New functionalities in the GROMOS biomolecular simulation software

Authors


Abstract

Since the most recent description of the functionalities of the GROMOS software for biomolecular simulation in 2005 many new functions have been implemented. In this article, the new functionalities that involve modified forces in a molecular dynamics (MD) simulation are described: the treatment of electronic polarizability, an implicit surface area and internal volume solvation term to calculate interatomic forces, functions for the GROMOS coarse-grained supramolecular force field, a multiplicative switching function for nonbonded interactions, adiabatic decoupling of a number of degrees of freedom with temperature or force scaling to enhance sampling, and nonequilibrium MD to calculate the dielectric permittivity or viscosity. Examples that illustrate the use of these functionalities are given. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2012

Introduction

During the past three decades, the GROMOS software for biomolecular simulation has been developed in the research group of van Gunsteren, first at the University of Groningen and since 1990 at the ETH.[1, 2] The functionalities of this molecular simulation software were summarized in three papers[3–5] for the GROMOS87 and GROMOS96 FORTRAN releases and for the new C++ version GROMOS05. The latter version has been under continuous development and a release to the scientific community occurred in 2011. A description of its functionalities can be found in the nine GROMOS manuals (http://www.gromos.net). Here, we briefly describe some of the new functionalities that have been added since 2005: electronic polarizability,[6, 7] an implicit surface area and internal volume solvation term[8] for the calculation of interatomic forces, functions required to use the GROMOS coarse-grained (CG) supramolecular force field,[9] a multiplicative switching function for nonbonded interactions, which replaces the one described in Ref.5, adiabatic decoupling as a means to enhance the sampling of a not too large subset of degrees of freedom of a system,[10] and nonequilibrium molecular dynamics (MD) methodology to compute the dielectric permittivity and relaxation[11] or viscosity. The architecture and implementation of the GROMOS C++ code is described elsewhere,[12] as are features with regard to biomolecular structure refinement based on experimental NMR or X-ray data,[13] to the computation of relative free energies,[14] and the analysis of MD trajectories using GROMOS.[15] The new simulation functionalities are illustrated using examples.

Methods

Polarizability

Electronic polarizability allows for a more accurate description of the nonbonded interactions in classical atomistic simulations and must be implemented in the software when using polarizable force fields.[6]

Several methods have been described in the literature that explicitly treat electronic polarization.[6, 16] GROMOS uses the charge-on-spring (COS),[17] also called Drude-oscillator[18] or shell,[19] model where an inducible pmb μi is modelled by attaching a massless, virtual site with a point-charge qmath image to the polarizable center i, via a spring with harmonic force constant kmath image,

equation image(1)

The inducible dipole μi adapts its size and direction according to its polarizability αi and the electric field Emath image at the COS of the polarizable center i, assuming isotropic αi and linear dependence of μi on Emath image, according to

equation image(2)

The charge at the polarizable center with position ri, which may have a permanent charge qi, is then (qiqmath image). Thus, the induced dipoles μi are represented by

equation image(3)

where rmath image is the position of the COS. In COS-based schemes in which the COS are not explicitly treated as additional degrees of freedom, the sum of the electrostatic and spring forces acting on any COS should be zero, and the virtual charge qmath image must be positioned such that

equation image(4)

with the force fmath image due to the spring given by

equation image(5)

and fmath image due to the (Coulombic) electric field at the COS Emath image) given by

equation image(6)

To satisfy eq. (4), the μi, that is, the rmath image, must be determined from the Emath image, where the field Emath image does not contain a contribution from the charge (qiqmath image) at ri. However, as the displacement |rmath imageri| of the COS from the polarizable center is nonzero on polarization, a better approximation of the ideal inducible dipole μi at site i would require rmath image to be determined from the electric field Ei at the polarizable center itself,[7]

equation image(7)

Using this approximation, the total force acting on the COS is only zero if

equation image(8)

which is usually not the case for the induced COS dipole due to the nonzero values for |rmath imageri|. By choosing qmath image large enough, |rmath imageri| adopts relatively small values, resulting in small differences between Ei and Emath image. On the other hand, the size of qmath image is limited to values for which |rmath imageri| is significant with respect to interatomic distances such that numerical precision is ensured when calculating, for example, interaction energies involving induced dipoles.[7, 17] The COS method has been used in combination with iterative procedures to minimize the energy of the COS with respect to its position rmath image, that is, solving eq. (4). In GROMOS qmath image is generally set to −8e.

