id: "3-0-0-0-0-0-0-K3P"
treename: "3-leaf star tree"
modelname: "Kimura 3-parameter"
rooted: "False"
leaves: "3"
n_cycles: "0"
level: "0"
internal_edges: "0"
dim: "10"
deg: "96"
np: "16"
nq: "16"
mu: "34"
e: "4"
M: "34"
E: "4"
sD: "-"
sd: "-"
MLdeg: "-"
EDdeg: "12673"
details:
  transition_matrix: |-
    $$M_j = \begin{pmatrix}
    a_i & b_i & c_i & d_i\\
    b_i & a_i & d_i & c_i\\
    c_i & d_i & a_i & b_i\\
    d_i & c_i & b_i & a_i
    \end{pmatrix}$$
  root_distributon: |-
    $\pi = \left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right)$
  fourier_parameters: |-
    $\left(x_i, y_i, z_i, t_i\right)$
  probability_parametrization: |-
    $\begin{align}
    p_{\texttt{ATT}} &= \frac{1}{4}(a_1d_2d_3 + a_2a_3d_1 + b_1c_2c_3 + b_2b_3c_1) \\[0.5em]
    p_{\texttt{ATG}} &= \frac{1}{4}(a_1c_3d_2 + a_2b_3d_1 + a_3b_2c_1 + b_1c_2d_3) \\[0.5em]
    p_{\texttt{ATC}} &= \frac{1}{4}(a_1b_3d_2 + a_2c_3d_1 + a_3b_1c_2 + b_2c_1d_3) \\[0.5em]
    p_{\texttt{ATA}} &= \frac{1}{4}(a_1a_3d_2 + a_2d_1d_3 + b_1b_3c_2 + b_2c_1c_3) \\[0.5em]
    p_{\texttt{AGT}} &= \frac{1}{4}(a_1c_2d_3 + a_2b_3c_1 + a_3b_2d_1 + b_1c_3d_2) \\[0.5em]
    p_{\texttt{AGG}} &= \frac{1}{4}(a_1c_2c_3 + a_2a_3c_1 + b_1d_2d_3 + b_2b_3d_1) \\[0.5em]
    p_{\texttt{AGC}} &= \frac{1}{4}(a_1b_3c_2 + a_2c_1d_3 + a_3b_1d_2 + b_2c_3d_1) \\[0.5em]
    p_{\texttt{AGA}} &= \frac{1}{4}(a_1a_3c_2 + a_2c_1c_3 + b_1b_3d_2 + b_2d_1d_3) \\[0.5em]
    p_{\texttt{ACT}} &= \frac{1}{4}(a_1b_2d_3 + a_2b_1c_3 + a_3c_2d_1 + b_3c_1d_2) \\[0.5em]
    p_{\texttt{ACG}} &= \frac{1}{4}(a_1b_2c_3 + a_2b_1d_3 + a_3c_1d_2 + b_3c_2d_1) \\[0.5em]
    p_{\texttt{ACC}} &= \frac{1}{4}(a_1b_2b_3 + a_2a_3b_1 + c_1d_2d_3 + c_2c_3d_1) \\[0.5em]
    p_{\texttt{ACA}} &= \frac{1}{4}(a_1a_3b_2 + a_2b_1b_3 + c_1c_3d_2 + c_2d_1d_3) \\[0.5em]
    p_{\texttt{AAT}} &= \frac{1}{4}(a_1a_2d_3 + a_3d_1d_2 + b_1b_2c_3 + b_3c_1c_2) \\[0.5em]
    p_{\texttt{AAG}} &= \frac{1}{4}(a_1a_2c_3 + a_3c_1c_2 + b_1b_2d_3 + b_3d_1d_2) \\[0.5em]
    p_{\texttt{AAC}} &= \frac{1}{4}(a_1a_2b_3 + a_3b_1b_2 + c_1c_2d_3 + c_3d_1d_2) \\[0.5em]
    p_{\texttt{AAA}} &= \frac{1}{4}(a_1a_2a_3 + b_1b_2b_3 + c_1c_2c_3 + d_1d_2d_3)
    \end{align}$
  equivalent_classes_prob: |-
    $\begin{align}
    \text{Class } &p_{\texttt{AAA}}:\ p_{\texttt{AAA}},\ p_{\texttt{CCC}},\ p_{\texttt{GGG}},\ p_{\texttt{TTT}} \\[0.5em]
    \text{Class } &p_{\texttt{AAC}}:\ p_{\texttt{AAC}},\ p_{\texttt{CCA}},\ p_{\texttt{GGT}},\ p_{\texttt{TTG}} \\[0.5em]
    \text{Class } &p_{\texttt{AAG}}:\ p_{\texttt{AAG}},\ p_{\texttt{CCT}},\ p_{\texttt{GGA}},\ p_{\texttt{TTC}} \\[0.5em]
    \text{Class } &p_{\texttt{AAT}}:\ p_{\texttt{AAT}},\ p_{\texttt{CCG}},\ p_{\texttt{GGC}},\ p_{\texttt{TTA}} \\[0.5em]
    \text{Class } &p_{\texttt{ACA}}:\ p_{\texttt{ACA}},\ p_{\texttt{CAC}},\ p_{\texttt{GTG}},\ p_{\texttt{TGT}} \\[0.5em]
    \text{Class } &p_{\texttt{ACC}}:\ p_{\texttt{ACC}},\ p_{\texttt{CAA}},\ p_{\texttt{GTT}},\ p_{\texttt{TGG}} \\[0.5em]
    \text{Class } &p_{\texttt{ACG}}:\ p_{\texttt{ACG}},\ p_{\texttt{CAT}},\ p_{\texttt{GTA}},\ p_{\texttt{TGC}} \\[0.5em]
    \text{Class } &p_{\texttt{ACT}}:\ p_{\texttt{ACT}},\ p_{\texttt{CAG}},\ p_{\texttt{GTC}},\ p_{\texttt{TGA}} \\[0.5em]
    \text{Class } &p_{\texttt{AGA}}:\ p_{\texttt{AGA}},\ p_{\texttt{CTC}},\ p_{\texttt{GAG}},\ p_{\texttt{TCT}} \\[0.5em]
    \text{Class } &p_{\texttt{AGC}}:\ p_{\texttt{AGC}},\ p_{\texttt{CTA}},\ p_{\texttt{GAT}},\ p_{\texttt{TCG}} \\[0.5em]
    \text{Class } &p_{\texttt{AGG}}:\ p_{\texttt{AGG}},\ p_{\texttt{CTT}},\ p_{\texttt{GAA}},\ p_{\texttt{TCC}} \\[0.5em]
    \text{Class } &p_{\texttt{AGT}}:\ p_{\texttt{AGT}},\ p_{\texttt{CTG}},\ p_{\texttt{GAC}},\ p_{\texttt{TCA}} \\[0.5em]
