LB~-43273

Molecular Thermodynamics for Salt-Induced

Protein Precipitation

I). E. Kuehner, H. W. Blanch and J. M. Prausnitz

Department of Chemical Engineering

University of California

and

Chemical Sciences Division

Lawrence Berkeley National Laboratory

University of California

Berkeley, CA 94720, U.S.A.

May 1999

This work was supported by the Director, Office of Energy Research, Office of Basic Energy

Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract Number

DE-AC03-76SFOO098.

ABSTRACT

A mokcukr-thermodynamic model is developid for salt-induced protein precipitation the

model considers an aquleous solution of globular protein molecules as a s:ystem of

interacting hard spheres in a continuum pseudo-solvent (water and salt ions). l%e protein

molecules interact in a manner described by a set of sphericaUy-symmetric two-body

potentials of mean force. These include screened CMombic repulsion, dispersion (van der

Wads) attraction, and ion-excluded-vohnne osmotic attraction. Specific chemical

interactions between proteins (e.g., hydrophobic interactions and self-association) am

modekd with a short-range, orientationdependent attractive square-weU potentkd. An

analytical equation of state is derived using the Random Phase Approximation with the

hard-sphere fluid as the reference system and a perturbation based on the protein-protein

overall potential of mean force, while the short-range specific chemical intemction is

incorporated through the S/WI’ perturbation theory of associating fluids. This equation of

state, similar in form to the well-known van der Waals equation of state, provides an

expression for the chemieal potential of the protein and determines liquid-liquid phase

equilibria. Critical and spinodal criteria for salt-induced protein precipitation are derived,

and the dependence of critical points and protein volubility on model parameters is

examined. Salting-out phase-equilibrium data are described for two globular proteins, hen-

egg-white lysozyme and bovine cz-chymotrypsin, in solutions of ammoniurh wdf~ at

25”C. For both proteins, the SAFT parameters M, &,P,and V are correlated from the

Iysozyme precipitation data of Coen, et al. (1995). Cikulated Cohn-Edsall slopes relating

the protein solubllity to the solution ionic strength are greater than those measured in

precipitation experiments with lysozyme and chymotrypsin, indicating that ionic--strength-

dependent parameters are required for quantitative application of this modei to these

proteins.

2

DISCLAIMER

This report was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.

DISCLAIMER

Portions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.

1. INTRODUCTION

Predpitation has long been used as an early step in the process of pufifjing proteins from

complex solutions {Green,1931;Green,1932). ‘I%&technique has been applied to the Kzovery

of proteins such as insulh, diagnostic and industrial enzymes, human growth hormone,

and interferon (McGregor,1983). Phase sepamtion in aqueous protein solutions is achieved

through addition of precipitating agents, such as inorganic salts at high concentrations,

nonionic polymers, poIyeketro@tes, and organic solve~ @ster et al., 1973; &iiiE et d., WW,

Shih et al., 1992 Rothstein, 1994). Salting-out of prokins provides one of the simplest

precipitation techniques and is used extensively in the laboratory and in the biotechnology

and pharmaceutical industries.

ln most previous studies, sait-induceclprecipitation of proteins and other ecdloids has been

understood as phase separation resuiting in a solid precipitated phase in equilibrium with a

liquid phse containing protein at its saturation concentration (Rosenbmn et aZ.,M@ Tmres

aadSan&er,HW7). TraditionaUy, quantitative charaeterizadon has been expressed through the

protein solubllity, defined as the protein concentration in the equilibrium liquid phase. For

example, the chssieal Cohn and E&all approach (E&W 1%3)correlates protein volubility,

S, with solution ionic strength, 1, in a simple exponential mannec logS = #l-all, where/3

and A are empirieaIIydetermined parameters specific to a given protein and salt. Recent

experimental results (Shih et al., l-, Coen, 19%) suggest that sak-induced protein

precipitation may also be appropriately described as a phase separation resulting in two

fluid phases: a supernatant fluid phase lean in protei~ in equilibrium with a pnxipitate

phase rich in protein and containing appreciable amounts of water and salt. Aeeording to

this view, the degree of precipitation is charackxized by the distribution coefficien~ K.,

defined as the ratio of the protein concentration in the prdpitate phase to that in the

supernatant phase. Various experimental equilibrium salting-out studies have shown that

protein partitioning can depend strongly on protein si% initkd concentration, and surface

chemistry; on eketrolyte identity and ionic stren@, and on solution pH (Mwa and

Timasheff,198% Arakawaand TkashefF, 1984). Speeific interactions (e.g., dirnerization or

h~gher oligomeriation, or hydrophobic interactions) may also play a major role in

governing protein-protein interactions in precipitating soIutions.

To establish a rational basis for designing a protein-pivcipitation process, it is useful to

develop a model to provide a theoretical i%ameworkfor interpretation and corre~ation of

3

protein-precipitation data. Melander and Horvath showed that the fi.mctional form of the

Cohn-Edsall equation maybe interpreted on the basis of solvophobic (MeIanderail HoNiith,

1977). Recent theoretical studies have been &xted at developing more fundamental

models to amount, on a mokxxdar IeveI, for diverse interactions between the various

components in the protein solution. For example, lWibadevanand Hall presented a model,

based on Barker-Henderson perturbation theory, for protein precipitation by a :nonionic

polymer (MahadevanandHall, 1990;MahadevanandHall, 1992). Vlachy, Blanch and ~%msnitz

describe a model for polymer-induced liquid-liquid phase separation in soIutions ofeolloids

and globular proteins, based on the random-phase approximation (Vlaehy et d., 1993).

However, these recent theoretical studies are eoncemed with aqueous solutions where the

electrolyte concentration is less than 0.1 molar. Experimental studies clearly show that

protein precipitation by salts typically requires electrolyte concentration in the range 1-10

molar.

This work presents a molecular-thermodynamic model for protein precipitation by

inorganic salts. Particular attention is given to highly concentrated salt solutions. The

procedure employed here represents the ternary solution (protein, electrolyte, and water) as

a one-component system containing globular protein molecules immersed in a continuous

pseudo-solvent. The pseudo--solvent is an aqueous salt solution. The effect of pseudo-

solvent chemistry (i.e., pH, salt type and ionic strength) on protein-protein intemctions is

taken into account through the strong infhience that it exerts on the foIlowing protein-

protein interactions: Cou[ombic (charge-charge) repulsion, dispersion (van der Waals)

attraction, ion-excluded-volume (or osmotic) attraction and interactions between exposed

hydrophobic groups on the surfaces of two or more protein molecules. Despite its

simplicity, the one-component representation has been successful in explaining some

experimental properties of colloidal dispersions (Grimsm, 1983) and globular-protein

soh.itions (VIaehyet al., 1993; Chiew et d., 1995; Kuehneres aL, 19%; Tavares and Sandier, lW,

Curtis et d., 1998). A powerful advantage of this representation is that final results 2ue given

by analytical equations based on statistical-mechanical theories.

Two-body potentials of mean force are discussed in Section 2. Section 3 presents a

derivation of the moIecular-thermodynamic equation-of-state model for protein solutions,

based on the random-phase approximation (RPA) and the statistical associating fluid theory

(SAFT’). ResuIts of model calculations and comparkn with experimental pnxipitation

data are presented in Section 4,

4

2. PROTEIN-PROTEIN POTENTIAL OF MEAN FORCE

In the one-component model, aqueous solutioti of globular proteins are represented by a

mormdkperse assembly of spherical non-interpenetrating spheres of diameter CP which

interact through effective solvent-dependent potentials of mean force (PMFs). The overall

potential of mean force, W(r), between protein molecules is given by the sum of five

potentials of mean force

W(r) = WA$(r)+ Wqq(r)+ Wdi#j + W...(r) + w,P(r,@ (1)

where r is the center-to-center distance between proteins. Here, Wh$r) is the hard-sphere

potential, WJr) is the repulsive screened electricdouble-layer potential of mean force,

WdiJr) is the attractive dispersion potential of mean force, and WoJr-) is the attractive

potential due to ion-excluded-volume effects. These four terms are spherically symmetric

potentials of mean force which depend on the center-~nter distance betwtxm proteins,

on the properties of the protein molecules, and on the solvent chemistry (e.g., pH and ionic

strength). WJr, co)represents attractive specific interactions between exposed groups on

the surfaces of the proteins these may include hydrophobic interactions and shoti-range

attractive electrostatic interactions. This potential depends not only on r but also on relative

molecular orientation, indicated by co.

