APM 504 – Applied Probability and Stochastic Processes
Spring 2015
Instructor: Dr. Jay Taylor, office: PSA 447; phone:
9652641; email: jetaylo6@asu.edu
Time: Tuesdays and
Thursdays 12:001:15
Location: LL 271
Office
Hours: Wednesdays 1:003:00 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 handinhand, 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 neutralismselectionism 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 `handson' 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.
Date 
Topic 
Reading 
13 Jan 
Overview 

15 Jan 
Probability: Interpretations and Basic Properties 

20 Jan 
Probability: Random Variables 

22 Jan 
The WrightFisher Model: Genetic Drift 
Durrett 1.2; notes 
27 Jan 
Kingman's Coalescent and Genealogies 
Durrett 1.2.11.2.2; notes 
29 Jan 
Discretetime Markov Chains 

3 Feb 
Continuoustime 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 McDonaldKreitman 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 