Adsorption Kinetics of Lysozyme on Silica at pH 7.4 - American

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Adsorption Kinetics of Lysozyme on Silica at pH 7.4: Correlation between Streaming Potential and Adsorbed Amount Jocelyne Ethe`ve and Philippe De´jardin* European Membrane Institute, UMR 5635 (ENSCM-UMII-CNRS), Universite´ Montpellier II, CC 047, 34095 Montpellier Cedex 5, France Received August 1, 2001. In Final Form: November 7, 2001 The adsorption of lysozyme in phosphate buffer (10-2 M + NaCl 10-2 M) at pH 7.4 on silica was studied at different flow rates. We measured simultaneously as a function of time (i) the streaming potential between the extremities of the capillary (internal diameter, 0.530 mm; length, 22 cm) and (ii) the amount adsorbed at x0 ) 14 cm from the entrance by means of 125I-radiolabeled proteins. We observed two regimes in the kinetics of adsorption, the first one being a transport-controlled process. They are separated at a critical interfacial concentration Γcr depending on the initial ζ0 potential of the interface, from 0.035 µg/cm2 at low density of charges (|ζ0| < 25 mV) to a mean value of 0.055 µg/cm2 for higher density of charges (26 mV < |ζ0| < 45 mV). A direct interpretation of those results concerns the electrostatic interactions: more molecules, which are slightly positively charged at pH 7.4, are adsorbed on the more negative surfaces. The ζ potential at the transition between both regimes (or final value) was dependent on the initial value ζ0: no charge reversal occurred when ζ0 ) -25 mV, while the interface became slightly positive when ζ0 ) -45 mV. While both adsorbed amount and streaming potential increased in the first regime, in the second regime streaming potential showed very small variations, if any, and interfacial concentration continued to increase. Hence, we focused mainly on the correlation between both quantities in the first regime. The study of the initial adsorption kinetics as a function of flow rate was in accordance with the presence of multimers (diffusion coefficient D ) 4.0 × 10-7 cm2 s-1; adsorption constant ka ≈ 4.0 × 10-3 cm s-1) in the shear zone near the wall. With these parameters, we performed numerical simulations to estimate the ratio between average and local interfacial concentrations, 〈Γ〉/Γ(x0). The variations of ∆ζ/〈Γ〉 versus time were then deduced. The data are also presented as ∆ζ/ζ0 as a function of 〈Γ〉 and analyzed according to the recent model of M. Zembala and Z. Adamczyk (Langmuir 2000, 16, 1593) for uniformly charged spheres. At low density of charges (ζ0 ≈ -25 mV) we observed ∆ζ ∝ 〈Γ〉 almost over the whole first regime, while a slight departure from linearity occurred above 0.3Γcr for higher density of charges. A semiquantitative analysis suggests that the molecules are oriented at the interface with their more positive parts facing the solid surface, the conformation of the molecules being different at small and high charge densities of silica.

1. Introduction Streaming potential measurements can be used as a tool to study protein adsorption1,2 or polymer adsorption,3-5 as this process leads to changes of the interface between solid and solution. Indeed, adsorption modifies the electrical state of the interface as well as the hydrodynamics near the surface, especially with adsorbed macromolecules, hence the possible detection of the adsorption process by electrokinetic methods like streaming potential. However, some assumptions are necessary to infer the adsorbed amounts only from the streaming potential measurements, while electrokinetics is connected at least partly with the protein side which is facing the solution and therefore could provide information about the conformation of the protein at the interface. The classical assumption to obtain the adsorbed amount as a function of time from streaming potential data is that adsorbance Γ is proportional to the variation |∆ζ| of the interface ζ potential, the constant of proportionality being given for instance by the ratio of the maxima of interfacial (1) Norde, W.; Rouwendal, E. J. Colloid Interface Sci. 1990, 139, 169-176. (2) Zembala, M.; Dejardin, P. Colloids Surf., B 1994, 3, 119. (3) Varoqui, R. Nouv. J. Chim. 1982, 6, 187. (4) Cohen-Stuart, M. A.; Waajen, F.; Dukhin, S. S. Colloid Polym. Sci. 1984, 262, 423-426. (5) Cohen-Stuart, M. A.; Fleer, G. J.; Lyklema, J.; Norde, W.; Scheutjens, J. Adv. Colloid Interface Sci. 1991, 34, 477-535.

concentration and |∆ζ| at saturated coverage. Often, the materials used for the two types of experiments, although as similar as possible in their chemical composition, are generally different in their geometry: for example,1 a flat plate for streaming potential and powder for the determination of the adsorbed amount. We present in this paper experiments where the streaming potential between the extremities of a single capillary is recorded during the flow of a protein solution, and simultaneously the adsorbed amount is also measured by means of radiolabeled molecules. Therefore, we expect to connect the two quantities measured on the same material through one single experiment. We choose the lysozyme as a model of a small “hard” protein positively charged at neutral pH. Its point of zero charge, or isoelectric point without specific ionic adsorption, is located at pH 11. Hence, we expect at pH 7.4 an electrostatic attraction between the protein and the wall silica capillary. Silica (pI ) 2) is negatively charged at neutral pH. There are numerous papers describing studies on the adsorption of lysozyme on various supports, with different methods: on glass1 with streaming potential and solution depletion, on Si(Ti)O26 with waveguide optical detection, on mica7 with atomic force microscopy, on silica SiO2 through neutron reflection8 and total internal reflection (6) Ball, V.; Ramsden, J. J. Colloids Surf., B 2000, 17, 81-94.

