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Langmuir 2002, 18, 8910-8918
Long Range Interactions between Apoferritin Molecules Marian Manciu and Eli Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received June 25, 2002. In Final Form: September 11, 2002 Recent light scattering experiments regarding the apoferritin molecules (the hollow shells of the ironstorage protein ferritin) indicated a surprising dependence of the repulsion between proteins on the electrolyte concentration (NaCH3COO). The second virial coefficient decreased to a value close to that corresponding to hard spheres for 0.15 M but increased to a very large value at 0.25 M. The results are difficult to be interpreted in the classical framework through the addition of double layer and hydration repulsive forces. While the double layer theory can predict the behavior of the virial coefficient at low electrolyte concentrations, only an abnormally large charge can explain the values of the virial coefficient at high ionic strengths. Alternatively, the traditional hydration force should increase with orders of magnitude between 0.15 and 0.25 M, to be consistent with experiment, and this is unlikely to happen. In this paper, it is shown that a unitary treatment of the repulsion (the double layer and the polarization-based hydration repulsions) might explain the unexpected values of the second virial coefficient and the corresponding long-ranged repulsion.
I. Introduction The interaction between charged particles immersed in an electrolyte solution is traditionally described by the Derjaguin-Landau-Verwey-Overbeek theory in terms of an attraction, due to the correlations between the instantaneous electronic dipoles of the particles (van der Waals force), and a screened Coulomb interaction, due to the charges on the particles’ surfaces (the double layer force).1 It is known that the theory is accurate only in a certain range of electrolyte concentrations (1.0 × 10-3 to 5 × 10-2 M), and a number of improvements were proposed for the calculation of the interactions, such as, for example, the accounting of the retardation2 and the field-theoretical treatment for the van der Waals interactions3 and the accounting of the dielectric saturation at high fields,4 the image forces,5 finite ion sizes,6 and the correlation between ions7 for the double layer interactions. A long time ago,Voet prepared stable sols of various metals (Pt, Pd) and salts (sulfides, halides) in highly concentrated solutions of sulfuric acid, phosphoric acid, and calcium chloride in water.8 Dilution with water induced their coagulation. More recent experiments revealed that amphoteric latex particles did not coagulate, even at high ionic strengths (above 1 M) of LiNO3.9 All the above experiments contradict the DLVO theory. The stability of some colloids at high electrolyte concentrations * Corresponding author. E-mail address: feaeliru@ acsu.buffalo.edu. Phone: (716) 645-2911/2214. Fax: (716) 645-3822. (1) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of stability of lyophobic colloids; Elsevier: Amsterdam, 1948. (2) Casimir, H. B.; Polder, D. Phys. Rev. 1948, 73, 360. (3) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. Adv. Phys. 1961, 10, 165. (4) Henderson, D.; Lozada-Casou, M. J. Colloid Interface Sci. 1986, 114, 180. (5) Jo¨nnson, B.; Wennerstro¨m, H. J. Chem. Soc., Faraday Trans. 1983, 79, 19. (6) Levine, S.; Bell, G. M. Discuss. Faraday Soc. 1966, 42, 69. Ruckenstein, E.; Schiby, D. Langmuir 1985, 1, 612. (7) Wennerstro¨m, H.; Jo¨nnson, B.; Linse, P. J. Chem. Phys. 1982, 76, 4665. Kjellander, R.; Marcelja, S. J. Chem. Phys. 1985, 82, 2122. (8) Voet, A. Thesis, Amsterdam, 1935. See also: Kruyt, H. R. Colloid Science; Elsevier: Amsterdam, 1952. (9) Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraday Discuss. Chem. Soc. 1978, 65, 156.
led to the conjecture that another repulsion (a non-DLVO one), due to the organization of the solvent around particles, should also be present. The existence of such a force was demonstrated both experimentally and theoretically for any solvent,10 being particularly strong for the polar ones. The force (called hydration force when the solvent is water) is partly responsible for the stability of multilayers of neutral lipid bilayers/water, a system that was extensively investigated because of its relevance to biology.11 One of the first quantitative models12 of the hydration force was based on the polarization of water molecules in the vicinity of a surface. Another model involving polarization, suggested by Gruen and Marcelja,13 was shown later to have some inconsistencies.14 Because Monte Carlo simulations15 indicated that the average polarization of water molecules oscillates in the vicinity of a surface, both polarization models have been contested. However, it was recently shown16 that the Schiby and Ruckenstein model12 can lead to an oscillatory profile of the polarization, if the water is assumed to be organized in icelike layers in the vicinity of the surface. This model can relate the strength of the hydration force to the surface dipole density, a dependence which was observed experimentally,17 and could also explain the restabilization of some colloids at high ionic strengths.18 The hydration repulsion is expected to be affected by the hydrogen bonding. When two surfaces approach each other, the increase of the free energy (due to the disruption of the hydrogen bonds) generates repulsion. Indeed, Monte (10) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Suface Forces; Plenum: 1987. (11) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. (12) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (13) Gruen, D. W. R.; Marcelja, S. Faraday Trans. 2 1983, 211 and 225. (14) Attard, P.; Wei, D. Q.; Patey, G. N. Chem. Phys. Lett. 1990, 172, 69. (15) Kjellander, R.; Marcelja, S. Chem. Scr. 1985, 25, 73; Chem. Phys. Lett. 1985, 120, 393. (16) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7582. (17) Simon, S. A.; McIntosh, T. J. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 9263. (18) Manciu, M.; Ruckenstein, E. Langmuir 2001, 17, 7061.
