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Mechanics of collagen fibrils: A two-scale discrete damage model
Author’s Accepted Manuscript Mechanics of collagen fibrils: a two-scale discrete damage model Kevin Linka, Mikhail Itskov
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S1751-6161(15)00354-9 http://dx.doi.org/10.1016/j.jmbbm.2015.08.045 JMBBM1622
To appear in: Journal of the Mechanical Behavior of Biomedical Materials Received date: 27 May 2015 Revised date: 14 August 2015 Accepted date: 18 August 2015 Cite this article as: Kevin Linka and Mikhail Itskov, Mechanics of collagen fibrils: a two-scale discrete damage model, Journal of the Mechanical Behavior of Biomedical Materials, http://dx.doi.org/10.1016/j.jmbbm.2015.08.045 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Mechanics of collagen fibrils: a two-scale discrete damage model Kevin Linka∗ and Mikhail Itskov Department of Continuum Mechanics, RWTH Aachen University Kackertstr. 9, 52072 Aachen Germany ∗ [email protected]
Abstract Collagen is one of the most important structural proteins of biological tissues. Recently, a new damage phenomena caused by a mechanical overloading of the tendon has been revealed on the collagen fibril level by means of scanning electron microscopy [39]. In order to describe this phenomena we propose in the present paper a constitutive damage model for single collagen fibrils. It utilizes a statistical framework in order to incorporate a physically motivated transition from the entropic- to the intrinsic-elasticity range of single tropocollagen molecules. In this two-scale approach a bridging relation between the tropocollagen and fibril level as well as a failure criterion after exceeding a critical stretch limit are introduced. The final constitutive relations are validated against experimental data available in literature for both, the reversible and the irreversible, stress-strain response. Keywords: collagen, helical molecules, discrete plasticity, soft biological tissues 1. Introduction Collagen fibrils are the main structural contributor to the strength and integrity of the extracellular matrix (ECM). They are assembled by a certain amount of covalent cross-linked tropocollagen (TC) molecules and form a very stiff network. Since the last decade much investigation towards understanding, prediction and numerical simulations of connective soft biological tissues has been carried out. Many new insights ranging from the macromolecular to the tissue/organ level have been reported. In particular, a strong
Preprint submitted to JMBBM
September 18, 2015
dependency between the hierarchical arrangement of collagen and mechanical properties of the tissue has been observed. The diversity in hierarchical arrangements of collagen is hereby seen as one of the main microstructural sources for the high variety of mechanical responses of individuals. This correlation between hierarchical collagen arrangement and mechanical properties opens new opportunities in modern medicine. For instance, a small amount of tissue obtained by a needle biopsy could deliver sufficient histological information for a patient specific material model. By this means, procedures like for example macroscopic in vivo testings used to specify phenomenological constitutive models could become dispensable. First studies of collagen fibrils and their mechanical properties were based on nano-scale experiments and molecular dynamics simulations [7, 25]. Thus, two models capturing the entropic and energetic part of the molecular response, have been proposed so far. Both concepts make use of the wormlike chain (WLC) model, which has successfully been applied to DNA molecules [28]. However, the transition zone from the entropic to the energetic regime is not physically described there. Furthermore, the models lack an exact description of the helical TC arrangement and thus describe the intrinsic elasticity of the helix in a phenomenological manner. In a further paper by Buehler [7], yielding beyond a critical stress is described. This approach has further been used in a continuum model by Tang et al. [36]. They separately considered collagen fibrils and the matrix part of a soft tissue and utilized an invariant based description of the strain energy function. A further crucial issue in the context of biological tissues is irreversible damage on the micro-level, often induced by cyclic loading. Recently, Veres et al. [39] reported breakage at discrete locations along single collagen fibrils after overstretching a steer tail tendon. The main source of this overload induced damage, referred by the authors to as discrete plasticity, is thought to be permanent opening of the TC triple helix and its transformation in a stable denaturated state. To the best of our knowledge, the irreversible response of the tendon caused by the discrete plasticity phenomenon of single collagen fibrils has not been addressed so far in the constitutive modeling. Only molecular dynamic (MD) simulations [13, 14], investigating the uncoiling of single TC molecules under different strain rates have been reported. Rupture of intramolecular hydrogen bonds due to uncoiling and straightening of the helical structure of collagen molecules has been predicted. Nevertheless, it is still in question, whether this mechanism is due to the discrete plasticity phenomena explored by Veres 2
et al. [39]. Following the same idea, a fibril model with linear elastic components has recently been proposed [26]. Damage is included by activation functions motivated by inter- and intramolecular structural changes. A primary focus of this work is thus to develop a fully histology based constitutive model of single collagen fibrils. The model should additionally disclose possible reasons for the high patient specific variety of the mechanical properties. A further objective of the present work is to incorporate structural changes caused by overstretching into the constitutive framework. As a result, a constitutive model capturing the mechanics of single TC molecules inside a cross-linked structure of collagen fibrils will be proposed. We begin with energetic relations for a single TC and apply a strain amplification concept to capture the native assembly of TCs forming collagen fibrils (Chap. 2). Then, in Chap. 3 the TC model is utilized to derive a strain energy function for a single fibril, where a statistical distribution of helix lengths is taken into account. Further, the model is extended to the case of irreversible deformations occurring due to overstretching of TC molecules. Chap. 4 and 5 are devoted to the derivation of the final constitutive equations and thermodynamic consistency of the model. Finally, in Chap. 6 model predictions are validated against experimental data available in literature for single collagen fibrils and irreversibly deformed tendon. 2. Mechanics of a single tropocollagen 2.1. Kinematics of a single tropocollagen Tropocollagen represents a triple helical arrangement of three coiled collagenprotein chains, linked together by an certain amount of hydrogen bonds. The protein strands are built up by three repeating protein bonds. For example, for a collagen-Type-I protein chain this is a Poly(Prolyl-HydroxyprolylGlycyl). In their origin environment TC molecules are assembled in a Dperiodical manner into so called light and dark zones, where the light zones correspond to a molecular gap in the packing [12]. In those gap regions the collagen molecules are in a disordered arrangement. Under tension molecules become aligned and start to straighten out. During this process, the molecular disorder reduces while the entropic force increases until the extended helix length Leh is reached, see Fig. 1. This observation agrees well both with the atomic force microscopy (AFM) [2] and the optical tweezers treatments [7]. The entropic behavior was additionally confirmed by a fitting of the WLC model [24, 28] to the observed experimental data. 3
Under increasing deformation the helix becomes uncoiled and the single protein strands stretch. This is referred to as energetic or intrinsic elasticity range. On the fibril level, this region is additionally characterized by a sliding between single TCs in the molecular packing [12]. This leads to a fuzzy pattern of the fibril, which in turns implies different lengths of single TCs in a fibril. However, the exact coupling between the twisting and stretching in this domain is still an open issue. In stretching tests of DNA one was able to attach a fluorescent avidin-coated rotor bead to the DNA molecule [15]. In this manner, the twist due to stretching by the rotation of these fluorescent beads was monitored. Due to similarities with DNA, we can assume a similar or even the same behavior for TCs. The third typical range in the force-extension behavior of TCs is mainly influenced by the applied boundary conditions. For simultaneous pulling on all three polypeptide strands, the MD simulations reported [21, 6] a very stiff response due to backbone stretching. In further MD simulations [37] closer reproducing native environments of TCs, a single strand of two crosslinked TCs was pulled. In this case, a plateau region of the force-extension diagram was observed. This plateau becomes larger if a strand is not directly connected to the cross-link and smaller for a strand with a direct coupling. 2.2. Strain energy decomposition of a tropocollagen molecule As discussed in the previous section, the strain energy in the low force regime of a TC arises from the thermally induced entropic elasticity. It has been shown [2, 5], that this energy can well be captured by the classical worm-like chain (WLC) model [24]. Accordingly [11, 44] Leh Ewlc = kB T
lp 2
∂t(s) 2 ∂s − Fwlc cos (Θ(s)) ds,
(1)
0
where the total energy of bending deformation due to thermal fluctuations is described by a vector t tangent to the arc length s of a filament subjected to an external force Fwlc . In addition, it is made use of the persistence length lp , a quantity displaying the bending stiffness. Using optical tweezers this parameter was determined for type I collagen to be 14.5 ± 0.5 nm [34, 31]. Further, Leh denotes the extended helix length while kB and T stand for the Boltzmann constant and the absolute temperature, respectively. Finally, the condition Leh lp as well as the inextensibility constraint t = 1 have to 4
be fulfilled. A force-extension relation resulting from (1) can be expressed in particular by the interpolation function by Marko and Sigga [28] as −2 rtc rtc kB T 1 1 Fwlc (rtc , Leh ) = 1− − + , (2) lp 4 Leh 4 Leh where the current end-to-end distance of a TC is given by the affine formulation rtc = Rtc λtc
(3)
in terms of the reference end-to-end distance Rtc and the TC-stretch λtc . In a higher force regime, when the external force is sufficient to suppress thermal fluctuations, the potential due to the intrinsic elasticity of the TC can be given by [23, 27] Πhx =
1 C 2 1 S ∗2 g θ ∗ ∗ θ + r + r − Fhx rtc , 2 Leh 2 Leh tc Leh tc
(4)
∗ where rtc = (rtc − Leh ) denotes the elongation after reaching the extended helix length Leh . Eq. (4) results from the linear theory of double stranded DNA and in addition to the potential of external force Fhx includes three terms necessary for a unique description of DNA [15, 17]. The first term is due to the twisting rigidity of a helix with the twist rigidity C, where θ denotes the winding angle. The second term describes the stretching of single protein chains with the stretch modulus S and the third term is a twiststretch coupling relation with the factor g. Using magnetic tweezers, Gore et al. [15] experimentally studied the double stranded DNA by applying a rotor bread tracking technique in order to determine g and even figured out a functional expression for the twist stretch coupling term. This expression has been incorporated to (4) by Gross et al. [17]. However, those experimental interventions are still missing for collagen. For this reason g will be treated here as a constant parameter obtained by solving the inverse problem. The force extension relation for the intrinsic elasticity part can be obtained by minimizing the potential (4) with respect to Rtc and θ, which yields
SC − g 2 (rtc − Leh ) . (5) C Leh The first fraction in this relation can be expressed in terms of the effective Fhx (rtc , Leh ) =
5
Young’s modulus of a TC in the energetic regime as SC − g 2 , (6) C Atc where the cross-section of a tropocollagen molecule is denoted by Atc . The bending and twist rigidity appearing in (2) and (5), respectively, can be connected to each other by the coiled coil theory for an elastic rod [29, 30, 43]. Accordingly, the following relation has been obtained [29] Etc =
C ≈2 kB T lp
(7)
by utilizing the Cosserat material frame to represent the three single protein strands of a TC molecule and equilibrium boundary conditions and applying condition. Eq. (7) allows us to reduce the number of material parameters. 2.3. Strain amplification Tropocollagen molecules receive their global stability by covalent cross-linking initialized by specific lysine and hydroxysine residues [22]. The cross-links are attached at the ends of the TCs, although the exact 3D spatial pattern of cross-linking bonds is still poorly understood [9]. In contrast, it is well known that the cross-link density uniquely determines the tensile strength of collagen fibrils [20, 18]. However, there are two kinds of immature, divalent cross-links types, which are thought to be strain-labile [39]. Recently, those cross-link types have been treated with NaBH4 in order to stabilize them and study their effect on mechanical behavior at the fibril and tissue level [39, 35]. It turned out that this treatment does not alter at least the mechanical response of single collagen fibrils. Even the discrete failure along single fibrils was not affected. The mechanical response of TC molecules spanned between cross-link connections is much weaker in comparison to the covalent bonded cross-links. Under even supra-physiological tension, first the TC elongates and intrahelical hydrogen bonds rupture before the cross-links considerably deform.This was also reported by Sasaki and Odajima [33], who revealed that the main elongation mechanism is the strain in the helix pitch. Thus, for the modeling purpose, we assume that the cross-links are ideally rigid and do not contribute to the strain energy of the system. They are considered to be responsible only for the internal integrity and load transmission between the TC molecules.
