Collective diffusion coefficient of proteins with hydrodynamic, electrostatic,
and adhesive interactions
Peter Prinsen and Theo Odijka兲
Complex Fluids Theory, Faculty of Applied Sciences, Delft University of Technology, 2628 BC Delft, The Netherlands
共Received 9 May 2007; accepted 18 July 2007; published online 20 September 2007兲
A theory is presented for C, the coefficient of the first-order correction in the density of the
collective diffusion coefficient, for protein spheres interacting by electrostatic and adhesive forces. An extensive numerical analysis of the Stokesian hydrodynamics of two moving spheres is given so as to gauge the precise impact of lubrication forces. An effective stickiness is introduced and a simple formula for C in terms of this variable is put forward. A precise though more elaborate
approximation for C is also developed. These and numerically exact expressions for C are
compared with experimental data on lysozyme at pH 4.5 and a range of ionic strengths between 0.05M and 2M. © 2007 American Institute of Physics.关DOI:10.1063/1.2771160兴
I. INTRODUCTION
Fick’s first law states that the particle flux is equal to the negative of the collective diffusion coefficient times the gra-dient of the particle concentration. For colloids or macromol-ecules in solution, this collective共also called cooperative or mutual兲 diffusion coefficient is often determined experimen-tally with the help of dynamic light scattering. If one ex-trapolates this coefficient to a vanishing concentration of par-ticles, it reduces to the single-particle diffusion coefficient since the interactions between the particles are presumably negligible then. At nonzero volume fractions, particle inter-actions, such as those of electrostatic and hydrodynamic ori-gin, influence the diffusion. At low enough concentrations, where three and higher body interactions may be disre-garded, the parameter C characterizes the departure from
the single-particle result.
The concentration dependence of the collective diffusion coefficient of proteins has been studied extensively in experi-ments, for example, in the cases of hemoglobin,1–5 bovine serum albumin,6–9 -lactoglobulin,10 ovalbumin,11 and lysozyme.12–22 On the theoretical side, a fair number of papers23–31 deal with the diffusion of interacting colloidal particles in solution. Apart from giving insight into the dif-fusion as such, the coefficientCis also important because it
could yield information about the complex pair interaction between protein molecules. Moreover, it has been argued that Cmay be an alternative parameter useful in diagnosing
un-der what conditions proteins would crystallize.17
In Ref.32, we approximated globular proteins in water with added monovalent salt by hard spherical particles that interact through a short-range attraction and a screened elec-trostatic repulsion. We appropriately replaced this system by one of spherical particles with sticky interactions only. At infinite dilution the effective stickiness is readily determined
by equating the respective second virial coefficients of the two systems. In the effective stickiness, part of the bare ad-hesion is balanced against the electrostatic repulsion.
In the next section, we formulate a theory for the coef-ficient C. We first introduce the interaction used previously
to compute protein solution properties32and give expressions for the effective stickiness. We then outline the formal ex-pression forCdue to Felderhof23in terms of the pair
poten-tial between two protein spheres and a hydrodynamic mobil-ity function. Although the latter has been studied often in the past, we present a more extensive numerical analysis in order to gain more insight into the asymptotics of the lubrication regime for two moving spheres very close to each other. The coefficient C is then computed in three ways: exactly via
numerics and in terms of two convenient approximations. In Sec. III, we compare these predictions for C with
experi-ment. A discussion of the results is given in the last section.
