Combining Alchemical Transformation with a Physical Pathway to Accelerate Absolute Binding Free Energy Calculations of Charged Ligands to Enclosed Binding Sites

Abstract

We present a new approach to more accurately and efficiently compute the absolute binding free energy for receptor–ligand complexes. Currently, the double decoupling method (DDM) and the potential of mean force method (PMF) are widely used to compute the absolute binding free energy of biomolecular complexes. DDM relies on alchemically decoupling the ligand from its environments, which can be computationally challenging for large ligands and charged ligands because of the large magnitude of the decoupling free energies involved. In contrast, the PMF method uses a physical pathway to directly transfer the ligand from solution to the receptor binding pocket and thus avoids some of the aforementioned problems in DDM. However, the PMF method has its own drawbacks: because of its reliance on a ligand binding/unbinding pathway that is free of steric obstructions from the receptor atoms, the method has difficulty treating ligands with buried atoms. To overcome the limitation in the standard PMF approach and enable buried ligands to be treated, here we develop a new method called AlchemPMF in which steric obstructions along the physical pathway for binding are alchemically removed. We have tested the new approach on two important drug targets involving charged ligands. One is HIV-1 integrase bound to an allosteric inhibitor; the other is the human telomeric DNA G-quadruplex in complex with a natural product protoberberine buried in the binding pocket. For both systems, the new approach leads to more reliable estimates of absolute binding free energies with smaller error bars and closer agreements with experiments compared with those obtained from the existing methods, demonstrating the effectiveness of the new method in overcoming the hysteresis often encountered in PMF binding free energy calculations of such systems. The new approach could also be used to improve the sampling of water equilibration and resolvation of the binding pocket as the ligand is extracted.

Graphical Abstract

INTRODUCTION

Accurate prediction of binding affinity from the first principles of statistical thermodynamics is not only essential for understanding the physics of molecular recognition¹ but is also important for supporting structure-based drug discovery.² Among the several statistical thermodynamics-based methods developed for computing the absolute binding free energy, including the double decoupling method (DDM),³ the potential of mean force method (PMF),⁴ Metadynamics,⁵ and the Binding Energy Distribution method (BEDAM),⁶ DDM has been widely applied to many small- to medium-sized ligands with reasonably good accuracy. In DDM, the ligand is decoupled from the solvated complex and from the aqueous solution to enter the vacuum state that has a much higher energy. Modeling charged systems using DDM is particularly challenging due to the large electrostatic free energy changes involved and long-range electrostatic effects.⁷ In addition to charged ligands, the convergence of absolute binding free energy for large ligands is also difficult using DDM because of the large Lennard-Jones decoupling free energies involved.

By using a physical pathway to connect the bound and unbound states, the PMF approach avoids many of the aforementioned difficulties in DDM.^7e,8 However, two problems in the standard PMF method have severely limited its utility. First, the method relies on a geometric pathway of ligand extraction that is free of steric obstructions from the receptor atoms. This creates problems for ligands with deeply buried groups because the ligand will clash with the receptor at some point during the pulling process resulting in highly overestimated binding free energy. The second problem with the standard method is that for some systems during the ligand extraction phase of the PMF calculation bulk waters can be blocked from entering and resolvating the pocket by the unbinding ligand. If there is no “side channel” to facilitate the water exchange with the bulk solution, water equilibration is compromised, and in addition, the partially dry binding pocket can collapse, further exacerbating the sampling problems and preventing the convergence of the absolute binding free energy calculation.

In this work, we combine alchemical transformation with a physical pathway to overcome these limitations in the existing PMF method and accelerate absolute binding free energy calculations of charged ligands to enclosed binding sites. The new thermodynamic cycle involves decoupling or softening the nonbonded interactions between the clashing atoms before reversibly extracting the ligand from the binding pocket and restoring the modified interactions afterward. As shown below, the modified thermodynamic cycle helps eliminate major steric clashes in the PMF calculation. We tested the AlchemPMF method on two drug targets involving charged ligands. The first is HIV-1 integrase, a target for antiviral therapy, in complex with a charged allosteric inhibitor.⁹ The second is the human telomeric G-quadruplex DNA, a target for anticancer therapy, bound with a natural product protoberberine.¹⁰ Compared with the standard PMF method, the new AlchemPMF yields more reliable absolute binding free energy estimates that converge better and, in these cases, are in closer agreement with experimental results.

METHODS

PMF Approach for Absolute Binding Free Energy

In the standard PMF approach, the absolute binding free energy $\mathrm{\Delta }{G}_{bind}^{°}$ is evaluated using the expression^7e,8

$$\begin{gathered} \mathrm{\Delta }{G}_{bind}^{°}=-k_{\text{B}}T\ln \frac{V}{V_0}\frac{Z_{RL,N}}{Z_{R+L,N}} \\ =-k_{\text{B}}T\ln \frac{V}{V_0}\frac{Z_{RL,N}}{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}\frac{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R+L(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}\frac{Z_{R+L(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R+L,N}} \end{gathered}$$ (1)

which is based on inserting intermediate states between the unbound and bound states. Here,

$$V_0=\frac{1}{C^{°}}=1660{\mathrm{\AA }}^2$$

corresponds to the standard concentration of C° = 1 M solution. V is the volume of the simulation box. Z_RL,N is the configuration integral of one receptor–ligand complex solvated by N water molecules. Z_R+L,N, represents a system containing the receptor R, the unbound ligand L, and N waters. Z_{RL(θ,ϕ,Θ,Φ,Ψ),N)} is the configuration integral of a bound complex RL in which the ligand external degrees of freedom, i.e., the polar angles (θ, φ) and three Euler angles (Θ, Φ, Ψ) (Figure 1), are harmonically restrained to their equilibrium values. Z_{R+L(r*,θ,ϕ,Θ,Φ,Ψ),N} represents a system in which the ligand L is subject to both the polar and orientational restraints (U_θ, U_ϕ, U_Θ, U_Φ, U_Ψ) and the harmonic distance restraint U_r* that restrains the ligand to r* in the bulk solution.

Figure 1.

Coordinate frame used to define the ligand orientation and position relative to the receptor. Here, three receptor atoms, P1, P2, and P3, and three ligand atoms, L1, L2, and L3, are selected. The ligand extraction path is defined by the vector r_P1‑L1; therefor, P1 and L1 should be chosen to minimize steric obstruction along the entire pulling pathway. There are no specific requirements in selecting the other atoms, P2, P3, L2, and L3. The polar coordinates used to define the ligand position relative to the receptor are the distance |r_P1‑L1| and two angles θ (P2–P1– L1) and φ (P3–P2–P1–L1). The ligand orientation is defined by the three Euler angles: Θ (P1–L1–L2), Φ (P1–L1–L2–L3), and Ψ (P2–P1–L1–L2).