6
The kinematical compatibility of the TC molecule in the reference and stretched state can thus be expressed by (see Fig. 2) (Lcl + Rtc ) λfb = Lcl λcl + Rtc λtc ,
(8)
where Lcl denotes the clamped part length of the TC molecule, while λcl is the corresponding stretch. Under the above assumption of the rigid cross-links λcl = 1, so that Lcl λfb − c , c= , (9) 1−c Rtc + Lcl where λfb is referred to as the fibril stretch. Accordingly, the fibril response becomes stiffer with the increasing cross-link density described by a parameter c considered as a constant. Note that (9) describe the strain amplification well known for filler reinforced rubbers [3, 4, 19]. λtc =
3. Mechanics of a single collagen-fibril 3.1. Elastic response of the fibril From the mechanical perspective a collagen fibril can be considered as a network of parallel aligned TCs, bonded together by cross-links. Initially, the TCs are in a disordered state due to their molecular kinks. A TC can be in an entropic or energetic energy state, depending solely on its current endto-end length rtc . Keeping in mind, that the cross-links do not contribute to the strain energy density, the latter one can be represented in terms of an entropic Ψs and energetic Ψe energy part as Ψfb = Ntc (Ψs + Ψe ) ,
(10)
where Ntc denotes the number of TCs inside a collagen fibril. By virtue of (2) and (5) these energy contributions can further be expressed by 1 Ψs (rtc , Leh ) = Vtc Ψe (rtc , Leh ) =
1 Vtc
rtc
rtc Fwlc d¯ rtc =
Rtc rtc
fwlc d¯ rtc ,
(11)
fhx d¯ rtc ,
(12)
Rtc rtc
Fhx d¯ rtc = Rtc
Rtc
7
where Vtc = Atc Leh
(13)
denotes the reference volume of a single TC and the force densities are exF(•) pressed by f(•) = , respectively. Vtc To account for the energy state change of a TC molecule under tension, a transition criteria is needed. Bozec and Horton [2] detected this at the transition force Ftr by discontinuities between the entropic and energetic regime in the force-extension responses of individual TC molecules. Let Ltr = Leh /ξtr be the corresponding length of the TC molecule, where ξtr > 1 denotes the transition ratio, visualized in Fig. 3 and discussed in the following. The transition from the entropic to the energetic state takes place when rtc =
Leh . ξtr
(14)
Thus, the transition force can be expressed by (2) as (see also Fig. 3) Ftr = Fwlc (Ltr , Leh ).
(15)
Now, we can formulate a strain energy function of a collagen fibril accounting for the energy transition between the entropic and energetic energy state. Due to the strong periodical assembly of fibrils observed in experiments [22], we can assume for modeling purposes that all TCs have the same initial end-to-end distance Rtc . In contrast, they can have a different extended helix length Leh . This observation is based on topographical interventions reported in [2]. There, a probability distribution function (PDF) of the extended helix lengths, denoted in the following by P (Leh ) was proposed. TCs shorter with respect to their extended helix length first pass over to the energetic regime. This is due to the fact, that all TCs are subject to the same stretch, but have a different length in the extended helix state. Certainly, the interval of activated TCs in the entropic regime Lseh decreases due to reaching the transition length Ltr by individual molecules, while the energetic regime increases Leeh (see right hand side of Fig. 4). This can be expressed in view
8
of (14) as
min (λtc ) = Leh Le ≤ Leh ≤ rtc ξtr ,
max s Leh rtc ξtr ≤ Leh ≤ Leh , Leh (λtc ) = Leeh
(16) (17)
where the minimal length in the system is set to be constant accordingly to (14) as min
Le = Rtc ξtr .
(18)
This ensures that the set Leeh is empty in the reference configuration. For ¯ tr = rtc ξtr . The strain energy the sake of simplification we further denote L function for a single collagen fibril can thus be expressed by ⎡ ⎤ ⎢ ⎥ Ψfb = Ntc ⎣ P (Leh )Ψs (rtc , Leh ) dLeh + P (Leh )Ψe (rtc ξtr , Leh ) dLeh ⎦, Lseh (λtc )
Leeh (λtc )
(19) where Ψs and Ψe are given by (11) and (12), respectively. The successive transition of TCs from the entropic to the energetic state under tension reduces the entropic energy contribution. Thus, at higher stretches the energetic response becomes dominant and (19) predicts the typical linear stiffening of the stress-strain response. 3.2. Micro-damage induced by cyclic loading Now, we are going to incorporate a damage formulation into the constitutive model described by the strain energy (19). Based on mechanical testings of overstretched steer tail tendon, Veres and colleagues [42, 39, 40, 38] revealed an irreversible change of the fibril structure. Employing scanning electron microscopy, they showed a kinked and even fibrillated structure at discrete locations along single collagen fibrils. They referred this phenomenon to as discrete plasticity and explained it by the permanent opening of the triplehelical structure of TC molecules once the helical structure is fully uncoiled and the single strands have reached their contour length (see Fig. 5). According to the experimental results [39] this micro-damage effect happens at discrete locations with irregular distances inbetween and only after gaining the uncoiled configuration of a TC. This observation is in line with the prob9
ability distribution of the extended helix lengths, since shorter TCs pass over to uncoiled state first and thus fail first as well. For the modeling purpose, we define the failure stretch relative to the extended helix state by λcr tc =
Lc ν, Leh
(20)
where ν denotes a failure stretch ratio (see also Fig. 6). This is an additional amount of stretch necessary to cause a permanent deformation to a TC molecule, once it has reached the uncoiled configuration (Lc ). The stretch needed to uncoil a TC from the extended helix state is described by the fraction in (20). Since in the natural state the TCs are in the helical conformation, the condition λcr tc > 1 has to be fulfilled. Once the shortest TC is , the minimal available extended helix lengths increases overstretched by λcr tc min
and thus the lower limit of the energetic interval Le does as well. Thus, we can write min
Le = Rtc ξtr max [λtc,m − λcr tc + 1, 1] ,
(21)
where λtc,m denotes the maximal TC stretch ever reached in the system. The underling model assumption of (21) is, that once a TC is stretched over the critical value λcr tc , it does not contribute more to the energy of the system. By this means the damage irreversibility is captured. min
Moreover, accordingly to (21) Le remains constant for λcr tc ≥ λtc,m , which is exactly the physiological loading range. In this range, all deformations min
are reversible up to λtc,m = λcr tc , applying any further loads increases Le and permanent damage starts. Hence, the irreversible limit of the transition ¯ tr cannot return to its origin and thus can be given as length L min ¯ (22) Ltr = max rtc ξtr , Le . 4. Constitutive modeling
Based on the strain energy (19), a constitutive equation for the fibrils can be derived. To this end, we will treat the fibril as a one-dimensional system, since its geometrical dimensions in the axial direction are much bigger than that ones of the cross-section. Taking the incompressibility condition into account, the constitutive equation for the fibril stress tfb can be written as 10
∂Ψfb − p λfb −1 , (23) ∂λfb where p stands for an arbitrary scalar parameter which can be obtained for a particular boundary value problem. Inserting (19) into (23) we thus obtain tfb =
⎡ ∂Ψfb ∂λtc ⎢ = Ntc Rtc ⎣ ∂λfb ∂λfb
P (Leh )fwlc dLeh +
Lseh (λtc )
P (Leh )fhx dLeh
Leeh (λtc )
¯ tr )Ψs (rtc , L ¯ tr ) , − ξtr P (L
(24)
where the latter part arises from the derivative of the integration limits due to the Leibniz integral rule. Furthermore, it is made use of the following identity ∂λtc 1 , = ∂λfb 1−c
(25)
resulting from the strain amplification concept (9). Assuming the tendon response results from the sum of single fibril contributions, we obtain in view of (23) the tendon stress ttd as ttd = Nfb tfb ,
(26)
where Nfb denotes the the number of fibrils inside a tendon. The assumption underlying this constitutive equation is, that the fibrils are not connected to each other and thus the existence of the proteoglycans is not accounted for. Their influence will be considered in further work and reported elsewhere. 5. Thermodynamic consistency By means of the internal variable λtc,m , appearing in (21) the strain energy Ψfb can be rewritten by ˆ fb (λfb , λtc,m ) . Ψfb := Ψ
(27)
Accordingly, the Clausius-Duhem inequality resulting from the second law of thermodynamics can be reduced to
11
−
∂ Ψfb λ˙ tc,m ≥ 0. ∂λtc,m
(28)
Under loading and reloading λtc,m = 0, while λtc,m > 0 in primary loading. Thus, (28) is satisfied if during the primary loading ∂Ψfb ≤ 0. ∂λtc,m
(29)
By using (19) this inequality can be expressed by ¯ tr )Ψe (rtc ξtr , L ¯ tr ) ≤ 0, −Rtc ξtr P (L ¯ tr ) = 0. which holds since Ψe (rtc ξtr , L
(30)
6. Experimental evaluation 6.1. Probability distribution of the extended helix length Up to now, we have not discussed a particular form of the PDF in (19). This distribution for the extended helix length was exposed by topographical interventions by Bozec and Horton [2]. They proposed the Gaussian distribution function which agrees well with the histogram data. Instead, we propose to use the Beta probability distribution function (PDF) of the following form (Leh − Rtc )α−1 P (Leh ; α, β) = α+β−1 max B(α, β) Leh − Rtc
max
Leh − Leh
β−1 ,
(31)
max
