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| '''Implicit solvation''' (sometimes known as '''continuum solvation''') is a method of representing [[solvent]] as a continuous medium instead of individual “explicit” solvent molecules most often used in [[molecular dynamics]] simulations and in other applications of [[molecular mechanics]]. The method is often applied to estimate [[Thermodynamic free energy|free energy]] of [[solution|solute]]-[[solvent]] interactions in structural and chemical processes, such as folding or [[conformational change|conformational transitions]] of [[proteins]], [[DNA]], [[RNA]], and [[polysaccharide]]s, association of biological macromolecules with [[ligand]]s, or transport of [[drugs]] across [[biological membrane]]s.
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| The implicit solvation model is justified in liquids, where the [[potential of mean force]] can be applied to approximate the averaged behavior of many highly dynamic solvent molecules. However, the interiors of [[biological membrane]]s or [[protein]]s can also be considered as media with specific [[solvation]] or [[dielectric]] properties. These media are continuous but not necessarily uniform, since their properties can be described by different analytical functions, such as “polarity profiles” of [[lipid bilayer]]s.<ref name="pmid11438731">{{cite journal | author = Marsh D | title = Polarity and permeation profiles in lipid membranes | journal = Proc. Natl. Acad. Sci. U.S.A. | volume = 98 | issue = 14 | pages = 7777–82 |date=July 2001 | pmid = 11438731 | pmc = 35418 | doi = 10.1073/pnas.131023798 | url =http://www.pnas.org/cgi/pmidlookup?view=long&pmid=11438731 | issn = 0027-8424 | format = Free full text |bibcode = 2001PNAS...98.7777M }}</ref> There are two basic types of implicit solvent methods: models based on [[accessible surface area]]s (ASA) that were historically the first, and more recent continuum electrostatics models, although various modifications and combinations of the different methods are possible.
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| The accessible surface area (ASA) method is based on experimental linear relations between [[Gibbs free energy]] of transfer and the [[surface area]] of a [[solution|solute]] molecule.<ref name="pmid326146">{{cite journal | author = [[Frederic M. Richards|Richards FM]] | title = Areas, volumes, packing and protein structure | journal = Annu. Rev. Biophys. Bioeng. | volume = 6 | issue = | pages = 151–76 | year = 1977 | pmid = 326146 | doi = 10.1146/annurev.bb.06.060177.001055 | url = | issn = 0084-6589 }}</ref> This method operates directly with free energy of [[solvation]], unlike [[molecular mechanics]] or [[electrostatic]] methods that include only the [[enthalpy|enthalpic]] component of free energy. The continuum representation of solvent also significantly improves the computational speed and reduces errors in statistical averaging that arise from incomplete sampling of solvent conformations,<ref name="pmid17030302">{{cite journal | author = Roux B, Simonson T | title = Implicit solvent models | journal = Biophys. Chem. | volume = 78 | issue = 1–2 | pages = 1–20 |date=April 1999 | pmid = 17030302 | doi = 10.1016/S0301-4622(98)00226-9| url = | issn = 0301-4622 }}</ref> so that the energy landscapes obtained with implicit and explicit solvent are different.<ref name="Zhou_2003">{{cite journal | author = Zhou R | title = Free energy landscape of protein folding in water: explicit vs. implicit solvent | journal = Proteins | volume = 53 | issue = 2 | pages = 148–61 |date=November 2003 | pmid = 14517967 | doi = 10.1002/prot.10483 | url = | issn = 0887-3585 }}</ref> Although the implicit solvent model is useful for simulations of biomolecules, this is an approximate method with certain limitations and problems related to parameterization and treatment of [[ionization]] effects.
