Simulating the Equation of State of Model Globular Proteins Adsorbed

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Simulating the Equation of State of Model Globular Proteins Adsorbed at a Surface Stephen R. Euston* and Md. Abu Naser School of Life Sciences, Heriot-Watt University, Edinburgh, EH14 4AS Scotland Received December 9, 2004. In Final Form: February 24, 2005 Monte Carlo computer simulation is used to follow the adsorption of a model globular and disordered protein at a hard surface and to simulate the surface equation of state for these molecules. The simulation utilizes the deformable globule model where the “protein” is treated as a collection of interacting subunits. Disordered globules are modeled as athermal molecules, whereas globular molecules have a strong attractive interaction between subunits. The surface equation of state is modeled by applying a known pressure to an adsorbed globule and following the changes in adsorbed conformation. Simulated equations of state for the disordered and globular molecules show features that are observed in experimental surface pressure versus area plots. In particular the simulated equations of state show “kinks” that correspond to regions where the adsorbed globules undergo conformational changes as they lift away from the surface in response to the increased pressure. The model proteins follow Bull’s equation at low surface pressures in a way that is broadly in line with results from experiment, and the changes in conformation as a function of surface pressure are in line with predictions by DeFeijter and Benjamins made using a soft-particle model for adsorbed proteins.

Introduction Proteins have a strong tendency to adsorb at the interface between an aqueous phase and a second nonaqueous phase such as a solid surface or a hydrophobic liquid surface. This is exploited, for example, in the manufacture of foods containing emulsified fats or oils, where the protein adsorbs to and stabilizes the fat droplets. The ability of proteins to form and stabilize emulsions is linked to the structure and conformation of the protein. Of the protein types used to stabilize emulsions random disordered proteins and globular protein are the most common. These are typified by the milk caseins (random coil) and milk whey proteins (globular). The conformation that proteins adopt when adsorbed at a surface depends on a number of factors. These are, in general, the same factors that control the conformational stability of the protein in solution (pH, ionic strength, temperature, etc.). Knowledge of protein conformation is important in a number of technological areas. The conformation of an adsorbed protein layer impacts the the colloidal stability of any particles to which they adsorb. Differences in the adsorbed conformations between globular and disordered proteins contribute to the observed differences in the shear and dilational rheology of adsorbed protein layers.1 The surface rheology of adsorbed proteins has been correlated with emulsion stability in a number of studies. Proteins are also being used increasingly in devices such as biosensors. In these, molecules such as enzymes are adsorbed onto surfaces and are then used for rapid testing of various analytes. If the efficiency of the assay is to be maintained the conformation of the enzyme must remain relatively unperturbed. Thus, an understanding of adsorbed protein conformation is important to the success of this technology. The detailed conformation of individual proteins at a surface cannot be elucidated directly in any great detail. * To whom correspondence should be addressed. E-mail: S. [email protected]. Tel: +44 131 451 3640. Fax: +44 131 4513009. (1) Dickinson, E. Colloids Surf., B 1999, 15, 161-176.

Techniques such as X-ray diffraction and NMR that are widely used to determine crystal structure or solution structure of proteins either cannot be used or have limited applicability to adsorbed proteins. The majority of information we have on adsorbed protein conformation has been inferred from indirect measurements. Early work in this field used surface pressure versus area isotherms to infer protein conformations.2 Recent research by Rodriguez-Patino and co-workers3 has used this approach to study adsorbed layer structural changes in more detail. More recently, neutron reflectance has been used by various researchers4-10 to infer the structure of proteins in adsorbed layers at air-water, oil-water, and solidliquid interfaces. An alternative approach to studying adsorbed protein conformation is to use computer simulation. The development of simulation methods for protein adsorption has been hindered by the structural complexity of even the most simple protein molecules. Of the proteins that are most often used to stabilize food colloids one can distinguish two types, the disordered proteins and the globular proteins. Different approaches have been taken to simulate these. Simulation models for adsorbed proteins have recently been reviewed elsewhere.11 (2) Graham, D. E.; Phillips, M. C. J. Colloid Interface Sci. 1979, 70, 427-438. (3) Rodriguez-Patino, J. M.; Sanchez, C. C.; Rodriguez-Nino, M. R. Food Hydrocolloids 1999, 13, 401-408. (4) Dickinson, E.; Horne, D. S.; Phipps, J. S.; Richardson, R. M. Langmuir 1993, 9, 242-248. (5) Lu, J. R.; Su, T. J.; Thomas, R. K.; Penfold, J.; Webster, J. J. Chem. Soc., Faraday Trans. 1998, 94, 3279-3287. (6) Lu, J. R.; Su, T. J.; Thomas, R. K. J. Colloid Interface Sci. 1999, 213, 426-437. (7) Su, T. J.; Lu, J. R.; Thomas, R. K.; Cui, Z. F.; Penfold, J. J. Colloid Interface Sci. 1998, 203, 419-429. (8) Lu, J. R.; Su, T. J.; Thirtle, P. N.; Thomas, R. K.; Rennie, A. R.; Cubitt, R. J. Colloid Interface Sci. 1998, 206, 212-223. (9) Su, T. J.; Lu, J. R.; Thomas, R. K.; Cui, Z. F.; Penfold, J. J. Phys. Chem. B 1998, 102, 8100-8108. (10) Lu, J. R.; Su, T. J.; Georganopoulou, D.; Williams, D. E. J. Phys Chem. B 2003, 107, 3954-3962. (11) Euston, S. R. Curr. Opin. Colloid Interface Sci. 2004, 19, 321327.

