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Collagen is a basic biopolymer usually found in animal bodies, but its mechanical property and behavior are not sufficiently understood so as to apply to effective regenerative medicine and so on. Since the collagen material is composed of many hierarchical structures from atomistic level to tissue or organ level, we need to well understand fundamental and atomistic mechanism of the collagen in mechanical response. First, we approach at exactly atomistic level by using all-atom modeling of tropocollagen (TC) molecule, which is a basic structural unit of the collagen. We perform molecular dynamics (MD) simulations concerning tensile loading of a single TC model. The main nature of elastic (often superelastic) behavior and the dependency on temperature and size are discussed. Then, to aim at coarse-graining of atomic configuration into some bundle structure of TC molecules (TC fibril), as a model of higher collagen structure, we construct a kind of mesoscopic model by adopting a simulation framework of beads-spring model which is ordinarily used in polymer simulation. Tensile or compression simulation to the fibril model reveals that the dependency of yield or buckling limit on the number of TCs in the model. Also, we compare the models with various molecular orientations in winding process of initial spiral of TC. The results are analyzed geometrically and it shows that characteristic orientational change of molecules increases or decreases depending on the direction and magnitude of longitudinal strain.

So far, materials engineering and science have mainly focused on the development of strong and tough materials by using relatively hard and rigid substances, such as metal or ceramics [

Collagen is known as a fundamental substance in human and animal bodies. In atomic scale, collagen is composed of narrow molecular chain called tropocollagen (TC), which is prospected as a novel and bio-originated element of materials. The structure of TC has been extensively studied theoretically and experimentally, but there has been little knowledge about mechanical behavior in connection with its relevant hierarchical structure. Therefore, in this study, by modeling structure of TC in atomic scale, we simulate numerically mechanical response of a single fiber of TC (TC fiber) and its bundles (TC fibrils). In particular, we will focus on mechanical flexibility, rigidity and shape-memory effect of TC fiber and fibrils. From experiment and theoretical approach, it is revealed that TC furnishes an unique triple helical structure (i.e. molecular chirality), where three long peptide chains (successively connected amino acid) are uniformly twisted each other and form fibrillar shape.

In this study, we will observe local and molecular twist angle of TCs when tensile or compressive deformation is applied to the structure and we will consider the origin of mechanical properties. So far, it has been a lot of researches concerning the deformation and strength of animal bones and other biological hard-tissue [

The point is that biological tissue or material is providing high performance such as ductility and high-strength using a multi-scale strategy. It is worth investigating to analyze the origin of stability and mechanical response of collagen from atomic scale to meso- or macroscopic scales by building up its hierarchical and multi-scale model, which is also relevant to most synthetic polymers. In considering the hierarchical structure of collagen fibrils [

Before we construct coarse-graining MD model, we report on the research to full-atomic (all-atom) modeling of TC fiber and TC fibrils. It is because the reliability of multi-scale view certainly depends on the smallest scale reality, i.e. atomistic structure and behavior. The all-atom TC model is developed from open digital data of molecular structure, that is, protein data bank (PDB). We will conduct the simulation of all-atom model of single TC fiber by using versatile MD software (NAMD) [

This paper is organized as follows. First, the hierarchical structure of collagen from the stage of microscopic TC molecules is explained and the conceptual ways of numerical modeling are introduced along our study. Secondly, the way of all-atom modeling of TC molecule is shown. The atomistic modeling can express the chirality of peptide chains and their real structure. Then, to exemplify all-atom simulation behavior, tensile and compressive loading tests of the single TC are shown and the mechanical properties obtained there are discussed.

Then, as a main body to study collagen in this study, a single TC molecule and its bundle are modeled by using coarse-graining method. The structures of the multiple fibers are constructed and are analyzed. This leads to further discussion on the behavior of collagen tissues with many hierarchical stages in reality, by scrutinizing mechanical properties such as Young’s modulus (tensile loading case) and buckling limit (compression case) depending on the number of components used those models. The interesting behavior associated with chirality change during deformation is clearly convinced. In particular, twisting and untwisting behavior are found from the simulation and they are mathematically characterized by using general geometrical parameters such as torsion and curvature of the line segments in each peptide changes, by means of the Frenet-Selet’s differential relations.

