pubs.acs.org/Langmuir © 2010 American Chemical Society
Kinetics of Fibrinogen Adsorption on Hydrophilic Substrates Zbigniew Adamczyk,† Jakub Barbasz,*,†,‡ and Michaz Ciesla‡ †
Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, 30-239 Krak ow, Niezapominajek 8, Poland, and ‡M. Smoluchowski Institute of Physics, Jagiellonian University, 30-059 Krak ow, Reymonta 4, Poland Received March 30, 2010. Revised Manuscript Received May 19, 2010
Irreversible side-on adsorption of fibrinogen, modeled as a linear chain of touching beads of various size, was studied theoretically using the random sequential adsorption (RSA) model. Numerical simulation of the Monte Carlo type enabled one to determine the dependence of the surface blocking function (available surface function) on the protein coverage. These numerical results were interpolated using analytical functions based on a polynomial expansion. The dependence of the jamming coverage on the size of the simulation area was also determined. By an extrapolation of these results to the infinite area size, the maximum surface concentration of fibrinogen for the side-on adsorption was determined to be 2.26 103 μm-2. This corresponds to a jamming coverage θ¥ of 0.29. It was shown that the blocking function can well be approximated in the limit of high coverage by the dependence C(θ¥ - θ)4. Using this interpolating expression, the kinetics of fibrinogen adsorption under convection and diffusion transport conditions were evaluated for various bulk concentrations of the protein. These kinetic curves were derived by numerically solving the mass transport equation in the bulk with the blocking function used as a nonlinear boundary condition at the interface. It was shown that our theoretical results are in agreement with experimental kinetic data obtained by AFM, ellipsometry, and other techniques for hydrophilic surfaces in the limit of low bulk fibrinogen concentration.
1. Introduction Understanding adsorption phenomena of bioparticles at solid/ liquid interfaces is of major significance for a variety of fields such as geophysics, material and food sciences, pharmaceuticals, cosmetics, and medical sciences. Especially significant are protein adsorption processes involved in blood coagulation, artificial organ failure, plaque formation, fouling of contact lenses and heat exchangers, and ultrafiltration or membrane filtration units. However, controlled protein deposition on various surfaces is a prerequisite to efficient separation and purification by chromatography and filtration for biosensing, bioreactors, and immunological assays. A quantitative theoretical analysis of bioparticle adsorption is complicated by the complex shapes of bioparticles, which are often characterized by significant elongation. Examples of such anisotropic bioparticles are straight or ring-shaped DNA,1-4 many proteins such as fibrinogen and fibrin,5 human serum albumin (HAS),6 bacteriophages, and viruses.7,8 Also, colloid and aerosol particle aggregates can appear in elongated shapes, such as fractal9-11 or linear structures,12 forming under the action of electric or magnetic fields.13,14 *Corresponding author. E-mail:
[email protected]. (1) Fiers, W.; Sinsheimer, R. L. J. Mol. Biol. 1962, 5, 424. (2) Vinograd, J.; Lebowitz, J. J. Gen. Physiol. 1966, 49, 103. (3) Kovaevic, R. T.; van Holde, K. E. Biochemistry 1977, 16, 1490. (4) Mansfield, M. L.; Douglas, J. F. Macromolecules 2008, 41, 5412. (5) Doolittle, R. F Annu. Rev. Biochem. 1984, 53, 195. (6) Haynes, C. A.; Norde., W. Colloids Surf., B 1994, 2, 517. (7) Bloomfield, V.; Dalton, W. O.; van Holde, K. E. Biopolymers 1967, 5, 135. (8) Carrasco, B.; Garcia de la Torre, J. Biophys. J. 1999, 75, 3044. (9) Lattuda, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 96. (10) Filipov, A. V. J. Colloid Interface Sci. 2000, 229, 84. (11) Binder, Ch.; Ch. Feichtinger, Ch.; Schmid, H.-J.; Thuerey, N.; Peukert, W.; Ruede, U. J. Colloid Interface Sci. 2006, 301, 155. (12) Geller, A. S.; Mondy, L. A.; Rader, D. J.; Ingber, M. S. J. Aerosol Sci. 1993, 24, 597. (13) Fermigier, M.; Gast, A. P. J. Colloid Interface Sci. 1992, 154, 522. (14) Vuppu, A. K.; Garcia, A. A.; Hayes, M. A. Langmuir 2003, 19, 8646.
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Despite considerable practical significance of anisotropic particle adsorption, theoretical results reported in the literature were obtained solely for the side-on adsorption of particles having regular (convex) shapes such as spheres,15,16 rectangles,17 ellipses, rectangles, discorectangles,18-20 and needles.21 A 3D (unoriented) adsorption regime of prolate and oblate spheroids was analyzed in ref 22. The kinetics of particle adsorption, the structure of monolayers formed on solid substrates, and the jamming coverage were determined in this work using the random sequential adsorption (RSA) model. RSA is a stochastic process in which objects (particles) are placed consecutively on a surface in such a way that they do not overlap with any previously adsorbed particles. They can adsorb only upon contacting an uncovered surface area of the interface (which, for the sake of convenience, is referred to later as the collector). Upon making the contact, the positions and orientations of particles remain unchanged in time, which corresponds to the conditions of localized and irreversible adsorption. Because adsorbed particles cannot come into contact with each other, their adsorption is completed after forming a monolayer, when there is no uncovered collector area available for particles. The coverage attained in this limit is called the jamming coverage and represents the most relevant parameter to be determined in RSA simulations. Most RSA calculations were performed for continuous surfaces,15-24 but there are also extensive results for lattice models25 where multilayer adsorption can occur. (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) 287.
