INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORETICAL APPROACH RESULTS CONCLUSIONS REFERENCES

Protein adsorption plays a major role in a variety of important technological and biological processes (Clerc and Lukosz, 1997; Denizli et al., 2000; Ghose and Chase, 2000; Hlady and Buijs, 1996; Montdargent and Letourneur, 2000; Shi and Ratner, 2000; Slomkowski, 1998; Topoglidis et al., 1998). For example, blood proteins tend to adsorb into surfaces of foreign materials. This is the first step on surface-induced thrombosis (Andrade and Hlady, 1986; Horbett, 1993; E. F. and S. 1993; Tanaka et al. 2000). A large number of biotechnological devices include surface-bound proteins, e.g., biosensors (Nyquist et al., 2000; Slomkowski et al., 1996; Sukhishvili and Granick, 1999; Zhou et al., 2000). Separation of proteins by chromatography involves the competitive adsorption of the particles (Wang 1993). The understanding of the fundamental factors that determine protein adsorption are imperative to improve our ability to design biocompatible materials and biotechnological devices. Moreover, protein adsorption is a very important fundamental problem that involves large competing energy scales and conformational statistics that may result in reversible and irreversible processes.

The adsorption of proteins on surfaces is a complex process. The adsorbing particles are large, and, thus, the surface-protein interactions are usually long range and the strength is many times the thermal energy. Further, due to the large size and the shape of the particles, the interactions between the adsorbed particles on the surface are nontrivial and can be strongly influentiated by the fact that the particles may undergo conformational changes upon adsorption (Billsten et al., 1995; Ishihara et al., 1998; Kondo and Fukuda, 1998; Nasir and McGuire, 1998; Norde and Giacomelli, 1999, 2000; Tan and Martic, 1990; Van Tassel et al., 1998; Gidalevitz et al., 1999). Actually, the kinetics and thermodynamics of protein conformational changes on the surface is a very complex subject and their understanding is at its early stages. The idea behind the work presented here is an attempt to formulate a molecular theoretical approach that can be applied to study both the equilibrium and the kinetic behavior of protein adsorption.

On experimental studies (Green et al., 1999; Malmsten, 1997), it has been observed that, when two or more kinds of proteins are present in solution, such as in blood plasma, the adsorption is the result of the competition between the time scale to reach the surface and the strength of the surface-protein interaction. For example, in blood plasma solutions of albumin, immunoglobulin-G (IgG) and fibrinogen (Fgn) in contact with a polystyrene surface, the initial adsorption is dominated by the smaller protein (albumin), which are also at larger concentrations in the bulk, to be later replaced by the larger proteins like IgG and Fgn. This sequential adsorption is called the Vroman sequence. In other experiments (Lassen and Malmsten, 1997), different adsorption patterns are observed when the surfaces are changed. On the hydrophobic PP-HMDSO (hexamethyldisiloxane), surface albumin and IgG dominate the adsorption. However, on hydrophilic PP-DACH (1,2-diaminocyclohexane) and PP-AA (acrylic acid) surfaces, Fgn is almost exclusively found on the surface. These experimental observations demonstrate that the incorporation of the solution conditions and the protein-surface interactions have to be considered for the proper understanding and description of the adsorption process.

One of the most important contributions to the understanding of the kinetics of protein adsorption is the random sequential adsorption (RSA) model (Feder and Giaever, 1980; Schaaf and Talbot, 1989). In this approach, the proteins are assumed to be rigid particles that interact only through excluded volume interactions. The particles are assumed to irreversibly adsorb to the surface, and, thus, they do not have translational degrees of freedom or desorption on the surface. This model has been very useful in understanding why the kinetics of protein adsorption do not follow the Langmuir predictions. Furthermore, the model has been extended to consider conformational changes, desorption, and the treatment of mixtures (Van Tassel et al., 1994, 1996, 1998). The main limitation of this model is that it is hard to include detailed molecular information of the proteins and the formulation is based on a kinetic approach.