It has been well known that collagen tissues which are found in animal bodies are one of fundamental biological and structural substances. Collagen constructs hard tissues of animal body such as bone, tendon and cornea. In viewing at microscopic region, it has a fibrillar or rod-shaped form [

It is known that the smallest structural unit of collagen has a molecular size, which is called tropocollagen (TC) [

The interesting point of the structure of TC is that three polypeptide chains are smoothly aligned and holding each other in right-handed helical manner (i.e. triple helical structure is given). In experimental investigation using micrograph, it is also revealed that many bundles of TCs are aligned in their longitudinal direction having an offset of one fourth in the length of TC. It is also assumed that TCs are firmly bound together by cross-linked mechanism, where two neighbor TCs are attached each other near their end regions.

Therefore, as crucial concept in multi-scale modeling of collagen, we should start from atomic scale, but the gap between each size scale is too large to build up straightforwardly. The larger scale directly relevant to organ size would be hard to model by the present atomistic modeling, just due to the limited degrees of freedom treatable in any cutting-edge computation equipment. So, it is also necessary to introduce some coarse-graining modeling which retains characteristics of atomistic- or peptide-level (molecular unit level) scale.

As shown above, collagen tissues are composed of TCs, which all have a fibrillar form. Each TC is constructed from atomic-scale peptide molecules. The purpose of this study is to understand the essential mechanism of mechanical response of TCs. Therefore, it is crucial to look into the structure of single TC from a microscopic scale to a mesoscopic scale which will include collective dynamics of TC fibrils. The best way to do this is that we build a physical model so as to observe the behavior of TC atom-by-atom, or possibly we go down to quantum mechanical level and conduct ab initio modeling and simulations.

So far, atomistic scale simulation of TC molecules in the collagen tissues has been extensively conducted by many researchers. For example, Monti et al. conducted MD analysis of supramolecular structures of collagen and investigated their stability in some solvent liquid molecules [

Solely by atomistic modelling, it is hard to understand the collective nature of TC and collagen fibrils since the gigantic data produced by molecular dynamics or atomistic simulations are to be beyond our control. But, at the same time, such small-sized model should not be thrown away because it has accuracy on dynamical behavior of TC. In such context, anyway, we will try to model TC in all-atom method and to see how it works, first. Simulation of a single TC model is constructed based on all-atom modeling as explained as follows [

Molecular dynamics code we use here is NAMD (Nanoscale molecular dynamics) [

A molecular model of TC is shown in

structural data designated for TC, named “1CAG” [

The computational model of TC having actual length, where “Gly-Pro-Pro” units are repeating until they are polymerized 355 times, is called “Long-TC” model. The length of “Long-TC” model is almost 300 nm that is just the same as that of real collagen fibrils. We will compare results of the “Long-TC” model with those obtained in the “Short-TC” model. The diameters of those two TC models are equal and 1.5 nm. It means two models have different aspect ratio though they are both micro-sized fibers.

Tensile loading simulation is conducted by using steered (or interactive) molecular dynamics (SMD) method [

Property | Unit | Value(s) | |
---|---|---|---|

Short-TC model | Long-TC model | ||

The number of atoms () | - | 1054 | 37,214 |

Length l_{x} × l_{y} × l_{z} | nm | 1.5 × 8.0 × 1.5 | 1.5 × 300 × 1.5 |

Pulling velocity (for SMD) | m/s | 10, 30, 50, 70, 100 | |

Spring constant (for SMD) | N/m | 6.97 | 0.697 |

Temperature T | K | 10, 100, 200, 300, 500 | |

Time increment Δt | fs/step | 1.0 | |

Total steps | - | 100,000 |

In concluding this section, we can confirm sufficient stability of single TC fibril and observe its sufficient resistance against the external tensile loading. Based on these microscopic and fundamental observations and understanding concerning the structure and the dynamics of TC molecule, we would like to move on to a larger-sized modeling, in which we can involve higher-level of structural hierarchy essential in collagen material.

