ARTICLE _

Classical Electrostatics in Biology and ChemistryBarry Honig* and Anthony Nicholls

A major revival in the use of classical electrostatics as an approach to the study of chargedand polar molecules in aqueous solution has been made possible through the develop-ment of fast numerical and computational methods to solve the Poisson-Boltzmannequation for solute molecules that have complex shapes and charge distributions. Graph-ical visualization of the calculated electrostatic potentials generated by proteins andnucleic acids has revealed insights into the role of electrostatic interactions in a wide range

of biological phenomena. Classical electrostatics has also proved to be a successfulquantitative tool yielding accurate descriptions of electrical potentials, diffusion limitedprocesses, pH-dependent properties of proteins, ionic strength-dependent phenomena,and the solvation free energies of organic molecules.

Electrostatic interactions play a centralrole in a variety of biological processes. Adetailed characterization of the strengthand nature of these interactions requires an

understanding of the physical chemicalproperties of molecules in aqueous solution.Although the principles underlying solventeffects on solute properties have been un-

derstood for many years, methods thattranslate these principles into accurate pre-

dictions of experimental observables were

generally not available. In the past fewyears, there have been significant advancesin this direction, motivated primarily byproblems of major biological as well as prac-

tical importance such as structure-baseddrug design and protein folding.

Understanding the properties of aqueoussolutions requires models of the solute, thesolvent, and the interactions betweenthem. In the widely used molecular me-

chanics approach, solute or solvent mole-cules or both are described in terms of a

"force field" where nonbonded van derWaals and electrostatic terms account formost of the details of intermolecular inter-actions. A proper description at the molec-ular level of solvent effects requires thecalculation of the mutual interactions of a

large number of molecules and the averag-

ing of these over many solvent configura-tions. The daunting computational require-ments of this approach have been partlyovercome through theoretical advances andby continuing enhancements in computa-tional power. Nevertheless, for many appli-cations, the explicit treatment of solventmolecules and mobile ions is not feasible.An alternative approach involves con-

tinuum or macroscopic models, in whichsolvent properties are described in terms ofaverage values. Continuum methods were

The authors are in the Department of Biochemistry andMolecular Biophysics, Columbia University, New York,NY 10032, USA.

*To whom correspondence should be addressed.

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widely used in the past but fell into disfavorwith the advent of modem simulation tech-niques. However, macroscopic solventmodels have experienced a resurgence inrecent years, partly because of computation-al and algorithmic advances that have madeit possible to describe solute molecules inatomic detail while treating only the sol-vent in terms of average properties. In par-

ticular, the importance of classical electro-statics as providing an important qualitativeparadigm and quantitative tool in structuralbiology, biochemistry, and chemistry hasbeen widely recognized.

The Poisson-Boltzmann Equation

The classical treatment of electrostatic in-teractions in solution is based on the Pois-son-Boltzmann equation (PBE)

V* [c(r)V * +(r)] - c(r)K(r)2sinh[kI(r)]

+ 4wpf(r)/kT= O (1)

where 4(r) is the dimensionless electrostat-ic potential in units of kT/q (k is the Boltz-mann constant, T is the absolute tempera-ture, and q is the charge on a proton), e isthe dielectric constant, and pf is the fixedcharge density (in proton charge units).The term K2 = 1/X2 = 8&rq2IlekT, where X

is the Debye length and I is the ionicstrength of the bulk solution. The variables+, £, K, and p are all functions of theposition vector r.

The second term of Eq. 1 accounts forsalt effects and is absent when no mobileions are present in the system (K = 0).Under this condition, Eq. 1 reduces to Pois-son's equation, which in turn reduces to

Coulomb's law when the dielectric constant

is uniform throughout space. However, be-cause water is much more easily polarized byan electric field than are most solutes, at

least two £'s are required to capture theunderlying physics of polar molecules inaqueous solution. The effect of having two

