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A Quasi-Equilibrium Theory of Protein Adsorption Jonathan Fernsler* Department of Physics, California Polytechnic State University, San Luis Obispo, California, United States Received August 19, 2010. Revised Manuscript Received November 11, 2010 A model is developed to describe the adsorption and desorption of proteins to and from a surface film under quasiequilibrium conditions. Starting from Fick’s first law of diffusion, an equation for the flux of molecules to a surface is derived assuming a gradient in the chemical potential from the bulk to the surface and a potential barrier due to an existing surface film. Protein molecules are modeled as components with varying surface areas to depict the different orientations of molecules with respect to the film. For concentrated solutions, formation of multilayer protein films is described by allowing components with small minimum surface areas. The thermodynamic analysis is based on Butler’s equation for the chemical potentials of the components of a Gibbs surface layer and a first-order model for the nonideality of the surface layer enthalpy and entropy. The model assumes reversible adsorption, consistent with globular proteins that show little denaturation or flexible-chain proteins that reversibly denature at the interface. The model predicts the behavior of five different experiments measuring film properties of the serum protein albumin in quasi-equilibrium and equilibrium conditions at over 2 orders of magnitude in concentration using a single set of parameters. This provides a new framework for analyzing interactions and adsorption of protein films. The key new features of this model are an extension of the classical Smoluchowski analysis to calculate the adsorption and desorption rate, a model of multilayers with decreased molecular areas to allow effective densities greater than a close-packed monolayer, and a concentration-dependent layer thickness.
Introduction Understanding protein adsorption is of great practical importance and has promoted many theoretical models of fluid films.1-21 One motivation is to understand the interactions between protein and lipid films in the context of biological systems. The behavior of proteins has significant differences from that of ordinary lowmolecular-weight surfactants and cannot be treated with the same models.1 However, it is this interaction between two different *Corresponding author. Department of Physics 52-D37, California Polytechnic State University, San Luis Obispo, CA 93407, USA. Fax: þ1 805 756 2439. E-mail:
[email protected]. (1) Miller, R.; Fainerman, V. B.; Makievski, A. V.; Kragel, J.; Grigoriev, D. O.; Kazakov, V. N.; Sinyachenko, O. V. Adv. Colloid Interface Sci. 2000, 86, 39–82. (2) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. Adv. Colloid Interface Sci. 2003, 106, 237–259. (3) Singer, S. J. J. Chem. Phys. 1948, 16, 872. (4) Frisch, H. L.; Simha, R. J. Chem. Phys. 1956, 24, 652. (5) Silberberg, A. J. Phys. Chem. 1962, 66, 1872. (6) Hoeve, C. A. J. J. Chem. Phys. 1966, 44, 1505. (7) Hesselink, F. T. J. Colloid Interface Sci. 1977, 60, 448. (8) Leermakers, F. A. M.; Atkinson, P. L.; Dickinson, E.; Horne, D. S. J. Colloid Interface Sci. 1996, 178, 681. (9) Cohen Stuart, M. A.; Fleer, G. J.; Lyklema, J.; et al. Adv. Colloid Interface Sci. 1991, 34, 477. (10) De Gennes, P.-G. Adv. Colloid Interface Sci. 1987, 27, 189. (11) De Feijter, J. A.; Benjamins, J. J. Colloid Interface Sci. 1982, 90, 289. (12) Douillard, R.; Daoud, M.; Lefebvre, J. J. Colloid Interface Sci. 1990, 139, 488. (13) Douillard, R.; Daoud, M.; Lefebvre, J.; et al. J. Colloid Interface Sci. 1994, 163, 277. (14) Lucassen-Reynders, E. H. Colloids Surf., A 1994, 91, 79. (15) Joos, P. Dynamic Surface Phenomena; VSP; Utrecht, 1999. (16) Joos, P.; Serrien, G. J. Colloid Interface Sci. 1991, 145, 291. (17) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1972, 41, 156. (18) Fainerman, V. B.; Miller, R.; Wustneck, N. J. Colloid Interface Sci. 1996, 183, 26. (19) Fainerman, V. B.; Miller, R. Proteins at Liquid Interfaces; Mobius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1998, 51. (20) Benjamins, J.; Cagna, A.; Lucassen-Reynders, E. H. Colloids Surf., A 1996, 114, 245. (21) Lucassen-Reynders, E. H.; Benjamins, J. In Food Emulsions and Foams: Interfaces, Interactions, and Stability; Dickinson, E.; Rodriguez Patino, J. M., Eds.; Royal Society of Chemistry: Cambridge, 1999; p 195.
148 DOI: 10.1021/la103316j
systems that leads to useful biomechanical or biochemical responses, or to disease. In the lungs, a lipid and protein film known as a lung surfactant is necessary to lower surface tension in the alveoli, the small spherical, liquid-coated chambers in which gas exchange occurs. However, a film of the blood serum protein albumin, which can be formed through damage or disease to the lungs, causes an electrostatic and steric barrier to subsequent surfactant adsorption. This can destroy the functional properties of lung surfactant: low surface tension during compression and rapid re-spreading after film collapse.22-26 Decreased surfactant adsorption can inactivate surfactant,27-31 which may play a role in Acute Respiratory Distress Syndrome.26,32-34 Proteins are essentially flexible-chain polymers that can adopt an enormous number of conformations, which allow them to have very different interactions in different environments. Although modeling every individual chemical interaction is presently (22) Holm, B. A.; Enhorning, G.; Notter, R. H. Chem. Phys. Lipids 1988, 1988, 49–55. (23) Taeusch, H. W. Biol. Neonate 2000, 77(suppl. 1), 2–8. (24) Holm, B. A.; Notter, R. H.; Finkelstein, J. N. Chem. Phys. Lipids 1985, 38, 287–298. (25) Holm, B. A.; Wang, Z.; Notter, R. H. Pediatrics Res. 1999, 46, 85–93. (26) Fernsler, J.; Zasadzinski, J. A. Langmuir 2009, 25, 8131–8143. (27) Zasadzinski, J. A.; Alig, T. F.; Alonso, C.; Bernardino de la Serna, J.; PerezGil, J.; Taeusch, H. W. Biophys. J. 2005, 89, 1621–1629. (28) Taeusch, H. W.; de la Serna, J. B.; Perez-Gil, J.; Alonso, C.; Zasadzinski, J. A. Biophys. J. 2005, 89, 1769–1779. (29) Yu, L. M. Y.; Lu, J. J.; Chiu, I. W. Y.; Leung, K. S.; Chan, Y. W. W.; Zhang, L.; Policova, Z.; Hair, M. L.; Neumann, A. W. Colloids Surf., B 2004, 36, 167–176. (30) Stenger, P. C.; Isbell, S. G.; Zasadzinski, J. A. Biochim. Biophys. Acta 2008. (31) Stenger, P. C.; Zasadzinski, J. A. Biophys. J. 2007, 92, 3–9. (32) Dahlem, P.; van Aalderen, W. M. C.; Bos, A. P. Paediatric Respir. Rev. 2007, 8, 348–362. (33) Nakos, G.; Kitsiouli, E. I.; Tsangaris, I.; Lekka, M. E. Intensive Care Med. 1998, 24, 296–303. (34) Ishizaka, A.; Matsuda, T.; Albertine, K.; Koh, H.; Tasaka, S.; Hasegawa, N.; Kohno, N.; Kotani, T.; Morisakai, H.; Takeda, J.; Nakamura, M.; Fang, X.; Martin, T.; Matthay, M.; Hashimoto, S. Am. J. Physiol. Lung Cell Mol. Physiol. 2004, 286, L1088–L1094.
