A comparison of pathway‐independent and pathway‐dependent methods in the calculation of conformational free enthalpy differences

Abstract

The multistep umbrella sampling method, which belongs to pathway‐dependent methods to calculate conformational free enthalpy differences, is used to calculate the free enthalpy difference between a right‐handed 2.7_10/12‐helix and a left‐handed 3₁₄‐helix of a hexa‐β‐peptide in methanol solution. The same conformational free enthalpy difference was previously investigated using pathway‐independent methods such as direct counting and enveloping distribution sampling. Our results show that the pathway‐dependent simulations are sensitive to the choice of the pathway and its parameter values. A pathway based on restraining distances of hydrogen‐bonding atom pairs shows poor sampling for two different values of the restraining force constant. Another pathway based on restraining backbone dihedral angles did smoothly sample the transition between the two helical conformations, but only with a proper choice of the restraining force constant. The results illustrate that if, and only if, a proper pathway and proper parameters are chosen, the multistep umbrella sampling can be almost 50 times more efficient than the pathway‐independent methods in this case. The analysis illustrates the advantages and pitfalls of the much used multistep umbrella sampling methodology.

Introduction

The calculation of free enthalpy differences is still one of the holy grails in computational chemistry.1, 2 Free enthalpy differences mainly fall into two categories.3 The first one is the free enthalpy difference between different molecules or Hamiltonians, that is, the alchemical free enthalpy. The other one is the free enthalpy difference between different conformations of a molecule, that is, the conformational free enthalpy.

Several methods have been applied4, 5 to calculate the free enthalpy difference between a right‐handed 2.7_10/12‐helix and a left‐handed 3₁₄‐helix of a hexa‐β‐peptide (Fig. 1) in methanol solution.6, 7 This is a perfect system for testing different methods to calculate conformational free enthalpy differences, because on one hand, it contains a simple polypeptide with only six residues, so the quality of the conformational sampling can be easily accessed. On the other hand, however, this is a challenging problem, because the conformational change involves a transition between a right‐handed helix and a left‐handed helix.

Figure 1.

Upper panel: Chemical formula of the β‐peptide studied: H ${}_2^{+}$‐β²‐HVal‐β³‐HAla‐β²‐HLeu‐β³‐HVal‐β²‐HAla‐β³‐HLeu‐OH. Lower panel: a right‐handed 2.7_10/12‐ helical and a left‐handed 3₁₄‐helical structure of this peptide.

The most straightforward way to calculate a conformational free enthalpy difference is direct counting of the number of trajectory structures sampled belonging to each conformation from a standard molecular dynamics (MD) simulation. The problem one may encounter when applying this simple method is that the sampling of the conformations of interest and the transitions between them is often not sufficient in an unbiased MD simulation.8

First of all, if there exist other stable conformations for the molecular system, much simulation time may be spent on irrelevant conformations. In such cases, the enveloping distribution sampling (EDS) method with two restraining potential energy functions focusing on the relevant conformations as the two end states can be applied (Fig. 2).9 Because the two EDS end states represent two different conformations, we refer to it as conformational EDS below. It was shown that the conformational EDS simulation focused much more on the two helical conformations than the standard MD simulations.4 This EDS methodology to compute conformational free enthalpy differences can be considered as a particular type of umbrella sampling using a single simulation (Fig. 2), with an automatically optimized EDS restraining potential energy term as a single biasing umbrella potential energy function. Only prior knowledge about essential features of the relevant end‐state conformations is required.

Figure 2.

Schematic representations of basic ideas of different methods to calculate the free enthalpy difference between conformations A and B.