10.1021/la026126m CCC: $22.00 © 2002 American Chemical Society Published on Web 10/15/2002
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Langmuir, Vol. 18, No. 23, 2002 8911
Carlo simulations15 indicated that a disruption of the hydrogen bonding occurs when two surfaces approach each other, and lattice models for the calculations of such interactions have been proposed.19 We consider, however, that an important contribution to the hydrogen bonding is already contained in the dipole-dipole interactions included in the polarization model. The restabilization of colloids at high ionic strengths observed experimentally20 (the coagulation rate having a maximum with increasing electrolyte concentration) could be explained in terms of the reassociation of charges on the interface and the formation of surface ion pairs (dipoles).18 While, at low ionic strengths, the repulsion decreases with increasing electrolyte concentration because of the electrostatic screening, at high ionic strengths, the reassociation increases the density of ion pairs on the surface, and this generates a strong repulsion. A restabilization of some colloidal systems at high ionic strength could be explained if the magnitude of the hydration force increases moderately with the addition of electrolyte.18 There are, however, some recent striking experimental results regarding the interaction between apoferritin molecules,21 which are more difficult to explain. Light scattering experiments21 on solutions of apoferritin molecules (the hollow shells of ferritin, an iron-storage globular protein) in an acetate buffer at pH ) 5.0 have indicated that the dimensionless second virial coefficient (the ratio between the second virial coefficient and the volume of the molecule) first decreases with increasing electrolyte concentration, becomes close to 4 (value corresponding to hard-core interactions alone) for an electrolyte concentration in the range 0.08-0.18 M, but increases markedly at higher ionic strengths, reaching a very large value (∼13) at 0.25 M. The dimensionless second virial coefficient is defined as ∞ 2 r (1 - exp(∫r)0
˜2 ) 3 B 2a3
4+
))
F(r) kT
dr )
∞ r2(1 - exp(∫r)2a
3 2a3
))
F(r) kT
dr (1)
where a ) 63.5 Å represents the radius of the spherical apoferritin molecules, r the distance between the centers of two particles, k the Boltzmann constant, T the absolute temperature, and F(r) the interaction potential between particles, which was assumed to be the sum between a hard core repulsion (F(r) ) ∞ for r < 2a) and the other interactions (double layer, hydration and van der Waals). The second virial coefficient is not very sensitive to the magnitude of the repulsion, as long as the interaction energy exceeds a few kT, but is extremely sensitive to the range of the interaction (the distance between the centers of the particles at which the interaction becomes comparable to kT). At low electrolyte concentrations, the experimental results can be explained within the DLVO theory. A wellknown approximation for the double layer interaction between weakly charged spheres, at constant surface charge, is1 (19) Attard, P.; Batchelor, M. T. Chem. Phys. Lett. 1988, 149, 206. (20) Molina-Bolivar, J. A.; Galisteo-Gonzalez, F.; Hidalgo-Alvarez, R. Phys. Rev. E 1997, 55, 4522. (21) Petsev, D. N.; Vekilov, P. G. Phys. Rev. Lett. 2000, 84, 1334.
(
FDL(r) )
)
r - 2a λD a 2 4π0 1 + r λD
(ne)2 exp -
(
)
(2)
where n is the number of charges on each protein, e the protonic charge, 0 the vacuum permittivity, the dielectric constant, and λD the Debye-Hu¨ckel length (λD ) (0kT/ 2e2CE)1/2, CE being the concentration of the 1:1 electrolyte). Using eqs 1 and 2, the experimental virial coefficient for CE ) 0.01 M could be obtained (neglecting the van der Waals interactions) by assuming n ) -38. This value is reasonable, since the isoelectric point of apoferritin is about 4.0 and the experiments were performed at pH ) 5.0.21 This is a consequence of the long-ranged double layer repulsion at low electrolyte concentrations. However, at 0.25 M the Debye-Hu¨ckel length is about 6.2 Å and the value B ˜ 2 ) 13 can be obtained only if the charge of each molecule would be of the order of |103e|, which is most unlikely. Can the conventional hydration force be responsible for this large value of B ˜ 2? If one assumes an exponential form (a commonly used approximation)11 for the hydration interaction between two planar surfaces separated by a distance D
( Dλ )
(D) ) A exp F planar H
(3a)
where λ is the hydration decay length and A is a preexponential constant, the interaction between two spheres of radius a becomes in the Derjaguin approximation
FH(r) ) πa
∞ r - 2a F planar (D) dD ) πaλA exp(∫r-2a H λ )
