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A New Coarse-Grained Model for Water: The Importance of Electrostatic Interactions Zhe Wu, Qiang Cui,* and Arun Yethiraj* Department of Chemistry and Theoretical Chemistry Institute, UniVersity of Wisconsin-Madison, 1101 UniVersity AVenue, Madison, Wisconsin 53706 ReceiVed: March 4, 2010; ReVised Manuscript ReceiVed: June 22, 2010
A new coarse-grained (CG) model is developed for water. Each CG unit consists of three charged sites, and there is an additional nonelectrostatic soft interaction between central sites on different units. The interactions are chosen to mimic the properties of 4-water clusters in atomistic simulations: the nonelectrostatic component is modeled using a modified Born-Mayer-Huggins potential, and the charges are chosen to reproduce the dipole moment and quadrupole moment tensor of 4-water clusters from atomistic simulations. The parameters are optimized to reproduce experimental data for the compressibility, density, and permittivity of bulk water and the surface tension and interface potential for the air-water interface. This big multipole water (BMW) model represents a qualitative improvement over existing CG water models; for example, it reproduces the dipole potential in membrane-water interface when compared to experiment, with modest additional computational cost as compared to the popular MARTINI CG model. I. Introduction Water plays a central role in many phenomena in chemistry and biology, and computational studies of water and aqueous solutions have become widespread. The most important part of a computer simulation is the choice of molecular model, and depending on the phenomena of interest a number of classes of models have been used. For systems where quantum effects are not deemed important, classical models have played an important role. In classical models, each molecule is usually modeled as a collection of sites that could, for example, be the O and H atoms, with empirical potentials between these sites. There has recently been a drive to develop coarse-grained (CG) models for water. In many applications, the behavior of the water is not of primary interest, but the water molecules constitute a significant fraction of the computational expense. One approach1 is to treat the water in an implicit fashion, where the water molecules are removed completely and replaced by an effective interaction between the remaining species. In many cases, this is too severe an approximation, and a model that is intermediate between atomistic and implicit solvent is of interest. CG models group several water molecules into a single unit, thus decreasing the computational cost of simulating explicit solvent, and allowing computations to sample the large timescales and length-scales of interest. In this work, we present a new CG model for water that we think has significant advantages over existing models. The most common CG philosophy2,3 is to group a number (3 or 4) of water molecules into a single site. The interaction potential between these sites is then parametrized to reproduce some desired physical properties of water. Generally, the interaction between these CG sites is short-ranged and has no electrostatic (or long-ranged) component, and this results in significant savings in computational time, while introducing hydrodynamic interactions in a natural fashion. Probably the most popular such model is that in the MARTINI force field,2,4 where each CG particle represents 4 water molecules, and * Corresponding author. E-mail:
[email protected] (Q.C.), yethiraj@ chem.wisc.edu (A.Y.).
interactions between sites are of the Lennard-Jones form. The model has been parametrized to reproduce the density and compressibility of bulk water. These CG models, however, introduce unphysical features when applied to interfaces, such as an interface between water and a lipid membrane. The most significant drawback of these nonelectrostatic water models is that the dipole potential at the membrane-water interface is of opposite sign to the results seen in experiment and atomistic simulations. For example, in the MARTINI model, the electrostatic potential in the interior of a DPPC bilayer (in water) is ∼-0.4 V, while the experimental value is +0.2-0.4 V.5,6 Atomistic simulations7,8 also give positive values for this dipole potential. This unphysical feature can be attributed to the nonelectrostatic nature of the MARTINI water potential;9 the electrostatic potential predicted by the MARTINI model is almost identical to that found at a bilayer-vacuum interface. As a consequence, the potential of mean force of charged amino acids at the membrane water interface predicted by the MARTINI model is very different from that obtained using atomistic simulations.10,11 Other water models, such as the soft sticky dipole model12,13 and the GB-EMP model,14 do reproduce some water electrostatic features, but they are also closer in character to atomistic models and thus offer little computational advantage. In this work, we develop a CG model for water by grouping 4 water molecules into one unit. The major departure of this work from previous parametrizations is that we calculate (from atomistic simulations) the properties of clusters of 4 water molecules and then fashion a new model based on these properties, with parameters optimized to reproduce experiment. Each unit contains three charged sites, and there is an additional nonelectrostatic soft interaction between central sites on different units, for which we use a modified Born-Mayer-Huggins15,16 (BMH) potential. We call it the big multipole water (BMW) model, and it reproduces the dipole moment and quadrupole moment tensor of 4-water clusters in atomistic simulations. The parameters of the model are optimized to reproduce several experimental quantities. The model is compatible with the
10.1021/jp1019763 2010 American Chemical Society Published on Web 07/27/2010
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Figure 1. Properties of 4-water clusters in the bulk and at the air-water interface for the SPC (rigid) and SWM4-NDP (polarizable) models: (a) distribution of magnitude of dipole moment, and (b) the element Qxx, of the quadrupole moment tensor (color code same as in (a)).