Second, if there is a high energy barrier between the two conformations of interest, the transitions between them may not occur frequently during an unbiased simulation, and thus the sampling efficiency would be low. In this case, enhanced sampling methods can be applied to improve the sampling efficiency.10 Soft‐core interactions were applied to smoothen the potential energy surface of the hexa‐β‐peptide to enhance conformational sampling.5 It was shown that soft‐core interactions did smoothen the potential energy surface and enhance the transitions between the two helices in the biased simulation, but the soft‐core perturbation introduced into the system was too big to recover the physical ensemble from the soft‐core simulation. Application of the EDS method could resolve this problem by constructing an EDS reference Hamiltonian with the physical peptide and the soft‐core peptide as the two end states (Fig. 2).5 Because the two EDS end states represent two different Hamiltonians, we refer to it as alchemical EDS. In the alchemical EDS simulations, the transitions between the two helices were significantly enhanced compared with the standard simulations, and the physical ensemble was correctly reproduced.5 Combined with different ways to smoothen the potential energy surface, this application of alchemical EDS offers a useful enhanced sampling technique, which strikes a proper balance between speeding up the conformational sampling and sampling relevant configurations. But, it may not always be trivial to determine which part of the Hamiltonian is inhibiting a proper conformational sampling of the system and so is to be modified to reduce the energy barrier between relevant conformations.

The conformational sampling can also be enhanced by lowering the solvent shear viscosity.4 In principle, such a method leaves the potential energy surface unchanged, and speeds up the diffusion of the solute, and therefore, enhances the sampling efficiency (Fig. 2).

All the methods mentioned above are pathway‐independent methods. That is, either unbiased or biased simulations were performed, in which transitions between different conformations occurred spontaneously. In pathway‐dependent methods to calculate conformational free enthalpy differences, a pathway is defined connecting the two end‐state conformations and intermediate states along the pathway are to be sampled, that is, the transitions are guided during the simulation instead of happening spontaneously as in the pathway independent methods. Such methods are widely used to calculate conformational free energy differences. Yet, their efficiency and accuracy critically depend on the pathway chosen. In this article, we apply the pathway dependent multistep umbrella sampling method to sample the transition from the 2.7_10/12‐helix to the 3₁₄‐helix, or from the 3₁₄‐helix to the 2.7_10/12‐helix. Four different pathways, of which two are based on restraining distances of hydrogen bonding atom pairs and the other two are based on restraining backbone torsional angles, are considered, and the sampling efficiency of the multistep umbrella sampling method is compared with that of the pathway independent methods mentioned.

Molecular Model and Computational Method

Multistep umbrella sampling

The λ‐dependent pathway based on distances of hydrogen bonding atom pairs is defined as

$$V_{US}^{dis}({\overrightarrow{r}}^N;\lambda )=V_{phys}({\overrightarrow{r}}^N)+\frac{1}{2}{K}_{rest}^{dis}{\displaystyle \sum _{i=1}^{N_{HB}}{\left(d_i({\overrightarrow{r}}^N)-(1-\lambda )d_i^{10/12}-\lambda d_i^{14}\right)}^2}$$ (1)

in which $V_{US}^{dis}$ is the potential energy of the umbrella sampling simulations, which depends on the Cartesian coordinates ${\overrightarrow{r}}^N$ of the N particles in the system and is the sum of the physical potential energy V _phys and an umbrella potential energy based on the restraining of distances d _i between hydrogen‐bonding atom pairs. $K_{rest}^{dis}$ is the restraining force constant, which is set to 100 and 5000 kJ mol⁻¹ nm⁻² for the pathways DIS_PATH_100 and DIS_PATH_5000, respectively. N _HB is the total number of restrained hydrogen bonding atom pairs. d _i is the atom‐atom distance of the hydrogen bonding atom pair i, $d_i^{10/12}$, and $d_i^{14}$ are the reference distances of the hydrogen bonding atom pair i for the 2.7_10/12‐helix and the 3₁₄‐helix, respectively (Table 1).

Table 1.

