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Kinetics of Protein Adsorption and Desorption on Surfaces with Grafted Polymers

Fang Fang ^*, Javier Satulovsky  and Igal Szleifer ^*

^* Department of Chemistry, Purdue University, West Lafayette, Indiana; and Department of Cell Biology and Physiology, Washington University School of Medicine, St. Louis, Missouri

Correspondence: Address reprint requests to I. Szleifer, Dept. of Chemistry, Purdue University, 560 Oval Dr., West Lafayette, IN 47907. Tel.: 765-494-5255; Fax: 765-494-5489; E-mail: igal{at}purdue.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MOLECULAR THEORY GENERALIZED DIFFUSION APPROACH... FREE ENERGY FUNCTIONAL MOLECULAR... REPRESENTATIVE RESULTS FOR GDA KINETIC THEORY (KA) REPRESENTATIVE RESULTS FOR KA DISCUSSION AND CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The kinetics of protein adsorption are studied using a generalized diffusion approach which shows that the time-determining step in the adsorption is the crossing of the kinetic barrier presented by the polymers and already adsorbed proteins. The potential of mean-force between the adsorbing protein and the polymer-protein surface changes as a function of time due to the deformation of the polymer layers as the proteins adsorb. Furthermore, the range and strength of the repulsive interaction felt by the approaching proteins increases with grafted polymer molecular weight and surface coverage. The effect of molecular weight on the kinetics is very complex and different than its role on the equilibrium adsorption isotherms. The very large kinetic barriers make the timescale for the adsorption process very long and the computational effort increases with time, thus, an approximate kinetic approach is developed. The kinetic theory is based on the knowledge that the time-determining step is crossing the potential-of-mean-force barrier. Kinetic equations for two states (adsorbed and bulk) are written where the kinetic coefficients are the product of the Boltzmann factor for the free energy of adsorption (desorption) multiplied by a preexponential factor determined from a Kramers-like theory. The predictions from the kinetic approach are in excellent quantitative agreement with the full diffusion equation solutions demonstrating that the two most important physical processes are the crossing of the barrier and the changes in the barrier with time due to the deformation of the polymer layer as the proteins adsorb/desorb. The kinetic coefficients can be calculated a priori allowing for systematic calculations over very long timescales. It is found that, in many cases where the equilibrium adsorption shows a finite value, the kinetics of the process is so slow that the experimental system will show no adsorption. This effect is particularly important at high grafted polymer surface coverage. The construction of guidelines for molecular weight/surface coverage necessary for kinetic prevention of protein adsorption in a desired timescale is shown. The time-dependent desorption is also studied by modeling how adsorbed proteins leave the surface when in contact with a pure water solution. It is found that the kinetics of desorption are very slow and depend in a nonmonotonic way in the polymer chain length. When the polymer layer thickness is shorter than the size of the protein, increasing polymer chain length, at fixed surface coverage, makes the desorption process faster. For polymer layers with thickness larger than the protein size, increases in molecular weight results in a longer time for desorption. This is due to the grafted polymers trapping the adsorbed proteins and slowing down the desorption process. These results offer a possible explanation to some experimental data on adsorption. Limitations and extension of the developed approaches for practical applications are discussed.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MOLECULAR THEORY GENERALIZED DIFFUSION APPROACH... FREE ENERGY FUNCTIONAL MOLECULAR... REPRESENTATIVE RESULTS FOR GDA KINETIC THEORY (KA) REPRESENTATIVE RESULTS FOR KA DISCUSSION AND CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Flexible polymer molecules grafted to surfaces or interfaces impose a steric barrier that can be tuned depending upon the polymer molecular weight, surface coverage, and type of chemical structure (1–3). These interactions are widely used in colloidal stabilization (4–6) and in the last few years have found application on the development of biocompatible materials and drug carriers (7–25). The basic idea is that the grafted polymer layer prevents nonspecific adsorption of proteins on the surface of the biocompatible material or drug carrier, reducing the immunological response (7,8,15,26–29). The understanding of the kinetics of protein adsorption and its reduction/prevention by grafted polymer layers is therefore very important for the design of materials interacting with biological systems. In this work we present a thorough theoretical study of the kinetics of protein adsorption on surfaces with grafted polymers, which complements our earlier work on both the thermodynamics and kinetics of protein adsorption (30–34).

Adsorption of proteins on surfaces is a complex process that involves very large energy scales and the ability of the proteins to change their conformation upon contact with the surface (26,35–38). Moreover, the timescale of the adsorption process can be very long and in many cases the adsorption is irreversible (31,33,39–41). It is important to differentiate between the equilibrium isotherms and the kinetics of the adsorption process. This is an important difference both in the practical applications of protein adsorption (or prevention of it) and in the fundamental studies of the understanding of the adsorption process. For example, in the design of biocompatible materials to be used for artificial organs it is important to completely prevent adsorption of proteins. Thus, thermodynamic control is necessary, meaning that for the given conditions, the equilibrium amount of proteins adsorbed on the surface is zero. On the other hand, drug carriers need to survive in the blood stream for the time necessary to deliver the drug to its target. In this case, control of the kinetics of adsorption, so that it is delayed beyond the timescale for drug delivery, is the necessary design criteria.

