OUTLINE

Solvent Effects and Water Models

Different Levels of Descr iptions/Treatments of L iquid Water and Solvent Effects

Water Models for Computer SimulationsTransferable Intermolecular Potentials (TIP3P, TIP4P, TIP5P)Stillinger model (ST2)Simple Point Charge models (SPC and SPC/E)Soft-Sticky Dipole potential (SSD)

Dielectr ic MediaStatistical descriptions: Clausius-Mossotti, Debye, Onsager equations; Lorentz-Debye-Sack theory

Treatment of Solvent Effects in Computer Simulations and Solvent Models in CHARMM

Inside CHARMM

Continuum Models for Simulations of Macromolecules in CHARMMSolvent Accessible Surface Area-based models (SASA, ASPENR, ASPENRMB)Generalized Born models (GENBORN, ACE, GBSW, GBMV, GBIM)Effective Energy Function model (EEF1)Screened Coulomb Potentials model (SCPISM)Poisson Boltzmann equation (PBEQ)Reference Interaction Sites Model (RISM)

Aqueous Solutions: Salts and Cosolvents

References

Appendix: Examples of Input Files

Sergio A Hassan, 11/2005

Solvent Effects and Water Models

The solvent controls physical and chemical proper ties of single molecules…

Structure, dynamics, thermodynamics of biopolymers (e.g., peptides, proteins, nucleic acid chains)

Chemical equilibrium, reaction mechanisms, electronic and vibrational spectra, etc. of smaller organic species (e.g., drugs, hormones, etc) (see [1])

…and their mutual interactions

protein-protein association/dissociation (kinetics, tightness of molecular complexes, aggregation, etc)

protein-drug binding (docking, affinities, etc)

enzyme-DNA interactions

enzymatic reaction (chemical mechanism, kinetics, thermodynamics, etc)

Etc.

Failure to account for solvent effects (or a deficient description of such effects) adversely affects results from simulation: wrong physics/chemistry wrong interpretation of structure-dynamic-function relationship

There are many kinds of solvents, this lecture focuses specifically on water and aqueous solutions

Gas phase (ε=1; not an approximation, just lack of solvation)

Statistical Potentials, Empirical Scoring Functions

Constant Dielectric (ε=2; 4; 15;…); Linear Dielectric (ε=2r; 4r; etc) with no solvation term (aka self energies)SAS energy functions (simplest model to include a solvation term)

GB (Generalized Born models: GB/SA, ACE, etc)

EEF1 (Effective Energy Function)PB (numerical solutions of Poisson-Boltzmann equation; several modifications)

SCPISM (Screened Coulomb Potentials model; based on theory of polar liquids)PDLD (Protein Dipole-Langevin Dipole model; other Semi-continuum or Semi-macroscopic Models)

QM-based continuum models (SCRF or Onsager cavity model, PCM, COSMO, EFP, SM5, etc)RISM (numerical solutions of Reference Interaction Sites Models; several extensions/approximations: 3D-RISM, DRISM, XRISM, etc)

Other integral equation formalisms

Statistical Mechanics Theory of Dielectric/Polar Liquids (Clausius-Mossotti, Debye, Onsager, Kirkwood, Lorentz-Debye-Sack, etc)

Point charges (ST2, SPC, TIP3P, TIP4P, TIP5P, SSD, etc. and their extensions/refinements)

Polarizable Models (PPC, GCPM, etc)

Semi-empirical QM approximations (MNDO, AM1, PM3, etc)

DFT (Density Functional Theory approach to QM)HF (Hartree-Fock Theory)Varying level of accuracy (basis sets, e.g., 6-31G basis set, 6-31G*, 6-31G** include atomic polarization; etc)

Limit of exact QM solution (exact solution of Schrödinger Equation)

CONTINUUM APPROXIMATIONS

(Implicit Solvent Models)

ATOMISTIC MODELS(Explicit Solvent Models)

QUANTUM MECHANICALDESCRIPTIONS

Several of these models/approximations are available in CHARMM or CHARMM/GAMESS

To be combined with different models of the solute (biomolecule): QM, QM/MM, point/rigid charges,polarizable models, fully atomistic (e.g., PAR22), increasing levels of coarse-graining (e.g., PAR19), etc