    \text{Class } &p_{\texttt{ATA}}:\ p_{\texttt{ATA}},\ p_{\texttt{CGC}},\ p_{\texttt{GCG}},\ p_{\texttt{TAT}} \\[0.5em]
    \text{Class } &p_{\texttt{ATC}}:\ p_{\texttt{ATC}},\ p_{\texttt{CGA}},\ p_{\texttt{GCT}},\ p_{\texttt{TAG}} \\[0.5em]
    \text{Class } &p_{\texttt{ATG}}:\ p_{\texttt{ATG}},\ p_{\texttt{CGT}},\ p_{\texttt{GCA}},\ p_{\texttt{TAC}} \\[0.5em]
    \text{Class } &p_{\texttt{ATT}}:\ p_{\texttt{ATT}},\ p_{\texttt{CGG}},\ p_{\texttt{GCC}},\ p_{\texttt{TAA}}
    \end{align}$
  fourier_parametrization: |-
    $\begin{align}
    q_{\texttt{TTA}} &= x_3t_1t_2 \\[0.5em]
    q_{\texttt{TGC}} &= y_3z_2t_1 \\[0.5em]
    q_{\texttt{TCG}} &= y_2z_3t_1 \\[0.5em]
    q_{\texttt{TAT}} &= x_2t_1t_3 \\[0.5em]
    q_{\texttt{GTC}} &= y_3z_1t_2 \\[0.5em]
    q_{\texttt{GGA}} &= x_3z_1z_2 \\[0.5em]
    q_{\texttt{GCT}} &= y_2z_1t_3 \\[0.5em]
    q_{\texttt{GAG}} &= x_2z_1z_3 \\[0.5em]
    q_{\texttt{CTG}} &= y_1z_3t_2 \\[0.5em]
    q_{\texttt{CGT}} &= y_1z_2t_3 \\[0.5em]
    q_{\texttt{CCA}} &= x_3y_1y_2 \\[0.5em]
    q_{\texttt{CAC}} &= x_2y_1y_3 \\[0.5em]
    q_{\texttt{ATT}} &= x_1t_2t_3 \\[0.5em]
    q_{\texttt{AGG}} &= x_1z_2z_3 \\[0.5em]
    q_{\texttt{ACC}} &= x_1y_2y_3 \\[0.5em]
    q_{\texttt{AAA}} &= x_1x_2x_3
    \end{align}$
  equivalent_classes_fourier: |-
    $\begin{align}
    \text{Class } &0:\ q_{\texttt{AAC}}, q_{\texttt{AAG}}, q_{\texttt{AAT}}, q_{\texttt{ACA}}, q_{\texttt{ACG}}, q_{\texttt{ACT}}, q_{\texttt{AGA}}, q_{\texttt{AGC}}, q_{\texttt{AGT}}, q_{\texttt{ATA}}, q_{\texttt{ATC}}, q_{\texttt{ATG}}, q_{\texttt{CAA}}, q_{\texttt{CAG}}, q_{\texttt{CAT}}, q_{\texttt{CCC}}, q_{\texttt{CCG}}, q_{\texttt{CCT}}, q_{\texttt{CGA}}, q_{\texttt{CGC}}, q_{\texttt{CGG}}, q_{\texttt{CTA}}, q_{\texttt{CTC}}, q_{\texttt{CTT}}, q_{\texttt{GAA}}, q_{\texttt{GAC}}, q_{\texttt{GAT}}, q_{\texttt{GCA}}, q_{\texttt{GCC}}, q_{\texttt{GCG}}, q_{\texttt{GGC}}, q_{\texttt{GGG}}, q_{\texttt{GGT}}, q_{\texttt{GTA}}, q_{\texttt{GTG}}, q_{\texttt{GTT}}, q_{\texttt{TAA}}, q_{\texttt{TAC}}, q_{\texttt{TAG}}, q_{\texttt{TCA}}, q_{\texttt{TCC}}, q_{\texttt{TCT}}, q_{\texttt{TGA}}, q_{\texttt{TGG}}, q_{\texttt{TGT}}, q_{\texttt{TTC}}, q_{\texttt{TTG}}, q_{\texttt{TTT}} \\[0.5em]
    \text{Class } &q_{\texttt{AAA}}:\ q_{\texttt{AAA}} \\[0.5em]
    \text{Class } &q_{\texttt{ACC}}:\ q_{\texttt{ACC}} \\[0.5em]
    \text{Class } &q_{\texttt{AGG}}:\ q_{\texttt{AGG}} \\[0.5em]
    \text{Class } &q_{\texttt{ATT}}:\ q_{\texttt{ATT}} \\[0.5em]
    \text{Class } &q_{\texttt{CAC}}:\ q_{\texttt{CAC}} \\[0.5em]
    \text{Class } &q_{\texttt{CCA}}:\ q_{\texttt{CCA}} \\[0.5em]
    \text{Class } &q_{\texttt{CGT}}:\ q_{\texttt{CGT}} \\[0.5em]
    \text{Class } &q_{\texttt{CTG}}:\ q_{\texttt{CTG}} \\[0.5em]
    \text{Class } &q_{\texttt{GAG}}:\ q_{\texttt{GAG}} \\[0.5em]
    \text{Class } &q_{\texttt{GCT}}:\ q_{\texttt{GCT}} \\[0.5em]
    \text{Class } &q_{\texttt{GGA}}:\ q_{\texttt{GGA}} \\[0.5em]
    \text{Class } &q_{\texttt{GTC}}:\ q_{\texttt{GTC}} \\[0.5em]
    \text{Class } &q_{\texttt{TAT}}:\ q_{\texttt{TAT}} \\[0.5em]
    \text{Class } &q_{\texttt{TCG}}:\ q_{\texttt{TCG}} \\[0.5em]
    \text{Class } &q_{\texttt{TGC}}:\ q_{\texttt{TGC}} \\[0.5em]
    \text{Class } &q_{\texttt{TTA}}:\ q_{\texttt{TTA}}
    \end{align}$
  invariants: |-
    Ideal generated by  $$ \begin{align}
    q_{\texttt{TTA}}q_{\texttt{GAG}}q_{\texttt{ACC}} - q_{\texttt{TCG}}q_{\texttt{GTC}}q_{\texttt{AAA}} \\
    -q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{GGA}}q_{\texttt{ACC}} \\
    q_{\texttt{GCT}}q_{\texttt{CAC}}q_{\texttt{AGG}} - q_{\texttt{GAG}}q_{\texttt{CGT}}q_{\texttt{ACC}} \\
    q_{\texttt{TTA}}q_{\texttt{CAC}}q_{\texttt{AGG}} - q_{\texttt{TGC}}q_{\texttt{CTG}}q_{\texttt{AAA}} \\
