Density functional theory with the omega B97X-D exchange-correlation functional together with implicit as well as explicit solvation is used to describe the reactions of the adenine and guanine purine bases on N,N',N ''-triethylenethiophosphoramide (thioTEPA), an alkylating agent used as an anticancer drug. This reaction is decomposed into (i) a nucleophilic addition and (ii) a proton "migration" that is mediated by the solvent molecules. The calculations reveal that the first step is rate determining and that the presence of an explicit water molecule to mediate the proton migration has a negligible role on the kinetics of the first step, so that the focus is set on the first step of the reaction. omega B97X-D calculations highlight (i) the activation energy (Gibbs free enthalpy) is smaller for imine nitrogens than amine nitrogens, (ii) for the imine functions, the activation energy is slightly smaller for adenine than for guanine together with a larger exergonicity for the alkylation by adenine, and (iii) among the amine nitrogens, the presence of stabilizing H-bonds in the case of exocyclic amines leads to smaller activation energy than for the endocyclic ones. The reaction mechanisms are unraveled by employing the bond evolution theory, combining the use of electronic localization functions, and their evolution along the intrinsic reaction coordinate, with Thom's catastrophe theory. These analyses, suitable for highlighting the populations of the major monosynaptic and disynaptic basins, show (i) the reaction with imine nitrogens begins by the cleavage of the C-N aziridine bond and is followed by the simultaneous formation of the new C-N bond and the disappearance of the nitrogen lone pair, (ii) the reaction with the nitrogen atom of an endocyclic or exocyclic amine proceeds first by the formation of the cross-linking C-N bond and then by the cleavage of the C-N aziridine bond and the disappearance of the nitrogen lone pair, and (iii) in case ii, this bond breaking and forming occur before the transition state, which has been correlated to the increased Gibbs enthalpy of activation with respect to the reaction with the nitrogen atom of imine functions.