Role of constraint in catalysis and high-affinity binding by proteins

Using a model for catalysis of a dynamic equilibrium, the role of constraint in catalysis is quantified. The intrinsic rigidity of proteins is shown to be insufficient to constrain the activated complexes of enzymes, irrespective of the mechanism. However, when minimization of the surface excess free energy of water surrounding a protein is considered, model proteins can be designed with regions of sufficient rigidity. Structures can be designed to focus surface tension or hydrophobic attraction as compressive stress. A monomeric structure has a limited ability to concentrate compressive stress and constrain activated complexes. Oligomeric or multidomain proteins, with domains surrounding a rigid core, have unlimited ability to concentrate stress, provided there are at least four domains. Under some circumstances, four is the optimum number, which could explain the frequency of tetrameric enzymes in nature. The minimum compressive stress in oligomers increases with the square of the radius. For tetramers of similar size to natural enzymes, this stress agrees reasonably well with that needed to constrain the activated complex. A similar principle applies to high affinity binding proteins. The models explain the trigonal pyramidal shape of fibroblast growth factor and provide a basis for interpretation of protein crystal structures.