atomistically informed mesoscale model of alpha-helical...

International Journal for Multiscale Computational Engineering, 7(3)xxx–xxx(2009)

Atomistically Informed Mesoscale Model ofAlpha-Helical Protein Domains

Jeremie Bertaud, Zhao Qin, & Markus J. Buehler∗

Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139, USA

ABSTRACT

Multiscale mechanical properties of biological protein materials have been the focal point ofextensive investigations over the past decades. In this article, we present the developmentof a mesoscale model of alpha-helical (AH) protein domains, key constituents in a varietyof biological materials, including cells, hair, hooves, and wool. Our model, derived solelyfrom results of full atomistic simulations, is suitable to describe the deformation and fracturemechanics over multiple orders of magnitude in time- and length scales. After validation of themesoscale model against atomistic simulation results, we present two case studies, in whichwe investigate, first, the effect of the length of an AH protein domain on its strength properties,and second, the effect of the length of two parallel AH protein domain arrangement on its shearstrength properties and deformation mechanisms. We find that longer AHs feature a reducedtensile strength, whereas the tensile strength is maximized for ultrashort protein structures.Moreover, we find that the shearing of two parallel AHs engenders sliding, rather than AHunfolding, and that the shear strength does not significantly depend on the length of the twoAHs.

KEYWORDS

hierarchical material, nanomechanics, materiomics, biological protein materials, fracture, de-formation, experiment, simulation, multiscale modeling

*Address all correspondence to [email protected].

1543-1649/08/$35.00 c© 2009 by Begell House, Inc. 1

2 BERTAUD, QIN AND BUEHLER

1. INTRODUCTION

Proteins constitute critical building blocks of life,forming biological protein materials (BPMs) such ashair, bone, skin, spider silk, and cells, which playan important role in providing essential mechan-ical functions to biological systems [1–5]. Flawsand failure of these materials can cause seriousdiseases and malfunctions in biological organisms,for example, due to misfolded protein structures.However, the fundamental deformation and failuremechanisms of biological protein materials remainlargely unknown, partly due to a lack of under-standing of how individual protein building blocksrespond to mechanical load and how they partici-pate in the function of the overall biological system.Significant advances in experimental, theoretical,and computational materials science have enableda deeper understanding of BPMs through the link-ing of structure-process-property (SPP). The mate-rial properties of biological materials have been thefocal point of extensive studies over past decades,leading to formation of a research field that connectsbiology and materials science, referred to as mate-riomics. Materiomics utilizes mechanistic insight,based on SPP relations in their biological context, toprovide a basis for understanding disease processes;to develop new approaches to treating genetic andinfectious diseases, injury, and trauma; and to en-hance engineered materials via translating materialconcepts from biology. Computational bottom-upmultiscale approaches provide methods for rapidscreening of SPP relations.

The behavior of materials, in particular, theirmechanical properties, are intimately linked to theatomic microstructure of the material. Whereascrystalline materials show mechanisms such as dis-location spreading or crack extension [6–8], biologi-cal materials feature molecular unfolding or sliding,with a particular significance of rupture of chemi-cal bonds such as hydrogen bonds (H-bonds), co-valent cross-links, or intermolecular entanglement.Additional mechanisms operate at larger lengthscales, where the interaction of extracellular mate-rials with cells and of cells with one another, dif-ferent tissue types, and the influence of tissue re-modeling (at longer timescales) become more evi-dent. The dominance of specific mechanisms is con-trolled by geometrical parameters, the chemical na-ture of the molecular interactions, and the struc-

tural arrangement of the protein elementary build-ing blocks across many hierarchical scales, fromnano to macro. The multiscale understanding ofhow molecular structures participate in macroscaledeformation of biological tissues remains an out-standing challenge, and multiscale computationalapproaches are believed to play a crucial role in ad-vancing this field.

