J. Phys. Chem. B 2007, 111, 2925-2931
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Simple Physics-Based Analytical Formulas for the Potentials of Mean Force for the Interaction of Amino Acid Side Chains in Water. 3. Calculation and Parameterization of the Potentials of Mean Force of Pairs of Identical Hydrophobic Side Chains Mariusz Makowski,†,‡ Emil Sobolewski,‡ Cezary Czaplewski,† Adam Liwo,† Stanisław Ołdziej,† Joo Hwan No,† and Harold A. Scheraga*,† Baker Laboratory of Chemistry and Chemical Biology, Cornell UniVersity, Ithaca, New York 14853-1301, and Faculty of Chemistry, UniVersity of Gdan´ sk, Sobieskiego 18, 80-952 Gdan´ sk, Poland ReceiVed: September 11, 2006; In Final Form: December 13, 2006
The potentials of mean force of homodimers of the molecules modeling hydrophobic amino acid side chains (ethane (for alanine), propane (for proline), isobutane (for valine), isopentane (for leucine and isoleucine), ethylbenzene (for phenylalanine), and methyl propyl sulfide (for methionine)) were determined by umbrellasampling molecular dynamics simulations in explicit water as functions of distance and orientation. Analytical expressions consisting of the Gay-Berne term to represent effective van der Waals interactions and the cavity term derived in paper 1 of this series were fitted to the potentials of mean force. The positions and depths of the contact minima and the positions and heights of the desolvation maxima, including their dependence on the orientation of the molecules, were well represented by the analytical expressions for all systems, which justifies use of such potentials in coarse-grain protein-folding simulations.
Introduction In paper 1 of this series,1 we derived a simple analytical expression for the cavity terms of the potentials of mean force of the association of hydrophobic solutes in water on the basis of an analysis of the number and context of water molecules in different parts of the solvation sphere and generalized it to spheroidal solute particles. In paper 2,2 we showed that the derived expression reproduces very well the features of the cavity part of the potentials of mean forces of homo- and heterodimers of simple spherical solutes in water. In this paper, we consider models of pairs of identical nonpolar amino acid side chains in water in order to replace the present knowledgebased side-chain-side-chain interaction potentials used in our united-residue UNRES force field3-10 with physics-based potentials. In the present UNRES force field,3-10 the side-chainside-chain interaction potentials were assigned Gay-Berne11 functional forms that take into account anisotropy of interactions, and their parameters were determined3 on the basis of fitting to correlation functions and side-chain contact energies determined from the Protein Data Bank (PDB).12 The Gay-Berne-type potential assumes that the interacting sites are ellipsoids of revolution (also termed spheroids). In this work, we determined the potentials of mean force of the following homodimers modeling pairs of identical nonpolar molecules, simulating nonpolar side chains: ethane-ethane (Et-Et, to model a pair of alanine side chains); propanepropane (Prp-Prp, to model a pair of proline side chains); isobutane-isobutane (iBut-iBut, to model a pair of valine side chains); isopentane-isopentane (iPen-iPen, to model a pair of leucine or isoleucine side chains); ethylbenzene-ethylbenzene (PhEt-PhEt, to model a pair of phenylalanine side chains); and * Corresponding author. E-mail:
[email protected]. Phone: (607) 2554034. Fax: (607) 254-4700. † Baker Laboratory of Chemistry and Chemical Biology. ‡ Faculty of Chemistry.
methyl propyl sulfide-methyl propyl sulfide (MePrpS-MePrpS, to model a pair of methionine side chains) as functions of distances between the molecules and their relative orientation by means of umbrella-sampling molecular dynamics (MD) simulations and subsequently fitted the analytical expressions to the PMFs. Except for proline, the CR atom is considered to be a part of a side chain, as in the UNRES model.3-10 Theory As in paper 2 of this series,2 we express the potential of mean force of a pair of nonpolar solutes (Wnn) in water by eq 1.
