INTRODUCTION

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A surface may be regarded as "molecularly flat" or "planar" with respect to a particular adsorbing macromolecule if the potential of interaction between adsorbate and surface is non-site-specific, and approximately independent of adsorbate position in the plane of the surface over distances that are within one to two orders of magnitude of the characteristic length of the adsorbate molecule. Reversible non-site-specific surface adsorption of proteins is of biological interest for several reasons, including the following. 1) The interior of a cell contains a variety of membranes and other structures presenting quasi-planar surfaces to which intracellular proteins may reversibly adsorb (Minton, 1990). Such adsorption is known to be linked to the catalytic activity of several enzymes (Kurganov, 1985) and may also be linked to the formation of multienzyme complexes (Minton, 1995). 2) The process of blood coagulation depends upon membrane adsorption-linked activation of clotting factors (Walker and Krishnaswamy, 1994). 3) The adsorption of protein onto the surface of various synthetic materials (polymers, ceramics, metals) under physiological conditions may determine the biocompatibility of those materials (Andrade, 1985). Thus models for the steady-state and kinetic properties of systems containing one or more soluble proteins interacting with a molecularly flat surface can provide important insight into the behavior of proteins (and synthetic materials) in various physiological milieux.

Kinetic and equilibrium models for irreversible and reversible adsorption of proteins to planar surfaces have been developed that take into account the surface area excluded to each other by molecules of adsorbed protein (Stankowski, 1983, 1984; Schaaf and Talbot, 1989; Talbot et al., 1994; Chatelier and Minton, 1996; Talbot, 1997; Minton, 1999; Ravichandran and Talbot, 2000). In the absence of attractive interactions between molecules of adsorbed protein, these models predict that the equilibrium association "constant" and the association rate "constant" should decrease monotonically and strongly with increasing fractional surface occupancy, i.e., that equilibrium adsorption should be strongly negatively cooperative. However, it has been demonstrated experimentally that at least under some conditions, proteins may relatively rapidly and reversibly adsorb onto a variety of surfaces at fractional surface occupancies as high as that corresponding to close packing of natively structured protein (Al-Malah et al., 1995). Moreover, depending upon experimental conditions and the particular protein and surface studied, protein adsorption isotherms may indicate positive cooperativity (Cutsforth et al., 1989; Heimburg and Marsh, 1995; Nygren, 1996) and "ideal" Langmuir-like behavior (Al-Malah et al., 1995; Spaargaren et al., 1995) as well as negative cooperativity. All of these types of equilibrium isotherm may be accounted for by a recently introduced model (Minton, 2000), hereafter referred to as part I. According to the model introduced in part I, surface area exclusion by adsorbate may be compensated to a variable extent by attractive interactions between adsorbate molecules leading to reversible formation of adsorbate clusters of varying size and shape. The purpose of the present communication is to extend this equilibrium model to treat the kinetics of adsorption.

In the following section we shall briefly recapitulate the essential assumptions and results of the equilibrium model of part I. Kinetic extensions are then introduced and the resulting rate equations presented, together with a description of the procedure used to solve the rate equations numerically. Next, the results of several model-simulated adsorption processes are presented and compared qualitatively with experimental results taken from the literature.

	EQUILIBRIUM MODEL FOR ADSORPTION WITH AREA EXCLUSION AND ADSORBATE CLUSTERING

TOP ABSTRACT INTRODUCTION EQUILIBRIUM MODEL FOR... KINETIC GENERALIZATION DEPENDENCE OF RATE COEFFICIENTS... CALCULATION OF KINETIC PROGRESS... RESULTS AND DISCUSSION APPENDIX REFERENCES

The following is a condensed summary of the equilibrium model introduced and described fully in part I, reviewed here to introduce subsequently used notation. Justification for assumptions and approximations introduced into the model are presented in part I.

