Kinetics of Adsorption, Desorption, and Exchange of Î±-Chymotrypsin

Biomacromolecules 2003, 4, 304-313

304

Kinetics of Adsorption, Desorption, and Exchange of r-Chymotrypsin and Lysozyme on Poly(ethyleneterephthalate) Tracked Film and Track-Etched Membrane Elena N. Vasina† and Philippe De´ jardin*,‡ Department of Molecular Physics, Kazan State University, Kremlevskaya St 18, Kazan 420008, Russia, and European Membrane Institute, UMR5635 (CNRS-ENSCM-UMII), CC047, Universite´ Montpellier II, 34095 Montpellier Cedex 5, F-34095 Montpellier, France Received September 6, 2002; Revised Manuscript Received November 20, 2002

Adsorption kinetics of 125I-radiolabeled R-chymotrypsin at pH 8.6 was studied in a laminar regime between two walls of poly(ethyleneterephthalate) tracked films and membranes. Adsorption kinetics in the presence of solution (10 µg/mL), desorption by rinsing with buffer, and the following exchange of proteins by flowing unlabeled solution were measured. At pH 8.6, R-chymotrypsin is almost neutral and can be mostly removed from the film surface, contrary to positive lysozyme adsorbed at pH 7.4. Results suggest that R-chymotrypsin is irreversibly adsorbed in pores, while desorption and exchange occur on membrane flat faces. A method is proposed to determine adsorption kinetics in the pores. Kinetics of desorption and exchange of R-chymotrypsin from the film surface can be described by stretched exponential functions in the examined time domain with the same exponent, β ≈ 0.62, which does not depend also on the former adsorption duration. However, the mean residence time at the interface is about 2.5 times greater in the presence of only the buffer than that in the presence of solution. This effect could be explained by a fast exchange at the arrival of unlabeled solution for a part of the adsorbed population. 1. Introduction Protein adsorption at solid-liquid interfaces1,2 is important in many fields such as biocompatible materials, especially hemocompatible materials3 or diagnostic kits,4 and enzymatic activity in mineral soils.5-12 Natural and artificial vessels adsorb proteins from blood. Changes of conformation or reactions at interfaces can induce a series of biochemical reactions. Generally, this type of phenomenon must be avoided since the coagulation and complement systems can be activated. Studies on model self-assembled organic monolayers on gold surfaces have shown that hydrophilization with ethylene glycol oligomers is the most effective to resist protein adsorption compared to hydrophobic residues;13 the application to biosensors has been proposed.14 Hydrophilization via polymer pretreatment of surfaces can inhibit or limit these phenomena in solid-phase diagnostics,4 in hemodialysis hollow fibers,15 for example, or on polymer surfaces.16 Such polymer pretreatment is generally based on copolymers containing poly(ethylene oxide) chains.17-24 In addition, adsorption of proteins at the interfaces modifies interfacial charge density, or electrokinetic potential, which can be estimated via streaming potential measurements.25-27 An appropriate pretreatment with polyelectrolytes such as poly(ethylenimine) is able to greatly reduce the activation of the phase contact system on sulfonated membranes.28 Conversely, adsorption of potentially nocuous proteins, like * To whom correspondence should be addressed. E-mail: [email protected]. Tel: +33 467 14 91 21. Fax: +33 467 14 91 19. † Kazan State University. ‡ CNRS, Universite ´ Montpellier II.

β2-microglobulin,29 in highly porous membranes can be desired for blood purification purposes.30 Hence studies were performed on the access of small proteins inside membranes31 with the external surface polymer pretreatment and without.32,33 For other applications, such as filtration of virus solutions, instead of using very porous but irregular membranes, other more regular and structured membranes can be more adapted. Their advantage is their well-defined structure, as the pores are essentially cylinders with a very low dispersion of section area. Most of these membranes are obtained by the technique of heavy ion beams on a polymer film to produce tracks, followed by chemical etching of the tracks. Adjustments of chemical and physical parameters in the etching process allow easy control of the sizes of pores.34-39 We used poly(ethyleneterephthalate) (PET) as a material of the “tracked” film and the “track-etched” membrane (TEM). Adsorption of proteins from solution, on walls of pores and on the external faces of the membrane, can modify membrane filtration properties.40 In addition, the conformational state of proteins at interfaces, or when removed from the interfaces to solutions, can be different from that of the native. An aggregation process can also occur, being induced by adsorption.41 Hence it is useful to examine adsorption properties of proteins on membranes. We chose R-chymotrypsin as a model “hard” protein, at pH 8.6, near the point of zero charge. Therefore, the electrostatic interaction component should be small. As the PET material is negatively charged, we considered lysozyme at pH 7.4, which is globally positive (the point of zero charge at pH 11). For lysozyme we can expect a significant contribution of electro-

10.1021/bm025668f CCC: $25.00 © 2003 American Chemical Society Published on Web 02/13/2003

Biomacromolecules, Vol. 4, No. 2, 2003 305

Kinetics of Adsorption, Desorption, and Exchange

static attractive forces to the adsorption process. We examined the adsorption, desorption, and homogeneous exchange processes (the same proteins, the same concentration). These processes are linked to possible conformational changes induced by the interface and are studied by other techniques: Fourier transform infrared spectroscopy,42,43 circular dichroismspectroscopy,anddifferentialscanningcalorimetry.44-46 Adsorption of some proteins has been previously studied on track-etched PET47,48 and polycarbonate49-52 membranes.53