Reference Values of the Distances Between Hydrogen Bonding Atom Pairs Used in the Distance Based Pathway

i	Hydrogen bonding pair	${\mathrm{d}}_{\mathrm{i}}^{10/12}\left[\text{nm}\right]$	${\mathrm{d}}_{\mathrm{i}}^{14}\left[\text{nm}\right]$
1	NH(3)…O(4)	0.20	0.36
2	NH(4)…O(1)	0.20	0.75
3	NH(6)…O(3)	0.20	0.75
4	NH(2)…O(4)	0.57	0.18
5	NH(3)…O(5)	0.68	0.18
6	NH(4)…O(6)	0.57	0.20

The λ‐dependent pathway based on backbone dihedral angles ξ _i is defined as

$$V_{US}^{dih}({\overrightarrow{r}}^N;\lambda )=V_{phys}({\overrightarrow{r}}^N)+\frac{1}{2}{K}_{rest}^{dih}{\displaystyle \sum _{i=1}^{N_{dih}}{\left({\xi }_i({\overrightarrow{r}}^N)-(1-\lambda ){\xi }_i^{10/12}-\lambda {\xi }_i^{14}\right)}^2}$$ (2)

in which $V_{US}^{dih}$ is the potential energy of the umbrella sampling simulations, which is the sum of the physical potential energy V _phys and an umbrella potential energy based on the restraining of dihedral angles ξ _i. $K_{rest}^{dih}$ is the restraining force constant, which is set to 5 and 100 kJ mol⁻¹ rad⁻² for the pathways DIH_PATH_5 and DIH_PATH_100, respectively. N _dih is the total number of restrained dihedral angles. ξ _i is the value of the dihedral angle i, ${\xi }_i^{10/12}$, and ${\xi }_i^{14}$ are the reference values of the dihedral angle i for the 2.7_10/12‐helix and the 3₁₄‐helix, respectively (Table 2).

Table 2.

Reference Values of Backbone Dihedral Angles ξ_i Used in the Dihedral Angle‐Based Pathway

i	Backbone dihedral angle	${\xi }_{\mathrm{i}}^{10/12}\left[\text{degree}\right]$	${\xi }_{\mathrm{i}}^{14}\left[\text{degree}\right]$
1	φ of residue 2	−105	−125
2	φ of residue 3	105	−125
3	φ of residue 4	−105	−125
4	φ of residue 5	105	−125
5	φ of residue 6	−105	−125
6–11	θ of residues 1−6	65	55
12	ψ of residue 1	−95	−145
13	ψ of residue 2	85	−145
14	ψ of residue 3	−95	−145
15	ψ of residue 4	85	−145
16	ψ of residue 5	−95	−145
17	ψ of residue 6	85	−145

Molecular model and simulation setup

The GROMOS force‐field parameter set 53A611 was used for the hexa‐β‐peptide, and the methanol solvent molecules were represented using a rigid three‐site model belonging to the standard GROMOS set of solvents.12 Aliphatic CH_n groups were treated as united atoms, both in the solute and solvent. Both termini were protonated. No counterion was used.

The folded structure of the 2.7_10/12‐helix and of the 3₁₄‐helix served as the initial structures. Minimum image periodic boundary conditions were applied based on a rectangular box. A minimum distance of 1.4 nm between any peptide atom and the closest box wall was enforced while solvating the peptide, resulting in 1123 methanol molecules. After a steepest descent energy minimization to remove close contacts between solute and solvent atoms, an equilibration scheme was performed which included sampling the atom velocities from a Maxwell distribution at 60 K and gradually raising the simulation temperature to 340 K, whereas decreasing the force constant of the position‐restraining potential energy term for the solute atoms from 2.5 × 10⁴ kJ mol⁻¹ nm⁻² to zero.

The simulations were performed at constant temperature, 340 K, and constant pressure, 1 atm, using the GROMOS11 simulation package.13, 14 The solute molecules and the methanol solvent were separately coupled to a temperature bath by means of weak coupling,15 using a coupling time of 0.1 ps. The pressure was calculated with a molecular virial and held constant by weak coupling15 to an external pressure bath with a coupling time of 0.5 ps, using an isothermal compressibility of 4.575 × 10⁻⁴ (kJ mol⁻¹ nm⁻³) ⁻¹. All bond lengths and the geometry of the methanol molecules were constrained using the SHAKE algorithm16 with a relative geometric accuracy of 10⁻⁴, allowing a time step of 2 fs in the leap‐frog algorithm to integrate the equations of motion. For the treatment of the nonbonded interactions, triple‐range cutoff radii of 0.8/1.4 nm were used. Interactions within 0.8 nm were evaluated every time step, the intermediate range interactions were updated every fifth time step and the long range electrostatic interactions beyond 1.4 nm were approximated by a reaction field force17 according to a dielectric continuum with a dielectric permittivity of 19.8, the value of the dielectric permittivity of the methanol model.12

