# Heterogeneity in effective size across the genome: effects on the Inverse Instantaneous Coalescence Rate (IICR) and implications for demographic inference under linked selection

Simon Boitard, Armando Arredondo, Lounès Chikhi, Olivier Mazet

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Simon Boitard, Armando Arredondo, Lounès Chikhi, Olivier Mazet. Heterogeneity in effective size across the genome: effects on the Inverse Instantaneous Coalescence Rate (IICR) and implications for demographic inference under linked selection. Rencontres Alphy/AIEM, UMR6553 ECOBIO, Mar 2022, Rennes, France. hal-03649541

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Submitted on 22 Apr 2022

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Distributed under a Creative Commons Attribution 4.0 International License
Heterogeneity in effective size across the genome:
effects on the Inverse Instantaneous Coalescence
Rate (IICR) and implications for demographic
inference under linked selection

Simon Boitard¹, Armando Arredondo²,
Lounès Chikhi³,⁴ & Olivier Mazet²

1. INRAE, Centre de Biologie et de Gestion des Populations (CBGP), Montpellier

2: INSA, Institut de Mathématiques de Toulouse (IMT), Toulouse

3: CNRS, Evolution et Diversité Biologique (EDB), Toulouse

4: Instituto Gulbenkian de Ciência (IGC), Oeiras, Portugal

Rencontres Alphy / AIEM, March 14-16, 2022

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## Genetic diversity shaped by

- Genetic Drift (Kimura, 1968; 1983).

- Population demography and structure (Nei et al, 1974; Wakeley, 1999; Wall et al, 2002).

- Linked selection (Hill and Robertson, 1966), including ...

- Selective sweeps (SW; Maynard Smith and Haigh, 1974; Gillespie, 1991).

- Background selection (BGS; Charlesworth et al, 1993).

- Balancing selection (BaIS; Ford 1975).

## Respective role of these forces? (Kern and Hahn 2018; Jensen et al 2019).

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- Linked selection **pervasive** (Elyashiv et al, 2016; Pouyet et al, 2018).

- Linked selection **biases demographic inference** (Ewing and Jensen, 2016; Schrider et al, 2016; Pouyet et al, 2018; Johri et al 2021)

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# The variable $N_e$ hypothesis

* Linked selection modelled by a change of $N_e$ (Hill and Robertson, 1966; Charlesworth, 2009):

* decreasing for BGS and SW.

* increasing for BaIS.

* Variable levels of $N_e$ genome-wide (Gossmann *et al*, 2011; Jimenez-Mena *et al*, 2016) reflecting variations of selection constraints.

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Characterize the effects of linked selection on genome-wide patterns of genetic diversity.

- Model linked selection through several classes of $N_e$.

- Study the genome-wide distribution of coalescence times ($T_2$).

- Consequences for demographic inference with PSMC (Li and Durbin, 2011) or MSMC (Schiffels and Durbin, 2014).

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■ Arbitrary evolution model.

■ $T_2$ coalescence time between two haploid sequences, in units of $2N$ generations.

■ Let $F(t) = \mathbb{P}(T_2 \le t)$, and $F'(t) = f(t)$.

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$$ \text{IICR}(t) = \frac{1 - F(t)}{f(t)} \quad (\text{Mazet et al, 2016}) $$

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* IICR $\Leftrightarrow T_2$ distribution.

* Can be obtained numerically (Rodriguez et al, 2018) for a given evolution model.

* Convenient because:

* inferred from genomic data by PSMC or MSMC.

* direct interpretation **under panmixia and neutrality** : population size history.

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■ PSMC does estimate the IICR but ...

■ ... the IICR does not reflect population size history.

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Mazet *et al* (2016), Chikhi *et al* (2018).

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■ K genomic classes with relative proportion $a_i$.