Different Levels of Descriptions/Treatments of Liquid Water and Solvent Effects

about 50 different models of water available to date

reviews in [2,3]

review in [4]

The utility of a biomolecular simulation depends on:

i) the accurate description of the physics of the system [i.e., the form of the Hamiltonian (or energy function, or force field), and the treatment of the system (classical, quantum, hybrid)]

ii) the accurate calculation of statistical/thermodynamic quantities [e.g., binding affinities, structure calculation, pKa shifts, relative population of a peptide conformers (e.g., random coil vs β-hairpin, α-helix, etc)]

In general:

1) Too approximate models ALLOWSfast computation which leads to good sampling which leads to good convergence and small statistical errorsBUTpoor description of the physicsTHENthermodynamic quantities unreliable

2) Too sophisticated models:ALLOWSgood description of the physicsBUTslow computation which leads to poor sampling which leads to poor convergence and large statistical errorsTHENthermodynamic quantities unreliable

In both cases caution is advised when interpreting results

Water Models for Computer SimulationsSeveral (~50) models developed to date, both for the liquid and the solid states (e.g., see [4] for recent review)

Examples of models based on classical mechanics ideas are (some available in CHARMM) :ST2, SPC, SPC/E, TIP3P, TIP4P, TIP5P, SSD; SSD/E; these are briefly described below

Transferable Intermolecular Potentials (TIP):

TIP3P (three-points TIP: three interaction sites centered on O and H nuclei; each site has a partial charge for computing Coulomb interactions; LJ interactions is accounted for through O-O nuclei interactions only; parameterization emphasizes properties of liquid water at ambient conditions, so it is more suitable for biomolecular simulations than other. Other models parameterized in less biologically relevant contexts; introduced in 1983 [5])

TIP4P (four points, similar to TIP3P but the center of negative charge is moved 0.15 Å towards the H sites, and along the line bisecting the HOH angle; this yields three centers for Coulomb interactions and one center for LJ interactions; improves quadrupole moment and structural properties of water; introduced in 1983 [5] but suggested in the ’30s)

TIP5P (five points, centers on O and H nuclei and two additional centers (partial charges) at the lone-pair sites; the functional form of the potential still the same as in TIP3P and TIP4P; improves several properties of water in a broad range of temperatures (from supercooled regime to well over 25 C), including self-diffusion, dielectric constant, etc; too early to say how it describes solvation of biomolecules; descendant of ST2, introduced in 2000 [6])

X and Y: rigid water moleculesVXY: interaction potential between X and Yε and σ: LJ parameters (associated to the O sites only)rOO: distance between the O centerssum runs over all the sites on X and Y (without double counting): 9 terms in TIP3P and TIP4P, 25 terms in TIP5P

�∈∈ �

�

�

�

��

�

�

��

���

−��

���

+=

YjXi OOOOij

jiYX rrr

qqV

612

, 4 σσε

ST2 (five-site model; point charges in O and H nuclei and electron lone pair orbitals; historical interest, developed by Stillinger and Rahman in 1974 in one of the first detailed MD simulations of liquid water [7])

SPC (Simple Point Charge model; similar to TIP3P, i.e., fixed point charges at O and H nuclei; reparameterization of partial charges to better describe polarization led to the extended version of the model, SPC/E; this improved several dynamic properties of liquid water; it has been modified and adapted in different ways, e.g., combining it with a fluctuating charges model to account for molecular polarizability, etc; originally introduced in 1981 [8])

SSD (Soft Sticky Dipole model; it has only one interaction site located at the molecule center of mass (the center of a dipole potential and a LJ core) and incorporates a tetrahedrally coordinated potential that controls the coordination of water molecules in the first hydration shell; extensions have been introduced to improve transport behavior and structural properties of bulk liquid, e.g., in the SSD/E model, etc; originally introduced in 1996 [9])

Etc.

…continued

(N=density, and α= polarizability of particles in the medium)

It is also valid for polar (µµµµ�0) liquids under high-frequency fields; in this case

Dielectric Media1) Macroscopic electrostatics: introductions to electrostatics in biomolecules usually begin with a review of macroscopic electrostatics concepts. This lecture will bypass such topic that may be found in many standard textbooks (see [10]).