During the last few years we have developed and applied a molecular theoretical approach to study the thermodynamics and kinetics of protein adsorption on surfaces with and without grafted polymers (21,24,30–33,42–45). The predictions from the theory are in excellent quantitative agreement for the adsorption isotherms of lysozyme and fibrinogen on surfaces with short- (31) and long-grafted polyethylene oxide (PEO) chains (21,32). In all these cases we studied systems in which the reduction of protein adsorption is due to the steric repulsion induced by the polymer layer, i.e., by flexible polymer and not by chemical modification of the surface, as in the case of high density self-assembled monolayers with functional end-groups (46). We have found that the grafted polymer layers properties that are optimal for thermodynamics control are different than those controlling the kinetic process (31,33). For example, for thermodynamic control of protein adsorption the polymer surface coverage is the most important factor in determining the reduction of protein adsorption (30). These predictions have been confirmed by experimental observations (16,47). For the kinetics of adsorption, though, we have predicted a very strong effect on molecular weight; however, its role for the equilibrium isotherms is only secondary (31,33).

In this article we present a kinetic theory that borrows from our previous work (31) and the insights learned from the theory of Halperin (48) and we develop a computational feasible molecular approach that enables the study of the whole kinetic process explicitly accounting for the deformation of the polymer-protein layer as the adsorption process takes place. The basic idea of the approach is to use the physical insights learned from the generalized diffusion approach to determine what the relevant steps are in the kinetic process. Then, we use the theoretical ideas of the Kramer-like approach developed by Halperin together with our molecular theory to construct a kinetic model that enables the study of both adsorption and desorption processes.

The next section, Molecular Theory, starts with a review of the generalized diffusion approach and the molecular theory that serves as basis for the kinetic approach. After that, we present the kinetic theory used, with examples of the kinetics of adsorption and desorption as a function of the grafted polymer chain length and surface coverage; this is then followed by our concluding remarks.

	MOLECULAR THEORY

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The system of interest here is composed of a surface of total area A spanning the x, y plane. The surface has N_g polymer molecules grafted at one of their ends (see Fig. 1). Each polymer has n_g segments, each of length l. The polymer-modified surface is put, at time t = 0, in contact with a solution containing proteins dissolved in water. The protein solution is characterized by a bulk density _p,bulk or equivalently a chemical potential µ_p,bulk. When the surface is put in contact with the solution the proteins "feel" anisotropic interactions, induced by the presence of the surface, which are the driving forces for the adsorption process.

View larger version (24K): [in this window] [in a new window]   FIGURE 1  Schematic representation of the system containing proteins in their native conformation dissolved in a low molecular-weight solvent, and in contact with a surface with grafted polymers. The large solid circles are the proteins and the small open circles are the solvent molecules. The strings of small solid circles, tethered to the surface, represent the grafted polymers. The z direction is defined perpendicular to the surface. The position of a protein, z', refers to the lowest point of the protein, whereas the volume that a protein contributes to z refers to the volume that the protein occupies between z and z + dz. The two rate coefficients represent the kinetic processes involved in the adsorption of proteins onto the surface with grafted polymers. The right of the figure represents schematically the potential of mean-force felt by the adsorbing/desorbing proteins (see text).

In this work we concentrate our attention only on cases in which the protein does not change its conformation upon adsorption. The generalization of the theory to more general cases has been presented elsewhere for bare surfaces (33). It will be shown in future work for surfaces with grafted polymers.

	GENERALIZED DIFFUSION APPROACH (GDA)

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The basic assumptions of separation of timescale present a natural scenario for the use of dynamical self-consistent theory or the dynamical density functional approach (31,49–51). This is effectively what we call the generalized diffusion approach (GDA), when the free energy used is from our molecular theory (33). We write a time-evolution equation for the density profile of the proteins with a diffusion equation of the form

(1)

with D_p the diffusion coefficient of the proteins. (Please note that, for simplicity, we assume that the diffusion coefficient is independent of position. The reason is that the kinetic slowdown induced by the interactions, through the chemical potential gradients, is the dominant effect in the cases of interest here.) The time- and position-dependent chemical potential is defined as

(2)

W/A represents the free energy density (per unit area) of a system of frozen configuration of the proteins. This means it is the minimal free energy with respect to the polymers and the solvent for the given distribution of proteins. It is convenient to write the chemical potential as the sum of an ideal term and a potential of mean-force U_mf(z;t), i.e., the nonideal contribution. Then

(3)

where k_B is the Boltzmann constant and T is the absolute temperature. Using the potential of mean-force definition, Eq. 3, on the diffusion equation, Eq. 1, we obtain

(4)

The first term is the ideal diffusion, whereas the second represents the motion due to the interactions between the proteins and all the other molecules in the system, including the surface. Consider for example the case in which, at time t = 0, a solution with homogeneously distributed proteins is put in contact with the surface. The main driving force for adsorption has to be the potential of mean-force, since there is no gradient of density. However, because proteins adsorb, the two terms contribute until the chemical potential (due to the balance of density and potential of mean-force) becomes constant and the new equilibrium with the adsorbed proteins is reached. Clearly, if there is a strong attraction between the proteins and the surface the new equilibrium corresponds to an inhomogeneous distribution of proteins, due to the anisotropic interaction potential that results from the presence of the surface.