The COS method treats the induced dipole moments via additional point charges only, which allows for an easy introduction of polarizability into schemes to compute long-range electrostatic forces, such as the reaction-field, Ewald-summation, particle-particle-particle-mesh (P3M), and Particle-Mesh-Ewald (PME) techniques or into a quantum-mechanical Hamiltonian for the electronic degrees of freedom of a (solute) molecule. The electrostatic potential ϕi at the polarizable centers i due to the monopoles and dipoles in the system can be expressed using Coulombic terms only

equation image(9)

where the positions of the N atoms and corresponding virtual charges are denoted by rN = (r1, r2, …, rN) and rN = (rmath imagermath image, …, rmath image), and εcs is the relative dielectric permittivity used for the model interactions. Because of the dependence of the rmath image on rN and rN via Ei in eq. (7), the relation between ϕi and the electric field Ei is given by

equation image(10)

when applying a Born–Oppenheimer-like iterative self-consistent field procedure, however, all rN are at every iteration step determined in the electric field due to all other qj and qmath image. When using a convergence criterion which minimizes ϕi with respect to the positions rmath image, the second term in eq. (10) is zero at convergence because ∂ϕi/∂rmath image = 0. Thus,

equation image(11)

The electrostatic part Vele of the potential energy can also be expressed in terms of Coulomb interactions. The only non-Coulombic term to be added to Vele is the self-polarization energy Vself, which in the COS model can be expressed in terms involving point charges as well

equation image(12)

with

equation image(13)

and

equation image(14)

Next, we consider the expression for the forces fi that act on (polarizable) atomic centers i

equation image(15)

Note again the dependence of the rmath image on the ri which appears in the second term on the right in eq. (15), which might adopt nonzero values because Vele not only contains terms due to the ϕi, the first two terms on the right in eq. (13), but also due to the ϕj, the last two terms on the right in eq. (13), and Vself, whereas when using eq. (7) only ϕi is minimized with respect to all rmath image. When nevertheless using assumptions in eqs. (4) and (8), eq. (15) reduces to

equation image(16)

From the assumptions in eqs. (4) and (8), we have

equation image(17)

and the reduced expression fmath image for the atomic forces becomes

equation image(18)

If the COS model is used with a reaction-field, a reaction-field correction term[11] is to be added to Vcoul

equation image(19)

where Rrf is the reaction-field cut-off and Crf is the reaction-field constant defined as[11]

equation image(20)

and κ is the inverse Debye screening length and εrf is the reaction-field dielectric permittivity outside the reaction-field cut-off radius Rrf. The reaction-field term makes the electrostatic force and interaction zero at the reaction-field cut-off distance. Using a cut-off radius plus reaction-field, the summation over j in eqs. (13) and (18) is over sites j inside the cut-off of site i, and omits the covalently bound nearest neighbors j of atom i that are excluded from the nonbonded interaction. These neighbors j of i must not be excluded in the summation over j in eq. (19) because they contribute to the reaction-field.[11] The last term in eq. (19) is a constant that is added to represent the self-interaction of the permanent, that is, non-COS, charges.

Off-atom site.

For certain polarizable models,[20, 21] it is necessary to add an additional virtual atomic interaction center M. The position of this virtual site M is defined in the plane of the three nonvirtual atoms i, j, and k,

equation image(21)

where γ is a constant, which determines diM = riM as a function of the distances dij = rij and dik = rik. For example, in a model for H2O, i would denote the oxygen atom and j and k the hydrogen atoms. The addition of the massless site M defined by eq. (21) does not introduce any extra degrees of freedom into the molecule in the calculation of the kinetic energy of the system. The “pseudo-force” fM that acts on the virtual-atom site M is redistributed to the nonvirtual atoms i, j, and k according to

equation image(22)

Damping of the polarization.

A problematic feature of most polarizable models, apart from their larger demand for computing power than nonpolarizable ones, is their tendency to allow for overpolarization in the presence of strong local electric fields, leading to a polarization catastrophe, and a too large static dielectric permittivity. There are several approaches to resolve this problem. Using the GROMOS force fields, a polarization catastrophe is avoided by having a large enough repulsive Lennard–Jones interaction between nonhydrogen atoms to ensure that dipole–dipole distances are larger than (4αmath image)1/6. In addition, the GROMOS simulation software allows the linear dependence of the induced dipole μind,i on the electric field Ei to be substituted by a sublinear dependence for large field strengths,[21] which can be achieved by making the polarizability αi electric field dependent. For example, αi is replaced by αD,i = αD,i(E) for large Ei using

equation image(23)

where pi is a polarization damping parameter. The induced dipole is then defined as

equation image(24)

This change in the expression for μmath image also influences the self-polarization contribution to the potential energy

equation image(25)

with Vself = ∑iVself,i, where i runs over all polarizable centers.