SCIENCE * VOL. 268 * 26 MAY 1995

dielectric constants is expressed in Eq. 1through the derivative of £(r) in the firstterm, which is nonzero only where £(r)varies. In all of the applications consideredbelow, this region corresponds to the mo-lecular surface, where there is a dielectric"discontinuity" between the low-£ soluteand the high-£ solvent.An intuitive understanding of the effect

of the shape of the solute-solvent interfacecan be obtained from the concept of in-duced surface charge. Poisson's equationcan be recast with a single dielectric model(Coulomb's law), where charge of polarityopposite to that of +(r) is induced at themolecular surface. An important example isthe simple case of a positive charge at thecenter of a low-c sphere, which induces anegative charge distributed over the surfaceof the sphere. The positive potential in thesolvent caused by the fixed positive chargewill be screened by the negative surfacecharge; the higher the solvent's value of c,the greater the surface charge, and hence,the greater the screening. Similarly, thefavorable interaction of the fixed positivecharge with its induced negative surfacecharge (which produces a "reaction field")is the basis of continuum solvation models.It is tempting to think of the induced sur-face charge as corresponding to an align-ment of water dipoles at the molecular sur-face producing an excess local charge, andindeed, this is part of the effect. However, itshould be realized that the surface chargedescription accounts for the polarization ofthe entire solvent.

The treatment of molecules in solutionused here follows the general treatmentdeveloped by Born (1), Kirkwood (2), andOnsager (3) in which a solute molecule isdescribed as a low-c "cavity" embedded ina medium with a different value of £.Because analytical solutions to Poisson'sequation are only available for simple geo-metric objects, all applications of the PBEto molecules in solution, until recently,required a simplifying assumption for theshape of the solute and its charge distri-bution. For example, small solutes andproteins were treated as spheres, DNA ascylinders, and membranes as planes. How-ever, numerical solutions to the PBE nowmake it possible to describe the shape ofthe solute in atomic detail while retaininga simplified "continuum" description ofonly the solvent.

Although the discussion so far has fo-cused on Poisson's equation, many of thesame concepts are applicable to the PBE as

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j;"P~

well. Under conditions where +(r) is smallrelative to kT, the sinh term can be lin-earized by setting sinh(4) = 4, yieldingthe linearized PBE (which reduces to De-bye-Huckel theory when no dielectric dis-continuity is present). In the linearizedPBE, the ion atmosphere behaves muchlike the high-£ solvent in that both re-spond linearly to the electrical potentialinduced by fixed charges. In cases wherethe nonlinear PBE must be applied [forexample, the ion atmosphere around high-ly charged macromolecules such as DNAor polylysine (4)1, some of the simplicityof linear response theory is lost.

Calculating Electrical Potentials

The molecular surface is defined as thecontact surface formed between the van derWaals envelope of the molecule and aprobe solvent molecule (5). All regions in-side the surface are assigned a low value of£ [-2 to 4 (6)1, whereas exterior regions areassigned the £ of water (-80). The chargedistribution of the molecule is usually rep-resented in the form of point charges locat-ed at atomic nuclei.

Numerical procedures to solve the PBEfirst require a discretization of some sort, forexample, finite difference methods thatmap variables onto a grid. A number ofanalytical and numerical techniques havebeen reported that can describe molecularsurfaces, even for complex molecules suchas proteins, in a few seconds of computertime [CPU (central processing unit) time]on standard processors (7).

The second step usually involves aniterative procedure, where an initial guessto a solution is refined successively. War-wicker and Watson reported the first nu-merical (finite difference) solution to thePoisson equation for a protein (8). Thisresult was followed by finite differencesolutions to the full linear (9, 10) andnonlinear PBEs (11). In recent years, avariety of numerical methods have yieldedincreasingly faster and more accurate al-gorithms based on grid-based (12, 13),boundary element (14), and finite ele-ment methods (15). A standard worksta-tion can now be used to calculate ¢(r) fora molecule in a solution of arbitrary ionicstrength in seconds to minutes of CPUtime, depending on the size of the mole-cule and the level of accuracy required ina particular application.

Patterns of 4(r) in Proteins

Perhaps the most extensive application ofcontinuum electrostatics in structural biol-ogy has been the visual representation of4(r) for proteins. [The plots of +(r) in thisarticle are calculated with the DelPhi pro-

gram (12, 16) and displayed with GRASP(17).] Unexpectedly, proteins generateunique ¢(r) patterns that, in many cases,have an important functional role. In addi-tion to the specific location of charged andpolar groups on the protein, the geometricshape of the molecular surface plays a cru-cial role in determining the form of the+(r). The effect of shape arises through theexistence of a boundary between the low-£and high-£ regions.