In the context of multi-scale simulation of materials, some coarse-graining methods of atomic simulations have been discussed extensively so far. For example, coarse graining molecular dynamics (CGMD) [

In this method, successive six units of peptide chain, each composed of three bases, Gly, Pro and Hyp in order (in all-atom model, Hyp was technically replaced by Pro, though), are grouped into one large massive bead (all relevant masses are concentrated onto the center of one particle). They are connected each other by inter-particle potential functions (like springs between masses). The potential functions are of intra-molecular and inter-molecular parts similar to all-atom modeling. The intra-molecular potential function represents the interaction inside a molecule and comprises two-body (corresponding to stretch and contraction) and three-body (corresponding to bending) functions. The inter-molecular potential function represents cross-linking (CL) between ends of collagen fibers, as well as ordinary inter-molecular interaction (those by Van der Waals or electrostatic bonding). For inter-molecular interactions, conventional Lennard-Jones (LJ) type function form is adopted. These potential functions are shown in Equations (1)-(4). The potential parameters of TC molecule we use here were suggested in the literature [

ϕ s t r = 1 2 k s t r ( r − r 0 ) 2 (1)

ϕ b e n d = 1 2 k b e n d ( θ i j k − θ 0 ) 2 (2)

ϕ L J = 4 ε { ( σ r ) 12 − ( σ r ) 6 } (3)

ϕ C L = β ϕ L J (4)

In the formulation of LJ-type potential, as usual, ε indicates a binding energy and σ corresponds to inter-particular equilibrium distance when interaction force changes from the repulsive to the attractive. In the intra-molecular functions, k s t r and k b e n d work as spring constants for the change of elongation ( r − r 0 ) and three-body angle θ i j k , respectively, where r 0 and θ 0 are equilibrium two-body distance and three-body angle, respectively. In the CL potential, β is an arbitrary constant for cross-linking (here we use β = 12.5 referring former studies [

The actual method to build up the structure of TC and its fibril is shown in _{x} = l_{y}). This is a basic model in this study and it is named a single fiber of TC. Then, several TCs are cross-linked and forms fibrillar structures. In arranging TC fibers, they are placed each other with an equilibrium separation, l_{int} not to produce strong force between TCs. In the parallel direction to the fibril axis, adding an offset distance D, TCs are placed in staggered manner, like real aggregates of TC fibrils as mentioned above. This model realizes a higher size of the structure of TC, and we would like to call it FTC (fibril of tropocollagen) model. In any models, periodic boundary condition is imposed just on the axial direction of TCs (in the direction of l_{z}) in order to exclude end-effect in stretching or contracting deformation. The number of TCs are from 2 to 16, and, for instance in the case 16 TCs, they are arrange 4 × 4 in staggered manner onto xy plane in parallel to z direction as shown the figure.

In order to discuss the effect of molecular orientation (MO) on properties, we modify the basic model by changing the pitch length in making helix of each TCs. All those models have different angles of chains with regard to longitudinal (loading) direction of the fiber. Those angles can be called orientation of molecules as for fibril direction, so we name those models orientation-angled fiber of tropocollagen (OFTC) models. Those models are specified by the single parameter, orientation angle φ, as shown in

Property | Unit | Value(s) |
---|---|---|

The number of particles (beads) | - | 1554 - 12,432 |

The number of TCs | - | 2 - 16 |

Length of single TC l_{z} | nm | 299 |

Length of TC fibril l_{z} | nm | 366 - 1304 |

Intermolecular distance l_{int} | nm | 2.97 |

Longitudinal gap D | nm | 6.7 |

Cross-link factor β | - | 12.5 |

Tensile/compressive strain rate | 1/s | 5 × 10^{7} |

Temperature T | K | 10, 100, 200, 300 |

Orientation angle φ | deg. | 0 - 35 |

Time increment Δt | fs | 1.0 |

clearly exhibit different behavior. The specimen seem to cause totally twist of TC structure.

In compressive loading, stress-strain relation shows a sudden drop. It is assumed that buckling event occurs at this point.

Let us consider the effect of the orientation of molecules inside TC fibril on loading behavior, by using OFTC models which are built with different molecular orientation (MO) angles inside TC fibril.

