Albumin adsorption onto pyrolytic carbon:A molecular mechanics approach

Sara Mantero,1 Daniela Piuri,1 Franco M. Montevecchi,1 Simone Vesentini,1 Fabio Ganazzoli,2

Giuseppina Raffaini2

1Dipartimento di Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy2Dipartimento di Chimica, Politecnico di Milano, Via Mancinelli 7, 20131 Milano, Italy

Received 3 August 2000; revised 7 June 2001; accepted 11 June 2001

Abstract: A number of implants of cardiac valve prosthe-sis, vascular prosthesis, and coronary stents present a pyro-lytic carbon interface to blood. Plasma protein adsorption isessential for the hemocompatibility of the implanted de-vices. This work quantitatively evaluates the molecular in-teraction force between a biomaterial surface (pyrolytic car-bon) and plasma protein (albumin) binding sites through asimplified molecular model of the interface consisting of (i)multioriented graphite microcrystallites; (ii) selected frag-ments of albumin; and (iii) a water environment. A numberof simplifying assumptions were made in the calculation:the albumin molecule was divided into hydrophobic andhydrophilic subunits (helices); an idealized clean, nonoxi-dized polycrystalline graphite surface was assumed to ap-proximate the surface of pyrolytic carbon. The interaction

forces between albumin helices and pyrolytic carbon sur-faces are evaluated from potential energy data. These forcesare decomposed into a normal and a tangential component.The first one is calculated using a docking procedure(F>tot MAX = 4.16 × 10−20 N). The second one (F\), calculatedby mean of geometric models estimating the energy varia-tion associated with the protein sliding on the material sur-face, varies within the range ±9.62 × 10−21 N. The molecularsimulations were performed using the commercial softwarepackage Hyperchem 5.0 (Hyperchem, Hypercube, Canada).© 2001 John Wiley & Sons, Inc. J Biomed Mater Res 59:329–339, 2002

Key words: molecular modeling; protein adsorption; albu-min; pyrolytic carbon; molecular mechanics

INTRODUCTION

Pyrolytic carbon is a biomaterial used in cardiacvalve prosthesis, vascular prosthesis, and coronarystents. Once immersed in the hematic flow, the bio-material interacts with plasma proteins, which form amonolayer adsorbed onto the surface through hydro-phobic interactions. The nature of these plasma pro-teins and their adsorption strength affect the im-planted devices’ hemocompatibility.1,2 Experimentalworks found that the adsorbed protein monolayer ismainly composed by albumin, which is held so tena-ciously that it cannot be easily removed,3–7 and de-vised experimental ways to measure these adhesionforces.8,9 Because of chemical and physical surfacecharacteristics, pyrolytic carbon shows excellent he-

mocompatibility. In biomedical devices it is found in agraphite microcrystalline state with multiorientedcrystallites and large boundary area.10,11 Albumincoats the biomaterial with a stable monolayer, inhib-iting the initiation of the coagulation response.

Spatial orientation and movements of proteins arealso influenced directly by the biomaterial topographyand mechanical force computation associated withprotein-surface interactions can provide a qualitativemethod to identify these interactions.

It is now widely recognized that molecular model-ing can play an important role in biomaterial designand in protein adsorption phenomena onto surfacesanalysis. Many aspects of protein adsorption at solid-liquid interface have been studied,12,13 and a unifiedtheory of intermolecular and surface force has beenwell established.14

The approach based on classical mechanics assumesthat the total interaction potential energy for proteinadsorption can be explicated as the sum of the elec-trostatic interaction, the dipolar attraction, the disper-sive attraction and the overlap repulsion (van derWaals contributions). Park calculated and comparedthe interaction potential energy between all possible

Correspondence to: S. Mantero; e-mail: mantero@biomed.polimi.it

Contract grant sponsor: Politecnico di Milano “Large ScaleComputing“ (LSC) program 1999–2001 for the “MolecularLevel Instrumentation for Biomaterial Interface Design”(BID) project.