Published on Web 12/02/2010
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impossible, several approaches have had success in describing the behavior of proteins, especially in equilibrium. Statistical models have successfully determined the distribution of flexible-chain polymers in a film and in the bulk by distinguishing between loops and tails of molecules.3-9,35,36 The enthalpy and entropy of these polymers can be determined to compute macroscopic properties like surface pressure (the reduction of surface tension from a clean interface), but only at low levels of adsorption. Furthermore, the calculations can be difficult and can only be accomplished numerically. These issues are especially severe for proteins that form a rigid and compact form at an interfacial film, such as albumin. A second approach to modeling protein films is to form a thermodynamic model based on a modification of the SzyszkowskiLangmuir equation for small area molecules, with additional terms to describe the nonideality of proteins.14-18,20,21,37 These models, although simple compared with the statistical theories, have been used to successfully predict bulk properties like the equation of state and the adsorption isotherm. Using a simple scaling theory for non-interacting particles, De Feijter and Benjamins11 successfully modeled low protein adsorptions. To model high adsorptions, the authors proposed soft and compressible macromolecules with a variable molecular area modeled as different protein components to represent the multiple conformations of adsorbed protein molecules. This approach is now widely used.14,16,18,21,37 Fernsler and Zasadzinski26 introduced a new experimental technique for measuring the properties of films at the air-water interface in quasi-equilibrium conditions on a Langmuir trough. Holding films at constant surface pressure Π while trough barriers were expanded or compressed, they found that the surface density Γ of films and the flux of molecules to the surface J were constant. This allowed a measurement of a quasi-equilibrium equation of state (Π vs Γ) and J simultaneously. Furthermore, they investigated these film properties over 2 orders of magnitude in the bulk concentration of surfactant. The equation of state for films composed primarily of small molecular weight surfactants was effectively the same for several orders of magnitude in concentration, as expected. However, films of albumin showed dramatically different behavior in the quasi-equilibrium equation of state at high concentrations and surface pressures.26 The surface pressure depends only on the surface layer of molecules, but their instrument measured a total surface density, which could include molecules in a bilayer or multilayer below the surface film. They hypothesized that this behavior is accounted for by the formation of a multilayer in the protein films dependent on the bulk concentration. This hypothesis is supported by ellipsometry experiments by McClellan et al.38 and X-ray experiments by Stenger et al.39,40 that show dramatic changes in film thickness where film thicknesses greater than the molecular dimensions were measured at high bulk concentrations. Furthermore, the standard equilibrium behavior of proteins shows strange behavior
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with rising concentration: at high concentrations, the surface pressure levels off, but the surface density continues to rise.21,41-45 This behavior seems to be in conflict with a general law of thermodynamics known as Gibb’s Law.46 Several successful models of the equilibrium behavior have accounted for this paradoxical behavior by supposing the formation of a protein bilayer in certain regimes of the surface pressure or surface density.2,12,13 In this paper, we propose a thermodynamic model to describe adsorption of soluble Gibbs-type protein films using a varying molecular surface area and a treatment of multilayers. We extend the classical Smoluchowski analysis for colloidal suspensions to derive an adsorption or desorption rate influenced by the presence of the surface film assumed to exist with quasi-equilibrium conditions (we used a similar derivation in a previous paper26). We use a set of surface and bulk potentials for molecular species with varying molecular surface areas based largely on the potentials proposed by Fainerman et al.2,47 We extend their work to explicitly model multilayers with a modification of the surface layer entropy and by allowing an expanded range of molecular surface areas (a minimum molecular surface area which is smaller than the true close-packed minimum allows multilayers with larger total surface densities). Finally, we take the minimum molecular surface area (which determines multilayer formation) as a continuously varying function of the bulk concentration. Our model has fewer free parameters than previous models and does not separate the adsorption behavior into separate regimes. Theoretical results using a single set of parameters are compared to experimental measurements in equilibrium and quasi-equilibrium conditions over 2 orders of magnitude bulk concentrations for the protein albumin.
Theory Adsorption and Surface Density. To attack this problem, let us now model the adsorption of a protein film to an interface by calculating the flux of proteins to the interface in the presence of an existing film. A glossary of terms can be found at the end for the reader’s reference. The initial adsorption of surfactant or protein leads to the formation of an energy barrier due to steric and electrostatic interactions; as in models of colloid stability, the energy barrier inhibits subsequent transport and adsorption to the interface, as well as the competitive displacement of one monolayer for another. We assume that a thermodynamic equilibrium exists between the surface film and the bulk, as in a Gibbstype layer. Irreversible processes cannot be described by the following arguments and require a kinetics theory. At high initial adsorptions, we expect a net flux transporting molecules away from the film, or desorption. Fick’s first law of diffusion states that the flux/area J to an interface located at z = 0 with a driving force due to a gradient in the chemical potential dμ/dz is48 J ¼ -
(35) Scheutjens, J. M. H. M.; Fleer, G. J. Adv. Colloid Interface Sci. 1982, 83, 361. (36) Fleer, G. J.; Scheutjens, J. M. H. M. Adv. Colloid Interface Sci. 1982, 16, 341. (37) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. Colloids Surf., A 1998, 143, 141–165. (38) McClellan, S. J.; Franses, E. I. Colloids Surf., B 2003, 28, 63–75. (39) Stenger, P. C.; Isbell, S. G., St; Hillaire, D.; Zasadzinski, J. A. Langmuir 2009, 25, 10045–10050. (40) Stenger, P. C.; Wu, G. H.; Miller, C. E.; Chi, E. Y.; Frey, S. L.; Lee, K. Y. C.; Majewski, J.; Kjaer, K.; Zasadzinski, J. A. Biophys. J. 2009, 97, 777–786. (41) Graham, D. E.; Phillips, M. C. J. Colloid Interface Sci. 1979, 70, 403. (42) Makievski, A. V.; Loglio, G.; Kragel, J.; Miller, R.; Fainerman, V. B.; Neumann, A. W. J. Phys. Chem. B 1999, 103, 9557.