Free energy calculation.

For free energy calculations using the coupling parameter approach, the Hamiltonian of the system has to be made dependent on a coupling parameter λ. Free energy differences between two states A (λ = 0) and B (λ = 1) are then obtained from a thermodynamic integration over λ of the averages of the derivative of the Hamiltonian with respect to λ. Several terms in the λ-dependent Hamiltonian will change compared to the nonpolarizable case.

The λ-dependent Vcoul term in eq. (12) reads

equation image(26)

and the λ-dependent, reaction-field correction term Vrf is

equation image(27)

where

equation image(28)

is an auxiliary function to simplify the formula, qmath image is the charge on atom i in state A(B), αC is the Coulombic soft-core parameter, n is an integer larger than zero, and Crf is defined in eq. (20). The derivatives with respect to λ are then

equation image(29)

and

equation image(30)

Using

equation image(31)
equation image(32)
equation image(33)

one gets for the λ-dependent Vself term

equation image(34)

The λ-derivative of this perturbed self-polarization potential energy is

equation image(35)

Implicit solute surface area and internal volume solvation term

To investigate the structure and dynamics of biological macromolecules in solution with MD simulation, typically an atomic-level representation is used. Inclusion of the solvent degrees of freedom then results in a huge number of degrees of freedom, greatly increasing the cost of the calculations. Complete neglect of the solvent effects, however, greatly limits the accuracy of the simulation.[22] For large-scale or computationally expensive applications such as drug screening, protein structure prediction, protein folding, computational engineering of peptides, and proteins, or aggregation of such molecules, a compromise between the speed of simulations done in vacuum and the accuracy of explicitly representing the solvent may be useful. Implicit solvent models, in which the effect of the solvent is modeled as a mean solvation term in the potential energy function of the solute, provide a means of including solvent effects at minimal cost.[23–26]

A widely used and simple class of implicit solvent models are the so-called solvent accessible surface area (SASA) models in which the local solute–solvent interactions are assumed to be proportional to the SASA of the solute atoms.[27–41] The SASA term should account for the free energy cost of forming a hard-sphere cavity in water. As it has been shown that the cavity creation work depends on both the solvent accessible volume and the SASA, a term proportional to the internal volume, defined as the sum of the volumes of all nonsolvent-exposed atoms, can be added that takes into account interactions between buried atoms and the solvent as well.[8] Another factor missing from existing SASA-based implicit solvent models is the inclusion of the favorable van der Waals interactions between the interior atoms of the solute and the solvent.[42–50] The GROMOS software contains functions that calculate the SASA of a solute according to the approximate expression of Hasel et al.[51] and the buried volume using an approximation[8] based on the SASA.

The solvation free enthalpy of a solute Gsolv may be partitioned into a solvent–solvent cavity term Gcav, a solute–solvent van der Waals term GvdW, and a solute–solvent electrostatic term Gelec

equation image(36)

Here, as in the original SASA model built on the GROMOS force field,[37] rather than calculate Gelec explicitly, the vacuum parameterization for the intrasolute interaction function is used: the partial charges of groups of atoms with nonzero total charge are changed such that these groups have zero charge, but their hydrogen-bonding capacity is maintained.[2] Gcav and GvdW are combined into a mean solvation term, such that the total interaction function for a protein solute containing N atoms with Cartesian coordinates rN = (r1, r2,…, rN) consists of two parts

equation image(37)

where Vphys(rN) is determined by a standard GROMOS force field for vacuum simulations.[2] The mean solvation term may be further split

equation image(38)

Each term in eq. (38) is calculated as a sum of atomic contributions. The first term is that of the original GROMOS SASA model[37]

equation image(39)

There are just two types of atom-specific solvation parameters σi, for hydrophilic (σmath image) and hydrophobic (σmath image) atoms.[8] The SASA of atom i, Ai, is defined using the approximate analytical expression of Hasel et al.[51]

equation image(40)

The total surface area of an isolated atom i with radius Ri accessible to a solvent probe atom with radius Rsolv is given by

equation image(41)

and the overlap reduction factor[52] bij for atoms i and j at a distance rij = ((rirj)2)1/2 is given by

equation image(42)

The parameters pi and pij were optimized by Hasel et al.[51] for Rsolv = 0.14 nm. Their mapping onto GROMOS atom types, along with the corresponding Ri values, are given in Ref.8.

