Tree 3-0-0-0-0-0-0

Evolutionary model

Jukes Cantor model with root distribution $\pi = \left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right)$.
The transition matrix associated with edge $i$ is of the form $$M_j = \begin{pmatrix} a_i & b_i & b_i & b_i\\ b_i & a_i & b_i & b_i\\ b_i & b_i & a_i & b_i\\ b_i & b_i & b_i & a_i \end{pmatrix}$$ and the Fourier parameters are $\left(x_i, y_i, y_i, y_i\right)$ for all edges $i$.

Summary

Probability parametrization

$\begin{align} p_{\texttt{ACG}} &= \frac{1}{4}(a_1b_2b_3 + a_2b_1b_3 + a_3b_1b_2 + b_1b_2b_3) \\[0.5em] p_{\texttt{ACC}} &= \frac{1}{4}(a_1b_2b_3 + a_2a_3b_1 + 2b_1b_2b_3) \\[0.5em] p_{\texttt{ACA}} &= \frac{1}{4}(a_1a_3b_2 + a_2b_1b_3 + 2b_1b_2b_3) \\[0.5em] p_{\texttt{AAC}} &= \frac{1}{4}(a_1a_2b_3 + a_3b_1b_2 + 2b_1b_2b_3) \\[0.5em] p_{\texttt{AAA}} &= \frac{1}{4}(a_1a_2a_3 + 3b_1b_2b_3) \end{align}$

Fourier parametrization

$\begin{align} q_{\texttt{CGT}} &= y_1y_2y_3 \\[0.5em] q_{\texttt{CCA}} &= x_3y_1y_2 \\[0.5em] q_{\texttt{CAC}} &= x_2y_1y_3 \\[0.5em] q_{\texttt{ACC}} &= x_1y_2y_3 \\[0.5em] q_{\texttt{AAA}} &= x_1x_2x_3 \end{align}$

Equivalent classes of probability parametrization

$\begin{align} \text{Class } &p_{\texttt{AAA}}:\ p_{\texttt{AAA}},\ p_{\texttt{CCC}},\ p_{\texttt{GGG}},\ p_{\texttt{TTT}} \\[0.5em] \text{Class } &p_{\texttt{AAC}}:\ p_{\texttt{AAC}},\ p_{\texttt{AAG}},\ p_{\texttt{AAT}},\ p_{\texttt{CCA}},\ p_{\texttt{CCG}},\ p_{\texttt{CCT}},\ p_{\texttt{GGA}},\ p_{\texttt{GGC}},\ p_{\texttt{GGT}},\ p_{\texttt{TTA}},\ p_{\texttt{TTC}},\ p_{\texttt{TTG}} \\[0.5em] \text{Class } &p_{\texttt{ACA}}:\ p_{\texttt{ACA}},\ p_{\texttt{AGA}},\ p_{\texttt{ATA}},\ p_{\texttt{CAC}},\ p_{\texttt{CGC}},\ p_{\texttt{CTC}},\ p_{\texttt{GAG}},\ p_{\texttt{GCG}},\ p_{\texttt{GTG}},\ p_{\texttt{TAT}},\ p_{\texttt{TCT}},\ p_{\texttt{TGT}} \\[0.5em] \text{Class } &p_{\texttt{ACC}}:\ p_{\texttt{ACC}},\ p_{\texttt{AGG}},\ p_{\texttt{ATT}},\ p_{\texttt{CAA}},\ p_{\texttt{CGG}},\ p_{\texttt{CTT}},\ p_{\texttt{GAA}},\ p_{\texttt{GCC}},\ p_{\texttt{GTT}},\ p_{\texttt{TAA}},\ p_{\texttt{TCC}},\ p_{\texttt{TGG}} \\[0.5em] \text{Class } &p_{\texttt{ACG}}:\ p_{\texttt{ACG}},\ p_{\texttt{ACT}},\ p_{\texttt{AGC}},\ p_{\texttt{AGT}},\ p_{\texttt{ATC}},\ p_{\texttt{ATG}},\ p_{\texttt{CAG}},\ p_{\texttt{CAT}},\ p_{\texttt{CGA}},\ p_{\texttt{CGT}},\ p_{\texttt{CTA}},\ p_{\texttt{CTG}},\ p_{\texttt{GAC}},\ p_{\texttt{GAT}},\ p_{\texttt{GCA}},\ p_{\texttt{GCT}},\ p_{\texttt{GTA}},\ p_{\texttt{GTC}},\ p_{\texttt{TAC}},\ p_{\texttt{TAG}},\ p_{\texttt{TCA}},\ p_{\texttt{TCG}},\ p_{\texttt{TGA}},\ p_{\texttt{TGC}} \end{align}$

