Molecular Nanosprings for Protein-Based Nanorobotics

Mustapha Hamdi 1, Antoine Ferreira 1 and Constantinos Mavroidis 21Laboratoire Vision et Robotique, ENSI Bourges-Université d’Orléans, 18000, Bourges, France.

2Computational BioNano Robotics Laboratory, Northeastern University, Boston, MA-02120, USA antoine.ferreira@ensi-bourges.fr , mavro@coe.neu.edu

This paper presents a molecular mechanics study using amolecular dynamics software (NAMD2) for characterization ofmolecular elastic joints for bio nanorobotic prototyping. Simpleprotein-like elastic joints elements for structural links have becarefully studied and simulated to understand the functionallimits of each one of them: fibronectin, titin, deca-alanine,fibrillin, topocollagen, tropomyosin, Rop and double-helicalDNA molecules. We simulated the restoring forces involvedunder various external mechanical stress (stretching,contraction, shearing, bending) to predict the type of forcespectra, reversibility, degrees of freedom and irreversible workthat may be expected from single-molecule proteinmanipulation experiments. Standard models for the elasticity ofunfolded proteins are based on entropic arguments incombination with enthalpic contributions.

I. INTRODUCTIONBio-nanorobotics is an emerging area of scientific and

technological opportunity bridging the fields of biology andnanotechnology. Developing nanomachines out of proteinselements requires the merging of two fields of researchapproaches: the inspiration by nature and biology(“biomimetics”) and the inspiration by large scale machinesand the traditional mechanisms and machine theory(“machine nanomimetics”). Both DNA and proteinmolecules possess a number of intrinsic characteristics thatmake them excellent candidates for the assembly of dynamicnanostructures and nanodevices. In this way proteins andDNA could act as motors, mechanical joints, transmissionelements, or sensors. Most research efforts are concentratedon enzymes that function as nanoscale biological motorssuch as kinesin [1], RNA polymerase [2], myosin [3], dynein[4], adenosine triphosphate (ATP) synthase [5], viral proteinlinear (VPL) motor [6] and DNA. These motors, which arecalled biomolecular motors, have attracted a great deal ofattention recently because they have high efficiency, theycould be self-replicating, and hence cheaper in mass usage,and they are readily available in nature. Biologicalmechanisms and joints with multi-degree of freedom are alsocurrently investigated in the scientific community. Aspotential transmission components in bionanorobotics, wecan note 2-dof molecular hinges based calix[4]arenes [7], -based DNA [8], linear ratchets mechanisms [9], molecularnanosprings using fibronectin [10] and titin [11] which canact as compliant joints in biomolecular robotic systems.Spring-like proteins are particularly interesting from theirelastic, resilient and stability characteristics: stretch and relaxwithout any net energy dissipation, can store and release

energy and rectify motion in physical nanomachines in a fullreversible way [12]. Protein flexibility refers to the capacityof the protein to experience dynamic changes inconformation under biological conditions.

In this project we are studying the development of proteinbased joint mechanisms. In order to draw analogies betweenmechanical protein methodologies and roboticsmethodologies, several protein-like elastic joints forstructural links have be carefully studied and simulated tounderstand their functional and elastic limits. Contributionfrom molecular dynamic (MD) simulations is important inorder to be able to understand the bio-nanomechanics ofproteins and develop dynamic and kinematic models to studytheir performances. Computational studies of differentclasses of proteins: α−helix bundle proteins class (rop,DNA), β-sheet class (fibronectin, titin,) and compound class(collagen, silk, actin) have been carried out. It allows topredict the type of force spectra, reversibility, degrees offreedom and irreversible work that may be expected frommolecular springs. Finally, quantitative models of biologicalspring mechanisms are developed and classified followingtheir nanomechanical performances.