    q_{\texttt{GTC}}q_{\texttt{CCA}}q_{\texttt{AGG}} - q_{\texttt{GGA}}q_{\texttt{CTG}}q_{\texttt{ACC}} \\
    -q_{\texttt{TCG}}q_{\texttt{CGT}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{CCA}}q_{\texttt{AGG}} \\
    -q_{\texttt{GTC}}q_{\texttt{CGT}}q_{\texttt{AAA}} + q_{\texttt{GGA}}q_{\texttt{CAC}}q_{\texttt{ATT}} \\
    q_{\texttt{TCG}}q_{\texttt{CAC}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{CTG}}q_{\texttt{ACC}} \\
    -q_{\texttt{GCT}}q_{\texttt{CTG}}q_{\texttt{AAA}} + q_{\texttt{GAG}}q_{\texttt{CCA}}q_{\texttt{ATT}} \\
    -q_{\texttt{TTA}}q_{\texttt{CGT}}q_{\texttt{ACC}} + q_{\texttt{TGC}}q_{\texttt{CCA}}q_{\texttt{ATT}} \\
    q_{\texttt{TGC}}q_{\texttt{GAG}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{AGG}} \\
    -q_{\texttt{TTA}}q_{\texttt{GCT}}q_{\texttt{AGG}} + q_{\texttt{TCG}}q_{\texttt{GGA}}q_{\texttt{ATT}} \\
    q_{\texttt{TGC}}q_{\texttt{GAG}}q_{\texttt{CCA}} - q_{\texttt{TCG}}q_{\texttt{GGA}}q_{\texttt{CAC}} \\
    -q_{\texttt{TTA}}q_{\texttt{GCT}}q_{\texttt{CAC}} + q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{CCA}} \\
    q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{CTG}} - q_{\texttt{TCG}}q_{\texttt{GTC}}q_{\texttt{CGT}} \\
    -q_{\texttt{TTA}}q_{\texttt{GAG}}q_{\texttt{CGT}} + q_{\texttt{TAT}}q_{\texttt{GGA}}q_{\texttt{CTG}} \\
    q_{\texttt{TTA}}q_{\texttt{TAT}}q_{\texttt{AGG}}q_{\texttt{ACC}} - q_{\texttt{TGC}}q_{\texttt{TCG}}q_{\texttt{ATT}}q_{\texttt{AAA}} \\
    -q_{\texttt{GTC}}q_{\texttt{GCT}}q_{\texttt{AGG}}q_{\texttt{AAA}} + q_{\texttt{GGA}}q_{\texttt{GAG}}q_{\texttt{ATT}}q_{\texttt{ACC}} \\
    q_{\texttt{TTA}}q_{\texttt{GGA}}q_{\texttt{CAC}}q_{\texttt{ACC}} - q_{\texttt{TGC}}q_{\texttt{GTC}}q_{\texttt{CCA}}q_{\texttt{AAA}} \\
    -q_{\texttt{TCG}}q_{\texttt{GCT}}q_{\texttt{CAC}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{GAG}}q_{\texttt{CCA}}q_{\texttt{ACC}} \\
    -q_{\texttt{CTG}}q_{\texttt{CGT}}q_{\texttt{ACC}}q_{\texttt{AAA}} + q_{\texttt{CCA}}q_{\texttt{CAC}}q_{\texttt{ATT}}q_{\texttt{AGG}} \\
    -q_{\texttt{TGC}}q_{\texttt{GAG}}q_{\texttt{CGT}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{GGA}}q_{\texttt{CAC}}q_{\texttt{AGG}} \\
    -q_{\texttt{TGC}}q_{\texttt{GAG}}q_{\texttt{CTG}}q_{\texttt{ACC}} + q_{\texttt{TCG}}q_{\texttt{GTC}}q_{\texttt{CAC}}q_{\texttt{AGG}} \\
    q_{\texttt{TTA}}q_{\texttt{GAG}}q_{\texttt{CCA}}q_{\texttt{AGG}} - q_{\texttt{TCG}}q_{\texttt{GGA}}q_{\texttt{CTG}}q_{\texttt{AAA}} \\
    q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{CCA}}q_{\texttt{AGG}} - q_{\texttt{TCG}}q_{\texttt{GGA}}q_{\texttt{CGT}}q_{\texttt{ACC}} \\
    q_{\texttt{TTA}}q_{\texttt{GAG}}q_{\texttt{CAC}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{CTG}}q_{\texttt{AAA}} \\
    q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{CAC}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{CGT}}q_{\texttt{ACC}} \\
    -q_{\texttt{TTA}}q_{\texttt{GCT}}q_{\texttt{CGT}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{GGA}}q_{\texttt{CCA}}q_{\texttt{ATT}} \\
    -q_{\texttt{TTA}}q_{\texttt{GCT}}q_{\texttt{CTG}}q_{\texttt{ACC}} + q_{\texttt{TCG}}q_{\texttt{GTC}}q_{\texttt{CCA}}q_{\texttt{ATT}} \\
    q_{\texttt{TCG}}q_{\texttt{GAG}}q_{\texttt{CGT}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{GCT}}q_{\texttt{CTG}}q_{\texttt{AGG}} \\
    -q_{\texttt{TTA}}q_{\texttt{GTC}}q_{\texttt{CGT}}q_{\texttt{AGG}} + q_{\texttt{TGC}}q_{\texttt{GGA}}q_{\texttt{CTG}}q_{\texttt{ATT}} \\
    q_{\texttt{GTC}}q_{\texttt{GAG}}q_{\texttt{CGT}}q_{\texttt{CCA}} - q_{\texttt{GGA}}q_{\texttt{GCT}}q_{\texttt{CTG}}q_{\texttt{CAC}} \\