Wnn ) EvdW + ∆Fcav
(1)
where EvdW is the van der Waals term and ∆Fcav is the cavity contribution to the potential of mean force of hydrophobic association. As in our earlier work on the UNRES model,3 we assume that united nonpolar side chains can be represented by ellipsoids of revolution (see Figure 1for illustration); consequently, the potentials of interactions have spheroidal symmetry. For the EvdW energy term, we use the Gay-Berne-type potential11 expressed by eq 2. It should be noted that, previously, we used eq 2 to express the complete side-chain-side-chain interaction potential in UNRES,3 but now we add the ∆Fcav term (eq 1).
[(
EvdW ) 4ij
σ0ij
) (
rij - σij + σ0ij
12
-
σ0ij
rij - σij + σ0ij
)] 6
(2)
where rij is the distance between the centers of the particles, σij is the distance corresponding to the zero value of EvdW for arbitrary orientation of the particles (σ0ij is the distance corresponding to the zero value of EvdW for the side-to-side approach of the particles), and ij (depending on the relative orientation
10.1021/jp065918c CCC: $37.00 © 2007 American Chemical Society Published on Web 02/27/2007
2926 J. Phys. Chem. B, Vol. 111, No. 11, 2007
Makowski et al. the van der Waals well depth and the parameter 0ij is the welldepth corresponding to the side-to-side orientation of the interacting particles. In this work, the parameters mentioned above were determined by least-squares fitting. The expression for ∆Fcav of spheroidal particles was derived in paper 1 of this series1 and is given by eq 11: 1
∆Fcav )
(2) (3) R(1) ij [(xλ)2 + Rij xλ - Rij ] 12 1 + R(4) ij (xλ)
(11)
with
x)
[
λ) 1Figure 1. Definition of variables describing the location of two spheroidal particles (i and j) with respect to each other. The vector uˆ (1) ij is the unit vector of the long axis of particle i, uˆ (2) ij is the unit vector of the long axis of particle j, rˆ ij is the unit vector of the vector pointing (2) from particle i to particle j, θ(1) ij and θij are the angles between the (2) vector rˆ ij and vectors uˆ (1) and u ˆ , respectively, and φij is the angle of ij ij counterclockwise rotation of the vector uˆ (2) about the vector rˆ ij from ij the plane defined by the vector uˆ (1) ˆ ij when looking from ij and vector r the center of particle j toward the center of particle i.
of the particles) is the van der Waals well depth. The dependence of ij and σij on the orientation of the particles is given by eqs 3-5 and eq 6, respectively.3
[
(2) ij ) 1 -
[
(2) (12) 0 (1) (2) ij ≡ (ω(1) ij ,ωij ,ωij ) ) ij ij ij
(3)
(1) (2) (12)2 -1/2 ] (1) ij ) [1 - χij χij ωij
(4)
]
(1)2 (2) (2)2 (1) (2) (1) (2) (12) χ′(1) ij ωij + χ′ij ωij - 2χ′ij χ′ij ωij ωij ωij
σij ) σ0ij 1 -
(2) (12)2 1 - χ′(1) ij χ′ij ωij
2
(5)
]
(1)2 (2) (2)2 (1) (2) (1) (2) (12)2 χ(1) ij ωij + χij ωij - 2χ ij χ ij ωij ωij ωij (2) (12)2 1 - χ(1) ij χij ωij
(6)
with
ˆ (1) ˆ ij ) cos θ(1) ω(1) ij ) u ij ‚r ij
(8)
ˆ (2) ˆ ij ) cos θ(2) ω(2) ij ) u ij ‚r ij
(9)
(1) (2) (1) (2) ω(12) ˆ (1) ˆ (2) ij ) u ij ‚u ij ) cos θij cos θij + sin θij sin θij cos φij (10)
where uˆ (1) ˆ (2) ij and u ij are unit vectors along the principal axes of the interacting sites (in this work identified with the CR-SC axes), rˆ ij is the vector linking the centers of the sites, rij is the distance between the side-chain centers (Figure 1), the param(2) eters χ(1) ij and χij are the anisotropies of the van der Waals (2) distance, the parameters χ′(1) ij and χ′ij are the anisotropies of
rij
xσ2i + σ2j
(12)
]
(1)2 (2) (2)2 (1) (2) (1) (2) (12) χ′′(1) ij ωij + χ′′ij ωij - 2χ′′ij χ′′ij ωij ωij ωij (2) (12)2 1 - χ′′(1) ij χ′′ij ωij
2
(13)
(2) (12) where the symbols ω(1) are defined by eqs ij , ωij , and ωij 8-10, rij is the distance between the centers of the particles, (2) χ′′(1) ij and χ′′ij are anisotropies pertaining to ∆Fcav, and σi and σj can be identified with the minimum distance between the (2) center of particle i or j, respectively. The parameters R(1) ij , Rij , (3) (4) and Rij , Rij , σi, σj, and the anisotropies are determined by least-squares fitting.