Some other studies have assumed that the adsorption kinetics is determined by the diffusion of the proteins to the surface (Iordanskii et al., 1996), whereas others assume that the dominant regime is the one controlled by a kinetic (activated) process (Chatelier and Minton, 1996; Minton, 1999). In a recent study, Cho et al. (1997) formulated a model in which both the diffusion and kinetic processes were included. Olson and Talbot (2000) studied the equilibrium and kinetics of adsorption of a polydisperse mixture. Each of these models has provided important insights toward the understanding of the adsorption process. However, none of them can describe both the equilibrium and kinetics of the adsorption process within the same molecular approach that can be applied for a large variety of experimental systems.

The theory that we use in this paper is based on the formulation of the free energy of the system. The minimization of the free energy provides the equilibrium state of the system, and, thus, we can study the protein adsorption isotherms. Furthermore, the free energy formulation enables the study of possible conformational changes of the protein on the surface. The equilibrium version of the theory for protein adsorption was originally formulated to study the ability of grafted polymer layers to prevent, or reduce, protein adsorption (Szleifer, 1997b). The predictions of the theory were shown to be in excellent quantitative agreement with experimental observations for the equilibrium adsorption isotherms of lysozyme on surfaces with grafted polyethylene oxide layers (McPherson et al., 1998; Satulovsky et al., 2000). The theory was later generalized to study the kinetics of the adsorption process in the same systems (Satulovsky et al., 2000). The basic idea in the dynamic version of the theory is to start with an equilibrium bulk system that, at time zero, is put in contact with a surface. The presence of the surface induces a distance dependent chemical potential of the proteins. The free energy of the new system is formulated, but instead of minimizing to obtain the new equilibrium state in the presence of the surface, the time evolution of the density of proteins is evolved with a diffusion-like equation, with the driving force being the gradient of chemical potentials arising from the sudden presence of the surface. These chemical potentials are obtained as derivatives of the time-dependent free energy with respect to the local density of proteins. Similar approaches were used for the adsorption of surfactants (Diamant and Andelman, 1996) and polymers (Fraaije, 1993; Hasegawa and Doi, 1997). Recently, it has been shown that this kind of dynamic equations can be derived for the time dependence of the density from density functional theory (Marconi and Tarazona, 1999).

In this paper, we are interested in using the same theoretical approach but to the study of protein adsorption on bare surfaces. The idea is to understand what are the parameters that determine the different dynamic regimes. Further, we are interested in studying in detail the effect of conformational changes on the kinetics of adsorption and also the adsorption of proteins mixtures.

The paper is organized as follows: the next section contains a description of the theoretical methodology, including a detailed presentation of the way the equations are solved. The following section present a variety of representative results. Finally, the last section includes our conclusions.

	THEORETICAL APPROACH

TOP ABSTRACT INTRODUCTION THEORETICAL APPROACH RESULTS CONCLUSIONS REFERENCES

In this section, we present our theoretical approach to study the equilibrium and kinetic properties of the adsorption of proteins to planar surfaces. We will present a general theoretical framework for the determination of equilibrium adsorption isotherms in the case of protein mixtures. The treatment explicitly includes the possibility that the proteins have many different configurations. The second part of this section presents the dynamic theory that we use to study the kinetics of protein adsorption.

After the presentation of the general thermodynamic and kinetic approaches, we will show the specific cases for which we present explicit calculations below. Namely, the adsorption of proteins that are assumed to have a single configuration in the bulk but that can undergo conformational changes upon contact with the surface and those assumed to be a mixture of proteins of different sizes for a variety of different bulk conditions and surfaces. Following the model, we present details on the numerical methodology used in solving the equilibrium and kinetic equations.

Equilibrium free energy

Consider a surface of total area A in contact with a protein solution, Fig. 1. The solution is composed by a mixture of proteins characterized by a bulk chemical potential µ, with i denoting the type of protein. Equivalently, we can represent the properties of the protein solution by the density of molecules . Each protein can be in any of its possible configurations. We denote the set of configurations of protein of type i by {_i}. Let us define by P(_i; z) the probability distribution function (pdf) of proteins of type i to be in configuration _i at distance z from the interface. The pdf can also be thought as the conditional probability that a protein of type i at distance z from the surface is in conformation _i.

