Accurate Determination of the Binding Free Energy for KcsA-Charybdotoxin Complex from the Potential of Mean Force Calculations with Restraints

Abstract

Free energy calculations for protein-ligand dissociation have been tested and validated for small ligands (50 atoms or less), but there has been a paucity of studies for larger, peptide-size ligands due to computational limitations. Previously we have studied the energetics of dissociation in a potassium channel-charybdotoxin complex by using umbrella sampling molecular-dynamics simulations, and established the need for carefully chosen coordinates and restraints to maintain the physiological ligand conformation. Here we address the ligand integrity problem further by constructing additional potential of mean forces for dissociation of charybdotoxin using restraints. We show that the large discrepancies in binding free energy arising from simulation artifacts can be avoided by using appropriate restraints on the ligand, which enables determination of the binding free energy within the chemical accuracy. We make several suggestions for optimal choices of harmonic potential parameters and restraints to be used in binding studies of large ligands.

Introduction

The in silico calculation of the binding free energies of ligands provides a bridge between structures of protein-ligand complexes and measurements of their interactions (1,2). Its successes and challenges have been well documented for a broad range of druglike molecules and small peptides, with a reported accuracy of a few kcal mol⁻¹ depending on the method used (3–6). The choices range from the faster docking algorithms through classical molecular dynamics to ab initio quantum mechanical simulations. A primary determinant of the optimal method for a particular problem is the trade-off between accuracy and available computational power.

Among the available methods, all-atom molecular dynamics (MD) is a common choice for systems where the region of interest is large and involves multiple specific interactions between binding partners. The power of MD calculations lies in their ability to explain the causes of a given interaction. For example, alchemical perturbation methods (1) have the ability to distinguish contributions from individual components (4). Path-based free energy methods, e.g., umbrella sampling/potential-of-mean-force (PMF) (7–10) and metadynamics (11), are able to derive connections between the mechanistic details and the underlying free energy surface, and are often used in description of dissociation processes. These additional descriptors provide valuable insights in understanding protein-ligand interactions, a topic of utmost importance in drug design (3).

Applications of these methods are relatively straightforward for small (<50 atoms) and relatively rigid compounds. Not surprisingly, most of the ligand binding studies in literature have focused on such small ligands. In contrast, few published studies exist for larger ligands of the size of small proteins. This is because simulation of large systems requires sampling of exponentially larger dimensions, and finding the experimentally relevant configurations can be highly challenging (5,12). This is particularly true for complex quantities such as entropic and solvation contributions to the free energy (13).

Among the larger ligands, toxin peptides that bind to ion channels are of particular interest because of their medical and pharmacological importance. Here we will focus on potassium channels because their crystal structures are readily available (14,15), which is an essential requirement for simulation studies. Potassium channels are potential targets for treatment of many pathologies involving T-cell immunology and neuronal function (16,17). Historically, the discovery of their function, physiology, and distribution has been aided by animal toxins (18), with a particular abundance of scorpion toxins studies. The latter are a class of disulfide-bonded peptides exhibiting a range of selectivity almost matching the variety of potassium channels (19). This ability to bind tightly to the targeted channel subtype, and not others, has also been recognized as a potential lead in pharmaceutical discovery. For the important class of the voltage-gated potassium (K_v) channels, the selectivity is commonly attributed to the presence of a functional dyad—with a central lysine plugging the pore—surrounded by a basic ring of residues (20).

Computational studies on the action of these peptide ligands are less numerous. A lack of experimental structures of bound complexes was a major problem, but this has been mostly ameliorated with increasing accuracy of the docking methods, which provide adequate binding poses (3,10,21). Several toxin-channel complexes have been studied using the method of docking and refinement with MD (22,23).

Progress on the main challenge of calculating the binding free energies of toxin ligands has been slower. The molecular mechanics-Poisson-Boltzmann surface area method has been used to calculate the relative binding free energies among different toxin configurations (22) or mutants (23). As far as we are aware, there has been only one attempt to calculate the absolute binding free energy of a toxin ligand—that of a charybdotoxin (ChTX) in complex with a potassium channel (24). In this study, we used the experimental structure of the potassium channel-ChTX complex (25) in umbrella sampling MD simulations (26), and determined the PMF for unbinding of ChTX.