II. DESIGN OF BIONANOMECHANISMS MIMICKINGBIOLOGICAL STRUCTURAL PROTEINS

New nanostructures (with dimensions of nanometers tomicrometers) using synthetic strategies have recentlyemerged. Inspired by biological systems, concepts of self-organization and self-assembly of building blocks allows theself-association (in series or in parallel) of particular patternsto form higher order organized biological nanomechanisms.Elementary biological structural components composed ofprotein-based nanosprings are of prime importance in theelasticity and stability of main biological functions at macro-scale, i.e., tendons, muscles, lung and skin. As illustration,Fig.1 shows some of these structural protein applications.Marine mussel produce byssal threads to attach to solidsurfaces that ressembles tiny tendons (Fig.1a). Each thread isstiff at one end and extensible at the other. The musselbyssus thread protein found in tendon exhibits powerfulelasticity characteristics, i.e., tendon is five times tougher andmore extensible than Achille’s tendon. Mussel thread is amicrocomposite fiber with a graded distribution of tensilemolecular elements: flexible near the mussel, strong and stiff

(a)

Fig. 1: Biological elastic microfilaments and fibers : (a) Structure of a mussel bysin striated muscle.

towards the substrate. Its elastic characteristics are composedof two types of macro proteins: proximal and distal proteins.Distal protein is composed of several domains such ascollagen (47kD), silk fibroin (20kD) and crosslinking sites(HIS tag). Proximal protein is mainly composed of elastindomains. Giant multidomain fibrillar protein titin (connectin)composing the sarcomeres of human striated muscles(Fig.1b) behaves the strong passive elasticity of muscle.Titin, a 1-µm-long protein found in striated musclemyofibrils, possesses unique elastic and extensibilityproperties due to the serial assembly of coiled titinmolecules. Titin is composed of ∼300 repeats of two types ofdomains, fibronectin type III-like (Fn-III) domains andimmunoglobulin-like (Ig) domains.

Just as muscles magnify forces and movements by usinggeometrical hierarchy mechanisms (Fig.1b), bio-inspirednanomechanisms using similar principles are investigated:cumulative nanoscale changes in elementary protein subunitsare amplified by their linear arrangement in filaments inorder to perform micro-scale mechanical work. These nano-scale fibrillar structures have equivalencies with macro-scalerobotic kinematic chains. The synthetic serial and/or parallelcomposition of protein-like springs allows to mimic thepassive elasticity of biomechanisms found in nature forapplication in new design concepts of bionanorobotic links.Figure 2a shows a bio-nanorobotic Stewart platformcomposed of protein components. Four biological elasticfilaments or fibers can be used as passive spring elements tojoin two platforms and form a single degree of freedomparallel platform that is actuated by three viral protein linearactuator (center) [13]. In Fig.2b, the VPL actuators hasstretched out and this results in the upward linear motion of

thesprusetha

Figmic

Thsynhomwitdisthisbeedyn

III

Wpro

Mussel

Proximalportion

Listalportion

Substratum

Attachmentplaque

COLLAGEN46kD

SILK5kD

PreCol-D

Mussel

(b)

sus thread protein and (b) giant multidomain fibrillar protein titin found

platform. The four protein fibers acting as mechanicalings are also stretched out. Their elastic behavior can bed as a passive control element or as the restitution forcet will bring the platform back to its original position.

. 2: Bio-nanorobotic paralell platform using four biological elasticrofilaments and three viral protein linear actuators.

e main challenge in bio-nanorobotic design involves thethetic formation of elastic fibrillar structures that areogeneous, structurally and kinematically well defined,

h unique mechanical properties combining a gradedtribution of tensile strength with good elasticity. Towards goal, different protein structures and assemblies haven simulated and modeled using steered molecularamics simulations (SMD).

. METHODS AND TECHNIQUES FOR STEERED MOLECULARDYNAMICS SIMULATION

e present in this section the simulated mechanicalperties of different molecular nanosprings for passive

40Ă

2µm

Sarcomere

Humanmuscle fiber

Titin Immunoglobulin-like (Ig) domain

Titin strand formed byhomologous domains

Molecularsprings

VPL actuators

(a) (b)

nanomechanisms when subjected to external forces. Themechanical characteristics (stretching, shearing or bending)are simulated through an Interactive Steered MolecularDynamics (SMD) system.

