Role of Hydrophobic Effect in the Salt-Induced Dimerization ofBovine b-Lactoglobulin at pH 3

Giuseppe GrazianoDipartimento di Scienze Biologiche ed Ambientali, Universita del Sannio, Via Port’Arsa 11, 82100 Benevento, Italy

Received 7 January 2009; revised 12 February 2009; accepted 16 February 2009

Published online 24 February 2009 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/bip.21174

This article was originally published online as an accepted

preprint. The ‘‘Published Online’’date corresponds to the preprint

version. You can request a copy of the preprint by emailing the

Biopolymers editorial office at biopolymers@wiley.com

INTRODUCTION

Bovine b-lactoglobulin, b-LG, very abundant in cow’s

milk, is a globular protein of 162 residues,1 with two

disulfide bridges (Cys66-Cys160 and Cys106-

Cys119), and a free SH group, Cys121. At neutral pH

the protein exists as a dimer, whose structure has

been solved by means of X-ray diffraction at 1.8 A resolu-

tion2 (PDB code, 1BEB). The secondary structure of each

monomer is dominated by the presence of nine b-strands

(labeled A-I) and one a-helix at the C-terminal end of the

molecule. The eight A-H b-strands constitute a b-barrel with

an interior cavity, suitable to host hydrophobic ligands such

as retinol.3 The dimeric structure is stabilized by a set of

interactions occurring across the interface, characterized by

the formation of an antiparallel two-stranded b-sheet for the

packing of the I strands of the two monomers.2 In addition,

there is the formation of several H-bonds between the resi-

dues of the external loop connecting the A and B strands of

each monomer; so, for instance, the side-chain of Asp33 of

one monomer is H-bonded to the main-chain NH group of

Ala34 of the other monomer, and vice-versa.2 It is worth not-

ing that the protein exists as a monomer at pH 3.0 and

below, and its monomeric structure (PDB code, 1DV9),

solved via NMR techniques, is very close to that of each sub-

unit of the dimer at neutral pH (i.e., the root-mean-square

deviation for backbone atoms amounts to 1.3 A).4

The monomer-dimer equilibrium of b-LG is characterized

by a dimerization constant KD 5 5 � 104 M21 at neutral pH,5

that is not a large value; thus several features of the process

have been studied in detail. It has been shown that modifica-

tion of the free SH group of Cys121, which is entirely buried

by the C-terminal a-helix, favors dissociation into mono-

mers.5,6 In addition, it was found that, while the b-LG mono-

mer is the stable species at pH 3.0 and low salt concentra-

tions, the dimer becomes the dominant species on increasing

salt concentration.7 In particular, Goto and coworkers,8 by

means of sedimentation equilibrium measurements, charac-

terized the effect of NaCl, NaClO4 and GuHCl on the dimer-

This article is dedicated to professor Lelio Mazzarella in theoccasion of his 70th birthday, who has become a model for thedevotion to science and work.

Role of Hydrophobic Effect in the Salt-Induced Dimerization ofBovine b-Lactoglobulin at pH 3

Correspondence to: Giuseppe Graziano; e-mail: graziano@unisannio.it

ABSTRACT:

b-Lactoglobulin is a dimeric protein around neutral pH,

but the monomer becomes the dominant species at pH 3.0

due to strong electrostatic repulsions between the positively

charged molecules. It has been found that the addition of

salts to water at pH 3.0 favors the dimerization of b-

lactoglobulin. In particular, the dimer is the dominant

species at 1M NaCl, 1M GuHCl, and 25 mM NaClO4

[Sakurai, Oobatake, and Goto, Protein Sci 2001, 10, 2325–

2335]. The effect of these salt conditions on the strength of

hydrophobic interaction has been calculated by means of a

simple but physically sound approach. The obtained

estimates indicate that: (a) the hydrophobic interaction

contribution is strengthened in 1M NaCl and 1M GuHCl

with respect to pure water, but not in 25 mM NaClO4; (b)

anion binding on the positively charged surface of protein

molecules has to be the major factor for the salt-induced

dimerization. # 2009 Wiley Periodicals, Inc. Biopolymers

91: 1182–1188, 2009.

Keywords: dimerization; hydrophobic interaction; cavity

creation; water accessible surface area; anion binding

VVC 2009 Wiley Periodicals, Inc.