What is the significance of mechanics of biolog-ical protein materials? The mechanical propertiesof biological materials have wide-ranging implica-tions for biology. In cells, for instance, mechanicalsensing is used to transmit signals from the envi-ronment to the cell nucleus or to control tissue for-mation and regeneration [1–5, 9]. The structural in-tegrity and shape of cells is controlled by the cell’scytoskeleton, which resembles an interplay of com-plex protein structures and signaling cascades ar-ranged in a hierarchical fashion. Bone and collagen,providing structure to our bodies, or spider silk,used for prey procurement, are examples of mate-rials that have incredible elasticity, strength, and ro-bustness unmatched by many synthetic materials,which can mainly be attributed to its structural for-mation with molecular precision. This transfer ofconcepts observed in biology into technological ap-plications and new materials design remains a bigchallenge with a potentially big payoff. In particu-lar, the combination of nanostructural and hierarchi-cal features into materials developments could leadto significant breakthroughs in the development ofnew materials that mimic or exceed the propertiesfound in biological analogs. The characterizationof material properties for biological protein materi-als may also play a crucial role in developing a bet-ter understanding of diseases. Injuries and geneticdiseases are often caused by structural changes inprotein materials (e.g., defects, flaws, changes to themolecular structure), resulting in failure of the mate-rial’s intended function. This approach enables oneto probe how mutations in structure alter the prop-erties of protein materials. In the case of osteoge-nesis imperfecta (brittle bone disease), for instance,molecular-scale models predict a softening of bone’sbasic collagen constituent [10]. These observationsmay eventually provide explanations of the molec-ular origin of certain diseases. Additionally, thesefindings provide evidence that material propertiesplay an essential role in biological systems, and thatthe current paradigm of focusing on biochemistry

International Journal for Multiscale Computational Engineering

ATOMISTICALLY INFORMED MESOSCALE MODEL 3

alone as the cause of diseases is insufficient. It isenvisioned that the long-term potential impact ofthis work can be used to predict diseases in the con-text of diagnostic tools by measuring material prop-erties, rather than focusing on symptomatic chemi-cal readings alone. Such approaches have been ex-plored for cancer and malaria, for instance [11, 12].

This article is focused on the analysis of AH pro-tein domains, as they appear, for instance, in inter-mediate filaments in the cell’s cytoskeleton. Figure 1displays the multiscale hierarchical structure of thisprotein network. The work reported here is focused

on the smallest of all elements in this protein mate-rial: the mechanics of a single AH (see Fig. 1, bot-tom).

1.1 Review: Multiscale Methods for ProteinMaterials

Multiscale simulation models for protein materialshave become increasingly popular in recent yearsand have enabled the direct link between experi-ment and theoretical bottom-up descriptions of ma-terials (Fig. 2). Here we provide a review of popular

FIGURE 1. Overview of different material scales, from nano to macro, here exemplified for the cellular proteinnetwork vimentin intermediate filaments. To understand their deformation and fracture mechanisms, it is crucial toelucidate atomistic and molecular mechanisms at each scale. Computational multiscale approaches thereby play acrucial role in transcending through multiple scales in length and time

Volume 7, Number 3, 2009


FIGURE 2. Overview of computational and experimental methods. Hierarchical coupling of different computationaltools can be used to traverse throughout a wide range of length- and timescales. Such methods enable one to providefundamental insight into deformation and fracture phenomena, across various time- and length scales. Handshakingbetween different methods enables one to transport information from one scale to another. Eventually, results of atom-istic, molecular, or mesoscale simulation may feed into constitutive equations or continuum models. While continuummechanical theories have been very successful for crystalline materials, biological materials require statistical theories(e.g., the Bell model discussed in Section 2.3 and related formulations) to describe elasticity and strength. Experimentaltechniques, such as the atomic force microscope, molecular force spectroscopy, nanoindentation, or magnetic/opticaltweezers, now overlap into atomistic and molecular approaches, enabling direct comparison of experiment and sim-ulation. Techniques such as X-ray diffraction, infrared spectroscopy, or nuclear magnetic resonance (NMR) provideatomic-scale resolution information about the 3-D structure of protein molecules and protein assemblies

multiscale approaches to describe the mechanics ofproteins in a multiscale setting.

Single-bead models are the most direct approachtaken for studying macromolecules. The term sin-gle bead derives from the idea of using single beads,that is, point masses, for describing each amino acidin a protein structure. The Elastic Network Model(ENM) [13], Gaussian Network Model [14], and Go-model [15] are well-known examples that are basedon such bead model approximations. These mod-els treat each amino acid as a single bead located atthe Cα position, with mass equal to the mass of theamino acid. The beads are connected via harmonicbonding potentials, which represent the covalentlybonded protein backbone. In Go-like models, anadditional Lennard-Jones–based term is includedin the potential to describe short-range nonbondedinteractions between atoms within a finite cutoffseparation. Despite their simplicity, these modelshave been extremely successful in explaining ther-mal fluctuations of proteins [16] and have also been

implemented to model the unfolding problem toelucidate atomic-level details of deformation andrupture that complement experimental results [17–19]. A more recent direction is coupling of ENMmodels with a finite element–type framework formechanistic studies of protein structures and assem-blies [20].

