Models_of_nucleotide_substitution

Models of nucleotide substitution are mathematical equations built to predict the probability (or proportion) of nucleotide change expected in a sequence.

Jukes and Cantor\'s one-parameter model

JC69 is the simplest of the models of nucleotide substitution. Jukes, T.H. and Cantor, C.R. (1969). "Evolution of protein molecules", in Munro, H.N. (editor): Mammalian protein metabolism. Academic Press, New York, 21-123.  The model assumes that all nucleotides has the same rate ($\backslash mu$) of change to any other nucleotides. The probability that any nucleotide $x$ remains the same from time 0 to time 1 is;

$P\_\{xx(1)\}\; =\; 1-3\backslash mu$

$P\_\{xx(t)\}$ must be read; probability (or proportion, in this case it is equivalent) that $x$ becomes $x$ at time $t$. For the probability that any nucleotide $x$ changes to any other nucleotide $y$ we write $P\_\{xy(t)\}$. The probability for time $t+1$ is;

$P\_\{xx(t+1)\}\; =\; (1-3\backslash mu)P\_\{xx(t)\}\; +\backslash mu(1-P\_\{xx(t)\})$

The second part of the equation denotes the probability that the nucleotide was changed from time 0 and 1, but then got back to its initial states on time 2. The model can be rewritten in a differential equation with the solution;

$P\_\{xx(t)\}\; =\; \backslash frac\{1\}\{4\}+\backslash frac\{3\}\{4\}e^\{-4\backslash mu\; t\}$

Or if we want to know the probability of nuleotide $x$ to change to nucleotide $y$;

$P\_\{xy(t)\}\; =\; \backslash frac\{1\}\{4\}-\backslash frac\{1\}\{4\}e^\{-4\backslash mu\; t\}$

With time, the probability will approach 0.25 (25%).

Kimura\'s two-parameters model

Mostly known under the name K80, this model was developed by Kimura in 1980 as it became clear that all nucleotides substitutions weren\'t occurring at an equal rate. Most often, transitions (changes between A and G or C and T) are more common than transversions.Kimura, M. (1980). "A simple method for estimating evolutionary rate of base substitution through comparative studies of nucleotide sequences". Journal of Molecular Evolution 16: 111-120. 

Further Reading

• Yang, Z. (2006). Computational Molecular Evolution. Oxford University Press. 

References

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