In his method of evolutionary parsimony, Lake [14] has highlighted the balanced transversion assumption. This assumption implies the constraints λAC = λAT , λGC = λGT , λCA = λCG, and λT A = λT G in the nucleotide substitution model with general transition rates.

Without further restrictions, infinitesimal balanced transversions do not imply finite-time balanced transversions. For example, the identity pAC (t) = pAT (t) may not hold. Prove that finite-time balanced transversions follow if the additional closure assumptions

λAG − λGA = λG − λA

λCT − λT C = λT − λC are made [1]. (Hint: Show by induction that the matrices Λk have the balanced transversion pattern for equality of entries.)