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* Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental Engineering, and Division of Biological Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
Correspondence: Address reprint requests to M. J. Buehler, Tel.: 626-628-4087; E-mail: mbuehler{at}MIT.EDU.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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p 
16 nm, confirming experimental results suggesting that tropocollagen molecules are very flexible elastic entities. We demonstrate that assembly of single tropocollagen molecules into fibrils significantly decreases their bending flexibility, leading to decreased contributions of entropic effects during deformation. The molecular simulation results are used to develop a simple continuum model capable of describing an entire deformation range of tropocollagen molecules. Our molecular model is capable of describing different regimes of elastic and permanent deformation, without relying on empirical parameters, including a transition from entropic to energetic elasticity. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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300 nm, roughly 1.5 nm in diameter (1
–7). Staggered arrays of TC molecules form fibrils, which arrange to form collagen fibers (1,2,8).
Collagen plays an important role in many biological tissues, including tendon, bone, teeth, cartilage, and in the cardiovascular system. Collagen is typically subjected to mechanical tensile load under physiological conditions (8–10).
Due to lack of understanding of nanomechanical properties of collagen, the source of the elasticity of individual TC molecules, fibrils, and collagen fibers remains controversial.
There are several reports of experimental studies focused on the mechanics of single tropocollagen fibers (7,15–20). However, tropocollagen molecules have rarely been studied using molecular dynamics (MD) studies. In a recent investigation, the mechanical properties of collagen fibers, using MD studies, focus on their Young's modulus (11) for tensile strains up to 4%. Another recent study was carried out using molecular mechanics, considering the stretching mechanics of tropocollagen molecules up to 10% deformation (5). Other studies focused on the stability of collagen molecules (12) or the effect of point mutations on the stability (13,14). Some researchers modeled collagen at the continuum scale, using techniques such as finite element modeling (15).
However, no studies of the large deformation behavior of tropocollagen molecules, including elastic and fracture regimes, have been reported. Further, most existing atomistic studies were limited to short sequences of tropocollagen proteins that did not exceed 10 nm, thus significantly below the length of molecules extracted from tissue, which typically features molecular lengths 300 nm (1–7). Atomistic investigations of the bending mechanics and the mechanics of long tropocollagen molecules at finite temperature are still missing. Such studies are vital to understand the effect of entropic contributions to elasticity and its competition with energetic terms.
Therefore, limited understanding exists about the transition from entropic to energetic elasticity, under which conditions either one dominates, and how properties change when individual TC molecules assemble into collagen fibrils (8). Understanding the nanomechanical properties of single TC molecules is essential to develop theories that link molecular and tissue properties of collagen.
Early experiments suggested that TC molecules are rigid, rod-shaped structures (16,17). However, based on hydrodynamic methods and transmission electron microscopy (TEM), it was found that TC molecules feature some flexibility, which can be measured by the persistence length. Hydrodynamic methods suggested a persistence length of 130–180 nm (18–20); TEM-based estimates (21) range between 40 nm and 60 nm. Recent experiments using optical tweezers suggested a much lower persistence length, with values ranging between 11 nm and 15 nm (22–24). Thus far, no satisfying explanation for these discrepancies has been provided. No molecular model of TC has been developed capable of describing different regimes of deformation.
Here we report hierarchical modeling consisting of sequential atomistic and molecular simulation that confirms optical tweezer results quantitatively, without relying on empirical parameters. We employ a mesoscale molecular model as a strategy to upscale the mechanical properties of single TC molecules to large timescales to enable capturing entropic effects. Our model is capable of carrying our molecular stretching experiments, with high precision, at timescales of tens of microseconds, approaching experimental deformation rates.
We find that when the TC molecule is stretched below its contour length (x < L), entropic elasticity governs elasticity. When the TC molecule is stretched close to and beyond its contour length (x > L), we observe a transition to energetic elasticity, featuring a two-regime hyperelastic or nonlinear elastic behavior due to unfolding and breaking of hydrogen bonds and stretching of the polypeptide backbone, followed by molecular fracture. We find that assembling TC molecules into fibrillar structures leads to rapid increase of the persistence length, rendering entropic elasticity less significant.
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MATERIALS AND METHODS |
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,26). We note that the focus of this article is not development of the mesoscale model. Instead, here we focus on applying the mesoscale model to describe deformation of ultra-long TC molecules.
However, extending earlier work (25), here we pay particular attention to the bending stiffness of TC molecules and its dependence on the loading rate. Therefore, we carry out additional molecular dynamics simulations of bending a short sequence of a TC molecule.