Figure 2 shows the thermodynamic cycle associated with eq 1: The first term

$\frac{Z_{RL,N}}{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}$

in the logarithm of eq 1 corresponds to the free energy of switching on the angular restraints (U_θ, U_ϕ, U_Θ, U_Φ, U_Ψ) on the bound ligand

$$\frac{Z_{RL,N}}{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}=e^{\mathrm{\Delta }{G}_{(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound}/k_{B}T}$$ (2)

In this work, $\mathrm{\Delta }{G}_{(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound}$ is computed using 14 λ values to gradually switch on the harmonic restraint in multiple λ windows, λ = 0.0, 0.01, 0.025, 0.05, 0.075, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 1.0. At each λ, the system is simulated for 0.8 ns, starting from the last simulated frame in the previous λ window. Here, $\mathrm{\Delta }{G}_{(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound}$ is then calculated using TI by numerically integrating ${\langle \frac{dU}{d\lambda }\rangle }_{\lambda }$ collected in each λ window.

Figure 2.

Thermodynamic pathway in the standard PMF method.

The second term

$$\frac{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R+L\left(r^{*},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }\right),N}}$$

is equal to the ratio of the probability of an angularly restrained bound complex and the probability of an unbound receptor–ligand system, in which the ligand location in the bulk and its orientation are restrained by $\left(U_{r^{*},U_{\theta },U_{\varphi },U_{\mathrm{\Theta }},U_{\mathrm{\Phi }},U_{\mathrm{\Psi }}}\right)$ We have previously shown that⁸

$$\begin{gathered} \frac{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R+L(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}=\frac{{\int }_{bound}{e}^{-[w(r)-w(r*)]/k_{B}T}dr}{{\left(\frac{2\pi k_{B}T}{k_r}\right)}^{1/2}} \\ =e^{w(r*)/k_{\text{B}}T}\times \frac{{\int }_{\mathit{bound}}{\text{e}}^{-w(r)/k_{\text{B}}T}dr}{{\left(\frac{2\pi k_{\text{B}}T}{k_r}\right)}^{1/2}} \end{gathered}$$ (3)

Here, w(r) is the 1D potential of mean force along the r_P1-L1 axis, which is computed using umbrella sampling in the presence of the angular restraints (U_θ, U_ϕ, U_Θ, U_Φ, U_Ψ). The numerator in the second term of the right-hand side of eq 3

$${\int }_{bound}{\text{e}}^{-w(r)/k_{\text{B}}T}dr$$

gives the effective range of ligand positional fluctuation along the radial direction, i.e., the r_P1-L1 axis in the bound state, while the denominator

$${\left(2\pi k_{\text{B}}T/k_r\right)}^{1/2}$$

measures the effective range of ligand positional fluctuation along the radial direction when harmonically restrained to an arbitrary location r* in the bulk solution by the force constant k_r. Since both the bound ligand and the ligand harmonically restrained near bulk location r* can be considered to undergo vibrations under harmonic potentials, we denote the second term of the right-hand side of eq 3 as

$$\frac{{\int }_{bound}{\text{e}}^{-w(r)/k_{\text{B}}T}dr}{{\left(\frac{2\pi k_{\text{B}}T}{k_r}\right)}^{1/2}}=e^{\mathrm{\Delta }{G}_{bound-bulk}^{vibr}/k_{\text{B}}T}$$

The term

$$\frac{Z_{R+L\left(r^{*},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }\right),N}}{Z_{R+L,N}}$$

in the right-hand side of eq 1 corresponds to the free energy of switching off the restraints $\left(U_{r^{*},U_{\theta },U_{\varphi },U_{\mathrm{\Theta }},U_{\mathrm{\Phi }},U_{\mathrm{\Psi }}}\right)$ on the unbound ligand and allowing it to occupy the standard volume 1/C° and rotate freely⁸

$$\frac{Z_{R+L(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R+L,N}}\approx \frac{r^{*2}\sin {\theta }_0\sin {\mathrm{\Theta }}_0{\left(2\pi k_{\mathrm{B}}\mathrm{T}\right)}^3}{8{\pi }^{2}V{\left(k_r{k}_{\theta }{k}_{\varphi }{k}_{\mathrm{\Theta }}{k}_{\mathrm{\Phi }}{k}_{\mathrm{\Psi }}\right)}^{1/2}}=e^{-\mathrm{\Delta }{G}_{restr}^{bulk}/k_{\mathrm{B}}T}$$ (4)

where θ ₀ and Θ₀ are the equilibrium values of the corresponding angles.

Combining eqs 1–4 gives $\mathrm{\Delta }{G}_{bind}^{°}$ in the standard PMF approach

$$\begin{gathered} \mathrm{\Delta }{G}_{bind}^{°}=-\mathrm{\Delta }{G}_{(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound}-w\left(r^{*}\right)-k_{\text{B}}T\ln \frac{{\int }_{bound}{e}^{-w(r)/k_{\text{B}}T}dr}{{\left(\frac{2\pi k_{\text{B}}T}{k_r}\right)}^{1/2}} \\ -k_{\text{B}}T\ln \frac{C^o{r}^{*2}\sin {\theta }_0\sin {\mathrm{\Theta }}_0{\left(2\pi k_{\text{B}}\text{T}\right)}^3}{8{\pi }^2{\left(k_r{k}_{\theta }{k}_{\varphi }{k}_{\mathrm{\Theta }}{k}_{\mathrm{\Phi }}{k}_{\mathrm{\Psi }}\right)}^{1/2}}=-\mathrm{\Delta }{G}_{(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound} \\ -w\left(r*\right)-\mathrm{\Delta }{G}_{bound-bulk}^{vibr}+\mathrm{\Delta }{G}_{restr}^{bulk} \end{gathered}$$ (5)

The derivation of eq 5 was given previously.^7e,8 In fact, $\mathrm{\Delta }{G}_{bind}^{°}$ has an equivalent expression

$$\begin{gathered} \mathrm{\Delta }{G}_{bind}^{°}=-\mathrm{\Delta }{G}_{\left(r_{site},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }\right)}^{bound}+[F_{\left(r_{\mathit{site}},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }\right)} \\ -F_{(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}]+\mathrm{\Delta }{G}_{restr}^{bulk} \end{gathered}$$ (5.1)

where $\mathrm{\Delta }{G}_{r_{site},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound}$ is the free energy of turning on the radial distance restraint $U_{r_{site}}$, which harmonically restrains the ligand to an arbitrary location r_site within the binding site, and also simultaneously switching on the angular restraints (U_θ, U_ϕ, U_Θ, U_Φ, U_Ψ) to restrain the orientations of the ligand relative to the receptor. F_{(r,θ,ϕ,Θ,Φ,Ψ)} is the free energy associated with the state in which the ligand is subject to both radial restraint U_r (centered on a position r) and angular restraints (U_θ, U_ϕ, U_Θ, U_Φ, U_Ψ). The free energies $F_{\left(r_{site},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }\right)}$ and $F_{\left(r^{*},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }\right)}$ are therefore the corresponding window free energies obtained from the WHAM (weighted histogram analysis method)¹¹ calculation with umbrella sampling along the radial direction r_P1-L1 (Figure 1). Their difference, $[F_{r_{site},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}-F_{r^{*}\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}]$, gives the reversible work of transferring an orientationally restrained ligand from r* to r_site. We have verified that eqs 5 and 5.1 yield essentially identical values of $\mathrm{\Delta }{G}_{bind}^{°}$.