2) Statistical Mechanics Treatment of Dielectr ics: Theory of Polar /Polar izable L iquids (see [11] for a formal treatment; and [2,12] for a discussions in connection with continuum approximations in solvated molecules): the goal of a statistical treatment is to connect microscopic properties of the molecules that compose a liquid with its macroscopic dielectric properties. Thus, unlike purely macroscopic concepts, a microscopic formalism is more suitable to describe electrostatic phenomena at the molecular level, i.e., in the biomolecular size scale.

Basic results:

The Clausius-Mossotti equationNonpolar (µµµµ=0) but polarizable (α�0) liquids: Nαπ

εε

34

21 =

+−

ε ∞εis replaced by (optical permittivity).

Eq.(1) connected for the first time a microscopic property of the liquid (α) to a macroscopic measurable quantities (ε). However, Eq.(1) has no dependence on temperature, so disagree with experiments. This was corrected by Debye.

�→k

kk NN αα , where k indicate the type of particles. For a multi-component medium same equation is obtained with

(1)

However, Eq.(2) yields a µ too small for water. This was corrected by Onsager, who reduced the magnitude of the directing field acting on the individual dipoles (local field); this self-orienting field is known as reaction field, so the correction is called Onsager reaction-field correction (for details see [13] and [11,12]).

(N=density, α= polarizability, and µ=permanent dipole moment; similarly generalized to a multi-component medium)

From Eq.(2) the magnitude of the dipole moment µµµµ can be obtained at any given temperature.

NKT ��

���

+=

+−

334

21 2µαπ

εε

The Debye equationPolar (µµµµ�0) and polarizable (α�0) liquids:

(2)

The Onsager equation( )( )

22

)2(2

94

++−=

∞

∞∞

εεεεεεπµ

KT

N

Agrees better with experimental data. However, its derivation (as all previous equations) neglects specific interactions between the particles of the medium (except their long range electrostatic interactions); then it better describes diluted systems.

Explicit interactions are needed for better describing denser systems. Some of these corrections were introduced by Kirkwood, what leads to the Kirkwood equation (not addressed here, see [11]).

All previous results are for homogeneous systems. A basic questions is: how these results change when a simple solute (e.g., a point charge or point dipole) is immersed in the medium. Although not discussed here, it is anticipated that these inhomogeneities lead to position dependent dielectric functions ε(r ) that approach bulk dielectric constant far from the solute. Such dielectric behavior can easily be derived with the Lorentz-Debye-Sack approximation (see detailed discussion in [12,14,15]).

Understanding the heterogeneity introduced by a simple point solute is a necessary first step to formulate a continuum approach for complex solutes (see SCPISM discussed later).

In the most general case the dielectric response of a medium (liquid or solid) is given by a time- and spatial-dependent tensor in the complex space, i.e.,

(3)

( )t,r

V V ~ V selfint +

Vint accounts for the effective interaction between particles (e.g., , for two charges separated a distance rij in a homogeneous medium of constant dielectric permittivity ε)

ijji rqq ε

Vself accounts for the solvation energy of individual particles, also known as self-energy, and exists even if there is onlyone particle in the system (e.g., for a single particle qi immersed in a homogeneous medium with constant dielectric permittivity ε, this term is given by the Born formula, i.e., , where R is the so-called Born radius of the particle (see [16-19] for discussions on assigning a physical meaning to R); additional terms are included to account for other, non-electrostatic effects, e.g., a cavity term, etc.)___________________________________________________________________________________________________________

The observation expressed in Eq.(4) is the basis of most continuum approaches available today (in CHARMM and other programs) for simulation of macromolecules.

Any model written in the form of Eq.(4), if properly parameterized, already accounts reasonable well for the main effects of solvation (although the devil is in the details, since solvation is a very complex phenomenon); at any rate, regardless of the physical content/rigor of a particular model, accounting for a Vself term is conceptually more appropriate than carrying out gas-phase calculations (or using only a Vint term) of a biological system.