It is assumed that an adsorbate molecule that is monomeric and thermodynamically ideal in solution may equilibrate with adsorbed monomer according to

(1)

where ₁ denotes the surface density of adsorbed monomer, ₁ the activity coefficient of adsorbed monomer, K the thermodynamic equilibrium constant for adsorption of monomer, c the concentration of (monomeric) adsorbate in solution, and c* an affinity-scaled solution concentration of adsorbate. It is also assumed that adsorbed monomer may self-associate reversibly to form adsorbed oligomers denoted by species i > 1, where the degree of oligomerization of oligomeric species i is denoted by n_i:

(2)

where _i and _i denote the activity coefficient and surface density of oligomeric adsorbate species i, respectively, and K_1,i the thermodynamic association equilibrium constant for the formation of one oligomer of species i from n_i monomers.

The activity coefficient of each adsorbed species depends upon the surface density, size, and shape of all species in a manner which is assumed to be calculable using the two-dimensional scaled particle theory of mixtures of convex hard particles (Talbot et al., 1994):

(3)

where a_i and s_i respectively denote the area and circumference of the "footprint" (projection on the plane of the surface) of adsorbed species i, and  _j, a  _ja_j, and s  _js_j.

Given a structural model for oligomer that specifies the size and shape of oligomer i, and the number and free energy of individual intermolecular contacts within oligomer i, the values of a_i, s_i, and K_1,i may be calculated for all i as described in part I. Given these quantities, Eqs. 1-3 may then be solved numerically for all _i and _i, and consequently the total amount of protein adsorbed per unit surface area (= n_ii) at equilibrium as functions of c*.

For purposes of developing a kinetic model we recast association equilibria in stepwise form. To simplify calculations we shall assume henceforth that n_i = i, i.e., all clusters with the same stoichiometry are structurally and energetically equivalent. Then Eq. 2 is equivalent to

(4a)

and the stepwise equilibrium association constant is given by

(4b)

where F_i denotes the standard state free energy change associated with the addition of adsorbed monomer to i-mer, R the molar gas constant, and T the absolute temperature.

	KINETIC GENERALIZATION

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Adsorbate cluster growth is assumed to proceed primarily by the reversible addition of single molecules of the adsorbing species, or "monomers," to preexisting clusters. (Diffusional mobility of clusters in the plane of the surface is expected to decrease strongly with increasing cluster size, so annealing of preexisting clusters to form larger clusters is assumed at this level of approximation to contribute negligibly to overall adsorption kinetics.) The formation of i + 1-mer from i-mer and monomer in solution may proceed via two mechanistically distinct pathways, depicted schematically in Fig. 1: 1) adsorption of soluble monomer to vacant surface (elementary process 1), followed by diffusion of adsorbed monomer to the edge of cluster species i and subsequent accretion (elementary process 2). This will be referred to as the "direct deposition + accretion pathway." 2) Deposition of soluble monomer onto and insertion into cluster species i (elementary process 3). This will be referred to as the "piggyback deposition pathway."

View larger version (12K): [in this window] [in a new window]   FIGURE 1   Schematic depiction of the kinetic pathways described in the text: 1) direct deposition of soluble monomer onto available surface; 2) accretion of adsorbed monomer onto cluster species i; 3) piggyback deposition of soluble monomer onto and incorporation into cluster species i.

Combination of these two pathways leads to the following set of rate equations:

(5a)

and for i > 1

(5b)

Since numerical calculations will be carried out for a finite set of clusters containing a maximum of imax protomers, it follows that for i = imax, Eq. 5b reduces to

(5c)

	DEPENDENCE OF RATE COEFFICIENTS UPON CLUSTER SIZE, SHAPE, AND EXCLUDED SURFACE AREA

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A general treatment of the effect of volume exclusion upon the kinetics of elementary bimolecular association and dissociation is presented in the Appendix.

Direct deposition of monomer (elementary process 1)

The transition state for this association, depicted schematically as T₁ in Fig. 1, requires near-contact approach of solute to the surface and thus corresponds to limiting case 1 of the Appendix. Hence

(6)

(7)

We note that the association rate coefficient corresponds to the product of a rate of an intrinsic rate coefficient times the probability (1/₁) that the monomer hitting the surface will hit an empty region large enough to accommodate its entire footprint (Talbot et al., 1994; Minton, 1999).