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| ==Accessible surface area-based method==
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| {{Main|Accessible surface area}}
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| The free energy of solvation of a [[solution|solute]] molecule in the simplest ASA-based method is given by:
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| :<math>
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| \Delta G_\mathrm{solv} = \sum_{i} \sigma_{i} \ ASA_{i}
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| </math>
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| where <math> ASA_{i}</math> is the [[accessible surface area]] of atom ''i'', and
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| <math> \sigma_{i}</math> is ''solvation parameter'' of atom ''i'', i.e. a contribution to the free energy of [[solvation]] of the particular atom i per surface unit area. The required solvation parameters for different types of atoms ([[carbon|C]], [[nitrogen|N]], [[oxygen|O]], [[sulfur|S]], etc.) are usually determined by a [[least squares]] fit of the calculated and experimental transfer free energies for a series of [[organic compound]]s. The experimental energies are determined from [[partition coefficient]]s of these compounds between different solutions or media using standard mole concentrations of the solutes.<ref name="Ben-Naim">{{cite book | title = Hydrophobic interactions | author = Ben-Naim AY |year = 1980 | publisher = Plenum Press | location = New York | isbn = 0-306-40222-X | page = | pages = | url = }}</ref><ref name="pmid7766825">{{cite journal | author = Holtzer A | title = The "cratic correction" and related fallacies | journal = Biopolymers | volume = 35 | issue = 6 | pages = 595–602 |date=June 1995 | pmid = 7766825 | doi = 10.1002/bip.360350605 | url =http://www.scholaruniverse.com/ncbi-linkout?id=7766825 | issn = 0006-3525 | format = Free full text }}</ref>
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| It is noteworthy that ''solvation energy'' is the free energy required to transfer a solute molecule from a solvent to “vacuum” (gas phase). This solvation energy can supplement the intramolecular energy in vacuum calculated in [[molecular mechanics]]. Therefore, the required atomic solvation parameters were initially derived from water-gas partition data.<ref name="pmid3472198">{{cite journal | author = Ooi T, Oobatake M, Némethy G, Scheraga HA | title = Accessible surface areas as a measure of the thermodynamic parameters of hydration of peptides | journal = Proc. Natl. Acad. Sci. U.S.A. | volume = 84 | issue = 10 | pages = 3086–90 |date=May 1987 | pmid = 3472198 | pmc = 304812 | doi = 10.1073/pnas.84.10.3086| issn = 0027-8424 | format = Free full text |bibcode = 1987PNAS...84.3086O }}</ref> However, the dielectric properties of proteins and [[lipid bilayer]]s are much more similar to those of nonpolar solvents than to vacuum. Newer parameters have therefore been derived from [[water]]-[[1-octanol]] [[partition coefficient]]s<ref name="pmid3945310">{{cite journal | author = Eisenberg D, McLachlan AD | title = Solvation energy in protein folding and binding | journal = Nature | volume = 319 | issue = 6050 | pages = 199–203 | year = 1986 | pmid = 3945310 | doi = 10.1038/319199a0 | url = | month = Jan | issn = 0028-0836 |bibcode = 1986Natur.319..199E }}</ref> or other similar data. Such parameters actually describe ''transfer'' energy between two condensed media or the ''difference'' of two solvation energies.
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| ==Poisson-Boltzmann==
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| {{Main|Poisson-Boltzmann equation}}
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| Although this equation has solid theoretical justification, it is computationally expensive to calculate without approximations. The [[Poisson-Boltzmann equation]] (PB) describes the electrostatic environment of a solute in a solvent containing [[ion]]s. It can be written in [[cgs]] units as:
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| :<math>
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| \vec{\nabla}\cdot\left[\epsilon(\vec{r})\vec{\nabla}\Psi(\vec{r})\right] = -4\pi\rho^{f}(\vec{r}) - 4\pi\sum_{i}c_{i}^{\infty}z_{i}q\lambda(\vec{r})e^{\frac{-z_{i}q\Psi(\vec{r})}{kT}}
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| </math>
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| or (in [[MKS system of units|mks]]):
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| :<math>
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| \vec{\nabla}\cdot\left[\epsilon(\vec{r})\vec{\nabla}\Psi(\vec{r})\right] = -\rho^{f}(\vec{r}) - \sum_{i}c_{i}^{\infty}z_{i}q\lambda(\vec{r})e^{\frac{-z_{i}q\Psi(\vec{r})}{kT}}
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| </math>