3.1. MethodologyMolecular dynamics (MD) was carried out using NAMD2

[14] with the CHARMM19 potential set. All simulationswere carried out at 300K, with temperature rescalingperformed every 10 timesteps. In more detail, the protocol togenerate a single simulation comprises four steps:

1. The first step is the energy minimization of thebiomolecular structure in order to remove any strong Vander Waals interactions that may exist which mightotherwise lead to unstable simulations. At this point, allproteins were solvated by a periodic box of dimensions(measuring 260×65×65 Å3) covering the module by at leastfive layers of water molecules with the TIP3P model. Theentire box of water is overlayed onto the protein and thosewater molecules that overlap the protein are removed.Energy re-minimization is realized in order to readjust thewater molecules to the protein molecule.

2. Then, an heating phase is initiated. Initial velocitiesat a low temperature are assigned to each atom of thesystem and Newton’s equations of motion are integrated topropagate the system in time. During the heating phase,initial velocities are assigned at a low temperature and thesimulation is started periodically, new velocities areassigned at a slightly higher temperature and the simulationis allowed to continue. The entire system gradually washeated up to 300 K in increments of 30 K for time intervalsof 1 ps, while leaving the box volume unchanged.

3. When the desired temperature is reached, theequilibration procedure consists to run the simulation untilthat the structure parameters, i.e., pressure, temperature andenergy, become stable with respect to time. Duringequilibration at a temperature of 300K, the water moleculescomposing the box were harmonically restrained to theiroriginal positions to maintain the shape of the waterbubble. All simulations were performed with a time step of1 femtosecond and a uniform dielectric constant of 1.

4. The final simulation step is to run the simulation in aproduction phase for the desired time length. However, thebiological processes that involve transitions from oneequilibrium state (native state) to another (conformationstate) in molecular springs are difficult to reproduce on MDtime scales, which today are still confined to the order oftens of nanoseconds. To address this issue, the applicationof external forces in a variety of ways may be used to guidethe system from one state to another, yielding newinformation about the mechanics of protein-basednanosprings.

3.2. Steered Molecular Dynamics

Steered molecular dynamics (SMD) simulations werecarried out with either a constant force applied to an atom(or a set of atoms) or by attaching a harmonic (spring-like)restraint to one or more atoms in the system. SMDsimulations were carried out by fixing one terminus of thedomain, and applying external forces to the other terminus.The forces were applied by restraining the pulled endharmonically to a restraint point and moving the restraintpoint with a constant velocity ν in the desired direction. Theprocedure is equivalent to attaching one end of a harmonicspring to the end of the domain and pulling on the other endof the spring. Force-induced unfolding processes withseveral choices of pulling positions were simulated in orderto characterize stretching, bending and shearing mechanicalproperties of the various α-helix and β-sheet biologicalsprings.

Fig.3: Basic concept of virtual environment and haptics technology forbio-nanorobot simulation.

The developed simulation system presented in Fig.1permits manipulation of bionanorobotic components in MDsimulations with real-time force feedback and 3D graphicaldisplay [16]. It consists of three primary components: ahaptic device controlled by a computer that generates theforce environment, a MD simulation for determining theeffects of force application, and a visualization program fordisplay of the results. Communication is achieved through anefficient protocol between the visualization program VisualMolecular Dynamics (VMD) [15] and the moleculardynamics program (NAMD2) running on single machine(can be extended to multiple machines). A force-feedbackPHANToM device measures a user’s hand position andexerts a precisely controlled force on the hand in order toapply different mechanical constraints, force and energyfields on the virtual model.

VMD: Virtual Molecular Dynamics NAMD: Molecular Dynamics

GHOST SDK

Communication Protocol

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Four-bundle α-helix ROP protein Double stranded DNA protein β-sheet titin protein

(a) (b) Fig. 4: Force curve and hysteresis curve done by forward stretching (with v=0.1 Å/sec) and relaxafour α-helix to coil stretching, (b) double-stranded DNA stretching and (c) titin Ig 27 stretching sim

IV. MECHANICAL SIMULATION OF BIOLOGICAL SPRINGS FORFILAMENT BIOMECHANISMS

We present in this section the simulated mechanicalproperties of different molecular nanosprings for filamentsand fibrils nanomechanisms when subjected to externalforces. A quantitative understanding of flexibility,extensibility, reversibility characteristics of passivebiomechanisms are presented.