( )1112 −−− εRqi

General phenomenological observation on the solvation energy V of a system:

(4)

Treatment of Solvent Effects in Computer Simulations and Solvent Models in CHARMM

{ } { } { }{ }

{ }χ+=++= ����

∈

+

∈∈∈

+

∈

+

Pji

nbbij

SjPi

nbij

Sji

nbbij

Pji

nbbij VVVVV

,,,

{ }�∈

≈≈Pji

ijPMFV

,φχ

{ }( )

{ } { } { }����∈∈∈∈

+=++≈Pji

ISMij

Pji

bij

Pjiij

nbij

Pji

bij VVVVV

,,,,φ

ISMijV

)( ijLJ

ijISM

ij rfVV +=

CHARMM forcefield (for point charges forcefield):

eliminate solvent degrees of freedom, represent them as a PMF and then approximate it by a pair-wise effective potential (corrections to this pair-wise simplification are needed to accurately represent the potential of mean force, as known from classical theory of liquids (see [20]); also Langevin dynamics simulation is more appropriate to account for the thermal collisions of (the absent) water with solvent-exposed atoms; see lecture on Dynamics)

continuum models (also known as ‘implicit solvent models’ or ISMs, as opposed to ‘explicit solvent models’ or ESMs) differ in the functional form of

the indexes b, nb and b+nb in Eqs.(5,7) denote the bonded, the non-bonded and the sum of both terms in the forcefield, respectively (see lecture on Forcefield).

The Lennard-Jones potential term in Eq.(7) is not usually modified. Therefore, in practice, the only modification in going from an ESM to an ISM is the change of the bare Coulomb potential to a new functional form f(rij) (the one that ultimately accounts implicitly for all the effects of the removed solvent), i.e.,

ijji rqq

LJijV nb

ijVin the

Different functional forms for f(r) have been coded into CHARMM. The CPU time needed to calculate V from Eq.(7) is expected to be (much) less than that using Eq.(1)...

Inside CHARMM

(5)

(6)

(8)

(7)

A note on Development and Benchmarking

ESM: Benchmarking of a water model involves comparing the theoretical results with the corresponding experimental data for a number of properties of water itself.

Some of the benchmarking properties are: water dipole moment, average configurational energy, compressibility coefficient, dielectric constant, density maximum as a function of temperature, self-diffusion coefficient, expansion coefficient, pair correlation function, etc. Testing and validating an ESM is a well-defined task.

ISM: Unlike ESM, benchmarking is far less straightforward because instead of comparing properties of water itself it involves calculating and comparing properties of a solute. I.e., the focus shifts from reproducing properties of water to reproducing properties of solutes (which is actually the main idea of an ISM).

This shift in target brings many additional problems because it is not always clear what properties of the solute better test an ISM. Structure, dynamics, kinetics, pKa shifts, and many chemical properties may be monitored in many different solutes. Some test may be insufficiently stringent to challenge the meaningfulness of a particular approach; while others may be too demanding for a model at particular stage of development, thus requiring further refinements before such problems can be addressed.

Continuum Models for Computer Simulations of MacromoleculesModels available in CHARMM (as of 08/05) are:

SASA, ASPENR, ASPENRMB, EEF1, GENBORN, ACE, GBSW, GBMV, GBIM, SCPISM, RISM, PBEQ

some of these models are briefly described below; refer to the respective documentation and references therein for details:

sasa.doc, aspenr.doc, aspenrmb.doc, eef1.doc, genborn.doc, ace.doc, gbsw.doc, gbmv.doc, gbim.doc, scpism.doc, rism.doc, pbeq.doc

___________________________________________________________________________________________________________

ASPENR, SASA: the simplest way to represent Eq.(4); based on atomic solvation parameters, i.e., the idea of assigning an energetic penalty/reward to the desolvation/solvation of a polar group, and to the solvation/desolvation of an apolar groups (see documentation for specifics of each model):

SASAi is the solvent accessible surface area of atom i. The quality of the results relies mainly on the quality of the parameters α, so these are purely empirical models (the SASA model uses a linear screening 2rij instead of a bare Coulomb term and neutralizes the charged side chains; besides it is parameterized for PAR19; see documentation for other differences between both). Fast for computer simulations. (ASPENRMB is an ‘extension’ of the ASPENR model to account for implicit membrane) [21].

