The Response of Genetic Correlations to Drift

Bob Week 🦠 KiTE Postdoc ~ Schulenburg Group 🪱 Kiel University

Response of Genetic Correlations to Drift

What are genetic correlations?

  • Between loci:
  • Between traits:

Why do genetic correlations matter?

  • Alter patterns of variation
  • Constrain adaptation

How do genetic correlations respond to drift?

  • Genetic correlations among loci become more extreme

What about quantitative characters?

  • Multivariate trait vector: \({\bf z}=\bigl(\begin{smallmatrix} z_1 \\ z_2 \end{smallmatrix}\bigr)\)
  • Genetic covariance structure ~ \({\bf G}\)-matrix: \(\bigl(\begin{smallmatrix} G_{11} & G_{12} \\ G_{12} & G_{22} \end{smallmatrix}\bigr)\)
    • Genetic correlation: \(\rho({\bf G})=G_{12}/\sqrt{G_{11}\,G_{22}}\)

    • How does \(\bf G\) respond to drift?
      • \(\mathbb E[{\bf G}_t]={\bf G}_0e^{-t/N_e}\), (Lande, 1979, 1980)
      • \(\rho(\mathbb E[{\bf G}_t])=\rho({\bf G}_0)\), constant correlation

The conventional view

  • Drift “shrinks” \(\bf G\)-matrices (Roff 2000)
    • No effect on genetic correlations between traits
    • Differences in orientation ~ selection, mutation, migration
    • … but variation around \(\mathbb{E}[{\bf G}_t]\) (Phillips et al, 2001; Steppan et al, 2002)

  • … still lacking neutral model (Mallard et al, 2024; Blomberg et al, 2025)

A Diffusion Approximation Approach

A new model of \(\bf G\)-matrix evolution

\[\frac{d{\bf G}}{dt}={\color{#c2185b} -\frac{v}{N_e}{\bf G}} + {\color{#5a4cc7}\sqrt{\frac{v}{N_e}}\sqrt{{\bf \Gamma}}:\frac{d{\bf B}}{dt}}\]

  • \(v=\) reproductive variance (~Gillespie)
  • \(N_e=\) effective population size
  • \({\bf Γ}={\bf G}\underline\otimes{\bf G}+{\bf G}\overline\otimes{\bf G}\)
  • \({\bf B}=\) matrix-valued Brownian motion
  • Deterministic part ~ shrinks \(\bf G\)
    • Agrees with Lande’s \(\mathbb E[{\bf G}_t]\)
  • Stochastic part ~ complicated …
    • study \(\rho\) to gain insights

Genetic correlations tend towards ±1

  • \(d\rho/dt\) from \(d{\bf G}/dt\) using chain rule: \(\frac {d\rho}{dt}=-\frac{v}{2N_e}\,\rho\,(1-\rho^2)+\sqrt{\frac{v}{N_e}\,(1-\rho^2)}\,\frac {dB}{dt}\)

Five replicates starting at \(\rho_0=0\), with \(v/N_e=0.001\)

Evolution of pdf for \(\rho\) with \(v/N_e=0.001\)

Effect of correlation on \(\bf G\)-matrix

Rate of attraction to ±1 set by \(v/N_e\)

alt

Averages of 200 replicates
Colors correspond to near ±1 thresholds
  • Interesting it does not depend on genetic vars \(G_{11},G_{22}\)
    • Evolution always depends on amount of genetic variation
  • How to test result?
    • sets of replicated mesocosms of clonal populations controlling for mutation
    • for each set impose different \(N_e\) using bottlenecks
    • time-series measurements of genetic correlations

Revised view

  • Drift alters orientation of \(\bf G\)-matrices
    • while also eroding variation
  • Stability of \(\bf G\)-matrix structure due to
    • mutation, migration, selection
  • Essentially complete opposite of previous view (Roff 2000)

Consequences for understanding evolution

McGlothlin et al (2018):

  • \(\bf G\)-matrice of Anolis lizards
  • Variation across lineages => evolution
  • Overall structure conserved
  • Explanations:
    • Plieotropy
    • Stabilizing selection
    • “Genetic drift is predicted to primarily influence G-matrix size…”

Fig 2 from McGlothlin et al (2018)

If drift explains small \(\bf G\)-matrices, why dont those matrices also have altered orientation?

More in the paper…

Thanks & Questions

Peter L. Ralph, Steve Krone, Hinrich Schulenburg, Patrick C. Phillips, Arne Traulsen, Jonas Wickman, Brendan Bohannan, Carola Peterson, Inga Hamerich, Amande de Azevedo-Lopez, Hildegard Uecker, many more …