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Evolutionary Models

Evolutionary Models. CS 498 SS Saurabh Sinha. Models of nucleotide substitution. The DNA that we study in bioinformatics is the end(??)-product of evolution Evolution is a very complicated process Very simplified models of this process can be studied within a probabilistic framework

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Evolutionary Models

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  1. Evolutionary Models CS 498 SS Saurabh Sinha

  2. Models of nucleotide substitution • The DNA that we study in bioinformatics is the end(??)-product of evolution • Evolution is a very complicated process • Very simplified models of this process can be studied within a probabilistic framework • Allows testing of various hypotheses about the evolutionary process, from multi-species data Source: Ewens and Grant, Chapter 14.

  3. Diversity in a population • There IS genetic variation between individuals in a population • But relatively little variation at nucl. level • E.g., two humans differ at the nucl. level at one in 500 to 1000 nucls. • Roughly speaking, a single nucleotide dominates the population at a particular position in the genome

  4. Substitution • Over long time periods, the nucleotide at a given position remains the same • But periodically, this nucleotide changes (over the entire population) • This is called “substitution”, i.e., replacement of the predominant nucl. for that position with another predominant nucl.

  5. Markov Chain to model substitution • Markov chain to describe the substitution process at a position • States are “a”, “c”, “g”, “t” • The chain “runs” in certain units of time, i.e., the state may change from one time point to the next time point • The unit of time (difference between successive time points) may be arbitrary, e.g., 20000 generations.

  6. Markov Chain to model substitution • A symbol such as “pag” is the probability of a change from “a” to “g” in one unit of time • When studying two extant species, the evolutionary model has to provide the joint probability of the two species’ data • Sometimes, this is done by computing probability of the ancestor, starting from one extant species, and then the probability of the other extant species, starting from the ancestor • If we want to do this, the evolutionary process (model) must be “time reversible”: P(x)P(x->y) = P(y)P(y->x)

  7. Jukes Cantor Model • Markov chain with four states: a,c,g,t • Transition matrix P given by:

  8. Jukes Cantor Model •  is a parameter depending on what a “time unit” means. If time unit represents more #generations,  will be larger •  must be less than 1/3 though

  9. Jukes Cantor Model • Whatever the current nucl is, each of the other three nucls are equally likely to substitute for it

  10. Understanding the J-C Model • Consider a transition matrix P, and a probability vector v (a row vector) • What does w = vP represent ? • If v is the probability distribution of the 4 nucls (at a position) now, w is the prob. distr. at the next time step.

  11. Understanding the J-C model • Suppose we can find a vector  such that P =  • If the probability distribution is , it will continue to remain  at future times • This is called the stationary distribution of the Markov Chain

  12. Understanding the J-C model • Check that  = (0.25, 0.25, 0.25, 0.25) satisfies  P =  • Therefore, if a position evolves as per this model, for long enough, it will be equally likely to have any of the 4 nucls! • This is the very long term prediction, but can we write down what the position will be as a function of time (steps) ?

  13. Spectral Decomposition • Recall that we found a  such that  P =  • Such a vector is called an “eigenvector” of P, and the corresponding “eigenvalue” is 1. • In general, if v P =  v (for scalar ), is called an eigenvalue, and v is a left eigenvector of P

  14. Spectral decomposition • Similarly, if P uT =  uT, then u is called a right eigenvector • In general, there may be multiple eigenvalues jand their corresponding left and right eigenvectors vjand uj • We can write P as

  15. Spectral decomposition • Then, for any positive integer, it is true that • Why is Pninteresting to us ? • Because it tells us what the probability distribution will be after n time steps • If we started with v, then Pnv will be the prob. distr. after n steps

  16. Back to the J-C model • We reasoned that  = (.25,.25,.25,.25) is a left eigenvector for the eigenvalue 1. • Actually, the J-C transition matrix has this eigenvalue and the eigenvalue (1-4), and if we do the math we get the spectral decomposition of P as:

  17. Back to the J-C model • So, if we started with (1,0,0,0), i.e., an “a”, the probability that we’ll see an “a” at that position after n time steps is: 0.25+0.75(1-4)n • And the probability that the “a” would have mutated to say “c” is: 0.25 - 0.25(1-4)n

  18. Substitution probability • As a function of time n, we therefore get • Pr(x -> y) = 0.25 + 0.75 (1-4)n if x = y • and = 0.25 - 0.25 (1-4)n otherwise • If n ->, we get back our (0.25, 0.25, 0.25, 0.25) calculation

  19. More advanced models • The J-C model made highly “symmetric” assumptions, in its formulation of the transition matrix P • In reality, for example, “transitions” are more common than “transversions” • What are these? Purine = A or G. Pyrimidine = C or T. Transition is substitution in the same category; transversion is substitution across categories • Purines are similarly sized, and pyrimidines are similarly sized. More likely to be replaced by similar sized nucl. • The “Kimura” model captures this transition/transversion bias

  20. Kimura model • This of course is the transition probability matrix P of the Markov chain • Two parameters now, instead of one.

  21. Kimura model • Again, one of the eigenvalues is 1, and the left eigenvector corresponding to it is  = (.25,.25,.25,.25) • So again, the stationary distribution is uniform • P(x -> x) = .25+.25(1-4)n+.5(1-2( +))n • P(x -> y) = .25+.25(1-4)n+.5(1-2( +))nif x is a purine and y is the other purine

  22. Even more advanced models • Get to greater levels of realism • Kimura model still has a uniform stationary distribution, which is not true of real data • One extension: purine to pyrimidine subst. prob. is different from pyrimidine to purine subst. prob. • This leads to a non-uniform stationary probability

  23. Felsenstein models Transition probability proportional to the stationary probability of the target nucleotide. Stationary distribution is (a, g, c, t)

  24. Reversible models • Many inference procedures require that the evolutionary model be time reversible • What does this mean?

  25. Reversible Markov Chain Looks like time has been reversed. That is, if we can find a  such that The models we have seen today all have this property. Source: Wikipedia

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