The SASA values Ai obtained in eq. (40) are used in a switching function g(A) to select the atoms of the solute that have Ai = 0 to contribute to the Vmath image(rN) term of eq. (38). This term accounts for the contribution of the interior solute atoms to the solvation free energy[45]

equation image(43)

The parameters Amath image and Amath image switch g from 1 to 0 for Amath imageAiAmath image, so that only atoms with AiAmath image contribute to the buried volume. The conditions for the choice of switching function are (with 0 ≤ Amath imageAmath image)

equation image(44)

The following switching function was chosen

equation image(45)

This differentiable approximation of a step function gives the value 1, if an atom has a very small or zero Ai, that is, the atom is in the middle of the solute, and a value of zero if an atom is at the surface, that is, Ai is large and thus the atom should not contribute to the buried volume term.

The choices of the parameters that can be freely adjusted in the SASA/VOL model, namely Amath image, Amath image, σmath image, and σmath image, are discussed in Ref.8 where the spatial derivatives of the equations constituting the mean solvation energy term, which are required to compute the forces, are given in the Supporting Information.

Coarse-grained supramolecular force field terms

For very large systems, for example, protein complexes in aqueous solution, it may be efficient to represent the many solvent degrees of freedom at a supramolecular level. For such a CG solvent, several solvent molecules are put into one CG bead, particle or CG molecule, leading to faster simulations due to a reduction of the degrees of freedom compared to fine-grained molecular models and simulations. A CG model for liquid water should represent van der Waals as well as electrostatic interactions. A CG water model[9] that is compatible with the GROMOS force fields has one van der Waals interaction site CP and two electrostatic interaction sites, CP and dipole particle (DP), and one such a CG water molecule represents five real, fine-grained water molecules, thereby allowing orders of magnitude faster simulations. The first interaction site, called CP, interacts with other CG beads through a pairwise Lennard–Jones and an electrostatic potential energy function. The second interaction site is a so-called DP which carries a charge opposite to that of the central site and interacts purely electrostatically, that is, has no Lennard–Jones interaction, with the charged sites of other CG beads. The nonbonded potential energy between two sites i and j depends on their distance rij and has three contributions

equation image(46)
equation image(47)
equation image(48)

where Rrf = Rc are the cut-off and reaction-field radii, C12(i,j) and C6(i,j) are the Lennard–Jones parameters, qi is the charge of particle i, and Crf is the Coulomb reaction-field constant defined in eq. (20).

The total nonbonded potential energy is obtained by summing over all ordered pairs (i,j), thereby taking into account the different nearest neighbor and cut-off radius exclusions,[11]

equation image(49)

where the last term is a constant added to represent the self-interaction of the charged particles. For atomistic models, the dielectric permittivity inside the cut-off sphere εcs = 1 whereas the CG model has εcs > 1, representing the dielectric response of the interior of a CG bead. The two interaction sites i and j of a bead that do not interact mutually through Vmath image and Vmath image are connected by an unconstrained bond with a half attractive quartic potential energy function

equation image(50)

for rij > rCP-DP, where rCP-DP sets a bound on the distance rij between particles i and j belonging to one CG bead. The use of a quartic instead of a harmonic attractive bond-length potential energy function makes the polarizability nonlinear[21] and has the advantage that overpolarization due to close distances between charges of different molecules is avoided as a result of the superlinear increase of the forces for longer bond lengths. The polarizability of the model resides in the free rotation of the DP particle around the central CP site and the oscillation of the bond connecting them, thus adjusting the dipole moment of a CG bead to the electric field of the environment.