10.1021/la046977p CCC: $30.25 © 2005 American Chemical Society Published on Web 03/24/2005

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In this study we have used the deformable globule model to investigate the structural transitions that occur in an adsorbed molecule as the surface pressure is increased. This model has been used previously to simulate the competitive adsorption of a mixture of compact and flexible polymers12,13 and the adsorption of protein-polysaccharide complexes.12,14 This model enables us to generate a surface equation of state for adsorbed globules and to relate structural changes to features in the equation of state. Our aim is to compare and contrast the behavior of compact deformable globular protein-like molecules and disordered molecules and to relate this to the observed experimental behavior of real proteins. Simulation Methodology Simulations were carried out on a cubic lattice of side 1000 lattice sites. Periodic boundary conditions were defined in the x- and y-coordinate directions. A protein molecule was simulated using a three-dimensional version of the deformable globule model proposed originally by Dickinson and Euston.12 The protein is defined as a collection of individual subunits, with 125 chosen for this work. Each subunit occupies a single lattice site. Unlike linear polymer molecules the subunits in a globule are not bound to each other in a linear chain via fixed bonds. They are held together by nonbonded attractive interactions. The individual subunits are allowed to move (1 lattice site each Monte Carlo (MC) step in each of the three coordinate directions. A move is accepted as long as three conditions are met. (i) Double occupancy of a lattice site does not occur. (ii) The move is energetically favorable (this is tested by summing pair interactions between subunits and using the Metropolis form of the MC method). (iii) All subunits remain connected to each other in the globule. The latter condition is tested using an algorithm designed by Stoddard15 for testing particle connectivity in aggregates. A simple pair potential represents the attractive interactions [U(r)] between subunits in the globule where U(r) is equal to a value s if the subunits occupy adjacent lattice sites and U(r) ) 0 if they do not. The interaction energy is measured in units of kT. The magnitude of the attractive subunit interaction s can be used to control the structure and flexibilty of the globule. Simulations on an isolated 125-subunit globule at varied s shows that the subunits adopt a compact globular structure for strongly attractive s. If the attractive interaction is lowered to less than about -1.4 to -1.3 kT, the attractive forces are not sufficient to overcome the effects of conformational entropy and the subunits adopt an open disordered conformation. At intermediate values of s the globule undergoes an unfolding transition (an order-disorder transition) for s values between -1.4 and -1.5. This behavior can be quantified by measuring the number of adjacent contacts that each subunit has with other subunits. A plot of normalized Nc versus -s is shown in Figure 1. The normalization factor used in Figure 1 is 26, the maximum number of neighbors that a single subunit can have on the cubic lattice. It is clear from this figure that as the attractive interactions between subunits decrease the globule starts to open up and adopts a more (12) Dickinson, E.; Euston, S. R. Adv. Colloid Interface Sci. 1992, 42, 89-148. (13) Dickinson, E.; Euston, S. R. J. Colloid Interface Sci. 1992, 152, 562-572. (14) Dickinson, E.; Euston, S. R. Food Hydrocolloids 1992, 6, 345357. (15) Stoddard, S. D. J. Comput. Phys. 1978, 27, 291-293.

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Figure 1. Unfolding transition for a 125-subunit globule. The number of nearest neighbors is summed for all 125 subunits and normalized by dividing by the maximum total neighbors [Nc(max)] for the cubic starting conformation. The fraction of nearest neighbors (Nc) is plotted as a function of subunit interaction energy (s). The values of s used in this study (-2.0, -1.5, and 0) are marked on the figure for reference purposes.