View larger version (38K): [in this window] [in a new window]   FIGURE 1   Schematic representation of the system containing a mixture of proteins dissolved in a low-molecular-weight solvent in contact with a surface. The filled circles are protein molecules with different sizes and the empty circles are solvent molecules. The z direction is defined perpendicular to the surface. The protein at position z' represents the molecules with their point of shortest distance with the surface being z'.

The relevant surface free energy density (per unit area) of the system (Rowlinson and Widom, 1982), assuming inhomogeneities in density only in the direction perpendicular to the surface, z, is given by

(1)

where the first and second terms represent the z-dependent translational (mixing) and the conformational entropy of the proteins, respectively. The third term is the intramolecular energy of the proteins. The fourth term includes the average interaction between the protein at z with the surface, U_ps(_i; z) is the interaction between the protein i in configuration _i with the surface. The fifth term is the protein-protein attractive interactions. _ij(|z  z'|) represents the strength of the interactions between protein in configuration _i at z and protein in configuration _j at z'. The sixth term is the chemical potential term necessary because we consider the surface in equilibrium with a bulk solution, i.e., the surface is in contact with a bath of proteins. The last two terms represent the solvent contribution, which include the translational (mixing) entropy and the chemical potential terms. _s(z) and µ_s represent the volume fraction at z and the chemical potential of the solvent molecules, respectively. Note that the argument of the first ln term in Eq. 1 contains the volume of the solvent to make the product dimensionless. Further, we will use v_s as the unit of volume throughout.

Inspection of Eq. 1 shows that the repulsions between the molecules are not included in the free energy expression. These interactions are accounted for by packing constraints. Namely, for each distance z from the surface, the volume available between z and z + dz is filled by the proteins or the solvent molecules. Thus, the volume constraint equation reads

(2)

where the first term represents the volume fraction that the proteins occupy at z, and the second term is the volume fraction of solvent. Note that the volume fraction of proteins includes the sum over all the molecules at different distances from the surface (z') that contribute volume to z. v(_i; z', z) dz' is the volume that the protein in configuration _i at z' occupies at z.

The next step is to determine the density of proteins and solvent as a function of z and the pdf of protein configurations. The systems free energy is a functional of (z), _s(z), P(_i; z). These quantities are found by minimization of the systems free energy, Eq. 1, subject to the packing constraints, Eq. 2. The minimization is carried out introducing a set of Lagrange multipliers, (z), to yield for the pdf of the protein configurations

(3)

where q_i(z) is the normalization constant that ensures for each z that  P(_i; z) = 1. The partition function is given by the sum over all the configurations of the exponential term in Eq. 3.

The density profile of proteins of type i is

(4)

and, for the solvent volume fraction, we have

(5)

The only unknowns are the Lagrange multipliers, which are obtained by replacing the explicit expressions for the pdf and density profiles, Eqs. 3, 4, and 5, into the constraint equation, Eq. 2. The explicit form of the equations solved will be described below for the specific model systems that we present in the Results section. The physical meaning of the Lagrange multipliers can be understood by looking at the expression for the solvent density profile, Eq. 5. Writing this expression in the form,

(6)

shows that the Lagrange multipliers are related to the (z-dependent) osmotic pressure necessary to keep the chemical potential of the solvent constant at all z.

The expressions for the density profiles and the pdf of the proteins enable us to understand what are the factors determining the equilibrium amount of protein adsorbed and the optimal adsorbed conformations. The partition of proteins as a function of the distance from the surface is determined by the thermodynamic equilibrium condition of constant chemical potential at all z. Thus, we can write Eq. 4 in the form

