A molecular approach to bioseparations: Proteinprotein andproteinsalt interactions

R.A. Curtis, L. LueSchool of Chemical Engineering and Analytical Science, The University of Manchester, PO Box 88, Sackville Street, Manchester M60 1QD UK

Abstract

Many bioprocess separations involve manipulating the solution conditions to selectively remove or concentrate a target protein. Theselectivity is determined by the thermodynamics of the protein solution, which are governed, at the molecular level, by proteinproteininteractions. Thus, to optimise these processes, one must understand the precise nature of these interactions, how they are affected bythe control parameters (e.g., temperature, ionic strength, pH), and how they relate to the solution properties of interest (e.g., proteinsolubility, protein stability, crystal nucleation rates). Recently, studies of proteinprotein interactions have been motivated by the discoveryof a crystallisation window, which places bounds on the proteinprotein attraction that is required for crystallisation to be possible. Thisreview focuses on experimental and theoretical studies of proteinprotein interactions and discusses the current gaps in understanding theseforces. The first models for these interactions had been based on DLVO theory, which has proven useful for understanding proteinproteininteractions in dilute electrolyte solutions. However, DLVO theory does not include such important effects as anisotropic interactions,solvation forces, and specific ion effects. Various approaches have been developed to include these phenomena although they are still notwell understood. One outstanding issue concerns specific ion effects, which play a crucial role in biological systems. An initial step inunderstanding these effects might be to first rationalize proteinsalt interactions. 2005 Elsevier Ltd. All rights reserved.

Keywords: Proteins; Phase equilibria; Crystallisation; Colloidal phenomena; Intermolecular interactions; Molecular thermodynamics

1. Introduction

At the molecular level, proteinprotein and proteinsaltinteractions control the structure and thermodynamics ofprotein solutions. Consequently, knowledge of these inter-actions, how they are affected by the solution conditions,and how they impact the bulk properties of the solution,can lead to key insights in designing and optimising manybioprocesses, including protein salting-out (Melander andHorvath, 1977; Coen et al., 1995), protein partitioning inaqueous two-phase polymer-based systems (Haynes et al.,1991, 1993), protein chromatography (Melander andHorvath, 1977; Vailaya and Horvath, 1998; Oberholzer andLenhoff, 1999), protein formulation (Chi et al., 2003) and

Corresponding author. Tel.: +441613064401; fax: +441613064399.E-mail address: r.curtis@manchester.ac.uk (R.A. Curtis).

the re-folding of proteins from inclusion bodies (Hoet al., 2003). A wealth of experimental and theoreticalstudies have been directed at understanding these interac-tions. Proteinprotein interactions have been probed using awide variety of experimental techniques, such as static anddynamic light scattering, osmometry, and small angle neu-tron scattering. The resulting experimental data can be usedas inputs in molecular thermodynamic models to predictprotein phase equilibria, protein crystallisation, and pro-tein aggregation (Vilker et al., 1981; Haynes et al., 1992;Taratuta et al., 1990). Proteinsalt interactions, which areaccessible through dialysis equilibrium experiments via thepreferential interaction parameter (Lee et al., 1979), can alsobe used to rationalise the behaviour of protein solutions.The first theoretical descriptions of protein salting-out werebased on modelling the effect of salt on the solvation freeenergy of the protein (Melander and Horvath, 1977). These

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theories were also used to calculate the retention time ofproteins in hydrophobic interaction chromatography. Moreimportantly, the influence of salt on protein stability canbe understood in terms of differences in the interactionsbetween the salt and the protein in the folded and unfoldedstates (Timasheff, 1991).

Recently, one of the main motivations of understandingproteinprotein interactions is based on the discovery of acrystallisation window for protein solutions. Only when theattractions between proteins fall within the crystallisationwindow can high quality crystals be obtained on reasonabletime scales. Much of the work in this area has focused onthe connection between proteinprotein interactions and thephase diagram and the relationship between protein crystalli-sation rates and the location on the protein phase diagram.

The majority of proteinprotein interaction models are de-rived from colloid science, where proteins are modelled ascharged, dielectric spheres. These models, based on DLVOtheory (Verwey and Overbeek, 1948), provide an adequatedescription for proteins in dilute aqueous electrolyte solu-tions, where the interactions are dominated by long-rangedelectrostatic forces. Despite the general success of DLVOtheory, there are many situations where it is unable to pro-vide even a proper qualitative description of protein solu-tion behaviour. One important example is protein salting-out, which is the precipitation of protein with the additionof salt. The ability of an ion to induce protein precipitationis strongly dependent on its type. The salting-out effective-ness is correlated with the ions position in the Hofmeisterseries (Hofmeister, 1888). DLVO theory is unable to quanti-tatively predict when salting-out will occur and, in addition,does not predict any dependence on salt type.

The remainder of this review is organised as follows. First,we discuss the use of osmometry as a tool to probe thegeneral behaviour of protein solutions. Then in Section 1.2,the concept of the potential of mean force is introduced,which provides a precise definition of proteinprotein in-teractions. The relationship between the pmf and the struc-ture and thermodynamics of the protein solution is given byMcMillanMayer theory, which is described in Section 1.3.

Section 2 provides a discussion of the general phase be-haviour of aqueous protein solutions, its dependence onthe form of the proteinprotein potential of mean force,and the crystallisation window. Section 2.1 reviews DLVOtheory and subsequently discusses its limitations in mod-elling proteinprotein interactions in moderately concen-trated electrolyte solutions. As mentioned earlier, one ofits major shortcomings is it cannot account for specificion effects, which are discussed in Section 2.2. It has alsobeen found that the proteinprotein interactions are prob-ably anisotropic; weak proteinprotein interactions shouldbe treated more like molecular recognition processes wherethe averaged proteinprotein interactions are only given bya few highly attractive configurations. These are discussedin Section 2.3. Furthermore, due to the short-ranged natureof most proteinprotein interactions, solvation forces need

to be considered. Solvation forces are a major driving forcein protein folding and protein complex formation and arecovered in Section 2.4.

Proteinprotein interactions result from overlapping lay-ers of solvent perturbation around the protein surfaces. Con-sequently, in order to understand proteinprotein interactionsin salt solutions, it is first necessary to understand the effectof salt on the protein solvation free energy. In Section 3, weprovide a general discussion of proteinsolvent interactions.Much of what is known about proteinsalt interactions hasbeen determined from extrapolating the results of solubilitystudies for model compounds in salt water. These studies aresummarised in Section 3.1. Section 3.2 describes the prefer-ential interaction parameter and its applications, such as inpredicting the effect of salt on protein stability (covered inSection 3.3) and protein salting-out (covered in Section 3.4).Because models of protein salting-out have been based onproteinprotein and on proteinsalt interactions, in Section3.5 we attempt to show the link between the two types of in-teractions, thereby gaining some insight into the Hofmeistereffects in proteinprotein interactions.

Finally, we conclude the review in Section 4 with a gen-eral discussion of the current outstanding problems and fu-ture directions in the molecular thermodynamics of proteinsolutions.

1.1. Osmotic pressure

One key measurement that probes the thermodynamic be-haviour of a protein solution is membrane osmometry. In thisexperiment, a salt solution containing protein is placed in achamber that is separated from a second chamber containingonly the salt solution. The two chambers are separated by amembrane that is impermeable to the protein, but permeableto the water and the salt. At equilibrium, the pressure of theprotein solution is higher than that of only the electrolytesolution. This difference in pressure is known as the osmoticpressure, , which can be expressed as an expansion in theexperimentally accessible protein mass concentration, cp,

cpRT= 1/Mp + B 2(T , w, s)cp

+ B 3(T , w, s)c2p + . (1)Here, R is the ideal gas constant, B 2 and B 3 are the sec-ond and third osmotic virial coefficients, respectively, T isabsolute temperature, and Mp is the molecular weight of theprotein. The chemical potential of component i is denotedby i , where i = w for water, i = s for salt, and i = p forprotein.

Experiments such as static light scattering or small an-gle neutron scattering have also been used to probe theosmotic pressure of protein solutions. More recently, a high-throughput chromatographic method has been used to mea-sure interactions between proteins in a mobile phase andproteins attached to a stationary phase (Tessier et al., 2002a).

Relating these measurements to values of B2 requires care asthe immobilized proteins must be at a very low surface cov-erage to avoid multi-body interactions (Teske et al., 2004).A less direct method of measuring protein behaviour is dy-namic light scattering, where hydrodynamic and thermody-namic contributions to protein behaviour are probed.

1.2. The potential of mean force

An extremely useful concept in interpreting and mod-elling the behaviour of protein solutions is the potential ofmean force (pmf). Physically, the n-body potential of meanforce is the work (free energy) required to bring n proteinmolecules from infinite separation to a fixed configurationin the solution. During this process, all the other moleculesin the system (e.g., water, salt, etc.) are allowed to freelymove. In general, the pmf is a many-body interaction thatdepends on the relative positions and orientations of the pro-tein molecules and is averaged over all positions of solventmolecules. In most approximations, however, it is treatedsimply as a pairwise additive potential.

The potential of mean force can be related to the structureof the solution. The two-body contribution to the potentialof mean force, w2, is directly related to the pair correlationfunction, g2, by McMillan and Mayer (1945)

w2(r,1,2; T , w, s)=kBT ln g2(r,1,2; T , w, s), (2)

where r is the distance between the centres of mass of theprotein molecules, 1 represents the orientation (parame-terised by Euler angles or quaternions) of protein 1, and 2is the orientation of protein 2, and kB is the Boltzmann con-stant.

Specifying the potential of mean force fixes the thermo-dynamic behaviour of the system. In other words, once thepotential of mean force is known, the structure and the ther-modynamics of the system can be determined, in principle.

The potential of mean force depends on the solution con-ditions. Changing such parameters as the temperature, saltconcentration, salt type, etc. will alter the pmf and, conse-quently, the thermodynamics of the system. Therefore, beingable to describe the proteinprotein pmf and its dependenceon solution conditions is crucial if one wants to understandthe behaviour of protein solutions.

