Kinetics of Protein Adsorption and Desorption on Surfaces with Grafted Polymers

Transcription

1 1516 Biohysical Journal Volume 89 Setember Kinetics of Protein Adsortion and Desortion on Surfaces with Grafted Polymers Fang Fang,* Javier Satulovsky, y and Igal Szleifer* *Deartment of Chemistry, Purdue University, West Lafayette, Indiana; and y Deartment of Cell Biology and Physiology, Washington University School of Medicine, St. Louis, Missouri ABSTRACT The kinetics of rotein adsortion are studied using a generalized diffusion aroach which shows that the timedetermining ste in the adsortion is the crossing of the kinetic barrier resented by the olymers and already adsorbed roteins. The otential of mean-force between the adsorbing rotein and the olymer-rotein surface changes as a function of time due to the deformation of the olymer layers as the roteins adsorb. Furthermore, the range and strength of the reulsive interaction felt by the aroaching roteins increases with grafted olymer molecular weight and surface coverage. The effect of molecular weight on the kinetics is very comlex and different than its role on the equilibrium adsortion isotherms. The very large kinetic barriers make the timescale for the adsortion rocess very long and the comutational effort increases with time, thus, an aroximate kinetic aroach is develoed. The kinetic theory is based on the knowledge that the time-determining ste is crossing the otential-of-mean-force barrier. Kinetic equations for two states (adsorbed and bulk) are written where the kinetic coefficients are the roduct of the Boltzmann factor for the free energy of adsortion (desortion) multilied by a reexonential factor determined from a Kramers-like theory. The redictions from the kinetic aroach are in excellent quantitative agreement with the full diffusion equation solutions demonstrating that the two most imortant hysical rocesses are the crossing of the barrier and the changes in the barrier with time due to the deformation of the olymer layer as the roteins adsorb/desorb. The kinetic coefficients can be calculated a riori allowing for systematic calculations over very long timescales. It is found that, in many cases where the equilibrium adsortion shows a finite value, the kinetics of the rocess is so slow that the exerimental system will show no adsortion. This effect is articularly imortant at high grafted olymer surface coverage. The construction of guidelines for molecular weight/surface coverage necessary for kinetic revention of rotein adsortion in a desired timescale is shown. The time-deendent desortion is also studied by modeling how adsorbed roteins leave the surface when in contact with a ure water solution. It is found that the kinetics of desortion are very slow and deend in a nonmonotonic way in the olymer chain length. When the olymer layer thickness is shorter than the size of the rotein, increasing olymer chain length, at fixed surface coverage, makes the desortion rocess faster. For olymer layers with thickness larger than the rotein size, increases in molecular weight results in a longer time for desortion. This is due to the grafted olymers traing the adsorbed roteins and slowing down the desortion rocess. These results offer a ossible exlanation to some exerimental data on adsortion. Limitations and extension of the develoed aroaches for ractical alications are discussed. INTRODUCTION Flexible olymer molecules grafted to surfaces or interfaces imose a steric barrier that can be tuned deending uon the olymer molecular weight, surface coverage, and tye of chemical structure (1 3). These interactions are widely used in colloidal stabilization (4 6) and in the last few years have found alication on the develoment of biocomatible materials and drug carriers (7 25). The basic idea is that the grafted olymer layer revents nonsecific adsortion of roteins on the surface of the biocomatible material or drug carrier, reducing the immunological resonse (7,8,15,26 29). The understanding of the kinetics of rotein adsortion and its reduction/revention by grafted olymer layers is therefore very imortant for the design of materials interacting with biological systems. In this work we resent a thorough theoretical study of the kinetics of rotein adsortion on surfaces with grafted olymers, which comlements our earlier work on both the thermodynamics and kinetics of rotein adsortion (30 34). Adsortion of roteins on surfaces is a comlex rocess that involves very large energy scales and the ability of the roteins to change their conformation uon contact with the surface (26,35 38). Moreover, the timescale of the adsortion rocess can be very long and in many cases the adsortion is irreversible (31,33,39 41). It is imortant to differentiate between the equilibrium isotherms and the kinetics of the adsortion rocess. This is an imortant difference both in the ractical alications of rotein adsortion (or revention of it) and in the fundamental studies of the understanding of the adsortion rocess. For examle, in the design of biocomatible materials to be used for artificial organs it is imortant to comletely revent adsortion of roteins. Thus, thermodynamic control is necessary, meaning that for the given conditions, the equilibrium amount of roteins adsorbed on the surface is zero. On the other hand, drug carriers need to survive in the blood stream for the time Submitted October 27, 2004, and acceted for ublication June 2, Address rerint requests to I. Szleifer, Det. of Chemistry, Purdue University, 560 Oval Dr., West Lafayette, IN Tel.: ; Fax: ; igal@urdue.edu. Ó 2005 by the Biohysical Society /05/09/1516/18 $2.00 doi: /biohysj

