PROTEINS: Structure, Function, and Genetics 39:76 – 81 (2000) Monte Carlo Simulation of Diffusion of Adsorbed Proteins V.P. Zhdanov1,2* and B. Kasemo1 1 Department of Applied Physics, Chalmers University of Technology, Göteborg, Sweden 2 Boreskov Institute of Catalysis, Russian Academy of Sciences, Russia ABSTRACT We present the results of threedimensional lattice Monte Carlo simulations of protein diffusion on the liquid-solid interface in a wide temperature range including the most interesting temperatures (from slightly below Tf and up to Tc, where Tf and Tc are the folding and collapse temperatures). For the model under consideration (27 monomers of two types), the temperature dependence of the diffusion coefficient is found to obey the Arrhenius law with the normal value (⬇10ⴚ2 ⴚ 10ⴚ3 cm2/s) of the preexponential factor. Proteins 2000;39:76 – 81. © 2000 Wiley-Liss, Inc. Key words: lattice model; Monte Carlo simulations; adsorption; diffusion; conformational changes; Arrhenius parameters INTRODUCTION Diffusion of proteins adsorbed at solid-liquid interfaces is of considerable intrinsic interest and also important for understanding the kinetics of protein adsorption. The latter process occurs via protein diffusion (in the bulk liquid) towards the interface, actual adsorption, and subsequent conformational changes resulting often in denaturation of the native protein structure.1 The details of simulations of these steps depend on the ratio of the rates of adsorption and diffusion of adsorbed proteins (for a review, see Zhdanov and Kasemo2). The conventionally used mean-field kinetic equations describing protein adsorption are implicitly based on the assumption that diffusion of proteins in the adsorbed overlayer is rapid compared to adsorption. The opposite adsorption regime when surface diffusion of proteins is slow is usually treated in the framework of the random sequential adsorption model. Experimental data on protein surface diffusion at the solid-liquid interface are limited. The possibility of this process was first addressed in the beginning of the eighties.3,4 Subsequent investigations (see Tilton et al.,5 Rabe and Tilton,6 Thompson et al.,7 Ramsden et al.,8 and a recent review by Tilton9) confirmed the occurrence of protein surface diffusion for several adsorbate/substrate systems. The results reported have been obtained in a narrow range of temperatures near room temperature. For example, the coefficients of bovine serum albumin diffusion on different surfaces were found5,6 to be in the range from 10⫺9 to 10⫺8 cm2/s. For proteins adsorbed on a lipid bilayer, the diffusion coefficient was measured7 to be of the order of 10⫺9 cm2/s. Diffusion coefficients obtained for © 2000 WILEY-LISS, INC. other systems9 where proteins were observed to be mobile are of the same order of magnitude, ⬃10⫺10–10⫺7 cm2/s. Attempts to get the Arrhenius parameters for diffusion of adsorbed proteins are in fact absent, because in a narrow range of temperatures the accurate measurements of these parameters are hardly possible (this obstacle does not however seem to be principal). Nevertheless, some of the available experimental data can be used to estimate the activation energy for diffusion and an order of magnitude of the preexponential factor. For example, the activation energy for bovine serum albumin diffusion on polyhexylmethacrylate was found6 to be about 30 kBT (⯝18 kcal/ mol). This estimate was based on the data obtained at temperatures between 10 and 30°C (with increasing temperature, the diffusion coefficient was reported to increase from 0.2 to 1.4 cm2/s). Looking through those data, one may conclude that the accuracy of the estimate is about 6 kBT. For the preexponential factor for diffusion, one can get Do ⯝ 105⫾3 cm2/s. This value is much larger than the “normal” value, Do ⫽ 10 ⫺ 2 ⫺ 10 ⫺ 3 cm2/s, corresponding to surface diffusion of atoms and simple molecules.12 Physically, protein surface diffusion is expected5,6,9 to occur via momentary disruptions of a fraction of weak bonds between amino-acids on the protein surface, and the substrate. This conceptual scheme of protein diffusion is qualitatively reasonable, but it does not specify the diffusion mechanism, because it is compatible at least with three modes of diffusion. If the structure of an adsorbed protein is stable (e.g., close to the native one), diffusion may occur via (i) “skating” or (ii) “rolling” of the whole molecule. (iii) If protein adsorption is accompanied by