where Leh denotes the maximal available extended helix length in the system and α and β represent the shape parameters. Further, B(α, β) is referred to as the beta function. All four parameters of the distribution (31) are obtained by a fitting against the PDF of [2] on the basis of the least-square method (see Fig 7). The values of these parameters are given in Table 1. The Beta PDF is defined over a limited interval and thus appears here to be advantageous in comparison to the normal distribution. By keeping α and β constant, we can utilize only the reference length Rtc and the maximal max
extended helix length Leh for later fitting purposes. These two quantities 12
defining the interval of the Beta distribution can be evaluated from histological data. To reduce the number of material parameters, we have additionally max
fixed the upper bound of the extended helix length Leh in accordance with the above fitting result and with a value reported in the literature [1, 34] to 310 nm. Hence, for fitting purposes only the reference end-to-end length Rtc is exploited. 6.2. Evaluation of the elastic response In this section, the constitutive model (23) is applied to two different sets of experimental data. These sets are obtained under physiological loading conditions of two single collagen fibrils. The fibrils were isolated from the sea cucumber and hydrated during uniaxial tension loading in a microelectromechanical system device [8]. The two tested fibrils had a significant difference in their dimensions, fibril (A) had a length of 20 μm and an average cross-section of 0.71 μm2 , while another fibril (B) had a length of 5 μm with an average cross section of 0.043 μm2 . For more details about the exact experimental method, the interested reader is referred to [8]. For the fitting of the constitutive model (23) to the experimental data of the two fibrils [8], the same material parameter set for TC molecules was used. Due to the repeating protein bond structure of the single polypeptide strand of a TC, material parameters as the transition ratio ξtr , the stretch modulus S and the coupling constant g are nearly the same for different kinds of fibrous tissues. The remaining material parameters, namely the reference end-to-end length Rtc and the number of TCs inside a fibril Ntc , have been adjusted to the experimental data sets for the fibrils A and B by a fitting procedure based on the least-square method. Since these two material parameters depend on the packing density of fibrils, thus they can differ for different individuals or kinds of tissues. The experimental results and model predictions are plotted as force stretch diagrams in Fig. 8. The corresponding material parameters for the TC molecules are given in Table 2, along with the the material parameters for the individual fibrils. Note, that the values for the TC molecules agrees especially well to the range of results reported in [2] and [33]. Further validation of our constitutive model of a single fibril against experimental data, can be derived from Fig. 9. There, the separate force contributions of the entropic and energetic part are compared to the absolute response of fibril A. As mentioned before, the entropic response is dominating in the low force range, 13
while the energetic contribution is negligible. This behavior is vice verse in the high force region [5, 2], which is in line with our results. 6.3. Evaluation of the inelastic response Finally, we validate predictive capabilities of the proposed irreversible damage model (21). Since uniaxial loading-unloading data about the inelastic stress-strain response of collagen fibrils, especially data reporting the phenomena of discrete plasticity are rare, we employed an experimental data set from the whole tendon. These data correspond to the steer tail tendon. There, the occurrence of micro-damage along the fibrils has been verified by SEM interventions after stretching the samples [39]. Although collagen fibrils form only a substructure of the tendon, from the mechanical point of view they represent the most important constituent [41, 12, 22, 16]. In comparison to the elastic formulation, one additional material parameter for the prediction of the inelastic stress-strain response, namely the critical TC stretch λcr tc is needed here. The material parameters for the inelastic tendon response obtained by fitting to the experimental data are given in Table 3. The TC material parameters S, g and ξtr were kept constant as obtained from the previous fit (see Table 2). The predicted stress-strain curve is plotted in Fig. 10. We observe good agreement between the model prediction and the experimental data except of the unloading path. This is due to the fact that some constituents of tendon like the proteoglycans are not considered in the present approach, but have a certain impact on the mechanical response [32, 10]. 7. Conclusion In this paper, we proposed a constitutive model of cross-linked tropocollagen molecules. The TC description is addressed by taking into account the wormlike chain model for the entropic elasticity range and a relation of the triple helix for the intrinsic elasticity part. The transition between the different energy ranges is physically motivated and described by a split integral relation (19). Further, a micro-damage mechanism has been incorporated in the fibril framework, which allows us to predict the irreversible structural change of TCs. The model includes only few material parameters for TC molecules and demonstrates good agreements with experimental data for two different collagen fibrils and for an inelastic steer tail tendon.
14
We are confident that the material parameters of the present model are sufficient to allow the prediction of patient specific material behavior. Indeed, one major reason for the individuality of the fibril responses is most likely the packing density of a fibril, which is captured in the proposed model and expressed in terms of the amount of TCs inside a fibril Ntc and the reference molecule length Rtc . Furthermore, these parameters could be identified by the electron microscopy and X-ray diffraction. Despite the fact, that the model predictions are well in line with the experimental data, there are still some open issues. For instance, in our approach the upper bound of the PDF was fixed and only the lower one was varied, according to values proposed in the literature [1]. Also the handling of the failure ratio ν, has further to be studied, since there are no experimental data available to estimate this parameter. For this reason, further experimental interventions, accounting for both, quantitative histological data and corresponding stress-strain data would be of great benefit. References [1] K. Beck and B. Brodsky. Supercoiled protein motifs: the collagen triplehelix and the alpha-helical coiled coil. Journal of structural biology, 122:17–29, 1998. [2] L. Bozec and M. Horton. Topography and mechanical properties of single molecules of type I collagen using atomic force microscopy. Biophysical journal, 88(6):4223–31, 2005. [3] F. Bueche. Molecular Basis for The Mullins Effect. Rubber Chemistry and Technology, 34(10):493–505, 1961. [4] F. Bueche. Mullins effect and rubber-filler interaction. Rubber Chemistry and Technology, 35(2):259–273, 1962. [5] M. J. Buehler. Nature designs tough collagen: explaining the nanostructure of collagen fibrils. Proceedings of the National Academy of Sciences of the United States of America, 103(33):12285–90, 2006. [6] M. J. Buehler. Nanomechanics of collagen fibrils under varying cross-link densities: atomistic and continuum studies. Journal of the mechanical behavior of biomedical materials, 1(1):59–67, 2008. 15
[7] M. J. Buehler and S. Y. Wong. Entropic elasticity controls nanomechanics of single tropocollagen molecules. Biophysical journal, 93(1):37–43, 2007. [8] S. J. Eppell, B. N. Smith, H. Kahn, and R. Ballarini. Nano measurements with micro-devices: mechanical properties of hydrated collagen fibrils. Journal of the Royal Society, Interface / the Royal Society, 3(6):117–21, 2006. [9] D. R. Eyre, M. A. Weis, and J. J. Wu. Advances in collagen cross-link analysis. 45(1):65–74, 2009. [10] G. Fessel and J. G. Snedeker. Evidence against proteoglycan mediated collagen fibril load transmission and dynamic viscoelasticity in tendon. Matrix biology : journal of the International Society for Matrix Biology, 28(8):503–10, 2009. [11] M. Fixman and J. Kovac. Polymer conformational statistics. III. Modified Gaussian models of stiff chains. The Journal of Chemical Physics, 58(4):1564, 1973. [12] P. Fratzl, K. Misof, I. Zizak, G. Rapp, H. Amenitsch, and S. Bernstorff. Fibrillar structure and mechanical properties of collagen. Journal of structural biology, 122(1-2):119–22, 1998. [13] A. Gautieri, M. J. Buehler, and A. Redaelli. Deformation rate controls elasticity and unfolding pathway of single tropocollagen molecules. Journal of the mechanical behavior of biomedical materials, 2(2):130–7, 2009. [14] A. Gautieri, S. Vesentini, A. Redaelli, and M. J. Buehler. Hierarchical structure and nanomechanics of collagen microfibrils from the atomistic scale up. Nano Letters, 11:757–766, 2011. [15] J. Gore, Z. Bryant, M. N¨ollmann, M. U. Le, N. R. Cozzarelli, and C. Bustamante. DNA overwinds when stretched. Nature, 442(1):836–839, 2006. [16] J. S. Graham, A. N. Vomund, C. L. Phillips, and M. Grandbois. Structural changes in human type I collagen fibrils investigated by force spectroscopy. Experimental cell research, 299(2):335–42, Oct. 2004.