10.1021/la011224d CCC: $22.00 © 2002 American Chemical Society Published on Web 02/08/2002

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fluorescence with a pH-sensitive fluorophore,9 and on polystyrenesulfonate latex10,11 with electrophoretic mobility, among many others.12-18 2. Experimental Section 2.1. Buffers and Chemicals. Lysozyme (Sigma L6876, lot no. 54H7045; 3× crystallized, dialyzed, and lyophilized) from hen egg white was used without further purification. The solutions were made from phosphate buffer 10-2 M with NaCl 10-2 M. The phosphate buffer was obtained from Na2HPO4‚7H2O (Merck, ref 1.06580) and NaH2PO4‚2H2O (Merck, ref 6345.1000) in the ratio (81/19), plus NaCl (Prolabo, ref 27810.295), in deionized water (MilliQ quality). The solution was filtered on Millex-HV 0.45 µm (Millipore, ref SLHV025LS). We used solutions at 5.0 µg/mL. 2.2. 125I Labeling of Lysozyme. Lysozyme was iodinated through the technique of Iodo-beads (Pierce, Rockford, IL). The iodination reagent N-chloro-benzene sulfonamide is covalently linked to polystyrene beads. We conditioned the beads (about 1 bead per 5 mg protein) in 0.1 M phosphate buffer. Leaving them in 0.5 mL, we added 1 mCi 125I (NaI, CIS-Bio, Gif/Yvette, France). Then 2.5 mL of the protein solution in 0.01 M phosphate buffer + 0.01 M NaCl was added, and the whole solution was removed for dialysis after 15 min. Dialysis was performed in front of a flowing buffer compartment and controlled by recording the radioactivity release. We found in every case a labeling yield close to 100%. Then the solution was stored in aliquots of 1.0 mL at -70 °C. The aliquots were quick-thawed at 25 °C by putting the aliquots in a thermostated circulating bath of water at 25 °C before preparing the solutions at final bulk concentration of 5 µg mL-1 for experiments. 2.3. Silica Surface Treatment. The silica capillaries, generally used for gas chromatography (Interchim, France), were 0.530 mm internal diameter. A few meters was first treated with a solution of sulfuric acid containing ammonium persulfate (80 °C, 1 h) and then rinsed with deionized water (Milli-Q Millipore) before the flow of a solution of H2O2/NH3 (50-60 min) and final rinsing with deionized water. The whole length was cut into separate parts of approximately 25 cm length and fixed in a small pipet with glue at its extremities, to allow an easy manipulation. Due to different time parameters in the treatment especially with the basic solution and also to different delays of several weeks between treatment and experiments, we got initial states for adsorption experiments with ζ0 between -23 and -45 mV. We observed in another work19 a slow drift of the charge density of silica with time as the interface is in a metastable state. 2.4. Conductivity of the Solution and Calculation of the ζ Potential. The ζ potential was estimated from the Smoluchowski equation,

ζ ) (dEs/dP)(ηκ0/)

(1)

where Es is the streaming potential, P is the pressure, η is the dynamic viscosity, κ0 is the specific conductivity of the solution, and is the dielectric constant of the solution. No correction for (7) Radmacher, M.; Fritz, M.; Hansma, H. G.; Hansma, P. K. Science 1994, 265, 1577-1579. (8) Su, T. J.; Lu, J. R.; Thomas, R. K.; Cui, Z. F.; Penfold, J. Langmuir 1998, 14, 438-445. (9) Robeson, J. L.; Tilton, R. D. Langmuir 1996, 12, 6104-6113. (10) Galisteo, F.; Norde, W. Colloids Surf., B 1995, 4, 389-400. (11) Galisteo, F.; Norde, W. Colloids Surf., B 1995, 4, 375-387. (12) Fraaije, J.; Murris, R. M.; Norde, W.; Lyklema, J. Biophys. Chem. 1991, 40, 303-315. (13) Su, T. J.; Green, R. J.; Wang, Y.; Murphy, E. F.; Lu, J. R.; Ivkov, R.; Satija, S. K. Langmuir 2000, 16, 4999-5007. (14) Ball, V.; Ramsden, J. J. J. Phys. Chem. B 1997, 101, 5465-5469. (15) Malmsten, M. Colloids Surf., B 1995, 3, 297. (16) Millesime, L.; Amiel, C.; Chaufer, B. J. Membr. Sci. 1994, 89, 223-234. (17) Burns, N. L.; Holmberg, K.; Brink, C. J. Colloid Interface Sci. 1996, 178, 116-122. (18) Claesson, P. M.; Blomberg, E.; Froberg, J. C.; Nylander, T.; Arnebrant, T. Adv. Colloid Interface Sci. 1995, 57, 161-227. (19) Mauprivez, O. Rapport DEA; Universite´ Louis Pasteur: Strasbourg, France, 1993.