denaturation, diffusion is expected to occur via rearrangements of the whole protein structure due to local elementary moves of amino-acids. Detailed simulations addressing these modes of diffusion of adsorbed proteins are lacking. In particular, it is not clear, for example, how large the preexponential factor for diffusion might be for the modes described. Some related simulations10,11 have been focused on surface diffusion of short-chain alkanes, but the results obtained in the latter studies are however not directly applicable to proteins because the models employed were in fact two-dimensional. To fill this gap, we Grant sponsor: TFR; Grant number: 281-95-782; Grant sponsor: NUTEK Biomaterials Consortium; Grant number: Contract No. 842496-09362. *Correspondence to: V.P. Zhdanov, Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia. E-mail: firstname.lastname@example.org Received 20 April 1999; Accepted 30 Septemer 1999 MONTE CARLO SIMULATIONS 77 present Monte Carlo (MC) simulations of diffusion of adsorbed proteins in the case when adsorption is accompanied by denaturation. MODEL Our work is based on the three-dimensional (3D) lattice model, which earlier has been widely used to study the kinetics of protein folding in solution (see a recent review by Shakhnovich13) and also denaturation of adsorbed proteins (see our review2). In the lattice approximation, a protein is schematically viewed as a linear sequence of N amino-acids. The structure of amino-acids is not analyzed explicitly. Instead, they are replaced by monomers, which are constrained (along the chain) to be nearest-neighbor on a 3D simple cubic lattice (each lattice site can be occupied at most once). The conformation of the chain is described by the monomer coordinates ri (i ⫽ 1, 2, . . . , N). The energy of a given conformation for an adsorbed protein is assumed to be the sum of the energies ⑀ij associated with topological contacts [a topological contact is formed whenever two nonbonded monomers i and j (兩i ⫺ j兩 ⱖ 3) are nearest neighbors] and the adsorption energies (⑀ia ⬎ 0), i.e. E⫽ 冘 兩i ⫺ j兩 ⱖ 3 ⑀ij ␦共rij ⫺ a兲 ⫺ 冘 ⑀ia ␦共zi兲, Fig. 1. Model protein-like molecule in the folded state. Open and filled circles show A and B monomers, respectively. (1) i where zi ⱖ 0 is the monomer coordinate perpendicular to the surface, a the lattice spacing, and ␦(0) ⫽ 1 and 0 otherwise. The monomers located in the first layer (with zi ⫽ 0) are assumed to interact with the surface. The interaction of the surface with monomers located in the layers with zi/a ⱖ 1 is neglected. All the interactions in Eq. (1) are effective because the protein-solvent interaction is not explicitly taken into account. Expression (1) is formally applicable both for biological and nonbiological polymers. In particular, it has been widely employed to study the statistics of adsorption of neutral nonbiological polymers14 (for adsorption of charged chains, see Kong and Muthukumar15). To apply expression (1) to proteins, one needs to introduce representative interaction energies ⑀ij resulting in a well-defined native state in the bulk of the solution. The results presented below have been obtained for a model protein molecule (Fig. 1) designed by Shakhnovich and Gutin.16 It contains 27 monomers of two types (A and B) with energies ⑀AA ⫽ ⑀BB ⫽ ⫺ 3 and ⑀AB ⫽ ⫺ 1 (for energy and temperature we use dimensionless units with kB ⫽ 1). In the bulk, this molecule is known to have a unique folded state with E ⫽ ⫺ 84. The probability of finding a protein in this state is considerable at temperatures below the folding temperature, T ⬍ Tf ⫽ 1.3.17,18 The folded state can be reached provided that the temperature is above the glass transition temperature, T ⬎ Tg ⫽ 1.1 (at T ⬍ Tg, a protein is typically trapped into one of the metastable states for a period which is much longer than any relevant time scale). With increasing temperature above Tf, the molecule is in the collapsed globular state at Tf ⬍ T ⬍ Tc ⫽ 2.0 (Tc is the collapse temperature) and then (at T ⬎ Tc) in the random-coil state. Adsorption and denaturation of this molecule have been explored in our earlier work19 in a wide range of temperatures from T ⫽ 1.2 ⬍ Tf up to T ⫽ 2.4 ⬎ Tc (during adsorption, the molecule structure is more or less compact even if T is slightly above Tc). The present simulations of surface diffusion of this molecule have been executed in the same range of temperatures for two sets of the a monomer-substrate interactions. The first one, ⑀A ⫽ 3 and