Numerical solutions to the PBE equationyield information not evident from thethree-dimensional (3D) structure alone. Ahard-sphere representation of acetylcholine-esterase (Fig. lA) (18) shows a preponder-ance of negative charges on the face of theprotein that borders the active site. How-ever, the existence of an intense electro-negative region centered in the active site(Fig. 1B) is not evident from the chargedistribution in Fig. 1A. The pattern is duein large part to the depth of the active sitepocket, which produces a dielectric dis-

Fig. 1. Surface representations of acetylcholineesterase. (A) Space filling model (18). Atoms col-ored red are the oxygens of negatively chargedcarboxylate groups, and those colored blue arethe nitrogens of positively charged basic groups.The substrate acetylcholine, which carries a netpositive charge, is shown in yellow in the bottomof the active site. (B) Surface potential, displayedwith GRASP (17). The molecular surface is colorcoded by electrostatic potential, as calculatedwith DelPhi (12) for the charges illustrated in (A).Potentials less than -1 OkT are red, those greaterthan 1 OkT are blue, and neutral potentials (0 kT)are white. Linear interpolation was used to pro-duce the color for surface potentials betweenthese values. The active site is clearly distinguish-able as a region of intense negative potential.

continuity with a distinctive shape.The effect of the dielectric discontinuity

is evident in Fig. 2, A and B, which illus-trates the +(r)'s involved in the formationof the trypsin-trypsin inhibitor complex(19). Although both molecules have a largepositive net charge, they form a very tightintermolecular complex. Whereas 4(r) fortrypsin with a single £ throughout space(Fig. 2A) would suggest that bovine pan-creatic trypsin inhibitor (BPTI) must dockinto an area of large positive potential(which is highly unlikely given its ownpositive charge), a two-e model (Fig. 2B)produces a significant local domain of neg-ative potential at the binding site.

Reports of protein structures often in-clude color-coded pictures representing so-

Fig. 2. Electrostatic potentials characteristic of thetrypsin-bovine pancreatic trypsin inhibitor (BPTI)complex. Secondary structure representations fortrypsin (left) and BPTI (right). The complex (49) has,for clarity, been partially separated along the axis ofinteraction (horizontal and parallel to the page).Also shown for BPTI are the positions of thecharged atoms for ionizable side chains, colorcoded as in Fig. 1A. The 4(r)'s were calculated forthe charged groups of trypsin alone and are illus-trated by a color-coded plane that slices throughthe binding site. The color code is the same as inFig. 1 B but with extrema of +-2kT. Contour linesare shown at -2, -1, 0, 1, and 2 kT (white, blue,green, red, and magenta, respectively) for poten-tials in this plane. The 4(r)'s were calculated (A)assuming a uniform dielectric constant of 80throughout space and (B) assuming an interior di-electric constant of 2 for trypsin and 80 elsewhere.

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lutions to the PBE. An important findingthat appears to be emerging from these stud-ies is that many intermolecular interactionsinvolve associations between surfaces withcomplementary +(r)'s. For example, DNAbinding proteins generally have regions ofpositive +b(r) at the DNA binding interface(20). An example is shown in Fig. 3A,which displays 4(r) at the surface of theprocessivity factor of Escherichia coli DNApolymerase III (21). This protein forms a"sliding clamp" that binds tightly but non-specifically to DNA. The protein has a largenegative net charge (-22 per dimer) butnevertheless displays a significant positive+(r) on the inner DNA binding surface.A neutral surface can also be functionally

significant and is suggestive of hydrophobicinteractions. Waksman et al. (22), in analyz-ing their structure of the Src SH2 domain incomplexed and peptide-free forms, noticedthat the binding site consisted of a neutralhydrophobic pocket, which binds isoleucine,and a region of positive +(r), which bindsphosphotyrosine (Fig. 3B). A striking exam-ple of a functionally significant zero poten-tial region is provided by the F1 adenosinetriphosphatase (ATPase) (23). The innersurfaces of its trimeric "wheel" and mono-meric "stalk" are essentially neutral, provid-ing a nonspecific, "greasy" axle for monomerrotation during ATP synthesis. In this sys-tem, the absence of charge in the interfacemay be essential because it would be ener-getically implausible for rotations to occur ifburied ion-pairs were broken in the process.