II. THEORY
A. Effective interaction
We model the globular proteins as spherical particles of radius a with a total charge Zq per particle that is uniformly distributed over its surface. Here q is the elementary共proton兲 charge. For convenience, we scale all distances by the radius
a and all energies by kBT, where kBis Boltzmann’s constant and T is the temperature. We approximate the interaction between two proteins by a steric repulsion plus a short-range attraction of scaled range ␦Ⰶ1 and constant absolute mag-nitude UA, and a far-field Debye-Hückel potential. The latter
describes the Coulomb repulsion that is screened due to the presence of monovalent salt of ionic strength I. The effective number of charges Zeffassociated with the far field is
com-puted in the Poisson-Boltzmann approximation共see Refs.32
and 33 for further details兲. The total interaction UT共x兲
be-tween the two particles with center-of-mass separation r is thus of the form
a兲Author to whom correspondence should be addressed. Mailing address:
P.O. Box 11036, 2301 EA Leiden, The Netherlands. Electronic mail: odijktcf@wanadoo.nl
UT共x兲 =
冦
⬁, 0艋 x ⬍ 2 UDH共x兲 − UA, 2艋 x ⬍ 2 +␦ UDH共x兲, x艌 2 +␦,冧
共1兲 x⬅ r a. 共2兲Here, the Debye-Hückel interaction is given by
UDH共x兲 = 2e −共x−2兲
x , 共3兲
where ⬅共Q/2a兲共Zeff/共1+兲兲2 and ⬅a. The Debye
length −1 is defined by 2⬅8QI and the Bjerrum length
by Q⬅q2/⑀kBT, which equals 0.71 nm in water at 298 K
共⑀is the permittivity of water兲;= 3.28a
冑
I, if the radius a isgiven in nanometers and the ionic strength I in moles/l. We suppose 1-1 electrolyte has been added in excess so I is the concentration of added salt only. We have derived an exact perturbative expression for the effective charge qZeff in the
Poisson-Boltzmann approximation,32 Zeff= Z − 2 6
冉
Q a冊
2冉
Z 1 +冊
3 e3E1共3兲. 共4兲Here, E1共x兲 is the exponential integral defined by E1共x兲 =兰x⬁dtt−1e−t. It turns out that the first-order correction to the
bare charge given by Eq.共4兲is almost always small for pro-teins or nanocolloids so Eq. 共4兲 is a convenient expression valid under a wide variety of conditions. However, Eq.共4兲is not useful for highly charged particles of colloidal size be-cause the correction term is not perturbative then.
Analysis of the Poisson-Boltzmann equation for a single sphere has a long history which we cannot discuss fully here. Mathematically rigorous work on the “condensation” of counterions on highly charged spheres was already presented some time ago.34–37A simple physical argument for conden-sation was advanced in Ref.38. Various approximations for the potential at largeas a function of the distance have also been proposed39–41but the most complete solution appears to have been derived by Shkel et al.42 using the method of multiple scales. It is straightforward to obtain Zefffrom the
latter共see Ref.43兲. The resulting expression for the effective
charge is quite accurate at all values of the bare charge pro-videdⲏ1. In the case of proteins at large, it happens to be numerically very close to the expansion given by Eq.共4兲 but a small disparity remains because the original solution42 is not expanded beyond O共−1兲.
We want to replace the system of particles interacting through the complicated interaction关Eq.共1兲兴 by a system of particles interacting through a simpler potential, the adhesive hard sphere共AHS兲 potential of Baxter44
UAHS共x兲 =
冦
⬁, 0艋 x ⬍ 2
ln 12/共2 +兲, 2 艋 x 艋 2 +
0, x⬎ 2 +.
冧
共5兲 Here,is a positive constant which signifies the strength of the effective adhesion and the limit ↓0 has to be taken appropriately after formal integrations. In order to replace the original system by this simpler system, we have to find
the correspondence between the parameterin the AHS po-tential and the parameters , , ␦, and UA in the original
interaction关Eq.共1兲兴. In this case, we do this by matching the respective second virial coefficients, which ensures that the free energy of the two systems at small concentrations are identical. We emphasize that in the general case, at arbitrary concentrations, we have to match the complete free energies of the respective systems;32,33it is then incorrect to focus on the second virials as has often been done in the past.
B. Stickiness parameter
We already determined the stickiness parameter in a previous paper.32 Here we reproduce the main results. The second virial coefficient B2is given by
B2=1 2
冕
Vdr共1 − e−U共r兲兲, 共6兲
where U共r兲 is the pair potential scaled by kBT and r is the
unscaled position vector connecting the centers of mass of the two particles. For the pair interaction of Eq.共1兲, B2 may
be expressed by
B2= B2HS共1 + 3
8J兲, 共7兲
where we introduce the following integrals:
J⬅
冕
2 ⬁ dxx2共1 − e−UT共x兲兲 ⬅ J 1−共eUA− 1兲J2, 共8兲 J1⬅冕
2 ⬁ dxx2共1 − e−UDH共x兲兲 ⯝4共+ 1/2兲 2冉
1 − ␣ 2冊
, 共9兲 J2⬅冕
2 2+␦ dxx2e−UDH共x兲 ⯝ 2␦冋
e−+冉
1 +␦ 2冊
2 e−关/共1+␦/2兲兴e−␦册
. 共10兲 Here, B2HS= 16a3/ 3 is the value of B2 if the proteins weresolely hard spheres and ␣=关e−−共1−兲兴/2. We equate Eq. 共7兲 with the second virial coefficient of the AHS model,
B2= B2HS
冉
1 −1
4
冊
, 共11兲which results in a stickness parametergiven by = − 2
3J. 共12兲
From Eqs.共1兲and共8兲we see how part of the original attrac-tion is compensated by repulsive electrostatics.