Accretion of adsorbed monomer to cluster (elementary process 2)

The transition state, depicted schematically as T₂ in Fig. 1, requires near-contact approach of adsorbed monomer to the periphery of the cluster and thus corresponds again to limiting case 1 of the Appendix. Assuming that the probability of reaction is the same at any point on the periphery of cluster species i, we obtain

(8)

Combining equation 8 with equilibrium relations 4, we obtain

(9)

Piggyback deposition (elementary process 3)

The transition state for this process is assumed to consist of a "piggyback complex" between monomer and the target cluster, depicted schematically as T₃ in Fig. 1. Unlike the transition states for the other two elementary processes, this transition state does not exclude additional volume to preexisting adsorbate clusters. Hence this process is an example of limiting case 2 of the Appendix, and nonideal effects resulting from area exclusion are expected to influence primarily the dissociation rather than the association rate coefficient.

The association rate is the product of two factors, the total rate R of soluble monomer hitting the surface per unit time and area, and the probability P(i) that a monomer hitting the surface will land on a cluster of species i. By comparison with process 1, we write

(10)

where J is a dimensionless constant expected to be of order unity. The introduction of J allows for differences between the orientational requirements for successful adsorption onto bare surface and successful piggyback deposition on a preexisting cluster. P(i) is the product of the joint probability that an approaching monomer will hit any cluster, P₁(i), and the conditional probability P₂(i) that a monomer hitting any cluster hits a cluster of species i. P₁(i) is just 1 minus the probability that an incoming monomer hits no cluster (1/₁), and P₂ is the ratio of the area covered by clusters of species i to the total area covered by all clusters (_ia_i/_ja_j). Thus the association rate coefficient is given by

(11)

Combining Eq. 11 with equilibrium relations 1 and 4, we obtain

(12)

Equations 6-12 may be further simplified by scaling all rate coefficients relative to k. Let k'_X = k_X/k, where X is any subscript. Then

(13)

(14)

(15)

(16)

(17)

(18)

The values of s_i, a_i, and F_i appearing in the above expressions are obtained from a structural/energetic model for clusters. In the present treatment, an i-meric cluster is generally represented as a hard convex particle with footprint area equal to i times the footprint area of adsorbed monomer

(19)

This assumption is equivalent to assuming that all monomers in the cluster are in direct contact with the surface. The stepwise free energy of monomer addition is taken as the product of the free energy of a single inter-protomer contact times the change in the number of contacts with addition of a monomer to an i-mer

(20)

In part I the circular cluster and linear cluster models were introduced as representing the most compact (least volume-excluding) and least compact (most volume-excluding) clusters possible for a given stoichiometry. The rationale for using two such highly simplified models for clusters is that even though these models are too simple to be realistic in and of themselves, one may argue with some confidence that any behavior exhibited qualitatively by both models is probably a consequence of the proposed underlying reaction mechanism in general, rather than any particular assumptions regarding cluster structure and energetics. The values of s_i and n_c,ii+1 obtained from these two models are given below.

Circular cluster model

(21)

(22)

Linear cluster model

(23)

(24)

	CALCULATION OF KINETIC PROGRESS CURVES

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We define the scaled time t* = kt. The rate equations expressed as d_i/dt* are identical to Eqs. 5a-c except that all rate coefficients are replaced by the corresponding scaled quantities presented in Eqs. 13-18. Given a model for clusters that specifies the values of a_i, s_i, and n_c,ii+1, and user-supplied values of the adjustable parameters c* (= Kc), k', J, and U_C, the scaled rate equations may be solved together with Eqs. 3 to yield the dependence of _i and _i on t*. These equations were solved for the circular and linear cluster models with i = 1 to 20 using the numerical differential equation solver ODE15s in MATLAB 5.3 (MathWorks, Natick, MA) with the initial condition _i(t* = 0) = 0 for all i. [MATLAB scripts are available upon request.] Calculated kinetic progress curves are presented as the dependence of fractional surface occupancy (t*)  _{i i}(t*)a_i upon log t*.

