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Kinetic Regimes of Protein Adsorption Claudio Calonder and Paul R. Van Tassel* Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, Michigan 48202 Received December 11, 2000. In Final Form: March 13, 2001 We use single-mode, optical waveguide lightmode spectroscopy to measure the mass of protein adsorbed at the waveguide/liquid interface at a frequency and precision sufficient to achieve sets of adsorption rate data (dΓ/dt, where Γ is the adsorbed protein density) that clearly delineate distinct diffusion- and reactionlimited kinetic regimes. These data suggest that the adsorption can be modeled by combining a particle approach at the surface with a boundary layer approach above the surface. The rate-limiting mechanisms responsible for the observed kinetic regimes appear as “resistors” in our transport/adsorption model.
Introduction The adsorption of proteins and other biomacromolecules at the liquid/solid interface is of critical importance in bioprocessing, biomaterials, and biosensing.1-3 Unfortunately, a broadly accepted theory for predicting the time dependence of the adsorbed amount remains elusive. Because the adsorption is often itself irreversible or accompanied by irreversible events such as changes in internal conformation, the protein (mono)layer can evolve to states away from equilibrium that could not, in general, be predicted by a thermodynamic approach.4 In these cases, a given property of the adsorbed layer, P, becomes a functional of the rate of adsorption, that is, P ) P[{r(t′), 0 e t′ e t}] where r(t) ) dΓ/dt, Γ being the protein mass per unit surface area. It is therefore imperative that a theoretical description accurately predict the rate of adsorption over all times. The importance of precise experimental adsorption rate data to the development of such a description cannot be overemphasized. Optical waveguide lightmode spectroscopy (OWLS)5-7 is a particularly sensitive technique for obtaining continuous measurements of the protein mass adsorbed at a waveguide/liquid interface. By focusing on a single guided mode, the data acquisition time can be as low as 2.9 s. It is this combination of frequency and precision that allows us to report adsorption rate data that show clearly delineated kinetic regimes, each characterized by a net adsorption rate controlled by a different mechanism. We proceed to incorporate these mechanisms into a theoretical model that accurately predicts the adsorption rates of two proteins over a range of concentrations. Experimental Section Our OWLS apparatus (Artificial Sensing Instruments, Zurich, Switzerland) consists of a flow cell whose bottom surface is a planar Si1-xTixO2 waveguide (x ≈ 0.2) with a thickness of ca. 200 nm coated onto a glass substrate. A polarized He-Ne laser beam (λ ) 632.8 nm) impinges, from below, on a diffraction grating etched into the waveguide surface. At a resonant angle θ, a guided * Corresponding author at
[email protected]. (1) Haynes, C. A.; Norde, W. Colloids Surf. B 1994, 2, 517. (2) Brash, J. L.; Horbett, T. A. ACS Symp. Ser. 1995, 602, 1. (3) 3.Biopolymers at Interfaces; Malmsten, M., Ed.; Surfactant Science Series; Marcel Dekker: New York, 1998; Vol. 75. (4) Ramsden, J. J. Chem. Soc. Rev. 1995, 24, 73. (5) Tiefenthaler, K.; Lukosz, W. J. Opt. Soc. Am. B 1989, 6, 209. (6) Ramsden, J. J. J. Stat. Phys. 1993, 73, 853. (7) Lukosz, W. Sens. Actuators B 1995, 29, 37.
mode of the transverse electric component of the light is established. Human plasma fibronectin or human serum albumin (Sigma Chemical) dissolved in 10 mM HEPES buffer ([NaCl] ) 100 mM, pH ) 7.4) is introduced continuously into the flow cell (of volume 70 µL) at a rate of 80 µL/min (shear rate ) 7.2 s-1). The adsorbed density is related to changes in the mode’s effective refractive index, NTE ) nair sin θ + lλ/Λ, via5
( )( )
nF2 - nC2 ∂NTE Γ ) ∆NTE nA + nC ∂tF
-1
∂nC ∂c
-1
(1)
where nair is the refractive index of air; l is the diffraction order; Λ is the spacing between grating elements; n and t are the refractive indices and thicknesses, respectively, of the solution (C), waveguide film (F), or adsorbed layer (A); and c is the bulk protein concentration.