Common electrostatic features within afamily of related proteins can be identifiedby "electrostatic homology modeling," inwhich the primary focus is on the 4(r) pat-terns rather than on specific residues or 3Dstructure similarities. One example is provid-ed by secretory phospholipases A2, whichhave common asymmetries in their 5(r)'sthat appear to correlate with their preferen-tial binding to negatively charged lipids(24). In another example, neurotrophic fac-tors that share the same low-affinity receptoras nerve growth factor (NGF) appear to havea remarkably similar positive patch at theputative binding site (25). Finally, in com-paring eukaryotic and bacterial DNA poly-merase processivity factors, Krishna et al.(26) found a highly similar 4(r) patterneven though the two proteins have verylow subunit sequence homology (15%).Moreover, the eukaryotic protein is a trimerof net charge -60, whereas the bacterialprotein is a dimer of charge -22 (Fig. 3A).Despite these differences, both proteins gen-erate almost identical positive 4(r) distribu-tions in the DNA binding region.

Patterns of surface potentials have alsoprovided a useful basis for interpreting bio-logical and biochemical data and can pointto residues that may play important func-

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tional roles. For example, Ryu et al. (27)observed that a negative 4(r) site on anotherwise positive face ofCD4 correspondedto a region implicated in cell fusion betweenHIV-infected cells and normal CD4-positivecells (Fig. 3C). MacDonald et al. (28), inanalyzing their structure of NGF, noticed apositive groove at one end of the dimer. Sixlysines line this groove, and their neutraliza-tion has a profound effect on the calculated4(r) for this surface (Fig. 3D). Using site-directed mutagenesis (SDM), Ibanez et al.(29) were then able to show that these ly-sines are essential for the binding affinity ofNGF to its "low-affinity" receptor.

The accuracy of the calculated 4(r)'s(for example, Figs. 1 through 3) can be

tested by SDM. One approach, pioneeredby Fersht and co-workers (30), is to useSDM to modify a charge at a particularsite on the protein surface and to measureshifts in pKa, that is, the proton affinity, ofacids and bases at other sites (Ka is theacid constant). The change in +(r) calcu-lated at these sites can then be related tothe shifts by the simple relation ApKa =A+/2.3RT (R being the gas constant).Loewenthal et al. (31) reported extensivestudies of charge-charge interactions ondifferent proteins as a function of ionicstrength and found that PB calculationsdone with the DelPhi program yieldedresults that were in impressive agreementwith experiment.

A

D

Fig. 3. Surface 4(r)'s of various proteins. (A) The DNA binding beta subunit (processivity factor) from Ecoli DNA polymerase Il (74). A DNA molecule has been modeled into the center of the dimer where,despite a net negative charge of -22, the electrostatic potential is positive. (B) The Src SH2 domain (22),which binds a phosphotyrosine-containing peptide (shown in a bond representation). The phosphoty-rosine binds in a distinctly positive pocket, while an isoleucine side chain from the peptide occupies a

similar but neutral depression. (C) A view of CD4 (27) illustrating a negative patch on the mainly positiveface of the protein that has been associated with cell fusion events. (D) Two views of the end of the NGFdimer (28) implicated in low-affinity receptor binding. On the left, the potential is that of the wild-typeprotein, whereas the potentials on the right were calculated with six lysine residues lining the groovetreated as uncharged. The lysines in question are necessary for low-affinity receptor binding (29).Potentials are from DelPhi, with dielectrics as in Fig. 2B. The color code is as in Figs. 1 and 2, but thevalues used to determine the saturations varied; white is always equal to neutral. The ionic strength is zeroin all cases except (A), where 0.1 M salt was assumed.

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How Proteins ExploitElectrostatic Potentials

In addition to their role in molecular rec-

ognition, numerous examples exist wherecalculated +(r)'s of proteins have been im-plicated in different functional roles. A par-

ticularly striking effect is the enhancementof the diffusion-limited association constant

of charged substrates (9, 32, 33).The most detailed studies of association

constants have been carried out on Cu,Znsuperoxide dismutase (SOD), which, de-spite its net negative charge, generates a

region of positive 4(r) near the active site(32). These 4.(r)'s are magnified in solutionthrough the active-site focusing effectsanalogous to those for trypsin (9) (Fig. 2).Electrostatic focusing by narrow clefts ap-

pears to be a general phenomenon and may

account, for example, for the production oflarge ion concentrations near the open-

ings of transmembrane channels. Brown-ian dynamics simulations of the diffusionprocess in SOD have demonstrated thatthe focused 4(r) enhances the associationrate of the negatively charged superoxidesubstrate (34, 35). The calculated ratesare in excellent agreement with the exper-

imental rate and its ionic strength depen-dence. Predictions of the effects of specificamino acid mutations on association rates

(35) have been confirmed by recent SDMstudies (36).

