A method for computing association rate constants of atomistically represented proteins under macromolecular crowding

Abstract

In cellular environments, two protein molecules on their way to form a specific complex encounter many bystander macromolecules. The latter molecules, or crowders, affect both the energetics of the interaction between the test molecules and the dynamics of their relative motion. In earlier work (Zhou and Szabo 1991 J. Chem. Phys. 95 5948-52), it has been shown that, in modeling the association kinetics of the test molecules, the presence of crowders can be accounted for by their energetic and dynamic effects. The recent development of the transient-complex theory for protein association in dilute solutions makes it possible to easily incorporate the energetic and dynamic effects of crowders. The transient complex refers to a late on-pathway intermediate, in which the two protein molecules have near-native relative separation and orientation, but have yet to form the many short-range specific interactions of the native complex. The transient-complex theory predicts the association rate constant as k(a) = k(a0)exp(-ΔG*(el)/k(B)T), where k(a0) is the 'basal' rate constant for reaching the transient complex by unbiased diffusion, and the Boltzmann factors captures the influence of long-range electrostatic interactions between the protein molecules. Crowders slow down the diffusion, therefore reducing the basal rate constant (to k(ac0)), and induce an effective interaction energy ΔG(c). We show that the latter interaction energy for atomistic proteins in the presence of spherical crowders is 'long'-ranged, allowing the association rate constant under crowding to be computed as k(ac) = k(ac0)exp[-(ΔG*(el) + ΔG*(c))/k(B)T]. Applications demonstrate that this computational method allows for realistic modeling of protein association kinetics under crowding.

Figure 1

Illustration of the insertion method. Each protein conformation is placed in the crowded solution, and the interaction energy with the crowders is calculated. The Boltzmann average then gives the transfer free energy.

Figure 2

Systems studied. (a) A model consisting of two spherical particles. (b) barnase-barstar complex. (c) TEM1-BLIP complex.

Figure 3

Results for R_p = R_c = 30 Å. (a) Comparison of the pair distribution function calculated from binning the number of pairs at various distances and that predicted from the insertion method. ϕ = 0.3061. (b) The effect of the crowder-induced interaction energy on the association rate constant of the spherical model and the reduction of the relative diffusion constant of the reactant particles by crowding.

Figure 3

Results for R_p = R_c = 30 Å. (a) Comparison of the pair distribution function calculated from binning the number of pairs at various distances and that predicted from the insertion method. ϕ = 0.3061. (b) The effect of the crowder-induced interaction energy on the association rate constant of the spherical model and the reduction of the relative diffusion constant of the reactant particles by crowding.

Figure 4

The effective interaction energy of the spherical reactants induced by crowders with R_c = 30 Å. (a) ϕ dependence of ΔG_c. R_p = 15 Å. (b) R_p dependence of ΔG_c. ϕ = 0.2447.

Figure 4

The effective interaction energy of the spherical reactants induced by crowders with R_c = 30 Å. (a) ϕ dependence of ΔG_c. R_p = 15 Å. (b) R_p dependence of ΔG_c. ϕ = 0.2447.

Figure 5

The effective interaction energy of the barnase-barstar pair induced by crowders with R_c = 30 Å. The two protein molecules are moved along the normal of the least-squares plane of the interface atoms, (a) ϕ dependence of ΔG_c, calculated by the insertion method, (b) Comparison of the ΔG_c results obtained by the insertion method (curves) and predicted by the GMFT (symbols).

Figure 5

The effective interaction energy of the barnase-barstar pair induced by crowders with R_c = 30 Å. The two protein molecules are moved along the normal of the least-squares plane of the interface atoms, (a) ϕ dependence of ΔG_c, calculated by the insertion method, (b) Comparison of the ΔG_c results obtained by the insertion method (curves) and predicted by the GMFT (symbols).

Figure 6

A schematic representation of the association process. Two protein molecules, represented by their electrostatic surfaces, diffuse to form the transient complex. Further tightening and rearrangement then lead to the native complex.

Figure 7

Comparison of the r dependences of ΔG_el and ΔG_c for the barnase-barstar pair. The two protein molecules are moved along the normal of the least-squares plane of the interface atoms. For calculating ΔG_el, the configurations with 2 Å ≤ r ≤ 7 Å were rotated slightly to avoid clashes between the protein molecules; ionic strength = 150 mM. The shaded region corresponds to the range of r sampled by the transient complex.

Figure 8

ϕ dependence of ΔG_c* for the barnase-barstar pair at three R_c values.

Figure 9

Schemes of protein association in (a) dilute and (b) crowded solutions. In (a) the down-hill energy surface arises from electrostatic attraction between the protein molecules. In (b) the crowder-induced interaction energy further tilts the energy surface, but the crowders also effectively increase the friction (indicated by shading) on the path to the transient complex.