User:Manudouz/sandbox/L95 model

L95, a model of DNA evolution proposed by Jean Lobry in 1995, is a general model under no-strand-bias conditions, i.e., when mutation and selection do have the same effect on each of the two complementary DNA strands. It incorporates Watson and Crick base pairing rules: the exchange rate from a nucleotide $$i$$ to another $$j$$ is equal to the rate from $$\bar{i}$$ (i.e., the complement of $$i$$) towards $$\bar{j}$$ (i.e., the complement of $$j$$). The model therefore reduces to six the number of exchange rates between nucleotides. Note that this model is not time-reversible.

L95 parameters consist of an equilibrium base frequency vector, $$\Pi = (\pi_T, \pi_C , \pi_A , \pi_G)$$, giving the frequency at which each base occurs at each site, and the rate matrix where exchangeabilities between pairing bases are equal. For example $$ \gamma $$ is the A $$ \rightarrow $$ G exchange rate, and it is equal to the exchange rate from T (the complementary base of A) to C (the complementary base of G): $$ \gamma = r(A \rightarrow G) = r(\bar{A} \rightarrow \bar{G}) = r(T \rightarrow C) $$. The rationale behind this is the fact that a mutation on one strand introduces a mismatch, prompting the occurrence / favoring a second mutation on the complementary strand to compensate for the first one / possibly prompting the DNA mismatch repair on the complementary strand:



\begin{matrix} & \mathsf{A} & & \mathsf{G} & & \mathsf{G} \\ \mathsf{(1)\ Initial\ pair:} & \parallel & \mathsf{\quad (2)\ A\ \rightarrow\ G\ \ change\ on\ the\ first \ strand:\ } & \nparallel & \mathsf{\quad (3)\ T\ \rightarrow\ C\ \ compensating\ change\ on\ the\ complementary\ strand:\ } & \parallel \\ & \mathsf{T} & & \mathsf{T} & & \mathsf{C} \\ \end{matrix} $$

Over long evolutionary times, the A $$ \rightarrow $$ G exchange rate would therefore equate the T $$ \rightarrow $$ C exchange rate.

Sueoka

By columns


$$ \qquad \ \mathsf{From} \qquad \quad \begin{matrix} \ \ \ \mathsf{T} & \qquad \qquad \qquad \qquad \ \ \mathsf{C} & \qquad \qquad \qquad \qquad \ \ \ \mathsf{A} & \qquad \qquad \qquad \qquad \ \ \mathsf{G} \\ \end{matrix} $$



Q = \begin{pmatrix} {-(\gamma\pi_C + \alpha\pi_A + \epsilon\pi_G)} & {\beta\pi_T} & {\alpha\pi_T} & {\delta\pi_T} \\ {\gamma\pi_C} & {-(\beta\pi_T + \delta\pi_A + \eta\pi_G)} & {\epsilon\pi_C} & {\eta\pi_C} \\ {\alpha\pi_A} & {\delta\pi_A} & {-(\alpha\pi_T + \epsilon\pi_C + \gamma\pi_G)} & {\beta\pi_A} \\ {\epsilon\pi_G} & {\eta\pi_G} & {\gamma\pi_G} & {-(\delta\pi_T + \eta\pi_C + \beta\pi_A)} \\ \end{pmatrix}

\ \ \mathsf{to}\ \ \begin{matrix} \mathsf{T} \\ \mathsf{C} \\ \mathsf{A} \\ \mathsf{G} \\ \end{matrix} $$

By rows | T C A G


$$ \qquad \qquad \qquad \qquad \ \mathsf{to} \qquad \quad \quad \begin{matrix} \ \ \ \ \mathsf{T} & \qquad \qquad \qquad \qquad \ \ \ \mathsf{C} & \qquad \qquad \qquad \qquad \ \ \ \mathsf{A} & \qquad \qquad \qquad \qquad \ \ \ \mathsf{G} \\ \end{matrix} $$



\mathsf{From}\ \ \begin{matrix} \mathsf{T} \\ \mathsf{C} \\ \mathsf{A} \\ \mathsf{G} \\ \end{matrix}

\qquad Q = \begin{pmatrix} {-(\gamma\pi_C + \alpha\pi_A + \epsilon\pi_G)} & {\gamma\pi_C} & {\alpha\pi_A} & {\epsilon\pi_G} \\ {\beta\pi_T} & {-(\beta\pi_T + \delta\pi_A + \eta\pi_G)} & {\delta\pi_A} & {\eta\pi_G} \\ {\alpha\pi_T} & {\epsilon\pi_C} & {-(\alpha\pi_T + \epsilon\pi_C + \gamma\pi_G)} & {\gamma\pi_G} \\ {\delta\pi_T} & {\eta\pi_C} & {\beta\pi_A} & {-(\delta\pi_T + \eta\pi_C + \beta\pi_A)} \\ \end{pmatrix}

$$

By rows | A G C T


$$ \qquad \qquad \qquad \qquad \ \mathsf{to} \qquad \quad \quad \begin{matrix} \ \ \ \ \mathsf{A} & \qquad \qquad \qquad \qquad \ \ \ \mathsf{G} & \qquad \qquad \qquad \qquad \ \ \ \mathsf{C} & \qquad \qquad \qquad \qquad \ \ \ \mathsf{T} \\ \end{matrix} $$



\mathsf{From}\ \ \begin{matrix} \mathsf{A} \\ \mathsf{G} \\ \mathsf{C} \\ \mathsf{T} \\ \end{matrix}

\qquad Q = \begin{pmatrix} {-(\gamma\pi_G + \epsilon\pi_C + \alpha\pi_T)} & {\gamma\pi_G} & {\epsilon\pi_C} & {\alpha\pi_T} \\ {\beta\pi_A} & {-(\beta\pi_A + \eta\pi_C + \delta\pi_T)} & {\eta\pi_C} & {\delta\pi_T} \\ {\delta\pi_A} & {\eta\pi_G} & {-(\delta\pi_A + \eta\pi_G + \beta\pi_T)} & {\beta\pi_T} \\ {\alpha\pi_A} & {\epsilon\pi_G} & {\gamma\pi_C} & {-(\alpha\pi_A + \epsilon\pi_G + \gamma\pi_C)} \\ \end{pmatrix}

$$