Atomistic model
For the full atomistic simulations, we use the classical force field CHARMM (27,28), implemented in the molecular dynamics (MD) program NAMD (29). The CHARMM force field (27,28) is widely used in the protein and biophysics community, and provides a basic description of the atomistic interactions in proteins. It is based on harmonic and anharmonic terms describing covalent interactions, in addition to long-range contributions describing van der Waals interactions and ionic (Coulomb) interactions as well as hydrogen-bonding.
We visualize the molecular structures using the visual molecular dynamics (30) program. We consider the atomic coordinates of a single TC molecule based on the crystal structure PDB ID 1QSU (31) with 1.75 Å resolution, a short sequence of a TC molecule with length L 8.4 nm. The 1QSU is a triple-helical type II collagen-like molecule with sequence (Pro-Hyp-Gly)4-Glu-Lys-Gly-(Pro-Hyp-Gly)5.
This model serves as a simple representation for the amino-acid sequence that makes up the sequence of longer tropocollagen molecules extracted from live tissue.
We use the psfgen program of NAMD (29) to add hydrogen atoms and to assign charges. The molecular model is set up according to pH 7, resembling conditions in experiment (24). No electrolytes are present in our atomistic setup.
We include crystallographic water in the simulations, and also add a skin of water of a few Ångströms surrounding the tropocollagen molecule. We note that electrostatic interactions can be significantly weakened by screening due to solvent molecules, which can lead to interactions that are weaker than van der Waals interactions and that decay over distances of several Ångströms (32).
Simulations with forces applied are carried out after energy minimization and finite temperature relaxation at T = 300 K. All simulations reported in this article are carried out at T = 300 K with temperature control (NVT ensemble), using a time step 
t = 1 fs. We apply forces to the molecule using an SMD scheme (33). Forces and fixed constraints are applied at the C
atom in each chain. Forces are applied on each of the three -chains equally. In the case of a fixed end, each C at the end of the molecule is fixed. No torsional loads are applied since the end of the tropocollagen molecule is free to rotate. This is essential to allow for proper unfolding of the molecular structure.
Forces in the SMD scheme are applied by adding a potential energy term to the Hamiltonian of the system,
 | (1) |
the pulling rate, 
the current position vector of the atom being pulled, 
the position vector of the atom at t = t0 (beginning of simulation), and 
the normalized direction vector in which the atom is being pulled.
To carry out the tensile experiments, we use a SMD spring constant kSMD = 10 kcal/mol/Å2, with 
Å/step.
Postprocessing of the simulations is done using the CMDF framework (34), a computational Python-based simulation environment capable of seamless integration of various file formats and computational engines, including molecular and crystal building tools. CMDF Python scripts are used to analyze and postprocess the simulation data, as required, for example, to compute statistical averages of force-displacement curves.
Fig. 1 depicts results of stretching calculations of a single TC molecule, based on the CHARMM model and the ReaxFF model (for details and simulation conditions, see (25)). Fig. 2 shows snapshots of the atomistic geometry of the tropocollagen molecule as it undergoes tensile deformation.
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8, underlying the significant stiffening effect at tensile deformation beyond 30%.
To determine the bending stiffness, we fix both ends of the TC molecules. This is achieved by fixing the terminal C atom in each of the three polypeptides. We apply a force in the center of mass of the TC molecule using steered molecular dynamics (using kSMD = 0.01 kcal/mol/Å2. To study the variation of bending stiffness with deformation rate, we simulate this three-point bending experiment at varying displacement rates, where 
0.01 to 0.1 m/s. The applied force is orthogonal to the axial direction of the TC molecule, leading to bending of the molecule upon increasing force. From the force-displacement data obtained by atomistic modeling, the bending stiffness EI is given by
 | (2) |
Fig. 3 depicts the bending stiffness as a function of displacement rate. The results suggest a linear relationship between the bending stiffness and the deformation rate. Extrapolation to zero deformation rate yields an estimate for the quasi-static bending stiffness, EI 1.247 x 10–29 Nm2.
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,5) carbon nanotube).
Molecular model
The full atomistic calculation results in feed parameters for the molecular model, representing TC molecules at a mesoscale, as shown schematically in Fig. 4. Our molecular model is implemented in LAMMPS (35), and parameters are identical to the model described in the literature (25,26). Here we chose a smaller bead-to-bead distance of 7 Å and adapt the bending stiffness to the extrapolated value for zero deformation rate (see Fig. 3). The effects of the solvent are treated implicitly in the model.