    -q_{\texttt{TTA}}q_{\texttt{TCG}}q_{\texttt{CGT}}q_{\texttt{CAC}} + q_{\texttt{TGC}}q_{\texttt{TAT}}q_{\texttt{CTG}}q_{\texttt{CCA}} \\
    -q_{\texttt{TTA}}q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{GAG}} + q_{\texttt{TCG}}q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{GGA}}
    \end{align} $$
  grobner_basis: |-
    Gröbner basis with respect to the ordering
    $ \texttt{degrevlex}\ [ q_{\texttt{TTA}}, q_{\texttt{TGC}}, q_{\texttt{TCG}}, q_{\texttt{TAT}}, q_{\texttt{GTC}}, q_{\texttt{GGA}}, q_{\texttt{GCT}}, q_{\texttt{GAG}}, q_{\texttt{CTG}}, q_{\texttt{CGT}}, q_{\texttt{CCA}}, q_{\texttt{CAC}}, q_{\texttt{ATT}}, q_{\texttt{AGG}}, q_{\texttt{ACC}}, q_{\texttt{AAA}} ] $
    and with elements  $$ \begin{align}
    q_{\texttt{TTA}}q_{\texttt{GAG}}q_{\texttt{ACC}} - q_{\texttt{TCG}}q_{\texttt{GTC}}q_{\texttt{AAA}} \\
    -q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{GGA}}q_{\texttt{ACC}} \\
    q_{\texttt{GCT}}q_{\texttt{CAC}}q_{\texttt{AGG}} - q_{\texttt{GAG}}q_{\texttt{CGT}}q_{\texttt{ACC}} \\
    q_{\texttt{TTA}}q_{\texttt{CAC}}q_{\texttt{AGG}} - q_{\texttt{TGC}}q_{\texttt{CTG}}q_{\texttt{AAA}} \\
    q_{\texttt{GTC}}q_{\texttt{CCA}}q_{\texttt{AGG}} - q_{\texttt{GGA}}q_{\texttt{CTG}}q_{\texttt{ACC}} \\
    -q_{\texttt{TCG}}q_{\texttt{CGT}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{CCA}}q_{\texttt{AGG}} \\
    -q_{\texttt{GTC}}q_{\texttt{CGT}}q_{\texttt{AAA}} + q_{\texttt{GGA}}q_{\texttt{CAC}}q_{\texttt{ATT}} \\
    q_{\texttt{TCG}}q_{\texttt{CAC}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{CTG}}q_{\texttt{ACC}} \\
    -q_{\texttt{GCT}}q_{\texttt{CTG}}q_{\texttt{AAA}} + q_{\texttt{GAG}}q_{\texttt{CCA}}q_{\texttt{ATT}} \\
    -q_{\texttt{TTA}}q_{\texttt{CGT}}q_{\texttt{ACC}} + q_{\texttt{TGC}}q_{\texttt{CCA}}q_{\texttt{ATT}} \\
    q_{\texttt{TGC}}q_{\texttt{GAG}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{AGG}} \\
    -q_{\texttt{TTA}}q_{\texttt{GCT}}q_{\texttt{AGG}} + q_{\texttt{TCG}}q_{\texttt{GGA}}q_{\texttt{ATT}} \\
    q_{\texttt{TGC}}q_{\texttt{GAG}}q_{\texttt{CCA}} - q_{\texttt{TCG}}q_{\texttt{GGA}}q_{\texttt{CAC}} \\
    -q_{\texttt{TTA}}q_{\texttt{GCT}}q_{\texttt{CAC}} + q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{CCA}} \\
    q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{CTG}} - q_{\texttt{TCG}}q_{\texttt{GTC}}q_{\texttt{CGT}} \\
    -q_{\texttt{TTA}}q_{\texttt{GAG}}q_{\texttt{CGT}} + q_{\texttt{TAT}}q_{\texttt{GGA}}q_{\texttt{CTG}} \\
    q_{\texttt{TTA}}q_{\texttt{TAT}}q_{\texttt{AGG}}q_{\texttt{ACC}} - q_{\texttt{TGC}}q_{\texttt{TCG}}q_{\texttt{ATT}}q_{\texttt{AAA}} \\
    -q_{\texttt{GTC}}q_{\texttt{GCT}}q_{\texttt{AGG}}q_{\texttt{AAA}} + q_{\texttt{GGA}}q_{\texttt{GAG}}q_{\texttt{ATT}}q_{\texttt{ACC}} \\
    q_{\texttt{TTA}}q_{\texttt{GGA}}q_{\texttt{CAC}}q_{\texttt{ACC}} - q_{\texttt{TGC}}q_{\texttt{GTC}}q_{\texttt{CCA}}q_{\texttt{AAA}} \\
    -q_{\texttt{TCG}}q_{\texttt{GCT}}q_{\texttt{CAC}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{GAG}}q_{\texttt{CCA}}q_{\texttt{ACC}} \\
    -q_{\texttt{CTG}}q_{\texttt{CGT}}q_{\texttt{ACC}}q_{\texttt{AAA}} + q_{\texttt{CCA}}q_{\texttt{CAC}}q_{\texttt{ATT}}q_{\texttt{AGG}} \\
    -q_{\texttt{TGC}}q_{\texttt{GAG}}q_{\texttt{CGT}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{GGA}}q_{\texttt{CAC}}q_{\texttt{AGG}} \\
    -q_{\texttt{TGC}}q_{\texttt{GAG}}q_{\texttt{CTG}}q_{\texttt{ACC}} + q_{\texttt{TCG}}q_{\texttt{GTC}}q_{\texttt{CAC}}q_{\texttt{AGG}} \\
    q_{\texttt{TTA}}q_{\texttt{GAG}}q_{\texttt{CCA}}q_{\texttt{AGG}} - q_{\texttt{TCG}}q_{\texttt{GGA}}q_{\texttt{CTG}}q_{\texttt{AAA}} \\
    q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{CCA}}q_{\texttt{AGG}} - q_{\texttt{TCG}}q_{\texttt{GGA}}q_{\texttt{CGT}}q_{\texttt{ACC}} \\
