APM 504 – Applied Probability and Stochastic Processes

Spring 2015

Instructor: Dr. Jay Taylor, office: PSA 447; phone: 965-2641; e-mail: jetaylo6@asu.edu
Time: Tuesdays and Thursdays 12:00-1:15
Location: LL 271
Office Hours: Wednesdays 1:00-3:00 in PSA 447; alternatively make an appointment or just stop by my office.
Text: Probability Models for DNA Sequence Evolution by Richard Durrett, 2'nd edition (Springer, 2008).

Course Description: Population genetics, probability theory and statistics are disciplines that have grown up hand-in-hand, with each field contributing in important ways to the development of the others. This is evidenced by the number of influential scientists who have worked at the interface of these subjects, including F. Galton, R. A. Fisher, G. Malecot, P. A. P. Moran, and J. F. C. Kingman. The objective of this course is to provide an introduction to this highly interdisciplinary subject. Mathematical topics covered will include discrete and continuous time Markov chains, branching processes, coalescent theory and genealogical processes, and diffusion approximations. Our goal will be to use these models and the powerful mathematical tools that accompany them to develop theory that makes quantitative predictions of the effects that demographic stochasticity, population structure, natural selection, mutation and recombination will have on genetic variation. Along the way, we will also explore some longstanding problems in evolutionary genetics such as the neutralism-selectionism debate and the evolution of sex. Particular attention will be given to the interplay between modeling and the development of statistical tools that can be used to analyze genetic sequence data.

Prerequisites: Students should be comfortable with basic calculus, matrix algebra and ordinary differential equations, and have some programming experience in a language such as C/C++, R or Matlab. No prior knowledge of genetics or evolutionary biology is required.

Practicals: There will be four to five practicals posted at the following link. These are intended to give you practice implementing some of the models introduced in this course so that you can get a `hands-on' feel for their behavior.

Project: Class participants will be expected to complete a research project that investigates a stochastic model of some biological process and to present their results in a written report of 5–10 pages. Topics can come from any area of the life sciences and the project can focus on theory, modeling or analysis of biological data. A short project proposal (1 page) is due on Feb. 19 and the final report is due on April 30. Group projects will require instructor approval.

Grading: Successful completion of the course requirements will guarantee an A.




13 Jan



15 Jan

Probability: Interpretations and Basic Properties

slides; Frank (2009)

20 Jan

Probability: Random Variables

slides; Fisher (1918)

22 Jan

The Wright-Fisher Model: Genetic Drift

Durrett 1.2; notes

27 Jan

Kingman's Coalescent and Genealogies

Durrett 1.2.1-1.2.2; notes

29 Jan

Discrete-time Markov Chains


3 Feb

Continuous-time Markov Chains


5 Feb

Infinite alleles model

Durrett 1.3; notes

10 Feb

Infinite alleles model

Durrett 1.3; notes

12 Feb

Infinite sites model

Durrett 1.4; notes

17 Feb

Infinite sites model

Durrett 1.4; notes

19 Feb

Moran model

Durrett 1.5; notes

24 Feb

Inference: site frequency spectrum

Durrett 2.1 – 2.2; notes

26 Feb

Neutrality tests: Tajima's D and other difference statistics

Durrett 2.3 - 2.4; notes

3 March

Neutrality tests: HKA test and McDonald-Kreitman test

Durrett 2.5 – 2.6; notes

5 March

Recombination: two loci

Durrett 3.1 – 3.2; notes

10 March

Spring Break – no class

12 March

Spring Break – no class

17 March

Recombination: linkage disequilibrium

Durrett 3.3; notes

19 March

Ancestral recombination graph

Durrett 3.4; notes

24 March

Recombination: estimation

Durrett 3.5 – 3.7; notes

26 March

Demography: effective population size

Durrett 4.4; notes

31 March

Demography: population dynamics and bottlenecks

Durrett 4.2 – 4.3; notes

2 April

Demography: fecundity variance

Durrett 4.1; notes

7 April

Demography: matrix migration models

Durrett 4.5; notes

9 April

Demography: the symmetric island model and fixation indices

Durrett 4.6 – 4.7; notes

14 April

Directional selection

Durrett 6.1

16 April

Balancing selection

Durrett 6.2

21 April

Background selection

Durrett 6.3

23 April

Muller's ratchet and the advantages of sex

Durrett 6.4

28 April

Genetic hitchhiking and selective sweeps

Durett 6.5 – 6.6

30 April

Recurrent sweeps

Durrett 6.7