1182 Biopolymers Volume 91 / Number 12

ization process of b-LG at pH 3.0, 20 mM glycine-HCl buffer.

At 208C, they found that the dimerization constant KD 5 1.8

3 105M21 in 1M NaCl, 6.0 3 105M21 in 1M GuHCl, and

2.1 3 106M21 in 25 mM NaClO4. These values indicate that:

(a) the dimer of b-LG is largely the dominant species at the

above salt concentrations; (b) GuHCl, notwithstanding its

denaturing action on globular proteins, at 1M concentration,

stabilizes the dimer to the same extent as NaCl; (c) NaClO4

proves to be much more effective than NaCl and GuHCl in

stabilizing the dimer.8

First of all it is necessary to provide an explanation for the

effect of pH on the monomer-dimer equilibrium of b-LG,

and secondly to develop an analysis of the effect of salts. b-

LG is an acid protein, with an isoelectric point of 4.6; on the

basis of amino acid sequence, it should possess a net charge

equal to 120 at pH 2.0, and 29 at pH 7.5. This means that,

on decreasing the solution pH from neutrality, two b-LG

monomers, having an increasing net positive charge, experi-

ence repulsive interactions that lead to dimer dissociation. In

addition, the twin salt-bridges at the dimer interface,

between the side-chains of Asp33 of one monomer and

Arg40 of the other monomer, could not exist at pH 3.0

because the carboxylic groups are normally protonated at

this pH value.

For the salt effects Goto and coworkers8 provided the fol-

lowing qualitative rationalization: (a) the anions of the salts,

Cl2 and ClO42, bind to the positively charged surface of b-LG

monomers, shielding the electrostatic repulsion among the

protein molecules; (b) in these conditions the hydrophobic

effect can promote the formation of dimers, whose specific

structure is determined by the feasible interactions across the

potential interface; (c) the anion binding mechanism is sup-

ported by the stronger effectiveness of ClO42 with respect to

Cl2 because this is in line with the electroselectivity series to-

ward anion-exchange resins. Since no quantitative estimate has

been provided, I have decided to try to develop a quantitative

analysis of the extra hydrophobic effect contribution due to the

presence of salts in order to test the validity of the rationaliza-

tion proposed by Goto and coworkers.

THEORETICAL BASISThe process of bringing two nonpolar solutes from a fixed

position at infinite separation to a fixed position at contact

distance in water or aqueous solution, at constant tempera-

ture and pressure, can be considered as the prototype hydro-

phobic effect, and was termed by Ben-Naim9 a pairwise

hydrophobic interaction, HI. I think that the dimerization of

b-LG can be treated as a special case of pairwise hydrophobic

interaction. According to classical statistical mechanics,9 the

associated Gibbs energy change can be separated in two

terms:

DGðHIÞ ¼ Eaðm-mÞ þ dGðHIÞ ð1Þ

where Ea(m-m) is the direct monomer-monomer interaction

energy, and does not depend on the presence of the solvent

and its nature; dG(HI) is the indirect part of the reversible

work to carry out the process, and accounts for the specific

features of the solvent in which dimerization occurs. Evalua-

tion of the dependence of DG(HI) upon the distance between

the two solute molecules gives rise to what is usually called

the potential of mean force. A general relationship connects

dG(HI) to the Ben-Naim standard solvation Gibbs energy of

dimer and monomer9:

dGðHIÞ ¼ DG�ðdimerÞ � 2 � DG�ðmonomerÞ ð2Þ

where DG� represents the Gibbs energy change associated

with the transfer of a solute molecule from a fixed position

in the ideal gas phase to a fixed position in a solvent, at con-

stant temperature and pressure.10 A rigorous and straightfor-

ward application of statistical mechanics allows the exact

splitting of DG� in two contributions11,12:

DG� ¼ DGc þ DGa ð3Þ

where DGc is the reversible work to create at a fixed position

in a solvent a cavity suitable to host the solute molecule, and

DGa is the reversible work to turn on the attractive interac-

tions between the solute molecule inserted in the cavity and

all the surrounding solvent molecules. It is worth noting that

Eq. (3) does not imply the additivity of independent contri-

butions: the turning on of attractive solute-solvent interac-

tions is conditional to the creation of a suitable cavity.12 By

using Eq. (3) in the definition of dG(HI), one obtains:

dGðHIÞ ¼ ½DGcðdimerÞ � 2 � DGcðmonomerÞ�þ ½DGaðdimerÞ � 2 � DGaðmonomerÞ� ð4Þ

The attractive interactions of a monomer with the sur-

rounding solvent molecules are likely to be proportional to the

accessible surface area13 of the monomer itself. Dimer forma-

tion causes a reduction in the total accessible surface area and

so a reduction in the total number of attractive interactions

with the surrounding solvent molecules. This means that the

second square bracket in Eq. (4) cannot be neglected in gen-

eral. However, the reduction in water accessible surface area

upon b-LG dimerization is a small fraction of the total; the

native monomeric form of b-LG has WASA 5 8100 A2,

whereas the DWASA upon dimerization amounts to 600 A2

Dimerization of Bovine b-Lactoglobulin 1183

Biopolymers

per monomer8 (i.e., the latter is 7.4% of the total). In this case,

it should be a reliable approximation to transform Eq. (4) in:

dGðHIÞ ffi DGcðdimerÞ � 2 � DGcðmonomerÞ ð5Þ

Equation (5) indicates that a quantitative estimate of

hydrophobic interaction can be obtained from the calcula-

tion of DGc in water or aqueous solution as a function of cav-

ity size and shape (note that DGc cannot be determined by

means of experimental measurements, it has to be calculated

by means of theoretical relationships or computer simula-

tions). The latter is a very difficult task especially for a large

solute such as a globular protein, and so some general

considerations are in order.

The creation of a cavity, at constant temperature, pressure

and number of particles, at a fixed position in a liquid, causes

an increase of the average volume of the system by a quantity

equal to the cavity volume. However, a region around the

cavity surface becomes inevitably inaccessible to solvent mol-

ecules because particles cannot overlap one another, and the

cavity region has to be void.14 One speaks of solvent excluded

volume effect because the centre of each solvent particle can-

not enter the shell region between the cavity van der Waals

surface and the solvent accessible surface of the cavity.

Clearly, the solvent excluded volume of a given cavity (that

will host a real solute molecule) cannot be described by the

sum of two-body interactions, because it is the consequence

of many-body interactions (i.e., the additivity principle does

not hold, or, more correctly, it would be meaningless in this

matter). In the case of a spherical cavity in water or aqueous

solution, the spherical shell accounting for the solvent

excluded volume can be readily and reliably approximated by

the water accessible surface area of the cavity:

WASAc ¼ 4pðrc þ rwÞ2 ð6Þ

where rc is the radius of the cavity, the spherical region void

of all parts of the solvent molecules, and rw is the effective

radius of a water molecule, usually equal to 1.4 A.13

In addition, as first recognized by Lum et al.,15 it is possi-

ble to normalize the DGc magnitude for WASAc and to

obtain a quantity, having the dimensions of surface tension,

that can be used to estimate the hydrophobic interaction

contribution from the knowledge of WASA buried upon

association. For instance, for the dimerization of b-LG, in

the assumption that the monomer structure does not change

upon dimer dissociation, as supported from structural

data,2,4 Goto and coworkers8 determined DWASA(dimeriza-

tion) 5 21200 A2 per dimer. According to the above consid-

erations, Eq. (5) can be re-written as:

dGðHIÞ ¼ ½DGcðwaterÞ=WASAc� � DWASAðdimerizationÞð7Þ

Since the WASA buried upon dimer formation is a nega-

tive quantity, while the quantity in square brackets is always

positive, the hydrophobic interaction contribution provides

a large and negative Gibbs energy change favorable to di-

merization. Note that: (a) it is not necessary to partition

DWASA(dimerization) according to the polar or nonpolar

nature of surface groups because the solvent excluded volume

effect is always operative; (b) a relation equivalent to Eq. (7)

was proposed and used by Chothia,16,17 in a heuristic manner,

to estimate the hydrophobic interaction contribution to protein

folding and protein-protein association. For a further deepening

on Eq. (7), see Appendix A.

To estimate the effect of salts on the strength of pairwise

hydrophobic interaction, it is necessary to calculate DGc also

in aqueous salt solutions, and to extend the validity of Eq.

