# Evolutionary and ecological processes determining properties of the $\bm G$-matrix

## Cite this dataset

Saether, Bernt-Erik; Engen, Steinar (2024). Evolutionary and ecological processes determining properties of the $\bm G$-matrix [Dataset]. Dryad. https://doi.org/10.5061/dryad.1vhhmgqz3

## Abstract

The $\bm G$-matrix is the matrix of additive genetic variances and covariances for a vector of phenotypes. Here we apply the classical theory for the balance between selection drift and mutations to find the contributions to $\bm G$ from each locus. The fitness is approximated by a linear function of phenotypes. Fluctuations in the environment generate variation in the coefficients of the fitness function. We show that the $\bm G$-matrix can be decomposed into 4 additive components generated by selection, drift, mutations and environmental fluctuations. Selection is on average counteracted by the other three processes included in Fisher's concept of the deterioration of the environment, in accordance with Frank's approximate conservation law proposing that the response to selection at stasis on average is canceled by effects of drift and mutations. The theory illustrates that Fisher's fundamental theorem cannot be used to accumulate selection through time to describe adaption unless the other effects are corrected for. Another implication of the analyses is that the factor loadings to the eigenvector of the $\bm G$-matrix with the least eigenvalue are likely to indicate which characters contributing the most to the fitness function. This is information notoriously difficult to obtain in natural populations.

## README: Evolutionary and ecological processes determining the properties of the $\bm G$-matrix

Steinar Engen, Department of Mathematical Sciences, Centre for Biodiversity Dynamics, NTNU, N-7491 Trondheim, Norway. Email: Steinar.Engen@ntnu.no

Bernt-Erik Sæther, Department of Biology, Centre for Biodiversity Dynamics, NTNU, N-7491 Trondheim, Norway. Email: Bernt-Erik.Sather@ntnu.no

We show that the additive genetic variance-covariance matrix $\bm G$ can be decomposed into 4 additive components generated by selection, drift, mutations and environmental fluctuations. Selection is on average counteracted by the other three processes included in Fisher's concept of deterioration of the environment, generating considerable changes in mean phenotypes.

The theory illustrates that neither Fisher's fundamental theorem nor Lande's classical gradient formula are sufficient for assessing adaptive changes through time unless the deteriorations are corrected for. This applies for populations at stasis, but also for populations that are subject to long term evolutionary changes. The theory also indicates several possible comparative studies for investigations of deteriorating effects.

Our analyses also suggest that the factor loadings to the eigenvector of the $\bm G$-matrix with the least eigenvalue will rather accurately indicate the relative contributions from different phenotype components to fitness. This is information notoriously difficult to obtain in natural populations.

The programmes are intended to produce the graphs to illustrate the implications of the analytical results derived in this paper. They are written by Steinar Engen and are in the Pascal language using the Delfie 6 dialect. The are no data to be read and no external procedures, and the parameters used are part of the program code. Further information is given in the beginning of the program code for each figure.

Figure 2

Figure 2 is an illustration of how two stochastic effects, random genetic drift (expressed by the effective population size N_e) and environmental fluctuations in the fitness function (expressed by a factor \theta that increases with increasing environmental fluctuations), may affect the average additive genetic variance of fitness at stasis defined at the average environment, which is given given by equation (4).

This program performs the computations leading to figure 2, showing for four different values of the effective population size (1000, 3000, 10000, and infinite) the additive genetic variance of fitness (denoted va) as function of the environmenal effect (denoted theta). The parameter values used are shown in the legend to figure 2.

Figure 3

Figure 3 demonstrates how environmental noise and genetic drift may affect the smallest eigenvalue lambda_k of the genetic variance-covariance matrix G. The smallest eigenvalue lambda_k is proportional to the additive genetic variance of fitness for a given fitness function, and the corresponding eigenvector is expected to have approximately same direction as the gradient of mean fitness (see Figure 5 for more details).

This program performs the computations leading to figure 3, showing for four different values of the effective population size (1000, 3000, 10000, and infinite) the smallest eigenvalue of G as function of the environmental effect (denoted theta). The parameter values used are shown in the legend to figure 3.

Figure 4

The average additive genetic variance of fitness will always increase with increasing environmental noise, but figure 4 is included to illustrate that each single components G[i,j] of the genetic variance-covariance matrix G may depend on the environmental noise and genetic drift in a rather unpredictable way, sometimes increasing and sometimes decreasing with increasing noise.

This program performs the computations leading to figure 4, showing for four different mutation rates and effective population sizes 1000 and infinite, the components G[1,1] and G[1,3] as well as the additive genetic variance of fitness, as function of the environmental effect (denoted theta). The parameter values used are shown in the legend to figure 4.

Figure 5

The eigenvector associated with the smallest eigenvalue lambda_k of genetic variance-covariance matrix G will have the same direction as the gradient b of the mean fitness when there are no deteriorations caused by mutations, drift or environmental fluctuations. Hence, this will determine the fitness function (up to some unknown factor). The graph illustrates how the directions deviate a little under deteriorations.

This program performs the computations leading to figure 5, showing, for four different mutation rates (u_B=u_b) and effective population sizes 1000 and infinity, the angle between the eigenvector associated with the smallest aigenvalue of G and the gradient of the fitness function, as function of the environmental effect (denoted theta). The parameter values used are shown in the legend to figure 5.

Figure 6

There will at stasis be an average response to selection, which is canceled by the average effects of the deterioration. This graph illustrates that the response to selection for different phenotype components depends in a rather unpredictable way on the different deteriorating effects.

This program performs the computations leading to figure 6, showing the response to selection as function of the environmental effect theta for 6 phenotype components. In the upper panel there are no mutations and no drift. In the medium panel there are mutations but no drift, and in the lower there are both mutations and drift. The parameter values used are shown in the legend to figure 6.

Figure 7

This graph illustrates that increasing average absolute values of the genetic effects, as well as increasing mutation rates, increase the average additive genetic variance of fitness. The genetic effects on the phenotype components at thousands of loci are simulated by a rather arbitrary model where the effects for homozygotes are taken from a distribution with zero mean and variance \sigma_a^2.

The program performs the computations leading to figure 7, showing the additive genetic variance of fitness as function of mutation rates (u_B=u_b) for different values of the variance \sigma_a^2 of the genetic effects on phenotype components, and effective population sizes 1000 and infinite. Parameter values are as given in the legend to figute 7.

## Funding

The Research Council of Norway, Award: 223257

European Research Council, Award: 15032023