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D dμ c RT dz
ð1Þ
where D is the diffusion constant for the adsorbate, T is the absolute temperature, R is ideal gas constant, and c is the solvate (43) Miller, R.; Fainerman, V. B.; Makievski, A. V. Colloids Surf., B 2004, 36, 123. (44) Kragel, J.; Wustneck, N.; Husband, F.; et al. Colloids Surf., B 1999, 12, 399. (45) Kragel, J.; O’Neill, M.; Makievski, A. V. Colloids Surf., B 2003, 31, 107. (46) Douillard, R.; Lefebvre, J.; Tran, V. Colloids Surf., A 1993, 78, 109. (47) Fainerman, V. B.; Miller, R. Colloid J. 2005, 67, 393–404. (48) Hiemenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1986.
DOI: 10.1021/la103316j
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molar concentration. The chemical potential of an ideal solution in an external potential, V(z), is μðzÞ ¼ RT ln x þ VðzÞ
ð2Þ
where x = c/cnet is the molar fraction, c is the molar concentration of protein, and cnet = c0 þ c is the total molar concentration of the solvate and solvent (denoted by subscript 0). Combining eqs 1 and 2 allows us to express a general diffusion equation J c dμ c RT dc RT dcnet dV ¼ ¼ þ D RT dz RT c dz cnet dz dz dc c dcnet c dV ¼ þ dz cnet dz RT dz
)
ð3Þ
Under steady-state conditions, the flux/area J is constant and eq 3 can be integrated by multiplying both sides by exp(V/RT)/cnet: -
J eV=RT eV=RT dc c V=RT dcnet c V=RT dV þ ¼ e e D cnet dz : cnet dz cnet dz cnet h i d c V=RT d ¼ e xeV=RT ¼ ð4Þ dz cnet dz
We consider the flux from the bulk at z = ¥ (denoted by superscript b) to an existing surface film at z = 0 (denoted by superscript s) with different external potentials. With these limits, we integrate eq 4 -
J D
Z
0 V=RT
e
¥
cnet
Z dz ¼
xðsÞ expðV ðsÞ =RTÞ
h i d xeV=RT
xðbÞ expðV ðbÞ =RTÞ
ð5Þ
The sum of the protein and solvent concentrations is the net concentration, cnet, which is dominated by the solvent, and is assumed to be a slowly varying function of z. We use an approximation similar to the classical Smoluchowski analysis49 where we assume the integration on the left side of eq 5 is dominated by the value of V at its maximum, Vmax, so we can Taylor-expand V(z) about its maximum at zmax and solve for the Gaussian integrand. Using this approximation, the integration of both sides of eq 5 is as follows: 2 !3 " # Z 0 2 2 J eV=RT 1 d V ðz z Þ max max 5 dz exp4 D cnet RT dz2 2 ¥ z ¼ zmax
pffiffiffi - J π eVmax =RT ðsÞ ðbÞ ¼ xðsÞ eV =RT - xðbÞ eV =RT ¼ 2Dp cnet jz ¼ zmax
ð6Þ
where p2 ¼ -
1 d2 Vmax 2RT dz2
can be viewed as the inverse square of the effective potential barrier width. Rearranging eq 6, we can solve for adsorption rate to the surface J ðbÞ ðsÞ Vmax 2Dp ðbÞ VRT ðsÞ VRT e- RT J ¼ pffiffiffi cnet x e - x e π
ð7Þ
Equation 7 describes the adsorption rate in the presence of an existing surface film and describes desorption by a change in sign (49) Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129–168.
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Figure 1. Cartoon of close-packed films of albumin molecules in the shape of ellipsoids of dimensions 4 4 14 nm3. (A) A monolayer film with the long axes of molecules parallel to the film surface has surface density Γ . (B) A monolayer film with the long axes oriented perpendicular to the film has surface density Γ^. (C) A bilayer of molecules with total surface density Γmm = N(s)/A = 2Γ^ where, in general, Γmm > Γ^ > Γ . (D) A mixed multilayer with different molecular orientations. )
-
of the flux. This equation can predict quasi-equilibrium adsorption rates of any molecular species, including mixtures of molecules, by specifying the potentials V(b), V(s), and Vmax. We now explore the form of these potentials for protein molecules adsorbing to a surface film. Equilibrium Behavior of Protein Adsorption. We define the Gibbs-type adsorbing film as the layer of solvent and solute molecules at a surface that are distinguished from the bulk phase by the Gibbs dividing surface. Adsorption of soluble proteins of this type has been well-studied.14-18,20,21,37 An expression proposed by Butler50 was generalized to macromolecules51 like proteins, and Lucassen-Reynders14,21 analyzed the effect of the size of mixed molecules on the entropy of the surface layer. Joos and Serrien first derived a relation for the adsorption protein with two conformations differing in molar surface areas.16 Fainerman et al. extended this approach for an arbitrary number of protein conformations,18,37 and Fainerman and Lucassen-Reynders derived the most general relations from the thermodynamic model of a two-dimensional solution.2,14,18,21,37 In this work, we modify the Fainerman et al. chemical potentials μi(s)for both the solvent and solute in order to predict the surface tension, surface density, and adsorption rate as functions of the bulk concentration of the protein in equilibrium and quasi-equilibrium conditions. A common description of the shape of an albumin molecule is a prolate ellipsoid with dimensions 4 14 nm2 (see Figure 1). The limits of the molecular surface area are described by parallel (see Figure 1A) and perpendicular (see Figure 1B) orientation to the surface. We will allow a range of n conformations modeled as protein components with partial molar surface areas, ωi (with units m2/mol), ranging from ωmin to ωmax. In addition to reorientation of proteins at the surface, it has been found experimentally21,30,31,38,41-45 and modeled theoretically2,12,13 that proteins may form bilayer films on surfaces at high concentrations and surface pressures. The total surface density is defined as Γ = N(s)/A, where N(s) is the number of moles of the protein in the entire surface film and A is the total surface area. This total surface density of a bilayer can be greater than the density of a close-packed monolayer (see Figure 1C). Experiments find that the measured thickness increases smoothly with concentration13,20,38,41 instead of discrete jumps from a monolayer to a bilayer; therefore, the protein film has a distribution of molecular orientations and film thicknesses (see Figure 1D), e.g., a multilayer. We propose a model allowing (50) Butler, J. A. V. Proc. R. Soc. London, Ser. A 1932, 138, 348. (51) Joos, P. Biochim. Biophys. Acta 1975, 375, 1.