In addition to their effects on diffusion,4(r)'s also play an important role in theenhancement of catalytic rates (37), con-

trol by phosphorylation (38), and determi-nation of redox potentials (39). A role forelectrostatic surface potentials that has onlyrecently been proposed for the bifunctionalenzyme thymidylate synthase-dihydrofo-late reductase is the channeling across theprotein surface of a substrate between thetwo active sites (40).

Calculating pKa's: pH Effects onBinding and Stability

The pKa's of ionizable groups in macromol-ecules can be significantly different thanthose of the isolated groups in solution.Shifts in pKa's between native and dena-tured states or between bound and freeforms of a complex give rise to pH effects onmacromolecular stability and on bindingconstants (41). For example, the acid dena-turation of proteins appears largely as theresult of a small number of amino acidswhose pKa's are shifted anomalously in thenative protein (42, 43).

In proteins, pKa's are typically calculatedas PKa shifts relative to the isolated aminoacid. The source of the shifts has tradition-ally been attributed to interactions amongionizable groups on the macromolecule; for

example, an acidic group will have its pKalowered through favorable interactionswith basic groups. However, hydrogenbonding interactions with nonionizablegroups and the degree of exposure to thebulk solvent have also been found to beimportant determinants of pKa'S (44). Forexample, an acidic group will have its PKaincreased if it is fully or partially removedfrom solvent, but the effect may be re-versed by strong hydrogen bonding inter-actions with other groups. Thus, a fulltreatment of pH-dependent effects re-quires that the detailed environment ofeach ionizable group be taken into ac-count. In addition to the need for struc-tural detail, the problem of calculatingpKa's is further compounded by the factthat a protein with N ionizable residueshas 2N possible ionization states. Because amoderately sized protein may contain asmany as 50 ionizable groups, methodshave been developed to reduce the num-ber of states that need to be considered.

Recent studies have addressed the com-binatorial problem and, in addition, haveused PB calculations to account for bothdesolvation effects and hydrogen bondinginteractions (45). Applications to a fewproteins have been reported, and the agree-ment with experimentally determined pKa'shas been good, with a few notable excep-tions. A limitation on the accuracy of suchcalculations arises from uncertainties as tothe nature of the conformational changesthat accompany changes in the ionizationstates of different residues. Nevertheless,the available methods now allow a detailedstudy of the structural origins of pH effectson protein stability (43, 46) and substratebinding (47). Moreover, they can be used togenerate structure-based predictions as tothe contribution of specific ionizable resi-dues to pH-dependent phenomena that canbe tested with SDM.

The methodology developed primarilyfor the calculation of pKa's in proteins hasmade it possible to solve a classical problemin physical organic chemistry. Dibasic anddiacidic organic compounds such as succin-ic acid and 1,3-diamino propane exhibittwo PKa's that generally have been attrib-uted to pKa shifts in one group caused by itsproximity to the charge on the other. Kirk-wood and Westheimer (48) approached thisproblem more than 50 years ago by usingclassical electrostatics and treating thesolute as a low-£ spherical cavity. Theresults were in qualitative agreement withexperiment but depended on the detailedmapping of the solute charges onto asphere. Quite recently, two groups havereinvestigated this problem using numeri-cal solutions to the PB equation, and theresults were in excellent agreement withexperiment (49).

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Salt Effects on Nucleic Acids

The large negative charge density generatedby the phosphodiester backbone of nucleicacids results in a significant concentrationof counterions in the vicinity of these mac-romolecules. For this reason, conformation-al changes and binding reactions involvingDNA and RNA are strongly dependent onsalt concentration (50). Salt effects on nu-cleic acids have been traditionally analyzedin terms of counterion condensation theory,which predicts that an essentially fixed con-centration of counterions is found near theDNA, independent of bulk salt concentra-tion (51). Manning (51) and Record andco-workers (52) have developed theoreticalmodels that have been extremely successfulin the interpretation of ligand binding equi-libria and conformational transitions of nu-cleic acids, based on simplified representa-tions of these molecules.

