Extending the IICR to multiple genomes and identification of limitations of some demographic inferential methods

Reconstructing the demographic history of populations and species is one of the greatest challenges facing population geneticists. Mazet et al. (2016) introduced, for a sample of size $k=2$ haploid genomes, a time- and sample-dependent parameter which they called the IICR (inverse instantaneous coalescence rate). Here we extend their work to larger sample sizes and focus on $T_k$, the time to the first coalescence event in a haploid sample of size $k$ where $k \geq 2$. We define the $\text{IICR}_k$ as the Inverse Instantaneous Coalescence Rate among $k$ lineages. We show that (i) under a panmictic population $\binom{k}{2}\times \text{IICR}_k$ is equivalent to $N_e$, (ii) the $ \text{IICR}_k$ can be obtained by either simulating $T_k$ values or by using the $Q$-matrix approach of Rodríguez et al. (2018) and we provide the corresponding Python and R scripts. We then study the properties of the $\binom{k}{2}\times \text{IICR}_k$ under a limited set of $n$-island and stepping-stone models. Besides, (iii) in structured models the $\binom{k}{2}\times \text{IICR}_k$ is dependent on the sample size and on the sampling scheme, even when the genomes are sampled in the same deme. For instance, we find that $\binom{k}{2}\times \text{IICR}_k$ plots for individuals sampled in the same deme will be shifted towards recent times with a lower plateau as $k$ increases. We thus show that (iv) the $\binom{k}{2}\times \text{IICR}_k$ cannot be used to represent “the demographic history” in a general sense, (v) the $\binom{k}{2}\times \text{IICR}_k$ can be estimated from real or simulated genomic data using the PSMC/MSMC methods, (vi) the MSMC2 method produces smoother curves that infer something that is not the $\binom{k}{2}\times \text{IICR}_k$, but are close to the $\binom{2}{2}\times \text{IICR}_2$ in the recent past when all samples are obtained from the same deme. Altogether we argue that the PSMC, MSMC and MSMC2 plots are not expected to be identical even when the genomes are sampled from the same deme, that none can be said to represent the “demographic history of populations” and that they should be interpreted with care. We suggest that the PSMC, MSMC and MSMC2 could be used together with the $\binom{k}{2}\times \text{IICR}_k$ to identify the signature of population structure, and to develop new strategies for model choice.

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