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an extended range of partial molar surface areas ωi smaller than ωmin, with a new minimum partial molar surface area ωmm (where mm denotes multilayer minimum). With this system, our partial molar surface areas range continuously from ωmm (i = 1) to ωmin (i = m) to ωmax (i = n). At higher concentrations and surface densities where thicker multilayers are observed, ωmm is lower (e. g., ωmm = ωmin/2 for the bilayer in Figure 1C). Let us use the conventions presented by Fainerman et al. for the most general relations for a thermodynamic model of a twodimensional solution2,47 in the following steps, which will result in an equation for the measurable equation of state. The chemical potentials of any component of a solution on the surface, μi(s), and in the bulk, μi(b)are as follows: ðsÞ
μi
ð0sÞ
¼ μi
ðsÞ ðsÞ
θP = Γω, where ω is the weight-average molar surface area defined as follows: ω ¼
ðsÞ
xj ðbÞ
μi
ð0bÞ
¼ μi
ðbÞ ðbÞ
ð9Þ
þ RT ln fi xi
where fi is the activity coefficient (fi = 1 at xi = 1), xi = Ni/ Σi=0nNi is the molar fraction (equivalent to the definition in the previous section, but with multiple conformations allowed), Ni is the number of moles of the ith component, γ is the surface tension, and μi(0s) and μi(0b) are constants describing the standard state of the surface and bulk, respectively. Components of the mixture cover the total surface area A as follows: ðsÞ
ðsÞ
ðsÞ
ω0 N0 þ ω1 N1 þ ω2 N2 þ 3 3 3 ¼ A
ð10Þ
and the In terms of the molar surface density Γi = fractions of surface coverage, θi = ωiΓi, from eq 10, we obtain ð11Þ
where a surface coverage equal to 1 can indicate a close-packed monolayer or multilayer. We can derive the general equations of state and adsorption isotherms for any surface-active component of a mixture from eqs 8 and 9. For an ideal solution, these equations have the following form:37,52 RT ðsÞ ðsÞ Π ¼ ln x0 þ ln f0 ω0 ðsÞ ðsÞ
ln
f i xi
ðbÞ
Ki x i
P
θj θ Γj Pj ¼ ¼ ðθi =ni Þ ω j Γi Γ0 þ Γ
ð15Þ
ig0
ðsHÞ
ln f0
ðsHÞ
¼ aP θP 2 ln fj
¼ aP nj θ0 2
ð16Þ
where ni = ωi/ω0, θ0 is the surface coverage of the solvent, θP = P ig0θi = 1 - θ0 is the total degree of the surface coverage with protein, and aP is the parameter of nonideality (Frumkin’s parameter). The entropy contribution to the activity coefficient, derived by Lucassen-Reynders,14 is as follows:
ln
ðsEÞ fj
" # X X ¼ 1 - nj ðθi =ni Þ þ ln nj ðθi =ni Þ ig0
ig0
¼ 1 - ωj ðΓ þ Γ0 Þ þ ln ωj ðΓ þ Γ0 Þ
ð17Þ
The general value of the activity coefficient fi(s) combines the two contributions of enthalpy and entropy ð13Þ
ðsÞ
Π¼0
- μ0(s))/ω0 where Π = γ0 - γ is the surface pressure, γ0 = (derived from eq 8) is the surface tension of the solvent, Ki = (xi(s)/ xi(b))|Π=0 is the constant of the adsorption equilibrium and (fi(s))|Π=0 is the activity coefficient for any surface-active compo(μ0(0s)
nent i, with this constant being the distribution at infinite dilution. We assume that xi(b)|Π=0 = xi(b) because the bulk molar fraction is unchanged by the formation of a surface film. These relations can describe a mixture of different components or a mixture of different states of the same component. Now, let us continue with the conventions presented by Fainermain et al. to relate the molar fractions to the experimenP tally measurable total surface density of the film Γ = i=1nΓi. The surface coverage of the entire protein film can be defined as (52) Fainerman, V. B.; Miller, R.; Mohwald, H. J. Phys. Chem. B 2002, 106, 809–819.
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nj
where we define the ratio ni = ωi/ω0. Equation 15 allows us to express the molar fractions xi(s) in eqs 8 and 9 in terms of the surface fractions (fraction of surface coverage) and surface densities of the components. Let us now estimate activity coefficients fi(s) in eqs 8 and 9.2,13 We divide the activity coefficients into two forms, enthalpy and entropy, where the enthalpy coefficient fi(sH) is determined by the intermolecular interaction and may be estimated in terms of the regular solution theory.53 The contribution for the solvent and the molecular component are as follows:
ð12Þ
ωi ðsÞ ðsÞ ðsÞ ¼ ln x0 þ ln f0 þ ln fi ω0
¼
ig0
Ni(s)/A
ω0 Γ0 þ ω1 Γ1 þ ω2 Γ2 þ 3 3 3 ¼ θ0 þ θ1 þ θ2 þ 3 3 3 ¼ 1
ð14Þ
The quantity ω depends on the total protein adsorption. Contributions from components with different molar surface areas vary with adsorption: at low surface coverage, molecules with large molar areas (near ωmax) dominate; at a high degree of coverage, multilayer films form and molecules with small effective molar areas (near ωm) dominate. The fraction of the surface coverage for component j differs from the molar fraction because of the range of molar surface areas
ð8Þ
þ RT ln fi xi - γωi
ω 1 Γ 1 þ ω 2 Γ2 þ 3 3 3 Γ
ln fi
ðsHÞ
¼ ln fi
ðsEÞ
þ ln fi
ð18Þ
We can substitute eqs 15-18 into eq 12 to produce the equation of state for the surface layer. Notice that this equation solves for the surface pressure using equations describing only the solvent -
Πω0 ¼ lnð1 - θP Þ þ θP ð1 - ω0 =ωÞ þ aP θP 2 RT
ð19Þ
Fainerman et al. chose a Gibbs dividing surface to define the separation between the surface film and the bulk, where surface density Γ is used instead of surface concentration c(s). Component i has ci(s) = Γi/δi, but the definition of xi(s) = ci(s)/cnet(s) in eq 15 δ is a constant for all components, so ci(s) = Γi/δ assumes that δi = P (s) and xi = (Γi/δ)/ (Γi/δ) = Γi/(Γ þ Γ0) for this approximation. We choose a dividing surface with the minimum protein dimension (53) Lucassen-Reynders, E. H. J. Phys. Chem. 1966, 70, 1777.
