Adsorption rate dependence on convection over a large length of a sensor to get adsorption constant and solute diffusion coefficient

Colloids and Surfaces B: Biointerfaces 76 (2010) 112–116

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Colloids and Surfaces B: Biointerfaces journal homepage: www.elsevier.com/locate/colsurfb

Adsorption rate dependence on convection over a large length of a sensor to get adsorption constant and solute diffusion coefficient Sylvie Noinville a,b , Jasmina Vidic b , Philippe Déjardin c,∗ a b c

Laboratoire de Dynamique, Interactions et Réactivité, UMR 7075, CNRS, Thiais, France Unité de Virologie et Immunologie Moléculaires, INRA, Domaine de Vilvert, 78352 Jouy-en-Josas, France Institut Européen des Membranes, Université Montpellier 2 (ENSCM, UM2, CNRS), UMR 5635, CC047, 2 Place Eugène Bataillon, F-34095 Montpellier Cedex 5, France

a r t i c l e

i n f o

Article history: Received 31 July 2009 Received in revised form 13 October 2009 Accepted 13 October 2009 Available online 23 October 2009 Keywords: BiaCore Ligand binding Adsorption kinetics Protein adsorption

a b s t r a c t We propose a representation of initial adsorption kinetic constant as a function of convection in a slit flow cell device, averaged over some restricted length of a wall acting as a sensor. The complete domain from transport-control to surface reaction control is included. The intercepts with axes give access to adsorption constant and solute diffusion coefficient. It is shown that, provided the close entrance is avoided, the function for the restricted length is very close to the function for local values. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Knowledge of interactions of proteins, antigens, antibodies, solutes, etc. with natural and artificial surfaces is important as it may help to classify specific or non-specific interactions of various types. Generally required high specificity of interactions in screening tests has to be associated with very low non-specific interactions on the background support. High specificity means high affinity and in most cases high adsorption kinetic constant. Then difficulty appears to take properly into account the contribution of transport in the observed kinetics to evaluate the constant really associated with the energy barrier. As mentioned earlier [1] the experimental determination of the adsorption constant ka has three major sources of difficulties: (i) mass transport, (ii) steric hindrance at the interface and (iii) determination of interfacial concentration. To avoid these difficulties, the authors [1] used the reactor system, where particles were introduced into the reactor under constant stirring. In that configuration, the data analysis [2] requires extrapolation to time zero and an incipient turbulence to achieve adequate transport. Conversely, another way is to use simple geometry of a slit or tube in laminar regime which helps to take into account quite properly the transport process. Study of kinetics as a function of convection and extrapolation to infi-

∗ Corresponding author. Tel.: +33 467149121; fax: +33 467149119. E-mail address: [email protected] (P. Déjardin). 0927-7765/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfb.2009.10.022

nite convection will provide the adsorption kinetic constant. We are considering here the case of solid functionalized sensors or flat surfaces in general submitted to the flow of the solution sample in well-controlled hydrodynamic conditions like in surface plasmon resonance [3–9], surface acoustic wave [10], optical waveguide lightmode spectroscopy [11–14], radioactivity [15–18], attenuated total reflection Fourier transformed infrared (ATR-FTIR) under flow [19] or fluorescence techniques [20]. Measurements of adsorption phenomena with time are then analyzed according to several models. Adsorption constant ka (or kon ) cannot be given by the initial experimental “constant” k = kexp = Cb−1 (dCsurf /dt)t=0 where Cb is the bulk solution concentration, Csurf the interfacial concentration and t the time. The contribution of transport – convection and solute diffusion – to k has to be taken into account for ka determination, the value of which being really of interest as related to the energy barrier to overcome in the adsorption process. Focusing on the early times is desirable as we are interested in the solute/surface interaction in absence of possible solute/solute interactions when surface becomes later more occupied. Then models can be developed independently of the adsorption constant determination. Moreover the determination of diffusion coefficient can be useful in mixtures or complex systems like serum as associations may occur. Contrary to studies considering a global analysis of the adsorption/desorption kinetics to deduce affinity constant, one procedure which was critically reviewed [21], the present study is limited to initial times and determination of the adsorption kinetic constant. If the functionalized surface of the sensor is actually a gel, the method still remains valid.