1.3. Relating pmf to thermodynamic properties

In most protein solutions, water makes up the majorityof the species present in the system. From a theoretical per-spective, one could treat all species in the system explic-itly, on an equivalent basis. However, the interactions be-tween water molecules are quite complex and play a majorrole in determining the behaviour of the solution. Althoughformal methods are available to handle these complexities,the approximations which must be made to make practical

calculations lead to theories that are either extremely com-putationally difficult or that are not sufficiently accurate.

On the other hand, the thermodynamic properties of purewater and water/salt solutions are well known through ex-perimental measurements. In addition, one can usually makereasonable approximations for the interactions between so-lute particles (e.g., proteins) immersed in water/salt solutions(the potential of mean force). These facts lead to the ideaof treating the solvent as a continuum background and ex-plicitly dealing with only the solute particles. McMillan andMayer (1945) provided a theoretical framework for dealingwith these types of systems.

McMillanMayer theory formalises the analogy betweenthe behaviour of a solute interacting through a solvent (e.g.,proteins in aqueous salt solutions) and molecules interactingin a vacuum. As a result, free energy models and equationof state descriptions of the behaviour of real fluids can bedirectly used to describe the behaviour of proteins in a solu-tion. The interaction potential between proteins is replacedby the potential of mean force. Similarly, the various ther-modynamic properties of the fluid are replaced by propertiesof the solution; for example, the fluid pressure is replacedby the osmotic pressure. The virial expansion for the os-motic pressure can be expressed in terms of protein numberconcentration, p:

kBT= p + B2(T , w, s)2p

+ B3(T , w, s)3p + , (3)

where Bi is the ith coefficient in the virial expansion. Thevirial coefficients reflect the effective interaction betweenprotein molecules. For instance, B2 is directly related to thetwo-body potential of mean force by McMillan and Mayer(1945)

B2 = 12

dr d1 d2

{

exp

[w2(r,1,2; T , w, s)

kBT

] 1

}. (4)

This is precisely the same relation found between the secondvirial coefficient of a dilute gas and the intermolecular po-tential. The osmotic second virial coefficient is related to theexperimentally obtained value by B 2 = B2(NA/M2p) whereNA is Avogadros number.

The phase behaviour of fluids has parallels for solu-tions. Below its critical temperature, a fluid may exhibit avapourliquid phase transition. Within the McMillanMayerframework, this corresponds to a phase transition betweentwo liquid phases: one phase dilute in protein, and the otherphase rich in protein.

2. Protein phase diagrams and proteinproteininteractions

The protein phase diagram is fundamental in understand-ing and manipulating the behaviour of proteins. Many dis-eases, such as neurodegenerative diseases (Koo et al., 1999),cataracts (Pande et al., 2001), and sickle-cell disease (SanBiagio and Palma, 1991), are linked to protein aggrega-tion/phase separation. Furthermore, the efficiency of a signif-icant number of biotechnologically relevant operations (e.g.,protein salting-out, aqueous polymer-based two-phase sepa-rations, protein crystallisation and formulation, and proteinrenaturation) is highly dependent on the relative location onthe phase diagram.

The phase diagrams of protein solutions exhibit a broadregion of liquidsolid coexistence, within which, there isa metastable area of liquidliquid coexistence, one liquidphase dilute in protein and the other concentrated in protein(see Fig. 1). This has been observed for lysozyme (Taratutaet al., 1990; Ishimoto and Tanaka, 1977; Broide et al., 1996;Muschol and Rosenberger, 1997; Grigsby et al., 2001) for -crystallins (Schurtenberger et al., 1989; Broide et al., 1991;Berland et al., 1992) and for hemoglobin (San Biagio andPalma, 1991). The quantitative details of the phase diagram,such as the temperature dependence of the protein solubil-ity, is controlled by the precise nature of the proteinproteinpotential of mean force. This, in turn, depends on such prop-erties as the protein size, shape, charge, or on the solutionconditions such as the salt concentration or type, pH, orpolymer concentration.

The phase diagrams of different proteins at different solu-tion conditions are, in general, quite quantitatively different.However, Guo et al. (1999) discovered that, if the proteinphase diagram is recast in terms of B2 instead of tempera-ture, the phase diagrams of many disparate systems could becollapsed on a single universal phase diagram. Expressed

Fig. 1. Schematic protein phase diagram where the optimal crystallisationregion is found near the metastable liquidliquid equilibrium.

in these terms, the liquid branch of the liquidsolid equilib-rium becomes independent of the solvent; once B2 is known,the approximate position on the phase diagram is known.

One important application of this universal phase dia-gram is in protein crystallisation. Determining the optimalconditions under which to crystallise proteins is especiallyimportant. It has been found that there are only small regionson the phase diagrams where crystals can be obtained onreasonable time scales (George and Wilson, 1994). To ob-tain crystals, one must be located in a window on the phasediagram bound by 8104

physics (Baxter, 1968). This potential is taken in the limitof short range where only two parameters are needed tocharacterise the potential, the hard sphere diameter and astickiness parameter. In this work, the location of the crys-tallisation window on the phase diagram was determinedand found to be located around the metastable liquidliquidcoexistence curve. The presence of the liquidliquid criticalpoint might be very important for protein crystallisationbecause density fluctuations are enhanced in this regionlowering the free energy barrier to the formation of criti-cally sized nuclei (ten Wolde and Frenkel, 1997; Talanquerand Oxtoby, 1998). However, most of the conditions withinthe crystallisation window correspond to temperatures be-low that at the liquidliquid critical point indicating thatthe presence of the light branch of the liquidliquid binodalcurve is important for crystallisation (Haas and Drenth,1998; Vliegenthart and Lekkerkerker, 2000). The enhance-ment of nucleation just outside the light branch of thebinodal might occur due to the formation of nuclei fromsmall droplets of density corresponding to the dense branchof the binodal (Galkin and Vekilov, 2000). Below the crys-tallisation window, the liquidliquid binodal is wider andmost conditions will fall within the binodal where the for-mation of large droplets of high-density is favoured overthe formation of the crystalline nuclei.

Experimentally, it has been found that the value of B2at the critical temperature, Tc, depends on the solutionconditions (Rosenbaum et al., 1999). For instance thevalue of B2(Tc)/d3p (where dp is the effective protein di-ameter) for lysozyme in 0.75 M ammonium sulphate isabout 5 whereas that for lysozyme in 5% w/v sodiumchloride is 12. Because it is very difficult to crystalliselysozyme in sulphate solutions, whereas chloride solutionsyield lysozyme crystals, it was suggested that nucleationcan occur more easily when there is a larger distance be-tween the liquidsolid line and the liquidliquid criticalpoint. This effect cannot be captured with models basedon centrosymmetric potentials that give B2(Tc)/d3p 6,where this value is insensitive to the form of the potential(Vliegenthart and Lekkerkerker, 2000). Thus, the relativeposition of the crystallisation window with respect to theliquidliquid critical point is independent of the details ofmodel centrosymmetric potentials. Because the enhance-ment of nucleation rates has been attributed to the proximityof the liquidliquid binodal, it is unlikely that these modelscan be used to explain differences in crytallisability forsolutions that fall within the window.

The dependence of B2(Tc)/d3p on solution conditions canbe explained using anisotropic potentials (Haas et al., 1999;Sear, 1999; Curtis et al., 2001; Kern and Frenkel, 2003). Forinstance, protein phase diagrams have been modelled usingWertheims theory for the fluid phases, where the proteinmolecules are modelled as hard spheres with sticky square-well sites located on the surfaces. The value of B2(Tc)/d3p in-creases with the number of sticky sites included in the model.These results are in semi-quantitative agreement of simula-

tion results where it was found that B2(Tc)/d3p decreases asthe directionality of the model potential is increased. Thefinding that proteinprotein interactions are better describedusing anisotropic interactions is not surprising because it isthese interactions that are required to fix the orientation ofthe protein in the crystal. Other work has shown that modelsbased on anisotropic interactions provide better predictionsof the width of the liquidliquid binodal (Lomakin et al.,1999), the entropy of crystallisation (Curtis et al., 2001) andprotein crystal nucleation rates (Dixit and Zukoski, 2000).

The formation of protein crystals can also be interruptedby a gel transition at high protein concentrations (Muscholand Rosenberger, 1997). This region of the phase diagramshould be avoided because the formation of the gel arrestscrystallisation, much more so than the formation of thedense liquid phase. The formation of a gel occurs when pro-tein molecules form a space spanning network, with proteinmolecules trapped in cages formed by other proteins. Thesecages are much tighter for short-ranged potentials becauseprotein molecules want to maximise attractive interactionsin the gel. Consequently, as the range of the potential isshortened, the gel phase becomes more stable with respect tothe dense liquid phase and eventually the liquidliquid crit-ical point can be submerged behind the gel transition (Foffiet al., 2002).

2.1. DLVO theory: Proteins as colloids

Determination of B2 can give valuable information aboutthe protein phase diagram and protein phase separations.However, B2 is very difficult to measure, consequently it hasbeen studied for only a few different proteins and, in manycases, for only a limited range of solvent conditions. Further-more, in cases where only a small amount of protein is avail-able, it is difficult to measure B2. For this reason, it is usefulto understand the molecular origins of the proteinproteininteractions. In the next section, we cover measurements ofthe proteinprotein interactions and descriptions of these in-teractions using models that include excluded volume, elec-trostatics, dispersion forces, and solvation forces.

The first models (Vilker et al., 1981; Haynes et al., 1992)of proteinprotein interactions were based on DLVO theory(Verwey and Overbeek, 1948), which was originally devel-oped for modelling the behaviour of lyophobic colloids. Inthis approach, the proteins are considered to be hard, polar-izable spheres of diameter, dp, with a net charge, Q, uni-formly distributed over their surface. Water is treated as astructureless, continuum dielectric medium with dielectricconstant, . The advantage of using DLVO theory is thatthere is an analytical solution for the two-body potential ofmean force, w2, which is given by the sum of three terms, anexcluded volume interaction, wex, an electric double-layerforce, welec, and a Hamaker dispersion interaction, wdisp,

w2(r)= wex(r)+ wdisp(r)+ welec(r). (5)

Due to the assumptions in modelling the electric-doublelayer force, DLVO theory is applicable for solutions withvery low salt concentrations, although it is often applied tosolutions with greater salt concentrations.