open structure. Around s ) -1.5 to -1.4 there is a rapid unfolding of the structure, and there is evidence that the structure becomes somewhat unstable and fluctuates between folded and open states. Qualitatively, the same behavior has been observed by Dinner and Karplus16 and Radke et al.17 for folded linear lattice chains. Changing s is equivalent to changing the temperature in the simulation. Thus, the transition from a compact to an open conformation can also be thought of as being equivalent to heat-induced unfolding of a globular protein. We are exploring this in a separate study investigating ensembles of deformable globules as a model for globular proteins undergoing heat-induced denaturation, aggregation, and gelation (Euston and Costello, unpublished data). In this study, though, we are using s to control the deformability and structure of the globule. This allows us to generate a range of globule structures from compact globular to open and disordered. Adsorption of the globule at a surface is simulated by defining a plane at zi ) 0 or zi ) 1001 at which the subunits can adsorb with a given attractive adsorption energy. For adsorption to occur a subunit must occupy a lattice site with a z coordinate in the plane adjacent to the adsorbing surface (z ) 1 or z ) 1000). The attractive adsorption energy is defined by the potential,

{

∞ z ) |zi - zs| < 1 U(z) ) i z ) |zi - zs| ) 1 0 z ) |zi - zs| > 1

(1)

and zs is the z coordinate of subunit s. Isolated deformable particles are allowed to adsorb at the surface under the influence of attractive i. Globule adsorption is simulated for three different values of subunit interaction, s ) -2.0, -1.5, and 0.0. At s ) -2.0 the globule is in a state that is well above the unfolding transition and so adopts a compact folded conformation in the bulk phase. At s ) -1.5 the globule is close to the unfolding region, and simulations carried out to construct Figure 1 suggest that the globule conformation in the bulk phase tends to be unstable, fluctuating between open and folded conformations. At s ) 0.0 (athermal) the full effects of conformational entropy take over, and the globule adopts a fully disordered conformation. Each type of globule is simulated at a range of attractive i up to -2.5. Simulations were carried out from a starting conformation where the subunits were arranged in a 5 × 5 × 5 cube positioned in (16) Dinner, A. R.; Karplus, M. J. Mol. Biol. 1999, 292, 403-419. (17) Anderson, R. E.; Pande, V. S.; Radke, C. J. J. Chem. Phys. 2000, 112, 9167-9185.

Modeling Adsorbed Proteins

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the bulk phase well away from the adsorbing surface. The system was equilibrated for 1 × 106 MC attempts to generate an equilibrium conformation in the bulk phase. This conformation was then moved closer to (but not touching) the surface, and the simulation continued for between 1 × 106 and 5 × 106 MC steps until equilibrium adsorbed conformations were generated at the surface. The adsorbed conformations were then sampled every 103 MC attempts over a simulation run of 106 MC attempts. To test whether the system was at equilibrium the internal energy, fraction of subunits bound to the surface, and number of subunit contacts within the globule (Nc) were followed, and equilibrium was assumed to have been reached when these quantities converged to constant values.

Figure 2. Fraction of subunits adsorbed to the surface (bound fraction) as a function of subunit adsorption energy i. The three curves correspond to (b) s ) -2.0; (3) s ) -1.5; and (9) s ) 0.0.

Modeling the Effect of Surface Pressure on Protein Conformation

The latter is defined as a weighted average thickness (L) given by the equation

In real systems proteins are rarely found adsorbed at a surface in isolation. Most often they will be one molecule in a dense adsorbed layer composed of many protein molecules. In this case the presence of other surrounding protein molecules exerts a pressure (a surface pressure) that has implications for the conformation that the protein is able to adopt in the adsorbed layer. We have carried out a direct simulation of surface pressure effects by applying a constant pressure in the x-y plane to an isolated adsorbed molecule. By coupling this with information on the area occupied at the surface by the adsorbed molecule, we can build an equation of state for the globules. The methodology used to do this is detailed below. Simulations were carried out at varied surface pressures using an algorithm similar to that proposed by Dickinson and Euston18 for determining the equation of state of cyclic lattice chains. A constant radial force is applied to the adsorbed globule in the x-y plane of the simulation box. No force is explicitly applied in the z plane normal to the surface. The applied force is toward a constant x-y line that runs through the center of the globule in the z direction, such that the force is zero at this center line. This line is calculated by averaging the x and y coordinates of each subunit in the globule before each MC step. An extra interaction energy increment (∆p) is included in the simulation that acts to promote movement of the subunits toward the center plane of the globule. The interaction energy increment is distance-dependent such that the increment at a distance r [p(r)] from the central plane is given by the equation

L)

1

z

∑iF(zi)

N i)l

(4)

The surface area over which the force is applied is then simply the surface area of a cylinder of height L and crosssectional circumference 2‚pi‚Rg(x, y), and the pressure is the force per unit area.