4.1. Biomechanisms using α−helical structures. 4.1.1. Four helix-bundle ROP protein: The repressor ofprimer protein (ROP) is a small, dimeric molecule consistingof two or four identical chains of 63 amino acids (Fig.4a).Each monomer consists of two α-helices connected by ashort turn and a seven-residue C-terminal tail. The twomonomers pack together as a fully antiparallel four helix-bundle. The bend region of Rop has attracted considerableinterest as a parallel molecular spring due to its stability andelasticity properties [17]. In the simulation, we constrainboth ends of two α-helices of the molecule to move onlyalong the z axis for stretching simulations and fix the shortturn. Fig.4a shows two distinct conformational transitions

that provoke thform approachextensions are Fig.4a. A usefgiven by the [18]. The WLrelationship beand the entropi

(= Tkf B

where kB is temperature, Alength, and L naturally fullywithout the neduring extensiosprings (see Fig

4.1.2. Double slong polymer mnucleotides. Sthelix, with two

stretching

relaxation

ROP

Dis

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e (n

m)

Time (pSec)

kstret

Distance (nm)

Forc

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relaxation

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Time (pSec)

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the B is the pis the reversied of an migh.6a).

trandedade up

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stretching

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Dis

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Time (pSec)500400300200100

10

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0

0

(c) structure when the stretching force

rsion of double α-helix to an e coil conformation. Both

ilar and can not be distinguoximation for spring constanible worm-like chain (WLC

el of entropic elasticity pree relative extension of a polyng force (f) through

]4/1))/1(4/1( 2 −−+ LzL ,

oltzmann constant, T is ersistence length, z is the en

length. The four α-helix strble in a wide domain of en external pulling force. Its t be modeled as four entropic

DNA protein: Chemically, Dof a linear series of subunits kly, DNA is usually found as wrapped around one another

600

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is zero: (a)

extended α-helixished int kstret is) model

dicts themer (z/L)

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ucture isxtensionbehavior parallel

NA is anown asa double(Fig.4b).

3x 104

α/β sheet fibrillin protein ouble-helix tropomyosin protein

(a) (b)

Fig. 5: Force curve and hysteresis curve done by forward stretching (with v=0.1 Å/sec) and relaxation of the structure when thefibrillin protein, (b) three α−helical tropocollagen protein and (c) double α−helical tropomyosin protein.

Assemblies of double-stranded DNA (ds-DNA) proteins canprovide strong elasticity and full reversibility. The water-DNA system was gradually heated over 7ps to 300K, andthen equilibrated with a thermal bath at 300K for another7ps. The dynamics of the DNA were performed with thedouble-stranded terminations stretched and the other endfixed. This elastic behavior is thus purely entropic [19]. Forvery low tension 1f pN≤ , the restoring force is provided by"entropic elasticity".

In the absence of any force applied to its ends, the DNA'sRMS end-to-end distance (chain length, L) is small comparedto its contour length defined as the maximum end-to-enddistance (maximum length, L0) and the chain enjoys a largedegree of conformational disorder. Stretching DNA reducesits entropy and increases the free energy. The correspondingforce f incre ses l arly as a Hooke’s law with theextension L:

00

,3

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fDNA

B <<≅ (2)

The length ADNA is known as the "thermal persistence length"of DNA and is of the order 50nm. For higher

forces ( 10 )f pN≥ , the end-to-end diand the elastic restoring force is duinternal structure of DNA. In this regimcurve can be approximated by two mused to describe the entropic elasticityjointed chain (FJC) model, the molecuorientationally independent Kuhn segmis a measure of chain stiffness. The rebehavior can be summarized in the forc

bfTk

Tkbf

Lz B

Btot−⎟⎟

⎠

⎞⎜⎜⎝

⎛= coth

defining the well-known LangevinEq.(3) gives the effective spring con

bTkk BFJC /3= . The terms Ltot represent

protein and f the stretching force. Thesegments by tension is described by Bo

k1stret

Forc

e (p

N)

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k

l

k

l

-1000

-500

500

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3000

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1500

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5 10 15 20 25 30

Distance (nm)

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stretching

Fibrillin

Time (pSec)

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Time (pSec)Time (pSec)

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of DNA. In the freelyle is made up of rigid,ents whose length, b,

sulting entropic elastice-extension relation:

(3)

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0 35 40 45 50 55 60Distance in nm

100 150 200 250 300 350 400

Stretching

Relaxation

Time (pSec)

Distance (nm)

Tropomyosin

ine

a the total length of the alignment ofltzmann distribution.