Methods Molecular dynamics simulations were carried out with the AMBER13 suite of programs, using the AMBER 7.0 force field.14 Each system was placed in a periodic box containing explicit water molecules in an amount corresponding to the experimental water density at 298 K. The box dimensions were 33 × 33 × 33 Å3 (for two ethane molecules), 34 × 34 × 34 Å3 (for two methyl propyl sulfide molecules), 35 × 35 × 35 Å3 (for two propane, isobutane, and isopentane molecules), and 36 × 36 × 36 Å3 (for two ethylbenzene molecules), respectively. These systems were used to model the effective interactions within homodimers of nonpolar side chains in peptides and proteins. The TIP3P water model15 was used in molecular dynamics (MD) simulations. The charges on the atoms of the solute molecules, needed for the AMBER 7.0 force field, were determined for each sidechain model by using a standard procedure,16 i.e., by fitting the point-charge electrostatic potential to the molecular electrostatic potential computed using the electronic wave function calculated at the restricted Hartree-Fock (RHF) level with the 6-31G* basis set. The program GAMESS17 was used to carry out quantummechanical calculations; the program RESP16 of the AMBER 7.0 package was used to compute the fitted charges. The charges and the AMBER atom types are shown in Figure 2 MD simulations were carried out in two steps. In the first step, each system was equilibrated in the NPT ensemble (constant number of particles, pressure, and temperature) at 298 K for 100 ps (picoseconds), and the integration step was 2 fs (femtoseconds). After equilibration, MD simulations were run in the NVT ensemble (constant number of particles, volume, and temperature) for 10 ns, and the integration step was 2 fs. A 9 Å cutoff for all nonbonded interactions, including electrostatic interactions, was imposed. For each system, a series of 20 windows of 10 ns simulations was run with different harmonic-restraint
PMF Formulas for Interaction of Side Chains in Water (3)
J. Phys. Chem. B, Vol. 111, No. 11, 2007 2927
Figure 2. Partial atomic charges (in electron charge units) of the ethane (a), propane (b), isobutene (c), isopentane (d), ethylbenzene (e), and methyl propyl sulfide (f) molecules calculated by using the RESP method16 on the basis of HF/6-31G* calculations carried out with GAMESS17 used in the calculations with the AMBER force field. The atoms are labeled with standard AMBER atom-type symbols, i.e., CT for an sp3 carbon atom, HC for a hydrogen atom connected to a carbon atom, and CA for an aromatic or an olefinic carbon atom.
potentials imposed on the distance between the atoms closest to the center of the mass of each of the molecules, as given by eq 14.
