Genotype-Phenotype Maps Maximizing Evolvability: Modularity Revisited

Abstract

The mechanisms translating genetic to phenotypic variation determine the distribution of heritable phenotypic variance available to selection. Pleiotropy is an aspect of this structure that limits independent variation of characters. Modularization of pleiotropy has been suggested to promote evolvability by restricting genetic covariance among unrelated characters and reducing constraints due to correlated response. However, modularity may also reduce total genetic variation of characters. We study the properties of genotype-phenotype maps that maximize average conditional evolvability, measured as the amount of unconstrained genetic variation in random directions of phenotypic space. In general, maximal evolvability occurs by maximizing genetic variance and minimizing genetic covariance. This does not necessarily require modularity, only patterns of pleiotropy that cancel on average. The detailed structure of the most evolvable genotype-phenotype maps depends on the distribution of molecular variance. When molecular variance is determined by mutation-selection equilibrium either highly pleiotropic or highly modular genotype-phenotype maps can be optimal, depending on the mutation rate and the relative strengths of stabilizing selection on the characters.

Appendices

Appendix 1

Here we present the derivation of the GP maps maximizing average conditional evolvability. The two models of pleiotropy are dealt with separately. We also present the maximization of the character conditional evolvability in the character model.

Character Model

Discrete Genetic Effects

In this part, we are concerned with discrete effects of genes, i.e., $b_{ij} \in \left\{ { - 1,0,1} \right\}$. Let the underlying genes have either a positive, a negative, or no effect on the trait, and let each trait be affected by at least one gene. We term modular loci (column vectors of B) as those affecting only one trait (i.e., (1,0), (−1,0), (0,1) or (0, −1)), synergistic loci as those affecting both traits in the same way ((1,1) or (−1, −1)), and antagonistic loci as those affecting both traits in opposite ways ((1, −1) or (−1,1)).

The squared average conditional evolvability $\bar{c}^{2}$ for two traits equals the sum of the products of determinants of all minors of B (a consequence of Cauchy-Binet formula). We will denote the 2 × 2 sub-matrices (minors) of B as B _i , where $i \in \left\{ {2, \ldots ,{{n\left( {n - 1} \right)} \mathord{\left/ {\vphantom {{n\left( {n - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \right\}$ and n is the number of genes.

$$ \begin{gathered} \bar{c}^{2} = Det\left[ {\mathbf{BB^{T}} }\right] = Det\left[ {\mathbf{B_1} } \right]Det\left[{\mathbf{B_1^T}} \right] + \cdots + Det\left[{\mathbf{B}}_{{{{n\left( {n - 1} \right)} \mathord{\left/ {\vphantom{{n\left( {n - 1} \right)} 2}} \right. \kern-\nulldelimiterspace}2}}} \right]Det\left[ {\mathbf{B}}_{{{{n\left( {n - 1} \right)}\mathord{\left/ {\vphantom {{n\left( {n - 1} \right)} 2}} \right.\kern-\nulldelimiterspace} 2}}} \right] \\ =\sum\limits_{i}^{{{{n\left( {n - 1} \right)} \mathord{\left/{\vphantom {{n\left( {n - 1} \right)} 2}} \right.\kern-\nulldelimiterspace} 2}}} {\left( {Det\left[ {\mathbf{B}}_{i} \right]} \right)^{2} } . \\ \end{gathered} $$

Following the solution of Hadamard’s maximum determinant problem, the maximum determinant of an m x m matrix with all entries $a_{ij} \le \left| 1 \right|$ equals $m^{m/2}$, which is two in the case of a 2 × 2 matrix. It can be furthermore shown (Brenner and Cummings 1972; Ehlich 1964) that if matrix elements are restricted to discrete values (−1, 0, 1), there exist four distinct matrices with the maximum determinant (=2). These are, written by row: {{−1, −1}, {1, −1}}, {{−1, 1}, {−1, −1}}, {{−1, 1}, {1, 1}}, {{1, 1}, {1, −1}}. As we are interested in maximum squared values of the determinant, we also consider the minimum determinant (−2). The sign of the determinant changes by exchanging the rows or columns of a matrix, which results in four more optimal matrices. Note that all of these matrices combine one column vector with synergistic and one column vector with antagonistic effects. Hence, the problem of maximum conditional evolvability of a matrix with more than two vectors reduces to the problem of combining the column vectors of B such that they result in most optimal combination of determinants of submatrices B _i . We present the values of the determinants of all four possible sub-matrices in Table 4.

Table 4 Summary of determinants of different types of 2 × 2 submatrices

The solution can be found by considering the general equation for combining the four groups of loci (antagonistic, synergistic, two modular) in different proportions and determining the proportions at which the average conditional evolvability is the highest.