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[17] P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite. Quantifying how DNA stretches, melts and changes twist under tension. Nature Physics, 7(9):731–736, 2011. [18] P. Hansen, T. Hassenkam, R. B. Svensson, P. Aagaard, T. Trappe, B. T. Haraldsson, M. Kjaer, and P. Magnusson. Glutaraldehyde cross-linking of tendon–mechanical effects at the level of the tendon fascicle and fibril. Connective tissue research, 50:211–222, 2009. [19] J. a. C. Harwood, L. Mullins, and a. R. Payne. Stress Softening in Natural Rubber Vulcanizates. Part II. Stress Softening Effects in Pure Gum and Filler Loaded Rubbers. Rubber Chemistry and Technology, 39:814–822, 1966. [20] R. C. Haut. The effect of a lathyritic diet on the sensitivity of tendon to strain rate. Journal of biomechanical engineering, 107(May 1985):166– 174, 1985. [21] P. J. in ’t Veld and M. J. Stevens. Simulation of the mechanical strength of a single collagen molecule. Biophysical journal, 95(1):33–9, July 2008. [22] K. E. Kadler, D. F. Holmes, J. a. Trotter, and J. a. Chapman. Collagen fibril formation. The Biochemical journal, 316 ( Pt 1:1–11, May 1996. [23] R. Kamien and T. Lubensky. Direct determination of DNA twist-stretch coupling. EPL (Europhysics Letters), 271, Nov. 1997. [24] O. Kratky and G. Porod. R¨ontgenuntersuchung gel¨oster fadenmolek¨ ule. Recueil des Travaux Chimiques des Pays-Bas, 68(12):1106–1122, 1949. [25] F. Maceri, M. Marino, and G. Vairo. A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement. Journal of biomechanics, 43(2):355–63, 2010. [26] M. Marino and G. Vairo. Influence of inter-molecular interactions on the elasto-damage mechanics of collagen fibrils: A bottom-up approach towards macroscopic tissue modeling. Journal of the Mechanics and Physics of Solids, 73:38–54, 2014. [27] J. Marko. Stretching must twist DNA. EPL (Europhysics Letters), 183, 1997. 17
[28] J. F. Marko and E. D. Siggia. 28(26):8759–8770, 1995.
Stretching DNA.
Macromolecules,
[29] S. Neukirch and G. V. D. Heijden. Geometry and mechanics of uniform n-plies: from engineering ropes to biological filaments. Journal of elasticity, pages 41–72, 2002. [30] S. Neukirch and G. van der Heijden. La triple helice: vers une mecanique du collagene. (2):183–188, 2002. [31] N. Rezaei, B. P. B. Downing, A. Wieczorek, C. K. Y. Chan, R. L. Welch, and N. R. Forde. Using Optical Tweezers to Study Mechanical Properties of Collagen. Spie, 8007:80070K–80070K–10, 2011. [32] S. Rigozzi, R. M¨ uller, A. Stemmer, and J. G. Snedeker. Tendon glycosaminoglycan proteoglycan sidechains promote collagen fibril slidingAFM observations at the nanoscale. Journal of biomechanics, 46(4):813– 8, 2013. [33] N. Sasaki and S. Odajima. Elongation mechanism of collagen fibrils and force-strain relations of tendon at each level of structural hierarchy. Journal of Biomechanics, 29(9):1131–1136, 1996. [34] Y.-L. Sun, Z.-P. Luo, A. Fertala, and K.-N. An. Direct quantification of the flexibility of type I collagen monomer. Biochemical and biophysical research communications, 295(2):382–6, 2002. [35] R. B. Svensson, H. Mulder, V. Kovanen, and S. P. Magnusson. Fracture mechanics of collagen fibrils: influence of natural cross-links. Biophysical journal, 104(11):2476–84, June 2013. [36] H. Tang, M. J. Buehler, and B. Moran. A constitutive model of soft tissue: from nanoscale collagen to tissue continuum. Annals of biomedical engineering, 37(6):1117–30, 2009. [37] S. G. M. Uzel and M. J. Buehler. Molecular structure, mechanical behavior and failure mechanism of the C-terminal cross-link domain in type I collagen. Journal of the mechanical behavior of biomedical materials, 4(2):153–61, 2011.
18
[38] S. P. U937 ticity 1–12,
Veres, E. P. Brennan-Pierce, and J. M. Lee. Macrophage-like cells recognize collagen fibrils with strain-induced discrete plasdamage. Journal of biomedical materials research. Part A, pages Mar. 2014.
[39] S. P. Veres, J. M. Harrison, and J. M. Lee. Cross-Link Stabilization Does Not Affect the Response of Collagen Molecules , Fibrils , or Tendons to Tensile Overload. (December):1907–1913, 2013. [40] S. P. Veres, J. M. Harrison, and J. M. Lee. Mechanically overloading collagen fibrils uncoils collagen molecules, placing them in a stable, denatured state. Matrix biology : journal of the International Society for Matrix Biology, 33:54–9, Jan. 2014. [41] S. P. Veres, J. M. Harrison, and J. M. Lee. Mechanically overloading collagen fibrils uncoils collagen molecules, placing them in a stable, denatured state. Matrix biology : journal of the International Society for Matrix Biology, 33:54–9, Jan. 2014. [42] S. P. Veres and J. M. Lee. Designed to Fail : A Novel Mode of Collagen Fibril Disruption and Its Relevance to Tissue Toughness. Biophysj, 102(12):2876–2884, 2012. [43] H. Wada and R. R. Netz. Stretching helical nano-springs at finite temperature. Europhysics Letters (EPL), 77(6):68001, Mar. 2007. [44] H. Yamakawa. Statistical-mechanics of wormlike chains. Pure and Applied Chemistry, 46(2):135–141, 1976.
19
Rtc
Leh
Lc
Figure 1: Illustration of a single TC molecule in the reference state referred to as the reference end-to-end distance Rtc , extended helix length Leh state and fully stretched state denoted by the contour length Lc .
Lcl
Rtc
Figure 2: Illustration of two TC molecules bonded together by cross-links in the reference configuration.
Fwlc
Ftr
1 ξtr
1
rtc Leh
Figure 3: Force versus extended helix length of the inextensible WLC formulation (2).
20
P(Leh)
P(Leh)
Leh
Leh L seh
a)
L eeh
L seh
b)
Figure 4: PDF of the extended helix length Leh motivated by topographical interventions reported in [2]: transition between the entropic- and intrinsic elastic regime for the a) reference configuration (rtc = Rtc ) and b) current configuration.
Leh
Lc
Figure 5: Illustration of intramolecular hydrogen bond rupture after overstretching the TC molecule. Reference state
Rtc
Ltr =
Leh ξtr
Leh
Lc
Entropic regime
Energetic regime
rtc
Lc ν Failed TCs
rtc = Rtc λtc
Figure 6: Visualization of the state evolution depending on the current end-to-end length rtc .
α 4.3
β 4.3
Rtc 42 nm
max
Leh 355 nm
Table 1: Beta distribution parameters obtained by fitting against the data of [2].
21
−3
x 10
8
Beta PDF Gauss PDF
eh
Probability density P(L )
7 6 5 4 3 2 1 0
0
50
100
150
200
250
300
350
400
Extended helix length Leh [nm]
Figure 7: Beta distribution fit against the normal distribution of the extended helix length from [2].
25
Experimental data Model prediction fibril A Model prediction fibril B
Force [μN]
20
15
10
5
0
1
1.2
1.4
1.6
1.8
Stretch λ [−]
2
2.2
2.4
Figure 8: Model predictions plotted against experimental data from the uniaxial tension data of two single collagen fibrils [8].
22
25
Experimental data Entropic force Energetic force Model prediction fibril A
Force [μN]
20
15
10
5
0
1
1.02
1.04
1.06
1.08
1.1
Stretch λ [−]
1.12
1.14
1.16
1.18
Figure 9: General force response of fibril A plotted versus the corresponding experimental data and its entropic and energetic contributions. 45
45
Experimental data Model prediction
35
35
30
30
25 20 15
25 20 15
10
10
5
5
0
1
1.05
1.1
Experimental data Entropic contribution Energetic contribution Model prediction
40
Stress P [MPa]
Stress P [MPa]
40
1.15
Stretch λ [−]
1.2
0
1.25
1
1.05
(a)
1.1
1.15
Stretch λ [−]
1.2
1.25
(b)
Figure 10: Model predictions compared against experimental data of an inelastic steer tail tendon response [40]: a) experimental evaluation of one loading cycle and b) loading response and single contributions of the entropic and energetic part, respectively
Fibril (A) Fibril (B)
Rtc 95 nm 23 nm
Ntc 3.6 × 106 2.4 × 106
S
g
ξtr
5285 pN
800 pNnm
1.08
Table 2: Material parameter set of a single TC molecule and for fibril (A) and (B), respectively.
23
Rtc 155.2 nm
Nfb Ntc 1.1 × 1012
λcr tc 1.06
S 5285 pN
g 800 pNnm
ξtr 1.08
Table 3: Specific material parameters for the inelastic response of steer tail tendon.
