Langmuir 2004, 20, 10599-10603
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Probing Macromolecular Adsorbed Layer Structure and History Dependence via the Interfacial Cavity Function Yanrong Tie, A. Pascal Ngankam, and Paul R. Van Tassel* Department of Chemical Engineering, Yale University, New Haven, Connecticut 06520-8286 Received August 17, 2004. In Final Form: September 6, 2004 Adsorbed layers of proteins and other macromolecules often relax structurally more slowly than they form, rendering layer growth an out-of-equilibrium process. We show here how the interfacial cavity function, Φ (the average Boltzmann factor for a single probe molecule), may be determined, using kinetic data available from optical waveguide lightmode spectroscopy, and used as a continuous, in situ measure of history dependent adsorbed layer structure. The increase of Φ observed with residence time for fibronectin and lysozyme layers suggests post-adsorption clustering on a time scale longer than that predicted by a surface diffusion model.
Introduction Adsorption of synthetic or biological macromoleculess a crucial event in a number of coating and separation processessoccurs due to favorable energetic (e.g., ionic, van der Waals, hydrogen bonding) and/or entropic (e.g., solvent and conformational effects) interactions. The time scale for post-adsorption intra- and intermolecular relaxation is typically much longer than that of the initial adsorption;1-11 thus, individual molecules attaching to the surface often find themselves interacting with a layer of previously adsorbed molecules whose overall structure is away from equilibrium. Quantifying nonequilibrium adsorbed layer structure is therefore very important to understanding and predicting adsorption.12 Statistical mechanical distribution functions are useful in quantifying adsorbed layer structure.13 These include the cavity-molecule joint distribution functions, Φ(m,n)(q b1, ..., b qm, b r1, ...,r bn), defined as the simultaneous probability of finding m adsorbed molecules at {q b1, ..., b qm} and n cavities free of adsorbed molecules at {r b1, ..., b rn}, where vector quantities include all relevant positional, rotational, and conformational information. The average of the one-body molecular distribution function over all b q1 is simply the adsorbed density; its in situ experimental measurement can be routinely made via optical and * Author to whom correspondence should be addressed. Phone: 203-432-8379. E-mail:
[email protected]. (1) Pefferkorn, E.; Jeanchronberg, A. C.; Varoqui, R. Macromolecules 1990, 23, 1735-1741. (2) Frantz, P.; Granick, S. Phys. Rev. Lett. 1991, 66, 899-902. (3) Dijt, J. C.; Stuart, M. A. C.; Fleer, G. J. Macromolecules 1994, 27, 3219-3228. (4) Sukhishvili, S. A.; Dhinojwala, A.; Granick, S. Langmuir 1999, 15, 8474-8482. (5) Norde, W.; Giacomelli, C. E. Macromol. Symp. 1999, 145, 125136. (6) Beverung, C. J.; Radke, C. J.; Blanch, H. W. Biophys. Chem. 1999, 81, 59-80. (7) Mubarekyan, E.; Santore, M. M. Macromolecules 2001, 34, 49784986. (8) Calonder, C.; Tie, Y.; Van Tassel, P. R. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 10664-10669. (9) Wertz, C. F.; Santore, M. M. Langmuir 2002, 18, 706-715. (10) Daly, S. M.; Przybycien, T. M.; Tilton, R. D. Langmuir 2003, 19, 3848-3857. (11) Tie, Y.; Calonder, C.; Van Tassel, P. R. J Colloid Interface Sci. 2003, 268, 1-11. (12) Van Tassel, P. R. In Encyclopedia of Polymer Science and Technology, 3rd ed.; Wiley-Interscience: New York, 2003; Vol. 5, pp 285-305. (13) Tarjus, G.; Schaaf, P.; Talbot, J. J. Stat. Phys. 1991, 63, 167202.
