pleiotropy

Martin G & Lenormand T 2006 A general multivariate extension of Fisher's geometrical model and the distribution of mutation fitness effects across species. Evolution 60:893-907.

• there may be a cost to phenotypic complexity although much weaker than previously suggested by earlier theoretical works

• the distribution of mutation phenotypic effects is also often assumed to be Gaussian
• we will use this assumption too

• a perhaps stronger assumption of Fisher's model is that a single mutation can potentially affect all the phenotypic traits of the organism
• universal pleiotropy

• Fisher's model only requires a description of the net phenotypic effect of all mutations averaged over modules and can thus accommodate modularity or partial pleiotropy

• given appropriate scaling, it is possible to account for some correlations between traits
• such scaling can only be performed for either selection, or mutation effects, not for both
• Waxman and Welch (2005) have proposed the first model to account for selective interactions between traits, but still neglecting mutational correlations
• one of the aims of this paper is to relax the symmetry assumptions for both mutation and selection

• we use a Gaussian distribution of mutation phenotypic effects
• this model of multivariate stabilizing selection and mutation is similar to that introduced by Zhang and Hill (2003) for the study of mutation-selection balance on a quantitative trait, but extended to account for beneficial mutations

• S is the n × n matrix of the selective effects of all traits
• diagonal elements in S measure the selection intensity on each trait
• nondiagonal elements measure selective interactions between trait pairs
• the distribution of mutant phenotypes around z_o is multivariate Gaussian with mean zero and covariance matrix M
• the model allows both for differences in mutational variances across traits and for mutational correlations between traits

• we chose the negative gamma distribution because it is the exact distribution of s corresponding to the simplest situation
• when all λ_i = λ are equal and s_o = 0, ƒ(s) is a negative gamma distribution with scale λ and shape n / 2
• n_e is the effective number of traits
• when the initial genotype is away from the optimum (s_o > 0), we use a similar approximation as above
• ƒ(s) becomes a "displaced gamma"
• the sum of a negative gamma and the constant s_o
• s = s_o − γ, where γ is approximately gamma distributed with scale α and shape β

• it is n_e, not n, that determines the rate of adaptation
• including heterogeneity between traits greatly reduces the cost of complexity by reducing the effective number of traits n_e

• the effective number of dimensions n_e increases with complexity as predicted by our model, but varies only by an order of magnitude from bacteria to fruit flies, and remains very small (0.2—2.5)

• the Fisher-Orr geometric approach may not be as unrealistic as it is sometimes suggested, provided phenotypic correlations are accounted for