C. General expression forC
For small volume fractionsof spherical particles, the collective diffusion coefficient DC may be written as
DC= D0共1 + C+ O共2兲兲, 共13兲
where D0 is the diffusion coefficient in the dilute limit. The
C=V+O+D+S+A. 共14兲
These terms have been studied for some time:23–26there is a virial correction because of a fluctuation in the osmotic pres-sure drives diffusion,
V= 3
冕
0 ⬁dxx2共1 − e−U共x兲兲, 共15兲
and four terms arising from the mutual friction between two hydrodynamically interacting spheres. An Oseen contribu-tion,
O= 3
冕
0 ⬁dxx共e−U共x兲− 1兲, 共16兲 and a dipolar contribution,
D= 1, 共17兲
express the long-range hydrodynamic interaction between two particles 1 and 2, whereas the short-range part of the hydrodynamic interaction comes into play in the term
S=
冕
2 ⬁dxx2e−U共x兲
冉
A12tt共x兲 + 2B12tt共x兲 −3x
冊
. 共18兲Finally, the modification of the single-particle mobility is expressed by
A=
冕
2 ⬁dxx2e−U共x兲共A11tt共x兲 + 2B11tt共x兲兲. 共19兲 Here, A11tt共x兲, A12tt共x兲, B11tt共x兲, and B12tt共x兲 are dimensionless hydrodynamic functions given in terms of the translational mobility matrix for two spheres centered at R1 and R2 共r=R1− R2兲 and acquiring velocities V1and V2as a result of
the forces F1 and F2acting on the spheres,
V1=11tt共1,2兲 · F1+12tt共1,2兲 · F2, 共20兲
V2=21tt共1,2兲 · F1+22tt共1,2兲 · F2. 共21兲
In the notation of Cichocki and Felderhof,45we have
11tt共1,2兲 = 1 6a
冋
ជជI+ A11 tt共r兲rr r2 + B11 tt共r兲冉
ជជI−rr r2冊
册
, 共22兲 12 tt共1,2兲 = 1 6a冋
A12 tt共r兲rr r2 + B12 tt共r兲冉
I ជជ−rr r2冊
册
, 共23兲whereis the viscosity of the solvent and Iជជis the unit tensor. The mobility tensors in Eq.共21兲are given by interchanging the labels in Eqs.共22兲and共23兲while taking into account the symmetry relations,
A12tt共r兲 = A21tt共r兲, B12tt共r兲 = B21tt共r兲. 共24兲
Recall that the particles have a hard-core interaction for
x⬍2 so exp−U共x兲 vanishes for x⬍2. We then sum Eqs. 共15兲–共19兲and conveniently rewrite Cas follows:
C= c0+ c1
冕
2 ⬁dxx2共1 − e−U共x兲兲 + R. 共25兲
The constant c0 equals the value C would adopt if the
spheres were hard but without any other interaction,
c0⬅ 3
冕
0 2 dxx2− 3冕
0 2 dxx + 1 +冕
2 ⬁ dxx2冉
h共x兲 −3 x冊
= 3 +冕
2 ⬁ dxx2冉
h共x兲 −3 x冊
. 共26兲Here, h共x兲 is the sum of scalar mobility functions,
h共x兲 ⬅ A11tt共x兲 + A12tt共x兲 + 2B11tt共x兲 + 2B12tt共x兲. 共27兲
The residual term R in Eq.共25兲depends on the actual inter-action,
R⬅
冕
2⬁
dxx2共e−U共x兲− 1兲共h共x兲 − h共2兲兲, 共28兲 though it would vanish if the interaction U was adhesive and purely of the Baxter-type关see Eq.共5兲兴. The second term on the right hand side of Eq.共25兲is proportional to the constant
c1⬅ 3 − h共2兲, 共29兲
and the integral is related to the second virial coefficient B2
by关see Eqs.共7兲 and共8兲兴
冕
2 ⬁ dxx2共1 − e−U共x兲兲 =8 3冉
B2 B2HS− 1冊
. 共30兲The resulting expression forCis
C= c0+
8c1 3
冉
B2
B2HS− 1
冊
+ R, 共31兲which we can evaluate once we know h共x兲 given by Eq.共27兲.