a ⑀B ⫽ 1, corresponds to the case when the interaction of A monomers with the surface is much stronger than that of B a a monomers. The second one, ⑀A ⫽ 3 and ⑀B ⫽ 2, represents the case when the A and B interactions with the surface are nearly equal. Physically, these two sets of interactions may qualitatively describe the situations when the protein structure in the adsorbed state depends primarily on hydrophilic and hydrophobic interactions. (i) If for example the surface is hydrophobic, the monomer-substrate interaction is not strongly affected by adsorbed water molecules (during protein adsorption these molecules can be easily pushed out of the first layer) and accordingly the interaction of the surface with hydrophobic monomers will be much stronger than that with hydrophilic monomers. This case can qualitatively be described by assuming the A monomers to be hydrophobic, the B monomers to be hydrophilic, and using the first set of the monomera a substrate interactions (⑀A ⫽ 3 and ⑀B ⫽ 1). (ii) If on the other hand the surface is hydrophilic, the monomersubstrate interaction is considerably weakened by adsorbed water molecules (these molecules can hardly be pushed out of the hydrophilic surface) and accordingly the difference of the interactions of different monomers with the substrate will not be very large. This case can qualitatively be described by assuming the A monomers to be hydrophilic, the B monomers to be hydrophobic, and using 78 V.P. ZHDANOV AND B. KASEMO Fig. 2. Energy (top panel) and numbers of monomers in the adsorbed overlayer and also in the two bulk layers closest to the surface (curves 1, 2, and 3 on the bottom panel)) in the course of diffusion of a model protein-like molecule with ⑀Aa ⫽ 3 and ⑀Ba ⫽ 1 at T ⫽ 1.2 (a) and 2.4 (b). Time is calculated in MCS (1 MCS is defined as 27 attempts to realize a monomer move). The kinetics have been calculated after the denaturation procedure which contained 107 and 105 MCS at T ⫽ 1.2 and 2.4, respectively. a the second set of the monomer-substrate interactions (⑀A a ⫽ 3 and ⑀B ⫽ 2). Initially, the molecule was considered to be adsorbed in the native folded state (Fig. 1) so that one of its sides contacts the surface. To realize denaturation of the molecule, we executed 105–107 MCS [1 MCS (MC step) is defined as 27 attempts to realize a monomer move]. After denaturation, we executed 2 ⫻ 105 ⫺ 2 ⫻ 107 MCS to find displacements of the molecule along the X and Y axes, ⌬x ⫽ ¥i ⌬xi/N and ⌬y ⫽ ¥i ⌬yi/N. Both procedures (denaturation and diffusion) were run for longer times at lower temperatures. The diffusion coefficient was calculated as ALGORITHM OF SIMULATIONS To simulate diffusion of adsorbed proteins, we have used the standard algorithm for end, corner and crankshaft monomer moves: A monomer is chosen at random. If it is an end monomer, then one of the neighboring lattice sites is also selected at random for an end move. If it is not an end bead, then, depending on the position of its neighbors along the chain, it can perform either a corner move or a crankshaft move (the direction of the latter move is selected at random). If the move chosen would violate the excluded volume constraint by moving the monomer to an occupied site, the trial ends. If there are no spatial constraints, the energies of the original and new configurations are calculated, and the move is realized with the probability given by the Metropolis rule [W ⫽ 1 for ⌬E ⱕ 0, and W ⫽ exp(⫺⌬E/T) for ⌬E ⱖ 0, where ⌬E is the energy difference of the final and initial states]. This algorithm does not include collective modes of motion and accordingly it can hardly be used to describe “skating” or “rolling” of a protein along the surface. The role of such modes is expected to be minor provided that adsorption is accompanied by denaturation because in the latter case diffusion may primarily occur via the rearrangements of the whole protein structure due to local moves. D ⫽ 关具共⌬x兲2典 ⫹ 具共⌬y兲2典兴/4⌬t, (2) where ⌬t is the diffusion time measured in MCS. To get the mean-square displacement, we used seven MC runs. RESULTS OF SIMULATIONS Using local elementary moves described above, one can observe protein diffusion only at T ⬎ Tg, because at T ⬍ Tg a protein is expected to be trapped into one of the metastable states and accordingly there will be no displacements of the whole molecule. The glass transition temperature Tg ⫽ 1.1 introduced in the section “MODEL” corresponds to the bulk. In general, the value of this temperature