2 Protein Adsortion/Desortion Kinetics 1517 necessary to deliver the drug to its target. In this case, control of the kinetics of adsortion, so that it is delayed beyond the timescale for drug delivery, is the necessary design criteria. During the last few years we have develoed and alied a molecular theoretical aroach to study the thermodynamics and kinetics of rotein adsortion on surfaces with and without grafted olymers (21,24,30 33,42 45). The redictions from the theory are in excellent quantitative agreement for the adsortion isotherms of lysozyme and fibrinogen on surfaces with short- (31) and long-grafted olyethylene oxide (PEO) chains (21,32). In all these cases we studied systems in which the reduction of rotein adsortion is due to the steric reulsion induced by the olymer layer, i.e., by flexible olymer and not by chemical modification of the surface, as in the case of high density self-assembled monolayers with functional end-grous (46). We have found that the grafted olymer layers roerties that are otimal for thermodynamics control are different than those controlling the kinetic rocess (31,33). For examle, for thermodynamic control of rotein adsortion the olymer surface coverage is the most imortant factor in determining the reduction of rotein adsortion (30). These redictions have been confirmed by exerimental observations (16,47). For the kinetics of adsortion, though, we have redicted a very strong effect on molecular weight; however, its role for the equilibrium isotherms is only secondary (31,33). In this article we resent a kinetic theory that borrows from our revious work (31) and the insights learned from the theory of Halerin (48) and we develo a comutational feasible molecular aroach that enables the study of the whole kinetic rocess exlicitly accounting for the deformation of the olymer-rotein layer as the adsortion rocess takes lace. The basic idea of the aroach is to use the hysical insights learned from the generalized diffusion aroach to determine what the relevant stes are in the kinetic rocess. Then, we use the theoretical ideas of the Kramerlike aroach develoed by Halerin together with our molecular theory to construct a kinetic model that enables the study of both adsortion and desortion rocesses. The next section, Molecular Theory, starts with a review of the generalized diffusion aroach and the molecular theory that serves as basis for the kinetic aroach. After that, we resent the kinetic theory used, with examles of the kinetics of adsortion and desortion as a function of the grafted olymer chain length and surface coverage; this is then followed by our concluding remarks. MOLECULAR THEORY FIGURE 1 Schematic reresentation of the system containing roteins in their native conformation dissolved in a low molecular-weight solvent, and in contact with a surface with grafted olymers. The large solid circles are the roteins and the small oen circles are the solvent molecules. The strings of small solid circles, tethered to the surface, reresent the grafted olymers. The z direction is defined erendicular to the surface. The osition of a rotein, z9, refers to the lowest oint of the rotein, whereas the volume that a rotein contributes to z refers to the volume that the rotein occuies between z and z 1 dz. The two rate coefficients reresent the kinetic rocesses involved in the adsortion of roteins onto the surface with grafted olymers. The right of the figure reresents schematically the otential of mean-force felt by the adsorbing/desorbing roteins (see text). The system of interest here is comosed of a surface of total area A sanning the x, y lane. The surface has N g olymer molecules grafted at one of their ends (see Fig. 1). Each olymer has n g segments, each of length l. The olymer-modified surface is ut, at time t ¼ 0, in contact with a solution containing roteins dissolved in water. The rotein solution is characterized by a bulk density r,bulk or equivalently a chemical otential m,bulk. When the surface is ut in contact with the solution the roteins feel anisotroic interactions, induced by the resence of the surface, which are the driving forces for the adsortion rocess. The basic idea to determine the time- and distance-deendent interactions between the surface and the roteins is to take advantage of the very different timescale for the diffusion of the rotein as comared to the fast local motions of the olymer monomers and the solvent molecules. Therefore, we can consider that for each configuration in sace of the roteins, the olymer and solvent can equilibrate around the larger, slower, articles. This assumtion is common to our generalized diffusion aroach that we resent here and the kinetic model in next section. We start reviewing the generalized diffusion aroach followed by the free energy functional molecular theory aroach. In this work we concentrate our attention only on cases in which the rotein does not change its conformation uon adsortion. The generalization of the theory to more general cases has been resented elsewhere for bare surfaces (33). It will be shown in future work for surfaces with grafted olymers. GENERALIZED DIFFUSION APPROACH (GDA) The basic assumtions of searation of timescale resent a natural scenario for the use of dynamical self-consistent theory or the dynamical density functional aroach (31,49 51). This is effectively what we call the generalized diffusion aroach (GDA), when the free energy used is from our

3 1518 Fang et al. molecular theory (33). We write a time-evolution equation for the density rofile of the roteins with a diffusion equation of the ¼ r m ðz; tþ ; (1) with D the diffusion coefficient of the roteins. (Please note that, for simlicity, we assume that the diffusion coefficient is indeendent of osition. The reason is that the kinetic slowdown induced by the interactions, through the chemical otential gradients, is the dominant effect in the cases of interest here.) The time- and osition-deendent chemical otential is defined as m ðz; tþ ¼ dw=a dr ðz; tþ : (2) W/A reresents the free energy density (er unit area) of a system of frozen configuration of the roteins. This means it is the minimal free energy with resect to the olymers and the solvent for the given distribution of roteins. It is convenient to write the chemical otential as the sum of an ideal term and a otential of mean-force U mf (z;t), i.e., the nonideal contribution. Then m ðz; tþ ¼k B T ln r ðz; tþ 1 U mf ðz; tþ; (3) where k B is the Boltzmann constant and T is the absolute temerature. Using the otential of mean-force definition, Eq. 3, on the diffusion equation, Eq. 1, we ðz; " 2 r ¼ D ðz; r ðz; mfðz; tþ : The first term is the ideal diffusion, whereas the second reresents the motion due to the interactions between the roteins and all the other molecules in the system, including the surface. Consider for examle the case in which, at time t ¼ 0, a solution with