Theoretical Section Following previous work,8-12 we treat the adsorbing proteins as hard particles capable of undergoing abrupt, irreversible, post-adsorption alterations in size and shape (representing a change in internal conformation or orientation). The rate of change of the surface density of unaltered (R) and altered (β) proteins can be expressed as
∂ΓR ) kac(z)0) Φ(z)0) - ksΓRΨ ∂t
(2)
∂Γβ ) ksΓRΨ ∂t where ka, kd, and ks are the rates of adsorption, desorption, and transition, respectively; c(z)0) is the protein concentration at the surface; Φ(z)0) is the fraction of the surface available to an incoming protein;13,14 and Ψ is the probability that the transition is not sterically blocked by one or more adsorbed protein molecules. (8) Van Tassel, P. R.; Viot, P.; Tarjus, G.; Talbot, J. J. Chem. Phys. 1994, 101, 7064. (9) Van Tassel, P. R.; Talbot, J.; Tarjus, G.; Viot, P. Phys. Rev. E 1996, 53, 785. (10) Van Tassel, P. R.; Viot, P.; Tarjus, G. J. Chem. Phys. 1997, 106, 761. (11) Van Tassel, P. R.; Guemouri, L.; Ramsden, J. J.; Tarjus, G.; Viot, P.; Talbot, J. J. Colloid Interface Sci. 1998, 207, 317. (12) Brusatori, M. A.; Van Tassel, P. R. J. Colloid Interface Sci. 1999, 219, 333. (13) Widom, B. J. Chem. Phys. 1966, 44, 3888. (14) Schaaf, P.; Talbot, J. Phys. Rev. Lett. 1989, 62, 175.
10.1021/la001734s CCC: $20.00 © 2001 American Chemical Society Published on Web 06/02/2001
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The experimental data presented below suggest that the surface description of eq 2 should be coupled to one of transport to the surface. We begin by considering a thin region of thickness d (on the order of a protein diameter) where protein surface interactions are appreciable. We neglect flow in this region because the relevant Peclet number, Pe ) v(z)d)d2/D, where v(z) is the flow velocity at a height z above the surface and D is the diffusivity, is on the order of 10-4. For z < d, the protein undergoes Brownian motion in the presence of a potential energy field due to the surface and previously deposited proteins.15-24 The flux through a plane in this layer is given by
[
J(z) ) D
]
∂c c(z) ∂u + Φ(z) ∂z ζ ∂z
D[c(d) - c(0)e-u(0)/kT]
∫0
de
u(z)/kT
Φ(z)
(4)
dz
where the dependence on the frictional coefficient is expressed through T and D via ζ ) kT/D (k is the Boltzmann constant and T is the absolute temperature). Protein flux through a barrier such as a grafted polymer layer has been previously treated using a mean-field approach.25,26 We note that, by carefully choosing Φ and u, eq 4 could, in principle, apply to this situation as well. Finally, we consider transport from the flowing solution by convective diffusion. Making the reasonable simplifying assumptions of (i) a linear velocity profile that begins at z ) d, (ii) negligible diffusion along the flow direction, and (iii) a concentration at z ) d that remains negligible until the onset of a quasi-steady-state diffusion-limited regime, we model the transport via
∂c ∂2c ∂c + a(z - d) ) D 2, t g 0, x g 0, z g d ∂t ∂x ∂z c(0,z,t) ) cB
3 c(x,z,t) 3 z - d 1 (z - d) ) cB 2 δ(x,t) - d 2 (δ(x,t) - d)3
(5)
(6)
