The G-Matrix and Multivariate Evolution
The genetic variance-covariance matrix (G-matrix) describes how genetic variation
is distributed across multiple traits in a population. It determines not just how much genetic
variation exists, but also how traits covary genetically.
The Lande Equation
The central equation of multivariate quantitative genetics is:
Δz̄ = G β
where Δz̄ is the change in mean trait values (evolutionary response),
G is the genetic variance-covariance matrix, and β is the
selection gradient (direction of selection in trait space).
Key Concepts
- G-matrix structure: Diagonal elements (G₁₁, G₂₂) are genetic variances
for each trait. Off-diagonal elements (G₁₂) are genetic covariances between traits.
- Selection gradient: The vector β points in the direction of strongest
selection. Positive values favor higher trait values, negative favor lower.
- Genetic constraints: If the G-matrix has low variance in some direction,
evolution is slow in that direction even if selection is strong.
- Genetic correlations: Positive G₁₂ means traits tend to evolve together.
Negative G₁₂ means they evolve in opposite directions.
- Principal components: The eigenvectors of G show the main axes of genetic
variation. Evolution is fastest along the direction of maximum genetic variance.
What the Visualization Shows
- Population cloud: Individual organisms plotted in 2D trait space
- G-matrix ellipse: Shows the shape and orientation of genetic variation.
Longer axes indicate more genetic variance in that direction.
- Selection gradient (blue arrow): Direction selection would push the
population if there were no genetic constraints
- Predicted response (green arrow): Direction population actually moves
according to Δz̄ = Gβ. Often differs from selection gradient!
- Population trajectory: Path of the population mean over generations
- Principal components (orange): Eigenvectors showing main axes of
genetic variation
Evolutionary Implications
The G-matrix creates genetic constraints that can slow or redirect evolution:
- Evolution is fast along directions of high genetic variance, slow along directions of
low variance, even if selection is strong
- Genetic correlations cause traits to evolve together as a package, creating
correlated responses to selection
- The population mean moves in the direction Gβ, not β—the G-matrix filters and redirects
the selection pressure
- If selection and genetic variation are misaligned, evolution can be slow or occur in
unexpected directions
Try different G-matrix structures and selection gradients to see how genetic architecture
shapes evolutionary trajectories!