مدل‌های سیر تکاملی دی‌ان‌ای

${\displaystyle {\displaystyle P(t)={\begin{pmatrix}p_{AA}(t)&p_{GA}(t)&p_{CA}(t)&p_{TA}(t)\\p_{AG}(t)&p_{GG}(t)&p_{CG}(t)&p_{TG}(t)\\p_{AC}(t)&p_{GC}(t)&p_{CC}(t)&p_{TC}(t)\\p_{AT}(t)&p_{GT}(t)&p_{CT}(t)&p_{TT}(t)\end{pmatrix}}}}$

${\displaystyle P(t+\tau )=P(t)P(\tau )\ }$

${\displaystyle {\mathbf {P} }(t)=(p_{A}(t),\ p_{G}(t),\ p_{C}(t),\ p_{T}(t))^{T}}$

${\displaystyle {\mathcal {E}}=\{A,\ G,\ C,\ T\}}$

${\displaystyle \mu _{x}=\sum _{y\neq x}\mu _{xy}}$

${\displaystyle p_{A}(t+\Delta t)=p_{A}(t)-p_{A}(t)\mu _{A}\Delta t+\sum _{x\neq A}p_{x}(t)\mu _{xA}\Delta t}$

${\displaystyle {\mathbf {P} }(t+\Delta t)={\mathbf {P} }(t)+Q{\mathbf {P} }(t)\Delta t}$

${\displaystyle Q={\begin{pmatrix}-\mu _{A}&\mu _{GA}&\mu _{CA}&\mu _{TA}\\\mu _{AG}&-\mu _{G}&\mu _{CG}&\mu _{TG}\\\mu _{AC}&\mu _{GC}&-\mu _{C}&\mu _{TC}\\\mu _{AT}&\mu _{GT}&\mu _{CT}&-\mu _{T}\end{pmatrix}}}$

${\displaystyle {\mathbf {P} }'(t)=Q{\mathbf {P} }(t)}$

${\displaystyle P(t)=\exp(Qt)}$

${\displaystyle {\mathbf {P} }(t)=P(t){\mathbf {P} }(0)=\exp(Qt){\mathbf {P} }(0)\,.}$

${\displaystyle Q{\mathbf {\Pi } }=Q{\mathbf {P} }(t)={\frac {d{\mathbf {P} }(t)}{dt}}=0\,.}$

${\displaystyle s_{xy}=s_{yx}\ }$

${\displaystyle \beta =1/\left(-\sum _{i}\pi _{i}\mu _{ii}\right)}$

${\displaystyle Q={\begin{pmatrix}{*}&{\mu \over 4}&{\mu \over 4}&{\mu \over 4}\\{\mu \over 4}&{*}&{\mu \over 4}&{\mu \over 4}\\{\mu \over 4}&{\mu \over 4}&{*}&{\mu \over 4}\\{\mu \over 4}&{\mu \over 4}&{\mu \over 4}&{*}\end{pmatrix}}}$

${\displaystyle P={\begin{pmatrix}{{1 \over 4}+{3 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}\\\\{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}+{3 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}\\\\{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}+{3 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}\\\\{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}-{1 \over 4}e^{-t\mu }}&{{1 \over 4}+{3 \over 4}e^{-t\mu }}\end{pmatrix}}}$

${\displaystyle P_{ij}(\nu )=\left\{{\begin{array}{cc}{1 \over 4}+{3 \over 4}e^{-4\nu /3}&{\mbox{ if }}i=j\\{1 \over 4}-{1 \over 4}e^{-4\nu /3}&{\mbox{ if }}i\neq j\end{array}}\right.}$

${\displaystyle {\displaystyle \nu ={3 \over 4}t\mu =({\mu \over 4}+{\mu \over 4}+{\mu \over 4})t}}$

${\displaystyle {\hat {d}}=-{3 \over 4}\ln({1-{4 \over 3}p})={\hat {\nu }}}$

${\displaystyle Q={\begin{pmatrix}{*}&{\kappa }&{1}&{1}\\{\kappa }&{*}&{1}&{1}\\{1}&{1}&{*}&{\kappa }\\{1}&{1}&{\kappa }&{*}\end{pmatrix}}}$

${\displaystyle {\displaystyle K=-{1 \over 2}\ln((1-2p-q){\sqrt {1-2q}})}}$

${\displaystyle Q={\begin{pmatrix}{*}&{\pi _{C}}&{\pi _{A}}&{\pi _{G}}\\{\pi _{T}}&{*}&{\pi _{A}}&{\pi _{G}}\\{\pi _{T}}&{\pi _{C}}&{*}&{\pi _{G}}\\{\pi _{T}}&{\pi _{C}}&{\pi _{A}}&{*}\end{pmatrix}}}$

${\displaystyle \beta =1/(1-\pi _{A}^{2}-\pi _{C}^{2}-\pi _{G}^{2}-\pi _{T}^{2})}$

