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Markov chains part 2

Markov chains part 2. Genetic Information. Gene – basic unit of genetic information. Genes determine the inherited characters. Genome – the collection of genetic information. Chromosomes – storage units of genes .

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Markov chains part 2

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  1. Markov chains part 2

  2. Genetic Information • Gene – basic unit of genetic information. Genes determine the inherited characters. • Genome – the collection of genetic information. • Chromosomes – storage units of genes. • DNA - is a nucleic acid that contains the genetic instructions specifying the biological development of all cellular forms of life

  3. Locus1 Possible Alleles: A1,A2 Locus2 Possible Alleles: B1,B2,B3 Chromosome Logical Structure • Locus – location of a gene/marker on the chromosome. • Allele – one variant form of a gene/marker at a particular locus.

  4. Human Genome Most human cells contain 46 chromosomes: • 2 sex chromosomes (X,Y): XY – in males. XX – in females. • 22 pairs of chromosomes named autosomes.

  5. Genotypes Phenotypes • At each locus (except for sex chromosomes) there are 2 genes. These constitute the individual’s genotype at the locus. • The expression of a genotype is termed a phenotype. For example, hair color, weight, or the presence or absence of a disease.

  6. GenotypesPhenotypes(example) • Eb- dominant allele. • Ew- recessive allele. genotypes phenotypes

  7. Population genetics • genetic structure of a population • alleles • genotypes group of individuals of the same species that can interbreed Patterns of genetic variation in populations Changes in genetic structure through time

  8. Describing genetic structure • genotype frequencies • allele frequencies rr = white Rr = pink RR = red An example of incomplete dominance

  9. Describing genetic structure • genotype frequencies • allele frequencies genotype frequencies: 200 white 500 pink 300 red 200/1000 = 0.2 rr 500/1000 = 0.5 Rr 300/1000 = 0.3 RR total = 1000 flowers

  10. Describing genetic structure • genotype frequencies • allele frequencies 200 rr 500 Rr 300 RR = 400 r = 500 r =500 R = 600 R allele frequencies: 900/2000 = 0.45 r 1100/2000 = 0.55 R total = 2000 alleles

  11. Fisher-Wright model • Genetic drift: the stochastic fluctuations in allele frequency due to random sampling in a finite population. • The Fisher-Wright model describes the process of genetic drift in a finite population. The model assumes: 1. M diploid organisms (N=2M alleles) 2. Monoecious reproduction (e.g. plants) 3. Non-overlapping generations 4. Random mating 5. No mutation 6. No selection • Given two alleles in a population, a and A, genetic drift describes their change in frequency over time.

  12. Fisher-Wright model This model assumes a fixed population of size N=2M genes composed of a and Aalleles (at one locus). The makeup of the next generation is determined by N independent binomial samples as follows: if the parent population consists of j alleles of type a and N −j alleles of type A, then each sample results in a or A with probabilities: respectively. Repeated selections are done with replacement. We thus generate a Markov chain where Xt is the number of a-genes in the tth generation. The state space contains the N+1 values {0, 1, 2, ...,N}.

  13. Exercise • Write a program to simulate the Fisher-Wright model in which the population is represented explicitlyas a vector of ones(a) and zeros(A). • Proceed by writing a function: function r = single_sim(N,j0,nGen) Where N=population size, j0=number of a-alleles in 1st generation, nGen=# of generations The output should look as follows:

  14. Exercise (continued) • Using the function single_sim, write a program to simulate the system 50 times, and estimate the probability that the a- allele reaches fixation (frequency = 1.0) after 35 generations with N=6 for j0=2,4 and 5. • Plot the first six simulations (you can use plot(Y) where Y is matrix). The result should look like this:

  15. Fisher-Wright model • The transition probability matrix for the system is: • Note that states 0 and N are absorbing in the sense that once Xt = 0 or N then no further state change is possible. We are interested in calculating the probability for a particular X0 that a or A will reach fixation, i.e. we will reach a population composed only of a- or A- alleles.

  16. Exercise • Write a function P = buildFWP(N) that constructs the Fisher-Wright transition probability matrix for arbitrary N. Take advantage of the function binomialPMF you have already written. Compute the corresponding matrix for N=6. • Use the eigenvectors method to find the stationary distribution(s). Do they make sense? • Raise P to a large power (e.g. 100). What does this tell us?

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