The Evolution of Canalization and Evolvability in Stable and Fluctuating Environments

Abstract

Using a multilinear model of epistasis we explore the evolution of canalization (reduced mutational effects) and evolvability (levels of additive genetic variance) under different forms of stabilizing and fluctuating selection. We show that the total selection acting on an allele can be divided into a component deriving from adaptation of the trait mean, a component of canalizing selection favoring alleles that epistatically reduce the effects of other allele substitutions, and a component of conservative selection disfavoring rare alleles. While canalizing selection operates in both stable and fluctuating environments, it may not typically maximize canalization, because it gets less efficient with increasing canalization, and reaches a balance with drift, mutation and indirect selection. Fluctuating selection leads to less canalized equilibria than stabilizing selection of comparable strength, because canalization then becomes influenced by erratic correlated responses to shifting trait adaptation. We conclude that epistatic systems under bounded fluctuating selection will become less canalized than under stabilizing selection and may support moderately increased evolvability if the amplitude of fluctuations is large, but canalization is still stronger and evolvability lower than expected under neutral evolution or under patterns of selection that shift the trait in directions of positive (reinforcing) epistasis.

Appendices

Appendix 1: Multiplicative Selection

Here we derive some base-line predictions for the long-term multiplicative fitness function under different types of fluctuating selection. Consider first Gaussian selection around a fluctuating optimum

$$ W\left( {z;t} \right) = k_{t} \,{\text{Exp}}\left[ { - s(z - \theta_{t} )^{2} } \right], $$

(14)

where θ _t is the value of the optimum in generation t and k _t an arbitrary time-dependent variable. The multiplicative relative fitness over T generations is then

$$ {\text{Exp}}\left[ {\sum\limits_{t} {Ln[k_{t} ] - \sum\limits_{t} {Ln[\overline{W} {\text{(}}t{\text{)}}] - sT(z - \overline{\theta } )^{2} - 2s(z - \overline{\theta } )\sum\limits_{t} {(\theta _{t} - \overline{\theta } ) - s\sum\limits_{t} {(\theta _{t} - \overline{\theta } )^{2} } } } } } \right] = Exp\left[ {\sum\limits_{t} {Ln\left[ {k_{t} } \right] - Ln\left[ {\overline{W} (t)} \right]} } \right]Exp\left[ { - sT(z - \overline{\theta } )^{2} + 2(z - \overline{\theta } )} \sum\limits_{{_{t} }} {(\theta _{t} - \overline{\theta } )/T + \sum\limits_{{_{t} }} {(\theta _{{_{t} }} - \overline{\theta } )^{2} /T} } \right], $$

(15)

where $\bar{\theta }$ is the mean optimum over the fluctuations. In the limit when T → ∞ this reduces to

$$ \mathop {Lim}\limits_{T \to \infty } \prod\nolimits_{t} {\frac{W(z;t)}{{\bar{W}(t)}}} = KExp\left[ { - sT(z - \bar{\theta })^{2} + Var[\theta ]} \right]\sim Exp\left[ { - sT(z - \bar{\theta })^{2} } \right], $$

(16)

where K is a constant and Var[θ] is the variance of the optimum over the fluctuations. Hence, the multiplicative relative fitnesses depend only on the average of the fluctuating optimum and not on the details of its distribution. They are the same with constant and varying optima, and we predict that the long-term effects of fluctuating gaussian optima should be similar to long-term stabilizing selection. If also the strength of selection is fluctuating, the situation is slightly more complicated. In this case

$$ \mathop {Lim}\limits_{T \to \infty } \prod\nolimits_{t} {\frac{W(z;t)}{{\bar{W}(t)}}} \sim Exp\left[ { - \bar{s}T\left( {\left( {z - \bar{\theta }} \right)^{2} + 2T\left( {z - \bar{\theta }} \right)Cov\left[ {s_{t} ,\theta_{t} } \right]} \right)} \right], $$

(17)

where $\bar{s}$ = Σ _t s _t /T is the average strength of selection. Hence, if there is a covariance between the changes of the optimum and the strength of stabilizing selection on that optimum, then this will induce a component of selection in the direction in which the deviances are associated with stronger selection. This shifts the long-term optimum to $\bar{\theta }$ + Cov[s _t , θ _t ]/$\bar{s}$. Note that this implies that there are no effects of fluctuations in the strength of selection if the optimum is fixed.

In this argument we can see that the extra stabilizing selection induced by variance in the selection gradient (Eq. 3) is compensated by the fluctuations shifting the population into more convex areas of the fitness landscape, which happens because the Gaussian fitness function is most concave at the optimum. If we consider quadratic stabilizing selection of the form

$$ W\left( {z;t} \right) = k_{t} \left( {1 - s(z - \theta_{t} )^{2} } \right), $$

(18)

the first and second derivatives are −2s(z − θ _t ) and −2s. Hence, from equation 1 we see that the second derivative of the multiplicative fitness function will be proportional to −2s − 4s ²Var[θ _t ], which implies that the strength of stabilizing selection will increase with the variance of fluctuations in the optimum (as long as the occurence of non-positive fitnesses is negligible).