(7) to write:

DdGðHIÞ ¼ f½DGcðsaltÞ � DGcðwaterÞ�=WASAcg3 DWASAðdimerizationÞ ð8Þ

This relationship, notwithstanding the simplifying

approximations involved in its derivation, is reliable from

the physico-chemical point of view, and is the cornerstone of

the present study.

RESULTSThe DGc quantity has been calculated in both water and

aqueous salt solutions by means of the analytical expressions

provided by scaled particle theory, SPT, for pure hard-sphere

fluids and hard-sphere fluid mixtures,18–21 neglecting the

pressure-volume term due to its smallness over the consid-

ered cavity size range when P 5 1 atm. To perform calcula-

tions, the experimental density of water and aqueous salt so-

lutions, at 258C and 1 atm, has been used; specifically, q 5

997 g L21 for water,22 1037 g L21 for 1M NaCl,23 999 g L21

for 25 mM NaClO4,24 and 1023 g L21 for 1M GuHCl.25 For

the effective hard-sphere diameter of water molecules, I have

used r(H2O) 5 2.8 A, the customary value,13,21 close to the

location of the first peak in the oxygen-oxygen radial distri-

bution function of liquid water at room temperature.26 For

the ions, the following effective hard-sphere diameters have

been used: r(Na1) 5 2.02 A, r(GuH1) 5 4.6 A or 5.0 A,

r(Cl2) 5 3.62 A, and r(ClO42) 5 4.8 A. These diameter

1184 Graziano

Biopolymers

values, except the two for the guanidinium ion, are in line

with the location of the first peak in the respective ion-oxy-

gen radial distribution functions.27,28 For the GuH1 ion, due

to its planar structure, there is not a clear-cut way to deter-

mine an effective hard-sphere diameter,29 and so I have

selected two r values that should define a reliable interval.

The obtained trends of DGc/WASAc as a function of the

cavity radius rc are reported for water (black line), 1M NaCl

(blue line) and 1M GuHCl with r(GuH1) 5 5.0 A (red line)

in Figure 1. The trend for 25 mM NaClO4 is not shown

because it is practically identical to that of water, while that

for 1M GuHCl with r(GuH1) 5 4.6 A is close to that for 1M

NaCl. All such trends possess a similar shape: (a) for small rc

values, the DGc/WASAc functions increase linearly with cav-

ity radius; (b) for larger rc values, the DGc/WASAc functions

are practically independent of cavity radius, reaching limit-

ing-plateau values. This dependence, first pointed out by

Lum et al.,15 has been confirmed by the results of computer

simulations using reliable water models30,31 (note, however,

that no-one of the five most popular and reliable water mod-

els, SPC, SPC/E, TIP3P, TIP4P and TIP5P, is able to exactly

reproduce the experimental temperature dependence of

water density32), and by the application of SPT, in both the

original version33 and the revised one.34 The finding that

DGc in water and aqueous salt solutions is directly propor-

tional to WASAc for sufficiently large cavity radii lends

support to the validity of Eqs. (7) and (8).

In addition, the limiting-plateau values in 1M NaCl and 1M

GuHCl are larger than those in water as shown in Figure 1,

whereas the limiting-plateau value in 25 mM NaClO4 is practi-

cally identical to that in water. Specifically, for rc 5 50 A

(i.e., the largest cavity radius considered in this study), DGc/

WASAc(in J mol21 A22) 5 348 in both water and 25 mM

NaClO4, 367 in 1M NaCl, and 364 or 376 in 1M GuHCl, using

r(GuH1) 5 4.6 A or 5.0 A, respectively. These numbers indicate

that it is more costly to create a cavity of a given size in 1M NaCl

or 1M GuHCl than in water or 25 mM NaClO4. This is largely a

consequence of the volume packing density of these aqueous so-

lutions.21 The volume packing density, n3 5 (p/6) �Sqi � ri3,

where qi is the number density, in molecules per A3, of the spe-

cies i and ri is the corresponding hard-sphere diameter, is the ra-

tio of the physical volume of a mole of liquid particles over the

molar volume of the liquid itself. In other words, n3 represent

the fraction of the molar volume that is physically occupied by

liquid particles. On increasing n3, the void volume in the liquid

decreases, the probability of finding molecular-sized cavities

decreases, and so the DGc magnitude increases.21 In fact, at 258Cand 1 atm, n3 5 0.3830 in water, 0.3835 in 25 mM NaClO4,

0.3934 in 1M NaCl, and 0.4019 or 0.4106 in 1M GuHCl, using

r(GuH1) 5 4.6 A or 5.0 A, respectively.