homogeneously distributed roteins is ut in contact with the surface. The main driving force for adsortion has to be the otential of mean-force, since there is no gradient of density. However, because roteins adsorb, the two terms contribute until the chemical otential (due to the balance of density and otential of mean-force) becomes constant and the new equilibrium with the adsorbed roteins is reached. Clearly, if there is a strong attraction between the roteins and the surface the new equilibrium corresonds to an inhomogeneous distribution of roteins, due to the anisotroic interaction otential that results from the resence of the surface. At this oint we should remark that the concet of chemical otential and otential of mean-force are defined as generalizations of the equilibrium (true thermodynamic) roerties (52,53). We refer to the time- and distance-deendent quantities in the same way as in the equilibrium cases. That is, they are formally defined in the same way, but do not corresond to the quantities for when the system is in true thermodynamics equilibrium. Instead, they corresond to the minimal free energy under the constraint of frozen distribution of rotein as given at that time by the dynamic equation. FREE ENERGY FUNCTIONAL MOLECULAR THEORY The understanding of the adsortion rocess 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 develoed to treat the structural and thermodynamic roerties of tethered olymer layers and later generalized to treat rotein adsortion on surfaces with and without grafted olymers (3,24,30 33,42 45,54 57). The basic idea is to write the free energy as a functional of the density of roteins and the conformational robability distribution of the grafted olymer chains and the roteins. To write the free energy we consider each molecular secies 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 aroximation. The resence of the surface induces an inhomogeneous distribution of all the molecular secies. Therefore, the meanfield felt by the molecules in each of their conformations is a function of the satial distribution of its units and the average distribution of all the other molecular secies. For simlicity we consider that the only inhomogeneous direction is the one erendicular to the surface, i.e., the z direction. We derive the free energy for a simle case of a mixture of olymers and roteins in which the solvent is equally good for both molecular secies. Further, we assume that the rotein can only be in its native conformation and does not change its conformation uon adsortion to the surface. The generalization to the cases in which conformational changes uon adsortion are considered (34), as well as different intermolecular interactions, has been treated elsewhere (30). The free energy er unit area of the equilibrium combined rotein-grafted olymer system is given by bw A ¼ s + P g ðgþ ln P g ðgþ g 1 1 Z N 0 Z N 0 r ðzþ½ln r ðzþv s ÿ 1 1 bu s ðzþÿbm Šdz r s ðzþ½ln r s ðzþv s ÿ 1Šdz; (5) where b ¼ 1/k B T. The first term reresents the conformational entroy of the tethered olymers, where s ¼ N /A is the olymer surface coverage and P g (g) (df) is the robability of finding a grafted olymer in conformation g. The second term is the rotein contribution, including: 1. A z-deendent mixing (translation) entroy with r (z) reresenting the rotein density rofile; v s is the solvent volume which is used as the unit of volume throughout.

4 Protein Adsortion/Desortion Kinetics The distance-deendent bare rotein-surface attraction, U s (z). 3. The chemical otential term to ensure equilibrium with the bulk, i.e., constant chemical otential of the roteins at all z with m ¼ m,bulk. The last term reresents the z-deendent mixing entroy of the solvent where r s (z) is the solvent density at z. Insection of Eq. 5 shows that the intermolecular reulsive interactions are not included. We assume that the reulsive interactions are of the excluded volume tye and thus we include them through acking constraints. That is, at each distance z from the surface the volume accessible to the molecules in the layer between z and z 1 dz is comletely occuied by a sum of contributions from the olymers, the roteins, and the solvent molecules. This is exressed in the form Z sæv g ðzþæ 1 r ðz9þv ðz; z9þdz91f s ðzþ ¼1; (6) where the first term is the volume fraction of olymers in layer z, with Æv g ðzþæ ¼ +PðgÞv g ðz; gþ being the average volume that a grafted olymer occuies at z. The second term is the volume fraction of rotein. This term includes the integral over z9 since we need to consider the contribution to the volume at z from roteins everywhere. The term v (z;z9) is the volume that a rotein with its oint of closest distance at z9 contributes to z (see Fig. 1). The last term is the solvent volume fraction with f s (z) ¼ r s (z)v s. The acking constraint exlicitly includes the size and shae of each of the molecular secies in the system, as well as the satial distribution of volume for each olymer conformation. We now can find the exlicit functional form of the df of chain conformations and the density rofiles of roteins and solvent by erforming a functional minimization of the free energy with resect to the olymer df, rotein, and solvent density rofiles subject to the acking constraints. The minimization is carried out by introducing Lagrange multiliers, b(z), associated with the acking constraints, to yield P g ðgþ ¼ 1 q g e ÿ R bðzþv gðz;gþdz ; (7) for the df of chain conformations, with q g being the grafted olymers artition function (normalization constant that ensures + g P g ðgþ ¼1), and r ðzþ ¼e bm e ÿbusðzþÿ R bðz9þv ðz9;zþdz9 ; (8) for the rotein density rofile. The equation for the rotein density rofile ensures that the chemical otential of the rotein is the same at all z as required for thermodynamic equilibrium. Finally, the solvent density rofile is given by f s ðzþ ¼e ÿbðzþvs : (9) The hysical meaning of the Lagrange multiliers can be seen in the exression of the solvent volume fractions rofile (Eq. 9). They reresent the local osmotic ressures. They actually measure the work to relace one unit of volume of solvent by one of olymer or rotein. As discussed elsewhere (3,30,33, 34), the lateral ressures are a measure of the average reulsive interaction at distance z from the surface. The