At this point we should remark that the concept of chemical potential and potential of mean-force are defined as generalizations of the equilibrium (true thermodynamic) properties (52,53). We refer to the time- and distance-dependent quantities in the same way as in the equilibrium cases. That is, they are formally defined in the same way, but do not correspond to the quantities for when the system is in true thermodynamics equilibrium. Instead, they correspond to the minimal free energy under the constraint of frozen distribution of protein as given at that time by the dynamic equation.

	FREE ENERGY FUNCTIONAL MOLECULAR THEORY

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The understanding of the adsorption process from one equilibrium state to a new one, requires the variation of the free energy, W. To this end, we use the molecular theory that we originally developed to treat the structural and thermodynamic properties of tethered polymer layers and later generalized to treat protein adsorption on surfaces with and without grafted polymers (3,24,30–33,42–45,54–57) . The basic idea is to write the free energy as a functional of the density of proteins and the conformational probability distribution of the grafted polymer chains and the proteins. To write the free energy we consider each molecular species exactly (within the chosen model to describe the molecular system) in terms of the intramolecular and surface interactions. The intermolecular interactions are considered within a mean-field approximation.

The presence of the surface induces an inhomogeneous distribution of all the molecular species. Therefore, the mean-field felt by the molecules in each of their conformations is a function of the spatial distribution of its units and the average distribution of all the other molecular species. For simplicity we consider that the only inhomogeneous direction is the one perpendicular to the surface, i.e., the z direction.

We derive the free energy for a simple case of a mixture of polymers and proteins in which the solvent is equally good for both molecular species. Further, we assume that the protein can only be in its native conformation and does not change its conformation upon adsorption to the surface. The generalization to the cases in which conformational changes upon adsorption are considered (34), as well as different intermolecular interactions, has been treated elsewhere (30).

The free energy per unit area of the equilibrium combined protein-grafted polymer system is given by

(5)

where ß = 1/k_BT. The first term represents the conformational entropy of the tethered polymers, where = N_p/A is the polymer surface coverage and P_g() (pdf) is the probability of finding a grafted polymer in conformation . The second term is the protein contribution, including:

1. A z-dependent mixing (translation) entropy with _p(z) representing the protein density profile; v_s is the solvent volume which is used as the unit of volume throughout.
   
2. The distance-dependent bare protein-surface attraction, U_ps(z).
   
3. The chemical potential term to ensure equilibrium with the bulk, i.e., constant chemical potential of the proteins at all z with µ_p = µ_p,bulk.
   

The last term represents the z-dependent mixing entropy of the solvent where _s(z) is the solvent density at z.

Inspection of Eq. 5 shows that the intermolecular repulsive interactions are not included. We assume that the repulsive interactions are of the excluded volume type and thus we include them through packing constraints. That is, at each distance z from the surface the volume accessible to the molecules in the layer between z and z + dz is completely occupied by a sum of contributions from the polymers, the proteins, and the solvent molecules. This is expressed in the form

(6)

where the first term is the volume fraction of polymers in layer z, with being the average volume that a grafted polymer occupies at z. The second term is the volume fraction of protein. This term includes the integral over z' since we need to consider the contribution to the volume at z from proteins everywhere. The term v_p(z;z') is the volume that a protein with its point of closest distance at z' contributes to z (see Fig. 1). The last term is the solvent volume fraction with _s(z) = _s(z)v_s. The packing constraint explicitly includes the size and shape of each of the molecular species in the system, as well as the spatial distribution of volume for each polymer conformation.

We now can find the explicit functional form of the pdf of chain conformations and the density profiles of proteins and solvent by performing a functional minimization of the free energy with respect to the polymer pdf, protein, and solvent density profiles subject to the packing constraints. The minimization is carried out by introducing Lagrange multipliers, ß(z), associated with the packing constraints, to yield

(7)

for the pdf of chain conformations, with q_g being the grafted polymers' partition function (normalization constant that ensures ), and

(8)

for the protein density profile. The equation for the protein density profile ensures that the chemical potential of the protein is the same at all z as required for thermodynamic equilibrium.

Finally, the solvent density profile is given by

(9)

The physical meaning of the Lagrange multipliers can be seen in the expression of the solvent volume fractions profile (Eq. 9). They represent the local osmotic pressures. They actually measure the work to replace one unit of volume of solvent by one of polymer or protein. As discussed elsewhere (3,30,33,34), the lateral pressures are a measure of the average repulsive interaction at distance z from the surface. The numerical values of the lateral pressures are determined by replacing the explicit expressions of the polymer pdf, Eq. 7, the protein density profile, Eq. 8, and the solvent density profile, Eq. 9, into the constraint equation, Eq. 6. The resulting equations require, as input, the single chain conformations of the polymer chains; the protein volume distribution; the polymer surface coverage; and the protein chemical potentials. For details on how the equations are solved numerically, see the Appendix.

At this point it is instructive to look at the expression of the protein chemical potential. At equilibrium it is given from Eq. 8 by

(10)

where we have defined the potential of mean-force U_mf(z) by

(11)

This quantity represents the effective interaction between a protein at z and the surface, averaged over all the degrees of freedom of the other molecules in the system. This quantity, actually its time-dependent counterpart, plays a key role in the kinetics of adsorption (see Eq. 2, above).

