Kinetic and Equilibrium Aspects of Adsorption and Desorption of Class

Article pubs.acs.org/Langmuir

Kinetic and Equilibrium Aspects of Adsorption and Desorption of Class II Hydrophobins HFBI and HFBII at Silicon Oxynitride/Water and Air/Water Interfaces Olga Krivosheeva,† Andra Deḋ inaite,̇ *,†,‡ Markus B. Linder,§,∥ Robert D. Tilton,⊥ and Per M. Claesson†,‡ †

School of Chemical Science and Engineering, Department of Chemistry, Division of Surface and Corrosion Science, KTH Royal Institute of Technology, Drottning Kristinas väg 51, SE-100 44 Stockholm, Sweden ‡ Chemistry, Materials and Surfaces, SP Technical Research Institute of Sweden, Box 5607, SE-114 86 Stockholm, Sweden § Biotechnology, VTT Technical Research Centre of Finland, Tietotie 2, FI-02044 VTT, Espoo, Finland ∥ School of Chemical Technology, Aalto University, Box 16100 FI-00076 AALTO, Espoo, Finland ⊥ Center for Complex Fluids Engineering, Departments of Chemical Engineering and Biomedical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States S Supporting Information *

ABSTRACT: Hydrophobins are relatively small globular proteins produced by filamentous fungi. They display unusual high surface activity and are implied as mediators of attachment to surfaces, which has resulted in high scientific and technological interest. In this work we focus on kinetic and equilibrium aspects of adsorption and desorption properties of two representatives of class II hydrophobins, namely HFBI and HFBII, at a negatively charged hydrophilic solid/water interface and at the air/water interface. The layers formed at the air/ liquid interface were examined in a Langmuir trough, whereas layers formed at the solid/liquid interface were studied using dual polarization interferometry (DPI) under different flow conditions. For comparison, another globular protein, lysozyme, was also investigated. It was found that both the adsorbed amount and the adsorption kinetics were different for HFBI and HFBII, and the adsorption behavior of both hydrophobins on the negatively charged surface displayed some unusual features. For instance, even though the adsorption rate for HFBI was slowed down with increasing adsorbed amount as expected from packing constraints at the interface, the adsorption kinetics curves for HFBII displayed a region indicating adsorption cooperativity. Further, it was found that hydrophobin layers formed under flow partly desorbed when the flow was stopped, and the desorption rate for HFBII was enhanced in the presence of hydrophobins in solution.

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INTRODUCTION

proteins has made them of high interest for different applications, for example, for protein immobilization,8 foam stabilization,9,10 and many others. Clearly, many of the natural functions and technical applications of these proteins depend on their interfacial properties. On the basis of the amino acid sequence, hydrophobins are divided into two classes, denoted class I and II. Class I representatives form highly insoluble films, whereas the films formed by proteins belonging to class II have been found to be more soluble. The most studied member of the class I hydrophobins is SC3, obtained from Schizophyllum commune. Among the class II hydrophobins, HFBI and HFBII from Trichoderma reesei are the most well-known proteins, and these are also the focus of the present investigation.

Many proteins are active at biological interfaces where they fulfill a diverse number of functions. One example is in the intricate biolubrication system that provides low friction in e.g. synovial joints,1 and another relates to hydration and protection of surfaces by the mucin glycoprotein family.2 Other proteins, like the mussel adhesive proteins,3,4 promote interactions with interfaces present outside the host organism. Thus, it is not surprising that protein adsorption and the properties of proteins at interfaces have been intensively studied for many years.5 Yet, since the protein family is very diverse, much remains to be understood about interfacial properties of specific proteins. In this work, we focus on a particular class of proteins, hydrophobins, which display unusually high surface activity.6 Hydrophobins are small proteins produced by filamentous fungi. They carry many different functions in the fungi, where they form coatings on spores, mediate the attachment to different surfaces, and reduce the water surface tension while the hyphae is growing.7 The remarkable surface activity of these © 2013 American Chemical Society