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| where <math>\epsilon(\vec{r})</math> represents the position-dependent dielectric, <math>\Psi(\vec{r})</math> represents the electrostatic potential, <math>\rho^{f}(\vec{r})</math> represents the charge density of the solute, <math>c_{i}^{\infty}</math> represents the concentration of the ion ''i'' at a distance of infinity from the solute, <math>z_{i}</math> is the valence of the ion, ''q'' is the charge of a proton, ''k'' is the [[Boltzmann constant]], ''T'' is the [[temperature]], and <math>\lambda(\vec{r})</math> is a factor for the position-dependent accessibility of position ''r'' to the ions in solution (often set to uniformly 1). If the potential is not large, the equation can be [[linearization|linearized]] to be solved more efficiently.<ref name="pmid12501158">{{cite journal | author = Fogolari F, Brigo A, Molinari H | title = The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology | journal = J. Mol. Recognit. | volume = 15 | issue = 6 | pages = 377–92 | year = 2002 | pmid = 12501158 | doi = 10.1002/jmr.577 | url = | month = Nov | issn = 0952-3499 }}</ref>
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| A number of numerical Poisson-Boltzmann equation solvers of varying generality and efficiency have been developed,<ref name="pmid16290441">{{cite journal | author = Shestakov AI, Milovich JL, Noy A | title = Solution of the nonlinear Poisson-Boltzmann equation using pseudo-transient continuation and the finite element method | journal = J Colloid Interface Sci | volume = 247 | issue = 1 | pages = 62–79 |date=March 2002 | pmid = 16290441 | doi = 10.1006/jcis.2001.8033 | url = | issn = 0021-9797 }}</ref><ref name="pmid15974723">{{cite journal | author = Lu B, Zhang D, McCammon JA | title = Computation of electrostatic forces between solvated molecules determined by the Poisson-Boltzmann equation using a boundary element method | journal = J Chem Phys | volume = 122 | issue = 21 | pages = 214102 |date=June 2005 | pmid = 15974723 | doi = 10.1063/1.1924448 | url = | issn = 0021-9606 |bibcode = 2005JChPh.122u4102L }}</ref><ref name="pmid11517324">{{cite journal | author = Baker NA, Sept D, Joseph S, Holst MJ, McCammon JA | title = Electrostatics of nanosystems: Application to microtubules and the ribosome | journal = Proc. Natl. Acad. Sci. U.S.A. | volume = 98 | issue = 18 | pages = 10037–41 |date=August 2001 | pmid = 11517324 | pmc = 56910 | doi = 10.1073/pnas.181342398 | url =http://www.pnas.org/cgi/pmidlookup?view=long&pmid=11517324 | issn = 0027-8424 | format = Free full text |bibcode = 2001PNAS...9810037B }}</ref> including one application with a specialized computer hardware platform.<ref name="pmid15942918">{{cite journal | author = Höfinger S | title = Solving the Poisson-Boltzmann equation with the specialized computer chip MD-GRAPE-2 | journal = J Comput Chem | volume = 26 | issue = 11 | pages = 1148–54 |date=August 2005 | pmid = 15942918 | doi = 10.1002/jcc.20250 | url = | issn = 0192-8651 }}</ref> However, performance from PB solvers does not yet equal that from the more commonly used generalized Born approximation.<ref name="pmid16540310">{{cite journal | author = Koehl P | title = Electrostatics calculations: latest methodological advances | journal = Curr. Opin. Struct. Biol. | volume = 16 | issue = 2 | pages = 142–51 |date=April 2006 | pmid = 16540310 | doi = 10.1016/j.sbi.2006.03.001 | url = | issn = 0959-440X }}</ref>
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| ==Generalized Born==
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| The ''Generalized Born'' (GB) model is an approximation to the exact (linearized) Poisson-Boltzmann equation. It is based on modeling the solute as a set of spheres whose internal dielectric constant differs from the external solvent. The model has the following functional form:
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| :<math>
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| G_{s} = \frac{1}{8\pi}\left(\frac{1}{\epsilon_{0}}-\frac{1}{\epsilon}\right)\sum_{i,j}^{N}\frac{q_{i}q_{j}}{f_{GB}}
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| </math>
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| where
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| :<math>
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| f_{GB} = \sqrt{r_{ij}^{2} + a_{ij}^{2}e^{-D}}
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| </math>
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| and
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| <math>
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| D = \left(\frac{r_{ij}}{2a_{ij}}\right)^{2}, a_{ij} = \sqrt{a_{i}a_{j}}
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| </math>