D. Hydrodynamics
The function h共x兲 was discussed by Batchelor46 in his theory of the diffusion of hard spheres. The sum A11tt+ A12tt pertains to the mobility of a pair of spheres moving in the direction of their line of centers, whereas B11tt + B12tt is related to their mobility when they move perpendicular to that line. 共Note that in Ref. 46, A11⬅A11tt + 1, B11⬅B11tt + 1, A12⬅A12tt,
and B12⬅B12tt兲. In the latter case, because the spheres are
couple-free, the spheres must rotate as the pair translates. At small separations 共x−2Ⰶ1兲, lubrication forces with a loga-rithmic singularity ln−1共x−2兲 are then expected to develop on general grounds.47 Goldman et al.48 proposed a form for the singularity which we will test below.
Batchelor46 computed h共2兲=1.312 on the basis of nu-merical work on the mobilities of touching spheres.49,50 Cichocki and Felderhof45 evaluated c0= 1.454关Eq. 共26兲兴 by
We assume that the interaction U共x兲 is of short range so we focus only on h共x兲 for x−2ⱗ1. First, we get an expres-sion for A11tt + A12tt as an infinite sum from the results of Stim-son and Jeffery52who expressed the hydrodynamic problem in terms of bispherical coordinates. 共Note that there is an error in their paper as pointed out in, for example, Ref.53in which one may find a similar expression for A11tt− A12tt in case one needs A11tt and A12tt separately兲. Calculating B11tt and B12tt is more involved. We use the numerical scheme by O’Neill and Majumdar,54 which is similar to that of Goldman et al.48 Note that there are a few typographical errors in Ref.54. In their Eq. 共3.9兲 d should be d1, the expression for v in Eq.
共4.1兲 should have a minus sign, ,, and in Eqs. 共4.3兲– 共4.5兲 should be replaced by c, c, and c, respectively, and sinh2兩兩 in Eq. 共5.10兲 should be sinh3兩兩. Also, to obtain
D2共An, Bn兲 关Eq. 共3.29兲兴 from D1共An, Bn兲 关Eq. 共3.28兲兴 the
signs of ␦n−1, ␦n, and␦n+1 should be reversed as well 共we
only checked the case of spheres of equal size兲. Their Table
Iis correct, however, for spheres of equal size, apart from the value for g12共1,0.1兲 which should read −0.1017 instead of −1.1017.
In order to investigate the regime of lubrication for a pair of spheres moving under the action of applied forces normal to their line of centers, we performed the numerical analysis down to r / a − 2 = 10−10which implies that 2⫻106 terms in
the series expansions are needed. We attempted to speed up the iteration by adapting the recurrence relationships intro-duced more recently by O’Neill and Bhatt55 for a sphere moving near a wall to the case of two spheres. However, this did not turn out to be useful as it is for the wall configuration.56One way of circumventing series expansions could be to elaborate on the trial functions initially used by Fixman in his variational theorem for the mobility matrix57 but we did not investigate this.
Goldman et al.48 were the first to give a comprehensive analysis of the mobility of a pair of identical spheres of ar-bitrary orientation. They numerically solved the Stokes and continuity equations using expansions in terms of bipolar coordinates to a very high order. For moving spheres whose line of centers is perpendicular to the applied force, the force
consists not only of a term arising from pure translation but also a term stemming from pure rotation of the spheres. The latter involves a torque on one sphere diverging as48
Tr⬃ 3 ln共x − 2兲
160⍀a3 共32兲
at very small separations, where ⍀ is its angular velocity. Equation共32兲was derived by extending the nontrivial lubri-cation theory of Ref. 58 in which inner and outer regions have to be matched. Equation 共32兲 ultimately leads to the following analytical expression for h共x兲 valid at small sepa-rations:
h共x兲 = h共2兲 − 0.476 66
ln共x − 2兲 + c2
+ O共x − 2兲. 共33兲
The coefficient 0.476 66 is computed from the numerical tables presented in Ref.48. We have added a constant c2 to
the logarithm because we expect the next higher order term in Eq.共32兲to be a constant, judging by the earlier analysis of the sphere-wall problem.58 In Fig.1we have fitted Eq.共33兲 to the numerical results discussed above, letting h共2兲 and c2
be adjustable. The intercept h共2兲=1.309 93 turns out to be close to the value of 1.312 quoted above for touching spheres which lends credence to the validity of the asymptotic ex-pression that we propose. Moreover, the resulting coefficient
c2= −4.694 and the concomitant shift in Eq.共32兲are
consis-tent with the numerical values of the torque Trat small
sepa-rations, as presented in Table 3 of Ref.48.
