The coalescent model is a powerful tool in the population geneticist’s toolbox. It traces the history of a sample back to its most recent common ancestor (MRCA) by looking at coalescence events between pairs of lineages. Starting from assumptions of random mating, selective neutrality, and constant population size, the coalescent uses a simple stochastic process that allows us to study properties of genealogies, such as the time to the MRCA and the length of the genealogy, analytically and through efficient simulation. Extensions to the coalescent allow us to incorporate effects of mutation, recombination, selection and demographic events in the coalescent model. A short introduction to the coalescent model can be found here and a longer, more detailed introduction can be read here.

However, coalescent analyses can be slow or suffer from numerical instability, especially for large samples. In a study published earlier this year in Theoretical Population Biology, CEHG fellow Ethan Jewett and CEHG professor Noah Rosenberg proposed fast and accurate approximations to general coalescent formulas and procedures for applying such approximations. Their work also examined the asymptotic behavior of existing coalescent approximations analytically and empirically.

## Computational challenges with the coalescent

For a given sample, there are many possible genealogical histories, i.e., tree topologies and branch lengths, which are consistent with the allelic states of the sample. Analyses involving the coalescent therefore often require us to condition on a specific genealogical property and then sum over all possible genealogies that display the property, weighted by the probability of the genealogy. A genealogical property that is often conditioned on is , the number of ancestral lineages in the genealogy at a time in the past. However, computing the distribution of is computationally expensive for large samples and can suffer from numerical instability.

## A general approximation procedure for formulas conditioning on

Coalescent formulas conditioning on typically involve sums of the form

For large samples and recent times, these computations have two drawbacks:

– The range of possible values for may be quite large (especially if multiple populations are being analyzed) and a summation over these values may be computationally expensive.

– Expressions for are susceptible to round-off errors.

Slatkin (2000) proposed an approximation to the summation in by a single term . This deterministic approximation was based on the observation that changes almost deterministically over time, even though it is a stochastic variable in theory. Thus we can write . From Figure 2 in the paper (reproduced here), we can see that this approximation is quite accurate. The authors prove the asymptotic accuracy of this approximation and also prove that under regularity assumptions, converges to uniformly in the limits of and . This is an important result since it shows that the general procedure produces a good approximation for both very recent and very ancient history of the sample. Further, the paper shows how this method can be used to approximate quantities that depend on the trajectory of over time, which can be used to calculate interesting quantities such as the expected number of segregating sites in a genealogy.

## Approximating for single populations

A difficulty with using the deterministic approximation is that often has no closed-form formula, and if one exists, it is typically not easy to compute when the sample is large.

For a single population with changing size, two deterministic approximations have previously been developed (one by Slatkin and Rannala 1997, Volz et al. 2009 and one by Frost and Volz, 2010, Maruvka et al., 2011). Using theoretical and empirical methods, the authors examine the asymptotic behavior and computational complexity of these approximations and a Gaussian approximation by Griffiths. A summary of their results is in the table below.

Method |
Accuracy |

Griffith’s approximation | Accurate for large samples and recent history. |

Slatkin and Rannala (1997), Volz et al. (2009) | Accurate for recent history and arbitrary sample size, inaccurate for very ancient history. |

Frost and Volz (2010), Maruvka et al. (2011) | Accurate for both recent and ancient history and for arbitrary sample size. |

Jewett and Rosenberg (2014) | Accurate for both recent and ancient history and arbitrary sample size, and for multiple populations with migration. |

## Approximating for multiple populations

Existing approaches only work for single populations of changing size and cannot account for migration between multiple populations. Ethan and Noah extend the framework for single populations to allow multiple populations with migration. The result is a system of simultaneous differential equations, one for each population. While it does not allow for analytical solutions except in very special cases, the system can be easily solved numerically for any given demographic scenario.

## Significance of this work

The extension of the coalescent framework to multiple populations with migration is an important result for demographic inference. The extended framework with multiple populations allows efficient computation of demographically informative quantities such as the expected number of private alleles in a sample, divergence times between populations.

Ethan and Noah describe a general procedure that can be used to approximate coalescent formulas that involve summing over distributions conditioned on or the trajectory of over time. This procedure is particularly accurate for studying very recent or very ancient genealogical history.

The analysis of existing approximations to show that different approximations have different asymptotic behavior and computational complexities. The choice of which approximation to use is therefore often a tradeoff between the computational complexity of the approximation and the likely behavior of the approximation in the parameter ranges of interest.

## Future Directions

As increasingly large genomic samples from populations with complex demographic histories become available for study, exact methods either become intractable or very slow. This work adds to a growing set of approximations to the coalescent and its extensions, joining other methods such as conditional sampling distributions and the sequentially markov coalescent. Ethan and Noah are already exploring applications of these approximate methods to reconciling gene trees with species trees. In the future, I expect that these and other approximations will be important for fast and accurate analysis of large genomic datasets.

## References

[2] Griffiths, R. C. (1984). Asymptotic line-of-descent distributions. *Journal of Mathematical Biology*, *21*(1), 67-75.

[3] Frost, S. D., & Volz, E. M. (2010). Viral phylodynamics and the search for an ‘effective number of infections’. *Philosophical Transactions of the Royal Society B: Biological Sciences*, *365*(1548), 1879-1890.

[4] Maruvka, Y. E., Shnerb, N. M., Bar-Yam, Y., & Wakeley, J. (2011). Recovering population parameters from a single gene genealogy: an unbiased estimator of the growth rate. *Molecular biology and evolution*, *28*(5), 1617-1631.

[5] Slatkin, M., & Rannala, B. (1997). Estimating the age of alleles by use of intraallelic variability. *American journal of human genetics*, *60*(2), 447.

[6] Slatkin, M. (2000). Allele age and a test for selection on rare alleles.*Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences*, *355*(1403), 1663-1668.

[7] Volz, E. M., Pond, S. L. K., Ward, M. J., Brown, A. J. L., & Frost, S. D. (2009). Phylodynamics of infectious disease epidemics. *Genetics*, *183*(4), 1421-1430.