An important outcome of this study was that application of the umbrella forces led to a permanent distortion of the toxin during the unbinding process, resulting in a large discrepancy in the binding free energy (24). Only after accounting for the work done in distorting the toxin, was it possible to reproduce the experimental result. Clearly, it would be preferable to avoid such simulation artifacts by maintaining the physiological conformations of the ligand via appropriate restraints, which is justified by the fact that ChTX has similar bound and unbound conformations. Therefore, in our study we construct new PMFs for the unbinding of ChTX with such restraints and further analyze the parameters required in setting up the PMF calculations. We also introduce a potential strategy to optimize the convergence and sampling efficiency in binding studies of large ligands.

Methods

Simulation details

Because construction of the simulation system was described in detail in our previous work (24), we will give a summary here for clarity. A mutant KcsA channel-ChTX complex was taken from the Protein DataBank database (PDB ID: 2A9H), and embedded into a POPE membrane, solvated with TIP3P water containing 100 mM KCl. The system contained ∼60,000 atoms, with the box dimensions 76 × 73 × 105 Å³. This assembly was gradually relaxed over 2 ns via stepwise decreases in positional restraints, first on side chains and then on backbone-atoms, using respective root-mean-square deviations (RMSDs) as criteria for proper equilibration. During relaxation, a minimum of 200 ps was specified at each step. A snapshot of the system after equilibration, indicating the ChTX residues involved in the binding is shown in Fig. 1. Cell dimensions were fixed in the xy direction after this equilibration. For testing purposes we have also performed MD simulations of ChTX (PDB ID: 2CRD) (27) in a water-filled box of dimensions 41 × 39 × 39 Å³.

Figure 1.

KcsA_mut-ChTX complex after relaxation in POPE membrane. Protein backbones are represented in cartoon format, with ChTX in yellow. Charged side chains are shown in stick format, with the most important members labeled. Potassium and chloride ions are shown as tan and aqua spheres, respectively. Lipid bilayer and water box are shown as surfaces. Image prepared using VMD.

All simulations in this study were conducted under the NpT conditions, maintained at 300 K and 1 atm via Langevin damping. Periodic boundary conditions were employed, and the long-distance electrostatic interactions were computed using the particle-mesh Ewald algorithm. The VMD software was used in construction of the simulation system and analysis of the results (28). The MD simulations were performed using the package NAMD (29) with the CMAP-corrected (30) CHARMM22 force field (31).

Umbrella sampling and PMF calculations

As in the previous work (24), we choose the channel axis as the reaction coordinate, which is aligned with the z axis. We select the center-of-mass (COM) of the main-chain backbone atoms (N, Cα, and C) in ChTX, measured relative to the COM of KcsA, as the collective variable most suitable for the description of the unbinding process. A reaction path extending 15 Å along the channel axis is sampled in 0.5 Å intervals—a choice designed to balance sampling density with available computational resources. Because full-protein PMFs are relatively rare in the literature, we also explore sampling efficiency of these choices by duplicating the trajectory for two umbrella potentials with k = 20 and 40 kcal/mol/Ų.

For each PMF, 31 umbrella windows are created at 0.5 Å intervals via steered MD simulations over the 15 Å path. To create a window, ChTX is pulled at the speed of 5 Å/ns for 100 ps, and then equilibrated for a further 200 ps before proceeding to the next. To enhance sampling around energetic barriers, additional windows were inserted postproduction between the existing windows—doubling the local resolution to 0.25 Å. The initial windows are numbered from 0 to 30, with 0 denoting the window for the bound complex. Extra windows inserted in between the initial ones are denoted by half-integers, e.g., the window between 6 and 7 is called 6.5.

Previously, we constructed two PMFs for the unbinding of ChTX using the umbrella potential parameters k = 20 and 40 kcal/mol/Ų with no additional restraints (24). Here they will be denoted by A₂₀ and A₄₀, where the subscript indicates the k value used (see Table 1 for a summary of parameters). In both cases, we observed tidal forces acting on ChTX as it was pulled from the binding site. These tidal forces are caused by the concurrent action of umbrella forces and attractive forces from the binding site on the toxin, distorting it via the opening of N-terminal residues and breaking of the V16-L20 hydrogen bond (see Fig. 11 in (24)). We deduced that the large discrepancies in the binding free energy with known results are due to irreversible work done in this distortion. To obtain a more accurate result, it is essential to prevent such distortions of the ligand during the umbrella sampling simulations.

