Strain rate induced toughening of individual collagen fibrils

Abstract

The nonlinear mechanical behavior of individual nanoscale collagen fibrils is governed by molecular stretching and sliding that result in a viscous response, which is still not fully understood. Toward this goal, the in vitro mechanical behavior of individual reconstituted mammalian collagen fibrils was quantified in a broad range of strain-rates, spanning roughly six orders of magnitude, from 10⁻⁴ to 35 s⁻¹. It is shown that the nonlinear mechanical response is strain rate sensitive with the tangent modulus in the linear deformation regime increasing monotonically from 214 ± 8 to 358 ± 11 MPa. More pronounced is the effect of the strain rate on the ultimate tensile strength that is found to increase monotonically by a factor of four, from 42 ± 6 to 160 ± 14 MPa. Importantly, fibril strengthening takes place without a reduction in ductility, which results in equivalently large increase in toughness with the increasing strain rate. This experimental strain rate dependent mechanical response is captured well by a structural constitutive model that incorporates the salient features of the collagen microstructure via a process of gradual recruitment of kinked tropocollagen molecules, thus giving rise to the initial “toe-heel” mechanical behavior, followed by molecular stretching and sustained intermolecular slip that is initiated at a strain rate dependent stress threshold. The model shows that the fraction of tropocollagen molecules undergoing straightening increases continuously during loading, whereas molecular sliding is initiated after a small fibril strain (1%–2%) and progressively increases with applied strain.

Collagenous tissues have a hierarchical structure with the nanoscale collagen fibrils representing the fundamental building blocks. Collagen fibrils result from the self-assembly of tropocollagen molecules, which are rod-like triple helices that are 300 nm long and 1–2 nm in diameter. Tropocollagen molecules pack together in a parallel, staggered arrangement with a periodicity of D = 67 nm,^1–3 generating a characteristic repeating pattern of 35-nm gap zones and 32-nm overlap zones corresponding to dark and bright regions in Transmission Electron Microscopy (TEM) images, inset in Fig. 1(a), respectively. The rate-dependent constitutive behavior of collagenous tissues has been studied extensively,^4–7 yet there is very limited information about the contribution of collagen fibrils to the viscous mechanical behavior of tissues. A limited number of experimental and computational studies have shown that collagen fibrils,^8–11 and their molecular constituents^12–15 exhibit viscoelastic behavior and significant and sustained damping.¹⁶ Therefore, they are expected to exhibit a strain rate dependent mechanical behavior. Initial evidence to that effect comes from studies of uncross-linked collagen fibrils stretched with the aid of an AFM cantilever inside phosphate-buffered saline (PBS), which showed that the elastic modulus increased by ∼30% after two orders of magnitude increase in strain rate.⁹ Because of experimental limitations, these prior studies^9,17 were restricted to a narrow range of strain rates (0.01–2 s⁻¹) and relatively small stretch ratios (λ <1.1), thus limiting the experimental evidence only to the regimes of molecular kink straightening and partial stretching of the tropocollagen molecules.^18–20 Furthermore, an experimentally calibrated constitutive modeling framework to simulate the viscous response of collagen fibrils is still missing.

FIG. 1.

(a) Stress–stretch ratio (σ–λ) curves from tests carried out at seven different strain rates between 10^–4 and 35 s⁻¹. The dashed arrow points to increasing strain rate. Insets: collagen fibril mounted on a MEMS device, and reconstituted mammalian collagen fibril with D-banding of 67 nm. (b) Stress vs stretch ratio of a fibril (wet diameter: 240 nm) tested in PBS at a nominal strain rate of 0.0026 s⁻¹. Three typical deformation regimes are distinguished: (i) an initial, nonlinear “toe-heel” regime (I), (ii) a linear regime (II), and (iii) a softening regime (III) extending to failure.

