Energetics of ProteinâInterface Interactions and Its Effect on Protein

Langmuir 1999, 15, 6991-7001

6991

Energetics of Protein-Interface Interactions and Its Effect on Protein Adsorption Tapashi Sengupta, Lev Razumovsky, and Srinivasan Damodaran* Department of Food Science, University of WisconsinsMadison, Madison, Wisconsin 53706 Received March 1, 1999. In Final Form: June 15, 1999 The kinetics of adsorption of several positively and negatively charged proteins at the air/water and triolein/water interfaces have been studied. It is shown that adsorption of proteins at these interfaces is not simply diffusion-controlled but is strongly influenced by the energetics of interaction of proteins with the interfaces. Generally, positively charged proteins experience an energy barrier for adsorption at the air-water interface and therefore exhibit adsorption rates an order of magnitude slower than their respective bulk diffusivities. In contrast, the negatively charged proteins exhibit an attraction toward the air/water interface and therefore their adsorption rates are either slightly higher or 1.5-2 times lower than their bulk diffusivities. At the triolein-water interface however, all proteins, except phosvitin, adsorbed at rates 1-2 orders of magnitude faster than their bulk diffusivities. The differences between adsorptivities of positively and negatively charged proteins at the air-water interface, and the differences between adsorptivities of all proteins at the air/water and triolein/water interfaces can be explained convincingly in terms of the energetics of interaction of proteins with the interfaces.

Introduction Protein adsorption at fluid-fluid and fluid-solid interfaces has been extensively studied in order to acquire a thorough understanding of the mechanism for adsorption of these biopolymers to such interfaces.1-13 One of the contentious issues that has risen out of these studies is whether protein adsorption at an interface is diffusioncontrolled. For instance, studies have shown that under quiescent conditions the rates of adsorption of certain negatively charged proteins6-9 to the air-water interface are slightly higher than their respective bulk diffusion coefficients, while the rate of adsorption of positively charged lysozyme8-11 to the same interface is about 1 order of magnitude smaller than its bulk diffusivity. On the other hand, recently we have shown that both negatively charged β-casein and bovine serum albumin and positively charged lysozyme adsorb to a planar triolein-water interface at a rate one to 2 orders of magnitude greater than their bulk diffusivities.2 Another study on protein adsorption at a liquid-solid interface has also shown that the rate of adsorption was strongly influenced by hydrophobic interactions between the protein and the solid surface, resulting in a rate enhancement over that of bulk diffusion coefficient.3 Yet, another study revealed that protein adsorption does not always correlate with surface * To whom correspondence should be addressed. Tel.: 608-2632012. Fax: 608-262-6872. E-mail: [email protected]. (1) Xu, S.; Damodaran, S. Langmuir 1992, 8, 2021. (2) Sengupta, T.; Damodaran, S. Langmuir 1998, 14, 6457. (3) Tilton, R. D.; Robertson, C. R.; Gast, A. P. Langmuir 1991, 7, 2710. (4) Young, B. R.; Pitt, W. G.; Cooper, S. L. J. Colloid Interface Sci. 1988, 125, 246. (5) Bazskin, A.; Lyman, D. J. J. Biomed. Mater. Res. 1980, 14, 393. (6) Xu, S.; Damodaran, S. J. Colloid Interface Sci. 1993, 157, 485. (7) Anand, K.; Damodaran, S. J. Agric. Food Chem. 1996, 44, 1022. (8) Sengupta, T.; Damodaran, S. J. Colloid Interface Sci. 1998, 206, 407. (9) Anand, K.; Damodaran, S. J. Colloid Interface Sci. 1995, 176, 63. (10) Xu, S.; Damodaran, S. J. Colloid Interface Sci. 1993, 159, 124. (11) De Feijter, J. A.; Benjamins J. In Food Emulsions and Foams; Dickinson, E., Ed.; Royal Society of Chemistry: London, 1987; pp 7285. (12) Razumovsky, L.; Damodaran, S. Langmuir 1999, 15, 1392. (13) Damodaran, S.; Xu, S. J. Colloid Interface Sci. 1996, 178, 426.

hydrophobicity of the solid phase.4 A general picture that emerges out of these studies is that an interface cannot be treated as an inert sink for proteins. Rather, it acts as an energy field, and the rate of adsorption of a protein at an interface is dictated by a delicate interplay between interaction of this energy field with the protein and diffusional forces generated by the concentration gradient.1 In a previous study,2 we have provided evidence for this hypothesis based on the interaction energy profiles generated for β-casein, bovine serum albumin (BSA), and lysozyme with the air-water and triolein-water interfaces. In the present study we investigated the adsorption behaviors of a wide spectrum of negatively and positively charged proteins at air-water and oil-water interfaces. We show that the positively charged proteins generally experience an energy barrier for adsorption at the airwater interface whereas it is not the case with the negatively charged proteins. On the other hand, both positively and negatively charged proteins are attracted to the triolein-water interface to the extent that the adsorption rates are about 1-2 orders of magnitude faster than their respective bulk diffusivities. These anomalies are attributed to differences in the energetics of interaction of positively and negatively charged proteins with the interfaces. For the computation of potential energy profiles as a protein approached an interface, we have considered the van der Waals interaction energy between the protein and the interface, the electrostatic interaction energy between the protein macroion and its image charge in the low dielectric phase, the hydrophobic interaction energy between the protein molecule and the interface, and finally the electrical double layer interaction energy due to overlapping of the double layers of the protein and the respective interface. Our calculations revealed that the dispersion contribution to the net van der Waals interaction energy is the most dominating factor in dictating the rate of protein adsorption to an interface. Materials and Methods Triolein (99%), lyophilized and salt-free bovine β-casein (min. 90%), lyophilized bovine serum albumin (96-99%), lyophilized bovine Rs1 casein (min. 85%), hen egg yolk phosvitin, lyophilized

10.1021/la990235s CCC: $15.00 © 1999 American Chemical Society Published on Web 07/30/1999

