Kinetics of Adsorption, Desorption, and Exchange ofr-Chymotrypsin and Lysozyme on Poly(ethyleneterephthalate)

Tracked Film and Track-Etched MembraneElena N. Vasina† and Philippe Dejardin*,‡

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 of125I-radiolabeledR-chymotrypsin at pH 8.6 was studied in a laminar regime betweentwo walls of poly(ethyleneterephthalate) tracked films and membranes. Adsorption kinetics in the presenceof solution (10µg/mL), desorption by rinsing with buffer, and the following exchange of proteins by flowingunlabeled solution were measured. At pH 8.6,R-chymotrypsin is almost neutral and can be mostly removedfrom the film surface, contrary to positive lysozyme adsorbed at pH 7.4. Results suggest thatR-chymotrypsinis irreversibly adsorbed in pores, while desorption and exchange occur on membrane flat faces. A methodis proposed to determine adsorption kinetics in the pores. Kinetics of desorption and exchange ofR-chymotrypsin from the film surface can be described by stretched exponential functions in the examinedtime domain with the same exponent,â ≈ 0.62, which does not depend also on the former adsorptionduration. However, the mean residence time at the interface is about 2.5 times greater in the presence ofonly the buffer than that in the presence of solution. This effect could be explained by a fast exchange atthe arrival of unlabeled solution for a part of the adsorbed population.

1. Introduction

Protein adsorption at solid-liquid interfaces1,2 is importantin many fields such as biocompatible materials, especiallyhemocompatible materials3 or diagnostic kits,4 and enzymaticactivity in mineral soils.5-12 Natural and artificial vesselsadsorb proteins from blood. Changes of conformation orreactions at interfaces can induce a series of biochemicalreactions. Generally, this type of phenomenon must beavoided since the coagulation and complement systems canbe activated. Studies on model self-assembled organicmonolayers on gold surfaces have shown that hydrophiliza-tion with ethylene glycol oligomers is the most effective toresist protein adsorption compared to hydrophobic residues;13

the application to biosensors has been proposed.14 Hydro-philization via polymer pretreatment of surfaces can inhibitor limit these phenomena in solid-phase diagnostics,4 inhemodialysis hollow fibers,15 for example, or on polymersurfaces.16 Such polymer pretreatment is generally based oncopolymers containing poly(ethylene oxide) chains.17-24 Inaddition, adsorption of proteins at the interfaces modifiesinterfacial charge density, or electrokinetic potential, whichcan be estimated via streaming potential measurements.25-27

An appropriate pretreatment with polyelectrolytes such aspoly(ethylenimine) is able to greatly reduce the activationof the phase contact system on sulfonated membranes.28

Conversely, adsorption of potentially nocuous proteins, like

â2-microglobulin,29 in highly porous membranes can bedesired for blood purification purposes.30 Hence studies wereperformed on the access of small proteins inside membranes31

with the external surface polymer pretreatment and with-out.32,33

For other applications, such as filtration of virus solutions,instead of using very porous but irregular membranes, othermore regular and structured membranes can be more adapted.Their advantage is their well-defined structure, as the poresare essentially cylinders with a very low dispersion of sectionarea. Most of these membranes are obtained by the techniqueof heavy ion beams on a polymer film to produce tracks,followed by chemical etching of the tracks. Adjustments ofchemical and physical parameters in the etching processallow 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 poresand on the external faces of the membrane, can modifymembrane filtration properties.40 In addition, the conforma-tional state of proteins at interfaces, or when removed fromthe interfaces to solutions, can be different from that of thenative. An aggregation process can also occur, being inducedby adsorption.41 Hence it is useful to examine adsorptionproperties of proteins on membranes. We choseR-chymo-trypsin as a model “hard” protein, at pH 8.6, near the pointof zero charge. Therefore, the electrostatic interactioncomponent should be small. As the PET material is nega-tively charged, we considered lysozyme at pH 7.4, which isglobally positive (the point of zero charge at pH 11). Forlysozyme we can expect a significant contribution of electro-

* To whom correspondence should be addressed. E-mail:Philippe.Dejardin@iemm.univ-montp2.fr. Tel:+33 467 14 91 21. Fax:+33 467 14 91 19.

† Kazan State University.‡ CNRS, Universite´ Montpellier II.