2.1 Hard-Sphere and Coulornbic Potentials

The hard-sphere and repulsive screened Coulombic potentials of mean force between

proteins, normalized with respect to the thermal energy kg, are represented by:

Wh,(r) = cu r< ap

=0 r> op

Wqq(r) (zPe)2 exp[+c(r - crP)]

‘=- r-(l+KOP 12)2kBT r> q

(2a)

(2b)

(2C)

5

where k~denotes Boltzma.nntsconstant, and T is the absolute tempeti, zPeis the charge

on the protein, where ZPis the protein valence and e is the elementary unit char,ge; 4=s0

represents the d~electric permhtivity of vacuum in units of C/Vin and c, is the ioNc-

strengthdependent dielectric constant of the salt-water continuum pseudo-solven~ K is the

Debye screening parameter, given by & = (2&ZVJ)/(q+#J’); NA is Avogadro’s numbe~

and ionic strength 1 = 0.5 z cmz~, where c. and ~ are the molar concentration and

valence of ion m, respectively.

The value of WqJr) depends on the ionic strength of the solution, as shown in Hgure 1,

where W#) is plotted as a function of the center-to-center distance between protein

molecules, r, normalized by the protein diameter, OP. In Figure 1, Op= 50~ and .ZP= 10.

As ionic strength increases, the electric double layer surrounding the protein molecule

compresses and the effective charge-charge interaction between proteins is screened. Also,

the protein valence, ZP,depends on protonation states of the various constituent amino acids

which have hydrogen-ion equilibria, hence the protein will have an effective net charge

which depends on the solution plil. Each protein has an isoelectric poin~ pI$ where the net

charge is zero; hence, at the pI, Wqq(r)= O. The net protein charge is a model parameter

obtained from independent experimental data.

The Coulombic potentkd of mean force accounts for long-range repulsive charge-charge

interactions between protein molecules present in dilute concentrations. The derivation of

this potential assumes that the charge of the protein is evenly distributed over the surface of

the molecule. In reality, protein surface charges, both positive and negative, are discrete

and are located at sites possessing hydrogen-ion equilibria. In the dilute-protein limi~

where the average separation between protein molecules is large (up to severai protein

diameters), the approximation of evenly-distributed surface charge is acmptable.

However, for concentrated-protein solutions (where precipitation or crystallization may

occur), the protein molecules am much more closely packed in solution, and shofi-range

attractive electrostatic intemctions between oppositely charged moieties on the protein

surface become more important, and this assumption is no longer valid. These specific

interactions are accounted for in WJr, co), as discussed below.

6

I

Furthermore, WJr) is derived within the Debye-Hiickel framework using a linearized

form of the Poisson-Bokzrnann equation for the electrostatic potential around the sphericai

charged protein molecukq this assumption is valid only in solutions where the ionic

strength is 0.1 M or Iess. Application of Wqq(r)to solutions with higher ionic strengths is

an extrapolation. However, for ionic strengths greater than approximately 0.5 M, this

potential is screened so strongly that it has oniy a negligible contribution to the total

potential of mean force. Higher-order electrostatic interactions, i.e., between charges and

permanent dipdes, are not considered here, since they are genemlly much weaker in

magnitude than Wqq(r)and are strongly screened by salt.

2.2 Dispersion Potential

Spontaneous electronic fluctuations within polyatomic molecules or surfaces give rise to

induced dipoles within the body, which interact with the induced dipoles in a neighboring

body. These van-der-Waals-type forces cause protein molecules to experience a net

attractive potential of mean force, also electrostatic in nature. For large bodies such as

protein molecules, the magnitude of this attractive force is significant. Following the

geometric derivation of Derjaguin, the unretarded attractivedispersion interaction WdiJr) is

given by the following expression @maker, 1937’):

(3)

where His the Munaker constant of the protein. This form of the dispersion potential has

no explicit dependence on ionic strength or pH. The intrinsic frequency of the elecbxmic

fluctuations width give rise to the dispersion force is much greater than the time constant

for rearrangement of ions in the double layer and in the solution between the protein

molecules. Therefore, dielectric screening of the dispersion attmction is assumed to be

neg@ibIe (1.melachvihandAdams, 1978), and any effect of ionic strength is contained within

the effective Hamaker constant, which is an adjustable modei parameter. Furthermore,

since His primarily a fimction of the electron density of the protein mokcules @aelaclwiIi,

1992),it is expected that most proteins have similar values of H since most have similar

electron densities (Nk, 1976), As discussed below, the effective Harnaker cunstant of hen-

egg-white lysozyme can be obtained from light-scattering m~urements. Combination of

7

I

Wgq(r)and WdiJr) form the basis of the I)ergajuin-Landau-Verwey-Overbeek (DLVO)

th~ry of colloidal stability (VerweyaudOverbeek,1!348).

2.3 Ion-Excluded-Volume Potential

h concentrated ekct.rolytesohitions, ions occupy a significant fraction of the total solution

volume. When two protein molecules approach conta@ ions are excluded from a region

between the protein particles; the geometry of this mechanism is shown in Figure 2. The

resulting imbalance in the local osmotic pressure exerted by the ions on the proteins gives

rise to a short-range attmctive force. The area of the plane of ion exclusion can be

calculated in terms of the diameter of the protein, OP,the average diameter of a salt ion,

GiOm,and the center-to-center separation, r. Multiplying this area by the osmotic pressure of

the corresponding protein-free salt solution, fla~~,yields the

respect to r, the ion-excluded volume potential, WO&), is

19*, AsakumandOosaw&1958X.

= o

where OP.= (~ + qon)/2 and the composition-weighted

force. When integmted with

given by (Asakuraand00sawiL

r< (OP+ q)

r2 (Op-1-Op) (4)

average ion diameter q.OX=

(~c.,~~ + ~..cQv, where v.a and Vanare the stoichiornetric coefficients of the cation and

anion, respectively, and v = v==,+ v=.. The osmotic pressure of the protein-free salt

solution is obtained from the literature in terms of the osm@.iccoefficient @O=~,which is a

measure of the non-idealify of the salt solution, by 1(0= = @O~m~~= @O,~@,kJ), where

the total ion number density, p=, is given in terms of the molar salt concentration, C$,by

P.= c~~v. WoJr) is a short-range attractive potential of mean force which becomes

significant at high ionic strength, as shown in Figure 3. In Figure 3, protein diameter OP=

30~. The salt is ammonium sulfate: hydra@d ionic diameters are UC.= 6.62A and O=.=

8

7.58A (Horvath,1985), and values of the osmotic coefficien~ @O~m,for ammonium sulfate

solutions were taken from the Iitemture (~egg et A, N?%)(see Appendix B). In the

derivation of equation (4), salt ions are considered uncharged hard spheres; electrostatic

effects are assumed to be taken into account in @Omand W,,(r).

Figure 4 shows the spherically-symmetric potential of mean force, tVV&), composed of

~JO + ~~i,P(O+ ~O,m(~),m a function of rfaP for several values of ammonium-sulfate

ionic strength. In Figure 4, UP= 30& ZP= 15 and H/kBT= 5, and the salt is ammonium

sulfate. At low ionic strength a repulsive barrier exists due to WJr). As ionic strength

increases, W&j diminishes due to dielectric screening, and the magnitude of the ion-

exch.tded-vohune potential WO$m(r)increases. For ionic strengths greater than 1.0 M,

Ww(r) is uniformly attractive with a steep attractive well near surface contac~ at extremely

high ionic strengths, tlds represents a strong driving force for protein aggregation and

precipitation.