The free energy functional and the pdf and density profiles just derived correspond to the equilibrium state of the system since they are found by the total minimization of the free energy functional. To determine the time-dependent quantities necessary to solve the generalized diffusion equation, Eq. 4, we use the separation of timescales mentioned above and consider a free energy with the same functional form as Eq. 5 but with a major modification. Following the assumption that the local motion of the polymers and that of the solvent are much faster than the proteins' motion, we can consider that for each given density profile of the proteins, the free energy contribution of the solvent and polymers is minimized. Therefore, we write the time-dependent free energy in the form

(12)

where the protein component is written without a chemical potential term; the time-dependent polymer pdf and solvent profile are obtained by minimization of the free energy; and the protein density profile is fixed, and is given by the dynamic equation, Eq. 4.

We need to minimize the free energy with respect to the pdf and the solvent density profile subject to the packing constraints. Following the same lines as in the equilibrium case, we introduce a time-dependent Lagrange multiplier, ß(z;t), which is associated with the time-dependent packing constraints. For the pdf and solvent-density profile expressions (identical to Eqs. 7 and 9, respectively), this leads to (z) being replaced by (z;t). For the determination of the lateral pressures, the main difference between the equilibrium and nonequilibrium cases is that, in the equilibrium case, the protein density profile is obtained in the minimization process and is therefore an explicit function, (z); but in the nonequilibrium case, _p(z;t) is an input. In general, the input comes from the diffusion equation. However, this does not have to be the case, as will be discussed in detail in the description of the kinetic approach.

The next step is to determine the time- and z-dependent chemical potential of the protein. This is obtained as a straightforward generalization of the equilibrium quantity (see Eqs. 10 and 11),

(13)

to obtain

(14)

and thus the time-dependent potential of mean-force is given by

(15)

in analogy to the equilibrium amount. Note that the bare surface-protein interaction is time-independent. All of the time-variation arises from the intermolecular interactions as expressed in the lateral repulsions (z;t), which result from the changes in the packing of the protein-polymers as the proteins move toward the adsorbing surface.

To solve the equilibrium and kinetics of protein adsorption we need to define the model system for the protein and the polymers. For the protein we use a very simple model that mimics the properties of lysozyme. We assume that the protein is spherical with a radius R_p = 1.5 nm. Furthermore, for U_ps(z), we use the potential calculated from atomistic simulations by Lee and Park (35) for lysozyme with polyethylene solid surfaces (see Fig. 2 below). For simplicity, we do not allow for conformational changes of the protein upon adsorption, and leave that for future work. For the polymer conformations, we use a rotational isomeric state model (58) in which each bond is allowed to have three isoenergetic states. This model is closely related to the one we have used to model PEO chains and the predictions are in excellent agreement with experimental observations (21,31,32,59). For each chain length that we study we use up to 2 x 10⁶ randomly generated self-avoiding polymer conformations. From each conformation generated we obtain v_g(z;), the volume that a chain in conformation has in the volume spanned between z and z + dz. These volume distributions are the input to solve the constraint equation, Eq. 6. Note that the set of conformations has all type of distributions of segments, v_g(z;), including highly stretched chains and mushroom-like conformations. The pdf determines the relative weight of each conformation for each different case and for different times. (The Appendix outlines how the equations are solved by discretization of the z direction; for more detail on the technical aspects for the equilibrium and dynamic solutions, see Refs. 21, 30, 31, 33, and 34.) In all the results presented below, we denote the dimensionless densities given by the product v_s simply by . To convert this quantity to the experimentally reported units of ng/cm² for lysozyme, one needs to multiply our reported values by 26,394. Otherwise, multiplying the reported values by 150 provides the area fraction occupied by the proteins.

View larger version (17K): [in this window] [in a new window]   FIGURE 2  (A) The initial potential of mean-force for different chain lengths: ng = 0 (bare surface, solid line); ng = 25 (dot-dashed line); ng = 50 (long-dashed line); ng = 75 (double-dot-dashed line); and ng = 100 (dotted line). The surface coverage is l2 = 0.01. (B) The inset presents the corresponding polymer volume fraction profiles at t = 0.

	REPRESENTATIVE RESULTS FOR GDA

TOP ABSTRACT INTRODUCTION MOLECULAR THEORY GENERALIZED DIFFUSION APPROACH... FREE ENERGY FUNCTIONAL MOLECULAR... REPRESENTATIVE RESULTS FOR GDA KINETIC THEORY (KA) REPRESENTATIVE RESULTS FOR KA DISCUSSION AND CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

In the case of no grafted polymers, the potential is purely attractive and it is given by U_ps(z). The presence of the grafted polymers introduces a repulsion whose range is equal to the thickness of the tethered layer (see inset of Fig. 2 for the polymer density profiles). For long enough polymers a repulsive barrier appears whose strength and position is a function of the polymer chain length. The amount of protein adsorbing at equilibrium depends on the potential of mean-force at contact. Even though the value of the potential at contact varies with the amount of proteins adsorbed (see below), the variation of U_mf(z = 0; t = 0) with polymer chain length shows the same dependence as the equilibrium amount of proteins adsorbed (30).

The kinetics of adsorption depends upon the timescale required by proteins to reach the surface. In the case of purely attractive potentials, as for no polymer and n_g = 25 on Fig. 2, the initial adsorption will be very fast and determined by the time that it takes the proteins to reach the range of the interactions. In this regime the surface acts as a strong attractive sink to the proteins. For the other cases shown, however, the initial adsorption is determined by the time that it takes the proteins to cross the repulsive barrier presented by the polymer layer. We can see in the potentials that both the range and magnitude of the repulsion increases with polymer chain length. Therefore, the initial adsorption will be slower as the polymer-chain length increases. Also, we do not expect a fast regime in the initial adsorption whenever the potential of mean-force at t = 0 shows a maximum.