(7)

which requires the chemical potential of the proteins at all z to be that of the bath, i.e., the value given by the bulk solution. The amount of protein of type i on the surface (z = 0) is determined by the value of the partition function on the surface, q_i(0). Thus, the partition function and the density at the surface, through Eq. 7, will be determined by the interplay between the interactions that increase the value of the partition function and those that reduce it. The attractive components (which increase q_i(0)) are the bare surface-protein interaction and the protein-protein van der Walls attractions. The repulsions (which decrease q_i(0)) are those determined by the pressure-volume-like term (PV), given by the product of the lateral pressures (z) by the volume of the protein as a function of z. This repulsive term is associated with the PV work necessary to bring the protein from the bulk solution to the surface. Thus, it is not enough to have a strong attractive interaction with the surface for a protein to preferentially adsorb, its volume distribution should be such that the repulsions are not too large. The same type of argument is obtained to explain the preferential adsorption of a given conformation. To this end, it is convenient to define the density of proteins at z in conformation _i, by multiplying the pdf of that conformation, Eq. 3, by the density of proteins of type i at z, Eq. 4, to obtain

(8)

This expression shows that the condition of equal chemical potential at all z has to be fulfilled for each protein configuration. Further, note that the value of constant chemical potential for each configuration is that of the bulk protein.

We can rewrite the equilibrium condition for each protein conformation in the form

(9)

where

(10)

is the potential of mean force (Chandler, 1987) between the protein, in conformation _i at distance z, and the surface. Namely, it is the work required to bring the protein in conformation _i from the bulk to the distance z from the surface. This way of writing the chemical potential enables the understanding of the factors that determine the type of conformation and protein that adsorbs on the surface, and it will be useful in the kinetic description presented in the next section. Note that the potential of mean force, and the last term in the solvent chemical potential Eq. 6, are the excess (or nonideal) contributions to the chemical potential.

Using the definition of the potential of mean force, we can see that the requirement of constant chemical potential, and thus what determines the amount of proteins in each conformation that are adsorbed, depends on the cost (or gain) of bringing a protein from the bulk solution to contact with the surface. There are four contributions that determine the potential of mean force. 1) The internal energy of the conformation. This term is independent of z. 2) The bare surface-protein interaction. This is usually a strongly attractive term. 3) The intermolecular repulsive interaction term. This term becomes more prominent as the density increases and therefore favors small densities at the surface. 4) The intermolecular attractive term, which favors large densities. The interplay between these contributions will determine the amount and type of conformation that will adsorb on the surface. Further, the manipulation of these contributions may lead to an enhanced (or decreased) adsorption and thus control of the amount and type of protein adsorbed (Szleifer, 1997a).

In the Results section, we will show explicit examples for how the interplay between the different interactions determines the optimal protein and conformation adsorbed. Further, we will discuss how this understanding can lead to the design of surfaces or conditions for optimal adsorption.

Equations of motion

We now treat the process of how the proteins in solution adsorb into the surface. Consider a solution containing a mixture of proteins at bulk densities (or equivalently chemical potential µ) dissolved in a low molecular-weight solvent. This homogeneous solution is in equilibrium, and, at time t = 0, is brought in contact with a layer of pure solvent that is in contact with a surface on the other end. The direction perpendicular to the surface is denoted as the z direction. A schematic view of the system is shown in Fig. 1.

The contact between the pure solvent and the protein solution induces the diffusion of the proteins toward the pure solvent. Further, the sudden presence of the surface implies that the proteins now feel an anisotropic interaction due to the bare protein-surface attractions. Therefore, the chemical potential of the proteins closer to the surface is not the same as that of the proteins in the bulk (far from the surface). The nonconstant chemical potential of the proteins as a function of z is the driving force for mass transport. Further, the protein-surface interaction and the motion toward the surface will depend upon the conformation of the protein.

The time evolution of the density of proteins of type i in conformation _i at distance z from the surface, (z, t), contains two contributions. The first one is the transport of the same conformation from neighboring distances. The second is from conformational changes of proteins at distance z from the surface. The transport can be described with a generalized diffusion equation, and the conformational changes can be written as kinetic master equations. The result is

(11)

where the first term represents the mass transport. D_i is the diffusion coefficient of proteins of type i in conformation _i, which is assumed to be composition independent; µ(z; t) is the time-dependent chemical potential, defined as an extension of the equilibrium quantity. Namely, we define

(12)

where W/A is the time-dependent free energy per unit area of the system. For the time-dependent free energy, we use the same expression as the equilibrium quantity, but the protein densities are not the ones that minimize the free energy but are given by the values obtained by the time-evolution equation.