The availability of solutions to the non-linear PBE (11, 53) and a general expres-sion to obtain electrostatic free energiesfrom the calculated 4(r)'s (54) now makesit possible to make quantitative predictionsof salt effects on the basis of 3D structures.The counterion concentration near nucleicacids is relatively insensitive to bulk saltconcentration. Nevertheless, variations inthis distribution near the surface of themolecule play a central role in determiningsalt effects on the conformational changesand binding reactions of nucleic acids. Inapplications to the problem of ligand bind-ing to DNA, the dependence of the equi-librium constant on monovalent ion con-centration was calculated for minor groovebinding drugs (55) and for a number ofproteins (56). In all cases, the calculatedsalt dependence of binding was in excellentagreement with experimental observation.Surprisingly, the major contribution wasnot the entropic release of counterions,which has been the standard view, but rath-er the changes in interaction of DNA withits ion atmosphere, which is a strong func-tion of bulk ion concentration. Althoughstudies of this type are still in their infancy,the results obtained so far indicate that it isnow possible to obtain accurate predictionsof nonspecific salt effects on the propertiesof highly charged polymers, even whenlarge salt concentrations are involved.

Solvation Free Energies

Solvation phenomena can be described atthe macroscopic level in terms of the dif-ference in the reaction field energy of acharge distribution upon being transferredfrom one dielectric medium to another. Thesimplest problem of this type concerns thettransfer free energy of an ion, which wastreated by Born in 1920 as a uniformly

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charged sphere (1). The vacuum-to-waterhydration free energies of both anions andcations can be reproduced quite accuratelywith the Born model if a physically mean-ingful set of cavity radii are used (57). Theremarkable success of the simple Born mod-el has led to a resurgence in the use ofcontinuum models to study solvation phe-nomena (58, 59). Electrostatic contribu-tions to solvation free energies (AGSO-v)with the PB method are in excellent agree-ment with those obtained from detailedmicroscopic solvent simulations (59, 60).Thus, classical electrostatics can reproducethe energetics of microscopic hydrogenbonding interactions of polar moleculeswith the aqueous phase.

In order to obtain values of AGso'V thatcan be compared with experimental values,both nonpolar and electrostatic contribu-tions must be considered. Nonpolar contri-butions can also be treated at the macro-scopic level through the use of surface ten-sion concepts; that is, relations betweenfree energy and surface area. The total freeenergy of transfer of a solute from the gasphase to water can be written as

AGSOIV = AGnp + AGes (2)

where AGes is the difference in electrostaticfree energy of the solute in the two phasesand AGnp corresponds to the free energy ofinserting a hypothetical nonpolar solute ofthe same size and shape as the present sol-ute into the solvent (61). It is frequentlyassumed, on the basis of earlier studies ofalkane-water partition coefficients (62),that AGnp is proportional to accessible sur-face area (63) and is given by

AGnp = yvwAAT + b (3)where yvw is a constant that may be viewedas representing the vacuum-water micro-scopic surface tension, AAT is the totalaccessible area of the solute, and b is aconstant (note that the area is total arearather than just nonpolar area because allpolar groups are assigned a charge of zero inthe transfer step). The constants b (fre-quently assumed to be zero) and yvw aregenerally obtained by fitting the gas phase-to-water partition coefficients of linear al-kanes (64) to a linear function of accessiblesurface area.

Atomic charges used in PB calculationsof solvation phenomena are typically ob-tained from molecular mechanics forcefields but can be treated as parameters aswell (65). The results obtained so far sug-gest that PB calculations can be used on asolute described in terms of standard atomicradii and charges to obtain AGSO'V valuesthat agree with experiment for a large num-ber of organic molecules. An alternative tousing atomic charges extracted from forcefields is to obtain the electronic distribution

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directly from quantum mechanics. Methodsare now available that combine quantummechanical calculations and continuumsolvent models to yield solvation free ener-gies that are in excellent agreement withexperiment (66).