numerical values of the lateral ressures are determined by relacing the exlicit exressions of the olymer df, Eq. 7, the rotein density rofile, Eq. 8, and the solvent density rofile, Eq. 9, into the constraint equation, Eq. 6. The resulting equations require, as inut, the single chain conformations of the olymer chains; the rotein volume distribution; the olymer surface coverage; and the rotein chemical otentials. For details on how the equations are solved numerically, see the Aendix. At this oint it is instructive to look at the exression of the rotein chemical otential. At equilibrium it is given from Eq. 8 by bm ¼ ln r ðzþ 1 bu mf ðzþ; (10) where we have defined the otential of mean-force U mf (z)by Z bu mf ðzþ ¼bU s ðzþ 1 bðz9þv ðz9; zþdz9: (11) This quantity reresents the effective interaction between a rotein at z and the surface, averaged over all the degrees of freedom of the other molecules in the system. This quantity, actually its time-deendent counterart, lays a key role in the kinetics of adsortion (see Eq. 2, above). The free energy functional and the df and density rofiles just derived corresond to the equilibrium state of the system since they are found by the total minimization of the free energy functional. To determine the time-deendent quantities necessary to solve the generalized diffusion equation, Eq. 4, we use the searation of timescales mentioned above and consider a free energy with the same functional form as Eq. 5 but with a major modification. Following the assumtion that the local motion of the olymers and that of the solvent are much faster than the roteins motion, we can consider that for each given density rofile of the roteins, the free energy contribution of the solvent and olymers is minimized. Therefore, we write the time-deendent free energy in the form bwðtþ A ¼ s + P g ðg; tþln Pðg; tþ 1 1 g Z N 0 Z N 0 ðf s z; tþ½ln f s ðz; tþÿ1šdz r ðz; tþ½ln r ðz; tþÿ1 1 U s ðzþšdz; (12) where the rotein comonent is written without a chemical otential term; the time-deendent olymer df and solvent rofile are obtained by minimization of the free energy; and the rotein density rofile is fixed, and is given by the dynamic equation, Eq. 4.

5 1520 Fang et al. We need to minimize the free energy with resect to the df and the solvent density rofile subject to the acking constraints. Following the same lines as in the equilibrium case, we introduce a time-deendent Lagrange multilier, b(z;t), which is associated with the time-deendent acking constraints. For the df and solvent-density rofile exressions (identical to Eqs. 7 and 9, resectively), this leads to (z) being relaced by (z;t). For the determination of the lateral ressures, the main difference between the equilibrium and nonequilibrium cases is that, in the equilibrium case, the rotein density rofile is obtained in the minimization rocess and is therefore an exlicit function, (z); but in the nonequilibrium case, r (z;t) is an inut. In general, the inut comes from the diffusion equation. However, this does not have to be the case, as will be discussed in detail in the descrition of the kinetic aroach. The next ste is to determine the time- and z-deendent chemical otential of the rotein. This is obtained as a straightforward generalization of the equilibrium quantity (see Eqs. 10 and 11),! bm ðz; tþ ¼ dwðtþ=a ; (13) dr ðz; tþ to obtain Z bm ðz; tþ ¼ln r ðz; tþ 1 bu s ðzþ 1 bðz9; tþv ðz9; zþdz9; (14) and thus the time-deendent otential of mean-force is given by Z U mf ðz; tþ ¼bU s ðzþ 1 bðz9; tþv ðz9; zþdz9; (15) in analogy to the equilibrium amount. Note that the bare surface-rotein interaction is time-indeendent. All of the time-variation arises from the intermolecular interactions as exressed in the lateral reulsions (z;t), which result from the changes in the acking of the rotein-olymers as the roteins move toward the adsorbing surface. To solve the equilibrium and kinetics of rotein adsortion we need to define the model system for the rotein and the olymers. For the rotein we use a very simle model that mimics the roerties of lysozyme. We assume that the rotein is sherical with a radius R ¼ 1.5 nm. Furthermore, for U s (z), we use the otential calculated from atomistic simulations by Lee and Park (35) for lysozyme with olyethylene solid surfaces (see Fig. 2 below). For simlicity, we do not allow for conformational changes of the rotein uon adsortion, and leave that for future work. For the olymer 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 redictions are in excellent agreement with exerimental observations FIGURE 2 (A) The initial otential of mean-force for different chain lengths: n g ¼ 0(bare surface, solid line); n g ¼ 25 (dot-dashed line); n g ¼ 50 (long-dashed line); n g ¼ 75 (double-dot-dashed line); and n g ¼ 100 (dotted line). The surface coverage is sl 2 ¼ (B) The inset resents the corresonding olymer volume fraction rofiles at t ¼ 0. (21,31,32,59). For each chain length that we study we use u to randomly generated self-avoiding olymer conformations. From each conformation g generated we obtain v g (z;g), the volume that a chain in conformation g has in the volume sanned between z and z 1 dz. These volume distributions are the inut to solve the constraint equation, Eq. 6. Note that the set of conformations has all tye of distributions of segments, v g (z;g), including highly stretched chains and mushroom-like conformations. The df determines the relative weight of each conformation for each different case and for different times. (The Aendix outlines how the equations are solved by discretization of the z direction; for more detail on the technical asects for the equilibrium and dynamic solutions, see Refs. 21, 30, 31, 33, and 34.) In all the results resented below, we denote the dimensionless densities given by the roduct rv s simly by r. To convert this quantity to the exerimentally reorted units of ng/cm 2 for lysozyme, one needs to multily our reorted values by 26,394. Otherwise, multilying the reorted values by 150 rovides the area fraction occuied by the roteins. REPRESENTATIVE RESULTS FOR GDA The kinetics of rotein adsortion is strongly affected by olymer-chain length. We have discussed these differences of molecular weight effects on the thermodynamics and kinetics of rotein adsortion in early work (31). However, we find it imortant to highlight the main oints again here as they hel in the understanding of the kinetic theory and results develoed below. The kinetic rocess that we study starts from a homogeneous solution of rotein, m (z) ¼ m, bulk for all z, that at time t ¼ 0 is brought into contact with