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| where <math>\epsilon_{0}</math> is the [[permittivity of free space]], <math>\epsilon</math> is the [[dielectric constant]] of the solvent being modeled, <math>q_{i}</math> is the [[electrostatic charge]] on particle ''i'', <math>r_{ij}</math> is the distance between particles ''i'' and ''j'', and <math>a_{i}</math> is a quantity (with the dimension of length) known as the ''effective Born radius''.<ref name="Still">{{cite journal | author = Still WC, Tempczyk A, Hawley RC, Hendrickson T |year = 1990 | title = Semianalytical treatment of solvation for molecular mechanics and dynamics | journal = J Am Chem Soc | volume = 112 | issue = 16 | pages = 6127–6129 | doi = 10.1021/ja00172a038 }}</ref> The effective Born radius of an atom characterizes its degree of burial inside the solute; qualitatively it can be thought of as the distance from the atom to the molecular surface. Accurate estimation of the effective Born radii is critical for the GB model.<ref name="Onufriev">{{cite journal | author = Onufriev A, Bashford D, Case DA | year = 2002 | title = Effective Born radii in the generalized Born approximation: The importance of being perfect | journal = J Comp Chem | volume = 23 | issue = 14 | pages = 1297–1304 | doi = 10.1002/jcc.10126 | pmid = 12214312 }}</ref>
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| ===GBSA===
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| GBSA is simply a Generalized Born model augmented with the hydrophobic solvent accessible surface area (SA) term. It is among the most commonly used implicit solvent model combinations. The use of this model in the context of [[molecular mechanics]] is known as MM/GBSA. Although this formulation has been shown to successfully identify the [[native state]]s of short peptides with well-defined [[tertiary structure]],<ref name="pmid16617376">{{cite journal | author = Ho BK, Dill KA | title = Folding Very Short Peptides Using Molecular Dynamics | journal = PLoS Comput. Biol. | volume = 2 | issue = 4 | pages = e27 |date=April 2006 | pmid = 16617376 | pmc = 1435986 | doi = 10.1371/journal.pcbi.0020027 | url =http://dx.plos.org/10.1371/journal.pcbi.0020027 | issn = 1553-734X | format = Free full text |bibcode = 2006PLSCB...2...27H }}</ref> the conformational ensembles produced by GBSA models in other studies differ significantly from those produced by explicit solvent and do not identify the protein's native state.<ref name="Zhou_2003" /> In particular, [[Salt bridge (protein)|salt bridge]]s are overstabilized, possibly due to insufficient electrostatic screening, and a higher-than-native [[alpha helix]] population was observed. Variants of the GB model have also been developed to approximate the electrostatic environment of membranes, which have had some success in folding the [[transmembrane helix|transmembrane helices]] of [[integral membrane protein]]s.<ref name="pmid14581194">{{cite journal | author = Im W, Feig M, Brooks CL | title = An Implicit Membrane Generalized Born Theory for the Study of Structure, Stability, and Interactions of Membrane Proteins | journal = Biophys. J. | volume = 85 | issue = 5 | pages = 2900–18 |date=November 2003 | pmid = 14581194 | pmc = 1303570 | doi = 10.1016/S0006-3495(03)74712-2 | url = | issn = 0006-3495 | bibcode=2003BpJ....85.2900I}}</ref>
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| ==Ad hoc fast solvation models==
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| Another possibility is to use ad hoc quick strategies to estimate solvation free energy. A first generation of fast implicit solvents is based on the calculation of a per-atom solvent accessible surface area. For each of group of atom types, a different parameter scales its contribution to solvation ("ASA-based model" described above).<ref name="pmid1304905">{{cite journal | author = Wesson L, Eisenberg D | title = Atomic solvation parameters applied to molecular dynamics of proteins in solution | journal = Protein Sci. | volume = 1 | issue = 2 | pages = 227–35 |date=February 1992 | pmid = 1304905 | pmc = 2142195 | doi = 10.1002/pro.5560010204 | issn = 0961-8368 | format = Free full text }}</ref>
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| Another strategy is implemented for the [[CHARMM]]19 force-field and is called EEF1.<ref name="pmid10223287">{{cite journal | author = Lazaridis T, Karplus M | title = Effective energy function for proteins in solution | journal = Proteins | volume = 35 | issue = 2 | pages = 133–52 |date=May 1999 | pmid = 10223287 | doi = 10.1002/(SICI)1097-0134(19990501)35:2<133::AID-PROT1>3.0.CO;2-N | url = http://www.sci.ccny.cuny.edu/~themis/eef1.pdf | issn = 0887-3585 | format = }} {{dead link|date=September 2009}}</ref> EEF1 is based on a Gaussian-shaped solvent exclusion. The solvation free energy is
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| :<math>
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| \Delta G_{i}^{solv} = \Delta G_{i}^{ref} - \sum_{j} \int_{Vj} f_i(r) dr
