Herein, we first address the question posed in the title by establishing the tautomerization trajectory via the double proton transfer of the adenine·guanine (A·G) DNA base mispair formed by the canonical tautomers of the A and G bases into the A*·G* DNA base mispair, involving mutagenic tautomers, with the use of the quantum-mechanical calculations and quantum theory of atoms in molecules (QTAIM). It was detected that the A·G ↔ A*·G* tautomerization proceeds through the asynchronous concerted mechanism. It was revealed that the A·G base mispair is stabilized by the N6H···O6 (5.68) and N1H···N1 (6.51) hydrogen bonds (H-bonds) and the N2H···HC2 dihydrogen bond (DH-bond) (0.68 kcal·mol−1), whereas the A*·G* base mispair—by the O6H···N6 (10.88), N1H···N1 (7.01) and C2H···N2 H-bonds (0.42 kcal·mol−1). The N2H···HC2 DH-bond smoothly and without bifurcation transforms into the C2H···N2 H-bond at the IRC = −10.07 Bohr in the course of the A·G ↔ A*·G* tautomerization. Using the sweeps of the energies of the intermolecular H-bonds, it was observed that the N6H···O6 H-bond is anticooperative to the two others—N1H···N1 and N2H···HC2 in the A·G base mispair, while the latters are significantly cooperative, mutually strengthening each other. In opposite, all three O6H···N6, N1H···N1, and C2H···N2 H-bonds are cooperative in the A*·G* base mispair. All in all, we established the dynamical instability of the А*·G* base mispair with a short lifetime (4.83·10−14 s), enabling it not to be deemed feasible source of the A* and G* mutagenic tautomers of the DNA bases. The small lifetime of the А*·G* base mispair is predetermined by the negative value of the Gibbs free energy for the A*·G* → A·G transition. Moreover, all of the six low-frequency intermolecular vibrations cannot develop during this lifetime that additionally confirms the aforementioned results. Thus, the A*·G* base mispair cannot be considered as a source of the mutagenic tautomers of the DNA bases, as the A·G base mispair dissociates during DNA replication exceptionally into the A and G monomers in the canonical tautomeric form. © 2013 Wiley Periodicals, Inc.