    q_{\texttt{TTA}}q_{\texttt{GAG}}q_{\texttt{CAC}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{CTG}}q_{\texttt{AAA}} \\
    q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{CAC}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{CGT}}q_{\texttt{ACC}} \\
    -q_{\texttt{TTA}}q_{\texttt{GCT}}q_{\texttt{CGT}}q_{\texttt{AAA}} + q_{\texttt{TAT}}q_{\texttt{GGA}}q_{\texttt{CCA}}q_{\texttt{ATT}} \\
    -q_{\texttt{TTA}}q_{\texttt{GCT}}q_{\texttt{CTG}}q_{\texttt{ACC}} + q_{\texttt{TCG}}q_{\texttt{GTC}}q_{\texttt{CCA}}q_{\texttt{ATT}} \\
    q_{\texttt{TCG}}q_{\texttt{GAG}}q_{\texttt{CGT}}q_{\texttt{ATT}} - q_{\texttt{TAT}}q_{\texttt{GCT}}q_{\texttt{CTG}}q_{\texttt{AGG}} \\
    -q_{\texttt{TTA}}q_{\texttt{GTC}}q_{\texttt{CGT}}q_{\texttt{AGG}} + q_{\texttt{TGC}}q_{\texttt{GGA}}q_{\texttt{CTG}}q_{\texttt{ATT}} \\
    q_{\texttt{GTC}}q_{\texttt{GAG}}q_{\texttt{CGT}}q_{\texttt{CCA}} - q_{\texttt{GGA}}q_{\texttt{GCT}}q_{\texttt{CTG}}q_{\texttt{CAC}} \\
    -q_{\texttt{TTA}}q_{\texttt{TCG}}q_{\texttt{CGT}}q_{\texttt{CAC}} + q_{\texttt{TGC}}q_{\texttt{TAT}}q_{\texttt{CTG}}q_{\texttt{CCA}} \\
    -q_{\texttt{TTA}}q_{\texttt{TGC}}q_{\texttt{GCT}}q_{\texttt{GAG}} + q_{\texttt{TCG}}q_{\texttt{TAT}}q_{\texttt{GTC}}q_{\texttt{GGA}}
    \end{align} $$
  coordinate_change: |-
    $\begin{align}
    q_{\texttt{TTA}} &= p_{\texttt{ATT}} + p_{\texttt{ATG}} + p_{\texttt{ATC}} + p_{\texttt{ATA}} - p_{\texttt{AGT}} - p_{\texttt{AGG}} - p_{\texttt{AGC}} - p_{\texttt{AGA}} - p_{\texttt{ACT}} - p_{\texttt{ACG}} - p_{\texttt{ACC}} - p_{\texttt{ACA}} + p_{\texttt{AAT}} + p_{\texttt{AAG}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{TGC}} &= p_{\texttt{ATT}} - p_{\texttt{ATG}} + p_{\texttt{ATC}} - p_{\texttt{ATA}} + p_{\texttt{AGT}} - p_{\texttt{AGG}} + p_{\texttt{AGC}} - p_{\texttt{AGA}} - p_{\texttt{ACT}} + p_{\texttt{ACG}} - p_{\texttt{ACC}} + p_{\texttt{ACA}} - p_{\texttt{AAT}} + p_{\texttt{AAG}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{TCG}} &= p_{\texttt{ATT}} + p_{\texttt{ATG}} - p_{\texttt{ATC}} - p_{\texttt{ATA}} - p_{\texttt{AGT}} - p_{\texttt{AGG}} + p_{\texttt{AGC}} + p_{\texttt{AGA}} + p_{\texttt{ACT}} + p_{\texttt{ACG}} - p_{\texttt{ACC}} - p_{\texttt{ACA}} - p_{\texttt{AAT}} - p_{\texttt{AAG}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{TAT}} &= p_{\texttt{ATT}} - p_{\texttt{ATG}} - p_{\texttt{ATC}} + p_{\texttt{ATA}} + p_{\texttt{AGT}} - p_{\texttt{AGG}} - p_{\texttt{AGC}} + p_{\texttt{AGA}} + p_{\texttt{ACT}} - p_{\texttt{ACG}} - p_{\texttt{ACC}} + p_{\texttt{ACA}} + p_{\texttt{AAT}} - p_{\texttt{AAG}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{GTC}} &= -p_{\texttt{ATT}} + p_{\texttt{ATG}} - p_{\texttt{ATC}} + p_{\texttt{ATA}} + p_{\texttt{AGT}} - p_{\texttt{AGG}} + p_{\texttt{AGC}} - p_{\texttt{AGA}} + p_{\texttt{ACT}} - p_{\texttt{ACG}} + p_{\texttt{ACC}} - p_{\texttt{ACA}} - p_{\texttt{AAT}} + p_{\texttt{AAG}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{GGA}} &= -p_{\texttt{ATT}} - p_{\texttt{ATG}} - p_{\texttt{ATC}} - p_{\texttt{ATA}} - p_{\texttt{AGT}} - p_{\texttt{AGG}} - p_{\texttt{AGC}} - p_{\texttt{AGA}} + p_{\texttt{ACT}} + p_{\texttt{ACG}} + p_{\texttt{ACC}} + p_{\texttt{ACA}} + p_{\texttt{AAT}} + p_{\texttt{AAG}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{GCT}} &= -p_{\texttt{ATT}} + p_{\texttt{ATG}} + p_{\texttt{ATC}} - p_{\texttt{ATA}} + p_{\texttt{AGT}} - p_{\texttt{AGG}} - p_{\texttt{AGC}} + p_{\texttt{AGA}} - p_{\texttt{ACT}} + p_{\texttt{ACG}} + p_{\texttt{ACC}} - p_{\texttt{ACA}} + p_{\texttt{AAT}} - p_{\texttt{AAG}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{GAG}} &= -p_{\texttt{ATT}} - p_{\texttt{ATG}} + p_{\texttt{ATC}} + p_{\texttt{ATA}} - p_{\texttt{AGT}} - p_{\texttt{AGG}} + p_{\texttt{AGC}} + p_{\texttt{AGA}} - p_{\texttt{ACT}} - p_{\texttt{ACG}} + p_{\texttt{ACC}} + p_{\texttt{ACA}} - p_{\texttt{AAT}} - p_{\texttt{AAG}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{CTG}} &= -p_{\texttt{ATT}} - p_{\texttt{ATG}} + p_{\texttt{ATC}} + p_{\texttt{ATA}} + p_{\texttt{AGT}} + p_{\texttt{AGG}} - p_{\texttt{AGC}} - p_{\texttt{AGA}} + p_{\texttt{ACT}} + p_{\texttt{ACG}} - p_{\texttt{ACC}} - p_{\texttt{ACA}} - p_{\texttt{AAT}} - p_{\texttt{AAG}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{CGT}} &= -p_{\texttt{ATT}} + p_{\texttt{ATG}} + p_{\texttt{ATC}} - p_{\texttt{ATA}} - p_{\texttt{AGT}} + p_{\texttt{AGG}} + p_{\texttt{AGC}} - p_{\texttt{AGA}} + p_{\texttt{ACT}} - p_{\texttt{ACG}} - p_{\texttt{ACC}} + p_{\texttt{ACA}} + p_{\texttt{AAT}} - p_{\texttt{AAG}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{CCA}} &= -p_{\texttt{ATT}} - p_{\texttt{ATG}} - p_{\texttt{ATC}} - p_{\texttt{ATA}} + p_{\texttt{AGT}} + p_{\texttt{AGG}} + p_{\texttt{AGC}} + p_{\texttt{AGA}} - p_{\texttt{ACT}} - p_{\texttt{ACG}} - p_{\texttt{ACC}} - p_{\texttt{ACA}} + p_{\texttt{AAT}} + p_{\texttt{AAG}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{CAC}} &= -p_{\texttt{ATT}} + p_{\texttt{ATG}} - p_{\texttt{ATC}} + p_{\texttt{ATA}} - p_{\texttt{AGT}} + p_{\texttt{AGG}} - p_{\texttt{AGC}} + p_{\texttt{AGA}} - p_{\texttt{ACT}} + p_{\texttt{ACG}} - p_{\texttt{ACC}} + p_{\texttt{ACA}} - p_{\texttt{AAT}} + p_{\texttt{AAG}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{ATT}} &= p_{\texttt{ATT}} - p_{\texttt{ATG}} - p_{\texttt{ATC}} + p_{\texttt{ATA}} - p_{\texttt{AGT}} + p_{\texttt{AGG}} + p_{\texttt{AGC}} - p_{\texttt{AGA}} - p_{\texttt{ACT}} + p_{\texttt{ACG}} + p_{\texttt{ACC}} - p_{\texttt{ACA}} + p_{\texttt{AAT}} - p_{\texttt{AAG}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{AGG}} &= p_{\texttt{ATT}} + p_{\texttt{ATG}} - p_{\texttt{ATC}} - p_{\texttt{ATA}} + p_{\texttt{AGT}} + p_{\texttt{AGG}} - p_{\texttt{AGC}} - p_{\texttt{AGA}} - p_{\texttt{ACT}} - p_{\texttt{ACG}} + p_{\texttt{ACC}} + p_{\texttt{ACA}} - p_{\texttt{AAT}} - p_{\texttt{AAG}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{ACC}} &= p_{\texttt{ATT}} - p_{\texttt{ATG}} + p_{\texttt{ATC}} - p_{\texttt{ATA}} - p_{\texttt{AGT}} + p_{\texttt{AGG}} - p_{\texttt{AGC}} + p_{\texttt{AGA}} + p_{\texttt{ACT}} - p_{\texttt{ACG}} + p_{\texttt{ACC}} - p_{\texttt{ACA}} - p_{\texttt{AAT}} + p_{\texttt{AAG}} - p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em]
    q_{\texttt{AAA}} &= p_{\texttt{ATT}} + p_{\texttt{ATG}} + p_{\texttt{ATC}} + p_{\texttt{ATA}} + p_{\texttt{AGT}} + p_{\texttt{AGG}} + p_{\texttt{AGC}} + p_{\texttt{AGA}} + p_{\texttt{ACT}} + p_{\texttt{ACG}} + p_{\texttt{ACC}} + p_{\texttt{ACA}} + p_{\texttt{AAT}} + p_{\texttt{AAG}} + p_{\texttt{AAC}} + p_{\texttt{AAA}}
    \end{align}$
  inverse_coordinate_change: |-
    $\begin{align}
    p_{\texttt{ATT}} &= \frac{1}{16}q_{\texttt{TTA}} + \frac{1}{16}q_{\texttt{TGC}} + \frac{1}{16}q_{\texttt{TCG}} + \frac{1}{16}q_{\texttt{TAT}} - \frac{1}{16}q_{\texttt{GTC}} - \frac{1}{16}q_{\texttt{GGA}} - \frac{1}{16}q_{\texttt{GCT}} - \frac{1}{16}q_{\texttt{GAG}} - \frac{1}{16}q_{\texttt{CTG}} - \frac{1}{16}q_{\texttt{CGT}} - \frac{1}{16}q_{\texttt{CCA}} - \frac{1}{16}q_{\texttt{CAC}} + \frac{1}{16}q_{\texttt{ATT}} + \frac{1}{16}q_{\texttt{AGG}} + \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{ATG}} &= \frac{1}{16}q_{\texttt{TTA}} - \frac{1}{16}q_{\texttt{TGC}} + \frac{1}{16}q_{\texttt{TCG}} - \frac{1}{16}q_{\texttt{TAT}} + \frac{1}{16}q_{\texttt{GTC}} - \frac{1}{16}