Question: Rate matrix Q with GTR+F+G4

[Eve]

A trying to reconstruct a tree within a bacteria CC group (expected to be very closely related). We removed expected recombinant sites, and worked with a core alignment.

We choosen a standard Model of substitution: GTR+F+G4 (we have not done model testing yet, we are just learning ...) .

We are not so surprized to observe that the G4 might be actually overfitting, because it seems we have a very low variation between heterogeneity rate classes if I understand well, wich we woud expect if we were working within a lineage/CC group. 

But we were wondering why the Rate matrix Q was not symetrical. 

Is it because of rounding/model approximation or scaling?  I see all diagonal values are negative. Could you explain? 

Please extract of output corresponding to this question bellow

Best regards

Eve

```

SEQUENCE ALIGNMENT

------------------

Input data: 25 sequences with 1441 nucleotide sites

Number of constant sites: 0 (= 0% of all sites)

Number of invariant (constant or ambiguous constant) sites: 0 (= 0% of all sites)

Number of parsimony informative sites: 481

Number of distinct site patterns: 226

SUBSTITUTION PROCESS

--------------------

Model of substitution: GTR+F+G4

Rate parameter R:

  A-C: 0.7237

  A-G: 5.0504

  A-T: 1.3350

  C-G: 0.2022

  C-T: 5.0167

  G-T: 1.0000

State frequencies: (empirical counts from alignment)

  pi(A) = 0.2086

  pi(C) = 0.2908

  pi(G) = 0.2811

  pi(T) = 0.2195

Rate matrix Q:

  A    -1.203    0.1316    0.8877    0.1832

  C    0.0944   -0.8185   0.03554    0.6886

  G    0.6588   0.03676   -0.8328    0.1373

  T    0.1741    0.9121    0.1758    -1.262

Model of rate heterogeneity: Gamma with 4 categories

Gamma shape alpha: 998.4-

 Category  Relative_rate  Proportion

  1         0.9601         0.25

  2         0.9894         0.25

  3         1.01           0.25

  4         1.041          0.25

Relative rates are computed as MEAN of the portion of the Gamma distribution falling in the category.

```

[Minh Bui]

Hi Eve,

Sorry for the slow response.

Q is not symmetric because q_ij = r_ij * pi_j, 

where r_ij is the symetric matrix R (the “Rate parameter R” in iqtree output), and pi_j is the state frequency vector.

Q is symmetric only if the model has equal state frequencies (e.g. the SYM model). See http://www.iqtree.org/doc/Substitution-Models#dna-models 

Note that this model still reversible, even though Q is not symmetric.

Hope that helps,

Minh

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[Eve]

Thanks a lot for you answer and for your time

All the best

Eve

[Minh Bui]

Hi Eve again,

Regarding your other questions: the diagonal elements are negative because sum of rows of Q matrix should be 0. This follows a standard Markov process. Therefore, I suggest you have a look at text book, such as the Phylogenetic Handbook: It has a nice chapter on the derivation of the Q matrix.

Cheers,

Minh

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