Fig. 6: Protein based nanomechanissprings take a coiled form to take aDifferent reversible spring nanometandem of four double-strand DNAβ-sheet proteins as serial link mecfilaments and (f) assembly of sandw

In the inextensible worm-likmolecule is treated as a flexsmoothly as a result of tResults, shown in Fig.4b, inmodel can describe the beh

Ig segment PEVKdomain

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A1

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)

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k1 k2 k3 k4

(a

ms generating re coiled conformachanisms have b proteins,. In (c) hanisms. In (e) ahich β-sheet silk

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storing force based on the mechanism of entropic elasticity. In the absence of extertion while extension or compression of the chain generates a restoring force due teen studied and modeled: (a) four α−helix Rop protein as a parallel spring nanand (d), we present superelastic structures based on (c) three-titin β-sheet proteinnd (f), we present serial structures with high tensile strength: (e) a pair of doubl proteins.

l (WLC) model, theength L that curvesation (see Equ.1).

ven though the FJCNA in the limit of

low and intermediate forces, it fails at highmodel, on the otherhand, provides an excemolecular elasticity at intermediate and models behave as a Hooke’s law for lowThe dsDNA is fully reversible as shown in

k3

Stretching forc

(c

(d)

(f

Hydrogenbonds

(b)

nal force, their entropico reduction of entropy.omechanism and (b) as and (d) three-fibrilline α−helix tropomyosin

forces. The WLCllent description ofhigh forces. Both stretching forces. Fig.4b. As shown

in Fig.6b, whenconsidering small extentions, a tandem of ds-DNA proteins can be modeled as a series of hookian springs.

4.2. Biomechanisms using β−bundle structures.

4.2.1.Immunoglobulin-like (Ig) domains. The behavior oftitin as a serial link entropic spring depends on the reversibleunfolding of individual Ig domains. Currently, I27 is the onlyI-band Ig with an experimentally solved structure [20] andhence has been selected for investigation. The Ig domain wasplaced in the center of the water box and equilibrated with athermal bath of 300°C. Stretching simulations were carriedout by fixing one terminus of the domain and applyingexternal forces to the other terminus. The simulation beganwith an equilibrated folded structure and was stopped when afully extended polypeptide was obtained. The extension ofI27 was performed with a pulling speed v=0.5Å/ns and wasstopped when the extension reached 33nm.The extensiondomain (Fig.4c) is divided into four sections: I. preburst atextension of 4nm during which the protein maintains β-sheetstructure and the external force remains smaller than 1500pN; II. major burst immediately after the preburst burst atextension of 8nm; III. post-burst at extension of 27 nmduring which the protein unravels; IV. pulling of fullyextended chain up-to an extension of 33nm. Othersimulations showed similar features of the unfolding processand force profiles with only small variations in force peakvalue and degree of extension at the force peak. Results ofstretching-relaxation curve (Fig.4c) show a good reversibilityof the protein motion when completely relaxed. Fig.6csuggests a mechanical portrait model. Its behavior duringextension might be modeled as series of elastic spring with aviscous element corresponding to the unfolding of theindividual I27 domain. Stretch would result first instraightening of the Ig domain chain (corresponding to thepreburst-part I) as an entropic spring. The tightly folded Igdomain might function as a “shock absorber” (parts II: majorburst and III: post-burst) by reversible unfolding only in thecase of extremely high stretching forces. This structureallows avoiding the complete rupture of the protein due tooverstretching.