q_{\texttt{GGA}} + \frac{1}{16}q_{\texttt{GCT}} - \frac{1}{16}q_{\texttt{GAG}} - \frac{1}{16}q_{\texttt{CTG}} + \frac{1}{16}q_{\texttt{CGT}} - \frac{1}{16}q_{\texttt{CCA}} + \frac{1}{16}q_{\texttt{CAC}} - \frac{1}{16}q_{\texttt{ATT}} + \frac{1}{16}q_{\texttt{AGG}} - \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{ATC}} &= \frac{1}{16}q_{\texttt{TTA}} + \frac{1}{16}q_{\texttt{TGC}} - \frac{1}{16}q_{\texttt{TCG}} - \frac{1}{16}q_{\texttt{TAT}} - \frac{1}{16}q_{\texttt{GTC}} - \frac{1}{16}q_{\texttt{GGA}} + \frac{1}{16}q_{\texttt{GCT}} + \frac{1}{16}q_{\texttt{GAG}} + \frac{1}{16}q_{\texttt{CTG}} + \frac{1}{16}q_{\texttt{CGT}} - \frac{1}{16}q_{\texttt{CCA}} - \frac{1}{16}q_{\texttt{CAC}} - \frac{1}{16}q_{\texttt{ATT}} - \frac{1}{16}q_{\texttt{AGG}} + \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{ATA}} &= \frac{1}{16}q_{\texttt{TTA}} - \frac{1}{16}q_{\texttt{TGC}} - \frac{1}{16}q_{\texttt{TCG}} + \frac{1}{16}q_{\texttt{TAT}} + \frac{1}{16}q_{\texttt{GTC}} - \frac{1}{16}q_{\texttt{GGA}} - \frac{1}{16}q_{\texttt{GCT}} + \frac{1}{16}q_{\texttt{GAG}} + \frac{1}{16}q_{\texttt{CTG}} - \frac{1}{16}q_{\texttt{CGT}} - \frac{1}{16}q_{\texttt{CCA}} + \frac{1}{16}q_{\texttt{CAC}} + \frac{1}{16}q_{\texttt{ATT}} - \frac{1}{16}q_{\texttt{AGG}} - \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AGT}} &= -\frac{1}{16}q_{\texttt{TTA}} + \frac{1}{16}q_{\texttt{TGC}} - \frac{1}{16}q_{\texttt{TCG}} + \frac{1}{16}q_{\texttt{TAT}} + \frac{1}{16}q_{\texttt{GTC}} - \frac{1}{16}q_{\texttt{GGA}} + \frac{1}{16}q_{\texttt{GCT}} - \frac{1}{16}q_{\texttt{GAG}} + \frac{1}{16}q_{\texttt{CTG}} - \frac{1}{16}q_{\texttt{CGT}} + \frac{1}{16}q_{\texttt{CCA}} - \frac{1}{16}q_{\texttt{CAC}} - \frac{1}{16}q_{\texttt{ATT}} + \frac{1}{16}q_{\texttt{AGG}} - \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AGG}} &= -\frac{1}{16}q_{\texttt{TTA}} - \frac{1}{16}q_{\texttt{TGC}} - \frac{1}{16}q_{\texttt{TCG}} - \frac{1}{16}q_{\texttt{TAT}} - \frac{1}{16}q_{\texttt{GTC}} - \frac{1}{16}q_{\texttt{GGA}} - \frac{1}{16}q_{\texttt{GCT}} - \frac{1}{16}q_{\texttt{GAG}} + \frac{1}{16}q_{\texttt{CTG}} + \frac{1}{16}q_{\texttt{CGT}} + \frac{1}{16}q_{\texttt{CCA}} + \frac{1}{16}q_{\texttt{CAC}} + \frac{1}{16}q_{\texttt{ATT}} + \frac{1}{16}q_{\texttt{AGG}} + \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AGC}} &= -\frac{1}{16}q_{\texttt{TTA}} + \frac{1}{16}q_{\texttt{TGC}} + \frac{1}{16}q_{\texttt{TCG}} - \frac{1}{16}q_{\texttt{TAT}} + \frac{1}{16}q_{\texttt{GTC}} - \frac{1}{16}q_{\texttt{GGA}} - \frac{1}{16}q_{\texttt{GCT}} + \frac{1}{16}q_{\texttt{GAG}} - \frac{1}{16}q_{\texttt{CTG}} + \frac{1}{16}q_{\texttt{CGT}} + \frac{1}{16}q_{\texttt{CCA}} - \frac{1}{16}q_{\texttt{CAC}} + \frac{1}{16}q_{\texttt{ATT}} - \frac{1}{16}q_{\texttt{AGG}} - \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AGA}} &= -\frac{1}{16}q_{\texttt{TTA}} - \frac{1}{16}q_{\texttt{TGC}} + \frac{1}{16}q_{\texttt{TCG}} + \frac{1}{16}q_{\texttt{TAT}} - \frac{1}{16}q_{\texttt{GTC}} - \frac{1}{16}q_{\texttt{GGA}} + \frac{1}{16}q_{\texttt{GCT}} + \frac{1}{16}q_{\texttt{GAG}} - \frac{1}{16}q_{\texttt{CTG}} - \frac{1}{16}q_{\texttt{CGT}} + \frac{1}{16}q_{\texttt{CCA}} + \frac{1}{16}q_{\texttt{CAC}} - \frac{1}{16}q_{\texttt{ATT}} - \frac{1}{16}q_{\texttt{AGG}} + \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{ACT}} &= -\frac{1}{16}q_{\texttt{TTA}} - \frac{1}{16}q_{\texttt{TGC}} + \frac{1}{16}q_{\texttt{TCG}} + \frac{1}{16}q_{\texttt{TAT}} + \frac{1}{16}q_{\texttt{GTC}} + \frac{1}{16}q_{\texttt{GGA}} - \frac{1}{16}q_{\texttt{GCT}} - \frac{1}{16}q_{\texttt{GAG}} + \frac{1}{16}q_{\texttt{CTG}} + \frac{1}{16}q_{\texttt{CGT}} - \frac{1}{16}q_{\texttt{CCA}} - \frac{1}{16