	RESULTS AND DISCUSSION

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Kinetic adsorption progress curves calculated using the models introduced here will be compared to a reference progress curve calculated according to ideal Langmuir surface binding kinetics (Langmuir, 1918), which assume the absence of any interaction (attractive or repulsive) between molecules of adsorbate:

(25)

To explore the kinetic contribution of each of the two alternate pathways proposed in the current model, we first present results obtained when one of the pathways is "switched off." In Fig. 2 results are shown for adsorption in the absence of piggyback deposition (cluster growth by accretion only), as calculated using the circular cluster model. The left-hand panel displays the calculated dependence of fractional surface occupancy  on log t*, and the right-hand panel displays the calculated dependence of adsorption rate upon . Curves a represent an ideal reference calculated according to Eq. 25 for the same value of c*, and curves b-e were calculated from the circular cluster model for varying rates of accretion, with fixed equilibrium conditions and fixed rate of monomer adsorption. Salient features of these simulations are as follows.

View larger version (17K): [in this window] [in a new window]   FIGURE 2   Kinetics of adsorption via the direct deposition + accretion pathway, calculated for the circular cluster model. Curves a are reference curves representing Langmuirian kinetics calculated according to Eq. 25 with c* = 3. Simulation parameters: c* = 3, Uc = 0, J = 0 throughout, and k = 0.001 (curves b), 0.1 (c), 1 (d), and 1000 (e).

First, when the rate of accretion is low compared to the rate of monomer adsorption (curves a) adsorption proceeds in two widely separated phases. The initial phase corresponds to the adsorption of monomer to a fractional surface occupancy corresponding to the equilibrium adsorption of a nonassociating monomer at solution concentration c* (part I). The second stage of adsorption corresponds to additional adsorption of monomer at a much slower rate, which is limited by the amount of free surface area (Fig. 3).

View larger version (14K): [in this window] [in a new window]   FIGURE 3   Time dependence of the rate of direct deposition, calculated for the simulation shown in Fig. 2, curves b.

Second, as the rate of accretion becomes larger the adsorption progress curve approaches an asymptotic limit (curve e in Fig. 2) in which all surface clusters are in instantaneous chemical equilibrium, and the overall rate of adsorption at any time depends upon the fraction of surface area made available for additional monomer adsorption by the equilibrium distribution of clusters at that time. It is evident upon inspection of the right-hand panel of Fig. 2 that in the absence of direct deposition, the rate of adsorption will always be less than the ideal (exponential) rate of adsorption because of decreasing available area for adsorption of additional monomer. The upward concave shape of the curve of d/dt* is a reflection of negative kinetic cooperativity, i.e., a condition in which occupancy of some fraction of the surface by adsorbate decreases the effective rate coefficient for subsequent adsorption.

The results of comparable calculations carried out on the linear cluster model are shown in Fig. 4. It is evident that the linear and circular cluster models yield qualitatively similar behavior, the main difference arising from the different levels of equilibrium adsorption of circular and linear clusters.

View larger version (16K): [in this window] [in a new window]   FIGURE 4   Kinetics of adsorption via the direct deposition + accretion pathway, calculated for the linear cluster model. Parameter values are as given in the caption to Fig. 2.

In Fig. 5 calculated results are shown for adsorption in the absence of accretion (cluster growth via piggyback deposition only), as calculated using the circular cluster model. When the rate of piggyback deposition is very small relative to the rate of direct deposition of monomer (curves b), adsorption becomes biphasic; the faster phase corresponds to the adsorption of nonassociating monomer, followed by growth of clusters at a much slower rate. This kinetic behavior closely resembles that of the accretion-only pathway, in the limit that accretion occurs over a much larger time scale than monomer deposition (cf. curves b of Fig. 2). The difference between the kinetic contribution of the two pathways becomes evident at higher rates of piggyback deposition. With increasing rate of monomer deposition, the adsorption progress curves steepens, first approaching the exponential behavior exhibited by the ideal adsorption curve (curves c), and then becoming even steeper (curves d), ultimately achieving a condition in which the rate of adsorption actually increases with increasing fractional occupancy at lower levels of surface occupancy (curves e). At the highest rate of deposition a small degree of kinetic overshoot is apparent. The upwardly convex shapes of curves d and e in the right-hand panel of Fig. 5 are indicators of positive kinetic cooperativity, i.e., a condition in which occupancy of some fraction of the surface by adsorbate increases the effective rate coefficient for subsequent adsorption.