The effect of solution pfl on Ww(r) is solely through the pH dependence of the protein

vaIence, ZP,in the scre,enedCoulombic repulsion potential of mean force. Hence, any pH

dependence of Win(r) diminishes as ionic strength increases. In dilute protein solutions at

low ionic strength, WV(r) describes well the pH dependence of the osmotic second virial

coefficient (Curtiset d., 1998)and of the average apparent difision coefficient of proteins in

aqueous solutions (Kuekeret d., 1997). However, protein-precipitation data in concentrated

salt solutions often show a significant dependence on solution pH (Shibef al., 199%Coen,

w!%) beyond that captured in Ww(r). As discussed below, the specific interaction

potential, W#, CO),contains parameters which are pH dependent and are obtained from

experimental data.

2.4 Specific Interactions

Specific interactions between proteins are known to occur in solutions where crystallization

or precipitation phase transitions occur. These additional forces include hydrophobic

interactions and short-ranged electrostatic interactions between surface groups, which

depend on the rekitive orientation of the two protein molecules. The hydrophobic

interaction between between exposed non-polar amino-acid residues on the surfaces of the

protein molecules is, in general, attractive, short-ranged, and orientationdependent.

9

Hydrophobic “bonds” are formed when hydrophobic groups on separate protein molecules

come into contact and may cause association or agjyegation of proteins. Similarly,

attractive charge-charge couplings between oppositely-charged groups on the surfaces of

two protein molecules, as well as salt bridges in solutions with divalent ions, can form in

concentrated protein solutions. Although short-rang~ these can affect protein volubility.

In this work, specific interactions are represented by a potential model used for associating

fluids (Jackson et aL, 1988). The shapes of these specdlc-interaction sites are idealized as

“circularpatehes” or “sphexica.1caps” located on the surface of the protein molecules. As

indicated in Figure 5, interaction potential WJr, @ between site A on the surface of protein

1 and site B on the surface of protein 2 is defined as an orientationdependent squw-well

potentiak

Wp(?-,co) &w—=. —

kBT kBTif r<rcand

=0 otherwise

Hence, two “patches” on two different paiticles

/)1 < (jJ,c~(j (+ s 02,=

((5)

are considered to form a “bond” if the

centers of the two particles are within a critical distance, rC, and if the two patches am

within the constraints of relative orientation, defined by the critical angles 61,Cand f?z,c. In

general, these quantities are not known. Fortunately, the thermodynamic properties of

systems with this type of interaction can be described in terms of quantities that are known

or can be estimated from physicochernical data for proteins, as described below.

3. LIQUID-STATE THEORY AND EQUATION OF STATE

Having established and defined pertinent potentials of mean force, it is necessary to

construct a molecular-thermodynamic model which relates these potentials to macroscopic

thwmmdynamicproperties. In this work, that model is based on perturbation theory. The

center-to-center spherically symmetrical Coulornkic, dispersion and ion-excluded-volume

interactions are incorpmatied into the model in the context of the mndom-phase

approximation (RPA), described in Section 3.1. The contribution of the orientation-

dependent specific interaction potential is included through a first-order perturbation theory

of associating ffuids, the statistical associating fluid theory (SM1.J, described in !Rxtion

10

3.2. The RPA-SAFW equation of state, which combines both these contributions, is

presented in Seetion 3.3.

3.1 Random-Phase Approximation

The Random-PhaseApproximation(WA) has been used previously to model the str!Klure

factorof colloid solutions (GrimsoIL 1983)and to describe liquid-liquid phase separation of

proteins due to addition of polymers (MahademnandHalL199QMihadevanandHall, 1992;Vlachy

et al., 1993). In the RPA, an assembly of hard spheres is used as the reference system,

while the spherkxdly symmetric interactions are treated as perturbations. Thermodynamic

properties of the system are obtained from liquid-state integral-equation theory. The basis

of this framework is the Ornstein-Zernike (02) equation describing two-body interactions

between particles in.a monodkperse, isotropic fluid (ChnsteinandZernike,1914):

In this

h(~l~ = ~~12)+ &(r13)h(r23)&23 (6)

formal representation, the interaction betsveen molecule 1 and molecule 2 in the

fluid, characterized by the total correlation ilmction l@IJ, is composed of two

contributions: the direct interaction, given by the direct correlation function c@J, and the

indwt interactions transmitted from 1 to 2 through interaction with one other molecule

(labeled molecule 3). Higher order terms, for instanm interactions between molecules 1

and 2 tmnsmitted through two other particles (i.e., molecules 3 and 4) are assumed

negligible. Therefore, the physical meaning of the 02 equation is that the total intemction

for a two-body interaction is composed of the direct correlation, plus the density-weighted

sum of 1-2 interactions tmnsmitted through all other particles 3. Here, p is the number

density (number of molecules per unit volume) of the molecular fluid. Through equation

(2.6) the intemctions between ali molecules in the fluid are coupled. In a monodkperse

isotropic fluid, rlz, rlj and rzj are replaced simply by r, the center-to-center distance

between molecules. The direct correlation function h(r) is related to the potential of mean

force between mokwuks through the following definition (McQuarrie,1976):

h(r) + 1 = g(r)= exp[-WY(r)/k8~ (7)

11

I

whereg(r) is the pair-correlation function, or radial distribution function, which rellatesthe

spatially-varying concentration of moiecules in the tlui~ p(r), to the average concmtration

p, by the following: p(r)= pg(r).

The connection of the 02 equation to thermodynamic properties of solutions comes from

performing a Fourier transformation on equation (6), resulting in

.

1- p(?(k) = -———&—1+ pzl(k)

(8)

where ~(k) and i$(k) represent the Fourier analogs of c(r) and h(r), respectively. By

Fourier-transforming in the limit k = O(which corresponds to taking a volume integral), tie

thermodynamic pressure and chemical potential may be obtained from the isothermal

compressibility relation for a one-component fluid

1- pe(o) ‘+%=$(:), (9)

The random-phase approximation consists of defining the direct correlation function, c@),

as

where ch,(r) is the dhect correlation fhnction for the hard-sphere reference system. This is

most valid in the limit of small Ww(r) and low mokcular concentmtion, p. Fourier

transformation of equation (10) in the k = O h-it and integmtion of equation (!?) with

respect to p yields the following expressions for the pressure and residual chemical

potential of protein in solution

(11)

(12)%es,RPA f% PU—=— -.

kBT kgT + kBT

12

where Ph$,and pk$are the pressure and chemical potential, respectively, of the hard-sphere

reference system. The contributio~ to the pressure and chemical potential from the

spherically-symmetric interactions am given by the terms in equations (11) and (12)

containing the RPA perturbation energy, U, defined as the volume integral of the

spherically-symmetric potential of mean force

Because U is an energy per unit density, it is independent of protein

depends only on potentials of mean force between protein molecuies.

(13)

concentration, U

The hard-sphere

reference terms are given by the standard Carnahan-Starling forms (&n&m andStarling,

“1970]

(14)

(15)

where the protein packing fraction q = (m/ 6)pr#. This theory is applied to proteins in

aqueous salt solutions within the context of the McMiliau-Mayersolution theory (Hill,1%0),

in which the potent.kdof mean force between particles immersed in a solvent is analogous

to the pair potential for non-ideal gases interacting through a vacuum.

3.2 Statistical Associating Fluid Theory

The contribution of the orientation-dependent specific interactions to the residual Helmholtz

energy and pressure of the system are evaluated using the first-order perturbation theory of

associating fluids formulated by Wertheim (wertheim,19s&vwertheim, 1986b),extended to

mixtures by Jackson and co-workers (JacksoIIet af., 1!?s8). At a given protein number

density, p, the StatisticalAssociating Fluid Theory (SAFT) gives the residual Helrnholtz

free energy of the associating system relative to that of the non-associating reference

13

system. The refenxwe system is an assembly of non-aggregating protein mokxxdes that

interact through the spherimliy-symmetric potential of mean force, Ww(r).