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| </math>
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| The reference solvation free energy of ''i'' corresponds to a suitably chosen small molecule in
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| which group i is essentially fully solvent-exposed. The integral is over the volume ''V<sub>j</sub>'' of
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| group ''j'' and the summation is over all groups ''j'' around ''i''. EEF1 additionally uses a distance-dependent (non-constant) dielectric, and ionic side-chains of proteins are simply neutralized. It is only 50% slower than a vacuum simulation. This model was later augmented with the hydrophobic effect and called Charmm19/SASA.<ref name="pmid11746700">{{cite journal | author = Ferrara P, Apostolakis J, Caflisch A | title = Evaluation of a fast implicit solvent model for molecular dynamics simulations | journal = Proteins | volume = 46 | issue = 1 | pages = 24–33 |date=January 2002 | pmid = 11746700 | doi = 10.1002/prot.10001 }}</ref>
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| ==Hybrid implicit/explicit solvation models==
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| It is possible to include a layer or sphere of water molecules around the solute, and model the bulk with an implicit solvent. Such an approach is proposed by M. J. Frisch and coworkers<ref name="isbn0-8412-2981-3">{{cite book | editor = Smith D | title = Modeling the hydrogen bond | publisher = American Chemical Society | location = Columbus, OH | year = 1994 | author = TA Keith, MJ Frisch | chapter = Chapter 3: Inclusion of Explicit Solvent Molecules in a Self-Consistent-Reaction Field Model of Solvation | isbn = 0-8412-2981-3 }}</ref> and by other authors .<ref name="pmid15470756">{{cite journal | author = Lee MS, Salsbury FR, Olson MA | title = An efficient hybrid explicit/implicit solvent method for biomolecular simulations | journal = J Comput Chem | volume = 25 | issue = 16 | pages = 1967–78 |date=December 2004 | pmid = 15470756 | doi = 10.1002/jcc.20119 }}</ref> <ref>{{Cite journal | author = Marini A, Muñoz-Losa A, Biancardi A, Mennucci B | title = What is Solvatochromism? | journal = The Journal of Physical Chemistry B | volume = 114 | pages = 17128 | year = 2010 | doi = 10.1021/jp1097487 | issue = 51}}</ref> For instance in Ref. <ref name=pmid15470756 /> the bulk solvent is modeled with a Generalized Born approach and the multi-grid method used for Coulombic pairwise particle interactions. It is reported to be faster than a full explicit solvent simulation with the [[Ewald summation|particle mesh Ewald]] (PME) method of electrostatic calculation.
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| ==Effects not accounted for==
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| ===The hydrophobic effect===
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| Models like PB and GB allow estimation of the mean electrostatic free energy but do not account for the (mostly) [[entropy|entropic]] effects arising from solute-imposed constraints on the organization of the water or solvent molecules. This is known as the [[hydrophobic effect]] and is a major factor in the [[protein folding|folding]] process of [[globular protein]]s with [[hydrophobic core]]s. Implicit solvation models may be augmented with a term that accounts for the hydrophobic effect. The most popular way to do this is by taking the solvent accessible surface area (SASA) as a [[Proxy (statistics)|proxy]] of the extent of the hydrophobic effect. Most authors place the extent of this effect between 5 and 45 cal/(Å<sup>2</sup> mol).<ref>{{cite journal | first = K.A. | last = Sharp | authorlink =| coauthors = A. Nicholls, R.F. Fine, B. Honig | year =1991 | month = | title = Reconciling the magnitude of the microscopic and macroscopic hydrophobic effects | journal = Science | volume = 252 | issue = 5002| pages = 106–109 | id =| doi =10.1126/science.2011744 | pmid = 2011744|bibcode = 1991Sci...252..106S }}</ref> Note that this surface area pertains to the solute, while the hydrophobic effect is mostly entropic in nature at physiological temperatures and occurs on the side of the solvent.
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| ===Viscosity===
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| Implicit solvent models such as PB, GB, and SASA lack the viscosity that water molecules impart by randomly colliding and impeding the motion of solutes through their van der Waals repulsion. In many cases, this is desirable because it makes sampling of configurations and [[phase space]] much faster. This acceleration means that more configurations are visited per simulated time unit, on top of whatever CPU acceleration is achieved in comparison to explicit solvent. It can, however, lead to misleading results when kinetics are of interest.