Let n _M1 , n _M2 , n _A , n _S be the counts of modular loci along the trait 1, modular loci along the trait 2, antagonistic and synergistic loci, respectively. Let x _M1 , x _M2 , x _A , and x _S be their respective proportions in the total number of loci n:

$x_{M1} = \frac{{n_{M1} }}{n},x_{M2} = \frac{{n_{M2} }}{n},x_{A} = \frac{{n_{A} }}{n},x_{S} = \frac{{n_{S} }}{n}$; so that $x_{M1} + x_{M2} + x_{A} + x_{S} = 1$. Then $\bar{c}^{2}$ can be written as the number of times a particular type of submatrix occurs, times the value of its squared determinant (see Table 4):

$$ \begin{gathered} \bar{c}^{2} = \hfill \\ \left( {\frac{{\left( {nx_{M1} } \right)^{2} - nx_{M1} }}{2}} \right)0^{2} + \left( {\frac{{\left( {nx_{M2} } \right)^{2} - nx_{M2} }}{2}} \right)0^{2} + \left( {\frac{{\left( {nx_{A} } \right)^{2} - nx_{A} }}{2}} \right)0^{2} + \left( {\frac{{\left( {nx_{S} } \right)^{2} - nx_{S} }}{2}} \right)0^{2} \hfill \\ + \left( {n^{2} x_{M1} x_{M2} } \right)1^{2} + \left( {n^{2} x_{M1} x_{A} } \right)1^{2} + \left( {n^{2} x_{M1} x_{S} } \right)1^{2} + \left( {n^{2} x_{M2} x_{A} } \right)1^{2} + \left( {n^{2} x_{M2} x_{S} } \right)1^{2} + \left( {n^{2} x_{S} x_{A} } \right)2^{2} \hfill \\ \end{gathered} $$

where n is a constant and can be neglected. Maximizing the equation above under the constraint $x_{M1} + x_{M2} + x_{A} + x_{S} = 1$ yields a single solution $x_{A} = x_{S} = 0.5$ and consequently $x_{M1} = x_{M2} = 0$. The maximum value of the function $\bar{c}^{2}$ equals n ²; therefore the maximum average conditional evolvability equals n. This shows that the single combination of loci giving the highest average conditional evolvability is the one in which half of the loci have antagonistic and the other half synergistic effects on the two traits, but none are modular. The average evolvability of such GP map is n, hence $\max \;\bar{c} = \bar{e}$. Also note that the average unconditional evolvability is at its maximum here, because all vectors attain their maximal length.

Continuous Genetic Effects

Here we show that the above is also the solution to the B matrix of continuous effects. We show that the values of the determinants in the above matrices are the maximal values and no other combination of effects will give higher average conditional evolvability.

Consider the type of submatrices above that yield a non-zero determinant. These are all submatrices combining loci with non-equal effects. With respect to the sign of the effect, we again classify the types of loci as before: modular, synergistic and antagonistic. For each submatrix with a non-zero determinant, involving two different types of loci, we show that the determinant (Det) will decrease, as the elements of the submatrix deviate from the limits of the interval [−1,1] by some amount $\varepsilon_{i}$, where 0 ≤ $\varepsilon_{i}$ ≤ 1.

Then for a submatrix combining a modular and an antagonistic vector:

$$ \left| \begin{gathered} \;\;\;0\quad - 1 + \varepsilon_{2} \hfill \\ 1 - \varepsilon_{1} \quad 1 - \varepsilon_{3} \hfill \\ \end{gathered} \right| = 1 - \varepsilon_{1} - \varepsilon_{2} + \varepsilon_{1} \varepsilon_{2} \Rightarrow 0 \le Det \le 1; $$

for a submatrix combining a modular and a synergistic vector:

$$ \left| \begin{gathered} \;\;\;0\;\;\,\quad 1 - \varepsilon_{2} \hfill \\ 1 - \varepsilon_{1} \quad 1 - \varepsilon_{3} \hfill \\ \end{gathered} \right| = \varepsilon_{1} + \varepsilon_{2} - \varepsilon_{1} \varepsilon_{2} - 1 \Rightarrow - 1 \le Det \le 0; $$

for a submatrix combining two modular vectors for different traits:

$$ \left| {\begin{array}{cc} {1 - \varepsilon _{1} } & 0 \\ 0 & {1 - \varepsilon _{3} } \\\end{array}} \right| = \varepsilon _{1} + \varepsilon _{2} - \varepsilon _{1} \varepsilon _{2} - 1 \Rightarrow - 1 \le Det \le 0 $$

and for a submatrix combining an antagonistic and a synergistic vector:

$$ \left| {\begin{array}{cc} {1 - \varepsilon _{1} } & -1+ \varepsilon _{3} \\ -1+ \varepsilon _{2} & {1 - \varepsilon _{4} } \\\end{array}} \right| = 2 - \varepsilon_{1} - \varepsilon_{2} - \varepsilon_{3} - \varepsilon_{4} + \varepsilon_{2} \varepsilon_{3} + \varepsilon_{1} \varepsilon_{4} \Rightarrow 0 \le Det \le 2. $$

Thus, submatrix determinants are maximized when $b_{ij} \in \left\{ { - 1,0,1} \right\}$, for which the solution has been derived above.

