scoring matrices computation from transition probability matrix

Question

[RT!37]

The transition probability matrix over time t is computed as P(t) = e ^Qt. In order to calculate a general set of transition probability matrices what type of t values can be used ( t = expected substitutions per site). How these transition probability matrices can be converted to scoring matrices to use in homolog searches/alignment ??

I am working with amino acids substitution models !!

Thanks !!