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This part of the tutorial presents some basics for readers who would be unfamiliar with the coalescence process.

What is coalescence?

Imagine a population of haploid individuals. They reproduce and die, and their children take their place. Even without considering natural selection, some individuals may randomly have few more children than others, transmitting a copy of their own genetic material to the following generation. Consequently, the genetic composition of the following generation may be slightly different from the previous one, even if no mutations happen. This is a well-known process that we intuitively understand forward in time:

Evolutionary process of whole population. Adapted from Irwin et al. (2016).

Task

  • Take a look at the previous figure.
  • Does it look like a coalescent tree? What is different?
  • Reconstruct the distribution of the number of children for few generations.
  • How does it affect the genealogical process and the population diversity?
  • What population parameter would you change in this example to attenuate this effect?

Modeling a process forward in time is not always relevant. For example, think about simulating data: we would need to simulate a whole population of individuals and their genetic material in order to simply sample a small fraction of the whole population. It’s not very convenient.

John Kingman developed the idea that it may actually be more elegant to track the lineages of a set of sampled gene copies backward in time until they find their common ancestor. The demographic processes (reproduction rate, migration…) stay the same, only the perspective changes: rather than following the genetic evolution of a whole population, we simply look at how the genetic diversity of the sample only has been shaped by the demographic history.

In this framework, a coalescence event is simply the replication of the DNA, viewed backward in time.

It means the two following views are equivalent:

Instead of following all the lineages inside a population from the past to the present, we track the past history only of the lineages that we sampled: it gives a clearer picture of the lineages of interest:

Evolutionary process of whole population. Adapted from Irwin et al. (2016).

Task

  • Does this tree better match your expectation of a coalescent tree?
  • Look closer at the number of branches: why do you think that usually coalescent trees are represented as binary trees?
  • WHat assumption do you make by choosing to track only the sampled lineages and not even consider the others?

Fifty shades of gene trees

We are generally interested in characteristics of the lineages trees that are generated under such demographic processes:

These quantities will be referred as genealogical properties: they are useful summaries of a genealogy (trees are quite complex structures when one thinks about it…)

Because reproduction is stochastic, we may never generate twice the exact same tree. That is, the genealogical properties don’t have a fixed value under a model, and they are instead better described by probability distributions: under a given demographic history, some values of these quantities may be more likely than others.

Along the years, mathematicians have produced a (large) number of formulas to describe these distributions (expected values and the dispersion around these values). Generally students are taught about the simplest models (e.g. Wright-Fisher with or without migration). It is important to keep in mind that a given formula (for example the expected time to the MRCA) is linked to a particular demographic model, that implies a number of assumptions.

Task

  • List the formulas you learned in class.
  • What quantities do they describe?
  • From what model have they been derived? List its assumptions.
  • What could be the most important consequences of an unmet assumption?

Inference

Different demographic models may differ by their ability to produce different shapes of gene trees.

Model choice can be straightforward if the biological system is well-known. Sometimes it can also be unclear if one model should be preferred over another, and selecting the model that allows to best represent the data can then become an important part of the inference process (model selection).

These models are generally parametrized, that is that slightly different parameter values (population size, migration rate, dispersal rate) may produce slightly different trees - different distributions of the genealogical properties. Such parameters are generally not known with precision, so identifying the parameters values that may have led to the observed genetic data become the main point of coalescence-based inference.

This a feasible because the characteristics of gene trees generated by a model grandly affect the genetic patterns of the sample when mutations are superimposed on the genealogies.

Maximum Likelihood estimation of the mean parameter of a model. Modifying the parameter of a model modifies the distribution of the data that can be generated under this model. In a coalescence setting, modifying the growth rate of a spatially explicit demographic model modifies the genealogical properties and the sample genetic diversity that depends on these genealogies. Finding both the model and the parameter that allows to best match observed patterns is the purpose of inference. The likelihood function is a way to describe the quality of the match. Animation borrowed from towardsdatascience.com.

Task

  • What other inference methods do you know?
  • In these methods, how is represented the match quality to the observations?

References

Irwin, K., Laurent, S., Matuszewski, S. et al. On the importance of skewed offspring distributions and background selection in virus population genetics. Heredity 117, 393–399 (2016). https://doi.org/10.1038/hdy.2016.58