6992

Langmuir, Vol. 15, No. 20, 1999

type I bovine R-lactalbumin (85%, calcium saturated), lyophilized bovine β-lactoglobulin, lyophilized salt-free grade V chicken egg ovalbumin (99%), and type IV chicken egg white conalbumin (98%) were obtained from Sigma Chemical Co. (St. Louis, MO). Also, lyophilized hen egg white lysozyme (95%), lyophilized bovine pancreatic R-chymotrypsin (42 units/mg solid), lyophilized bovine pancreatic trypsin (13 000 units/mg of solid), lyophilized pepsin (2100 units/mg of solid) from porcine stomach, lyophilized bovine heart cytochrome c (95%), type III-ss histone-I from calf thymus, bovine liver catalase aqueous solution (38 000 units/mg solid), and lyophilized bovine pancreatic RNAase A were obtained from Sigma Chemical Co. (St. Louis, MO). Methyl oleate (99%), sodium cyanoborohydride (95%), ultrapure Na2HPO4, and NaH2PO4 and NaCl were purchased from Aldrich Chemical Co. (Milwaukee, WI). 14C-Formaldehyde, with a specific radioactivity of 10 mCi/ mmol, was purchased from New England Nuclear Co. (Boston, MA). Extreme care was taken in purifying water for adsorption studies. A Milli-Q Ultrapure water purification system (Millipore Corp., Bedford, MA) with a Qpak1cartridge package was used to purify the water. The resistivity of the water used was 18.2 mΩ cm. The surface tension of this water at 25 °C was 71.9 ( 0.1 mN/m. Radiolabeling of proteins was carried out by the reductive methylation of amino groups using sodium cyanoborohydride and 14C-formaldehyde, as described elsewhere.1 The concentrations of proteins were calculated based on their extinction coefficients of 1% solution at the appropriate wavelengths.13-15 The extinction coefficient of histone-I was determined by the Lowry method16 to be 1.69 for 1% solution. The wavelength used for most proteins was 280 nm; exceptions were 281 nm for lysozyme, 277.5 nm for RNAse A, 282 nm for R-chymotrypsin, 278 nm for catalase and histone-I, 279 nm for BSA, and 275 nm for phosvitin. Apparatus Used for Adsorption Studies at the OilWater Interface. A new noninvasive methodology employing the radiotracer technique for monitoring the adsorption of 14Clabeled proteins at a planar triolein-water interface2 was used in this study. The entire system for protein adsorption studies consisted of a Plexiglass housing placed on a vibration-free surface. The temperature inside the housing was maintained at 25 ( 2 °C by circulating water from a thermostated water bath, and humidity was maintained constant by placing reservoirs of water inside the housing as well as by spreading wet towels. The system was set up such that the surface tension as well as the protein concentration at the interface could be monitored simultaneously during the entire course of the experiment. The surface tension was measured by the Wilhelmy plate technique using a ST 9000 surface tensiometer (Nima Technology, Ltd., Coventry, England), interfaced with an IBM computer. A thin sand-blasted platinum plate was used as the sensor. A Teflon trough having inner dimensions of 17.45 cm length, 5.5 cm width, and 4 cm depth was used as the Langmuir trough. One side of the trough had a small hole (1 mm diameter) capped tightly with a septum for injecting protein into the bulk phase. In each experiment, 350 mL of solution consisting of 20 mM phosphate buffer (pH 7.0) adjusted to an ionic strength of 0.1 M with NaCl was used. A very thin (3 mm diameter, 12.7 mm length) Tefloncoated magnetic stir bar was placed in the center of the trough prior to spreading the triolein film on the water surface. A 1000 Å oil film was layered on the water surface as described in detail elsewhere.2 The oil film remained stable over the entire duration of protein adsorption lasting for 15-24 h.2 The film was allowed to equilibrate for another hour or until no further change in surface tension occurred. The surface tension of pure water was lowered by 18.9-20.9 mN/m, after equilibration of the oil film on the water surface at 25 °C.2 To initiate protein adsorption, a known volume (0.5-2.0 mL) of the protein stock solution was injected through the hole in the side of the trough, without disturbing the oil film. The final bulk protein concentration was (14) CRC Handbook of Biology and BiochemistrysSelected data for Molecular biology, 2nd ed.; Sober, H. A., Ed.; CRC Press: Cleveland, 1973. (15) Bhown, A. J. Handbook of Proteins; A&M Publications: Birmingham, AL, 1990. (16) Lowry, O. H.; Rosebrough, N. J.; Farr, A. L.; Randall, R. J. J. Biol. Chem. 1951, 193, 265.

Sengupta et al. 1.5 × 10-4% (w/v). The interfacial pressure and surface counts per minute measurements at the oil-water interface were started simultaneously as soon as the injection of the protein was completed. At the same time, gentle stirring of the protein solution by a magnetic stir bar was initiated. The speed of the stir bar was preadjusted to about 60 rotations per minute in order to avoid ripples on the triolein film. The stir bar was moved very slowly along the length of the trough for proper mixing of the injected protein by moving the magnetic stirrer beneath the Plexiglass platform. To the naked eye, the oil-covered water surface remained perfectly calm under this gentle stirring motion. The stirring was continued only for the first 15 min, after which it was stopped. The protein concentration at the oil-water interface was monitored by measuring surface radioactivity using a rectangular gas proportional counter with a Mylar window (8 × 4 cm) from Ludlum Measurements, Inc. (Sweetwater, TX). The counts per minute (cpm) were integrated using a rate meter (model 2200, Ludlum Measurements, Inc., Sweetwater, TX) and printed out on a strip chart recorder interfaced with the rate meter. The gas proportional counter was suspended over the middle of the trough at a distance of about 4-5 mm from the water surface, without touching the trough. A carrier gas composed of 98.0% argon and 2.0% propane was passed continuously through the gas proportional counter at a rate of 20 mL/min. The counts per minute measurements were made at 1 min intervals for the first hour of the experiment, followed by measurements at 10 min intervals during the rest of the experiment. Calibration curves required to convert counts per minute readings at the oil-water and airwater interfaces into protein surface concentrations (mg/m2) were constructed, as described elsewhere.2 The calibration curve for background bulk radioactivity correction was also generated with Na214CO3 in the bulk solution.2 The surface cpm of the equilibrated oil film before radiolabeled protein injection, along with the background bulk cpm when the protein was injected, was taken as the net background radioactivity. This background cpm was subtracted from the actual surface cpm values collected during protein adsorption at the oil-water interface. The background-corrected cpm was then divided by the slope of the cpm versus surface radioactivity calibration curve to obtain the surface radioactivity (µCi/m2) at each time interval. The transient surface concentration of the protein (mg/m2) was obtained by dividing the instantaneous surface radioactivity (µCi/m2) by the specific activity of the protein (µCi/mg). The adsorption of proteins at the air-water interface was performed as described elsewhere.6

Results Figures 1A and 2A show the rate of accumulation of negatively and positively charged proteins at the trioleinwater interface, respectively. Figures 1B and 2B depict the variation of interfacial pressures (i.e., due to protein adsorption only, with respect to zeroed surface pressure for triolein-film on water) with square-root-of-time for the same sets of negatively and positively charged proteins, respectively. In all experiments with the triolein-water interface, an initial lag period of 9-20 min was observed for the proteins to start adsorbing at the oil-water interface. This lag period has been shown to be the result of the initial mixing time required for the bulk solution to become homogeneous following the injection of proteins into it.2 The initial lag period has been subtracted from the adsorption curves shown in Figures 1A and 2A. According to Ward and Tordai,17 the concentration of a protein (Γ) accumulated at an interface at any time t by means of diffusive transport from the bulk solution is given by

Γ ) 2Cb(Dbt/π)1/2

(1)

where Db is the bulk diffusion coefficient of proteins. This equation was derived under the assumption that bulk (17) Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946, 14, 453.