Multiplicative switching function

When using a cut-off radius for nonbonded interactions, the error due to the discontinuity of the force at the cut-off can be masked by the use of a switching function that smoothly switches the force and energy to zero at the cut-off distance between two particles. To be able to switch or shift arbitrary functions of r, that is, not only negative powers of r, such as soft-core or reaction-field terms, one should use a multiplicative switching function that serves as a prefactor of the total nonbonded interaction function Vnb(r), and avoid the use of additive switching functions[5] that are added to each term of the nonbonded interaction function. Thus, in GROMOS the nonbonded interaction may be smoothly switched off at Rc according to

equation image(51)

The function S(r) equals 1 for rRs, where Rs is the switching radius, and equals 0 for rRc, where Rc is the cut-off radius. In the range RsrRc it should satisfy the conditions

equation image(52)
equation image(53)

where the notations S = dS/dr and S = d2S/dr2 are used for the derivatives. This is because we have

equation image(54)

and

equation image(55)

A switching function that satisfies conditions in eq. (53) and has simple coefficients in its first derivative form, which is used to calculate forces, is

equation image(56)

with

equation image(57)

and

equation image(58)

The constants A, B, and C can be chosen such that conditions in eq. (52) are satisfied and will only depend on Rs and Rc. We find

equation image(59)

which leads to

equation image(60)
equation image(61)
equation image(62)

for the range RsrRc,

equation image(63)
equation image(64)

This switching function replaces the one described in Ref.5.

Adiabatic decoupling of a subset of degrees of freedom to enhance sampling

If one is interested in artificially enhancing the sampling of a particular, small subset of the degrees of freedom of a system, one may adiabatically decouple their motion from the other degrees of freedom in the system and then use high temperature or force scaling to better sample the subset of degrees of freedom.[10]

A particular set of 3Nh degrees of freedom can be adiabatically decoupled from the other 3Nl degrees of freedom of the system by increasing the mass of the Nhh type particles,

equation image(65)

with sm ≫ 1 in such a way that the transfer of kinetic energy between the decoupled degrees of freedom through momentum conserving collisions becomes very small. sm has to be chosen in such a way that the lightest mass of the Nh atoms, usually a hydrogen, is made significantly larger than the heaviest mass of the Nl atoms. It is therefore recommended to choose sm ≥ 100. Here, we have denoted the masses of the Nhh type particles collectively by mh and the ones of the Nl remaining atoms collectively by ml. This decoupling is more easily achieved[10] if NhNl. The nondynamic equilibrium properties of the system remain unaltered by this change of mass of a subset of degrees of freedom.

To enhance the sampling of the h type degrees of freedom, their temperature, potential energy, or force can be scaled.[10] Applying temperature scaling, the temperature Th of the h type degrees of freedom is set to

equation image(66)

where β = (kBT)−1 and kB is the Boltzmann constant. The Hamiltonian for a system of Nhh type particles and Nll type particles is then

equation image(67)

where Kh(ph) and Kl(pl) are the kinetic energy of the h type and the l type particles, respectively, Vhh(xh) is the potential energy between h type particles, Vll(xl) is the potential energy between l type particles and Vhl(xh,xl) that between h and l type particles. In the adiabatically decoupled limit, the partition function of the l type particles is

equation image(68)

and the Hamiltonian of the h type degrees of freedom can therefore be written as

equation image(69)

The average value of a property Q(xh) at temperature Tl, 〈Qmath image, can then be calculated from 〈Qmath image of Q(xh) at Th using the standard unbiasing formula,

equation image(70)

where the ensemble averages are over the h type degrees of freedom, and

equation image(71)

where the ensemble average in the denominator is over the l type degrees of freedom[10] at Tl.

Applying force scaling,[10] the force acting on a h type degree of freedom is scaled by a factor sV,

equation image(72)

where fhh is the force exerted by the other h type degrees of freedom and fhl is the force exerted by the l type degrees of freedom.

The Hamiltonian, in analogy to eq. (69), can be written as

equation image(73)

with Zl(xh; βl) defined as in eq. (71).

The average value of a property Q(xh) at Tl, 〈Qmath image, can then be calculated from 〈Qmath image of Q(xh) at βh = βlsV using the standard unbiasing formula eq. (70).

External forces and nonequilibrium simulation

External electric field.

