User:Manudouz/sandbox/Models of codon evolution

A first model of codon evolution.

A second model of codon evolution, either restricted (a single nucleotide substitution between codons) or unrestricted (up to three nucleotide substitutions between codons).

More generalized models.

Requirement of accurate codon-based alignments: use of amino acid-aware alignment of DNA sequences.

Codon exchangeabilities

The $P(t)$ matrix

Specifically, if $i$ and $j$ are the codons respectively containing the nucleotide triplets $i_{1}i_{2}i_{3}$ and $j_{1}j_{2}j_{3}$, then codon exchangeabilities can be expressed by the transition matrix

$P(t)={\big (}P_{ij}(t){\big )}$ where each individual entry, $P_{ij}(t)\$ refers to the probability that codon $i$ will change to codon $j$ in time $t\$.

Example: We would like to model the substitution process between codons in a continuous-time fashion. The corresponding $61\ X\ 61\$ transition matrix will look like:

$P(t)={\begin{pmatrix}p_{TTT\to TTT}(t)&p_{TTT\to TTC}(t)&\ldots &p_{TTT\to j_{1}j_{2}j_{3}}(t)&\ldots &p_{TTT\to GGA}(t)&p_{TTT\to GGG}(t)\\p_{TTC\to TTT}(t)&p_{TTC\to TTC}(t)&\ldots &p_{TTC\to j_{1}j_{2}j_{3}}(t)&\ldots &p_{TTC\to GGA}(t)&p_{TTC\to GGG}(t)\\\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\p_{i_{1}i_{2}i_{3}\to TTT}(t)&p_{i_{1}i_{2}i_{3}\to TTC}(t)&\ldots &p_{i_{1}i_{2}i_{3}\to j_{1}j_{2}j_{3}}(t)&\ldots &p_{i_{1}i_{2}i_{3}\to GGA}(t)&p_{i_{1}i_{2}i_{3}\to GGG}(t)\\\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\p_{GGA\to TTT}(t)&p_{GGA\to TTC}(t)&\ldots &p_{GGA\to j_{1}j_{2}j_{3}}(t)&\ldots &p_{GGA\to GGA}(t)&p_{GGA\to GGG}(t)\\p_{GGG\to TTT}(t)&p_{GGG\to TTC}(t)&\ldots &p_{GGG\to j_{1}j_{2}j_{3}}(t)&\ldots &p_{GGG\to GGA}(t)&p_{GGG\to GGG}(t)\end{pmatrix}}$

The codon $i_{1}i_{2}i_{3}$ where each of the $i_{1}$, $i_{2}$, and $i_{3}$ is a nucleotide $A$, $C$, $G$ or $T$.

The $Q$ matrix

"The rate at which each particular allowed substitution occurs is proportional to the (equilibrium) frequency $\pi _{j_{1}j_{2}j_{3}}$ of the codon (j) being changed to."

$Q={\begin{pmatrix}q_{TTT\to TTT}&q_{TTT\to TTC}&\ldots &q_{TTT\to j_{1}j_{2}j_{3}}&\ldots &q_{TTT\to GGA}&q_{TTT\to GGG}\\q_{TTC\to TTT}&q_{TTC\to TTC}&\ldots &q_{TTC\to j_{1}j_{2}j_{3}}&\ldots &q_{TTC\to GGA}&q_{TTC\to GGG}\\\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\q_{i_{1}i_{2}i_{3}\to TTT}&q_{i_{1}i_{2}i_{3}\to TTC}&\ldots &q_{i_{1}i_{2}i_{3}\to j_{1}j_{2}j_{3}}&\ldots &q_{i_{1}i_{2}i_{3}\to GGA}&q_{i_{1}i_{2}i_{3}\to GGG}\\\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\q_{GGA\to TTT}&q_{GGA\to TTC}&\ldots &q_{GGA\to j_{1}j_{2}j_{3}}&\ldots &q_{GGA\to GGA}&q_{GGA\to GGG}\\q_{GGG\to TTT}&q_{GGG\to TTC}&\ldots &q_{GGG\to j_{1}j_{2}j_{3}}&\ldots &q_{GGG\to GGA}&q_{GGG\to GGG}\end{pmatrix}}$

$q_{i_{1}i_{2}i_{3}\to j_{1}j_{2}j_{3}}=\left\{{\begin{array}{ccccc}0&{\mbox{ if i or j differ by two or more substitutions }}\\\pi _{j}&{\mbox{ if i and j differ by one synonymous transversion }}\\\pi _{j}.\kappa &{\mbox{ if i and j differ by one synonymous transition }}\\\pi _{j}.\omega &{\mbox{ if i and j differ by one non-synonymous transversion }}\\\pi _{j}.\kappa .\omega &{\mbox{ if i and j differ by one non-synonymous transition }}\\\end{array}}\right.$

Transitions / transversions.

A codon model with three layers.