Protein-Interface Interactions

Figure 1. Variation of (A) surface concentration, Γ, and (B) surface/interfacial pressure of negatively charged proteins with time during their adsorption to the triolein-water interface: 4, R-lactalbumin, 9, β-lactoglobulin, O, ovalbumin, b, conalbumin, 1, phosvitin; 3, catalase; s, β-casein; ‚‚‚, R-casein; 0, pepsin; ], BSA.

concentration of protein prevails at infinite distance from the interface and the concentration at the subsurface layer (just below the interface) is zero due to instantaneous adsorption. The apparent diffusion coefficients of proteins were calculated from the initial slopes of the Γ vs xt curves, which are listed in Table 1. The bulk diffusivities of the proteins are also listed in Table 1. An examination of the data in Table 1 reveals that the apparent diffusivities at the triolein-water interface are almost 1-2 orders of magnitude higher than the bulk diffusivities, indicating

Langmuir, Vol. 15, No. 20, 1999 6993

Figure 2. Variation of (A) surface concentration, Γ, and (B) surface/interfacial pressure of positively charged proteins with time during their adsorption to the triolein-water interface: 4, R-chymotrypsin; 9, cytochrome c; ×f3s, lysozyme; O, trypsin; 3, RNAase A; ‚‚‚, histone I.

that besides diffusion other extremely long-range attractive forces, emanating probably from the oil phase, are acting to enhance the adsorption of proteins to this interface. A similar effect was observed during the adsorption of BSA onto spread lecithin monolayer at the air-water interface.18 Also, the equilibrium concentrations of proteins, Γeq, at the same interface are much higher than those usually obtained at the air-water interface. (18) Cho, D.; Narsimhan, G.; Franses, E. I. Langmuir 1997, 13, 4710.

6994


Sengupta et al.

Table 1. Selected Experimental Data from Surface Concentration and Surface Pressure Curves of Positively and Negatively Charged Proteins at the Air-Water and the Triolein-Water Interfaces protein β-casein BSA Rs1-casein R-lactalbumin, Ca2+ sat β-lactoglobulin ovalbumin conalbumin pepsin phosvitin, pH 7 pH 2 pH 7 catalase lysozyme cytochrome c RNAase A trypsin R-chymotrypsin histone- I (from calf thymus)

interface A/W O/W A/W O/W A/W O/W A/W O/W A/W O/W A/W O/W A/W O/W A/W O/W A/W A/W O/W A/W O/W A/W O/W A/W O/W A/W O/W A/W O/W A/W O/W A/W O/W

Da, cm2/s 10-7

5.94 × 139.2 × 10-7 6.1 × 10-7 197.9 × 10-7 11.3 × 10-7 158.6 × 10-7 7.6 × 10-7 130.8 × 10-7 3.8 × 10-7 140.4 × 10-7 2.8 × 10-7 239.2 × 10-7 2.3 × 10-7 114.3 × 10-7 0.17 × 10-7 8.11 × 10-7 0.04 × 10-7 2.06 × 10-7 9.39 × 10-7 0.26 × 10-7 43.4 × 10-7 1.5 × 10-7 120 × 10-7 1.2 × 10-7 53.8 × 10-7 0.047 × 10-7 15.5 × 10-7 0.48 × 10-7 113.7 × 10-7 5.7 × 10-7 162.3 × 10-7 4.21 × 10-7 30.53 × 10-7

Db, cm2/s 6.1 ×

10-7 b

6.1 × 10-7 c 9.1 × 10-7 b 10.6 × 10-7 b 7.8 × 10-7 b 7.4 × 10-7 c 5.3 × 10-7 c 9.35 × 10-7 d 5.77 × 10-7 e,f 4.1 × 10-7 c 11.6 × 10-7 c 13.0 × 10-7 c 11.7 × 10-7 c 9.3 × 10-7 g 10.2 × 10-7 d 5.13 × 10-7 d

Γeq, mg/m2

surface pressure, mN/m

2 6.64 0.91 4.13 1.68 5.69 0.77 2.93 1.1 3.69 .83 3.76 0.96 5.07 0.26 0.73 0.1 1.25 3.53 .45 4.85 0.72 2.97 0.46 2.28 0.22 1.87 0.65 2.43 0.71 2.96 1.64 5.39

19.5 12.9 13.0 12.0 18.1 14.5 16.5 13.7 20.1 14.8 8.5 8.0 10.3 12.0 7.6 9.2 -0.25 -0.60 7.3 2.1 15 5.8 8.96 3.5 13.8 1.4 7.9 0.8 13 4.5 12 4.3 13.5

a D , D , and Γ represent apparent diffusivity, bulk diffusivity, and surface/interface concentration of proteins, respectively. a b eq 40. c Reference 41. d Reference 14. e Reference 35. f Reference 36. g Reference 42.

Figures 1B and 2B also show that no induction period exists in the development of further interfacial pressure at the triolein-water interface, which is totally reasonable since a condensed oil film already exists on the water surface. The interfacial pressure and interfacial cpm develop almost simultaneously, contrary to that at the air-water interface. The interfacial pressures obtained with different proteins at the triolein-water interface after ∼20 h, are listed in Table 1. Figures 3A and 4A show the rate of accumulation of negatively and positively charged proteins at the airwater interface, while Figures 3B and 4B depict the corresponding variation of surface pressures during protein adsorption with the square-root-of-time, respectively. The air-water adsorption curves were carried out by the pouring method.1,2,6-10,12,13 In this case the protein solution was first poured into the trough, the surface was cleaned of proteins by sweeping with a capillary tube attached to an aspirator, and then the protein from the bulk phase was allowed to adsorb to the interface. Since the solution was homogeneous from the very beginning, there should not be any mixing related lag time in the air-water adsorption curves. However, lag times in the range of 60 min for lysozyme, 80 min for ribonuclease A, 36 min for cytochrome c, and 5 min for catalase were observed. No lag times were noted in the adsorption curves of trypsin, R-chymotrypsin, histone-I, and pepsin. Also, with the exception of R-chymotrypsin and histone-I, the apparent diffusion coefficients (Da) of these proteins at the air-water interface are at least an order of magnitude smaller than their respective bulk diffusivities (Db), indicating that some repulsive forces are acting to retard the transport of proteins to this interface. The Da values of R-chymotrypsin and histone-I are only about 1.2-2-