Table 1.

Summary of the parameters used for all PMFs presented in this work

Set	T (ns)	k (kcal/mol/Å2)	Conformational restraints (kcal/mol/Å2)	Notes
A20	3.4	20	None	From Chen and Kuyucak (24)
A40	2.4	40	None	From Chen and Kuyucak (24)
B20	5.6	20	10 (V16–L20 Cα atoms at 4.2 Å)	From this study
C40	5.4	40	10 (V16–L20 Cα atoms at 4.2 Å)	From this study
			10 (T3–C35 side-chain atoms at 4.8 Å)

T denotes the total simulation time per window and k is the umbrella potential parameter. Data for PMF-production and analyses are taken from the last 2 ns of each simulation window.

We have identified two pairs of residues which could protect ChTX from the tidal forces when restrained—a hydrogen bond between V16 and L20 on the far end of a α-helix, and a hydrophobic association between the N-terminal T3 and one of the disulphides C35. Restraint forces are applied to the C_α atoms in the former case, and the side-chain atoms (C_γ-S) in the latter. These will be referred to as the helical and N-terminal restraints, respectively. The PMF constructed using only the helical restraint is called B₂₀ and the PMF constructed using both the helical and N-terminal restraints is called C₄₀ (see Table 1 for parameters). Initial coordinates for these PMFs are chosen from an unperturbed simulation of the complex, and subjected to an additional 500-ps equilibration under their respective restraints before production. The restraint parameters in column four of Table 1 are fitted to target values derived from a 10-ns simulation of bulk ChTX.

Positional data on the COM of ChTX are collected at each timestep and analyzed via the weighted histogram analysis method (32). To check the convergence, PMFs are created from overlapping 2-ns segments of data separated by 0.4 ps (see Fig. S1 in the Supporting Material). These are inspected for evidence of convergence and poor sampling—indicated by spikes on the PMF surface and gaps in coordinate histograms. A set is considered to have converged if several successive data produce comparable well-depths and topology (this is evaluated by visual inspection using an arbitrary tolerance of 0.5 kcal/mol).

The binding constant is estimated by invoking a one-dimensional approximation and integrating the PMF, W(z), along the z axis

$$K_{eq}=\pi R^2{\int }_{b.s.}^{bulk}{e}^{-W\left(z\right)/k_{\text{B}}T}dz,$$ (1)

where the radius in the factor πR² measures the cross-sectional area of the binding pocket and is determined from the transverse fluctuations of the ChTX COM in the binding pocket as R = 0.7 Å. Test calculations show that allowing R to vary with z gives essentially the same answer (see the Supporting Material). The absolute binding free-energy of ChTX is determined from

$$\Delta G_b=-k_{B}T\ln \left(K_{eq}{C}_0\right),$$

where C₀ is the standard concentration of 1 M/L.

Influence of restraints on the PMF

The helical and N-terminal restraints are applied to the residues on the opposite side of the binding site, which are not directly involved in the binding of ChTX. Thus, intuitively one does not expect the restraints employed here to have any effect on the constructed PMFs. To confirm this assertion, we have calculated the correction to the binding free energy due to the restraints. From the thermodynamic cycle shown in Fig. 2, this can be written as

$$\Delta G_b=\Delta G_b^{\ast }+\Delta G_{rest}\left(\mathrm{bulk}\right)-\Delta G_{rest}\left(b.s.\right),$$ (2)

where ΔG_b^∗ is the PMF obtained using restraints and ΔG_rest = ΔG_{free → rest} is the free energy cost of restraining the ligand calculated in both bulk and at the binding site. Following Cecchini et al. (12), the latter quantities can be calculated via thermodynamic integration where the harmonic restraint is relaxed from the applied value of k_i to 0—in effect calculating the free energy of relaxation and inferring its opposite from

$$\Delta G_{rest\to free}={\int }_{k_i}^0\frac{\partial G_{rest}}{\partial k}dk,\frac{\partial G_{rest}}{\partial k}=\frac{1}{2}{\langle {\left|X-X_0\right|}^2\rangle }_k,$$ (3)

where X – X₀ represent the difference of the restrained coordinates from the target values.

Figure 2.

