When explaining substitution models, the substitutions are expressed as Q matrix. Why is the sum of the elements in a row zero?
Debopam Ghosh
Mathematically it implies that the unit vector, i.e. the vector of all 1's, lies in the null space of this Q matrix; this implies some sort of conservation, or balance, e.g charge/mass balance during transition from one compartment to another in a closed multi compartment system. Look for such balance/conservation laws satisfied by the Q matrix in your problem.
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Guido Ehrmann
Hello,
this theory or better mathematical method is based on the rule that all matrix-lines are zero, otherwise the method wouldn't work.
Best regards
Guido
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Kengo Shibuya
A can change to C, G, or T not to foreign X. [Closed system] The probabilities of A changing to C, A changing to G, and A changing to T are the same. [Randomness] When 3a A's are lost in a given time, 1a C's, 1a G's, and 1a T's are generated simultaneously. [Conservation of total number (A+C+G+T) in this term: -3a+a+a+a=0.] The above is also true for interchanged A, C, G, and T. [Symmetry]
I'm not a biologist, but mathematically the matrix makes me feel as follows:
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