Introduction

Protein adsorption at solid-liquid interfaces (Andrade 1985; Brash and Horbett 1995) is important in many fields such as hemocompatible materials (Leonard et al. 1987), diagnostic kits (Malmsten et al. 1996), enzymatic activity (Quiquampoix and Ratcliffe 1992; Servagent-Noinville et al. 2000) and environmental hazards in mineral soils (Vasina et al. 2005). Natural and artificial vessels adsorb proteins fromblood. Conformational changes or reactions at interfaces can induce series of biochemical reactions. In general, this type of phenomenon must be avoided as coagulation and complement

Philippe Déjardin, Elena N. Vasina: European Membrane Institute, UMR 5635 (ENSCM-UMII-CNRS), Université Montpellier 2, CC 047, 34095 Montpellier Cedex 5, France, E-mail: [email protected], [email protected]

Principles and Practice Proteins at Solid-Liquid Interfaces Philippe Déjardin (Ed.) © Springer-Verlag Berlin Heidelberg 2006

systems can be activated. Application to biosensors has been proposed (Mar et al. 1999). Surface hydrophilization via polymer pretreatment can inhibit or limit these phenomena in solid-phase diagnostics (Malmsten et al. 1996), in hemodialysis hollow fibers (Yan et al. 1992) or on polymer surfaces (Lee et al. 1990). Such polymer pretreatment is generally based on copolymers that contain poly (ethylene oxide) chains (Holmberg et al. 1997; Lee et al. 1989; Nitschke et al. 2000; Price et al. 2001; Tirelli et al. 2002; Wu et al. 2000; Xu and Marchant 2000) or phosphorylcholine moeities (Huang et al. 2005; Ishihara et al. 1999a, b; Iwasaki et al. 2003; Kojima et al. 1991; Nakabayashi and Williams 2003; Nakabayashi and Iwasaki 2004; Ueda et al. 1991; Ye et al. 2005; see Chaps. 10-12). In addition, protein adsorption modifies the interfacial charge density, or the electrokinetic potential at the interface, which can be deduced from streaming potential measurements. This technique was used also to study adsorption kinetics (Ethève and Déjardin 2002; Norde and Rouwendal 1990; Zembala and Déjardin 1994).

One fundamental parameter contributing to the analysis of protein-solid surface interactions is the adsorption kinetic constant ka, which is related to the energy barrier the protein molecule has to overcome during adsorption mechanism. Determination of the adsorption constant, however, is not straightforward: The initial adsorption process can be controlled by transport or interfacial reaction as the two extreme limits; in addition, any intermediate case can exist where both the transport and the interfacial reaction have to be taken into account and their interplay accurately described. Under well-controlled laminar flow conditions, for instance, in a channel (Fig. 1) or tube with a radius much larger than the diffusion layer thickness, the experimental adsorption kinetic constant k(x)iscomparedto the Lévêque constant kLev(x) corresponding to a fully transport-controlled process in a rectangular channel, where x is the distance from the channel entrance to the observation point. Such comparison gives a qualitative estimation of the role of transport in the overall adsorption process.

The rate of adsorption in the presence of low-concentration solutions, when the steady-state of the concentration profile C(x,y) has been established, can be written as:

where r is the interfacial concentration, t is the time, D is the diffusion coefficient, cb is the bulk solution concentration; ka is the adsorption kinetic constant at the interface, c(x, 0) is the solution concentration at distance x from the channel entrance and at y = 0, and k(x) is the kinetic constant of the overall process at x.

Fig. 1. Channel with rectangular section. Flow occurs in the x direction. The distance to the wall is given byy, channel height by b, and fluid velocity profile by v(y) = yy(1 -y/b), where y is the wall shear rate b\ \z

Fig. 1. Channel with rectangular section. Flow occurs in the x direction. The distance to the wall is given byy, channel height by b, and fluid velocity profile by v(y) = yy(1 -y/b), where y is the wall shear rate

In case of the fully transport-controlled process (C(x, 0) = 0), the kinetic constant k at distance x from the entrance of the channel depends only on diffusion through the solute diffusion coefficient D and convection through the wall shear rate y. According to Lévêque (1928) its expression is kLev(x) = 0.538 (D2y/x)1/3, a relationship that was also derived later (Levich 1962). When adsorption is controlled only by the interfacial reaction k ~ ka and practically does not depend on x.