At each of the 11 equidistant λ‐values in the multistep umbrella sampling simulations, the system was equilibrated for 1 ns followed by 1 ns of production. The final structure of the simulation of one λ‐value served as the starting structure for the next λ‐value. The statistical uncertainties were estimated using block averaging18.

Analysis

In all simulations, trajectory coordinates and energies were saved every 500 steps for analysis. Atom‐positional root‐mean‐square deviations (RMSD) were calculated after translational superposition of the solute centers of mass and rotational least‐squares fitting of the atomic coordinates of all backbone atoms (N,CB,CA,C) except for those in the N‐ and C‐ terminal residues of the β‐peptide. The ideal right‐handed 2.7_10/12‐helix was defined as the energy‐minimized structure derived from NMR experiments,6 and the ideal left‐handed 3₁₄‐helix was defined through the backbone torsional‐angle values −180.0°, −154.7°, 64.3°, and −135.9° for ω (CA‐C‐N‐CB), φ (C‐N‐CB‐CA), θ (N‐CB‐CA‐C), and ψ (CB‐CA‐C‐N), respectively, followed by an energy minimization in vacuum.

Results and Discussion

Time evolution of backbone atom‐positional RMSD with respect to an ideal right‐handed 2.7_10/12‐helix and an ideal left‐handed 3₁₄‐helix, respectively, in the multistep umbrella sampling simulations with the pathway DIS_PATH_100 starting from the 2.7_10/12‐ helix is shown in the upper panel of Figure 3. Only the production periods, that is, the last 1 ns of all λ‐values, were taken into account. The conformation of the peptide did change from the 2.7_10/12‐helix to the 3₁₄‐helix during the umbrella sampling simulations. The transition, however, is not smooth, that is, there is a rapid change from the 2.7_10/12‐helix to the 3₁₄‐helix in the simulations with λ = 0.5 and 0.6. The multistep umbrella sampling simulations starting from the 3₁₄‐helix displayed in the lower panel of Figure 3 show a similar picture, that is, there is a rapid change from the 3₁₄‐helix to the 2.7_10/12‐helix in the simulations with λ = 0.6.

Figure 3.

Time evolution of backbone atom‐positional RMSD with respect to an ideal 2.7_10/12‐helix (green) and an ideal 3₁₄‐helix (red), respectively, in the multistep umbrella sampling simulations using the pathway DIS_PATH_100 starting from the 2.7_10/12‐helix (upper panel) and the 3₁₄‐helix (lower panel). At each λ‐value 1 ns equilibration and 1 ns sampling were performed. Only the latter trajectory structures are displayed.

The two simulations shown in Figure 3 share the same pathway, with different directions though. Therefore, λ = 0.0 of one umbrella sampling simulation is equivalent to λ = 1.0 of the other one, and vice versa. If the transition occurs at λ‐value, say 0.6, in one umbrella sampling simulation, it should occur at λ‐value 0.4 in the other simulation. The results indicate that these two simulations with the same pathway DIS_PATH_100 are not consistent. This inconsistency reflects poor convergence of the simulations, so rather different values of the free enthalpy difference between the two helices would be obtained from these two simulations.

If the force constant of the distance restraining potential energy function is increased to 5000 kJ mol⁻¹ nm⁻² in the multistep umbrella sampling, that is, DIS_PATH_5000, no transition between the two conformations did occur during the two simulations with different starting structures (Fig. 4). Together with the simulations with DIS_PATH_100, these results suggest that the pathway defined using the distances of hydrogen bonding atom pairs is not a good one for the transition between these two different, right‐ and left‐handed, helical conformations of the hexa‐β‐peptide, because the pathway involves a high energy barrier between the two helices. When the restraining force constant is big, the restraining will lead to high energy structures. When the force constant is small, the peptide tends to stay in one helical conformation, and the transition may occur randomly at some intermediate λ‐value (Fig. 3).