Actually, this explanation is not entirely correct, as

emphasized by the fact that the limiting-plateau value in 1M

NaCl is larger than that in 1M GuHCl with r(GuH1) 5 4.6

A, even though the latter solution has the larger n3 number.

A complete analysis has to take into account not only the

total fraction of void volume in a liquid, but also the size of

the liquid particles, because the diameters of the latter deter-

mine the average size of the pieces in which the total void

volume is partitioned.21,35 Keeping n3 fixed, the DGc magni-

tude increases on decreasing the diameters of liquid particles.

Indeed, even though water has the smallest volume packing

density of all the other pure liquids, it shows the largest DGc

values due to the small size of its molecules.35–37

What is more important for the present study is that the

limiting-plateau values of DGc/WASAc, when inserted in Eq.

(8), allow a quantitative estimation of the change in the

strength of hydrophobic interaction caused by the presence

of salts for the dimerization of b-LG. More correctly, as

emphasized by the curves in Figure 1, for cavities with rc �20 A or larger, the difference between the DGc/WASAc func-

tions (in J mol21 A22) remains practically constant: 19

between 1M NaCl and water, 16 or 28 between 1M GuHCl

and water, using r(GuH1) 5 4.6 A or 5.0 A, respectively,

and zero between 25 mM NaClO4 and water. Using DWASA

(dimerization) 5 21200 A2, the obtained DdG(HI) values

are: zero in 25 mM NaClO4, 222.8 kJ mol21 in 1 M NaCl,

219.2 or 233.6 kJ mol21 in 1M GuHCl, using r(GuH1) 5

4.6 A or 5.0 A, respectively. The extra hydrophobic interac-

tion contribution due to salts is large negative in the case of

1M NaCl and 1M GuHCl, but it does not exist in the case of

25 mM NaClO4. The increase in the strength of hydrophobic

FIGURE 1 Plot of DGc/WASAc as a function of the cavity radius

rc for water (black line), 1M NaCl (blue line), and 1M GuHCl with

r(GuH1) 5 5.0 A (red line). SPT calculations were performed

using the experimental density of water and aqueous salt solutions

at 258C and 1 atm; see text for further details.

Dimerization of Bovine b-Lactoglobulin 1185

Biopolymers

interaction in 1M NaCl determined by means of simple SPT

calculations agrees with the results of detailed molecular dy-

namics (MD) simulations by Garde and coworkers.38

According to the above DdG(HI) numbers, the b-LG dimer

formation would be less favored in 25 mM NaClO4 than in

1M NaCl or 1M GuHCl, in complete contrast with experi-

mental data.8 Because Eq. (8) is physically sound, the correct

rank order of the three salt solutions in promoting the b-LG

dimerization has to be caused by other factors.

To gain perspective, it is useful to calculate the Gibbs

energy change associated with b-LG dimer formation by

means of the fundamental thermodynamic relation DGD 5

2RT � lnKD. Using the KD values reported in the Introduc-

tion section, one obtains that, at 208C and pH 3.0, DGD (in

kJ mol21) 5 229.5 in 1M NaCl, 232.4 in 1M GuHCl, and

235.5 in 25 mM NaClO4. Moreover, by assuming that in

water, at 208C and pH 3.0, KD 5 1 3 1023M21 (i.e., the

molar concentration of dimer is a very small fraction of the

total protein concentration in solution), it would result DGD

5 16.8 kJ mol21. From the thermodynamic point of view,

the salt effect, at 208C and pH 3.0, could be quantified in

DDGD (water ) salt) 5 246.3 kJ mol21 in 1M NaCl, 249.2

kJ mol21 in 1M GuHCl, and 252.3 kJ mol21 in 25 mM

NaClO4. By comparing the latter values with the above

DdG(HI) estimates, it is clear that the other factors promot-

ing b-LG dimerization at pH 3.0 in the presence of salts pro-

vide the major contribution to the Gibbs energy balance,

especially in the case of 25 mM NaClO4.