The application of an external electric field enables a simple and fast way to probe the static dielectric properties of a system like the dielectric permittivity ε(0) or the Debye relaxation time τD.[11]

If a constant homogeneous external field Eext is applied to a system, for examples along the z-axis,

equation image(74)

an additional force on particle i is present,

equation image(75)

where Eext,rf describes the contribution to the electric field Eext,cs inside the cut-off sphere due to the polarization Pext,rf in the dielectric continuum outside the cut-off sphere that is induced by the difference in dielectric permittivity inside,εcs, and outside, εrf, the cut-off sphere. This contribution depends on the shape of the cut-off region.[53] For a spherical region, we have

equation image(76)

where the polarization induced around the cut-off cavity in the dielectric continuum by the external field is

equation image(77)

The electric field Eext that is applied to the system will induce a polarization P in the system,

equation image(78)

where ε(0) is the zero-frequency or static dielectric permittivity of the molecular model. The polarization P can be calculated from the total dipole moment M of the system and its volume V,

equation image(79)

if ∑math imageqi = 0. If the latter condition is not satisfied, M and thus P will be origin dependent. For a homogeneous external field, eq. (74) along the z-axis we then find

equation image(80)

where 〈Mz〉 is the average dipole moment of the volume V in the z-direction in the simulation.

It is also possible to obtain a value of τD from averaging over a set of nonequilibrium MD simulations that start from a Boltzmann distributed set of initial configurations and velocities and in which a homogeneous static external electric field Eext is switched on at t = t0. On switching on Eext along the z-axis at t = t0, the z-component Mz of M will increase from its initial value Mz(t0), which values are Gaussian distributed around Mz = 0, to a steady state value Mz(t = ∞). For a Debye dielectric medium, this build up will be exponential,

equation image(81)

The value of 〈Mz (t = ∞)〉math image will be larger for larger Emath image, but different field strengths Emath image should yield the same τM, as long as Emath image is not too small and not too large. The relation between τM and τD is

equation image(82)

which is a generalization for κ ≥ 0 of the relation for κ = 0 given by Neumann.[54] The value of ε(0) can be calculated using eq. (80), in which 〈Mz〉 = 〈Mz(t = ∞)〉math image.

Sinusoidal applied acceleration field.

In the GROMOS software, a sinusoidal, periodic applied acceleration field[55] can be used to calculate the viscosity of a liquid.

The Navier–Stokes equation provides a continuum description of the motion of a fluid with mass density ρ,

equation image(83)

in which on the left side, the material derivative of the velocity field describes the unsteady and convective accelerations per volume, and on the right side the first term corresponds to the pressure gradient, the second to the viscosity, and f represents additional forces that can be imposed on the system. If an external shear-stress acceleration field a is applied, the equation becomes

equation image(84)

Assume that the acceleration field a is applied in the x-direction and has a sinusoidal dependence on the z-coordinate of the particle i such that

equation image(85)

where k = 2π/lz, A is the applied acceleration amplitude, lz is the box length in the z direction, zi is the z-coordinate of particle i and n an integer with n > 0. As the Navier–Stokes equation constitutes a continuum description, the approach is only valid in the long wave length limit, where the wavelength of the external field is large compared to the molecular dimensions.[56] Thus, the natural choice is to take n equal to 1. The box size should also be chosen in such a way that lz is big compared to the size of the individual molecules. Given the nature of the perturbation, the average velocities along y- and z-directions (uy and uz) vanish, and therefore, no pressure gradient in the x-direction is present, which simplifies eq. (84), leading to an equation for ux(z) of the form

equation image(86)

At the steady-state

equation image(87)

and the steady-state solution of eq. (86) is given by

equation image(88)

By applying an acceleration field of the form of eq. (85) with n = 1 in eq. (88) and assuming that ux(z) = 0 at t = 0, the resulting velocity field has the form

equation image(89)

where

equation image(90)

and where the relaxation time τr is given by

equation image(91)

For t sufficiently larger than τr one has

equation image(92)

The calculation of V can be easily performed on the fly, for example, at every few time steps, using

equation image(93)

where vi,x(t) is the velocity of particle i in the acceleration direction x. By monitoring the time series of V(t), an equilibration part is observed, which has to be excluded from the averaging. From the average of V(t), one can calculate the viscosity using eq. (90). Error estimates can be obtained from the fluctuations of V(t) and applying block averaging. The amplitude A of the applied acceleration should not be chosen too small, which would lead to a low signal-to-noise ratio, and not too big, which would lead to a disturbance of the equilibrium properties of the liquid, temperature drift, and nonlinear responses.[56]

Examples of the use of the simulation software

Mixtures of polarizable water and polarizable ethylene glycol.