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The model of fully transport-controlled process in a slit was analyzed by Lévêque [22], who was considering the analogous problem of heat transfer. Therefore we will denote the fully transportcontrolled local constant as kLev (x).



kLev = 0.538

D2  x

1/3

(1)

where D is the solute diffusion coefficient,  is the wall shear rate which characterizes convection and x is the distance to the slit entrance. When integrating such expression over the distance from entrance to position x, we define the average kinetic constant as:

 kLev  = 0.808

D2  x

1/3 (2)

The ratio between the numerical coefficients is 3/2. Later Levich [23] found the same kind of expression, although he mentioned some discrepancy between the numerical factors. We could not find however any difference between both results. The previous expressions can be considered as the limit expressions for ka → ∞. Considering finite ka values for local [24] and average [25] constants leads to much more complex expressions (see Appendix A). Therefore we looked for approximate accurate simple relations in the full range of ka based on the limit expressions of the exact solution in both limits of the transport-controlled and the interfacial reaction controlled processes [26,27]. Moreover, we adopted the formulation of k/ka as a function of k/kLev which leads to a synthetic compact representation where interfacial depletion is directly visualized [27] and the intercepts with axes provide the adsorption constant and the solute diffusion coefficient. The method was applied for local constant k(x) and the average constant k. However, in practice, the detection is carried out over some predefined length between two positions x1 and x2 . In the present paper we extend the method to the treatment of data obtained in such conditions. In Section 1 we recall briefly the expressions relative to local and average kinetic constants. We present afterwards the method to deduce the graph relative to the integration of the kinetic constant over a limited domain [x1 ; x2 ] where x1 is not zero, before Section 5.

Fig. 1. Functions y = f(u) (straight line) and Y = F(U) (dashed line). Dotted line is the diagonal y = 1 − u and Y = 1 − U.

Fig. 2. Illustration of notations relative to different averages. The arrow shows the flow in the slit. Detection is carried out between positions x1 and x2 .

1% around u = 0.8. When considering the average value k between entrance and position x, the relation corresponding to Eq. (3) −1 −1 and we determined, following the same is k = ka−1 + kLev  method [26], the function Y = F(U) = (U − 1)(AU − 1)/(BU + 1) with Y = k/ka and U = k/kLev  and obtained A = 0.2031 and B = −0.2728 with a maximal error of 0.03% around U = 0.8. The functions f(u) and F(U) are represented in Fig. 1 with the approximation y = 1 − u corresponding to Eq. (3).

2. Representations of local and average kinetic constants When solving the partial differential equations relative to the diffusion–convection-reaction coupling in laminar conditions in a slit [24], the result appears as k as a function of ka and kLev . From a practical point of view it would be preferable to obtain the required ka constant as a function of k and kLev . Let us consider the approximation where the diffusion layer thickness is assumed to be independent of ka and equal to the Lévêque value ıLev with kLev = D/ıLev . With C(x,yw ) the solution concentration at position x and distance yw to the wall where yw = 0, we have kCb = ka C(x,0) = D(Cb − C(x,0))/ıLev , or: −1 k−1 = ka−1 + kLev

(3)

This type of relation can be viewed as the total resistance (or time) being the sum of the two contributions of reaction and transport. Analysis of the exact solution provides two linear approximations −1 in the two limits of control by transport (k−1 = kLev + 0.684ka−1 )

−1 and interfacial reaction (k−1 = 0.827kLev + ka−1 ) which are similar to Eq. (3) with different numerical coefficients. We determined then [26] the function y = f(u) where y = k/ka and u = k/kLev satisfying those both limits under the form of f(u) = (u − 1)(au − 1)/(bu + 1). We obtained a ≈ 0.4517 and b ≈ −0.6247 with a maximal error of

Fig. 3. Schematic representation of the determination of the coordinates (U2 ,Y2 ) corresponding to an arbitrary position x2 from a previously known point (U1 ,Y1 ) by means of the relations between the tangents of angles ˛1 and ˛2 , tan ˛ = kLev /ka .