24
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PII: DOI: Reference:
S1751-6161(15)00354-9 http://dx.doi.org/10.1016/j.jmbbm.2015.08.045 JMBBM1622
To appear in: Journal of the Mechanical Behavior of Biomedical Materials Received date: 27 May 2015 Revised date: 14 August 2015 Accepted date: 18 August 2015 Cite this article as: Kevin Linka and Mikhail Itskov, Mechanics of collagen fibrils: a two-scale discrete damage model, Journal of the Mechanical Behavior of Biomedical Materials, http://dx.doi.org/10.1016/j.jmbbm.2015.08.045 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Mechanics of collagen fibrils: a two-scale discrete damage model Kevin Linka∗ and Mikhail Itskov Department of Continuum Mechanics, RWTH Aachen University Kackertstr. 9, 52072 Aachen Germany ∗ [email protected]
Abstract Collagen is one of the most important structural proteins of biological tissues. Recently, a new damage phenomena caused by a mechanical overloading of the tendon has been revealed on the collagen fibril level by means of scanning electron microscopy [39]. In order to describe this phenomena we propose in the present paper a constitutive damage model for single collagen fibrils. It utilizes a statistical framework in order to incorporate a physically motivated transition from the entropic- to the intrinsic-elasticity range of single tropocollagen molecules. In this two-scale approach a bridging relation between the tropocollagen and fibril level as well as a failure criterion after exceeding a critical stretch limit are introduced. The final constitutive relations are validated against experimental data available in literature for both, the reversible and the irreversible, stress-strain response. Keywords: collagen, helical molecules, discrete plasticity, soft biological tissues 1. Introduction Collagen fibrils are the main structural contributor to the strength and integrity of the extracellular matrix (ECM). They are assembled by a certain amount of covalent cross-linked tropocollagen (TC) molecules and form a very stiff network. Since the last decade much investigation towards understanding, prediction and numerical simulations of connective soft biological tissues has been carried out. Many new insights ranging from the macromolecular to the tissue/organ level have been reported. In particular, a strong
Preprint submitted to JMBBM
September 18, 2015
dependency between the hierarchical arrangement of collagen and mechanical properties of the tissue has been observed. The diversity in hierarchical arrangements of collagen is hereby seen as one of the main microstructural sources for the high variety of mechanical responses of individuals. This correlation between hierarchical collagen arrangement and mechanical properties opens new opportunities in modern medicine. For instance, a small amount of tissue obtained by a needle biopsy could deliver sufficient histological information for a patient specific material model. By this means, procedures like for example macroscopic in vivo testings used to specify phenomenological constitutive models could become dispensable. First studies of collagen fibrils and their mechanical properties were based on nano-scale experiments and molecular dynamics simulations [7, 25]. Thus, two models capturing the entropic and energetic part of the molecular response, have been proposed so far. Both concepts make use of the wormlike chain (WLC) model, which has successfully been applied to DNA molecules [28]. However, the transition zone from the entropic to the energetic regime is not physically described there. Furthermore, the models lack an exact description of the helical TC arrangement and thus describe the intrinsic elasticity of the helix in a phenomenological manner. In a further paper by Buehler [7], yielding beyond a critical stress is described. This approach has further been used in a continuum model by Tang et al. [36]. They separately considered collagen fibrils and the matrix part of a soft tissue and utilized an invariant based description of the strain energy function. A further crucial issue in the context of biological tissues is irreversible damage on the micro-level, often induced by cyclic loading. Recently, Veres et al. [39] reported breakage at discrete locations along single collagen fibrils after overstretching a steer tail tendon. The main source of this overload induced damage, referred by the authors to as discrete plasticity, is thought to be permanent opening of the TC triple helix and its transformation in a stable denaturated state. To the best of our knowledge, the irreversible response of the tendon caused by the discrete plasticity phenomenon of single collagen fibrils has not been addressed so far in the constitutive modeling. Only molecular dynamic (MD) simulations [13, 14], investigating the uncoiling of single TC molecules under different strain rates have been reported. Rupture of intramolecular hydrogen bonds due to uncoiling and straightening of the helical structure of collagen molecules has been predicted. Nevertheless, it is still in question, whether this mechanism is due to the discrete plasticity phenomena explored by Veres 2
et al. [39]. Following the same idea, a fibril model with linear elastic components has recently been proposed [26]. Damage is included by activation functions motivated by inter- and intramolecular structural changes. A primary focus of this work is thus to develop a fully histology based constitutive model of single collagen fibrils. The model should additionally disclose possible reasons for the high patient specific variety of the mechanical properties. A further objective of the present work is to incorporate structural changes caused by overstretching into the constitutive framework. As a result, a constitutive model capturing the mechanics of single TC molecules inside a cross-linked structure of collagen fibrils will be proposed. We begin with energetic relations for a single TC and apply a strain amplification concept to capture the native assembly of TCs forming collagen fibrils (Chap. 2). Then, in Chap. 3 the TC model is utilized to derive a strain energy function for a single fibril, where a statistical distribution of helix lengths is taken into account. Further, the model is extended to the case of irreversible deformations occurring due to overstretching of TC molecules. Chap. 4 and 5 are devoted to the derivation of the final constitutive equations and thermodynamic consistency of the model. Finally, in Chap. 6 model predictions are validated against experimental data available in literature for single collagen fibrils and irreversibly deformed tendon. 2. Mechanics of a single tropocollagen 2.1. Kinematics of a single tropocollagen Tropocollagen represents a triple helical arrangement of three coiled collagenprotein chains, linked together by an certain amount of hydrogen bonds. The protein strands are built up by three repeating protein bonds. For example, for a collagen-Type-I protein chain this is a Poly(Prolyl-HydroxyprolylGlycyl). In their origin environment TC molecules are assembled in a Dperiodical manner into so called light and dark zones, where the light zones correspond to a molecular gap in the packing [12]. In those gap regions the collagen molecules are in a disordered arrangement. Under tension molecules become aligned and start to straighten out. During this process, the molecular disorder reduces while the entropic force increases until the extended helix length Leh is reached, see Fig. 1. This observation agrees well both with the atomic force microscopy (AFM) [2] and the optical tweezers treatments [7]. The entropic behavior was additionally confirmed by a fitting of the WLC model [24, 28] to the observed experimental data. 3
Under increasing deformation the helix becomes uncoiled and the single protein strands stretch. This is referred to as energetic or intrinsic elasticity range. On the fibril level, this region is additionally characterized by a sliding between single TCs in the molecular packing [12]. This leads to a fuzzy pattern of the fibril, which in turns implies different lengths of single TCs in a fibril. However, the exact coupling between the twisting and stretching in this domain is still an open issue. In stretching tests of DNA one was able to attach a fluorescent avidin-coated rotor bead to the DNA molecule [15]. In this manner, the twist due to stretching by the rotation of these fluorescent beads was monitored. Due to similarities with DNA, we can assume a similar or even the same behavior for TCs. The third typical range in the force-extension behavior of TCs is mainly influenced by the applied boundary conditions. For simultaneous pulling on all three polypeptide strands, the MD simulations reported [21, 6] a very stiff response due to backbone stretching. In further MD simulations [37] closer reproducing native environments of TCs, a single strand of two crosslinked TCs was pulled. In this case, a plateau region of the force-extension diagram was observed. This plateau becomes larger if a strand is not directly connected to the cross-link and smaller for a strand with a direct coupling. 2.2. Strain energy decomposition of a tropocollagen molecule As discussed in the previous section, the strain energy in the low force regime of a TC arises from the thermally induced entropic elasticity. It has been shown [2, 5], that this energy can well be captured by the classical worm-like chain (WLC) model [24]. Accordingly [11, 44] Leh Ewlc = kB T
lp 2
∂t(s) 2 ∂s − Fwlc cos (Θ(s)) ds,
(1)
0
where the total energy of bending deformation due to thermal fluctuations is described by a vector t tangent to the arc length s of a filament subjected to an external force Fwlc . In addition, it is made use of the persistence length lp , a quantity displaying the bending stiffness. Using optical tweezers this parameter was determined for type I collagen to be 14.5 ± 0.5 nm [34, 31]. Further, Leh denotes the extended helix length while kB and T stand for the Boltzmann constant and the absolute temperature, respectively. Finally, the condition Leh lp as well as the inextensibility constraint t = 1 have to 4
be fulfilled. A force-extension relation resulting from (1) can be expressed in particular by the interpolation function by Marko and Sigga [28] as −2 rtc rtc kB T 1 1 Fwlc (rtc , Leh ) = 1− − + , (2) lp 4 Leh 4 Leh where the current end-to-end distance of a TC is given by the affine formulation rtc = Rtc λtc
(3)
in terms of the reference end-to-end distance Rtc and the TC-stretch λtc . In a higher force regime, when the external force is sufficient to suppress thermal fluctuations, the potential due to the intrinsic elasticity of the TC can be given by [23, 27] Πhx =
1 C 2 1 S ∗2 g θ ∗ ∗ θ + r + r − Fhx rtc , 2 Leh 2 Leh tc Leh tc
(4)
∗ where rtc = (rtc − Leh ) denotes the elongation after reaching the extended helix length Leh . Eq. (4) results from the linear theory of double stranded DNA and in addition to the potential of external force Fhx includes three terms necessary for a unique description of DNA [15, 17]. The first term is due to the twisting rigidity of a helix with the twist rigidity C, where θ denotes the winding angle. The second term describes the stretching of single protein chains with the stretch modulus S and the third term is a twiststretch coupling relation with the factor g. Using magnetic tweezers, Gore et al. [15] experimentally studied the double stranded DNA by applying a rotor bread tracking technique in order to determine g and even figured out a functional expression for the twist stretch coupling term. This expression has been incorporated to (4) by Gross et al. [17]. However, those experimental interventions are still missing for collagen. For this reason g will be treated here as a constant parameter obtained by solving the inverse problem. The force extension relation for the intrinsic elasticity part can be obtained by minimizing the potential (4) with respect to Rtc and θ, which yields
SC − g 2 (rtc − Leh ) . (5) C Leh The first fraction in this relation can be expressed in terms of the effective Fhx (rtc , Leh ) =
5
Young’s modulus of a TC in the energetic regime as SC − g 2 , (6) C Atc where the cross-section of a tropocollagen molecule is denoted by Atc . The bending and twist rigidity appearing in (2) and (5), respectively, can be connected to each other by the coiled coil theory for an elastic rod [29, 30, 43]. Accordingly, the following relation has been obtained [29] Etc =
C ≈2 kB T lp
(7)
by utilizing the Cosserat material frame to represent the three single protein strands of a TC molecule and equilibrium boundary conditions and applying condition. Eq. (7) allows us to reduce the number of material parameters. 2.3. Strain amplification Tropocollagen molecules receive their global stability by covalent cross-linking initialized by specific lysine and hydroxysine residues [22]. The cross-links are attached at the ends of the TCs, although the exact 3D spatial pattern of cross-linking bonds is still poorly understood [9]. In contrast, it is well known that the cross-link density uniquely determines the tensile strength of collagen fibrils [20, 18]. However, there are two kinds of immature, divalent cross-links types, which are thought to be strain-labile [39]. Recently, those cross-link types have been treated with NaBH4 in order to stabilize them and study their effect on mechanical behavior at the fibril and tissue level [39, 35]. It turned out that this treatment does not alter at least the mechanical response of single collagen fibrils. Even the discrete failure along single fibrils was not affected. The mechanical response of TC molecules spanned between cross-link connections is much weaker in comparison to the covalent bonded cross-links. Under even supra-physiological tension, first the TC elongates and intrahelical hydrogen bonds rupture before the cross-links considerably deform.This was also reported by Sasaki and Odajima [33], who revealed that the main elongation mechanism is the strain in the helix pitch. Thus, for the modeling purpose, we assume that the cross-links are ideally rigid and do not contribute to the strain energy of the system. They are considered to be responsible only for the internal integrity and load transmission between the TC molecules.