The excluded volume interaction, wex, is simply given bya hard sphere potential, which is infinite for configurationswhere protein molecules overlap and vanishes otherwise.The diameter of the protein corresponds to the diameter of asphere that has approximately the same volume of the crys-talline protein, which can be determined from the crystalcoordinates. In addition, the potential includes a layer of im-penetrable solvent surrounding the protein of thickness, ,which controls the distance of closest approach for the pro-teins. Because the proteinprotein interactions are strongestfor the closest separations, B2 calculations are very sensi-tive to the choice of . has been approximated as about3 A, which corresponds roughly to the diameter of a watermolecule. This value has also been determined from pro-tein diffusion coefficient measurements using dynamic lightscattering (Eberstein et al., 1994).

The electric double-layer force, welec, arises from the elec-trostatic interactions between the colloids (e.g., proteins) andelectrolytes present in the solution. Within DLVO theory, thedescription of this electrostatic interaction is based on thePoissonBoltzmann equation where the salt ions are treatedas point charges. There are different forms of the double-layer potential, based on the various approximations used tosolve the PoissonBoltzmann equation. The most commonapproximation is based on low electrolyte concentration andlow surface potential expansion, and is given by Vilker et al.(1981)

welec(r)= Q2

r

e(rdp)

(1 + dp/2)2, (6)

where is the inverse Debye length which controls the rangeof attraction. For Eq. (6) to be valid, dp 1, this cor-responds to about an ionic strength of between 0.01 M and0.1 M. The net charge on the protein can be determined us-ing the intrinsic pKas of the individual amino acids via theHendersonHasselbach equation or, alternatively, using thehydronium ion titration data. The latter method is preferredbecause pKas are perturbed due to the surrounding proteinand the presence of salt. It has been assumed the charge isuniformly distributed over the surface of the protein. As thisis not necessarily the case with proteins, it is sometimes nec-essary to include corrections due to the asymmetric chargedistributions of the proteins. These include a chargedipoleinteraction and a dipoledipole interaction. These latter in-teractions are generally attractive (Coen et al., 1995; Vilkeret al., 1981; Haynes et al., 1992; Velev et al., 1998).

The Hamaker dispersion potential is given by

wdisp(r)=AH6

[2

s2+ 2

s2 4 + lns2 4s2

],

where s = 2r/dp, AH is the Hamaker constant and deter-mines the magnitude of the proteinprotein dispersion inter-action. Because all proteins have similar compositions theyalso have similar Hamaker constants. These values are onthe order of 3 kBT (Roth et al., 1996). This dispersion po-tential is based on a continuum approximation which breaksdown for very small separations where the potential becomesinfinitely attractive at contact. Due to this divergence, theamount of dispersion attraction included in the model is verysensitive to the choice of . As a first approximation, the dis-persion interaction is independent of ionic strength becausethe time scale for electronic fluctuations is much smallerthan the time needed for the reorganisation of the salt ions.

The pH and ionic strength trends of the proteinproteininteractions as predicted using DLVO theory are based onthe electric double-layer force. Proteinprotein interactionsbecome more attractive as the salt concentration is increaseddue to screening of the repulsive double-layer force, whichis also reduced when the net charge of the protein is de-creased by changing the pH toward the isoelectric pH of theprotein. These qualitative trends predicted by DLVO theoryhave been verified for small proteins such as lysozyme (seeFig. 2a, where lysozyme pI 11) (Eberstein et al., 1994;Velev et al., 1998; Muschol and Rosenberger, 1995; Tardieuet al., 1999), chymotrypsin (Coen et al., 1995; Haynes etal., 1992), chymotrypsinogen (Velev et al., 1998), ribonu-clease A (Tessier et al., 2002a; Boyer et al., 1996, 1999),-crystallins (Bonnete et al., 1997), BSA (Vilker et al.,1981; Tessier et al., 2002a), subtilisin (Pan and Glatz, 2003),ovalbumin (Curtis et al., 2002a), and myoglobin (Tessieret al., 2002a), and for larger proteins including apoferritin(Petsev et al., 2000), aspartyl transcarbamase (Budayovaet al., 1999), urate oxidase (Bonnete et al., 2001), and -crystallins (Finet and Tardieu, 2001). Under the solutionconditions where the electric double-layer force is small,the proteinprotein interactions are determined by the ex-cluded volume of the protein and the attractive Hamakerdispersion interaction. However, as discussed below, thereare some forces that are not accounted for by DLVOtheory, which are believed to be significant, especiallywhen long-range double-layer forces are screened and themolecular/chemical nature of the protein surface cannot beignored.

The applicability of DLVO theory has been further testedby comparing the model parameters (dp, AH , Q, and )fit to experimental data with physically realistic values ofthose parameters. For example, Eberstein et al. (1994) useddynamic light scattering to study aqueous sodium chloridesolutions with lysozyme in a pH 4.2, 0.10 M sodium ac-etate buffer. The concentration of sodium chloride in thesolutions was varied from 0.05 to 1.4 M and the experi-mental data were fit to a potential of mean force modelgiven by Eq. (5). A Hamaker constant of 7.7 kBT and a netcharge of 6.4e gave the best fits of the model to the ex-perimental data. Similar results were obtained by Muscholand Rosenberger (1995) who investigated fitting the data

0 0.1 0.2 0.3 0.4 0.5-10

0

10

20

30

40

0 2 4 6 8-20

-15

-10

-5

0

5

0 0.2 0.4 0.6 0.8 1-5

-4

-3

-2

-1

0

1

2

0 0.5 1 1.5 2 2.5 3-8

-6

-4

-2

0

2

B 2

(10

-4 m

L m

ol/g

2 )

Ionic Strength (M)

(a) (b)

(c) (d)

Fig. 2. Protein osmotic second virial coefficients plotted as a function of ionic strength in various aqueous electrolyte solutions. (a) Lysozyme in solutionsof sodium chloride at pH 4.5 (circles), pH 6 (squares), or pH 9 (triangles) (Velev et al., 1998). (b) Myoglobin at pH 9 (triangles) (Tessier et al., 2002a),BSA at pH 6.2 (circles) (Asanov et al., 1997), ovalbumin at pH 7 (squares) (Curtis et al., 2002a) in solutions of ammonium sulfate. (c) Solutions oflysozyme at pH 4.5 with potassium isothiocyanate (open circle) (Curtis et al., 2002a), with sodium isothiocyanate (closed circle), with sodium nitrate(squares), with sodium chloride (diamonds), or with sodium acetate (triangles) (Bonnete et al., 1999). (d) Lysozyme in solutions of magnesium bromideat pH 7.8 (circles) (Tessier et al., 2002b) and ovalbumin in solutions of magnesium chloride at pH 7 (squares) (Curtis et al., 2002a).

using different expressions for the electric double-layer po-tential. They found that the fit parameters were very sensi-tive to the pmf model indicating that it is difficult to assignphysical meaning to the fit parameters, Q and AH . Never-theless, other studies have obtained reasonable fits of ex-perimental B2 data to DLVO theory for solutions of oval-bumin (AH = 3 kBT , Curtis et al., 2002a), of BSA (AH =3 kBT , Moon et al., 2000), and of subtilisin (AH =5.1 kBT ,Pan and Glatz, 2003).

Velev et al. (1998) compared lysozymelysozyme inter-actions to those for chymotrypsinogen. In agreement withprevious work, the lysozymelysozyme interactions couldbe accurately modelled using the pmf model given by Eq.(5). However, at a pH greater than 5.3, chymotrypsinogeninteractions are more attractive at low salt concentration thanat high salt concentration indicating the presence of attrac-tive electrostatic interactions. This effect was attributed tothe large dipole moment of chymotrypsinogen. The extraelectrostatic attraction was included in the model by usingdipoledipole and dipolecharge potentials. These latter in-teractions were also used to describe the pH dependence ofthe proteinprotein interactions in aqueous electrolyte so-lutions of chymotrypsin (Coen et al., 1995; Haynes et al.,1992) and of BSA (Vilker et al., 1981).

One of the major approximations in DLVO theory isthe use of the PoissonBoltzmann equation to describe

the electrostatic interactions in the colloidal system. ThePoissonBoltzmann theory treats the ions in solution aspoint charges, which interact with each other and with theprotein (colloid) only through a mean electrostatic potential.The theory does not account for effects such as correlationsbetween ions, non-electrostatic interactions between ions(e.g., excluded volume interactions, dispersion interactions,etc.), and image charge interactions between the electrolytesand the colloidal particles.

Correlations between ions can lead to behaviour thatis qualitatively different from that predicted by thePoissonBoltzmann equation. One important example isthe electrostatic contribution to the potential of meanforce between two similarly charged surfaces. Bell andLevine (1958) demonstrated that models based on thePoissonBoltzmann equation always lead to repulsive in-teractions between similarly charged surfaces. However, byincluding ionion correlations within the hypernetted chainintegral equation, Patey (1980) and later Kjellander andMarcelja (1984), demonstrated the theoretical possibilityof an attraction with sufficiently high surface charges onthe colloidal particles. Computer simulation studies on therestricted primitive model (Torrie and Valleau, 1980, 1982;Guldbrand et al., 1984; Valleau et al., 1991) confirmed thepresence of an attractive interaction. Computer simulationshave indicated that the potential of mean force between

two similarly charged hard spheres immersed in an elec-trolyte solution can be attractive (Gronbech-Jensen et al.,1998; Wu et al., 1999). More surprisingly, the potential ofmean force between two oppositely charged sphere can berepulsive (Wu et al., 2000). This effect has significant im-plications to the stability and phase behaviour of colloidalsolutions and cannot be captured by the PoissonBoltzmannequation.