Figure 1. Schematic representation of the experimental setup. The flow occurs by applying a gas pressure. Pressure sensors and electrodes connected to an electrometer give access to differential pressure and streaming potential; the radioactivity detector (γ radiation) gives access to the adsorbed amount. the surface conductivity was introduced given the relatively high electrolyte concentration and large capillary radius with respect to the double-layer thickness.20 Experiments were performed at temperatures of 23-25 °C. Values of the viscosity and dielectric constant at the temperature of the experiment were introduced in eq 1 from the data for water (Handbook of Chemistry and Physics). We measured the conductivity of the buffer as a function of the temperature and found the relation κ0 ) 1.134 + 0.059T, where T is in °C and κ0 is in mS/cm. Although the Smoluchowski equation is relative to a uniformly charged surface, we used it in the treatment of the kinetic data, where a gradient of interfacial concentration most probably occurs, to deduce an “apparent” average ζ potential. If we neglect d2ψ/dx2 with respect to d2ψ/dy2, where ψ is the electrostatic potential, x is the distance from the capillary entrance, and y is the distance to the wall, this average can be defined as (1/L)∫L0 ζ(x) dx. Previous theoretical work in the literature21,22 suggests that this approximation is valid in many cases of periodic axial patches of different interfacial charges. However, the hydrodynamic perturbation of the adsorbed particles was not taken into account in these works. Anyway, we will analyze our results within this approximation and use in the following text for sake of simplicity ζ potential instead of apparent average ζ potential. 2.5. Experimental Study. The measurement cell, presented schematically in Figure 1, consisted of a capillary connected to two Ag/AgCl electrode compartments filled with buffer. The entry is connected to the reservoirs of buffer or solution. A cylindrical γ radioactivity detector (NaI-Tl) having a slit of thickness 12 mm and length 6.5 cm is positioned near the middle of the capillary. The whole system is in a thermostated box placed in an earthed Faraday cage. For the determination of the adsorbed amount, the calibration is given directly by the sudden increase (drop) in radioactivity at the arrival (replacement by buffer) of solution.23 The solution flow was driven by the hydrostatic pressure between the entry and exit containers plus the nitrogen pressure applied upward. Two pressure sensors (Honeywell, Microswitch 150PC) connected to the two extremities of the horizontal capillary measured the pressure gradient. An electrometer of input resistance higher than 2 × 1014 Ω (Keithley 617) is used to measure the streaming potential established by the flowing solution, while a microcomputer collected the data from the electrometer, the balance, and the pressure and temperature sensors. A second microcomputer acquires the radioactivity data. The ζ potential of the bare capillary in contact with buffer solution was obtained from the linear slope dEs/dP. Decrease of the pressure can be monitored easily by switching off the connection to the nitrogen cylinder under pressure. The pressure variation is then created by the decreasing level of liquid in the (20) Erickson, D.; Li, D. Q.; Werner, C. J. Colloid Interface Sci. 2000, 232, 186-197. (21) Anderson, J.; Idol, W. Chem. Eng. Commun. 1985, 38, 93-106. (22) Cohen, R. R.; Radke, C. J. J. Colloid Interface Sci. 1991, 141, 338-347. (23) Boumaza, F.; Dejardin, P.; Yan, F.; Bauduin, F.; Holl, Y. Biophys. Chem. 1992, 42, 87-92.

Adsorption of Lysozyme on Silica

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Figure 2. Example of an experimental recording: (top left graph) plots of streaming potential (b, left axis) and differential pressure (line, right axis) as a function of time; (bottom left graph) mass as a function of time when pressure is varied initially before being constant; (top right graph) after data treatment, streaming potential as a function of differential pressure with linear fits (before adsorption, full line; after adsorption and rinsing, dashed line); (bottom right graph) flow rate as a function of differential pressure before adsorption and after rinsing with a linear fit (dashed line) and the theoretical variation for a diameter of 530 µm (full line). entrance container associated with the decrease of the gas pressure above the liquid.19 The decreasing rate of pressure can be analytically estimated from the experimental parameters (see Appendix 1). In the presence of the flowing protein solution at constant pressure gradient, the potential is corrected by the asymmetry potential of the electrodes. The acquisition from the balance and pressure sensors allows the plot of mass and pressure as a function of time, hence the flow rate versus the differential pressure. One can verify whether the measurements, according to Poiseuille law relative to laminar flow, give the expected capillary radius (Figure 2). We obtained generally a few percent of variation around the value claimed by the manufacturer, which is satisfactory for our purposes. Anyway, an additional pressure drop should occur at the extremities of the capillary.

3. Results and Discussion Figure 3 shows typical results of experiments obtained for small and high charge density of silica. The existence of two regimes was a feature common to all the results: a first initial steep rise in streaming potential (or ζ potential) and interfacial concentration before a second regime where the increase of interfacial concentration is much slower and the streaming potential becomes constant or presents a very small (positive or negative) variation. The change of regime occurs at the “critical” points (Γcr, tcr) and (ζcr, t′cr) for interfacial concentration and streaming potential, respectively. The corresponding critical times are of the order of 1 min. Figure 4a,b focuses on the variations at initial times. While streaming potential, or apparent ζ potential, does not vary linearly in the first regime, the local adsorbed amount does. Let us examine first the two quantities separately before trying to connect them. 3.1. Streaming Potential. As ∆ζcr ) ζcr - ζ0 is a priori a function of two parameters, wall shear rate and initial

Figure 3. Typical variation of the ζ potential increase (O) and adsorbed amount (1) as a function of time showing clearly the two regimes of adsorption, separated at the points (Γcr,tcr) and (ζcr,t′cr): (a) ζ0 ) -39 mV, γ ) 1590 s-1, T ) 25.6 °C; (b) ζ0 ) -25 mV, γ ) 3550 s-1, T ) 23.5 °C (bottom).

value ζ0, it is useful to determine which one is the more pertinent. In Figure 5a, ∆ζcr is plotted as a function of