© 2001 John Wiley & Sons, Inc.

orientation of four proteins (lysozyme, trypsin, immu-noglobulin Fab, and hemoglobin) and five neutralpolymer surfaces. The results show that among allthese contributions, the dispersion interaction is themajor force holding the protein on the polymersurface.12,13,15

The aim of the present work is to propose a meth-odology to quantitatively evaluate the interactionforce between a biomaterial surface and an adsorbedprotein in water. As an example, albumin adsorptionon pyrolytic carbon is analyzed through an elemen-tary molecular model of the biointerface. The adsorp-tion is studied by computer simulation at the atomisticlevel and through a multiscale mechanical analysis.

Three constituents make up the molecular modeladopted in the present exploratory study. The first oneis an ideal atomic model of a small fragment of pyro-lytic carbon consisting of a few parallel planes of car-bon atoms. Note that a clean graphite surface has thegreat advantage that electrostatic interactions due toeither free charges or to permanent dipoles are miss-ing. Therefore, the interaction is predominantly due tovan der Waal’s interactions. This fact, which has anobvious computational appeal in this preliminary,methodological study, is clearly an oversimplificationin simulating real pyrolytic carbon.

The second constituent, the albumin molecule,16 isdivided into hydrophilic and hydrophobic units (he-lices) in the computer simulations. We consider thattwo such helices separately approach the carbon at-oms in two different orientations, either from above,with their axis parallel to the planes, or from one side,with their axis perpendicular to the planes. The thirdand final constituent is made up of water moleculesfor simplicity. Any other solute is ignored in this pre-liminary study, in particular, the possible presence ofions. This choice is obviously related to the difficultproblem of dealing with long-range electrostatic inter-actions in computer simulations with periodic bound-ary conditions.17 These simulations permit us to ob-tain quantitatively the interaction forces between rep-resentative hydrophobic and hydrophilic helices andthe carbon planes in water. The results are used toestimate the interaction force of the whole albuminthrough a simplified representation of the protein ad-sorption and a mechanical analysis to elucidate theadsorption phenomenon. It has to be expected that theinteraction force of albumin with a real pyrolytic car-bon surface would be larger than the one estimated byour procedure, because we neglected electrostatic, andin particular dipolar interactions with the partiallyoxidized surface.

The comparison between the different interactionenergies calculated in the present computer simula-tions showed a more favorable value for adhesionwhen the hydrophobic helix interacts with the surface.Therefore, the normal component of the interaction

force was estimated using two simple biointerfacemodels composed by the hydrophobic helix in waterand a differently oriented ideal pyrolytic carbon sur-face. Such force was evaluated from the interactionenergy as a function of the distance between the centerof mass of the helix and the planes. From the samesimulation results, the tangential component of theinteraction force between albumin and the surfaceof ideal pyrolytic carbon was also derived throughtwo geometric models that account for moleculartranslation.

The results allow quantifying the total interactionforce between an albumin molecule and the carbonsurface in terms of the interaction of the hydrophobichelix. According to the proposed procedure, normaland tangential forces between the albumin moleculeand the surface of ideal pyrolytic carbon were calcu-lated using a multiscale approach, based on atomisticcomputer simulations and continuum mechanicsmethods.

MATERIALS AND METHODS

The biointerface model

The biointerface model includes three elements:

● Model of ideal pyrolytic carbon as a random aggrega-tion of domains measuring 50 × 25 × 25 Å.4 The gen-eration of the model proceeds from the atomic level tomicrocontinuum. The domains are made of two or sixsmall graphite parallel planes measuring about 50 × 25Å.2 This optimal geometrical conformation was foundusing a molecular mechanics (MM+ forcefield), runningthe conjugate gradient algorithm up to a gradient betterthan 10−3 (kcal/mol Å). Figure 1 shows two atomicmodels of ideal pyrolytic carbon domains. Two parallelcarbon planes compose the first domain, and six carbonplanes form the second one. The figure shows also thedomain’s surface facing albumin.