��=≠

+=N

iii

N

ji ij

jiij SASA

r

qqrf

121)( α (9)

GENBORN, ACE, GBSW, GBMV, GBIM: these and other similar models belong to the family of so-called generalized Born (GB) models, being off-springs of the original GB/SA model proposed in [22]. The GB equation in [22] is given by (εw is dielectric constant of water):

the main differences among these and other GB models reported in the literature is in the way the Born radii αι of atom i are obtained. Different techniques have been proposed mostly to attain two goals: faster computation and/or higher accuracy in reproducing PB solvation energies.

GB models are not analytical approximations of the PB equation (as often claimed); Eq.(10) has not been derived but was empirically proposed in [22]. However, these models contain enough flexibility in the functional form [Eq.10] to allow for accurate fitting of PB results (the target of their parameterization). In this regards GB models are also empirical approaches, similar to SASA and ASPENR models, but with more elaborated functional forms of the interaction and solvation terms [Eq.(10) is a different way to represent Eq.(4)].

SASArf

qqrf

N

ji ijGB

ji

wij α

ε+��

���

−= �

=1, )(11

21)(

2/1

2

222

4exp)(

���

�

���

�

��

��

�

−+≡

ij

ijijijijGB

rrrf

ααwhere jiij ααα ≡and

(10)

EEF1: Effective Energy Function model based on a Gaussian solvent-exclusion approach [23]. It proposes an efficient method to evaluate the solvation/desolvation penalty of groups in the molecule. It uses a linear dielectric function to screen Coulomb interactions, at the time that neutralizes ionic side chains. Its functional form is given by:

where , and S is obtained from the definition

with I referring to a groups in the molecule (see [23] for details). Coefficients α are given by

The model allows very fast computation, uses the polar hydrogen (PAR19) force field, and has been used successfullyin many applications. It has recently been adapted to use with implicit membrane, which led to the development of the IMM1 model; it has also been improved to describe interactions between ionizable side chains, that were too strong in the original development; this improvement led to the EEF1.1 model (see eef1.doc for details).

� �≠ =

∆+=N

ji

M

I

slvI

ij

jiij G

r

qqrf

122

1)(ε

�≠

−∆=∆M

IJJIJI

refI

slvI VrSGG )(

��

�

�

��

�

�

��

���

−−−=2

2 exp)(4I

III

RxxxS

λαπ

Ifree

II G λπα /2∆=

In the equations above ∆Gref and ∆Gfree are a ‘reference solvation free energy’ and the ‘solvation free energy of the isolated (free) group’, respectively; RI and VJ are the van der Waals radius and the volume of groups I and J, respectively,while λΙ is a correlation length associated to the group I.

(11)

SCPISM: based on a superposition of screened Coulomb potentials (SCP) which sigmoidal functional form is determined from the LDS approximation of polar liquids (see [14,24]). The function f in this case is given by

1)exp(1

1)( −−+

+=ijij

wij r

rDακ

εwhere the screening function is

�≠

−+=N

kiikiiiBi rCBAR )exp(,and the Born radius of atom i is calculated from the shielding created by all its surrounding atoms, i.e.,

The term φSIF is a term that account for solvent induced forces (SIF) that are non-electrostatic in nature (it exists even when all q=0, in which case it is a hydrophobic term [25]); these forces are generated by the microscopic nature of the solvation shells. To account for these effects a formal derivation of φSIF is needed, which is not currently available. So the SCPISM incorporates these SIF effects by: i) setting φSIF=αSASA (cavity term), and ii) simultaneously adjusting the strength of hydrogen-bonding (HB) interactions in solution. This approach, although incomplete, accounts for the effects of SIF on the interaction energies between donor and acceptor groups, as discussed in [26]. These groups are particularly affected by the structure of the hydration shells because of the water HB network that tends to bridge them (see [27] for details).

The model can be derived from basic physical considerations, which leads to screening functions D(r) appearing in Eq.(12) instead of dielectric functions ε(r), as it would be expected. The effective position-dependent screening function D(r ) (obtained from the distance-dependent functions D(r)) permeates all of the space without actually introducing a dielectric boundary between the solute and the solvent (neither internal/external dielectric constants).