The thermodynamic cycle for evaluating the effect of conformational restraints on the binding of ChTX. The unrestrained and restrained paths are shown at the top and bottom, respectively.

We have applied this method to the N-terminal restraint because it is more mobile and hence more likely to have an effect on the PMF compared to the helical restraint. To complete the thermodynamic cycle in Fig. 2, we conduct thermodynamic integration of both the bound system and bulk. For the former, we use the initial umbrella-sampling window of the bound complex; and for the latter, ChTX in a box of water is used. For both calculations, we initialize the T3–C35 restraint at k_i = 10 kcal/mol/Ų, and create windows for both systems at the k values, 5, 2, 1, 0.5, 0.2, 0.1, 0.05, 0.02, and 0.01 kcal/mol/Ų over successive 50-ps simulations (10 integration windows in total). Each window is simulated for 2 ns, and the first nanosecond is discarded as equilibration.

Results and Discussions

Effect of conformational restraints

ChTX maintains similar conformations at both the bound and bulk end-states, which also corresponds to the conformations found in the NMR coordinates 2A9H.pdb and 2CRD.pdb, respectively. Thus, any deformations occurring during umbrella sampling simulations must be transient, otherwise they will have an adverse effect on the PMF. In our previous work we have shown that the RMSD of ChTX—after it was pulled from the binding site—was significantly larger compared to the bulk or complex values because it was distorted during this process (24). Here we perform similar conformational tests for the new PMFs. In Fig. 3 we present the RMSD values of ChTX obtained from each umbrella window of the PMFs, B₂₀ and C₄₀, and compare them with those of A₄₀ (A₂₀ results are similar to A₄₀ and therefore not shown to avoid cluttering). Note that the N-terminal residues (1–5) have been excluded from the RMSD in order to focus exclusively on the changes occurring in the main body of the toxin. It is seen that the traces of B₂₀ converge at ∼2.25 Å, similar to those of A₄₀, whereas the traces of C₄₀ remain near the bulk value of ∼1.5 Å.

Figure 3.

RMSD of ChTX backbone atoms with respect to the NMR structure recorded for each umbrella window of the three PMFs, A₄₀, B₂₀, and C₄₀. Residues 1–5 from the N-terminal have been excluded to show the dynamics of the main toxin mass. The trajectory C₄₀ is terminated at window 20 because it has already reached bulk status (see Fig. 5).

These observations imply that nonnative N-terminal conformations invariably induce an altered configuration in the main body. The toxin conformations in set A₄₀ were distorted by opening of the N-terminal, followed by breaking of an intrahelical hydrogen bond V16–L20 and subsequent nonnative contacts with the N-terminal. This transformation corresponds to a spike between windows 2–4 of set A₄₀ (Fig. 3), which is also reflected by a large kinetic barrier in the PMF (discussed below). We find an analogous transformation in B₂₀, where the N-terminal departs in the initial windows 2–4, and similar distortions happen as ChTX is pulled from the binding site. This occurs despite the presence of a helical restraint in B₂₀, indicating that the hydrogen bond rearrangement is not the only cause of our PMF inaccuracy. Inspection of the ChTX structures in the critical initial PMF windows show that sets A₄₀ and B₂₀ differ physically from C₄₀. Opening of the N-terminal in the former leads to invasion of the ChTX core with water and concomitant deformation of its backbone. Thus, the conformational tests indicate that both the helical and N-terminal restraints are required to maintain a nativelike ChTX backbone during PMF production.

Considering that ChTX is a rather small and stable peptide (stabilized by three disulfide bonds), distortion of peptide-ligands by tidal forces is expected to be a common problem in construction of PMFs. Therefore, use of appropriate restraints is likely to be an important consideration for accurate prediction of binding free energies. Conformational tests such as above help to identify potential problems with ligand distortion and suggest strategies to counter them before investing heavily in PMF production. Although calculation of the free energy cost of such deformations may be feasible (24), this repair-and-rescue method introduces its own simulation errors, which are more difficult to enumerate. Alternatively, an RMSD-based restraint may be used, which will eliminate the need for a-posteriori restraint matching.