MARTINI CG lipid model,4 and the MARTINI/BMW model gives the correct membrane potential when compared to experiment. The rest of this Article is organized as follows. Systematic studies of 4-water clusters in atomistic simulations are presented in section II, the BMW model is discussed in section III, predictions of the model for experimental quantities are compared to other models in section IV, and conclusions are presented in section V. II. Water Clusters in Atomistic Simulations We group 4 water molecules into a single coarse-grained unit, as is done in the MARTINI force-field. Water molecules form a percolating hydrogen-bonded network with strong local interactions. Because each molecule has approximately 3.5 hydrogen bonds,17 it seems reasonable to group 4 molecules into a single CG unit; one might anticipate weaker interactions between these CG units. Our choice of 4 water molecules per CG unit turns out to be convenient because the electrostatic properties of these clusters are similar for bulk water, the air-water interface, and salt solutions (see below). We calculate the properties of clusters as follows. For each water molecule, we define a cluster as that molecule plus its three nearest neighbors. We carry out simulations of atomistic models, and from the trajectories we calculate the average properties of clusters. The dipole moment, b µ, is
f µ)
∑ qiri
(1)
i
where qi and ri are the charge and position of site i with the central site as the origin, ri ) {rix, riy, riz}, where rix, riy, and riz are the Cartesian coordinates of site i, and the sum is over all of the sites in the cluster. The elements of the quadrupole moment tensor Q are
QRβ )
∑ qi(3riRriβ - ri2δRβ)
(2)
i
where R,β ) x, y, or z. The coordinate is defined with the O atom of the central water as the origin. The central water lies in the xz plane, and the z axis corresponds to its bisector for the H-O-H angle. The importance of quadrupole moment in determining the air-water potential has been discussed in
previous studies,18-20 which in part motivated our inclusion of quadrupole moment in developing the CG water model. The distribution of the magnitude of dipole moment and the elements of the quadrupole moment tensor are very similar in a number of different atomistic models in the bulk and at the air-water interface (for the air-water interface, we include water molecules within 2.5 Å of the interface). We consider seven popular water models: SPC,21 SPC/E,22 TIP4P/2005,23 TIP5P,24 TIP3P,25 SWM4-DP,26 and SWM4-NDP,27 which include models with fixed charges as well as polarizable ones. The distribution of the magnitude of cluster dipole moments and components of the quadrupole moment tensor are insensitive to the models and are similar for bulk water and the air-water interface. Figure 1a and b compares the distribution of | µ| and Qxx, respectively, for two of these models (SPC and SWM4NDP). Similar results are obtained with other models (see the Supporting Information). The electrostatic properties of the 4-water clusters are insensitive to the model and bulk water or water at the air-water interface. The reason the cluster properties are similar at the air-water interface (when compared to the bulk) is that the interface tends to reorient the clusters rather than change their dipole or quadrupole moments. The cluster dipole and quadrupole moments are also insensitive to salt concentration. We perform simulations with two salts (NaCl, CsI) at 4 M concentration with the SWM4-DP28 model. Even for 4 M salt, that is, 7 water molecules per ion on average, the distribution of multipoles is indistinguishable from bulk water (see the Supporting Information). It is only at very short distances from the ions, less than 2 Å, that the cluster multipole moments are perturbed (solutions with 1 M NaCl, CsI, CaCl2). This is not an issue because CG models for ions usually are meant to include the first solvation shell of water within the ion CG particles. The cluster properties are therefore robust, and we can conclude that (i) the cluster electric multipoles are shared features in various atomistic models, in bulk, interface, and salt solutions; (ii) the large dipole and quadrupole terms are sufficient to represent the electrostatic interactions in the clusters; and (iii) the orientation preference in the cluster dipoles is the origin for interface potential. Therefore, a CG model can be constructed merely to reproduce bulk behavior and should be able to predict interface properties as well. III. Big Multipole Water Model In the big multipole water (BMW) model, we coarse-grain a 4-water cluster into a single unit, as depicted in Figure 2. The
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Wu et al. TABLE 2: Parameters in the BMW Model
Figure 2. A 4-water cluster is coarse grained into a three-site model with θ ) 120°, l ) 1.2 Å, q ) 1e-.