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| Viscosity may be added back by using [[Langevin dynamics]] instead of [[Hamiltonian mechanics|Hamiltonian dynamics]] and choosing an appropriate damping constant for the particular solvent.<ref>Tamar Schlick (2002). ''Molecular Modeling and Simulation: An Interdisciplinary Guide'' Interdisciplinary Applied Mathematics: Mathematical Biology. Springer-Verlag New York, NY, ISBN 0-387-95404-X</ref> Recent work has also been done developing thermostats based on fluctuating hydrodynamics to account for momentum transfer through the solvent and related thermal fluctuations. <ref> Yaohong Wang, Jon Karl Sigurdsson, Paul J. Atzberger (2012). ''Dynamic Implicit-Solvent Coarse-Grained Models of Lipid Bilayer Membranes : Fluctuating Hydrodynamics Thermostat'' arXiv:1212.0449, http://arxiv.org/abs/1212.0449. </ref> One should keep in mind, though, that the folding rate of proteins does not depend linearly on viscosity for all regimes.<ref name="pmid12868108">{{cite journal | author = Zagrovic B, Pande V | title = Solvent viscosity dependence of the folding rate of a small protein: distributed computing study | journal = J Comput Chem | volume = 24 | issue = 12 | pages = 1432–6 |date=September 2003 | pmid = 12868108 | doi = 10.1002/jcc.10297 }}</ref>
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| ===Hydrogen bonds with solvent===
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| Solute-solvent [[hydrogen bond]]s in the first [[solvation shell]] are important for solubility of organic molecules and especially [[ions]]. Their average energetic contribution can be reproduced with an implicit solvent model.<ref name="pmid21438609">{{cite journal | author = Lomize AL, Pogozheva ID, Mosberg HI | title = Anisotropic solvent model of the lipid bilayer. 1. Parameterization of long-range electrostatics and first solvation shell effects | journal = J Chem Inf Model | volume = 51 | issue = 4 | pages = 918–29 |date=April 2011 | pmid = 21438609 | doi = 10.1021/ci2000192 | pmc=3089899}}</ref><ref name="pmid21438606">{{cite journal | author = Lomize AL, Pogozheva ID, Mosberg HI | title = Anisotropic solvent model of the lipid bilayer. 2. Energetics of insertion of small molecules, peptides and proteins in membranes | journal = J Chem Inf Model | volume = 51 | issue = 4 | pages = 930–46 |date=April 2011 | pmid = 21438606 | doi = 10.1021/ci200020k | url = | pmc=3091260}}</ref>
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| ==Problems and limitations==
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| All implicit solvation models rest on the simple idea that nonpolar atoms of a [[solution|solute]] tend to cluster together or occupy nonpolar media, whereas polar and charged groups of the solute tend to remain in water. However, it is important to properly balance the opposite energy contributions from different types of atoms. Several important points have been discussed and investigated over the years.
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| ===Choice of model solvent===
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| It has been noted that wet [[1-octanol]] solution is a poor approximation of proteins or biological membranes because it contains ~2M of water, and that [[cyclohexane]] would be a much better approximation.<ref name="Radzicka">{{cite journal | author = Radzicka A., Wolfenden R. | year = 1988 | title = Comparing the polarities of the amino acids: side-chain distribution coefficients between the vapor phase, cyclohexane, 1-octanol, and neutral aqueous solution | url = | journal = Biochemistry | volume = 27 | issue = | pages = 1664–1670 }}</ref> Investigation of passive permeability barriers for different compounds across lipid bilayers led to conclusion that 1,9-decadiene can serve as a good approximations of the bilayer interior,<ref name="pmid11920749">{{cite journal | author = Mayer PT, Anderson BD | title = Transport across 1,9-decadiene precisely mimics the chemical selectivity of the barrier domain in egg lecithin bilayers | journal = J Pharm Sci | volume = 91 | issue = 3 | pages = 640–6 |date=March 2002 | pmid = 11920749 | doi = 10.1002/jps.10067| url = | issn = 0022-3549 }}</ref> whereas [[1-octanol]] was a very poor approximation.<ref name="pmid3735402">{{cite journal | author = Walter A, Gutknecht J | title = Permeability of small nonelectrolytes through lipid bilayer membranes | journal = J. Membr. Biol. | volume = 90 | issue = 3 | pages = 207–17 | year = 1986 | pmid = 3735402 | doi = 10.1007/BF01870127| url = | issn = 0022-2631 }}</ref> A set of solvation parameters derived for protein interior from [[protein engineering]] data was also different from octanol scale: it was close to [[cyclohexane]] scale for nonpolar atoms but intermediate between cyclohexane and octanol scales for polar atoms.<ref name="Lomize_2002">{{cite journal | author = Lomize AL, Reibarkh MY, Pogozheva ID | title = Interatomic potentials and solvation parameters from protein engineering data for buried residues | journal = Protein Sci. | volume = 11 | issue = 8 | pages = 1984–2000 |date=August 2002 | pmid = 12142453 | pmc = 2373680 | doi = 10.1110/ps.0307002 | issn = 0961-8368 | format = Free full text }}</ref> Thus, different atomic solvation parameters should be applied for modeling of protein folding and protein-membrane binding. This issue remains controversial. The original idea of the method was to derive all solvation parameters directly from experimental [[partition coefficient]]s of organic molecules, which allows calculation of solvation free energy. However, some of the recently developed electrostatic models use ''ad hoc'' values of 20 or 40 cal/(Å<sup>2</sup> mol) for ''all'' types of atoms. The non-existent “hydrophobic” interactions of polar atoms are overridden by large electrostatic energy penalties in such models.