V)
1 k(r - roi )2 2
(14)
where k is a force constant [assumed to be 2 kcal/(mol Å2)], r is the inter-particle distance, and r°i is the center of the restraint in the ith simulation (window); the values of r° were 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, and 13 Å. A total number of 50 000 configurations were collected for each window. To determine the potentials of mean force (PMFs) of the systems studied, we processed the results of all restrained MD simulations for each system by using the Weighted Histogram Analysis Method (WHAM).18,19 For a given system, four(2) dimensional histograms in rij, θ(1) ij , θij , and φij (Figure 1) were constructed. The ranges and bin sizes were 3.5 Å e rij e 13 Å, or 5 Å e rij e 13 Å, depending on the system, with one bin side distance 0.2 Å, with angles 0° e θ(1) ij e 180° with bin side 60°, with angles 0° e θ(2) e 180° with bin side 60°, and with ij angles -180° e φij < 180° with bin side 60°. Consequently, each distance corresponded to 54 bins, each containing counts from different orientations of the molecules. For illustration, the PMFs are plotted vs distance for the “side-to-side” (parallel), “edge-to-edge” (linear), and “side-to-edge” (perpendicular) orientations; these orientations are depicted schematically in Figure 3. Fitting of analytical formulas to the PMFs was
Figure 3. Illustration of the (a) side-to-side, (b) edge-to-edge, and (c) side-to-edge orientation of two spheroidal particles. The lines represent the long axes of the spheroids. The orientation variables (see Figure 1 (2) for definition) are as follows: θ(1) ij ) 90°, θij ) 90°, and φij ) 0° or (1) (2) (1) (2) 180° (a); θij ) 0°, θij ) 0°, or θij ) 0°, θij ) 180°, or θ(1) ij ) 180°, (1) (2) (1) θ(2) ij ) 0°, or θij ) 180°, θij ) 180°, and φij undefined (b); θij ) 90°, (2) (1) (2) (1) (2) θij ) 180°, or θij ) 180°, θij ) 90°, or θij ) 90°, θij ) 0°, or θ(1) ij ) 0°, θ(2) ij ) 90°, and φij undefined (c).
accomplished by minimizing the sum of the squares of the differences between the PMF values computed from analytical formulas and determined from MD simulations (Φ) defined by eq 15 by using the Marquardt method.20
min Φ(y) ) x
∑i [WMD(ri,θ(1)ij ,θ(2)ij ,φij) 2 (2) Wanal(ri,θ(1) ij ,θij ,φij;y)] (15)
(2) where WMD(ri,θ(1) ij ,θij ,φij) is the PMF value determined by (2) analsimulations for distance rij and orientation (θ(1) ij ,θij ,φij); W (1) (2) (ri,θij ,θij ,φij;y) is the analytical approximation to the PMF at
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Makowski et al.
Figure 4. The PMF curves for ethane (a), propane (b), isobutane (c), isopentane (d), ethylbenzene (e), and methyl propyl sulfide (f) homodimers. The dashed, dotted, and dot-dashed lines correspond to PMFs determined for the side-to-side (Figure 3a), edge-to-edge (Figure 3b), and side-toedge (Figure 3c) orientation, respectively, obtained by MD simulations. The solid lines correspond to the analytical approximation to the PMFs, with coefficients determined by least-squares fitting (eq 15) of the analytical expression to the PMF determined by MD simulations, with the Gay-Berne potential (eq 2) to represent the van der Waals interactions between the solute molecules and eq 11 to represent the cavity potential.
distance rij calculated with parameters given by the vector y, whose components are the adjustable parameters of eqs 2-6 and 11-13. Results and Discussion The PMFs of the pairs considered, together with curves computed with the fitted analytical expressions, are plotted as functions of the distance between the centers of the molecules in Figure 4a-f. Dashed, dot-dashed, and dotted lines in Figure
4a-j refer to the “side-to-side”, “edge-to-edge”, and “side-toedge” orientations, respectively. It can be seen that all PMFs exhibit fine structure with minima and maxima present. The plots are characteristic of the PMFs of two hydrophobic particles interacting in water.21-25 Each plot possesses a deep contact minimum (occurring at the shortest distances), a solvent-separated minimum, and a desolvation maximum separating the two minima. Because the number of data per four-dimensional bin is much smaller than those used
PMF Formulas for Interaction of Side Chains in Water (3)
J. Phys. Chem. B, Vol. 111, No. 11, 2007 2929
TABLE 1: Parameters of EvdW (eq 2) and ∆Fcav (eq 11) Determined by Minimization of the Function Defined by eq 15a systemb