Thermodynamic Pathway in AlchemPMF

When applying the standard PMF approach in systems with a more enclosed binding pocket, the steric clash between certain atoms in the ligand and the receptor will create high energy barriers during the umbrella sampling simulation for computing PMF (Figure 3). Such high free energy barriers will be difficult to converge because the protein residues involved in the clash with the unbinding ligand will not have sufficient time to sample the more open conformations, which would provide a clear pathway, within the tens of nanosecond time scale of the umbrella sampling. In addition, in some systems where there is no side channel to allow water to diffuse into the binding pocket, the unbinding ligand can block the entry of the bulk waters from resolvating the partially empty cavity, as illustrated in Figure 3B.

Figure 3.

High energy clash between the ligand and the enclosed binding pocket as the ligand is extracted. Water is shown in the ball-and-stick model.

To avoid the high energy atomic clashes, we modify the thermodynamic cycle as shown in Figure 4. First, identify the group of ligand and receptor atoms at the mouth of the pocket that will clash with each other along the unbinding pathway. Second, alchemically turn off or soften (see below) the interaction between the clashing atoms in the ligand and those in the receptor, as represented by the step A → B in Figure 4. In certain systems, where there are no side channels to allow water exchange with the bulk solvent, the interactions between these clashing atoms (of both the ligand and the receptor) and water are also turned off or softened, which will make the binding pocket more accessible to bulk waters as the ligand is being pulled out. All other interactions are left intact. The decoupling (softening) step A → B (Figure 4) is realized by using a number of alchemical λ values. Finally, the ligand is pulled out along the chosen pathway, which is now free from steric obstruction because the clashing atoms are decoupled. In addition, bulk waters can now freely enter the binding pocket (Figure 4B and C), allowing the binding pocket to be properly resolvated during pulling. The absolute binding free energy $\mathrm{\Delta }{G}_{bind}^{°}$ now includes the free energy for decoupling the clashing atoms in the bound state (step A → B, Figure 4), as well as a term for turning on their interactions with water in the dissociated state (step D → E, Figure 4). That is, eq 1 is modified by inserting two additional intermediate states, i.e.

Figure 4.

Thermodynamic pathway in AlchemPMF.

$$\begin{gathered} \mathrm{\Delta }{G}_{bind}^{°}=-k_{\text{B}}T\ln \frac{V}{V_0}\frac{Z_{RL,N}}{Z_{R+L,N}} \\ =-k_{\text{B}}T\ln \frac{V}{V_0}\frac{Z_{RL,N}}{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}\frac{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R_{\prime }{L}_{\prime }(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}\frac{Z_{R_{\prime }{L}_{\prime }(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R_{\prime }+L_{\prime }(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}\frac{Z_{R_{\prime }+L_{\prime }(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R+L(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}\frac{Z_{R+L(r*,\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R+L,N}} \\ =-\mathrm{\Delta }{G}_{(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound}-\mathrm{\Delta }{G}_{decouple}^{bound}-w\left(r*\right)-\mathrm{\Delta }{G}_{bound-bulk}^{vibr}-\mathrm{\Delta }{G}_{recouple}^{bulk}+\mathrm{\Delta }{G}_{restr}^{bulk} \end{gathered}$$ (6)

Here R′ and L′ represent the alchemically modified receptor and ligand in which the interactions between their clashing atoms are turned off or softened.

$$\mathrm{\Delta }{G}_{decouple}^{bound}=-k_{\text{B}}T\ln \frac{Z_{RL(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}{Z_{R_{\prime }{L}_{\prime }(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }),N}}$$

is the free energy of alchemically decoupling those clashing atoms in the bound state.

$$\mathrm{\Delta }{G}_{recouple}^{bulk}=-k_{\mathrm{B}}T\ln \frac{Z_{R_{\prime }+L_{\prime }\left(r^{*},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }\right),N}}{Z_{R+L\left(r^{*},\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi }\right),N}}$$

is the free energy of recoupling those clashing atoms when the ligand is unbound. Both $\mathrm{\Delta }{G}_{decouple}^{bound}$, and $\mathrm{\Delta }{G}_{recouple}^{bulk}$ can be obtained using FEP or TI.

In this work, we apply the AlchemPMF method to compute the absolute binding free energies of two receptor–ligand complexes, the HIV-1 integrase:BI224436 complex and the DNA G-quadruplex:EPI complex. For both systems, the decoupling of the clashing atoms in the ligand in the binding pocket is performed using a series of alchemical λ values. Eleven λ values are used to turn off the charges on the clashing atoms: λ _Coulomb = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. In order to maintain the same net charge on the ligand, it is necessary to simultaneously transfer the net charge on the clashing atoms to the neighboring ligand atoms. Seventeen λ values are used to turn off the Lennard-Jones interaction involving the clashing atoms: λ_LJ = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.94, 0.985, and 1.0. The Lennard-Jones decoupling is performed using the soft-core potential in GROMACS. The charge interaction is switched off first, followed by the decoupling of the Lennard-Jones. The resulting decoupling free energies

$$\mathrm{\Delta }{G}_{decoupl\_Coulomb}={\int }_{{\lambda }_{Coulomb}=0}^{{\lambda }_{Coulomb}=1}{\langle \frac{\mathit{dU}}{d{\lambda }_{Coulomb}}\rangle }_{{\lambda }_{Coulomb}}d{\lambda }_{Coulomb}$$

and

$$\mathrm{\Delta }{G}_{decoup{l}_{-}LJ}={\int }_{{\lambda }_{LJ}=0}^{{\lambda }_{LJ}=1}{\langle \frac{dU}{d{\lambda }_{LJ}}\rangle }_{{\lambda }_{LJ}}d{\lambda }_{LJ}$$

are computed using TI. Here, a 10 ns decoupling simulation is performed at each λ, with the first 5 ns treated as equilibration and the last 5 ns used to compute the decoupling free energies. A total of approximately a 280 ns simulation is used to compute a pair of ΔG_decoupl__Coulomb and ΔG_decoupl__LJ values. For each ligand, three sets of independent decoupling simulations are performed with different initial velocities to estimate the statistical error.