● Albumin protein model. The molecular structure ofcrystalline albumin was obtained from the Protein DataBank (http://www.rcsb.org/pdb/index.html). Data in-clude structural details, the amino acid sequence, thesubdivision into helices, and the Cartesian coordinatesof the nonhydrogen atoms.7 Figure 2 shows the albu-min molecule as obtained from the Protein Data Bank.

● Simplified physiologic environment model. A box ofwater was assumed to represent the plasmatic compo-nent in which the helices are diluted. Figure 3 shows thewater molecules optimized in a local energy minimumby running molecular mechanics (Amber forcefield) us-ing the conjugate gradient algorithm up to a gradientbetter than 10−3 (kcal/mol Å).

Afterwards, we selected two subunits (helices), namely ahydrophobic (A5C) and a hydrophilic (A1A) helix in zwit-terion form shown in Figure 4. Further studies were carried

330 MANTERO ET AL.

out on these helices interacting with the carbon planes indifferent orientations.

Figure 5 shows the four molecular models of the investi-gated biointerface:

(a) hydrophilic helix with its axis parallel to the carbonplanes in water;

(b) hydrophobic helix with its axis parallel to the carbonplanes in water;

Figure 2. The albumin protein molecule imported fromProtein Data Bank, the international repository for the pro-cessing and distribution of 3-D macromolecular structuredata. Figure 3. The 1200 water molecules box.

Figure 1. Domain of two graphite parallel planes (a) and domain of six graphite parallel planes (b) measuring about 50 ×25 Å.2 They are characterized by real interatomic and interplanar distances. Carbon-carbon bond distance is 1.41 Å if the twocarbons belong to the same hexagon, and 2.46 Å if the two carbons belong to different hexagons on the same plane, 3.5 Å beingthe distance between two different planes of the same domain.

331ALBUMIN ADSORPTION AND MOLECULAR MECHANICS

(c) hydrophilic helix with its axis perpendicular to thecarbon planes in water; and

(d) hydrophobic helix with its axis perpendicular to thecarbon planes in water.

The biointerfaces composed by helices parallel and perpen-dicular to the carbon planes are designated as BC1 and BC2,respectively (see Fig. 5). The energy optimization in water,which allows calculating the interaction potential energy,

was performed with the Amber forcefield using periodicboundary conditions. Energy minimization was carried outwith the conjugate gradient algorithm, up to a gradient>10−3 (kcal/mol Å).

The normal component of the interaction force

In order to quantify the normal component of the inter-action force between albumin and the carbon surfaces, wecalculated the interaction potential energy as a function ofthe distance between the center of mass of the helix and thesurface in BC1 and BC2 biointerfaces. The helix center ofmass was shifted vertically from 0.5 to 0.9 nm, with respectto the plane, with increments <0.02 nm. For each positionof the center of mass an equilibrium optimization wasperformed, and the interaction potential energy value wascalculated.

The interaction potential energy as a function of the dis-tance between the center of mass of the helix and the carbonplane was subsequently fitted to an analytical equation. Twodifferent equations for BC1 and BC2 biointerfaces wereadopted, both forms having a theoretical justification.

We interpolated the BC1 potential energy with a function

Figure 5. Visualization of the four biointerfaces after a molecular static energy optimization. Hydrophilic helix with its axisparallel to the carbon planes (a); hydrophobic helix with its axis parallel to the carbon planes (b); hydrophilic helix with itsaxis perpendicular to the carbon planes (c); hydrophobic helix with its axis perpendicular to the carbon planes (d).

Figure 4. The atomic structure of the hydrophobic helixA5C: LEU-LYS-GLU-CYS-CYS-GLU (a) and of the hydro-philic helix A1A: SER-GLU-VAL-ALA-HIS-ARG-PHE-LYS-ASP (b).