Next, we derive an expression for the residual term given by Eq.共28兲. First, we propose an initial estimate h0共x兲
for h共x兲. We have plotted the numerical values of h共x兲 as a function of x in Fig.2. As a result of the lubrication regime,
h has a maximum, as displayed in the inset. However, h共x兲 is
only a strongly varying function for x⬍2.04. We therefore simply force a linear fit to the data for h at x = 2.1, 2.2, and 2.3,
TABLE I. Values of the actual charge Z of hen-egg-white lysozyme共from Ref.60兲, the effective charge Zeff关see Eq.共4兲兴, the lowered effective charge Z¯
= Zeff− 1, and dimensionless interaction parametersandas a function of the ionic strength I. The pH equals 4.5 andhas been calculated using the lowered
effective charge Z¯ . R has been calculated from Eq.共39兲,from Eq.共12兲,C共via兲 from Eq.共36兲, andC共direct兲 from Eq.共38兲. In all cases approximations
for J and K given by Eqs.共8兲–共10兲,共40兲,共43兲, and共44兲were used. The computation of the numerically exactCis explained in the text.
I共M兲 Z Zeff ¯Z R
C
Via Direct Exact
h0共x兲 ⬇ 1.3670 − 0.4745共x − 2兲. 共34兲
We then insert this estimate into Eq.共28兲and add a correc-tion term so as to derive an expression for R accurate enough for our purposes,
R⬇ − 0.147
冉
B2 B2HS− 1冊
+ 0.4745冕
2 ⬁ dxx2共x − 2兲共1 − e−U共x兲兲 + 9⫻ 10−4共1 − e−U共2兲兲. 共35兲The first term on the right comes from the fact that the linear interpolation gives h0共2兲=1.3670, whereas the real value is h共2兲=1.312. Since the interaction usually does not change
appreciably for 2⬍x⬍2.04, it is straightforward to write an
estimate for the error—the third term—owing to the devia-tion of Eq.共34兲from the exact function h共x兲 共see inset in Fig.
2兲. In our case the error term turns out to be an order of
magnitude smaller than the first two terms.
E. Determination ofC
It is clear from Eq.共28兲that R would vanish if the actual interaction were a pure AHS potential. If we then insert Eq.
共11兲 into Eq.共31兲, we obtain26 C= c0−
2c1
3. 共36兲
Inspection of the various terms in Eq. 共35兲reveals that R is often much smaller than unity when the interaction is given by Eq. 共1兲. Hence, a possibly convenient approximation to the coefficient Cis from Eq. 共12兲,
C= c0+ c1J = c0+
8c1
3
冉
B2
B2HS− 1
冊
, 共37兲where J may be evaluated numerically or approximately with the help of Eqs.共8兲–共10兲.
The full expression for the dynamical coefficient is writ-ten as
C= c0+ c1J + R, 共38兲
using Eqs.共11兲and共31兲. Now R from Eq.共35兲is reexpressed as
R⬇ − 0.055J + 0.4745K − 9 ⫻ 10−4共eUA−− 1兲, 共39兲
in view of Eqs. 共1兲 and 共3兲. Here we have introduced the function K for which we derive a convenient approximation,
K⬅
冕
2 ⬁ dxx2共x − 2兲共1 − e−UT共x兲兲 ⬅ K 1−共eUA− 1兲K2, 共40兲 where K1⬅冕
2 ⬁ dxx2共x − 2兲共1 − e−UDH共x兲兲 共41兲 and K2⬅冕
2 2+␦ dxx2共x − 2兲e−UDH共x兲. 共42兲In the same spirit as in Ref. 32, we approximate
x共1−e−UDH共x兲兲⬇2e−共x−2兲− 2␣2e−2共x−2兲, with ␣=关e−
−共1−兲兴/2. We then have K1⬇
共+ 1兲共4 −␣兲
3 , 共43兲
where we have neglected the small term␣2/ 23. In the case
of lysozyme at pH 4.5, the deviation of Eq. 共43兲 from the exact result is smaller than about 3% for I艌0.05M and smaller than about 1% for I艌0.3M. For the second integral we use the trapezoid approximation 兰22+␦dxg共x兲⬇12␦关g共2兲
+ g共2+␦兲兴, ␦Ⰶ1, and we neglect a factor 共1+␦/ 2兲2,
FIG. 1. The hydrodynamic function h plotted in terms of the variable s = −1 /共ln共x−2兲−4.694兲. Squares denote results from the numerical analysis to the accuracy as explained in the text. The straight line signifies the function
h = 0.476 66s + 1.309 93.
