Simple 'qk-models' of genetic robustness reduce a genotype-phenotype (GP) map to two quantities: the probability qq that a mutation leaves the phenotype unchanged, and the phenotypic neighborhood k\mathbf{k} of phenotypes reachable by single mutations. Hardy's unification of these models (published as 'How does robustness affect evolvability?', Evolution, 2025) showed that evolvability is maximized at intermediate robustness when the neighborhood is smaller than the phenotype space, and that this logic transfers from mutational to environmental perturbation - but it explicitly left open how these two kinds of robustness interact with other GP-map properties that the qk abstraction cannot represent: criticality (robustness of robustness), phenotypic plasticity, cryptic genetic variation, environmental memory, and architectural design features such as modularity, excitation, scale-free degree distributions, and feedback/feed-forward network motifs. The open problem is to build and analyze GP-map models rich enough to exhibit these properties and to characterize how multiple kinds of robustness and these other map properties jointly shape the evolution of evolvability. A 2026 Trends in Genetics review independently lists the robustness-plasticity interplay and the coevolution of cryptic variation, plasticity, and mutational robustness among the field's foremost outstanding questions.

Background

A genotype-phenotype (GP) map is the correspondence between genetic sequences (genotypes) and the observable traits they produce (phenotypes); RNA secondary structure and gene regulatory networks are standard model examples. Mutational robustness is the probability that a random mutation leaves the phenotype unchanged. Evolvability is the capacity of a population to generate heritable phenotypic variation that selection can act on - for discrete phenotypes, often measured as the speed of evolving a newly adaptive phenotype. Mutationally robust phenotypes typically form large neutral networks (connected sets of genotypes mapping to the same phenotype) that populations drift across, thereby accessing mutationally adjacent, phenotypically novel genotypes. Two simple modeling traditions quantified how robustness and evolvability relate. Ancel Meyers, Ancel and Lachmann (2005) studied genotypes under selection that fluctuates between environmental states, where each genotype has both a mutational and an environmental robustness (the probability that an environmental change leaves the phenotype unchanged): optimal mutational robustness is tuned to the rate of environmental change. Draghi, Parsons, Wagner and Plotkin (2010) showed that when not every phenotype is reachable by a single mutation, evolvability is non-monotonic in mutational robustness qq and maximized at intermediate qq - robustness both shields deleterious mutations and lets populations accumulate on neutral networks from which novelty is reachable. Hardy (bioRxiv 2024, published 2025 in Evolution) consolidated these into 'qk-models', where a GP map is abstracted to qq (probability a mutation is phenotypically neutral) and k\mathbf{k} (the set of KK alternative phenotypes accessible by single mutations), and showed the core insights survive when qq, k\mathbf{k}, and the mutational network itself co-evolve, and that the same logic transfers to environmental (migration-like) perturbation. The paper's closing outlook observes that this abstraction necessarily omits properties known to affect evolvability in real developmental systems: criticality (the robustness of robustness itself, i.e., how sensitively a map's robustness responds to perturbation, studied in gene-regulatory-network models near dynamical criticality), phenotypic plasticity (environmentally induced phenotypic change) and the cryptic genetic variation it can conceal and later expose to selection, environmental memory (the ability of evolving systems to 'remember' and generalize from past selective environments, in the sense of learning-theoretic views of evolution, in which a developmental system stores and recalls previously selected phenotypes; Watson et al., 2014; Kouvaris et al., 2017), and architectural design features - modularity (quasi-independent subnetworks), excitation (the propagation and amplification of perturbations through connected network elements), scale-free degree distributions (a few highly connected nodes), and network motifs such as feedback and feed-forward loops (recurring small wiring patterns that regulate the flow of signals through the network). How these properties interact with each other and with multiple kinds of robustness, and which design features determine robustness, evolvability, and memory, is not captured by the qk abstraction and, per the source and subsequent reviews, remains uncharacterized.

Problem Statement

In models of genotype-phenotype maps that go beyond the simple qk abstraction - maps that can exhibit any of the following properties (not necessarily all of them): criticality (robustness of robustness), the capacity for phenotypic plasticity and the accumulation of cryptic genetic diversity, the ability to remember and generalize from past selective environments, and architectural design features such as modularity, excitation, scale-free degree distributions, and network motifs like feedback and feed-forward loops - characterize the interplay between multiple kinds of robustness (e.g., mutational and environmental robustness) and these other map properties. The overarching objective is a predictive account, generalizing the unified qk-model results (intermediate-robustness optima for evolvability, preferential selection on the phenotypic neighborhood), of how the several kinds of robustness and the other map properties interact - which design features determine robustness, evolvability, and memory, and how their interactions, rather than each property in isolation, shape the evolution of evolvability. The question concretely includes the following sub-questions, which the source passage and later reviews flag as open directions: under which conditions mutational and environmental robustness co-evolve or decouple; how plasticity and cryptic genetic variation mediate the robustness-evolvability relationship; and how network architecture and criticality modify the non-monotonic robustness-evolvability relationship established in simple models. A resolution is a predictive account, established on at least one concrete genotype-phenotype map that goes beyond the qk abstraction and exhibits at least two of the properties listed above (counting the capacity for phenotypic plasticity and the accumulation of cryptic genetic diversity as distinct properties, and counting each named architectural design feature as a distinct property), of how more than one kind of robustness and those properties interact with one another to shape the evolution of evolvability; a result that treats a single property, or a single kind of robustness, without relating it to the others - and likewise the resolution of any one of the sub-questions above - is a partial contribution that does not by itself resolve it.

References

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