Effects of Solvation onStability and Binding

The existence of complementary chargedsurfaces is a good indicator of associationinterfaces between different macromole-cules. However, this does not necessarilyimply a favorable electrostatic contributionto the free energy of association. Indeed, inmany cases that have been studied, electro-static interactions between groups of oppo-site charge actually oppose binding (67).The effect may be understood from the factthat the formation of, for example, an ionpair at an interface between two macromol-ecules involves the removal of the chargedmoieties from water. The desolvation costsassociated with this process appear in gen-eral to be larger than the pairwise Coulom-bic interaction energy. The point was firstmade in a biological context by Parseghian(68), who argued that the removal of twooppositely charged ions from the aqueousphase that then form an ion pair in a lipidbilayer was costly in a free energy sensebecause the loss of favorable solvation in-teractions would not be compensated byCoulombic attraction. The validity of thisargument has been subsequently verified bymore detailed analysis (69).An identical argument suggests that elec-

trostatic interactions destabilize folded pro-teins even when there is a net Coulombicattraction between surface groups. Indeed, astudy of the pH dependence of protein sta-bility showed that ionizable groups destabi-lize proteins at all pH values (45). Recently,Hendsch and Tidor (70) found that saltbridges appear in general to destabilize pro-teins because of incomplete Coulombiccompensation of desolvation effects.

The same type of argument appears toapply to hydrogen bonds as well. The elec-trostatic free energy loss associated with theremoval of hydrogen bonding donor andacceptor groups from water is not generallycompensated by the formation of a hydro-gen bond in an interface or in the interiorof a protein (71). If the anecdotal evidenceavailable so far is verified, the rather globalimplication is that electrostatic interactionstend to favor the unfolded states of macro-molecules and the unbound state of com-plexes. This would imply that the thermo-dynamic driving force for most processes inaqueous solution results from "nonpolar"interactions such as the hydrophobic effectand close packing. In this model, electro-statics in biological systems confer specific-

SCIENCE * VOL. 268 * 26 MAY 1995

ity and play a role in the generation ofunique structures (70, 72). In other words,once charged and polar groups have beenremoved from water, they must arrange toform ion pairs or hydrogen bonds so as tominimize the free energetic cost of theirremoval from water. A corollary of thisargument is that the presence of polargroups in interfaces or in the interior ofmacromolecules has a destabilizing effect,reducing association constants and unfold-ing free energies.

Conclusions and FutureProspects

Continuum electrostatics as a tool for theanalysis of molecular structure and functionin solution has seen a major renaissance inrecent years. Progress has been due in partto computational developments such as fastnumerical solutions to the PB equation, fastalgorithms to represent molecular surfaces,and new visualization techniques. However,equally important has been the develop-ment of methods to map molecular proper-ties onto the language of continuum elec-trostatics and the recognition that manysolvent effects can be treated accuratelywith a model that ignores the microscopicdescription of solvent molecules.

Graphical visualization of the electro-static properties of macromolecular struc-tures is having significant impact on thefield of structural biology. Continuum elec-trostatics has also proved to be an extremelyuseful quantitative tool, allowing the accu-rate prediction of the magnitude of electro-static effects. These developments offer thehope of a complete and accurate method todescribe the properties of molecules inaqueous solution, although much remainsto be done before this goal is realized. Amajor problem results from the fact thatalmost all applications of continuum meth-ods have been to fixed structures. This isclearly adequate for many problems, butthere are many other cases for which dy-namical effects will have to be considered.The inclusion of continuum methods indynamical studies, such as the direct incor-poration of PB solutions into molecular me-chanics force fields, has been attempted in anumber of cases but remains a significantand important theoretical challenge (73).

Overall, the greatest impact of continu-um methods results from the fact that theyoffer an intuitively simple yet physicallycomplete model of molecules in solution.Fairly reliable predictions, both quantita-tive and qualitative, can be obtained withonly moderate computational expense andare available to researchers on standardworkstations. Equally important, generalphysical principles based on classical elec-trostatics are emerging that can be of con-

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siderable value in the design and interpre-tation of experimental results. These devel-opments have been made possible by theadvent of fast computers and the applica-tion of sophisticated numerical methods,which have revived classical theory as acritical tool in the study of contemporaryproblems.

REFERENCES AND NOTES

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75. We are grateful to K. A. Sharp and R. V. Sampognafor their careful reading of the manuscript. This paperwas supported by grants from the National ScienceFoundation (BIR 9207256 and MCB 9304127), theOffice of Naval Research (NOOO1 4-93-1-0405), andthe National Institutes of Health (GM41371 and GM30518). A gift from the Upjohn Company to supportthe development of the GRASP program is alsogratefully acknowledged.

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