332 MANTERO ET AL.

which is a sum of two terms: (1) the overall attraction energybetween a nondeformable sphere (the modeled helix, in ourcase) and a plane and (2) the repulsive component of theLennard-Jones potential energy.18

EVdW = −H6 Fa

d+

a2a + d

+ lnS d2a + dDG + 4« FSd0

d D12G(1)

where H is Hamaker constant, a is sphere radius, d is sphere-plane distance, d0 is sphere-plane reference distance, and « isequilibrium energy.

Since electrostatic and/or dipolar interactions with theideal pyrolytic carbon surface are missing, we interpolatedthe BC2 potential energy with the Lennard-Jones potentialenergy:

EVdW = 4« FSd0

d D12

− Sd0

d D6G (2)

where d is sphere-plane distance, d0 is sphere-plane refer-ence distance, and « is equilibrium energy.

Afterwards, we calculated the interaction force for BC1and BC2 biointerfaces, respectively:

­EVdW

­d= FVdW ~d! =

H6 F 4a3

d2~2a + d!2G − 48« Fd012

d13 G (3)

­EVdW

­d= FVdW ~d! = 4« F−12

~d0!12

~d!13 + 6~d0!6

~d!7 G (4)

The tangential component of the interaction force

The tangential component of the interaction force betweenan albumin helix and the ideal pyrolytic carbon surface wasobtained using two monodimensional geometric models(M1 and M2) to represent the tangential movement of thehelix on the material surface.

Models M1 and M2 take in account the characteristics ofthe different carbon surfaces represented in BC1 and BC2biointerfaces:

● The carbon surface of the BC1 biointerface shows al-most perfect isotropic characteristics: during every trans-lation on the surface, the helix faces similar materialstructure’s properties, as shown in Figure 6.

● The carbon surface of the BC2 biointerface shows ortho-tropic characteristics. In fact, during a translation paral-lel to the planes the protein faces the same structuralproperties of the material [Fig. 7(a)]. Conversely, duringa translation perpendicular to the planes the proteinfaces different structural properties of the material,because adjacent planes are shifted by a half hexagon[Fig. 7(b)].

The geometric model M1 defines all possible moleculartranslations in BC1 biointerfaces and the molecular transla-tions parallel to the planes in BC2 biointerface. All carbonatoms pairs (Ai, Bi) can be represented by pseudo-atoms (Ci)halfway located between them, as shown in Figure 8. Duringtranslation the helix interacts with all pseudo-atoms of thematerial surface.

The geometric model M2 defines the helix translation per-pendicular to the planes in BC2 biointerfaces. Each plane canbe represented by a carbon pseudo-atom (Ai). The shiftedplane distribution (each plane is translated by a half hexa-gon compared with the adjacent ones) is described by asequence of alternating pseudo-atoms, as shown in Figure 9.The helix interacts with all carbon pseudo-atoms during itstranslation.

In both models the helix is treated as a particle concen-trated in its center of mass. For M1 and M2 models wesimulated three mechanical states:

● Stable equilibrium state, where the sum of all normal andtangential components of the interaction force betweenthe particle and the pseudo-atoms is zero.

● Instable equilibrium state, where again the sum of all nor-mal and tangential components of the interaction forcebetween the particle and the pseudo-atoms is zero, butany perturbation moves the molecular system towardthe stable equilibrium state.

● Nonequilibrium state, where only the sum of all normalcomponent of interaction force between the particle andthe pseudo-atoms is zero. The particle needs applica-tion of an external tangential force to be kept in firmposition onto the material surface.

All mechanical states were described by simple trigonomet-ric equations as a function of the distance between the par-ticle and the surface plane.

Figure 7. Some possible helix translations on the graphitedomain surface of BC2 interface (a) and the carbon atoms’spatial position used to describe the material surface in in-terface BC2 (b).