Hinrichsen, E. L.; Feder, J.; Jossang, T. J. Stat. Phys. 1986, 44, 793. Schaaf, P.; Talbot, J. J. Chem. Phys. 1989, 91, 4401. Vigil, R. D.; Ziff, R. M. J. Chem. Phys. 1989, 91, 2599. Viot, P.; Tarjus, G.; Ricci, S. M.; Talbot, J. J. Chem. Phys. 1992, 97, 5212. Ricci, S. M.; Talbot, J.; Tarjus; Viot, P. J. Chem. Phys. 1992, 97, 5219. Ricci, S. M.; Talbot, J.; Tarjus; Viot, P. J. Chem. Phys. 1994, 101, 9164. Viot, P.; Tarjus, G.; Ricci, S. M.; Talbot, J. Physica A 1992, 191, 248. Adamczyk, Z.; Weronski, P. J. Chem. Phys. 1996, 105, 5562. Senger, B.; Voegel, J. C.; Schaaf, P. Colloids Surf., A 2000, 165, 255. Talbot, J.; Tarjus, G.; van Tassel, P. R.; Viot, P. Colloids Surf., A 2000, 165,
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It is interesting to mention that these RSA calculations have been performed for convex particle shapes for which the thermodynamic (equilibrium) results obtained from Boublik’s theory26 are applicable in the limit of low coverage. To our knowledge, there are no theoretical results reported in the literature for more complicated particle shapes, especially for concave particles. Such results would be of considerable practical interest because the shapes of proteins and bioparticles can be well reflected in terms of the bead model, which leads to inherently concave particles. This approach was originally formulated by Kirkwood and Riseman27 to calculate hydrodynamic resistance tensors of anisotropic particles. According to this model, the real shape of a molecule is replaced by an array of monodisperse spheres. Later on, this model was extended to nonidentical (polydisperse) spheres28 and extensively used to calculate hydrodynamic properties of proteins, viruses, other bioparticles,29-31 and cyclic polymers.32,33 Recently, this bead model was used to calculate hydrodynamic resistance tensors and diffusion coefficients of elongated macromolecules, forming linear chains, circles, and other shapes, and to determine the flow-past monolayers of particles distributed randomly over interfaces.34 Therefore, the goal of this article was to derive, using the continuous-surface RSA model, the available surface function (often called the surface blocking function) and the jamming limit in 2D for side-on adsorption for a rigid, concave particle modeled as a linear chain of touching spheres. The results obtained for this object will be exploited for a quantitative prediction of the adsorption kinetics of fibrinogen on solid substrates, which is a protein that plays a fundamental role in blood clotting.5 Because of its essential biological significance, fibrinogen conformations were extensively studied over the decades starting from the pioneering work of Hall and Slayter35 (HS). From the electron microscope micrographs of fibrinogen deposited on mica, the size and dimensions of the molecule were established. According to these authors, the fibrinogen molecule has a colinear, trinodular shape with a total length of 47.5 nm. The two equal end domains are spherical in shape and have a diameter of 6.5 nm, and the middle domain has a diameter of 5 nm. These domains were connected by cylindrical rods with a diameter of 1.5 nm. Numerous studies have been carried out since then with the purpose of refining the HS model of fibrinogen using electron microscopy36-38 and atomic force microscopy (AFM).39-42 For example, Fowler and Erickson,36 using more refined shadowing methods, confirmed the HS model of fibrinogen, showing that its length is 45 ( 2.5 nm. (25) Evans, J. W. Rev. Mol. Phys. 1993, 65, 1281. (26) Boublik, T. Mol. Phys. 1975, 29, 421. (27) Kirkwood, J. G.; Riseman, J. J. Chem. Phys. 1948, 16, 565. (28) Riseman, J.; Kirkwood, J. G. J. Chem. Phys. 1950, 18, 512. (29) Bloomfield, V.; van Holde, K. E.; Dalton, W. O. Biopolymers 1967, 5, 149. (30) Harding, S. E. Biophys. Chem. 1995, 55, 69. (31) Harding, S.E. Protein Hydrodynamics. In Protein Structure: A Comprehensive Treatise; Allen, G., Ed.; JAI Press: 1999; Vol. 2, pp 271-305. (32) Garcia Bernal, J. M.; Garcia de la Torre, J. Biopolymers 1980, 19, 751. (33) Garcia de la Torre, J.; Rodes, V. J. Chem. Phys. 1983, 79, 2454. (34) Adamczyk, Z.; Sadlej, K.; Wajnryb, E.; Nattich, M.; Ekiel-Je_zewska, M. L., Bzawzdziewicz, J. Adv. Colloid Interface Sci., 2010, 153, 1-29. (35) Hall, C. E.; Slayter, H. S. J. Biophys. Biochem. Cytol. 1959, 5, 11. (36) Fowler, W. E.; Erickson, H. P. J. Mol. Biol. 1979, 134, 241. (37) Veklich, Y. I.; Gorkun, O. V.; Medved, L. V.; Nieuwenhuizen, W.; Weisel, J. W. J. Biol. Chem. 1993, 268, 13577. (38) Weisel, J. W.; Philips, G. N.; Cohen, C. Nature 1981, 289, 263. (39) Ortega-Vinuesa, J. L.; Tengvall, P.; I. Lundstrom, I. Thin Solid Films 1998, 324, 257. (40) Sit, P. S.; Marchant, R. E. Thrimb Haemost. 1999, 82, 1053. (41) Marchin, K. L.; Berrie., C. L. Langmuir 2003, 19, 9883. (42) Toscano, A.; Santore, M. M. Langmuir 2006, 22, 2588.
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Weisel et al.38 proposed a more detailed model, derived from electron micrographs of fibrinogen crystals, in which the end domains have been replaced by an array of three subdomains of various sizes. A few degrees of bending of the molecule was also suggested. It should be noted, however, that these observations have been carried out under high vacuum or dry conditions, where the fibrinogen molecule was likely to be significantly dehydrated. Also, the fine details of its conformation could not be resolved, such as the presence of the side arms of the fibrinogen molecule, the so-called C domains.5 A similar conformation of fibrinogen dimensions has been confirmed in AFM studies of fibrinogen molecules on various solid substrates.39-42 It seems that the original HS model, because of its simplicity, is the most suitable for numerical simulations. It reflects well the basic geometrical features of the molecule, which could affect the jamming coverage and the ASF function. Therefore, in this work we used a slightly refined version of the HS model. To increase the calculation efficiency, especially by performing overlapping tests, the cylindrical rod domains was replaced by arrays of touching beads of various diameters. Also, the molecular length was increased slightly to account for the hydration effects. It should be noted that besides the practical significance of predicting fibrinogen adsorption, our calculations have interesting repercussions in basic science because they represent the first attempt to obtain quantitative data for concave particles.
