Directional selection can drive the evolution of modularity in complex traits

Abstract

Modularity is a central concept in modern biology, providing a powerful framework for the study of living organisms on many organizational levels. Two central and related questions can be posed in regard to modularity: How does modularity appear in the first place, and what forces are responsible for keeping and/or changing modular patterns? We approached these questions using a quantitative genetics simulation framework, building on previous results obtained with bivariate systems and extending them to multivariate systems. We developed an individual-based model capable of simulating many traits controlled by many loci with variable pleiotropic relations between them, expressed in populations subject to mutation, recombination, drift, and selection. We used this model to study the problem of the emergence of modularity, and hereby show that drift and stabilizing selection are inefficient at creating modular variational structures. We also demonstrate that directional selection can have marked effects on the modular structure between traits, actively promoting a restructuring of genetic variation in the selected population and potentially facilitating the response to selection. Furthermore, we give examples of complex covariation created by simple regimes of combined directional and stabilizing selection and show that stabilizing selection is important in the maintenance of established covariation patterns. Our results are in full agreement with previous results for two-trait systems and further extend them to include scenarios of greater complexity. Finally, we discuss the evolutionary consequences of modular patterns being molded by directional selection.

Keywords: G-matrix; phenotypic correlations; pleiotropy; quantitative genetics; variational modularity.

Fig. 1.

Pictorial representation of the simulation scheme. Dynamics for a single individual in the population. In this example, two loci (y, $m=2$) control the additive values for two traits (x, $p=2$), with the pleiotropic B matrix connecting these two levels. To obtain the phenotype (z), additive values are added to Gaussian noise (e). Phenotypic values are then used to attribute a fitness to each individual, according to a Gaussian selection surface defined by the peak θ and covariance matrix ω. Mating pairs are sampled with a probability proportional to their fitness, and gametes formed by sampling one allele from each locus, along with their pleiotropic effects expressed in the B matrix. Mutation can alter the values of each element of y with probability μ by an amount drawn from a Gaussian distribution with a mean of 0 and variance σ; and mutation can also add or remove arrows connecting y to the additive values in x with probability ${\mu }_{\mathbf{B}}$. Mutation acts before the formation of gametes but after selection.

Fig. 2.

Populations under divergent directional selection. (A) Ratio of average within- and between-module phenotypic correlations (AVG Ratio) under increasing correlated peak movement rate. The AVG Ratio increases with peak movement rate, indicating more differentiation of within- and between-module correlations. Variance of the AVG Ratio within populations for a given peak movement rate increases with the rate because between-module correlations get progressively smaller, leading to a divergence in AVG Ratio. (B) Percentage of variation explained by each eigenvalue. Each color represents an eigenvalue, ordered from top to bottom. As the divergent peak movement rate increases, variation associated with dissociation between modules increases (the second principal component, a contrast between modules; SI Appendix). (C) Mean and directional autonomy for various peak movement rates. Directional autonomy is measured in the direction of each module. Mean autonomy is an average using 1,000 random vectors in all directions. As the peak movement rate increases, directional autonomy rises, indicating more variation in the direction of selection. All plot points show mean values, and error bars span 2.5% and 97.5% quantiles for 10 populations under each peak movement rate, after 10,000 generations (population size $N_e=5,000$, per locus mutation rate $\mu =5\times {10}^{-4}$, pleiotropic effects mutation rate ${\mu }_{\mathbf{B}}={10}^{-4}$, number of traits $p=10$, number of loci $m=500$, per locus mutation variance $\sigma =0.02$, environmental variance $V_e=0.8$, selection surface variance $V_{\mathit{\omega }}=10$, selection surface within-module correlation $r_{\mathit{\omega }}=0.8$, rate of peak movement per generation ${\mathrm{\Delta }}_{\theta }=0.0001-0.004$).

Fig. 3.

Average correlation within and between modules under corridor selection. Faster peak movement promotes a higher differential between correlations in each group. Traits under directional selection become more correlated, whereas traits under stabilizing selection maintain the initial mean correlation. Between-module correlations fall proportional to peak movement rate. Points are mean values, and error bars span 2.5% and 97.5% quantiles for 10 populations under each peak movement rate after 10,000 generations. Parameter values are as in Fig. 2.

Fig. 4.

AVG Ratio under drift and uncorrelated and correlated stabilizing selection. The initial population had undergone divergent directional selection, so all populations start from a highly modular pattern. Mean values and 95% confidence intervals for 100 populations under each regime are given. Populations under drift quickly lose any modularity, whereas populations under stabilizing selection are able to maintain it for many generations. On average, correlated stabilizing selection maintains modularity at the same level established by directional selection, whereas under uncorrelated stabilizing selection, modularity decays to a lower level. Parameter values are as in Fig. 2, and ${\mathrm{\Delta }}_{\theta }=0.004$ for the initial population.