We next consider the influence of the N-terminal restraint on calculated binding free energies. If the restrained residues have significant interactions with the channel, then we expect the restraints to have a different energy of release at the two PMF end-points. Carrying out the thermodynamic integrations described in Methods, we find that the free energy of N-terminal release from 10 kcal/mol/Ų are –1.47 kcal/mol in bulk, and –1.43 kcal/mol at the binding site. Equivalence of the relaxation energies between these end-points show that this restraint is not linked with active residues. From Eq. 2, the correction to the binding free energy obtained from the restrained PMF amounts to

$$\Delta G_{rest}\left(\mathrm{bulk}\right)-\Delta G_{rest}\left(\mathrm{site}\right)=1.47-1.43=0.04\mathrm{kcal}/\mathrm{mol},$$

which is completely negligible when compared to the accuracy of the calculations.

Sampling effectiveness

The distribution of ChTX center-of-mass follows a well-defined Gaussian distribution over the 2-ns sampling period for all windows (see the Supporting Material for details). Sufficient overlap of successive windows are required to interpolate PMFs from individual windows and obtain a reliable free energy profile. In Fig. 4, we show the percentage overlap of the ChTX COM coordinate between adjacent windows. For reference, we also indicate the percentage overlap between two normalized Gaussian distributions with widths σ, and separated by a distance d, which is given by

$$\%\text{overlap}=100\times \left[1-\text{erf}\left(d/\sqrt{8}\sigma \right)\right].$$ (4)

As expected from Eq. 4, this metric has a direct dependence on the umbrella potential strength, with k = 20 PMFs averaging 14 ± 1% overlap and k = 40 PMFs averaging 3.5 ± 0.3% overlap. These values are slightly smaller than the overlaps calculated from Eq. 4, which is again due to binding forces. Near the tail-end of the trajectories, all sets converge toward the Gaussian values, implying that each PMF satisfactorily reaches bulklike conditions. The inverse nature of the relationship between window overlap and force constant k is illustrated further in Fig. 4 c, which also highlights the influence of local perturbations at the critical windows 4 and 5. Clearly, an informed choice of umbrella potential parameters is essential for obtaining sufficient overlaps between adjacent windows. To help with this, we have solved Eq. 4 for constant overlaps of 5% and 10%, and plotted the resulting relationship between d and k in Fig. 4 d. Parameter sets that are too far from the indicated band are likely to be either inefficient (below the band) or poorly sampled (above the band).

Figure 4.

Overlap between successive sampling windows for each trajectory classed according to k. The dotted lines in panels a and b indicate the overlap of Gaussian distributions from Eq. 4. (a) PMFs A₂₀ and B₂₀, (b) PMFs A₄₀ and C₄₀. Windows 21–30 for A and B trajectories are similar in behavior to windows 16–20, thus omitted for brevity. (c) Replicates of windows 4 and 5 of set C₄₀ simulated at different k to highlight the inverse relationship between the force constant k and the window coverage. (d) The relationship between window separation d and force constant k obtained from Eq. 4 assuming a constant overlap of 5% and 10%. Parameters used in the current PMFs are indicated with a plus-sign and a cross.

We note peculiar dips in overlaps (Fig. 4, a and b), occurring at windows 6–7 in set A₂₀, 12–13 in B₂₀, 2–6 in A₄₀ and 3–5 in C₄₀, which are correlated with the location of a critical transition in the unbinding process (discussed later). These are the locations where additional windows are introduced to improve sampling. Compared to the original PMFs, these augmentations can change the final depth by up to 1 kcal mol⁻¹. For example, the extra window inserted between windows 5 and 6 in C₄₀ raises the energy well by ∼0.5 kcal/mol. This illustrates that there must be a minimum overlap between adjacent windows in order to obtain an accurate PMF, which a straightforward protocol may not achieve. For our particular settings, we derived an value of ∼2% as a minimum coverage level for the PMF C₄₀, requiring additional windows between 3 and 6.

Based on the above observations, we can make optimal choices for the initial sampling windows which will maximize computational efficiency. One such strategy is to use ∼1 kcal mol⁻¹ per residue of the ligand for the force constant while adjusting the window separation such that consecutive overlaps converge toward 5%. From Eq. 4, this corresponds to

$$d\sim 4\sqrt{k_{\text{B}}T/k}.$$

The first metric scales the restraint in proportion to ligand mass so that the distribution will be sufficiently sampled in simulation runs while the second choice includes a leeway in window overlaps to compensate for reaction barriers. These suggestions are based on the characteristics of our C₄₀ results, where we calculate an ideal separation of 0.48 Å (close to a window separation of 0.5 Å) and convergence within 5 ns, including equilibration time. We note that three additional sampling windows were necessary in our example of C₄₀. This implies that an increase in window density may be prudent where significant barriers are observed, say, in trial steered MD trajectories before production.