2. Simulation Procedure The model of the fibrinogen used in our calculations is shown schematically in Figure 1. As can be seen, the real shape of the molecule was replaced by a string of 23 colinear touching spheres of various diameters. The two external spheres have a diameter of 6.7 nm, and the central sphere has a diameter of 5.3 nm. All remaining 20 spheres have a diameter of 1.5 nm. The overall length of the molecule is 48.7 nm, and its cross-sectional area Sg equals 128 nm2 (Table 1). Simulations of the side-on adsorption of such molecules were carried out for interfaces having a square shape with a size of Lc Lc, which varied between 5 103 - 5 104 nm. At the periphery of the collector, the no-penetration boundary conditions have been applied (i.e., the molecule could adsorb only if its entire crosssection was within the simulation area). This type of boundary condition can well mimic the widely studied, patterned surface case.43-45 Numerical simulations have been carried out according to the continuous, random sequential adsorption (RSA) model used previously for modeling the irreversible adsorption of spherical15,16,23,24 and ansisotropic particles.17-22 The main features of the numerical procedure used in our work are the following: (i)
A virtual particle (a model fibrinogen molecule) was created by choosing at random its position and orientation (measured between its symmetry axis and the collector boundary) over the simulation area. (ii) The overlapping test was carried out for the nearest neighbors of the virtual particle by determining (43) Adamczyk, Z.; Barbasz, J.; Nattich, M. Langmuir 2008, 24, 1756. (44) Maury, P. A.; Reinhouldt, D. N.; Huskens, J. Curr. Opin. Colloid Interface Sci. 2008, 13, 74. (45) Adamczyk, Z.; Nattich, M.; Barbasz, J. Adv. Colloid Interface Sci. 2009, 2, 147-148.
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Figure 1. Schematic representation of the bead model of fibrinogen. Table 1. Calculated Constants for Fibrinogen and a Spherocylinder of the Same Length
whether the surface-to-surface distance between any of the spheres of different particles was not smaller than zero. The efficiency of the overlapping test was increased using a subsidiary matrix containing information on previously adsorbed particles in the vicinity of the virtual one. (iii) If there was no overlap, then the virtual particle was irreversibly adsorbed. Its position and orientation remained unchanged during the course of calculations, which reflected the postulate of irreversible and localized adsorption. (iv) If there was an overlap, a new adsorption attempt was made that was uncorrelated with previous attempts. The surface concentration of fibrinogen monolayers generated in these simulations was calculated as N ¼
Np Sc
ð1Þ
where N p is the number of molecules and Sc = Lc2 is the collector area. The dimensionless coverage of fibrinogen was calculated as Θ ¼ Sg N
ð2Þ
The ASF (blocking) function was determined for a fixed fibrinogen coverage from the constitutive dependence ASFðΘÞ ¼
Nsucc Natt
ð3Þ
3. Results and Discussion 3.1. Calculations of the ASF Function. The principal aim of calculations performed in this work was to determine the ASF function of fibrinogen for the entire range of coverage. This was realized, by generating according to the above RSA procedure, fibrinogen molecule populations characterized by the desired coverage Θ. Snapshots of monolayers derived in theses simulations are shown in Figure 2 for various fibrinogen coverages, equal to 0.05 (a), 0.10 (b), and 0.29 (c) (collector size 500 500 nm2). For each monolayer, the ASF function was determined according to the procedure described above. Averages from 10 collectors were taken, typically having a size of 5000 per 5000 nm for the lower coverage range, Θ < 0.05. For higher coverage, the size of the collector was typically 2000 per 2000 nm. It was determined from these calculations that for the lower coverage range the ASF function could be well approximated by the polynomial expansion ASF ¼ 1 - C1 Θ þ C2 Θ2 þ 0ðΘÞ3
where C1 = 16.3, and C2 = 82. As can be seen in Figure 3, the expansion given in eq 5 well reflects the exact ASF function derived from simulations for Θ < 0.05 (for ASF < 0.3). These expansion coefficients have major significance because they are directly related to the second (B2) and third (B3) virial coefficients of the equilibrium (reversible) fibrinogen monolayer via the following equations:19,46 1 B2 ¼ C1 ¼ 8:15 2
Natt f ¥ where Nsucc is the number of successive attempts to adsorb the fibrinogen molecule over the interface precovered to the degree of Θ and Natt is the overall number of attempts. The dimensionless adsorption time in an individual adsorption run was defined as
ð5Þ
ð6Þ
1 2 B3 ¼ C1 2 - C2 ¼ 33:9 3 3
ð4Þ
For the sake of convenience, these constants are collected in Table 1. When the virial coefficients are known, the 2D pressure Π and the chemical potential of fibrinogen μ can be expressed for the
where Natt is the total number of adsorption attempts in the run.
(46) Adamczyk, Z. Particles at Interfaces: Interactions, Deposition, Structure; Elsevier/Academic Press: Amsterdam, 2006.
τ ¼
Natt Sg Sc
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In this way, one obtains B2 ¼ 3:77 B3 ¼ 6:56
ð10Þ
Consequently, in the case of the spherocylinder, the constants in the low-coverage expansion of ASF are C1 = 7.54 and C2 = 18.6. As can be noticed, these constants are much smaller than for fibrinogen, which suggests that replacing the real shape of fibrinogen by the spherocylinder is not adequate. The determination of the ASF for the higher coverage range is more tedious because for long simulation times the available surface area exists in the form of isolated targets with very small surface areas.15,23,24 Moreover, in the case of anisotropic particles such as fibrinogen, these targets are selective because a specific orientation of the adsorbing particles is required to fit the available collector areas.24 These aspects have been discussed in detail by Viot et al.18 It was shown by these authors that in the case of the side-on adsorption of axis-symmetric particles the rate of filling such selective targets is proportional to h4, where h is the characteristic linear dimension of the target.18 This allowed them to specify the asymptotic equation for the kinetics of particle adsorption in the limit of large times Θ ¼ Θ¥ - C¥ τ - 1=3
Figure 2. Snapshots of fibrinogen monolayers derived from RSA for various coverage θ: (a) θ = 0.05,(b) θ = 0.10, and (c) θ = 0.29 (jamming).