The model may be adapted for use with polarizable force fields because the LDS theory allows on-the-fly calculations of the dielectric screenings D(r), as discussed [14].

� �≠ =

+��

���

�−+=

N

jiSIF

N

i BiBi

i

ijij

jiij RDR

q

rrD

qqrf φ

1 ,,

2

1)(

121

)(21)( (12)

PBEQ: Poisson-Boltzmann (PB) equation approach. The Poisson equation (zero ionic strength) is a fundamental equation of electrostatics at any level of description (micro or macroscopic). On the other hand the PB equation (non-zero ionic strength) can be derived under statistical mechanics considerations regarding the ion atmosphere around a solute. A large body of literature on the PB equation is available, both in basic textbooks and in the specialized literature (e.g., [28,29]).

Some basic terminology follows: The PB equation is given by

�=

+=∇⋅∇−m

iiii kTqq

1))(exp()()}()({ rrrr φηρφε

where qi and ηi are the charge and concentration of the ith ionic species; φ(r ) is the electric potential at point r in the space. At sufficiently low ionic concentrations [28,29] the linear form is obtained, i.e., , with ρ(r ) and ε(r ) being the explicit (solute) charge density and the dielectric function at position r within the system.

The Debye-Hückel screening length κ is given by ; e is the electron charge, and I is the ionic strength, defined as

)()()()}()({ 2 rrrrr ρφκεφε =+∇⋅∇−

)(2 22 rεκ kTIe=�= 2/)( 2eqI iiη

The PB equation can be solved numerically (several methods available) to obtain the electric potential φ(r ) and the electrostatic component of the solvation energy in a molecular system of arbitrary shape. PB is not a continuum representation of solvent effects, but a continuum representation of electrostatic effects.

(13)

In practice, to solve the PB equation on a biomolecule certain assumptions are made (these are not related to the fundamental nature of the equation, but needed for practical calculations).

Assign internal (εεεεi) and external (εεεεw) dielectr ic constants (εi=1, 2, 4, 15, 40,…and other values have been used to reproduce experimental data; the problem of assigning an internal dielectric constant to a protein is not trivial because the protein interior is highly heterogeneous)

Assign atomic charges and radii (special set of charges/radii has been derived to reproduce experimental solvation energies of small organic molecules with particular PB solvers, e.g., PARSE set)

Define a dielectr ic boundary (solvent accessible surfaces are commonly used. In this case, a probe radius has to be chosen. Instead of a sharp dielectric boundary, a smooth dielectric transition between the protein and the solvent has also been used to improve results. This also avoids artifacts in the calculation of forces at the dielectric interface in MD simulations (note that dielectric theory of liquids shows that the dielectric profile increases smoothly with the distance from the solute, and approaches bulk solvent permittivity only at long distances; the theory also shows that the shape of the transition depends on the charge/dipole of the solute)

In certain systems/applications (particularly when charged groups are located at the solvent/protein interface, i.e., as most often found in proteins) the quantitative results obtained from a PB solver may be very sensitive to these assumptions.

So, despite the fundamental nature of the PB formalism cautions is advised before drawing conclusions based on actual calculations (or when comparing results from different PB calculations because the sets of parameters used may have been different). An analysis of the sensitivity of the results to changes in PB parameters should be carried out prior to calculations.

…continued

RISM: numerical solution of the RISM equations for very simple systems (Reference Interaction Sites Model); see theory in [20,32]. A formally robust method, but extremely impractical for simulation of biosystems. Coded as an independent program and adapted to CHARMM code later on (discussed here for completeness but unlikely you will ever use it)

The RISM equation are analogous to the original Ornstein-Zernike equation [20] for intermolecular site-site Correlations g(r), and given by (for multi-component system)

where ρρρρ(a); h(a,b)=g(a,b)-1; c(a,b); and ωωωω(a,b) (a and b are molecular sites in the solute-solvent system), so ρρρρ(a) is a linear arrays and h, c and ωωωω are bi-dimensional arrays); * is denotes convolution. Once g’s are known [by solving Eq.(14), I.e., using the RISM model in CHARMM], many properties of the system can in principle be calculated, e.g., PMF, chemical potential, etc

To solve Eq.(14) an additional equation is needed (because the number of unknowns, h and c) are larger than the number of equations available. Several closure equations have been proposed to help solving Eq.(14) (e.g., HNC, PY, PYA, etc.; refer to the bibliography for details). The module as coded in CHARMM allows for a maximum of two solutes, with less than 20 atoms each.

hcch ∗∗+∗∗= (14)

Simulations in Aqueous Solutions: Salts

Introducing ions and co-solvents in the system brings a new set of problems in computer simulations of macromolecules.