is known to depend on smoothness of the energy landscape. Rough energy landscapes resulting in high values of Tg occur in problems in which there are many competiting MONTE CARLO SIMULATIONS 79 Fig. 4. Arrhenius plots for surface diffusion of the model protein molecule with ⑀Aa ⫽ 3 and ⑀Ba ⫽ 1. The average statistical error in the results is shown by the size of the data points. Fig. 3. Mean-square displacement of the model protein molecule as a function of time for ⑀Aa ⫽ 3 and ⑀Ba ⫽ 1, at T ⫽ 1.2 (a) and 2.4 (b). The inserts exhibit typical protein structures during diffusion. The monomers contacting the surface [20 and 17 beads in cases (a) and (b), respectively] are located in the first layer (in the plane of the figure). The other monomers are located in the second layer. interactions20 (in analogy with the spin glass theory, this competition is sometimes called “frustration”). For adsorbed proteins, due to frustration, Tg might be higher than that for the bulk. For our model parameters, this effect seems to be negligible, because for reasonable lengths of MS runs (up to 2 ⫻ 107 MCS) we were able to observe diffusion at temperatures (e.g., T ⫽ 1.2) slightly above than that corresponding to the glass transition in the bulk. The data obtained for the first set of the monomera a substrate interactions (⑀A ⫽ 3 and ⑀B ⫽ 1) are displayed in Figures 2– 4. In this case, the monomers are located (Fig. 2) primarily in the first and second layers both at low and high temperatures (T ⫽ 1.2 and 2.4, respectively). During diffusion, the fluctuations in the number of monomers in these layers are seen to be appreciable and accordingly the changes of the protein structure are considerable. The typical mean-square displacements (Fig. 3) are proportional to time as it should be for diffusion. The dependence of the diffusion coefficient on temperature is found to be linear in the Arrhenius coordinates (Fig. 4), i.e., it can be represented in the standard form, D ⫽ Do exp(⫺Ea/T), where Do ⫽ 0.18 ⫻ 10 ⫾ 0.2 a2/MCS and Ea ⫽ 12.6 ⫾ 0.6 are the preexponential factor and activation energy, respectively. Near Tg, one could expect deviations from the Arrhenius behavior because the dynamics of relaxation of the protein structure in this case is non-exponential19 (see also relevant simulations21,22 of the relaxation of adsorbed nonbiological polymers), but in our case the Arrhenius representation seems to be applicable down to Tg. To convert the preexponential factor calculated in MC units into real units, we can compare the value obtained with that corresponding to an ideal random walk of a single monomer. In the latter case, the prexponential factor calculated in MC and real units is respectively given by 0.25 a2/MCS and Do ⬇ 10 ⫺ 2 ⫺ 10 ⫺ 3 cm2/s (the latter “normal” value directly follows from the transition state theory12). The fact that for our model the preexponential factor calculated in the MC units is close to 0.25 indicates that in real units the preexponential factor should be close to the normal value, i.e., Do ⬇ 10 ⫺ 2 ⫺ 10 ⫺ 3 cm2/s. For the second set of the monomer-substrate interaca a tions (⑀A ⫽ 3 and ⑀B ⫽ 2), the dependence of the protein structure on temperature is stronger. At temperature below Tf, almost all the monomers are located in the first layer (Fig. 5a). The occupation of the second layer is considerable at higher temperatures (Fig. 5b). The former finding is quite different compared to that obtained for the first set of the monomer-substrate interactions (cf. Figs. 2a and 5a). Despite this difference, the Arrhenius parameters found for the second set of the interactions, Do ⫽ 0.15 ⫻ 10⫾0.2 a2/MCS and Ea ⫽ 12.6 ⫾ 0.6, are very close to those obtained for the first set. 80 V.P. ZHDANOV AND B. KASEMO Fig. 5. As Figure 2 for ⑀Ba ⫽ 2. CONCLUSION Our MC simulations indicate that for the model under consideration the preexponential factor for diffusion of adsorbed proteins is close to the normal value (about 0.25a2/MCS in MC units or about 10⫺2 ⫺ 10⫺3 cm2/s in real units) typical of monomer diffusion. The rationalization of the results obtained is not straightforward. Taking into account that the preexponential factor found is close to 0.25, one might suggest that diffusion is limited by the movement of certain monomers, e.g., by corner moves. The maximum change in energy involved in such moves is 12 (4 bonds of strength 3 are broken). The latter value is close to the apparent activation energy (12.6) obtained in simulations. We are