The recoupling of the clashing atoms is performed when the ligand is pulled out of the binding pocket and resides in the bulk location. The same set of λ values is used in the recoupling phase as is in the decoupling phase described above. The only difference is that here the Lennard-Jones interactions are turned on first, followed by turning on the Coulomb interactions. The statistical errors are estimated in the same way as those in the decoupling step.

In this work, we have also experimented with decoupling only 50% of the Lennard-Jones interactions between the clashing atoms and the environment to see if a partial softening of the collision will be sufficient to reduce the barriers to pulling. Here, the Coulomb interactions are still completely turned off using 11 λ: λ_Coulomb = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. The following λ values are used to turn off 50% of the Lenard-Jones interactions involving the clashing atoms: λ_LJ = 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5. The same set of λ values are used in the recoupling phase when the ligand is extracted from the binding pocket and resides in the bulk solution.

Umbrella Sampling Simulation

Umbrella sampling is used to compute the 1D PMF w(r), where r = |r_P1-L1| (Figure 1). For the systems studied here, between 24 and 30 umbrella windows are used to cover the full range of the distance space using a single force constant k_r = 1000 kJ mol⁻¹ nm⁻². In each umbrella window, a reasonable starting configuration is generated by first running a short 1.2 ns umbrella sampling simulation starting from the last snapshot of the short umbrella sampling simulation in the previous umbrella window. Next, between 20 and 30 ns production umbrella sampling simulations are run in parallel in each window starting from the initial configurations generated above.

The WHAM program implemented by Grossfield is used to calculate the potential of mean force.¹² We run five independent umbrella sampling simulations to estimate the statistical uncertainties in the calculated PMF. Thus, a total of approximately a 2.1 μs simulation time is used to compute PMF for a single receptor–ligand complex.

Trial and error are used to identify the ligand pulling pathways for the PMF calculation. First, the ligand atom L1 and receptor atom P1 are chosen to define axis r_P1-L1. Then, in Discovery Studio (DS) Visualizer (Biovia, Inc.), the ligand is translated manually along this axis with a fixed orientation relative to the receptor. If major collisions occur with the nearby receptor atoms, (shown as red dashed line in the DS Visualizer), a different set of P1-L1 will be tried. One or more low barrier paths may be identified after a few tries. As an example, in the case of the HIV-1 integrase:BI-224436 complex, two such low free energy paths are found (Figures S1 and S2, Supporting Information). PMF result reported here uses the lowest free energy path.

DDM Calculation Setup

In DDM^3,13 method, two legs of decoupling simulations are performed. In one leg, a positionally restrained ligand is decoupled from the binding site; in the other, an unbound ligand is decoupled from the bulk solution. The intermolecular Coulomb interaction is decoupled first, followed by the Lennard-Jones decoupling. The absolute binding free energy $\mathrm{\Delta }{G}_{bind}^{°}$ is

$$\mathrm{\Delta }{G}_{bind}^{°}=-\mathrm{\Delta }{G}_{restr}^{bound}-\mathrm{\Delta }{G}_{decoupl}^{bound}+\mathrm{\Delta }{G}_{decoupl}^{bulk}+\mathrm{\Delta }{G}_{restr}^{gas}$$ (7)

Here $\mathrm{\Delta }{G}_{bound}^{restr}$ is the free energy of switching on the ligand–receptor distance restraint on a fully coupled receptor–ligand complex. The magnitude of this term is typically small and can therefore be estimated by using the Zwanzig formula,³

$$\mathrm{\Delta }{G}_{restr}^{bound}=k_{\text{B}}T\ln {\langle e^{U_{restr}(\xi )/k_{\text{B}}T}\rangle }_{U_{restr}}$$

in which ξ, the position of the ligand center relative to the binding site, is harmonically restrained by U_restr (ξ)

In eq 7, $\mathrm{\Delta }{G}_{decoupl}^{bound}$ is the free energy of decoupling the ligand from its environment, with the ligand restrained by U_restr (ξ) to remain in the binding site

$$\mathrm{\Delta }{G}_{decoupl}^{bound}=\mathrm{\Delta }{G}_{decoupl-Coulomb}^{bound}+\mathrm{\Delta }{G}_{decoupl-LJ}^{bound}$$ (7.1)

where $\mathrm{\Delta }{G}_{decoupl-Coulomb}^{bound}$ and $\mathrm{\Delta }{G}_{bound-LJ}^{bound}$ are computed using TI. $\mathrm{\Delta }{G}_{decoupl}^{bulk}$ in eq 7 is the free energy of decoupling a free ligand in solution from its solvent environment

$$\mathrm{\Delta }{G}_{decoupl}^{bulk}=\mathrm{\Delta }{G}_{decoupl-Coulomb}^{bulk}+\mathrm{\Delta }{G}_{decoupl-LJ}^{bulk}$$ (7.2)

which is also calculated using TI.

Lastly, $\mathrm{\Delta }{G}_{gas}^{restr}$ in eq 7 is the free energy of switching on the harmonic distance restraint U_restr (ξ) on the fully decoupled, gas-phase ligand; this term has an analytical expression

$$\mathrm{\Delta }{G}_{restr}^{gas}=-k_{\text{B}}T\ln \frac{1}{V_0}{\left(\frac{2\pi k_{\text{B}}T}{k_{\xi }}\right)}^{3/2}$$ (7.3)

where k_ξ is the force constant of the ligand–receptor distance restraint. In this work, k_ξ = 1000 kJ/nm² mol.

Thus, the absolute binding free energy $\mathrm{\Delta }{G}_{bind}^{°}$ in DDM is written as

$$\mathrm{\Delta }{G}_{bind}^{°}=-\mathrm{\Delta }{G}_{restr}^{bound}+\mathrm{\Delta }\mathrm{\Delta }{G}_{Coulomb}+\mathrm{\Delta }\mathrm{\Delta }{G}_{LJ}+\mathrm{\Delta }{G}_{restr}^{gas}$$ (8)

where $\mathrm{\Delta }\mathrm{\Delta }{G}_{Coulomb}=-\mathrm{\Delta }{G}_{decouple-Coulomb}^{bound}+\mathrm{\Delta }{G}_{decoupl-Coulomb}^{bulk}$, and $\mathrm{\Delta }\mathrm{\Delta }{G}_{LJ}=-\mathrm{\Delta }{G}_{decoupl-LJ}^{bound}+\mathrm{\Delta }{G}_{decoupl-LJ}^{bulk}$.