lower-coverage range via the series expansions46 Π ¼
kT ½Θ þ B2 Θ2 þ B3 Θ3 þ 0ðΘ4 Þ::: Sg
ð7Þ
3 μ ¼ μo þ kT ln Θ þ kT½2B2 Θ þ B3 Θ3 þ 0ðΘ3 Þ::: 2 where k is the Boltzmann constant, T is the absolute temperature, and μo is the reference potential. It is interesting to compare these results with analogous data for a spherocylinder having the same length and diameter (48.7 and 6.7 nm, respectively), where the aspect ratio parameter equals 7.27. In this case, the second virial coefficient is given by the exact analytical formula derived by Boublik26 C1 ¼ 2B2 ¼ 1 þ γp
ð8Þ
where γp = P2/4π Sg is the shape parameter and P is the perimeter of the particle measured along its geometrical cross-section. On the other hand, the C2 constant and the third virial coefficient for a spherocylinder can be calculated from the scaling particle theory (SPT),26 which predicts the approximate equation C2 ¼
1 þ γp þ 2γp 2 2
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ð9Þ
ð11Þ
where Θ¥ is the true jamming coverage and C¥ is the constant, which can be determined from the RSA simulations. Therefore, eq 11 can be used for the efficient extrapolation of results obtained for long but finite simulation times to infinite times, which are impractical to attain. It has been confirmed in our simulations that eq 11 approximates the numerical results well for τ -1/3 < 0.1 (τ > 103) and various collector sizes. An example of such a kinetic run in the case of the 1000 per 1000 nm collector is shown in Figure 4. As can be seen, the dependence of the fibrinogen coverage θ on τ -1/3 can be well approximated by a straight line with a slope of C¥ = 0.332. The extrapolation of this line to τ -1/3 = 0 gives 0.30 for the jamming coverage of fibrinogen for this collector size. It is also interesting that the relative error connected with this extrapolation is well below 1%. In a similar way, the jamming coverage for other collector sizes was determined. The results are shown in Figure 5 in the form of the dependence of the extrapolated jamming coverage of fibrinogen Θ0 ¥ on the d/Lc parameter (where d is the fibrinogen length). As can be seen, this relationship can be well fit by a straight line with a slope of 1.087 and a zero value of 0.29. Therefore, this can be treated as the true jamming coverage for the side-on adsorption of fibrinogen on a collector of infinite dimensions. By knowing the jamming limit, one can easily calculate the surface concentration of fibrinogen (i.e., the number of molecules per unit area), which is a quantity that is convenient for the interpretation of experimental results. This is so, because, contrary to the coverage, the surface concentration is independent of the choice of the cross-sectional area of the molecule. The measured concentration can also be determined directly by atomic force microscopy (AFM) or by electron microscopy. The jamming surface concentration N¥ is given by N¥ ¼
Θ¥ Sg
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Figure 3. Dependence of the ASF function on the fibrinogen coverage in the low-coverage limit. The solid points denote the exact results derived from numerical simulations, the dashed line shows the analytical results derived from the first-order expansion, and the dashed-dotted line shows the results derived from the second-order expansion (eq 5).
Figure 4. Dependence of the fibrinogen coverage θ on τ -1/3 (high-coverage limit, collector size 1000 1000 nm2). The solid points denote results derived from numerical simulations, and the solid line represents the linear fitting function (i.e., θ = θ¥- C¥τ-1/3).
Using the above value of Θ¥, one obtains for fibrinogen N¥ = 2.27 103 μm-2. In the case of the spherocylinder, N¥ = 1.61 103 μm-2.18 As can be noticed, the number of fibrinogen molecules per unit area under the jamming state is significantly larger (by 40%) than in the case of the spherocylinder. This is a direct consequence of the concave shape of fibrinogen molecules, enabling an apparent penetration of two molecules into each other. Once the jamming coverage is determined, one can use eq 11 to formulate the limiting expression for the ASF function for the higher-coverage range. This can be most directly achieved by noting that the ASF function is connected to the rate of particle adsorption through the constitutive dependence23,24 dΘ ð13Þ ASF ¼ dτ 11938 DOI: 10.1021/la101261f
Using eq 11, one can show that the ASF function for fibrinogen in the high-coverage range assumes the form ASF ¼ C¥0 ð1 - ΘÞ4
ð14Þ C¥0 =1/(3Θ¥2 C¥3 )=
where Θh = Θ/Θ¥ is the reduced coverage and 6.53 10-2. By noting that the C¥0 constant is rather small, one can predict from eq 14 that the ASF function for fibrinogen decreases rapidly for coverage approaching the jamming limit. Thus, for Θh = 0.8, ASF = 1.04 10-4 and for Θh = 0.9, ASF = 6.53 10-6. This shows that fibrinogen adsorption at higher coverage, close to the jamming state, becomes very ineffective. By knowing the exact expressions for ASF in the limits of low and high coverage, one can formulate an interpolating function in Langmuir 2010, 26(14), 11934–11945
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Figure 5. Dependence of the extrapolated jamming coverage of fibrinogen Θ0 ¥ on the 1/L parameter derived from simulations of the kinetic dependencies such as shown in Figure 4 (b). The dashed line shows the linear interpolation function.