b

Reference

fold slower than their Db. The Da of the negatively charged proteins, such as ovalbumin, β-lactoglobulin, R-lactalbumin, and conalbumin to the air-water interface, are either slightly higher or approximately 1.5-2.5 times slower than their respective bulk diffusivities. Again, the adsorption behaviors of phosvitin, catalase, and pepsin proved to be exceptions. Phosvitin did not adsorb to the air-water interface at all,13 while pepsin and catalase adsorbed at rates which were an order of magnitude smaller than their bulk diffusivities. The Da of all these proteins at the airwater interface, as well as their equilibrium surface concentrations, are listed in Table 1. At the air-water interface, there is always a time lag between the development of surface pressure and commencement of surface cpm, evident from Figures 3B and 4B. This time lag is defined as the induction period. The induction period originates from the fact that surface pressure starts developing only when the surface protein concentration reaches around 0.4-0.7 mg/m2, when cohesive interactions among protein molecules set in.12 The presence of an induction period in the air-water adsorption curves is a universal phenomenon, while it is totally absent in the oil-water adsorption curves. The surface pressures, obtained with different proteins at the air-water interface after ∼20 h, are listed in Table 1. Discussion The results from protein adsorption experiments carried out at the air-water and triolein-water interfaces clearly indicate that diffusion is not the rate-controlling step in the adsorption process. Hydrodynamic interactions of proteins with interfaces19 also cannot account for the variations in the rates of adsorption of proteins obtained


Figure 3. Variation of (A) surface concentration, Γ, and (B) surface pressure of selected negatively charged proteins with time during their adsorption to the air-water interface: O, β-lactoglobulin; 4, R-lactalbumin; s, ovalbumin; 1, conalbumin; 0, catalase; [, pepsin.

at different interfaces. If hydrodynamic interactions are considered during the approach of a globular protein toward a free surface, as treated by Brenner,19 the diffusivities of all proteins should be reduced with respect to their bulk diffusivities near the interface, irrespective of their charge. This is because modification of the friction factor in the Stokes-Einstein equation is only related to the radius of the protein and its distance of separation from the interface. This retardation in bulk diffusivity (19) Brenner, H. Chem. Eng. Sci. 1961, 16, 242.

Langmuir, Vol. 15, No. 20, 1999 6995

Figure 4. Variation of (A) surface concentration, Γ, and (B) surface pressure of positively charged proteins with time during their adsorption to the air-water interface: 3, R-chymotrypsin; O, cytochrome c; s, lysozyme; 2, trypsin; 0, RNAase A; 9, histone I.

may explain the slightly lower (∼2-3 times smaller) diffusivities obtained for globular proteins at the airwater interface but cannot explain the 1 order of magnitude smaller diffusion coefficients obtained for positively charged proteins at the air-water interface. Also, hydrodynamic interactions cannot explain the 1-2 orders of magnitude faster diffusivities measured for all proteins at the oil-water interface. Although the net charge of a protein appears to influence its adsorption behavior at the air-water interface (via mostly repulsive interactions for positively charged proteins, and attractive interactions for negatively charged proteins with this interface), it has

6996


Sengupta et al.

very little effect on the adsorption kinetics of proteins at the oil-water interface. All proteins experience attractive interactions with the oil-water interface, irrespective of their charge. The enhancement or retardation in the rate of adsorption of proteins over their bulk diffusivities must therefore arise due to an attractive or repulsive interaction between the protein and the interface, respectively. Previously, Xu and Damodaran hypothesized1 that protein adsorption at an interface is fundamentally driven by a chemical potential gradient between the subsurface and the interface. The chemical potential gradient profile of a protein as it approaches the interface is not simply related to the concentration gradient between the bulk and the interface but is also influenced by the potential energy of interaction between the protein and the interface. The total chemical potential of the protein at any distance from the interface can be regarded as

µ ) µideal + µconf + µhΦ + µvdW + µelec

[ ]

3 - 2 -2κd Q2 e 4(4π03)(d + R) 3 + 2

Eim )

(2)

where µ is the net chemical potential and µideal, µconf, µhΦ, µelec, and µvdW are the contributions from concentration gradient, entropic (conformational), hydrophobic, electrostatic, and van der Waals interactions, respectively. The interaction energy profiles generated for β-casein, lysozyme, and BSA during their approach toward the airwater or triolein-water interfaces8 confirmed the above hypothesis. Extension of this analysis to a wider spectrum of proteins shows that although both negatively and positively charged proteins behave similarly at the triolein-water interface, they behave differently at the airwater interface. Interaction Energy Calculations. It is known that the hydrophobic force is the driving force for adsorption of any amphiphilic molecule, such as a small surfactant or a protein molecule, from the bulk solution to an interface. The hydrophobic interaction energy (in kJ/mol) between a protein and an interface has been modeled by the interaction energy between a flat nonpolar surface and a spherical nonpolar particle of radius R (in nanometers) at a distance d from the surface, given by

Ehφ ) -84R e-d/d0

ing the higher dielectric phase from a lower dielectric phase generates an image charge in the latter. The repulsive Coulombic interaction between an ion and its image charge dictates the distance of its closest approach toward an interface. The repulsive electrostatic interaction energies between proteins and interfaces depend on the charge density of the proteins; it assumes the highest value for highly negatively charged phosvitin at pH 7.0 for both interfaces. This electrostatic interaction energy between a protein macroion of charge Q, radius R, dielectric constant 3, at a distance d from the interfacial plane (separating water and a nonpolar medium such as air or triolein phase of dielectric constant 2), and the image charge of -Q(2 - 3)/(2 + 3) generated at a distance d on the other side of the interface is given by23

where 0 is the permittivity of free space. The exponential term takes into account the screening effect of the ionic atmosphere around the charge.24 The molar polarization values of the respective amino acids and different heteroatomic groups required for the calculation of static dielectric constants of proteins are listed partly in our previous article8 and partly in Appendix A of this article. The molecular properties required to calculate the electrostatic interaction energy are listed in Appendix B. The Lifshitz-van der Waals interaction between a spherical particle of radius R, excess polarizability R1, dielectric susceptibility ε1, placed in a medium of dielectric susceptibility ε3 (the aqueous phase), and situated at a distance d from the plane surface of a second medium of dielectric susceptibility ε2 (air or oil phase) as a function of the imaginary part of frequency iξn, is given by Israelachvili23 as

EvdW ) EDebye-Keesom + ELondon-dispersion )-

(20) Israelachvili, J. N.; Pashley, R. Nature 1982, 300, 341. (21) Claesson, P. M.; Blom, C. E.; Herder, P. C.; Ninham, B. W. J. J. Colloid Interface Sci. 1986, 114, 234. (22) Lee, B.; Richards, F. M. J. Mol. Biol. 1971, 55, 379.