View larger version (17K): [in this window] [in a new window]   FIGURE 5   Kinetics of adsorption via the piggyback deposition pathway, calculated for the circular cluster model. Curves a are reference curves representing Langmuirian kinetics calculated according to Eq. 25 with c* = 3. Simulation parameters: c* = 3, Uc = 0, k = 0 throughout, and J = 0.001 (curves b), 0.3 (c), 1.0 (d), and 3.0 (e).

The results of comparable calculations carried out on the linear cluster model are shown in Fig. 6. As in the case of the accretion pathway calculations summarized in Figs. 2 and 4, the linear and circular cluster models yield qualitatively similar behavior, the main difference arising from the different levels of equilibrium adsorption of circular and linear clusters.

View larger version (16K): [in this window] [in a new window]   FIGURE 6   Kinetics of adsorption via the piggyback deposition pathway, calculated for the linear cluster model. Parameter values are as given in the caption to Fig. 5.

Under circumstances such that the rate of direct deposition per unit area of cluster was comparable to the rate of monomer adsorption per unit area of available free surface, the overall rate of adsorption would be expected to be insensitive to the fractional occupancy of surface, and that under such conditions one might observe quasi-exponential adsorption kinetics. In Fig. 7 an example is presented of a calculated adsorption progress curve exhibited by a highly nonideal system that displays almost perfectly exponential adsorption kinetics. In the absence of additional data, an investigator observing such time dependence might attribute it to the lack of interaction between molecules of adsorbate. However, the weight-average degree of adsorbate oligomerization (cluster size), calculated from this model according to

(26)

increases substantially with increasing fractional surface occupancy, as shown in Fig. 8. The observed kinetics should properly be termed "pseudo-Langmuirian" rather than "quasi-Langmuirian," as the latter term connotes near-absence of interadsorbate interactions rather than the reality of substantial but compensating attractive and repulsive interactions.

View larger version (13K): [in this window] [in a new window]   FIGURE 7   (Left) Points: calculated adsorption progress curve for circular cluster model with c* = 3, Uc = 0, k = 10, J = 0.4. Curve: best least-squares fit of Eq. 25 (Langmuir kinetics). (Right) Difference between circular cluster model simulation and best fit of single exponential to results shown in the left-hand panel.

View larger version (12K): [in this window] [in a new window]   FIGURE 8   Weight-average degree of cluster polymerization Pw, calculated from the simulation of Fig. 7, plotted as a function of fractional surface occupancy .

The experimental literature on kinetics of protein adsorption contains examples of negative kinetic cooperativity (Ramsden, 1993; Ramsden et al., 1994; Wahlgren et al., 1995), quasi- or pseudo-Langmuirian kinetics (Ramsden et al., 1994; Spaargaren et al., 1995), and positive kinetic cooperativity (Nygren, 1993, 1994; Ball and Ramsden, 1997). Previous models taking into account the effect of excluded surface area on adsorption kinetics (Talbot et al., 1994; Sild et al., 1996; Minton, 1999; Ravichandran and Talbot, 2000) have focused exclusively on direct deposition of monomer, neglecting clustering and hence the possibility of piggyback deposition. The present treatment indicates that in the absence of piggyback deposition, cluster formation can reduce but not eliminate the negative kinetic cooperativity arising from area exclusion by adsorbate. The present model predicts that pseudo-Langmuirian or positively cooperative adsorption progress curves are possible only when a substantial fraction of total adsorption proceeds via piggyback deposition of monomer on existing clusters.

To the best of the author's knowledge, the kinetic model presented here is the only quantitative mechanistic model so far proposed that can account for the experimentally observed positive kinetic cooperativity cited above, and it is the only model that can account for Langmuir-type adsorption kinetics when the surface density of adsorbate approaches that of a two-dimensional close-packed array of adsorbate in the limit of long time. Until now, almost all quantitative studies of protein adsorption kinetics have been carried out using techniques that monitor the average amount of protein adsorbed per unit surface area as a function of time. Using techniques such as electron microscopy or atomic force microscopy, it may be possible to directly observe clusters of adsorbed protein if and when they are present (see, for example, Nygren and Stenberg, 1990; Schwartz et al., 1992), and to quantify cluster size and shape distribution as a function of the time of exposure of the surface to supernatant protein solution.