For a protein having M equivalent exposed spec~tic-intenwtion sites intemcting via the

potential given in equation (5], the first-order perturbation theory yields the following

contribution to the rksidual Helmholtz free energy due to hydrophobk association (~’acksonet

d., 1988):

a,e~,~,JkBT=M{lnX + (1-X)/2] (16)

Here, X represents the fraction of specific-interaction sites which do not participate in a

bonding interaction

-1 + Jmpxx=

2MpA —(17)

where

A = g(q>[exp(s#Bq - lIoV

and

(18)

3[1- c0s(f34.)][1- COS(~2,C)l(~ -Q‘v= .—

23T(19)

‘P

where g(@ is the radial distribution function of the non-aggregating system, and s~P

represents the characteristic energy of the specific interaction. Parameter V corresponds to

the volume of interaction of a site, expressed as a fraction of the totaI protein volume. The

fimction g(@ for non-aggregating proteins (which interact through the sphericaliy-

syrnmetric potentiai of mean force) is estimated using the EXP approximation (Kcmiorand

Jedrzejekj1985}

(20)

.14

where g~$(@ is given by the contact value of the Camahan-Starling expression for the

hard-sphere radiaI distribution functiom

1-?7/2gh,(cp) = —

(1-q)s(21)

The first-order perturbation theory of association assumes that the interactions between

specific-interaction sites on different molecules are independent and that no ring structures

are formed. The standard thermodynamic relation P = -(&4/ilV’)~ yiekis the associating-

fiuid contributicmto the total thermodynamic pressmw

(22)

The associating-ffuid contribution to the residual chemical potential of protein is obtained

from equations (16) and (22):

y=+.++%(:)&)}”l%es,amoc

3.3 Equation of State for Associating Protein Molecules

(23)

Combining equations (11) - (23) using the thermodynamic relation p,JkJ = a,JkBT +

P/pkBT - 1 yields the RPA-SAFI’ equation of state and expressions for the residuaI

Hehnholtz free energy and excess chemical potential of aqueous protein solutions:

15

{

p-p” q(8-9q+3q2)+ PU ~M ~X; 1-X (~_~ . ~\—=—. .kBT (1-q)s kBT 2 )(‘lx 2 MT

+hl,f)

(26)

Here, the ideal-gas chemical potential pi8/kBT= y*/kJ + inp, where p“ is a function only

of temperatunx p“ lkBT= ln(A3);the deBroglie wavelength A = ~~-, where h is

Planck’s constant and m is the mass of the protein molecule.

The number of specific-interaction sites, M, the interaction energy, &~P,and the intmaetion

volume, V, win be estimated from relevant physicoehemiea.1data for the protein of interest

or correlated from a reasonably small set of precipitation data, as described below.

4. PHASE EQUXLIBRHJM IN PROTEIN SOLUTIONS

4.1 Critical Points andl S pinodal and Coexistence Curves

The Spinodd criterion for a one-component sohdion is (Rausnitzet al., 1986):

-(#)T=(;)T=o (27)

where A is the Hehnh&z free energy of protein and V is system volume. At the critical

point, in addition to equation (27), the following criterion also applies:

-($)T=($]T=OThe partial differentials of the pressure with respect to density for the RPA-SAFT equation

of state (24) are derived in Appendix A. At equilibrium, protein concentrations in the

supematant and dense-fluid phases are calculated from equations (24) and (26) based on

the classical equilibrium editions:

16

~s=pd (2%)

P,= Pd (29b)

Here, subscripts s and d denote the equilibrium supernatant and dense protein phases,

respectively. In equation (29a), p“ eaneels out.

Figure 6 shows thermodynamic pressure ealcukt.edfiom the RPA-SAFI’ equation of state,

normalized by @~, as a function of protein packing fkaction, q, for several values of

ionic strength, For this case, the parameters are CP= 50& HJk# = 10, M =6, qJkJ’ =

2, V= 0.02, and the saIt is ammonium sulfate. For ionic strength below the critical poin~

a unique P-q relationship exists. However, for higher ionic strengths there are two values

of q having the same pressure, corresponding to two coexisting equilibrium liquid phases.

The additional constraint of equality of chemical potentials, equation (29a), defines the

aetwdvalues of the equilibrium phase compositions.

Figure 7 shows @pied spinodal and coexistence curves for the RPA-SAFT’ equation of

state. The parameters in this figure are the same as for Figure 6. The died ionic strength

for liquid-liquid phase equilibrium, 1=,is 2.22 M in this case, and the critical protein

volume ilaction,. qC,isO.171. For solutions with ionic strength greater than lCand protein

concentrations which correspond to values of q which lie between the two branches of the

sphmdal curve (where ilP/~q <0 and the solution is thermodynamically unstable), the

solution separates into two liquid phases having protein Coneentmtions q. and qd, in the

supernatant and dense phase, respectively, obtained from the coexistence curve at the

specified solution ionic strength.

Figures $(a) and 8(b) show the dependence of the critical ionic stre&th, 1=,on the SAFT

parametem M, s~pand V, and on the protein diameter, Op,respectively, for a hypothetical

protein in ammonium sulfate solution. (h Figure 8(a), V is represented as V% = V“100,

for convenience in graphing.) For these ealcuiations, the base set of parameters is the same

as in Figures 6 and 7; the pammeter indicated in the figure legend was varied, while all the

other parameters were held constant at the values given above. 13gure 8(a) shows that as

17

the value of any of the SAFT parameters increases, attractive specific interactions increase,

and hence the critical ammonium sulfate ionic strength required to induce precipitation

decreases. The strength of the specific interaction, E,P,and the number of interaction sites,

M, have a stronger effect on 1=than does the interaction volume, V. The protein diameter,

OP,has the strongest effect on IC: for a protein with diameter of 30~, 1== 9 M, while for a

60~ protein, only 1 M armnonium-sulf~ ionic strength is required to cause phase

separation. This is due entirely to ion-excluded-volume forces. Consider Figure 2 as the

protein diameter increases, the plane from which ions are excluded also increases in area

when the two protein molecules approach surface contact. Hence, there is a larger area

over which the osmotic pressure imbakmceacts, increasing the attractive force at any given

ionic strength and resulting in a lower Z=required to induce precipitation. This phenomenon

-- that larger proteins precipitate at lower ionic strength than smaller proteins -- has been

previously observed in experimental salting-out protein-precipitation studies (~~bertsson,

198@Shih et al., 199%Coen, 1995).

For precipitating systems, two quantities of primary interest in comparing

predictions to experimental phase-equilibrium data w the supernatant

model

protein

concentration, often equated with the protein apparent volubility, S, expressed herein terms

of the supernatant-phase protein packing fractio~ q=(recall, q, = m/6p,~~, where ,pXis the

protein number concentration in the supernatnnt phase); and the equilibrium partition

coefficient, K,, defined as K=’=q~q~, where q~ is the equilibrium protein packing fraction

in the dense (p~cipitated) phase. Figures 9 through 12 examine the effect of varying the

SAFI’ parameters and the :proteindiameter on calculated phase-equilibrium compositions.

Again, the base set of parameters is the same as in Figures 6, 7 and 8; the parameter

indicated in the figure legend was varied, while all the other parameters were held l-cnstant

at the values given above. In Figures 9 and 10, an increase in ionic strength results in a

decrease in q,, indicating greater partitioning into the dense phase. Furthermore, as any of

the SAFT parameters increases at a given ionic strength, the magnitude of the attractive

specific interactions increases and the equilibrium supernatant protein packing fkaction

decreases. As shown in Figure 10(b), protein diameter, o,, has the most significant eff’t

on protein volubility, and interaction volume, V, has the weakest influence.

18

Figures 11 and 12 show that the equilibrium prom-n partition coefficient, K., increases

exponentially with increasing ionic strength. ASshown in Figures 9 and 10, increasing the

values of the SAIW pmameters increases the amount of attraction between protein

molecules and results in ligher protein partition coefficients. Similarly, larger proteins

show much greater partitioning than smaller proteins at any given ionic strength above the

critical value. Also, l%gures 11 and 12 show that the RPA-SAFI’ model has the power to

describe equilibrium protein partition coefficients with vabs from 1 (near the critical point)

up to values on the order of Id, which corresponds to essentially complete protein

precipitation, using ~onable values of model parameters at experimentally relevant salt

concentrations.