TABLE 1: Average Dipole Moment (in Units of debye) and Elements of the Quadrupole Moment Tensor (units of debye · Å) in the BMW Model As Compared to Clusters of Atomistic Watera µ atomistic cluster BMW model a
6.0 6.0
Qxx 9.0 9.0
Qyy -7.0 -7.2
εr
ε (kJ/mol)
rm (nm)
f
a
1.3
1.521
0.761
5.871
7.953
undesirable in the larger (as compared to atomistic) CG water models. A modified form of the Born-Mayer-Huggins (BMH)15,16 potential (eq 4) is soft and provides a good fit to the PMF. We therefore use this interaction to describe nonelectrostatic interactions. The BMH potential has four parameters (ε, rm, f, and a): ε and rm are the depth and position of the potential minimum, and f and a determine the softness of the repulsion and shallowness near the minimum. The LJ form is recovered for f ) 0. We modify the BMH potential to obtain a LJ repulsion at short distance.
Qzz -2.0 -1.8
V(r) )
{ () ()
rm 6 6 - f rm 12 + r f 6 - f 12 r 1- a 12 f r exp a 1 a rm ε
[(
The off-diagonal elements of Q are zero in both cases.
)]}
(4)
For the electrostatic interactions, we use a dielectric screening coefficient, εr, that is different from unity. This is necessary in CG models because the response to an external electric field is less effective than in atomistic models. We use εr as an adjustable parameter to reproduce experimental data for the permittivity. Electrostatic interactions in the new CG model are calculated using the Particle-Mesh-Ewald (PME)33,34 method. From preliminary studies, in agreement with previous work,35 we find that including the PME is crucial to obtain correct electrostatic properties at the air-water interface. Figure 3. The potential of mean force (PMF) between the central sites of 4-water cluster (SPC). The solid line is a fit using a modified form of the Born-Mayer-Huggins potential (with ε ) 0.783 kJ/mol, rm ) 0.604 nm, f ) 5.842, a ) 7.952).
central (blue) site has a charge of -2q, and the other sites have a charge of +q. The rigid topology allows one to employ efficient simulation schemes (SETTLE29 or SHAKE30) while also reproducing the electrostatic properties of water clusters (see Table 1). A nonelectrostatic (van der Waals) interaction is introduced between the central site of different CG units. To estimate the form of the interaction, we calculate the potential of mean force (PMF) between water clusters as follows. Cluster configurations are taken randomly from an atomistic (SPC) water simulation trajectory and placed at different separations (r) of central sites (in vacuum). The total energy is calculated for each orientation (Ω), and the PMF, W(r), is given by
W(r) ) -kBT ln〈exp(-U(r, Ω)/kBT)〉
(3)
where kB is Boltzmann’s constant, T is the temperature, and the average is over configurations and orientations. The PMF (see Figure 3) is quite soft, suggesting that a simple Lennard-Jones interaction may not suffice. A soft interaction is necessary because clusters in reality can penetrate each other. A Lennard-Jones interaction would result in long-ranged oscillatory pair correlation functions,31,32 which is particularly
IV. Results and Discussion A. Parameterization of Model and Comparison to Experiment and Simulations. There are five parameters in the BMW model: ε, rm, f, a, and εr, and we use five properties of water, namely, the density, isothermal compressibility, permittivity in bulk, surface tension, and interface potential at the air-water interface, to obtain these parameters. The BMW model is first parametrized using a trial-and-error scheme. When all five properties are within a reasonable range as compared to experiment, a steep descent optimization is performed to ensure an accurate fitting of both electrostatic and nonelectrostatic features. The values of the optimal parameters are given in Table 2. We note that the optimized dielectric screening coefficient, εr, is fairly close to 1, which indicates that the majority of the electrostatic properties of bulk water are explicitly described with the multipole moments of our model. The BMW predictions are compared to other models and experimental data in Table 3. For these properties, results obtained using the BMW model are in excellent agreement with experiment. For