π)

∆p 2‚pi‚Rg(x, y)L

(5)

Globules were simulated at three values of the intersubunit interaction, s ) -2.0, -1.5, and 0.0, and an adsorption energy of i ) -1.0 per subunit for all globules. Each globule was simulated over a range of applied surface pressures. The mean surface area occupied per molecule was determined from the mean radius of gyration, and a surface equation of state (surface pressure vs area per molecule) plot was constructed. Results and Discussion

In other words, ∆p (which is equivalent to the slope of the interaction energy-distance plot) is a measure of the force applied to the globule. This can be converted to a pressure if the surface area over which it is applied can be calculated. The surface area is estimated by assuming that the globule is a cylinder, the radius of which is given by the radius of gyration of the globule in the plane parallel to the surface (the x-y coordinate direction) and the height of which is calculated from the segment density profile for the globule in the z-direction [F(z)], normal to the surface.

Conformation of Isolated Adsorbed Globules. In Figure 2 the adsorption of globules of varied degrees of flexibility is followed by plotting the fraction of subunits bound to the surface as a function of adsorption energy. The degree of flexibility clearly influences the adsorption behavior of the globules. For the disordered globule with s ) 0.0 no adsorption is detected at values of -i below 0.5. This has been observed previously by Lal et al.19 who found that for linear polymer chains adsorption does not occur below an adsorption energy of about -i ) 0.45. This is a consequence of chain conformational entropy, which acts to exclude the chain from the vicinity of the surface where the numbers of conformations it can occupy are reduced. At adsorption energies above -i ) 0.5, the enthalpic contribution of the surface-subunit interaction compensates for loss of chain entropy and adsorption occurs. As -i increases above 0.5 the bound fraction remains relatively low until -i g 0.75. Presumably, for 0.5 e -i e 0.75 chain entropy plays a significant role in determining adsorption behavior. Above -i ≈ 0.75 there is a rapid increase in the bound fraction, although even at -i ) 2.5 there is still a significant fraction of the subunits (10%) not bound to the surface. If we now consider the case of a compact adsorbing globule with -s ) 2.0, we observe very different adsorption behavior. Compact globules have a significant bound

(18) Dickinson, E.; Euston, S. R. Mol. Phys. 1989, 66, 865-886.

(19) Clark, A. T.; Lal, M. Discuss. Faraday Soc. 1978, 65, 227.

p(r) )∆pr

(2)

The force exerted on the globule is simply the derivative of the applied interaction energy with respect to distance

F)-

d(∆pr) dp(r) )) -∆p dr dr

(3)

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Figure 3. Typical adsorbed conformations for an athermal (s ) 0) globule at different subunit adsorption energies (i). (a) i ) -0.5; (b) i ) -0.75; (c) i ) -1.0; and (d) i ) -1.5.

fraction even at low adsorption energies of -i ) 0.1. As -i is increased up to 1.0, the rate of increase of the bound fraction is much slower than for the athermal disordered globule. It would appear that for -i e 1.0 these globules adsorb as a soft particle and remain relatively intact at the surface. Consequently, chain entropy plays little part in the adsorption behavior, and adsorption at low i is possible. This adsorption behavior is typical of particle adsorption rather than chain adsorption.20 As the strength of the surface-subunit interaction is increased further above -i ) 1.0, the rate at which the bound fraction changes starts to increase. This is indicative of a greater spreading of the globule at the surface. Above -i ≈ 1.5 the bound fraction is higher than that of the athermal globule at the same i. Above -i ) 1.75 almost all subunits are bound to the surface. The shape of the bound fraction curve for -s ) 2.0 is representative of a molecule that adsorbs initially as a compact globular particle-like molecule, spreads slowly at the surface at low i, and then, when i is strong enough to overcome the internal interactions, unfolds at the surface. The high bound fraction at relatively high i is a consequence of the strong attraction between subunits. As the globule unfolds and collapses onto the surface, it becomes more favorable energetically to form a two-dimensional condensed monolayer where there is a balance between surface-subunit and subunit-subunit interactions, rather than forming a three-dimensional globule where only subunit-subunit interactions are optimized. This is evident in Figure 5 where representative adsorbed conformations of a globule at s ) -2.0 are plotted for different values of i. These can be compared to conformations for the athermal globule (Figure 3). The globular form of the adsorbed compact molecule is clearly seen at low i (Figure 5a), and the transition to a collapsed monolayer is equally evident at high i. This is not observed for the athermal, open globule (Figure 3). There is experimental evidence to support the formation of ordered structures in a protein-adsorbed layer. Lefevre and Subirade21 have observed the formation of intramolecular β-sheet structure between different β-lactoglobulin molecules adsorbed at an oil-water interface. When the protein adsorbs, the native secondary structure unfolds during surface denaturation and then reforms intramolecularly. (20) Dickinson, E.; Pelan, E. G. Mol. Phys. 1991, 74, 1115-1124. (21) Lefevre, T.; Subirade, M. J. Colloid Interface Sci. 2003, 263, 59-67.

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Figure 4. Typical adsorbed conformations for a globule close to the unfolding transition (s ) -1.5) at different subunit adsorption energies (i). (a) i ) -0.5; (b) i ) -0.75; (c) i ) -1.0; and (d) i ) -1.5.

