# 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.[1] The model assumes that all nucleotides has the same rate (${\displaystyle \mu }$) of change to any other nucleotides. The probability that any nucleotide ${\displaystyle x}$ remains the same from time 0 to time 1 is;

${\displaystyle P_{xx(1)}=1-3\mu }$

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

${\displaystyle P_{xx(t+1)}=(1-3\mu )P_{xx(t)}+\mu (1-P_{xx(t)})}$

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;

${\displaystyle P_{xx(t)}={\frac {1}{4}}+{\frac {3}{4}}e^{-4\mu t}}$

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

${\displaystyle P_{xy(t)}={\frac {1}{4}}-{\frac {1}{4}}e^{-4\mu t}}$

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.[2]