the density and air-water interface potential (see the Supporting Information for a more detailed comparison), the predictions of the BMW model and atomistic models are similarly good (we do not compare the model to experiments for the air-water interface potential because the experimental value is indirect and not widely accepted). For the dielectric permittivity, the BMW model predicts a slightly lower value (74), while the TIP3P model predicts a higher value (82) when compared to experiment. Both models are superior to the TIP4P/ 2005 model. For the compressibility and the surface tension, the BMW model is not as accurate as the TIP4P/2005 but far
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TABLE 3: Water Properties in Bulk and Interface at 300 K BMW exp.46 MARTINIa TIP3P47 TIP4P/200523 3
density (kg/m ) 1047 compressibility 3.3 (×10-5 bar-1) surface 77 tension (dyn/cm) dielectric 74 permittivity -0.6 air-water interface potential (V) a
997 4.6
900 2.6
1002 6.4
998 4.7
72
33
55
6848
82
60
-0.6
-0.5
78
With a time step of 10 fs42 and 10% antifreezing particles.
superior to the MARTINI model. Overall, the results from the BMW are at least comparable to those of the atomistic models. The BMW is comparable in computational efficiency to the MARTINI model (with the same computational protocols) and about 2 orders of magnitude more efficient than atomistic simulations. For example, in a system with 128 MARTINI DPPC lipids and 2000 water (atomistically 8000) particles simulated in GROMACS 4.0.5,36 it takes 6 min per ns (on an Intel Core2 Quad 2.40Ghz machine) with the BMW model, while standard MARTINI (both with 20 fs time step) takes ∼1 min for the same calculation on the same machine. These may be compared to atomistic simulations (CHARMM force field37 in GROAMCS) for the same system, which takes about 1 day to simulate 1 ns. Note that the efficiency difference between BMW and MARTINI largely comes from the following: a factor of 3 due to the three-site topology of BMW versus a single particle in MARTINI, and a factor of 2 from PME (with a spacing of 0.2 nm) in BMW versus the shift scheme in MARTINI for the electrostatic treatment. In other words, other technical details, such as adopting the somewhat complex BMH functional form for the nonelectrostatic component, do not contribute significantly to the speed of calculations. The relatively modest increase in the computational cost (a factor of 6) of BMW over the original MARTINI makes it a preferable CG model for problems that are sensitive to the treatment of electrostatics. B. Bilayer-Water Interface. To combine the BMW model for water with the MARTINI model for the lipids, we change only the interactions between the BMW water (center site) and the MARTINI charged particle types (Qda, Qd, Qa, Q0). In standard MARTINI charged types, the nonelectrostatic interaction is parametrized to implicitly include the charge-dipole interaction between the charged groups and the water particles. Because explicit electric multipoles are present in the BMW model, these VDW interactions are changed from supra attractive (εij ) 5.6 kJ/mol) to almost attractive (εij ) 4.5 kJ/mol), while maintaining the LJ form of cross term interaction. With this minor modification, the structural properties of the membrane are not affected when MARTINI water is replaced with BMW water. Take DPPC bilayers (at T ) 325 K) for example, the area per lipid is 64.2 Å2 and thickness is 39.8 Å, in agreement with MARTINI and experimental results.4,38 The density profiles of the various components are also very similar (not shown). Importantly, the BMW model when combined with the MARTINI lipid force field gives improved values for the membrane electrostatic properties. Figure 4 depicts the electrostatic potential across the DPPC bilayer. The contribution from the BMW water is significant and of opposite sign to that from the bilayer. These two contributions cancel, and the
Figure 4. Dipole potential for MARTINI DPPC lipid bilayers with BMW water (325 K). The blue curve is the contribution from water screening, while the green one is from the membrane; the total potential is shown in black. The bilayer center is located at 0 nm.
