Protein-protein bond formations, such as antibody-antigen complexation or aggregation of protein monomers into dimers and larger aggregates, occur with bimolecular rate constants on the order of 10' M-l's-', which is only 3 orders of magnitude slower than the diffusion-limited Smoluchowski rate. However, since the protei-protein bond requires rotational alignment to within a few angstroms of tolerance, purely geometric estimates would suggest that the observed rates might be 6 orders of magnitude below the Smoluchowski rate. Previous theoretical treatments have not been. solved for the highly specific docking criteria of proteinprotein association-the entire subunit interface must be aligned within 2 A of the correct position. Several studies have suggested that diffusion alone could not produce the rapid association kinetics and have postulated "lengthy collisions"and/or the operation of electrostatic or hydrophobic steering forces to accelerate the association. In the present study, the Brownian dynamics simulation method is used to compute the rate of association of neutral spherical model proteins with the stated docking criteria. The Brownian simulation predicts a rate of 2 x 10' M-'s-' for this generic protein-protein association, a rate that is 2000 times faster than that predicted by the simplest geometric calculation and is essentially equal to the rates observed for protein-protein association in aqueous solution. This high rate is obtained by simple diffusive processes and does not require any attractive or steering forces beyond those achieved for a partially formed bond. The rate enhancement is attributed to a diffusive entrapment effect, in which a protein pair surrounded and trapped by water undergoes multiple collisions with rotational reorientation during each encounter.The association of protein molecules to form dimers or larger complexes is characterized by second-order rate constants (7) and cytochrome b5 (8) (k2 varies from 107 to 109, with the faster rates at low ionic strength). These very fast reactions are the results of strong attractive coulombic forces that highly favor formation ofthe productive reaction complexes (8-11). Since rates of k2 = 0.5-5 x 106 M-1-s-1 are achieved by many protein associations, including the very general reaction of antibodies with protein antigen, this range appears to represent the typical rate for proteins associating and docking at the precise orientation for bond formation, without any special steering forces.When one considers the steric specificity of the bond connecting protein subunits, this rate seems incredibly fast.If the proteins were spheres of 18 A radius (typical of a small protein), and if the spheres associated with every contact, without regard to orientation, the diffusion-limited association rate constant would be given by the Smoluchowski (12) rate constant, k2 = 7 x 109 M-1 s-1. That the observed rates are substantially slower than the diffusion-limited encounter of spheres is easily explained as being due to steric specifici...

A method is developed and tested for extracting diffusion-controlled rate constants for condensed phase bimolecular reactions from Brownian dynamics trajectory simulations. This method will be useful when highly detailed model systems are employed, such as those required to explore the complicated range of interactions between enzymes and their substrates. The method is verified by comparing with exact analytical results for simple cases of spheres with uniform reactivity subject to various centrosymmetric Coulombic and Oseen slip hydrodynamic interactions. The utility of the method is illustrated for more complicated cases involving anisotropic reactivity and rotational diffusion.

Brownian dynamics computer simulations of the diffusional association of electron transport proteins cytochrome c (cyt c) and cytochrome c peroxidase (cyt c per) were performed. A highly detailed and realistic model of the protein structures and their electrostatic interactions was used that was based on an atomic-level spatial description. Several structural features played a role in enhancing and optimizing the electron transfer efficiency of this reaction. Favorable electrostatic interactions facilitated long-lived nonspecific encounters between the proteins that allowed the severe orientational criteria for reaction to be overcome by rotational diffusion during encounters. Thus a "reduction-in-dimensionality" effect operated. The proteins achieved plausible electron transfer orientations in a multitude of electrostatically stable encounter complexes, rather than in a single dominant complex.

The reduction of wild-type yeast iso-1-ferricytochrome c (ycytc) and several mutants by trypsin-solubilized bovine liver ferrocytochrome b5 (cytb5) has been studied under conditions in which the electron-transfer reaction is bimolecular. The effect of electrostatic charge modifications and steric changes on the kinetics has been determined by experimental and theoretical observations of the electron-transfer rates of ycytc mutants K79A, K'72A, K79A/K'72A, and R38A (K' is used to signify trimethyllysine (Tml)). A structurally robust Brownian dynamics (BD) method simulating diffusional docking and electron transfer was employed to predict the mutation effect on the rate constants. A realistic model of the electron-transfer event embodied in an intrinsic unimolecular rate constant is used which varies exponentially with donor-acceptor distance. The BD method quantitatively predicts rate constants over a considerable range of ionic strengths. Semiquantitative agreement is obtained in predicting the perturbing influence of the mutations on the rate constants. Both the experimentally observed rate constants and those predicted by BD descend in the following order: native ycytc > K79A > K'72A > K79A/K'72A. Variant R38A was studied at a different ionic strength than this series of mutations, and the theory agreed with experiment in predicting a smaller rate constant for the mutant. In all cases the predicted effect of mutation was in the correct direction, but not as large as that observed. The BD simulations predict that the two proteins dock through essentially a single domain, with a distance of closest approach of the two heme groups in rigid body docking typically around 12 A. Two predominant classes of complexes were calculated, the most frequent involving the quartet of cytb5/ycytc interactions, Glu48-Arg13, Glu56-Lys87, Asp60-Lys86, and heme-Tml72, having an average electrostatic energy of -13.0 kcal/mol. The second most important complexes were of the type previously postulated (Salemme, 1976; Mauk et al., 1986; Rodgers et al., 1988) with interactions Glu44-Lys27, Glu48-Arg13, Asp60-Tml72, and heme-Lys79 and having an energy of -6.4 kcal/mol. The ionic strength dependence of the bimolecular reaction rate was well reproduced using a discontinuous dielectric model, but poorly so for a uniform dielectric model.

The theory of diffusion-influenced reactions is extended to cases where the reactivity of the species fluctuates in time (e.g., the accessibility of a binding site of a protein is modulated by a gate). The opening and closing of the gate is assumed to be a stationary Markov process [i.e., it is described by the kinetic scheme (open) a⇄b (closed)]. When the reaction is described by suitable boundary conditions, by solving the appropriate reaction-diffusion equations, it is shown that the stochastically gated association rate constant (kSG) is given by k−1SG=k−1∞ + [a−1 b(a+b)κ̂u(a+b)]−1, where κ̂u(s) is the Laplace transform of the time-dependent rate constant of the ungated problem and k∞ is the corresponding steady-state rate constant. The limits when the relaxation time for gate fluctuations is larger or smaller than the characteristic time for diffusion are considered. The relation to previous work is discussed. The theory is applied to three models: (i) a gated sphere, (ii) a gated disk on an infinite plane (e.g., a channel in a membrane), and (iii) a gated localized axially symmetric reactive site on the surface of a spherical macromolecule.