}q_{\texttt{CAC}} - \frac{1}{16}q_{\texttt{ATT}} - \frac{1}{16}q_{\texttt{AGG}} + \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{ACG}} &= -\frac{1}{16}q_{\texttt{TTA}} + \frac{1}{16}q_{\texttt{TGC}} + \frac{1}{16}q_{\texttt{TCG}} - \frac{1}{16}q_{\texttt{TAT}} - \frac{1}{16}q_{\texttt{GTC}} + \frac{1}{16}q_{\texttt{GGA}} + \frac{1}{16}q_{\texttt{GCT}} - \frac{1}{16}q_{\texttt{GAG}} + \frac{1}{16}q_{\texttt{CTG}} - \frac{1}{16}q_{\texttt{CGT}} - \frac{1}{16}q_{\texttt{CCA}} + \frac{1}{16}q_{\texttt{CAC}} + \frac{1}{16}q_{\texttt{ATT}} - \frac{1}{16}q_{\texttt{AGG}} - \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{ACC}} &= -\frac{1}{16}q_{\texttt{TTA}} - \frac{1}{16}q_{\texttt{TGC}} - \frac{1}{16}q_{\texttt{TCG}} - \frac{1}{16}q_{\texttt{TAT}} + \frac{1}{16}q_{\texttt{GTC}} + \frac{1}{16}q_{\texttt{GGA}} + \frac{1}{16}q_{\texttt{GCT}} + \frac{1}{16}q_{\texttt{GAG}} - \frac{1}{16}q_{\texttt{CTG}} - \frac{1}{16}q_{\texttt{CGT}} - \frac{1}{16}q_{\texttt{CCA}} - \frac{1}{16}q_{\texttt{CAC}} + \frac{1}{16}q_{\texttt{ATT}} + \frac{1}{16}q_{\texttt{AGG}} + \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{ACA}} &= -\frac{1}{16}q_{\texttt{TTA}} + \frac{1}{16}q_{\texttt{TGC}} - \frac{1}{16}q_{\texttt{TCG}} + \frac{1}{16}q_{\texttt{TAT}} - \frac{1}{16}q_{\texttt{GTC}} + \frac{1}{16}q_{\texttt{GGA}} - \frac{1}{16}q_{\texttt{GCT}} + \frac{1}{16}q_{\texttt{GAG}} - \frac{1}{16}q_{\texttt{CTG}} + \frac{1}{16}q_{\texttt{CGT}} - \frac{1}{16}q_{\texttt{CCA}} + \frac{1}{16}q_{\texttt{CAC}} - \frac{1}{16}q_{\texttt{ATT}} + \frac{1}{16}q_{\texttt{AGG}} - \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AAT}} &= \frac{1}{16}q_{\texttt{TTA}} - \frac{1}{16}q_{\texttt{TGC}} - \frac{1}{16}q_{\texttt{TCG}} + \frac{1}{16}q_{\texttt{TAT}} - \frac{1}{16}q_{\texttt{GTC}} + \frac{1}{16}q_{\texttt{GGA}} + \frac{1}{16}q_{\texttt{GCT}} - \frac{1}{16}q_{\texttt{GAG}} - \frac{1}{16}q_{\texttt{CTG}} + \frac{1}{16}q_{\texttt{CGT}} + \frac{1}{16}q_{\texttt{CCA}} - \frac{1}{16}q_{\texttt{CAC}} + \frac{1}{16}q_{\texttt{ATT}} - \frac{1}{16}q_{\texttt{AGG}} - \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AAG}} &= \frac{1}{16}q_{\texttt{TTA}} + \frac{1}{16}q_{\texttt{TGC}} - \frac{1}{16}q_{\texttt{TCG}} - \frac{1}{16}q_{\texttt{TAT}} + \frac{1}{16}q_{\texttt{GTC}} + \frac{1}{16}q_{\texttt{GGA}} - \frac{1}{16}q_{\texttt{GCT}} - \frac{1}{16}q_{\texttt{GAG}} - \frac{1}{16}q_{\texttt{CTG}} - \frac{1}{16}q_{\texttt{CGT}} + \frac{1}{16}q_{\texttt{CCA}} + \frac{1}{16}q_{\texttt{CAC}} - \frac{1}{16}q_{\texttt{ATT}} - \frac{1}{16}q_{\texttt{AGG}} + \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AAC}} &= \frac{1}{16}q_{\texttt{TTA}} - \frac{1}{16}q_{\texttt{TGC}} + \frac{1}{16}q_{\texttt{TCG}} - \frac{1}{16}q_{\texttt{TAT}} - \frac{1}{16}q_{\texttt{GTC}} + \frac{1}{16}q_{\texttt{GGA}} - \frac{1}{16}q_{\texttt{GCT}} + \frac{1}{16}q_{\texttt{GAG}} + \frac{1}{16}q_{\texttt{CTG}} - \frac{1}{16}q_{\texttt{CGT}} + \frac{1}{16}q_{\texttt{CCA}} - \frac{1}{16}q_{\texttt{CAC}} - \frac{1}{16}q_{\texttt{ATT}} + \frac{1}{16}q_{\texttt{AGG}} - \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \\[0.5em]
    p_{\texttt{AAA}} &= \frac{1}{16}q_{\texttt{TTA}} + \frac{1}{16}q_{\texttt{TGC}} + \frac{1}{16}q_{\texttt{TCG}} + \frac{1}{16}q_{\texttt{TAT}} + \frac{1}{16}q_{\texttt{GTC}} + \frac{1}{16}q_{\texttt{GGA}} + \frac{1}{16}q_{\texttt{GCT}} + \frac{1}{16}q_{\texttt{GAG}} + \frac{1}{16}q_{\texttt{CTG}} + \frac{1}{16}q_{\texttt{CGT}} + \frac{1}{16}q_{\texttt{CCA}} + \frac{1}{16}q_{\texttt{CAC}} + \frac{1}{16}q_{\texttt{ATT}} + \frac{1}{16}q_{\texttt{AGG}} + \frac{1}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}}
    \end{align}$
