Large Deformation Mechanisms, Plasticity, and Failure of an Individual Collagen Fibril With Different Mineral Content

ABSTRACT Mineralized collagen fibrils are composed of tropocollagen molecules and mineral crystals derived from hydroxyapatite to form a composite material that combines optimal properties of both constituents and exhibits incredible strength and toughness. Their complex hierarchical structure allows collagen fibrils to sustain large deformation without breaking. In this study, we report a mesoscale model of a single mineralized collagen fibril using a bottom‐up approach. By conserving the three‐dimensional structure and the entanglement of the molecules, we were able to construct finite‐size fibril models that allowed us to explore the deformation mechanisms which govern their mechanical behavior under large deformation. We investigated the tensile behavior of a single collagen fibril with various intrafibrillar mineral content and found that a mineralized collagen fibril can present up to five different deformation mechanisms to dissipate energy. These mechanisms include molecular uncoiling, molecular stretching, mineral/collagen sliding, molecular slippage, and crystal dissociation. By multiplying its sources of energy dissipation and deformation mechanisms, a collagen fibril can reach impressive strength and toughness. Adding mineral into the collagen fibril can increase its strength up to 10 times and its toughness up to 35 times. Combining crosslinks with mineral makes the fibril stiffer but more brittle. We also found that a mineralized fibril reaches its maximum toughness to density and strength to density ratios for a mineral density of around 30%. This result, in good agreement with experimental observations, attests that bone tissue is optimized mechanically to remain lightweight but maintain strength and toughness. © 2015 The Authors. Journal of Bone and Mineral Research published by Wiley Periodicals, Inc. on behalf of American Society for Bone and Mineral Research (ASBMR).


Derivation of parameters for mineral interactions
The interfibrillar mineral is modeled by an FCC lattice. Since hydroxyapatite is an ionic crystal, 21 the complex interactions between the ions are approximated by a Lennard-Jones potential Eq.

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(6). The equilibrium distance between two particles is given by: For an FCC lattice, it corresponds to the nearest neighbor's distance which is: The bulk modulus is given by: With the elastic modulus and , the Poisson ratio.

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Assuming that elasticity is due to energetic contributions and that hydroxyapatite satisfies the 30 Cauchy-Born rules, the shear modulus of the FCC crystal can be expressed as (1) : With the volume of a unit cell which, for a FCC lattice equals to = 0 3 /4 since there are 4 32 atoms per unit cell. 33 Combining Eq. (11), (13) and (14) and using the relationship = 8 3 µ , we obtain: The cutoff radius of the LJ potential is chosen to be 20% larger than the nearest neighbor 35 distance. 36 Interactions between collagen and apatite are also represented by a LJ potential. The 37 equilibrium distance is assumed to be equal to a single tropocollagen molecule's radius. At 38 equilibrium, a collagen particle will be located at the apex of a square pyramid with a base 39 formed by 4 hydroxyapatite atoms (Fig. S1A). The base diagonal is equal to the FCC lattice 40 ( 0 ) and its height corresponds to the desired equilibrium distance . The equilibrium 41 distance 0 between collagen and mineral particle is determined by geometric relations and 42 is found according to Eq. (10). 43 The energy parameter and the cutoff radius are computed by direct comparison 44 between full atomistic and coarse-grained simulation of collagen-hydroxyapatite adhesion.

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Steered molecular dynamic (SMD) have been used to explore the shear strength of collagen-46 mineral composite (Figs. S1B and C). To overcome the loading rate dependence, the shear 47 strength has been computed for several loading rates ranging from 1 to 20 m/s and extrapolated 48 to a quasi-static loading (0 m/s) (Fig. S1D). Using an iterative scheme, the coarse-grained 49 parameters and are selected to match the quasi-static shear strength.

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1.2 Fibril model geometry 52 The geometry of the coarse-grained models is based on full atomistic simulation of mineralized 53 collagen previously reported (2) . Further details on the development of the mineralized collagen 54 fibril full atomistic model can be found in references (2,3)  represent most of the total mineral in bone tissue (5,6) . We create fibril models including up to 60 45% in weight of intrafibrillar mineral, which represents the upper limit of the density naturally 61 found in bone.

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The conversion from full atomistic to coarse-grained is performed in two steps. For the protein  Only the mineral beads within the fibril are kept (Fig. 2).