4.2 IIen-Egg-White Lysozyme Precipitation

Figure 13 shows experimental precipitation data (C= 1995) for hen-egg-white Iysozyme

(points) in ammonium-sulfate solutions, for pll between 5 and 8 and ionic strength

between 5 and 9 mokd, aiong with values of K. cdcuhtedfrom the RPA-SAFT equation of

state for the same solutions (dotted lines). While the experimental equilibrium partition

coefficient increases with ionic strength, no dependence on pH is observed at any ionic

strength shown in Figure 13. The potential-of-mean-force model incorporates pH

dependence only through the screened Coulombic repulsion potential, Wqq(r). For ionic

strengths above 1 mokd, the contribution of long-range macromolecular Coulombic

repulsion to the total spherically-symmetric potential of mean force, Win(r), is negligible

due to dielectric screening; therefore, the RPA contribution to the thermodynamic properties

of the system is independent of pl%

However, equilibrium partitioning at high ionic strength does depend on the solution pH

for most proteins. This includes lysozyme at pH c 5, Iysozyme K=increases significantly

with decreasing pH at all ionic strengths [data not shown; see [Coen,199s}]. Chym@rypsin

partition coefficient also increases with decreasing pm .as shown in Figure 16(u). There

are several possible physiochemical reasons for this observation. Partial denaturation in

acidic conditions may make the protein more prone to precipitate. As the protein denatures,

its eff’ve hard-sphere diameter DPincreases. As shown in Figure 12(b), for a given

ionic strength, larger proteins precipitate more readily than smaller proteins. Also, with

partial denaturation comes the possibility of greater solvent exposure of hydrophobic

amino-acid groups which are normally sequestered in the interior of the protein, when in its

19

native conformation. Increased hydrophobic interactions can enhanee protein aggregation

and precipitation. Furthermore, at low pH, sigrtii%antanion binding may occur (chrr, 1953;

Przybyeienand Bailey, 1989), altering the surface chemistry and attenuating short-range

ekdmstatic interactions. In high-ionic-strength solutions, there may be little free water of

hydration available to the protein molecides. Desolation of the protein can lead to

eonformational shifts and changes in surface chemistry, which may provide an ionic-

strength dependence to protein partitioning beyond what is accounted for in the spherically-

syrnmetric potential of mean force. Whether this change favors”increased or deereased

partitioning depends on the specific protein and salt under consideration. In the context of

the RPA-SN?T description of salting-out phase equilibrium, ali these protein- and salt-

spedlc effeets are accounted for by correlating pH- andor ionic-strength-dependent values

of the SAFT parameters from a set of experimental precipitation data.

In applying the RPA-SAIW model to precipitation&a of the type shown in Figure 13,

physical insight is required in choosing appropriate values of the SAFI’ pammeters M, E~p,

and V. Here, M is assigned the value of 6 for Iysozyme, enabling the formation c)f large,

dense’aggregates; V is taken to be 0.03, or 3%of the totalmolecularvohune per interaction

site. The specific-interaction energy, e~P,is Wowed to vary and is regressed from the

experimental precipitation data. The potential-of-mean-force parameters for lysozyme are

taken from earlier work (Kuehneret d., 1997] the Hamaker constant (which depends on

neither pH nor ionic strength), H/kBZ’= 8.9, and the protein net valence, ZPis obtained

from extrapolation of lysozyme titration curves titration cnrves (Kuehneret uZ.,1998).

Figure 14 shows the values of the speeific-interaction energy, S,P, regressed from the

average values of the equilibrium protein partition eoefficien~ JKe,=vg,shown in Figure 13.

S$Pdecreases in a linear fashion with increasing ionic strength. This is an empirical

regression and reflects the “true” interactions oeeurring in this system only to the extent that

the physics included in the RPA-SA17T model are correct. Whhin these physical

assumptions, Figure 14 implies that as ionic strength increases, ion-excluded-volume

forces grow rapidly enough that a decr~e in the specific-interaction energy is required to

—

1Inaddition,speeificioneffeetsonproteinipteraetionsandsoluhilityhavebeenwellstudiedandanalyzedtheoretically(llofmeister,1888;Grr, 1953;Carr,1953;Melanderandllorvath,1~, AmkawaandTimasheff,198%ArakawaandTimasheff,1984ArakawaandTiieff, 1985;Ries-KauttandDucruix,1989;Ries-Kauttand Ducruix,1994 Corm,1995;CurdSet d., 1998) but m not discussedin the contextof this work,whereammoniumWfate is the only salt examined

20

describeaccurately the experimental equilibrium protein partition coefficient in ammonium-

sulfate solutions.

Figure 15 examines the ionic-strength dependence of the equilibrium protein pardtion

coefficient of lysozyme in ammonium-sulfate solutions at pH 8.0. Figure 15@) shows

experimental values of JC.in logarithmic scale, along with calculated partition coefficients

for two different vttlues of &~P,where S$Pis held constant with respect to ionic strength.

Both the experimental and the calculated values of ~~ foilow classical Cohn-Edsall behavior

@dsdl, 1943),where the protein volubility (expressed here as ~~) exhibits au exponential

dependence on the ionic strength. However, the slope of the experimental line is smaller

than that of either of the two calculated lines. The values of Ife calculated using q~k~ =

4.0 match the experimental values better at lower ionic strengths but overpredict

partitioning as ionic strength approaches”9 mokd. Conversely, the values of E= calculated

using qJkBT = 3.8 match the experimental values well at high ionic strength, but

underpredict partitioning at lower ionic strengths. This subtlety is not easiiy seen in Figure

15(b), where the same values of K. are displayed on a linear scale. However, Figure 15(b)

shows the sensitivity of Calculatedvalues of K. to &.pat high ionic strength more effectively

than does Figure 15(a). Figure 15(b) also illustrates that when high degrees of

precipitation are desired (which would presumably be one of the potential applications of

such a model), it is more important to fit modeI parameters to high-ionic-strength data thark

to Iow-ionic-stmmgthdata. In order to make calculated values fit experimental values more

exactly over the whole range of salt concentration%s~Pmust be allowed to vary with ionic

strength.

4.3 Bovine a-Chyinotrypsin Precipitation

Bovine a-chymotrypsin is a serine protease involved in the degradation of other proteins in

the digestive process. It is composed of 245 amino acids in three chains joined by disulfide

linkages between cysteine residues. As with Iysozyme, chymotrypsin’s structure has been

chamcterized by X-ray crystallography in its native conformation, it is a compact globular

molecule, slightly prolate, with axes 5,1x40x 40& giving an equivalent spherical diameter

Op= 43.4A; it has molecular weight 24S00 Daand isoelectric point 8.3 ($lkryer,1988).

21

Figure 16(a) shows the dependence of the equilibrium protein partition coefficient lie, on

solution pH and ionic strength, for precipitation of a-chymotrypsin in ammonium-sulfate

solutions. For chyrqotrypsin, p=ipitation is significantly enhanced as pH decneases,

whereas K=for lysozyme showed no pH dependence. 33gure M(b) shows calculated and

experimental values of K. in logarithmicformat. For the RPA-SAFI’ calculations with

chymotrypsin, M was again taken as 6. Chymotrypsin is larger than Iysozyme (IvIW =

14,500 D% UP= 34.4A). Hence, the volume fraction of each specific-interaction site in

chymotrypsin was chosen to be slightly smaller than in lysozyme; here, V = 0.02,

comesponding to 2~0 of the protein volume per site. The Hamaker constant of

chymotrypsin, H/kBT= 10, a value similar to that of lysozyme. The Hamaker constant

depends primarily on the electron density of the macromolecule and is therefore expected to

have similar vaiues for most small globukw proteins QWr,1%’6). The net valeu& of

chymotrypsin was obtained from titration experiments (Fergg,w%). Figure 16(b) shows

that, as is the case for lysozyme precipitation in ammonium-sulfate solutions, the RPA-

SAFTmodel gives caicuiatedCohn-Edsall slopes for chymotrypsin that are slightiy higher

than the experimental values. Consequently, wdues of qP which vary with ionic strength,

as well as with pH, must be used to correlate the data of Coen (Coen, 19%) for

chyrnotrypsin precipitation with quantitative accuracy.