This model assumes that the whole population Γ0 at time t0 at the interface can be desorbed. However, there could be a stabilization step at the interface.44,46,56-60 Let us consider two populations: one reversibly adsorbed, i.e., Γr, and another one irreversibly adsorbed, i.e., Γir. At time t0 Γ0 ) Γr0 + Γir0

(7)

dΓir /dt ) kirΓr

(8)

Γb ) Γb∞ + (Γ0 - Γb∞) exp[-a(t - t0)]

(9)

Let us assume

2. Theoretical Basis Let us examine the adsorption process in a laminar flow in a slit of width w and thickness b. The field of fluid velocity ν(y) is ν ) γy(1 - y/b)

(1)

where γ is the wall shear rate (unit time-1) and y is the distance to one wall. The flow rate is given by Q)γ

b2w 1 -dp 3 ) wb 6 12η dx

( )

(3)

where Cb is the bulk concentration of the solution, Γ is the interfacial concentration, t is time, and k is the kinetic constant of the overall process. In the case of the fully transport-controlled process, the rate of adsorption at distance x from the entrance depends only on transport parameters, e.g., diffusion coefficient D of the solute, and the wall shear rate. Its expression according to Le´veˆque54 is kLev(x) ) 0.538

( ) D2γ x

with a ) kd + kir and Γ0 and Γb∞ being the interfacial concentrations at t0 and infinity, respectively. The superscript b means that the removal of proteins from the surface occurs in the presence of buffer. The same kind of results is obtained in the presence of proteins in solution, where the exchange process is described by the kinetic constant ke

(2)

where η is the viscosity of the fluid and (-dp/dx) is the pressure gradient along the x direction of the slit. The initial adsorption process can be controlled by the transport or the interfacial reaction. The general expression may be written as ∂Γ(x,t) ) k(x)Cb ∂t

then

1/3

(4)

When the adsorption process is controlled by the interfacial reaction with kinetic constant ka, k ) ka. More complex expressions describe the continuous passage from one limit to the other.55 In the case of desorption we assume no dependence on x for the kinetic constant. Moreover, for the domain of x which was examined in our experiments, we expect that during the desorption and exchange there was no strong dependence of the interfacial concentration on x. The simplest way consists of writing a desorption rate proportional to the interfacial concentration with kinetic constant kd dΓ/dt ) -kdΓ

(5)

Γ ) Γ0 exp[-kd(t - t0)]

(6)

whose solution is

dΓr /dt ) -(kd + keCb + kir)Γr

(10)

Γ ) Γ∞ + (Γ0 - Γ∞) exp[-b′(t - t0)]

(11)

Then

with b′ ) kd + keCb + kir and Γ0 and Γ∞ being the interfacial concentrations at t0 and infinity, respectively. More sophisticated models can be proposed.61-64 Of course, these foregoing descriptions are rough approximations of the reality, as, for instance, to get these expressions for Γ(t), the kinetic constants are assumed not to vary with surface occupation. In some cases, we found, however, that such an assumption was reasonable.56 The foregoing analysis shows at least that even in a single-exponential description, the experimental time constant has several contributions and it is not often justified to reduce it to only one component. The consideration of several different states at the interface, rather than a single reversible one, would lead to a sum of exponential functions. Possible spreading or change in surface occupation of adsorbed molecules is also not included.65 Considering a continuous distribution of relaxation times leads to a possible global description by a stretched exponential function (eq 12), as in the analysis of the proteins’ fluorescence decay profile66 or of their internal diffusive motions:67 Γ ) Γ0 exp[-(t/τ)β]

(12)

where Γ0 is the interfacial concentration at time t ) 0 and τ is some effective relaxation time. According to that model, the distribution of relaxation times leading to the observed stretched exponential function has a width characterized by the exponent β: the wider the distribution, the larger the departure of β from one. The mean relaxation time 〈τdistr〉 of the distribution68 is connected to the effective relaxation time τ (eq 12) by 〈τdistr〉 ) τβ-1G(β-1), where G is the usual gamma function. The mean residence time of the proteins

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Vasina and De´ jardin

at the interface, after the removal process has begun, is given by

∫0∞t exp(-(t/τ)β) dt G(2/β) 〈t〉 ) ∞ )τ ∫0 exp(-(t/τ)β) dt G(1/β)

(13)

The graphs of 〈t〉/τ, 〈τdistr〉/τ, and 〈t〉/〈τdistr〉 as functions of β are given in Appendix 1; they emphasize the high values of 〈t〉/〈τdistr〉 when the width of the distribution increases (small β values). The term “residence time” is valid if the readsorption process is negligible. Otherwise, eq 13 represents only the upper limit of the true residence time. Desorption of colloidal particles from a water-glass interface, observed in a parallel plate channel, has been also described by a stretched exponential function.69 Such description is applicable according to several models, one of them is the aging model. That model could describe a protein stabilization process at the interface. The potential energy U is increasing with time from U0 accordingly to the ansatz70 U(t) ) U0 + U1 ln(1 + t/τ0). The desorption kinetics takes then the expression70

