Stochastic Eco-Evolutionary Dynamics of Multivariate Traits

Bob Week 🦠 KiTE Postdoc ~ Schulenburg Group 🪱 Kiel University

🌐bobweek.github.io 🦋@bweek.bsky.social

Consequences of random genetic drift

• Erodes heritable variation
• Increases genetic correlations/linkage

• Fundamental principle of evolutionary biology

What about quantitative characters?

• Multivariate trait vector: \({\bf z}=\left(\begin{smallmatrix} z_1 \\ \vdots \\ z_d \end{smallmatrix}\right)\)
• Genetic covariance structure ~ \({\bf G}\)-matrix: \(\left(\begin{smallmatrix} G_{11} & \cdots & G_{1d} \\ \vdots & \ddots & \vdots \\ G_{d1} & \cdots & G_{dd} \end{smallmatrix}\right)\)
  • Genetic correlation: \(\rho_{ij}({\bf G})=G_{ij}/\sqrt{G_{ii}\,G_{jj}}\)
    
    
  • We should know how \(\bf G\) responds 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 (different previous slide!)

The conventional view

• Drift “shrinks” entries of \(\bf G\)-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 stochastic neutral model (Mallard et al, 2024; Blomberg et al, 2025)

A Diffusion Approximation Approach

• Individual-Based Model:
  • Traits \({\bf z}\in {\mathbb R}^d\), population \({\cal P}_t({\bf z})\)
  • Gaussian mutation
    • \({\bf z}'\sim{\mathrm{MVN}}_d({\bf z},{\bf M})\)
    • NOT infitesimal model!!
  • \(W({\cal P}_t,{\bf z})=\) fitness function

• Diffusion Limit:
  • Rescale turnover by factor \(k\)
  • Rescale population \({\cal P}_t^{(k)}({\bf z})\to \nu_t({\bf z})\)
  • Growth rate \(m(\nu_t,{\bf z})=\lim_k F_k(W)\)
    • \(k[W^{1/k}({\cal P}_t^{(k)},{\bf z})-1]\to m(\nu_t,{\bf z})\)

Why Use Diffusion Approximations?

• Nice analytical properties
• Many models -> common diffusion limit
  • Previous slide:
    • Birth-death process
    • <-> Poisson offspring count
  • Extracts important features
  • Uncover biological principles

The Diffusion Limit

Moment Dynamics (not closed)

\(\text{mean fitness:}\quad \bar m=\tfrac{1}{n}\int_{\mathbb{R}^d}m(\nu,{\bf z})\,\nu({\bf z})\,\mathrm d{\bf z}\)

\[ \text{total abundance:}\quad \mathrm dn={\color{#c2185b}\bar m \, n}\,\mathrm dt + {\color{#5a4cc7}\sqrt{v\,n}\,\mathrm dB_n} \]

\[ \text{mean trait (}d\text{-dim):}\quad \mathrm d\bar{\bf z}={\color{#c2185b}\textrm{Cov}(m,{\bf z})}\,\mathrm dt + {\color{#5a4cc7}\sqrt{\tfrac v n {\bf P}}\,\mathrm d{\bf B}_{\bar{\bf z}}} \]

\[\begin{multline} \text{trait variance (}d\times d\text{):}\quad \mathrm d{\bf P}=\big[{\color{#3c8500}{\bf M}}+{\color{#c2185b}\textrm{Cov}(m,({\bf z}-\bar{\bf z})({\bf z}-\bar{\bf z})^\top)}{\color{#5a4cc7}-\tfrac v n {\bf P}}\big]\,\mathrm dt \\ + {\color{#5a4cc7}\sqrt{\tfrac v n ({\bf K}-{\bf P}\otimes{\bf P})}:\mathrm d{\bf B}_{\bf P}} \end{multline}\]

\[ \text{skew tensor }(d\times d\times d)\text{:}\quad \mathrm d{\bf S}=\dots, \quad \text{kurtosis tensor }(d\times d\times d\times d)\text{:}\quad \mathrm d{\bf K}=\dots,\quad \dots\]