K2⬇ 2␦2exp
冋
− e−␦1 +␦/2
册
. 共44兲For lysozyme at pH 4.5 with ␦= 0.079共see below兲, this ap-proximation deviates less than about 5% from the exact value for I艌0.05M and less than about 3% for I艌0.2M.
III. COMPARISON WITH EXPERIMENT
We compare our predictions ofC as a function of the
ionic strength I with experimental results for lysozyme at room temperature and at a pH of about 4.5. The added salt is NaCl and in most cases, a small amount of Na acetate has been added as buffer. The reason for choosing lysozyme un-der these conditions is that we have previously evaluated the range and strength of the short-range attraction32and a lot of experimental data on the collective diffusion coefficient are available in the literature共see Fig.3兲.
Lysozyme has a moderate aspect ratio of about 1.5 and we approximate it by a sphere of radius a = 1.7 nm.59 The dimensionless parameter is then given by = 5.58
冑
I,where the ionic strength I is given in moles/l, and = 0.209共Z¯/共1+兲兲2. Here we follow our discussion in Ref. 32 and use the adjusted charge on the lysozyme sphere
Z
¯ =Zeff− 1 instead of the effective charge Zeff. Values of Z, Zeff, and Z¯ as a function of ionic strength can be found in
TableIas well as the corresponding quantitiesand. For the range ␦ and strength UA of the attraction we use ␦
= 0.079 and UA= 3.70, which were computed on the basis of
a wide variety of data on the second virial coefficient.32 We next employ three methods to predict C
theoreti-cally. In the first method, we compute by equating the respective second virial coefficients of Sec. II B关see Eqs.共8兲 and 共12兲兴. We then calculate C from Eq. 共36兲 using c0
= 1.454 and c1= 1.688. In the second method, we use Eq.共38兲
to determine C, where R is evaluated with the help of Eq. 共39兲. In both cases the approximations for J and K given by Eqs.共8兲–共10兲,共40兲,共43兲, and共44兲were used共see TableIand Fig.3兲. Note that there are no free parameters so the curves
in Fig.3 are predictions. For comparison, we also calculate Cfrom Eq.共14兲exactly, that is, by performing the integrals
in Eqs. 共15兲–共19兲numerically with the help of a highly ac-curate interpolation formula for h共x兲 共see TableI兲. Finally, in
Fig.3, we have also plotted data ofCmeasured by several
experimental groups.
IV. DISCUSSION
In Sec. II E we have outlined two approximate methods to calculateC. As one can see from TableI, both the direct
method incorporating an approximation for the residual R and the method relying solely on the stickiness via the second virial yield results that are often close to the exact numerical computations. The direct method is, of course, somewhat more accurate. Themethod breaks down below 0.2M. Note that in the important regime I⬎0.2M pertaining to protein crystallization, R is much smaller than the absolute magnitude of C. This may explain whyC is a useful
pa-rameter to characterize the onset of crystallization.17 We also note in Fig.3 that Cdecreases monotonically
with increasing ionic strength or effective attraction. The friction per particle becomes less as the chance of two spheres clustering together is enhanced, yet this is offset by the decrease in the osmotic pressure driving the diffusion, for numerical reasons 关see Eqs.共25兲 and共29兲, the residual R is merely a perturbation兴.
In Fig.3, it is clear that there is a large degree of scatter which may be attributed to the systematic variation in sets of data from various groups, especially at large ionic strengths 共I⬎0.4M兲. We do not know what is the cause of this. In one experiment,13we do observe that there is considerable scatter in a plot of the diffusion coefficient versus the protein solu-bility, which might explain the extreme downturn of several data in Fig. 3 at about 0.5M. Figure 3 also shows that our predicted curves lie fairly neatly in the midst of the swarm of data. We emphasize again that we have no adjustable param-eters in our calculations except for a slight downward adjust-ment of the effective charge共see also the discussion in Refs.