{ [( ) ]}

Γ ) Γ0 exp -

k0τ0 t +1 β τ0

β

-1

(14)

where, in this case, the exponent β is related to the depth of the additional potential energy at the interface U1 through the relation β ) 1 - U1/kBT

(15)

and k0 is the initial desorption kinetic constant, kB is the Boltzmann constant, and T is the temperature. The two limits of eq 14 are t , τ0, Γ ) Γ0 exp(-k0t)

(16a)

( )

t . τ0, Γ ) Γ0 exp[-(t/τ)β]; τ ) τ0

β k0τ0

1/β

(16b)

Finally, it should be outlined that the transport is an important parameter, as the molecules leaving the interface have some probability of being adsorbed again. Taking into account the readsorption process, it was shown for static conditions (where convection does not occur) that the description by a stretched exponential function is possible, but the β exponent depends on the time domain considered.71,76 However, one should keep in mind that, in the present work, convection occurs and probably modifies the contribution of readsorption when compared to static conditions. 3. Materials and Methods 3.1. Proteins. R-Chymotrypsin and lysozyme were purchased from Sigma. According to the provider, R-chymotrypsin (reference C4129) was from bovine pancreas, 3× crystallized from 4× crystallized chymotrypsinogen, dialyzed, and lyophilized. It was used as such. Lysozyme (reference L6876) was from chicken egg white, 3× crystallized, dialyzed, and lyophilized. The proteins were radiola-

beled with 125I by means of the Iodo-Beads technique; final dialysis was performed to remove remaining free iodide ions. R-Chymotrypsin was used in solution in tris(hydroxymethyl)aminomethane buffer, pH 8.6. R-Chymotrypsin has the size 5.1 × 4 × 4 nm3, the mass72 25300 g/mol, and the point of zero charge at pH 8.1.73 Other authors determined as the isoelectric point11 pI ) 8.6. According to these references, at pH 8.6 where the adsorption proceeds, R-chymotrypsin is not globally charged or slightly negative. Lysozyme solutions were in 10-2 M phosphate buffer + 10-2 M NaCl, pH 7.4. As the isoelectric point of lysozyme is pI ) 11, the protein is globally positive, although it was observed71 that in phosphate buffer the point of zero charge shifts from pH 11 to pH 9 due to the adsorption of phosphate ions on the protein surface. The solutions were filtered on Millex-HV, 0.45 µm (Millipore, reference SLHV025LS). 3.2. Tracked Film and Membrane. The tracked film and the membrane had a thickness of 15 µm. The tracked film had a track density of 1.63 × 108 tracks cm-2. The PET membrane had the pore density F ) 1.5 × 108 pores cm-2, with hydrodynamic pore diameter dh ) 0.20 µm. Panels a, b, and c of Figure 1 illustrate the density of pores, the pore size, and the membrane porous structure, respectively. Dimensions of the holes at the membrane external surface differ (Figure 1b), as heavy ion beams had some angular distribution (Figure 1c) around the normal to the surface of the initial PET film. The order of magnitude of the fractional part of the membrane external area occupied by the openings of pores can be estimated as Fdh2 ≈ 6%. Considering the larger average opening at the surface (0.3 µm) leads to 12%. Image analysis provides f ) 11.6% for this fraction, and this value is used for our data interpretation. Anyway, the surface fraction not occupied by the openings of pores is close to 90%. The external surfaces were characterized by streaming potential measurements in the presence of 10-2 M KCl; we found that the zeta potential ζ ) -29 ( 1 mV and -32 ( 2 mV for the tracked film and TEM, respectively. Measurements were performed under continuous monotonic decreasing pressure drop, as described in Appendix 2. The surface morphology of the samples, covered with platinum, was examined with a field emission gun scanning electron microscope, Hitachi S4500 (Tokyo, Japan). 3.3. Experimental Setup. A cell of adequate dimensions 4 cm × 11 cm × 1 cm (to be inserted in a detector of gamma radioactivity) consisted of two poly(methyl methacrylate) plates, where between them two sheets of either the membrane or the tracked film and two paraffin films (Parafilm “M”, Chicago) as a spacer were pressed. There was no flow through the membranes. The slit was 1 cm wide and 250 µm thick. The laminar flow through the slit corresponded to the wall shear rate γ ) 300 s-1. Two lead shields limited the detection to the radiolabeled material present in the central part of the cell, between 2.15 and 7.3 cm from the slit entrance. In such conditions, the diffusion layer thickness δ ) (1/0.54)(Dx/γ)1/3, for a solute of diffusion coefficient D ≈ 10-6 cm2 s-1 and at the distance from the slit entrance x ) 5 cm, is 47 µm, which is about one-third of the slit half-thickness. Hence, we could expect that there was no significant depletion of the bulk solution concentra-



Figure 2. Kinetics of adsorption (Cb ) 10 µg/mL; 10-2 M Tris buffer, pH 8.6), then desorption in the presence of buffer, and final exchange (unlabeled R-chymotrypsin, Cb ) 10 µg/mL) of R-chymotrypsin on the PET tracked film. Duration of the first adsorption step: 10 min (9, lower record) and 30 min (O, upper record). (Insert) Initial adsorption kinetics.

generally used in the presence of buffer, even if the physical phenomenon is also an interfacial exchange process, i.e., between the protein and the buffer components. Experiments occurred at T ) 25 °C. 4. Results and Discussion