Received: December 14, 2012 Revised: January 28, 2013 Published: January 28, 2013 2683

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Table 1. Molecular Weight (Mw), Dimensions, Hydrodynamic Radius (RH), Radius of Gyration (Rg), Isoelectric Point (pI), and Apparent Diffusion Constant (Dapp) of the Proteins Used in This Study proteins HFBI HFBII lysozyme

Mw (kDa)

dimensions (nm)

RH (10−9 m) b

16

7.5 7.216 14.325

2.4 × 2.7 × 3.024 3.0 × 3.0 × 4.526

Rg (10−9 m) a

1.8 1.5b 2.027

1.4 1.2a 1.428

Dapp (10−10 m2 s−1)

pI 23

5.7 6.723 11.329

1.2 1.4 1.1

a

These values were calculated23 from the crystal structures of HFBI30 and HFBII.24 bThe hydrodynamic radius for both hydrophobins were calculated by using the relationship Rg = 0.778RH valid for spheres, which is close the relation between Rg and RH suggested by Mereghetti and Wade.31

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HFBI and HFBII are rather small proteins, consisting of ∼100 amino acids. The amino acid sequences for each of the proteins have been identified and found to share 63% homology.11 The overall molecular structure is rigid because of four disulfide bridges. Moreover, the hydrophobin molecules are amphiphilic with distinct hydrophilic and hydrophobic domains, where the latter is not “hidden” inside the molecule but exposed to the solvent. Because of this feature, watersoluble HFBI and HFBII molecules tend to form aggregates in solution in a concentration-dependent manner.12 Following the function of hydrophobins in nature, namely reduction of water surface tension,13 a significant amount of research has been focused on the air/liquid interfacial behavior of these proteins. It was found that the aggregates formed in the bulk tend to break and form a monomolecular film at this interface, with thickness 2.8 and 2.4 nm for HFBI and HFBII, respectively, as determined by in situ X-ray diffraction measurements.14 Several attempts have been made to transfer Langmuir−Blodgett and Langmuir−Schaefer films onto hydrophilic (mica) and hydrophobic (graphite) solid supports15 and to measure the layer thickness by means of AFM. It turned out that the film thickness of HFBI was 1.3 and 2.8 nm on mica and graphite, respectively. The difference between layer thickness measured at these two interfaces suggested different protein conformations on these solid surfaces.16 However, circular dichroism studies of HFBI and HFBII at the air/liquid interface did not suggest large conformational changes.17 Less is known about the properties of adsorbed layers, rather than transferred films, of hydrophobins at liquid/solid interfaces. Adsorption of HFBI and HFBII to both hydrophilic and hydrophobic surfaces has, however, been reported.18,19 These studies revealed that the adsorption rate to a hydrophilic silica is slow by comparison to that for a silanized silica, but the sensed mass detected by QCM was higher on the pure silica surface. Moreover, it was shown that adsorption of HFBI to a hydrophobic surface resulted in monolayer formation, similar to the air/liquid interface.20 Another important question that arises from the protein adsorption to the solid surfaces is the stability of the layers formed over time. This is of particular importance for hydrophobins, since they have been found to be good candidates for surfaces modification.21,22 Although a significant body of work has been done to further the knowledge of the interfacial properties of hydrophobins, still their intricate behavior is not fully understood. This is particularly true for hydrophobins at hydrophilic surfaces. In this report we address the following questions: (i) Do the adsorption properties of HFBI and HFBII differ on hydrophilic surfaces? (ii) Does the liquid flow rate affect the adsorption of hydrophobins? (iii) If the answer is yes, then how is the layer affected by a change in liquid flow rate? (iv) How is the desorption of the layers formed by HFBI and HFBII affected by removal of the protein from solution?

