A quantitative theory of the evolution of evolvability and its relation to mutational robustness in gene-network genotype-phenotype maps

Draghi and Wagner (2009) showed by simulation that fluctuating natural selection increases the evolvability of populations of model gene regulatory networks, even under recombination, and that in their model evolvability rises while mutational robustness falls. They explicitly left open the conversion of this qualitative result into a quantitative and general theory: a fully quantitative analysis of how population size, period of environmental change, mutation rates and step size, and network size influence the evolution of evolvability, going beyond the source's one-factor parameter slices; the generality of simple structural predictors of evolvability such as network excitation; delineation of the conditions under which robustness promotes rather than impedes evolvability; scaling to large networks with fully converged developmental dynamics; and integration with the evolution of mutation- and recombination-rate modifiers. Later work quantitatively resolved the population-genetics-level part of the robustness-evolvability question, but the full account, including the architecture-dependent sign of the robustness-evolvability relationship and the generality of structural predictors, remains open.

Background

Evolvability is the capacity of a population to produce heritable, adaptive phenotypic variation; whether and how natural selection can increase this capacity is a long-standing controversy in evolutionary biology. Evolvability is rooted in the genotype-phenotype (GP) map, the rule relating genotypes to the phenotypes they generate. When many genotypes yield the same phenotype (degeneracy), mutations can be neutral; epistasis (interactions among the effects of mutations at different sites) and pleiotropy (single mutations affecting several traits) then shape how much selectable variation a genotype produces. Mutational robustness is the tolerance of a phenotype to mutation, and a neutral network is a connected set of genotypes mapping to the same phenotype.

An influential laboratory for these questions is the Wagner gene-network model (Wagner 1996; Siegal and Bergman 2002; Aldana et al. 2007). A genotype is a network of K genes in which every gene can regulate every other (and itself) with a weight $w_{ij}\in[-1,1]$. Gene expression is binary: starting from all genes active, the state vector $S(t)$ evolves by the rule that gene $i$ is active at time $t+1$ if and only if $\sum_j w_{ij}S_j(t)>0$. The developmental trajectory reaches either a fixed point or a cycle; a network with a fixed point is called stable, and its phenotype is the equilibrium expression pattern $S(\infty)$. With phenotypic distance $d$ (the number of expression states mismatching an optimal phenotype) and mismatch cost $s$, fitness is $(1+s)^{-d}$; cycling networks are assigned $d=K+1$. Mutations alter connection weights by an amount uniform in $[-m,m)$ at rate $\lambda$, and add or delete connections at rate $\lambda_A$. This architecture creates a highly degenerate, epistatic GP map.

Prior work in this model family showed that selection can increase mutational robustness (Wagner 1996; Siegal and Bergman 2002; Aldana et al. 2007). In parallel, studies of RNA secondary structures found the opposite-level pattern: genotype (sequence) robustness antagonizes evolvability, whereas phenotype (structure) robustness enhances it, because phenotypes with large neutral networks border more alternative phenotypes (Wagner 2008). Draghi and Wagner (2009), building on their earlier developmental model (Draghi and Wagner 2008), measured evolvability by $E_T$: expand a genotype into a clone population, evolve it for $T$ generations toward each alternative optimal phenotype, and record the normalized mean progress. They found that periodically changing environments increase evolvability across a broad parameter range, that recombination does not prevent this, and that a simple structural statistic, network excitation $E_{\mathrm{net}}=\sum_{i,j} w_{ij}$ (the balance of activating to repressing regulatory influences), predicts a substantial fraction of the variation in evolvability among random networks. Strikingly, evolvability in their simulations increased together with volatility (the fraction of mutations altering a stable phenotype) and with decreasing mutational robustness, contradicting studies in which robustness promotes evolvability. The paper closes by stating several open issues, and these jointly define the present problem.

Problem Statement

The problem is to convert Draghi and Wagner's (2009) qualitative demonstration into a predictive and general quantitative account of the evolution of evolvability in gene-network GP maps, as the source itself formulates its open agenda. Concretely, the unresolved targets are: (i) a fully quantitative analysis of how population size $N$, the period $P$ of environmental change, the weight-mutation rate $\lambda$, and the other parameters varied in the source's Appendix 1 (the topology-mutation rate $\lambda_A$, the mutation step size $m$, and network size $K$) influence the evolution of evolvability, going beyond the source's one-factor parameter slices; interactions that the source's own method could not judge, such as that between period and population size, remain an open aspect of this analysis; (ii) an assessment of whether simple structural statistics of the GP map, above all network excitation $E_{\mathrm{net}}=\sum_{i,j} w_{ij}$, predict evolvability beyond the source's own model, testing the source's suggestion that GP-map 'connections with evolvability might be simple and intuitive'; (iii) a delineation of the conditions, in parameter ranges and GP-map architectures, under which mutational robustness promotes rather than impedes evolvability, unifying the negative correlation observed in the source's network model with the positive correlations reported for RNA and protein maps; (iv) an establishment of how these results scale to large networks (large $K$) with fully converged developmental dynamics rather than truncated trajectories; and (v) an integration of the evolution of GP-map structure with the concurrent evolution of mutation-rate and recombination modifiers into a coherent theory of evolvability. These five components are the open directions separately stated in the source's Discussion, and each is independently addressable; partial progress exists on components (iii) and (v), but none of the components is closed.