Figure 4.

Time evolution of backbone atom‐positional RMSD with respect to an ideal 2.7_10/12‐helix (green) and an ideal 3₁₄‐helix (red), respectively, in the multistep umbrella sampling simulations using the pathway DIS_PATH_5000 starting from the 2.7_10/12‐helix (upper panel) and the 3₁₄‐helix (lower panel). At each λ‐value 1 ns equilibration and 1 ns sampling were performed. Only the latter trajectory structures are displayed.

The results of the simulations with the pathway DIH_PATH_5 are shown in Figure 5. The RMSD time series exhibit rather similar features as for the simulations with DIS_PATH_100. Again, the two simulations are not consistent. However, if the force constant of the dihedral angle restraining potential energy function is increased to 100 kJ mol⁻¹ rad⁻², the picture changes completely (Fig. 6). The transition from one helix to the other is smooth, and the two simulations shown in Figure 6 are consistent, that is, they are almost mirror images of each other. Therefore, the pathway based on backbone dihedral angles is a good one for the two helices. Yet, a proper force constant needs to be chosen. If the force constant is too small, the peptide would stay in one of the helical conformations, and the transition may occur randomly at some intermediate λ‐value, as in the DIS_PATH_100 simulations. Only when a proper force constant is chosen, the intermediate states along the pathway of the two helices are sampled sufficiently.

Figure 5.

Time evolution of backbone atom‐positional RMSD with respect to an ideal 2.7_10/12‐helix (green) and an ideal 3₁₄‐helix (red), respectively, in the multistep umbrella sampling simulations using the pathway DIH_PATH_5 starting from the 2.7_10/12‐helix (upper panel) and the 3₁₄‐helix (lower panel). At each λ‐value 1 ns equilibration and 1 ns sampling were performed. Only the latter trajectory structures are displayed.

Figure 6.

Time evolution of backbone atom‐positional RMSD with respect to an ideal 2.7_10/12‐helix (green) and an ideal 3₁₄‐helix (red), respectively, in the multistep umbrella sampling simulations using the pathway DIH_PATH_100 starting from the 2.7_10/12‐helix (upper panel) and the 3₁₄‐helix (lower panel). At each λ‐value 1 ns equilibration and 1 ns sampling were performed. Only the latter trajectory structures are displayed.

We note that only 11 × (1 + 1) = 22 ns of simulation time was used in the multistep umbrella sampling simulations with the pathway DIH_PATH_100. If we would reduce the equilibration period of each λ‐value to 0.1 ns, only 12.1 ns would be necessary, that is, the computational cost would be almost 50 times less compared with the pathway independent methods to compute the same conformational free enthalpy difference, of which the direct counting method using 500 ns unbiased simulations did not reach convergence.5 This is because in the simulations using pathway independent methods, transitions between different conformations may be rare, and many such transitions need to occur to obtain the correct population ratio between the different conformations. In the pathway dependent methods, multistep umbrella sampling in this case, the transition from one conformation to the other is forced along a λ‐dependent pathway, and more importantly, intermediate states along the pathway are sampled instead of many transitions between the end conformations. However, to obtain reliable free enthalpy differences, intermediate states should be sufficiently sampled as well, which requires a well designed pathway, and properly chosen restraining parameters.

From the DIH_PATH_100 simulations, the free enthalpy difference between the two helices can be calculated in a thermodynamic integration manner using

$$\Delta G_{BA}={\displaystyle {\int }_{{\lambda }_A}^{{\lambda }_B}<\frac{\partial V_{US}^{dih}}{\partial \lambda }{>}_{\lambda }d\lambda }$$ (3)

in which A and B refer to the end states in the multistep umbrella sampling simulations with the pathway DIH_PATH_100, which are representative for the 2.7_10/12‐helix and the 3₁₄‐helix, respectively.