For charged ligands, the finite-size, periodic simulation box can introduce a non-negligible error on the calculated electrostatic charging free energies $\mathrm{\Delta }{G}_{decoupl\_Coulomb}^{bound}$ and $\mathrm{\Delta }{G}_{decoupl\_Coulomb}^{bulk}$. Such errors can be corrected by using a scheme developed by Rocklin and coworkers^7a that includes the following terms: periodicity-induced net-charge interactions, periodicity-induced net-charge undersolvation, discrete solvent effects, and residual integrated potential effects. Three Poisson–Boltzmann calculations (PB) of electrostatic potentials for different combinations of receptor and ligand charges are required for the calculation of the residual integrated potential. Here, the APBS program¹⁴ is used for the PB calculations. Taken together, the DDM expression of $\mathrm{\Delta }{G}_{bind}^{°}$ is

$$\mathrm{\Delta }{G}_{bind}^{°}=-\mathrm{\Delta }{G}_{restr}^{bound}+\mathrm{\Delta }\mathrm{\Delta }{G}_{Coulomb}+\mathrm{\Delta }\mathrm{\Delta }{G}_{LJ}+\mathrm{\Delta }{G}_{elec\_corr}^{finite\_size}+\mathrm{\Delta }{G}_{restr}^{gas}$$ (9)

In this work, the Coulomb decoupling is performed using 11 λ values: λ_Coulomb = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0; the Lennard-Jones decoupling is carried out using 17 λ values: λ_LJ = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.94, 0.985, and 1.0. The resulting decoupling free energies are calculated using TI

$$\mathrm{\Delta }{G}_{decoupl\_Coulomb}={\int }_{{\lambda }_{Coulomb}=0}^{{\lambda }_{Coulomb}=1}{\langle \frac{\mathit{dU}}{d{\lambda }_{\mathit{Coulomb}}}\rangle }_{{\lambda }_{Coulomb}}d{\lambda }_{Coulomb}$$

and

$$\mathrm{\Delta }{G}_{decoupl\_LJ}={\int }_{{\lambda }_{LJ}=0}^{{\lambda }_{LJ}=1}{\langle \frac{dU}{d{\lambda }_{LJ}}\rangle }_{{\lambda }_{LJ}}d{\lambda }_{LJ}$$

To avoid instabilities at the end point, the soft-core potential implemented in GROMACS is used for the Lennard-Jones decoupling. Here, a 20 ns decoupling simulation is performed at each λ, with the first 10 ns treated as equilibration and the last 10 ns used to compute free energy. Thus, calculating one absolute binding free energy value uses a total of approximately a 1.12 μs simulation time. For each ligand, three independent DDM runs are performed to estimate the statistical error.

MD Simulation Setup

HIV-1 Integrase:BI-224436 Complex

In this work, the MD free energy simulations were performed using the GROMACS program 2016–3.¹⁵ BI-224436 (Figure 5) is known to bind strongly at the catalytic core dimer interface of HIV-1 integrase.^9b The structure of BI-224436 in complex with HIV-1 integrase was obtained by using the crystal structure of the integrase bound with a similar ligand BI-D¹⁶ as the template input to the Glide docking program (Schrodinger, Inc.)¹⁷ The resulting bound structure features the key interactions of the bidentate hydrogen bonds between the carboxylate group on the ligand and the backbone amide hydrogens of E170/H171, which are the hallmark of this class of the allosteric inhibitors of HIV-1 integrase.^16,18 The AMBER parm99ILDN force field¹⁹ is used to describe the HIV-1 integrase; the Amber GAFF²⁰ and the AM1-bcc charge model²¹ are used to model the ligand BI-224436. A TIP3P water²² box is used to solvate the protein–ligand complex. The distance between the solute and the nearest wall of the box is ≥10 Å. To ensure that the simulation boxes have zero net charge, one Cl⁻ and one Na⁺ are added to the solvent boxes containing the protein–ligand complex and that containing the ligand, respectively. The electrostatic interactions were treated using the particle-mesh Ewald (PME)²³ method with a real space cutoff of 10 Å and a grid spacing of 1.0 Å. MD simulations were performed in the NPT ensemble with a time step of 2 fs.

Figure 5.

BI-224436 in the allosteric site of the HIV-1 integrase catalytic core dimer. The tricyclic group (circled) is decoupled in the AlchemPMF calculation.

The following three protein atoms and three ligand atoms (Figure 1) are chosen to define the ligand pulling axis r_P1‑L1 and the five angles (θ, ϕ, Θ, Φ, Ψ) that determine the ligand orientation: P1 (Y99–CA), P2 (A98–CA), P3 (Q95–O), L1 (C5), L2 (C17), and L3 (C3). The range of the pulling distance is 9.325 Å ≤ |r_P1‑L1| ≤ 21.675 Å. The value of r* used in eqs 5 and 6 for computing $\mathrm{\Delta }{G}_{bind}^{°}$ is 21.7 Å. At this distance, the PMF w(r) is essentially flat. As discussed in previous reports,^7e,8 the estimated binding free energy is insensitive to the precise choice of r*, as long as it is in the bulk region. The equilibrium values of the five angles are θ₀ = 48.8°,ϕ₀ = −56.58°, Θ₀ = 50.94°, Φ₀ = −145.78°, and Ψ₀ = 164.25°. A single force constant k_r = 1000 kJ mol⁻¹ nm⁻² is used for the distance restraints in all the umbrella sampling windows. The force constants used in the angular restraints are k_θ = k_ϕ = k_Θ = k_Φ = k_Ψ = 1000 kJ rad⁻¹ mol⁻¹.

Human Telomeric DNA G-Quadruplex:Epiberberine Complex

The starting structure for the MD simulations is the solution NMR structure of epiberberine (EPI) in complex with the hybrid-2 human telomeric G-quadruplex (PDB entry: 6ccw).¹⁰ The MD simulations are performed using the GROMACS program 2016–3.¹⁵ The AMBER parm99bsc0²⁴ force field is used to model the G-quadruplex DNA in aqueous solutions. EPI is modeled by the Amber GAFF parameters set²⁰ and the AM1-bcc charge model.²¹ A truncated octahedral box containing TIP3P²² water molecules is used to solvate the ligand–DNA complexes. The solvent box is set up to ensure that the distance between solute atoms from nearest walls of the box is at least 15 Å. Here, 37 K⁺ and 15 Cl⁻ ions are added to the solvent box to maintain charge neutrality and an ionic concentration of 0.1 M, which is the salt concentration used in the experiment.¹⁰ The electrostatic interactions were computed using the particle-mesh Ewald (PME) method²³ with a real space cutoff of 10 Å and a grid spacing of 1.0 Å. A 2 fs time step is used for MD simulations. Before the production run, the system is equilibrated in a number of steps. First, the energy-minimized system is heated from 0 to 300 K in 200 ps, while the solute atoms are restrained with a force constant k = 10 kcal/Ų mol. Then the system is equilibrated for 8 ns with decreasing harmonic restraints: 2 ns with k = 5 kcal/Ų mol, 2 ns with k = 1 kcal/Ų mol, 2 ns with k = 0.25 kcal/Ų mol, and 2 ns with k = 0.05 kcal/Ų mol. The production MD is run in the NPT ensemble using the Langevin thermostat and isotropic position scaling for constant pressure.