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Figure 4. (a) Variation of the increment in ζ potential, normalized to its maximal value ∆ζcr, in the first regime at various wall shear rates: (9) 608 s-1; (b) 2530 s-1; ([) 6110 s-1; (2) 9550 s-1. (b) Linear variation of the local interfacial concentration as a function of time in the first regime for the same wall shear rates as in (a).

wall shear rate. Among all the data, it is possible to select two groups: one corresponds to ζ0 ) -24.8 ( 1.3 mV, and the other to ζ0 ) -35.7 ( 0.5 mV. In the examined range of convection, from the overall data one can infer that the wall shear rate is a parameter much less pertinent than the initial charged state of the capillary. We reported in Figure 5b the dependence of ζcr and ∆ζcr with ζ0. The data show that the higher the initial negative density of charge, the larger the variation of ζ potential, and also the easier it is to obtain a charge reversal at the interface. We obtained d∆ζcr/dζ0 ) -1.8 in the examined range. If ∆ζcr depends primarily on ζ0 rather than on wall shear rate, the initial variation (dζ/dt)0 should show its dependence on both parameters. The simplest model available to check the influence of convection and diffusion is the Le´veˆque model where the transport-controlled process leads to an adsorption rate depending on the 1/3 power of the wall shear rate γ. To take into account the influence of the electrical state of the bare interface, we used a “normalized” variable ζ/∆ζcr in Figure 6. The experimental data agree with a linear dependence of (dζ/dt)0/∆ζcr with γ1/3. This is in accordance with the assumption of the connected variations of ζ potential and the interfacial concentration, which will be discussed below. In the second regime, the experiments showed slightly positive or negative variations of the ζ potential with time. The mean value is essentially zero. 3.2. Adsorbed Amount. The function Γcr(γ,ζ0) is illustrated in Figure 7. The wall shear rate seems not to be a pertinent parameter, while ζ0 may play a role. The dispersion of the data however does not allow any conclusion to be drawn with confidence. The change of regime occurs at a mean critical interfacial concentration


Figure 5. (a) Variation ∆ζcr ) ζcr - ζ0 of the apparent ζ potential at the change of regime as a function of wall shear rate, for different initial values ζ0 of the bare silica. (b) Variation ∆ζcr (b) and ζcr (O) at the change of regime as a function of the initial value ζ0 of the bare silica.

Figure 6. (O) Initial slope (1/∆ζcr)(dζ/dt)0 as a function of γ1/3 suggesting the importance of the transport and the correlation between ζ potential and adsorbed amount in the first regime. γ is the wall shear rate. (b) Initial slopes (1/∆ζcr)(dζ/dt) in the second regime.

0.055 µg/cm2, well below the crowding limits. Based on dimensions of 4.5 × 3 × 3 nm3, the jamming limit for the ellipsoid model in a “side-on” orientation24 is 0.13 µg/cm2. The hexagonal close packing limit is 0.21 µg/cm2. For an “end-on” orientation, the respective limits are 0.19 and 0.31 µg/cm2. That small value could be linked to an important effect of the ionic strength. Such an effect was indeed observed in the adsorption behavior of lysozyme25 on silica at pH 7.0 in 10-2 M phosphate buffer by the total internal reflection fluorescence technique using the intrinsic fluorescence of the tryptophanes. The change of regime occurred at 0.2 µg cm-2 without any salt added, while in (24) Vigil, R. D.; Ziff, R. M. J. Chem. Phys. 1989, 91, 2599-2602. (25) Buijs, J.; Hlady, V. J. Colloid Interface Sci. 1997, 190, 171-181.


Figure 7. Adsorbed amount Γcr at the change of regime of adsorption as a function of wall shear rate, for different initial values ζ0 of the bare silica.

the presence of 150 mM NaCl this concentration was about 0.02 µg cm-2. Our experiment with 0.05 µg cm-2 and 10 mM NaCl is positioned between these two extreme conditions. It suggests that in phosphate buffer the effect of ionic strength with NaCl becomes effective at rather low values and that the attractive interaction substrate/ protein is more pertinent than the repulsive lateral interactions between adsorbed proteins to interpret the data. This is in accordance with what was observed (0.130.17 µg/cm2) in 5 mM triethanolamine buffer pH 7.4, with a fluorescent label9 at much smaller wall shear rates (332 s-1). In addition, one can suggest a specific role for the phosphate anions as the shift of null electrophoretic mobility of lysozyme from 11 (point of zero charge), observed in triethanolamine buffer,26 to 9 in phosphate buffer was attributed to the specific adsorption of phosphate anions on the protein.27 This phenomenon would still decrease the electrostatic attraction substrate/protein based on the global charge of the protein. Besides, it was recently suggested that phosphate ions compete with albumin for adsorption sites on silica.28 Let us look now at the initial adsorption kinetics. In Figure 4b is represented the adsorption kinetics for four different wall shear rates. There was a linear variation of the adsorbed amount as a function of time approximately until the critical value was reached. Figure 8 shows the plot of the inverse of the experimental initial kinetic constant k as a function of γ-1/3, where γ is the wall shear rate. k ) Cb-1(dΓ/dt)0 where Cb is the bulk concentration, Γ is the interfacial concentration, and we assumed the reaction was of first order with respect to bulk concentration. Following the model taking into account the coupling between convection, diffusion, and interfacial reaction29 (Appendix 2), we should observe as a good approximation a linear relation between k-1 and γ-1/3, the slope leading to the diffusion coefficient of lysozyme and the intercept to ka-1, inverse of the interfacial adsorption constant. In any case, near the completely transport-controlled regime k-1 ) 0.684ka-1 + kLev-1 where kLev is the kinetic constant relative to a completely transport-controlled process. kLev ) 0.54(D2γ/x)1/3 where D is the diffusion coefficient and x is the distance from the entrance. The fit of the data by a straight line leads to a small negative ordinate at the (26) Lucas, D.; Rabiller-Baudry, M.; Michel, F.; Chaufer, B. Colloids Surf., A 1998, 136, 109-122. (27) Rabiller-Baudry, M.; Chaufer, B. J. Chromatogr., B 2001, 753, 67-77. (28) Docoslis, A.; Rusinski, L. A.; Giese, R. F.; van Oss, C. J. Colloids Surf., B 2001, 22, 267-283. (29) Dejardin, P.; Le, M. T.; Wittmer, J.; Johner, A. Langmuir 1994, 10, 3898-3901.