In this respect the anion binding mechanism, claimed by

Goto and coworkers,8 is surely operative at acid pH, and its

effectiveness proves to be inversely proportional to the anion

charge density. Since the ClO42 ion is larger than the Cl2 ion, it

has a smaller charge density and exerts a weaker attraction on

surrounding water molecules. Therefore, ClO42 ions, being

weakly hydrated, are much more prone than Cl2 ions to bind

on the positively charged surface of b-LG. Experimental data,

coupled to the present DdG(HI) estimates, especially that for

25 mM NaClO4, point out that an efficient screening of posi-

tive charges due to anion binding on the surface of b-LG

monomers is the major factor to promote dimerization. It is

worth noting, however, that the approach leading to Eq. (8)

works qualitatively well in the case of 1M NaCl and 1M

GuHCl, and is also able to rationalize the experimental datum

that b-LG dimerization induced by NaCl at pH 3.0 is an

exothermic process8 (for more details, see Appendix B).

DISCUSSIONA first point is related to the complete neglect of water struc-

ture reorganization, or better water-water H-bond reorganiza-

tion in the theoretical relationships devised to estimate the

hydrophobic interaction contribution. Apart from strictly sta-

tistical mechanical considerations, this is a simple consequence

of the finding that the switching off of H-bonds in detailed

water models has negligible effects on the calculated potential

of mean force between two nonpolar solutes.39,40 The funda-

mental role is played by the solvent excluded volume effect

because an almost complete enthalpy-entropy compensation

holds for the water-water H-bond reorganization.41,42

The second point is concerned with the claim by several

authors15,30,31,34 that the limiting-plateau value of DGc/ASAc

has to correspond to the experimental air-liquid surface ten-

sion of the considered liquid, because the creation of a suffi-

ciently large molecular cavity should be assimilated to the

creation of an air-liquid interface. Calculations with the orig-

inal version of SPT, using customary values for the effective

hard-sphere diameters of several liquids, did not find any

correlation between the obtained limiting-plateau values and

the experimental air-liquid surface tension data.33 In the

present case, at 258C and 1 atm, the SPT-calculated DGc/

WASAc limiting-plateau estimates, 348 and 367 J mol21 A22

in water and 1M NaCl, respectively, are markedly smaller

than the experimental air-liquid surface tension values, 433

and 443 in J mol21 A22, respectively. It is worth noting that

the SPT results are in line with those of detailed free energy

perturbation calculations based on MD simulations of water

and several organic liquids.43,44 It appears that the creation

of a microscopic (i.e., molecular-sized) cavity in a liquid is

not strictly related to the creation of a macroscopic air-liquid

interface because the former process is ruled by the solvent

excluded volume effect that is not operative in the latter. This

view seems to be also supported by experimental measure-

ments on superhydrophobic surfaces.45

A third point concerns the finding that GuHCl at 1M con-

centration increases the strength of hydrophobic interaction

notwithstanding its well-known denaturing action on the

native structure of globular proteins. This SPT result, already

pointed out in a previous study,46 is in complete agreement

with the conclusion emerged from the MD simulations per-

formed by Thirumalai and coworkers.47 The latter were able

to show that the denaturing action of GuHCl is due to the

ability of the guanidinium ion to directly bind on the protein

surface and to destroy both charge-charge interactions and

H-bonds.47 This direct interaction mechanism is entirely

compatible with the finding that the magnitude of the work

of cavity creation is larger in aqueous GuHCl solutions than

in water. Similarly, the recent result of MD simulations48 that

GuHCl inhibits the onset of dewetting between nanosepa-

rated hydrophobic plates should not be considered as a

demonstration that GuHCl decreases the strength of hydro-

1186 Graziano

Biopolymers

phobic interaction by reducing the Gibbs energy cost of cav-

ity creation. Such a finding is a further indication that

GuHCl directly interacts with hydrophobic surfaces, due to

its planar and weakly hydrated structure and its delocalized pelectrons, thus favoring solvent-separated configurations of

the two plates.

In conclusion, the devised analysis of the hydrophobic

effect role in the salt-induced dimerization of b-LG at pH 3.0

indicates that: (a) the hydrophobic interaction contribution

is strengthened in 1M NaCl and 1M GuHCl with respect to

pure water, but not in 25 mM NaClO4; (b) anion binding on

the positively charged surface of protein molecules, by effec-

tively shielding repulsive electrostatic interactions, is the

major factor for the salt-induced dimerization.