6 Protein Adsortion/Desortion Kinetics 1521 a layer of ure solvent of thickness d that has a solid surface located at z ¼ 0. The surface may have grafted olymers in it at surface coverage s and the olymer chain length is n g. The resence of the surface induces an interaction field in the z direction that generates gradients of chemical otential in the rotein. These manifest themselves in varying otentials of mean-force, that at time t ¼ 0 reresent the driving force of the roteins, together with the gradient of the rotein density, to adsorb on the surface. Fig. 2 shows the otentials of meanforce at time t ¼ 0 for a variety of olymer chain lengths, all at the same surface coverage. In the case of no grafted olymers, the otential is urely attractive and it is given by U s (z). The resence of the grafted olymers introduces a reulsion whose range is equal to the thickness of the tethered layer (see inset of Fig. 2 for the olymer density rofiles). For long enough olymers a reulsive barrier aears whose strength and osition is a function of the olymer chain length. The amount of rotein adsorbing at equilibrium deends on the otential of meanforce at contact. Even though the value of the otential at contact varies with the amount of roteins adsorbed (see below), the variation of U mf (z ¼ 0; t ¼ 0) with olymer chain length shows the same deendence as the equilibrium amount of roteins adsorbed (30). The kinetics of adsortion deends uon the timescale required by roteins to reach the surface. In the case of urely attractive otentials, as for no olymer and n g ¼ 25 on Fig. 2, the initial adsortion will be very fast and determined by the time that it takes the roteins to reach the range of the interactions. In this regime the surface acts as a strong attractive sink to the roteins. For the other cases shown, however, the initial adsortion is determined by the time that it takes the roteins to cross the reulsive barrier resented by the olymer layer. We can see in the otentials that both the range and magnitude of the reulsion increases with olymer chain length. Therefore, the initial adsortion will be slower as the olymer-chain length increases. Also, we do not exect a fast regime in the initial adsortion whenever the otential of mean-force at t ¼ 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 otentials of mean-force resented in Fig. 2. However, once we integrate the first time-ste the distribution of roteins changes and thus the otential of mean-force will also change. That is, it will corresond to the minimal free energy of the olymer-solvent mixture in the frozen new configuration of the roteins. Therefore, we exect the otentials of mean-force to be a strongly varying function of time. The comlete kinetic behavior is resented in Fig. 3, where the amount of rotein on the surface is lotted as a function of time. The figure shows the kinetics of adsortion for different cases: one in which there are no olymers grafted on the surface, with the rest reresenting surfaces with grafted olymers with the same surface coverage, but different molecular weights. FIGURE 3 Variation of the amount of rotein adsorbed as a function of time for surfaces with (and without) grafted olymers. The lines are the results from the generalized diffusion aroach. The lines with symbols are the results from the kinetic theory aroach. In all cases, sl 2 ¼ The bulk rotein volume fraction is f, bulk ¼ The different chain lengths are denoted in the figure. The time-deendent adsortion for no-olymer on the surface and short-chain-length-grafted olymer shows a very fast early regime, in which the surface acts as a sink to the roteins due to the strong attractions between the surface and the roteins (see otentials in Fig. 2). After a certain amount of roteins adsorb, there is a very shar slowdown, during which the kinetic rocess is dominated by barrier crossing. In the case of n g ¼ 50 we see in Fig. 2 the resence of a kinetic barrier even at the beginning of the adsortion rocess. Thus, Fig. 3 shows that the kinetics of adsortion does not have a fast regime but it is dominated at all times by barrier crossing. Actually, the height of the barrier and the range of the otential of mean-force change as the adsortion roceeds. Fig. 4 shows the otential of mean-force for four different stages of the adsortion for n g ¼ 50. The height of the barrier increases, and it moves toward the surface as more roteins adsorb. Furthermore, the range of the reulsive interaction increases. This is the result of the deformation of the olymer layer as the roteins adsorb. The changes in the structure of the olymer layer and the volume fraction rofile of the roteins at the same stages of the adsortion are also shown in Fig. 4. There is a clear ush of the olymer segments close to the surface to move toward the solvent as the roteins adsorb. This is due to the need of the rotein to have enough room on the surface to adsorb. The main message of the figure is that to roerly describe the kinetic rocess, the deformation of the olymer-rotein layer in the vicinity of the surface has to be taken into account. This is the contribution resonsible for the very large variation of the otential of mean-force, and thus the rate of adsortion, with time. Another imortant result from Fig. 4 is the rofile of the rotein volume fraction as a function of time. It is clear that