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δ = 4 nm, like the parallel orientation in Figure 1A where molecules have molar area ωmax. This choice minimizes error in accounting for solvent molecules, which have smaller dimensions than proteins. Let us now depart from the Fainerman et al. model to estimate the error in this approach for a protein multilayer (with δi 6¼ δ) and derive the adsorption isotherm (Γ vs c) for each component j. The surface pressure depends only on the properties of the surface layer itself, but the adsorption isotherm depends on the properties of the entire film. We assume the protein molar volume Vi = ωiδi = ωmaxδ is constant regardless of molecular orientation or layer thickness. The surface coverage using the volume and concentration is θmm ¼
n X
n n X X ¼ ðΓi =δÞðωi δi Þ ¼ Γi ωi ðδi =δÞ
ðsÞ ci Vi
i¼1
¼
n X
i¼1
i¼1
Γi ωi ðωmax =ωi Þ ¼
i¼1
n X
Γi ωmax ¼ Γωmax
ð20Þ
i¼1
while the Fainerman et al. coverage assuming a monolayer with constant δi = δ is θP. This results in a correction to the surface coverage Δθ ¼ θmm - θP ¼ ðωmax - ωÞΓ
ð21Þ
We propose that this modifies the entropy in eq 17 where the first term in square brackets describes the reduction in surface coverage if all molecules (solvent and protein) were replaced with component j (molar surface area ωj). Because of the multilayer, there is effectively a total surface coverage 1 þ Δθ available to molecules (there is normally always a total surface coverage of 1) ln
¼ ½ð1 þ ΔθÞ - ωj ðΓ þ Γ0 Þ þ ln ωj ðΓ þ Γ0 Þ
ðsEÞ fj
ð22Þ
Because the solvent molecules do not form multilayers, we use the previous formulation for f0(s) in eq 17 to solve for the activity coefficients for the solvent, which was also used to derive eq 18. Let us now derive the adsorption isotherm for each state j of the protein molecules with bjcj = Kjxj(b) where bj is the adsorption equilibrium constant for component j and c = cj is the bulk concentration of protein (which is the same for all protein components).2 We solve for bjc = Kjxj(b)on the left side of eq 13 to derive the adsorption isotherm ðbÞ
b j c ¼ Kj x j
ðsÞ ðsÞ ðsÞ xj exp½ - ωj =ω0 ln f0 þ ln x0
ðsÞ
¼ h ¼
fj i
ðsÞ
fj
Π¼0
ω 0 Γj ðθ0 Þωj =ω0
exp½ - 2aP ðωj =ω0 ÞθP þ Δθ
ωΓj
bP c ¼
ðθ0 Þωj =ω
exp½ - 2aP ðωj =ωÞθP þ Δθ
ωΓj
¼
ð1 - ωΓÞωj =ω
exp½ - 2aP ðωj =ωÞθP þ Δθ
ð24Þ
where we substituted for surface coverage of the protein, θP, and the solvent, θ0, using the above definitions. Note that eq 24, the adsorption isotherm, is effectively calculating the concentration as a function of total surface density, c(Γ), which is satisfied when the surface density is at its equilibrium value for that bulk concentration, Γ = Γe. However, we cannot predict these quantities yet, without knowledge of the distribution of molar surface densities and their relationship to the weight-average molar surface area, ω. Let us now derive the distribution of protein components. Despite our modification of the adsorption isotherm (eq 24), the results will be the same as those shown by Fainerman et al. We can use eq 24 to solve for the molar surface density, Γi, and find the ratio of molar surface densities of two components, Γi/Γj, and use this to find the distribution of states Γi ¼ ð1 - ΓωÞðωi - ωj Þ=ω exp½2ap Γðωi - ωj Þ Γj
ð25Þ
Let us now use eq 25 to find the relative molar surface density, Γj, in terms of the measurable total molar surface density, Γ, and the weight-average molar surface area, ω Γj ¼ Γ
Γj Γj =Γ1 ¼ Γ n n P P Γi Γi =Γ1
i¼1
¼ Γ
i¼1
ð1 - ΓωÞðωj - ωmm Þ=ω exp½2aP ðωj - ωmm Þ
n P
i¼1
ð1 - ΓωÞðωj - ωmm Þ=ω exp½2aP ðωj - ωmm Þ
ð26Þ
where we used the definition of Γ = Σi=1nΓi and ω1 = ωmm. We need to be able to compute weight-average molar surface area, ω, as a function of Γ and our model parameters, to use our theory to predict experimentally measurable quantities. We can use eq 14 to solve for ω and use the ratios of relative molar surface densities, eq 25 n P
ð23Þ
where we used eqs 16, 19, 20, and 21 to solve for fj(s) and [fj(s)]Π=0 (with Γ = Δθ = 0), eq 15 was used to solve for xj(s), and eqs 15 and 18 for the exponential term. We use a range of partial molar surface areas from ωmm (multilayers) to ωmax (parallel monolayer) (see Figure 1). We choose an increment of ω0 between components ωj = ω1 þ (j - 1)ω0 where ω1 = ωmm and ωn = ωmm þ (n - 1)ω0 = ωmax and j varies from 1 to n. This choice leads to n . 1, so the use of the increment ω0 has almost no effect on our results compared with a continuously varying ωj. Each component of the protein has the same adsorption equilibrium constant, bj = bP (where bj is the constant for the jth state), P with total adsorption equilibrium constant equal to b = bj = nbp = [(ωmax - ωmm)/ω0]bp. For reasons not fully understood, the 152 DOI: 10.1021/la103316j
formulation of eq 23 leads to predictions of surface pressure rising much more steeply with concentration than experimentally observed.54-56 A much better agreement with experiment proposed by Fainerman et al. can be obtained by substituting the weight-average molar surface area, ω, for the solvent area, ω02
ω ¼
i¼1 n P
i¼1 n P
¼
n P
Γi ω i Γi
¼
Γi ωi =Γ1
i¼1 n P
i¼1
Γi =Γ1
ωi ð1 - ΓωÞðωj - ωmm Þ=ω exp½2aP ðωj - ωmm Þ
i¼1 n P
i¼1
ð1 - ΓωÞðωj - ωmm Þ=ω exp½2aP ðωj - ωmm Þ
ð27Þ
We now have equations that predict the relationships between the measurable equilibrium quantities of bulk protein concentration, (54) Abramzon, A. A.; Gromov, E. V. Colloid J. 1969, 31, 131. (55) Lankveld, J. M. G.; Lyklema, J. J. Colloid Interface Sci. 1972, 41, 454. (56) De Feijter, J. A.; Benjamins, J. J. Colloid Interface Sci. 1981, 81, 91.
