The efficient integration of binding, catalysis, and multiple turnovers remains a challenge in building enzyme models. We report that systematic derivatization of polyethylene imine (PEI) with alkyl (C2-C12), benzyl, and guanidinium groups gives rise to catalysts ('synzymes') with rate accelerations (k(cat)/k(uncat)) of up to 10(4) for the intramolecular transesterification of 2-hydroxypropyl-p-nitrophenyl phosphate, HPNP, in the absence of metal. The synzymes exhibit saturation kinetics (K-M approximate to 250 mu M, k(cat) approximate to 0.5 min(-1)) and up to 2340 turnovers per polymer molecule. Catalysis can be specifically and competitively inhibited by anionic and hydrophobic small molecules. The efficacy of catalysis is determined by the PEI derivatization pattern. The derivatization reagents exert a synergistic effect, i.e., their combinations increase catalysis by more than the sum of each single modification. The pH-rate profile for k(cat)/K-M is bell shaped with a maximum at pH 7.85 and can be explained as a combination of two effects that both have to be operative for optimal activity: K-M increases at high pH due to deprotonation of PEI amines that bind the anionic substrate and k(cat) decreases as the availability of hydroxide decreases at low pH. Thus, catalysis is based on substrate binding by positively charged amine groups and the presence of hydroxide ion in active sites in an environment that is tuned for efficient catalysis. Inhibition studies suggest that the basis of catalysis and multiple turnovers is differential molecular recognition of the doubly negatively charged transition state (over singly charged ground state and product): this contributes a factor of at least 5-10-fold to catalysis and product release.