Similar effects happen with fluctuating directional selection. Consider first a fitness function of the form

$$ W\left( {z;t} \right) = k_{t} \left( {1 + s_{t} z} \right), $$

(19)

where s _t fluctuates such that the population experience no net long-term directional selection. In this case the long-term multiplicative function has a second derivative proportional to $- \Upsigma_{t} s_{t}^{2} /\left( {1 + s_{t} z} \right)^{2}$, which implies that the strength of stabilizing selection will increase with increasing fluctuations in s _t . The optimum will depend on the distribution of s _t , but if this has mean zero, the optimum will be at z = 0. With exponential directional selection (W(z; t) = Exp[s _t z]), which is inherently convex, the multiplicative fitness function will be flat if the s _t average to zero.

A slightly different result follows from an alternating linear fitness function of the type

$$ W\left( {z;t} \right) = 1 + s_{t} \left( {z - z_{t} } \right), $$

(20)

where $\bar{z}_{t}$ is the mean of z in generation t. This form ensures that the selection gradient is always equal to s _t and is hence a model of a fluctuating linear selection gradient. In this case the mean fitness in each generation is always unity, so that the cumulative fitness function is

$$ \prod\nolimits_{t} {\frac{W(z;t)}{{\bar{W}(t)}}} = \prod\nolimits_{t} {\left( {1 + s_{t} \left( {z_{t} - \bar{z}_{t} } \right)} \right)} = Exp\left[ {\Upsigma_{t} Ln\left[ {1 + s_{t} \left( {z_{t} - \bar{z}_{t} } \right)} \right]} \right] $$

(21)

By assuming no net directional selection in the sense that $\bar{s}$ = 0, that selection within each generation is weak in the sense that |s_t(z − $\bar{z}_{t}$)| < 1, and considering only the two first terms in a series expansion of the logarithm we get

$$ \prod\nolimits_{t} {\frac{W(z;t)}{{\bar{W}(t)}}} \sim Exp\left[ { - \tfrac{1}{2}T\sigma_{s}^{2} (z - \bar{z})} \right], $$

(22)

where $\sigma_{s}^{ 2}$ is the temporal variance of the selection coefficient and $\bar{z} = \Upsigma_{t} {{\bar{z}_{t} } \mathord{\left/ {\vphantom {{\bar{z}_{t} } T}} \right. \kern-0pt} T}$ is the mean phenotype over the history of the population. Hence, the multiplicative fitness landscape approximates normalizing selection around the average mean phenotype during the period we consider. This resembles “normalizing” selection (Travis 1989) favoring the current mean. Even if there is no net directional selection, slight random changes in the mean will be preserved, so that the mean can slowly drift away from its ancestral value in a Brownian-motion-like manner. To understand how this normalizing selection arises, consider a situation in which the selection alternates every generation between positive and negative, but is always equally strong such that the mean shifts back and forth between two values m ₁ and m ₂. In this situation phenotypic values intermediate between m ₁ and m ₂ will always have higher fitness than the current mean.

Appendix 2: Selection on a Decanalizing Gene Substitution

In this appendix we compute the fitness effects of a canalizing or decanalizing substitution at a locus, x. The phenotype is

$$ z = z_{r} + \Upsigma_{i} {}^{i}y + \Upsigma_{i} \Upsigma_{j > i} {}^{ij}\varepsilon {}^{i}y{}^{j}y + \Upsigma_{i} \Upsigma_{j > i} \Upsigma_{k > j} {}^{ijk}\varepsilon {}^{i}y{}^{j}y{}^{k}y + \cdots + {}^{g \to x}f\,x = z_{x = 0} + {}^{g \to x}f\,x $$

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where x is the reference effect of the substitution, z _x=0 is the phenotype when x = 0, and the epistasis factor, ^g→x f = 1 + Σ _i ^ixε ⁱ y + Σ _i Σ _j>i ^ijxε ⁱ y ^j y +···, describes the epistatic effects on x from the rest of the genome, $g = \{ {}^{1}y, \ldots ,{}^{n}y\}$. Hence, the term ^g→x fx describes the effect of the substitution in the genetic background of g. We will study a substitution with effect x = 1. This sets a scale and entails no loss of generality. We allow epistasis of any order. Assuming a quadratic fitness function W(z) = 1 − s(z − θ)², the change from x = 0 to x = 1 gives a fitness change of

$$ {\text{E}}_{g} \left[ {W\left( z \right)|x = 1} \right] - {\text{E}}_{g} \left[ {W\left( z \right)|x = 0} \right] = - s{\text{E}}_{g} \left[ {2{}^{g \to x}f\left( {z_{x = 0} - \theta } \right) + {}^{g \to x}f^{2} } \right], $$