Using more than one bead per amino acid pro-vides a more sophisticated description of proteinmolecules. In the simplest case, the addition of an-other bead can be used to describe specific side-chain interactions in proteins [21]. Higher-levelmodels, for instance, four- to six-bead descriptions,capture more details by explicit or united atom de-scription for backbone carbon atoms, side chains,and carboxyl and amino groups of amino acids[22, 23]. Even coarser-level multiscale modelingmethods have been reported more recently, appliedto model biomolecular systems at larger time- andlength scales. These models typically employ su-peratom descriptions that treat clusters of amino



acids as beads. In such models, the elasticity of thepolypeptide chain is captured by simple harmonicor anharmonic (nonlinear) bond and angle terms.These methods are computationally quite efficientand capture shape-dependent mechanical phenom-ena in large biomolecular structures [24], and canalso be applied to collagen fibrils in connective tis-sue [25] as well as mineralized composites such asnascent bone [26]. In this article, we develop such acoarse-level description of AH protein domains.

1.2 Outline of This Article

The plan of this article is as follows. In Sec-tion 2, we begin with a presentation of the com-putational setup. This includes a detailed descrip-tion of the mesoscopic model formulation, the fit-ting procedure, and validation. We present a reviewof strength models for protein domains, with a focuson descriptions derived from the Bell model. Sec-tion 3 is dedicated to a discussion of computationalresults obtained using the mesoscale model, hereapplied to study the length dependence of strengthproperties of AH protein domains in single and par-allel arrangements. We conclude in Section 4 with adiscussion and an outlook for future research.

2. COMPUTATIONAL METHODS

In the following sections, we describe our atomistic-based multiscale simulation approach used to de-velop a mesoscale description of AH protein do-mains. The setup of the coarse-grained model forAH protein domains is based on the geometry of anAH, which features a linear array of turns or con-volutions; during rupture, any one of these convo-lutions ruptures, as reported in earlier simulationstudies [27]. In our model, every convolution is rep-resented by one bead so that an entire AH is repre-sented by a linear combination of multiple beads.

2.1 Coarse-Graining Approach

To achieve the coarse-grained description, the en-tire sequence of amino acids that makes up the AHstructure is replaced by a collection of beads thatare linked to their neighbors by a single-type bond,as shown in Fig. 3. An AH consists of a series ofconvolutions, where each features 3.6 residues. Thisprotein secondary structure is stabilized through the

FIGURE 3. Schematic of the coarse-graining procedure,in which we replace the full atomistic representation bya mesoscopic bead model. A pair of beads represents oneturn in the alpha-helix (AH; also called a convolution) andthus 3.6 residues (each bead also has the correspondingmass). In the atomistic representation, the folded statesof the turns are stabilized by the presence of 3.6 H-bondsbetween turns. In the mesoscopic bead model, this is rep-resented by using a double-well potential to describe theenergy landscape under bond stretching. Different AHchains interact with a Lennard-Jones potential to describeintermolecular adhesion

presence of H-bonds between the O atom of residuen and the N atom of residue n + 4 (there are 3.6 H-bonds between turns, on average).

In our coarse-grained model, we aim at capturingthe main structural and energetic features of an AHprotein domain. Full atomistic molecular dynam-ics (MD) simulations have confirmed that H-bondsbreak in clusters of three to four (corresponding toone convolution) for pulling velocities lower than0.3 m/s [27]. Thus, to capture the role of the con-voluted structure on AH protein strength, the sizeof one bead (the smallest unit of our system) rep-resents one convolution of the AH, with the corre-sponding mass that reflects all atoms that are repre-sented by that bead. Our model is set up to studyAH protein domain strength for pulling velocitieslower than 0.3 m/s.



The beads interact according to a bond poten-tial and an angle potential. We choose a double-well bond potential to capture the existence of twoequilibrium states for a convolution, correspond-ing to the folded and unfolded configuration (seeFig. 4 for the energy landscape and snapshots ofatomistic geometries of the folded and unfoldedstates). The model does not involve explicit solvent;rather, the effect of solvent on the breaking dynam-ics of AH convolutions is captured in the effectivedouble-well potential. Through this formulation,the bond potential can describe the microscopic de-tails of the rupture mechanism of the 3.6 H-bondsbetween each convolution under force, and the tran-sition from the folded states to the unfolded statesof convolutions through an energy barrier that sep-arates the two states. Yet the description is suffi-ciently coarse so that it enables a significant com-putational speedup and efficiency compared with

FIGURE 4. Double-well profile of the bond stretchingpotential in the bead model, representing the energy land-scape associated with unfolding of one convolution. Thenumerical values of the equilibrium states (x0 and x1), en-ergy barriers (Eb and Er), and the transition state xtr areobtained from geometric analysis of the AH geometry aswell as full atomistic simulations. The transition state (lo-cal energy peak of the potential) corresponds to the break-ing of the 3.6 H-bonds between two convolutions of theAH. After failure of these weak bonds, the convolutionunfolds to a second equilibrium state with a larger inter-bead distance. Under further loading, its covalent bondsbegin to be stretched, which leads to a second increase ofthe potential at large deformation

the full atomistic description. To characterize inter-actions between AH protein domains in a parallelarrangement, we use a Lennard-Jones (LJ) potentialbetween pairs of beads of different strands. This de-scribes the adhesion energy between AHs.