3kTR

)-

∞′

[

ε1(iξn) - ε3(iξn)

∑ n)0 ε (iξ ) + 2ε (iξ )

2d3

(3)

The hydrophobic energy decays exponentially with a characteristic decay length (d0) of 1 nm in the range of 0-10 nm,20 and then more gradually further out with a decay length of 4.5 nm.21 Moreover, since the surface of a protein consists of an uneven distribution of both hydrophilic and hydrophobic patches, only 50% of the interaction energy in eq 3 was considered to be valid. This follows from the fact that approximately 50% of the solvent accessible surface area of most globular proteins has been shown to be hydrophobic.22 Phosvitin is an exception, with only 10% nonpolar amino acid residues.13 Therefore, only 10% of the interaction energy in eq 3 was considered for this protein. The dimensionless hydrophobic interaction energy (Ehφ/kT) profiles of all proteins with both air-water and triolein-water interfaces are attractive, ranging in values from -5 to -60 at a distance of 0.5 nm from the interface. The larger the size of the protein (such as catalase), the higher the attractive hydrophobic interaction energy, as reflected from eq 3. It is also known from electrostatics that an ion in a higher dielectric phase located near an interface separat-

(4)

1

n

3

n

][

]

ε2(iξn) - ε3(iξn)

ε2(iξn) + ε3(iξn) (5)

AR3 3d3

Equation 5 can be split up into the zero-frequency term and terms pertaining to frequencies corresponding to n g 1, in the following manner,

[ ][ ] ∑[ ][

3kTR3 1 - 3

2 - 3

e-2κd + 2 + 4d 1 3 2 3 3 ∞ ε (iξ ) ε (iξ ) ε2(iξn) - ε3(iξn) 3kTR 1 n 3 n

EvdW ) -

2d3

3

n)1

ε1(iξn) + 2ε3(iξn) ε2(iξn) + ε3(iξn)

A ) Aξ)0(2κd)e-2κd + Aξgξ1

]

(6a) (6b)

The first term in eq 6a is the zero frequency term and constitutes contributions from the Debye-Keesom inter(23) Israelachvili, J. N. Intermolecular and Surface forces, 2nd ed.; Academic Press Limited: London, 1992; pp 74-77, 181-183, 282-286. (24) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolyte Solutions, 3rd ed.; Reinhold Publishing Corporation: New York, 1958; pp 89-91.


actions, which involve interactions between induced dipoles and permanent dipoles and those between permanent dipoles and permanent dipoles. This term is expressed in terms of the static dielectric constants and is always negative, constituting an attractive contribution to the van der Waals energy. The prime on the summation symbol indicates that the zero frequency term is weighted by a factor of 1/2. In electrolyte solutions, the zero frequency term is further reduced by a factor of e-2κd due to the screening atmosphere of ions.25 The second term in eq 6a is due to London dispersion contribution arising from interaction between induced dipoles and induced dipoles. The dispersion contribution can be either positive or negative, depending on the susceptibilities of the three phases. The London dispersion interactions between the protein molecules and air are always repulsive, whereas those between protein molecules and oil are always attractive. The Debye-Keesom interactions between the protein molecules and the oil-water or air-water interface are always attractive. The London dispersion interaction profile and the net van der Waals interaction profile for each protein with the air-water and oil-water interfaces are shown in Figures 5A, 5B, 6A, and 6B, respectively. The effective Hamaker constants based on Lifshitz theory (eq 6b) have been calculated at a distance of 0.5 nm for van der Waals interactions at the two interfaces and are listed in Appendix B. Finally, the contribution from the interaction between the electrical double layers of the protein molecule and the interface is included in the overall potential energy profile of each protein. An interface, whether it is airwater or oil-water, possesses a net charge and thereby a net surface potential. This charge is neutralized by a diffuse layer of ions around the interface, containing an excess of ions bearing charge opposite to that of the interface. The interface and the diffuse layer of ions constitute the so-called electrical double layer. The electrical force field at the air-water interface arises from the preferred orientation of water molecules at the interface, with the oxygen atoms outermost on the surface and the hydrogen atoms oriented inward from the interface in the bulk liquid.26-28 The value of this surface potential has been taken to be +120 mV.8 This value of surface potential appeared to be the best fitted parameter for generating potential energy profiles in our previous study8 and also in this study for a spectrum of proteins. The oil-water interface usually bears a negative charge and thereby a negative surface potential due to the preferential binding of anions to the oil-water interface via favorable attractive dispersion interactions with the oil phase.29 The surface potential of the triolein-water interface has been assumed to be -13 mV.8 The protein molecule has a net positive or negative charge on its surface (a positive or negative surface potential) and thereby possesses an electrical double layer itself. The surface potentials of the protein and the interfaces decay exponentially into the bulk, obeying the Poisson-Boltzmann equation. The Debye-Hu¨ckel approximation30 for low surface potentials (25) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: London, 1976; Chapters 3 and 4. (26) Weyl, W. A. J. Colloid Sci. 1951, 6, 389. Good, R. J. J. Phys. Chem. 1957, 61, 810. Fletcher, N. H. Philos. Mag. 1962, 7, 255. (27) Jarvis, N. L.; Schieman, M. A. J. Phys. Chem. 1968, 72, 74. (28) Frumkin, A. Electrochim. Acta 1960, 2, 351. Kochurova, N. N.; Rusanov, A. I. J. Colloid Interface Sci. 1980, 81, 297. Brodskaya, E. N.; Eriksson, J. C.; Laaksonen, A.; Rusanov, A. I. J. Colloid Interface Sci. 1996, 180, 86. (29) Ninham, B. W.; Kurihara, K.; Vinogradova, O. I. Colloids Surf., A 1997, 123, 7.