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3. Representation with kinetic constant averaged between arbitrary positions

5. Discussion

Let us consider two positions x1 and x2 . Detection is performed as an averaged signal over the distance x = x2 − x1 . The ratio r = x1 /x2 can be small, for instance 1/5 in typical surface plasmon experiments. It is indeed necessary to find a compromise between a well-defined position and good sensitivity of detection, with the condition of a depletion layer thickness much smaller that the half width of the channel to simplify the data interpretation. We will determine Y12 (U12 ) with Y12 = k12 /ka and U12 = k12 /kLev 12 which becomes y(u) and Y(U) when r = 1 and r = 0, respectively. The subscript “12” means that the average is taken between positions x1 and x2 (Fig. 2). 3.1. Evaluation of (U2 ,Y2 ) from (U1 ,Y1 )

−1 − A − r 1/3 tan ˛1 +



(U12 − 1)(a1 U12 − 1) k12 = ka (b1 U12 + 1)(b2 U12 + 1)

(8)

To deduce wall shear rate  from flow rate Q, let us recall that Q in a slit of width w and height h is linked to  by Q = (h2 w/6). The average kLev 12 is given by:



kLev 12 = 0.808 with

Let us consider the very accurate approximation Y = F(U) and define the angle ˛ (Fig. 3) from tan ˛ = Y/U = kLev /ka . Given the supposed known result for the case x = x1 , say a point A1 (U1 ,Y1 ), the point corresponding to another value x2 of x, all other parameters fixed, will be related to A1 by tan ˛2 = r1/3 tan ˛1 . The point (U2 ,Y2 ) is at the intercept of the curve Y = F(U) with the straight line Y = r1/3 U tan ˛1 . U2 =

From the previous analysis, it can be concluded that although the experimental signal may correspond to integration over a large part of the sensor, the function Y12 (U12 ) is generally better approximated by the local function f1 .



x12 =

D2  x12

x2 − x1 2/3

x2

2/3

− x1

1/3

(9a)

3 = x2

 1 − r 3

(9b)

1 − r 2/3

It can be verified that, when x1 → x2 or r → 1, the limit of kLev 12 is the expected local expression k(x2 ) as x12 → (3/2)3 x2 . An alternative to Eq. (9a) is to use the local expression with prefactor 0.538

2

(1 + A + r 1/3 tan ˛1 ) + 4(−A + Br 1/3 tan ˛1 )

2(−A + Br 1/3

(4)

tan ˛1 )

3.2. Evaluation of Y12 and U12 From the relation: obtained Y12 = k12 /ka as Y12 =

k12 = (x2 − x1 )−1 (x2 k2 − x1 k1 )

Y2 − rY1 1−r

we

(5)

From the similar relation kLev 12 = (x2 − x1 )−1 (x2 kLev 2

− x1 kLev 1 ), and ki = Ui kLev i (i = 1, 2) we obtained U12 = k12 /kLev 12 as: U12 =

U2 − r 2/3 U1 1 − r 2/3

(6)

The full curve Y12 = F12 (U12 ) for a given r value is obtained from a scanning of U1 in the range [0, 1] with Y1 = F(U1 ). Afterwards (U2 ,Y2 ) is evaluated using Eq. (4) and Y2 = F(U2 ), finally Y12 and U12 by Eqs. (5) and (6).

 = k (x /) and [x]12 = x12 (2/3)3 . We define U12 12 12 2/3 U12 D . Then Eq. (8) becomes:

k12 = ka

 − D2/3 )(a U  − D2/3 ) (U12 1 12  + D2/3 )(b U  + D2/3 ) (b1 U12 2 12

1/3

/0.808 =

(10)

It can be used for a two parameters fit (D and ka ) to experimen . In such a representation (Fig. 5), the tal k12 as a function of U12 intercept with the ordinate axis is ka , while the intercept with the abscissa axis is D2/3 . Alternatively, if the bulk concentration Cb is not known, plot of initial adsorption rate under the form of dm/dt12 for instance, where m is the mass of adsorbed solute per unit area (for instance ␮g cm−2 ), as a function of dm/dt12 (x12 /)1/3 /0.808 will provide the intercepts ka Cb and D2/3 Cb , where Cb is expressed as mass per unit volume (for instance ␮g cm−3 ).