4.2.2. Fibrillin proteins. Fibrillins form the structuralframework of an essential class of extracellular microfibrilsthat endow dynamic connective tissues with long-rangeelasticity (skin, lung, eye) [21]. Mechanical extensioninduces a conformational change with an apparent decreasein randomly coiled regions of the protein and a relativeincrease in α-helical regions. As we can see in Fig.5a, aseries of three unfolding events that results ultimately in thegeneration of a stabilized tensioned polymer. Uponrelaxation it appears that the protein spontaneously refolds tothe original state, indicated by a return to the native state.WLC model fails in this case for small amount of stretchingforces. It can be represented by a spring-stiff pair. Forlengths below the length (lk), spring constant of the protein isk1

stret , whereas for lengths greater than (lk), spring constant is

k2stret. As the tension increases, the protein extends according

to the unfolding events until to reach the length lk (as aspring component), and then the folded protein extendsaccording to its stiffness (as a stiff component). Whensubjected to external mechanical bending force, the fibrillinprotein behaves as a linearly elastic stiff spring. Suchproteins are very interesting in compliant molecular linksconstituted by fibrillin-rich microfibrills. In this structure, theindividual microfibrills acts as relatively stiff elasticpolymers. Based on simulation studies (with v=1 Å/ns andk=10 pN/Å), it was concluded that individual isolatedmicrofibrills were reversibly extensible (see Fig.5a)although the mechanism of this elasticity is unknown. Themechanical elastic model can be represented as a seriesmodel of spring-string components (Fig.6d).

4.3. Biomechanisms using compounds helix-bundlestructures.

4.3.1. Triple-stranded helix tropocollagen protein.Collagens are a family of stuctural proteins of the

extracellular matrix with one or more triple-helical domains.We studied here the triple-stranded helix tropocollagen (TP)macromolecule. The TC is a rigid rod of definite lengthcomposed of three helical peptide strands thatare equal inlength. The collagen structural stability is provided by anorderly intramolecular hydrogen bond network [22]. Todetermine the elasticity of the tropocollagen molecule at thenanoscale, Fig.5b gives the force-extension characteristic.We verify the linear response, on average, when themolecule was extended 2.6-3.3Å beyond its initial length,representing 2.0-2.5% of the molecular strain. For higherextensions (>3.3Å), high values of forces are reached (3×104

pN) for small extension values 3.5-6.5 nm leading to highstiffness constants (>106 pN/nm). Figure 5 b shows clearlythat the reversibility of the molecular spring is stronglyperturbed by the destruction (during stretching) andreorganization (during relaxation) of the internal hydrogenbonds. The TP can be modeled as three strings in parallel(stiff spring) developing its stress when completely unfolded.When a stretching force F is applied, the system isconsidered homogeneous, each line bears a third of force andeach spring in each line carries the same force. Then thestrain is equally distributed within the lines and the springs.Since the TP protein is almost inelastic, it works as a stop-length fibre, preventing further strain when reaching itsoverstretched limits. This redistribution of the hydrogenbonds guarantees a constant mean of interactions needed tokeep a reversible pathway when the stretching effect isinterrupted.

4.3.2. Double-stranded helix myosinThe nanomechanical properties of the coiled-coils of myosinare fundamentally important in understanding muscleassembly and contraction [23]. The molecular mechanicssimulations of coiled-coil extension (shown in Fig.5c)demonstrates a soft hookian spring (∼10 pN/nm) for small

extensions (40-45 nm) preceding the final exponential phaseleading to stiff springs modeled by WLC model of elasticity.The length Amyosin known as the "thermal persistence length"of myosin has been chosen on the order 40nm. In this case,the FJC model of elasticity fails. An important considerationin analyzing the force spectra of stretched tropomyosin isthat serial assembly using carbon atom attachments aremodeled by simple WLC springs connected in parallel.However, serial configuration using hydrogen bondattachments (see Fig.6e) behaves as a symmetric orasymmetric molecular network of interconnected soft andbreakable springs (hydrogen bonds). For some compoundsof proteins, the exponential fit of the data is necessary butnot sufficient to claim non-entropic nature of the elasticresponse. Alternate models have been also developedcomposed of molecular network composed of interconnectedsprings.