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c, surface pressure, Π, and total molar surface density, Γ, for multilayer protein films. Quasi-Equilibrium Adsorption Rate. Let us now use the same chemical potentials derived above to predict the total molar protein adsorption rate J in quasi-equilibrium conditions, which Fernsler and Zasadzinski have measured in a previous experiment.26 Equation 7 predicts the molar adsorption rate with a selection of external potentials as defined in eq 2. We will assume that the surface density is not at equilibrium with the bulk concentration, Γ 6¼ Γe, but the distribution of states Γi is still determined by eq 23. We can derive the molar adsorption rate Ji for the ith component of the protein with a choice of external potentials Vi(b) (the bulk external potential), Vi(s) (the surface (the potential barrier). A comparexternal potential), and Vmax i ison of eqs 8 and 9 to eq 2 allows us to define the surface and bulk external potentials for component i of the protein ðsÞ
ð0sÞ
¼ μi
Vi
ðbÞ
Vi
ðsÞ
þ RT ln fi - γω
ð0bÞ
¼ μi
ðbÞ
þ RT ln fi
ð28Þ ð29Þ
We believe that the maximum potential barrier occurs in the interfacial region of the film where bulk and surface properties intersect. An approximate description of the potential barrier could be formed by a combination of potentials at the surface and bulk, described by eqs 28 and 29. We propose that the potential barrier is the sum of the bulk and surface potentials to describe their overlapping influences in the interfacial region ðsÞ
ðbÞ
Vimax ¼ Vi þ Vi
ð30Þ
We substitute eqs 28-30 into eq 7 to solve for the molar adsorption rate for the ith component ðsÞ ðbÞ 2Dp ðbÞ ðsÞ Ji ¼ pffiffiffi cnet xi e - Vi =RT - xi e - Vi =RT π 2Dp ðΓ þ Γ0 Þ ðbÞ 1 ð0sÞ ðsÞ xi exp½ μi þ RT ln fi - γωi ¼ pffiffiffi δ RT π 1 ðsÞ ð0bÞ ðbÞ μ þ RT ln fi ð31Þ - xi exp½ RT i where we approximate the net molar concentration of solvent and proteins as the net molar surface density in the film with thickness δ. We use the equation for adsorption equilibrium and eqs 8 and 9 to solve for bulk molar fraction of the ith component " # ðbÞ bP c x ðbÞ xi ¼ ðbÞ ¼ bP c iðsÞ xi Π ¼ 0 Ki " n o n ðbÞ ð0bÞ ðbÞ ðsÞ ð0sÞ ¼ bP c exp 1=RT μi - μi - RT ln fi - μi - μi - RT ln
ðsÞ fi
þ γω
ðbÞ
- RT ln fi
#
þ γ0 ω
Π¼0
ð0sÞ ð0bÞ ¼ bP c exp 1=RT μi - μi
Π¼0
ωi ωi ωi expðaP þ 1 - Þ ð32Þ ω0 ω0 ω0
where the two terms in the second and third lines in {} brackets are the molar fractions, xi(b)|Π=0 and xi(s)|Π=0, which are simplified by fact that the bulk and surface chemical potentials are equal, μi(s) = μi(b), when Π = 0 (γ = γ0), and there is no surface film. Langmuir 2011, 27(1), 148–157
The {} brackets in the fourth line is the surface activity coefficient fi(s)|Π=0 from eqs 16, 17, and 20. Using eqs 15, 16, 17, 20, and 30 in eq 29, we solve for the following: 2Dp 1 bi c ωi ð1 - θP Þωi =ω0 expð2aP θP Ji ¼ pffiffiffi ω π δ ω0 1 ð0bÞ ðbÞ μi þ RT ln fi - ΔθÞ - Γi exp½ RT
ð33Þ
where we use the approximation that the bulk activity coefficients are independent of the surface layer fi(b) = fi(b)|Π=0. We use the substitution of ω for ω0 and bi = bP used in eq 22 for the adsorption isotherm2,47 to find our adsorption rate for the ith component 2Dp 1 1 ð0bÞ μ Ji ¼ pffiffiffi exp½ RT i π δ b c ωi P ðbÞ ωi =ω ð1 - ΓωÞ þ RT ln fi expð2aP θP - ΔθÞ - Γi ω ω bP c ωi ð1 - ΓωÞωi =ω expð2aP θP - ΔθÞ - Γi where ¼ v ω ω 2Dp 1 1 ð0bÞ ðbÞ μi þ RT ln fi v ¼ pffiffiffi exp½ ð34Þ RT π δ The quantity v, which has units s-1 and is effectively a rate constant, is not known for a particular system, but we will assume that v is essentially independent of the surface properties and use it as an empirical constant in our model. The first term in square brackets in eq 34 represents the adsorption rate of molecular species i, and is a positive quantity, and the second term describes desorption of species i and is negative. Note that in general there is both adsorption and desorption of species occurring for the given conditions. When the calculated adsorption rate, Ji is positive, it indicates that adsorption is dominant over desorption and the film surface density is increasing; when the adsorption rate is negative, desorption is dominant, resulting in a loss of the surface film density. When the adsorption rate is set to zero, Ji = 0, eq 34 reduces to the adsorption isotherm, eq 22, which means that the system is in equilibrium with balanced adsorption and desorption. Finally, we cannot experimentally measure the adsorption rates of individual components, but the total adsorption rate for the entire protein is measurable: J ¼
n X
Ji
ð35Þ
i¼1
Interpretation of the total adsorption rate calculated from eq 35 is similar to the above analysis of eq 34.
Results and Discussion Multilayer Adsorption Model. We now predict relationships between measurable properties of bulk concentration c, total surface density Γ, surface pressure Π, and adsorption rate J as a function of our input parameters, ω0, ωmax, ωmm, b, and v. We make the following important assumption in our analysis: the layer thickness, and hence ωmm, is a function of the concentration c. We have developed a software package using MATLAB to compute and fit the model to experimental data (Miller et al. use an iterative method60). In most cases, we will use Γ as the independent variable and calculate other quantities from it. Equation 27 is a transcendental equation for the weight-average molar surface density ω, which we calculate as a function of Γ and DOI: 10.1021/la103316j
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our input parameters, using a bounded minimization routine in MATLAB with bounds from 0 < ω < 1/Γ. We use eq 26 to calculate the distribution of molar surface densities Γi and eq 21 to determine the increased surface coverage Δθ due to multilayers. Finally, we calculate the measurable quantities of surface pressure Π (eq 19), bulk protein concentration c (eq 24), and adsorption rate J (eqs 34 and 35) as a function of Γ. Let us investigate multilayer formation on the equation of state by lowering ωmm in Figure 2A. When ωmm ≈ ωmax, there is a small range of molar surface areas and, we reproduce the Frumkin equation of state with surface density asymptotically approaching the close-packed density, Γ = 1/ωmm ≈ 1/ωmax, as the surface pressure diverges. A smaller ωmm allows a range of areas and can be used to describe protein monolayers, and a very small ωmm describes the formation of multilayers, which produces the two inflection points in the equation of state seen in Figure 2A (e.g., Γ ≈ 2.5 mg/m2 and Γ ≈ 7 mg/m2 for ωmm = 50 105 mol/m2). The distribution of states i shows that, at low total surface densities, Γ = 1/ωmax, the distribution is nearly uniform, but at high densities, Γ = 1/ωmm, the distribution is heavily weighted toward small i with small molar surface areas near ωmm (see Figure 2B). The inflection points in the equation of state result from close-packing being achieved in these two regimes. Comparison to Experiment. Let us test our model on the protein, bovine serum albumin, with molecular weight 65 kDa. The experimental data was collected on a Langmuir trough which measured surface pressure Π, the total surface density Γ using a quantitative BAM method, and the adsorption rate J for different bulk concentrations.26 J and Γ were found to be constant when Π was held fixed (using a feedback mechanism to expand or contract the trough); therefore, these conditions should be a good test of our quasi-equilibrium model. Our model relies on the known parameters of concentration c and temperature T = 295 K, and we use a value for ωmax = 750 105 m2/mol computed from results measuring the thickness of an adsorption layer.41 The models are a function of the following unknown parameters: ω0, ωmm, aP, v, and b. The standard equilibrium behavior of proteins shows strange behavior with rising concentration, which is not seen in standard surfactant monolayers: at high concentrations, the surface pressure levels off, but the surface density continues to rise.21,41-45 Several successful models have proposed the formation of a protein bilayer in certain regimes of the concentration to account for this behavior.2,12,13,47 The Fainerman et al. model2,47 continuously changed the average molar surface area above a certain critical concentration, to produce a modified equation of state and adsorption isotherm in this regime. In Fernsler and Zasadzinski’s adsorption experiments, the quasi-equilibrium equation of state Γ(Π) was nearly identical regardless of concentration for monolayer films of small molecular weight surfactants,26 but Γ(Π) depended strongly on concentration for the protein albumin (see Figure 3A). The true equation of state is computed by measuring single values of equilibrium Γe and Πe for each concentration. Fernsler and Zasadzinski26 hypothesized that different values of Γ measured at the same Π (see Figure 3A) were a result of multilayers forming at higher bulk concentrations. Our equation of state (eq 19) depends on parameters ω0, ωmm, and aP which we fit using a weighted least-squares minimization using the MATLAB function “fminsearch”. To account for the thickening layer at higher concentrations, we take the parameter ωmm, which depends on the layer thickness, as fixed for experiments performed at a particular concentration, but adjustable between different concentrations. This is analogous to the Fainerman et al. modification of the average molar surface ω at higher concentration. 154 DOI: 10.1021/la103316j
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Figure 2. (A) Equation of state from eq 18 plotted using ω0 =
2 105 mol/m2, ωmax = 400 105 mol/m2, and aP = 1. Curves with different values of ωmm are shown in different line styles, representing rigid molecular monolayers (high wmm) through multilayers (low wmm). (B) Distribution of states, Γi, from values of 1 < i < n, where i = 1 is the state with minimum molar surface area, ω1 =ωmm = 200 105 m2/mol and i = n has the maximum molar surface area, ωn= ωmax = 400 105 m2/mol. At low Γ = 1/ωmax, the distribution of states is uniform, and at high Γ = 1/ωmm, the distribution is weighted toward low i with small molar surface areas.