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The model of the fibril is built to exhibit five gap/overlap regions along its length which The tropocollagen molecule and crystal structure are generated following the protocol described 96 in (8) . We focused on the interaction of the collagen molecule with the (010) surface which is 97 dominant in bone apatite platelets, due to the growth-directing effect of the collagen matrix (9) .

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The surface structure has been assumed stoichiometric as described in (10) since it has been 99 observed experimentally in several studies (11,12) . Lennard-Jones and Coulomb interactions are 100 computed with a switching function that ramps the energy and force smoothly to zero starting 101 with 8 Å and cutting off at 10 Å. To model the interaction between collagen and apatite, we 102 used the extended CHARMM force field as presented in (2,8) . We use SMD for pulling the  (Fig. S2). We also compare strain levels inside the molecule by measuring bond 132 deformation in the backbone of the collagen (Fig. 1B-C). The strain distribution matches 133 qualitatively and quantitatively between the two modalities. The collagen molecule shows a 134 stick slip behavior as observed previously between two collagen molecules (16) . controversial. Some studies proposed that after being formed in the gap region of the fibrils, the 141 crystals continue to grow and penetrate into the overlap region of the collagen fibrils (9,(17)(18)(19) .

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Others suggested that some, if not most of the crystals, must be present between the fibrils, . In a recent study using full atomistic simulation, we showed that the fibril's gap 147 region was able to accommodate at least 40% mineral (2) . In this study, mineral distribution is for strength and ultimate strain (Fig. S3).

Comparison with previous models 196
The results presented in this study complete our previous simulation analysis. When compared 197 to the original full atomistic model, the coarse-grained model gives similar trends for the stress-198 strain curve and gap/overlap length ratio (2) . However, the coarse-grained model is significantly The present study complements our previous analysis of nascent bone using a two-dimensional 205 mesoscale model (33) . Here, we develop a larger finite size sample which can capture the diverse 206 deformation mechanisms taking place during the tensile deformation of the fibril until failure.

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By varying amounts of mineral we were able to explore the role of apatite crystals in collagen's 208 fibril structure.

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Similar to cross-links, the mineral plays the role of a bridging agent that allows the whole 222 structure to take better advantage of a single molecule's properties. The adhesion forces 223 introduced by mineral are weaker than cross-links but reformable since the link is mainly due 224 to electrostatic interactions. These interactions allow a stick-slip deformation process enabling 225 larger energy dissipation compared to cross-linked molecules (Fig. S4B). However, bone tissue 226 contains both mineral and cross-links, as shown in figure 8. High mineral density as represented 227 in this study prevents shearing of the fibril, which is the source of deformation of regime III in 228 cross-linked fibrils (34) . Similarly, high cross-link density prevent the slippage of molecules' 229 termini, which arise during regime II in non-cross-linked mineralized fibrils. Combining cross-230 links and mineral therefore have an influence on the third deformation regime, leading to a 231 lower failure strain. (Fig. S4B). 232 When compared to bone tissue experiments performed at the nanoscale, the fibrils present 233 similar stiffness but higher strength (35,36) . Several reason could explain the discrepancy in 234 strength. First, the model focus on a single mineralized collagen fibril. Compared to bone tissue, 235 the model does not contain extrafibrillar mineral which is likely to play an important role on 236 the tissue mechanics (6,37) . Furthermore, the model has been developed for dry interface between 237 collagen and mineral. Adding some water would reduce the adhesion forces between the two 238 components and would favor their sliding and significantly reduce the strength of the system 239 (38) . Finally, the model created here represent a perfect fibril and does not include any defects 240 that are present in biological samples. 1m/s (14) . Besides, the parameters derived for the model have been extrapolated for quasi-static 259 loading (Fig. S3D). Furthermore, no viscosity has been taken into account in the mesoscale 260 model, which make the model less sensitive to time dependent phenomenon. Indeed, the shear 261 strength between collagen and mineral in the coarse-grained framework does not exhibit 262 significant variation for strain rate ranging from 0.01 to 20 m/s (Fig. S3D). Therefore, we 263 believe that the deformation mechanisms highlighted here are not significantly altered by the 264 large strain rates employed and we expect the results to be in agreement with observation made 265 experimentally at lower strain rate. To limit the effect of viscosity, we assume that the interface 266 between collagen and mineral does not contain water. This assumption is likely to overestimate