Figure 1. SEM pictures of the PET track-etched membrane: (a) illustration of pore density; (b) openings of pores at the surface; (c) internal structure showing the angular distribution of the cylindrical pores. The scale bars indicate (a) 8.57 µm, (b) 375 nm, and (c) 3.75 µm.

tion in the slit central plane of symmetry. The flow of solution or buffer was imposed by a syringe pump (Harvard PHD2000) and controlled by data acquisition from a balance (Mettler PM2000). The radioactivity data were acquired with software provided by Canberra Co. and processed by SigmaPlot. The principle of the adsorption measurement is that, in flow conditions, the passage from a buffer to a radiolabeled solution leads to a radioactivity jump relative to a volume of the solution, which provides the calibration as far as geometry of the system is known.74 Then the subsequent increase is relative to the adsorption process on the walls. The reverse occurs when an unlabeled solution (or buffer) flows after a radiolabeled one: there is a drop in radioactivity followed by a stable value if no exchange (or desorption) occurs, while a decreasing radioactivity is recorded if radiolabeled proteins are removed from the interface, in the presence of a flowing buffer (desorption) or unlabeled solution (exchange). The term “desorption” is

4.1. Adsorption of r-Chymotrypsin at pH 8.6. 4.1.a. Adsorption on Tracked Film. Figure 2 presents the interfacial concentration of R-chymotrypsin on the PET tracked film versus time for successive flows of the labeled solution (concentration Cb ) 10 µg/mL in Tris buffer, pH 8.6), Tris buffer, and unlabeled solution of the same protein at the same Cb. The two experiments presented in the insert of Figure 2 indicate the level of reproducibility of the results for the initial 10 min of adsorption. Whatever the initial delay of adsorption before rinsing, 10 or 30 min, the variations remain qualitatively the same (Figure 3): not all molecules are irreversibly adsorbed, and the presence of molecules in the solution increases the rate of removal from the interface, as seen from the variation of the slope dΓ/dt when the unlabeled solution flows after the buffer. Hence, the dynamic process of exchange between an adsorbed protein and a protein in the solution is very efficient and greatly reduces the mean residence time of a protein molecule on the surface. On the basis of R-chymotrypsin dimensions45 5.1 × 4 × 4 nm3, the close-packed monolayer of proteins, modeled as ellipsoids with the circular section or radius of 2 nm, corresponds to 0.30 µg/cm2, which is close to the experimental value after 30 min of adsorption. However, a real plateau is not reached, and roughness of the support can contribute to an underestimation of the true surface available for the proteins. We estimated roughly a diffusion coefficient of R-chymotrypsin from the diffusion coefficient of lysozyme75 (1.06 × 10-6 cm2 s-1) corrected from the ratio of masses to a power of 1/3, according to the Stokes-Einstein law. For a wall shear rate of 300 s-1 and diffusion coefficient D ) 8.7 × 10-7 cm2 s-1, the theoretical value of Le´veˆque corre-

308



Table 1. Apparent Kinetic Constants for R-Chymotrypsin, k (Initial Adsorption), and the Removal Constants in the Presence of Buffer, krb (from a Single-Exponential Fit) and in the Presence of Solution, krs (from a Single-Exponential Fit)a PET support

adsorption time (min)

kb × 104 (cm s-1)

krb × 103 (s-1)

krs × 103 (s-1)

Γ∞,b (µg cm-2)

Γ∞,s (µg cm-2)

tracked film

10

1.29 ( 0.02

1.76 ( 0.18

.141 ( 0.002

.056 ( 0.001

tracked film membrane

30 30

1.39 ( 0.08 0.85 ( 0.05

1.15 ( 0.08 0.9 ( 0.2

1.65 ( 0.13c 0.82 ( 0.03d (1.28 ( 0.16)e 0.46 (0.01 0.55 ( 0.12

.215 (.003 0.83 ( 0.01

.078 ( 0.001 0.70 ( 0.02

a Γ b ∞,b and Γ∞,s are the interfacial concentration at infinite times given by the exponential fits in the desorption and exchange steps, respectively. To be compared with the fully transport-controlled value 〈kLev〉 ) 2.02 × 10-4 cm s-1, with D ) 8.7 × 10-7 cm2 s-1 and γ ) 300 s-1. c Over the first 30 min. d Over 1 h. e Second exchange (labeled solution); strictly speaking, it concerns arrival of molecules at the interface.

Figure 3. Relative variations of the interfacial concentration of labeled R-chymotrypsin on the PET tracked film after 10 (3, ]) and 30 min (O, 9) of adsorption in the presence of the unlabeled solution at 10 µg/mL in 10-2 M Tris buffer, pH 8.6. The graphs show the similar relative variations in the dynamic processes. Comparison with data obtained for lysozyme (b) in 10-2 M phosphate buffer + 10-2 M NaCl, pH 7.4; Cb ) 10 µg/mL; adsorption step, 30 min.

