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Maximum Mutational Robustness in Genotype-Phenotype Maps Follows a Self-similar Blancmange-like Curve

Mohanty, Vaibhav and Greenbury, Sam F. and Sarkany, Tasmin and Narayanan, Shyam and Dingle, Kamaludin and Ahnert, Sebastian E. and Louis, Ard A. (2023) Maximum Mutational Robustness in Genotype-Phenotype Maps Follows a Self-similar Blancmange-like Curve. . (Unpublished)

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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 organised 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 (GP) maps for RNA secondary structure and the 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.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Mohanty, Vaibhav0000-0003-1475-4228
Greenbury, Sam F.0000-0003-4452-2006
Dingle, Kamaludin0000-0003-4423-3255
Ahnert, Sebastian E.0000-0003-2613-0041
Louis, Ard A.0000-0002-8438-910X
Additional Information:The copyright holder for this preprint is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. The authors thank Nora Martin and Akshay Jaggi for helpful discussions. V.M. was supported by a Marshall Scholarship and by award numbers 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. 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 comprised of 1 to k^ℓ 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 GitHub link in ref. [42]. The authors have declared no competing interest.
Funding AgencyGrant Number
Marshall Aid Commemoration CommissionUNSPECIFIED
NIH Predoctoral FellowshipT32GM007753
NIH Predoctoral FellowshipT32GM144273
NSF Graduate Research FellowshipUNSPECIFIED
Simons FoundationUNSPECIFIED
Google Faculty Research AwardUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Gatsby Charitable FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20230316-181920000.5
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:120124
Deposited By: George Porter
Deposited On:22 Mar 2023 16:56
Last Modified:22 Mar 2023 16:56

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