In Table 1, the fit values of aP, ω0, and ωmm, are listed for the four concentrations measured. Fit values of aP and ω0 are similar, but ωmm is significantly different. We use the average value aP ≈ 1 and ω0 = 2.50 105 m2/mol from the four experiments for all future predictions of the model, which is similar to values found by Fainerman et al.2,47 We now fit only ωmm (see Table 1 final column) and plot Γ(Π) in Figure 3A. Note that higher concentrations produced smaller fits for ωmm as expected for the formation of multilayers. At the highest concentration, the maximum range of between ωmm and ωmax differs by approximately a factor of 12, which is accounted for by the change in orientation (parallel or perpendicular to the surface is approximately a factor of 4) and the formation of multilayers (up to a maximum of about three allowed layers at the highest concentration measured). Our model has an ensemble of orientational and layer thickness states so the weight-average layer thickness (which can be measured in experiment) is not as thick as the maximum allowed thickness. The error bars on the experimental Γ(Π) data were computed assuming a (7% accuracy in Γ from the quantitative BAM technique,26 with larger errors expected at low and high Π because of shortened collection time (we computed this additional error assuming that the error was inversely proportional to the data collection time). Surface pressure measured Π to (0.1 mN/m accuracy, but calibration error could shift Π by approximately (1 mN/m. The weighted fits are within the error in almost all measured data, which confirms our model; and the assumption of a Langmuir 2011, 27(1), 148–157
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Article Table 2. Fit of J vs Π Using Equations 34 and 35 and Previous Fit Parameters Described in Table 1 concentration
v (10-6 s-1)
b (108 m3/mol)
v (10-6 s-1) with b = 0.392 108 m3/mol
0.02 mg/mL 0.2 mg/mL 2 mg/mL 10 mg/mL
11.9 22.7 11.4 14.8
0.87 0.12 0.27 0.19
14.6 6.4 8.6 7.8
Figure 4. Adsorption rate J measured as a function of the surface
Figure 3. (A) Quasi-equilibrium equation of state for the protein albumin (molecular weight 65 kDa) in buffer solution.26 Π was held fixed on a Langmuir trough and Γ measured; different-colored data points have different bulk concentrations. Colored solid curves are theoretical fits to corresponding colored data using different ωmm for different concentrations, but otherwise a single set of parameters (see Table 1). (B) Equilibrium equation of state Πe vs Γe, for albumin from data for J = 026 in black circles and theory curve calculated using the parameters in Table 1 and an interpolation for ωmm. Table 1. Fit of Π vs Γ Using Equation 18 concentration
aP
0.02 mg/mL 0.2 mg/mL 2 mg/mL 10 mg/mL average
0.85 1.56 1.17 1.33 ∼1
ωmm ωmm (105 m2/mol) ω0 (105 m2/mol) (105 m2/mol) using averages 2.18 2.27 2.57 3.00 2.50
123.1 37.7 59.8 61.5
130 78 61 58
concentration-dependent layer thickness and cross correlation was not observed between fitting parameters, which indicated we achieved a unique fit. We model the equilibrium equation of state Γe vs Πe in Figure 3B (we use an interpolation of ωmm as described below). The model accurately predicts the equilibrium data from Fernsler and Zasadzinski.26 Other than a scaling factor between different experimental techniques, our model is consistent with the Fainerman data,21,41,43,57 including the characteristic nearly flat dependence of Γe on Πe at Πe = 15 mN/m (our measurements were closer to those of McClellan38). Let us fit the remaining parameters b and v to the adsorption rate experiments. The experimental data measured J(Π), so we use a bounded minimization routine in MATLAB, “fzero”, to calculate Γ(Π) (the reverse of eq 18) where the maximum bound of (57) Benjamins, J.; De Feijter, J. A.; Evans, M. T. A. Faraday Discuss. Chem. Soc. 1978, 59, 218.
Langmuir 2011, 27(1), 148–157
pressure, Π, for the protein albumin (molecular weight 65 kDa) from previously published data26 shown with data points and error bars. Π was held fixed on a Langmuir trough and J was measured; different-colored points have different bulk concentrations of albumin. Colored solid curves are theoretical fits to corresponding colored data using different ωmm for different concentrations, but otherwise a single set of parameters (see Table 1 and b = 0.392 108 m3/mol and v = 9.3 10-6 s-1).