The last two terms in the kinetic equation, Eq. 11, represent the time-dependent conformational changes. There is a gain and a loss term. The gain term arises from all the conformations '_i that can undergo a conformational change to configuration _i. The last term represents the conformational change from _i to any possible configuration. The constants k('_{i  i}) represent the intrinsic rate of conformational change of the protein from '_i to _i. Namely, it is the rate associated with the conformational change of the protein in the presence of pure solvent. The factor ('_{i  i}; z) represents the effect of the intermolecular and surface interactions to the rate of conformational change from '_i to _i. This term can be interpreted as the probability of finding the necessary space for the conformation to change from '_i to _i, modulated by the appropriate energetic gain or loss. This probability is related to the work necessary to change the conformation in the given environment. In the terms defined in the previous section, this quantity will be the Boltzmann factor of the interaction difference between the two conformations in the given environment at z and t. This quantity is readily obtained from the theory by using the third term in Eq. 10 with the temporal densities obtained from the dynamic equations. Note that this term will depend very strongly on the density distribution, and, therefore, will be a function of time. We will show some explicit examples below.

The boundary conditions to solve the dynamics equation is that the gradient of chemical potential at the surface (and in the bulk solution) is zero. Namely,

(13)

This boundary condition at z = 0 is, in reality, the condition that the molecules cannot diffuse behind the surface, i.e., to negative values of z.

At this point, it is important to emphasize the difficulties associated with treating realistic proteins. Eq. 11 requires the knowledge of the rate of change of the protein conformations from one to another. This is a formidable task, considering the fact that even the conformational space of real proteins cannot be properly sampled with the techniques and computer resources available today (Chan and Dill, 1998; Scheraga, 1996; Yue et al., 1995; Brooks et al., 1998). Thus, we need to use simplified models. However, these simplified models are based on the behavior of real proteins. For example, in many cases, proteins in bulk exist in a small set of conformations that are close to the native structure. Thus, the description of a single conformation of the protein in bulk is a reasonable approximation. There is clear experimental evidence that proteins undergo conformational changes upon adsorption on surfaces and interfaces (Billsten et al., 1995; Ishihara et al., 1998; Kondo and Fukuda, 1998; Nasir and McGuire, 1998; Norde and Giacomelli, 1999, 2000; Gidalevitz et al., 1999; Tan and Martic, 1990; Van Tassel et al., 1998). There are two kinds of configurational changes that can happen upon adsorption. One of them corresponds to the denaturation of the protein from the native configuration to a random coil. In the second, the protein undergoes a conformational change to a very small subset of conformations that are as unique as the native configuration but with a different structure. Recent extensive calculations in a simple model system strongly suggests that the second one is the most common case for solid surfaces (R. Abdulla, Jr. and I. Szleifer, manuscript in preparation). The calculations presented below correspond to this second case. It is important to emphasize that the rate constants and the protein conformations are input to the theory. Thus, even in the case of multiple adsorbed configurations, if those data are available, the kinetic theory can be applied without any major additional complications.

To understand the time-dependent adsorption, it is useful to look at each of the contributions separately. We start with the mass transport part. The driving force for this motion is the gradient in (time-dependent) chemical potentials. We can use the analog of Eq. 9 for the time-dependent chemical potential to obtain

(14)

Replacing this expression into the transport part of the equation of motion, we obtain

(15)

The first term in the rhs of the equation is the regular diffusion term and it arises from the ideal term in the free energy. The fact that we explicitly consider the interactions between the molecules and between the proteins and the surface results in the additional term to the transport equation. Thus, the motion of the proteins is driven by the effective interactions between the particles and the surface. The time scale for the diffusion process will depend on the explicit form of the potential of mean force, U_mf(_i; z; t). As we will show, this quantity undergoes dramatic changes as a function of time, and, thus, the adsorption process changes character.