not however quite sure that this reasoning is correct because in our simulations the monomers are typically located in two layers (Fig. 3). For such configurations, the maximum change in energy during the corner moves is 9 (3 bonds are broken). A more plausible explanation seems to be that diffusion is controlled by a combination of monomer moves complicated by spatial constraints. The preexponential factor obtained in our simulations is higher than that ( ⯝ 0.25/N) predicted for diffusion in a random-coil state but much lower compared to that estimated for bovine serum albumin diffusion on polyhexylmethacrylate (see the Introduction). The former is not surprising because in our simulations the typical structures of adsorbed protein molecules are rather compact even at high temperatures (see e.g., Fig. 3b for T ⫽ 2.4). The latter might indicate that conformational changes in adsorbed bovine serum albumin are qualitatively different compared to those predicted by our model (the structure of adsorbed bovine serum albumin is much more stable than that in our simulations). Finally, we may conclude that diffusion of adsorbed proteins merits additional theoretical studies. In particular, it is of interest to execute simulations for the same model but with other dynamics of elementary moves or explore other modes of diffusion including collective moves. ACKNOWLEDGMENT V.P. Zh. is grateful for the Waernska Guest Professorship at Göteborg University. REFERENCES 1. Ramsden JJ. Kinetics of protein adsorption. In: Malmsten M, editor. Biopolymers at interface. New York: Marcel Dekker, 1998. p 321–361. 2. Zhdanov VP, Kasemo B. Monte Carlo simulations of the kinetics of protein adsorption. Surf Rev Lett 1998;5:615– 634. 3. Michaeli I, Absolom DR, van Oss CJ. Diffusion of adsorbed proteins within the plane of adsorption. J Coll Interf Sci 1980;77: 586 –587. 4. Burghardt TP, Axelrod D. Total internal reflection/fluorescence photobleaching recovery study of serium albumin adsorption dynamics. Biophys J 1981;33:455– 467. 5. Tilton RD, Robertson CR, Gast AP. Lateral diffusion of bovine serium albumin adsorbed at the solid-liquid interface. J Coll Interf Sci 1990;137:192–203. 6. Rabe TE, Tilton RD. Surface diffusion of adsorbed proteins in the vicinity of the substrate glass transition temperature. J Coll Interf Sci 1993;159:243–245. MONTE CARLO SIMULATIONS 7. Thompson NL, Pearce KH, Hsieh HV. Total internal-reflection fluorescence microscopy application to substrate-supported planar membranes. Eur Biophys J 1993;22:367–378. 8. Ramsden JJ, Bachmanova GI, Archakov AI. Kinetic evidence for protein clustering at a surface. Phys Rev E 1994;50:5072–5076. 9. Tilton RD. Mobility of biomolecules at interfaces. In: Malmsten M, editor. Biopolymers at interface. New York: Marcel Dekker, 1998. p 363– 407. 10. Raut JS, Fichthorn KA. Diffusion mechanism of short-chain alkanes on metal substrates: unique molecular features. J Chem Phys 1998;108:1626 –1635. 11. Hjelt T, Herminghaus S, Ala-Nissila T, Ying SC. Dynamics of chainlike molecules on surfaces. Phys Rev E 1998;57:1864 –1872. 12. Zhdanov VP. Elementary physicochemical processes on solid surfaces. New York: Plenum; 1991. 314 p. 13. Shakhnovich EI. Protein design: a perspective from simple tractable models. Fold Des 1998;3:R45–R58. 14. Sumithra K, Bacemgaer A. Adsorption of random copolymers: a scaling analysis. J Chem Phys 1999;110:2727–2731; and references therein. 81 15. Kong CY, Muthukumar M. Monte Carlo study of adsorption of a polyelectrolyte onto charged surfaces. J Chem Phys 1998;109:1522– 1529; and references therein. 16. Shakhnovich EI, Gutin AM. Engineering of stable and fast-folding sequences of model proteins. Proc Natl Acad Sci USA 1993;90: 7195–7199. 17. Socci ND, Onuchic JN. Folding kinetics of proteinlike heteropolymers. J Chem Phys 1994;101:1519 –1528. 18. Socci ND, Onuchic JN. Kinetic and thermodynamic analysis of proteinlike heteropolymers: Monte Carlo histogram technique. J Chem Phys 1995;103:4732–1528. 19. Zhdanov VP, Kasemo B. Monte Carlo simulation of denaturation of adsorbed proteins. Proteins 1998;30:168 –176. 20. Bryngelson JD, Onuchic JN, Socci ND, Wolynes PG. Funnels, pathways, and the energy landscape of protein folding: a synthesis. Proteins 1995;21:167–195. 21. Chakraborty AK, Adriani PM, Glassy relaxation at polymer-solid interfaces. Macromolecules 1992;25:2470 –2473. 22. Shaffer JS. Computer simulation of homopolymer and copolymer adsorption dynamics. Macromolecules 1994;27:2987–2995.