2.2. Specific ion effects: The Hofmeister series

One of the deficiencies of the PoissonBoltzmann equa-tion is that ions are treated as point charges and consequentlylose their identity. However, many types of biological inter-actions depend on the specific ion type. The effect of the ionis usually related to its position in an empirical series, basedon some measurable effect. One such series is the Hofmeisterseries (Hofmeister, 1888), which was originally developedas a ranking of the salting-out effectiveness of various ionsfor globular proteins. The effectiveness is much strongerfor anions, where this series is given by in decreasing orderSO24 >HPO

24 >CH3COO

>Cl>Br> I>SCNand that for cations is given by Li+>Na+ K+>NH+4>Mg2+. Another series that characterises specific ion ef-fects is the lyotropic series (Israelachvili, 1992), whereions are ranked according to their interactions with water.High lyoptropic series ions interact with water stronglyleading to large hydration numbers. These ions are termedkosmotropes because the water molecules surrounding theions are structured relative to bulk water. Alternatively, lowlyotropic series ions are termed chaotropes because wateris less structured around these ions. The lyotropic seriesis correlated with measurements such as the JonesDoleviscosity B coefficients or molal surface tension incre-ments, both of which reflect the affinity of the ion for water(Collins and Washabaugh, 1985).

In general, the Hofmeister series is similar to the lyotropicseries. However, there are exceptions; for example, multi-valent cations are kosmotropes (high lyotropic series ions),but have low salting-out effectiveness and are consequentlylow on the Hofmeister series.

As salt concentration is raised within the salting-outregion, the protein solubility decreases as a result of salt-induced proteinprotein attractions. In this region, the solu-bility also depends on the type of salt according to itspositionin the Hofmeister series. Two distinctive trends have beenobserved. Generally, if the pH of the solution is above the pIof the protein, traditional salting-out behaviour is observedwhere the effectiveness of the ion at reducing the proteinsolubility increases with the ions position in the ascendingHofmeister series. One explanation for this phenomena isthat the salt ions sequester water molecules and preventthem from forming favourable hydrogen bonds with theprotein surface (Collins, 2004). Consequently, proteins pre-fer to form intermolecular interactions between themselves

instead of with water resulting in an effective proteinproteinattraction. This type of behaviour is illustrated in Fig. 2b,where B2 is plotted versus ionic strength for ovalbumin (Cur-tis et al., 2002a), myoglobin (Tessier et al., 2002a), or BSA(Tessier et al., 2002a; Asanov et al., 1997) in aqueous solu-tions of ammonium sulphate, a popular salting-out agent.

If the pI of the protein is above the pH of the solution, theeffectiveness of the anion at inducing proteinprotein at-traction follows the reverse lyotropic series (Pan and Glatz,2003; Rieskautt and Ducruix, 1989; Finet et al., 2004).This reverse lyotropic series effect has been investigatedthoroughly, for solutions containing lysozyme (Curtis etal., 2002a; Bonnete et al., 1999; Rieskautt and Ducruix,1989) (see Fig. 2c), where it was found that the chaotropicanion, SCN, is an excellent crystallisation agent. On theother hand solutions containing the kosmotropic anion,SO24 , rarely yielded lysozyme crystals. Differences inlysozymelysozyme interactions as a function of salt typeare observed at low salt concentration ( 0.05 M). This isin contrast to traditional salting-out behaviour where theeffect of the salt type on protein solubility is not observeduntil much greater salt concentrations ( 1.0 M) (Curtiset al., 2002a). The reverse lyotropic series effect has beenattributed to the formation of an insoluble lysozymeSCNcomplex due to strong binding interactions between thechaotropic anions and the positively charged arginine groupson the lysozyme surface (Rieskautt and Ducruix, 1989).

In some cases, proteins are salted-in by concentratedsalt solutions. For aqueous sodium chloride solutions con-taining apoferritin, as salt concentration is initially raised,the proteinprotein interactions become more attractive dueto screening the electric double-layer repulsion (Petsev et al.,2000). However, the proteinprotein attraction is at a maxi-mum for a salt concentration of 0.15 M. Above this value, theproteinprotein interactions become much more repulsive.Similar effects have been observed from proteinproteininteraction measurements for solutions containing eitherlysozyme or ovalbumin dissolved in aqueous solutions con-taining magnesium salts (see Fig. 2d) (Curtis et al., 2002a;Tessier et al., 2002b; Grigsby et al., 2000) and for solutionsof malate dehydrogenase dissolved in various salt solutions(Costenaro et al., 2002). Also, similar trends have been ob-served in solutions of divalent cations with solubility studiesof lysozyme (Broide et al., 1996; Grigsby et al., 2001; Be-nas et al., 2002), of BSA or of -lactoglobulin (Arakawa etal., 1990). In these studies, the protein solubility has a mini-mum at intermediate salt concentrations. These effects havebeen attributed to repulsive hydration forces which occurwhen heavily hydrated cations bind to negatively chargedsurfaces (Israelachvili, 1992). The origin of these forcesis still not clear although it is expected that they originatefrom the energy required to disrupt the hydrogen bondingnetwork around charged or polar surfaces. The strength andthe range of the hydration force was found to scale withthe hydration number of the bound cation which correlateswith the position of the cation in the lyotropic series. It is

interesting to note that the binding of well-hydrated ionsto the protein surface leads to repulsive hydration forces,whereas the binding of chaotropic anions leads to attractiveforces.

For a few large oligomeric proteins, proteinprotein at-traction is not observed over the entire range of concen-trated salt solutions. This observation is attributed to thenon-compact structure of these proteins, resulting in a re-duced proteinprotein dispersion interaction because the di-electric constant of the protein interior is similar to that ofwater (Budayova et al., 1999; Finet and Tardieu, 2001).

Although many different phenomena have been observedto follow the Hofmeister series, the physical mechanism thatleads to the series is still poorly understood (Kunz et al.,2004). This is partly due to the sheer diversity of the phe-nomena. Recently, it has been proposed that the specificion effects on proteinprotein interactions may be recon-ciled by first understanding the effect of salt on the sur-face tension of water (Piazza and Pierno, 2000). Salts havepositive surface tension increments due to the desorption ofions from the airwater interface; ions experience repulsiveimage charge interactions at the airwater interface due tothe relatively low dielectric constant of air. The effect ofsalt on the surface tension of water was first captured byOnsager and Samaras (1934) by incorporating the imagecharge interactions into the PoissonBoltzmann equation.In general, however, the actual surface tension increment isstrongly dependent on the type of salt and not just its va-lency, an effect that cannot be explained simply by imagecharges (which only distinguish ions by their valency). Al-ternatively, specific ion effects can be captured in models byincluding dispersion interactions between ions and betweenions and interfaces (Bostrm et al., 2001; Tavares et al.,2004). Specific ion effects also become more pronouncedat high salt concentrations, where electrostatic interactionsbecome highly screened and dispersion interactions becomedominant.

Bostrm and coworkers have used dispersion interactionsto model the influence of salt type on the net charge oflysozyme. The dispersion interactions between the proteinand the ions become more attractive with increasing ion po-larisability leading to preferential adsorption of the morepolarisable ions (Bostrm et al., 2003). Note that anionsare generally more polarisable than cations, and the po-larizability of an ion generally increases with its positionin the descending order of the lyotropic series. This pref-erential interaction of the protein with various ions canlead to either an enhancement or a displacement of H+ions from the surface of the protein. The fractional dis-sociation of acid and base groups on the surface of theprotein, and thus the net charge of the protein, is deter-mined by the surface pH. Incorporating these effects into thePoissonBoltzmann equation, Bostrm et al. were able toexplain the observed variation of the surface charge of egg-white lysozyme with salt concentration, as well as with salttype.

2.3. Anisotropic interactions

Under most conditions relevant to bioseparations, or inphysiological salt solutions, the electric double-layer poten-tial will be effectively screened and the proteinprotein in-teractions will be generally short-ranged. In this case, thedetails of the proteinprotein interactions will depend on thechemical nature of the protein surface and on the proteinshape and surface roughness. Most likely, the interactionswill be anisotropic due to the protein surface heterogeneity;it is these interactions that are important in fixing the orien-tation of a protein molecule in the crystal.

Recent investigations have focused on the effect of surfaceanisotropy on the proteinprotein interactions using morerealistic models for proteins where the shape of the proteinis determined from the crystal structure coordinates (Nealet al., 1999; Asthagiri et al., 1999; Elcock and McCammon,2001). The key finding was that the net proteinprotein in-teractions are determined from a few highly attractive orien-tations marked by a high degree of surface complementaritymuch like the types of interactions involved in molecularrecognition events between proteins (Neal et al., 1999). Theprimary contribution to the energy of the highly attractiveconfigurations comes from the dispersion force; electrostaticinteractions contribute to the energy of these interactions,but do not determine the different orientations that contributeto B2. It was postulated that attractive electrostatic inter-actions occur due to the charge asymmetry that occurs inthe dominant configurations. The importance of anisotropicinteractions were confirmed by showing that the substitu-tion of a single amino acid located in a crystal contact canlead to large changes in values of B2 for bacteriophage T4lysozyme (Chang et al., 2000). Here, the orientations of theproteins in the crystals have been identified as the attractiveconfigurations sampled by the protein in the solution.

The all-atomistic models of proteinprotein interactionsare very sensitive to the surface properties of proteins. Inthe model used by Elcock and McCammon, the value ofB2 is very sensitive to the protonation states of the chargedresidues located on the complementary surfaces; these pro-tonation states could change upon complexation. This sen-sitivity of the model to the protein surface properties is ex-pected because slight alterations in protein surface charge(Chang et al., 2000) or surface hydrophobicity (Curtis et al.,2002b; Ho and Middelberg, 2003) can lead to large changesin the values of B2. These changes cannot be accounted forby using pmf models based on centrosymmetric potentials,where the large changes in values of B2 would be accountedfor by unrealistic changes to the fit parameters such as AH ,Q, or .