Multivariate Normal Approximation (closed)

\[ \text{total abundance (1-dim):}\quad \mathrm dn={\color{#c2185b}\bar m \, n}\,\mathrm dt + {\color{#5a4cc7}\sqrt{v\,n}\,\mathrm dB_n} \]

\[ \text{mean trait (}d\text{-dim):}\quad \mathrm d\bar{\bf z}={\color{#c2185b}{\bf P}\big(\nabla_{\bar{\bf z}}\,\bar m-\overline{\nabla_{\bar{\bf z}}\,m}\big)}\,\mathrm dt + {\color{#5a4cc7}\sqrt{\tfrac v n {\bf P}}\,\mathrm d{\bf B}_{\bar{\bf z}}} \]

\[\begin{multline} \text{trait variance (}d\times d\text{):}\quad \mathrm d{\bf P}=\big[{\color{#3c8500}\bf M}+{\color{#c2185b}2\,{\bf P}(\nabla_{\bf P}\,\bar m-\overline{\nabla_{\bf P}\,m})\,{\bf P}}{\color{#5a4cc7}-\tfrac v n {\bf P}}\big]\,\mathrm dt \\ + {\color{#5a4cc7}\sqrt{\tfrac v n (\mathbf{P}\overline\otimes \mathbf{P}+\mathbf{P}\underline\otimes \mathbf{P})}:\mathrm d{\bf B}_{\bf P}} \end{multline}\]

Multivariate Normal + Decomposed Traits

Traits decompose \({\bf z}={\bf g}+{\bf e}\) with iid \({\mathrm{Cov}}({\bf e})={\bf E}\) so that \({\bf P}={\bf G}+{\bf E}\)

\[ \text{total abundance:}\quad \mathrm dn={\color{#c2185b}\bar m \, n}\,\mathrm dt + {\color{#5a4cc7}\sqrt{v\,n}\,\mathrm dB_n} \]

\[ \text{mean trait (}d\text{-dim):}\quad \mathrm d\bar{\bf z}={\color{#c2185b}{\bf G}\big(\nabla_{\bar{\bf z}}\,\bar m-\overline{\nabla_{\bar{\bf z}}\,m}\big)}\,\mathrm dt + {\color{#5a4cc7}\sqrt{\tfrac v n {\bf G}}\,\mathrm d{\bf B}_{\bar{\bf z}}} \]

\[\begin{multline} \text{genetic variance (}d\times d\text{):}\quad \mathrm d{\bf G}=\big[{\color{#3c8500}{\bf M}}+{\color{#c2185b}2\,{\bf G}(\nabla_{\bf G}\,\bar m-\overline{\nabla_{\bf G}\,m})\,{\bf G}}{\color{#5a4cc7}-\tfrac v n {\bf G}}\big]\,\mathrm dt \\ + {\color{#5a4cc7}\sqrt{\tfrac v n (\mathbf{G}\overline\otimes \mathbf{G}+\mathbf{G}\underline\otimes \mathbf{G})}:\mathrm d{\bf B}_{\bf G}} \end{multline}\]

• this is a particularly kewt consolidation *^-^*

How to use?

Deterministic/selection:

• \(m(\nu,{\bf z})\) arbitrary
• Examples:
  • Logistic growth + stabilizing selection
    • \(m(\nu,{\bf z})=r-\tfrac 1 2 {\bf z}^\top{\bf \Psi}{\bf z}-c\,n\)
  • Lotka-Volterra
    • \(m_i=r-\sum_j\alpha_{ij}n_j\)
  • Coevolution
    • \(m({\bf z}_i,{\bf z}_j)_i=\)
  • Rugged ?

Stochasticity/drift:

• Numerical integration: github.com/bobweek/multi-mtgl
• Analytical approach:
  • Ito’s Formula
    • Heuristics

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

\[\mathrm d{\bf G}={\color{#c2185b} -\tfrac{1}{N_e}{\bf G}\,\mathrm d t} + {\color{#5a4cc7}\sqrt{\tfrac{1}{N_e}({\bf G}\underline\otimes{\bf G}+{\bf G}\overline\otimes{\bf G})}:\mathrm d{\bf B}}\]

• \(N_e=v/n\) effective population size
• Deterministic part ~ shrinks \(\bf G\)
  • Agrees with Lande’s \(\mathbb E[{\bf G}_t]\)
• Stochastic part ~ complicated …
  • study \(\rho_{ij}=G_{ij}/\sqrt{G_{ii}G_{jj}}\) to gain insights

Obtaining dynamics of genetic correlations

(and dealing with stochastic terms)

• Use ito’s formula on \(\rho=f({\bf G})\)

\[ \mathrm d\rho=(\nabla_{\bf G}f)^\top\mathrm d{\bf G}+\tfrac 1 2 (\mathrm d{\bf G})^\top (\mathrm H_{\bf G}f)(\mathrm d {\bf G}) \]

Genetic correlations tend towards ±1

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

Five replicates starting at \(\rho_0=0\), with \(1/N_e=0.001\)

Evolution of pdf for \(\rho\) with \(1/N_e=0.001\)

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?
• Adding recombination doesn’t solve the problem … add drosphila slide at end for people who ask this question

More in the paper…

Thanks & Questions

Steve Krone, Peter L. Ralph, Hinrich Schulenburg, Patrick C. Phillips, Arne Traulsen, Jonas Wickman, Brendan Bohannan