Throughout the discussion in the Results section, we refer to two distinct dynamic regimes. We call them diffusion-controlled regime and kinetic (or activated) regime. The diffusion-controlled regime refers to the dynamic processes that are dominated by the first term in the rhs of Eq. 15. This will be the "ideal" diffusion driven exclusively by the gradient of densities. We also include in this regime the "driven" diffusion, which represents the motion that arises from the bare surface-protein interactions. The kinetic or activated regime is the one dominated by the nonideal contribution to the chemical potential arising from the intermolecular interactions. This term contains in it any kinetic barriers that appear in the system due to the repulsive interactions between molecules.

At this point, it is important to emphasize one of the main differences between our approach and the purely kinetic approaches that can be found in the literature. Even for the pure transport process, our theory describes the adsorption and desorption process at once. We do not need to include an explicit term that considers the possibility of desorption. Furthermore, according to our theory, there is only one elementary time scale measured by the diffusion constant. The different time scales for adsorption and desorption will depend upon the time and z dependence of the potential of mean force. Further, our approach warrants the approach to equilibrium. However, some types of irreversible adsorption can also be treated within the same framework, because, in that case, the time scale of the adsorption process will be slow in the experimental time scale.

It should be noted that, although we have emphasized the advantages of our approach, there are many limitations as well. The main one that we comment upon here is that the lateral dynamics (within a given z) are assumed to be instantaneous as compared to the diffusion to the surface, Namely (x, y, z; t) = (z; t) for all x, y. Although recent Brownian dynamic simulations have shown that this is a reasonable approximation (Ravichandran and Talbot, 2000), it is important to keep its limitation in sight. Additional important limitations will be discussed in the Conclusions section.

The second contribution to the time-dependent adsorption, see Eq. 11, arises from the ability of the molecules to undergo conformational changes. As mentioned above, this is a rather complex and yet barely understood process. Thus, we will use a simple model to understand the effect of conformational changes on the kinetics of adsorption. This will be the case in which the conformational change can only occur upon contact of the protein with the surface.

Eq. 11 shows the need to provide the rate constants for conformational transformations   ' and '  . However, because the system will reach thermodynamic equilibrium, only one is needed. The ratio of the rate constants is proportional to the product of the ratio of the conformation populations and the ratio of the repulsive factors at equilibrium.

In the next subsection, we describe in detail the model systems that we will study and the parameters used in the calculations. Further, we present the explicit sets of equations that we solve and the numerical methodology used.

Model systems

We consider a set of simple systems to apply the theory developed above for the study of the thermodynamic and kinetic properties of protein adsorption. We study two different kinds of systems. The first is a binary mixture of model proteins. Both proteins are modeled as spherical in shape and they differ in size and in their interactions with the surface. These proteins can exist in a single configuration even when they are adsorbed on the surface. The motivation to study this mixture is to understand competitive adsorption in which the proteins differ in size and surface interaction. Namely, we want to understand the underlying physical process that is responsible for the Vroman sequence (Green et al., 1999). Further, we are interested in the general properties of competitive adsorption and under what condition one should expect adsorption of one or the other species. Thus, we chose a model that contains the minimal ingredients to study these effects, without too many complications that may cloud the physical origin of the observed behavior.

The mixtures are composed by two protein-like particles. Because both particles can exist in only one configuration, we take U_int(_i) = 0. The larger particle is the same size as our previous model for lysozyme (McPherson et al., 1998). Namely, it is a particle with a radius of 15 Å. The potential of interaction between this protein and the surface is shown in Fig. 2. The distance dependence of the protein-surface interaction is taken from the atomistic calculations of the interactions between lysozyme and hydrophobic surfaces as calculated by Lee and Park (1994). However, the strength of the attraction is taken to be of the original calculated one. The reason for this choice is that the extensive kinetic calculations that will be shown in the next subsection are less computationally demanding for a weaker potential. Furthermore, we have found that the predictions of the kinetic and thermodynamic behavior is qualitatively the same, and, therefore, we can perform more systematic studies with the weaker attractive potential.