PMF and binding energy of ChTX

PMF results for the new trajectories B₂₀ and C₄₀, together with the unrestrained A₂₀ and A₄₀, are shown in Fig. 5. The most prominent feature of B₂₀ and C₄₀ is that the free energy wells are significantly higher compared to those in the old set. This is reflected in the binding free energies calculated from the PMFs (see Table 2). The new values are much improved compared to the previous results from set A and reproduce the experimental value within the chemical accuracy of 1 kcal/mol.

Figure 5.

KcsA-ChTX PMFs along the backbone-COM coordinate of ChTX for the four trajectories considered. The z distance is measured relative to the COM of KcsA. C₄₀ represents the most accurate PMF as it reproduces the binding free energy within the accuracy of the calculations without conformational distortions.

Table 2.

Binding free energies obtained from the integration of PMFs

Set	Binding energies (kcal/mol)
A20	–17.1 ± 0.9∗
A40	–16.8 ± 2.3∗
B20	–8.7 ± 1.7
C40	–7.6 ± 1.1
Expt.	–8.3†

Standard errors are determined by calculating PMFs for five nonoverlapping segments of the simulation data and retrieving the standard deviation of the binding energy predictions.

^∗Does not include correction for conformational distortion of ∼9 kcal/mol (24).

^†Converted from K_D = 900 nM (25).

The C₄₀ PMF has a well-formed free energy well, whose width (1.5 Å) is comparable to expected deviations from unrestrained simulations of the bound ChTX (not shown). Comparison of the unrestrained trajectory A₄₀ with the restrained C₄₀ will shed more light on differences in appearance. The malformed free energy well in A₄₀ is spatially matched to corresponding peaks in the backbone RMSD (see Fig. 3). Here, ChTX experiences its initial distortion under the tidal forces, which are caused by the strong umbrella forces pulling it out while ChTX maintains its contacts with channel residues. The steep increase in the A₄₀ PMF disappears in C₄₀ once this distortion is prevented via application of restraints. Thus, the A₄₀ PMF does not reflect the native free energy surface. This comparison confirms our previous proposal of strong tidal forces as the source of the ligand distortion and the ensuing discrepancy in the binding free energy.

The B₂₀ PMF possesses a large (4 Å) free energy well, containing a minor barrier at z ∼32.5 Å. We correlate this feature with the N-terminal departure in the initial four windows, with the minor barrier representing the activation energy required to extract the strand from its native position. As remarked earlier (compare to Fig. 3), opening of the N-terminal in B₂₀ leads to a permanent deformation of the ChTX backbone. The work done in this process can be estimated from the binding energies in Table 1 as 1 kcal/mol, which is much smaller than the helical distortion energy of 8 kcal/mol. Despite the good agreement of the binding free energy with experiment, we must reject this result in favor of C₄₀ because B₂₀ is also tainted by unphysical distortions.

Returning to the discussion of PMFs (Fig. 5), the general features of k = 20 and 40 PMFs conform at the bulk-end of the reaction path. This is an expected result, because in the absence of direct contacts, ChTX can be well-approximated by a charged object reacting to the mean electric field near KcsA. This region is bounded by a principal shoulder (z ∼38 Å for k = 20 and z ∼37 Å for k = 40 calculations), where we can identify the final dissociation of contacts in the physical trajectories. The unique energy surface for each PMF is interesting and will be revisited when we seek to connect the PMFs with the kinetics of unbinding.

Analysis of the unbinding process

A detailed picture of the unbinding process can be obtained by analyzing the trajectory data from the PMF windows. Because similar analysis has been performed for the unrestrained PMFs previously (24), we focus on the trajectories B₂₀ and C₄₀ here. The three quantities we consider for this purpose are:

• 1.Trajectory of the N atom of the K27 side chain which inserts into the pore.
• 2.The N-O distances between the peripheral ChTX and KcsA residues that make contact in the bound complex.
• 3.Orientations of ChTX obtained from its dipole moment.