304 Biomacromolecules 2003,4, 304-313

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

static attractive forces to the adsorption process. We exam-ined the adsorption, desorption, and homogeneous exchangeprocesses (the same proteins, the same concentration). Theseprocesses are linked to possible conformational changesinduced by the interface and are studied by other tech-niques: Fourier transform infrared spectroscopy,42,43circulardichroismspectroscopy,anddifferentialscanningcalorimetry.44-46

Adsorption of some proteins has been previously studied ontrack-etched PET47,48 and polycarbonate49-52 membranes.53

2. Theoretical Basis

Let us examine the adsorption process in a laminar flowin a slit of widthw and thicknessb. The field of fluid velocityν(y) is

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

where η is the viscosity of the fluid and (-dp/dx) is thepressure gradient along thex direction of the slit.

The initial adsorption process can be controlled by thetransport or the interfacial reaction. The general expressionmay be written as

whereCb is the bulk concentration of the solution,Γ is theinterfacial concentration,t is time, andk is the kineticconstant of the overall process. In the case of the fullytransport-controlled process, the rate of adsorption at distancex from the entrance depends only on transport parameters,e.g., diffusion coefficientD of the solute, and the wall shearrate. Its expression according to Le´veque54 is

When the adsorption process is controlled by the interfacialreaction with kinetic constantka, k ) ka. More complexexpressions describe the continuous passage from one limitto the other.55

In the case of desorption we assume no dependence onxfor the kinetic constant. Moreover, for the domain ofx whichwas examined in our experiments, we expect that during thedesorption and exchange there was no strong dependenceof the interfacial concentration onx. The simplest wayconsists of writing a desorption rate proportional to theinterfacial concentration with kinetic constantkd

whose solution is

This model assumes that the whole populationΓ0 at timet0 at the interface can be desorbed. However, there could bea stabilization step at the interface.44,46,56-60 Let us considertwo populations: one reversibly adsorbed, i.e.,Γr, andanother one irreversibly adsorbed, i.e.,Γir. At time t0

Let us assume

then

with a ) kd + kir and Γ0 and Γ∞b being the interfacial

concentrations att0 and infinity, respectively. The superscriptb means that the removal of proteins from the surface occursin the presence of buffer. The same kind of results is obtainedin the presence of proteins in solution, where the exchangeprocess is described by the kinetic constantke

Then

with b′ ) kd + keCb + kir andΓ0 andΓ∞ being the interfacialconcentrations att0 and infinity, respectively.

More sophisticated models can be proposed.61-64 Ofcourse, these foregoing descriptions are rough approxima-tions of the reality, as, for instance, to get these expressionsfor Γ(t), the kinetic constants are assumed not to vary withsurface occupation. In some cases, we found, however, thatsuch an assumption was reasonable.56 The foregoing analysisshows at least that even in a single-exponential description,the experimental time constant has several contributions andit 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 ofexponential functions. Possible spreading or change insurface occupation of adsorbed molecules is also notincluded.65 Considering a continuous distribution of relax-ation times leads to a possible global description by astretched exponential function (eq 12), as in the analysis ofthe proteins’ fluorescence decay profile66 or of their internaldiffusive motions:67

whereΓ0 is the interfacial concentration at timet ) 0 andτis some effective relaxation time. According to that model,the distribution of relaxation times leading to the observedstretched exponential function has a width characterized bythe exponentâ: the wider the distribution, the larger thedeparture ofâ from one. The mean relaxation time⟨τdistr⟩ ofthe distribution68 is connected to the effective relaxation timeτ (eq 12) by ⟨τdistr⟩ ) τâ-1G(â-1), whereG is the usualgamma function. The mean residence time of the proteins

Γ0 ) Γr0 + Γir0 (7)

dΓir /dt ) kirΓr (8)

Γb ) Γ∞b + (Γ0 - Γ∞

b) exp[-a(t - t0)] (9)

dΓr /dt ) -(kd + keCb + kir)Γr (10)

Γ ) Γ∞ + (Γ0 - Γ∞) exp[-b′(t - t0)] (11)

Γ ) Γ0 exp[-(t/τ)â] (12)

ν ) γy(1 - y/b) (1)

Q ) γ b2w6

) 112η(-dp

dx )wb3 (2)

∂Γ(x,t)∂t

) k(x)Cb (3)

kLev(x) ) 0.538(D2γx )1/3

(4)

dΓ/dt ) -kdΓ (5)

Γ ) Γ0 exp[-kd(t - t0)] (6)

Kinetics of Adsorption, Desorption, and Exchange Biomacromolecules, Vol. 4, No. 2, 2003 305

at the interface, after the removal process has begun, is givenby

The graphs of⟨t⟩/τ, ⟨τdistr⟩/τ, and⟨t⟩/⟨τdistr⟩ as functions ofâ are given in Appendix 1; they emphasize the high valuesof ⟨t⟩/⟨τdistr⟩ when the width of the distribution increases(small â values). The term “residence time” is valid if thereadsorption process is negligible. Otherwise, eq 13 repre-sents only the upper limit of the true residence time.