piezoelectric methods. Ex situ measurement of averages over the two-body molecular distribution function are also possible via electron microscopy.14 Distribution functions involving cavities have yet to be experimentally reported, yet they are a promising potential source of information on macromolecular adsorbed layer structural evolution. In this paper, we introduce a means to determine, continuously during adsorbed layer formation, the one-body interfacial cavity function. Our method requires kinetic adsorption and desorption data as input; we obtain these here using optical waveguide lightmode spectroscopy, but other optical methods such as surface plasmon resonance and piezoelectric methods such as quartz crystal vibrational analysis could also be used. Being a measure of the interaction between an adsorbed layer and an incoming molecule, the one-body cavity function is a valuable metric of history-dependent structure useful in quantifying postadsorption events such as aggregation and conformational change. Theory We define the one-body interfacial cavity function as the average Boltzmann factor for a single adsorbate in the vicinity of a surface containing previously adsorbed molecules:
Φ(0,1)(r b) ) 〈e-U(rb|rb1,...,rbN)/kT〉
(1)
In eq 1, k is the Boltzmann constant, T is the absolute temperature, and U is the potential energy of a single macromoleculeswhose positional, rotational, and conformational state is described by the vector b rsinteracting with an adsorbed layer of N molecules whose structure is bN. The brackets indicate an given by the vectors b r1, ...,r average taken over b r1, ...,r bN and weighted by the (generally nonequilibrium) N-molecule probability density. In the case of purely hard-core repulsive interactions, the cavity function is just the probability of finding a cavity in which the probe molecule may be inserted. Φ(0,1)(r b) relates to other common functions via ln Φ(0,1)(r b) ) -c(1)(r b) ) δ(F/kT)/δF(1)(r b) ) µex(r b)/kT where c(1)(r b) is the one-body direct correlation function, F[F(1)(r b)] is the Helmholtz free energy functional, µex(r b) is the local excess chemical potential, and F(1)(r b) is the one-body density distribution.15 In what follows, we focus on the average (14) Alaeddine, S.; Nygren, H. Colloids Surf., B 1995, 5, 227-240.
10.1021/la047944c CCC: $27.50 © 2004 American Chemical Society Published on Web 10/23/2004
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value of the one-body cavity function, defined as Φ ) ∫Φ(0,1)(r b) dr b/∫dr b. At low bulk concentration, the equilibrium surface concentration of nonadsorbed molecules is the product of the one-body cavity function and the bulk concentration. Therefore, the rate of adsorption (in the absence of transport limitations) follows the general relation11
dΓ dt
∑i kd,iΓi
) kacbΦ -
(2)
where Γ is the density of adsorbed macromolecule, t is the time, ka is the adsorption rate constant, cb is the bulk concentration, and kd,i and Γi are, respectively, the desorption rate constant and the density of adsorbed molecules in the i’th state (states represent, e.g., different orientations, conformations, degrees of association, etc.). Thus, Φ may be determined experimentally from accurate measures of the adsorption rate and adsorption/desorption parameters. Since Φ is expected to be analytic in Γ, an estimate of ka may be obtained by extrapolating the linear region of the dΓ/dt versus Γ curve (during adsorption), over values where transport limitations are negligible, to Γ ) 0.11,16 Better accuracy may be achieved by fitting (again, nontransport-limited) data to a nonlinear model, as described further below. The desorption parameters appearing in eq 2 are most directly measured by replacing the protein solution by a pure buffer solution, differing only by the absence of the protein. It is straightforward to show that11
( ) d
( ) dΓ dt
dΓ
)-
∑i k2d,iΓi
(3)
∑i kd,iΓi
cb)0
so linear regions in a dΓ/dt versus Γ plot are approximations of {kd,i} and the intercepts of extrapolated linear regions yield the {Γi}. For very simple systems, the cavity function may be estimated using a liquid state theory. For example, when the adsorbing molecules interact with one another as hard disks and rapidly arrange in an equilibrium structure for a given adsorbed density (either via surface diffusion or successive adsorption and desorption), the cavity function, Φ′, may be approximated using the scaled particle theory (SPT) 17,18 as
Φ′ ) (1 - θ) exp
[
]
-3θ + 2θ2 (1 - θ)2
(4)
where θ ) Γa/m is the fractional coverage and a and m are, respectively, the effective area and mass of a single protein molecule. The SPT may also be used to estimate the cavity function following surface aggregation. Assuming (i) monodisperse clusters of n adsorbed molecules and (ii) clusters to behave as an equilibrium hard disk fluid, the cavity function is the following:19
[
Φ′ ) (1 - θ) exp -
2θ - (2 - n-1/2)θc 1-θ
]
θ - (1 - n-1)θc + n-1(n1/2 - 1)2θc(θ - θc) (1 - θ)2
(5)
In eq 5, θc is the fractional area covered by the clusters. (In a multistep experiment, this would be the coverage following the first desorption step, θc ) θ(1d) ) Γ(1d)a/m.) The SPT is less accurate for highly asymmetric mixtures. In the limit of large n, it is therefore more accurate to express Φ′ as
(
Φ′ ) (1 - θc)Φ′eq5 θnon-cluster )
)
θ - θc 1 - θc
(6)
where θnon-cluster is the coverage due to molecules not residing in clusters. Experiment Optical waveguide lightmode spectroscopy (OWLS) is a highly sensitive method for measuring macromolecular adsorption kinetics at the solid-liquid interface20-23 and can yield data of sufficient precision for analysis in terms of models such as eq 2.16,24 Detection is based on excitation of guided modes via polarized laser light directed upon a grating coupler. Incoupling occurs when N ) nair sin(R) + lλ/Λ, where N is the effective refractive index of the guided mode, nair is the refractive index of air, R is the resonant incoupling angle between the light beam and the waveguiding film, l ()1) is the defraction order, λ ()632.8 nm) is the wavelength of the light, and Λ ()1/2400 mm) is the defraction grating period. Adsorbed density is related to changes in N, as described previously.25 Our OWLS instrument (MicroVacuum, Budapest, Hungary) is composed of a parallel plate flow cell whose bottom surface is an OW 2400 Sensor Chip (MicroVacuum), consisting of a planar Si1-xTixO2 waveguide (x ) 0.25 ( 0.05), of thickness ca. 200 nm and refractive index 1.77 ( 0.03, coated onto a glass substrate. The flow cell/ sensor chip assembly rests on the rotating head of a precision goniometer.