0ij [kcal/mol]
σ0ij [Å]
c χ(1) ij
c χ′(1) ij
c χ′′(1) ij
σi [Å]
R(1) ij [kcal/mol]
R(2) ij [kcal/mol]
R(3) ij [kcal/mol]
c R(4) ij
Et-Et Prp-Prp iBut-iBut iPen-iPen PhEt-PhEt MePrpS-MePrpS
0.0005 0.0102 0.0023 0.0107 0.0589 0.0556
6.40 4.71 7.05 6.20 4.76 4.39
0.283 0.349 0.158 0.322 0.581 0.587
0.503 0.535 0.267 0.546 0.914 0.892
0.158 0.170 0.105 0.171 0.157 0.206
7.46 7.81 8.89 10.36 10.83 9.59
16.56 49.93 149.76 135.50 0.838 9.13
-0.077 -0.530 -0.557 -0.589 2.88 -0.270
0.779 0.467 0.448 0.424 2.84 0.657
31.70 5.71 5.84 14.37 33.09 33.07
a The calculated van der Waals energy was based on the Gay-Berne-type potential (eq 2), and the calculated cavity creation energy term was obtained from eq 11. The parameters are for the best fitting. b Abbreviations: Et, ethane; Prp, propane; iBut, isobutane; iPen, isopentane; PhEt, ethylbenzene; MePrpS, methyl propyl sulfide. c Dimensionless.
in determining distance-dependent PMFs,2,21-28 where the data are collected in one-dimensional bins, the solvent-separated minima are only weakly determined. The deepest contact minima in the PMFs are observed for the side-to-side orientation, and the shallowest ones for the edge-to-edge orientation. The contact minima occur at the shortest distances for the side-toside and at the longest distances for the edge-to-edge orientation. The positions of the desolvation maxima occur at different distances between the centers of the interacting particles, as do those of the contact minima; however, their heights remain constant within the simulation error. For this reason, the multiplicative factor R(1) ij in eq 11 does not depend on orientation, whereas the van der Waals well depth, and ij, in the GayBerne potential (eqs 3-5) do. The orientation dependence of the positions of the contact minima and desolvation maxima arises, because the distance from the center of a molecule to its surface varies with its orientation with respect to the other molecule. The contact minima are deeper for the side-to-side orientation than for the edge-to-edge or side-to-edge orientations because more water is eliminated from the hydration shell upon formation of the side-to-side homodimer. The contact minima of the pairs studied have depths of about -1.0 kcal/mol (for the side-to-side orientation) except for the iPen-iPen system for which the depth reaches about -1.4 kcal/mol. The increase of the depth of the contact minimum with the size of the contact surface of the hydrocarbon molecules is consistent with the results of an earlier study by Ne´methy and Scheraga29 who calculated the free energies of hydrophobic interactions between nonpolar amino acid side chains on the basis of the free energies of transfer of hydrocarbon molecules from the organic to the aqueous phase. These researchers found that the decrease of the free energy is caused by the fact that more water molecules are eliminated from the solvation sphere of larger hydrocarbon molecules upon dimer formation. The differences between the PMF at the contact minimum and the desolvation maximum are about 1.5 kcal/mol. However, it can be seen from Figure 4 that the contact minima are wider for the pairs of larger and, consequently, more hydrophobic molecules (iPen (Figure 4d), PhEt (Figure 4e), MePrpS (Figure 4f)) than for those of Et (Figure 4a) and Prp (Figure 4b). The differences between the PMF at the contact minimum and the first maximum appear in the following order: side-to-side, side-to-edge, and edge-to-edge orientation of the side chains, being about 1.5 kcal/mol, 1.25 kcal/mol, and 1.0 kcal/mol, respectively. For the iBut-iBut pair (Figure 4c) the PMF curves computed for the three orientations are closer to each other than those for the remaining systems, because the iBut molecule is the most symmetric among the systems studied in this work. The results of fitting the PMFs, using the analytical expressions given by eq 1 with components defined by eqs 2 and 11, respectively, are also shown in Figure 4 (solid lines); the fitted