The mathematical expression for the total energyof the system is given by

E = ET + EB + ES (1)

where ET is the total tensile energy, EB is the to-tal bending energy, and ES is the total intermolec-ular interaction energy. The total bending energy isgiven by the sum over all triples of beads,

EB =∑

triplets

φB(x) (2)

The angle potential is given by

φB(θ) =12KB(θ− θ0)2 (3)

where KB relates to the bending stiffness of themolecule EI , with θ as the interbead angle (intriplets of atoms) and θ0 as the equilibrium angle.The bending stiffness parameter KB is given by

KB =3EI

x0(4)

with x0 denoting the equilibrium bead distance,which corresponds to the equilibrium distance ofone folded convolution, and EI as the bending stiff-ness of the alpha helix. The total tensile energy isgiven by the sum over all pair-wise interactions,

ET =∑

pairs

φT (x) (5)

The double-well bond potential φT (x) is given by(see also Fig. 4 for a schematic)

φT (x)=

Eb

x4b

(x−xtr)2(x−xtr−

√2 · xb

)

× (x−xtr+

√2 · xb

)+Eb, x<xtr

Er

x4r

(x− xtr)2(x−xtr−

√2 · xr

)

× (x−xtr+

√2 · xr

)+Eb, xtr≤x

(6)

The first equilibrium with reaction coordinate x0

(first potential minimum) corresponds to the folded



state of one turn of an AH under no force. Thetransition state (energy barrier Eb), with positionxtr (peak of the potential between two wells), cor-responds to the breaking of the 3.6 H-bonds be-tween two turns of the AH. After failure of theseweak bonds, the turn unfolds to a second equilib-rium state. This corresponds to the second potentialminimum with a larger interbead distance, x1. Un-der further loading, its backbone bonds begin to bestretched, which leads to a second increase of thepotential. The parameters xb and Eb represent thedistance and energy barrier required to unfold oneconvolution. Similarly, xr and Er correspond to therefolding process.

The total intermolecular interaction energy ES isgiven by the sum over all pair-wise interactions be-tween beads of different AH protein domains:

ES =∑

pairs

φS(x) (7)

The adhesion potential φS is described by a LJ po-tential:

φS(x) = 4 · ε ·[(σ

x

)12

−(σ

x

)6]

(8)

The energy minimum ε of the LJ potential corre-sponds to the adhesion energy per convolution be-tween two AHs in equilibrium divided by 3.51 (fac-tor that takes into account the fact that we do not usea cutoff distance for the LJ potential, and thereforethe adhesion energy for each bead is the same as theadhesion energy estimated from full atomistic sim-ulation); and x is the distance between mesoscaleparticles. The zero-crossing distance σ is linked tothe equilibrium bead distance d0 (equilibrium spac-ing between two AHs) by

d0 = 21/6 · σ (9)

2.2 Parameter Fitting: Linking Atomistic andMesoscale

The parameters are determined through a fittingprocedure against geometric properties of AHs aswell as full atomistic MD simulation results in ex-plicit solvent. We fit the energy barrier measuredfrom MD simulation to the energy barrier in themesoscale model formulation. The mass of eachbead corresponds to the approximate average massof one turn or 3.6 residues, leading to 400 amu.

The two parameters of the angle potential areintroduced in Eq. (3). The value of the equilib-rium angle θ0 is 180◦, based on the geometry of theAH structure. The bending stiffness parameter KB