(3b) For a decay length of 14.4 Å (as selected in ref 21), a ) 63.5 Å and T ) 300 K, a value B ˜ 2 ∼ 13 could be obtained for A ) 5 × 10-3 J/m2. These values of λ and A are, however, too large, since the values for the hydration decay length reported recently are closer to 2 Å.11 Assuming that λ ) 14.4, independent of the electrolyte concentration, the value of the parameter A has to decrease by 2 orders of magnitude to provide the low value B ˜ 2 ∼ 4.5 (close to the value corresponding to the hard-core repulsion). As noted above, a small increase of A by a few percent with increasing electrolyte concentration is not unexpected (due to the charge recombination), but what mechanism could explain the 102 fold increase of A when the ionic strength changes from 0.15 to 0.25 M? If one assumes a decay length λ ) 2 Å and a preexponential factor A ) 0.05 J/m2 (which are typical values for lipid bilayers),11 one obtains B ˜ 2 ∼ 4.9. However, the preexponential factor must increase by more than 11 orders of magnitude to provide B ˜ 2 ∼ 13, which is, of course, unreasonable. To summarize the difficulties, if the hydration repulsion would be responsible for the long-ranged repulsion of apoferritin protein at high ionic strengths (0.25 M), a theory of the hydration force should be able to predict a large decay length at large separations and explain why such a repulsion was not observed in the interactions between two phospholipid bilayers. Second, it should also explain why a repulsion with a much shorter decay length (about 2 Å) has been typically determined in the latter experiments. Third (and more importantly), it should
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Langmuir, Vol. 18, No. 23, 2002
explain why there is such a strong increase of the repulsion when the electrolyte concentration changes from 0.15 to 0.25 M. As already noted, the restabilization of some colloids at high ionic strength found in previous experiments20 can be explained in the traditional framework of the additivity between double layer and hydration forces, by a slight increase of the hydration repulsion caused by the increase in surface ion pair (dipole) density with electrolyte concentration. However, the increase in repulsion due to this mechanism is much too low to explain the strong increase of the second virial coefficient. To have an idea about the range of the repulsion required to provide such a high virial coefficient, it should be noted that, if the hard-core repulsion, infinite in magnitude, is extended with 15 Å (above the 2a separation), B ˜ 2 increases from 4 to only 5.6. If the range of the hard-core repulsion is extended with 30 Å, B ˜ 2 increases to 7.55, while 60 Å leads to 12.8. From these simple estimations one can infer that the repulsion needed to explain the measured second virial coefficient for apoferritin molecules should have a much longer range than that typically observed for the traditional hydration force. A new theory for the hydration force was proposed recently by Paunov et al. and used to explain the existence of a minimum of B ˜ 2 with increasing electrolyte concentration.22 However, the predicted interactions vanish for distances that exceed two hydration diameters (about 14 Å), while the hydration force necessary to explain the high values of B ˜ 2 determined experimentally requires a much longer range. The purpose of this article is to show that a recent model,23 which provided a unitary treatment of the double layer and hydration forces, can satisfy all the above requirements and might explain the high values of the second virial coefficient of apoferritin at high ionic concentrations. The total repulsive force is described in this model by two equations for the electrical potential ψ and the average dipole moment m of a water molecule. The latter is no longer assumed to be proportional to the macroscopic electric field, as in the traditional theory, but depends also on the field generated by the neighboring dipoles. In the linear approximation, the solutions for both ψ and m are linear superpositions of functions with two distinct decay lengths. At low electrolyte concentrations, one of the decay lengths is close to the Debye-Hu¨ckel length, which is characteristic for the traditional double layer repulsion, and the other is close to a length characteristic for the traditional hydration repulsion. However, at high electrolyte concentrations, the two decay lengths characteristic of the system are markedly different from the latter two, with one of them being above 14.9 Å at any ionic strength and the other being small, below 1.67 Å. The large length is responsible for a long-ranged repulsion. At small separation distances, the terms with a shorter decay length become dominant, in agreement with the experiments on the hydration force between neutral lipid bilayers. The most important feature of this model consists of the nonadditivity of the hydration and double layer repulsions. The “double layer-like” repulsion is generated by the charge on the surface, while the “hydration-like” repulsion is generated by the surface ion pairs (dipoles), which are formed through the reassociation of the surface charges with counterions. These dipoles orient the neighboring water molecules in a direction opposite to that (22) Paunov, V. N.; Kaler, E. W.; Sandler, S. I.; Petsev, D. N. J. Colloid Interface Sci. 2001, 240, 640.