resulting dipole potential for the DPPC bilayer is +0.3 V, which is in quantitative agreement with experiment and polarizable atomistic model simulation results (+0.2-0.4 V).5,6,8 Note that the oscillation of the potential at the interface, although not readily detected with experiments, is also generally seen in calculations from atomistic models.7 V. Conclusions We present a new coarse-grained model for water. The philosophy is to group clusters of 4 water molecules into one CG unit. Its electrostatic properties are obtained from an analysis of clusters taken from atomistic simulations. The form of nonelectrostatic interaction between the CG units is determined from the potential of mean force between clusters. The parameters in the model are fitted to experimental data, and the resulting model accurately reproduces experimental data for the density, compressibility, permittivity, surface tension, and air-water interface potential. The model is only 6 times more expensive than the nonelectrostatic MARTINI water (with the same integration time step) and 2 orders of magnitude more efficient than atomistic models. When combined with the MARTINI lipid force field, the model gives a lipid interface dipole potential that is in quantitative agreement with experiment and polarizable atomistic simulation results, without affecting the lipid properties that the MARTINI model describes well. This result emphasizes the importance the water electrostatics at the bilayer surface. We therefore anticipate that the BMW model will provide a more reliable picture of heterogeneous aqueous systems, which makes it useful in not only CG simulations but also multiscale simulations that involve both atomistic and CG components.39 Although the CG model successfully reproduces many water properties, there are several limitations that should be kept in mind. First, the model has been parametrized for fluid phase at one temperature (300 K). It is not clear how the model can be used to study crystals where the clusters have to be different, especially in the coordination number. Thus, in its present form, the model is not recommended for studying ice. The freezing
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point is below 200 K due to the soft interaction, which is convenient for biological models because (unlike MARTINI) artificial freezing agents do not have to be incorporated. Similarly, the model is not suited for providing a physical description for the vapor phase, including that near hydrophobic regions of proteins and lipids, which requires explicitly representing single molecules rather than clusters. Finally, explicit hydrogen-bonding interactions are absent in the model. These drawbacks are common to all CG models, however, and one has to be careful to choose the model carefully for the application of interest. Acknowledgment. This research has been supported by the National Science Foundation (CHE-0957285 to Q.C. and CHE0717569 to A.Y.). Computational resources from the National Center for Supercomputing Applications at the University of Illinois and the Centre for High Throughput Computing (CHTC) at UW-Madison are greatly appreciated. Supporting Information Available: Additional figures. This material is available free of charge via the Internet at http:// pubs.acs.org. Appendix: Simulation Details for All of the Atomistic Models and the New BMW Models A. Atomistic Simulations. Simulations for the SPC, SPC/ E, TIP3P, TIP4P-2005, and TIP5P models are performed using GROMACS 3.3.3,40 and simulations for the SWM4-DP and SWM4-NDP models are performed using CHARMM,41 following standard simulation procedures. Bulk simulations are performed in the NPT ensemble, and the duration of these simulations is 6 ns. Simulations of the air-water interface are performed for 20 ns in the NVT ensemble with a slab of water along the z-axis. For the SPC model, the box size is 5 × 5 × 15 nm with a slab thickness of 5 nm, and for the SWM4-DP model, the box size is 3 × 3 × 9 nm with a slab thickness of 3 nm. Simulations with added salt are performed for 1-4 M NaCl and CsI (with the SWM4-DP model) and 1 M CaCl2 with the TIP5P/OPLS model (there is no Ca2+ model for SWM4DP water). B. BMW Model Simulations. Simulations with the BMW model are performed using GROMACS 4.0.536 with a time step of 20 fs; the fluctuation of the total energy of a 1 ns NVE simulation is around 17% of the ones in kinetic or potential energy when large neighbor list is applied (for benchmark, we use nlist ) 1 and rlist ) 1.7 nm, but for the sake of efficiency we suggest nlist ) 10 and rlist ) 1.4 nm), which fulfills the requirement for the reliable integration of equations of motion (less than 20%, following suggestions in the literature42,43). For the NVT and NPT ensembles, the Berendsen scheme with τT ) 0.2 ps and τP ) 2 ps is employed for temperature and pressure coupling. SETTLE algorithm29 is applied to constrain all “bonds”, and all intraunit interactions are excluded. Electrostatic interactions are calculated using PME with a spacing of 0.2 nm. The BMH potential is implemented via a user defined table with a switching function from ron ) 1.0 nm to roff ) 1.4 nm as given in eq s1. The rest of the calculation details follow the standard MARTINI protocol.