2.4. Solvation forces

Because of the similarities between weak proteinproteininteractions and the specific proteinprotein interactions

that stabilise biologically relevant complexes, recent workhas focused on using methods similar to those used inprotein docking algorithms for calculation of the weakproteinprotein interactions (Elcock and McCammon,2001). One of the interactions that is not included in DLVOtheory, but is a major driving force in protein complex for-mation, is the hydrophobic interaction (Horton and Lewis,1992; Jones and Thornton, 1996; Janin and Chothia, 1990).The hydrophobic interaction is a type of solvation force.The structure of water is perturbed near the protein surfacewhere water molecules cannot hydrogen bond freely withother water molecules. A solvation force occurs when thelayers of solvent perturbation overlap (Israelachvili, 1992).This force could be either repulsive or attractive dependingon the chemical nature of the interacting surfaces. If the in-teracting surfaces contain non-polar groups, then the forceis referred to as hydrophobic and is generally attractive. Onthe other hand, the term, hydration forces, is used to referto the solvation interaction between surfaces containing po-lar or charged groups. These forces are generally repulsivebecause energy is required to dehydrate polar or chargedgroups.

Thermodynamically, the interaction between a solute andthe solvent is given by the solvation free energy, which isthe work to transfer a solute from an ideal gas phase intothe solvent. A solutesolute solvation force is related to thechange in the solvation free energy of a solute due to theproximity of a second solute. If the solutesolvent interac-tion is sufficiently short-ranged, the solvation force can beapproximated by the area that has been removed from thesolvent due to the solutesolute interaction multiplied bythe solvation free energy of the solute. This is known as thesolvent accessible surface area (SASA) approach, which iscommonly used in protein folding and protein docking al-gorithms (Horton and Lewis, 1992; Chothia, 1974; Eisen-berg and McLachlan, 1986). Within the SASA method, thesolvation free energy, Gsolv, is approximated by

Gsolv =i

iAi . (7)

The sum is over all different atomic groups, i, Ai is the areaof group i that is exposed to the solvent, and i is the atomicsolvation parameter of group i. The atomic solvation param-eters are determined by fitting model compound solubilitydata, where it is assumed that the free energy to bury a sur-face group is similar to the free energy of transferring thatgroup from a non-polar phase to water. The applicability ofEq. (7) is based on the additivity approximation which statesthat the solvation free energy of the entire protein is given bythe sum of the solvation free energies of the surface groups(Eisenberg and McLachlan, 1986; Hermann, 1972). This ap-proximation was based on the results of solubility studiesof alkane molecules where it was found that the solvationfree energies scaled with the alkane chain length. However,because the interaction between polar groups and the sur-rounding water molecules is longer-ranged then one solvent

layer, the additivity approximation might not necessarily bevalid (Wang and BenNaim, 1997; Chalikian et al., 1994).There is also evidence that the interaction between surfaceswith small hydrophobic patches is different from that withlarge hydrophobic patches. The crossover between the twotypes of interactions occurs at a length scale of approxi-mately 1 nm (Lum et al., 1999).

The SASA approach has been included in models forcalculating weak proteinprotein interactions (Elcock andMcCammon, 2001; Curtis et al., 2002b; Ho and Middel-berg, 2003). In the pmf model of Elcock and McCammon(Elcock and McCammon, 2001), both the short-range dis-persion force and the solvation force were modelled using aSASA potential with a surface-averaged solvation parame-ter. This surface free energy parameter was fit to the B2 datafor solutions of lysozyme in sodium chloride. Interestingly,the fit parameter was similar to the surface free energy ob-tained from transferring apolar compounds from nonpolarsolvents into water. An alternative approach to model solva-tion forces was used by Asthagiri et al. (1999), where a hy-brid Hamaker/LennardJones model was used to calculatethe short-range attractive interactions. Here, the Hamakerdispersion potential was used to describe the proteinproteininteraction for large separations. For close separations, aLennardJones potential is used to calculate the interac-tions between groups on opposite protein molecules witha surface-to-surface separation of less than 3 A. The cut-off corresponds to excluding one solvent molecule from be-tween the closely separated surfaces.

3. Proteinsolvent interactions

As discussed above, the interface between the protein andthe solvent plays an important role in the stability of theprotein. If the interaction between the solvent and proteinis favourable, the protein will prefer to remain in solutionsurrounded by the solvent. However, if the interaction be-tween the protein and the solvent is unfavourable, the pro-tein molecules will prefer to form proteinprotein contactsresulting in an effective proteinprotein attraction.

The solvation free energy of a compound can be decom-posed into the sum of two contributions

Gsolv =Gcav +Gint, (8)

where Gcav is the work to create a cavity in the solvent, andGint is the work of turning on the interactions between thesolute and the solvent molecules.

The work required to create very large cavities (large com-pared to molecular length scales) can be readily estimatedfrom the bulk properties of the solvent. For cavity sizes muchless than 100 nm, this work is given by

Gcav A, (9)

where is the surface tension of the solvent, and A is thesurface area of the cavity. This form leads one to considerthat the solvation of the protein molecules can, in principle,be related directly to the surface area of the protein andthe macroscopic surface tension of the solvent. From this, itwould follow that the effect of salt on Gcav is related to thesurface tension increment of the salt.

However, the use of Eq. (9) is not well justified when thesize of the cavity becomes of the order of the correlationlengths present in the solvent. The work of cavity forma-tion is then related to the probability that there will sponta-neously be no solvent molecules in a volume of the shape ofthe cavity. According to scaled-particle theory (Reiss et al.,1959; Pierotti, 1965; Stillinger, 1973) or information theory(Hummer et al., 1996, 1998), this probability is related tothe volume of the cavity, not the exposed surface area ofthe cavity as given in Eq. (9). Lum and coworkers (1999)were able to develop a theory that successfully interpolatesbetween the small cavity expressions given by informationtheory and the limiting large cavity expression given by themacroscopic surface tension. Thus, the work to form a mi-croscopic cavity, Gcav, can be interpreted as a curvature-dependent surface tension,

Gcav = RA, (10)where R is a factor to correct for the curvature of the inter-face. Note that Eq. (10) must be applied with care, as the ef-fect of salt on the microscopic surface free energy, R, is notnecessarily related to its effect on the bulk surface tension,, as the molecular origins of the microscopic and macro-scopic surface tensions are different. However, assuming thetwo effects are related has proven to be a good starting pointin interpreting the properties of proteins.

3.1. Solubility studies of model compounds

Most knowledge of proteinsalt interactions has been de-termined from solubility/partitioning studies of model com-pounds where the effects of salt on the different atomicgroups of the protein can be clearly delineated (Baldwin,1996). The effectiveness of an ion at salting-out alkanes iscorrelated with the molal surface tension increment of thesalt, and thus, the effectiveness follows the lyotropic se-ries. In addition to salting-out non-polar groups on a pro-tein, ions also interact favourably with the peptide groups.The effectiveness of ion salting-in increases with the posi-tion of the ion in the descending order of the Hofmeisterseries (Robinson and Jencks, 1965; Nandi and Robinson,1972; Schrier and Schrier, 1967). Salting-in is primarily dueto favourable electrostatic interaction between the ion andthe large dipole moment of a peptide bond; however, thecause of ion specificity is not clear (Baldwin, 1996). Somebelieve that the iondipole interaction is specific (Nandiand Robinson, 1972), whereas others believe the iondipoleinteraction is non-specific, and the specificity arises from

unfavourable interactions of the ions with nearby non-polargroups (Hamabata and von Hippel, 1973). The latter expla-nation can be used to understand the salting-in interactionof chaotropic anions. However, divalent cations also salt-inthe peptide group, although it is expected that these ionswill interact unfavourably with non-polar groups due to thehigh surface tension increment of these ions. The salting-ininteraction of divalent cations has been attributed to the hy-drogen bonding interactions of the heavily hydrated cationwith the peptide dipole (Collins, 2004). As a consequenceof this interaction, divalent cations are high on the lyotropicseries, but low on the Hofmeister series.

In addition, there are specific interactions between the saltions and the charged groups of peptides/proteins. These in-teractions are stronger between similar ions (i.e., where bothions are either kosmotropes or both are chaotropes) then be-tween dissimilar ions (Collins, 1997). Thus, the binding abil-ity of monovalent anions to protein charges should followthe reverse Hofmeister series because the positively chargedguanidinium and imizadolium groups are chaotropic. How-ever, this electrostatic interaction is stronger if the ion is di-valent versus monovalent, as with the case of SO24 (Nandiand Robinson, 1972). The opposite trend is observed for theinteraction of cations with the protein charged groups. Thepositively charged carboxylate ions form very strong inter-actions with kosmotropic cations, such as Na+ or the diva-lent cations Mg2+ or Ca2+. The strong interactions betweenthe divalent cations and the charged groups of proteins playvery important biological roles.

3.2. Preferential interaction parameter measurements

The solvent environment directly surrounding a proteinis different than that in the bulk solution, due to pref-erential interactions between the protein surface and thevarious solvent components. The difference of the salt com-position in the region next to the protein surface and ofthat in the bulk solution is experimentally determinable interms of the proteinsalt preferential interaction parameter,(ms/mp)T,w,s , where mi is the molality of compo-nent i (Timasheff, 1998; Timasheff and Arakawa, 1988).Favourable proteinsalt interactions are characterised by apositive value of (ms/mp)T,w,s . For this case, there isan excess of salt near the surface of the protein, relativeto the bulk solution. If (ms/mp)T,w,s < 0, the solventlayer around the protein is depleted of salt. The perturbationof the protein chemical potential by the addition of salt isrelated to the preferential interaction parameter by(

Gsolvms

)T ,P,mp

=(

msmp

)T ,w,s

(sms

)T ,P,mp

, (11)

where P is pressure. Accordingly, the protein chemical po-tential is determined from the proteinsalt interactions. Ifthe proteinsalt interactions are favourable, the addition ofsalt will decrease the solvation free energy of the protein.

Whereas, if the salt is preferentially excluded from the do-main of the protein, addition of salt will raise the proteinsolvation free energy.