is linked to molecular parameters as described byEq. (4) and therefore can be determined from fullatomistic simulations of bending studies of AH pro-tein domains (we use the results reported in [28]).We find that KB = 21.589 kcal/mol/rad2. Theparameters of the tensile double potential are in-troduced in Eq. (6) (see also Fig. 4). We find thatx0 = 5.4 A for the equilibrium bead distance ofthe folded state, which corresponds to the length ofone folded convolution. The distance xb betweenthe folded state equilibrium and the transition statecorresponds to the distance to break 3.6 H-bonds,which leads to the unfolding of the convolution, andthe parameter Eb is the corresponding energy bar-rier (full atomistic MD simulations [27] have con-firmed that H-bonds in convolutions indeed breakin clusters). These two parameters are determinedfrom fitting against full atomistic simulations of ten-sile loading experiments of AH domains, for a rangeof pulling rates below 0.3 m/s [27]. We find thatxb = 1.2 A and Eb = 11.1 kcal/mol by fitting thetheoretical strength model described in the next sec-tion to these full atomistic studies. The equilibriumbead distance of the unfolded state, x1, is deter-mined by fitting the mesoscopic force-strain curveagainst the atomistic simulation results at large de-formation. The parameter describes at what strainlevels an AH convolution is completely unfoldedand when further strain leads to significant stiff-ening due to stretching of the protein backbone.Therefore we adjust x1 so that the angular point be-tween the plateau regime and the backbone stretch-ing regime occurs at the same strain as in the atom-istic simulation curve. We find that x1 = 10.8 A,which corresponds to twice the length of the foldedstate. Last, the energy barrier E1 to refold a so-calledbroken convolution must be smaller than Eb sincethe folded state is the most favorable state for a con-volution in equilibrium. On the basis of a sugges-tion put forth in [29], we determine that E1 = 0.6Eb

(it is noted, however, that the resulting mechanicalproperties of the AH are insensitive to variations ofchoices of E1, as long as E1 < Eb).

The two parameters of the intermolecular poten-tial are introduced in Eqs. (8) and (9). The parame-ter ε is obtained from the measurement of the min-



imum of the adhesion energy per convolution be-tween two AHs in equilibrium, and the parame-ter d0 is obtained from the equilibrium spacing be-tween two AHs. They are determined from fullatomistic simulations, as shown in Fig. 5. We con-sider two identical AHs in parallel. Each AH con-tains 100 glutamine amino acids, with an approxi-mate length of 150 A. The two AHs are kept paral-lel to each other at a distance of d by fixing all theCα atoms in each chain, while all other atoms (in-cluding side chains) are free to move. We use theCHARMM-19 force field with excluded volume im-plicit solvation model (Effective Energy Function 1,EEF1) [30, 31]. Full atomistic simulations are carriedout with the Chemistry at HARvard Macromolecu-lar Mechanics (CHARMM) package. We considera series of molecular arrangements with systemati-cally increasing intermolecular distance d; for eachcase, we record the potential energy after energyminimization. Figure 5 shows the results of the ad-hesive properties between two AHs, where we sys-tematically change the intermolecular spacing be-tween two parallel AHs. The MD studies reveal that

FIGURE 5. Full atomistic model with implicit sol-vent calculation with CHARMM, used here to identifythe intermolecular adhesion between two parallel AHs.This simulation reveals that the adhesion energy is γ =1.262 kcal/mol/A, that is, 6.815 kcal/mol per convolu-tion, with an equilibrium spacing of 12.1 A. The full atom-istic simulation is carried out for a AH sequence made ofglutamine amino acids

the adhesion energy is γ = 1.262 kcal/mol/A (cor-responding to the minimum of the adhesive energyfunction), reached at d0 = 12.1 A equilibrium spac-ing, leading to a LJ distance parameter σ = 10.8 A.The adhesion energy per convolution is linked tothe LJ parameter ε, which is determined to be ε =6.815 kcal/mol.

The complete set of parameters of the mesoscopicmodel and their physical meanings are summarizedin Table 1.

2.3 Theoretical Modeling of Alpha-HelixRupture: Chemomechanical Strength Model

Here we briefly review a theoretical strength modelthat can be applied to describe the mechanical be-havior of AH protein domains: the Bell model. TheBell model is a simple phenomenological modelthat describes the frequency of failure of reversiblebonds [32]. The concept of reversibility means thatan individual bond can break under no force if onewaits a sufficiently long time, and that it can reformspontaneously. Such bonds may be associated withelectrostatic, van der Waals, or H-bond interactions(as they apply to AHs). The frequency of failure,also called the dissociation rate or off rate, is definedas the inverse of the bond lifetime and is used as aconcept to describe the dynamical behavior of suchbonds.

Bell’s theory explains the force dependence of theoff rate and thus shows the significant role of me-chanical force in biological chemistry. For instance,this theory can be applied to describe the forced un-binding of biological adhesive contacts such as ad-hesion of cells to cells. Bell’s theory is an exten-sion of the transition state theory for reactions ingases developed by Eyring and others [33]. Inspiredalso by Zhurkov’s work on the kinetic theory of thestrength of solids [34], Bell predicted for the firsttime that the off rate of a reversible bond, whichis the inverse of the bond lifetime, increases whensubjected to an external force f . Indeed, the rup-ture of bonds occurs via thermally assisted cross-ing of an activation barrier Eb, which is reduced byf · xb as the applied force f increases, xb being thedistance between the bound state and the transitionstate. Thus the Bell off-rate expression is

k = ω0 exp(−Eb − f · xb

kBT

)(10)