7 1522 Fang et al. by solving a set of couled nonlinear equations with thousands of terms in each one, i.e., the constrained free energy minimization (see Aendix). The calculations for long chain lengths are ractically imossible; as an examle, the calculation for n g ¼ 50 resented in Fig. 3 takes many days of comuter time. Moreover, there are cases where the time evolution is so slow that we cannot reach the equilibrium state (see, e.g., n g ¼ 100 in Fig. 3). Therefore, a more ractical aroach is needed, such as the one resented next. FIGURE 4 (A) The otential of mean-force at four different stages of the adsortion for the case of n g ¼ 50 shown in Fig. 3. (B) The volume fraction rofile of grafted olymers, and the volume fraction rofile of roteins (inset) corresonding to the same four stages shown in A. the rotein is found on the surface and then deleted from the other regions where the grafted olymer is, u to the bulk. This deletion is the result of the large effective reulsion felt by the roteins due to the grafted olymers. This two-state tye of rotein structure serves as the basis of the kinetic model that we resent below. Based on this result, we can assume that the otential of mean-force, and in articular its maximum and value at contact, deends on the structure of the olymer-rotein layer, when the roteins are only at the surface. Therefore, alying the free energy functional molecular theory aroach, we can calculate the otentials of mean-force at all ossible adsorbed densities from zero to the equilibrium value. The descrition of the kinetics with the generalized diffusion model enables a rather detailed molecular descrition of the adsortion rocess with its associated structural changes. The roblem is that the solution of the numerical equations is very demanding. First, the kinetic rocess requires the solution of the differential equations over 16 orders-of-magnitude in time. Second, at each time-ste the determination of the chemical otential gradients is obtained KINETIC THEORY (KA) Adsortion kinetics We find the most roblematic cases to solve with the above aroach are those in which the barriers in the otential of mean-force are very high. These are also the cases in which we find a two-state rotein distribution. Thus, we resent a kinetic model in which the rate-determining ste is the crossing of the barrier. This means that we look at a kinetic equation for the transition of roteins between the bulk and the adsorbed state given ads ðtþ ¼ k ads bulk ÿ k des ðtþr ads ðtþ; (16) where the first term on the right-hand side of Eq. 16 is the gain term associated with the increase of roteins on the surface due to the adsortion, and the second is the loss term due to the desortion of the roteins. The adsortion and desortion are activated rocesses, as deicted qualitatively in Fig. 1. Therefore, the kinetic constants should have the form of k ads ðtþ ¼k 1 e ÿbdu adsðtþ ; k des ðtþ ¼k 2 e ÿbdu desðtþ ; (17) where DU ads (t) ¼ U mf (z ¼ z max ;t) U mf (bulk) is given by the difference in the otential of mean-force between the maximum of the barrier height and the bulk. The desortion Boltzmann factor, DU des (t) ¼ U mf (z ¼ z max ;t) U mf (z ¼ 0;t), is the difference between the otential at the maximum and that of the adsorbed state, i.e., the otential at contact with the surface (see Fig. 1). Based on the full solution of the diffusion equation we assume that the otential of mean-force deends on the structure of the olymer-rotein layer, when the roteins are only at the surface. Therefore, we can calculate the otentials of mean-force at all ossible adsorbed densities from zero to the equilibrium value, meaning that we solve the equilibrium roblem for the olymer-solvent with a fixed amount of rotein r ads on the surface. This allows us to calculate U mf (z;r ads ), which we will use to determine the necessary energies for the Boltzmann factors in Eq. 17. Once we know the otential of mean-force as a function of r ads and the distance from the surface, z, we can determine the height of the otential barrier and the otential at contact. Using our choice of zero for the otential in the bulk, U mf (bulk) ¼ 0, we

8 Protein Adsortion/Desortion Kinetics 1523 know the three values of the otential of mean-force needed to calculate the rate coefficients. The next ste is to determine the reexonential factors in the rate coefficients. To this end we use the ideas of Halerin (48), who calculated the initial rate of adsortion of roteins on surfaces with grafted olymers using an extension of Kramer s theory of chemical reactions. We follow his aroach but we exlicitly include the time variation of the arameters determining the reexonential factor, thus allowing the comlete treatment of the adsortion rocess from its initial condition u to the aroach to equilibrium. According to Kramer s theory, the reexonential factor is given by k 1 ¼ k 2 ¼ D (18) al (see derivation in the Aendix), where D is the diffusion constant; a is the width of the otential at a distance k B T below the maximum; and L is the distance a rotein in the bulk state has to travel to reach the barrier maximum, which can be aroximated by the thickness of the olymer layer. Both a and L deend uon the molecular structure of the olymer layer. Therefore, they also deend on time through the changes in structure of the combined olymer-rotein layer as a function of the amount of adsorbed roteins. As we have done with the barrier of the otential and its value at contact, we can determine a and L as a function of the amount of rotein adsorbed. Thus, we can have the imlicit deendence of the kinetic coefficients k ads and k des on time through their exlicit deendence on the amount of rotein adsorbed. Note that the rate coefficients as defined fulfill microscoic reversibility at all times., i.e., k ads ðtþ k des ðtþ ¼ eÿbðu mf ðz¼0;tþÿu mf ðbulkþþ : (19) This result is consistent with our local equilibrium aroximation. The symmetry arises from the aroximations used in the derivation of the flux (see the Aendix for details and discussion). Other choices for the reexonential factors will not affect any of the results resented for the adsortion, since the kinetics of adsortion is dominated by the flux toward the surface. Furthermore, as it will be shown below, the excellent agreement between the redictions of the KA and the full GDA suorts the validity of this aroximation. Fig. 4 demonstrates the very large changes that the olymerrotein layer structure undergoes through the adsortion rocess. Thus, it is clear that the roer quantification of the kinetic rocess requires the consideration of the exlicit density-deendence, and thus the imlicit time-deendence, of the quantities U mf (z ¼ z max ;r ads (t)), U mf (z ¼ 0;r ads (t)), a (r ads (t)), and L(r ads (t)). Examles of the first two quantities can be clearly seen in Fig. 4. We now discuss the