Figure 5. Typical adsorbed conformations for a compact globule (s ) -2.0) at different subunit adsorption energies (i). (a) i ) -0.5; (b) i ) -0.75; (c) i ) -1.0; and (d) i ) -1.5.

The spreading of adsorbed globules at the surface can also be followed by looking at the components of the radius of gyration of the molecule in the x-y coordinate plane, parallel to the surface. This is defined by eq 6,

Rg(x, y) )

1

N

∑[((xc - xi)2 + (yc - yi))2]1/2

Ni)1

(6)

where xc and yc are the mean x and y coordinates averaged over all N subunits. In Figure 6, Rg(x, y) is plotted against i. At low i both the compact and open globules adopt adsorbed conformations with Rg(x, y) close to that of the globule in solution. For the compact globule Rg(x, y) increases slowly up to -i ) 1.5 and then increases rapidly up to an approximately constant value. This transition to a higher plateau value of Rg(x, y) corresponds to the point at which i becomes strong enough to overcome internal attractions in the globule, and it unfolds and flattens at the surface. For the athermal globule, a similarly shaped Rg(x, y) versus i plot is found. Here, however, the transition from an open adsorbed conformation to a flattened adsorbed conformation occurs at much lower -i (≈ 0.5 to


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Figure 6. Plot of radius of gyration in the x-y plane [Rg(x, y)] as a function of subunit adsorption energy (i). The three curves correspond to (b) s ) -2.0; (3) s ) -1.5; and (9) s ) 0.0.

0.75), since there are no internal interactions holding the globule together and only conformational entropy needs to be overcome. The plateau value of Rg(x, y) for -i g 0.75 is higher than for the compact globule, since there are no attractive inter-subunit interactions to promote condensation of the globule at the surface. We have so far described two extreme cases for globule structure, athermal and compact. Intermediate values of s exhibit a behavior part way between that for athermal and compact globules. For s ) -1.5 the bound fraction as a function of i shows features typical of a chain molecule. Conformational entropy ensures that there is a critical value of i below which no adsorption occurs. This value of i is lower than for the athermal globule (-i < 0.25). Similarly, there is no obvious unfolding transition as is observed with the compact globules (s ) -2.0). Since the s ) -1.5 globule is already close to the unfolding transition (Figure 1) unfolding may occur at low i values. We can compare the conformations adopted by the s ) -1.5 globules (Figure 4) with the athermal and compact globules (Figures 3 and 5). At all i values the s ) -1.5 globule adopts conformations that are more open than for the compact globule but more compact than for the athermal globule (i.e., an intermediate behavior). The Rg(x, y) data shows that like the athermal globule there is a transition from a relatively compact to a spread adsorbed conformation as -i is increased above 0.5. The picture we can build up of deformable globule adsorption is one where the internal interactions of the globule play a large role in the unfolding behavior of the globule at a surface. Athermal globules behave like typical chain molecules where adsorption is opposed by chain entropy. Compact globules, where chain entropy is negligible, have behavior more like that of rigid particles at low adsorption energies but show a surface unfolding transition at higher i. Globules with s values between that of the athermal and compact cases show a behavior intermediate between the two. Protein molecules display a similar range of flexibilties of structure. The intermolecular forces holding together the tertiary structure of the protein control this. Some proteins such as lysozyme have a rigid structure, while others such as the milk caseins are unstructured, behave as random coils in solution, and are highly flexible. Many other globular proteins are intermediate in their behavior. The conformation that the model protein adopts depends on the balance between inter-subunit interactions and surface subunit interactions. Stronger inter-subunit interactions give more rigid molecules that adopt, in general, more compact conformations at the surface. Stronger surfacesubunit interactions promote surface unfolding of the model proteins. An analogous behavior is inferred for proteins from experiment. Lu et al.7,8 have shown that lysozyme adopts a more spread out conformation at a

Figure 7. Simulated equation of state for the athermal (s ) 0.0) globule. The equation of state is plotted for three different ranges of area occupied per adsorbed molecule. Features that are discussed later in the text (πBull and πL) and the surface pressures corresponding to typical adsorbed conformations (ad) detailed in Figure 14 are marked on the plots.