6
The kinematical compatibility of the TC molecule in the reference and stretched state can thus be expressed by (see Fig. 2) (Lcl + Rtc ) λfb = Lcl λcl + Rtc λtc ,
(8)
where Lcl denotes the clamped part length of the TC molecule, while λcl is the corresponding stretch. Under the above assumption of the rigid cross-links λcl = 1, so that Lcl λfb − c , c= , (9) 1−c Rtc + Lcl where λfb is referred to as the fibril stretch. Accordingly, the fibril response becomes stiffer with the increasing cross-link density described by a parameter c considered as a constant. Note that (9) describe the strain amplification well known for filler reinforced rubbers [3, 4, 19]. λtc =
3. Mechanics of a single collagen-fibril 3.1. Elastic response of the fibril From the mechanical perspective a collagen fibril can be considered as a network of parallel aligned TCs, bonded together by cross-links. Initially, the TCs are in a disordered state due to their molecular kinks. A TC can be in an entropic or energetic energy state, depending solely on its current endto-end length rtc . Keeping in mind, that the cross-links do not contribute to the strain energy density, the latter one can be represented in terms of an entropic Ψs and energetic Ψe energy part as Ψfb = Ntc (Ψs + Ψe ) ,
(10)
where Ntc denotes the number of TCs inside a collagen fibril. By virtue of (2) and (5) these energy contributions can further be expressed by 1 Ψs (rtc , Leh ) = Vtc Ψe (rtc , Leh ) =
1 Vtc
rtc
rtc Fwlc d¯ rtc =
Rtc rtc
fwlc d¯ rtc ,
(11)
fhx d¯ rtc ,
(12)
Rtc rtc
Fhx d¯ rtc = Rtc
Rtc
7
where Vtc = Atc Leh
(13)
denotes the reference volume of a single TC and the force densities are exF(•) pressed by f(•) = , respectively. Vtc To account for the energy state change of a TC molecule under tension, a transition criteria is needed. Bozec and Horton [2] detected this at the transition force Ftr by discontinuities between the entropic and energetic regime in the force-extension responses of individual TC molecules. Let Ltr = Leh /ξtr be the corresponding length of the TC molecule, where ξtr > 1 denotes the transition ratio, visualized in Fig. 3 and discussed in the following. The transition from the entropic to the energetic state takes place when rtc =
Leh . ξtr
(14)
Thus, the transition force can be expressed by (2) as (see also Fig. 3) Ftr = Fwlc (Ltr , Leh ).
(15)
Now, we can formulate a strain energy function of a collagen fibril accounting for the energy transition between the entropic and energetic energy state. Due to the strong periodical assembly of fibrils observed in experiments [22], we can assume for modeling purposes that all TCs have the same initial end-to-end distance Rtc . In contrast, they can have a different extended helix length Leh . This observation is based on topographical interventions reported in [2]. There, a probability distribution function (PDF) of the extended helix lengths, denoted in the following by P (Leh ) was proposed. TCs shorter with respect to their extended helix length first pass over to the energetic regime. This is due to the fact, that all TCs are subject to the same stretch, but have a different length in the extended helix state. Certainly, the interval of activated TCs in the entropic regime Lseh decreases due to reaching the transition length Ltr by individual molecules, while the energetic regime increases Leeh (see right hand side of Fig. 4). This can be expressed in view
8
of (14) as
min (λtc ) = Leh Le ≤ Leh ≤ rtc ξtr ,
max s Leh rtc ξtr ≤ Leh ≤ Leh , Leh (λtc ) = Leeh
(16) (17)
where the minimal length in the system is set to be constant accordingly to (14) as min
Le = Rtc ξtr .
(18)
This ensures that the set Leeh is empty in the reference configuration. For ¯ tr = rtc ξtr . The strain energy the sake of simplification we further denote L function for a single collagen fibril can thus be expressed by ⎡ ⎤ ⎢ ⎥ Ψfb = Ntc ⎣ P (Leh )Ψs (rtc , Leh ) dLeh + P (Leh )Ψe (rtc ξtr , Leh ) dLeh ⎦, Lseh (λtc )
Leeh (λtc )
(19) where Ψs and Ψe are given by (11) and (12), respectively. The successive transition of TCs from the entropic to the energetic state under tension reduces the entropic energy contribution. Thus, at higher stretches the energetic response becomes dominant and (19) predicts the typical linear stiffening of the stress-strain response. 3.2. Micro-damage induced by cyclic loading Now, we are going to incorporate a damage formulation into the constitutive model described by the strain energy (19). Based on mechanical testings of overstretched steer tail tendon, Veres and colleagues [42, 39, 40, 38] revealed an irreversible change of the fibril structure. Employing scanning electron microscopy, they showed a kinked and even fibrillated structure at discrete locations along single collagen fibrils. They referred this phenomenon to as discrete plasticity and explained it by the permanent opening of the triplehelical structure of TC molecules once the helical structure is fully uncoiled and the single strands have reached their contour length (see Fig. 5). According to the experimental results [39] this micro-damage effect happens at discrete locations with irregular distances inbetween and only after gaining the uncoiled configuration of a TC. This observation is in line with the prob9
ability distribution of the extended helix lengths, since shorter TCs pass over to uncoiled state first and thus fail first as well. For the modeling purpose, we define the failure stretch relative to the extended helix state by λcr tc =
Lc ν, Leh
(20)
where ν denotes a failure stretch ratio (see also Fig. 6). This is an additional amount of stretch necessary to cause a permanent deformation to a TC molecule, once it has reached the uncoiled configuration (Lc ). The stretch needed to uncoil a TC from the extended helix state is described by the fraction in (20). Since in the natural state the TCs are in the helical conformation, the condition λcr tc > 1 has to be fulfilled. Once the shortest TC is , the minimal available extended helix lengths increases overstretched by λcr tc min
and thus the lower limit of the energetic interval Le does as well. Thus, we can write min
Le = Rtc ξtr max [λtc,m − λcr tc + 1, 1] ,
(21)
where λtc,m denotes the maximal TC stretch ever reached in the system. The underling model assumption of (21) is, that once a TC is stretched over the critical value λcr tc , it does not contribute more to the energy of the system. By this means the damage irreversibility is captured. min
Moreover, accordingly to (21) Le remains constant for λcr tc ≥ λtc,m , which is exactly the physiological loading range. In this range, all deformations min
are reversible up to λtc,m = λcr tc , applying any further loads increases Le and permanent damage starts. Hence, the irreversible limit of the transition ¯ tr cannot return to its origin and thus can be given as length L min ¯ (22) Ltr = max rtc ξtr , Le . 4. Constitutive modeling
Based on the strain energy (19), a constitutive equation for the fibrils can be derived. To this end, we will treat the fibril as a one-dimensional system, since its geometrical dimensions in the axial direction are much bigger than that ones of the cross-section. Taking the incompressibility condition into account, the constitutive equation for the fibril stress tfb can be written as 10
∂Ψfb − p λfb −1 , (23) ∂λfb where p stands for an arbitrary scalar parameter which can be obtained for a particular boundary value problem. Inserting (19) into (23) we thus obtain tfb =
⎡ ∂Ψfb ∂λtc ⎢ = Ntc Rtc ⎣ ∂λfb ∂λfb
P (Leh )fwlc dLeh +
Lseh (λtc )
P (Leh )fhx dLeh
Leeh (λtc )
¯ tr )Ψs (rtc , L ¯ tr ) , − ξtr P (L
(24)
where the latter part arises from the derivative of the integration limits due to the Leibniz integral rule. Furthermore, it is made use of the following identity ∂λtc 1 , = ∂λfb 1−c
(25)
resulting from the strain amplification concept (9). Assuming the tendon response results from the sum of single fibril contributions, we obtain in view of (23) the tendon stress ttd as ttd = Nfb tfb ,
(26)
where Nfb denotes the the number of fibrils inside a tendon. The assumption underlying this constitutive equation is, that the fibrils are not connected to each other and thus the existence of the proteoglycans is not accounted for. Their influence will be considered in further work and reported elsewhere. 5. Thermodynamic consistency By means of the internal variable λtc,m , appearing in (21) the strain energy Ψfb can be rewritten by ˆ fb (λfb , λtc,m ) . Ψfb := Ψ
(27)
Accordingly, the Clausius-Duhem inequality resulting from the second law of thermodynamics can be reduced to
11
−
∂ Ψfb λ˙ tc,m ≥ 0. ∂λtc,m
(28)
Under loading and reloading λtc,m = 0, while λtc,m > 0 in primary loading. Thus, (28) is satisfied if during the primary loading ∂Ψfb ≤ 0. ∂λtc,m
(29)
By using (19) this inequality can be expressed by ¯ tr )Ψe (rtc ξtr , L ¯ tr ) ≤ 0, −Rtc ξtr P (L ¯ tr ) = 0. which holds since Ψe (rtc ξtr , L
(30)
6. Experimental evaluation 6.1. Probability distribution of the extended helix length Up to now, we have not discussed a particular form of the PDF in (19). This distribution for the extended helix length was exposed by topographical interventions by Bozec and Horton [2]. They proposed the Gaussian distribution function which agrees well with the histogram data. Instead, we propose to use the Beta probability distribution function (PDF) of the following form (Leh − Rtc )α−1 P (Leh ; α, β) = α+β−1 max B(α, β) Leh − Rtc
max
Leh − Leh
β−1 ,
(31)
max