Equivalent classes of Fourier parametrization

$\begin{align} \text{Class } &0:\ q_{\texttt{AAC}}, q_{\texttt{AAG}}, q_{\texttt{AAT}}, q_{\texttt{ACA}}, q_{\texttt{ACG}}, q_{\texttt{ACT}}, q_{\texttt{AGA}}, q_{\texttt{AGC}}, q_{\texttt{AGT}}, q_{\texttt{ATA}}, q_{\texttt{ATC}}, q_{\texttt{ATG}}, q_{\texttt{CAA}}, q_{\texttt{CAG}}, q_{\texttt{CAT}}, q_{\texttt{CCC}}, q_{\texttt{CCG}}, q_{\texttt{CCT}}, q_{\texttt{CGA}}, q_{\texttt{CGC}}, q_{\texttt{CGG}}, q_{\texttt{CTA}}, q_{\texttt{CTC}}, q_{\texttt{CTT}}, q_{\texttt{GAA}}, q_{\texttt{GAC}}, q_{\texttt{GAT}}, q_{\texttt{GCA}}, q_{\texttt{GCC}}, q_{\texttt{GCG}}, q_{\texttt{GGC}}, q_{\texttt{GGG}}, q_{\texttt{GGT}}, q_{\texttt{GTA}}, q_{\texttt{GTG}}, q_{\texttt{GTT}}, q_{\texttt{TAA}}, q_{\texttt{TAC}}, q_{\texttt{TAG}}, q_{\texttt{TCA}}, q_{\texttt{TCC}}, q_{\texttt{TCT}}, q_{\texttt{TGA}}, q_{\texttt{TGG}}, q_{\texttt{TGT}}, q_{\texttt{TTC}}, q_{\texttt{TTG}}, q_{\texttt{TTT}} \\[0.5em] \text{Class } &q_{\texttt{AAA}}:\ q_{\texttt{AAA}} \\[0.5em] \text{Class } &q_{\texttt{ACC}}:\ q_{\texttt{ACC}},\ q_{\texttt{AGG}},\ q_{\texttt{ATT}} \\[0.5em] \text{Class } &q_{\texttt{CAC}}:\ q_{\texttt{CAC}},\ q_{\texttt{GAG}},\ q_{\texttt{TAT}} \\[0.5em] \text{Class } &q_{\texttt{CCA}}:\ q_{\texttt{CCA}},\ q_{\texttt{GGA}},\ q_{\texttt{TTA}} \\[0.5em] \text{Class } &q_{\texttt{CGT}}:\ q_{\texttt{CGT}},\ q_{\texttt{CTG}},\ q_{\texttt{GCT}},\ q_{\texttt{GTC}},\ q_{\texttt{TCG}},\ q_{\texttt{TGC}} \end{align}$

Minimal generating set of the vanishing ideal

Ideal generated by $$ -q_{\texttt{CGT}}^2q_{\texttt{AAA}} + q_{\texttt{CCA}}q_{\texttt{CAC}}q_{\texttt{ACC}} $$

Gröbner basis of the vanishing ideal

Gröbner basis with respect to the ordering $ \texttt{degrevlex}\ [ q_{\texttt{CGT}}, q_{\texttt{CCA}}, q_{\texttt{CAC}}, q_{\texttt{ACC}}, q_{\texttt{AAA}} ] $ and with elements $$ \begin{align} -q_{\texttt{CGT}}^2q_{\texttt{AAA}} + q_{\texttt{CCA}}q_{\texttt{CAC}}q_{\texttt{ACC}} \end{align} $$

Additional information

Probability $\to$ Fourier

$\begin{align} q_{\texttt{CGT}} &= \frac{1}{3}p_{\texttt{ACG}} - \frac{1}{3}p_{\texttt{ACC}} - \frac{1}{3}p_{\texttt{ACA}} - \frac{1}{3}p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em] q_{\texttt{CCA}} &= -\frac{1}{3}p_{\texttt{ACG}} - \frac{1}{3}p_{\texttt{ACC}} - \frac{1}{3}p_{\texttt{ACA}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em] q_{\texttt{CAC}} &= -\frac{1}{3}p_{\texttt{ACG}} - \frac{1}{3}p_{\texttt{ACC}} + p_{\texttt{ACA}} - \frac{1}{3}p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em] q_{\texttt{ACC}} &= -\frac{1}{3}p_{\texttt{ACG}} + p_{\texttt{ACC}} - \frac{1}{3}p_{\texttt{ACA}} - \frac{1}{3}p_{\texttt{AAC}} + p_{\texttt{AAA}} \\[0.5em] q_{\texttt{AAA}} &= p_{\texttt{ACG}} + p_{\texttt{ACC}} + p_{\texttt{ACA}} + p_{\texttt{AAC}} + p_{\texttt{AAA}} \end{align}$

Fourier $\to$ Probability

$\begin{align} p_{\texttt{ACG}} &= \frac{3}{4}q_{\texttt{CGT}} - \frac{3}{8}q_{\texttt{CCA}} - \frac{3}{8}q_{\texttt{CAC}} - \frac{3}{8}q_{\texttt{ACC}} + \frac{3}{8}q_{\texttt{AAA}} \\[0.5em] p_{\texttt{ACC}} &= -\frac{3}{8}q_{\texttt{CGT}} - \frac{3}{16}q_{\texttt{CCA}} - \frac{3}{16}q_{\texttt{CAC}} + \frac{9}{16}q_{\texttt{ACC}} + \frac{3}{16}q_{\texttt{AAA}} \\[0.5em] p_{\texttt{ACA}} &= -\frac{3}{8}q_{\texttt{CGT}} - \frac{3}{16}q_{\texttt{CCA}} + \frac{9}{16}q_{\texttt{CAC}} - \frac{3}{16}q_{\texttt{ACC}} + \frac{3}{16}q_{\texttt{AAA}} \\[0.5em] p_{\texttt{AAC}} &= -\frac{3}{8}q_{\texttt{CGT}} + \frac{9}{16}q_{\texttt{CCA}} - \frac{3}{16}q_{\texttt{CAC}} - \frac{3}{16}q_{\texttt{ACC}} + \frac{3}{16}q_{\texttt{AAA}} \\[0.5em] p_{\texttt{AAA}} &= \frac{3}{8}q_{\texttt{CGT}} + \frac{3}{16}q_{\texttt{CCA}} + \frac{3}{16}q_{\texttt{CAC}} + \frac{3}{16}q_{\texttt{ACC}} + \frac{1}{16}q_{\texttt{AAA}} \end{align}$