- Engineering and Physical Sciences Research Council
- Royal Society
- Gatsby Charitable Foundation
- Marshall Aid Commemoration Commission
- Marshall Scholarship