Likewise tensile case, the smaller the MO angle φ is, the larger the buckling stress become. In TC fibers, when the MO angle is small, all the molecules tend to align to longitudinal direction. In this situation, the directions of covalent bonding which will produce the largest bonding energy among interactions prone to coincide with tensile or compressive axis. Consequently, for smaller φ, the resistance against axial loading becomes larger.

Chirality can be defined as twist angle along a TC molecule. In OFTC models, each of line segment connecting neighbor beads is continuously identified along respective peptide chains. When three peptide chains around the outer surface of that cylinder is recognized, the angle with respect to the center line of the cylinder is regarded as the orientation angle. By calculating these angles in three dimension space, we will be able to evaluate the degree of local twist of molecular chain inside TC structure. We try to use a mathematical handling by differential geometry as follows. A set of equations shown in Equation (5) are known as Frenet-Serret formulas [

∂ N / ∂ s = − κ T + τ B , ∂ N / ∂ s = κ N , ∂ B / ∂ s = − τ N , (5)

where the vector T is tangential to the connecting vector between neighboring particles (unit tangent vector). The vector N is always perpendicular to T (unit principal normal vector), and the vector B (unit binormal vector) is perpendicular to both T and N. And scalars κ and τ are called curvature and torsion, respectively. These scalar factors, κ and τ, are calculated from positions of particles inside chiral chains in our MD simulations. We can propose that they quantitatively estimate the degree of chirality (twisting) and its change for respective chains.

In particular, value of torsion τ indicates a degree of twist of the three-dimensional curve. We will monitor the change of τ, or alternatively (1/τ) which means torsion radius, during deformation.

Δ ( 1 / τ ) = 1 / τ ( ε ) − ( 1 / τ ( 0 ) ) ,

where τ ( 0 ) and τ ( ε ) are torsion values before deformation and at deforming. Note that all plots in

Initial values of torsion radius 1 / τ ( 0 ) are not equivalent because molecular orientation (MO) angles in making each models are different. Moreover, these radii also alter during deformation. Overall trend is that torsion radius increases for tensile loading and decreases for compressive loading. It is understood that tensile deformation of the specimen helps all molecules to align in parallel direction

to the tensile axis and the torsion value τ is prone to decrease during that aligning. On the other hand, the torsion value τ increases during compressive deformation. That is because positions of beads tend to move in circumferential direction rather than in longitudinal direction during compression. This mechanism is also valid for buckling in compression though it occurs suddenly.

As shown in strain range ε = ± 0 ~ 0.05 in the graphs of

In this study, mechanical properties of collagen are studied from the simulation of a microscopic and biological polymer material called tropocollagen (TC) by using atomic simulations constructed with multi-scale point of view. The simulations are two-fold: the first one is constructed by using all-atom modeling and molecular dynamics (MD) method, where we observed a precise behavior and mechanical properties of a single TC molecule having a unique triple-helical structure made up of polypeptide chains. Secondly, as main simulation method, a TC molecule and its fibril are modeled by adopting coarse-grained framework to MD simulations, retaining chain and cross-linking structures by using beads-spring model and their force field. Elastic moduli and buckling limits are compared as for the size of the fibril and for temperature conditions. Besides, it is recognized that molecular chirality (geometrically evaluated as torsion around the axis of tensile or compressive loading) of each peptide chains related closely to mechanical state of total TC fibril. If a coarse-graining method can be directly built upon the results obtained all-atom modeling and can be connected larger continuum-level analysis like finite element analysis (FEA), such simulation will clarify the total mechanics of collagen material in all sizes. We could make the basis for that.

This work is partly supported by Nippon Steel Corporation. The authors acknowledge their support here and appreciate it. The authors also appreciate those people who helped us to make computation model and computation results as former students in our laboratory: T. Shirahana, T. Imanishi and T. Suzuki, Faculty of Engineering Science/Graduate School of Science and Engineering, Kansai University.

The authors declare no conflicts of interest regarding the publication of this paper.

Saitoh, K.-I., Sato, T., Takuma, M. and Takahashi, Y. (2020) Molecular Dynamics Study of Collagen Fibrils: Relation between Mechanical Properties and Molecular Chirality. Journal of Biomaterials and Nanobiotechnology, 11, 260-278. https://doi.org/10.4236/jbnb.2020.114017