Q and P Matrices

Evolutionary analyses of sequences are conducted on a wide variety of time scales.

Thus, it is convenient to express these models in terms of the instantaneous rates of change between different states. This representation of the model is typically called the model's Q Matrix.

SubstitutionModels.Q — Function.

Generate a Q matrix for a NucleicAcidSubstitutionModel, of the form:

\[Q = \begin{bmatrix} Q_{A, A} & Q_{A, C} & Q_{A, G} & Q_{A, T} \\ Q_{C, A} & Q_{C, C} & Q_{C, G} & Q_{C, T} \\ Q_{G, A} & Q_{G, C} & Q_{G, G} & Q_{G, T} \\ Q_{T, A} & Q_{T, C} & Q_{T, G} & Q_{T, T} \end{bmatrix}\]

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If we are given a starting state at one position in a DNA sequence, the model's Q matrix and a branch length expressing the expected number of changes to have occurred since the ancestor, then we can derive the probability of the descendant sequence having each of the four states.

This transformation from the instantaneous rate matrix (Q Matrix), to a probability matrix for a given time period (P Matrix), is described here.

SubstitutionModels.P — Function.

Generate a P matrix for a NucleicAcidSubstitutionModel, of the form:

\[P = \begin{bmatrix} P_{A, A} & P_{A, C} & P_{A, G} & P_{A, T} \\ P_{C, A} & P_{C, C} & P_{C, G} & P_{C, T} \\ P_{G, A} & P_{G, C} & P_{G, G} & P_{G, T} \\ P_{T, A} & P_{T, C} & P_{T, G} & P_{T, T} \end{bmatrix}\]

for specified time

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