The consequences of Eq. (11) are similar to the predictionsof the Gibbs adsorption isotherm. According to the Gibbsadsorption isotherm, the change in the macroscopic surfacetension of an airsalt/water interface is given by the amountof adsorption or desorption of the ions at the interface. Saltshave positive surface tension increments because they areexcluded from the interface due to the formation of imagecharges. Arakawa and Timasheff (1982,1984a) (Arakawa etal., 1990) have shown that the preferential exclusion of ionsfrom the proteinsolvent interface can also be predicted fromthe salt molal surface tension increment. They determinedthe curvature correction factor, R, by equating measuredproteinsalt preferential interaction parameters to the valuespredicted using Eqs. (10) and (11)

(msmp

)calcT ,w,s

=RNAA(/ms)T,P,mp(s/ms)T,P,mp

. (12)

For solutions containing strongly excluded salts (such asNa2SO4), the curvature correction factor ranges from R=0.5to 0.7. The obtained values of R are in semi-qualitative agree-ment with the values predicted using the theory of Lum andcoworkers (1999). Eq. (12) is based on the assumption thatthe surface free energy of a microscopic cavity is related tothe macroscopic surface tension. The result that the exclu-sion of the salt from a microscopic protein surface can bepredicted from the macroscopic surface tension incrementis intriguing and needs further studies. Factors that mightaccount for this effect are the influence of image charges onthe macroscopic and microscopic surface tensions of aque-ous salt solutions.

Eq. (12) provides an upper bound on the preferential ex-clusion of the salt. The difference between the calculatedvalue of the proteinsalt preferential interaction parameterand the measured value is attributed to preferential bindingof the salt to the protein. The amount of compensation orbinding is found to follow the reverse lyotropic series; themonovalent anion SCN has a large preferential bindingto proteins, whereas Cl is weakly excluded (Arakawa andTimasheff, 1982) from proteins. Strong proteinsalt prefer-ential binding is also observed with solutions of divalentcations such as Mg2+ or Ca2+, even though salts of theseions tend to have high surface tension increments (Arakawaand Timasheff, 1984a; Arakawa et al., 1990). These resultsare in agreement with the solubility data of model com-pounds where the preferential interactions are usually at-tributed to the interaction of the salt ion and the peptide bond.Another finding is that the preferential interactions are ad-ditive. For example, generally, MgSO4 is preferentially ex-cluded from protein surfaces, whereas MgCl2 is only weaklyexcluded. The Mg2+ ion can overcome the weak exclusionof the Cl ions, however, it cannot overcome the strong pref-erential exclusion of the kosmotropic SO24 anion. While the

preferential exclusion does not depend on the pH or the saltconcentration, the preferential binding interactions do. Thepreferential binding of MgCl2 to proteins increases with in-creasing salt concentration and increasing pH indicating thatthe charge density of the protein affects the binding. Thisbehaviour has also been observed with solutions of ArgHCland lysozyme or BSA (Kita et al., 1994).

3.3. Protein stabilisation

Measurements of preferential interaction parameters haveprovided valuable insights into various reactions includingprotein stabilisation, protein solubility, self-assembly of sub-unit systems, binding of ligands to proteins, and proteincomplex formation (Timasheff, 1998). These investigationsare based on the Wyman linkage relation which relates theequilibrium constant for a reaction to the preferential inter-actions between the solvent and the reacting system and be-tween the solvent and the product (Wyman, 1964; Tanford,1969; Aune et al., 1971). When applied to protein stabilisa-tion by salt, the Wyman linkage relation is given by(

lnKuf

s

)T ,P,mp

=(

msmp

)dT ,w,s

(

msmp

)nT ,w,s

,

(13)

where Kuf is the equilibrium constant for the protein unfold-ing reaction, and superscripts d and n denote the denatured(unfolded) and native (folded) protein. Protein stabilisationis determined by the difference of the preferential interac-tions of the salt with the native state and with the denaturedstate. Stabilisation is favoured if the preferential binding ofthe salt to the native state is greater or the preferential ex-clusion is lesser than that of the unfolded protein.

The proteinsalt interactions can only be measured withrespect to one of the states; generally the interactions aremeasured with the native state and then inferred for the de-natured state (Timasheff, 1998). Preferential exclusion ofthe salt due to the surface tension increment effect is a non-specific interaction between the salt and the protein surface;if this effect dominates the proteinsalt interaction, the saltstabilises the protein because the unfolded state presents alarger surface than the folded state, thus more salt is ex-cluded from the unfolded state. However, in cases wherepreferential exclusion is counteracted by preferential bind-ing of the salt to the protein surface, the solvation interac-tions of the denatured protein cannot be inferred from thoseof the native protein. This is because proteinsalt bindinginteractions depend on the chemical nature of the proteinsurface which is different for the unfolded and the foldedprotein. For these cases, both protein stabilisation or proteindestabilisation are possible. For instance, most guanidiniumsalts destabilise proteins due to extensive binding betweenthe guanidinium ions and the peptide groups exposed uponprotein unfolding. For this reason, salts such as GuaHCland GuaSCN are used as denaturants. However, GuaSO4 is

a protein stabiliser because the exclusion of the SO24 ionovercomes the binding interactions of the guanidinium ion(Arakawa and Timasheff, 1984b).

3.4. Protein salting-out

In the same manner that salts prevent proteins from expos-ing surface to the solvent, they can encourage burying thesurfaces in the formation of proteinprotein contacts like theones that occur in protein precipitates. For this reason, strongsalting-out agents are good stabilisers of protein structure.Note that this correlation is not as strong for weakly inter-acting salts, because in this case the chemical nature of theprotein surface that is buried is different from that surfacewhich is exposed during unfolding.

Melander and Horvath (1977) were the first to link proteinsalting-out to the molal surface tension increment. In theirwork, the precipitated phase was treated as a pure phase andproteinprotein interactions were neglected, in which casethe protein solubility is determined from the equilibriumcondition

cp = lp = igp +Gsolv + RT ln Sp, (14)where superscripts c, l, and ig are used to denote the solidphase, the liquid phase and the standard state ideal gas, re-spectively, and Sp is protein solubility. According to thismodel, the solubility of the protein is determined from thevariation of the protein solvation free energy with respect tosalt concentration. At low salt concentration, salting-in oc-curs because there are favourable interactions between thecharged protein and its electric double layer. As salt con-centration is increased further, the solubility goes through amaximum, after which the solubility decreases because thesalt raises the solvation free energy of the protein accordingto its molal surface tension increment.

Experimentally, it has been found that protein crystalscontain a considerable amount of solvent. Consequently,the protein solubility is determined by the difference of theproteinsalt interactions in the fluid phase and those in thesolid phase according to

d ln Spdms

= 1RT

[(Gsolvms

)lT ,P,mp

(

pms

)cT ,P,mp

].

(15)

Arakawa et al. (1990) determined (p/ms)cT ,P,mp

from experimental measurements of protein solubilityand of (ms/mp)lT ,w,s . They found that variations in

(p/ms)cT ,P,mp

parallelled changes in (Gsolv/ms)lT ,P,mpindicating that the proteinsalt interactions in the liquidphase were representative of those interactions in the solidphase. The absolute values of the proteinsalt interactionsin the precipitated phase were smaller because there is lesssurface exposed to the solvent in this phase, consequentlysalting-out always occurs if salt is preferentially excluded.

However, salts such as MgCl2 can have a salting-in ef-fect because preferential binding to the protein will favoursolubilization.

3.5. Linking proteinprotein and proteinsalt interactions

Protein stabilisation and protein solubility can be un-derstood in terms of proteinsalt interactions. However,these interactions are not currently included in pmf mod-els. Timasheff and Arakawa (1988) proposed an expressionfor the proteinprotein association based on the Wymanlinkage relation for a monomerdimer equilibrium wherethe difference in the preferential interactions of the saltwith the dimer and with the monomers determines theeffect of salt on the proteinprotein association constant.Because a protein dimer has less solvent exposed surfacethan that of two monomers, the absolute magnitude of(ms/mp)T,w,s will be larger for the monomers than forthe dimer. Consequently, preferential exclusion of the saltfrom the protein surface will favour protein association.On the other hand, a preferential proteinsalt binding willresult in an effective proteinprotein repulsion, where theprotein prefers to be surrounded by the solvent rather thanmaking a proteinprotein contact.

Using the Wyman linkage relation for protein associationprovides a method for linking preferential interaction param-eters to proteinprotein interactions, but is only useful whenthe end-states are well delineated (e.g., a monomerdimerequilibrium). A more appropriate approach to linking theseinteractions might be by using the solvent accessible surfacearea potential where the proteinsalt preferential interactionparameters can be linked to the atomic solvation parame-ters using Eqs. (7) and (11). This approach was followedby Curtis et al. (2002a) who fit a surface-averaged atomicsolvation parameter to B2 data as a function of salt concen-tration for lysozyme in solutions of (NH4)2SO4 or NaCl,and for ovalbumin in solutions of KSCN or (NH4)2SO4.In all cases, semi-quantitative agreement was obtained be-tween the salt increment of the fit atomic solvation parameterand the molal surface tension increment of the salt indicat-ing this method could provide a first approximation at con-necting experimentally measured proteinsalt interactions toproteinprotein interactions.

It is interesting to compare the proteinprotein inter-actions of ovalbumin and of lysozyme in solutions ofKSCN. Very little attraction is observed between ovalbuminmolecules in concentrated solutions of KSCN, whereas thelysozymelysosyme interactions are very attractive in solu-tions dilute in KSCN. The difference in these behaviours isrelated to the binding affinity of SCN to protein surfaces(Curtis et al., 2002a). If binding of the ion to the proteinsurface is weak, the potential of mean force is determinedby the work to remove the solvent. When the ion is SCN,the force is only weakly attractive due to the balance in thepreferential exclusion of the salt with the anion preferential

binding. However, if the ion binding is strong, the forceis determined from the potential of mean force betweenthe proteinanion complexes. When the bound anion isSCN, the forces between the complexed proteins are moreattractive than those between the uncomplexed proteins.Here, the positively charged residues are identified as thestrong ion-binding sites and the peptide groups as the weakion-binding sites.