four quantities in more detail, since they will rovide insightful hysical information on just what the roles of surface coverage and olymer-chain length are in the kinetic rocess. Fig. 5 shows the maximum in the otential of mean-force as a function of the amount of adsorbed roteins for a variety of olymer-chain lengths. Also shown is the case of the bare surface. For short chain lengths (including the bare surface, n g ¼ 0), a barrier larger than the thermal energy aears only after a finite amount of roteins adsorb (see Fig. 5, inset, and the exlanation 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 roteins adsorbed. The variation of the maximum of the otential of mean-force with density reveals which one is the dominant contribution in determining the kinetic barriers: the olymer, the rotein, or both. In all the regimes where the maxima are arallel, then, it is the rotein that determines the variation of the otential. Note the reason that curves are arallel, and not identical, is that there is a background contribution of the olymer layer which is strongly deendent on chain length. For the regions in which the curves are not arallel, the olymer contribution is dominant. Thus, there is a very different maxima at low rotein densities for the longest olymer chain lengths shown, where the olymer effect on the kinetics is large. Interestingly, for the two longest chain lengths shown, the curves become identical for large amounts of roteins adsorbed. This imlies that at these late stages of the adsortion the roteins see no difference between the two chain lengths. That the region close to the surface is basically identical, is because both chain lengths are long enough such that only FIGURE 5 (A) The maximum in the otential of mean-force as a function of the density of roteins adsorbed on surfaces with grafted olymers for different chain lengths: n g ¼ 0(bare surface, solid line); n g ¼ 15 (large dotsdashed line), n g ¼ 25 (dot-dashed line); n g ¼ 30 (double-dash-dotted line); n g ¼ 40 (large dot-solid line); n g ¼ 50 (long-dashed line); and n g ¼ 100 (dotted line). In all cases, sl 2 ¼ The thin-dashed line at bumf max ¼ bðumf bulk 1 ktþ marks the minimum otential that a kinetic barrier must resent. (B) The inset resents the minimum density of rotein that has to be adsorbed for the formation of a kinetic barrier as a function of grafted olymer-chain length.

9 1524 Fang et al. a finite art of them is affected by the adsortion rocess. In other words, the degree of deformation is identical for both surfaces. Thus, we see a very interesting chain-length effect. First, when the olymer layer thickness is longer than the size of the rotein, the equilibrium amount of roteins adsorbed is indeendent of molecular weight (34). Second, the initial rate of adsortion deends very strongly on the olymer-chain length; for examle, the difference in the maximum of the otential of mean-force between n g ¼ 50 and n g ¼ 100 at t ¼ 0is10k B T, imlying a ratio of rate constants of the order of 10 ÿ5. Third, once the adsortion is advanced, the maximum of the otential barrier for adsortion becomes identical for these chain lengths. The kinetic aroach cannot be directly alied to the initial adsortion kinetics if there is no barrier in the otential of mean-force at t ¼ 0. This is the case for the olymer chains shorter than 35 segments, as shown in Fig. 5. In the absence of kinetic barriers, the solution of the GDA resented in the revious section is not comutationally demanding. However, when the adsortion 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 ¼ 25 in Fig. 3). Thus, for ractical uroses, it is imortant to have an amount of density adsorbed (as a function of olymer-chain length), showing a kinetic barrier larger than the thermal energy (see Fig. 5, inset). The use of this grah is that, for the surface coverage shown in the region below the curve, the kinetics of adsortion is obtained from the full solution of the GDA, whereas above the curve the ractical aroach is to use the KA. Clearly, the different regimes deend on the surface coverage of olymer and the secific rotein studied. However, once those arameters are defined, we can calculate a curve, such as that shown in Fig. 5 s inset, to find the roer aroach to aly in each case. Fig. 6 shows the values of the width of the steric barrier at t ¼ 0 and at the end of the adsortion rocess as a function of grafted olymer-chain length. There is a strong deendence of a on n g at t ¼ 0 due to the secific structure of the olymer layer and its changes with the molecular weight of the olymer. The width of the otential barrier, however, is almost indeendent of the molecular weight of the olymer at the end of the adsortion rocess, and it is much smaller than its initial value. The data resented for the initial stage of adsortion in Fig. 6 resents the result for olymers shorter than 35 segments searately from the result for chains longer than 35 segments. The reason is that, below this molecular weight, there is no barrier in the otential of mean-force at t ¼ 0 for the surface coverage shown (see Fig. 5, inset). The very large change in a from the initial value to the end of the adsortion rocess is a reflection of the change in the shae of the otential as the adsortion rocess takes lace. At the initial stages, the barrier is dominated comletely by the grafted olymers. However, as the concentration of roteins on the surface increases, the barrier becomes more dominated by the contribution of the adsorbed roteins. Therefore, FIGURE 6 The width of the kinetic barrier, a, before adsortion (solid line with circles) and after adsortion (dashed line with squares) as a function of chain length for surfaces with grafted olymers at sl 2 ¼ The solid line with diamonds reresents the width of the kinetic barrier for the shorter olymer chains at the moment the kinetic barrier is just formed, i.e., Umf max ¼ Umf bulk 1kT; after some roteins have been adsorbed. a becomes indeendent of the molecular weight of the olymer, because the maximum narrows, and its shae is due mostly to the rotein and the olymer segments interacting with it, as discussed above. The last variable that we need to discuss is the thickness of the olymer-rotein layer L and its deendence on the amount of rotein