S(r) )
2 2 (roff - r2)2 · (roff + 2r2 - 3r2on)
ron
2 (roff - r2on)3 e r e roff
(s1)
The compressibility, χ, surface tension, γ, and electrostatic potential, Φ(z), are calculated via
χ)
〈(∆V)2〉 kBTV
γ ) Lz[-(Pxx + Pyy)/2 + Pzz]/2
(s2)
(s3)
and
Φ(z) - Φ(0) ) -
4π εεr
∫0z ∫0z' F(z'') dz'' dz'
(s4)
where V is the volume in an NPT simulation, Pij is the ij component of the pressure tensor, Lz is the height of the simulation cell, and F(z) is the charge density as a function of z. The permittivity is calculated from the potential of mean force between ions in water. There is a small difference in its definition of the permittivity in atomistic and CG models. A dielectric screening coefficient, εr > 1, is introduced into the BMW model to compensate for the missing response from detailed electronic degrees of freedom to external field. Therefore, the electric susceptibility has a static part induced by the screening factor εr, and a dynamic part caused by the reorientation of the CG multipoles. Because the dielectric permittivity comes physically from both components, we calculate the permittivity from the PMF between charged ions rather than by following the standard Kirkwood scheme.44 Umbrella sampling is applied to restrain the charged particles, and the Jacobian factor associated with the particle separation was added45 to the PMF. The long-range part of the PMF is fitted to obtain the permittivity for the CG (BMW) model. References and Notes (1) Feig, M.; Brooks, C. L., III. Curr. Opin. Struct. Biol. 2004, 14, 217–224. (2) Marrink, S. J.; de Vries, A. H.; Mark, A. E. J. Phys. Chem. B 2004, 108, 750. (3) Shelley, J. C.; Shelley, M. Y.; Reeder, R. C.; Bandyopadhyay, S.; Klein, M. L. J. Phys. Chem. B 2001, 105, 4464. (4) Marrink, S. J.; Risselada, H. J.; Yefimov, S.; Tieleman, D. P.; de Vries, A. H. J. Phys. Chem. B 2007, 111, 7812. (5) Shapovalov, V. L.; Kotova, E. A.; Rokitskaya, T. I.; Antonenko, Y. N. Biophys. J. 1999, 77, 299. (6) Clarke, R. J. AdV. Colloid Interface Sci. 2001, 89, 263. (7) Siu, S. W. I.; Va´cha, R.; Jungwirth, P.; Bo¨ckmann, R. A. J. Chem. Phys. 2008, 128, 125103. (8) Harder, E.; MacKerell, A. D.; Roux, B. J. Am. Chem. Soc. 2009, 131, 2760. (9) Lin, J. H.; Baker, N. A.; McCammon, J. A. Biophys. J. 2002, 83, 1374. (10) Monticelli, L.; Kandasamy, S. K.; Periole, X.; Larson, R. G.; Tieleman, D. P.; Marrink, S. J. J. Chem. Theory Comput. 2008, 4, 819. (11) Vorobyov, I.; Li, L.; Allen, T. W. J. Phys. Chem. B 2008, 112, 9588–9602. (12) Orsi, M.; Haubertin, D. Y.; Sanderson, W. E.; Essex, J. W. J. Phys. Chem. B 2008, 112, 802–815. (13) Liu, Y.; Ichiye, T. J. Phys. Chem. 1996, 100, 2723. (14) Golubkov, P. A.; Ren, P. J. Chem. Phys. 2006, 125, 064103. (15) Fumi, F. G.; Tosi, M. P. J. Phys. Chem. Solids 1964, 25, 31. (16) Fumi, F. G.; Tosi, M. P. J. Phys. Chem. Solids 1964, 25, 45. (17) Kumar, R.; Schmidt, J. R.; Skinner, J. L. J. Chem. Phys. 2007, 126, 204107. (18) Wilson, M. A.; Pohorille, A.; Pratt, L. R. J. Chem. Phys. 1988, 88, 3281–3285. (19) Wilson, M. A.; Pohorille, A.; Pratt, L. R. J. Chem. Phys. 1989, 90, 5211–5213. (20) Harder, E.; Roux, B. J. Chem. Phys. 2008, 129, 234706.
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