A general methodology that allows the estimation of maximum rates of enzymatic reactions is described. For a typical mechanism of an enzymatic reaction, the rate is a function of kinetic parameters which are unknown but required to obey certain constraints. Specifically, the ratio of forward to backward rate constants must be consistent with the equilibrium constant, and the rate of each bimolecular reaction-step must be less than the rate of collision of the two reactant species. If additional information is available on the reaction rate, more constraints can be introduced. By maximizing the rate expression with respect to the kinetic parameters, subject to all applicable constraints, a first-principles upper bound is obtained for the reaction rate. If the reaction rate is actually known, the methodology can alternatively estimate an extremum for the concentration of the enzyme, a substrate, or a product. Simple thermodynamic arguments could also provide bounds for concentrations or the direction (but not the magnitude) of the rate, by examining only the overall transformation of reactants to products and completely ignoring the mechanism. The collision-limit treatment proposed here exploits basic internal characteristics of enzymatic reaction mechanisms to predict better bounds for the concentrations and the thermodynamically allowable maximum magnitude of the rate.