MATERIALS AND METHODS

Proteins. HFBI and HFBII were extracted and purified according to the procedure described earlier.12 The proteins were stored in darkness under moisture-free conditions. Lysozyme from chicken egg white, 95% purity, was purchased from Sigma-Aldrich and used as received. The molecular characteristics of the proteins used in this study are summarized in Table 1. Langmuir Trough. Surface pressure vs area isotherm measurements, as well as stability experiments where the change in area under a constant surface pressure was monitored at the air/liquid interface, were performed using a KSV 5000 instrument (KSV Instruments, Finland). It consists of a Teflon trough with two moving barriers and a platinum Wilhelmy plate to monitor the surface pressure change with an accuracy of ±0.01 mN m−1. The trough is equipped with a temperature control unit. It is enclosed in a cabinet kept free of dust by particle-filtered horizontal laminar air flow and mounted on an antivibrational table. Water was purified by employing a Milli-ROPls unit connected to a Milli-Q plus 185 system and filtered through a 0.2 μm Millipak filter at 25 °C. The resistivity of the water was 18.2 MΩ cm, and the organic content was less than 3 ppb. This water, which has a pH of 5.5−6, was used as subphase in all experiments, and the temperature was maintained at 20 °C. Before each experiment the purity of the water surface was tested by compressing the barriers to their final positions, whereby the surface pressure was not allowed to increase by more than 0.1 mN m−1. If it did, the trough was cleaned and the experiment was started over. HFBI and HFBII proteins were dissolved in water to a concentration of 0.1 mg mL−1, and 1 mL of these solutions was carefully spread on the subphase surface using a 1 mL syringe. The compression experiments were performed at the speed of 3 mm min−1 until the final barrier position was reached. The same compression rate was used in the relaxation experiments, until the target surface pressure (30 mN m−1) was achieved. After this the system was programmed to maintain the desired surface pressure. Dual Polarization Interferometry (DPI). The adsorption and desorption experiments at the solid/liquid interface were carried out with an AnaLight 4D instrument from Farfield Group Ltd. (Manchester, UK). This is an evanescent wave technique that measures changes in the transverse magnetic (TM) and transverse electric (TE) polarization state of light. The theory behind this method, as well as the principles of operation and evaluation of thickness and refractive index, have been described in detail elsewhere.32,33 Briefly, a laser beam is focused on two horizontally stacked slab waveguides. The lower one is a reference waveguide, and the upper one is the sensing waveguide on which adsorption occurs during the experiment. Light propagates through the two waveguides due to total internal reflection, and the interference Young’s fringes that appear after the reference and sensing beams have exited the waveguides are recorded in the far-field. Adsorption of proteins on the sensing waveguide surface will affect the evanescent field and therefore cause a phase shift in the interference patterns, which carries information on the adsorption process. Since two polarization states of light are used, both thickness and refractive index of the adsorbed layer can be determined. The adsorbed amount, Γ, was evaluated from the layer thickness, df, and refractive index of the film, nf, using the de Feijter formula:34 2684

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df (nf − nb) dn/dc

Article

(1)