Desorption of colloidal particles from a water-glassinterface, observed in a parallel plate channel, has been alsodescribed by a stretched exponential function.69 Such de-scription is applicable according to several models, one ofthem is the aging model. That model could describe a proteinstabilization process at the interface. The potential energyU is increasing with time fromU0 accordingly to the ansatz70

U(t) ) U0 + U1 ln(1 + t/τ0). The desorption kinetics takesthen the expression70

where, in this case, the exponentâ is related to the depth ofthe additional potential energy at the interfaceU1 throughthe relation

and k0 is the initial desorption kinetic constant,kB is theBoltzmann constant, andT is the temperature. The two limitsof eq 14 are

Finally, it should be outlined that the transport is animportant parameter, as the molecules leaving the interfacehave some probability of being adsorbed again. Taking intoaccount the readsorption process, it was shown for staticconditions (where convection does not occur) that thedescription by a stretched exponential function is possible,but the â exponent depends on the time domain consid-ered.71,76 However, one should keep in mind that, in thepresent work, convection occurs and probably modifies thecontribution of readsorption when compared to static condi-tions.

3. Materials and Methods

3.1. Proteins.R-Chymotrypsin and lysozyme were pur-chased from Sigma. According to the provider,R-chymot-rypsin (reference C4129) was from bovine pancreas, 3×crystallized from 4× crystallized chymotrypsinogen, dia-lyzed, and lyophilized. It was used as such. Lysozyme(reference L6876) was from chicken egg white, 3× crystal-lized, dialyzed, and lyophilized. The proteins were radiola-

beled with125I by means of the Iodo-Beads technique; finaldialysis 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 size5.1 × 4 × 4 nm3, the mass72 25300 g/mol, and the point ofzero charge at pH 8.1.73 Other authors determined as theisoelectric point11 pI ) 8.6. According to these references,at pH 8.6 where the adsorption proceeds,R-chymotrypsinis not globally charged or slightly negative. Lysozymesolutions were in 10-2 M phosphate buffer+ 10-2 M NaCl,pH 7.4. As the isoelectric point of lysozyme is pI ) 11, theprotein is globally positive, although it was observed71 thatin phosphate buffer the point of zero charge shifts from pH11 to pH 9 due to the adsorption of phosphate ions on theprotein surface. The solutions were filtered on Millex-HV,0.45 µm (Millipore, reference SLHV025LS).

3.2. Tracked Film and Membrane.The tracked film andthe membrane had a thickness of 15µm. The tracked filmhad a track density of 1.63× 108 tracks cm-2. The PETmembrane had the pore densityF ) 1.5 × 108 pores cm-2,with hydrodynamic pore diameterdh ) 0.20µm. Panels a,b, and c of Figure 1 illustrate the density of pores, the poresize, and the membrane porous structure, respectively.

Dimensions of the holes at the membrane external surfacediffer (Figure 1b), as heavy ion beams had some angulardistribution (Figure 1c) around the normal to the surface ofthe initial PET film. The order of magnitude of the fractionalpart of the membrane external area occupied by the openingsof pores can be estimated asFdh

2 ≈ 6%. Considering thelarger average opening at the surface (0.3µm) leads to 12%.Image analysis providesf ) 11.6% for this fraction, and thisvalue is used for our data interpretation. Anyway, the surfacefraction not occupied by the openings of pores is close to90%. The external surfaces were characterized by streamingpotential measurements in the presence of 10-2 M KCl; wefound that the zeta potentialú ) -29 ( 1 mV and-32 (2 mV for the tracked film and TEM, respectively. Measure-ments were performed under continuous monotonic decreas-ing pressure drop, as described in Appendix 2. The surfacemorphology of the samples, covered with platinum, wasexamined with a field emission gun scanning electronmicroscope, Hitachi S4500 (Tokyo, Japan).