an analogous way as previously done for spherocylinders and ellipses19 2
3
ASF ¼ ð1 þ a1 Θ þ a2 Θ þ a3 Θ Þð1 - ΘÞ4
ð15Þ
By matching this expression with the known asymptotic forms for low and high coverage, the a1 - a3 coefficients can be calculated as a 1 ¼ - C1 Θ ¥ þ 4 a2 ¼ C2 Θ2¥ þ 4a1 - 6
ð16Þ
a3 ¼ C¥0 - ð1 þ a1 þ a2 Þ Using the previously determined constants, one can calculate that a1 = -0.727, a2 = -2.01, and a3 = 1.80 The analytical results derived from the interpolating formula (eq 15) are compared in Figure 6 with the numerical results derived from the RSA simulations. As can be seen, the interpolating function describes the exact numerical data for the entire range of coverage reasonably well. The slight deviation observed for higher coverage is caused by fluctuations appearing in numerical simulations due to the finite number of fibrinogen molecules within the simulation area. Therefore, one can conclude that because of its closed form and proper asymptotics for high and low coverage, the interpolating function given by eq 15 is especially convenient to use in numerical calculations of fibrinogen adsorption as discussed next. 3.2. Calculations of Fibrinogen Adsorption Kinetics. It should be noted, however, that the above-determined ASF function describes the kinetics of an idealized adsorption process of 2D objects, which are generated directly at the interface. It is not considered how they are transferred from the bulk to this point. Accordingly, in the case of an irreversible adsorption, the kinetic equation can be formulated as23,24 dΘ ¼ ka no ASFðΘÞ dτf Langmuir 2010, 26(14), 11934–11945
ð17Þ
where τf is the dimensionless time, ka is the formal adsorption constant, and no is the concentration of particles at the interface. Within the framework of the RSA model, τf cannot be related to the physical time and the adsorption and desorption constant cannot be predicted a priori. Therefore, to apply eq 17 to real systems, one has to use the generalized RSA model considered in ref 47. In this approach, the 3D motion of particles within the monolayer is considered, which results in a generalized concept of the ASF function. It is dependent not only on the surface coverage but also on the distance of the particle from the interface. However, in the limit of vanishing distance between the particle and the interface, the generalized ASF function approaches the standard ASF function derived above. Physically, this means that because of adsorbed particles a steric barrier at the interface is formed, whose height is governed by the standard ASF function and whose extension is equal to particle dimensions (adsorption layer thickness). Using this concept, the kinetic equation describing irreversible adsorption assumes the form46,47 1 dΘ ¼ ja ¼ ka nðδa Þ hðΘÞ ASF ð18Þ Sg dt where t is the real physical time, ja is the adsorption flux, ka is the physical adsorption and desorption constants, n(δa) is the number concentration of particles at the adsorption boundary layer of thickness δa, and ASF is the generalized function averaged over the adsorption boundary layer. It is interesting that the adsorption constant in eq 18 can be expressed in terms of physical parameters characterizing the system such as the particle diffusion coefficient, specific energy distribution, depth of the primary minimum, and energy barrier height.46,48 For the barrier-less adsorption regime, the adsorption constant is given by48 D 1 ka ¼ 1 δa δa ð19Þ 1 þ ln 2 δm (47) Adamczyk, Z.; Senger, B.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1999, 110, 3118. (48) Adamczyk, Z. J. Colloid Interface Sci. 2000, 220, 477.
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Figure 6. Dependence of the ASF function of fibrinogen on normalized coverage θ/θ¥. The points denote exact results derived from numerical simulations, and the solid line shows the analytical results derived from the fitting function given by eq 15. Table 2. Characteristic Kinetic Data for Fibrinogen Adsorptiona cb (ppm)
nb (cm-3)
K (1)
hκa(1)
tch (s) convection
t0 ch (s) convection
tch (s) diffusion
t0 ch (s) diffusion
1 1.77 1012 1.35 103 3.00 105 4.41 103 1.28 103 9.27 105 7.80 104 13 3 4 2 2 3 10 1.77 10 1.35 10 3.00 10 4.41 10 1.28 10 9.27 10 7.80 102 100 1.77 1014 1.35 103 3.00 103 4.41 101 1.28 101 9.27 101 7.80 1.35 103 3.00 102 4.41 1.28 0.927 7.80 10-2 1000 1.78 1015 a -3 -4 -1 -7 2 -1 -7 -8 T = 293 K, F = 1.35 g cm , Mw = 340 000, kc = 10 cm s , D = 2.1 10 cm s . δa = 6.7 10 cm, δm = 5 10 cm, Sg = 1.28 10-5 cm2] (128 nm2). ka = (D/δa)(1 þ 1/2 ln δa/δm). hκa = (1/δa)Sgnb(1 þ 1/2 ln δa/δm ). K = ka/kc. tch = (1/Sg)kcnb, t0 ch = Θ¥tch(convection). tch = (1/D)(Sgnb)2, t0 ch = Θ¥2tch (diffusion).
where δm is the minimum distance between the molecule and the interface and D is the diffusion coefficient. By knowing ka, one can calculate dimensionless constants ka and K characterizing fibrinogen adsorption under diffusionand convection-controlled transport conditions, respectively, (Appendix) ka ka ¼ Sg Dnb
K ¼
ka ¼ kc
D 1 δa δa kc 1 þ ln 2 δm
ð20Þ
where kc is the bulk transfer rate constant. Considering that for fibrinogen D = 2.1 10-7cm2 s-1 (this value was determined experimentally at T = 293 K49), δa = 6.7 nm (6.7 10-7cm), δm = 0.5 nm (the minimum distance between the fibrinogen molecule and the interface), and Sg = 128 nm2, one can predict that the ka constant attains large values (Table 2). Thus, for the 1 ppm bulk concentration of fibrinogen (Table 2), which corresponds to nb = 1.78 1012 cm-3, we have ka = 3 105. For nb = 1000 ppm, which is the highest value used in experimental measurements, nb=1.78 1015 cm-3 and ka = 3 102. Analogously, by using eq 20, one can calculate by assuming the typical value of kc = 1 10-4 cm s-1 that K = 1.35 102 (Table 2). For kc = 1 10-6 cm s-1, which corresponds to the lowest limit of practically occurring situations, one has K = 1.35 105. By knowing the adsorption and coupling constants, one can evaluate fibrinogen explicitly adsorption kinetics from eq A4 derived in the Appendix for convection-controlled transport. Results of these calculations are shown Figure 7 as the dependence 11940 DOI: 10.1021/la101261f
of the fibrinogen coverage on the reduced time t/tch = t/Sgkcnb. These results were obtained by a numerical integration of eq A4 using the ASF function expressed in eq 15 for various values of coupling constant K ranging from 1.35 103 to 1.35 105. As can be seen, for higher values of K, which are typical of fibrinogen adsorption at low-intensity convection, the adsorption kinetics remain linear for a broad range of dimensionless time, with the coverage approaching 0.9 times the jamming coverage. This indicates unequivocally that the surface blocking effects played a negligible role in comparison with the bulk transport effects. However, for coverage approaching the jamming limit, the kinetic curves shown in Figures 7 were well reflected by the limiting results, derived from eq A6. It is also interesting that the results derived from the standard RSA model (dashed line in Figure 7), which neglects coupling with bulk transport, deviate significantly from the results derived from the extended model. Analogous results obtained in the case of diffusion-controlled transport are shown in Figure 8. These theoretical kinetic data were derived by numerical integration of the equation set (eqs A12-A14) using the implicit Crank-Nicholson finite difference scheme described previously.48 It is to be noted, however, that in the case of diffusion transport the most convenient coordinate system for plotting results is the coverage versus the square root of the reduced time (t/tch)1/2, where tch = 1/D(Sgnb)2. Similar to convection transport, the results shown in Figure 8 indicate that for higher values of ka typical of fibrinogen adsorption for the low-concentration range (1-10 ppm) its adsorption kinetics remained linear with respect to (t/tch)1/2, in accordance with eq A17, until the coverage approached 0.9 times the jamming coverage. This indicates that the surface transport resistance stemming from blocking effects played a negligible role in comparison with the bulk transport resistance. It is interesting Langmuir 2010, 26(14), 11934–11945
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Figure 7. Kinetics of fibrinogen deposition on the dimensionless time calculated for the convection-controlled deposition regime. The solid line denote theoretical results calculated from eq A4 using the ASF function expressed by eq 15 for various values of the coupling constant: (1) K = 1.35 105, (2) K = 1.35 104, and (3) K = 1.35 103;(4) the dashed line shows the results calculated using the standard RSA model in eq 17 for K = 1.