where nb is the bulk refractive index of the aqueous solution (1.333 for water). The refractive index increment (dn/dc) used in the calculation was 0.182 mL mg−1, which is a typical value for proteins.35,36 A threelayer model, surface−adsorbed layer−bulk solution, was used in the evaluation process, and each of these layers is assumed to be homogeneous. Thus, the optical evaluation model is the same as normally used for evaluating e.g. ellipsometry data. Before each adsorption experiment the chip was calibrated by using a two-step procedure, in which two solutions of known refractive index were passed through the channels. The DPI chip calibration is described in more detail by Coffey et al.37 Protein adsorption was in most cases studied at two flow rates 5 and 50 μL min−1, but in some cases also at 25 μL min−1. Eventual changes in protein layer properties due to interruption of the flow were investigated with and without protein present in the aqueous layer present above the surface. In both cases the surface was first saturated with protein at a flow rate of 50 μL min−1. In the first case the flow was stopped with the protein remaining in the solution above the surface, whereas in the second case the protein solution was replaced by water before the flow was stopped. All experiments were carried out at 20 ± 0.001 °C as controlled by the temperature control unit of the DPI instrument. Surfaces Preparation. The adsorption studies were performed using hydrophilic silicon oxynitride DPI chips (Farfield Scientific). For surface potential determination silicon dioxide and silicon oxynitride wafers were used. All surfaces were cleaned with a 50/50 (v/v) mixture of methanol (BDH, AnalaR) and 37% hydrochloric acid (Merck) for 15 min followed by 30 min treatment with a 2% Deconex 20 NS surfactant-free solution (Fisher Scientific). After these cleaning steps the surface was left overnight in Milli-Q water. Electrokinetic Analyzer. The ζ-potential for silicon oxynitride and silicon dioxide wafers was measured by an electrokinetic analyzer SurPASS (Anton Paar GmbH, Graz, Austria). This technique has been described in detail elsewhere.38,39 Briefly, an electrolyte solution with laminar flow is forced across a channel formed by two identical plates. This generates a pressure difference and a streaming current (as well as streaming potential), which are measured.40 In our studies 1 mM KCl solution was used and two rectangular slides of silicon oxynitride with size 20 mm × 10 mm were mounted parallel to each other and separated by a 100 ± 5 μm gap in the measuring cell. For comparison, the ζ-potential for planar silica surfaces with the same dimensions was also determined. For smooth surfaces the Helmholtz−Smoluchowski approach can be applied to calculate the ζ-potential:38,39 ζ=

IsληL ΔPεε0H

Figure 1. ζ-potential of silicon oxynitride (filled circles) and of silica (unfilled circles) wafers as a function of pH. The values were recorded in 1 mM KCl. The dotted and dashed lines are guides for eyes for SiON and SiO2, respectively.

the effect of having hydrophobins present in the stagnant solution. HFBI and HFBII at Air/Water Interfaces. The surface pressure (Π)−area (A) isotherms and the relaxation isotherms (loss of monolayer area) for HFBI and HFBII films at a constant surface pressure of 30 mN m−1 are shown in Figure 2. The surface pressure−area isotherm for HFBII is more expanded than that of HFBI (Figure 2a), which demonstrates stronger repulsive interactions between HFBII molecules at large areas per molecule. For HFBII a smooth increase in the surface pressure, starting at 500 Å2 per molecule is noted, while for HFBI a steeper isotherm that commences at 290 Å2 per molecule is found. The area per molecule for HFBI extrapolated from the steep part of the isotherm at high surface pressure to zero equals 260 Å2. This is in a good agreement with the previously reported values for HFBI (258 Å2).16 A similar value for HFBII is difficult to obtain due to the expanded nature of the surface pressure vs area isotherm. The surface pressure at which the film relaxation was performed, 30 mN m−1, was chosen since no film collapse was observed, and the mean area per molecule is similar for the two hydrophobins (around 220 and 230 Å2 for HFBI and HFBII, respectively) at this surface pressure. The experiments were carried out at the constant surface pressure, and the changes in area were continuously recorded. The transport of molecules from the layer can be described as consisting of two steps. First, the molecules move from the adsorbed layer to the subsurface region immediately below the layer, and then the molecules diffuse from the subsurface region to the bulk. Either of these two steps can be rate limiting. In case the transport from the layer to the subsurface region is rate limiting, then the area relaxation is given by

(2)

where ε is the dielectric constant of the solvent, ε0 the permittivity of vacuum, η the viscosity of the medium, Is the streaming current, ΔP the pressure difference, and λ the specific electric conductivity of the electrolyte solution in the channel; L and H are the channel length and widths, respectively. The silicon oxynitride and silicone dioxide wafers were cleaned in exactly the same way as the DPI chips and stored under water before the measurements. The pH dependence of the ζ-potential for these two surfaces is shown in Figure 1. We note the similarity of the ζpotential for silica and silicon oxynitride at pH values below 7. For both surfaces the isoelectric point is found at around pH 3.