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Figure 8. Inverse of the apparent kinetic constant of adsorption as a function of the power (-1/3) of the wall shear rate γ: (full line) results from simulation with the adsorption kinetic constant ka ) 4 × 10-3 cm s-1 and diffusion coefficient D ) 4.0 × 10-7 cm2 s-1, those two values determined by a least-squares method; (short-dashed line) linear regression; (long-dashed line passing through the origin) Le´veˆque model with diffusion coefficient from the literature D ) 1.0 × 10-6 cm2 s-1 emphasizing a lower experimental value.

origin, due probably to a curvature effect and/or to the linear approximation. To take into account the curvature effect, an improved analysis was performed using numerical simulations and adopting the least-squares method for a two-parameter fit. The velocity field v was of the Poiseuille type,

r R 12 R

v)γ

2

[ ( )]

(2)

where γ is the wall shear rate, r is the distance to the axis, and R is the capillary radius. The mass conservation law contains the sum of the diffusion and convection terms. Diffusion in the direction of the capillary axis can be neglected.

(

)

∂C ∂2C 1 ∂C ∂C )D 2 + -v ∂t r ∂r ∂x ∂r

(3)

where D is the diffusion coefficient, x is the distance from the capillary entrance, and C(x,r,t) is the concentration. (∂C/∂r)r)0 ) 0. The adsorption kinetic constant was ka, defined by

∂Γ(x,t) ∂C ) kaC(x,R,t) ) -D ∂t ∂r

( )

r)R

(4)

We obtained ka ) 4.0 × 10-3 cm s-1 and D ) 4.0 × 10-7 cm2 s-1. This value of D is too small with respect to what appears in the literature30 (1.06 × 10-6 cm2 s-1). The departure from this reference is outside the experimental error (see in Figure 8 the Le´veˆque model with D ) 10-6 cm2/s), hence the possibility of multimers or aggregates in solution. This value is close to the “effective” one determined on glass by applying directly the Le´veˆque model.1 However, this low value comes from a mistake in the numerical prefactor of 3/2 in the expression of the adsorption rate1,31 and hence of (3/2)3/2 ) 1.84 for the diffusion coefficient. Using a correct prefactor leads in fact almost to the expected diffusion coefficient. As the (30) Bauer, D. R.; Opella, S. J.; Nelson, D. J.; Pecora, R. J. Am. Chem. Soc. 1975, 97, 2580-2582. (31) Elgersma, A. V.; Zsom, R. L. J.; Lyklema, J.; Norde, W. Colloids Surf. 1992, 65, 17-28.

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interaction with the surface is put in our model through ka, this low apparent value of D should be connected to the state of the solution. There are several previous studies mentioning the association of lysozyme molecules in solution. It was shown many years ago by deuterium nuclear magnetic resonance32 that the lysozyme is predominantly dimerized at pH 7.5. The self-association of lysozyme was the subject of other papers,33 especially in connection with the crystallization process,34,35 the dissociation induced by substrates,36 the binding with saccharides,37 and the influence of stirring at high temperature.38 At pH 6.4, lysozyme in stirring solutions showed a very slow aggregation process even at 25 °C.39 However, under the conditions of our experiments, the quantitative dimerization in solution is unlikely because the total concentration is very small, 5 µg/mL or 0.35 µM. For example,33 at 30 °C in water with a dimerization constant K of 143 M-1 the molar ratio dimer/monomer would be ≈0.5 × 10-4. However, the pH is a crucial point and our value pH 7.4 is larger than the pHs in these works. At a higher pH (pH 8.0; 10-2 M (NH4CO3/NH4CO2NH2)), in a study6 on adsorption of lysozyme on Si(Ti)O2, the self-association phenomenon was presented as a factor influencing the adsorption state at the interface where concentration is high, but the presence of dimers in solution was estimated to be negligible. The previously mentioned27 adsorption of phosphate anions on the protein might suggest an easier self-association of molecules as their own overall charge is decreased by this phenomenon. 3.3. Relation between Streaming Potential and Adsorbed Amount. We observed that in the first regime of adsorption the local interfacial concentration varies linearly with time over a time domain larger than the one observed for ζ potential (Figure 4a,b), hence a ratio ∆ζ/Γ which is not constant. The difference between the two types of variations with time could be related to an interfacial concentration averaged over the length of the tube which probably does not vary linearly with time and/ or to the influence of the concentration profile Γ(x) on the streaming potential, x being the distance from entrance of the capillary. Therefore, we estimated the ratio 〈Γ〉/Γ(x0), where x0 ) 14 cm and brackets mean averaging over the capillary length, from numerical simulations with the diffusion coefficient and adsorption kinetic constant determined previously (Figure 9).