The ligand pulling axis r_P1‑L1 and the five angles (θ, ϕ, Θ, Φ, Ψ) that determine the ligand orientation are defiend by P1 (G16–O5′), P2 (G16–N9), P3 (G16–N2), L1 (C15), L2 (C4), and L3 (C6). The range of the pulling distance is 9 Å ≤ | r_P1‑L1 | ≤ 25.5 Å. The value of r* used in eqs 5 and 6 for computing $\mathrm{\Delta }{G}_{bind}^{°}$ is 26.5 Å. The equilibrium values of the five angles are θ₀ = 48.77°, ϕ₀ = 39.07°, Θ₀ = 148.99°, Φ₀ = 175.23°, and Ψ₀ = −129.73°. A single force constant k_r = 1000 kJ mol⁻¹ nm⁻² is used for the distance restraint in all the umbrella sampling windows. The force constants used in the angular restraints are k_θ = k_ϕ = k_Θ = k_Φ = k_Ψ = 1000 kJ rad⁻¹ mol⁻¹.

RESULTS

AlchemPMF Significantly Accelerates Convergence of the Calculated $\mathbf{\Delta }{\mathit{G}}_{\mathit{bind}}^{°}$ in the HIV-1 Integrase:BI-224436 Complex

We first apply the AlchemPMF method to compute the absolute binding free energies for the charged ligand BI-224436 that binds at an allosteric site in the HIV-1 integrase catalytic core domain (CCD) dimer, a target that has attracted significant interests in recent years for developing antiviral therapies.^{9b,c,16,18,25} BI-224436, a potent allosteric integrase inhibitor and once a clinical candidate^9a,26 (Figure 5), carries a negative charge at pH 7. In this ligand, the calculation of $\mathrm{\Delta }{G}_{bind}^{°}$ by using the regular PMF method (eq 5) is hampered by the presence of the bulky tricyclic group, which is enclosed in the binding pocket (Figure 5). The tricyclic group, which is involved in steric hindrance with the binding site residues during ligand extraction, causes the potential of mean force w(r) to be overestimated: see Figure 6. As a result, $\mathrm{\Delta }{G}_{bind}^{°}$ computed using the simple PMF converges slowly: with 30 ns of simulation in each umbrella sampling window (total simulation time: 24 × 30 = 720 ns), the PMF-calculated $\mathrm{\Delta }{G}_{\text{bind}}^{°}=-13.77\pm 2.46\text{kcal}/\text{mol}$, compared with the experimental result $\mathrm{\Delta }{G}_{bind}^{°}=-10.33\text{kcal}/\text{mol}$ from Surface Plasmon Resonance (SPR) measurements (Figure 7).

Figure 6.

Potential of mean force w(r) of BI-224436 binding obtained from the simple PMF and AlchemPMF methods using the last 10 ns of simulation trajectories.

Figure 7.

Computed absolute binding free energy as a function of simulation time using the simple PMF approach and AlchemPMF.

Using the new method AlchemPMF, we first alchemically modify the ligand in the bound state by decoupling the nonbonded interactions between the tricyclic group (Figure 5) of the ligand with both the protein and water molecules in the system and then run umbrella sampling to compute PMF of pulling the modified ligand out of the binding pocket. Note that other interactions, including the ligand intramolecular interaction with the tricyclic group, are not modified. When the ligand is in the bulk solution, we recouple the tricyclic group to its environment. Finally, $\mathrm{\Delta }{G}_{bind}^{°}$ is computed by using eq 6. As shown from Figure 7, $\mathrm{\Delta }{G}_{bind}^{°}$ computed from AlchemPMF converges significantly faster: using the last 10 ns simulation data, $\mathrm{\Delta }{G}_{bind}^{°}$ computed by AlchemPMF has smaller error bars than those computed using the standard PMF method (Figures 6 and 7). In addition, $\mathrm{\Delta }{G}_{bind}^{°}$ computed by AlchemPMF is also in closer agreement with the experimental value. Figure 6 shows PMF w(r) computed using the simple PMF versus AlchemPMF. Compared with PMF for the unmodified ligand, the free energy cost along the same pulling pathway is substantially smaller for the alchemically modified ligand. Clearly, alchemically softening the tricyclic group in the ligand helps to eliminate the high energy steric clash with the receptor in the course of ligand pulling. Table 1 gives the breakdown of the total binding free energy in the two methods. Compared with the standard PMF result, the much lowered reversible work −w(r*) from AlchemPMF is properly compensated by adding back the free energy difference of coupling the tricyclic group of the ligand in the complex and that in the bulk solution, resulting in a more precise estimate of $\mathrm{\Delta }{G}_{bind}^{°}$.

Table 1.

Components of Absolute Binding Free Energy for HIV-1 Integrse:BI224436 Association Calculated Using the Simple PMF Method and AlchemPMF

Component	Simple PMF	AlchemPMF
−w(r*)	−19.31 ± 2.48	−5.81 ± 1.12
$\mathrm{\Delta }{G}_{restr}^{bulk}$	9.73	9.73
$-\mathrm{\Delta }{G}_{(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound}$	−4.23 ± 0.15	−4.23 ± 0.15
$-\mathrm{\Delta }{G}_{bound-bulk}^{vibr}$	0.05 ± 0.03	−0.11 ± 0.03
$-\mathrm{\Delta }{G}_{decouple}^{bound}-\mathrm{\Delta }{G}_{recouple}^{bulk}$	0	−10.03 ± 0.2
$\mathrm{\Delta }{G}_{bind}^{\circ }$a	−13.77 ± 2.46	−10.44 ± 1.15
$\mathrm{\Delta }{G}_{bind}^{°}(expt)$b	−10.33	−10.33

^a$\mathrm{\Delta }{G}_{bind}^{\circ }$ was obtained using the last 10 ns trajectory segment in each of the umbrella sampling windows.