In each case, the quantity in question is determined from the average of 2 ns of data for each PMF window.

Trajectory of the K27 amide (Fig. 6) provides a pictorial summary of the unbinding process. Considering C₄₀ first, the K27 amide resides in the filter during windows 0–8, making hydrogen bonds with the carbonyl oxygens of Y78 and G79. In window 9, K27 transfers from the filter to the neighboring D80(D) side chain and displaces R34 from this site. Its detachment from D80(D) occurs over several windows, following the general direction of the reaction coordinate. The K27 amide follows a rather different path in B₂₀, partly due to the lower force constant used and partly due to toxin distortion. The effect of these influences show up in K27's longer residence time, moving from the filter in window 12 to make contact with D80(B) on the opposite side. This allows R34 to remain attached to D80(D) until window 16 when K27 swings back to the D80(D), only to detach again in the next window.

Figure 6.

Trajectories of the K27 amide superimposed on a cutaway view of KcsA, showing the oxygens it interacts with during the unbinding process. The coordinates of the nitrogen for each window was averaged over the sampling period and shown (dot), which are connected to form a smooth line. Where possible, dots have been labeled with window numbers. The trajectory of B₂₀ (red) and C₄₀ (blue). The critical windows (12 in B₂₀ and 9 in C₄₀) are shown (black dot).

We next consider the behavior of the peripheral contacts (Fig. 7). These include three charge contacts, D80(C)-R25, D80(D)-R34, and D80(B)-K11, and a hydrophobic contact, Y82(D)-Y34. The average N-O separations in initial windows hint at a ranking of contact strengths, with strong binding from R25 and R34, and a weaker binding from K11 (and other peripheral lysines not shown here). We attribute this to the additional hydrogen bonds that arginine can make with KcsA, because the channel conformation exposes a number of backbone oxygens as well as D80 around the pore. As other potassium channels share this conformation, it is probable that peripheral arginines will generally find lower energetic states than lysines. The evidence presented in Figs. 6 and 7 is in agreement with the general consensus on the binding motifs of αKTX scorpion toxins (19).

Figure 7.

N-O contact distances between the important functional residues of ChTX and KcsA versus ChTX position along the reaction coordinate, measured between the ChTX headgroups and the oxygens of the aspartate residues. The last panel shows coupling between hydrophobic tyrosine residues and gives the distance between the COM of respective benzene rings. Error bars represent standard deviation of the sample. B₂₀ is shown in light shading (red in color) and C₄₀ in dark shading (blue in color).

The N-O distances in Fig. 7 provide complementary information for the unbinding of ChTX. In C₄₀, R34 detaches first from D80(D) (window 9, ∼35 Å) and vacates it for K27, as described above. This is followed by detachment of R25 from D80(C) in window 10, which returns briefly in window 13. K11 is weakly bound to D80(B)—it detaches in the first window but also comes back briefly in window 10. The large changes in the N-O distances in windows 9–11 are clearly associated with rotations of ChTX. The B₂₀ results follow a similar trend to those of C₄₀ with some delay due to the weaker umbrella forces. We note, in particular, R34 which does not fully detach from D80(D) until window 15.

While the contribution of hydrophobic interactions to the binding free energy is relatively small compared to charge interactions, they appear to persist longer. As an example, we show the Y82(D)-Y36 side-chain distances (Fig. 7), which remain in contact over the entire width of the bound regions in the PMF. Their detachment at ∼37–39 Å are marked by respective shoulder regions.

Following the discussions in Figs. 6 and 7, we can distinguish three distinct phases in the unbinding process (for brevity, we consider only the more realistic C₄₀ case):

• 1.Bound locale (windows 0–8 or 31 < z < 35): Here ChTX is tightly bound to the channel via its numerous charge contacts, e.g., K27 is in the filter, and R25 and R34 are firmly anchored to the respective D80 residues. This effectively restricts the toxin's rotational freedom such that similar behavior is observed across all trajectories. This is reflected in the dipole orientations, which show little variation from the original NMR value (Fig. S3 a in the Supporting Material).
• 2.Transition states (windows 9–14 or 35 < z < 38): To break out, ChTX must undergo some rotation and remove K27 from the filter. This occurs by rotation of ChTX to the left, which results in detachment of R34 from D80(D) and its replacement by K27. In this region, contacts of K27/R25/R34 triad with respective D80 residues are gradually broken. This corresponds to increasing rotational freedom in the toxin (Fig. S3 b).
• 3.Electrostatic region (windows 15–20 or 38 < z): By this region, all direct contacts have been severed, and residual electrostatic interactions have been screened by incoming waters. Under weak electrostatic attraction, the toxin is more or less rotationally free. This is reflected by large changes in contact distances (Fig. 7) and dipole moment orientations (Fig. S3 c).