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The bottom subplot in Fig. 4 depicts how the load is applied to the long tropocollagen molecule using the SMD scheme (33). The outermost particle of the molecule is kept fixed during the simulation, and a slowly increasing force is applied on the right end. The SMD spring constant is chosen to be kSMD = 1 kcal/mol, and we simulate deformation at a pulling velocity of 0.025 m/s (for the SMD formulation, see Eq. 1; the SMD model in the atomistic and molecular model have the same formulation). The continuously increasing force slowly stretched the molecule from a convoluted initial shape to a straight configuration, followed by further extension and fracture. During the simulation, we record snapshots of magnitude of force applied and corresponding end-to-end distance of the molecule. This force-extension data is averaged into bins after the simulation is over to obtain a smooth curve. We find that further reduction of loading rate shows negligible changes in the force-extension curve.
We consider a stretching experiment of a TC molecule with L = 301.7 nm. The time step used in molecular modeling is t = 20 x 10–15 s, and we reach a total simulation time of ttot 20 x 10–6 s. Each mesoscale particle has a mass of m = 1358.7 amu, accounting for the mass of the TC segment it represents plus a skin of water molecules in a radius of 3 Å.
All calculations are carried out in three dimensions.
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RESULTS AND DISCUSSION |
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100 nm. During the initial regime (Regime I), the molecule loses its entangled structure, while the applied forces remain relatively low. Once the contour length is reached (x = L), (Regime II) uncurling of the triple helix, (Regime III) stretching of covalent bonds in the individual polypeptides, and (Regime IV) rupture of the TC molecule occurs, followed by a sharp drop of the forces to zero.
The qualitative behavior of TC molecules under stretch is similar to recent experimental studies carried out on collagen fibrils that reach forces on the order of several µN (36), also showing a significant hyperelastic stiffening effect.
Fig. 5 b depicts a detailed view into Regime I, including a quantitative comparison with experiment (23,24). The plot reveals very good agreement, even though the deformation rate in MD is still much higher.
The elastic properties of materials can be expressed as partial derivatives of the free energy density with respect to strain 
. Based on the free energy density AV = (U – TS)/V (where V stands for the system volume), the scalar stress 
= 
AV/. At constant temperature, the entropic regime is characterized by U/ 0. In contrast, during the energetic regime, S/ 0.
Changes in the internal energy U can be correlated with changes in the average internal tensile strain in the TC molecule (d(x)/dx), where
 | (3) |
 | (4) |
). The result of d/dx(x) shown in the inlay of Fig. 5 a confirms that the regime x < 280 nm is dominated by entropic elasticity since d/dx 0.
For x 
280 nm, increases slowly, leading to a change of internal energy that contributes to the elastic response. We find that 1.2% when x 
L. These changes indicate the onset of a smooth transition from entropic to energetic elasticity at x = L.
Energetic elasticity dominates when x > L and leads to a significantly stiffer response. Entropic contributions to the elastic response decay quickly and approach zero at x 317 nm, suggesting that energetic terms dominate for x > 317 nm. This is confirmed by considering that d/dx approaches a constant value (see inset of Fig. 5 a).
By fitting the Marko-Siggia estimate of the worm-like chain (WLC) model (37,38) to the MD results (for F < 11 pN), we determine an effective persistence length p 16 nm, with L = 301.7 nm.
Even though the WLC model is a reasonable approximation for small forces, it ceases to hold when significant deformation occurs and the molecule is stretched toward its contour length (see Fig. 7 a, curve fWLC – f1).
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= L/p, noting that p EI.
Assembly of TC molecules into fibrils leads to strongly increased bending stiffness, thus reducing the significance of entropic elasticity. Fig. 6 depicts the bending stiffness as a function of the number of molecules in a fibril, suggesting a power-law scaling EI np, where p 2.5. Elastic beam theory predicts p = 3, confirming the results of our numerical model.
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). This effect is not accounted for in the present article. To implement the three-point bending test, we fix two outermost left and right particles in the lowest molecule. We apply a slowly increasing force in the center particle of the top molecule using the SMD scheme.
Collagen fibrils have diameters of several tens of nanometers and consist of hundreds of aligned TC molecules. Since EI–p, < 1 for n > 10—suggesting that the elasticity in collagen fibrils includes significant energetic contributions, particularly in the regions of dense molecular packing. We have confirmed the effect of increasing EI by direct MD calculations that clearly show the reduction in entropic elasticity, leading to much lower forces in the stretching regime below the contour length (x < L). We note that while the bending stiffness increases significantly after assembly, our model predicts that the tensile stiffness or Young's modulus decreases continuously with increasing number of molecules.