parameters of the expressions for EvdW (eq 2) and ∆Fcav (eq 11) PMF components are collected in Table 1. It can be seem from Figure 4a-f that, except for PhEt and MePrpS, the analytical expression fits both the region of the contact minimum and that of the desolvation maximum very well. The solventseparated minimum is not represented well, in contrast to our earlier results for the methane dimer.2 However, as mentioned above, the solvent-separated minimum is not accurately determined from the simulations because of noise in the data and is so shallow that its reproduction in the side-chain-side-chain interaction potential does not seem to be crucial for constructing a mesoscopic force-field. For ethylbenzene (the model of the phenylalanine side chain) and methly propyl sulfide (the model of the methionine side chain), the analytical expressions underestimate the difference between the depth of the contact minima at the side-to-side and edge-to-edge orientations and do not reproduce the desolvation maxima well. This might be due to the fact that the benzene ring is planar and might not be well represented by a particle of spheroidal symmetry even when the angle of rotation about its axis is averaged out, and the sulfur atom bears some negative charge (Figure 2) making the molecule of methyl propyl sulfide polar; this could give rise to a lower than spheroidal symmetry of the PMF. Nevertheless, given the inaccuracy of the AMBER force field used to determine the PMFs and given the fact that the UNRES force field is aimed at reproducing protein structures at low resolution, the quality of the fit seems to be sufficiently satisfactory. The EvdW and ∆Fcav components of the fitted PMFs of all systems studied are shown in Figure 5. It can be seen that ∆Fcav has a desolvation maximum for all systems and the desolvation maximum shifts to greater distances when the orientation is changed from side-to-side through side-to-edge to edge-to-edge. Conclusions We determined the potentials of mean force of six pairs of molecules modeling hydrophobic side chains in water as functions of distance and orientation of the molecules and fitted an approximate expression to the PMFs composed of the anisotropic Gay-Berne potential11 to represent van der Waals interactions and a cavity term proposed in paper 1 of this series.1 Both components of the approximate PMFs depend on distance and orientation of the spheroidal particles. We demonstrated that the analytical expression introduced to approximate the free energy of the interaction of nonpolar side chains fits the PMFs reasonably well, suggesting that it is a good candidate for the physics-based mean-field potentials of side-chain-side-chain interactions in our mesoscopic UNRES force field3-10 for the simulation of the structure and dynamics of proteins. Our present work is focused on developing and parametrizing similar analytical expressions to describe the interactions that involve polar and charged side chains.
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Figure 5. Plots of the energy components of the analytical approximations to the PMFs for (a) ethane, (b) propane, (c) isobutane, (d) isopentane, (e) ethylbenzene, and (f) methyl propyl sulfide homodimers. Lower curves: ∆Fcav (eq 11). Upper curves: EvdW (eq 2). Solid lines: side-to-side orientation (Figure 3a). Dot-dashed lines: edge-to-edge orientation (Figure 3b). Dashed lines: side-to-edge orientation (Figure 3c).
Acknowledgment. This work was supported by grants from the U.S. National Institutes of Health (GM-14312), the U.S. National Science Foundation (MCB05-41633), the National Institutes of Health Fogarty International Center Grant TW007193, and the Polish Ministry of Science and Informatization (1 T09A 099 30). Mariusz Makowski was supported by a grant from the “Homing” program of the Foundation for Polish Science (FNP). This research was conducted by using the resources of (a) our 818-processor Beowulf cluster at the Baker Laboratory of Chemistry and Chemical Biology, Cornell University, (b) the National Science Foundation Terascale Computing System at
the Pittsburgh Supercomputer Center, (c) our 45-processor Beowulf cluster at the Faculty of Chemistry, University of Gdan´sk, (d) the Informatics Center of the Metropolitan Academic Network (IC MAN) in Gdan´sk, and (e) the Interdisciplinary Center of Mathematical and Computer Modeling (ICM) at the University of Warsaw. References and Notes (1) Makowski, M.; Liwo, A.; Scheraga, H. A. J. Phys. Chem. B 2007, 111, 2910. (2) Makowski, M.; Liwo, A.; Maksimiak, K; Makowska, J.; Scheraga, H. A. J. Phys. Chem. B. 2007, 111, 2917.
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