TABLE 1. Summary of parameters of the mesoscale model, derived from geometrical analyses and atomisticsimulationsa

Parameters Numerical valuesEquilibrium bead distance of the folded state x0 [A] 5.4Equilibrium bead distance of the unfolded state x1 [A] 10.8Distance between folded state and transition state xb [A] 1.2LJ distance parameter between beads of two AHs, σ [A] 10.8Energy barrier between folded state and transition state Eb [kcal/mol] 11.1Energy barrier between unfolded state and transition state Er [kcal/mol] 6.7Energy minimum between beads of two AHs, ε [kcal/mol] 6.815Equilibrium angle θ0 [deg] 180Bending stiffness parameter KB [kcal/mol/rad2] 21.589Mass of each mesoscale bead [amu] 400a Table corresponds to Eqs. (1)–(9) as well as to the discussion presented throughout Section 2.2.Abbreviations are as follows: AH, alpha-helix; LJ, Lennard-Jones.

where ω0 is the natural vibration frequency andkBT is the thermal energy. The force f0 = Eb/xb

represents the force needed for the energy barrierto vanish completely. This expression can be rear-ranged to yield an expression of strength as a func-tion of pulling velocity v:

f(v, Eb, xb) =kBT

xbln(v)−kBT

xbln (xbω0)+

Eb

xb(11)

with v = ∆x/∆t as the pulling speed at which a pro-tein structure is deformed. We will use Eq. (11) inthe following sections in a comparison between thetheoretical model and simulation results (for furtherdetails regarding this model and its derivation, werefer the reader to [27]).

3. COMPUTATIONAL RESULTS

The first step of the application studies is the val-idation of the mesoscale results against full atom-istic simulation results. Figure 6 presents the vali-dation of our mesoscale model by direct compari-son of the strength of AH protein domains with fullatomistic results of the rupture mechanics of an AHprotein domain. The straight line in this plot cor-responds to the predictions by the theoretical Bellmodel (Eq. (11)), discussed in Section 2.3.

Figure 7 depicts the entire force-strain curve fora stretching experiment on the 14-bead mesoscopicmodel of an AH with a length of 70.2 A, at a tem-perature of 300 K and a pulling rate of 0.1 m/s. Thecurve shows the three typical regimes observed in

FIGURE 6. Validation of the mesoscopic model bycomparison with full atomistic results (MD results takenfrom [27]). The plot shows the rate dependence of the un-folding force for both the mesoscopic and atomistic mod-els. The mesoscale model is in very good agreement withthe full atomistic simulations, validating the fitting of themesoscopic bond potential. The figure further illustratesthat the mesoscale model is capable of reaching muchslower pulling rates than those accessible to full atomisticsimulation studies, here shown for the slowest pullingspeed of 0.0001 m/s

full atomistic simulations: an elasticity regime atlow strain, an energy dissipation regime that corre-sponds to the unfolding of the 13 bonds (13 peaks onthe curve), and the subsequent regime of stretchingof the backbone bonds. The peaks of force, whichcorrespond to the AH rupture force, fit atomistic re-



FIGURE 7. Entire force-strain curve for a stretchingexperiment on the 14-bead mesoscopic model of an AH(with a length of 70.2 A at a temperature of 300 K anda rate of 0.1 m/s). The mesoscale curve shows the threetypical regimes observed in full atomistic simulations: anelasticity regime (before the first peak occurs), an energydissipation regime, which corresponds to the unfolding ofthe 13 bonds (corresponding to the 13 peaks on the curve),and the regime of stretching of the backbone bonds. Thepeaks in the mesoscale model, corresponding to the rup-ture forces, agree very well with the rupture force mea-sured from full atomistic simulation. We note that thefull atomistic simulation curve presents only small fluc-tuations since the atomistic system has about 3,000 timesmore particles (atoms) than the mesoscale model, andthus a larger number of interactions contribute to thedamping of force relaxation during unfolding of convo-lutions. The dotted line approximates the increase of rup-ture force with increasing strain. In agreement with atom-istic simulations reported earlier [28], the third regime setsin at approximately 135% strain

sults very closely. Both full atomistic and mesoscalemodels predict an initial rupture force, referred toas the angular point, of approximately 350 pN, witha slight increase as the strain is increased. The re-sults shown in Fig. 6 illustrate a key advantage ofthe coarse-grained model in reaching much longertimescales than what could be achieved in full atom-istic simulations (the current limit in MD simula-tions is 0.01 m/s, whereas we have easily reached a100-fold increase in accessible timescales using ourmesoscale model). Experimental results of stretch-ing and breaking single AH domains [35, 36] (with alength of less than 100 A) report forces between 140and 240 pN during unfolding, corresponding to theforce level predictions at ultraslow pulling speeds.