Manciu and Ruckenstein
produced by the electric field generated by the surface charge (due to the dissociated groups), because they expose an opposite charge to the water molecules. The increase of the electrolyte concentration in the acetate buffer increases the amount of Na+ ions adsorbed on the acidic sites of the surface of apoferritin, and this lowers the surface charge but enhances the surface dipole density. At a particular concentration, the electric fields generated by the surface charge and the surface dipole densities in the neighboring water molecules compensate each other to a high extent and the repulsion passes through a deep minimum. At low ionic strengths, the charges dominate the interaction, while, at high ionic strengths, the surface dipoles dominate. It will be shown below that this approach can explain the strong variation of the second virial coefficient over a relatively narrow range of electrolyte concentrations. II. Theoretical Framework II.A. Surface Association-Dissociation Equilibria. As already noted, we suggest that the behavior of the second virial coefficient of the apoferritin in acetate buffer is due to the adsorption of Na+ ions upon the negative sites of the protein surface, which depends on the concentration of the Na+ ions in the liquid in the vicinity of the surface. In what follows, the adsorption of acetate ions upon the positive sites or of neutral Na+-CH3COOpairs on the neutral sites of the protein surface will be neglected and it will be assumed that only the dipoles of the ion pairs formed through the association of Na+ to the acidic sites of the surface polarize the neighboring water molecules. The equilibrium constants for the association-dissociation of H+, Na+, and OH- to the apoferritin acidic and basic groups, respectively, are not precisely known. We will try to estimate them as follows: The apoferritin protein has NA ) 624 acidic and NB ) 576 basic amino acid residues on its surface and an isoelectric point of ∼4.0.24 The dissociation equilibria of the acidic and basic sites, at negligible electrolyte concentrations, are25
xNA a (1 - x)NA + [H+]S
(4a)
yNB a (1 - y)NB + [OH-]S
(4b)
where (1 - x) and (1 - y) represent the fractions of the dissociated acidic and basic sites and [H+]S and [OH-]S represent the concentrations of hydrogen and hydroxyl ions, respectively, in the liquid in the vicinity of the surface. At equilibrium one can write
KH )
(1 - x)[H+]S x
(5a)
and
(1 - y)[OH-]S KOH ) y
(5b)
where KH and KOH are the dissociation constants of the acidic and basic sites, respectively. The total charge q on the surface, obtained using eqs 5a and b, is given by (23) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 7584. (24) Petsev, D. N.; Thomas, B. R.; Yau, S.-T.; Vekilov, P. G. Biophys. J. 2000, 78, 2060. (25) Prieve, D. C.; Ruckenstein, E. J. Theor. Biol. 1976, 56, 205.
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Langmuir, Vol. 18, No. 23, 2002 8913
(
)
q ) -e((1 - x)NA - (1 - y)NB) ) NB NA + (6) e - + [H ]S [OH-]S +1 +1 KH KOH Since at the isoelectric point q ) 0, the surface potential ψS ) 0 and the concentrations of H+ and OH- in the liquid are the same in the vicinity of the surface and in the bulk. In this case, eq 6 leads to a relation between the dissociation constants:
KOH )
10-10 mol/L NB 10-4 NA - NB NA KH NA
(7)
Because the dissociation constants must be positive, the condition KOH > 0 in eq 7 provides an upper bound for KH:
KH
a is the distance from the center of the particle and ψS is the surface potential. The surface potential is related to the surface charge density through the expression
(
)
∂ψ(r′) 1 1 q |r′)a ) ψS + )2 ∂r a λ 4πa 0 D
(13)
Using eqs 7, 10, 11, and 13, one can determine the charge on an apoferritin molecule at any pH, provided that the values of the dissociation constants KH and KNa are known. Further, using eqs 1 and 2, one can evaluate the second virial coefficient for the interaction between two particles at constant surface charge. The dissociation constant of sodium (from the acidic sites of apoferritin) is not accurately known; a reasonable value might be 0.2 M, which is compatible with the values for the Na+ and an amino acid25 (see Figure 3 of ref 25). The charge of apoferritin molecules is plotted in Figure 1 as a function of electrolyte concentration, for three pairs of values of KH and KOH compatible with the isoelectric point (eq 7) and for KNa ) 1.0, 0.2 and 0.04 M. In some cases, the molecule, which is negatively charged at very low electrolyte concentrations, becomes positively charged at higher ionic strength, because of the adsorption of Na+ ions. The change of the charge sign might constitute an appealing explanation for the abnormal behavior of the second virial coefficient. Indeed, the double layer repulsion first decreases with the electrolyte concentration, vanishes at a concentration corresponding to neutral molecules, and then increases when the charge becomes positive. However, the charge at 0.25 M is much smaller than that required to reach B ˜ 2 ) 13. Also, a large final positive charge implies that the molecules become neutral at concentrations less than 0.01 M (see Figure 1) and not at about 0.10-0.15 M, the location of the minimum determined (26) Hunter, R. J. Foundations of Colloid Science; Oxford Science Publications: 1987.