Γ = 1/ωmm is used. We use a weighted least-squares fit to the data using eqs 26, 27, 34, and 35 to calculate J, with parameters shown in Table 2. Using the average b = 0.392 108 m3/mol and refitting the data for the single parameter v (see Table 2, final column), we obtain a final averaged value of v = 9.3 10-6 s-1, which was used to compute the modeled data shown in Figure 4. The theoretical curves were computed with a single set of parameters for all experiments, except parameter ωmm, which is a function of the concentration (see values in Table 1). The model satisfactorily predicts J over a range of over 2 orders of magnitude in bulk concentration, where the theoretical curves lie within the error bars in most cases. J is positive and large at low Γ and Π, which indicates that adsorption dominates for a sparsely populated film; J is small and negative at high Γ and Π, which indicates that desorption dominates in dense films. Adsorption is linearly proportional to bulk concentration (the first term in eq 34). Desorption (the second term in eq 34) is linearly proportional to the surface density, so the desorption is predicted not to diverge even near close-packed density. This has significant effects on the nature of these films: our model predicts that rapidly compressed films can achieve surface densities and surface pressures significantly higher than equilibrium, due to the relatively slow rate of desorption for high density films. The model modestly underestimates desorption at high surface pressures: better agreement with the high pressure desorption data can be found assuming a different potential barrier, = μ(0s) - μ(0b) - γωi, but at the sacrifice of accuracy such as Vmax i i i describing lower pressure adsorption data. Let us model the adsorption isotherm using the parameters described above. Because our model assumes that layer thickness is a function of concentration, we must use a changing parameter ωmm. We use a cubic spline routine in MATLAB (“interp1” with DOI: 10.1021/la103316j
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volume by 1.5-219 and we assume a packing efficiency of 74% for cubic or hexagonal packing: the estimated molar volume with these effects is V = 0.14-0.19 m3/mol. Using our interpolated values for ωmm, we compute the δave(c) in Figure 5B (we used the upper range of hydration for the plot). The approximation produces a layer thickness that is in good agreement with averages of the data of McClellan et al.38 (see Figure 5B) and Graham and Phillips.59 Note that, at high concentrations, the average layer thickness is calculated and measured to be only a little over that of a perpendicularly aligned monolayer or a parallel-aligned multilayer using the equilibrium surface density, Γe. However, the distribution of molar surface areas in our model predicts some at ∼ωmm, which will produce regions with a full bilayer or greater in thickness. Furthermore, a compressed albumin film can reach surface pressures and densities significantly greater than the equilibrium values (see Figure 3) where we predict a full bilayer or greater in thickness. Finally, the layer thickness scales with the log of the concentration as expected from standard thermodynamic considerations.
Conclusion
Figure 5. Minimum molar surface area ωmm was interpolated between the points in Table 1 to calculate the entire range of concentrations. (A) The adsorption isotherm for albumin (molecular weight 65 kDa) with Πe from data for J = 026 plotted as black circles, and from data published from Warriner et al.58 plotted as red circles. The theoretical curve was determined using the parameters from Table 1 and b = 0.392 108 m3/mol and v = 9.3 10-6 s-1, where (B) black circles are averages of layer thickness of albumin films versus concentration c from McClellan et al.38 and theory curve is the approximated average layer thickness δave. The layer thickness is approximately linear with log c.
method “spline”) to interpolate our parameter ωmm over a range of concentrations and use eqs 18, 21, 24, 26, and 27 to find the weight-average molar surface area ω, total surface density Γ, equilibrium surface pressure Πe, and concentration c. We plot the adsorption isotherm Πe vs c in Figure 5A. The theory satisfactorily predicts the data from our adsorption rate experiments (Πe is interpolated where J = 0) and previous results by Warriner et al.58 and Graham et al.59 over 5 orders of magnitude in concentration. The adsorption isotherms calculated by drop shape methods, such as that in Makievski,42 are shifted to higher concentrations by approximately an order of magnitude (but are otherwise identical) due to a decrease in the protein concentration in the drop due to its adsorption on the drop surface. The same interpolation routine was used to generate the equilibrium equation of state in Figure 3B. Finally, let us estimate the average thickness of the protein layer δave using the volume of the protein molecules V and the weight-average surface density, δave = V/ω. For albumin, we assume an ellipsoid, with dimensions 4 4 14 nm3, so the volume is V = 4π/3(2 2 7 nm3). Hydration effects increase the (58) Warriner, H. E.; Ding, J.; Waring, A. J.; Zasadzinski, J. A. Biophys. J. 2002, 82, 835–842. (59) Graham, D. E.; Phillips, M. C. J. Colloid Interface Sci. 1979, 70, 415. (60) Miller, R.; Fainerman, V. B.; Aksenenko, E. V.; Leser, M. E.; Michel, M. Langmuir 2004, 20, 771–777.
156 DOI: 10.1021/la103316j
In summary, the proposed thermodynamic model of protein adsorption satisfactorily predicts the behavior of five experiments conducted on the protein albumin over several orders of magnitude in concentration, and in equilibrium and quasi-equilibrium conditions. This model should predict behavior of other globular proteins that show little denaturation at interfaces or flexiblechain proteins that reversibly denature at interfaces. The model uses a set of four fixed parameters to describe all these experiments with a single concentration-dependent parameter describing the formation of multilayers. The success of our model supports our hypothesis of the formation of protein multilayers at high concentrations and compressions. An accurate treatment of protein multilayers is important to understand these films and may have a significant effect on interactions in lipid-protein films. In particular, we are interested in the competitive adsorption of protein films with low-molecular-weight surfactant films, in which a protein multilayer may present a larger barrier to adsorption than a monolayer. We are working to extend our treatment of quasi-equilibrium adsorption to low-molecularweight surfactants composed of more than one species to model lung surfactant films. These models may then predict the competitive adsorption of the two films, which could lead to a greater understanding of these and other mixed lipid and protein films. Acknowledgment. We thank Professor J. Zasadzinski for his helpful comments and ideas for further applications of this theory work.
Glossary J z μ D T R V x c cnet V(s) V(b)
protein flux to a surface distance from surface chemical potential diffusion constant absolute temperature ideal gas constant external potential molar fraction molar concentration molar concentration of all solvent and solvate surface potential bulk potential Langmuir 2011, 27(1), 148–157
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V max p i μi(s) μi(b) Γi Γ N (s) A ωi ωmin ωmax ω ωmm ω0 n μi(0s) μi(0b) fi(s) fi(b)
Article
maximum potential a constant related to barrier width component index chemical potential of surface component chemical potential of bulk component molar surface density of a component protein molar surface density number of moles of protein at surface area of surface molar area of a component minimum protein molar area maximum protein molar area weight average molar area protein multilayer minimum molar area molar area of solvent number of protein components chemical potential of surface standard state chemical potential of bulk standard state surface activity coefficient bulk activity coefficient
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γ γ0 Π θi θP K ni fi (sH) fi (sH) aP δi V Δθ bi b Γe Πe Ji v δave
surface tension surface tension of clean interface surface pressure surface coverage of component protein surface coverage coefficient of adsorption equilibrium ratio of molar surface areas surface enthalpy contribution to activity coefficient surface entropy contribution to activity coefficient parameter of nonideality (Frumkin’s parameter) thickness of component at surface molar volume surface coverage correction for multilayers to entropy adsorption equilibrium constant of a component total adsorption equilibrium constant equilibrium surface density equilibribum surface pressure flux to the surface of a component adsorption rate constant average thickness of the protein layer
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