〈Γ〉 )

∫0L Γ(x) dx

1 L

(5)

We used a simple model with a Langmuir excluded surface factor (1 - Γ/Γsat), Γsat being the adsorbed amount at saturation assumed to be equal to Γcr. Anyway, this model should be valid at the beginning of adsorption where we expect a high value of 〈Γ〉/Γ(x0), while toward the end of the adsorption when the transport no longer controls the adsorption process, whatever the exact mechanism, (32) Wooten, J. B.; Cohen, J. S. Biochemistry 1979, 18, 4188-4191. (33) Nesmelova, I. V.; Fedotov, V. D. Biochim. Biophys. Acta: Protein Struct. Mol. Enzymol. 1998, 1383, 311-316. (34) Price, W. S.; Tsuchiya, F.; Arata, Y. J. Am. Chem. Soc. 1999, 121, 11503-11512. (35) Boue, F.; Lefaucheux, F.; Robert, M. C.; Rosenman, I. J. Cryst. Growth 1993, 133, 246-254. (36) Zehavi, U.; Lustig, A. Biochim. Biophys. Acta 1969, 194, 532. (37) Studebaker, J.; Sykes, B.; Wien, R. J. Am. Chem. Soc. 1971, 93, 4579. (38) Colombie, S.; Gaunand, A.; Rinaudo, M.; Lindet, B. Biotechnol. Lett. 2000, 22, 277-283. (39) Caussette, M.; Planche, H.; Delepine, S.; Monsan, P.; Gaunand, A.; Lindet, B. Protein Eng. 1997, 10, 1235-1240.

Figure 9. Results of simulation of adsorption for three wall shear rates (from bottom to top: 608, 3560, and 10350 s-1 and Γsat ) 0.05, 0.037, and 0.05 µg cm-2, respectively) in the experimental geometry with the parameters ka and D determined from the variation of the initial experimental kinetic constant with the convection. The excluded surface factor is of Langmuir type. Local (full line) and average (dashed line) interfacial concentration are shown as a function of time. Note (i) the linear variation over a long range of time of the local interfacial concentration contrary to the average concentration and (ii) the time step ∆t0 between the two initial variations. The extrapolation to Γ ) 0 from the linear variation of the local interfacial concentration leads to a time where the average concentration takes still a significant value.

the experimental value Γ(x0) should be a very good approximation of the average value 〈Γ〉. It can be seen that the local interfacial concentration varies linearly over a long time, as found experimentally, although an excluded surface factor of Langmuir type was put into the model. This comes from the upstream-downstream coupling between the rapid maximal surface occupation upstream which reduces the boundary layer thickness downstream. It is as if the entry of the capillary was approaching the examined point with elapsed time. Some abrupt acceleration can be even simulated and sometimes observed experimentally, as in the adsorption of high molecular weight kininogen on silica.40 According to these simulations, the experimental data of streaming potential were shifted by ∆t0 with respect to the data of local interfacial concentration. ∆t0 can be estimated as the time necessary for a fluid particle entering the capillary at distance δ from the wall, δ being the boundary diffusion layer thickness at x ) x0, to reach the distance x0 by convection. ∆t0 ≈ 0.54(x/γ)2/3D-1/3. Figure 11 illustrates the variations of ∆ζ/〈Γ〉 as a function of time determined in such a way, compared to ∆ζ/Γ(x0). In the second regime, we expect the concentration profile to be quite smooth unless quasi-independent of x. It is clear that the accumulation of protein does not lead to big changes in streaming potential. For some experiments, we observed even opposite variations of ζ potential and adsorbed amount. Therefore, the situation is more complex than in the first regime and we wish not to go deeply in the different possibilities of interpretation. The mechanisms could be the building up of a multilayer structure and/or some conformation changes at the interface. Hence, at least for this system, because of the existence of a second regime where the correlation between (increasing) adsorbed amount and (constant) streaming potential seems not trivial, it is hazardous to rely on a direct proportionality between the two entities ∆ζ and 〈Γ〉 (Figure 10) choosing for instance the coefficient as the ratio of their final (possibly steady-state) values, as (40) Le, M. T.; Dejardin, P. Langmuir 1998, 14, 3356-3364.


Figure 10. Examples of the ratio of the ζ potential variation over the average (full symbols) or local (open symbols) interfacial concentration as a function of time, deduced from the experimental local interfacial concentration and numerical simulations: (1) γ ) 9550 s-1, ζ0 ) -35.5 mV, T ) 25.6 °C; (b) γ ) 2530 s-1, ζ0 ) -40.9 mV, T ) 23 °C.