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Manciu and Ruckenstein
experimentally. We performed nondenaturating electrophoretic measurements of apoferritin in an acetate buffer, and the results indicated that the molecules do not become positively charged at any acetate concentration tested (up to 0.25 M). Consequently, the large repulsion at high ionic strength cannot be due to a large positive charge acquired by the apoferritin molecules. II.B. Electrostatic Repulsion between Molecules. In the traditional double layer theory, the polarization is assumed to be proportional to the macroscopic electric field. However, this assumption is valid only when the field varies sufficiently slowly with the distance. When the variation is stronger, the interactions between neighboring dipoles become important and have to be accounted for. In addition, the dipoles present on a surface generate an electric field, which polarizes the neighboring water molecules, which in turn generate electric fields, and so on. The overlap of the polarization fields generates a repulsion when two surfaces approach each other. When both dipoles and charges are present on the surface, the traditional double layer and hydration repulsions are no longer independent, because both depend on the polarization. A unitary treatment of the interaction was presented recently23 and will only be summarized here. The average polarization of a water molecule, m(z), between two identical, charged parallel plates separated by a distance 2d is related to the macroscopic electric field, E(z), and to the local field produced by the neighboring dipoles via23
∂2m(z) m(z) ) 0v0( - 1)E(z) + 0v0( - 1)C1∆2 (14) ∂z2 where 0 is the vacuum permittivity, is the bulk dielectric constant, v0 is the volume occupied by one water molecule, ∆ is the distance between the centers of two adjacent icelike layers, and C1 ) 1.827/4π0′′l3 is an interaction coefficient, with ′′ denoting the dielectric constant for the interaction between neighboring water molecules and l the distance between the centers of two adjacent water molecules. Equation 14, coupled with the Poisson equation
∂2ψ(z) ∂z2
where z is measured from the middle distance between the parallel plates. The constants a j 1 and a j 2 are related to a1 and a2 through
( (
1 λ12 λD2
(18a)
a j 2 ) a20v0λ2
1 λ22 λD2
(18b)
and λ1 and λ2 are the characteristic lengths of the interaction, given by
λ1,2 )
(
(λ
2 D
(
( )
(15)
where the first term on the right side was obtained by assuming Boltzmann distributions for the ions, constitutes a system of differential equations for m(z) and ψ(z). For small surface potentials, hence in the linear approximation, the above system of equations becomes 2
∂ ψ(z) 2
∂z
∂2m(z) λm2 2 ∂z
)
1 ∂m(z) ψ+ 2 0v0 ∂z λD
∂ψ(z) ) m(z) + 0v0( - 1) ∂z
2
account for the symmetry of the system, can be written for ψ and m:
() ()
() ()
z z + a2 cosh λ1 λ2
(17a)
z z +a j 2 sinh m(z) ) a j 1 sinh λ1 λ2
(17b)
ψ(z) ) a1 cosh
)
1/2
(19)
()
()
λD2 d d + a2λ2 sinh ) σ λ1 λ2 0
(20a)
while the average polarization of the first water layer from the surface is given by
[ (
( )
(
)(
1 (1 - 0v0( - 1)C0) × λ12 λD2 d d-∆ - 0v0( - 1)C1 sinh + sinh λ1 λ1 0v0( - 1) d 1 sinh + a2 λ20v0 2 - 2 × λ1 λ1 λ λ
a1 λ10v0
( )]
(
[ (
( )
(
))
d-∆ λ2
))
2
D
)
d - 0v0( - 1)C1 × λ2 0v0( - 1) d + sinh ) λ2 λ2
(1 - 0v0( - 1)C0) sinh sinh -
where λm2 ≡ 0v0( - 1)C1∆2. The following solutions, which
1/2
)) )
At low ionic strengths, λ1 = λD and λ2 = λH ) λm/x, which indicates that the double layer is not affected much by the local interaction due to neighboring dipoles. The presence of a surface dipole density can either increase or decrease the repulsion, depending on the dipole’s orientation. Selecting v0 ) 30 Å3, ) 80, and ′′ ) 1 for the dielectric constant for the interaction between neighboring water molecules, l ) 2.76 Å for the distance between the centers of two adjacent water molecules, and ∆ ) 3.68 Å,16 one obtains λH ) 1.67 Å,23 which is in agreement with the value determined experimentally for neutral lipid bilayers in water.11 However, at high ionic strength, λ1 and λ2 are markedly different from λD and λH, respectively. For CE f ∞ (λD f 0), λ1 f λm ) 14.89 Å, which indicates the presence in the system of an interaction with a much longer range than those of the traditional hydration or double layer. To obtain the solution of the system of eqs 16, two boundary conditions are needed. The overall neutrality leads to the following relation between potential and surface charge density σ:
(16a)
(16b)
2
(
+ λm2 ( λD4 + λm4 + 2λD2λm2 1 -
a1λ1 sinh 2ecE 1 ∂m(z) eψ ) sinh + 0 kT 0v0 ∂z