APPENDIX A

On the Validity of Eq. (7)

It would be important to show that the magnitude of DGc/

WASAc does not depend on the cavity shape to support the

general validity of Eq. (7). I have performed a set of DGc cal-

culations, by means of the appropriate SPT relationship,49 for

spherocylindrical cavities in a hard-sphere fluid (i.e., the lat-

ter has the experimental density of water at 258C and 1 atm,

and the sphere diameter is 2.80 A). By keeping fixed the cav-

ity volume, it resulted that: (1) WASAc(spherocylinder) 5

4p(a 1 rw)2 1 2pl(a 1 rw) increases markedly on increasing

the cylindrical length-to-radius ratio, l/a, of the spherocylin-

der; (2) DGc increases with the l/a ratio, but to a lesser extent

than WASAc; (3) thus, the DGc/WASAc ratio decreases on

increasing the l/a ratio, and reaches the maximum value in

the case of a spherical cavity. Therefore, it is not true that the

DGc/WASAc ratio is independent of the cavity shape.

However, I have verified that it changes to a little extent

passing from a sphere to spherocylinders with a l/a ratio up

to about 40. Specifically, for a cavity volume of 30427 A3,

corresponding to a sphere of 19.366 A radius, DGc/WASAc(in

J mol21 A22) 5 327 for the sphere, 323 for the spherocylin-

der with a 5 10 A and l 5 83.5 A, 309 for the spherocylinder

with a 5 6 A and l 5 261 A, and 291 for the spherocylinder

with a 5 4 A and l 5 600 A. The sphere is very different in

shape from the spherocylinder with l/a 5 261/6 5 43.5, but

the change in DGc/WASAc is small, amounting to 5.5% of the

value for the spherical cavity. This analysis confirms that:

(a) DGc depends on the solvent excluded volume of the

cavity that is reliably approximated by WASAc; (b) the value

of the DGc/WASAc ratio obtained for spherical cavities can

be considered to hold also for cavities that are very different

in shape from a sphere. Clearly, the DGc dependence on

WASAc should be a fundamental ingredient for the collapse

of polypeptide chains in the folding process.

APPENDIX B

On the Exothermicity of b-LG Dimerization

Induced by Salts

Goto and coworkers,8 by determining the dimerization con-

stant at different temperatures, found that the b-LG dimer-

ization induced by salts at pH 3.0 is exothermic. I would like

to show that the process of bringing together the two cavities

hosting the two b-LG molecules is exothermic in 1M NaCl.

Starting from Eq. (8), the enthalpy change due to pairwise

hydrophobic interaction is given by:

DdHðHIÞ ¼ f½DH cðsaltÞ�DH cðwaterÞ�=WASAcg3 DWASAðdimerizationÞ ðB1Þ

According to SPT, DHc is proportional to the thermal

expansion coefficient a of the solvent, and, at 258C and 1

atm, a (in K21 1023) 5 0.257 for water, and 0.325 for 1M

NaCl.21 Calculations indicate that the [DHc(salt) 2

DHc(water)]/WASAc quantity is practically constant, equal to

22 J mol21 A22, for cavities with rc � 20 A or larger. There-

fore, inserting the latter value in Eq. (B1), it results DdH(HI)

5 226.4 kJ mol21, in qualitative agreement with the experi-

mental datum DHD 5 250.4 kJ mol21.

In this respect, it is important to note that21,33: (a) DHc

does not account for the solvent excluded volume effect, but

solely for the structural reorganization of solvent molecules

upon cavity creation; (b) the DHc contribution is perfectly

compensated for by a corresponding entropy term so that

DGc is always purely entropic. At 258C and 1 atm, DHc is a

positive quantity (i.e., cavity creation is endothermic), mark-

edly smaller than DGc in both water and 1M NaCl, and

DHc(1M NaCl) [ DHc(H2O), indicating that the structural

reorganization upon cavity creation occurs to a greater extent

in 1M NaCl than in water.21 Clearly, bringing together two

cavities and so reducing the total WASAc leads to an energy

gain that is larger in 1M NaCl than in water.

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Reviewing Editor: Laurence Nafie

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