where generally d , δ. Inserting eq 6 into an integrated form of eq 5, defining the new variable τ ) 4Dt/(45Dx/ 2a)2/3, and writing δ(x,τ) ) (45Dx/2a)1/3/f(τ), one obtains the following equation for f
(3)
where ζ is a frictional coefficient, u(z) is the translationally averaged potential energy, and Φ(z) is the fractional lateral available area (u and Φ possess implicit time dependences). Assuming a steady-state flux through this thin layer, we obtain
J)
shear rate at the surface, and cB is the bulk protein concentration. It is convenient to consider a boundary layer thickness, δ(x,t), above which the protein concentration is constant, and at which the z derivative of the protein concentration vanishes. We approximate c in terms of δ and z as
f - f4 df ) f(0) ) 0 dτ f - 2τ
(7)
whose analytical solution is τ(f) ) [1 - (1 - f 3)2/3]/(2f 2). We note that δ diverges at t ) 0 and at x ) infinity, as it should. We derive the overall rate equations by assuming steady-state concentrations at z ) 0 and z ) d. Specifically, this involves expressing the time derivatives of c(z)0) and c(z)d) in terms of the relevant fluxes (taken from eqs 2, 4, and 6) and then setting these derivatives equal to zero. We also make use of the fact that eq 6 can be written, during the quasi-steady state where c(z)d) begins to increase (very slowly, we assume) from near zero, as [c(x,z) - c(z)d)]/[cB - c(z)d)]. (In other words, one assumes that the steady-state concentration profile of eq 5 is reached while c(z)d) ) 0, after which c(z)d) and c(z)0) increase very slowly.) The resulting overall rate equations are as follows
∂ΓR ) ∂t 2 δ(x,t) 3
D
cB +D
[
∫0
-1
u(z)/kT
eu(0)/kT dz + Φ(z) kaΦ(0)
-
de
kdΓR
kaΦ(0) 2 δ(x,t) + D-1 1 + u(0)/kT 3 D e
u(z)/kT de
∫0
Φ(z)
]
- ksΓRΨ (8)
dz
∂Γβ ) ksΓRΨ ∂t
where x is the distance from the inlet along the direction of the flow (parallel to the surface), z is the vertical distance from the surface (for z > d, c also depends on x), a is the
We note that the three terms in the first denominator of eq 8 can be thought to represent “resistors” in series accounting for bulk diffusion, Brownian motion near the surface, and surface reaction, respectively. Distinct kinetic regimes, characterized by a net adsorption rate controlled by one of these mechanisms, are predicted when one of the terms is larger than the other two.
(15) Kramers, H. A. Physica 1940, 7, 284. (16) Adamczyk, Z.; Dabros, T.; Czarnecki, J.; van de Ven, T. G. M. Adv. Colloid Interface Sci. 1983, 19, 183. (17) Schaaf, P.; De´jardin, P. Colloids Surf. 1987, 24, 239. (18) Schaaf, P.; Johner, A.; Talbot, J. Phys. Rev. Lett. 1991, 66, 1603. (19) Tarjus, G.; Viot, P. Phys. Rev. Lett. 1992, 68, 2354. (20) Semenov, A. N.; Joanny, J. F. J. Phys. II 1995, 5, 859. (21) Wojtaszcyk, P.; Avalos, J. B.; Rubi, J. M. Europhys. Lett. 1997, 40, 299. (22) Cohen Stuart, M. A.; Hoogendam, C. W.; de Keizer, A. J. Phys. Condens. Matter 1997, 9, 7767. (23) Faraudo, J.; Bafaluy, J. Europhys. Lett. 1999, 46, 505. (24) Adamczyk, Z.; Senger, B.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1999, 110, 3118. (25) Sotulovski, J.; Caragnano, M. A.; Szleifer, I. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 9037. (26) Carignano, M. A.; Szleifer, I. Colloids Surf. B 2000, 18, 169.