Trait Model

The B matrix in the trait model is characterized by constant vector length irrespective of orientation, thus the effects of loci per trait depend on the angles of these vectors in phenotypic space:

$$ {\mathbf{B}} = \left( {\begin{array}{*{20}c} {\frac{\sin \theta }{{\left| {{\mathbf{b}}_{1} } \right|}}} & {\frac{{\sin \left( {\theta + x} \right)}}{{\left| {{\mathbf{b}}_{2} } \right|}}} \\ {\frac{\cos \theta }{{\left| {{\mathbf{b}}_{1} } \right|}}} & {\frac{{\cos \left( {\theta + x} \right)}}{{\left| {{\mathbf{b}}_{2} } \right|}}} \\ \end{array} } \right), $$

where $\left| {{\mathbf{b}}_{j} } \right|$ is the length of the substitution-effect vector at jth locus, which we assume in the following to be unit length in all vectors; θ is the angle of the first vector from some reference vector (e.g., trait axis 1), and x is the angle between the two vectors, so that −π < x < π. Average conditional evolvability for a two-gene system is then

$$ \bar{c} = Det\left[ {\mathbf{B}} \right] = \sin \theta \cos \left( {\theta + x} \right) - \cos \theta \sin \left( {\theta + x} \right) = \sin x. $$

The solution for this system is obvious: the average conditional evolvability is maximized when x = ±π/2, and it is irrelevant how the two vectors are oriented with respect to the trait axes (θ cancels out in the equation). It follows that the degree of pleiotropy is irrelevant in this model, as long as the two vectors are orthogonal.

Extending the model to multiple genes, we again use the observation that the squared average conditional evolvability is the sum of the squared determinants of all possible two-gene sub-matrices. In terms of between-vector angles, this means that average conditional evolvability equals the square root of the sum of the squared sinus function of angles between all possible pairs of locus vectors. Note that the initial reference angle θ always cancels out, meaning that the orientation of the genetic effect vectors relative to the phenotypic axes is irrelevant. This also means that the degree of pleiotropy is not the essential criterion for optimization.

Due to rotational invariance in the trait model, we can expect different equally optimal solutions when optimizing ${{n\left( {n - 1} \right)} \mathord{\left/ {\vphantom {{n\left( {n - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$ angles simultaneously. To characterize these solutions we consider that, because in the trait model the length of the effect vector is not affected by its orientation relative to the trait axes, the total genetic variance (evolvability) is constant. In terms of the G matrix this means that the trace of the matrix $\left( {{\rm trace}\left[ {\mathbf{G}} \right] = \lambda_{1} + \lambda_{2} } \right)$ is constant. For the two-trait system $\bar{c} = \sqrt {\lambda_{1} \lambda_{2} }$, which is maximized when $\lambda_{1} = \lambda_{2} = {{{\rm trace}\left[ {\mathbf{G}} \right]} \mathord{\left/ {\vphantom {{{\rm trace}\left[ {\mathbf{G}} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}$, and therefore when the genetic covariance of G equals zero.

Note that the maximum is at $\max \bar{c} = \frac{{{\rm trace}\left[ {\mathbf{G}} \right]}}{2} = \frac{{G_{11} + G_{22} }}{2} = \bar{e}.$

Multiple equally optimal solutions for a B matrix fulfill this condition (Fig. 2b). The number of trigonometric variables representing vectors in B precludes defining an exhaustive set of solutions. Given the vectors of equal length, the set of solutions includes e.g., all B matrices in which each vector has an orthogonal counterpart, however the way such pairs of vectors are arranged relative to other pairs is arbitrary. Complete modularity is a special case of this vector distribution, in which all pairs of vectors point into two identical directions, forming two orthogonal bundles. Further distributions of vectors for which $\bar{c} = \bar{e}$ are those with equally spaced vectors (“Equally-spaced vector arrangement”; Fig. 2b), or equally spaced bundles of vectors, etc.

The condition G ₁₂ = 0 does not specify how the single covariance contributions add up. In general, all solutions where negative and positive covariance of loci cancels out are equally optimal as those where no covariance is generated. In reality the vectors can be of different lengths, contributing differently to (co)variance. The equilibrium is then reached not by the equal numbers of vectors generating positive and negative variance, but by balancing their contributions. Thus, the amount of variance allocated at certain angles is also relevant.