^bFrom SPR measurements performed by the Kvaratskhelia lab.^7e

We also computed the absolute binding free energy of the HIV-1 integrse:BI224436 complex using the standard DDM method, (Table 2). Here, since ligand BI224436 carries a net charge, to correct for the error caused by the finite size, periodic solvent box, we used the procedure developed by Rocklin et al.^7a which involves three separate Poisson–Boltzmann electrostatic potential calculations using the APBS program.¹⁴ Using the same sampling time of 20 ns per window, the DDM-calculated $\mathrm{\Delta }{G}_{bind}^{°}=-12.49\text{kcal}/\text{mol}$ shows a slightly larger deviation from the experimental value of −10.33 kcal/mol compared with the result of $\mathrm{\Delta }{G}_{bind}^{°}=-11.51\text{kcal}/\mathrm{mol}$ calculated by AlchemPMF.

Table 2.

Binding Free Energy Components of HIV-1 Integrse:BI224436 Computed from DDM^a

ΔΔGCoulomb	ΔΔGLJ	$\mathbf{\Delta }{\mathit{G}}_{\mathit{b}\mathit{i}\mathit{n}\mathit{d}}^{°}\mathbf{\Delta }{\mathit{G}}_{\mathit{r}\mathit{e}\mathit{s}\mathit{t}\mathit{r}}^{\mathit{g}\mathit{a}\mathit{s}}$	$\mathbf{\Delta }{\mathit{G}}_{\mathit{e}\mathit{l}\mathit{e}\mathit{c}\_\mathit{c}\mathit{o}\mathit{r}\mathit{r}}^{\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{e}\_\mathit{s}\mathit{i}\mathit{z}\mathit{e}}$	$-\mathbf{\Delta }{\mathit{G}}_{\mathit{r}\mathit{e}\mathit{s}\mathit{t}\mathit{r}}^{\mathit{c}\mathit{o}\mathit{u}\mathit{p}\mathit{l}\mathit{e}\mathit{d}}$	$\mathbf{\Delta }{\mathit{G}}_{\mathit{b}\mathit{i}\mathit{n}\mathit{d}}^{°}$
−2.42 ± 1.23	−11.39 ± 0.48	3.23 ± 0	−1.88 ± 0.03	−0.03 ± 0.01	−12.49 ± 0.75

^aUnits are kcal/mol. See eq 9 in Methods for the definitions of the different binding free energy terms.

AlchemPMF Leads to More Reliable Estimates of $\mathbf{\Delta }{\mathit{G}}_{\mathit{bind}}^{°}$ for the Human Telomeric DNA G-Quadruplex:- Epiberberine Complex

We also applied AlchemPMF to study the binding of epiberberine (EPI) with the human telomeric G-quadruplex DNA (Figure 8),¹⁰ an attractive target for developing anticancer drugs.²⁷ In this complex, the ligand in the binding site is significantly more enclosed compared with the complex of HIV-1 integrase:BI-224436 (Figure 5). As a result, the application of the standard PMF method resulted in a highly overestimated binding free energy, $\mathrm{\Delta }{G}_{bind}^{°}=-34.61\pm 4.88\text{kcal}/\text{mol}$ (Table 3). Experimentally, the standard binding free energy was determined to be $\mathrm{\Delta }{G}_{bind}^{°}(expt)=-10.7\text{kcal}/\text{mol}$.²⁸ Using the AlchemPMF method, we identify the ligand atoms involved in the clash with the G-quadruplex (Figure 8, circled) and decouple these atoms from the environment before performing ligand extraction in umbrella sampling. This leads to a much flatter w(r) from umbrella sampling simulation (Figure 9). After accounting for the free energy of decoupling the clashing atoms in the bound state and that in the dissociated state, the AlchemPMF calculation yields $\mathrm{\Delta }{G}_{bind}^{°}=-15.03\pm 0.74\text{kcal}/\text{mol}$, which is a much more reliable estimate of the absolute binding free energy than the result from the simple PMF method (see Table 3, which gives the free energy components computed using the two methods). It is noted that while AlchemPMF has resulted in much improved $\mathrm{\Delta }{G}_{bind}^{°}$, it is still about −4.3 kcal/mol overestimated compared with the experimental result. This is because the DNA reorganization free energy ΔG_reorg has not been fully accounted for in our calculations. Experimentally, it has been shown that the binding of EPI induces extensive rearrangement in the binding site region of the DNA; a very large conformational change can be seen by comparing the NMR structure of the apo DNA²⁹ and EPI-bound structure.¹⁰ This conformational transition from the holo to apo structures is not adequately sampled in our nanosecond time scale simulations, which is largely responsible for the overestimation in $\mathrm{\Delta }{G}_{bind}^{°}$ from AlchemPMF. From our calculation, the EPI-induced reorganization free energy ΔG_reorg for the holo to apo transition is expected to be quite large:¹⁰ comparing the AlchemPMF-calculated $\mathrm{\Delta }{G}_{bind}^{°}$ with the experimental value suggests that ΔG_reorg is about 4 kcal/mol. We note that the effect of reorganization could be treated systematically by imposing conformational restraints in the bound state and subsequently releasing them in the dissociated state. Without using restraints, the reorganization is an uncontrolled effect. Since the focus of the current work is using AlchemPMF to address the atomic clash problem in computing PMF, the sampling of the holo to apo conformational transition will be addressed separately.

Figure 8.

EPI (spheres) buried in the 5′ pocket of the human telomeric G-quadruplex. (Left) Clashing atoms (circled) that are decoupled in the AlchemPMF calculation.

Table 3.

Absolute Binding Free Energy Components for EPI-G4 Complex Calculated Using the Simple PMF Approach and AlchemPMF

Component	Simple PMF	AlchemPMF
−w(r*)	−42.55 ± 4.92	−10.69 ± 0.82
$\mathrm{\Delta }{G}_{restr}^{bulk}$	9.79	9.79
$-\mathrm{\Delta }{G}_{(\theta ,\varphi ,\mathrm{\Theta },\mathrm{\Phi },\mathrm{\Psi })}^{bound}$	−2.57 ± 0.20	−2.57 ± 0.20
$-\mathrm{\Delta }{G}_{bound-bulk}^{vibr}$	0.77 ± 0.004	0.04 ± 0.06
$-\mathrm{\Delta }{G}_{decouple}^{bound}-\mathrm{\Delta }{G}_{recouple}^{bulk}$	0	−11.59 ± 0.10
$\mathrm{\Delta }{G}_{bind}^{°}$a	−34.61 ± 4.88	−15.03 ± 0.74
$\mathrm{\Delta }{G}_{bind}^{°}(expt)$b	−10.7	−10.7

^a$\mathrm{\Delta }{G}_{bind}^{°}$ was obtained using the last 10 ns trajectory segment in each of the umbrella sampling windows.