) )
a j 1 ) a10v0λ1
()
( )]
1 p v0( - 1) ′ 2π ANa + ∆′2 π
(
)
3/2
(20b)
where p is the dipole moment normal to the surface of a surface dipole, ANa is the area corresponding to one surface dipole, ′ is the dielectric constant for the interaction between a dipole and the adjacent water molecules, and
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Langmuir, Vol. 18, No. 23, 2002 8915
∆′ is the distance between the center of a dipole and the center of the first water layer. The interaction coefficients C0 and C1 are given by C0 ) -3.766/4π0′′l3 andC1 ) 1.827/ 4π0′′l3, where l is the distance between the centers of two adjacent water molecules in ice.16 We will assume, as before,23 that the free energy of the surface layer formed by the surface dipoles and the water molecules between them is independent of the distance between the two plates. Consequently, the free interaction energy, per unit area, of two planar plates separated by a distance D ) 2d is composed of a chemical term Fch, an entropic term due to the mobile counterions, and a term due to the electrostatic fields:23
Fflat ) Fch +
(
0
∫-d λ
1 2
d
2
(ψ) +
2
D
[(
])
)
2 ∂ψ m ∂ψ C1∆ ∂2m 0 m 2 ∂z v0 ∂z v0 ∂z
dz (21)
For interaction at constant surface charge, Fch ≡ 0, since the (fixed) surface charge is not in thermodynamic equilibrium with the medium, while, at constant surface potential, Fch ) -2σψS, where ψS is the surface potential.1 When neither σ nor ψS is constant, the change in chemical free energy, per unit area, from infinite separation to a distance D ) 2d between plates is given by
∫∞ ψS(η) dσ(η)
Fch(D) - Fch(∞) ) -2
D
(22)
The latter expression can be derived by decomposing the trajectory of ψS versus σ in a succession of small changes at constant σ followed by changes at constant ψS. At constant σ, the change in chemical energy is zero, while, at constant surface potential, the change in chemical energy is -2ψS dσ. The interaction between two identical spherical particles of radius a, separated by a distance r between their centers, is calculated using a variant of the Derjaguin approximation (see Appendix)
Fsph(r) )
r (F(D) - F(∞))(r - D) dD ∫r-2a
π 2
(23)
where the concentration of ions in liquid in the vicinity of the surface (eqs 11) is calculated using the renormalized surface potential ψS′ (see Appendix)
ψS′ )
ψS λD 1+ a
(
)
(24)
with ψS being the surface potential for planar surfaces calculated from the system (eq 16) subjected to the boundary conditions (eq 20). II.C. van der Waals Attraction. The van der Waals attraction between the hollow shells of the ferritin can be easily computed in the pairwise summation approximation. Let us consider a large sphere B made up of a small sphere b and a spherical shell S. The interaction free energy FBB between two large spheres can be written (assuming pairwise interactions) as
FBB ) F(b+S)(b+S) ) Fbb + FbS + FSb + FSS
(25)
The interaction between a large sphere B and a small one b can be separated into
FBb ) F(b+S)b ) Fbb + FbS
(26)
Figure 2. Ratio between van der Waals attraction of compact and hollow spheres of external radius a ) 63.5 Å, at various distances, as a function of the thickness of the shell.
Using eqs 25 and 26, one can calculate the interaction between two spherical shells
FSS ) FBB + Fbb - 2FBb
(27)
by taking into account that the van der Waals attraction between two spheres of radii a1 and a2, separated by a distance r between their centers, is given by26
FvdW ) -
Ha1a2 6(r - (a1 + a2))(a1 + a2) 1
1+
2a1a2
(
(r - (a1 + a2))(a1 + a2)
(
)
[
+
1+
1 + r - (a1 + a2) 2(a1 + a2)
r - (a1 + a2)
+
2(a1 + a2)
(r - (a1 + a2))(a1 + a2) (r - (a1 + a2))(a1 + a2) ln × a1a2 2a1a2
(
1+
r - (a1 + a2) 2(a1 + a2)
)]
(r - (a1 + a2))(a1 + a2) (r - (a1 + a2))2 1+ + 2a1a2 4a1a2
(28)
where H is the Hamaker constant. In Figure 2, the ratio between the van der Waals interactions of hollow and compact spheres of radius a ) 63.5 Å is plotted, at various distances of closest approach (r - 2a), as a function of the shell thickness. While the outer radius of the apoferritin molecule (a ) 63.5 Å) can be accurately determined, for example by dynamic light scattering,21 the inner radius (of the hollow shell) is more difficult to estimate. The iron cores of the ferritin have a maximum diameter of the order of 80 Å, while the six hydrophobic channels have a length of about 12 Å.27 In what follows the value 20 Å will be employed for the average thickness of the shell. The Hamaker (27) Ford, G. C.; Harrison, P. M.; Rice, D. W.; Smith, J. M. A.; Treffry, A.; White, J. L.; Yariv, J. Philos. Trans. R. Soc. London 1984, B304, 551.