${\displaystyle P_{ij}(\nu )=\left\{{\begin{array}{cc}e^{-\beta \nu }+\pi _{j}\left(1-e^{-\beta \nu }\right)&{\mbox{ if}}i=j\\\pi _{j}\left(1-e^{-\beta \nu }\right)&{\mbox{ if}}i\neq j\end{array}}\right.}$

${\displaystyle Q={\begin{pmatrix}{*}&{\kappa \pi _{C}}&{\pi _{A}}&{\pi _{G}}\\{\kappa \pi _{T}}&{*}&{\pi _{A}}&{\pi _{G}}\\{\pi _{T}}&{\pi _{C}}&{*}&{\kappa \pi _{G}}\\{\pi _{T}}&{\pi _{C}}&{\kappa \pi _{A}}&{*}\end{pmatrix}}}$

${\displaystyle \beta ={\frac {1}{2(\pi _{A}+\pi _{G})(\pi _{C}+\pi _{T})+2\kappa [(\pi _{A}\pi _{G})+(\pi _{C}\pi _{T})]}}}$

${\displaystyle P_{AA}(\nu ,\kappa ,\pi )=\left[\pi _{A}\left(\pi _{A}+\pi _{G}+(\pi _{C}+\pi _{T})e^{-\beta \nu }\right)+\pi _{G}e^{-(1+(\pi _{A}+\pi _{G})(\kappa -1.0))\beta \nu }\right]/(\pi _{A}+\pi _{G})}$

${\displaystyle P_{AC}(\nu ,\kappa ,\pi )=\pi _{C}\left(1.0-e^{-\beta \nu }\right)}$

${\displaystyle P_{AG}(\nu ,\kappa ,\pi )=\left[\pi _{G}\left(\pi _{A}+\pi _{G}+(\pi _{C}+\pi _{T})e^{-\beta \nu }\right)-\pi _{G}e^{-(1+(\pi _{A}+\pi _{G})(\kappa -1.0))\beta \nu }\right]/\left(\pi _{A}+\pi _{G}\right)}$

${\displaystyle P_{AT}(\nu ,\kappa ,\pi )=\pi _{T}\left(1.0-e^{-\beta \nu }\right)}$

${\displaystyle \pi _{G}=\pi _{C}={\pi _{GC} \over 2}}$

${\displaystyle \pi _{A}=\pi _{T}={(1-\pi _{GC}) \over 2}}$

${\displaystyle Q={\begin{pmatrix}{*}&{\kappa (1-\pi _{GC})/2}&{(1-\pi _{GC})/2}&{(1-\pi _{GC})/2}\\{\kappa \pi _{GC}/2}&{*}&{\pi _{GC}/2}&{\pi _{GC}/2}\\{(1-\pi _{GC})/2}&{(1-\pi _{GC})/2}&{*}&{\kappa (1-\pi _{GC})/2}\\{\pi _{GC}/2}&{\pi _{GC}/2}&{\kappa \pi _{GC}/2}&{*}\end{pmatrix}}}$

${\displaystyle d=-h\ln(1-{p \over h}-q)-{1 \over 2}(1-h)\ln(1-2q)}$

${\displaystyle {\displaystyle Q={\begin{pmatrix}{*}&{\kappa _{1}\pi _{C}}&{\pi _{A}}&{\pi _{G}}\\{\kappa _{1}\pi _{T}}&{*}&{\pi _{A}}&{\pi _{G}}\\{\pi _{T}}&{\pi _{C}}&{*}&{\kappa _{2}\pi _{G}}\\{\pi _{T}}&{\pi _{C}}&{\kappa _{2}\pi _{A}}&{*}\end{pmatrix}}}}$

${\displaystyle Q={\begin{pmatrix}{-(\alpha \pi _{C}+\beta \pi _{A}+\gamma \pi _{G})}&{\alpha \pi _{C}}&{\beta \pi _{A}}&{\gamma \pi _{G}}\\{\alpha \pi _{T}}&{-(\alpha \pi _{T}+\delta \pi _{A}+\epsilon \pi _{G})}&{\delta \pi _{A}}&{\epsilon \pi _{G}}\\{\beta \pi _{T}}&{\delta \pi _{C}}&{-(\beta \pi _{T}+\delta \pi _{C}+\eta \pi _{G})}&{\eta \pi _{G}}\\{\gamma \pi _{T}}&{\epsilon \pi _{C}}&{\eta \pi _{A}}&{-(\gamma \pi _{T}+\epsilon \pi _{C}+\eta \pi _{A})}\end{pmatrix}}}$

${\displaystyle {\begin{aligned}\alpha =r(T\rightarrow C)=r(C\rightarrow T)\\\beta =r(T\rightarrow A)=r(A\rightarrow T)\\\gamma =r(T\rightarrow G)=r(G\rightarrow T)\\\delta =r(C\rightarrow A)=r(A\rightarrow C)\\\epsilon =r(C\rightarrow G)=r(G\rightarrow C)\\\eta =r(A\rightarrow G)=r(G\rightarrow A)\end{aligned}}}$

${\displaystyle {{n^{2}-n} \over 2}+n-1={1 \over 2}n^{2}+{1 \over 2}n-1.}$