hydrophobic surface8 than it does at a hydrophilic surface.7 One of the driving forces for adsorption at a surface is displacement of surface-adjacent water into the bulk phase by hydrophobic protein groups, a driving force that is lower at a hydrophilic surface. Consequently, protein-surface interactions will be enhanced at hydrophobic surfaces and, as observed in experiments and predicted by our simulations, a protein will spread more. Qualitatively, these simulation results are the same as those that have been observed in an earlier model based on a cyclic lattice chain approach to modeling globular protein adsorption.12-14,22 Simulated Equation of State. Macromolecules are rarely found in isolation at surfaces, and such a situation has little technological significance for the stabilization of biological colloids. In a dense adsorbed layer the presence of surrounding macromolecules exerts a force or pressure on a globule. We have simulated the surface equation of state for deformable globules using the method detailed in section 2.2. Figures 7-9 are surface equations of state for athermal (s ) 0, Figure 7, s ) -1.5, Figure 8) and compact (s ) -2.0, Figure 9) globules. We can compare the simulation results to those obtained from simple theories and make observations on the similarity between simulated and experimental equations of state. Typically, experimental π-a curves are obtained by compression of adsorbed protein layers in a Langmuir trough. Our model can be thought of as a simulated Langmuir trough experiment. Bull23 proposed a simple equation of state for proteins (eq 7) that is a two-dimensional analogue of the threedimensional van der Waals equation of state.

p(a - a0) ) kT

(7)

where a is the surface area available per molecule and a0 is the co-area of the adsorbed protein. (22) Dickinson, E.; Euston, S. R. J. Chem. Soc., Faraday Trans. 2 1990, 86, 805-809. (23) Bull, H. B. J. Biol. Chem. 1950, 185, 27.

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Figure 8. Simulated equation of state for the s ) -1.5 globule. The equation of state is plotted for three different ranges of area occupied per adsorbed molecule. Features that are discussed later in the text (πBull) and the surface pressures corresponding to typical adsorbed conformations (a-d) detailed in Figure 15 are marked on the plots.

Figure 10. Plots of πa versus π to test the fit of the simulation results to Bull’s equation (eq 7). The values of π (πBull) at which Bull’s equation becomes invalid are marked on the plots. (a) s ) 0; (b) s ) -1.5; and (c) s ) -2.0.

Figure 9. Simulated equation of state for the compact (s ) -2.0) globule. The equation of state is plotted for three different ranges of area occupied per adsorbed molecule. Features that are discussed later in the text (πBull and πL) and the surface pressures corresponding to typical adsorbed conformations (a-d) detailed in Figure 16 are marked on the plots.

The surface equation of state is a plot of the twodimensional pressure at the surface as a function of the area occupied by a protein molecule at the surface. The simulated equations of state share the same features and also have some features in common with experimental results. At low π, π is independent of area occupied (a). This corresponds to a region where there is no inter-

action between protein molecules. In this region, Bull’s equation is expected to be valid. Bull’s equation (eq 7) implies that a plot of πa versus π should be a straight line. We can check for our simulation results how closely the simulation data fit this requirement. This is an indicator of the limits over which the adsorbed deformable particles act as impenetrable, nondeformable molecules. Parts a-c of Figure 10 are plots of πa versus π for the three types of globules. All globules show some rigid particle behavior. Not surprisingly the compact globules act as rigid particles up to higher applied surface pressures than do the more open globules. This is a simple reflection of the more compact conformation they adopt at the surface. From Figure 10 we can estimate the surface pressures (πBull) above which Bull’s equation is no longer valid (πa vs π is nonlinear) as being equal to 5 × 10-6 at s ) 0, 1.8 × 10-4 at s ) -1.5, and 1.5 × 10-3 at s ) -2.0. The pressure (πBull) is marked on the equation of state (Figures 7-9). At π < πBull the area occupied is


