# Population Genetics

#### Authors: Adam Griffin, Jere Koskela, Felipe Medina Aguayo Supervisors: Paul Jenkins, Dario Spanó

Modelling the genetic diversity of a population has been a great focus of study for some time; trying to model the genetic evolution forwards in time and trying to infer ancestry backwards in time to find a most recent common ancestor. One can apply these models in several areas, particularly in biology:

• Mapping the evolution and mutation of strains of infective diseases, and predicting the future levels of mutation.
• Tracking the geneology of a human population through comparison of the DNA of various individuals.
• Inferring the effects of large-scale events on the genetic diversity of a population

In particular, for our project we look at a finite-alleles model. Here we look at a single locus (possibly even a single gene) within the genome of a population, and measure which of a finite number of types this locus has. The type of this locus is passed on to offspring, but can also mutate during an individual's lifetime.

### The SLFV Process

In our project, we look at a closed haploid population within a spatial environment, such that reproduction events are dependent not only on the allelic type of the indivdual, but its location within the spatial setting. In particular we modify the Spatial $$\Lambda$$-Fleming-Viot process and the Spatial $$\Lambda$$ -Coalescent (Etheridge, Barton, Véber), and simulate sample populations driven by these processes.

### Sequential Monte Carlo

We would like to be able to perform liklihood inference on certain parameters in the model. Following on from an algorithm by Griffiths,Tavaré, we look at using SMC with dyamic stopping time resampling.

### Approximate Bayesian Calculation

As an alternative approach, we also try to perform parameter inference through ABC which instead focuses on just looking at an appropriate statistic of the data. Here we also include inference results for test data.

These population genetics models stretch back to work by Fisher, Wright on populations with discrete generations, and Kingman's coalescent model in the 1980s.

Below left: The Wright-Fisher Model
Below right: the Coalescent model (Kingman)

## References:

• Barton, N.H., Etheridge, A.M., & Véber, A. (2010) A new model for evolution in a spatial continuum. Electron. J. Probab. 15, p. 162-216
• M.A. Beaumont (2003) Estimation of population growth or decline in genetically monitored populations, Genetics 164: 11391160
• R.C. Griffiths & S. Tavaré (1994) Simulating Probability Distributions in the Coalescent, Theoretical Population Biology 46, p. 131-159
• P. Jenkins (2012) Stopping-Time Resampling and Population Genetic Inference under Coalescent Models, Stat. App in Genetics and Molecular Biology, Vol 11:Iss 1, Article 9.
• A. Véber & A. Wakolbinger (2012) The spatial Lambda-Fleming-Viot process: an event-based construction and a lookdown representation, Preprint, arXiv:1212.5909 [math.PR]

#### Acknowledgements

We acknowledge the help of our project co-ordinators Dr Paul Jenkins and Dr Dario Spanó.
This project was done through MASDOC, funded by EPSRC grant number EP/HO23364/1.