Calculations of protein-ligand binding entropy of relative and overall molecular motions

Abstract

In the context of virtual database screening, calculations of protein-ligand binding entropy of relative and overall molecular motions are challenging, owing to the inherent structural complexity of the ligand binding well in the energy landscape of protein-ligand interactions and computing time limitations. We describe a fast statistical thermodynamic method for estimation the binding entropy to address the challenges. The method is based on the integration of the configurational integral over clusters obtained from multiple docked positions. We apply the method in conjunction with 11 popular scoring functions (AutoDock, ChemScore, DrugScore, D-Score, F-Score, G-Score, LigScore, LUDI, PLP, PMF, X-Score) to evaluate the binding entropy of 100 protein-ligand complexes. The averaged values of binding entropy contribution vary from 6.2 to 9.1 kcal/mol, showing good agreement with literature. We calculate positional sizes and the angular volume of the native ligand wells. The averaged geometric mean of positional sizes in principal directions varies from 0.8 to 1.4 Å. The calculated range of angular volumes is 3.3−11.8 rad². Then we demonstrate that the averaged six-dimensional volume of the native well is larger than the volume of the most populated non-native well in energy landscapes described by all of 11 scoring functions.

Appendix

This appendix derives the partition functions of overall and relative rotational motions. It follows the method described by Kubo [83]. The Hamiltonian of a rotator is

$${\begin{array}{l}H=\frac{1}{2I_1\sin^2\theta}\left((p_{\varphi}-p_{\psi}\cos\theta)\cos\psi -p_{\theta}\sin\theta\sin\psi \right)^2+\\\frac{1}{2I_2\sin^2\theta} \left((p_{\varphi}-p_{\psi}\cos\theta)\sin\psi +p_{\theta}\sin\theta\cos\psi \right)^2+\frac{1}{2I_3}p_{\psi}^2\end{array}}$$

(12)

where I ₁, I ₂, I ₃ are the principal moments of inertia of the molecular; (θ, φ, ψ) are Euler angles of rotational motions; (p _θ, p _φ, p _ψ) are the corresponding canonical conjugate moments. The classical formula for the partition function of free rotations is

$${Z_{rot}=\frac{1}{(2\pi\hbar)^3\sigma}\int\limits_0^{\pi} d\theta\int\limits_0^{2\pi} d\psi \int\limits_0^{2\pi} d\varphi\int\limits_{-\infty}^{\infty} dp_{\theta}\int\limits_{-\infty}^{\infty} dp_{\psi}\int\limits_{-\infty}^{\infty}dp_{\varphi}\exp\left(-\frac{H}{k_BT}\right),}$$

(13)

where σ is the order of symmetry of the molecule; ${\hbar}$ is the Planck’s constant. The Hamiltonian in Eq. (12) can be rewritten as

$${\begin{array}{c}H=\frac{1}{2}\left(\frac{\sin^2\psi}{I_1}+\frac{\cos^2\psi}{I_2}\right)\left( p_{\theta}+\left(\frac{1}{I_2}-\frac{1}{I_1}\right)\frac{\sin\psi\cos\psi}{\sin\theta((\sin^2\psi)/I_1+(\cos^2\psi)/I_2)}(p_{\varphi}-p_{\psi}\cos\theta)\right)^2+\\\frac{1}{2I_1I_2\sin^2\theta((\sin^2\psi)/I_1+(\cos^2\psi)/I_2)}(p_{\varphi}-p_{\psi}\cos\theta)^2 +\frac{1}{2I_3}p_{\psi}^2\end{array}}$$

(14)

Substituting Eq. (14) into Eq. (13) and using the Gaussian integral formula

$${\int\limits_{-\infty}^{\infty}\exp\left(-a(x+b)^2\right)dx=\int\limits_{-\infty}^{\infty}\exp\left(-ax^2\right)dx=\left(\frac{\pi}{a}\right)^{1/2},}$$

(15)

we can perform the integrations over the canonical conjugate moments in Eq. (13). Integration over p _θ gives

$${(2\pi k_BT)^{1/2}\left(\frac{\sin^2\psi}{I_1}+\frac{\cos^2\psi}{I_2}\right)^{-1/2}}$$

(16)

Integration over p _φ gives

$${(2\pi k_BTI_1I_2)^{1/2}\sin\theta\left(\frac{\sin^2\psi}{I_1}+\frac{\cos^2\psi}{I_2}\right)^{1/2}}$$

(17)

Integration over p _ψ gives

$${(2\pi k_BTI_3)^{1/2}}$$

(18)

Eqs. (16, 17, 18) and (13) yield the partition function as

$${Z_{rot}=(2\pi k_BT)^{3/2}(I_1I_2I_3)^{1/2}\frac{1}{(2\pi\hbar)^3\sigma}\int\limits_0^{\pi}\sin\theta d\theta \int\limits_0^{2\pi} d\psi \int\limits_0^{2\pi}d\varphi =\frac{(2k_BT)^{3/2}(\pi I_1I_2I_3)^{1/2}}{\sigma\hbar^3}}$$

(19)

The sine of the polar angle in Eq. (19) appears as a result of integrations over the canonical conjugate moments. Thus, the partition function of restricted relative rotations is proportional to

$${\int\limits_{\theta_1}^{\theta_2} \sin\theta d\theta\int\limits_{\psi_1}^{\psi_2} d\psi\int\limits_{\varphi_1}^{\varphi_2}d\varphi=[\cos\theta_1-\cos\theta_2)][\varphi_2-\varphi_1][\psi_2-\psi_1]}$$

(20)