However, a possible source of entropic elasticity in collagen fibrils are hydroxyproline-deficient sequences (labile domains) that represent more floppy parts of a collagen fibril with reduced molecular density (6–8,39), as they occur, for instance, in the gap regions. The mechanism studied in this article may thus relate to the transition from less ordered to more ordered molecule states, as discussed in the literature (6,7), where it was suggested that disappearance of "thermally activated kinks in the collagen molecules" may account for the entropic elasticity in fibrils. Thus, the entropic regime found in this article may account for the initial regime of stress-strain behavior for fibril strains up to 8%, shown in Fig. 6 in Misof et al. (6).
In summary, the toe region may relate to Regime I (entropic elasticity), and the heel region may relate to Regime II (energetic elasticity) (6,7). The qualitative shape of the force-extension curves showing the strong progressive increase of the tangent slope with strain is found in both our simulations and experiments.
However, since the length of labile domains in collagen fibrils is typically limited to 60 residues (or 20 nm), quantifying their relative contribution to elasticity of a collagen fibril remains unclear. We leave further investigations of this topic to future research.
Further, variations of the molecule properties along the molecule axis may change the stretching mechanics of single tropocollagen molecules. This is because the variation of sequence along the axial direction leads to locally varying bending stiffnesses and therefore modifications in the local persistence length. Such effects could be analyzed by carrying out studies similar to the one presented here, by considering different amino-acid sequences and then mapping them onto a axial variation of nanomechanical properties. We emphasize that our molecular modeling approach can, in principle, easily be adapted to account for such effects.
We use the MD results to develop a continuum model for the elasticity of TC molecules. The force as a function of distance (F(x)) is given by
 | (5) |
,38)
 | (6) |
 | (7) |
0, f1(x) 
for x L (due to the divergence of the WLC model), and f2(x) = 0 for x > > L, but f2 reaches a finite value for x L. Note that 
Both functions are determined numerically by MD (see Fig. 7). For L = 301.7 nm, xbreak = 452.6 nm, x1 = 39.0 nm, k(0) = 51.7 pN/nm, k(1) = 294.8 pN/nm, and p = 16 nm. The predictions of our model agree with the numerical results shown in Fig. 5.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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p 16 nm. Our force-stretching curves are in quantitative agreement with results obtained by optical tweezer experiments (see Fig. 5 b) (23,24), which suggested persistence lengths between 11 and 15 nm.
We note that our estimate of p from the force-stretching curve is larger than the value obtained by considering
 | (8) |
This difference could be attributed to coupling between bending and tension deformation modes that are not captured in the WLC theory, but included in our dynamical calculations, leading to an increased persistence length.
This suggests that p determined from force-extension curves in both molecular dynamics and experiment (23,24) represents an effective persistence length. Calculation of the persistence length from the tangent correlation function confirms that 
suggesting that p 30...45 nm. Interestingly, this is close to experimental TEM results obtained by optical methods (21) that are based on a similar geometrical analysis scheme. These observations may help to explain the discrepancies noted in earlier studies (23,24).
An alternative reason for the large variations in persistence lengths in experiment (18–20) is based on the idea that not a single molecule, but an assembly of several molecules has been stretched, effectively increasing EI that leads to an increased persistence length (24). This possibility is not inconsistent with our model (see Fig. 6).
The observed transition from an entropic to an energetic elastic regime found here may correspond to experimental observations of progressive increase in the tangent slope of the stress-strain curve during stretching of collagen fibrils (6,7). Our results confirm that the transition from less-ordered to more-ordered molecule states as discussed in the literature (6,7) can lead to the characteristic mechanical behavior.
Our results illustrate a possible strategy to develop molecular or mesoscale models of complex biological systems, considering various mechanisms of deformation, ranging from entropic to energetic elasticity, considering fracture mechanics of individual molecules. Similar methods could be used for other biopolymers with similar geometry. Future studies of mechanics of individual and ensembles of TC molecules could help to shed light on assembly processes or help to clarify the mechanisms during cellular production of collagen molecules. Further, the effect of sequence changes along the molecular axis is an important subject of study.
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ACKNOWLEDGEMENTS |
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FOOTNOTES |
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Submitted on December 6, 2006; accepted for publication February 22, 2007.
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The resulting bending stiffness for vanishing deformation rate is used as input parameter for the mesoscale molecular model.