References

1. Draghi, J. and Wagner, G. P. (2009). The evolutionary dynamics of evolvability in a gene network model. Journal of Evolutionary Biology 22(3): 599-611. DOI: 10.1111/j.1420-9101.2008.01663.x. https://doi.org/10.1111/j.1420-9101.2008.01663.x
2. Draghi, J. and Wagner, G. P. (2008). Evolution of evolvability in a developmental model. Evolution 62(2): 301-315. DOI: 10.1111/j.1558-5646.2007.00303.x. https://doi.org/10.1111/j.1558-5646.2007.00303.x
3. Wagner, A. (1996). Does evolutionary plasticity evolve? Evolution 50(3): 1008-1023. DOI: 10.1111/j.1558-5646.1996.tb02342.x. https://doi.org/10.1111/j.1558-5646.1996.tb02342.x
4. Siegal, M. L. and Bergman, A. (2002). Waddington's canalization revisited: Developmental stability and evolution. Proceedings of the National Academy of Sciences 99(16): 10528-10532. DOI: 10.1073/pnas.102303999. https://doi.org/10.1073/pnas.102303999
5. Aldana, M., Balleza, E., Kauffman, S. and Resendiz, O. (2007). Robustness and evolvability in genetic regulatory networks. Journal of Theoretical Biology 245(3): 433-448. DOI: 10.1016/j.jtbi.2006.10.027. https://doi.org/10.1016/j.jtbi.2006.10.027
6. Wagner, A. (2008). Robustness and evolvability: a paradox resolved. Proceedings of the Royal Society B 275(1630): 91-100. DOI: 10.1098/rspb.2007.1137. https://doi.org/10.1098/rspb.2007.1137
7. Draghi, J. A., Parsons, T. L., Wagner, G. P. and Plotkin, J. B. (2010). Mutational robustness can facilitate adaptation. Nature 463: 353-355. DOI: 10.1038/nature08694. https://doi.org/10.1038/nature08694
8. Steiner, C. F. (2012). Environmental noise, genetic diversity and the evolution of evolvability and robustness in model gene networks. PLoS ONE 7(12): e52204. DOI: 10.1371/journal.pone.0052204. https://doi.org/10.1371/journal.pone.0052204
9. Pinho, R., Borenstein, E. and Feldman, M. W. (2012). Most networks in Wagner's model are cycling. PLoS ONE 7(4): e34285. DOI: 10.1371/journal.pone.0034285. https://doi.org/10.1371/journal.pone.0034285
10. Rünneburger, E. and Le Rouzic, A. (2016). Why and how genetic canalization evolves in gene regulatory networks. BMC Evolutionary Biology 16: 239. DOI: 10.1186/s12862-016-0801-2. https://doi.org/10.1186/s12862-016-0801-2
11. Mayer, C. and Hansen, T. F. (2017). Evolvability and robustness: A paradox restored. Journal of Theoretical Biology 430: 78-85. DOI: 10.1016/j.jtbi.2017.07.004. https://doi.org/10.1016/j.jtbi.2017.07.004
12. Payne, J. L. and Wagner, A. (2019). The causes of evolvability and their evolution. Nature Reviews Genetics 20(1): 24-38. DOI: 10.1038/s41576-018-0069-z. https://doi.org/10.1038/s41576-018-0069-z
13. Kumawat, B., Lalejini, A., Acosta, M. M. and Zaman, L. (2025). Evolution takes multiple paths to evolvability when facing environmental change. Proceedings of the National Academy of Sciences 122(1): e2413930121. DOI: 10.1073/pnas.2413930121. https://doi.org/10.1073/pnas.2413930121
14. Hardy, N. B. (2025). How does robustness affect evolvability? Evolution 79(10): 1996-2006. DOI: 10.1093/evolut/qpaf116. https://doi.org/10.1093/evolut/qpaf116