independent of π and the globules behave as impenetrable particles. At surface pressures above πBull, relatively large increases in π lead to small changes in a. In this region we can also see plateaus in the π-a curves, and this is particularly noticeable for the compact s ) -2.0 globule. These features have been observed in experimental π-A curves2 and are attributed to the formation of loop segments where parts of the protein detach from the surface and protrude into the bulk phase. This allows the area occupied by the molecule to decrease, but at constant π. At higher π, a tends toward a minimum area which corresponds to a fully elongated conformation. In experimental curves, a maximum π is reached called the collapse pressure. At this point the adsorbed protein layer responds to increasing protein density by “squeezing-out” some of the protein molecules. Thus, the layer relaxes to a constant pressure state. This is not observed in the simulations, which is a consequence of there being only a single molecule in the simulation. De Feijter and Benjamin24 have discussed at length the problems with Bull’s equation. They have proposed that many of the problems might be overcome by thinking of a globular protein as a soft particle which is able to change shape in response to changes in surface concentration/ surface pressure. De Feijter and Benjamins were able to make some general predictions as to how a protein might behave at high surface coverage using the so-called scaled particle theory of fluids originally proposed by Helfand et al.25 They predicted that for the globular protein ovalbumin the radius of the protein in the plane of the surface remained constant until the surface coverage reached saturated monolayer coverage and decreased above this as surface concentration increased. The disordered, nonglobular protein β-casein was predicted to behave somewhat differently. They predicted that the radius R of the protein changed over a wider range of surface concentration, and from their results it would appear that if disordered proteins exhibit a rigid disk behavior it must be at very low surface concentrations. Figure 11 shows the mean radius of the simulated globules in the plane parallel to the surface. To compare simulated globule conformations with the predictions of De Feijter and Benjamins24 we can calculate the mean radius of the globule in the plane parallel to the surface [Rg(x, y)]. However, we have no way of determining the surface coverage for our simulated system. Consequently, we have plotted Rg(x, y) as a function of π in Figure 11. π is related to the surface coverage Γ for adsorbed proteins, although not necessarily linearly. The simulation results in Figure 11 are in general agreement with the predictions of De Feijter and Benjamins. All simulated globules exhibit a region where Rg(x, y) is independent of π. This region extends over a wider range of π values for the compact globule than it does for the s ) -1.5 globule and the disordered athermal globule. This is analogous to the compact globule behaving in a manner similar to ovalbumin, while the disordered globules behave more like β-casein. For all globules Rg(x, y) is almost constant up to π ) πBull. Above this Rg(x, y) decreases as the globules become compressed. We have also followed changes in the globule conformation by plotting the fraction of subunits adsorbed to the surface (ν, Figure 12) and the mean extension of the (24) DeFeijter, J. A.; Benjamins, J. J. Colloid Interface Sci. 1982, 90, 289-292. (25) Helfand, E.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1961, 34, 1037.

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Figure 11. Radius of gyration [Rg(x, y)] in the x-y plane plotted as a function of log(π). (a) s ) 0; (b) s ) -1.5; and (c) s ) -2.0. Values of πBull and πL and the surface pressures corresponding to typical adsorbed conformations (a-d) detailed in Figures 14-16 are marked on the plots.

Figure 12. Bound fraction of adsorbed subunits plotted as a function of log(π). (a) s ) 0; (b) s ) -1.5; and (c) s ) -2.0. Values of πBull and πL and the surface pressures corresponding to typical adsorbed conformations (a-d) detailed in Figures 14-16 are marked on the plots.

globule normal to the surface (L, Figure 13). For values of π e πBull both ν and L are only weakly dependent on π. These results are consistent with a globule adsorbed into a layer at low surface coverage where there is little interaction between globules, and, thus, the conformation remains unperturbed. This is confirmed by the typical globule conformations plotted in Figures 14-16. Figures

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Figure 15. Typical adsorbed conformations for the s ) -1.5 globule at different values of the applied pressure (π). The corresponding surface pressures, Rg(x, y), bound fractions, and mean extensions for these conformations are marked in Figures 8 and 11-13. (a) log(π) ) -3.75; (b) log(π) ) -3.10; (c) log(π) ) -2.58; and (d) log(π) ) -1.93.

Figure 13. Extension of the globule perpendicular to the adsorbing surface plotted as a function of log(π). (a) s ) 0; (b) s ) -1.5; and (c) s ) -2.0. Values of πBull and πL and the surface pressures corresponding to typical adsorbed conformations (a-d) detailed in Figures 14-16 are marked on the plots.

Figure 16. Typical adsorbed conformations for the s ) -2.0 globule at different values of the applied pressure (π). The corresponding surface pressures, Rg(x, y), bound fractions, and mean extensions for these conformations are marked in Figures 9 and 11-13. (a) log(π) ) -2.84; (b) log(π) ) -2.39; (c) log(π) ) -2.13; and (d) log(π) ) -1.91.

Figure 14. Typical adsorbed conformations for the athermal (s ) 0) globule at different values of the applied pressure (π). The corresponding surface pressures, Rg(x, y), bound fractions, and mean extensions for these conformations are marked in Figures 7 and 11-13. (a) log(π) ) -5.32; (b) log(π) ) -4.21; (c) log(π) ) -3.98; and (d) log(π) ) -3.90

14a, 15a, and 16a are conformations for globules adsorbed at a surface where π is just below πBull. These conformations describe globules where the conformation is not perturbed by the surface pressure. As π increases above πBull the globules are subject to the equivalent of an increase in the surface density of globules (Figures 14b, 15b, and 16b). This leads to a decrease in the surface area occupied by the individual globule and a decrease in Rg(x, y) (Figure 11). However, these changes are not the same for all types of globule. For the compact