32and33兲. The model is thus not inconsistent with the
ex-perimental data though we will have to await more experi-ments under conditions which are better controlled before one may reach a more definitive conclusion. In a similar vein, it is not possible to claim that the neglect of electrolyte friction assumed here is entirely warranted.
In summary, we have approximated proteins by spherical particles interacting by a hard core and electrostatic repulsion together with a short-range attraction. An analysis of the FIG. 3. Experimental data and theoretical predictions ofCfor lysozyme as
a function of the ionic strength I at a pH of about 4.5. Black squares, Nyström and Johnsen共Ref.12兲 4.0, 25 °C; gray squares, Mirarefi and
Zu-koski共Ref.13兲, pH 4.6; white squares, Mirarefi and Zukoski 共Ref.13兲, pH
4.6; black diamonds, Muschol and Rosenberger共Ref.14兲, pH 4.7, 20 °C;
gray diamonds, Zhang and Liu共Ref.15兲, pH 4.5, 20 °C; white diamonds,
Skouri et al.共Ref.16兲, pH 4.6, 20 °C; black triangles, Eberstein et al. 共Ref. 17兲, pH 4.2, 20 °C; gray triangles: Leggio et al. 共Ref.18兲 pH 4.75, 25 °C;
white triangles, Price et al.共Ref.19兲, pH 4.6, 25 °C; black circles,
Annun-ziate et al.共Ref.20兲, pH 4.5, 25 °C; gray circles, Annunziata et al. 共Ref. 20兲, pH 4.5, 25 °C; and white circles, Retailleau et al. 共Ref.21兲, pH 4.0. In
two-particle statistics and hydrodynamics leads to a reason-able prediction of the ionic strength dependence of the linear coefficientC. At high ionic strengths, when B2 is negative,
the residual R is relatively small so there is then an interest-ing direct relationship betweenC and B2 关Eq. 共37兲兴 which
could be tested experimentally.
1S. S. Alpert and G. Banks, Biophys. Chem. 4, 287共1976兲. 2A. P. Minton and P. D. Ross, J. Phys. Chem. 82, 1934共1978兲. 3R. S. Hall, Y. S. Oh, and C. S. Johnson, Jr., J. Phys. Chem. 84, 756
共1980兲.
4K. J. LaGattuta, V. S. Sharma, D. F. Nicoli, and B. K. Kothari, Biophys.
J. 33, 63共1981兲.
5S. Beretta, G. Chirico, D. Arosio, and G. Baldini, J. Chem. Phys. 106,
8427共1997兲.
6I. Tinoco, Jr. and P. A. Lyons, J. Phys. Chem. 60, 1342共1956兲. 7T. Raj and W. H. Flygare, Biochemistry 13, 3336共1974兲.
8G. D. J. Phillies, G. B. Benedek, and N. A. Mazer, J. Chem. Phys. 65,
1883共1976兲.
9J. L. Anderson, F. Rauh, and A. Morales, J. Phys. Chem. 82, 608共1978兲. 10C. Le Bon, T. Nicolai, M. E. Kuil, and J. G. Hollander, J. Phys. Chem. B
103, 10294共1999兲.
11S. J. Gibbs, A. S. Chu, E. N. Lightfoot, and T. W. Root, J. Phys. Chem.
95, 467共1991兲.
12B. Nyström and R. M. Johnsen, Chem. Scr. 22, 82共1983兲. 13A. Y. Mirarefi and C. F. Zukoski, J. Cryst. Growth 265, 274共2004兲. 14M. Muschol and F. Rosenberger, J. Chem. Phys. 103, 10424共1995兲. 15J. Zhang and X. Y. Liu, J. Chem. Phys. 119, 10972共2003兲.
16M. Skouri, J. P. Munch, B. Lorber, R. Giegé, and S. Candau, J. Cryst.
Growth 122, 14共1992兲.
17W. Eberstein, Y. Georgalis, and W. Saenger, J. Cryst. Growth 143, 71
共1994兲.
18C. Leggio, L. Galantini, E. Zaccarelli, and N. V. Pavel, J. Phys. Chem. B
109, 23857共2005兲.
19W. S. Price, F. Tsuchiya, and Y. Arata, J. Am. Chem. Soc. 121, 11503
共1999兲.
20O. Annunziata, D. Buzatu, and J. G. Albright, Langmuir 21, 12085
共2005兲.
21P. Retailleau, M. Riès-Kautt, A. Ducruix, L. Belloni, S. J. Candau, and J.