The necessity to use polarizability is illustrated with a water–ethylene glycol mixture, described in Ref.57. As shown in Figure 1, using different nonpolarizable force fields for ethylene glycol (blue lines) in water the maximum in the experimental solvation free enthalpies ΔGsolv of argon as a function of the ethylene glycol mole fraction xEG is not reproduced. The introduction of polarizability in the ethylene glycol and water models (red lines) leads to a qualitative reproduction of the experimental data. The separation into solute–solvent enthalpic contributions ΔHuv and entropic contributions TΔSuv reveals that the occurrence of a maximum in ΔGsolv is entropic in nature.

Figure 1.

Solvation free enthalpies ΔGsolv (upper panel), solute–solvent interaction enthalpies ΔHuv (middle panel) and solute–solvent interaction entropies TΔSuv (lower panel) of argon in ethylene glycol–water mixtures at different mole fractions of ethylene glycol (xEG).[57]

Mixtures of coarse-grained DMSO or MeOH and coarse-grained water

The application of coarse-graining is demonstrated with mixtures of water[9] and methanol (MeOH), and water[9] and dimethylsulfoxide (DMSO). For the CG DMSO systems a cubic box with edge length of 6.82 nm was filled with 2100 DMSO beads leading to a density of 1096 kg/m3, corresponding to the density of liquid DMSO at 298 K and 1 atm.[58] For the CG MeOH systems a cubic box with edge length of 6.82 nm was filled 1875 MeOH beads leading to a density of 787 kg/m3, corresponding to the density of liquid MeOH at 298 K and 1 atm.[58]

For all systems, the temperature was weakly coupled[59] to a bath of 298 K with a relaxation time of 0.1 ps. The pressure was weakly coupled[59] to a bath of 1 atm with a relaxation time of 0.5 ps. The isothermal compressibility was set to the experimental value[60] of 7.513 × 10−4 (kJ mol−1 nm−3)−1 for water, 8.718 × 10−4 (kJ mol−1 nm−3)−1 for pure DMSO, 7.5 × 10−4 (kJ mol−1 nm−3)−1 for pure MeOH, and linear combinations for the mixtures. The nonbonded van der Waals and electrostatic interactions were calculated using a cutoff radius of 2 nm. The long range electrostatic interactions beyond the cut-off of 2 nm were represented by a reaction field[54, 61] with εRF = 78.5 for water, 46 for DMSO, 32.6 for MeOH, and linear combinations for the mixtures.

The equations of motion were integrated using the leap-frog algorithm with a time step of fs 2. The velocities of the atoms at the beginning of the simulation were assigned from a Maxwell distribution at 298 K. During the runs, configurations of the system were saved every 2 ps.

The parameters of the CG DMSO and the MeOH models are given in Table 1, the ones for the water model in Ref.9. For the mixtures the Lennard–Jones parameters were calculated using Cmath image = equation image and Cmath image = equation image. Starting configurations for the mixtures were obtained by converting a fraction of the DMSO and the MeOH beads, respectively, into water beads.[9] The mixtures were simulated for 4.5 ns after 0.1 ns of equilibration.

Table 1. Parameters of the coarse-grained models of DMSO and MeOH
ModelmCP umDP uqCP er0 nmKb 106 mol−1 nm−4σLJ nmεLJ kJ mol−1εCS
  1. mCP, mass of the CP particle of the bead; mDP, mass of the DP particle of the bead; qCP, charge of the CP particle of the bead; r0, ideal distance between CP and DP; Kb, force constant between CP and DP; σLJ, Lennard–Jones parameter; εLJ, Lennard–Jones parameter; and εCS, dielectric permittivity inside the cut-off sphere.

DMSO78.2588780.6420.2200.5422.442.5
MeOH64.1696640.560.22.00.562.12.5

The behavior of mixtures of CG water with CG DMSO and MeOH are shown in Figure 2 and Figure 3, respectively. The CG models are able to reproduce the general trends of the experimental data for the density, the excess volume, and the dielectric permittivity. For the excess potential energy of mixing the CG model predicts the opposite trend of the experimental data, which indicates that the geometric-mean combination rule for obtaining the Lennard–Jones parameters between unlike beads should be modified.[63]

Figure 2.

Thermodynamic properties of DMSO:H2O mixtures at 298 K and 1 atm, as a function of the mole fraction of DMSO, XDMSO, from MD simulations using the CG model of DMSO described in Table 1 and the CG model of H2O from[9]: (A) densities ρ, (B) dielectric permittivities ε(0), (C) excess volume of mixing ΔVOLmix, and (D) excess potential energy of mixing ΔUmix. Experimental data are shown with circles and a dashed line, and results of the CG simulations are shown with triangles and a solid line.