⎛A⎞ ln⎜ ⎟ = −kt ⎝ A0 ⎠

■

(3)

where A0 is the total area covered by protein right after compression to the target surface pressure, A is the corresponding area at time t, and k is the transport coefficient. This equation arises due to the constant state of the adsorbed layer (constant area per molecule due to constant surface pressure) and the constant chemical potential difference between protein in the surface layer and in bulk solution (the concentration of protein in bulk solution is essentially zero throughout the experiment due to the large subphase volume).

RESULTS AND DISCUSSION In this section we first discuss hydrophobins at air/water interfaces with particular focus on stability of layers held under constant surface pressure. Next, we describe the adsorption of hydrophobins to hydrophilic and negatively charged silicon oxynitride, with emphasis on equilibrium layer properties and adsorption kinetics. Finally, we discuss desorption of the layers formed under flow when kept under stagnant conditions and 2685

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Figure 2. (a) Surface pressure (Π)−area (A) isotherms of HFBI (dashed line) and HFBII (dotted line) on water. (b) Relaxation data plotted as ln(A/A0) versus square root of time, for HFBI (●) and HFBII (▲) at a constant surface pressure of 30 mN m−1.

Figure 3. Adsorbed amount of (a) HFBI and (b) HFBII on silicon oxynitride as a function of time. The data were obtained at two different flow rates 5 μL min−1 (filled symbols) and 50 μL min−1 (open symbols). The protein concentration was 0.01 mg mL−1 in all experiments.

Table 2. Mean Values of Thickness, Refractive Index, and Adsorbed Amount of HFBI and HFBII on Silicon Oxynitridea thickness (nm) −1

protein

flow rate (μL min )

HFBI

5 50 5 50

HFBII

max 1.7 1.7 2.3 2.3

± ± ± ±

0.2 0.2 0.1 0.2

rinse 1.7 1.6 2.0 2.5

adsorbed amount (mg m−2)

refractive index ± ± ± ±

max

0.2 0.1 0.1 0.2

1.412 1.418 1.442 1.454

± ± ± ±

0.009 0.015 0.004 0.004

rinse 1.413 1.414 1.438 1.445

± ± ± ±

0.010 0.013 0.004 0.008

max 0.75 0.83 1.34 1.51

± ± ± ±

0.04 0.21 0.04 0.04

rinse 0.75 0.72 1.13 1.29

± ± ± ±

0.04 0.18 0.03 0.01

a The protein concentration was 0.01 mg mL−1, and the values obtained after rinsing were taken after 20 min of rinsing with a flow rate of 50 μL min−1.

At the air/water interface the surface excess Γ is constant at the constant surface pressure; therefore, integration of eq 4 yields

If the desorption rate of HFBI and HFBII would be determined by this step, then the experimental data plotted as ln(A/A0) versus t would result in a linear curve. However, this was not the case (see Supporting Information). The analytical solution for the case when the diffusion from the subsurface region to bulk is rate limiting has been provided by Minassian-Saraga41 and is ⎛ Dapp ⎞1/2 ⎛ Dapp ⎞1/2 d(ΓA) = −⎜ ⎟ (Cs − C b)A ≈ −⎜ ⎟ CsA dt ⎝ πt ⎠ ⎝ πt ⎠