Figure2illustratestheinterfacial region ofdepletioninsolutionobtained by numerical simulations. For a given solute diffusion coefficient, the higher the adsorption kinetic constant, ka, the larger the interfacial depletion, and the larger the distance from the channel entrance, the larger the thickness, Ô, of the depletion layer and the depletion magnitude at the surface. Hence the crossover length Lco (Eq. 5c) was introduced in the complete treatment (Déjardin et al. 1994), contrary to the simpler case of the rotating disk where Ô is constant (Coltrin and Mitchell 2003; Levich 1962). We define the depletion d(x) at the interface:

As the steady-state adsorption rate is related to the slope of C(x, y) at the wall (Eq. 1):

Fig.2. Stationary concentration profiles normalized to the bulk concentration Cb, for D = 6.0X10-7 cm2 s-1, and wall shear rate 1,000 s-1. Channel height is 100 ^m. Left Three-dimensional graph C(x,y) from the entrance to distance x = 2 cm. Right Concentration profile at x = 2 cm, with the tangent to the profile at the wall (dashed line, left scale), and parabolic velocity profile with the tangent at the wall (dash-dotted line, right axis). a ka = 1.0X10-4 cm s-1; b ka = 5.0X10-4 cm s-1; c ka = 1.0X10-2 cm s-1

Fig.2. Stationary concentration profiles normalized to the bulk concentration Cb, for D = 6.0X10-7 cm2 s-1, and wall shear rate 1,000 s-1. Channel height is 100 ^m. Left Three-dimensional graph C(x,y) from the entrance to distance x = 2 cm. Right Concentration profile at x = 2 cm, with the tangent to the profile at the wall (dashed line, left scale), and parabolic velocity profile with the tangent at the wall (dash-dotted line, right axis). a ka = 1.0X10-4 cm s-1; b ka = 5.0X10-4 cm s-1; c ka = 1.0X10-2 cm s-1

we obtain:

The Lévêque limit where depletion is complete corresponds to d ^ l, kLev(x) = D/ôLev(x) and ¿Lev(x) = 1.859(Dx/y)1/3. When approaching this limit, k(x) << ka, while for the opposite one of a negligible depletion k(x) ^ ka.

The Damkohler number, Da = ka/<v>, where <v> is the average velocity of the fluid, is often used as a criterion to separate the domains of control by the interfacial reaction (Da << 1) and by transport (Da » 1), especially in studies concerning porous media (Adler and Thovert 1998). There is a direct connection between the ratio ka/kLev and this number. The difference originates from the introduction of the diffusion coefficient. Other expressions of Da are used that also take it into account (Bizzi et al. 2002; Coltrin and Mitchell 2003) for gas phases: Da = ka/(D/s), where S is the diffusion layer thickness. In the present problem, we can define a Damkohler number Da(x) = ka/kLev(x), with kLev(x) = D/SLev(x).

As summarized earlier (Docoslis et al. 1999), the experimental determination of the adsorption kinetic constant ka has three major sources of difficulties: (1) mass transport, easier to take into account with the simple geometries of rectangular channels or circular tubes, (2) steric hindrance at the interface, which can be limited by using low-concentration solutions, and (3) determination of low interfacial concentration, which requires very sensitive techniques, usually using radioactive or fluores-cently labeled molecules like in the total internal reflection fluorescence (TIRF) technique (Britt et al. 1998; Buijs et al. 1998; Kelly and Santore 1995; Malmsten et al. 1996; Rebar and Santore 1996; Robeson and Tilton 1996; Wertz and Santore 2002). Sophisticated optical methods such as surface plasmon resonance (SPR; Mar et al. 1999) and optical waveguide lightmode spectroscopy (OWLS; Calonder and Van Tassel 2001; Hook et al. 2002; Ramsden et al. 1995; Chaps. 1-2) provide the means to measure low interfacial concentrations without labeled molecules.

The following data treatment can be useful for the SPR, reflectometry, ellipsometry, and OWLS techniques if the experimental flow cells have rectangular channel geometries. For instance, in OWLS experiments, a chamber tightened with an O-ring joint is not adapted to the present formulation as the channel width varies continuously and strongly near the inlet, which leads to a nonconstant wall shear rate. In this case the Lévêque limit formula is no longer applicable. The cell has to be adapted (Chap. 1) to achieve a constant width. Moreover, the height of the channel should be large enough to provide a linear velocity profile in the depletion layer, which is assumed in the Lévêque derivation, and its extension to finite ka (Déjardin et al. 1994).

For example, in Fig. 2 the channel height of 100 ^m is too small to fulfill the criterion when D = 6.0x 10-7 cm2 s-1 and y = 1,000 s-1. Another technique is the impinging jet on a flat substrate, which was also used in combination with TIRF to study the deposition of latex particles (Goransson and Tragardh 2000). Here we do not consider this kind of geometry.

In the first part of the present work we recall derivation of the simple accurate approximations, providing an easy way to treat quantitative data; thus, one can deduce the protein diffusion coefficient D and protein interfacial adsorption kinetic constant ka from experimental initial constant k as a function of the variable (k/kLev)D2/3, where kLev/D2/3 does not depend on D.

In the second part, we present the variation of the interfacial potential as a function of surface coverage and suggest the possible importance of protein-protein dipolar attractive interactions to create domains with an antiferroelectric order at high coverage.

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