Comparison of Figures 15 and 16 shows that Iysozyme precipitates at lower amrnonium-

sulfate ionic strength tharchymotrypsin, although lysozyme is the smzdler of tlm two

proteins. Normally larger proteins are observed to salt out at iower ionic strength than

smaller proteins (Albertsson,1986 Rothstein,1994);this observation is also reflected in RPA-

SAFT calculations, as shown in Figure 12(b). Attmctive speciilc interactions (or, at leas~

interactions not included in W~(r), the spherically symmetric potential of mean force) in

lysozyme solutions must be responsible for lysozyme’s lower sohdility in ammonium

sulfate. This is highlighted by the difference in the correlated values of the specific-

interaction energies: for chymotrypsin, 6,JkBT-1, whilefor lysozyme, qJkBT -4. These

additional forces may arise from, for example, diffenmces in the extent of hydrophobic

interaction, diffemmes

precipitating solutions.

in protein-salt interactions, and/or conformationrd changes in

22

5. CONCLUSION

An approximate statistical-mechanical equation-&f-statemodel has been derived to describe

salt-induced protein precipitation. In tlds model, protein-protein interactions are quantified

through potentials of mean force (PI@%) which account for electrostatic repulsion,

dispersion attraction, ion-excluded-volume attractiq and hydrophobic interactions.

Thermodynamic properties of the system are derived using two perturbation theorkx

spherically symmetric interactions are described by the random phase approximation

(RPA), and orientationdependent attmctive specific interactions (e.g., hydrophobic

interactions, short-range electrostatic interactions or self-association) are accounted for

using the statistical associating fluid theory (SAFT’).

The equation of state has been applied to solutions of hypothetical prot&s in aqueous

solutions of ammonium sulfate. Calculated phase-equilibrium results show that both the

supematant-phase protein concentration, q., and the equilibrium protein partition

coefficien~ K., exhibit exponential dependence on ionic strength, C&n and E&all @&all,

1943)were the first to make this observation experimentally and to correlate salting-out

solubilities for a variety of proteins using a simple, purely empirical exponential formukq

successfully describing this phenomenon is one of the first requirements of a

thermodynamic model for protein precipitation. Protein partitioning is most sensitive to

protein size, with larger proteins being more prone to precipitate than smaller proteins, in a

solution of any given ionic strength. Furthermore, attractive specific interactions may play

an important mle in the pmipitation of proteins at high salt concentrations; the SAITf

contribution to precipitation is a strong fi.mctionof the magnitude of the specific-interaction

energy, &,P. It has been shown in principle that using the RPA-SAIW equation of state,

either ion-excluded-volume or specific-intenwtion forces, or both, can be used to interpret

protein-precipitation data.

The RPA-SAFI’ equation of state has been applied to real protein-precipitation data for hen-

egg-white lysozyme and bovine rwchymotrypsin in concentrated ammonium-sulf~

solutions. Potential-of-mean force panirneters for hen%gg-white lysozyme from previous

work (Kueb.aeret ul., 1997)are used to calc~ate the RPA contributions to the thermodynamic

pressure and chemical potential of protein in precipitating solutions. Precipitation of both.

proteins is described well using six specific-interacdon sites. Quantitative agreement

between experimental and calculated values of the protein equilibrium partition Coefficient

23

~~, is obtained by using values of the specific-interaction energy, s~P,which depend on

ionic strength for Iysozyme and on both ionic strength and pH for chymotrypsim In

general, the strength of attractive specific interactions was found to be greater for lysozyme

than for chymotrypsin at aIl solution conditions.

In summary, the RPA-SA171’equation of state provides simple analytic expressions for the

thermodynamic properties of soluble and precipitating solutions of globular proteins. It

includes contributions from a variety of intermolecular interactions which depend on

solution pH and ionic strength, two variables often used to control protein precipitation in

practice. Wxth five adjustable parameters which may be correlated from a reasonable

number of light-scattering, titration and protein-precipitation experiments, the RPA.-SAFI’

equation of state provides quantitative description of equilibrium protein partitioning in

precipitation systems. In the future, it is expected that incorporation of additional significant

features -- e.g., specific protein-ion interaction effects, multi-component expressions

describing the properties of mixtures of proteins, and application to protein crystallization –

will significantly enhance the utility of this type of molecular-thermodynamic model in

biochemical eng@eering applications.

24

APPENDIX A Spinodal and Critical Points in Aqueous Protein Solutions

For a one-component fluid, the spinodal criterion i~

-($)T=(:)T=Z+T(;;T=O

where A is the Hehnholtz free energy of protein and V is system volume; the packing

fraction of protein q = (z / 6)p0~, where p is the protein number density and OPis the

protein effeetive hard-sphere diameter. The eompressibility ftior, Z, may be written as the

sum of three contributions:

Z = P/pkBT= Zh=+ Zm. -t-Z_.c (A.2)

whertx.

wz =—“ 2kBT

(A.3)

(A.4)

(A.5]

where U is the R.PAperturbation energy (per unit density), kBis Boltzrnann’s constant and T

is absolute temperature. The association parameters M and X represent the number of

equivalent bonding sites on the protein molecule and the fraction of unbound sites,

respectively, where X is given by:

-l+&zi@x=”

2MpA{A*6)

25

and

.

(.A.7)

Diiexentiationwith respectto q yieldx

dz~ 2(2+2r/-q2)—=—.t?q (1-q)4

(M-?)

(,?!.9)

(,4.10)

where the constant-temperaturenotationhas been omiaed for convenience on the density

derivatives. The derivatives of X with respeet to q are given by

(z“”*~-x) (A.11)

where

J311 —-

‘~-2-q+q(A.13)

26

(A.14)

Thus, at the spinodal poin~ equation (A. 1) becomes:

At the critical point the additional constraint

.($),=($),=O

is imposed. Equation(A.16) is equivalent@.

Again, differentiation with respect to q yields

(#zk 4(5 +2q -92)

‘= (AJ5dr/2

(A.16)

(A.17)

(A.18)

(A.19)

27

(A.20)

where

2h@AJ f283 ,J~ l-2MpA

‘(1 +4MpA)3~2 [ % ()1+ 4MpA

and

d2J 6 22..— —=== (2-@3 ‘fL?q

Combining, the additional critical criterion (A.17) beeomes:

(A.21)

(A.22)

Given a potentiaf of mean force and values for the association-potential parameters, as

discussed in the body of this chapter, equations (A. 15) and (A.23) are used to calculate

spinodal and critical points in aqueous protein solutions.

28

/4PPENI)IX B:

In evahmting the

Osmotic Coefficients for (NHJ2S04 Solutions

ion-excluded volume poten’kal of mean force, WOSm(r),the osmotic

pressure, I&, of the corresponding pmtein-free salt solution is required. The ideal and

real osmotic pressures of an aqueous electrolyte solution are related by

(B.1)

where the ideal osmotic pressure ix

%= %NA(%.,+ V.J+BT (B.2)

where c. is the molar concentration of salt, ZV~is Avogadro’s number, V@,and v=. are the

stoichiometric coefficients of the cation and anion, respectively, kB is Boltmmmn’s

constant, and T is absolute temperature.

Osmotic coefficients for ammonium-sulfate solutions at T = 2981S are reported in the

literature (Clegget & 1996)and are shown in Figure B.1. For use in calculating WOw(r),the

data of Clegg and coworkers were fit to a sixth-order polynomial fimction, yielding

n om = 0.000142m6 - 0.002788m5 + 0.021813nz4-0.0881 18rn3(B.3)

+0.204384m2 - 0.26135Fn+ 0.768580

with a high degree of accuracy (R2= 0.998).

29

NOTATION

a

ci~r)

Ci

e

g&r)

h$r)H

h

?