Appendix 2

Here we show that a modular vector arrangement and an equally spaced vector arrangement yield maximal average conditional evolvability in the trait model.

Modular Arrangement

The average conditional evolvability can be calculated for a general model with orthogonal bundles of vectors by grouping the between-vector angles into three groups: between vectors within the bundle 1, between vectors within the bundle 2, and the angles between vectors of the opposite bundles. The sine of angles of the first two groups equal 0, and the sine of the between-group angles all equal 1. Hence:

$$ \bar{c}^{2} = 2\frac{n}{4}\left( {\frac{n}{2} - 1} \right)\sin^{2} 0 + \left( \frac{n}{2} \right)^{2} \sin^{2} \frac{\pi }{2} = \left( \frac{n}{2} \right)^{2} \Rightarrow \bar{c} = \frac{n}{2}. $$

This is the same evolvability for the fully modular GP map as in the character model. Note that all vectors contribute the same amount of variance. In a case of the modular two-trait system ${n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2} = \bar{e}$.

Equally-Spaced Vector Arrangement

In this case the B matrix has a uniform distribution of vectors around 2π (the angles between neighboring vectors are equal). Let n be the number of vectors (angles), then each angle between neighbor vectors is 2π/n. The squared average conditional evolvability (=the sum of the squared sine between all vector-pairs) is then

$$ \begin{aligned} \bar{c}^{2} & = \left( {n - 1} \right)\sin^{2} \frac{2\pi }{n} + \left( {n - 2} \right)\sin^{2} 2\frac{2\pi }{n} + \cdots + \left( {n - \left( {n - 1} \right)} \right)\sin^{2} \left( {\left( {n - 1} \right)\frac{2\pi }{n}} \right) \\ & = \sum\limits_{j = 1}^{n - 1} {\left( {n - j} \right)\sin^{2} \left( {j\frac{2\pi }{n}} \right)} \\ & = \sum\limits_{j = 1}^{n - 1} {\frac{1}{2}\left( {n - j} \right)\left( {1 - \cos \left( {2j\frac{2\pi }{n}} \right)} \right)} \\ & = \frac{1}{2}\left[ {\frac{{n\left( {n - 1} \right)}}{2} + \frac{n}{2}} \right] \\ & = \frac{{n^{2} }}{4}, \\ \end{aligned} $$

and therefore, $\bar{c} = {n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2} = \bar{e}.$ The last is true because the B matrix is composed of n columns each with two elements, a sine and a cosine of the same angle α. The sum of their squares is always one, and the average evolvability equals half the sum across n such columns.

Appendix 3

Here we show that the trait and character models of pleiotropy are equivalent at full pleiotropy, i.e., when in a trait model, $\theta = s\left( {\pi /4} \right)$, so that s $\in${1, 2,…,8}).

House-of-Cards Approximation

Considering the contribution of a single locus j to the genetic variance of a character 1. In the character model, the jth B matrix column is ${\mathbf{b}}_{j} = \left( \begin{gathered} \pm 1 \hfill \\ \pm 1 \hfill \\ \end{gathered} \right),$ contributing the following variance to the character 1: $G_{11j} = \frac{4u}{{s_{1} + s_{2} }}.$ In the trait model, the corresponding column is ${\mathbf{b}}_{j} = \left( \begin{gathered} \pm \cos \theta \hfill \\ \pm \sin \theta \hfill \\ \end{gathered} \right),$ contributing the following amount of variance to the character 1: $G_{11j} = \frac{{4u\cos^{2} \theta }}{{s_{1} \cos^{2} \theta + s_{2} \sin^{2} \theta }}$ (where θ is the angle between the mutational effect vector and the axis of character 1; see Fig. 1).

Gaussian Approximation

In the character model, with locus-vector ${\mathbf{b}}_{j} = \left( \begin{gathered} \pm 1 \hfill \\ \pm 1 \hfill \\ \end{gathered} \right),$ the variance contribution to the character 1 is $G_{11j} = \frac{{\sigma_{m} }}{{\sqrt {s_{1} + s_{2} } }};$ whereas in the trait model of pleiotropy, the locus vector ${\mathbf{b}}_{j} = \left( \begin{gathered} \pm \cos \theta \hfill \\ \pm \sin \theta \hfill \\ \end{gathered} \right)$ contributes $G_{11j} = \frac{{\sigma_{m} \cos^{2} \theta }}{{\sqrt {s_{1} \cos^{2} \theta + s_{2} \sin^{2} \theta } }}$ to the variance of the trait 1. These contributions are equivalent when $\theta = \pi /4$.