3.3. Experimental Setup.A cell of adequate dimensions4 cm× 11 cm× 1 cm (to be inserted in a detector of gammaradioactivity) consisted of two poly(methyl methacrylate)plates, where between them two sheets of either themembrane or the tracked film and two paraffin films(Parafilm “M”, Chicago) as a spacer were pressed. Therewas no flow through the membranes. The slit was 1 cm wideand 250 µm thick. The laminar flow through the slitcorresponded to the wall shear rateγ ) 300 s-1. Two leadshields limited the detection to the radiolabeled materialpresent in the central part of the cell, between 2.15 and 7.3cm from the slit entrance. In such conditions, the diffusionlayer thicknessδ ) (1/0.54)(Dx/γ)1/3, for a solute of diffusioncoefficientD ≈ 10-6 cm2 s-1 and at the distance from theslit entrancex ) 5 cm, is 47µm, which is about one-thirdof the slit half-thickness. Hence, we could expect that therewas no significant depletion of the bulk solution concentra-

⟨t⟩ )∫0

∞t exp(-(t/τ)â) dt

∫0

∞exp(-(t/τ)â) dt

) τG(2/â)

G(1/â)(13)

Γ ) Γ0 exp{-k0τ0

â [( tτ0

+ 1)â- 1]} (14)

â ) 1 - U1/kBT (15)

t , τ0, Γ ) Γ0 exp(-k0t) (16a)

t . τ0, Γ ) Γ0 exp[-(t/τ)â]; τ ) τ0( âk0τ0

)1/â(16b)

306 Biomacromolecules, Vol. 4, No. 2, 2003 Vasina and Dejardin

tion in the slit central plane of symmetry. The flow ofsolution or buffer was imposed by a syringe pump (HarvardPHD2000) and controlled by data acquisition from a balance(Mettler PM2000). The radioactivity data were acquired withsoftware provided by Canberra Co. and processed bySigmaPlot. The principle of the adsorption measurement isthat, in flow conditions, the passage from a buffer to aradiolabeled solution leads to a radioactivity jump relativeto a volume of the solution, which provides the calibrationas far as geometry of the system is known.74 Then thesubsequent increase is relative to the adsorption process onthe walls. The reverse occurs when an unlabeled solution(or buffer) flows after a radiolabeled one: there is a drop inradioactivity followed by a stable value if no exchange (ordesorption) occurs, while a decreasing radioactivity isrecorded if radiolabeled proteins are removed from theinterface, in the presence of a flowing buffer (desorption)or unlabeled solution (exchange). The term “desorption” is

generally used in the presence of buffer, even if the physicalphenomenon is also an interfacial exchange process, i.e.,between the protein and the buffer components. Experimentsoccurred atT ) 25 °C.

4. Results and Discussion

4.1. Adsorption of r-Chymotrypsin at pH 8.6. 4.1.a.Adsorption on Tracked Film. Figure 2 presents theinterfacial concentration ofR-chymotrypsin on the PETtracked film versus time for successive flows of the labeledsolution (concentrationCb ) 10 µg/mL in Tris buffer, pH8.6), Tris buffer, and unlabeled solution of the same proteinat the sameCb.

The two experiments presented in the insert of Figure 2indicate the level of reproducibility of the results for theinitial 10 min of adsorption. Whatever the initial delay ofadsorption before rinsing, 10 or 30 min, the variations remainqualitatively the same (Figure 3): not all molecules areirreversibly adsorbed, and the presence of molecules in thesolution increases the rate of removal from the interface, asseen from the variation of the slope dΓ/dt when the unlabeledsolution flows after the buffer. Hence, the dynamic processof exchange between an adsorbed protein and a protein inthe solution is very efficient and greatly reduces the meanresidence time of a protein molecule on the surface.

On the basis ofR-chymotrypsin dimensions45 5.1 × 4 ×4 nm3, the close-packed monolayer of proteins, modeled asellipsoids with the circular section or radius of 2 nm,corresponds to 0.30µg/cm2, which is close to the experi-mental value after 30 min of adsorption. However, a realplateau is not reached, and roughness of the support cancontribute to an underestimation of the true surface availablefor the proteins. We estimated roughly a diffusion coefficientof R-chymotrypsin from the diffusion coefficient of lysozyme75

(1.06× 10-6 cm2 s-1) corrected from the ratio of masses toa power of1/3, according to the Stokes-Einstein law. For awall shear rate of 300 s-1 and diffusion coefficientD ) 8.7× 10-7 cm2 s-1, the theoretical value of Le´veque corre-

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 cylindricalpores. The scale bars indicate (a) 8.57 µm, (b) 375 nm, and (c) 3.75µm.