Figure 6. Some possible helix translations on the graphitedomain surface of BC1 interface (a) and the pseudo-atoms’spatial position used to describe the material surface in in-terface BC1 (b).

333ALBUMIN ADSORPTION AND MOLECULAR MECHANICS

In order to calculate the tangential force in the nonequi-librium state for both M1 and M2 we used, for each point topoint interaction, the analytic Equations (3) and(4), respectively. Finally, we calibrated the calculation of thetangential component with two coefficient K1 and K2:

K1 =dequilibrium

dsimulation

where dequilibrium is the distance between the particle andthe plane at equilibrium, calculated using M1 and M2models, whereas dsimulation is the distance between theparticle and the plane at equilibrium calculated bymolecular simulation; and

K2 =Fequilibrium 60%

Fsimulation 60%

where Fequilibrium60% is the force calculated for a 60%normal displacement from equilibrium state calculatedusing M1 and M2 models and Fsimulation60% is the forcecalculated for a 60% normal movement from equilibriumstate calculated by Equation (3) for BC1 biointerface andEquation (4) for BC2 biointerface.

The albumin interaction force

In order to calculate the total interaction force between theideal pyrolytic carbon surface and the whole albumin, wedetermined a weight for its helices as a function of theiramino acid polarity and identified the helices with the samepolarity as those used in simulating the BC1 and BC2 bio-interfaces. Because simulation results confirm that, as ex-pected, the albumin adsorption on the carbon planes is gov-erned by the hydrophobic interactions, we report in Table Ithe hydrophobic adhesion domains for albumin and theamino acid sequence of the hydrophobic helices that com-pose the four interacting domains for one of the two iden-tical asymmetrical submolecules (1E, 2E, 3E, and 4E).

Figure 10 shows a different perspective of the adsorptionof albumin onto ideal pyrolytic carbon, of the orientation ofthe graphitic domains and of the related geometrical modelsM1 and M2.

The normal force for each adhesion site (F>i) calculated forBC1 and BC2 was averaged to take into account the randomorientation of the ideal pyrolytic carbon domains

F⊥i =FBC1 + FBC2

2(5)

Figure 9. The geometric model and the trigonometric equations used to describe the interaction force in nonequilibriumstate for model M2.

Figure 8. The geometric model and the trigonometric equations used to describe the interaction force in nonequilibriumstate for model M1.

334 MANTERO ET AL.

Hence, the total normal component of force in the configu-ration of best adhesion of albumin on the ideal pyrolyticcarbon surface (assuming all domains to interact with thesurface) is

F⊥tot = k ? F⊥i (6)

where k is the number of adhesion sites.In order to calculate the tangential component of the in-

teraction force (F\tot) between albumin and ideal pyrolyticcarbon, we assumed that one half of the helices interacts inthe same way as in BC1 biointerface and the other half in thesame way as in BC2 biointerface.

In the first case the force is always calculated using modelM1, and in the second case the force is the weighted averageof that calculated for M1 and M2.

RESULTS AND DISCUSSION

The energy optimization performed on hydrophilicand hydrophobic helices interacting with ideal pyro-lytic carbon planes both in a parallel and in a perpen-dicular orientation shows that the energy of the hy-drophobic helix is much more affected by the interac-tion than the hydrophilic helix. For this reason wewere able to simplify the further analysis by takinginto account only one reference hydrophobic helixA5C (LEU-LYS-GLU-CYS-CYS-GLU) interacting withthe carbon planes to describe the BC1 and the BC2biointerface.

Figure 11 shows the calculated potential energy forthe BC1 and BC2 biointerfaces as a function of thedistance between the center of mass of the hydropho-bic helix and the planes.

The BC1 potential energy was fitted to Equation (1),with the values of the parameters as follows: H = 1.23× 10−16 J; a = 0.15 nm; d0 = 0.41 nm; and « = 3.12 × 10−19

J, whereas the BC2 potential energy was fitted toEquation (2), with the final parameters: d0 = 0.51 nm

Figure 10. A perspective of albumin molecule’s adhesionon a volumetric element of pyrolitic carbon (a) and the geo-metrical model related to the pyrolitic carbon domain ori-entations (b).