The solution of the generalized diffusion equation is obtained by integrating Eq. 1 with the initial condition mentioned above, and by using the potentials of mean-force presented in Fig. 2. However, once we integrate the first time-step the distribution of proteins changes and thus the potential of mean-force will also change. That is, it will correspond to the minimal free energy of the polymer-solvent mixture in the frozen new configuration of the proteins. Therefore, we expect the potentials of mean-force to be a strongly varying function of time. The complete kinetic behavior is presented in Fig. 3, where the amount of protein on the surface is plotted as a function of time. The figure shows the kinetics of adsorption for different cases: one in which there are no polymers grafted on the surface, with the rest representing surfaces with grafted polymers with the same surface coverage, but different molecular weights.

View larger version (24K): [in this window] [in a new window]   FIGURE 3  Variation of the amount of protein adsorbed as a function of time for surfaces with (and without) grafted polymers. The lines are the results from the generalized diffusion approach. The lines with symbols are the results from the kinetic theory approach. In all cases, l2 = 0.01. The bulk protein volume fraction is p, bulk = 0.001. The different chain lengths are denoted in the figure.

Fig. 4 shows the potential of mean-force for four different stages of the adsorption for n_g = 50. The height of the barrier increases, and it moves toward the surface as more proteins adsorb. Furthermore, the range of the repulsive interaction increases. This is the result of the deformation of the polymer layer as the proteins adsorb. The changes in the structure of the polymer layer and the volume fraction profile of the proteins at the same stages of the adsorption are also shown in Fig. 4. There is a clear push of the polymer segments close to the surface to move toward the solvent as the proteins adsorb. This is due to the need of the protein to have enough room on the surface to adsorb. The main message of the figure is that to properly describe the kinetic process, the deformation of the polymer-protein layer in the vicinity of the surface has to be taken into account. This is the contribution responsible for the very large variation of the potential of mean-force, and thus the rate of adsorption, with time.

View larger version (17K): [in this window] [in a new window]   FIGURE 4  (A) The potential of mean-force at four different stages of the adsorption for the case of ng = 50 shown in Fig. 3. (B) The volume fraction profile of grafted polymers, and the volume fraction profile of proteins (inset) corresponding to the same four stages shown in A.

The description of the kinetics with the generalized diffusion model enables a rather detailed molecular description of the adsorption process with its associated structural changes. The problem is that the solution of the numerical equations is very demanding. First, the kinetic process requires the solution of the differential equations over 16 orders-of-magnitude in time. Second, at each time-step the determination of the chemical potential gradients is obtained by solving a set of coupled nonlinear equations with thousands of terms in each one, i.e., the constrained free energy minimization (see Appendix). The calculations for long chain lengths are practically impossible; as an example, the calculation for n_g = 50 presented in Fig. 3 takes many days of computer time. Moreover, there are cases where the time evolution is so slow that we cannot reach the equilibrium state (see, e.g., n_g = 100 in Fig. 3). Therefore, a more practical approach is needed, such as the one presented next.

	KINETIC THEORY (KA)

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Adsorption kinetics
We find the most problematic cases to solve with the above approach are those in which the barriers in the potential of mean-force are very high. These are also the cases in which we find a two-state protein distribution. Thus, we present a kinetic model in which the rate-determining step is the crossing of the barrier. This means that we look at a kinetic equation for the transition of proteins between the bulk and the adsorbed state given by

(16)

where the first term on the right-hand side of Eq. 16 is the gain term associated with the increase of proteins on the surface due to the adsorption, and the second is the loss term due to the desorption of the proteins.

The adsorption and desorption are activated processes, as depicted qualitatively in Fig. 1. Therefore, the kinetic constants should have the form of

(17)

where U_ads(t) = U_mf(z = z^max;t) – U_mf(bulk) is given by the difference in the potential of mean-force between the maximum of the barrier height and the bulk. The desorption Boltzmann factor, U_des(t) = U_mf(z = z^max;t) – U_mf(z = 0;t), is the difference between the potential at the maximum and that of the adsorbed state, i.e., the potential at contact with the surface (see Fig. 1).

Based on the full solution of the diffusion equation we assume that the potential of mean-force depends on the structure of the polymer-protein layer, when the proteins are only at the surface. Therefore, we can calculate the potentials of mean-force at all possible adsorbed densities from zero to the equilibrium value, meaning that we solve the equilibrium problem for the polymer-solvent with a fixed amount of protein _ads on the surface. This allows us to calculate U_mf(z;_ads), which we will use to determine the necessary energies for the Boltzmann factors in Eq. 17. Once we know the potential of mean-force as a function of _ads and the distance from the surface, z, we can determine the height of the potential barrier and the potential at contact. Using our choice of zero for the potential in the bulk, U_mf(bulk) = 0, we know the three values of the potential of mean-force needed to calculate the rate coefficients.