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 onthe PET tracked film. Duration of the first adsorption step: 10 min(9, lower record) and 30 min (O, upper record). (Insert) Initialadsorption kinetics.

Kinetics of Adsorption, Desorption, and Exchange Biomacromolecules, Vol. 4, No. 2, 2003 307

sponding to a completely transport-controlled process iskLev

) 2.0× 10-4 cm s-1. The initial adsorption kinetic constantk is smaller but of this order of magnitude for both thetracked film and the track-etched membrane, therefore,showing the importance of transport at initial times for bothmaterials. With an observed mean value of 1.35× 10-4 cms-1, the adsorption kinetic constant can be estimated55 to be3.55× 10-4 cm s-1 for the tracked film. For the membrane,k is smaller, and henceka(TEM) ) 1.31 × 10-4 cm s-1.Figure 4 presents especially the desorption and exchangeprocesses after 10 min of adsorption, the data being fittedby simple exponential functions.

After the exchange of the radiolabeled material byunlabeled proteins in solution over about 1 h, we shifted toa flow of the radiolabeled solution. This last step shows thatthe exchange process continues to be very efficient after thepreceding one. Let us note that, at the beginning of the firstexchange step, the surface is not entirely covered. Thus, animportant fraction of molecules of the solution hits directlythe bare interface. Conversely, in the second exchange step,the probability that a molecule of the solution will reach anadsorbed molecule at the interface for their exchange is muchhigher. In the first exchange step we detect molecules leavingthe interface, while in the second one, we count molecules

arriving at the interface. Table 1 presents values of constantsdetermined for a single-exponential fit to the data of thedifferent experiments.

The slower kinetics of desorption and exchange when theadsorption time increases from 10 to 30 min suggests thatthere is an interfacial stabilization process, which progres-sively makes it more difficult to remove proteins. This isalso in agreement with the limit value of the interfacialconcentration (when time goes to infinity), which increaseswith the adsorption time in the first step. It should be noted,however, that the description by the single-exponential decayis probably a too simple approximation of the whole process.Especially, we underestimate the initial slope at each changeof solutions, and we cannot exclude the possibility of largerdesorbed or exchanged amounts over much longer times.

Considering an initial distribution of relaxation times68 ora continuous stabilization process70 of the protein at theinterface, the description of the exchange and the desorptiondata by the stretched exponential functionΓ/Γ0 ) exp{-[(t- t0)/τ]â} can be proposed after the first 3-4 min ofdesorption and exchange (Figure 5 and Table 2).

We found close values ofâ for both the desorption andexchange process and an average relaxation time, which ismore than twice longer for the desorption than that for theexchange. The interpretation of such exchange experimentaldata is not easy, as transport is an important parameter,especially under static conditions where the transport canlead to describe polymer exchange data by stretched expo-

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 supportadsorptiontime (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 1.65 ( 0.13c .141 ( 0.002 .056 ( 0.0010.82 ( 0.03d

(1.28 ( 0.16)e

tracked film 30 1.39 ( 0.08 1.15 ( 0.08 0.46 (0.01 .215 (.003 .078 ( 0.001membrane 30 0.85 ( 0.05 0.9 ( 0.2 0.55 ( 0.12 0.83 ( 0.01 0.70 ( 0.02

a Γ∞,b and Γ∞,s are the interfacial concentration at infinite times given by the exponential fits in the desorption and exchange steps, respectively. b Tobe 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 labeledR-chymotrypsin on the PET tracked film after 10 (3, ]) and 30 min(O, 9) of adsorption in the presence of the unlabeled solution at 10µg/mL in 10-2 M Tris buffer, pH 8.6. The graphs show the similarrelative variations in the dynamic processes. Comparison with dataobtained for lysozyme (b) in 10-2 M phosphate buffer + 10-2 M NaCl,pH 7.4; Cb ) 10 µg/mL; adsorption step, 30 min.

Figure 4. Interfacial concentration of labeled R-chymotrypsin on thePET tracked film versus time. The initial adsorption step of 10 min isnot presented. The start points of three successive flows are indicatedby arrows: (3) Tris buffer (10-2 M, pH 8.6), desorption; (O) solutionof unlabeled R-chymotrypsin, first exchange; (0) solution of labeledR-chymotrypsin, second exchange. Lines correspond to the fit by anexponential variation toward a nonzero steady-state final value.