sponding to a completely transport-controlled process is kLev ) 2.0 × 10-4 cm s-1. The initial adsorption kinetic constant k is smaller but of this order of magnitude for both the tracked film and the track-etched membrane, therefore, showing the importance of transport at initial times for both materials. With an observed mean value of 1.35 × 10-4 cm s-1, the adsorption kinetic constant can be estimated55 to be 3.55 × 10-4 cm s-1 for the tracked film. For the membrane, k is smaller, and hence ka(TEM) ) 1.31 × 10-4 cm s-1. Figure 4 presents especially the desorption and exchange processes after 10 min of adsorption, the data being fitted by simple exponential functions. After the exchange of the radiolabeled material by unlabeled proteins in solution over about 1 h, we shifted to a flow of the radiolabeled solution. This last step shows that the exchange process continues to be very efficient after the preceding one. Let us note that, at the beginning of the first exchange step, the surface is not entirely covered. Thus, an important fraction of molecules of the solution hits directly the bare interface. Conversely, in the second exchange step, the probability that a molecule of the solution will reach an adsorbed molecule at the interface for their exchange is much higher. In the first exchange step we detect molecules leaving the interface, while in the second one, we count molecules

Figure 4. Interfacial concentration of labeled R-chymotrypsin on the PET tracked film versus time. The initial adsorption step of 10 min is not presented. The start points of three successive flows are indicated by arrows: (3) Tris buffer (10-2 M, pH 8.6), desorption; (O) solution of unlabeled R-chymotrypsin, first exchange; (0) solution of labeled R-chymotrypsin, second exchange. Lines correspond to the fit by an exponential variation toward a nonzero steady-state final value.

arriving at the interface. Table 1 presents values of constants determined for a single-exponential fit to the data of the different experiments. The slower kinetics of desorption and exchange when the adsorption time increases from 10 to 30 min suggests that there is an interfacial stabilization process, which progressively makes it more difficult to remove proteins. This is also in agreement with the limit value of the interfacial concentration (when time goes to infinity), which increases with the adsorption time in the first step. It should be noted, however, that the description by the single-exponential decay is probably a too simple approximation of the whole process. Especially, we underestimate the initial slope at each change of solutions, and we cannot exclude the possibility of larger desorbed or exchanged amounts over much longer times. Considering an initial distribution of relaxation times68 or a continuous stabilization process70 of the protein at the interface, the description of the exchange and the desorption data by the stretched exponential function Γ/Γ0 ) exp{-[(t - t0)/τ]β} can be proposed after the first 3-4 min of desorption and exchange (Figure 5 and Table 2). We found close values of β for both the desorption and exchange process and an average relaxation time, which is more than twice longer for the desorption than that for the exchange. The interpretation of such exchange experimental data is not easy, as transport is an important parameter, especially under static conditions where the transport can lead to describe polymer exchange data by stretched expo-



Figure 5. Check of the description by the stretched exponential function Γ/Γ0 ) exp{-[(t - t0)/τ]β} with t - t0 > 4 min for proteins removal from the tracked film in the presence of the buffer, desorption (O) and then in the presence of the solution, exchange (0); solution concentration Cb ) 10 µg/mL. R-Chymotrypsin had been previously adsorbed from the Tris buffer (Cb ) 10 µg/mL) during 30 min.

Figure 6. Desorption (O) followed by the exchange after the initial 30 min adsorption for the PET tracked film. The lines correspond to fits by the function assuming a stabilization process (eq 14); the function behaves as a stretched exponential at long times. Key: after 0.5 h, extrapolated curve of rinsing by the buffer (dashed line); the experimental exchange (vertical marks); the true bimolecular exchange (]) defined as the difference between the experimental exchange curve and the supposed contribution of the desorption by the buffer.

Table 2. Fitting Parameters of Desorption and Exchange (Cb ) 10 µg/mL) of R-Chymotrypsin (onto the PET Tracked Film) with the Function Γ ) Γ0 exp{-[(t - t0)/τ]β}a protein/material R-chymotrypsin/ PET tracked film

adsorption duration (min) 10 30

expt

β

desorption exchange desorption exchange

0.61 0.64 0.64 0.62

τ (h) 〈t 〉 (h) 2.7 1.24 3.3 1.25

7.9 3.1 8.3 3.5

a These parameters show the close values of the β exponent for all experiments, while the characteristic time τ is twice shorter for the exchange process (for the chosen solution concentration of 10 µg/mL) than that for the desorption in the presence of only the buffer. 〈t 〉 is the residence time averaged over the stretched exponential decay function, provided that such a law is valid until complete desorption or exchange and the readsorption process is neglected.

nential functions whose exponent is not constant.71 The geometry of the support is also important,76 and the point is the readsorption term: in an exchange or desorption process, molecules leaving an interface have a nonzero probability of arriving again at the interface. In static conditions, the concentration of such molecules can become important and vary with time, hence a slowdown of the decay rate with respect to a single-exponential description (β < 1) and a variation of the β exponent if one considers different time domains. In flowing systems, without a recirculation loop, hopefully the importance of this term may be minimized by convection56,69 but can play a role by the transport from upward to downward. This aspect has yet to be examined. Another possibility, as mentioned above, is to consider a distribution68 of relaxation times: the wider the distribution, the larger the departure of β from one. Concerning the two close values of the β exponent, this result could be connected to the importance of desorption at long times compared to exchange in the exchange step. Indeed, when the curve of desorption is extrapolated to the exchange time domain, the contribution of the exchange at long times appears to be negligible compared to the desorption part. That is illustrated in Figure 6 for the experiment after 30 min of adsorption. Hence, one may suggest that some small population is really exchanged at the arrival of the unlabeled solution, but then the contribution of the bimolecular process becomes