The next step is to determine the preexponential factors in the rate coefficients. To this end we use the ideas of Halperin (48), who calculated the initial rate of adsorption of proteins on surfaces with grafted polymers using an extension of Kramer's theory of chemical reactions. We follow his approach but we explicitly include the time variation of the parameters determining the preexponential factor, thus allowing the complete treatment of the adsorption process from its initial condition up to the approach to equilibrium.

According to Kramer's theory, the preexponential factor is given by

(18)

(see derivation in the Appendix), where D is the diffusion constant; is the width of the potential at a distance k_BT below the maximum; and L is the distance a protein in the bulk state has to travel to reach the barrier maximum, which can be approximated by the thickness of the polymer layer. Both and L depend upon the molecular structure of the polymer layer. Therefore, they also depend on time through the changes in structure of the combined polymer-protein layer as a function of the amount of adsorbed proteins. As we have done with the barrier of the potential and its value at contact, we can determine and L as a function of the amount of protein adsorbed. Thus, we can have the implicit dependence of the kinetic coefficients k_ads and k_des on time through their explicit dependence on the amount of protein adsorbed.

Note that the rate coefficients as defined fulfill microscopic reversibility at all times., i.e.,

(19)

This result is consistent with our local equilibrium approximation. The symmetry arises from the approximations used in the derivation of the flux (see the Appendix for details and discussion). Other choices for the preexponential factors will not affect any of the results presented for the adsorption, since the kinetics of adsorption is dominated by the flux toward the surface. Furthermore, as it will be shown below, the excellent agreement between the predictions of the KA and the full GDA supports the validity of this approximation.

Fig. 4 demonstrates the very large changes that the polymer-protein layer structure undergoes through the adsorption process. Thus, it is clear that the proper quantification of the kinetic process requires the consideration of the explicit density-dependence, and thus the implicit time-dependence, of the quantities U_mf (z = z_max;_ads (t)), U_mf (z = 0;_ads (t)), (_ads(t)), and L(_ads(t)). Examples of the first two quantities can be clearly seen in Fig. 4. We now discuss the four quantities in more detail, since they will provide insightful physical information on just what the roles of surface coverage and polymer-chain length are in the kinetic process.

Fig. 5 shows the maximum in the potential of mean-force as a function of the amount of adsorbed proteins for a variety of polymer-chain lengths. Also shown is the case of the bare surface. For short chain lengths (including the bare surface, n_g = 0), a barrier larger than the thermal energy appears only after a finite amount of proteins adsorb (see Fig. 5, inset, and the explanation below). However, for long enough chain length (n_g 40 for the surface coverage shown in the figure), there is a barrier even when there are no proteins adsorbed. The variation of the maximum of the potential of mean-force with density reveals which one is the dominant contribution in determining the kinetic barriers: the polymer, the protein, or both. In all the regimes where the maxima are parallel, then, it is the protein that determines the variation of the potential. Note the reason that curves are parallel, and not identical, is that there is a background contribution of the polymer layer which is strongly dependent on chain length. For the regions in which the curves are not parallel, the polymer contribution is dominant. Thus, there is a very different maxima at low protein densities for the longest polymer chain lengths shown, where the polymer effect on the kinetics is large.

View larger version (18K): [in this window] [in a new window]   FIGURE 5  (A) The maximum in the potential of mean-force as a function of the density of proteins adsorbed on surfaces with grafted polymers for different chain lengths: ng = 0 (bare surface, solid line); ng = 15 (large dots-dashed line), ng = 25 (dot-dashed line); ng = 30 (double-dash-dotted line); ng = 40 (large dot-solid line); ng = 50 (long-dashed line); and ng = 100 (dotted line). In all cases, l2 = 0.01. The thin-dashed line at marks the minimum potential that a kinetic barrier must present. (B) The inset presents the minimum density of protein that has to be adsorbed for the formation of a kinetic barrier as a function of grafted polymer-chain length.

The kinetic approach cannot be directly applied to the initial adsorption kinetics if there is no barrier in the potential of mean-force at t = 0. This is the case for the polymer chains shorter than 35 segments, as shown in Fig. 5. In the absence of kinetic barriers, the solution of the GDA presented in the previous section is not computationally demanding. However, when the adsorption mechanism crosses over to be dominated by the crossing of the barrier, then we need to switch to the KA (see, e.g., n_g = 25 in Fig. 3). Thus, for practical purposes, it is important to have an amount of density adsorbed (as a function of polymer-chain length), showing a kinetic barrier larger than the thermal energy (see Fig. 5, inset). The use of this graph is that, for the surface coverage shown in the region below the curve, the kinetics of adsorption is obtained from the full solution of the GDA, whereas above the curve the practical approach is to use the KA. Clearly, the different regimes depend on the surface coverage of polymer and the specific protein studied. However, once those parameters are defined, we can calculate a curve, such as that shown in Fig. 5's inset, to find the proper approach to apply in each case.

Fig. 6 shows the values of the width of the steric barrier at t = 0 and at the end of the adsorption process as a function of grafted polymer-chain length. There is a strong dependence of on n_g at t = 0 due to the specific structure of the polymer layer and its changes with the molecular weight of the polymer. The width of the potential barrier, however, is almost independent of the molecular weight of the polymer at the end of the adsorption process, and it is much smaller than its initial value. The data presented for the initial stage of adsorption in Fig. 6 presents the result for polymers shorter than 35 segments separately from the result for chains longer than 35 segments. The reason is that, below this molecular weight, there is no barrier in the potential of mean-force at t = 0 for the surface coverage shown (see Fig. 5, inset).

