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Evolution of evolvability in rapidly adapting populations

Nature Ecology & Evolution (2024)Cite this article

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Abstract

Mutations can alter the short-term fitness of an organism, as well as the rates and benefits of future mutations. While numerous examples of these evolvability modifiers have been observed in rapidly adapting microbial populations, existing theory struggles to predict when they will be favoured by natural selection. Here we develop a mathematical framework for predicting the fates of genetic variants that modify the rates and benefits of future mutations in linked genomic regions. We derive analytical expressions showing how the fixation probabilities of these variants depend on the size of the population and the diversity of competing mutations. We find that competition between linked mutations can dramatically enhance selection for modifiers that increase the benefits of future mutations, even when they impose a strong direct cost on fitness. However, we also find that modest direct benefits can be sufficient to drive evolutionary dead ends to fixation. Our results suggest that subtle differences in evolvability could play an important role in shaping the long-term success of genetic variants in rapidly evolving microbial populations.

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Fig. 1: Modelling indirect selection on evolvability in rapidly adapting asexual populations.
Fig. 2: Interference between linked mutations enhances indirect selection for evolvability.
Fig. 3: Trade-offs between direct and indirect selection.
Fig. 4: Indirect selection on a continuous distribution of fitness effects.
Fig. 5: Incorporating deleterious mutations and modifiers with finite mutational horizons.

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Data availability

Fitness measurements and confidence intervals in Fig. 1a were obtained from the Supplementary Information of ref. 8. Simulation results in the remaining figures are available in the accompanying source data files. Source data are provided with this paper.

Code availability

Source code for forward-time simulations, numerical calculations and figure generation are available via Github (https://github.com/bgoodlab/evolution_of_evolvability).

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Acknowledgements

We thank D. Wong for useful discussions and S. Walton, O. Ghosh and Z. Liu for comments and feedback on the manuscript. This work was supported in part by the Alfred P. Sloan Foundation (FG-2021-15708) and a Terman Fellowship from Stanford University. B.H.G. is a Chan Zuckerberg Biohub San Francisco Investigator.

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Authors and Affiliations

  1. Biophysics Program, Stanford University, Stanford, CA, USA

    James T. Ferrare

  2. Department of Applied Physics, Stanford University, Stanford, CA, USA

    Benjamin H. Good

  3. Department of Biology, Stanford University, Stanford, CA, USA

    Benjamin H. Good

  4. Chan Zuckerberg Biohub – San Francisco, San Francisco, CA, USA

    Benjamin H. Good

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Conceptualization: J.T.F. and B.H.G.; theory and methods development: J.T.F. and B.H.G.; analysis: J.T.F. and B.H.G.; writing: J.T.F. and B.H.G.

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Correspondence to Benjamin H. Good.

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Extended data

Extended Data Fig. 1 Examples of epistatic fitness landscapes that satisfy the minimal model in Fig. 1.

(a) The evolvability modifier in Fig. 1b can be viewed as the lowest order term in a general macroscopic epistasis expansion (left, SI Section 3.1). Different fitness landscapes can produce the same macroscopic behavior. (b,c) Examples of highly epistatic fitness landscapes that satisfy the simple model above. (b) A ‘maximally epistatic’ landscape of branching uphill paths, which generalizes the model in refs. 79,80. Each step k = 1, …, K of a given path can access M ≪ L beneficial mutations; all other genotypes have fitness zero. (c) A fitness landscape formed by a non-linear combination of two global phenotypes, for example, stability, \(\Phi (\,{\overrightarrow{g}})\), and activity, \(\Psi (\,{\overrightarrow{g}})\). Individual mutations can affect both traits simultaneously (right). Stabilizing mutations can act like modifier alleles by potentiating the fitness benefits of mutations that would destabilize the protein on their own (left). In particular, a strongly stabilizing mutation can allow K ≈ ϕm/∣ϕℓ∣ new mutations to accumulate before their effects on stability become important. See SI Section 3.1 for more details.

Extended Data Fig. 2 Deleterious mutations and indirect selection for robustness.