4. Application to different values of r Fig. 4a shows the results for fixed x2 and increasing x1 . r is between 1/5 and 1.0. It can be seen that even for a quite small value of r, the function F12 (U12 ) is very close to the local function f(u) and far away from the average function F(U). This is due to the diffusion layer thickness varying as x1/3 . However, looking more closely (Fig. 4b), the function y = f(u) is not the limit of F12 (U12 ) as x1 → x2 . It is indeed known that the function f(u) overestimates the exact result by less than 1%, while the function F(U) for averages is much more accurate. That could explain the higher accuracy of the limit of F12 (U12 ) as x1 → x2 compared to the local approximation f(u). Therefore, a better approximation under the form y = f1 (u) was determined by introducing the additional condition f1 (0) = −0.1596 (see Appendix A). f1 (u) =

(u − 1)(a1 u − 1) (b1 u + 1)(b2 u + 1)

(7)

The numerical coefficients are: a1 = 0.556055, b1 = −0.680691 and b2 = −0.0483712. Such function can be used in practice as a fit to experimental data even for small values of r.

Fig. 4. (a) From bottom to top: diagonal y = 1 − u (dotted line), Y = F(U) (dashed line) corresponding to x1 = 0, Y12 (U12 ) for x2 = 2.0 mm and x1 = 0.4, 0.8, 1.2, 1.6, 2.0 and the upper limit x1 = x2 according to the approximation y = f(u); (b) idem with focus on the central part which shows that f(u) is an overestimation of the exact result when x1 = x2 . x1 = 1.6 mm and 2.0 mm are undistinguishable from the improved function f1 (u)—see text.

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Fig. 6. Ratio  of the mean [x]12 , needed when using local expression of Lévêque constant, to the approximation (x1 + x2 )/2 as a function of r = x1 /x2 (dashed line); same ratio to power 1/3 (full line). (0) = 0.59; ((0))1/3 = 0.84. Fig. 5. Illustration of the function f1 applied to k12 as a function of 1.238k12 (x12 /)1/3 for a two parameters fit, diffusion coefficient D and adsorption constant ka , provided by the intercepts with the axes. Between parentheses, the extension to adsorption rate dm/dt12 when bulk concentration Cb is not known. tan ˛ is varying as wall shear rate to power 1/3. Crossing of the curve with the diagonal (dash-dotted line) corresponds to the case ka = kLev .

In practice, study of initial kinetics as a function of convection with fixed domain [x1 ; x2 ] will provide data k12 as a function of wall shear rate . Increasing flow rate corresponds to increasing angle ˛ with tan ˛ ∼  1/3 , from control by transport providing essentially the diffusion coefficient to control by interfacial reaction providing essentially the adsorption kinetic constant. However, at small shear rates, attention has to be paid that the diffusion layer thickness be smaller than the half height of the channel, for the model includes the condition of linear velocity variation with distance to the wall in the depletion layer. This condition ensures also the independence between the reactions at both walls. With h the height of the channel and the condition ıLev < (1/3)h/2, the lower boundary for wall shear rate is  > 1390(Dx/h3 ). We performed experiments with beta(2)microglobulin (buffer HBS-EP, 10 mM HEPES, pH 7.4, NaCl 150 mM, EDTA 3 mM, 0.005% surfactant P20) as analyte and surface sensor functionalized with corresponding antibodies (BiaCore 3000; one unit RU corresponds [5] to 10−4 ␮g cm−2 ). For such surface plasmon resonance cell with distances x1 = 0.4 mm and x2 = 2.0 mm, r = 1/5 and x12 = 0.36 cm.  = 0.880k  −1/3 . Flow rate was between With k12 in cm s−1 , U12 12 0.002 and 0.1 mL min−1 . However, the small gap of the channel (20 ␮m) needs high flow rate to satisfy the condition of depletion layer thickness smaller than the half height of the channel. The diffusion coefficient measured by light scattering in neutral conditions [28] is 12.4 × 10−7 cm2 s−1 with Stokes radius 1.73 nm. With the choice of ıLev < (1/3)h/2, the lower boundary value for wall shear rate is 4.3 × 104 s−1 at distance x2 = 2.0 mm. This is a quite large value which requires high pressure. The upper limit above which the leaks may appear is about 5 × 104 s−1 . As tan ˛ ∼  1/3 , the angular domain defined by these two limits is very small and can be a source of large errors. Over the domain 5 × 104 –5 × 103 s−1 with 5 wall shear rates, we obtained the underestimated D value of (7.5 ± 0.7) × 10−7 cm2 s−1 and the overestimated ka = (3.5 ± 0.3) × 10−3 cm s−1 , as the depletion layer thickness at the smallest wall shear rates becomes the same order of magnitude than the height of the channel. In that case the adsorption process was also dependent of what happened at the other interface. These results show however the feasibility of the model but impose using a larger cell height to extend the range of validity to smaller wall shear rates, as observed in previous adsorption studies with radiolabeled molecules [16]. In that work, the local expression was used with [x]12 approximated by (x1 + x2 )/2. Given