TABLE IHydrophobic Domains (1E, 2E, 3E, and 4E)

Domain HelixesHelixes AminoAcid Number Helixes Amino Acid Sequence

1E 1E.1 11 GLU-PRO-GLU-ARG-ASN-GLU-CYS-PHE-LEU-GLN-HIS2E 2E.1 19 LEU-LEU-GLU-CYS-ALA-ASP-ASP-ARG-ALA-ASP-LEU-ALA-LYS-TYR-ILE-CYS-GLI-ASN-

GLN2E.2 6 LEU-LYS-GLU-CYS-CYS-GLU2E.3 10 LEU-LEU-GLU-LYS-SER-HIS-CYS-ILE-ALA-GLU

3E 3E.1 15 PRO-GLN-ASN-LEU-ILE-LYS-GLN-ASN-CYS-GLU-LEU-PHE-GLU-GLN-LEU4E 4E.1 24 LYS-ARG-MET-PRO-CYS-ALA-GLU-ASP-TYR-LEU-SER-VAL-VAL-LEU-ASN-GLN-LEU-

CYS-VAL-LEU-HIS-GLU-LYS-THR

Boldface = reference helix.

Figure 11. Equilibrium potential energy as a function ofthe distance between helix center of mass and planes for BC1and BC2 biointerfaces.

335ALBUMIN ADSORPTION AND MOLECULAR MECHANICS

and « = 9.56 × 10−20 J. The normal components of theinteraction force for BC1 and BC2 biointerfaces aredescribed by Equations (3) and (4), respectively, withthese parameters’ values.

Figure 12 shows the normal component of the in-teraction forces as a function of the distance betweenthe helix center of mass and the planes for BC1 andBC2 biointerfaces. The tangential component of inter-action force was calculated considering the helix as aparticle as described in Materials and Methods.

For the tangential displacement of the helices withrespect to the carbon planes, we calculated the point topoint interaction force (between the particle and asingle pseudo-atom of the material) using Equation (3)with the parameters calculated for model M1 and us-ing Equation (4) with the parameters calculated formodel M2.

Tables II and III report the normal and tangentialcomponents of the interaction force calculated in nor-mal equilibrium state between the particle and 16 ma-terials’ pseudo-atoms for model M1 and between theparticle and 28 materials’ pseudo-atoms for model M2.The total tangential component of the interaction forcewas calculated by the vector sum among the point topoint force described in Table II and calibrated by K2coefficient (5.39) for model M1 and by the vector sumamong the point to point force described in Table IIIcalibrated by K2 coefficient (10.1) for model M2 (thefinal values are shown in the last column of Table IIand Table III).

Figures 13 and 14 show the total tangential compo-nent of the interaction force for models M1 and M2 asa function of the distance between the particle and theplane passing through the pseudo-atoms of the mate-

TABLE IINormal and Tangential Component of Point-to-Point Interaction Force Calculated by Eq. (3) for Model M1a

d [nm] i di [nm] ai F⊥i [N] F\i [N] F\ [N]