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constant of the apoferritin is not known; the Hamaker constants reported for proteins cover a rather large range, from 0.06kT for R-crystallin to about 10kT for R-chymotrypsin and bovine serum albumin.28 In the present calculations, the value H ) 5.0kT will be used. III. Second Virial Coefficient of Apoferritin Molecules In what follows, the model presented will be employed to calculate the dependence of the dimensionless second virial coefficient of the apoferritin molecules. Assuming that the surface dipolar density is generated only by the Na+ ions adsorbed on the surface, the area corresponding to an ion is ANa ) (4πa2/wNA). As already emphasized, one of the characteristic decay lengths of the system is larger than λm ) 14.9 Å at any electrolyte concentration; hence, a relatively long-ranged interaction is always present in the system, even at high ionic strengths. The magnitude of this interaction per unit area depends on the boundary conditions and is generally small. Apoferritin is, however, a large protein, and since in the Derjaguin approximation the interaction is proportional to the radius, the interaction is of the order of kT at large separations. Another particularity of the apoferritin is its hollow nature, which implies a relatively small van der Waals attraction. For other proteins, the attraction might overcome the weak long-ranged repulsion, leading to a small second virial coefficient at high ionic strengths. The electrostatic repulsion is calculated by combining eqs 10 and 11 for the charge (using the renormalized surface potential given by eq 24) with the system of eqs 16a and b solved with the boundary conditions (eqs 20a and b). For all separation distances D ) 2d a solution was obtained for planar surfaces by successive approximations and the corresponding free energy per unit area was calculated using eqs 21 and 22. The free energy of the repulsive interaction between two spherical apoferritin molecules was calculated using eq 23, and the van der Waals interaction was obtained from eqs 27 and 28. Figure 3a presents the interaction free energy (in kT units) as a function of the closest approach distance r 2a for various electrolyte concentrations. The following values have been employed for the parameters: KH ) 1 × 10-4 M, KOH ) 1.18 × 10-10 M, KNa ) 0.4 M, T ) 300 K, ) 80, ′′ ) 1, (p/′) ) 5 D, ∆′ ) 1.5 Å, l ) 2.76 Å, v0 ) 30 Å3, H ) 5kT, and 20 Å for the thickness of the ferritin shell. The repulsion first decreases with increasing electrolyte concentration and then attains a minimum around 0.1 M, after which it increases. This behavior is a result of the adsorption of Na+ ions on the acidic sites. While the total charge of the protein decreases with increasing ionic strength, the strong dipole moment of the adsorbed Na+ orients the water molecules and generates a repulsion. The repulsive free energy (without the van der Waals attraction) is plotted in Figure 3b for various separations against the electrolyte concentration. The repulsion has a strong minimum at an ionic strength of about 0.08 M. The reason for this unexpected behavior is the nonadditivity of the double layer and hydration repulsions.23 When the effect of the charge is dominant, the formation of the surface dipoles decreases the free energy, because the surface dipoles and the surface charges orient the neighboring water molecules in opposite directions. Similarly, when the effect of the surface dipoles becomes dominant, the recombination of charges increases the repulsion, by (28) Broide, M. L.; Tominc, T. M.; Saxowsky, M. D. Phys. Rev. E 1996, 53, 6325.
Figure 3. (a) Interaction energy between two apoferritin molecules (in kT units) as a function of the distance of closest approach, for various electrolyte concentrations. (b) Repulsive interaction as a function of electrolyte concentration, at various distances of closest approach between particles.
increasing the number of surface dipoles and decreasing the charge. For the values of the parameters employed, about 64% of the acidic sites are occupied by Na+ ions at 0.25 M, a value which is compatible with the increase of the apoferritin mass determined by light scattering.22 For the values of the parameters employed (a relatively large Hamaker constant), the potential barrier is only a few kT or less; hence, the apoferritin should coagulate at almost all the concentrations studied. Since experiment shows that the proteins did not coagulate, another repulsion should be present, at least al low separation distances. This repulsion, while essential for the stability of the system, did not affect much, because of its short range, the behavior of the second virial coefficient. In the calculation of the second virial coefficient, it was assumed that the distance of closest approach between apoferritin proteins cannot be less than 8 Å. This value leads to a dimensionless second virial coefficient for the hard spheres repulsion of 4.8 instead of 4. The dimensionless second virial coefficient calculated from eq 1 for various values of (p/′) is compared in Figure 4 with the experimental results (circles) of ref 21. There is a minimum in the repulsion at about 0.1 M. At higher ionic strengths, the accumulation of Na+ ions via association with the acidic sites of the surface increases the surface dipole density and decreases the surface charge, leading to a higher repulsion. The value of (p/′) ) 5.5 D, which provides a good agreement between the theory and
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Figure 5. Geometry of two interacting spherical particles.
Figure 4. Dimensionless virial coefficient of the interaction, calculated as a function of the electrolyte concentration for various (p/′) ratios, compared with the experimental results of ref 22 (circles). The crosses represent the experimental results when the partial dissociation of the sodium acetate is taken into account.
experiment, is reasonable, since the water dipole is about 1.85 D and the dipole formed by two elementary charges separated by 1 Å is about 5 D. Good agreement with experiment can be obtained when the equilibrium constants KH and KOH are varied in a rather large range (over 1 order of magnitude), if KNa and (p/′) are suitably chosen. Small changes in the values of the Hamaker constant, shell thickness, or cutoff distance for the virial integration do not affect essentially the behavior of the second virial coefficient. While the values of the parameters chosen are reasonable, the results presented in this article should be considered as qualitative only, for the reasons outlined below. The results presented in ref 21 are, to our knowledge, the only experiments with 1:1 electrolytes in which such a strong increase of the repulsion with ionic strength was observed. If, for CE ) 0.25 M, one would disregard the last experimental point from the Debye plot, which provided the second virial coefficient (Figure 1b in ref 21), the corresponding B ˜ 2 would be about 6 instead of 13. In addition, the sodium acetate in the buffer is not completely dissociated. In water, sodium acetate forms both inner and outer sphere complexes. In the former, the ions are directly bound, while in the latter, Na+ and CH3COOions are separated by one or more water molecules. Raman spectroscopy identifies only the inner complexes as associated species, while potentiometry identifies both inner and outer complexes as associated species.29 Using the dissociation constants provided by ref 29, one can infer that, at room temperature, the inner sphere complexes are almost completely dissociated for the whole range of electrolyte concentrations (0.01-0.25 M) investigated here. However, some of the Na+ and CH3COO- remain bound in neutral pairs (outer sphere complexes) and do not participate as free charges to the electrostatic screening; hence, the concentration of truly dissociated Na+ (CE) ions differs slightly from the NaCH3COO buffer concentration. The results of our estimations for the truly dissociated Na+ are presented as crosses in Figure 4. In addition, the large dipoles of the outer sphere complexes can affect the interactions, particularly when adsorbed on the surface of apoferritin. The simplified (29) Fournier, P.; Oelkers, E. H.; Gout, R.; Pokrovski, G. Chem. Geol. 1998, 151, 69.