308 Biomacromolecules, Vol. 4, No. 2, 2003 Vasina and Dejardin

nential functions whose exponent is not constant.71 Thegeometry of the support is also important,76 and the point isthe readsorption term: in an exchange or desorption process,molecules leaving an interface have a nonzero probabilityof arriving again at the interface. In static conditions, theconcentration of such molecules can become important andvary with time, hence a slowdown of the decay rate withrespect to a single-exponential description (â < 1) and avariation of theâ exponent if one considers different timedomains. In flowing systems, without a recirculation loop,hopefully the importance of this term may be minimized byconvection56,69 but can play a role by the transport fromupward to downward. This aspect has yet to be examined.Another possibility, as mentioned above, is to consider adistribution68 of relaxation times: the wider the distribution,the larger the departure ofâ from one. Concerning the twoclose values of theâ exponent, this result could be connectedto the importance of desorption at long times compared toexchange in the exchange step. Indeed, when the curve ofdesorption is extrapolated to the exchange time domain, thecontribution of the exchange at long times appears to benegligible compared to the desorption part. That is illustratedin Figure 6 for the experiment after 30 min of adsorption.

Hence, one may suggest that some small population isreally exchanged at the arrival of the unlabeled solution, butthen the contribution of the bimolecular process becomes

negligible. Considering the model where the desorptionconstant varies with time also leads to such a descriptionfor long times.70 Then the exponentâ is related to theadditional potential energyU1 at the interface through eq15. The fit of eq 14 to the ensemble of the data fromt ) t0provides different values of the exponent, i.e., 0.34 and 0.47for desorption which follows 10 and 30 min of adsorption,respectively, suggesting for our systemU1/kBT ≈ 0.66-0.53.

4.1.b. Adsorption on Track-Etched Membrane: Com-parison with Data Obtained for Tracked Film. Anexperiment 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 membranethan on the film because of access into the pores.

In such a short delay, a near-constant value of theinterfacial concentration is not reached, and the adsorptionis still occurring with a high rate even after 30 min. Theinternal areaA per unit area of one external surface facecan be estimated from pore densityF, pore radiusr, andmembrane thicknessh asA ) 2πrhF. For the track-etchedmembrane used here, withr ) 0.1 µm, A ≈ 14. Thus, after30 min of flowing solution, the average interfacial concentra-

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

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

protein/materialadsorption

duration (min) expt â τ (h) ⟨t ⟩ (h)

R-chymotrypsin/ 10 desorption 0.61 2.7 7.9PET tracked film exchange 0.64 1.24 3.1

30 desorption 0.64 3.3 8.3exchange 0.62 1.25 3.5

a These parameters show the close values of the â exponent for allexperiments, while the characteristic time τ is twice shorter for theexchange process (for the chosen solution concentration of 10 µg/mL)than that for the desorption in the presence of only the buffer. ⟨t ⟩ is theresidence time averaged over the stretched exponential decay function,provided that such a law is valid until complete desorption or exchangeand the readsorption process is neglected.

Figure 6. Desorption (O) followed by the exchange after the initial30 min adsorption for the PET tracked film. The lines correspond tofits by the function assuming a stabilization process (eq 14); thefunction behaves as a stretched exponential at long times. Key: after0.5 h, extrapolated curve of rinsing by the buffer (dashed line); theexperimental exchange (vertical marks); the true bimolecular ex-change (]) defined as the difference between the experimentalexchange curve and the supposed contribution of the desorption bythe buffer.

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 ofthe unlabeled solution of R-chymotrypsin (the same buffer, the sameconcentration) on the PET track-etched membrane (O) and PETtracked film (0).

Kinetics of Adsorption, Desorption, and Exchange Biomacromolecules, Vol. 4, No. 2, 2003 309

tion, about 1/15) 0.07 µg cm-2, is still far from thetheoretical maximal value, even without introducing anyroughness factor. In the presence of the buffer (desorption)or the unlabeled solution (exchange), the fraction of theinitially adsorbed amount that was removed from theinterface was very small, compared to that on the trackedfilm. The analysis of the data shows, however, that theabsolute amounts, desorbed and exchanged, are very similarto what was measured with the tracked film (Table 3).

Hence it suggests that, for the membrane, in fact, weobserve the removal of proteins from the external face, asthe openings of pores represent the small fraction of theexternal surface area. The proteins adsorbed into the poresare strongly stabilized at the interface and cannot be removedby 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 invokedto explain slow processes as we can estimate the diffusiontime over the membrane thickness 15µm to be of the orderof 1 s for a solute of diffusion coefficient 10-6 cm2 s-1.