Figure 7. Adsorption kinetics of R-chymotrypsin (10-2 M Tris buffer, pH 8.6; Cb ) 10 µg/mL) followed by rinsing with buffer and flow of the unlabeled solution of R-chymotrypsin (the same buffer, the same concentration) on the PET track-etched membrane (O) and PET tracked film (0).

negligible. Considering the model where the desorption constant varies with time also leads to such a description for long times.70 Then the exponent β is related to the additional potential energy U1 at the interface through eq 15. The fit of eq 14 to the ensemble of the data from t ) t0 provides different values of the exponent, i.e., 0.34 and 0.47 for desorption which follows 10 and 30 min of adsorption, respectively, suggesting for our system U1/kBT ≈ 0.66-0.53. 4.1.b. Adsorption on Track-Etched Membrane: Comparison with Data Obtained for Tracked Film. An experiment of the same kind was performed on the TEM (Figure 7) with the 30 min adsorption time. As expected, there is a larger amount of protein adsorbed on the membrane than on the film because of access into the pores. In such a short delay, a near-constant value of the interfacial concentration is not reached, and the adsorption is still occurring with a high rate even after 30 min. The internal area A per unit area of one external surface face can be estimated from pore density F, pore radius r, and membrane thickness h as A ) 2πrhF. For the track-etched membrane used here, with r ) 0.1 µm, A ≈ 14. Thus, after 30 min of flowing solution, the average interfacial concentra-

310



Table 3. Γ1, Γ2, and Γ3: Interfacial Concentrations (µg cm-2) of the Labeled Protein after the First Step (0.5 h Labeled Solution; Concentration Cb ) 10 µg/mL), Second Step (+0.5 h Buffer), and Third Step (+0.5 h Unlabeled Solution; Concentration Cb ) 10 µg/mL), Respectivelya protein

PET support

Γ1

Γ2

Γ3

Γ4

(Γ1 - Γ2)[/Γ1]

(Γ2 - Γ3)[/Γ1]

R-chymotrypsin

tracked film membrane tracked film membrane

0.30 0.93 0.85 1.90

0.22 0.85 0.8 1.85

0.13 0.75 0.75 1.80

0.07

0.08 [0.27] 0.08 [0.086] 0.05 [0.059] 0.05 [0.026]

0.09 [0.30] 0.10 [0.11] 0.05 [0.059] 0.05 [0.026]

lysozyme

a Γ : after an additional 0.5 h flow of the unlabeled solution. Γ in µg cm-2 of one face of the tracked film or membrane. The variations normalized to 4 Γ1 are given in square brackets in the two last columns.

Figure 8. Schematic presentation of the PET track-etched membrane. Only the protein molecules adsorbed onto the external faces can be removed by the buffer (or exchanged by other protein molecules). For R-chymotrypsin at pH 8.6, 27% of that population was removed by flowing the buffer during 30 min.

tion, about 1/15 ) 0.07 µg cm-2, is still far from the theoretical maximal value, even without introducing any roughness factor. In the presence of the buffer (desorption) or the unlabeled solution (exchange), the fraction of the initially adsorbed amount that was removed from the interface was very small, compared to that on the tracked film. The analysis of the data shows, however, that the absolute amounts, desorbed and exchanged, are very similar to what was measured with the tracked film (Table 3). Hence it suggests that, for the membrane, in fact, we observe the removal of proteins from the external face, as the openings of pores represent the small fraction of the external surface area. The proteins adsorbed into the pores are strongly stabilized at the interface and cannot be removed by buffer as well as by molecules of the protein solution. This model is schematically presented in Figure 8. The transport from and into the pores cannot be invoked to explain slow processes as we can estimate the diffusion time over the membrane thickness 15 µm to be of the order of 1 s for a solute of diffusion coefficient 10-6 cm2 s-1. Within this description, we analyzed as a function of time the difference between adsorbed amounts of R-chymotrypsin observed on the tracked film and on the membrane. Taking into account that the openings of pores occupy the fraction f (11.6%) of the membrane surface Γmembrane ) (1 - f )Γfilm + f Γpore

(17)

Figure 9 shows that Γmembrane - (1 - f )Γfilm is a linear function of time, except at short times.

Figure 9. Dependence of Γmembrane - (1 - f )Γfilm versus time (f is the fraction of the PET membrane surface occupied by the openings of pores) for R-chymotrypsin (9) (Cb ) 10 µg/mL; 10-2 M Tris buffer, pH 8.6; wall shear rate, 300 s-1) and lysozyme (O) (Cb ) 10 µg/mL; 10-2 M phosphate buffer + 10-2 M NaCl, pH 7.4; wall shear rate, 300 s-1) for the PET membrane and PET tracked film.