previously proposed.1 Even for a monolayer of uniformly oriented proteins without specific intermolecular interactions, we would expect a linear relation at small coverage with a deviation from linearity at higher coverage. This behavior was analyzed recently by M. Zembala and Z. Adamczyk41 from a theoretical point of view, and their model was applied to the system of mica plates covered with latex particles bearing different charge densities. We could expect the adsorbed molecules to be relatively isolated from each other on the surface. Hence, the variation in streaming potential ∆Es should vary linearly with the interfacial concentration Γ assumed constant all along the plate. The flow field around a sphere positioned in a simple shear flow can be solved analytically.42,43 With charged interfaces and the hydrodynamic and electric interactions plate/sphere, the problem is more complex and requires a numerical solution of the PoissonBoltzmann and Navier-Stokes equations. The electrostatic potential ψ and local charge density F can be calculated numerically by using the finite-difference method.44 Yet our system is still more complex. The lysozyme is not a sphere and does not bear a uniform surface charge density. Moreover, the interfacial concentration is not constant all along the capillary and the Debye-Hu¨ckel double-layer thickness (1.7 nm) is of the same order of magnitude as the dimensions of the molecules (3-4.5 nm). Therefore, a direct quantitative interpretation is questionable as these characteristics do not correspond to the model, and the discussion is limited to qualitative arguments. The data of ζ/ζ0 (or Es/Es0) are put together on the same plot (Figure 11) as a function of the average interfacial concentration. It should be kept in mind that the data correspond to a low-coverage θ. With θ/〈Γ〉 ) (Avogadro number × section/molar mass) ≈ 5 (µg/cm2)-1, 〈Γ〉 ) 0.05 (µg/cm2) would correspond to θ ) 0.25. Moreover, with R being the radius of the molecule (and the assumptions D ∼ R-1, molecular mass ∼ R3, molecular section ∼ R2, and thus θ/〈Γ〉 ∼ D), aggregates of diffusion coefficient D roughly 40% of the isolated molecule would lead to a still lower coverage θ ) 0.10. For a high density of charge, until a large fraction (50%) of the critical interfacial concentration all the data are put together on a same (41) Zembala, M.; Adamczyk, Z. Langmuir 2000, 16, 1593-1601. (42) O’Neill, M. Chem. Eng. Sci. 1968, 23, 1293. (43) Goren, S.; O’Neill, M. Chem. Eng. Sci. 1971, 26, 325. (44) Warszynski, P.; Adamczyk, Z. J. Colloid Interface Sci. 1997, 187, 283-295.

Langmuir, Vol. 18, No. 5, 2002 1783

Figure 11. The ζ potential over its initial value ζ0 as a function of the average adsorbed amount, showing clearly the two regimes with almost linear variations in the first regime, and a constant value of ζ/ζ0 above a critical interfacial concentration. The data come from experiments at different values of ζ0 and wall shear rates. The three upper curves correspond to low values of ζ0. The others, with a continuous line for fit (ζ/ζ0 ) -22〈Γ〉 + 98〈Γ〉2), correspond to high values of ζ0. The sharp variations at the change of regime for some experiments may result from a lack of precision in experimental data rather than from a fundamental aspect. The dashed line corresponds to the proposed exponential variation (ref 41) (y ) eax + bx) with the same initial slope and curvature as the second-order polynomial fit.

curve. The slope at the origin is -22 cm2/µg. It is even less for the surfaces of low charge density. The “equivalent” particle within the model mentioned before41 would have the same sign as the surface. Anyway, according to these authors, we have the following relation at low coverage for the streaming potential Es relative to the initial one Es0:

Εs/Es0 ≈ 1 - [|Ci0| - Cp0(ζp/ζi)]θ

(6)

where θ is the surface coverage, |Ci0| ) 10.21, and Cp0 ) 6.51. Therefore, no variation with coverage is expected for the ratio of the potentials of particle and bare interface (ζp/ζi) ) 1.57. Assuming the crude model of a protein particle with “uniform” ζp and section 9 to 13.5 nm2, θ/Γ ) 3.8-5.7 (µg/cm2)-1; hence, the equivalent (ζp/ζi) is positive and close to 1. This would be in accordance with a molecule exposing its more positive parts near the solid surface where transport of charges by convection is low, while other less positive or even slightly negative parts would be facing the bulk solution. However, as the double-layer thickness (κ-1) is comparable to the molecule size (a), it could mean also a weak hydrodynamic screening of the uncovered silica/buffer double layer, which is not properly taken into account in eq 6. Recently,45 it was proposed that |Ci0| ) 6.0 for κa ) 0.5, which leads to a smaller but still positive value for (ζp/ζi). It should be kept in mind again that proteins are not uniformly charged spheres. For example, the exponential model41 does not work as shown in Figure 11. On the contrary, we observed with lysozyme a strong limitation to an almost neutral interface when additional molecules are adsorbed above a critical concentration. Concerning the second regime, we cannot exclude the possibility of multilayer buildup, especially because of observation of an abrupt decrease of the streaming potential at the beginning of desorption when solution is replaced by buffer, suggesting that a small amount of molecules far away from the interface influence strongly the streaming potential by their hydrodynamic effect. A (45) Zembala, M.; Adamczyk, Z.; Warszyski, P. Colloids Surf., A 2001, 195, 3-15.

1784



study on the structure of lysozyme adsorbed at the silicawater interface in similar conditions (phosphate buffer 2 × 10-2 M, pH 7), using the specular neutron reflection, led the authors8 to propose the constitution of multilayers in the presence of a solution at a concentration of 1 mg/ mL. Their conclusion of side-on multilayers is against the side-on/end-on process to accumulate more molecules at the interface.18 However, they did not observe such a multilayer at a smaller concentration, which is still 6 times higher than the one we used in our experiments. The existence of multilayers was also suggested recently.6 The deposition of aggregates was also observed by neutron reflectivity from an experiment on the adsorption of lysozyme (Cb ) 10 µg/mL; 10-2 M phosphate buffer, pH 7.4) on a polyelectrolyte brush.46

between the upward and downward compartments, the latter being at the atmospheric pressure and placed on the pan of a balance, whose data are acquired with time. Because of flow, there is simultaneous decrease of gas pressure and liquid level difference. The recipient on the balance contains a small beaker where the buffer or solution arrives and overflows into the recipient; therefore, we can estimate the level of liquid downward to be constant. Let ∆P be the pressure difference between the capillary extremities, Pup be the pressure into the upward compartment, Pdown be the atmospheric pressure above the downward compartment, ∆h be the level difference of liquid, F be the mass per unit volume of liquid, and g be the gravity constant.