0.524 0 0.4846 0.124 −1.90 × 10−18 −2.30 × 10−19 −8.00 × 10−21

1 0.5135 0.358 −1.50 × 10−19 −5.60 × 10−20

2 0.5668 0.558 5.96 × 10−19 3.72 × 10−19

3 0.6385 0.718 5.32 × 10−19 4.64 × 10−19

4 0.7231 0.843 3.41 × 10−19 3.83 × 10−19

5 0.8166 0.941 2.05 × 10−19 2.81 × 10−19

6 0.9163 1.018 1.23 × 10−19 2.00 × 10−19

7 1.0204 1.081 7.60 × 10−20 1.42 × 10−19

8 1.1277 1.130 4.83 × 10−20 1.02 × 10−19

9 1.2373 1.172 3.15 × 10−20 7.48 × 10−20

10 1.3486 1.206 2.12 × 10−20 5.55 × 10−20

11 1.4614 1.235 1.46 × 10−20 4.19 × 10−20

12 1.575 1.260 1.03 × 10−20 3.21 × 10−20

13 1.6899 1.282 7.41 × 10−21 2.5 × 10−20

14 1.8052 1.301 5.43 × 10−21 1.96 × 10−20

15 1.9211 1.318 4.05 × 10−21 1.57 × 10−20

16 2.0375 1.332 3.06 × 10−21 1.26 × 10−20

ad, vertical equilibrium distance between the particle and the plane passing through the material’s pseudo-atoms calibratedby coefficient K1 = 0.92; di, distance between the pseudo-atom i and the particle i, ai are described in Figure 8.

Figure 12. Normal component of interaction forces for BC1 and BC2 as a function of the distance of the helix center of massfrom the reference plane.

336 MANTERO ET AL.

rial. The tangential stress on an albumin molecule in-teracting with the ideal pyrolytic carbon surface wasfinally obtained by taking the ratio between the totaltangential force and the adhesion molecular area:

t =F\

Amol

where Amol ∼ 50 nm2 is the assumed interacting mol-ecule surface roughly inferred from the overall mo-lecular size (see Fig. 2).

Figure 15 shows the tangential surface stress ofmodel M1 as a function of the particle position.

We remark that the tangential stress values in stableand instable equilibrium state is zero for all particle-

reference plane distance and the tangential stress val-ues of nonequilibrium state decrease with increasingdistance between the particle and the reference plane.

The normal component of the interaction force inbest adhesion configuration of an albumin moleculeon the ideal pyrolytic carbon surface were calculatedusing Equations (5) and (6), with the forces calculatedfor biointerfaces BC1, BC2 and models M1, M2.

Figures 16 and 17 show the normal the tangentialcomponent of the total interaction force as a functionof the distance between the center of mass of the hy-drophobic sites of albumin and the ideal pyrolytic car-bon surface.

The periodic pattern of tangential component of thetotal interaction force corresponds to the alternation ofthe potential energy surface of albumin tangentialequilibrium state and nontangential equilibrium state.

TABLE IIINormal and Tangential Component of Point-to-Point Interaction Force Calculated by Eq. (4) for Model M2a

d [nm] i j di,j [nm] ai,j F⊥i,j [N] F\i,j [N] F\ [N]

0.647 0 0 0.547 0.325 −6.67 × 10−19 −2.25 × 10−19 4.16 × 10−20

0 1 0.681 0.259 3.65 × 10−19 9.69 × 10−20

2 0 0.738 0.791 1.82 × 10−19 1.84 × 10−19

2 1 0.842 0.673 9.18 × 10−20 7.32 × 10−20

4 0 1.017 1.036 1.71 × 10−20 2.89 × 10−20

4 1 1.095 0.925 1.22 × 10−20 1.62 × 10−20

6 0 1.330 1.170 2.05 × 10−21 4.85 × 10−21

6 1 1.391 1.077 1.83 × 10−21 3.40 × 10−21

8 0 1.658 1.253 3.54 × 10−22 1.07 × 10−21

8 1 1.707 1.175 3.56 × 10−22 8.51 × 10−22

10 0 1.994 1.307 8.11 × 10−23 3.01 × 10−22

10 1 2.034 1.241 8.76 × 10−23 2.56 × 10−22

11 0 2.163 1.329 4.23 × 10−23 1.71 × 10−22

11 1 2.201 1.267 4.67 × 10−23 1.49 × 10−22

12 0 2.333 1.347 2.31 × 10−23 1.01 × 10−22

12 1 2.368 1.289 2.59 × 10−23 8.97 × 10−23

13 0 2.504 1.362 1.31 × 10−23 6.18 × 10−23

13 1 2.537 1.308 1.50 × 10−23 5.58 × 10−23

14 0 2.676 1.376 7.71 × 10−23 3.90 × 10−23

14 1 2.706 1.325 8.94 × 10−23 3.56 × 10−23

ad, vertical equilibrium distance between the particle and the plane passing through the material’s pseudo-atoms calibratedby coefficient K1 = 1.02; di,j, distance between the pseudo-atom i,j and the particle i,j, ai,j are described in Figure 9.

