Modelling complex traits with ancestral recombination graphs
Mar 7, 2025
Overview
The ancestral recombination graph (ARG) describes the evolutionary relationship between
genetic materials in the presence of recombination and drift
The ancestral recombination graph (ARG) describes the evolutionary relationship between
genetic materials in the presence of recombination and drift
The full probabilistic process is complicated
In this work, we condition on the realized ARG, resulting a sequence of local trees
What is the conditional distribution of a trait given the trees?
Since the genealogy is fixed, the only randomness that remains is mutation
\[ \text{Trait} \mid \text{Local trees} \quad \sim \quad ? \]
How do we get traits?
\[
\mathbf{y}_1=\boldsymbol{\beta}_1+\boldsymbol{\beta}_2, \;
\mathbf{y}_2 = \boldsymbol{\beta}_2, \;
\mathbf{y}_3 = \boldsymbol{\beta}_2
\]
\[
\mathbf{y}_1=\boldsymbol{\beta}_4, \;
\mathbf{y}_2=\boldsymbol{\beta}_4, \;
\mathbf{y}_3 = \boldsymbol{\beta}_3
\]
Consider a local tree that spans over a region
We get trait values by adding up effect sizes (\(\boldsymbol{\beta}\))
- \(\mathbf{y}_n = \mathbf{G}_{n1} \boldsymbol{\beta}_1 + \mathbf{G}_{n2} \boldsymbol{\beta}_2\)
- \(\mathbf{y}_n = \mathbf{G}_{n3} \boldsymbol{\beta}_3 + \mathbf{G}_{n4} \boldsymbol{\beta}_4\)
Branch-centric view of trait transmission
Inherit a branch first, then a mutation
- Sample \(1\) inherits edges \(1-4\) and \(4-5\)
- Sample \(2\) inherits edges \(2-4\) and \(4-5\)
- Sample \(3\) inherits edge \(3-5\)
Branch’s effect \(=\) Sum of mutations’ effect
- Effect of \(4-5 = 0\) (1st realization)
- Effect of \(4-5 = \boldsymbol{\beta}_4\) (2nd realization)
Branch effect is a random variable!
How do we weigh branches of the ARG?
\[
\small l_e:\text{ length in time} \quad s_e:\text{ span in base pairs}
\] 
\[
\small l_e:\text{ length in time} \quad s_e:\text{ span in base pairs}
\] 
\[ \small \mathrm{Var}(\mathbf{u}_e) \quad \propto \quad \text{Number of mutations} \quad \propto \quad \text{Area}=l_es_e \]
Genetic value covariance
\(\mathbf{y}_1 = \mathbf{u}_{13} + \mathbf{u}_{34} \quad \text{and} \quad \mathbf{y}_2 = \mathbf{u}_{23} + \mathbf{u}_{34}\)
\[ \mathrm{Cov}(\mathbf{y}_1, \mathbf{y}_2) = \mathrm{Cov}(\mathbf{u}_{13} + \mathbf{u}_{34}, \mathbf{u}_{23} + \mathbf{u}_{34}) = \mathrm{Cov}(\mathbf{u}_{34}, \mathbf{u}_{34}) = \mathrm{Var}(\mathbf{u}_{34}) \propto t_3 - t_2 \]
\[ \mathrm{Cov}(\mathbf{y}_1, \mathbf{y}_2) = \mathrm{Cov}(\mathbf{u}_{13} + \textcolor{red}{\mathbf{u}_{34}}, \mathbf{u}_{23} + \textcolor{red}{\mathbf{u}_{34}}) = \mathrm{Cov}(\textcolor{red}{\mathbf{u}_{34}}, \textcolor{red}{\mathbf{u}_{34}}) = \mathrm{Var}(\textcolor{red}{\mathbf{u}_{34}}) \propto t_3 - t_2 \]
Heritability is ill-defined in ARG-LMM
\[ \small \text{Heritability: }h_g^2 = \frac{\mathrm{Var}(\mathbf{g}_n)}{\mathrm{Var}(\mathbf{y}_n)} =\frac{\mathrm{Var}(\mathbf{g}_n)}{\mathrm{Var}(\mathbf{g}_n)+\mathrm{Var}(\boldsymbol{\varepsilon}_n)} \] This applies to all individuals \(\small n \in \{1,\ldots,N\}\)
However, all individuals have a different amount of genetic variance (except haploids) \[ \small \mathrm{Var}(\mathbf{g}_n) = \mathrm{Var}(\mathbf{h}_{n1}+\mathbf{h}_{n2}) = \mathrm{Var}(\mathbf{h}_{n1}) + \mathrm{Var}(\mathbf{h}_{n2}) + 2\mathrm{Cov}(\mathbf{h}_{n1},\mathbf{h}_{n2}) \]
\[ \small \mathrm{Var}(\mathbf{g}_n) = \mathrm{Var}(\mathbf{h}_{n1}+\mathbf{h}_{n2}) = \mathrm{Var}(\mathbf{h}_{n1}) + \mathrm{Var}(\mathbf{h}_{n2}) + 2\textcolor{red}{\underbrace{\mathrm{Cov}(\mathbf{h}_{n1},\mathbf{h}_{n2})}_{\text{Self-relatedness}}} \]
We can’t define a single quantitity \(h^2_g=\frac{\textcolor{red}{\mathrm{Var}(\mathbf{g}_n)}}{\textcolor{red}{\mathrm{Var}(\mathbf{g}_n)} + \mathrm{Var}(\boldsymbol{\varepsilon}_n)}\) for everyone
Polygenic prediction is constrained by demography
Some people are less genetically variable than others
Some people are harder to predict genetically than others
Some populations are inherently harder to predict!
The matrix-vector product algorithm
The algorithm needs to pass mutations to the correct samples
A naive approach is to push the mutations down to the leaves every time
Wait until the subtree’s topology changes due to edge insertion/deletion
The wrong recipient will receive the mutations if we procrastinate further
Fitting REML \(\mathcal{O}(n_s^3) \; \Rightarrow \; \mathcal{O}(n_s+n_t\log n_s)\)
\(n_s\): number of samples, \(n_t\): number of trees
Runtime for variance component estimation
The runtime scales linearly with respect to the number of individuals (genome length = \(10^8\))
Best linear unbiased prediction (BLUP)
We measured the accuracy of polygenic scores computed from \(\textsf{tslmm}\)
Training and testing on two non-overlapping groups embedded in the same tree sequence
True trees are better, but inferred trees are not too behind!
Thank you for listening
Link to (Lehmann et al. 2025), \(\textsf{tslmm}\) preprint coming soon
Collaborators: Nathaniel Pope (Oregon), Jerome Kelleher (Oxford), Gregor Gorjanc (Edinburgh), and Peter Ralph (Oregon)
Becareful of pseudoreplication
- Non-overlapping samples are not independent
- Everyone shares some amount of mutational history
- Parameter estimates (e.g. variance components) are correlated
This is also the very reason why BLUP works
We are all correlated!
Missing heritability?
ARG-LMM variance component only reflects mutational variability
- Pedigree-based heritability captures Mendelian segregation and mutation is ignored
- ARG-LMM’s generative model only has mutation and no segregation
- Why compare quantities stemming from different random forces? (Zhang et al. 2023)
Estimation quality of variance components
Simulated trees
Inferred (tsinfer+tsdate) trees