Results and Discussion In Figure 1a, we show OWLS measurements of lysozyme adsorption from a 0.1 mg/mL solution in 10 mM HEPES buffer ([NaCl] ) 100 mM, pH ) 7.4) flowing at a surface shear rate of 2 s-1. When a pure buffer solution replaces the protein solution, desorption occurs. We show desorption curves beginning at several initial adsorbed densities. (For clarity, we show just a single initial adsorption curve. Others differ by no more than a few percent from this curve.) In two cases, we show measurements of adsorption during a reapplication of the lysozyme solution. We show the shifted traces of these two curves (dashed lines) and find one of them to lie significantly above the corresponding initial step curve, indicating a change in adsorbed layer structure, at a given density, between first and second step adsorption. In Figure 1b, we show the rate of adsorption determined by taking the numerical derivative of the first and second step adsorption branches shown in Figure 1a. The initial, monotonically increasing part of (15) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (16) Calonder, C.; Van Tassel, P. R. Langmuir 2001, 17, 4392-4395. (17) Reiss, H.; Frisch, H. L.; Liebowitz, J. L. J. Chem. Phys. 1959, 31, 369-380. (18) Lebowitz, J. L.; Helfand, E.; Praestgaard, E. J. Chem. Phys. 1965, 43, 774-779. (19) Brusatori, M. A.; Van Tassel, P. R. J. Colloid Interface Sci. 1999, 219, 333-338. (20) Nellen, P. M.; Tiefenthaler, K.; Lukosz, W. Sens. Actuators 1988, 15, 285-295. (21) Lukosz, W.; Clerc, D.; Nellen, P. M.; Stamm, C.; Weiss, P. Biosens. Bioelectron. 1991, 6, 227-232. (22) Ramsden, J. J. J. Stat. Phys. 1993, 73, 853-877. (23) Voros, J.; Ramsden, J. J.; Csucs, G.; Szendro, I.; De Paul, S. M.; Textor, M.; Spencer, N. D. Biomaterials 2002, 23, 3699-3710. (24) Ramsden, J. J. Phys. Rev. Lett. 1993, 71, 295-298. (25) Tiefenthaler, K.; Lukosz, W. J. Opt. Soc. Am. B 1989, 6, 209220.
Probing Adsorbed Layer Structure and History Dependence
Figure 1. (a) The adsorbed density versus time for lysozyme in HEPES buffer onto Si(Ti)O2 during multistep experiments. A single initial adsorption branch, beginning from Γ ) 0, is shown. Several desorption branches, begun at various times and measured during buffer rinses, are also shown. In two experiments, a second adsorption step is shown. (Traces of these curves are reproduced and compared to the uninterrupted curve.) In one of these cases, first- and second-step adsorption differ significantly, suggesting a structural transition on a time scale of 25 min. (b) The rate of adsorption versus adsorbed density for first- and second-step adsorption. (c) The rate of desorption versus adsorbed density. The desorption rate constants and partial densities (see eq 1) may be determined from the slopes and intercepts of the linear regions, respectively, as approximated by the solid lines.
