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This example is taken from the excellent textbook Biological Sequence Analysis: probabilistic models of proteins and nucleic acids by Durbin, Eddy, Krogh and Mitchison. CpG islands are regions of the genome with a higher than normal percentage of C and G bases adjacent to each other. The usual percentage of adjacent CG bases in the genome is about 1%, but in CpG islands that percentage is over 6%. The reason that C followed by G is relatively rare in The "p" in "CpG" refers to the phosphodiester bond between the cytosine and the guanine, and serves to distinguish it from the C and G pairing on the double stranded DNA helix. CpG islands are bioogically intersting because they are in or near 40% of the promoters in mammalian genes and 70% in human promoter genes. CpG islands vary in length between 300 and 3000 basepairs. Thus fixed-length consensus sequence based approaches do not work well for detecting them. Effective identification of of CpG islands can aid in localizing genes in eukaryotes. CpG island detection also serves as an excellent problem to illustrate the power of Markov models.
We will consider two problems.
We will construct generative models of CpG islands. A generative model produces strings, and the model parameters are tuned to reflect the characteristics of CpG islands.
The simplest probabilistic generative DNA sequence model associates a probability with the occurrence of each base: P(A), P(C), P(G) and P(T) such that these probabilities all sum to 1. For H. influenzae, these probabilities are P(A) = 0.3, P(C) = 0.2, P(G) = 0.2, and P(T) = 0.3. To generate a sequence based on this model, we first choose the length L of the sequence that we wish to construct. Then we draw bases for each position based on the discrete distribution above, as shown in the code fragement below.
i = 1;
while i less-than-or-equal-to L doS[i] = a base drawn from the discrete probability distribution [0.3,0.2,0.2,0.3](for A,C,G,T)
i = i+1end
This model does not capture interdependencies between bases. It assumes that the choice of base in each position of the generated sequence is independent of the bases surrounding it. A more complex model of DNA sequences can be constructed using the theory of Markov chains. In Markov chains, the probability of observing a base at a given position in a sequence is conditioned on the bases preceding it. Thus, Markov chains can model local correlations among the nucleotides. A Markov chain of order 1 assumes that the probability of a base at position i is dependent only on the base at position i - 1. A first order Markov chain can be specified by a probability matrix as shown below.
A | C | G | T | |
---|---|---|---|---|
A | 0.6 | 0.2 | 0.1 | 0.1 |
C | 0.1 | 0.1 | 0.8 | 0.0 |
G | 0.2 | 0.2 | 0.3 | 0.3 |
T | 0.1 | 0.8 | 0.0 | 0.1 |
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