In view of these open questions, the strain-rate dependent mechanical behavior of individual mammalian collagen fibrils was investigated in this study in the broad range of strain rates 10⁻⁴–10² s⁻¹, spanning timescales that are relevant to physical activities and injury of collagenous tissues.^21–24 The true strain rate varies by a small amount during testing, Fig. 1(b), due to the nonlinear constitutive response of collagen fibrils, which, however, is minute compared to the six orders of magnitude range of applied strain rate. As shown in Fig. 1(b), the engineering strain rate varies at most by a factor of two in an entire σ–λ curve, with the minimum value reported herein as the applied strain rate. On average, three fibrils were tested at each strain rate. Lyophilized type I collagen from calfskin was synthesized according to Williams et al.²⁵ by dissolving collagen extractions in acetic acid and subsequently reconstituting for several hours to form nanoscale collagen fibrils. The detailed synthesis protocol is described in Ref. 16. The reconstituted collagen fibrils had the characteristic D-band structure,¹ inset in Fig. 1(a), average dry diameter of 134 ± 54 nm and lengths up to a few hundreds of micrometers, as measured via a Scanning Electron Microscope (SEM). When immersed in PBS, the fibrils swelled to two times their dry diameter, as measured by in situ AFM;²⁶ therefore, the dry fibril diameters, measured after testing via SEM, were multiplied by 2 to obtain the in vitro diameters, spanning the range 140–528 nm. Then, individual collagen fibrils were isolated from PBS solutions with a micromanipulator and mounted onto a Microelectromechanical System (MEMS) device²⁷ for microscale uniaxial tension experiments. The gauge length of the test specimens was 30 μm. The insets in Fig. 1(a) show a fiber specimen mounted on the grip tips of a MEMS device and a high-resolution TEM image (not the same specimen as the one on the MEMS device) of the characteristic D-band structure. The entire MEMS device was immersed in PBS for testing. Independent measurements of force and fibril extension (engineering strain) were resolved via Digital Image Correlation (DIC) that was applied to high resolution optical images obtained during testing via a 40× water immersion lens. The DIC calculations provided subpixel displacement resolution corresponding to ∼25 nm increments of fibril stretching, which are smaller than 0.1% of the initial fibril gauge length.²⁷

The experimental stress–stretch ratio (σ–λ) curves at strain rates of 10⁻⁴–35 s⁻¹, Fig. 1(a), exhibited three distinct regimes of deformation, Fig. 1(b). Regime I consists of the nonlinear “toe-heel” segment of the σ–λ curve.^18–20 The gap regions of the collagen fibril D-band structure are thought to contain nanoscale kinks because of reduced packing density and greater flexibility of tropocollagen molecules due to lower proline and hydroxyproline content.²⁸ Straightening of molecular kinks in the “toe” section of regime I leads to gradual recruitment of more tropocollagen molecules to carry an applied force. In σ–λ curves, this mechanism is manifested as stiffening, thus causing the gradual transition to regime II that represents the linear segment of the curves. Finally, the strain rate increases gradually into regime III where softening takes place due to increased molecular sliding and gradual failure of molecular cross-links. Reconstituted collagen fibrils lack mature cross-links; therefore, there is no stiffening regime before failure.

Kinks in the D-band gap regions are present along the entire length of a fibril and are successively straightened during loading.^18–20 When a significant fraction of molecules are straightened, the collagen fibril follows a (relatively) linear σ–λ response (regime II) as tropocollagen molecules undergo stretching²⁸ along with molecular sliding.²⁹ The slope of this segment of an σ–λ curve is defined as the tangent modulus of the fibril, Fig. 1(b). Finally, softening takes place before failure, regime III, which is different from the post-stiffening softening regime observed in human patellar tendon fibrils that contain mature trivalent crosslinks.³⁰ The softening behavior of the reconstituted calf skin collagen fibrils tested in this work is similar to that of native rat tail tendon fibrils,³⁰ as both types of collagen possess only immature divalent cross-links of weak strength. In this case, molecular slip dominates over backbone stretching either due to failure of these immature cross-links³⁰ or due to low cross-link density.³¹ As a result, further load transfer between individual tropocollagen molecules is hindered by molecular slip, thus leading to a decreasing tangent modulus with the increasing strain, as inferred from Fig. 1(b). Additionally, it has been suggested that shear force transfer between tropocollagen molecules takes place by a process of rupture and reformation of multiple water-mediated hydrogen bonds between tropocollagen molecules. At the single tropocollagen molecule level, this process produces a serrated rather than a smooth force–displacement response, analogously to stick-slip motion.^32,33 Finally, repeated intermolecular slip and rupture of immature cross-links lead to localized formation of nanoscale voids and lower density regions,³⁴ hence resulting in necking and eventual rupture.