Langmuir, Vol. 15, No. 20, 1999 6997

Figure 5. Dimensionless London dispersion interaction and net van der Waals interaction energy profiles of negatively charged (A) and positively charged (B) proteins at the airwater interface.

has been assumed to be valid in this work. Due to the asymmetrical shape of proteins and nonuniform distribution of charges on the molecules, the assumption of constant surface potential of the protein surface is more appropriate than that of constant surface charge.8 As the protein approaches the interface, the double layer of the protein begins to overlap with that of the interface resulting in an attractive or repulsive force depending on the sign of the charges on the protein and the interface. The surface potentials of the proteins, and their respective charges, are given in Appendix B. The protein-planar interface electrical double layer interaction was modeled by a sphere-plane interaction using Derjaguin’s approximation of a sphere to be composed of infinitesimal rings parallel to the plane31 and Debye-Hu¨ckel approximation for low surface potentials.30 The details of (30) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Marcel Dekker: New York, 1986; Chapter 12. (31) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier Publishing Company: New York, 1948.

6998


Figure 6. Dimensionless London dispersion interaction and net van der Waals interaction energy profiles of negatively charged (A) and positively charged (B) proteins at the trioleinwater interface.

the calculations are given in our previous article.8 Since Derjaguin’s method holds for only short distances of separation (i.e., ∼3 κ-1 32), the computations were done for distances of separation less than 5 nm between the protein and the plane. The electrical double layer interaction energy between negatively charged proteins and the positively charged air-water interface or that between positively charged proteins and the negatively charged oil-water interface is always attractive, ranging from -7 kT to -35 kT at a distance of 0.5 nm from the interface. However, the electrical double layer interaction energy between proteins and interfaces having the same sign of surface potential, changes from repulsive in nature at larger distances of separation to attractive in nature at distances of separation less than 0.5 nm. This occurs due to the fact that two surfaces are interacting under constant but widely differing potentials of the same sign.32 Under such circumstances, the surface charge of proteins changes from positive to negative, due to crowding of counterions and their subsequent adsorption onto the protein surface. Finally, the sum of the van der Waals, electrostatic image charge, hydrophobic, and electrical double layer (32) Bell, G. M.; Peterson, G. C. J. Colloid Interface Sci. 1972, 41, 542.

Sengupta et al.

Figure 7. Dimensionless total interaction energy profiles of negatively charged (A) and positively charged (B) proteins at the air-water interface.

interaction energies have been carried out to generate the net potential energy profile for all proteins with the air-water and oil-water interfaces. The net interaction energy profiles for negatively and positively charged proteins at the air-water interface are shown in parts A and B of Figure 7, respectively. Similarly, the net interaction energy profiles for negatively and positively charged proteins at the oil-water interface are illustrated in parts A and B of Figure 8, respectively. The most striking feature of these potential energy curves is the existence of an energy barrier in the profiles of all positively charged proteins at the air-water interface, with the exception of histone-I. In contrast, the potential energy profiles of most negatively charged globular proteins exhibit no energy barrier. Phosvitin, catalase, and pepsin are the only negatively charged proteins showing deviation from this general trend. The experimentally observed rates of adsorption of all positively charged proteins at the air-water interface are indeed significantly lower than their respective bulk diffusion coefficients, validating the existence of an energy barrier in their respective energy profiles. The highly


Langmuir, Vol. 15, No. 20, 1999 6999 Table 2. Values of Energy Minima, Maxima, and Energy Barriers for Several Proteins at the Air-Water Interface

protein

secondary energy minimum/kT at d, Å

-9.80 at 7 Å from interface cytochrome c -7.76 at 7 Å RNAase A -9.21 at 8 Å trypsin -12.9 at 7 Å R-chymotrypsin -12.5 at 6 Å catalase -19.18 at 10 Å pepsin -19.2 at 8 Å conalbumin -22.07 at 8 Å phosvitin, pH 7 -2.57 at 28 Å phosvitin, pH 2 -6.73 at 9 Å

lysozyme

Figure 8. Dimensionless total interaction energy profiles of negatively charged (A) and positively charged (B) proteins at the triolein-water interface.

positively charged histone-I proved to be the only exception, adsorbing at a rate almost similar to its bulk diffusivity, which is again well corroborated by its energy profile at the air-water interface. The 1.5-2.5 times retardation in the rates of negatively charged globular proteins can be very well attributed to the hydrodynamic interactions near the interface19 or factors related to their structural changes accompanying the adsorption process at distances very close to the interface. In general, conformation changes will take place in proteins at the interfacial plane or during its approach toward the interfacial plane. These changes will further alter the net interaction profile between the proteins and the interfaces. However, because of lack of information on conformational changes at or near an interface, its contribution to the net potential energy profile cannot be determined either theoretically or experimentally.

energy maximum/kT at d, Å

energy barrier/kT at d, Å

3.45 at 2 Å from interface 1.77 at 2 Å infinite -1.65 at 3 Å -7.45 at 3 Å 1.33 at 5 Å -15.1 at 4 Å -20.21 at 5 Å infinite infinite

13.25 at 2 Å from interface 9.53 at 2 Å infinite 11.25 at 3 Å 5.05 at 3 Å 20.51 at 5 Å 4.1 at 4 Å 1.86 at 5 Å infinite infinite

A completely different picture is obtained at the oilwater interface. The net interaction energy profiles for all proteins at this interface, irrespective of charge and structure, are attractive. The experimentally determined rates of adsorption of all proteins at the oil-water interface are about 1-2 orders of magnitude faster than their respective bulk diffusion coefficients, confirming the validity of the theoretically determined attractive energy profiles. Phosvitin is the only exception to this rule, exhibiting an energy barrier to adsorption. At the air-water interface, the interaction energy profiles of lysozyme, cytochrome c, trypsin, and R-chymotrypsin possess a secondary energy minimum, an energy maximum, and a primary energy minimum, respectively (Figure 7B). The energy barriers at the airwater interface for these positively charged proteins are mainly attributable to the repulsive dispersion interactions between the proteins and the interface. The magnitude of the barrier, however, also depends on the values of the adjustable parameters chosen for theoretical calculations. These parameters are mainly wavelengths for maximum absorption (e.g., 287 nm for lysozyme, 284 nm for catalase), surface potential of water as +120 mV, and a specific volume of 0.73 mL/g for pepsin. Justifications for these assumptions are discussed elsewhere.8 These values yield net energy profiles with finite barriers below 15 kT (except for catalase where the barrier is 20 kT), which is still reasonable for actual adsorption of proteins to take place by leaking over the barrier. The energy barriers for various proteins are listed in Table 2. The energy profile of RNAase A for the air-water interface showed no primary minimum, only a secondary minimum of -9.21 kT at 8 Å from the interface and an infinite energy maximum. This agreed very well with the experimental data, which showed negligible adsorption of RNAase A at the air-water interface. For other positively charged proteins, the magnitude of retardation in the rates of adsorption reflected in a semiquantitative manner the magnitudes of their energy barriers. Although lysozyme and RNAase A are very similar to each other in charge, shape, size, and bulk diffusivities, these two proteins exhibited quite different lag times for the commencement of adsorption. Moreover, since an infinite energy barrier is predicted theoretically for RNAase A, it is very likely that this protein should not adsorb to the air-water interface at all. However, the experimental data show that RNAase A does adsorb to the air-water interface, albeit to a small extent of 0.22 mg/m2 (Figure 3B), indicating that besides these energy calculations another factor is equally, if not more, important for protein adsorption. The structure of the protein and its kinetics of unfolding at or very close to the interface may significantly affect the net chemical potential of the

