ewensnewguanajuato

Ewensnewguanajuato - Mathematics Population Genetics Introduction to the Stochastic Theory Guanajuato March 2009 Warren J Ewens Genes are of

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Mathematics Population Genetics. Introduction to the Stochastic Theory Guanajuato March 2009 Warren J Ewens
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Genes are of different types (= different “alleles” = different colors). We assume initially that at the gene locus of interest there are only two possible alleles, usually denoted (and denoted in the handout notes) as A 1 and A 2 . To be colorful, in both sense of the word, we sometimes refer to these as the “red” allele and the “green” allele respectively.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
The individual shown is A 1 A 2 (= red / green). The other two possibilities are (of course) A 1 A 1 (=red / red) and A 2 A 2 (= green / green). We next consider the entire population (of genes) at this locus, and discuss the evolution of the A 1 and A 2 allelic frequencies. Although these lectures (and slides) concern the stochastic theory of population genetics, we first consider (briefly) some simple aspects of the deterministic theory.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Hardy-Weinberg frequencies Genotype: A 1 A 1 A 1 A 2 A 2 A 2 Frequencies: x 2 2x(1-x) (1-x) 2 (eqn. (6)) Fitnesses w 11 w 12 w 22 (eqn. (8)) or 1 + s 1 + sh 1 (eqn. (9)) or 1 – s 1 1 1 – s 2 (eqn. (10))
Background image of page 6
x ' – x sx (1- x ) { x + h (1-2 x )} (eqn. (11)) d x /d t sx (1- x ) { x + h (1-2 x )} (eqn. (12)) (eqn. (13)) t x x s x x x h h d x x x ( , ) [ ( ) { ( ) } ] 1 2 1 1 1 2 1 2 = - + - -
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Markov chain theory Standard results are given in the notes in equation (20) - absorption probabilities, equation (21) - mean absorption times, equations (24)-(28) – conditional processes, equation (32) – stationary distribution equation (34) – reversibility.
Background image of page 8
We use Markov chain theory to discuss the case where random changes in these frequencies occur from one generation to the next. We first consider the cases where there are no complicating features such as selection, mutation, two sexes, etc. Even for this very simple situation, there are MANY possible stochastic models describing these changes, (with greater or lesser accuracy). The first one that we consider is the “simple” Wright-Fisher model. This is a model of pure binomial sampling. It assumes a diploid population size that is constant over time at the value N , with non-overlapping generations, and no complicating features.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Since only two alleles ( A 1 and A 2 ) are allowed, and since the population size is assumed to be constant (= N individuals = 2 N genes), it is sufficient to focus on the number of A 1 genes in any generation. In generation t , this number is denoted by X ( t ). Thus number of A 2 genes in generation t is since the number of green genes is automatically 2 N X ( t ). The binomial random sampling assumption implies that the Markov chain model for the number of ‘red” genes in the population is as shown on the following slide.
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The “simple” Wright-Fisher model (eqn. (35)) N j i N i N i j N ij j N j p i t X j t X 2 , 2, , 1 , 0 , 2 1 2 2 2 } ) ( | ) 1 ( { ob Pr = - = - = = = +
Background image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

Page1 / 77

Ewensnewguanajuato - Mathematics Population Genetics Introduction to the Stochastic Theory Guanajuato March 2009 Warren J Ewens Genes are of

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online