The slope is 4.8 × 10-4 µg cm-2 s-1 and corresponds to an experimental apparent constant of 4.8 × 10-5 cm s-1. If we wish to determine a kinetic constant relative to the surface of pores, the obtained value of the experimental apparent constant has to be divided by f. Then we obtain kpore ) 4.1 × 10-4 cm s-1, which is larger than the transport-limited value (2.0 × 10-4 cm s-1); hence the true adsorption constant at the surface of pores cannot be estimated directly. Fine analysis would require taking into account the fact that around the pores the concentration is high as the flat external surface here is quickly occupied; hence the transport-limited expression of Le´veˆque54 (eq 4) does not provide the true higher boundary value for the kinetic constant. 4.2. Adsorption of Lysozyme at pH 7.4. The three steps, adsorption (30 min), desorption (30 min), and exchange (30 min), for lysozyme at pH 7.4 in the 10-2 M phosphate buffer + 10-2 M NaCl are presented in Figure 10. Contrary to R-chymotrypsin, lysozyme molecules are strongly adsorbed on both supports, and only a small part of them can be desorbed or exchanged. However, the total adsorbed amount is higher for lysozyme compared to R-chymotrypsin (Table 3), about twice more on the membrane and three times more on the tracked film; similar values were already observed.53 This suggests formation of aggregates or multilayers at the interface.77-79 Moreover, the rate of adsorption on the tracked film, at the end of the first step, is far from zero, contrary to that of R-chymotrypsin when the near plateau of the adsorbed amount is observed. These characteristics suggest a strong interaction of lysozyme with the surface, which could be



Figure 10. Adsorption kinetics of lysozyme (Cb ) 10 µg/mL; 10-2 M phosphate buffer + 10-2 M NaCl, pH 7.4) followed by rinsing with the buffer and flow of the unlabeled solution of lysozyme (the same buffer, the same concentration) on the PET track-etched membrane (O) and PET tracked film (0).

related to the electrostatic interactions between the positive protein and the negative PET surface. As well as for R-chymotrypsin, we note here that the desorbed and exchanged amounts were the same on the membrane and on the tracked film, also suggesting that lysozyme molecules adsorbed in the pores are strongly stabilized. The precision in the determination of kinetic parameters, especially the kinetic constants of desorption and exchange, is poor. For the film, the initial adsorption kinetic constant [(2.3 ( 0.6) × 10-4 cm s-1] was found to be close to the transport limited value of Le´veˆque (2.31 × 10-4 cm s-1 with diffusion coefficient D ) 1.06 × 10-6 cm2 s-1). However, the precision does not allow an estimation of the interfacial adsorption kinetic constant ka to be given. For the tracketched membrane, the experimental k ) (1.8 ( 0.05) × 10-4 cm s-1 would correspond to the average55 ka ≈ 7. × 10-4 cm s-1 for pores and membrane external surface. For the adsorption of lysozyme we also analyzed a time dependence of Γmembrane - (1 - f )Γfilm; the result is presented along with the R-chymotrypsin data in Figure 9. The estimated kinetic constant relative to the surface of pores is kpore ) 5.6 × 10-4 cm s-1 and larger than the Le´veˆque limit; as for R-chymotrypsin this discrepancy could be attributed to the experimental conditions which do not match the boundary conditions of the Le´veˆque model. 5. Conclusions Adsorption kinetics of R-chymotrypsin on the PET tracked film and PET track-etched membrane show different behaviors: a near plateau for the tracked film after 30 min, while still a high rate of adsorption on the membrane due to access into the pores. R-Chymotrypsin can be desorbed in the presence of the buffer and exchanged by the same protein, reducing its mean residence time at the interface. We obtained a better description of the kinetics of removal from the interface by a stretched exponential function rather than by a singleexponential one. The exponent β was constant and near 0.62 for both the desorption and the exchange, while the mean residence time in the presence of solution (Cb ) 10 µg/mL)

is 2.5 times shorter than that in the presence of only the buffer. We can observe a slight effect of the former adsorption step duration on the mean residence time, which could be related to a process of stabilization at the interface with time. However, no conclusion could be drawn, as measurements over much longer times are necessary to clarify the aspect of complete or partial removal of the proteins. The comparison of data obtained for the film and membrane suggests that, for the membrane, removed proteins come from the external surface and that the proteins at the surface of pores are strongly adsorbed. This is a consequence of the tracking which creates new chemical functions and leads along with the following etching to a different surface state. The difference between the two nonlinear variations of the interfacial concentration for the membrane and the tracked film is a linear function of time over a wide range, which corresponds to the adsorption into the pores which represent a huge area available for proteins of the solution. Acknowledgment. We thank J. Ethe`ve (EMI) for the radiolabeling of the proteins, D. Cot (EMI) for performing SEM pictures, R. Souard and P. Montels (mechanical workshop of EMI) for manufacturing the cell, P. Travo (CRBM, Montpellier) for image analysis, and A. Nechaev and B. Mchedlishvili (Shubnikov Institute of Crystallography, RAS, Moscow) for providing the PET supports. E.N.V. benefited from INTAS, Grant YSF 00-133. This work was performed within the framework of a bilateral collaboration between the CNRS (France) and Kazan State University (Russia), project 12889. Appendix 1 For a stretched exponential decay function, the mean residence time is given by

∫0∞t exp(-(t/τ)β) dt 〈t〉 ) ∞ ∫0 exp(-(t/τ)β) dt

(A1-1)

After defining a new variable u ) (t/τ)β, the denominator integral becomes τ β

∫0∞ u[(1/β)-1] exp(-u) du ) (τ/β)G(1/β)

(A1-2)

