A complex individual selection surface for two traits. Phenotypic trait values for trait one are on the x-axis and those for trait two are on the y-axis. In this way of depicting a selection surface, altitude represents fitness and changes in altitude are indicated by the contour lines. Selection drives trait values toward the nearest peak (peaks are shown in blue in this case) and away from the valleys (shown in red). See below for more information.
Heritability and the Breeder’s Equation
Individuals have many traits, such as body size, coloration, shape, and even susceptibility to disease. Most of these traits are determined by a combination of genetic factors and environmental effects. For example, an individual raised in a particularly nutritious environment could grow to a larger size than an individual raised in a nutrient-poor environment. This difference would be due to environmental effects. On the other hand, two individuals in the same environment could grow to different sizes as a consequence of their genes. Such differences would be due to genetic effects.
The proportion of the phenotypic variation due to additive genetic effects for a single trait is described by the heritability (h2), which ranges from 0 to 1. If the heritability is 1, then an offspring is exactly intermediate between the two parents. For a trait with a heritability of 0, the offspring would be no more similar to its own parents than to a random individual from the population. As a measure of the amount of trait variation in the population that is due to genetic differences among individuals, the heritability is very important to evolutionary biology. Most importantly, the heritability describes how a trait will respond to selection. Traits with more genetic variation respond more rapidly to natural or artificial selection, and the response to selection is given by the breeder’s equation:
The Multivariate Breeder’s Equation
The G-matrix is important because organisms have multiple traits and some of these traits are related at the level of the genome. For example, several traits may be affected by the same set of genes, and selection on one of the traits would result in changes in the others. These traits would be genetically correlated with one another. The G-matrix describes these genetic correlations. For two traits, the G-matrix looks like this:
The additive genetic variances for trait one and trait two are given by G11 and G22, and G12 is the additive genetic covariance. The covariance is similar to the correlation, except correlations are standardized to take values between -1 and +1 while covariances are not. Thus, the G-matrix tells us how much genetic variation each trait has, as well as the extent to which the traits are genetically correlated with one another. The importance of the G-matrix can be seen by its role in the multivariate breeder’s equation:
In this equation, Δz is a vector of changes in mean trait values, and β is a vector of selection gradients for the traits (similar to selection differentials). Here is the equation with the G-matrix and vectors written out for the two trait case:
The subscripts indicate the trait, so Δz1 is the change in the mean of trait one, Δz2 is the change in the mean of trait two, β1 is the selection gradient for trait one, and β2 is the selection gradient for trait two. For people unfamiliar with linear algebra, the relationship can be rewritten as a system of two equations in the two-trait case just by following the rules of matrix and vector multiplication. Here are the equations in regular algebra:
From these equations, it is very easy to see why the G-matrix is important. The change in the trait one mean, for example, depends partially on the trait one selection differential multiplied by the genetic variance for the trait and partly on the genetic covariance between trait one and trait two multiplied by the selection differential on trait two. Thus, selection on trait two can cause a correlated response to selection on trait one if the two traits are genetically correlated.
Visualizing the G-matrix
The role of the G-matrix in evolution is easier to understand if we depict it graphically. The phenotype of each individual is partially due to genetic effects and partially due to environmental effects. If we consider each individual’s genetic value, which can be thought of as the average trait value of that individual’s offspring raised in random environments, then we can plot these genetic values. The genetic values are also called breeding values, and in the two-trait case it is useful to plot breeding values of each individual for trait one and trait two on a scatter plot. Such a plot is shown here. Each point represents the breeding value for trait one (x-axis) and trait two (y-axis) for a single individual from the population.
In the above graph, the traits are uncorrelated. The values of G11 and G22 would be positive, but G12 would be zero. When G12 is zero, the multivariate breeder’s equation reduces to the univariate version, and the traits evolve independently from one another. However, traits often are genetically correlated, as shown on the next graph.
In the graph above, trait one exhibits a positive genetic correlation with trait two, so these traits would not evolve independently. If selection acted on trait one, trait two also would change in value even without any direct selection on trait two. For example, if large values of trait one were favored by natural selection, then trait two would increase as well over time because individuals with large values of trait one happen to have large values of trait two. This would produce a correlated response to selection. This genetic correlation can be graphically depicted in an intuitively pleasing way by drawing an ellipse around 95 percent of the breeding values in the population.
If the ellipse is round or its axes coincide with the x-axis and y-axis, then the traits are not correlated and the traits will evolve independently.
However, if the ellipse is tilted, then the traits are correlated and selection on one trait will result in a correlated response to selection on the other trait. The G-matrix will constrain the trajectory of evolution in this case.
Evolution on a Selection Surface
We can understand selection by considering the relationship between trait values and fitness. Absolute fitness can be defined as the average number of offspring that survive to reproductive age per adult individual of a given phenotype. In evolutionary biology, we usually use relative fitness, which is the absolute fitness divided by the mean fitness of the individuals in the population. For two traits, we can plot relative fitness as a function of trait values to obtain an individual selection surface. In the two trait case, the selection surface is conveniently represented as a topography, with trait one and trait two on the x- and y-axes and elevation representing relative fitness. In the graph below, the optimum indicates the phenotype with the highest fitness, and relative fitness drops off, as indicated by the contour lines, as the phenotype departs from the optimum.
Evolution on this selection surface can be understood easily by depicting the G-matrix as an ellipse with its center at the current population mean. When the G-matrix is round, the two traits evolve independently and the population evolves directly toward the optimum, as shown below.
When the G-matrix is not round, however, evolution proceeds most quickly along the long axis of the ellipse, so the G-matrix constrains the evolutionary trajectory. Under these circumstances, the population takes a curved path to the optimum and it sometimes takes a very long time for the population to reach the optimum. If the optimum is moving, even slowly, the population may never reach the optimum. The graph below shows this situation.
Obviously, we’ve simplified a lot of things and left out many details on this page, but more information about the G-matrix can be found in a number of review articles. See Required Reading for more information. To see how the G-matrix affects the evolutionary process and how the G-matrix itself evolves, go to our See It Evolve page.