⎛A⎞ ⎛ Dapp ⎞1/2 Cs 1/2 ln⎜ ⎟ = −2⎜ t ⎟ ⎝ π ⎠ Γ ⎝ A0 ⎠

In Figure 2b, ln(A/A0) is plotted as a function of t1/2 for HFBI and HFBII. For both hydrophobins the linear decrease as expected from eq 6 was observed, which shows that the desorption kinetics is controlled by diffusion from the subphase to the bulk. The subphase concentrations for HFBI and HFBII can be calculated from the slopes of these curves, and they equal 0.47 and 0.16 μg mL−1, respectively. This shows that HFBI molecules have lower affinity to the air/water interface by comparison to HFBII. HFBI and HFBII at Solid/Water Interfaces: Effect of Flow Rate. The adsorption of HFBI and HFBII under two volumetric flow rates, 5 and 50 μL min−1, was studied using DPI, and the data obtained are shown in Figure 3. The rinse

(4)

where Dapp is the apparent diffusion constant, Γ is the surface excess, and Cs is the subphase concentration. The apparent diffusion constant was calculated by using the Einstein−Stokes equation:

Dapp =

kT 6πηRH

(6)

(5)

where k is the Boltzmann constant, T the absolute temperature, and η the viscosity of the medium. 2686

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Adsorption Kinetics of HFBI and HFBII. A close look at the data presented in Figure 3 shows a significant difference in the behavior of HFBI and HFBII. The first 60 s of their adsorption at the flow rate 50 μL min−1 is illustrated in Figure 4. The solid lines represent the linear increase of the surface

arrows indicate the end of the adsorption step and the start of rinsing with pure water. As expected, an increase in flow rate results in faster initial adsorption. However, no large difference in adsorption is observed at later stages. A major difference between the two hydrophobins is that HFBII adsorbs to a significantly larger amount than HFBI. Rinsing with pure water resulted in removal of loosely attached proteins, in all experiments, except for HFBI at 5 μL min−1 flow rate where no desorption was observed upon rinsing. The mean values and standard deviations of layer thickness, refractive index, and adsorbed amount for the HFBI and HFBII at the two different flow rates are summarized in Table 2. These values were calculated based on at least two experiments. As shown in Table 2, the flow rate during adsorption has a rather limited effect on the layer properties of HFBI and HFBII. However, there is a significant difference between HFBI and HFBII. Previously, Linder et al.18 concluded that the behavior of these two proteins on silica QCM crystals was very similar, leading to the formation of rigid layers with similar mass. It should be noted, though, that the mass determined by QCM includes the mass of the adsorbed protein as well as that of hydrodynamically coupled water. The latter quantity can be significant due to water trapped between adsorbed species.42 For instance, for lysozyme adsorbed on silica the mass of trapped water has been determined to amount to 75% of the mass sensed by QCM, and for human serum albumin on the same surface the hydrodynamically coupled water contributed to 90% of the sensed mass. Considering this important difference between the mass detected by QCM and an optical technique such as DPI, we conclude that the adsorbed amount of HFBII on silicon oxynitride (and likely also on silica) is significantly higher than that of HFBI, whereas the water content in the layer is higher for HFBI. Other important characteristics of the adsorbed layers are thickness and refractive index. As shown in Table 2, the thickness of the layers formed by HFBI and HFBII equals 1.7 and 2.5 nm, respectively. The thickness values confirm that neither of the hydrophobins form multilayers on the surface. Further, based on the refractive indexes, the HFBII forms a more compact adsorbed layer than HFBI (see Table 2), which supports the suggestion above that the HFBI layer contains more water. The refractive index of most proteins is close to 1.6,42 but this value is much higher than what is reported in the literature for adsorbed protein films measured by means of optical methods.43 This is a consequence of the large amount of water that normally is contained within the protein layer.44 The average area per adsorbed hydrophobin molecule was calculated from the adsorbed amounts using the equation σ=

Mw ΓNA

Figure 4. First 60 s of the adsorption curves for HFBI (●) and HFBII (▲) at the flow rate 50 μL min−1. The lines were calculated based on the first 10 s of the adsorption process.