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where E _g refers to expectations taken over the reference effects of the loci in g. Assuming linkage equilibrium, this can be written

$$ \begin{gathered} {\text{E}}_{g} \left[ {W\left( z \right)|x =1}\right] - {\text{E}}_{g} \left[ {W\left( z \right)|x = 0}\right] =- s{\text{E}}_{g} \left[ {2^{g \to x} f\left( {z - \theta} \right)+\,{^{g \to x}f^{2}} \left( {1 - 2\bar{x}} \right)} \right]\hfill \\\quad = - 2s\left( {{\bar{\text{z}}} - \theta}\right){\text{E}}_{g} \left[ {{}^{g \to x}f} \right] -{2}s{\text{Cov}}_{g} \left[ {z, {^{g \to x}f}} \right] - s\left( {1-2\bar{x}} \right)\left( {{\text{E}}_{g} \left[ {^{g \to x}f}\right]^{2} + {\text{Var}}_{g} \left[ {^{g \to x}f}\right]}\right) \hfill \\ \quad = - 2s\left( {{\bar{\text{z}}} -\theta }\right){\text{E}}_{g} \left[{}^{g \to x}f \right]- {2}s{\text{Cov}}_{g}\left[ z_{x = 0} ,{}^{g \to x}f \right] -s{\text{Var}}_{g} \left[ {{}^{g\to x}f} \right] - s\left( { 1-2\bar{x}} \right){\text{E}}_{g}\left[ {{}^{g \to x}f} \right]^{2}, \hfill \\\end{gathered} $$

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where the last step uses Cov _g [z, ^g→x f] = Cov _g [z _x=0  + ^g→x f x, ^g→x f] = Cov _g [z _x=0, ^g→x f] + $\bar{x}$Var _g [^g→x f].

In the bilinear case we can write

$$ {\text{Cov}}_{g} \left[ {z_{x = 0} ,{}^{g \to x}f} \right] = \sum_{i} {}^{ix}\varepsilon {\text{E}}_{g} \left[ {^{g \to x} f} \right]{}^{i}v, $$

(26a)

$$ {\text{Var}}_{g} \left[ {{}^{g \to x}f} \right] = \Upsigma_{i} {}^{ix}\varepsilon^{2} \,{}^{i}v, $$

(26b)

where ⁱ v = Var[ⁱ y], and ^g→x f = 1 + Σ _j ^ijε ^j y. This yields Eq. 7 in the main text. Returning to the general multilinear case, we simplify the equations by using $\bar{g} = \{ {}^{1}\bar{y}, \ldots ,{}^{n}\bar{y}\}$ and x = 0 as reference genotype. Then

$$ {\text{Cov}}_{g} \left[ {z_{x = 0} ,{}^{g \to x}f} \right] = \Upsigma_{i} {}^{ix}\varepsilon {}^{i}v + \Upsigma_{i} \Upsigma_{j > i} {}^{ijx}\varepsilon {}^{ij}\varepsilon {}^{i}v{}^{j}v + \Upsigma_{i} \Upsigma_{j > i} \Upsigma_{k > j} {}^{ijkx}\varepsilon {}^{ijk}\varepsilon {}^{i}v{}^{j}v{}^{k}v + \cdots , $$

(27a)

$$ {\text{Var}}_{g} \left[ {{}^{g \to x}f} \right] = \Upsigma_{i}^{ix} \varepsilon^{2} \,{}^{i}v + \Upsigma_{i} \Upsigma_{j} {}^{ijk}\varepsilon^{2}\, {}^{i}v{}^{j}v + \Upsigma_{i} \Upsigma_{j} \Upsigma_{k} {}^{ijkx}\varepsilon^{2} \,{}^{i}v{}^{j}v{}^{k}v + \cdots , $$

(27b)

and then

$$ \begin{gathered} {\text{E}}_{g} \left[ {W\left( z \right)|x = 1} \right] - {\text{E}}_{g} \left[ {W\left( z \right)|x = 0} \right] \hfill \\ \quad = - 2s\left( {\bar{z} - \theta } \right) - { 2}s{\text{Cov}}_{g} \left[ {z_{x = 0} ,{}^{g \to x}f} \right] - s{\text{Var}}_{g}\left[ {{}^{g \to x}f} \right] - s\left( { 1- 2\bar{x}} \right) \hfill \\ \quad = - 2s\left( {\bar{z} - \theta } \right) - s\Upsigma_{i} {}^{ix}\varepsilon \left( {^{ix} \varepsilon + 2} \right){}^{i}v - s\Upsigma_{i} \Upsigma_{j} {}^{ijx}\varepsilon \left( {^{ijx} \varepsilon + {}^{ij}\varepsilon } \right){}^{i}v^{j} v \hfill \\ \quad - s\Upsigma_{i} \Upsigma_{j} \Upsigma_{k} {}^{ijkx}\varepsilon \left( {^{ijkx} \varepsilon + {^{ijk}\varepsilon} /2} \right){}^{i}v{}^{j}v{}^{k}v - s\left( {{\text{Higher - order}}\;{\text{terms}}} \right) - s\left( {1 \, - 2\bar{x}} \right), \hfill \\ \end{gathered} $$

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where the higher-order terms have the form Σ _J ^Jxε(^Jxε + ^Jε/2 ^k−2) Π _j∈J ^j v, where J is the set of all k-tuples of indices from g. Restricting to pairwise epistasis yields Eq. 8 in the main text.