(a) A generalization of the simplified model in Fig. 5a, where the modifier can also shift the typical fitness cost sd. (b, c) Fixation probability of a robustness-enhancing modifier with \({U}_{d}^{{\prime} } < {U}_{d}\) (and all other parameters are held fixed). Symbols denote the results of forward time simulations for N = 108, sb = 10−2, and Ub = 10−5, while solid lines denote our theoretical predictions in SI Section 6. Panel (b) shows that the purgeable mutations approximation holds across a broad range of fitness costs, with the dashed line marking the predicted transition to quasi-neutrality (∣sd∣ ≈ v/xc). Panel (c) shows that selection for increased robustness is relatively weak unless Ud ≳ sb (dashed line). (d) Fixation probability of a modifier that imposes a tradeoff between robustness and evolvability by increasing the strength of selection on beneficial and deleterious mutations simultaneously. Results are shown for sb = ∣sd∣ = s and Ud = 10−2, with the remaining parameters the same as panel b. Since Ud ≫ Ub, this example shows that strong selection for evolvability can occur for modifiers reduce the average fitness effect of mutations (\(\Delta \overline{s}\propto {U}_{b}\Delta s-{U}_{d}\Delta s < 0\)). (e) Fixation probability of modifier that enhances robustness and evolvability at the same, by shifting mutations from deleterious to beneficial \(({U}_{d}-{U}_{d}^{{\prime} }={U}_{b}^{{\prime} }-{U}_{b})\). Symbols denote results of forward-time simulations with sd = − 10−2 and Ud = 10−2, with the remaining parameters the same as panel (b). Lines denote our theoretical predictions in the absence of deleterious mutations (\({U}_{d}={U}_{d}^{{\prime} }=0\)). This example shows that enhancements in evolvability are weighted more strongly than comparable increases in robustness, even when nearly all new mutations are deleterious (Ub ≪ Ud).

Source data

Extended Data Fig. 3 Relaxing the assumption that modifiers permanently change the mutation spectrum.

An alternative version of the model in Fig. 5c, where the modifier reverts to an evolutionary dead-end (μm(s) = 0) after K mutations. Symbols denote the results of forward-time simulations for N = 108, sb = 10−2, and Ub = 10−5, while the line denotes our theoretical predictions for the minimal modifier model in Fig. 1b (that is K = ∞). Even in this extreme case, our minimal modifier model (solid line) remains highly accurate for moderate values of K, and as little as K = 1 in the quasi-sweeps regime. This demonstrates that large populations can only ‘see’ across the fitness landscape for \(\approx {x}_{cm}/{s}_{b}^{{\prime} }\) additional mutations (SI Section 7.2).

Source data

Extended Data Fig. 4 Selection for evolvability in the presence of diminishing returns epistasis.

(a, b) A simple model of global diminishing returns epistasis motivated by the empirical example in ref. 58 (SI Section 7.3). The fitness effects of new mutations shrink as the population adapts (panel a), leading to a decelerating rate of adaptation over time (panel b). Points denote the results of forward-time simulations for the distribution of fitness effects \(\mu (s|\overrightarrow{g})={U}_{b}\cdot \delta (s-{\tilde{s}}_{b}\cdot {e}^{-X(\overrightarrow{g})/\theta })\), with Ub = 10−5, \({\tilde{s}}_{b}=1{0}^{-1}\), θ = 0.2, and N = 107; points are connected by solid lines to aid visualization. (c, d) The fixation probability of an evolvability modifier that arises at the beginning of the inset in panel a, where the fitness trajectory is still decelerating. Green symbols in (c) show a selection-strength modifier with the same diminishing returns schedule as the background population (θm ≈ θ), while the blue symbols show an alternate example where the modifier avoids future diminishing returns once it arises (θm ≈ ∞). The green line illustrates the predictions from the ‘adiabatic’ approximation in SI Section 7.3, demonstrating that the permanent modifier model [\({\mu }_{m}(s|\overrightarrow{g})\approx {\mu }_{m}(s)\)] provides a good approximation when the local selection strengths are properly renormalized. The blue line shows the predictions from our heuristic analysis in SI Section 7.3, which accounts for the additional benefits that accrue for the modifier lineage when θm ≫ θ (panel d). This example illustrates that the evolvability advantages that accrue from large differences in diminishing returns epistasis can drive modest deviations from our existing theory when θ grows close to xc.

Source data

Extended Data Table 1 Table of mathematical symbols. Definitions of mathematical symbols used in the main text, along with locations where they are used
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Ferrare, J.T., Good, B.H. Evolution of evolvability in rapidly adapting populations. Nat Ecol Evol (2024). https://doi.org/10.1038/s41559-024-02527-0

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