excess at the very beginning of the adsorption process, whereas at longer adsorption times (symbol curves) clear deviations from the initial adsorption rate are observed. The slopes of the lines (dΓ/dt)t→0 for HFBI and HFBII equal 0.014 and 0.023 mg m−2 s−1, respectively. As shown in Figure 4, the rate of HFBI adsorption decreases after about 15 s, which is a consequence of hindrance imposed by already adsorbed proteins. More interestingly, the opposite behavior is observed for HFBII. In this case the rate of adsorption increases after about 15 s. Thus, the probability of HFBII adsorption increases when some HFBII molecules already have adsorbed. This is a sign of cooperativity and indicates association within the monomolecular thick layer at the solid−liquid interface. As the coverage of HFBII increases further, the adsorption process slows down again (after about 40 s) as the adsorbed proteins sterically hinder the HFBII molecules from reaching the surface. The weak cooperativity in the adsorption of HFBII is observed in all kinetic data for this protein (see Supporting Information). Theoretical Calculations of the Initial Adsorption Rate. In general, the protein adsorption process can be considered to consist of several steps: (i) transport of the molecule from the bulk to the interface, (ii) attachment to the surface, and (iii) rearrangement on the surface.5 Either of the first two stages can determine the rate of the initial adsorption, and the first step can be described by using general transport equations once the hydrodynamic conditions and molecular dimensions are known.46 In case all molecules that reach the surface also adsorb, then the flux of the protein to the surface equals the adsorption rate (dΓ/dt). Experimentally, these values can be obtained from the initial slope of the adsorption isotherms (dΓ/dt)t→0 (Figure 4, solid lines). Analytically, it can be calculated by using the Leveque equation:47

(7)

where σ is the area per molecule, Mw the molecular weight, Γ the adsorbed amount of HFBI and HFBII after 20 min of rinsing, and NA Avogadro’s number. From the calculation, the area per molecule for HFBI is 14 nm2 and the one for HFBII is 9 nm2, which is much larger than the molecular dimensions of these proteins. Previously, it was shown by Aumaitre45 that the smallest area occupied by a single HFBII molecule equals 3.47 nm2. Thus, we can conclude that no tightly packed monolayer of the hydrophobins are formed at the silicon oxynitride/water interface.

2 ⎞1/3 2 ⎞1/3 ⎛ ⎛ dΓ(x) 1 ⎜ γDapp ⎟ 1 ⎜ 2FDapp ⎟ = C0 = C0 dt G(4/3) ⎜⎝ 9x ⎟⎠ G(4/3) ⎜⎝ 3bw 2x ⎟⎠

(8) 2687

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Figure 5. Initial adsorption rate of HFBI (left) and HFBII (right) versus wall share rate to the power of 1/3. The solid lines represent the experimental data, whereas the dashed and dotted lines represent predictions of the maximum adsorption rate from the integrated Leveque (eq 11) and Glaser (eq 10) equations, respectively.

Figure 6. Adsorbed amount of HFBI (dashed line), HFBII (dotted line), and lysozyme (solid line) as a function of square root of time without and with the corresponding protein present in the solution above the surface.

where G is the gamma function (the normal symbol Γ for the gamma function is avoided to prevent confusion with the adsorbed amount that also is denoted by Γ), and G(4/3) equals 0.893, Dapp is the apparent diffusion constant, C0 is the protein concentration, γ is the wall share rate, and x is the distance from the flow cell inlet to the observation point. The second equality arises from the relation between wall shear rate and volumetric flow (F) in a rectangular cell with given thickness (b) and width (w) under laminar flow conditions: γ=

6F b2 W

dΓ dt

(11)

In earlier DPI work where eq 10 was used it was found that the observed adsorption rate was slightly larger than the maximum theoretical one, and this was explained as being due to sample polydispersity. We instead suggest that the use of eq 11, rather than eq 10, would remove this inconsistency. The DPI cell volume is 0.98 μL, and to infuse the solution in the entire cell with the flow rates 5, 25, and 50 μL min−1 takes around 11.8, 2.4, and 1.2 s, respectively. Under these conditions laminar flow is established (Reynolds number