General effects/observations: Ions in the liquid compete with the solute in polarizing and reorienting water molecules in the bulk phase. Ions in bulk also interact directly with the solute. Ions in a solute hydration shell interact directly with said solute. Ions in a solute hydration Shell perturb the structure of water in the hydration shell. All these effects coexists and must be reasonably represented in a simulation to draw sensible conclusions.

Ion diffusion is slower than neutral species. Ion diffusion decreases (with respect to the infinite dilute limit) as the ionic concentration increases (the solution becomes more structured). In general ion diffusion decreases in the hydration shells of solutes and in constrained geometries.

E.g., the diffusion constant of ions in bulk water at 25 C (in the infinite dilute limit) are in the order of 10-9 m2/s [30] (e.g., Na+~1.3, K+~1.8 Cl–~1.7; compare with the neutral species Na~5.4, K~2.6 and Cl~1.5; in units of m2/s). To allow ions to diffuse within a cubic simulation box of size ~(50 Å)3 would require nanosecond time scale at the very least. For comparison, the measured self-diffusion coefficient of pure water at the same temperature is D~2.3 10-9 m2/s (see [31] for a comparison of D’s calculated with several water models).

Therefore, the time scale of the simulation is critical to guarantee convergence of calculated quantities, and to have a reasonable representation of a real system [all possible states are explored in an experiment (as average in time and/or average on ensemble within the sample); this statistical representation is difficult to attain in a molecular simulation unless that long and/or multiple simulations are carried out and/or the solute is sufficiently simple (a single methane molecule is a simple solute; a large protein with charged solvent-exposed groups is a fairly complex solute)].

Caution is advised when designing a simulation protocol in aqueous salt, particularly when the goal is trying to achieve a ‘mechanistic’ insight.

There is no reliable way to calculate effects of salts in continuum solvent models. The obvious way would be the PB method, i.e., to solve the PB equation in its linear (low ion concentration) or non-linear form. This, however, does not account for explicit ion-solute interactions in the hydration shells [27] (which may be the most important control mechanism in many biological systems).

…continued

References (minimalist)

[1] C. Reichardt; Solvents and Solvent Effects in Organic Chemistry, Wiley-VCH, Weinheim, 2002

[2] Molecular Interactions in Solution: An Overview of Methods Based on Continuous Distribution of the Solvent; J. Tomasi and M. Perisco; Chem. Rev., 94, 2027 (1994)

[3] Quantum Mechanical Continuum Solvation Models; J. Tomasi, B. Mennucci, and R. Cammi; Chem. Rev.105, 2999 (2005)

[4] A Reappraisal of What We Have Learnt during Three Decades of Computer Simulations on Water; B. Guillot; J. Mol. Liq 101, 219 (2002)

[5] Comparison of Simple Potential Functions for Simulating Liquid Water; W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein; J. Chem.Phys. 79, 926 (1983)

[6] A Five-site Model for Liquid Water and the Reproduction of the Density Anomaly by Rigid, Nonpolarizable Potential Functions; M. W. Mahoney and W. L. Jorgensen; J. Chem. Phys. 112, 8910 (2000)

[7] Improved Simulation of Liquid Water by Molecular Dynamics, F. H. Stillinger and A. Rahman; J. Chem. Phys.60, 1545 (1974)

[8] H. J. Berendsen, J. P. M. Postma, W. F. van Gunsteren, and J. Hermans. Intermolecular Forces, Reidel, Dordrecht, Ed. B. Pullman,1981, p.331

[9] Soft Sticky Dipole Potential for Liquid Water: a New Model; Y. Liu and T. Ichiye, J. Phys. Chem. 100, 2723 (1996)

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