Applying the same procedure for the numerator leads to (τ2/β)G(2/β) and eq 12, where G(x) is the usual gamma function. Figure 11 shows the ratio G(2/β)/G(1/β) versus β over the range 0.2-1. Appendix 2 We determine here the expected variation of a flow rate with time with the experimental setup used in this work, which is relative to a horizontal slit. The description below includes the case of a capillary as well. The experiment starts with some pressure of a gas above the upward compartment. The gas is contained in a closed volume. From the practical point of view, this compartment is initially connected to a

312



through the capillary. Assuming that the container with the gas has constant section area S, we have ∆h ) ∆h0 - (∆V/S)

(A2-5)

where ∆h0 is the initial level difference and the downward level is kept constant by overflowing. If overflowing does not occur, given the sections Sup and Sdown of the upward and downward compartments, respectively, we use S defined by 1 1 1 ) + S Sup Sdown Figure 11. Graphs of ratios G(2/β)/G(1/β) ) 〈t 〉/τ (solid line), (1/β)/ G(1/β) ) 〈τdistr〉/τ (dashed line), and βG(2/β)/G2(1/β) ) 〈t 〉/〈τdistr〉 (dashed-dot line) as functions of β, where G(x) is the usual gamma function of x.

gas bottle whose pressure is fixed approximately to a desired value. Then the connection between the compartment and bottle is closed. At this time no flow of solution occurs. When solution or buffer is flowing through the capillary or slit, the pressure difference between its extremities is given by the gas pressure difference and the level difference of solution between the upward and downward compartments, the latter being at the atmospheric pressure and placed on the plateau of a balance, whose data are acquired with time. Because of flow, there is a 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 may overflow into a larger recipient placed on the balance; then we can estimate the level of liquid downward to be constant. If not, the variation of pressure during the flow is larger. Let ∆P be the pressure difference between the capillary or slit extremities, Pup the gas pressure into the upward compartment, Pdown the atmospheric pressure above the downward compartment, ∆h the level difference of liquid, Fl the mass per unit volume of liquid, and g the gravity constant. ∆P ) Pup - Pdown + ∆hFlg

(A2-1)

For a laminar Poiseuille flow, the flow rate is given by dV/dt ) ∆PR

(A2-2)

with R ) πR4/8ηL R ) wb3/12ηL

(capillary)

(A2-3)

(slit)

where η is the viscosity of the liquid, L is the slit or capillary length, R is the radius of the capillary, and dV is the volume of liquid flow or gas volume increase during time dt. For a slit, w is the width and b is the thickness. The equation for perfect gas gives Pup ) (P0V0)/V

(A2-4)

where P0 and V0 are the 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

(A2-6)

From these four preceding equations, we get the differential equation dV/dt ) R(A/V + B + CV)

(A2-7)

A, B, and C being constants A ) P0V0 B ) Flg(∆h0 + V0/S) - Pdown

(A2-8)

C ) -Flg/S The solution of the differential equation can be given as time as a function of V:80 R(t - t0) )

∫VV A + BVV+ CV 2 dV ) F(V) - F(V0)

(A2-9)

0

Let us introduce q ) B2 - 4AC, to write down F(V) F(V) )

|

|

|

|

2CV + B - xq B 1 1 ln A + BV + CV 2 ln 2C 2C xq 2CV + B + xq (A2-10)

The foregoing expression of t as a function of V can be used to fit the experimental data, mass versus time, as Fl ) ∆m/ ∆V and to obtain, from parameter R, a value of the thickness b of the slit, to be compared to the expected one. References and Notes (1) Andrade, J. Surface and Interfacial Aspects of Biomedical Polymers; Andrade, J. D., Ed.; Plenum Press: New York, 1985; Vol. 2. (2) Brash, J.; Horbett, T. A. Proteins at Interfaces II: Fundamentals and Applications; American Chemical Society: Washington, DC, 1995; Vol. 602. (3) Leonard, E.; Turitto, V.; Vroman, L. Blood in Contact with Natural and Artificial Surfaces; New York Academy of Sciences: New York, 1987; Vol. 516. (4) Malmsten, M.; Lassen, B.; Holmberg, K.; Thomas, V.; Quash, G. J. Colloid Interface Sci. 1996, 177, 70-78. (5) Quiquampoix, H.; Ratcliffe, R. G. J. Colloid Interface Sci. 1992, 148, 343-352. (6) Quiquampoix, H.; Staunton, S.; Baron, M. H.; Ratcliffe, R. G. Colloids Surf., A 1993, 75, 85-93. (7) Quiquampoix, H.; Abadie, J.; Baron, M. H.; Leprince, F.; Ratcliffe, R. G.; Staunton, S. Abstr. Pap.-Am. Chem. Soc. 1994, 207, 223COLL. (8) Staunton, S.; Quiquampoix, H. J. Colloid Interface Sci. 1994, 166, 89-94. (9) Quiquampoix, H.; Abadie, J.; Baron, M. H.; Leprince, F.; MatumotoPintro, P. T.; Ratcliffe, R. G.; Staunton, S. In Proteins at Interfaces II; American Chemical Society: Washington, DC, 1995; Vol. 602, pp 321-333. (10) Leprince, F.; Quiquampoix, H. Eur. J. Soil Sci. 1996, 47, 511-522.

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Kinetics of Adsorption, Desorption, and Exchange of Î±-Chymotrypsin

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