Next, we illustrate the application of this modelto AH protein domains with three distinct lengths,L: 5, 14, and 81 beads (thus corresponding to L =21.6 A, 70.2 A, and 432 A lengths, respectively, orequivalently, 4, 13, and 80 convolutions). Figure 8shows the strength obtained from our model forthese three different lengths. We observe that therupture strength decreases as the length increases.The first force peak of the five-convolution systemis approximately 500 pN and is thus almost twiceas high as the 81-convolution structure, where thestrength approaches 250 pN. We note that the sim-ulation of the system with 432 A length would nothave been possible with a full atomistic simulation(the atomistic simulation for a 70.2 A protein struc-ture took several weeks of computational time). Wehave used the simulation results to fit an empiricalequation of the form

f(L) = a ln(L/L0) + b (12)

where we find a = −74.463 pN, b = 689.08 pN,and L0 = 1 A. Extrapolating the strength valuespredicted from Eq. (12) to ultralong protein lengthsyields a strength of 125 pN at a length of 2, 000 A

FIGURE 8. Strength properties of AHs with differ-ent lengths, ranging from L = 21.6 A to L = 432 A,at a 0.1 m/s pulling rate. The results illustrate that thestrength decreases as the length of the AH increases. Thisbehavior also explains the continuous strengthening ef-fect observed in Fig. 7. As more turns are broken at in-creasing strain, the force required to break the increas-ingly short AH segments increases. The continuous lineshows a fit to the data obtained from mesoscale simula-tions (see Eq. (12) for the equation and numerical fittingparameters)



(= 200 nm), thus only one-fourth of the strengthat 21.6 A. In summary, our model predicts thatthe strength of AHs decreases as the length in-creases. Notably, this weakening behavior with in-creasing molecular lengths also explains the contin-uous strengthening effect observed in Fig. 7 (indi-cated using the dotted line). This is because as moreturns are broken at increasing strains, the force re-quired to break the increasingly short AH segmentsincreases. We leave further investigation of thislength-scale effect to future studies.

We present a second application of our meso-scopic model to study the shear strength of twoparallel AH proteins with the same three lengths:21.6 A, 70.2 A, and 432 A. Figure 9 shows a

FIGURE 9. Shear strength of an assembly of two AHs.(a) Schematic of the coarse-grained model set up for theshearing of two aligned AH protein domains. One strandhas its end fixed and the second strand is pulled in the op-posite direction at a constant velocity v = 0.1 m/s. (b) En-tire force strain curve for a shearing experiment on themesoscopic model of two parallel AH protein domains(here we use 14 beads per AH strand, a pulling veloc-ity of 0.1 m/s, and a temperature of 300 K). The shearstrain is defined as the ratio between the extension and theequilibrium distance between two AH strands. This plotshows three different regimes: first, an elasticity regime,as we start shearing; then a plateau regime, which corre-sponds to the sliding of the pulled strand along the fixedstrand; and last, a zero-force regime, once the two strandsseparate. The average of the force value over the plateauregime gives the mean shear strength (indicated with thehorizontal red dashed line)

schematic of the coarse-grained model for theshearing of two AH protein domains in parallel(Fig. 9(a)). The extremity of one strand is fixed,and the second strand is pulled in the opposite di-rection at a constant velocity v = 0.1 m/s. Fig-ure 9(b) depicts the entire force-strain curve for ashearing experiment on the mesoscopic model oftwo parallel AH protein domains (here we use alength of 14 beads for each AH strand, a pulling ve-locity of 0.1 m/s, and a temperature of 300 K). Theshear strain is defined as the ratio between the ex-tension and the equilibrium distance between twoAH strands. This plot shows three distinct regimes:first, an elasticity regime, as the shearing begins;then a plateau regime, which corresponds to thesliding of the pulled strand along the fixed strand;and last, a zero-force regime, once the two strandsseparate. The average of the force value over theplateau regime provides an estimate for the meanshear strength. Figure 10 shows snapshots duringshearing of two AHs: a short and a long AH struc-ture. The predominant deformation mechanism isshearing, with only a few unfolded convolutions ap-pearing in the long case (red segments). This behav-ior is expected since the shear resistance of 83 pN islower than the rupture strength of AHs, which ap-proaches values of more than 200 pN (see the resultsin Fig. 8). Figure 11 shows the shear strength prop-