^bDetermined from fluorescence measurements.²⁸

Figure 9.

PMF of EPI-G4 binding computed using standard PMF and AlchemPMF.

We also calculated the absolute binding free energy of the EPI-G4 complex using the DDM method (Table 4). Here, since the ligand carries a net charge, the DDM calculation also involves the calculation to correct for the error in the electrostatic charging free energies due to the finite size, periodic solvent box.^7a As shown in Table 4, the DDM result of $\mathrm{\Delta }{G}_{bind}^{°}=-16.62\text{kcal}/\text{mol}$ shows a larger deviation from the experimental value of −10.7 kcal/mol compared with the result of $\mathrm{\Delta }{G}_{bind}^{°}=-15.03\text{kcal}/\text{mol}$ obtained from AlchemPMF (Table 3).

Table 4.

Binding Free Energy Components of EPI-G4 Complex Calculated from DDM^a

ΔΔGCoulomb	ΔΔGLJ	$\mathbf{\Delta }{\mathit{G}}_{\mathit{r}\mathit{e}\mathit{s}\mathit{t}\mathit{r}}^{\mathit{g}\mathit{a}\mathit{s}}$	$\mathbf{\Delta }{\mathit{G}}_{\mathit{e}\mathit{l}\mathit{e}\mathit{c}\_\mathit{c}\mathit{o}\mathit{r}\mathit{r}}^{\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{e}\_\mathit{s}\mathit{i}\mathit{z}\mathit{e}}$	$-\mathbf{\Delta }{\mathit{G}}_{\mathit{r}\mathit{e}\mathit{s}\mathit{t}\mathit{r}}^{\mathit{c}\mathit{o}\mathit{u}\mathit{p}\mathit{l}\mathit{e}\mathit{d}}$	$\mathbf{\Delta }{\mathit{G}}_{\mathit{b}\mathit{i}\mathit{n}\mathit{d}}^{°}$
−0.3 ± 0.05	−18.99 ± 0.50	3.23	−0.52 ± 0.01	−0.03 ± 0.01	−16.62 ± 0.45

^aUnites are kcal/mol. See eq 9 in Methods for the definitions of the different binding free energy terms.

DISCUSSION

In this work, we described a modified thermodynamic cycle, AlchemPMF, to overcome the main limitation arising from ligand–receptor steric clashes when computing absolute binding free energy using the standard PMF approach. In AlchemPMF, the ligand atoms (and receptor atoms) that are potentially involved in the steric clash are alchemically decoupled to reduce the impact severity of the clash thereby significantly improving the convergence of the calculation of the reversible work for ligand extraction. The value of PMF of the unbound state relative to the bound state thus obtained becomes artificially low, which must be accounted for by including in the final binding free energy expression the free energies of decoupling in the bound state and recoupling in the dissociated state. The key to the success of AlchemPMF relies on the assumption that the computed free energy of decoupling a few ligand atoms involved in steric clashes converges much faster than converging PMF for ligand extraction in the presence of steric clash. To properly converge the latter would require sampling reversibly the large conformational opening of the binding pocket to provide a clear pathway, which is generally very challenging, as it may require identifying and activating specific collective motions leading to the pocket opening, which could involve many receptor atoms. Our results on the HIV-1 integrase:BI-224436 complex and the human telomeric DNA G-quadruplex:Epiberberine complex suggest that AlchemPMF provides an effective way that avoids sampling of the reversible opening of the binding pocket.

In fact, even a partial softening of the interaction involving the clashing atoms can lead to substantial improvement in some systems where the buried ligand is not extensive, such as the HIV-1 integrase:BI-224436 complex. We have compared the convergence in $\mathrm{\Delta }{G}_{bind}^{°}$ for the HIV-1 integrase:BI-224436 binding from simple PMF and AlchemPMF, in which the Lennard-Jones interaction between the ligand tricyclic group and the environment is weakened by 50% in the bound state (Methods). (Their Coulomb interaction is still fully turned off as in the regular AlchemPMF.) Using the last 10 ns of the umbrella sampling data, the partial softening AlchemPMF yields $\mathrm{\Delta }{G}_{bind}^{°}=-10.40\pm 1.47\text{kcal}/\text{mol}$. Compared with the standard PMF-estimated $\mathrm{\Delta }{G}_{bind}^{°}=-13.77\pm 2.46\text{kcal}/\text{mol}$, the partial decoupling of the clashing atoms leads to a better converged absolute binding free energy. We note that in this system the ligand is not extensively buried, and therefore, a partial weakening of the interaction between the potentially sterically hindered groups is sufficient to yield improved result.

In this work, we also compared the absolute binding free energies computed using DDM with those estimated using the new AlchemPMF. For both test cases studied here, we found that AlchemPMF yields better agreement with the experimental results, but the improvement over DDM is small, e.g., approximately 1–1.6 kcal/mol. This is mainly because DDM already performs well for the two relatively small ligands studied here. Since the limitation of DDM in computing absolute binding free energy will be more pronounced in large ligands, whose alchemical decoupling is very difficult to converge, we expect to see more significant advantages of the AlchemPMF method over DDM in treating the binding of larger-sized ligands.

In the two test systems studied in this work using AlchemPMF, we observed rapid equilibration of water molecules in the binding cavity, since in these two systems there exists side channels that allow water exchange during ligand extraction. However, in cases where no side channels exist to allow water exchange, certain “gate-keeper” atoms in the receptor could also be alchemically softened to facilitate water exchange and resolvation of the binding pocket, as illustrated in Figure 4. In addition to the scheme of Figure 4, the recently published Hamiltonian Simulated Annealing of Solvent (HSAS)³⁰ method may be combined with the thermodynamic cycle of AlchemPMF to accelerate the water equilibration in PMF calculations of the absolute binding free energy.

CONCLUSIONS

The PMF method is a powerful method for computing absolute binding free energy,^4a,31 especially in treating binding problems involving charged ligands.^7e,8 However, the PMF method is not suitable when the binding site is significantly enclosed; this is because a converged calculation of PMF depends on the existence of a ligand extraction pathway that is free of steric obstruction from receptor atoms. To overcome this major limitation in the standard PMF method, we develop a novel method called AlchemPMF by combining the alchemical transformation with a physical pathway to remove steric obstructions in the calculation of PMF. Our test results on the binding of the charged ligands to HIV-1 integrase and the human telomeric G-quadruplex DNA demonstrate that AlchemPMF leads to significantly improved absolute binding free energy estimates and smaller statistical errors compared with those given by the standard PMF method. The new method is expected to allow a broader range of binding problems to be treated accurately, including protein–peptide, protein–protein, and protein–DNA complexes, which involve larger and more complex binding interfaces compared with those in the protein–small molecule complexes.