Ke

NA

P’

r

sT

uv

W(r)

Zj

Helmholtzenergy, J/mol ‘

directcorrekitionfimction of&pair

concentration of species i, moi/L

elementary charge, 1.602x10-19C

radial distribution function of ~ pair

total correlation function of ij pair

Hnmaker’s eonstang J

Planck’s constant, 6.6252x10-~ J-see

ionic strenglh of solution, mol/L

distribution coefficient

Bokzmann’s COllS@ll~1.3804x10-23J/K

molecuiarmass of species i

number of attractive specific-interaction sites per protein molecule

Avogadro’s number, 6.023x1023 mol-1

thermodynamic pressure, Pa

center-t.o-eenterdistance, A

protein solubiiity, mol/L

absolute temperature, K

RPA perturbation energy, J-JJmol

specific-interaction volume flaction

potential of mean force

valence of species i

Greek Symbols

P Cohn-E&all salting-out intercept

E, relative permittivity

% permitivity of vacuum, UVm

&SP association energy

~ protein packing fraction

e angle

30

K

A

Pi

q.

q-on

@ osm

Debye length, ~-1

Cohn-Edsall salting out slope ~

chemical potential

stoichiometriccoefficient of species i

real, idealosmotic pressureof protein-freesalt solution, Pa

number concentration of species i

diameter of species i, A

composition-weighted average hydrated ion diameter

osmotic coefficient of protein-tie salt solution

31

l“%—--r-I’’’”~”’’P’”~” “l’ ’i ’”l’’’’l ’i ’’l’’” -1

0.8

0.61**

●*.●*

●.●..,.

●

●.●

●.,..Ionic Strength ~]

------0.10--- 0.20...........0.50— 1.00

●

L ●

(),4 3, ●’**%.,,\ ●

\ “.\ ‘.

\ ‘*‘.\ ‘.0.2 &

\ -..\ .-

\ ‘.:.- % ...%.. - ●..- -.. .* -----+. --=...... ----------------

++...... . ,....... . -rrl-1+++~

1.0 1.1 1.2 1.3 1.4 1.5

rfap

Fignrel: De~ndence oftiescrwnedC oulombicp kntidofmemf orm,~q~(r),on center-to-centerseparationbetween proteinmonomers,r, andon solution ionicstreng@ 1, in aqueous ammonium sulfate solutions.

Protein parameters: 0“ = 50A, z =10Salt parameter Ocat-- &62~, 2Cat= 1

crm= 7.58& zm = -2

32

/- \\

/-—-. . \/

/. .%%......%..%.. ..-..:.:-:.:-:.:.:--------.

..........................%-...%.................................. .

\...................../.... .. ... ..... ,..:.:.:.:.:.:.:.:.:.:%-.:.>:.:.~:.:.:.:.:.:.:.:.}:.:..

~.:.:.:.+~.:.y,:.:.:.:.:.:.:.:.~..:.:.:.~.:.... .. ......... .. ... ..... ... .. .....

I%...--...%%........%..%..-.....%..........%..%%..%%%........................................................... ................................................................/..............................:.:.:.:.:.:-:-:.:.:-:.:.%:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:-:.:.:.:.:.>.

f.........................................................................................-./............................ ..............%%..%..%...... .........

\........................................................... %::::>,:::::::j$j$j;::::::y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....%....-....... ........ ..............%%..%%%%.... ......... .............%..............................w.%.. ..:-:.:.:.x+:.:.~.:.:.:.:.:.:.:.:.:.:. :.~.y..:.:.:.>:-y..:.:.:.:.:

. ............ .................. .......................... ..... ............-,%..=-...... .... ...... ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

\

..:.:.:+:++:.:.:.:.:.:.:.:.:.:..- . : ..... ................. ..... ... ..:.:.:.:.~.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:. :.:.x+:.:.:.:.:.:........ ... ... ............. ... ... .>:.:.~.>:.:.:.>: by..:.:.:.:.:.:.:. :.:.:.:.~.:.:.:.:.~.:.:.:.-

..................... . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . ............ ...... ...... .... ...... ...... .. ........... ............ .. .......-./. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.............. ..... ......... .. ..... ..... .... .................

\

........................................ .......... .........%:.:.:.:.:.:.:.:.:.:.:.:.:.:.*:.:,:.:.:.:..

..%.. %%....%..-.........%....%..%-----.............................+.+y..

\

\/

/\

/

\ 0.- - --

Fi~re 2: Geometry of the ion-excluded-volumeattractionmechanism.

33

0KI I I 1 I 1 I I I I

---------- --------------------- ---------- ----------- ----- ....------*-”’ .-. -”

-.*. - .9.~.- ------ #“@ /“----- ----

6 -. ...- 0“ 0“------ /- .“

-1: -------- ~‘Hw ./=’~---@“,0 ‘/”.4* /’0“ .

-2 p ,“4”●/”

● ,/’/.z /“k

-3

t

-4

t

/Ionic Strength ~

/’ — U.1O,! ------0.50/’4 --- 1.00/’ -.-.-.2.00

/4 ---5.00./“ ---- 10.0de/ 1

AL-_Lllllllfltllllldi.oo 1.05 1.10 1.15 1.20

I-hp

Figure 3: Dependence of the ion-excluded-volume potential of mean force, Worn(r),on center-to-center separation between protein monomers, r, and on solution ionicstrength, 1, in aqueous ammonium sulfate solutions.

Protein parameters: oP = 30f$$tdt parameters: ~C=t= 6.62A, Zcat = 1

am= 7.58A, Zm= -2

34

2 p......

L

.

L

m

L. ------ 5.0-6 ~’

~-.-. -” 10.0

-t**

-8 ~l’I’I’ ’I ’1 l’IIi ’’II fl’’’I’” I ““’ {’11’ 1’’’ ’1’1’151.0 1.2 1.4 1.6 1.8 2.0

Figure 4: Dependence of the spherically symmetricpotentialof mean force,Wwm(r),on center-to-center separation between protein monomers, r, and onsolution ionic strength, I,inaqueous ammonium sulfate solutions.

Protein Parameters: cP=30& =15, H/kBT=5&%ltparameter~ cr@=6.6 ,zc~=l

- 7.58& zm = -2Um-

35

specific interaction sites

A /’

r12

\-

Figure 5: Representation of speeific-interaction “siteson the protein surface.

36

.

I 1 I 1“”1 I I 1 I I I I I i l“’’I’”’’’ 1’’”:.“,*

0.5#,“

Ionic Strength ~~ ,’,,’,

------ 1.0 ,..0.4

E— 2’22--- 95 /’-,’

,’.--- .“

,’‘

/

.

.’●.....

.’..“.

.’t“.

.“#-

.-.“

.“.0

●**“

.“ ------.

O.ofi’’’l’’’’ l’’’’ l’’’’ l’” ‘1’’’’ 1’’”+0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

F~ure 6: Thermodynamic pressure, normalized by ap3/kBT,as a function of proteinpacking fraction, q, forthreedifferent values ofionic strength. Here, thecritical valueof ionic strength, IC,is 2.22M.

Protein parameters: OP= 50& H/kBT= 10, M= 6, s~kBT= 2, V= 0.02Salt pararnete~. o==,= 6.62& Zcat= 1

Om= 7.58A, zm = -2

37

k''''''''''''''''''i''''''''''''''''''''1'''''''''''''''''2■ Critical Point

------Spinadal Curve— Coexistence Curve

0.1

0.0

L 1 -1

Ltt,tiL[●..

‘.‘..

.... . .

“-----------

-“--------= ------- ---

. . .

-.—0 1 2 3 4“5 6

1, [mol/liter]

Figure 7: Typical critical point and spinodal and coexistence curves calculated fromthe RPA-SAFI’ eqnation of state.

Protein parameters: ~P= 50& H/k#= 10, M= 6, s#kBT= 2, V= 0.02Salt parameters: u’,~l= 6.62& Zcat= 1

o-m= 7,58& zan= -2

38

Ic ~[rnoI/Iiter] k

2 :

1 : [a) ‘....=...A

...0 “’lil’’l’”’’l’”l-02468

- :~k~... ,-.-.. ---v%%..%=..+.>..%..-* ~“. -..........“.....\l

M &,.kBT,V%

()~30 35 40 45 50 55 60

up [A]

Figure 8 Dependence of critical ionic strentiti, 1=,om(a) SAFT Pammeters M, s,~kBT,and V% = VQ1OO(b) protein diameter iYP

39

‘:K-

L \ d

10-5 Ll&LdLd4ukkd

5678910

I [mol/Iiter]

%~*-3

104

I (b)

\\\\\

\

lo-5 l'';'"'''''''''''~' '''''''''''''''''''''''''''''l2 4 6 8 10

I [mollliter]

Figure 9: Dependence of equilibriumproteinpackingfractionin the supernatantphase,q., on Sm pmete~: (a) ~K and (b) s,~kBT in aqueous ammonium sulfate solutions.

