Dear Sergei,

Page 13 of the hyphybook says that these two ratios are close but not equal and this is true for my data using the MG model. I would like to hear or see a reference to a more detailed explanation of the difference between these two ratios.  My intuition is that the nt substitution and equilibrium frequency values in the Q matrix attempt to account for (normalize for) changes that are due simply to background nt substitution rates.  However, dS is based partly on the codon frequencies.  In contrast, in the MG Q matrix the frequencies used are the nt freqs at each codon position.  Is this part of the reason for the difference in the ratios?  The nt substitution rates do not seem to be part of the dS calculation; is this another reason for the difference?  And then there's the T/S which seems rough.  I guess one way to ask is, is there, in principle, a way to calculate dN and dS such that dN/dS = rateN/rateS.  For the purposes of calculating omega in HYPHY though, I take it that rateN/rateS is the answer and dN/dS is some kind of ad hoc approximation?

-Danny