model presented here has neglected the influence of the dipoles of H+ and OH- ions associated with the surface and the effect of the sodium acetate neutral pairs adsorbed on the surface. It should be again emphasized that the Derjaguin approximation is not accurate at large separations, particularly at low electrolyte concentrations, when the Debye-Hu¨ckel length is not sufficiently small compared to the protein radius. At very low ionic strengths (0.01 M), the surface potential is large and the linear approximation employed here is not accurate. The main point of the article is that the outlined theory can explain the strong increase in repulsion with increasing electrolyte concentration. This is a result of the nonadditivity of the repulsions generated by the surface charge and surface dipole densities. IV. Conclusions Recent light scattering experiments on apoferritin proteins in an acetate buffer at pH 5.0 revealed an unexpected behavior of the second virial coefficient, which passed through a minimum when the electrolyte concentration was about 0.15 M but increased to a high value at a concentration of 0.25 M. The results are incompatible with the traditional theoretical framework, which assumes that the total repulsion is the sum between a “double layer repulsion” due to the charges on the interface and a “hydration repulsion” due to the structuring of the solvent near the surface. The results could have been explained by the traditional double layer theory, if the apoferritin would have acquired an unexpectedly large positive charge at 0.25 M. This possibility was, however, ruled out by nondenaturating electrophoretic experiments. The result could also have been explained by postulating that the hydration provides a very long range exponential repulsion, with a decay length about 10 times larger than what is generally accepted. Even in this case, it would still have been difficult to explain why the repulsion increased by orders of magnitude from a concentration of 0.15 M to 0.25 M. In this paper, we presented an alternate explanation for the unexpected behavior of the virial coefficient, based on our previous unitary treatment of the electrostatic repulsion between particles immersed in a polar solvent, which combined the double layer and the hydration forces into a single repulsive force.23 The main features of the model are the existence of two characteristic decay lengths for the interaction (a small one, important at small separations, and a large one, which is relevant at large separations) and the nonadditivity of the effects of surface charges and surface dipoles, which in the present case tend to orient the water molecules neighboring the surface in opposite directions. The number of Na+ ions adsorbed on the surface increases with the electrolyte concentration, decreasing the surface charge but increasing the surface dipole density. At low electrolyte concentrations, the charge effect is dominant and the interaction is well approximated by the traditional DLVO theory. At high ionic strength, the surface dipoles are mainly responsible
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for the repulsion. At intermediate concentrations, the opposite fields generated by the charges and the surface dipoles in the vicinity of the surface compensate each other to some extent, and consequently the repulsion has a strong minimum. Acknowledgment. We are indebted to Prof. M. Ettinger for performing the nondenaturating electrophoretic measurements on apoferritin and to Prof. G. Frens for drawing our attention to the experimental results discussed in ref 8. Appendix The interactions between spherical particles can be calculated, in the Derjaguin approximation, in terms of the interactions between planar surfaces, by decomposing the spherical surface in small areas (which can be considered locally planar). The free energy of interaction, corresponding to each small piece, is assumed to be equal to the product between the area of the piece and the interaction free energy per unit area of some virtual infinite planar surfaces, located at the same separation distance D (see Figure 5). The total free energy of interaction between identical spherical particles of radius a is the integral over the surface of one particle:26
Fsph(r) )
∫0 (Fflat(D) - Fflat(∞))2πy dy ≈ ∞ πa∫r-2a(Fflat(D) - Fflat(∞)) dD
radii (compared to both the Debye-Hu¨ckel length and the distance of closest approach, r - 2a). Because (see Figure 5) y2 ) a2 - [(r - D)/2]2 w 2y dy ) ((r - D)/2) dD, eq A.1 becomes
Fsph(r) )
with r being the distance between the centers of the spheres. The approximation is accurate for large particle
(A2)
Another difficulty in using the Derjaguin approximation arises when the charge is related to the surface potential via various ionic equilibria. Indeed, in the linear approximation, the surface potential is related to the surface charge density, σ, of a single planar surface by
∂ψflat(z) ψS,flat σ |z)0 ) )0 ∂z λD
(A3)
while for a spherical particle (eq 13)
(
)
1 1 σ ) ψS,sph + 0 a λD
(A4)
To account for the effect of the radius of the particles, an approximate renormalized surface potential ψS′
ψ S′ )
a
(A1)
∫rr-2a(Fflat(D) - Fflat(∞))(r - D) dD
π 2
ψS,flat λD 1+ a
(
)
(A5)
will be used to calculate the ion density in the vicinity of the surface. Both corrections (in eqs A2 and A5) become negligible at high ionic strengths, since then (λD/a) , 1. LA026126M