For protein solutions containing divalent cations, prefer-ential binding of cations to proteins occurs over broad rangesof pH and salt concentration. This preferential binding isassociated with repulsive hydration forces; these forces arerepulsive because the binding of strongly hydrated cationsto proteins leads to increased protein hydration.

Two general trends can be deduced from the resultsof preferential interaction parameter measurements. If themeasured preferential interaction parameter is only dueto the preferential exclusion of the salt, the effect of theion is to dehydrate the protein surface resulting in aneffective proteinprotein attraction, this effect has beencorrelated with the surface tension increment of the salt.However, if the protein surface chemistry is altered by ionbinding, the salting-out behaviour follows the reverse ly-otropic series. In this case, the binding of heavily hydratedcations to the protein surface is associated with repulsiveproteinprotein interactions, whereas the interactions be-tween proteinchaotropic anion complexes are attractive.

4. Conclusions

Recent studies of proteinprotein interactions have beenfocused at determining the origin of the crystallisation win-dow. Analysis of protein phase diagrams and those gener-ated using model potentials have shown that the this windowsurrounds a metastable liquidliquid binodal, near which thecrystallisation rate is enhanced due to large density fluctua-tions. However, crystallisation is not guaranteed for locationsin the window. One possible reason is that the relative posi-tion of the liquidliquid binodal to that of the window is not aconstant, but depends on the form of the proteinprotein po-tential. This effect has not been accounted for by using cen-trosymmetric potentials. However, it has been shown that therelative position of the binodal in the crystallisation windowdepends on the anisotropy of the proteinprotein potential.The next step in developing a more accurate crystallisationdiagnostic is to link the model anisotropic potentials to theactual proteinprotein interactions. However, currently, themolecular origins of proteinprotein interactions are poorlyunderstood.

One of the major challenges in understanding proteinprotein interactions is to explain the specific ion effect. Theproteinprotein interactions are determined by the balancebetween the preferential exclusion and the preferential bind-ing of the salt to the protein. Both these proteinsalt inter-actions are dominated by Hofmeister effects. For instance,

the strength of the anion interactions with the peptide dipolefollows the reverse Hofmeister series, while the interactionbetween the charged groups on the protein surface and thecounter-ions depends on the nature of each ion in the inter-acting pair. The effect of ion binding on the proteinproteininteractions also depends on the nature of the ion. The bind-ing of high lyotropic series cations to the protein surface islinked to repulsive hydration forces whereas the binding ofchaotropic anions to the protein surface is associated withattractive proteinprotein interactions.

By including ionprotein and ionion dispersion poten-tials in their model, Bostrm and coworkers have been ableto rationalize various specific ion effects, such as the depen-dence of the protein surface charge on salt type. However,a substantial part of the influence of salt on proteinproteininteractions is related to the interaction of the ion with wa-ter. For instance, the salting-out effect of kosmotropic saltshas been attributed to the ability of the ions to prevent wa-ter from forming favourable hydrogen bonds with the po-lar groups on the protein surface and, more generally, tothe structure-making ability of the ion. In these cases, itis likely that water cannot be treated as a continuum, butits discrete nature needs to be taken into account. Becausethe salting-out of proteins is often correlated with the mo-lal surface tension increment of the salt, understanding themolecular origin of surface tension might give insight intothe origins of proteinprotein interactions and their relationto the Hofmeister series. A recent issue of Current Opinionin Colloid and Interface Science (2004, vol. 9) was devotedto uncovering Hofmeister effects in a whole range of exper-iments. Once these interactions are determined, it should bepossible to include these effects in more realistic models ofthe proteinprotein interactions.

References

Arakawa, T., Timasheff, S.N., 1982. Preferential interactions of proteinswith salts in concentrated-solutions. Biochemistry 21 (25), 6545.

Arakawa, T., Timasheff, S.N., 1984a. Mechanism of protein salting in andsalting out by divalent-cation saltsbalance between hydration andsalt binding. Biochemistry 23 (25), 5912.

Arakawa, T., Timasheff, S.N., 1984b. Protein stabilization anddestabilization by guanidinium salts. Biochemistry 23 (25), 5924.

Arakawa, T., Bhat, R., Timasheff, S.N., 1990. Preferential interactionsdetermine protein solubility in 3-component solutionsthe MgCl2system. Biochemistry 29 (7), 1914.

Asakura, S., Oosawa, F., 1954. Interaction between two bodies immersedin a solution of macromolecules. Journal of Chemical Physics 22, 1255.

Asanov, A.N., DeLucas, L.J., Oldham, P.B., Wilson, W.W., 1997.Interfacial aggregation of bovine serum albumin related tocrystallization conditions studied by total internal reflectionfluorescence. Journal of Colloid and Interface Science 196 (1), 6273.

Asherie, N., Lomakin, A., Benedek, G.B., 1996. Phase diagram of colloidalsolutions. Physical Review Letters 77 (23), 4832.

Asthagiri, D., Neal, B.L., Lenhoff, A.M., 1999. Calculation of short-rangeinteractions between proteins. Biophysical Chemistry 78 (3), 219.

Aune, K.C., Goldsmith, L.C., Timasheff, S.N., 1971. Dimerization ofalpha-chymotrypsin. 2. Ionic strength and temperature dependence.Biochemistry 10 (9), 1617.

Baldwin, R.L., 1996. How Hofmeister ion interactions affect proteinstability. Biophysical Journal 71 (4), 2056.

Baxter, R.J., 1968. Percus-Yevick equation for hard spheres with surfaceadhesion. Journal of Chemical Physics 49 (6), 2770.

Bell, G.M., Levine, S.M., 1958. Statistical thermodynamics ofconcentrated colloidal solutions. 2. General theory for the double layerforces on a colloidal particle. Transactions of the Faraday Society 54,785.

Benas, P., Legrand, L., Ries-Kautt, M., 2002. Strong and specific effects ofcations on lysozyme chloride solubility. Acta Crystallographica SectionD-Biological Crystallography 58, 1582.

Berland, C.R., Thurston, G.M., Kondo, M., Broide, M.L., Pande, J., Ogun,O., Benedek, G.B., 1992. Solidliquid phase boundaries of lens proteinsolutions. Proceedings of the National Academy of Sciences of USA89 (4), 1214.

Bonnete, F., Malfois, M., Finet, S., Tardieu, A., Lafont, S., Veesler, S.,1997. Different tools to study interaction potentials in gamma-crystallinsolutions: relevance to crystal growth. Acta Crystallographica SectionD-Biological Crystallography 53, 438.

Bonnete, F., Finet, S., Tardieu, A., 1999. Second virial coefficient:variations with lysozyme crystallization conditions. Journal of CrystalGrowth 196 (24), 403.

Bonnete, F., Vivares, D., Robert, C., Colloch, N., 2001. Interactions insolution and crystallization of aspergillus flavus urate oxidase. Journalof Crystal Growth 232 (14), 330.

Bostrm, M., Williams, D.R.M., Ninham, B.W., 2001. Surface tensionof electrolytes: specific ion effects explained by dispersion forces.Langmuir 17 (15), 4475.

Bostrm, M., Williams, D.R.M., Ninham, B.W., 2003. Specific ion effects:the role of co-ions in biology. Europhysics Letters 63 (4), 610.

Boyer, M., Roy, M.O., Jullien, M., 1996. Dynamic light scattering studyof precrystallizing ribonuclease solutions. Journal of Crystal Growth167 (12), 212.

Boyer, M., Roy, M.O., Jullien, M., Bonnete, F., Tardieu, A., 1999. Proteininteractions in concentrated ribonuclease solutions. Journal of CrystalGrowth 196 (24), 185.

Broide, M.L., Berland, C.R., Pande, J., Ogun, O.O., Benedek, G.B., 1991.Binary-liquid phase separation of lens protein solutions. Proceedingsof the National Academy of Sciences of the United States of America88 (13), 5660.

Broide, M.L., Tominc, T.M., Saxowsky, M.D., 1996. Using phasetransitions to investigate the effect of salts on protein interactions.Physical Review E 53 (6B), 6325.

Budayova, M., Bonnete, F., Tardieu, A., Vachette, P., 1999. Interactionsin solution of a large oligomeric protein. Journal of Crystal Growth196 (24), 210.

Chalikian, T.V., Sarvazyan, A.P., Breslauer, K.J., 1994. Hydration andpartial compressibility of biological compounds. Biophysical Chemistry51 (23), 89.

Chang, R.C., Asthagiri, D., Lenhoff, A.M., 2000. Measured and calculatedeffects of mutations in bacteriophage T4 lysozyme on interactions insolution. ProteinsStructure Function and Genetics 41 (1), 123.

Chi, E.Y., Krishnan, S., Randolph, T.W., Carpenter, J.F., 2003. Physicalstability of proteins in aqueous solution: mechanism and driving forcesin non-native protein aggregation. Pharmaceutical Research 20 (9),1325.

Chothia, C., 1974. Hydrophobic bonding and accessible surface-area inproteins. Nature 248 (5446), 338.

Coen, C.J., Blanch, H.W., Prausnitz, J.M., 1995. Salting-out of aqueousproteins-phase-equilibria and intermolecular potentials. A.I.Ch.E.Journal 41 (4), 996.

Collins, K.D., 1997. Charge density-dependent strength of hydration andbiological structure. Biophysical Journal 72 (1), 65.

Collins, K.D., 2004. Ions from the Hofmeister series and osmolytes: effectson proteins in solution and in the crystallization process. Methods 34,300.

Collins, K.D., Washabaugh, M.W., 1985. The Hofmeister effect and thebehavior of water at interfaces. Quarterly Reviews of Biophysics 18(4), 323.

Costenaro, L., Zaccai, G., Ebel, C., 2002. Link between proteinsolventand weak proteinprotein interactions gives insight into halophilicadaptation. Biochemistry 41 (44), 13245.

Curtis, R.A., Blanch, H.W., Prausnitz, J.M., 2001. Calculation of phasediagrams for aqueous protein solutions. Journal of Physical ChemistryB 105 (12), 2445.