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| ===Solid-state applications===
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| Strictly speaking, ASA-based models should only be applied to describe ''solvation'', i.e. energetics of transfer between [[liquid]] or uniform media. It is possible to express van der Waals interaction energies in the [[solid]] state in the surface energy units. This was sometimes done for interpreting [[protein engineering]] and [[ligand binding]] energetics,<ref name="pmid1553543">{{cite journal | author = Eriksson AE, Baase WA, Zhang XJ, Heinz DW, Blaber M, Baldwin EP, Matthews BW | title = Response of a protein structure to cavity-creating mutations and its relation to the hydrophobic effect | journal = Science | volume = 255 | issue = 5041 | pages = 178–83 |date=January 1992 | pmid = 1553543 | doi = 10.1126/science.1553543| url = | issn = 0036-8075 |bibcode = 1992Sci...255..178E }}</ref> which leads to “solvation” parameter for [[aliphatic]] carbon of ~40 cal/(Å<sup>2</sup> mol),<ref name="pmid11297670">{{cite journal | author = Funahashi J, Takano K, Yutani K | title = Are the parameters of various stabilization factors estimated from mutant human lysozymes compatible with other proteins? | journal = Protein Eng. | volume = 14 | issue = 2 | pages = 127–34 |date=February 2001 | pmid = 11297670 | doi = 10.1093/protein/14.2.127| url =http://peds.oxfordjournals.org/cgi/pmidlookup?view=long&pmid=11297670 | issn = 0269-2139 | format = Free full text }}</ref> which is 2 times bigger than ~20 cal/(Å<sup>2</sup> mol) obtained for transfer from water to liquid hydrocarbons, because the parameters derived by such fitting represent sum of the hydrophobic energy (i.e. 20 cal/Å<sup>2</sup> mol) and energy of van der Waals attractions of aliphatic groups in the solid state, which corresponds to [[heat of fusion|fusion enthalpy]] of [[alkanes]].<ref name="Lomize_2002" /> Unfortunately, the simplified ASA-based model can not capture the "specific" distance-dependent interactions between different types of atoms in the solid state which are responsible for clustering of atoms with similar polarities in protein structures and molecular crystals. Parameters of such interatomic interactions, together with atomic solvation parameters for the protein interior, have been approximately derived from [[protein engineering]] data.<ref name="Lomize_2002"/> The implicit solvation model breaks down when solvent molecules associate strongly with binding cavities in a protein, so that the protein and the solvent molecules form a continuous solid body.<ref name="Lomize_2004">{{cite journal | author = Lomize AL, Pogozheva ID, Mosberg HI | title = Quantification of helix–helix binding affinities in micelles and lipid bilayers | journal = Protein Sci. | volume = 13 | issue = 10 | pages = 2600–12 |date=October 2004 | pmid = 15340167 | pmc = 2286553 | doi = 10.1110/ps.04850804 | issn = 0961-8368 | format = Free full text }}</ref> On the other hand, this model can be successfully applied for describing transfer from water to the ''[[fluid]]'' lipid bilayer.<ref name="Lomize_2006">{{cite journal | author = Lomize AL, Pogozheva ID, Lomize MA, Mosberg HI | title = Positioning of proteins in membranes: A computational approach | journal = Protein Sci. | volume = 15 | issue = 6 | pages = 1318–33 |date=June 2006 | pmid = 16731967 | pmc = 2242528 | doi = 10.1110/ps.062126106 | issn = 0961-8368 | format = Free full text }}</ref>
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| ===Importance of extensive testing===
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| More testing is needed to evaluate the performance of different implicit solvation models and parameter sets. They are often tested only for a small set of molecules with very simple structure, such as hydrophobic and amphiphilic [[alpha helix|α-helices]]. This method was rarely tested for hundreds of protein structures.<ref name="Lomize_2006"/>
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| ===Treatment of ionization effects===