adsorbed. As it is well known from olymer brushes, the thickness of the layer varies linearly with the molecular weight of the olymer (1,57). What is interesting is that even after there is adsortion of roteins the change in the thickness of the layer is very small (results not shown). This is because the height of the olymer 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 ractical uroses we can use the same L at all times; i.e., that for t ¼ 0. Desortion kinetics We next consider the case in which, once the system reaches equilibrium, the rotein solution in contact with the surface is washed-out and thus the adsorbed roteins in the grafted olymer layer are in contact with ure water. The equilibrium state will be such that all the roteins leave the surface since there is an infinite entroic gradient due to the zero concentration of roteins in solution. The kinetics of the rocess can, in rincile, 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 desortion in the ads ¼ÿk des ðtþr ads ðtþ; (20)

10 Protein Adsortion/Desortion Kinetics 1525 where there is no gain term because the solution is roteinfree. The desortion rate coefficient is given by k des ðtþ ¼ D ar eÿbdu des (21) (see derivation in the Aendix), where the energy difference is as given following Eq. 17, and we have used the Kramer aroach for the reexonential factor. Note that instead of L we have R in the denominator of the reexonential factor. This is because the maximum of the otential of mean-force is located at a distance from the surface of the order of the rotein size, as shown in Fig. 4. Thus, the desortion rocess measures the rotein going from the surface to the maximum in the otential, i.e., a distance R from the surface. Interestingly, the width of the otential around the maximum for the desortion rocess is indeendent of chain length. However, it is a function of the amount of rotein adsorbed and olymer surface coverage. The determination of a and DU 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 alied for the adsortion rocess (e.g., see Fig. 5). Integration methodology The equation for the adsortion and desortion kinetics, as derived from the KA, requires four arameters as a function of time. They are the otential of mean-force at contact, U mf (z ¼ 0, t); the maximal value of the otential of meanforce, U mf (z ¼ z*,t); the width of the otential of mean-force of 1 k B T below the maximum, a(t); and the thickness of the film, L(t). The fifth quantity, U mf (bulk), is indeendent of time. From the four quantities we have shown that the thickness of the film (or the radius of the rotein for the desortion rocess) does not vary with time, and therefore we take its value at time t ¼ 0. For the other three quantities we know their values as a function of the amount of rotein on the surface, which we tabulate before starting the kinetic calculations. Thus, we start the integration at time t ¼ 0 where we know all the necessary values and we integrate the kinetic equations one time-ste. The integration gives the value of the density of adsorbed roteins at the new time. We use this density value to find the three arameters, from the tabulated values, and integrate the kinetic equations another ste. 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-ste until we reach the equilibrium state. REPRESENTATIVE RESULTS FOR KA The first question that arises relates to the quality of the results obtained from the kinetic theory as comared to the diffusion aroach. Fig. 3 shows the redicted 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 aroximate aroach is very good. The shae of the adsortion curves and the magnitudes are very well reroduced for all grafted olymer-chain lengths and in the whole range of time in which a barrier is resent. This imlies that the hysical mechanism for rotein adsortion on surfaces with grafted olymers is indeed what we have assumed; that is, once there is a kinetic barrier, the time-determining ste is that of crossing the barrier. However, it is imerative to exlicitly include the variation of the arameters as the roteins adsorb to roerly describe the whole kinetic rocess i.e., the olymer-rotein layer deformation determines the shae of the time-deendent adsortion. It is imortant to emhasize the difference in the comutational effort necessary to solve the KA as comared to the GDA. For the case of n g ¼ 50 there is a factor of 10 6 between the two calculations. Furthermore, there are many cases in which the GDA needs to be integrated with a very small time-ste, and therefore the calculations cannot be comleted at all (see, e.g., the curve for n g ¼ 100 in Fig. 3). However, for the KA, the calculations are very simle; in essence, for each time-ste, the solution is that of a simle first-order differential equation. The reliability of the results from the KA gives us confidence to aly it where the GDA is not ractical due to the comutational limitations. Thus, we now study the effect of olymer-chain length and surface coverage. Fig. 7 shows the kinetics of adsortion for three different surface coverages of olymers 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 adsortion is almost indeendent of olymer chain but deends on surface coverage. However, the kinetic rocess slows, by orders of magnitude, with both chain length and surface coverage. The variation of the kinetic rocess with grafted oly mer-chain length is different for each of the three surface coverages shown. For the smallest surface coverage resented, there are large differences in the initial adsortion time; however, for the three chain lengths, the roteins reach their equilibrium-adsorbed amount at more or less the same time. For the two larger surface coverages, this is not the case. The longer the olymer-chain length, the slower the whole adsortion rocess becomes. Further, as the surface coverage increases, the differences at the latter stages of the adsortion rocess become larger. As the surface coverage of grafted olymer increases, the chain molecules become more stretched. This results in several effects on the rotein adsortion. First, the layer is more rotein-resistant, because there is less room for the roteins to adsorb. Second, the barriers for adsortion become larger, and therefore dislay slower adsortion kinetics. Third,