Curtis, R.A., Ulrich, J., Montaser, A., Prausnitz, J.M.,Blanch, H.W., 2002a. Proteinprotein interactions in concentratedelectrolyte solutionsHofmeister-series effects. Biotechnology andBioengineering 79 (4), 367.

Curtis, R.A., Steinbrecher, C., Heinemann, A., Blanch, H.W., Prausnitz,J.M., 2002b. Hydrophobic forces between protein molecules in aqueoussolutions of concentrated electrolyte. Biophysical Chemistry 98 (3),249.

Daanoun, A., Tejero, C.F., Baus, M., 1994. Van-der-Waals theory forsolids. Physical Review E 50 (4), 2913.

Dixit, N.M., Zukoski, C.F., 2000. Crystal nucleation rates for particlesexperiencing short-range attractions. Applications to proteins. Journalof Colloid and Interface Science 228 (2), 359.

Eberstein, W., Georgalis, Y., Saenger, W., 1994. Molecular-interactionsin crystallizing lysozyme solutions studied by photon-correlationspectroscopy. Journal of Crystal Growth 143 (12), 71.

Eisenberg, D., McLachlan, A.D., 1986. Solvation energy in protein foldingand binding. Nature 319 (6050), 199.

Elcock, A.H., McCammon, J.A., 2001. Calculation of weakproteinprotein interactions: the pH dependence of the second virialcoefficient. Biophysical Journal 80 (2), 613.

Finet, S., Tardieu, A., 2001. Alpha-crystallin interaction forces studiedby small angle X-ray scattering and numerical simulations. Journal ofCrystal Growth 232 (14), 40.

Finet, S., Skouri-Panet, F., Casselyn, M., Bonnete, F., Tardieu, A., 2004.The Hofmeister effect as seen by SAXSs in protein solutions. CurrentOpinion in Colloid and Interface Science 9 (12), 112.

Foffi, G., McCullagh, G.D., Lawlor, A., Zaccarelli, E., Dawson, K.A.,Sciortino, F., Tartaglia, P., Pini, D., Stell, G., 2002. Phase equilibriaand glass transition in colloidal systems with short-ranged attractiveinteractions: application to protein crystallization. Physical Review E65 (31), 031407/1.

Galkin, O., Vekilov, P.G., 2000. Control of protein crystal nucleationaround the metastable liquidliquid phase boundary. Proceedings ofthe National Academy of Sciences of the United States of America 97(12), 6277.

Gast, A.P., Hall, C.K., Russel, W.B., 1983. Polymer-induced phaseseparations in non-aqueous colloidal suspensions. Journal of Colloidand Interface Science 96 (1), 251.

George, A., Wilson, W.W., 1994. Predicting protein crystallization from adilute-solution property. Acta Crystallographica Section D-BiologicalCrystallography 50, 361.

Grigsby, J.J., Blanch, H.W., Prausnitz, J.M., 2000. Diffusivities oflysozyme in aqueous MgCl2 solutions from dynamic light-scatteringdata: effect of protein and salt concentrations. Journal of PhysicalChemistry B 104 (15), 36453650.

Grigsby, J.J., Blanch, H.W., Prausnitz, J.M., 2001. Cloud-pointtemperatures for lysozyme in electrolyte solutions: effect of salt type,salt concentration and pH. Biophysical Chemistry 91 (3), 231.

Gronbech-Jensen, N., Beardmore, K.M., Pincus, P., 1998. Interactionsbetween charged spheres in divalent counterion solution. Physica A261, 74.

Guldbrand, L., Jnsson, B., Wennerstrm, H., 1984. Electrical double-layer forcesa Monte-Carlo study. Journal of Chemical Physics 80,2221.

Guo, B., Kao, S., McDonald, H., Asanov, A., Combs, L.L., Wilson, W.W.,1999. Correlation of second virial coefficients and solubilities usefulin protein crystal growth. Journal of Crystal Growth 196 (24), 424.

Haas, C., Drenth, J., 1998. The proteinwater phase diagram and thegrowth of protein crystals from aqueous solution. Journal of PhysicalChemistry B 102 (21), 4226.

Haas, C., Drenth, J., Wilson, W.W., 1999. Relation between the solubilityof proteins in aqueous solutions and the second virial coefficient ofthe solution. Journal of Physical Chemistry B 103 (14), 2808.

Hagen, M.H.J., Frenkel, D., 1994. Determination of phase diagrams forthe hard-core attractive Yukawa system. Journal of Chemical Physics101 (5), 4093.

Hamabata, A., von Hippel, P.H., 1973. Model studies on effects ofneutral salts on conformational stability of biological macromolecules.2. Effects of vicinal hydrophobic groups on specificity of binding ofions to amide groups. Biochemistry 12 (7), 1264.

Haynes, C.A., Carson, J., Blanch, H.W., Prausnitz, J.M., 1991.Electrostatic potentials and protein partitioning in aqueous 2-phasesystems. A.I.Ch.E. Journal 37 (9), 1401.

Haynes, C.A., Tamura, K., Korfer, H.R., Blanch, H.W., Prausnitz,J.M., 1992. Thermodynamic properties of aqueous alpha-chymotrypsinsolutions from membrane osmometry measurements. Journal ofPhysical Chemistry 96 (2), 905.

Haynes, C.A., Benitez, F.J., Blanch, H.W., Prausnitz, J.M., 1993.Application of integral-equation theory to aqueous 2-phase partitioningsystems. A.I.Ch.E. Journal 39 (9), 1539.

Hermann, R.B., 1972. Theory of hydrophobic bonding. 2. Correlationof hydrocarbon solubility in water with solvent cavity surface-area.Journal of Physical Chemistry 76 (19), 2754.

Ho, J.G.S., Middelberg, A.P.J., 2003. The influence of molecular variationon protein interactions. Biotechnology and Bioengineering 84 (5), 611.

Ho, J.G.S., Middelberg, A.P.J., Ramage, P., Kocher, H.P., 2003. Thelikelihood of aggregation during protein renaturation can be assessedusing the second virial coefficient. Protein Science 12 (4), 708.

Hofmeister, F., 1888. Zur lehre von der wirkung der salze. Archiv forExperimentelle Pathologie und Pharmakologie 24, 247.

Horton, N., Lewis, M., 1992. Calculation of the free-energy of associationfor protein complexes. Protein Science 1 (1), 169.

Hummer, G., Garde, S., Garca, A.E., Pohorille, A., Pratt, L.R., 1996. Aninformation theory model of hydrophobic interactions. Proceedings ofthe National Academy of Science USA 93, 8951.

Hummer, G., Garde, S., Garca, A.E., Paulaitis, M.E., Pratt, L.R.,1998. Hydrophobic effects on a molecular scale. Journal of PhysicalChemistry B 102 (51), 10469.

Ilett, S.M., Orrock, A., Poon, W.C.K., Pusey, P.N., 1995. Phase behaviorof a model colloidpolymer mixture. Physical Review E 51 (2), 1344.

Ishimoto, C., Tanaka, T., 1977. Critical behavior of a binary mixture ofprotein and salt water. Physical Review Letters 39 (8), 474.

Israelachvili, J., 1992. Intermolecular and Surface Forces. Academic Press,London.

Janin, J., Chothia, C., 1990. The structure of proteinprotein recognitionsites. Journal of Biological Chemistry 265 (27), 16027.

Jones, S., Thornton, J.M., 1996. Principles of proteinprotein interactions.Proceedings of the National Academy of Sciences of the United Statesof America 93 (1), 13.

Kern, N., Frenkel, D., 2003. Fluidfluid coexistence in colloidal systemswith short-ranged strongly directional attraction. Journal of ChemicalPhysics 118 (21), 9882.

Kita, Y., Arakawa, T., Lin, T.-Y., Timasheff, S.N., 1994. Contribution ofthe surface free energy perturbation to proteinsolvent interactions.Biochemistry 33 (50), 15178.

Kjellander, R., Marcelja, S., 1984. Correlation and image charge effectsin electric double layers. Chemical Physics Letters 112, 49.

Koo, E.H., Lansbury, P.T., Kelly, J.W., 1999. Amyloid diseases: abnormalprotein aggregation in neurodegeneration. Proceedings of the NationalAcademy of Sciences of the United States of America 96 (18), 9989.

Kunz, W., Lo Nostro, P., Ninham, B.W., 2004. The present state of affairswith Hofmeister effects. Current Opinion in Colloid and InterfaceScience 9 (12), 1.

Lee, J.C., Gekko, K., Timasheff, S.N., 1979. Measurements of preferentialsolvent interactions by densimetric techniques. Methods in Enzymology61 (Part H), 26.

Lekkerkerker, H.N.W., Poon, W.C.K., Pusey, P.N., Stroobants, A., Warren,P.B., 1992. Phase behavior of colloid + polymer mixtures. EurophysicsLetters 20 (6), 559.

Lomakin, A., Asherie, N., Benedek, G.B., 1999. Aeolotopic interactionsof globular proteins. Proceedings of the National Academy of Sciencesof USA 96 (17), 9465.

Lomba, E., Almarza, N.G., 1994. Role of the interaction range in theshaping of phase-diagrams in simple fluidsthe hard-sphere Yukawafluid as a case-study. Journal of Chemical Physics 100 (11), 83672.

Lum, K., Chandler, D., Weeks, J.D., 1999. Hydrophobicity at small andlarge length scales. Journal of Physical Chemistry B 103 (22), 4570.

McMillan, W.G., Mayer, J.E., 1945. The statistical thermodynamics ofmulticomponent systems. Journal of Chemical Physics 13, 276.

Melander, W., Horvath, C., 1977. Salt effects on hydrophobic interactionsin precipitation and chromatography of proteins: an interpretation ofthe lyotropic series. Archives of Biochemistry and Biophysics 183 (1),200.

Moon, Y.U., Curtis, R.A., Anderson, C.O., Blanch, H.W., Prausnitz,J.M., 2000. Proteinprotein interaction