7000


protein at a finite distance from the interface. In the case of RNAase A it is possible that after each unsuccessful collision of the protein with the air-water interface, the protein may incrementally unfold, and after a critical extent of unfolding it may adsorb to the interface. Histone-I is the only positively charged protein which showed a rate of adsorption at the air-water interface similar to its bulk diffusivity. Although enough precautions were taken to clean this protein from any associated lipids (filtration through Sephadex G-50 gel, followed by extensive dialysis), contamination might still have been a problem and therefore the data should be treated with caution. However, the energy profile calculated for histone-I does agree with the experimental trend. The inner core of this globular protein is extremely hydrophobic, flanked by two extended highly basic hydrophilic regions around it.33 The calculated relative permittivity of this protein reflects this high hydrophobicity, which in turn also affects the interaction energies between the protein and the interfaces. Among the negatively charged proteins, pepsin, catalase, conalbumin, and phosvitin exhibit energy barriers at the air-water interface. These values are given in Table 2. Phosvitin, a highly hydrophilic protein with only 10% nonpolar amino acid residues (at positions 205-216) providing a hydrophobic patch,13 exhibits an infinite energy barrier and a secondary minimum of -2.57 kT at a distance of 28 Å from the air-water interface. However, at pH 2.0, the charge of the protein decreases significantly from -179 to -15, and the protein conformation changes significantly from a highly extended random coil-type to a compact, globular protein-like with 67% β-sheet and 11.5% R-helix.13 The calculated energy profile for this protein at pH 2.0 shows a secondary minimum of -6.73 kT at a distance of 9 Å from the interface, followed by an infinite energy barrier. In calculating the energy profile for phosvitin at pH 2.0, two assumptions have been made due to the lack of experimental data at this pH:34 the specific volume of the protein was taken as 0.7 mL/g (similar to most globular proteins) instead of the anhydrous specific volume of 0.545 mL/g (for the elongated protein at pH 7.035,36) and a radius of 20.64 Å was calculated based on this specific volume and the calculated molecular weight from the amino acid sequence. Although this calculated radius of phosvitin at pH 2.0 represents the minimum value that it can assume (since hydration and asymmetry factors have been neglected), a larger radius will provide the same trend in the net energy profile with a more shallow secondary minimum at a farther distance from the interface. The experimental data show that although phosvitin has an infinite barrier at both pH 7.0 and 2.0, whereas it does not adsorb to the air-water interface at pH 7.0, it does adsorb at pH 2.0. This must be related to its minimum distance of approach to the air-water interface under these two pH conditions. Because phosvitin can approach as close as 5 Å from the interface at pH 2.0, it is likely that it can anchor to the interface through binding of some of the hydrophobic side chains. This may not be possible at pH 7.0 because the closest distance of approach to the interface is only about 28 Å (Figure 7A). A situation, similar to pH 2.0 at the air-water interface, may hold for phosvitin at the oilwater interface at pH 7.0, owing to which it adsorbed at a slightly faster rate than its bulk diffusion rate despite (33) Cole, R. D. Int. J. Peptide Res. 1987, 30, 433. (34) Grizzuti, K.; Perlman, G. E. J. Biol. Chem. 1970, 245, 2573. (35) Joubert, F. J.; Cook, W. H. Can. J. Biochem. Physiol. 1958, 36, 399. (36) Taborsky, G.; Mok, C. J. Biol. Chem. 1967, 242, 1495.

Sengupta et al. Table 3

groups

molecular weight

molar polarization, cm-3

serine phosphate threonine phosphate heme group in cytochrome c Ca2+ Fe2+ CO32Fe protoporphyrin IX in catalase NADPH acetyl

166.1 180.1 612a 40 52 60 610a 741a 43

33.97 39.8 158.98 2.78b -2.53b 9.97b 166.88 168.34 10.25

a

Reference 37. b Reference 38.

the presence of a calculated energy barrier between the protein and the interface. Conclusion Adsorption kinetics of proteins at the air-water or oilwater interface can be semiquantitatively explained from the interaction energy profiles of the proteins, as they approach these interfaces. Experiments reveal that the rate of adsorption of proteins to an interface is not simply diffusion-controlled but is governed by the interaction forces prevailing between the protein and the interface. The relative permittivity of the protein molecule or the dielectric constant appears to be more important in these calculations than the net charge of the protein or its hydrophobicity, with a few exceptions such as phosvitin. This is because of the fact that the dispersion interaction between the protein and the interface dominates the net potential energy profile, with only small contributions from the other interactions. The dispersion interaction between a protein and air is always repulsive, while that between a protein and oil is always attractive. On the other hand, the Debye-Keesom interactions between a protein and both interfaces are always attractive. Therefore, the net van der Waals interaction energy between the protein and the air-water interface changes from repulsive at long distances to attractive at very short distances, exhibiting a maximum close to the interface. The sum of the van der Waals, electrostatic, hydrophobic, and the electrical double layer interaction energies between the proteins and each interface predict quite well the adsorption behaviors of proteins at both interfaces. The most interesting trend unveiled is that, excepting histone-I, all positively charged proteins experience energy barriers during their adsorption to the air-water interface, which correlates well with an order of magnitude slower rates of adsorption compared to their respective bulk diffusion coefficients. Most of the negatively charged proteins (except pepsin, catalase, and phosvitin), however, adsorb to the air-water interface at rates either slightly faster or only 1.5-2.5 times slower than their bulk diffusion coefficients, which agrees well with their attractive energy profiles. The slight retardation may very well arise from the hydrodynamic interactions close to the interface. The rates of adsorption of proteins to the oil-water interface are almost 1-2 magnitudes faster than their bulk diffusivities, which is expected from the large attractive interactions between the protein and the oil phase. Phosvitin proves to be the only exception. In this respect, the theoretical calculations seem to predict quite well the behavior of proteins at the two interfaces. These calculations could even apparently explain the anomalous behaviors of phosvitin and histone-I at the two interfaces. Therefore, this work clearly establishes the important role