■ Class $i$ evolves under the WF model with $\lambda_i N$ diploids.

$$T_2^i \sim \mathcal{E}(\mu_i), \quad \mu_i = 1/\lambda_i$$

■ Genome-wide distribution of $T_2$

$$f(t) = \sum_{i=1}^{K} a_i \mu_i e^{-\mu_i t}$$

■ IICR

$$IICR(t) = -\frac{1-F(t)}{f(t)} = \frac{\sum_{i=1}^{K} a_i e^{-\mu_i t}}{\sum_{i=1}^{K} a_i \mu_i e^{-\mu_i t}}.$$

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- $IICR(t)$ increasing (backward in time)
  → spurious signal of **population size decline**.

- Starting value:

$$IICR(0) = \frac{1}{\sum_{i=1}^{K} \frac{a_i}{\lambda_i}}$$

- $IICR(t) \rightarrow \lambda_{max}$ as $t \rightarrow +\infty$

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■ $\lambda_1 = 0.1$ (BGS/SW), $\lambda_2 = 1$ (neutral).

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→ BGS/SW impact recent past IICR.

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- $\lambda_1 = 1$ (neutral), $\lambda_2 = 10$ (BalS).

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→ BalS impacts intermediate to **ancient past** IICR, even if in **small proportion**.

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$$IICR(t) = \frac{\sum_{i=1}^{K} a_i(1 - F_i(t))}{\sum_{i=1}^{K} a_i f_i(t)}$$

$f_i(t)$ pdf of $T_2$.

- Genuine population size changes.

- Population structure (stationary or non stationary).

- Transient (rather than recurrent) selection.

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Example 1: stationary *n* island model

■ $n = 10$ islands, two classes: $\lambda_1 = 0.1$, $\lambda_2 = 1$.

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→ Selection effect weaker than under panmixia for $M \le 1$.

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i

2
Example 2: human evolution model (Mazet et al, 2016)

■ $n = 10$, **non stationary** $M$ (4 epochs).

■ $+ \lambda$ distribution estimated by Gossmann et al (2011) (**red**).

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→ **Main effect** of selection in the **ancient past**.

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* Under panmixia, **linked selection** leaves a **signature of population decline** from the largest genomic $N_e$ (with PSMC or MSMC).

* BGS/SW impact recent past IICR, BalS ancient past.

* Larger effect of BalS, even if in much lower proportions.

* Linked selection partly **masked by population strucure**.

* Ref: Genetics, 220(3), iyac008.

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* **Joint inference of demography and linked selection parameters?**

* Approximate model of linked selection, but :

* **fast** IICR evaluation.

* **flexible** modelling: several forms of selection, population structure, demographic changes, transient or fluctuating selection ...

* Model genomic variations of mutation rate, gene flow ...

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PhD positions in population genomics available in September 2022.

- Inference in spatial models.

- Inference from genomic time series.

- Genomic offset.

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# Why BGS does not leave a signal of expansion?

* The decline signature comes for the variability of $N_e$ along the genome, not considered by

* single locus models (Zeng and Charlesworth, 2011; Walczak et al, 2012).

* models with **regular and short scale** alternance of neutral and selected loci (Johri et al, 2021).

* Variable $N_e$ hypothesis less appropriate for samples properties like the **Site Frequency Spectrum** (Ewing and Jensen, 2016).

* excess of singletons in BGS vs neutral models (Charlesworth et al, 1993).

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# IICR prediction and PSMC estimation

- Panmictic model with $\lambda_1 = 0.1$, $\lambda_2 = 1$.

- PSMC applied to genomes, simulated with variable $N_e$ (class 1, class 2, class 1 ...).

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→ consistent for **large scale** $N_e$ **variations.**

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$\lambda_1 = 0.1$ (BGS/SW), $\lambda_2 = 1$ (neutral), $\lambda_3 = 3$ (BalS).

$a_3 = 0.01$

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$a_1 = 0.5$

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→ stronger effect of BalS vs BGS/sweeps for the same proportion.

→ intermediate plateau depending on $a_1$ and $a_3$

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Realistic $N_e$ distributions: humans

Gossmann *et al* (2011): $N_e$ distribution assumed log-normal and estimated from polymorphism and divergence data.

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→ long term 6-fold decline from $t = 10$.

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