DOI: 10.5061/dryad.1vhhmgqz3
title: Evolutionary and ecological processes determining properties of the $\bm G$-matrix

description: 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. 
description: # 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](mailto: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](mailto: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.