where Leh denotes the maximal available extended helix length in the system and α and β represent the shape parameters. Further, B(α, β) is referred to as the beta function. All four parameters of the distribution (31) are obtained by a fitting against the PDF of [2] on the basis of the least-square method (see Fig 7). The values of these parameters are given in Table 1. The Beta PDF is defined over a limited interval and thus appears here to be advantageous in comparison to the normal distribution. By keeping α and β constant, we can utilize only the reference length Rtc and the maximal max
extended helix length Leh for later fitting purposes. These two quantities 12
defining the interval of the Beta distribution can be evaluated from histological data. To reduce the number of material parameters, we have additionally max
fixed the upper bound of the extended helix length Leh in accordance with the above fitting result and with a value reported in the literature [1, 34] to 310 nm. Hence, for fitting purposes only the reference end-to-end length Rtc is exploited. 6.2. Evaluation of the elastic response In this section, the constitutive model (23) is applied to two different sets of experimental data. These sets are obtained under physiological loading conditions of two single collagen fibrils. The fibrils were isolated from the sea cucumber and hydrated during uniaxial tension loading in a microelectromechanical system device [8]. The two tested fibrils had a significant difference in their dimensions, fibril (A) had a length of 20 μm and an average cross-section of 0.71 μm2 , while another fibril (B) had a length of 5 μm with an average cross section of 0.043 μm2 . For more details about the exact experimental method, the interested reader is referred to [8]. For the fitting of the constitutive model (23) to the experimental data of the two fibrils [8], the same material parameter set for TC molecules was used. Due to the repeating protein bond structure of the single polypeptide strand of a TC, material parameters as the transition ratio ξtr , the stretch modulus S and the coupling constant g are nearly the same for different kinds of fibrous tissues. The remaining material parameters, namely the reference end-to-end length Rtc and the number of TCs inside a fibril Ntc , have been adjusted to the experimental data sets for the fibrils A and B by a fitting procedure based on the least-square method. Since these two material parameters depend on the packing density of fibrils, thus they can differ for different individuals or kinds of tissues. The experimental results and model predictions are plotted as force stretch diagrams in Fig. 8. The corresponding material parameters for the TC molecules are given in Table 2, along with the the material parameters for the individual fibrils. Note, that the values for the TC molecules agrees especially well to the range of results reported in [2] and [33]. Further validation of our constitutive model of a single fibril against experimental data, can be derived from Fig. 9. There, the separate force contributions of the entropic and energetic part are compared to the absolute response of fibril A. As mentioned before, the entropic response is dominating in the low force range, 13
while the energetic contribution is negligible. This behavior is vice verse in the high force region [5, 2], which is in line with our results. 6.3. Evaluation of the inelastic response Finally, we validate predictive capabilities of the proposed irreversible damage model (21). Since uniaxial loading-unloading data about the inelastic stress-strain response of collagen fibrils, especially data reporting the phenomena of discrete plasticity are rare, we employed an experimental data set from the whole tendon. These data correspond to the steer tail tendon. There, the occurrence of micro-damage along the fibrils has been verified by SEM interventions after stretching the samples [39]. Although collagen fibrils form only a substructure of the tendon, from the mechanical point of view they represent the most important constituent [41, 12, 22, 16]. In comparison to the elastic formulation, one additional material parameter for the prediction of the inelastic stress-strain response, namely the critical TC stretch λcr tc is needed here. The material parameters for the inelastic tendon response obtained by fitting to the experimental data are given in Table 3. The TC material parameters S, g and ξtr were kept constant as obtained from the previous fit (see Table 2). The predicted stress-strain curve is plotted in Fig. 10. We observe good agreement between the model prediction and the experimental data except of the unloading path. This is due to the fact that some constituents of tendon like the proteoglycans are not considered in the present approach, but have a certain impact on the mechanical response [32, 10]. 7. Conclusion In this paper, we proposed a constitutive model of cross-linked tropocollagen molecules. The TC description is addressed by taking into account the wormlike chain model for the entropic elasticity range and a relation of the triple helix for the intrinsic elasticity part. The transition between the different energy ranges is physically motivated and described by a split integral relation (19). Further, a micro-damage mechanism has been incorporated in the fibril framework, which allows us to predict the irreversible structural change of TCs. The model includes only few material parameters for TC molecules and demonstrates good agreements with experimental data for two different collagen fibrils and for an inelastic steer tail tendon.
14
We are confident that the material parameters of the present model are sufficient to allow the prediction of patient specific material behavior. Indeed, one major reason for the individuality of the fibril responses is most likely the packing density of a fibril, which is captured in the proposed model and expressed in terms of the amount of TCs inside a fibril Ntc and the reference molecule length Rtc . Furthermore, these parameters could be identified by the electron microscopy and X-ray diffraction. Despite the fact, that the model predictions are well in line with the experimental data, there are still some open issues. For instance, in our approach the upper bound of the PDF was fixed and only the lower one was varied, according to values proposed in the literature [1]. Also the handling of the failure ratio ν, has further to be studied, since there are no experimental data available to estimate this parameter. For this reason, further experimental interventions, accounting for both, quantitative histological data and corresponding stress-strain data would be of great benefit. References [1] K. Beck and B. Brodsky. Supercoiled protein motifs: the collagen triplehelix and the alpha-helical coiled coil. Journal of structural biology, 122:17–29, 1998. [2] L. Bozec and M. Horton. Topography and mechanical properties of single molecules of type I collagen using atomic force microscopy. Biophysical journal, 88(6):4223–31, 2005. [3] F. Bueche. Molecular Basis for The Mullins Effect. Rubber Chemistry and Technology, 34(10):493–505, 1961. [4] F. Bueche. Mullins effect and rubber-filler interaction. Rubber Chemistry and Technology, 35(2):259–273, 1962. [5] M. J. Buehler. Nature designs tough collagen: explaining the nanostructure of collagen fibrils. Proceedings of the National Academy of Sciences of the United States of America, 103(33):12285–90, 2006. [6] M. J. Buehler. Nanomechanics of collagen fibrils under varying cross-link densities: atomistic and continuum studies. Journal of the mechanical behavior of biomedical materials, 1(1):59–67, 2008. 15
[7] M. J. Buehler and S. Y. Wong. Entropic elasticity controls nanomechanics of single tropocollagen molecules. Biophysical journal, 93(1):37–43, 2007. [8] S. J. Eppell, B. N. Smith, H. Kahn, and R. Ballarini. Nano measurements with micro-devices: mechanical properties of hydrated collagen fibrils. Journal of the Royal Society, Interface / the Royal Society, 3(6):117–21, 2006. [9] D. R. Eyre, M. A. Weis, and J. J. Wu. Advances in collagen cross-link analysis. 45(1):65–74, 2009. [10] G. Fessel and J. G. Snedeker. Evidence against proteoglycan mediated collagen fibril load transmission and dynamic viscoelasticity in tendon. Matrix biology : journal of the International Society for Matrix Biology, 28(8):503–10, 2009. [11] M. Fixman and J. Kovac. Polymer conformational statistics. III. Modified Gaussian models of stiff chains. The Journal of Chemical Physics, 58(4):1564, 1973. [12] P. Fratzl, K. Misof, I. Zizak, G. Rapp, H. Amenitsch, and S. Bernstorff. Fibrillar structure and mechanical properties of collagen. Journal of structural biology, 122(1-2):119–22, 1998. [13] A. Gautieri, M. J. Buehler, and A. Redaelli. Deformation rate controls elasticity and unfolding pathway of single tropocollagen molecules. Journal of the mechanical behavior of biomedical materials, 2(2):130–7, 2009. [14] A. Gautieri, S. Vesentini, A. Redaelli, and M. J. Buehler. Hierarchical structure and nanomechanics of collagen microfibrils from the atomistic scale up. Nano Letters, 11:757–766, 2011. [15] J. Gore, Z. Bryant, M. N¨ollmann, M. U. Le, N. R. Cozzarelli, and C. Bustamante. DNA overwinds when stretched. Nature, 442(1):836–839, 2006. [16] J. S. Graham, A. N. Vomund, C. L. Phillips, and M. Grandbois. Structural changes in human type I collagen fibrils investigated by force spectroscopy. Experimental cell research, 299(2):335–42, Oct. 2004.