Within this description, we analyzed as a function of timethe difference between adsorbed amounts ofR-chymotrypsinobserved on the tracked film and on the membrane. Takinginto account that the openings of pores occupy the fractionf (11.6%) of the membrane surface

Figure 9 shows thatΓmembrane - (1 - f )Γfilm is a linearfunction of time, except at short times.

The slope is 4.8× 10-4 µg cm-2 s-1 and corresponds toan experimental apparent constant of 4.8× 10-5 cm s-1. Ifwe wish to determine a kinetic constant relative to the surfaceof pores, the obtained value of the experimental apparentconstant has to be divided byf. Then we obtainkpore ) 4.1× 10-4 cm s-1, which is larger than the transport-limitedvalue (2.0× 10-4 cm s-1); hence the true adsorption constantat the surface of pores cannot be estimated directly. Fineanalysis would require taking into account the fact thataround the pores the concentration is high as the flat externalsurface here is quickly occupied; hence the transport-limitedexpression of Le´veque54 (eq 4) does not provide the truehigher 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 (30min), for lysozyme at pH 7.4 in the 10-2 M phosphate buffer+ 10-2 M NaCl are presented in Figure 10. Contrary toR-chymotrypsin, lysozyme molecules are strongly adsorbedon both supports, and only a small part of them can bedesorbed or exchanged.

However, the total adsorbed amount is higher for lysozymecompared toR-chymotrypsin (Table 3), about twice moreon the membrane and three times more on the tracked film;similar values were already observed.53 This suggests forma-tion of aggregates or multilayers at the interface.77-79

Moreover, the rate of adsorption on the tracked film, at theend of the first step, is far from zero, contrary to that ofR-chymotrypsin when the near plateau of the adsorbedamount is observed. These characteristics suggest a stronginteraction of lysozyme with the surface, which could be

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 0.30 0.22 0.13 0.07 0.08 [0.27] 0.09 [0.30]membrane 0.93 0.85 0.75 0.08 [0.086] 0.10 [0.11]

lysozyme tracked film 0.85 0.8 0.75 0.05 [0.059] 0.05 [0.059]membrane 1.90 1.85 1.80 0.05 [0.026] 0.05 [0.026]

a Γ4: 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 toΓ1 are given in square brackets in the two last columns.

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

Γmembrane) (1 - f )Γfilm + f Γpore (17)

Figure 9. Dependence of Γmembrane - (1 - f )Γfilm versus time (f isthe fraction of the PET membrane surface occupied by the openingsof 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.

310 Biomacromolecules, Vol. 4, No. 2, 2003 Vasina and Dejardin

related to the electrostatic interactions between the positiveprotein and the negative PET surface. As well as forR-chymotrypsin, we note here that the desorbed and ex-changed amounts were the same on the membrane and onthe tracked film, also suggesting that lysozyme moleculesadsorbed 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 thetransport limited value of Le´veque (2.31× 10-4 cm s-1 withdiffusion coefficientD ) 1.06× 10-6 cm2 s-1). However,the precision does not allow an estimation of the interfacialadsorption kinetic constantka to be given. For the track-etched membrane, the experimentalk ) (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 theadsorption of lysozyme we also analyzed a time dependenceof Γmembrane- (1 - f )Γfilm; the result is presented along withtheR-chymotrypsin data in Figure 9. The estimated kineticconstant relative to the surface of pores iskpore) 5.6× 10-4

cm s-1 and larger than the Le´veque limit; as forR-chymo-trypsin this discrepancy could be attributed to the experi-mental conditions which do not match the boundary condi-tions of the Leveque model.

5. Conclusions

Adsorption kinetics ofR-chymotrypsin on the PET trackedfilm and PET track-etched membrane show different behav-iors: a near plateau for the tracked film after 30 min, whilestill a high rate of adsorption on the membrane due to accessinto the pores.

R-Chymotrypsin can be desorbed in the presence of thebuffer and exchanged by the same protein, reducing its meanresidence time at the interface. We obtained a betterdescription of the kinetics of removal from the interface bya stretched exponential function rather than by a single-exponential one. The exponentâ was constant and near 0.62for both the desorption and the exchange, while the meanresidence time in the presence of solution (Cb ) 10 µg/mL)

is 2.5 times shorter than that in the presence of only thebuffer. We can observe a slight effect of the formeradsorption step duration on the mean residence time, whichcould be related to a process of stabilization at the interfacewith time. However, no conclusion could be drawn, asmeasurements over much longer times are necessary toclarify the aspect of complete or partial removal of theproteins.