View larger version (10K): [in this window] [in a new window]   FIGURE 6  The width of the kinetic barrier, , before adsorption (solid line with circles) and after adsorption (dashed line with squares) as a function of chain length for surfaces with grafted polymers at l2 = 0.01. The solid line with diamonds represents the width of the kinetic barrier for the shorter polymer chains at the moment the kinetic barrier is just formed, i.e., after some proteins have been adsorbed.

The last variable that we need to discuss is the thickness of the polymer-protein layer L and its dependence on the amount of protein adsorbed. As it is well known from polymer brushes, the thickness of the layer varies linearly with the molecular weight of the polymer (1,57). What is interesting is that even after there is adsorption of proteins the change in the thickness of the layer is very small (results not shown). This is because the height of the polymer layer is not a very sensitive function of the local changes of the molecular organization in the region closed to the grafting surface. Thus, for practical purposes we can use the same L at all times; i.e., that for t = 0.

Desorption kinetics
We next consider the case in which, once the system reaches equilibrium, the protein solution in contact with the surface is washed-out and thus the adsorbed proteins in the grafted polymer layer are in contact with pure water. The equilibrium state will be such that all the proteins leave the surface since there is an infinite entropic gradient due to the zero concentration of proteins in solution. The kinetics of the process can, in principle, be studied with both the GDA and the KA. However, we found that the time evolution with the GDA is so slow that no calculations can be carried out. Therefore, we need to use the KA. We can write the kinetic equation for desorption in the form

(20)

where there is no gain term because the solution is protein-free. The desorption rate coefficient is given by

(21)

(see derivation in the Appendix), where the energy difference is as given following Eq. 17, and we have used the Kramer approach for the preexponential factor. Note that instead of L we have R in the denominator of the preexponential factor. This is because the maximum of the potential of mean-force is located at a distance from the surface of the order of the protein size, as shown in Fig. 4. Thus, the desorption process measures the protein going from the surface to the maximum in the potential, i.e., a distance R from the surface. Interestingly, the width of the potential around the maximum for the desorption process is independent of chain length. However, it is a function of the amount of protein adsorbed and polymer surface coverage. The determination of and U_des as a function of time is obtained from the knowledge of these quantities as a function of the amount of density adsorbed, along the same lines as the KA is applied for the adsorption process (e.g., see Fig. 5).

Integration methodology
The equation for the adsorption and desorption kinetics, as derived from the KA, requires four parameters as a function of time. They are the potential of mean-force at contact, U_mf(z = 0, t); the maximal value of the potential of mean-force, U_mf(z = z*,t); the width of the potential of mean-force of 1 k_BT below the maximum, (t); and the thickness of the film, L(t). The fifth quantity, U_mf(bulk), is independent of time. From the four quantities we have shown that the thickness of the film (or the radius of the protein for the desorption process) does not vary with time, and therefore we take its value at time t = 0. For the other three quantities we know their values as a function of the amount of protein on the surface, which we tabulate before starting the kinetic calculations. Thus, we start the integration at time t = 0 where we know all the necessary values and we integrate the kinetic equations one time-step. The integration gives the value of the density of adsorbed proteins at the new time. We use this density value to find the three parameters, from the tabulated values, and integrate the kinetic equations another step. We continue this iteration of finding the new density, obtaining the value of the kinetic coefficients at the new density and integrating a new time-step until we reach the equilibrium state.

	REPRESENTATIVE RESULTS FOR KA

TOP ABSTRACT INTRODUCTION MOLECULAR THEORY GENERALIZED DIFFUSION APPROACH... FREE ENERGY FUNCTIONAL MOLECULAR... REPRESENTATIVE RESULTS FOR GDA KINETIC THEORY (KA) REPRESENTATIVE RESULTS FOR KA DISCUSSION AND CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
The first question that arises relates to the quality of the results obtained from the kinetic theory as compared to the diffusion approach. Fig. 3 shows the predicted kinetic curves for both cases. In the case of the KA, the curves are calculated in the valid region as denoted in the inset of Fig. 5. The agreement between the full calculations and the more approximate approach is very good. The shape of the adsorption curves and the magnitudes are very well reproduced for all grafted polymer-chain lengths and in the whole range of time in which a barrier is present. This implies that the physical mechanism for protein adsorption on surfaces with grafted polymers is indeed what we have assumed; that is, once there is a kinetic barrier, the time-determining step is that of crossing the barrier. However, it is imperative to explicitly include the variation of the parameters as the proteins adsorb to properly describe the whole kinetic process—i.e., the polymer-protein layer deformation determines the shape of the time-dependent adsorption.

It is important to emphasize the difference in the computational effort necessary to solve the KA as compared to the GDA. For the case of n_g = 50 there is a factor of 10⁶ between the two calculations. Furthermore, there are many cases in which the GDA needs to be integrated with a very small time-step, and therefore the calculations cannot be completed at all (see, e.g., the curve for n_g = 100 in Fig. 3). However, for the KA, the calculations are very simple; in essence, for each time-step, the solution is that of a simple first-order differential equation.