View larger version (10K): [in this window] [in a new window]   FIGURE 2   The distance dependence of the attractive interaction between one large protein particle and the surface in a binary mixture of model proteins. The interactions are measured in units of  = 1/kT, the distances are measured in units of  = D/5, where D is the diameter of the largest protein that we model.

The small particle has a radius that is that of the large protein. The distance dependence of the attractive interaction between the surface and the small protein is the same as that shown in Fig. 2. However, we vary the strength of the attraction in a wide range of values, as will be explicitly shown in the Results section. We assume that the solvent is equally good to both proteins. Thus, we model the intermolecular, protein-protein and protein-solvent interactions as purely repulsive. Namely, _ij = 0 for all _ij.

Some comments are needed here. The choice of purely repulsive interactions implies that all the attractive intermolecular interactions are the same and not that they are absent in the system. One can question the validity of this approximation merely on the basis of colloidal interactions, where it is known that the strength of the attractive interactions between particles is a function of the size of the particles (Israelachvili, 1991). We have carried out some calculations for both the kinetic and the thermodynamic properties of mixtures of model proteins where the attractive interaction was explicitly considered. We found that, unless we are close to a phase separation region, i.e., the two-phase region where the mixture separates into two solutions with different miscibilities for the proteins, the qualitative results are very similar to the ones obtained for the athermal (good solvent) systems. Therefore, we decided to concentrate our attention on these simpler systems.

The equations necessary to study the kinetic and thermodynamic behavior of the mixtures just defined are obtained from the general equations derived above. Because there are no conformational changes, only the densities of the proteins as a function of the distance from the surface (and time) are relevant quantities. The density of particles of type i at equilibrium is given by

(16)

where v_i(z, z') dz' is the volume that the protein (sphere) of type i, with its point of closest distance to the surface at z, occupies at z', and U(z) is the attraction between the surface and the protein (sphere) of type i shown in Fig. 2 or its appropriate modification (see above). R_i is the radius of the protein of type i. To determine the Lagrange multipliers, (z'), we need to solve the constraint equations, which for the binary mixture considered here, is (see Eq. 2)

(17)

which is solved by replacing Eq. 16 for each density and then by discretization of the z direction into finite elements. The volumes v_i(z', z) dz' are given by the cross-sectional area of the sphere at z when the bottom of the sphere is at z'. Namely, v_i(z', z) dz' = {R  [R_i  (z'  z)]²} dz'. The discrete version is obtained by integrating the cross-sectional area over the thickness of the discrete layer. The solution of these equations is straightforward, and, from them, we obtain the equilibrium adsorption isotherms. The bulk conditions of the solution are introduced in the chemical potentials, µ, which are explicitly given by

(18)

where V is the total volume of the protein and is the bulk volume fraction of the solvent. Eq. 18 is obtained from Eq. 16 by considering (z)v_s = _bulkv_s = ln and U(bulk) = 0.

It should be noted that, due to the volume-constraint equations, we have reduced the number of independent thermodynamic variables by one. Namely, we cannot vary the volume of the system at a fixed number of proteins and solvent molecules. Therefore, we do not have absolute chemical potentials, but the chemical potential of the protein is, in reality, an exchange chemical potential that measures the work related with changing V/v_s solvent molecules by one protein molecule of type i. Although we do not explicitly write the chemical potentials as exchanges, it should be clear that this is the quantity that we are calculating throughout this work. Further, for the same reason, the value of the chemical potential of the solvent is not a relevant quantity and therefore is not needed (Carignano and Szleifer, 1994), or, in other words, the chemical potentials of the proteins and the lateral pressures are measured with respect to the solvent chemical potential.