In Figures 1 and 2, we show measurements of the mass density of protein adsorbed onto the waveguide surface (Γ) as a function of time and of the rate of adsorption (dΓ/dt) as a function of surface density for each protein at different bulk concentrations. We make at least three independent measurements for each curve shown and find the standard deviation of the absolute rate curves to be always less than 2% and that of the derivative curves (in the peak region) to be always less than 10%. Our use of single-mode OWLS allows for rate curves that, for the first time, contain a sufficient number of points and are of sufficient smoothness (even at low surface coverage) to
c(x,d,t) ) 0 c(x,z,0) ) 0
Results
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Figure 1. (a) Mass density of human plasma fibronectin adsorbed to the SiTiO2 waveguide as a function of time and (b) rate of adsorption as a function of adsorbed density as measured using single-mode OWLS. Adsorption data from bulk concentrations of 10 (×), 50 (+), and 100 (O) µg/mL are shown. Also shown are the predictions of eq 8 (continuous line).
Figure 2. (a) Mass density of human serum albumin adsorbed to the SiTiO2 waveguide as a function of time and (b) rate of adsorption as a function of adsorbed density as measured using single-mode OWLS. Adsorption data from bulk concentrations of 10 (×) and 100 (O) mg/mL are shown. Also shown are the predictions of eq 8 (continuous line).
clearly delineate transport- and reaction-limited regimes. [Note that we use the term reaction to mean the formation of attractive bonds between the protein and the surface. These bonds can be physical (as we expect is the case in our experiments) or chemical.] The former occurs initially with diffusion from the flowing solution to the surface being the governing mechanism. This regime continues until a maximum rate of adsorption is reached. The latter occurs over longer times, with adsorption to an increasingly blocked surface being the governing mechanism. The positive curvature seen in this regime is fully consistent with the uniform filling of (surface) space by adsorbing objects27,28 and its previous experimental observation for proteins29 and polyelectrolytes.30 Although the presence of kinetic regimes can be inferred from plots of Γ versus t, as shown here and elsewhere,31,32 their identification and quantification is greatly facilitated by a dΓ/dt versus Γ representation. We also show predictions of eq 8 in Figures 1 and 2. The observed kinetic regimes are well-described by this model, (27) Evans, J. W. Rev. Mod. Phys. 1993, 65, 1281. (28) Talbot, J.; Tarjus, G.; Van Tassel, P. R.; Viot, P. Colloids Surf. A 2000, 165, 287. (29) Ramsden, J. J. Phys. Rev. Lett. 1993, 71, 295. (30) Geffroy, C.; Labeaub, M. P.; Wong, K.; Cabane, B.; Cohen Stuart, M. A. Colloids Surf. A 2000, 172, 47. (31) Cohen Stuart, M. A.; Fleer, G. J. Annu. Rev. Mater. Sci. 1996, 26, 463. (32) Bijsterbosch, H. D.; Cohen Stuart, M. A.; Fleer, G. J. Macromolecules 1998, 31, 9281.
Calonder and Van Tassel
and good quantitative agreement is achieved, especially in the case of the rate data. We determine the model parameters in the following way. The apparent initial rate of adsorption, kae-u(z)0,Γ)0)/kT, and the (assumed linear) change in u with Γ are given by a least-squares fit of a line to the linear region of the reaction-limited regime of the rate curves (this involves expanding Φ and e-u(z)0,Γ)/kT to first order in Γ). kd is determined by separately measuring the rate of desorption during a buffer rinse. Values of D and σR are taken from the literature.33,34 The rate of alteration, ks, and the diameter of the protein in the altered state, σβ, are determined by manually optimizing a leastsquares fit over the entire data range. With exception of cB, the same parameters are used in all of the curves for a given protein. The shear rate is the same for both proteins. The parameters employed are given in Table 1. Discussion We employ a simple two-state picture in which a protein adsorbs reversibly in an initial “R” state and then undergoes a conversion to an irreversibly bound “β” state. These two states represent different internal conformations or spatial orientations and are meant to represent an average of many “microscopic” states of a real protein. Early observations of (i) an increase in the degree of surface contact upon post-adsorption transition,35,36 (ii) the possibility of steric blockage of the transition by neighboring molecules,35,37-40 and (iii) desorption rates dependent upon the degree of transition41-46 have led to this simple twostate picture,47 which, in turn, provides a nice qualitative explanation of experimentally observed features such as partial reversibility and history dependence.10,48,49 Although synthetic polymers also undergo transitions upon adsorption, these transitions are typically of a more continuous nature. A model in which adsorbing particles spread continuously upon adsorption, without steric constraint, has been proposed for these systems.50,51 A novel aspect of this work is the measurement and theoretical description of a transient-diffusion-limited regime characterized by an increasing rate of adsorption from an initial value of zero. In certain flow arrangements, this transient period can be extremely short and, therefore, (33) Hermans, J. In Plasma Fibronectin: Structure and Function; McDonagh, J., Ed.; Marcel Dekker: New York, 1986; Chapter 4. (34) Kurrat, R.; Ramsden, J. J.; Prenosil, J. E. J. Chem. Soc., Faraday Trans. 1994, 90, 587. (35) Morrissey, B. M.; Stromberg, R. R. J. Colloid Interface Sci. 1974, 46, 152. (36) Norde, W.; Lyklema, J. J. Colloid Interface Sci. 1978, 66, 266. (37) Schaaf, P.; De´jardin, P.; Johner, A.; Schmidt, A. Langmuir 1987, 3, 1128. (38) Norde, W. Favier, J. P. Colloids Surf. 1992, 64, 87. (39) Kondo, A. Mihara, J. J. Colloid Interface Sci. 1996, 177, 214. (40) Maste, M. C. L.; Norde, W.; Visser, A. J. W. G. J. Colloid Interface Sci. 1997, 196, 224. (41) Soderquist, M. E.; Walton, A. G. J. Colloid Interface Sci. 1980, 75, 386. (42) Jennisson, H. P. J. Colloid Interface Sci. 1986, 111, 570. (43) Mura-Galelli, M. J.; Voegel, J.-C.; Behr, S.; Bres, E. F.; Schaaf, P. Proc. Nat. Acad. Sci. U.S.A. 1991, 88, 5557. (44) Wahlgren, M.; Arnebrant, T.; Lundstro¨m, I. J. Colloid Interface Sci. 1995, 175, 506. (45) Ball, V.; Bentaleb, A.; Hemmerle, J.; Voegel, J.-C.; Schaaf, P. Langmuir 1996, 12, 1614. (46) Buijs, J.; Hlady, V. J. Colloid Interface Sci. 1997, 190, 171. (47) Lundstro¨m, I. Prog. Colloid Polym. Sci. 1985, 70, 76. (48) Krisdhasima, V.; McGuire, J.; Sproull, R. J. Colloid Interface Sci. 1992, 154, 337. (49) Wahlgren, M.; Arnebrant, T.; Lundstro¨m, I. J. Colloid Interface Sci. 1995, 175, 506. (50) Pefferkorn, E.; Elaissari, A. J. Colloid Interface Sci. 1990, 138, 187. (51) van Eijk, M. C. P.; Cohen Stuart, M. A. Langmuir 1997, 13, 5447.
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Table 1. Parameters Used to Apply the Model Description of Eq 8 to the Experimental Data in Figures 1 and 2 parameter D m σR kae-u(z)0)/kT kd du(z)0)/dΓ ks σβ
description diffusion coefficient protein mass protein diameter initial adsorption rate desorption rate change in potential energy at surface with protein density transition rate protein diameter following transition
determined from
fibronectin value
albumin value
10-7
literature33,34 literature33,34 intercept of extrapolated linear region independent measurement slope of linear region
2.36 × 8.63 × 10-13 µg 13.2 nm 1.39 × 10-4 cm/s 6.0 × 10-3 s-1 1.3 × 10-20 cm4/s2
6.1 × 10-7 cm2/s 1.13 × 10-13 µg 5.0 nm 1.64 × 10-5 cm/s 9.2 × 10-5 s-1 7.0 × 10-23 cm4/s2
fitting parameter fitting parameter
1.35 × 10-3 s-1 1.03 × σR
7.0 × 10-3 s-1 1.05 × σR
literature33,34
Figure 3. Examples of rate curves based on our model given by eq 8 (particle model with transport, continuous line), a Langmuir model (dotted line), and a version of our model without the transport terms given by eq 2 (particle model without transport, dashed line). To make a fair comparison, desorption and spreading are neglected (kd ) ks ) 0).