The comparison of data obtained for the film andmembrane suggests that, for the membrane, removed proteinscome from the external surface and that the proteins at thesurface of pores are strongly adsorbed. This is a consequenceof the tracking which creates new chemical functions andleads along with the following etching to a different surfacestate. The difference between the two nonlinear variationsof the interfacial concentration for the membrane and thetracked film is a linear function of time over a wide range,which corresponds to the adsorption into the pores whichrepresent a huge area available for proteins of the solution.

Acknowledgment. We thank J. Ethe`ve (EMI) for theradiolabeling of the proteins, D. Cot (EMI) for performingSEM pictures, R. Souard and P. Montels (mechanicalworkshop of EMI) for manufacturing the cell, P. Travo(CRBM, Montpellier) for image analysis, and A. Nechaevand 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 wasperformed within the framework of a bilateral collaborationbetween the CNRS (France) and Kazan State University(Russia), project 12889.

Appendix 1

For a stretched exponential decay function, the meanresidence time is given by

After defining a new variableu ) (t/τ)â, the denominatorintegral becomes

Applying the same procedure for the numerator leads to(τ2/â)G(2/â) and eq 12, whereG(x) is the usual gammafunction. Figure 11 shows the ratioG(2/â)/G(1/â) versusâover the range 0.2-1.

Appendix 2

We determine here the expected variation of a flow ratewith time with the experimental setup used in this work,which is relative to a horizontal slit. The description belowincludes the case of a capillary as well. The experiment startswith some pressure of a gas above the upward compartment.The gas is contained in a closed volume. From the practicalpoint of view, this compartment is initially connected to a

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

⟨t⟩ )∫0

∞t exp(-(t/τ)â) dt

∫0

∞exp(-(t/τ)â) dt

(A1-1)

τâ ∫0

∞u[(1/â)-1] exp(-u) du ) (τ/â)G(1/â) (A1-2)

Kinetics of Adsorption, Desorption, and Exchange Biomacromolecules, Vol. 4, No. 2, 2003 311

gas bottle whose pressure is fixed approximately to a desiredvalue. Then the connection between the compartment andbottle is closed. At this time no flow of solution occurs. Whensolution or buffer is flowing through the capillary or slit,the pressure difference between its extremities is given bythe gas pressure difference and the level difference of solutionbetween the upward and downward compartments, the latterbeing at the atmospheric pressure and placed on the plateauof a balance, whose data are acquired with time. Because offlow, there is a simultaneous decrease of gas pressure andliquid level difference. The recipient on the balance containsa small beaker where the buffer or solution arrives and mayoverflow into a larger recipient placed on the balance; thenwe 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 capillaryor slit extremities,Pup the gas pressure into the upwardcompartment,Pdown the atmospheric pressure above thedownward compartment,∆h the level difference of liquid,Fl the mass per unit volume of liquid, andg the gravityconstant.

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

with

whereη is the viscosity of the liquid,L is the slit or capillarylength,R is the radius of the capillary, and dV is the volumeof liquid flow or gas volume increase during time dt. For aslit, w is the width andb is the thickness. The equation forperfect gas gives

whereP0 andV0 are the pressure and volume of gas in theupward compartment at time zero. The variation of gasvolume V corresponds to the volume of liquid flowing

through the capillary. Assuming that the container with thegas has constant section areaS, we have

where∆h0 is the initial level difference and the downwardlevel is kept constant by overflowing. If overflowing doesnot occur, given the sectionsSup and Sdown of the upwardand downward compartments, respectively, we useSdefinedby

From these four preceding equations, we get the differentialequation

A, B, andC being constants

The solution of the differential equation can be given as timeas a function ofV:80

Let us introduceq ) B2 - 4AC, to write downF(V)

The foregoing expression oft as a function ofV can be usedto fit the experimental data, mass versus time, asFl ) ∆m/∆V and to obtain, from parameterR, a value of the thicknessb of the slit, to be compared to the expected one.

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Figure 11. Graphs of ratios G(2/â)/G(1/â) ) ⟨t ⟩/τ (solid line), (1/â)/G(1/â) ) ⟨τdistr⟩/τ (dashed line), and âG(2/â)/G2(1/â) ) ⟨t ⟩/⟨τdistr⟩(dashed-dot line) as functions of â, where G(x) is the usual gammafunction of x.

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R ) πR4/8ηL (capillary) (A2-3)

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