Figure 8. Kinetics of fibrinogen deposition with respect to the dimensionless time calculated for the diffusion-controlled deposition regime. The solid line denotes theoretical results calculated by solving the diffusion equations (eqs A12-A14) using the fibrinogen ASF function for various values of the adsorption constant: (1) κha = 3 104, (2) κha = 3 103, and (3) κha = 3 102; (4) the dashed line shows the results calculated using the standard RSA model, assuming that the initial flux is reduced by the ASF(θ) factor.
that the results derived from the standard RSA model (dashed line in Figure 8) deviate significantly from the results derived from the extended model. No well-pronounced linear regime appears in this case, and the coverage attained for (t/tch)1/2 = 2 is 0.60 times the jamming coverage. To gain more insight into the fibrinogen adsorption kinetics, it is useful to estimate the characteristic relaxation time tch used above as a scaling variable for various protein concentrations and mass transfer rates. These data are also shown in Table 2. As can be noticed, the low concentration of fibrinogen (1 ppm) this time equals 4.41 103 s (1.2 h) for convection-controlled transport, whereas in the case of diffusion it is much higher, equal to Langmuir 2010, 26(14), 11934–11945
92.7 105 s (257 h). This estimation unequivocally shows that for low fibrinogen concentration, convection is more efficient than diffusion. However, for fibrinogen concentration equal to 100 ppm, the relaxation time for convection-controlled transport is reduced to 44.1 s and for pure diffusion it is 9.27 s, which indicates that the convection effect becomes less pronounced. 3.3. Comparison with Experimental Data. It is also useful to compare our results with experimental measurements reported in the literature. The data presented by Toscano and Santore42 seem to be especially valuable; they determined by direct AFM enumeration the surface concentration of fibrinogen N as a function of time on substrate surfaces, which were etched microscope DOI: 10.1021/la101261f
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Figure 9. Dependence of the surface concentration of fibrinogen N (μm-2) on the reduced time t/tch. The points denote experimental results
obtained by Toscano and Santore42 under the convection-controlled transport conditions (parallel-plate channel) for cb = 25 ppm at pH 7.4 using the direct AFM enumeration of the number of fibrinogen molecules per unit area (tch = 986 s), with circles showing the results for acidetched glass slides and triangles showing the results for silicon wafer pieces. The solid line denotes our theoretical results derived from eq A4 for K = 7.5 103, and the dashed line denotes the theoretical results obtained from the standard RSA model (K = 1 for no coupling).
slides. These kinetic measurements were performed under convection-controlled transport conditions in the parallel-plate channel for fibrinogen concentration cb = 25 ppm at pH 7.4. A linear dependence of N on the adsorption time was observed for up to 70 s with N attaining 560 μm-2. As mentioned by these authors, for higher values of N a direct determination of the surface concentration of fibrinogen by AFM was not feasible. Hence, the kc constant calculated from these experimental data was 1.8 10-5 cm s-1, tch = 1/Sgkcnb = 986 s, and K = 7.5 103. These experimental data are compared with our theoretical predictions in Figure 9. As can be seen, theoretical results derived from the generalized RSA model well reflect the experimental data for the entire range of reduced time t/tch, attaining a value of 0.06. The straight-line dependence of the surface concentration of fibrinogen on the adsorption time indicates that its adsorption was irreversible and that the transport kinetics was limited by the bulk transfer rate rather than the surface blocking effects. It is also interesting that the experimental results shown in Figure 9 are not properly reflected in terms of the standard RSA model (dashed line in Figure 9) or in terms of any heuristic model, where it is assumed that the actual flux to the interface is reduced by the factor stemming from the Langmuir function or similar blocking functions. Other experiments concerning fibrinogen adsorption on solid substrates were performed using indirect experimental techniques. This limits their utility for the direct verification of our adsorption model. Interesting experiments of this kind have been reported by Ortega-Vineusa et al.,39 who measured, using ellipsometry, the amount of fibrinogen adsorbed on a silicon wafer under various pH and ionic strength conditions. The optical signal was then converted into the mass of fibrinogen per unit area (mg m-2) using standard theoretical methods. It is to be remembered, however, that by using ellipsometry one can determine the integrated amount of adsorbed protein only, rather than its surface concentration. In these experiments, fibrinogen adsorption occurred in essence under diffusion transport conditions, although gentle stirring of the suspension was also applied. This exerted practically no 11942 DOI: 10.1021/la101261f