4. Conclusion

∆P ) Pup - Pdown + ∆hFg

For the system lysozyme/silica in phosphate buffer 10-2 M + NaCl 10-2 M, pH 7.4, we observed a critical concentration Γcr which separates two regimes of adsorption. Γcr ≈ 0.055 µg cm-2 corresponds to an average occupation of one molecule per 49 nm2. This quite low interfacial concentration could be connected to a strong effect of the ionic strength25 and/or to the specific interactions of phosphate ions with the protein26,27 or the surface.28 From a thorough analysis of the variation of the initial kinetic constant with the flow rate, we estimated the adsorption constant ka ≈ 4.0 × 10-3 cm s-1 and diffusion coefficient 4.0 × 10-7 cm2 s-1. This last value is too low for corresponding to an isolated molecule. As a first approximation, we assumed a direct relation between the streaming potential between the extremities of the capillary and the average interfacial concentration. From the values of ka and D, it was possible to perform numerical simulations and infer the average concentration along the capillary from the experimental local interfacial concentration. In the second regime, the increase in the interfacial concentration with almost no variation in streaming potential suggests a different behavior. Some changes of conformation of the adsorbed molecules with or without possible additional layers which would not greatly influence the streaming potential can be suggested. At 〈Γ〉 < Γcr, the streaming potential varies smoothly with the surface occupation with a slope d(∆ζ/ζ0)/d〈Γ〉 ≈ -22 cm2/µg for high charge density, while it becomes abruptly zero for Γ > Γcr. Following a recent theoretical model,41 this result is equivalent to the adsorption of spheres of the same sign as the surface. This apparent discrepancy would suggest molecules oriented at the interface with their more positive parts facing the solid surface. The overall variation cannot be described by the sum of a linear and an exponential function.41 Appendix 1 We determine here the expected variation of flow rate with time with the experimental setup used in this work. The capillary is horizontal. The experiment starts with some pressure of gas above the upward compartment. The gas is contained in a closed volume. From a practical point of view, this compartment is initially connected to a gas bottle whose pressure is fixed approximately to the desired value. Then the connection between compartment and bottle is closed. At this time, no flow of solution occurs. When solution or buffer is flowing through the capillary, the pressure difference between its extremities is given by the gas pressure and the level difference of solution (46) Tran, Y.; Auroy, P.; Lee, L. T.; Stamm, M. Phys. Rev. E 1999, 60, 6984-6990.

(A1-1)

For a laminar Poiseuille flow, the flow rate is given by

dV/dt ) ∆P (πR4/8ηL)

(A1-2)

where η is the viscosity of the liquid, L is the length of the capillary, R is its radius, and dV is the volume of liquid flow or gas volume increase during time dt. The equation for a perfect gas gives

Pup ) (P0V0)/V

(A1-3)

where P0 and V0 are pressure and volume of gas in the upward compartment at time zero. The variation of gas volume V corresponds to the volume of liquid flowing through the capillary. Assuming the gas to be in a container of constant section area S, we have

∆h ) ∆h0 - (∆V/S)

(A1-4)

where ∆h0 is the initial level difference, and the downward level is kept constant by overflowing. From these four preceding equations, we get the differential equation

dV/dt ) A/V + B + CV

(A1-5)

with A, B, and C being constants:

A ) (πR4/8ηL)P0V0

(A1-6)

B ) (πR4/8ηL)[Fg(∆h0 + V0/S) - Pdown] C ) (πR4/8ηL)[-Fg/S] The solution of the differential equation can be given as time as a function of V.47

t)

∫VV A + BVV + CV2dV ) F(V) - F(V0)

(A1-7)

0

Let us introduce X ) A + BV + CV2 and q ) B2 - 4AC, to write F(V):

F(V) )

(

)

2CX + B - xq 1 B 1 ln(X) ln 2C 2C xq 2CX + B + xq

(A1-8)

Appendix 2 We recall here briefly the equations29 of the model of steady-state convection-diffusion-interfacial reaction in the case of a flat surface with a semi-infinite medium (y > 0). The velocity field is


v ) γy

Langmuir, Vol. 18, No. 5, 2002 1785

(A2-1)

is the stationary profile of concentration. One boundary condition is

where γ is the wall shear rate and y is the distance to the plane. The steady-state condition is given by

0)D

( )

∂2C ∂C -v ∂x ∂y2

(A2-2)

yf∞

(A2-3)

The adsorption kinetic constant ka is defined via the steady rate of adsorption:

where D is the diffusion coefficient of the solute, x is the distance from the (infinitely large) slit entrance, and C(x,y) (47) Handbook of Chemistry and Physics, 51st ed.; Weast, C., Ed.; CRC Press; Cleveland, OH, 1970-71; p A-169.

∂C )0 ∂y

∂Γ(x,t) ∂C ) kaC(x,0) ) -D ∂t ∂y

( )

LA011224D

y)0

(A2-4)

Adsorption Kinetics of Lysozyme on Silica at pH 7.4 - American

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