each curve represents a transport-limited adsorption regime and the subsequent, monotonically decreasing part represents a surface-limited adsorption regime. Clearly, only the latter should be used to calculate Φ. In Figure 1c, we show the numerical derivative of the desorption branches (i.e., during the buffer rinse) of data from Figure 1a. An initial transport limited regime is apparent (reading from right to left), followed by two distinct desorption regimes. On the basis of eq 3, we may attribute these two quasi-linear portions (as approximated by the solid lines in Figure 1c) to distinct adsorbed states with differing kd,i. Adsorption in three distinct states, one of them irreversible (i.e., with kd,i ) 0), therefore represents a minimal description of the lysozyme system. In Figure 2, we show Φ′ ) Φ/Φ(Γ ) 0) for lysozyme and fibronectin adsorption from 0.1 mg/mL HEPES buffer
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Figure 2. The cavity functions for (a) lysozyme and (b) fibronectin versus adsorbed density during first and second step adsorption. The differences between initial and subsequent adsorption are measures of the dependence of the adsorbed layers on their histories. Also shown are theoretical predictions using eqs 4, 5, and 6. In part (b), a prediction of second-step adsorption is made in which the coverage of clustered proteins has been increased by a factor 1.9 to account for conformational change.
solutions, at shear rates of 2 s-1, onto Si(Ti)O2 as calculated using eq 2 with surface-limited adsorption rate data. Values of {kd,i} and {Γi}, as functions of density from which desorption begins, are calculated via eq 3 from plots such as those in Figure 1c.11 For other adsorbed densities, these parameters are determined via simple interpolation. We find Φ′ to decrease with Γ, as one would expect, for both proteins. Interestingly, in neither case do the curves approach zero and in the case of fibronectin, an increase is observed for Γ > 0.25 µg/cm2. We also show Φ′ for a second adsorption step, i.e., during reapplication of the protein solution following a buffer rinse. For both proteins, the cavity function is much larger (at a given density) during the second adsorption step, indicating an adsorbed layer more attractive to incoming molecules. We discuss below the cause of this history dependence. In Figure 2, we compare cavity functions predicted for hard disks within the scaled particle theory (eq 4) to those determined experimentally from eq 2. (Predictions based on the random sequential adsorption model could also be used.26,27 Over the density range considered here, these two approaches yield very similar results.) Specifically, we perform a regression over the first few data points yielding optimal values of ka and a/m. These values are as follows: lysozyme, ka ) 1.15 × 10-4 cm/s, a/m ) 2.75 cm2/µg; fibronectin, ka ) 9.5 × 10-5 cm/s, a/m ) 1.6 (26) Schaaf, P.; Talbot, J. J. Chem. Phys. 1989, 91, 4401-4409. (27) Schaaf, P.; Talbot, J. Phys. Rev. Lett. 1989, 62, 175-178.
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cm2/µg. Clearly, the SPT/disk approximation works well for the entire initial adsorption step for lysozyme and for the initial step for fibronectin up to a density of about 0.2 µg/cm2. Lysozyme is a small, rigid, and nearly spherical protein; it is therefore not surprising that the SPT/disk model yields an accurate cavity function. Fibronectin is a large, conformationally labile, anisotropic protein; the assumptions of eq 4 begin to break down for this system at higher density. Noting that Φ′, but not dΓ/dt, increases with fibronectin density beyond Γ ) 0.22 µg/cm2, we conclude this nonmonotonic behavior to be due to the growing presence of numerous small pockets, bounded by previously attached molecules, into which adsorption may take place and from which desorption is relatively fast. Structural rearrangements occurring over long times increase the density of these pockets. It is important to note that a treatment of the fibronectin system accounting for shape anisotropy and energetic heterogeneity would not generally predict an increase in Φ′ with density. Its presence here suggests a given molecule’s tendency to desorb to be sensitive to its immediate environment and post-adsorption structural rearrangements to yield pockets of highly reversible adsorption. Slow-to-relax adsorbed layers engender a strong dependence on system history.1-5,7-11 It is therefore interesting to compare cavity functions for systems of identical composition with different histories. We show in Figure 2 the cavity functions measured along a second adsorption step. These are systematically higher than those of the initial step. Three possible explanations for this increase are substrate heterogeneity, biospecificity, and surface aggregation. We eliminate the first two possibilities by noting the nearly identical slopes of the rate versus density curve (e.g., Figure 1b) during first and second step adsorption and the similar increase in Φ′ when a different protein is used during the second step (data not shown). Aggregation among adsorbed molecules is thus the most likely explanation, and the cavity function values reported here provide important structural information on this event. If one assumes second-step adsorption to begin onto a surface filled with monodisperse clusters of n molecules, the SPT/disk approximation may again be used to estimate the cavity function (via eqs 5 and 6). In Figure 2, we show these estimates for several cluster sizes along with experimental values determined from eq 2. The behavior of the lysozyme system is captured fairly well in this treatment, assuming a cluster size of 70. For the fibronectin system, the initial slope of Φ′ is greater (in magnitude) than predicted. In fact, this slope is closer to that predicted for n ) ∞. This indicates the fractional area covered by the clusters, θc, is likely greater than the coverage following the desorption step, θ(1d). In turn, this indicates a degree of conformational spreading accompanying cluster formation. We find eq 6 with θc/θ(1d) ) 1.9 to give a good prediction of the measured second step Φ′ for the fibronectin system, indicating an average degree of spreading of 90%. Although a more detailed model description could be obtained by computer simulation,28 the good agreement with experiment observed here suggests the basic physics of aggregation to be well captured within this coarse-grained analysis. Finally, the time scale over which the history dependence occurs may be determined by systematically varying the time between the beginning of the first and second adsorption steps. As we show in Figure 1a, the adsorption curve differs beyond experimental uncertainty (here
Tie et al.
