Realistic protein-protein association rates from a simple diffusional model neglecting long-range interactions, free energy barriers, and landscape ruggedness

Abstract

We develop a simple but rigorous model of protein-protein association kinetics based on diffusional association on free energy landscapes obtained by sampling configurations within and surrounding the native complex binding funnels. Guided by results obtained on exactly solvable model problems, we transform the problem of diffusion in a potential into free diffusion in the presence of an absorbing zone spanning the entrance to the binding funnel. The free diffusion problem is solved using a recently derived analytic expression for the rate of association of asymmetrically oriented molecules. Despite the required high steric specificity and the absence of long-range attractive interactions, the computed rates are typically on the order of 10(4)-10(6) M(-1) sec(-1), several orders of magnitude higher than rates obtained using a purely probabilistic model in which the association rate for free diffusion of uniformly reactive molecules is multiplied by the probability of a correct alignment of the two partners in a random collision. As the association rates of many protein-protein complexes are also in the 10(5)-10(6) M(-1) sec(-1) range, our results suggest that free energy barriers arising from desolvation and/or side-chain freezing during complex formation or increased ruggedness within the binding funnel, which are completely neglected in our simple diffusional model, do not contribute significantly to the dynamics of protein-protein association. The transparent physical interpretation of our approach that computes association rates directly from the size and geometry of protein-protein binding funnels makes it a useful complement to Brownian dynamics simulations.

Figure 1.

Simple model of binding dynamics. Attractive short-range forces produce a funnel in the free energy landscape leading into the native complex. Once the molecules descend several kT into the funnel, they are effectively captured and binding occurs rapidly. In our simple model, the rate of association is approximated by the rate of free diffusion into a reactive zone in phase space, as indicated schematically in the XY plane of the drawing. To compute the rate of association, we need first to determine the dimensions of the reactive zone, and second, to compute the rate of free diffusion into this zone. A more general model would include long-range (electrostatic) interactions that would bias the diffusion process toward the funnel entrance.

Figure 2.

The axes and angles relevant to the reaction condition, equation 1. The angles θ_A and θ_B measure how close the center of each reactive patch (coinciding with the respective body-fixed z-axis) is to the center-to-center vector (dashed line). The Euler angles δφ and δχ denote relative torsion angles of the two body-fixed coordinate systems (x_A, y_A, z_A) and (x_B, y_B, z_B). For the sake of easier visualization of these two angles, the origin of the x_A and y_A axes (belonging to the coordinate system of sphere A) has been shifted such as to coincide with the origin of the coordinate system of sphere B. Our reaction condition, equation 1, requires near-optimal alignment; that is, all angles θ_A, θ_B, δφ, and δχ must be below given limits.

Figure 3.

Mapping of translational diffusional association in a short-range attractive potential onto the problem of free diffusion with an absorbing region. A Lennard–Jones potential with parameters appropriate to protein–protein association (ɛ = 10 kcal/mole, σ = 3 Å, R₁ = 40 Å) is shown in the figure, together with the corresponding free diffusion “capture radius” R₀. The capture radius is defined as the radius for which the free diffusion rate k_on⁽⁰⁾ is equal to the association rate in the presence of U(r), k_on⁽¹⁾. Evaluating U(r) at the capture radius yields the energy drop ΔE ≡ U(∞) − U(R₀) ≈ 0.3 kcal/mole (i.e., of order kT); this is quite insensitive to large changes in ɛ and σ.

Figure 4.

Mapping of the problem of diffusion in a potential onto that of free diffusion with an absorbing region for rotational diffusion on a spherical surface. We use an attractive Gaussian potential U(θ) = −ɛ exp[−(σ θ)²] with ɛ = 10 kcal/mole and σ = π, where the latter corresponds to a (half) width of the potential of , a reasonable assumption for a short-range potential. Equating the resulting association rate (equation 4) with the rate for free diffusion in the presence of an absorbing region at θ = θ ₀ (equation 5), we obtain θ ₀ ≈ 33° for the width of the absorbing region, corresponding to an energy drop of ΔE ≡ U(π) − U(θ ₀) ≈ 0.4 kcal/mole.

Figure 5.

Free energy funnels around the native structure. The energy ℰ and rmsd (A), the energy ℰ and the angular deviations θ_A⁰ (B), and δχ₀ (C) are shown for a set of randomly perturbed structures of the protein–protein complex 1FIN. States of lower energy are seen to be associated with smaller angles, suggesting that the angles are a reasonable measure for the deviation from the correctly complexed structure. The two parallel lines represent the two energy cutoffs ℰ_c = ℰ_av and ℰ_c = ℰ_av − 5kT, where ℰ_av is the average energy of the five lowest energy complexes with an rmsd above 10 Å. The vertical lines in the plots indicate the resulting angular constraints θ_A⁰ and δχ₀ corresponding to ℰ_c = ℰ_av (dashed line) and ℰ_c = ℰ_av − 5kT (dotted/dashed line).

Figure 5.

Free energy funnels around the native structure. The energy ℰ and rmsd (A), the energy ℰ and the angular deviations θ_A⁰ (B), and δχ₀ (C) are shown for a set of randomly perturbed structures of the protein–protein complex 1FIN. States of lower energy are seen to be associated with smaller angles, suggesting that the angles are a reasonable measure for the deviation from the correctly complexed structure. The two parallel lines represent the two energy cutoffs ℰ_c = ℰ_av and ℰ_c = ℰ_av − 5kT, where ℰ_av is the average energy of the five lowest energy complexes with an rmsd above 10 Å. The vertical lines in the plots indicate the resulting angular constraints θ_A⁰ and δχ₀ corresponding to ℰ_c = ℰ_av (dashed line) and ℰ_c = ℰ_av − 5kT (dotted/dashed line).