effect on the transport conditions because the diffusion was much faster for the fibrinogen concentration applied, equal to 70 ppm, as estimated above. For this fibrinogen concentration, tch = 190 s and ka= 4 103. In Figure 10, the experimental results obtained by OrtegaVineusa et al.39 at pH 4 and I = 4 10-3 are plotted in the form of the dependence of the fibrinogen coverage, expressed in mg m-2, on the time. Fibrinogen adsorption under this pH condition was irreversible and occurred in a side-on mode. The solid line in Figure 10 denotes our theoretical results derived from the extended RSA model by numerically solving eqs A12-A14, assuming the maximum coverage pertinent to hydrated fibrinogen to be equal to 1.6 mg m-2. As can be seen, the theoretical results reflect the experimental data reasonably well, especially for longer times. The deviation observed at shorter times is probably due to experimental error associated with ambiguity in defining the initial conditions for the adsorption process. Nevertheless, these results confirmed the main feature of adsorption predicted by our model (i.e., an abrupt increase in the amount adsorbed), characterized by the transition time tch = 1/D(Sgnb)2, which cannot be derived from previously applied models. Also, the maximum adsorbed amount of fibrinogen correlates well with the amount predicted theoretically in this work for a hydrated fibrinogen forming a side-on monolayer. Kinetic runs similar to those shown in Figure 10 were observed by Bai et al.,50 who studied fibrinogen adsorption on various substrates such as stainless steel, a nickel-titanium alloy, and pure titanium. These experiments were carried out for pH 7.4, I = 0.15 M (buffer) at a temperature of 320 K. The amount of adsorbed fibrinogen, expressed in ng cm-2 = 100 mg m-2, was determined using ex situ wavelength-dispersive spectroscopy (WDS). It the case of the stainless steel substrate, the maximum coverage of fibrinogen, attained after an adsorption time of ca. 900 s, was 1.40 mg m-2. In the case of the nickel-titanium alloy, (49) Wasilewska, M.; Adamczyk, Z.; Jachimska, B. Langmuir 2009, 25, 369. (50) Bai, Z.; Filiaggi, M. J.; Dahn, J. R. Surf. Sci. 2009, 603, 839.
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Figure 10. Dependence of the surface concentration of fibrinogen expressed in mg m-2 on the time t. The points denote experimental results
obtained by Ortega-Vineusa et al.39 under diffusion-controlled transport conditions for cb = 70 ppm, pH 4, and I = 4 10-3 (tch = 190 s) using ellipsometry. The solid line denotes our theoretical results derived from eqs A12-A14 for κha = 4 103 .
the maximum coverage was 1.70 mg m-2. These values of the maximum coverage of fibrinogen and the characteristic adsorption time are in reasonable agreement with the theoretical prediction obtained in this work. On the basis of this experimental evidence, one can conclude that for hydrophilic surfaces fibrinogen adsorption occurs according to the side-on mechanism with the maximum amount of adsorbed protein equal to 1.4 -1.7 mg m-2 (depending on the degree of hydration). The kinetics of this process are governed by the bulk transport rate, except for coverage closely approaching the jamming limit. It should be noted, however, that for other pH values, ionic strengths, and fibrinogen concentrations above 100 ppm, higher values of the maximum adsorbed amount have been determined experimentally in the above works.39,42,50 This is probably caused by the three main factors: (i)
reversible adsorption of fibrinogen forming unoriented multilayers in the higher bulk concentration range; (ii) adsorption of fibrinogen in the form of aggregates, which is likely for pH close to the isoelectric point as demonstrated in ref 49; and (iii) roughness of the substrate surfaces, which can be the case for oxidized substrates such as titanium or silicium. These factors are to be taken into account by interpreting the experimental results if deviations from the above-determined limiting value of the adsorbed amount of fibrinogen appear.
4. Concluding Remarks The bead model of fibrinogen applied in this work enabled one to determine the surface blocking function (ASF), which is pertinent to the side-on adsorption of the protein. An analytical expression, given by eq 15, was formulated that well approximates the ASF function over the entire range of coverage, including the jamming limit Θ¥. It was also found that Θ¥ = 0.29, corresponding to a surface concentration of fibrinogen of N = 2.27 103 μm-2. Langmuir 2010, 26(14), 11934–11945
This is 40% larger than in the case of a spherocylinder model of fibrinogen. This value of the jamming coverage corresponds to the amount of adsorbed fibrinogen equal to 1.4 -1.7 mg m-2, depending on the degree of hydration. Using the above blocking function, we evaluated the kinetics of fibrinogen adsorption in terms of the dimensionless constants K for convection-controlled transport and ka for diffusion-controlled transport. Because both constants assume values much larger than unity, typically 103-105 for the fibrinogen concentration range met in experiments, adsorption kinetics is bulk-transport-controlled over a broad coverage range. However, for fibrinogen coverage approaching the jamming limit, adsorption kinetics can be approximated by the asymptotic formula Θ ¼ Θ¥ - C¥ ðKτÞ - 1=3 ¼ Θ¥ - 0:332ðSg ka nb tÞ - 1=3 This equation can be used to calculate the true jamming limit of fibrinogen by extrapolation of the coverage Θl determined for long adsorption time tl using the relationship Θ¥ ¼ Θl þ C¥ ðSg ka nb tl Þ - 1=3 It was shown that our theoretical model was adequate for the interpretation of experimental data obtained using various techniques, for hydrophilic substrates, and for fibrinogen bulk concentration below 100 ppm. Acknowledgment. This work was supported by PBZ/MNiSW/ 07/2006/43 and by the COST D43 action special grant COST/12/ 2007.