Discussion of the free energy surfaces

It is apparent from both the PMFs (Fig. 5) and the unbinding processes (Figs. 6 and 7) that the measured free energy profiles differ substantially depending on the choice of umbrella force constant k. This dependence was not observed in previous cases of ions or small, rigid ligands. Therefore, some explanation of this phenomenon is warranted. We have given an illustration of this phenomenon in the Supporting Material, using schematic examples. In summary, when a flexible ligand is pulled from a binding site via umbrella forces, it stretches and stores some elastic energy which depend on the force constant k used. A weak umbrella force allows more stretching of the ligand and vice versa, for a strong force. Thus the shorter reaction distances observed in the k = 40 PMFs is a direct result of the stronger COM restraint employed. In contrast, the weaker force constant in the k = 20 PMFs not only allows longer contacts between the charged residues but also postpones the return of the elastic energy stored in the toxin until much larger separations.

Under this interpretation, a PMF obtained using the backbone-COM as the reaction coordinate is not a direct representation of the native free energy surface, because it does not contain interacting residues. The kinetic barriers containing information on dissociation of interacting residues are modified by the internal degrees of freedom. However, as long as the dynamical processes linking the ligand's internal coordinates to the reaction coordinate are approximately energy-conserving, the errors introduced on the binding free energy calculations will be negligible.

We stress that the most important information derived from a PMF is the binding free energy, which is an observable quantity. The PMF itself is not an observable, and therefore its apparent k-dependence is of a lesser concern. As long as no dissipative work is done and the ligand conformations are well conserved at endpoints, all PMFs should give a consistent binding free energy prediction. The A₂₀ and A₄₀ PMFs in Fig. 5 furnish an example of this—despite having substantially different free energy profiles, the two binding free energies (Table 2) are consistent within error. Therefore, the choice of k is an entirely practical matter and should be driven by concerns to optimize the production process. Comparing the k = 20 and 40 PMFs in Fig. 5, we see that doubling of the k value reduces the required path length by almost half, which reduces computation time in the same proportion. This benefit is mitigated by reduced sampling area and potential need for extra windows. Nevertheless, the stiffer force constant is closer to optimal parameters for our case.

Conclusions

This work was motivated by our previous study of unbinding of the toxin ChTX from the KcsA potassium channel using umbrella sampling simulations, where a large discrepancy in the binding free energy was found. This was related to distortion of ChTX by tidal forces—a simulation artifact caused by the applied umbrella forces overcoming the hydrophobic-core interactions that preserve ligand integrity. The main aim of this work was, therefore, to find a minimal set of restraints that would prevent such distortions of ChTX, and thereby reduce the discrepancy in free energy calculations. We have shown that using only two harmonic restraints, it is possible to achieve this aim and reproduce the binding free energy of ChTX within chemical accuracy. We have also shown that the use of these restraints do not have any effect on the calculated free energies. Our results demonstrate the possibility of accurate calculation of the binding free energies of peptide-ligands, which could have important ramifications in, e.g., development of drug leads from peptide candidates.

Over the course of this article, we have analyzed the properties of PMFs obtained from the COM reaction-coordinates, and found a set of optimized parameters with which umbrella sampling techniques can be scaled-up for large peptide-ligands. For the case of KcsA_mut-ChTX interactions presented here, we find that a COM restraint value of ∼1 kcal/mol/Ų per residue and window separations of 0.5 Å are near-optimal values to balance sampling density versus necessary window numbers. Although this is an initial example of a methods optimization, we believe that it provides a useful benchmark for other similarly-sized ligands. We emphasize the importance of wisely balancing competing factors k-σ and k-sampling time in production, and monitoring simulations closely. An accurate PMF over a large collective variable (i.e., COM coordinate) may be all-but-impossible to achieve if these checks are not in place.