Theory and simulation of diffusion-influenced, stochastically gated ligand binding to buried sites

Abstract

We consider the diffusion-influenced rate coefficient of ligand binding to a site located in a deep pocket on a protein; the binding pocket is flexible and can reorganize in response to ligand entrance. We extend to this flexible protein-ligand system a formalism developed previously [A. M. Berezhkovskii, A, Szabo, and H.-X. Zhou, J. Chem. Phys. 135, 075103 (2011)] for breaking the ligand-binding problem into an exterior problem and an interior problem. Conformational fluctuations of a bottleneck or a lid and the binding site are modeled as stochastic gating. We present analytical and Brownian dynamics simulation results for the case of a cylindrical pocket containing a binding site at the bottom. Induced switch, whereby the conformation of the protein adapts to the incoming ligand, leads to considerable rate enhancement.

Figure 1

Gating situations studied in the present paper. (a) A buried binding site that switches between inert and reactive conformations. (b) A buried binding site that has gated access. The gate is at the entrance of the binding pocket. (c) The gated access situation, with the gate midway along the binding pocket. (d) A binding pocket that switches between an open lid/inert binding-site state and a closed lid/reactive binding-site state.

Figure 2

Comparison of analytical (solid lines) and BD simulation (symbols) results for the rate coefficient in the constant-linear potential of Eq. 2.29. Results at four values of L₁/a are shown in different colors according to the key in the figure. Other parameters are: κ₀ = 0.1D/a; L/a = 5; and βV₀ = –3.

Figure 3

$k_{\mathrm{G}}^{\mathrm{ss}}$ (scaled by $k_{\mathrm{E}}^{\mathrm{ss}}=4Da$) for a gating binding site under either induced switch (red curve) or indifferent switch (blue curve). The indifferent-switch scenario has a constant potential V₀ = V_eff in the binding pocket. Parameters are as follows: κ₀ = ∞; L/a = 5; p_a/p_r = 0.01; p_Ia/p_Ir = 10; e^−βVa = 10³; and e^−βVr = 1.

Figure 4

Comparison of analytical (solid lines) and BD simulation (symbols) results for the rate coefficient of binding to a gating disk. Results at three values of (a²/D)^1/2ω_– are shown in different colors according to the key in the figure. Other parameters are: κ₀ = D/a; and (a²/D)^1/2ω₊ = 100.

Figure 5

The rate constant $k_{\mathrm{I};\mathrm{G}}^{\mathrm{ss}}$ (scaled by $k_{\mathrm{I}}^{\mathrm{ss}}=D\sigma e^{-\beta V_{\mathrm{eff}}}∕L$) for the interior problem in the gating binding-pocket situation. The indifferent-switch scenario is represented by e^{−β(Vc – Vo)} = 1. Results at four values of ν_IL = (L²/D)^1/2ω_I, representing the ratio of the diffusional timescale and the conformational transition timescale, are shown in different colors according to the key in the figure. Other parameters are: κ₀ = ∞; p_c/p_o = 0.01; e^−βVeff = 10.