globules (s ) -2.0) this is accompanied by a significant decrease in ν (Figure 12) and a relatively slow increase in L (Figure 13) with π. Evidently a compact globule responds to increased surface pressure in this range by altering its conformation to one that occupies less space at the surface and extends further into the bulk phase away from the interface. This behavior persists until π reaches a critical value we denote as πL. At this point there is a rapid expansion of the globule dimension normal to the surface (L increases rapidly) and a rapid decrease in Rg(x, y). Conformations c and d in Figure 16 typify the globule conformations on either side of this transition point. Above πL the globule adopts an elongated conformation, extended in the plane normal to the interface (Figure 16d). For the athermal globule, as π is increased above πBull the mean Rg(x, y) also decreases as the molecule has less space to occupy at the surface (Figure 11). Similarly the bound fraction of subunits also decreases, although unlike s ) -2.0 where there is a definite change in the behavior of ν with π at πBull, for the athermal globule ν appears to respond to changes in π in the same way regardless of whether π is above or below πBull. Also unlike the compact globule the values of L for the athermal globule do not


show a significant increase as π is increased above πBull. If we compare typical conformations for athermal globules in the region up to πL (Figure 14) we note that the globule adopts a flattened, open conformation (Figure 14a,b). There is a tendency, however, for the conformation to become more compact at the surface as π increases, which is consistent with the data in Figures 11 and 12. Like the compact globule, the athermal globule also exhibits a transition at πL above which there is a rapid decrease in Rg(x, y) and ν and a rapid increase in L. It would appear that athermal globules also respond to changes in π (or surface coverage) by decreasing the area they occupy at the surface. Unlike the compact globules, however, because athermal globules have a more open conformation they are able to become more compressed before the conformation starts to “lift” away from the surface. This occurs close to πL (Figure 14c). The s ) -1.5 globule exhibits a surface behavior intermediate between the compact and athermal cases. For π < πBull the conformations (Figure 15a) are similar to those for the athermal case. Below πBull these globules are insensitive to increases in π (Figures 11-13). As π increases above πBull the globule adopts more compact conformations (Figure 15b,c), but the globule does not show significant extension normal to the surface (Figure 13). For the s ) -1.5 globule it is not possible to define a value for πL from Figure 13. Rather there is a continuous, steady increase in L as π increases up to π ≈ 0.01. Similarly, there are no obvious plateaus in the π-a curve (Figure 7b) as is seen with the compact and athermal globules. Features in Figures 14-16 can be correlated with features in the simulated equation of state (Figures 7-9). In general, two transition points can be identified. These correspond to the value of π above which Bull’s equation is no longer valid (πBull) and the value of π (πL) above which there is a rapid increase in L as the globule starts to elongate normal to the surface. We can plot the values of πBull and πL on the simulated equations of state (Figures 7-9) to see how these changes in conformation are reflected in the π versus a plot. For all simulated equations of state πBull appears to correspond to a transition point at which π starts to increase rapidly. This is the value of π or area occupied a at which a saturated monolayer is formed and the globule would begin to “feel” the presence of surrounding globules, with the resulting increase in pressure leading to structural changes. πL on the other hand corresponds to a plateau in the π-a curves. This represents the part of the equation of state at which the globule is squeezed such that it starts to lift off the surface. The result is a relaxation in the adsorbed layer as the globule conformation changes, the

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surface area of the globule decreases, and, thus, the pressure remains constant rather than increases. In effect this represents a yield point for the globule conformation where the perturbing forces from the external pressure exceed the internal forces holding the globule together, and it yields and flows. Conclusions The simulation model we have developed is able to reproduce features observed in experimental surface equations of state and to give some insight into the molecular and conformational changes that might lead to these transitions. The model also highlights possible differences in the way in which compact globular and disordered molecules respond to changes in the surface pressure. These results help to explain some of the observed and inferred differences between the surface behavior of globular and random coil proteins. However, because we have used a single globule and because we apply a centro-symmetric force field to simulate the pressure change we are unable to model the collapse pressure in these simulations. Thus, part of the simulated equation of state in Figures 7-9 may be an artifact of our simulation method. Currently, we are working on a multiparticle version of the simulation. In this version we do not need to explicitly apply a pressure force field, but we will vary π by changing the surface area of the interface available for the adsorbed globules. Then we can measure the average area occupied by the globule and use the results from Figure 7 for isolated globules as a calibration curve from which π can be determined. In this way we can construct a surface equation of state that we hope will simulate the collapse transition. In the absence of experimental techniques that are able to probe the detailed conformation of adsorbed protein, computer simulation represents our best current opportunity to study these structures in detail. We recognize that our model is very simplified, but it still reproduces many of the features expected of protein conformational change in adsorbed layers. The limitations of the model, though, are such that it cannot represent completely the full structural richness of real proteins. A comprehensive model awaits a full atomistic simulation of adsorbed protein, a goal we are working toward. Acknowledgment. The Engineering and Physical Sciences Research Council (EPSRC) is acknowledged for providing funds to purchase a Sunblade 1000a Unix workstation, and Heriot-Watt University computer centre is acknowledged for use of a Dell Poweredge Server. LA046977P

Simulating the Equation of State of Model Globular Proteins Adsorbed

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