Figure 3.

Thermodynamic properties of MeOH: H2O mixtures at 298 K and 1 atm, as a function of the mole fraction of MeOH, XMeOH, from MD simulations using the CG model of MeOH described in Table 1 and the CG model of H2O from[9]: (A) densities ρ, (B) dielectric permittivities ε (0), (C) excess volume of mixing ΔVOLmix, and (D) excess potential energy of mixing ΔUmix. Experimental data are shown with circles and a dashed line, and results of the CG simulations are shown with triangles and a solid line.

Properties from nonequilibrium simulations of polarizable water

The nonequilibrium methods were tested with a box of polarizable COS/D water.[21]

For the calculation of the dielectric permittivity, a cubic box with an edge length of 3.1057 nm was filled with 1000 COS/D[21] water molecules, resulting in a density of 997 kg/m3, corresponding to the density of liquid water at 298 K and 1 atm.[60] For the calculation of the viscosity, the box was extended in the z-direction by a factor 5, containing 5000 water molecules. The geometry of the water molecules was constrained by applying the SHAKE algorithm[62] with a relative geometric tolerance of 10−4 on the OH bond length and on the intramolecular HH distance.

The temperature was weakly coupled[59] to a bath of 298.15 K with a relaxation time of 0.1 ps. The box to calculate the viscosity was simulated at constant volume, for the other, the pressure was weakly coupled[59] to a bath of 1 atm with a relaxation time of 0.5 ps. The isothermal compressibility was set to the experimental value[60] of 7.513 × 10−4 (kJ mol−1 nm−3)−1. The nonbonded van der Waals and electrostatic interactions were calculated using triple-range cutoff radii of 0.8/1.4 nm. The short range interactions were calculated every time step by updating the molecular pair list for distances smaller than the first cutoff radius of 0.8 nm. For the intermediate range of distances between 0.8 and 1.4 nm, the pairlist was only updated every fifth timestep and at the same time the interaction was calculated and kept unchanged between these updates. The long range electrostatic interactions beyond the outer cutoff of 1.4 nm were represented by a reaction field[54, 61] with εRF = 78.5 for water. The equations of motion were integrated using the leap-frog algorithm with a time step of 2 fs. The velocities of the atoms at the beginning of the simulation were assigned from a Maxwell distribution at 298 K. During the runs, configurations of the system were saved every 0.2 ps for the dielectric permittivity and every 1 ps for the viscosity.

The electric field in the z-direction was set to Emath image = 0, 0.01, 0.025, 0.0375, 0.5, 0.625, 0.75, 0.1, 0.15, and 0.2 e nm−2. For each value, the box was simulated for 5.8 ns after an equilibration phase of 0.2 ns.

The amplitude of the acceleration field was set to A = 0.001, 0.002, 0.005, 0.01, 0.02, 0.04, 0.05, 0.06, 0.07, 0.08, and 0.1 kJ mol−1 nm−1 u−1, and for each value, a simulation of 2.9 ns was performed after 0.1 ns of equilibration.

The dependence of the polarization in z-direction for different field strength in the same direction is shown in Figure 4. The static dielectric permittivity ε(0) of the polarizable COS/D water model derived from the slope, ignoring large field strengths, is 77.3 ± 0.6 from a total of 34.8 ns of simulation. This is closer to the experimental value[58] of 78.5 than the previously reported[21] ε(0) = 69.8 ± 6 derived from a total of 50 ns of simulation using the box dipole moment fluctuation formula.

Figure 4.

Dependence of the polarization averaged over time 〈Pz〉 on the applied electric field Emath image for COS/D[21] water. The circles are the results from the simulation, the line from linear regression for which the filled cicles were ignored since the system is saturated for larger field strengths.

The viscosity was determined to be 1.86 cP (see Fig. 5) for the region in which the perturbation is small enough not to disturb the system but strong enough to converge within reasonable time. The value found is about twice as high as the experimental value[58] of 0.89 cP.

Figure 5.

Viscosity η as a function of the amplitude A of the applied external sinusoidal acceleration. The solid line indicates the assumed minimum of the curve used to determined the viscosity η of COS/D[21] water.

Acknowledgements

The authors thank Markus Christen, Haibo Yu, Chris Oostenbrink, Moritz Winger, and Merijn Schenk for fruitful discussions.

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