For the kinetic equations, we can write for protein of type i,

(19)

where the time-dependent chemical potential is given by

(20)

and the time-dependent Lagrange multipliers are obtained from the time-dependent constraint equation,

(21)

The procedure to integrate the equations of motion, Eq. 19, is to start with the initial condition of a homogeneous (very low,  = 10¹⁰) density for z  L, and, for z > L, the proteins are at bulk density and do not change that density over time. This is to represent a flow cell (Calonder and Van Tassel, 2001). At t = 0, the surface-protein interactions are turned on. Then, using Eq. 20 for each protein, one obtains the chemical potential profiles that are needed for a time iteration of the densities. After the densities for the new time are obtained, Eq. 21 is used for the time-dependent Lagrange multipliers so that the new chemical potentials can be obtained to perform the next time iteration. This procedure is continued until all the chemical potentials are the same, which corresponds to the new equilibrium condition.

The very low density used in the closed vicinity of the surface, instead of pure solvent, is for numerical convenience. Further, the diffusion of the proteins from the bulk into the pure solvent region can be calculated analytically and added to the time-dependent adsorption that we calculate. However, the time scale of this process is so fast, compared to the processes calculated here, that its inclusion does not change any of the behavior presented.

An experimentally measurable quantity that we can calculate at equilibrium and as a function of time is the surface tension. The thermodynamic potential that we use in deriving the theory is exactly the free energy per unit area that corresponds to the surface tension (Rowlinson and Widom, 1982) when the bulk value is subtracted. We use the same excess free energy to calculate the dynamic surface tension. This is given (for both equilibrium and dynamic surface tension), by Eq. 1, which, for the binary mixture just presented, becomes

(22)

where the values at equilibrium (t  ) provide the thermodynamic surface tension.

The second system on which we report calculations is aimed at looking at the effect that surface-induced conformational transitions of the protein have on the equilibrium and kinetic process of adsorption.

The bulk solution is composed by spherical model proteins with a radius R = 15 Å, which interact with the surface with the potential shown in Fig. 3. Upon contact with the surface, the protein may undergo a conformational change to a configuration that we call pancake. This conformation has the shape of a disk with a height equal to the diameter of the spherical conformation. The cross-sectional area of the disk is such that the volume of the protein is the same in the spherical and in the pancake configurations. The attraction of the pancake conformation with the surface is larger than that of the sphere. The motivation for studying this case is that, if the pancake conformation would not be more favorable on the surface, there will be no reason for the protein to undergo the conformational change upon contact with the surface. It is important to note that this type of configurational change, from a sphere-like conformation to a more disk-like one, can be related to the conformational changes observed experimentally in studies of lysozyme adsorption (Billsten et al., 1995).

View larger version (10K): [in this window] [in a new window]   FIGURE 3   The distance dependence of the attractive interaction between the surface and one spherical protein for the case of proteins that may undergo conformational changes upon contact with the surface. Units are as in Fig. 2.

As in the case of the binary mixture, we assume that _ij = 0 for all _ij. Further, because there is only one relevant energy difference, we can take U_int(_i) = 0. Recall that the protein is allowed to change its configuration only at z = 0. The difference U_sph-s(0)  U_pan-s(0) contains in it any difference in the internal energy between the two configurations.

The equations that are solved for the equilibrium system are

(23)

for all z, and there is an additional equation for the pancake conformation,

(24)

where U_pan-s(0) is the pancake-surface attraction. The equation for the density of pancake conformations is only at z = 0 because this configuration is assumed to exist only upon contact of the protein with the surface.

The constraint equations to determine the lateral pressures for z  h are

(25)

and, for z > h,

(26)

Again, as described above, these equations are solved by discretization of the z direction.

The kinetic equations for the sphere configuration are, for z  0,

(27)

and, for z = 0,

(28)

with the time-dependent chemical potential of the sphere given by

(29)

The dynamic equation for the pancake configuration contains no mass transport component because it can only exist on the surface and as a transformation from an already adsorbed spherical conformation. Thus, we have

(30)

where for both Eqs. 28 and 30, the blocking functions, are given by

(31)

with the repulsive contribution to the potentials of mean force given by

(32)

for the pancake, and

(33)

for the sphere, where R is the radius of the spherical protein.

The intrinsic rates of conformational change, k(pan  sph) and k(sph  pan) are input for the theory. However, due to the condition of thermodynamic equilibrium, we only need to provide one. The equilibrium condition from which the constant is determined is

(34)

where the equilibrium values