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[17] P. Gross, N. Laurens, L. B. Oddershede, U. Bockelmann, E. J. G. Peterman, and G. J. L. Wuite. Quantifying how DNA stretches, melts and changes twist under tension. Nature Physics, 7(9):731–736, 2011. [18] P. Hansen, T. Hassenkam, R. B. Svensson, P. Aagaard, T. Trappe, B. T. Haraldsson, M. Kjaer, and P. Magnusson. Glutaraldehyde cross-linking of tendon–mechanical effects at the level of the tendon fascicle and fibril. Connective tissue research, 50:211–222, 2009. [19] J. a. C. Harwood, L. Mullins, and a. R. Payne. Stress Softening in Natural Rubber Vulcanizates. Part II. Stress Softening Effects in Pure Gum and Filler Loaded Rubbers. Rubber Chemistry and Technology, 39:814–822, 1966. [20] R. C. Haut. The effect of a lathyritic diet on the sensitivity of tendon to strain rate. Journal of biomechanical engineering, 107(May 1985):166– 174, 1985. [21] P. J. in ’t Veld and M. J. Stevens. Simulation of the mechanical strength of a single collagen molecule. Biophysical journal, 95(1):33–9, July 2008. [22] K. E. Kadler, D. F. Holmes, J. a. Trotter, and J. a. Chapman. Collagen fibril formation. The Biochemical journal, 316 ( Pt 1:1–11, May 1996. [23] R. Kamien and T. Lubensky. Direct determination of DNA twist-stretch coupling. EPL (Europhysics Letters), 271, Nov. 1997. [24] O. Kratky and G. Porod. R¨ontgenuntersuchung gel¨oster fadenmolek¨ ule. Recueil des Travaux Chimiques des Pays-Bas, 68(12):1106–1122, 1949. [25] F. Maceri, M. Marino, and G. Vairo. A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement. Journal of biomechanics, 43(2):355–63, 2010. [26] M. Marino and G. Vairo. Influence of inter-molecular interactions on the elasto-damage mechanics of collagen fibrils: A bottom-up approach towards macroscopic tissue modeling. Journal of the Mechanics and Physics of Solids, 73:38–54, 2014. [27] J. Marko. Stretching must twist DNA. EPL (Europhysics Letters), 183, 1997. 17
[28] J. F. Marko and E. D. Siggia. 28(26):8759–8770, 1995.
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[29] S. Neukirch and G. V. D. Heijden. Geometry and mechanics of uniform n-plies: from engineering ropes to biological filaments. Journal of elasticity, pages 41–72, 2002. [30] S. Neukirch and G. van der Heijden. La triple helice: vers une mecanique du collagene. (2):183–188, 2002. [31] N. Rezaei, B. P. B. Downing, A. Wieczorek, C. K. Y. Chan, R. L. Welch, and N. R. Forde. Using Optical Tweezers to Study Mechanical Properties of Collagen. Spie, 8007:80070K–80070K–10, 2011. [32] S. Rigozzi, R. M¨ uller, A. Stemmer, and J. G. Snedeker. Tendon glycosaminoglycan proteoglycan sidechains promote collagen fibril slidingAFM observations at the nanoscale. Journal of biomechanics, 46(4):813– 8, 2013. [33] N. Sasaki and S. Odajima. Elongation mechanism of collagen fibrils and force-strain relations of tendon at each level of structural hierarchy. Journal of Biomechanics, 29(9):1131–1136, 1996. [34] Y.-L. Sun, Z.-P. Luo, A. Fertala, and K.-N. An. Direct quantification of the flexibility of type I collagen monomer. Biochemical and biophysical research communications, 295(2):382–6, 2002. [35] R. B. Svensson, H. Mulder, V. Kovanen, and S. P. Magnusson. Fracture mechanics of collagen fibrils: influence of natural cross-links. Biophysical journal, 104(11):2476–84, June 2013. [36] H. Tang, M. J. Buehler, and B. Moran. A constitutive model of soft tissue: from nanoscale collagen to tissue continuum. Annals of biomedical engineering, 37(6):1117–30, 2009. [37] S. G. M. Uzel and M. J. Buehler. Molecular structure, mechanical behavior and failure mechanism of the C-terminal cross-link domain in type I collagen. Journal of the mechanical behavior of biomedical materials, 4(2):153–61, 2011.
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[38] S. P. U937 ticity 1–12,
Veres, E. P. Brennan-Pierce, and J. M. Lee. Macrophage-like cells recognize collagen fibrils with strain-induced discrete plasdamage. Journal of biomedical materials research. Part A, pages Mar. 2014.
[39] S. P. Veres, J. M. Harrison, and J. M. Lee. Cross-Link Stabilization Does Not Affect the Response of Collagen Molecules , Fibrils , or Tendons to Tensile Overload. (December):1907–1913, 2013. [40] S. P. Veres, J. M. Harrison, and J. M. Lee. Mechanically overloading collagen fibrils uncoils collagen molecules, placing them in a stable, denatured state. Matrix biology : journal of the International Society for Matrix Biology, 33:54–9, Jan. 2014. [41] S. P. Veres, J. M. Harrison, and J. M. Lee. Mechanically overloading collagen fibrils uncoils collagen molecules, placing them in a stable, denatured state. Matrix biology : journal of the International Society for Matrix Biology, 33:54–9, Jan. 2014. [42] S. P. Veres and J. M. Lee. Designed to Fail : A Novel Mode of Collagen Fibril Disruption and Its Relevance to Tissue Toughness. Biophysj, 102(12):2876–2884, 2012. [43] H. Wada and R. R. Netz. Stretching helical nano-springs at finite temperature. Europhysics Letters (EPL), 77(6):68001, Mar. 2007. [44] H. Yamakawa. Statistical-mechanics of wormlike chains. Pure and Applied Chemistry, 46(2):135–141, 1976.
19
Rtc
Leh
Lc
Figure 1: Illustration of a single TC molecule in the reference state referred to as the reference end-to-end distance Rtc , extended helix length Leh state and fully stretched state denoted by the contour length Lc .
Lcl
Rtc
Figure 2: Illustration of two TC molecules bonded together by cross-links in the reference configuration.
Fwlc
Ftr
1 ξtr
1
rtc Leh
Figure 3: Force versus extended helix length of the inextensible WLC formulation (2).
20
P(Leh)
P(Leh)
Leh
Leh L seh
a)
L eeh
L seh
b)
Figure 4: PDF of the extended helix length Leh motivated by topographical interventions reported in [2]: transition between the entropic- and intrinsic elastic regime for the a) reference configuration (rtc = Rtc ) and b) current configuration.
Leh
Lc
Figure 5: Illustration of intramolecular hydrogen bond rupture after overstretching the TC molecule. Reference state
Rtc
Ltr =
Leh ξtr
Leh
Lc
Entropic regime
Energetic regime
rtc
Lc ν Failed TCs
rtc = Rtc λtc
Figure 6: Visualization of the state evolution depending on the current end-to-end length rtc .
α 4.3
β 4.3
Rtc 42 nm
max
Leh 355 nm
Table 1: Beta distribution parameters obtained by fitting against the data of [2].
21
−3
x 10
8
Beta PDF Gauss PDF
eh
Probability density P(L )
7 6 5 4 3 2 1 0
0
50
100
150
200
250
300
350
400
Extended helix length Leh [nm]
Figure 7: Beta distribution fit against the normal distribution of the extended helix length from [2].
25
Experimental data Model prediction fibril A Model prediction fibril B
Force [μN]
20
15
10
5
0
1
1.2
1.4
1.6
1.8
Stretch λ [−]
2
2.2
2.4
Figure 8: Model predictions plotted against experimental data from the uniaxial tension data of two single collagen fibrils [8].
22
25
Experimental data Entropic force Energetic force Model prediction fibril A
Force [μN]
20
15
10
5
0
1
1.02
1.04
1.06
1.08
1.1
Stretch λ [−]
1.12
1.14
1.16
1.18
Figure 9: General force response of fibril A plotted versus the corresponding experimental data and its entropic and energetic contributions. 45
45
Experimental data Model prediction
35
35
30
30
25 20 15
25 20 15
10
10
5
5
0
1
1.05
1.1
Experimental data Entropic contribution Energetic contribution Model prediction
40
Stress P [MPa]
Stress P [MPa]
40
1.15
Stretch λ [−]
1.2
0
1.25
1
1.05
(a)
1.1
1.15
Stretch λ [−]
1.2
1.25
(b)
Figure 10: Model predictions compared against experimental data of an inelastic steer tail tendon response [40]: a) experimental evaluation of one loading cycle and b) loading response and single contributions of the entropic and energetic part, respectively
Fibril (A) Fibril (B)
Rtc 95 nm 23 nm
Ntc 3.6 × 106 2.4 × 106
S
g
ξtr
5285 pN
800 pNnm
1.08
Table 2: Material parameter set of a single TC molecule and for fibril (A) and (B), respectively.
23
Rtc 155.2 nm
Nfb Ntc 1.1 × 1012
λcr tc 1.06
S 5285 pN
g 800 pNnm
ξtr 1.08
Table 3: Specific material parameters for the inelastic response of steer tail tendon.
24