the precision of the measurements and r value of 0.45 corresponding to x1 = 2.5 cm and x2 = 5.5 cm, this approximation was valid as severe corrections are needed only for very small values of r (Fig. 6). 6. Conclusion We previously proposed [26] a simple representation of the full range of kinetics from control by transport (constant kLev ) to control by surface reaction (constant ka ) for slit geometry by the plot of local y = k/ka and average Y = k/ka as a function of u = k/kLev and U = k/kLev , respectively, between 0 and 1. As in many experimental setups the signal recording is relative to a large part of the wall length, say between positions x1 and x2 , in order to increase the detection sensitivity, we studied the influence of such length over the analogous function Y12 = k12 /ka as a function of U12 = k12 /kLev 12 which should be more adapted to interpretation. As a very accurate approximation Y = F(U) was available for the average kinetic constants over the whole length of the wall, we used it to build the analogous functions relative to the averages over restricted domains. We found that, provided the close entrance domain is avoided, the function Y12 (U12 ) is very similar to the local function y(u) and far away from the average function Y(U). Besides, we improved the previously published approximation of y(u) by adding a third parameter in order to have the right second derivative at the origin. It can be expected that such representation should help to characterize the reactions at interfaces as it leads to the determination of diffusion coefficient and adsorption constant by taking into account properly the transport contribution. The sole consideration of initial adsorption rate at different wall shear rates avoids the introduction of some model to describe the progressive occupation of the surface till the final equilibrium value. Conversely, the adsorption constant can be put then as constant in such models with different levels of approximation or in complete numerical simulations. Appendix A. We recall briefly the previous results [24]. We consider the stationary regime of adsorption with laminar convection of a solution of bulk concentration Cb along x-direction. The velocity field of the fluid near the wall is  = y, where y is the distance to the wall and  the wall shear rate. For a solute diffusion coefficient D and concentration C(x,y), we have the following relations: D

∂2 C(x, y) ∂C(x, y) = y ∂x ∂y2

C(0, y) = Cb

(A1a) (A1b)

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lim C(x, y) = Cb

y→∞



D

(A1c)



∂C(x, y) ∂y

= ka C(x, 0)

(A1d)

y=0

The adsorption rate is defined as: ∂Csurf (x, t) = ka C(x, 0) = k(x)Cb ∂t

(A2)

with The solution for k(x) is: k(x) = ka g(X), X = [(ka /kLev )/ (2/3)]3 where  (n) is the usual Gamma function [29].(A3)g(X) = e−X + G(2/3, X) − G(1/3, X); G(n, X) = 1 e−X  (n)

X 0

z n−1 ez dz (1/3) ≈ 2.67894;

 (2/3) ≈ 1.35412;

 (1/3) (2/3) = 2/31/2 . From the first terms of series expansion of k(x)/ka in powers of x = (ka /kLev ) around x = 0 (interfacial control), say 1 + c1 x + c2 x2 (c1 = −0.826993; c2 = 0.604118), we deduce the first terms of series expansion in powers of u = (k/kLev ): k = 1 + c1 ka

 k  kLev

 k 2

+ (c2 − c12 )

kLev

(A4)

−1 + 0.684ka−1 Close to the control by the transport, we have k−1 = kLev or:



k k = 1.461 1 − ka kLev



(A5)

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