The reliability of the results from the KA gives us confidence to apply it where the GDA is not practical due to the computational limitations. Thus, we now study the effect of polymer-chain length and surface coverage. Fig. 7 shows the kinetics of adsorption for three different surface coverages of polymers and three different chain lengths. The figure shows that increasing both the chain length and the surface coverage results in slower kinetics. Interestingly, as we have shown elsewhere (30,31,34), for the three chain lengths shown, the equilibrium adsorption is almost independent of polymer chain but depends on surface coverage. However, the kinetic process slows, by orders of magnitude, with both chain length and surface coverage.

View larger version (9K): [in this window] [in a new window]   FIGURE 7  The amount of proteins adsorbed as a function of time for different grafted polymer-surface coverages: (A) l2 = 0.01, (B) l2 = 0.02, and (C) l2 = 0.03. The different curves in each graph represent different chain lengths: ng = 50 (solid line), ng = 100 (dashed line), and ng = 150 (dotted line). In all cases, the bulk protein-volume fraction is p, bulk = 0.001.

As the surface coverage of grafted polymer increases, the chain molecules become more stretched. This results in several effects on the protein adsorption. First, the layer is more protein-resistant, because there is less room for the proteins to adsorb. Second, the barriers for adsorption become larger, and therefore display slower adsorption kinetics. Third, the polymer molecules have a large degree of stretching due to the interpolymer repulsions; therefore, the degree of polymer layer deformation upon protein adsorption is smaller than for lower . Thus, variation of the effect of polymer chain length on the kinetics as a function of time is much less pronounced at high than at lower surface coverage. The result is that the adsorption kinetics looks the same at different stages (for different chain lengths) and the lag between the curves is maintained throughout the adsorption. Then, we see that for l² = 0.01, there is a difference in the initial adsorption time of six orders-of-magnitude between n_g = 50 and n_g = 100, but less than one order-of-magnitude to reach equilibrium. On the other limit for l² = 0.03, we find a difference of seven orders-of-magnitude in time for chain lengths 50 and 100, which is maintained until the system reaches thermodynamics equilibrium.

The question that arises, however, is what is the significance of treating systems in which the equilibrium is reached in 10¹⁴ s, or equivalently, 10⁶ years? The reason for showing these results is that unless we do the calculation we do not know the timescale for adsorption. The calculations with the KA are rather simple, and we can gain insights into why the timescale is so long and how chain-length variations have different behavior at different surface coverage. Further, if we just perform an equilibrium calculation we obtain that there is a finite adsorption even for n_g = 150 and l² = 0.03. This suggests that such a high surface coverage should not be enough, say, for a biocompatible material to completely prevent protein adsorption. However, the timescale for even the initial adsorption is predicted to be so large that for all practical purposes, this coating of the surface should completely prevent protein adsorption for all relevant experimental times.

In reality, we do not need to calculate the whole kinetic process, to see that the timescale of the adsorption process is infinitely slow for practical purposes. This is one of those cases where the initial adsorption time is all we need. If the time for initial adsorption is short enough within the experimental timescale, then we can perform the whole calculation. To show the dependence of timescales on molecular weight and surface coverage, Fig. 8 displays the maximum of the potential of mean-force before any adsorption take place as a function of polymer surface coverage for three different molecular weights of polymer. The maximum is found to have a close-to linear dependence on surface coverage, and the slopes depend on polymer molecular-weight.

View larger version (13K): [in this window] [in a new window]   FIGURE 8  (Top) The maximum potential of mean-force as a function of surface coverage when no proteins are adsorbed for different chain lengths: ng = 50 (solid line), ng = 100 (long-dashed line), and ng = 150 (dotted line). (Middle) The thickness of the grafted polymer layer, L (dot-dashed line with diamonds), the width of the kinetic barrier, (long-dashed line with squares), and the product L (solid line with circles), before adsorption as a function of polymer surface coverage for ng = 50. (Bottom) The initial rate constant kads(t = 0) as a function of surface coverage for the three chain lengths shown in the top panel. The thin dotted line marks the rate that corresponds to a time constant of 10 h, i.e., 36,000 s.

L

L

The initial rate constants are shown in the bottom graph of Fig. 8 as a function of surface coverage for three polymer chain lengths. There is a very sharp decrease of the constant with surface coverage. The figure also includes a line for a rate that corresponds to a time constant of = 1/k_ads = 10 h. This is an arbitrary cutoff, but it is shown to demonstrate how, given a desired timescale for prevention of adsorption, one can use the figure as a design tool in terms of the molecular weight and surface coverage necessary to graft on the surface. From the figure, one can see that a longer chain length requires a much smaller surface coverage. Interestingly, if we would use the equilibrium adsorption as a design tool we will need the same surface coverage for the three molecular weights, since the isotherms are independent of molecular weight for the range of chain lengths shown in Fig. 8 (30).

Fig. 9 shows the amount of protein on the surface as a function of time for the desorption process. The starting point of the desorption is the equilibrium achieved under the conditions shown in Fig. 3. The desorption kinetics shown correspond to the cases of surfaces without grafted polymer and a variety of surfaces with grafted polymers of different chain length, all at the same surface coverage. The initial time for desorption is very long in all cases. Further, the timescale for the initial desorption of proteins from the surface is much longer than the initial time for adsorption (compare Fig. 9 with Fig. 3). These results are due mostly to the strong protein-surface attraction upon contact, which leads to strong kinetic barriers for desorption.