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| Ionization of charged groups has been neglected in continuum [[electrostatic]] models of implicit solvation, as well as in standard [[molecular mechanics]] and [[molecular dynamics]]. The transfer of an ion from water to a nonpolar medium with [[dielectric constant]] of ~3 (lipid bilayer) or 4 to 10 (interior of proteins) costs significant energy, as follows from the [[Max Born|Born]] equation and from experiments. However, since the charged protein residues are ionizable, they simply lose their charges in the nonpolar environment, which costs relatively little at the neutral [[pH]]: ~4 to 7 kcal/mol for Asp, Glu, Lys, and Arg [[amino acid]] residues, according to the [[Henderson-Hasselbalch equation]], ''ΔG = 2.3RT (pH - pK)''. The low energetic costs of such ionization effects have indeed been observed for protein mutants with buried ionizable residues.<ref name="pmid1747370">{{cite journal | author = Dao-pin S, Anderson DE, Baase WA, Dahlquist FW, Matthews BW | title = Structural and thermodynamic consequences of burying a charged residue within the hydrophobic core of T4 lysozyme | journal = Biochemistry | volume = 30 | issue = 49 | pages = 11521–9 |date=December 1991 | pmid = 1747370 | doi = 10.1021/bi00113a006| url = | issn = 0006-2960 }}</ref> and hydrophobic α-helical peptides in membranes with a single ionizable residue in the middle.<ref name="pmid12641459">{{cite journal | author = Caputo GA, London E | title = Cumulative effects of amino acid substitutions and hydrophobic mismatch upon the transmembrane stability and conformation of hydrophobic alpha-helices | journal = Biochemistry | volume = 42 | issue = 11 | pages = 3275–85 |date=March 2003 | pmid = 12641459 | doi = 10.1021/bi026697d | url = | issn = 0006-2960 }}</ref> However, all electrostatic methods, such as PB, GB, or GBSA assume that ionizable groups remain charged in the nonpolar environments, which leads to grossly overestimated electrostatic energy. In the simplest [[accessible surface area]]-based models, this problem was treated using different solvation parameters for charged atoms or Henderson-Hasselbalch equation with some modifications.<ref name="Lomize_2006"/> However even the latter approach does not solve the problem. Charged residues can remain charged even in the nonpolar environment if they are involved in intramolecular ion pairs and H-bonds. Thus, the energetic penalties can be overestimated even using the Henderson-Hasselbalch equation. More rigorous theoretical methods describing such ionization effects have been developed,<ref name="pmid9615168">{{cite journal | author = Schaefer M, van Vlijmen HW, Karplus M | title = Electrostatic contributions to molecular free energies in solution | journal = Adv. Protein Chem. | volume = 51 | issue = | pages = 1–57 | year = 1998 | pmid = 9615168 | doi = 10.1016/S0065-3233(08)60650-6| url = | issn = 0065-3233 | series = Advances in Protein Chemistry | isbn = 978-0-12-034251-8 }}</ref> and there are ongoing efforts to incorporate such methods into the implicit solvation models.<ref name="pmid15051331">{{cite journal | author = García-Moreno E B, Fitch CA | title = Structural interpretation of pH and salt-dependent processes in proteins with computational methods | journal = Meth. Enzymol. | volume = 380 | issue = | pages = 20–51 | year = 2004 | pmid = 15051331 | doi = 10.1016/S0076-6879(04)80002-8 | url = | issn = 0076-6879 | series = Methods in Enzymology | isbn = 978-0-12-182784-7 }}</ref>
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| ==See also==
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| {{columns-list|2|
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| * [[Polarizable continuum model]]
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| * [[COSMO solvation model]]
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| * [[Molecular dynamics]]
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| * [[Molecular mechanics]]
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| * [[Water model]]
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| * [[Force field (chemistry)|Force field]]s in chemistry
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| * [[Force field implementation]]
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| * [[Poisson's equation]]
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| * [[Accessible surface area]]
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| * [[List of software for molecular mechanics modeling]]
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| }}
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| ==References==
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| {{reflist|2}}
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| {{DEFAULTSORT:Implicit solvation}}
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| [[Category:Molecular modelling]]
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| [[Category:Computational chemistry]]
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| [[Category:Molecular dynamics]]
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| [[Category:Protein structure]]
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