4.3.2. β-sheet silk fibroin protein. Silk is a protein produced by the silkworm Bombyx mori

for the construction of the cocoon and also a large number ofspider families for the manufacture of web. The silk fibresare made up of a protein called fibroin [24]. The protein isconstructed from layers of antiparallel beta pleated sheetswhich run parallel to the silk fibre axis (see Fig.6f). Relaxedand extended stretching simulations show the large change inlength that can occur with extension. Single molecules offibroin silk are composed of tandem repeats of the entireensemble, consisting of repetitive β-sheet sequences plusnon-repetitive spacer sequences. Even if the elasticity offibroin silk is due to β-strands acting as molecular springs,these springs are not necessary Hookian but functionprimarily as entropic springs. We are carrying out simulatedpulling and stretching, using steered molecular dynamics, toinvestigate the molecular origins of the unique characteristicsof the β-sheet sequences in fibroin silk filalements. In such acase, filament can be modeled as a collection of chainsjoined by crosslinks (Fig.6f).

V. DISCUSSION AND CONCLUSION

To summarize the performances of the differentmolecular springs, Table 1 makes a performance comparisonand classification of protein-based elastic joints. These bio-springs are classified following their mechanicalcharacteristics: number of dof, elasticity, reversibility,displacement, linearity, relaxation time (see Table1). It isclear that we have treated only idealized cases in thismodeling study and that we are using a very simpleenvironmental representation. However, the present studyshows clearly some important properties of molecularsprings. All characterized molecular springs exhibit aconformational bistability of latched and unlatched stateswith different elastic features. The assembly of molecularsprings features interesting mechanical properties. (i)Magnification of movements by a serial geometrical

hierarchy of proteins depends strongly of the nature of thespring. For structural smartness, assemblies built from α-helical units tend to be more readily reversible thanstructures dominated by β-strand units. This difference arisesprincipally from the relative importance of hydrogen-bonding and hydrophobic contributions to structural stability.β-Sheet structures have extensive interpeptide or interchainhydrogen bonds, which add great rigidity and irreversibilityto such structures (i.e. the rigidity of silk fibroin is derived inlarge part from hydrogen bonding). The use of stretchyproteins in serial links, such as elastin or spider’s silk, stretchand relax without any net energy dissipation. These proteinsare highly resilient with smooth rebound when the elasticprotein is relaxed. The fibtoin silk has tensile strength of ~1Gpa, comparable in strength to Kevlar or steel. The otherproteins presented in Table 1 dissipate energy as heat in theprocess of stretching and relaxing making them less resilient.(ii) Magnification of force by parallel structure are welladapted to “large force-small displacement” spring-likeelastic joint, i.e., α-helix bundles proteins. Through thedatabase of Table 1, different kinematic and dynamicequivalences between macro and nanomechanisms have beendrawn in order to allow prototyping of “open-loop” or“closed-loop” kinematic chains with adaptable stiffness andlarge displacements.

I. REFERENCES

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Table1: Comparison of elasticity characteristics of some protein–like nanosprings.

Proteins type LengthChange

Spring Stiffness

Stretching Bending Shearing

RelaxationTime

Energyvariation

Reversibilityerror

Elasticitymodel

∆l/l kstret

(pN/nm)kbend

(pN/nm)kshear

(pN/nm)τ

(sec)∆Ε

(Kcal/mol) (%)

Deca-alanine 1.305 ∼ 440 ∼ 32 ∼ 710 9.04 975 ∼ 30 HookianFibrillin 4.17 ∼ 40 ∼ 250 ∼ 100 0.163 1835 ∼ 10 HookianFibronectin Fn-III10 8.483 ∼ 850 ∼ 360 ∼ 60 0.1 6355 ∼ 5 WLCImmunoglobulin I27 5.769 ∼ 875 ∼ 925 ∼ 280 0.1 4778 ∼ 5 WLCTopocollagen 1.707 ∼ 10000 N.A. N.A. 0.365 1211 ∼ 24 WLCRop 1.2 ∼ 40 ∼ 190 ∼ 240 0.48 705 ∼ 5 HookianDNA 1.476 ∼ 200 ∼ 450 ∼ 220 0.192 3018 <5 FJC Silk N.A. ∼ 1000 N.A. N.A. 0.1 1790 ∼ 5 WLCTropomyosin N.A. ∼745 N.A. N.A. 0.1 N.A. ∼ 5 WLC and

FJC

N.A: Not Available