Figure 13. Tangential component of interaction force formodel M1 as a function of the distance of the particle fromthe reference plane.

Figure 14. Tangential component of interaction force formodel M2 as a function of the distance of the particle fromthe reference plane.

337ALBUMIN ADSORPTION AND MOLECULAR MECHANICS

CONCLUSIONS

The adsorption of albumin in water onto an idealpyrolytic carbon surface was investigated through amultiscale approach based on atomistic computersimulations and multiscale mechanics procedures. Inthis preliminary study, for practical reasons, basicallyrelated to computer and code limitations, we adopteda simplified model of the biointerface. First, in thesimulations we only considered a few subunits of al-bumin, consisting of a hydrophilic and of a hydropho-bic helix. Second, the biomaterial surface was taken asformed by a random orientation of nonoxidized andsmooth graphite crystallites, simulating in turn anidealized, defect-free pyrolytic carbon surface. The in-dividual graphite crystallites were simply describedby a small number of carbon planes (2 to 6). This num-ber was shown to be adequate to describe a graphitesurface,19 although it should be pointed out that thesurface (in-plane) polarizability of graphite is not ac-counted for in current force fields. On the other hand,the use of such a model for the biomaterial is neces-sarily a simplified picture of pyrolytic carbon, which ismade of a large distribution of graphite microdomainsin random orientation with a hierarchy of grain struc-

tures. Moreover, surface modifications due to oxida-tion may be relevant in real samples, but are not easilyincluded in atomistic simulations. Third, the physi-ological environment was considered as made up ofwater molecules only, neglecting the presence of ions.This assumption is obviously related to the difficultproblem of dealing with long-range electrostatic (Cou-lomb) interactions in computer simulations when pe-riodic boundary conditions are used. In this case,more advanced (and time-consuming) methods mustbe adopted, such as for instance the Ewald sumstechnique.17

Fourth, we do not include any electrostatic interac-tions between the helices and the carbon planes.

In this preliminary work a multiscale approach (hy-drophobic site–hydrophobic helix–albumin) was de-veloped into a hierarchical set of models to estimatethe interaction force at the interface between albuminand pyrolytic carbon.

Standard simulation tools (molecular mechanics, inparticular) provide the basis for a mechanical analysis.The force values we obtain have the correct order ofmagnitude, whereas the fitted parameters in Equa-tions (1) and (2) roughly show the expected values. Anumber of simplifying assumptions have been intro-duced to ease the calculation, many of them ask forfuture refinements. We know that unfolding appearsonce the albumin molecule approaches the interfacesurface in order to expose the hydrophobic sites to theinterface; a detailed model for unfolding will affect themedium/long range force evaluation, in particular inthe presence of ions that modify the ionic strength ofthe solution, as previously pointed out. On the otherhand, the real carbon surface offers a far less regularshape than we assumed, which clearly affects theshort-range force evaluation. Despite these shortcom-ings, the approach proposed here offers a frame ca-pable of incorporating future more realistic assump-tions. This capability depends mostly on specific ex-perimental data and on computer power large enoughto model larger protein portions.

Figure 15. Shear stress t surface for model M1 as a functionof the distance between the particle and the reference plane.

Figure 16. Normal component of total albumin interactionforce.

Figure 17. Tangential component of total albumin interac-tion force.

338 MANTERO ET AL.

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