FIGURE 10. Dynamical mechanisms of shearing twoAHs for (a) a short length of 70.2 A (14 beads) and for(b) a long length of 432 A (81 beads). The predominantdeformation mechanism is sliding, with some unfoldedconvolutions appearing in the long case (visualized by redsegments)



FIGURE 11. Shear strength properties of two paral-lel AHs with different lengths ranging from L = 21.6 Ato L = 432 A, at a 0.1 m/s pulling rate. For differentlengths, the plot shows the mean shear strength obtainedfrom the force strain curve of the shearing experiment.The results illustrate that the shear strength does not sig-nificantly depend on the length, as opposed to the tensilestrength, where the mean shear strength value is approx-imately 83 pN. The constant shear strength behavior maybe explained by a simple theoretical model based on vari-ation of surface energy (model described in Eq. (13)). Wenote that for this range of lengths, the shear strength isbelow the tensile strength (which is > 200 pN; see Fig. 8),and thus we do not observe unfolding of the whole AHstrand as shear deformation occurs

erties of two parallel AHs obtained with three differ-ent lengths ranging from L = 21.6 A to L = 432 A,at a 0.1 m/s pulling rate. The results illustrate thatthe shear strength does not significantly depend onthe length as opposed to the tensile strength. On av-erage, the mean shear strength value is about 83 pN.

The shear strength may be estimated by a simpletheoretical model based on the variation of the to-tal surface energy of the system as the two AHs aresheared. During the shearing experiment, once fail-ure due to intermolecular shear initiates, the pulledAH slides relative to the fixed AH, and thus the sur-face of contact between the two strands decreases.Therefore the total surface energy of the system in-creases as the strands are being separated. Thus, fora shear extension ∆L, the work done by the pullingforce f should equal the increase of surface energyof the system:

f ·∆L = γ ·∆L (13)

where γ is the adhesion energy per length asmeasured from full atomistic simulations (γ =1.262 kcal/mol/A, dimension of force). Equa-tion (13) leads to

f = γ (14)

According to this model, the theoretical shearstrength equals 87.6 pN. The theoretical value of theshear strength is constant and thus agrees with theexistence of a shear strength plateau regime, as ob-served in the mesoscopic simulations. Moreover,the theoretical value of the shear strength approx-imately equals the shear strength value measuredfrom simulations (83 pN). Therefore this simple the-oretical model seems to be relevant to describe theshear strength between parallel AHs. The small dif-ference between the numerical values of theory andsimulation may be explained by the fact that oursimple theoretical model does not include the effectof the pulling rate, which might shift the measuredshear strength to larger values.

4. DISCUSSION AND CONCLUSION

The coarse-grained model enables us to simulatethe dynamics of large systems over a large rangeof length and timescales. The model is capableof reaching timescales of several microseconds andlonger, with a quantitative accuracy comparablewith full atomistic MD simulations. Such relativelylong simulations can be carried out within severaldays of computational time (on a single Intel XeonCPU). In comparison, MD simulations of the dy-namical behavior at fractions of microseconds cantake weeks and months of computational time (evenon a large parallelized simulation setup). This re-flects a considerable speedup due to the coarse-graining approach, while the model is still capa-ble of describing the small- and large-deformationforce-strain response characteristics (e.g., softeningat ≈ 10% strain and stiffening at ≈ 135% strain) aswell as strength values quite accurately. The appli-cation to structures of different lengths and arrange-ments (see the results shown in Figs. 8–11) illus-trates the unique ability of our mesoscale model todescribe how structure (that is, size, length, geome-try) influences strength properties and mechanismsof deformation. Earlier attempts at describing therupture mechanics of protein domains with coarsemesoscale models have not included such a level of



detail and have thus not been capable of describinghow structure and strength properties are linked.Our results show that intermolecular sliding is thedominating mechanism of deformation, suggestingthat AHs remain intact during deformation of AHassemblies. Further investigations to elucidate theeffects of varying amino acid sequences on this be-havior will be necessary to understand better howthe chemical structure and overall mechanical de-formation mechanisms and strength properties arerelated.

Potential applications of the coarse-grained AHprotein domain model presented here could be fur-ther studies of length-scale effects on AH strengthand elasticity, and the effects of hierarchical ar-rangements of AH-based protein domains. Fur-ther studies could focus on larger variations oftimescales (e.g., to extend to even slower pullingspeeds) and the development of similar formula-tions for coiled-coil proteins or larger-level hierar-chical protein structures.

ACKNOWLEDGMENTS

This research was supported by the Air Force Of-fice of Scientific Research (AFOSR). Additional sup-port from NSF, the Office of Naval Research (ONR),and DARPA is greatly acknowledged. J.B. acknowl-edges support from the Schoettler Fellowship Pro-gram at the Massachusetts Institute of Technology.

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