Langmuir, Vol. 15, No. 20, 1999 7001

Table 4. Physical Properties of Proteins Used in van der Waals, Electrostatic, Electrical Double-Layer Energy Calculations, and Calculation of Hamaker Constants at Air-Water and Oil-Water Interfaces protein β-casein BSA Rs1-casein R-lactalbumin β-lactoglobulin ovalbumin conalbumin pepsin phosvitin, pH 7 phosvitin, pH 2 catalase lysozyme cytochrome c RNAase A trypsin R-chymotrypsin histone I

∞/s 2.7/3.107 2.717/3.1317 2.717/3.1378 2.723/3.1387 2.672/3.0691 2.692/3.093 2.718/3.135 2.717/3.135 3.093/3.773 2.8387/3.309 2.743/3.164 2.7847/3.2328 2.7658/3.1989 2.8123/3.274 2.7498/3.1805 2.751/3.1839 2.669/3.0616

sp vol., mL/g RH, Å 0.742e 0.734f 0.725e 0.735e 0.751e 0.748f 0.732f 0.73 0.545g 0.73 0.733h 0.703f 0.72f 0.692f 0.719h 0.717h 0.76i

35.79 35.79 24.00 19.50 20.10 29.50 41.19 23.35 37.83 20.64 53.25 19.7 16.79 18.66 23.35 21.40 41.86

FP/FNP, g/mL

charge

ψ2, mV

-13e

A ×1021 J, (a/w) A ×1021 J, (o/w)

mol wt

1.534/0.903 -14.08 23 980 1.547/0.860 -17j -26.18b 66 306j 1.5445/0.899 -21 e -45.83 23 612e 1.559/0.876 -2.6e -8.06 14 174e 1.5635/0.871 -11e -32.42 18 362e 1.586/0.905 -8.5 -12.96 45 000 1.555/0.881 -5 -5 77 776 1.617/0.877 -37.5 -85.76 34 470 2.092/0.851 -179 -175.5 31 671k 2.092/0.851 -15 -42.3 1.5464/ 0.9126 -13.44 c -7.31 250 000 1.545/0.886 +9 +34.40 14 315l 1.512/0.887 +9.5 +49.18 12 292g 1.547/0.9038 +6 +30.89 13 691 1.5759/0.839 +7.5 +19.68b 23 348 1.5741/ 0.8502 +8 +20.98 25 210d 1.4652/ 0.7727 +47 +38.4 22 130g

e

-0.19 -0.84 -0.49 -0.89 +0.67 +0.06 -0.75 -1.05 -12.81 -4.45 -0.572 -1.6 -1.62 -4.06 -1.48 -1.41 +0.30

3.89 4.06 3.98 4.08 3.67 3.83 4.04 4.11 7.14 4.06 4.30 4.29 4.89 4.24 4.23 3.76

a / , relative permittivities at infinite and zero frequencies; F /F , densities of polar and nonpolar groups in proteins, A is the Hamaker ∞ s P NP constant, RH is the hydrodynamic radius derived from the bulk protein diffusivity values, and ψ2 represents the surface potential of the b proteins in mV. Surface potentials are based on crystalline radius of 29.345 Å for BSA, 17.17 Å for lysozyme, 14.25 Å for cytochrome c, 14.3 Å for RNAase A, and 21.4 Å for trypsin. c Surface charge of catalase was calculated from its pI value of 6.7 in ref 15, and amino acid sequence in PDB. d Molecular weights of other proteins were calculated from amino acid sequences obtained from the protein data bank. e Reference 40. f Reference 43. g Reference 14. h Reference 45. i Reference 46. j Reference 47. k Reference 13. l Reference 44.

of the dispersion part of the van der Waals interactions in controlling the adsorption behavior of proteins, and probably other polymers as well, to an interface. Acknowledgment. This study was partially supported by the US Department of Agriculture National Research Initiative Competitive Grants Program (Grant No. 9635500-3318) and the National Science Foundation (Grant No. BES-9712197). Appendix A: Molar Polarization Values

LA990235S

The molar polarization values of amino acid residues have been given in our previous article.8 The molar polarization values of only the heteroatomic groups and modified amino acid residues have been calculated here, and are listed in Table 3. Appendix B: Frequency-Dependent Dielectric Susceptibility Calculations The dispersion interactions can be either positive or negative, depending on the susceptibilities of the three phases. The frequency, ξn, is given by the following relation39

ξn ) 2πnkT/p,

n ) 1, 2, 3, ... ∞

rad/s) were considered in the present calculations for the proteins. Equation 6b shows the contributions to the Hamaker constant, A, from all frequencies. The frequency-dependent dielectric susceptibilities of different system constituents are calculated based on the procedure followed in our previous article.8 The relative permittivity, static dielectric constant, and densities of polar and nonpolar groups of each protein, based on its amino acid sequence, are listed in Table 4.

(7)

where p is Planck’s constant. The first nonzero frequency is 2.45 × 1014 rad/s at 25 °C, which is in the infrared region.39 The frequencies until the far ultraviolet (1017

(37) Stryer, L. Biochemistry; W. H. Freeman and Company: San Francisco, CA, 1975. (38) Tessman, J. R.; Kahn, A. H. Phys. Rev. 1953, 92, 890. (39) Parsegian, V. A.; Gingell, D. In Recent Advances in Adhesion; Lee, L. H., Ed.; Gordon and Breach Science Publishers: London, 1973; pp 153-192. (40) Swaisgood, H. E. In Developments in Dairy Chemistry; Fox, P. F., Ed.; Elsevier Applied Science: London, 1986; pp 1-60. (41) Kauzmann, W.; Kuntz, I. D. In Advances in Protein Chemistry; Academic Press: New York, 1974; Vol. 28, p 239. (42) Creighton, T. E. Proteins: Structure and Molecular Properties; W. H. Freeman and Company: New York, 1993. (43) Gekko, K. In Water Relationships in Food; Levine, H., Slade, L., Eds.; Plenum Press: New York, 1991; pp 753-771. (44) Pethig, R. Dielectric and Electronic Properties of Biological Materials; John Wiley and Sons: Chichester, 1979; Chapter 2-3. (45) Bigelow, J. J. Theor. Biol. 1967, 16, 187. (46) Teller, D. C.; Kinkade, J. M.; Cole, R. D. Biochem. Biophys. Res. Commun. 1965, 20, 739.52. (47) Brown, J. R. In Albumin Structure, Function and Uses; Rosenoen, V. M., Oratz, M., Rothschild, M. A., Eds.; Pergamon Press: Oxford, 1977; pp 27-51.

Energetics of ProteinâInterface Interactions and Its Effect on Protein

Recommend Documents

Energetics of ProteinâInterface Interactions and Its Effect on Protein