assumed to be 3%) after a 25 min, but not a 21 min, period in the lysozyme system. A similar experiment on the fibronectin system shows an altered adsorption curve after a 60 min, but not a 55 min, period. We interpret the shortest time at which history dependence occurs as a characteristic time of surface restructuring, in this case, due to surface aggregation. If surface diffusion were rate limiting, one might expect a time of order 100A/Ds, where A is the projected area of a single adsorbed protein molecule and Ds is its surface diffusivity. Using Ds ) 10-10 cm2/s (a lower bound based on previously reported values 29,30), the diffusion-limited time scales are roughly 0.065 and 1.4 s for lysozyme and fibronectin, respectively. Clearly, the aggregation we observe here is not diffusion limited and, based on the results of Figure 2, likely requires an accompanying conformational transition. We pause for a moment to discuss some limitations of eq 2, the starting point in our determination of the onebody cavity function. Again, eq 2 is applicable only in the absence of transport limitations. In plots such as Figure 1b, we clearly observe a crossover from transport-limited adsorption (characterized by a monotonically increasing adsorption rate) to surface-limited adsorption (characterized by a monotonically decreasing adsorption rate). Using a boundary layer treatment of convective-diffusion, the time to reach a steady flux to a rapidly adsorbing surface is t ) (45Dx/2a)2/3/8D, where D is the diffusivity, x is the distance from the inlet at which adsorption is measured, and a is the shear rate.16 For the systems investigated here, t ≈ 60 s. Since no steady-state transport-limited regime is observed (this would appear as a flat line on the rate versus density curve of Figure 1b), this represents an upper bound to the time required to reach surface-limited adsorption, after which meaningful measurements of the cavity function may be made. A second limitation concerns inaccessible cavities, i.e., those regions near the surface where energetic interactions are favorable but where accessibility is limited by large energy barriers. A cavity function determined via eq 2 using experimental data generally neglects the contribution of inaccessible cavities, whereas one determined using the definition of eq 1 would include their contribution. This difference can be reconciled if one assumes the average in eq 1 to be constrained to accessible regions. A third limitation to eq 2 is an assumption of adsorption (desorption) that is first (zeroth) order in bulk concentration. This assumption becomes false in cases of cooperative adsorption/desorption, and while adherence to eq 2 would still yield an accurate cavity function, the resulting rate constants would become concentration-dependent. We present here an experimental determination of the interfacial cavity function, Φ, from kinetic adsorption data. The kinetic parameters ka, {kd,i}, and {Γi}, as defined in eq 2, are required for this analysis and we provide prescriptions for their determination here and elsewhere.11 To most consistently compare our results to predictions from the SPT/disk model, we treat ka (along with protein area per mass) as a fitting parameter in this work. This is not to imply that the experimental determination of Φ is in any way model dependent. Indeed, ka may also be estimated by extrapolating the linear, surface limited adsorption rate, as shown in Figure 1b, to zero density. (28) Pugnaloni, L. A.; Dickinson, E.; Ettelaie, R.; Mackie, A. R.; Wilde, P. J. Adv. Colloid Interface Sci. 2004, 107, 27-49. (29) Tilton, R. D.; Gast, A. P.; Robertson, C. R. Biophys. J. 1990, 58, 1321-1326. (30) Tilton, R. D.; Robertson, C. R.; Gast, A. P. J. Colloid Interface Sci. 1990, 137, 192-203.
Probing Adsorbed Layer Structure and History Dependence
Conclusion The one-body cavity function, Φ′, is a sensitive measure of adsorbed layer structure. We show here how Φ′ may be determined via kinetic measurements and use it to understand adsorbed layer structural evolution and history dependence.
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Acknowledgment. We gratefully acknowledge the financial support of the National Science Foundation (CTS-9733310) and the National Institutes of Health (R01-EB00258). LA047944C