Appendix As discussed in refs 46 and 47, the generalized blocking function ASF can be well approximated by the standard ASF derived from RSA simulations, especially for the high coverage limit, which is precisely the range of particular interest for the experimental DOI: 10.1021/la101261f
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viewpoint. By assuming this, the expression for the adsorption flux (eq 18) becomes ja ¼
1 dΘ ka nðδa Þ ASFðΘÞ Sg dt
ðA1Þ
Equation A1 has major significance because it can be used as the general boundary condition for bulk transport governed by diffusion, convection, or sedimentation.46 Moreover, several solutions of practical significance can be derived from this equation in the limiting cases of quasi-stationary transport, driven by convection, or in the case of high coverage, approaching the jamming state. In the former case of convection transport, the particle concentration at the edge of the adsorption layer n(δa) remains in quasi-equilibrium with the surface coverage, which results from the fact that the relaxation time of the bulk transport is much shorter than the time of surface coverage variations.46 Under these circumstances, the concentration can be expressed by the equation nðδa Þ ¼
1 nb K ASFðΘÞ þ 1
ðA2Þ
where K = ka/kc represents the dimensionless coupling constants and kc is the bulk transfer rate constant, known in analytical form for many types of flows and interface configurations.46 Using eq A2, the constitutive expression for the adsorption flux, eq A1 assumes the explicit form K ASFðΘÞ ja ¼ kc nb ðK - 1ÞASFðΘÞ þ 1 Equation A3 can be integrated to the useful form of Z Θ ðK - 1ÞASFðΘ0 Þ þ 1 0 t ¼ τ dΘ ¼ 0 tch K ASFðΘ Þ 0
ðA3Þ
ðA4Þ
where tch = 1/Sgkcnb is the characteristic time of particle monolayer formation under convection transport conditions (Table 2) and τ is the dimensionless time. Equation A4 represents the general solution for the particle deposition kinetics under convection-driven transport. It can be explicitly evaluated by numerical integration techniques if the ASF function is known in an analytical form, for example, as the series expansion given above for fibrinogen. It is useful to derive some limiting forms of eq A4. In the case of large K, where the inequality (K - 1)ASF(Θ) . 1 is fulfilled, eq A4 reduces to the simple linear form Θ ¼
t tch
¼ τ
ðA6Þ
As can be noticed, this equation has an analogous form as previously derived eq 11, which indicates that in the limit of long adsorption time the overall transport of particles is governed solely by the ASF function (surface transport limitation). 11944 DOI: 10.1021/la101261f
Additionally, eq A6 can be used to calculate the true jamming limit by the extrapolation of kinetic data obtained for a long but finite adsorption time. Denoting the coverage attained after such a long adsorption time tl by Θl, one can calculate the jamming coverage from the dependence Θ¥ ¼ Θl þ C¥ ðSg ka nb tl Þ - 1=3
ðA8Þ
It is interesting that this correction to the jamming coverage, which vanishes very slowly with its bulk concentration, is proportional to nb-1/3. In the case of diffusion-controlled transport, the expression for the flux (eq A1) cannot be integrated directly because the flux from the bulk to the interface and n(δa) vary with time. To calculate the coverage explicitly, one has to solve the bulk mass transport equation, which assumes in the case of 1D transport the following form46 ∂n ∂2 n ¼ D 2 ∂t ∂z
ðA9Þ
where n is the particle number concentration a distance z from the interface and D is the diffusion coefficient of the particle. The boundary conditions for eq A7 in the bulk assume the form of n f nb for z f ¥
ðA10Þ
However, the boundary condition at the edge of the adsorption layer δa should reflect the continuity of the particle flux, which can be expressed using eq A1 in the form of -D
∂n ¼ ka nðδa Þ ASFðΘÞ at z ¼ δa ∂z
ðA11Þ
Because eq A11 is nonlinear with respect to the coverage, the entire boundary value problem described by eqs A9-A11 also becomes nonlinear and can be solved only numerically, using, for example, the implicit finite-difference methods as discussed in ref 48. However, useful limiting solutions for particle coverage can be derived from eq A11 in the case of large adsorption constants and for long times, where the surface blocking effects become ratelimiting. To analyze this limiting form in more detail, it is useful to transform eqs A9-A11 to the dimensionless form by substituting τ = t/tch, z = z/L, n = n/nb, where L = 1/Sgnb and tch = L2/D. In this way, one obtains
ðA5Þ
In the case of ASF , 1/K (high-coverage range), where the ASF for fibrinogen can be approximated by eq 14, one can integrate eq A4 analytically to the form Θ ¼ Θ¥ - C¥ ðKτÞ - 1=3 ¼ Θ¥ - C¥ ðSg kc nd tÞ - 1=3
Moreover, by comparing eq A6 with eq 11, one can relate the formal simulation time τ with the physical adsorption time via the constitutive dependence t ¼ Sg kc nb t ðA7Þ τ ¼ K tch
-
∂n ∂2 n ¼ 2 ∂τ ∂z
ðA12Þ
n f 1 for z f ¥
ðA13Þ
∂n ¼ ka nðδa Þ ASFðΘÞ at z ¼ δ a ∂z
ðA14Þ
where n(δha) is the reduced particle concentration at z = δha, δha = δa/2a, and ka ¼
ka tch L
ðA15Þ
is the dimensionless adsorption constant. Langmuir 2010, 26(14), 11934–11945
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Hence, in the case of irreversible adsorption, where ka . 1, one can deduce from eq A14 that n(δa) f 0 (because the diffusion flux and adsorption flux should remain finite). Accordingly, the boundary condition at n = δha simplifies to n ¼ 0 at z ¼ δ a
ðA16Þ
With this boundary condition, one can solve eq A12 analytically, which results in the following expression for the particle coverage. Θ ¼ 2ðτ=πÞ1=2 ¼ 2ðt=πtch Þ1=2
ðA17Þ
As can be noticed, in this case, the coverage of particles is a linear function of the square root of the time rather than the time, as was the case for convection-controlled transport. The surface concentration of particles is given by the expression N ¼ 2ðDt=πÞ1=2 nb
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ðA18Þ
However, for long adsorption times, where the particle coverage attains the jamming limit, the particle concentration at the adsorption layer edge becomes equal to the bulk concentration, hence n(δha) f 1. In this case, the overall particle adsorption rate is governed by the transport in the adsorption layer. We can therefore integrate eq A14, which assumes the following form: Θ ¼ Θ¥ - C¥ ðka τÞ - 1=3 ¼ Θ¥ - C¥ ðSg ka nb tÞ - 1=3
ðA19Þ
This equation has an analogous form to eq A6, which was derived previously for convection-controlled transport. As previously reported, eq 19 can be used to extrapolate the results obtained for long times to infinite adsorption time in order to calculate the true jamming coverage.
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