${〈\partial V/\partial \lambda 〉}_{\lambda }$ as a function of λ for the two DIH_PATH_100 simulations is shown in Figure 7. The sequence of λ‐values in the simulation starting from the 3₁₄‐helix is inverted to make the comparison easier. They give similar results, which is consistent with the RMSD results shown in Figure 6. ΔG_10/12,14 = ΔG_10/12 − ΔG₁₄ values calculated through Eq. (3) using trapezoid integration from the two simulations are 2.6 ± 12.1 and 2.1 ± 7.6 kJ mol⁻¹ starting from the 2.7_10/12‐helix and the 3₁₄‐helix, respectively. These results are consistent with the ones, that is, 1.9 ± 1.2 and 5.1 ± 0.8 kJ mol⁻¹, obtained from 500 ns unbiased MD simulations using the same definitions for the two helical conformations and initial structures.5 Note that to compare our current results with those obtained from pathway independent simulations, these ΔG_10/12,14 values do not include reweighting to unbiased end states. And the different sign of the ΔG_10/12,14 values obtained here compared with the previous results in Ref. 5 is probably due to a different definition of the end‐state helical conformations. The results of Ref. 4 are based on another force field, GROMOS 45A3, and on a different definition of the end‐state helices as well, so cannot directly be compared. The relatively large statistical uncertainties obtained for the umbrella sampling simulations are probably due to the relatively short simulation time for each λ value. Further calculation of ΔG_10/12,14 values with a structural criterion for the two helical conformations would require WHAM19 or MBAR analysis.20

Figure 7.

${〈\partial V/\partial \lambda 〉}_{\lambda }$ As well as statistical uncertainties as a function of λ in the multistep umbrella sampling simulations with the pathway DIH_PATH_100, starting from the 2.7_10/12‐helix (black) and from the 3₁₄‐helix with inverted sequence of λ‐values (red).

Conclusions

In this article, using a right‐handed 2.7_10/12‐helix and a left‐handed 3₁₄‐helix of a hexa‐β‐peptide in methanol solution as the test system, we applied the multistep umbrella sampling method, that is, a pathway‐dependent method, to sample the transitions between these two helices, and compared its performance with pathway‐independent methods used for the same system in previous studies.4, 5

To obtain reliable conformational free enthalpies, pathway‐independent methods require sufficient sampling of the relevant end‐state conformations, and many transitions between them, whereas in the pathway‐dependent methods, the transition is forced along a pathway and intermediate states are to be sampled. Therefore, in case of slow or rare transitions between conformations of interest in an unbiased or pathway independent MD simulation, pathway dependent methods are much more efficient than pathway independent methods. However, it is not always trivial to determine the right pathway connecting the end‐state conformations, and to choose proper biasing parameters allowing a sufficient sampling of intermediate states, not even for a simple system with only six residues as is shown in this work.

We note that the methods called “pathway dependent” by us do require the definition of a pathway connecting the two end states. Yet, assuming infinite sampling the resulting free enthalpy difference will be pathway independent. In the computational practice, however, one rarely approaches infinite sampling. This implies that the efficiency of pathway dependent methods becomes dependent on the choice of pathway. Such a choice generally involves a rather simple linear or non‐linear function of a coupling parameter and some geometric quantities regarding the system of interest. The chosen pathway may or may not correspond to a low free enthalpy pathway. High energy or low entropy barriers may hamper its sampling efficiency and thus the precision of the obtained free enthalpy differences. A pathway that happens to lie close to a low free enthalpy one will lead to enhanced efficiency, but may in practice not easily be identified.

In summary, pathway‐dependent methods can be applied to calculate conformational free enthalpies in cases where transitions between relevant conformations do not occur frequently in a standard MD simulation, and can reduce the computational cost significantly compared with pathway independent methods. However, attention should be paid to the choice of pathway and its parameter values. Unfortunate choices of pathway or parameters will lead to poor sampling, and in turn to an unreliable value for the free enthalpy difference.

Acknowledgment

The authors would like to thank Chris Oostenbrink for useful discussions. We thank Ron Levy for his many contributions to the development of methodology to simulate bio‐molecular systems and calculate free energy differences.