negligible.16,31 More generally, the duration of this regime might be long enough to merit explicit consideration. Here, we observe a transient period of about 30 s during which 20-30% of the mass is adsorbed. Although rarely accounted for in theoretical approaches (a recent exception is ref 52), it is clear that treatment of this regime is needed to predict our experimental rate curves. In Figure 3, we present a general comparison of our model with a Langmuir model and with a version of our model that neglects transport effects (i.e., eq 2). These other approaches fare poorly at short times, and the Langmuir approach, as noted earlier,29 additionally fails to predict the long-time positive curvature characteristic of protein adsorption systems. Evaluation of eq 8 requires approximate expressions for the functions u(z,FR,Fβ), Φ(z,FR,Fβb), and Ψ(FR,Fβ). Φ|z)0, the probability of finding an empty region on the surface to house a protein in the R state, and Ψ, the conditional probability that a protein in the R state has sufficient space to undergo a transition to the β (altered) state, can be determined using the scaled particle theory53-55 for particles with disk-shaped surface projections, as described previously.12 In principle, the scaled particle theory (52) Fang, F.; Szleifer, I. Biophys. J. 2001, in press. (53) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959, 31, 369. (54) Lebowitz, J. L.; Helfand, E.; Preastgaard, E. J. Chem. Phys. 1965, 43, 774. (55) Talbot, J.; Jin, X.; Wang, N.-H. L. Langmuir 1994, 10, 1663.
cm2/s
can also be used to determine Φ|z>0 for proteins modeled as certain idealized geometric objects. The potential energy, u, can be determined by considering electrostatic and van der Waals contributions.56-58 We find, however, that, for reasonable functional values of u and Φ, the resistance due to Brownian motion near the surface (z > 0) never becomes important in any of the systems reported here. This is consistent with our experimental observation of only two distinct kinetic regimes. A Brownian-motionlimited regime could occur in systems with larger adsorbing particles or higher activation energy barriers. Although not a perfect description, our model predicts both absolute and derivative experimental data to a fairly high degree of accuracy (to an accuracy greater than the standard deviation between successive measurements for short to intermediate times). This is achieved using only two true adjustable parameters, ks and σβ, to fit all of the curves for a given protein (see Table 1). At longer times, the model mildly (but systematically) underestimates the absolute data. We attribute this to a breakdown, at longer protein-surface contact times, of the assumption of protein as hard particle. We are currently investigating the specific events leading to this discrepancy.59 Practical scenarios demand a quantitative understanding of the protein adsorption process over diverse time scales. In biosensing applications, for example, one desires to know the bulk concentration of a protein in terms of its adsorbed amount within seconds after exposure. In biomaterials applications, one desires to know the amount and state of adsorbed protein over days. Theoretical descriptions must account, in a simple yet realistic way, for the important mechanisms that control the adsorption kinetics over these and intervening time scales. An accurate modeling of early events is also important because, as a result of irreversible aspects, they can significantly impact the state of the adsorbed layer at long times. Adsorption rate data that clearly delineate kinetic regimes, like those introduced here, are extremely useful in revealing important mechanisms and verifying subsequent theoretical expressions. Acknowledgment. The authors acknowledge the financial support of the National Science Foundation (CAREER Award CTS-9733310) and the National Institutes of Health (Grant R01-GM59487). LA001734S (56) Roth, C. M.; Lenhoff, A. M. Langmuir 1993, 9, 962. (57) Johnson, C. A.; Wu, P.; Lenhoff, A. M. Langmuir 1994, 10, 3705. (58) Roth, C. M.; Lenhoff, A. M. Langmuir 1995, 11, 3500. (59) Calonder, C.; Tie, Y.; Van Tassel, P. R., manuscript submitted.