P. Munch, Europhys. Lett. 46, 154共1999兲.
22J. Narayanan and X. Y. Liu, Biophys. J. 84, 523共2003兲. 23B. U. Felderhof, J. Phys. A 11, 929共1978兲.
24C. van den Broek, F. Lostak, and H. N. W. Lekkerkerker, J. Chem. Phys.
74, 2006共1981兲.
25T. Ohtsuki and K. Okano, J. Chem. Phys. 77, 1443共1982兲. 26B. Cichocki and B. U. Felderhof, J. Chem. Phys. 93, 4427共1990兲.
27U. Genz and R. Klein, Physica A 171, 26共1991兲.
28D. N. Petsev and N. D. Denkov, J. Colloid Interface Sci. 149, 329
共1992兲.
29N. D. Denkov and D. N. Petsev, Physica A 183, 462共1992兲.
30D. N. Petsev, N. D. Denkov, and K. Nagayama, Chem. Phys. 175, 265
共1993兲.
31J. Gapinski, A. Wilk, A. Patkowski, W. Häußler, A. J. Banchio, R.
Pecora, and G. Nägele, J. Chem. Phys. 123, 054708–1共2005兲.
32P. Prinsen and T. Odijk, J. Chem. Phys. 121, 6525共2004兲. 33P. Prinsen and T. Odijk, J. Chem. Phys. 125, 074903共2006兲. 34M. A. Lampert and R. S. Crandall, Chem. Phys. Lett. 68, 473共1979兲. 35M. A. Lampert and R. S. Crandall, Phys. Rev. A 21, 362共1980兲. 36N. Martinov, D. Ouroushev, and E. Chelebiev, J. Phys. A 19, 1327
共1986兲.
37G. V. Ramanathan, J. Chem. Phys. 88, 3887共1988兲.
38S. Alexander, P. M. Chaikin, P. Grant, G. J. Morales, P. Pincus, and D.
Hone, J. Chem. Phys. 80, 5776共1984兲.
39W. C. Chen and P. N. Sen, J. Chem. Phys. 77, 2042共1982兲.
40H. Ohshima, T. W. Healy, and L. R. White, J. Colloid Interface Sci. 90,
17共1982兲.
41G. Weisbuch and M. Gueron, J. Phys. Chem. 85, 517共1981兲. 42I. A. Shkel, O. V. Tsodikov, and M. T. Record, Jr., J. Phys. Chem. B 104,
5161共2000兲.
43M. Aubouy, E. Trizac, and L. Bocquet, J. Phys. A 36, 5835共2003兲. 44R. J. Baxter, J. Chem. Phys. 49, 2770共1968兲.
45B. Cichocki and B. U. Felderhof, J. Chem. Phys. 89, 1049共1988兲. 46G. K. Batchelor, J. Fluid Mech. 74, 1共1976兲.
47G. K. Batchelor and J. T. Green, J. Fluid Mech. 56, 375共1972兲. 48A. J. Goldman, R. G. Cox, and H. Brenner, Chem. Eng. Sci. 21, 1151
共1966兲.
49M. D. A. Cooley and M. E. O’Neill, Proc. Cambridge Philos. Soc. 66,
407共1969兲.
50A. Nir and A. Acrivos, J. Fluid Mech. 59, 209共1973兲. 51D. J. Jeffrey and Y. Onishi, J. Fluid Mech. 139, 261共1984兲.
52M. Stimson and G. B. Jeffery, Proc. R. Soc. London, Ser. A 111, 110
共1926兲.
53H. Brenner, Chem. Eng. Sci. 16, 242共1961兲.
54M. E. O’Neill and S. R. Majumdar, Z. Angew. Math. Phys. 21, 164
共1970兲.
55M. E. O’Neill and B. S. Bhatt, Q. J. Mech. Appl. Math. 44, 91共1991兲. 56M. Chaoui and F. Feuillebois, Q. J. Mech. Appl. Math. 56, 381共2003兲. 57M. Fixman, J. Chem. Phys. 76, 6124共1982兲.
58A. J. Goldman, R. G. Cox, and H. Brenner, Chem. Eng. Sci. 22, 637
共1967兲.
59V. Mikol, E. Hirsch, and R. Giegé, J. Mol. Biol. 213, 187共1990兲. 60D. E. Kuehner, J. Engmann, F. Fergg, M. Wernick, H. W. Blanch, and J.