Protein parameter% a.= 50& H/kBT= 10, M= 4 fin (b)], c~kBT= 2 [in (a)], V= ().02Salt parameters: tiCat= 6.62fi, Zc=t= 1

.

q.m= 7.58& za~= -2

#

~o-l

~~-2

10-3

104

k’i ’’’’ l’’t ’l ’’’’ t’’’ ’l’’’’l’” ‘I{a

“$x&\\

I(a)

+ 0.01+ 0.02+ 0.03+ 0.04 1

-1~o-51,1, ,,, l*, *, l,, ,,!, t,,i,, ,,1, ,,,1,[ . .....

3456789

I [molfiter]

1o-1

10-2

10-3

104

E’I’’’l l’’’ l’’’ l’’’ l’’’l”a

I [mollliter~

Figure IO: llependenceofequiiibriump~teinpaeking fractioninthesupematantphase,%.!on: (@ s~ P-@r V$ad (b) ~pIn Wueous -o~~ s~fite so~utio~s”

Protein parameters: cP = 50~ [in (a)], H/kBT= 10, M= 4, e~kBT= 2, V= 0.02[in(b)]Saitparameterx aWl=6.62&zc@=l

..= 7.58A, zm = -20

41

Ke

Ke

105

104

103

102

101

10°

104

103

102

101

# Ii, ,, lt, t, 1,1, ,1, ,,,l,,, ,1,

45678910

1’ [molfiter]

11/~/&~~kBT

+ 1.0+ 2.0+ 3.0

(b)+ 4.0+ 5.0

I [mol/liter]

Figure 11: Dependence of protein equilibrium partition coeffkient, K,, on SAFI’parameters: (a)lf, and(b) s@/k#’in aqueous ammonium sulfate solutions.

- 50.& H/kBT= 10, M = 4 [in (b)], s~kBT= 2 [in(a)], V= (11.02Protein parameters: oP-Salt parameters: UCat= 6.62A, zCaf= 1

- %58& zm = -2O=n–

42

105

104

103Ke

102

101

I“’’J’* ’’l ’’’’ l’ ’’’I’” ‘1’’”1’

vq 4-0.01

4-0.02+ 0.03+ 0.04+ 0.05

e

(a)

10°tll,,,llll ill, tllllli llltlll*litlll3456789

I [rnoMiter]

105I * 1 1 I 8 I 4 I 1 1’ a I 1 1 t I i 1 * I I 1

CTp[A].(b)

104

Ke 103102

101

10° Ffillll 111 11111 Illll*llll2 “4681012

1 [molfliter]

Figure 12: Dependence ofprotein equilibrium partition cokfflcien6iK@on(a) S.AIWparameter V, &d(b) OPin aqueous ammonium sulfate solutions.

Protein parameters: aP = 50~ [in (a)], H/kBT= 10, M = 4, s,~kBT= 2, V= 0.02 @ (b)]%ltpammeters: oCat=6.62&zcaf=l

Om= 7.58A, Z@ = -2

43

10w8~’’’ l’’’’’’’’’4’’’’’’}’’I’’’’’’’’’i’’i’46~

4 -

2 -

N)08 :

6 -

4 -

2

108~

6 -

4 -

2 -

r

* ●✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎●

✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎✎☛❞ 9me.avg= 257

Q..........................................................o ~ I=8m.......... .............. .

u ~,w = 90.2

A.........................................................A .,......_..-...........~ j:=:= 4(-J8

A

nu..........................................................

n u ...”...--”-”a”-......”a$’;:= 14.1

*..........................* ........................e .....--~...............g ~:::= 6.34

4

5 6 7 8 9

pH

Figure K%Dependence of the equilibrium protein partition coefficient, K,, on pHand ionic strength for Iysozyme precipitating in solutions of ammoniumsulfateat 298K.

44

4.4

4.2

4.0

3.8

3.6i- -i

Fi,ll,l,l i,l,l,l,,l J,fllllil nil, Jill, il,l,l, d5 6 7 8 9

1 [mold]

Figure 14: Dependence of specific-interaction energy, QkBT, on solution ionic strengthfor precipitation of hen-egg-white iysozyme in solutions of ammonium sulfate at 298K.

Ammonium sulfate am= 6.62A, Om= 7.58A; zC=t= +1, Zm= -2Lysozymc OP= 34.4& H/kBT= 8.9, M = 6, V= 0.03

45

Ke

Nloop, ,’’, ,,, ,,l,,,, ,,,,,,,6 ~4 - 0 Experimental (pH8)’ 0

........ e#BT=3.g 02- ‘ ..*4 ...

--- &JkBT= 4.0 # ..0 .....100 r / ‘ ‘.8......

6 ~ / ..”.0 /’

4- ~6 ..”..’”& ..2- / -----

& ...

10? * ‘ ....””..0 .,# ..”.6: ● , ‘ .....””.

4- # ~/..’.-....’...

2-(a)

400

t

● Experimental (pH 8).,-.....~~kRT= 3.8

300 --- %&7w= 4.04

4

4

I

f .64 ---

r...

4....

...t

I ~-

f --:

/ ..:

# -----

‘:~5“6 ‘7 8 9

1 [mold]

Figure 15: Experimentalandcalculated values of the equilibriumprotein partitioncoeffkknt, K~,as a function of ionic strengthfor hen-egg-white Iysozyme precipitatingin solutions of ammoniumstifate at 298& in logarithmic(a) and linear(b) scale.

Ammonium sulfate Oal = 6.62A, C3a= 7.58A; Zcat= +1, Zm= -2Lysozyme OP= 34.4~, H/kBT= 8.9, M = 6, &~kBT= 3,8, V= 0.03

46

Ke

Ke

250L, ,1, ,,, ,,, ,, I,,,,I ,,, ,,, ,,, l,,,,iJ

200

150

100

50

0

t

-a- pH 4.0-o- pH7.O-+ pH8.3

,m...............-,..-.. {

7.5 8.0 8.5 9.0 9.5 10.0 10.5

I [mokil]

100o k,,,,,,,,,,,,,,,,,,,, ,,, ,,, ,,, ,,, ,q6

I

4

2

10064

2

10

..-....””“.”..’.m...-.

.....”-.ii”...-......4”’ A.....

w...”----#..-....#.-...-”s Experimental calculated

6

[

.----...- * pH4.o

4–.-.E~.kBT=l.()

● pH7.O — q.JkBT= ().8

2 ;; A pH8.3 ---- sJkBZ’= 0.6 !

11’’l ’”’’’l ’’’’ l””~ “’’1 ’’’’ 1’’”117.5 8.0 8.5 9.0 9.5 10.0 10.5

1 [mokd]

l?igure 16: Dependenceoftheproteinequilibrium partitioncoefficienGK,,onionicstm@.handpHfor cz-chymo~sinin solutionsofammonium suHat~at298K.

(a) Experimental data displayed in linear format(b) experimental and calculated values of K=displayed in logarithmic format.

Ammonium sulfate a~~ = 6.62A, Om= 7.58A; ~a, = +1, zm = -2a-chymotrypsin UP= 43.4~, H/kBT= 10, M= 6, V= 0.02

47

0.72

0.70

0.68

0.66

O*64

0.62

lil:llllfll![ltlllllllllll [llllllllllllfllJllJjll’11 Illly

*..--..

-1

Data of Clegg, et al.

Polynomial fit 1#y

.’,

J..’

●

.“.“.

.’.. 1.0

o 1 2 3 4 5 6

Cs [mola.1]

Figure B.1: Osmotic cwfficients inpureaqueous solutions ofmotim sulfateat 298.15K.

48

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