The σ–λ response shown in Fig. 1(a) clearly supports that the mechanisms governing the aforementioned three regimes of deformation depend on the strain rate. Softening is reduced at higher strain rates as the σ–λ curves “straighten” for strain rates larger than ∼0.1 s⁻¹, Fig. 1(a). At higher strain rates, the collagen fibrils exhibit increased tangent modulus and ultimate tensile strength, as demonstrated by the statistically significant increase in the tangent modulus [68% increase, p-value (between groups) < 4 × 10⁻⁴, Fig. 2(a)] and the ultimate tensile strength [280% increase, p-value (between groups) < 8 × 10⁻³, Fig. 2(b)] with applied strain rate. Statistical comparisons between the means of different sample groups were made using a one-way analysis of variance (ANOVA), with p-values <0.05 indicating a significant difference between the mean group values. The increase in ultimate tensile strength was the most dramatic, from 42 ± 6 to 160 ± 14 MPa, Fig. 2(b), namely, by a factor of four without compromising the ultimate tensile strain, Fig. 1(a). These results are consistent with prior indirect measurements of the effect of strain rate on collagen fibrils: x ray diffraction studies⁷ showed that the extent of straightening of molecular kinks in the gap regions and the ratio of the local stretch ratio in the gap regions to the total stretch ratio, (λ_gap/λ), of a fibril decrease with the increasing strain rate. The authors in Ref. 7 concluded that higher strain rates lead to increased tangent modulus due to stretching of the tropocollagen molecules and intermolecular cross-links, while at lower strain rates gradual removal of kinks in the gap regions, followed by the stretching of molecules and intermolecular cross-links, leads to a lower stiffness. Compared to prior experimental reports on collagen fibrils, the tangent modulus values found in the present study, varying due to strain rate between 214 ± 8 and 358 ± 11 MPa, Fig. 2(a), are similar to the moduli reported for acid-separated type I, bovine Achilles tendon, collagen (200–600 MPa),^9,35 enzyme(trypsin)-separated rat patellar tendon collagen fibrils (326 ± 112 MPa),³⁶ and collagen fibrils from sea cucumber dermis (470 ± 410 MPa).³⁷

FIG. 2.

(a) Tangent modulus derived from the linear regime of σ–λ curves. (b) Ultimate tensile strength vs applied strain rate. The error bars represent one standard deviation. The circles represent the actual experimental data.

The hypothesis that the experimentally established strain rate dependent behavior, Fig. 1(a), is due to gradual straightening of molecular kinks in the gap regions, is tested with the aid of a model developed in the past to explain collagenous tissue mechanics,^38–40 which was modified to capture the salient features of the collagen fibril structure [inset, Fig. 3(a)]. In this model, the fibrils are comprised of N parallel collagen molecules. Thus, the model does not capture the staggered arrangement of the triple helix molecules, rather distributes the kinked molecules along the entire length of a fibril. The gap (less dense) regions, e.g., TEM image of fibril in the inset of Fig. 1(a), are represented by kinks, which must be straightened before a tropocollagen molecule can bear tensile load. According to this statistical model, there is a stretch ratio threshold, ${\lambda }_s^{\left(i\right)}$ for the ith tropocollagen molecule, to straighten and begin bearing load. The molecular stretch ratio to reach straightening, ${\lambda }_s^{\left(i\right)}$, is related to the molecular straightening strain, $e_s^{\left(i\right)}$, as⁴¹

$${\lambda }_s^{\left(i\right)}=\sqrt{1+2e_s^{\left(i\right)},}$$ (1a)

which in this work is assumed to follow a Weibull distribution:

$$e_s^{\left(i\right)}={\beta }_s{\left[-\ln \left(1-P_s^{\left(i\right)}\right)\right]}^{1/{\alpha }_s},$$ (1b)

where α_s > 0 and β_s > 0 are the shape and scale parameters of the Weibull Probability Density Function (PDF), respectively.

FIG. 3.

(a) Fibril σ–λ response for three strain rates spanning the experimental range. The inset table shows the calculated model parameters. Under uniaxial tension, first molecular kinks gradually straighten in the gap regions ①, followed by stretching of collagen triple helices ②. Intermolecular sliding ③ is initiated at σ₀, leading to softening (conceptual fibril schematics were adapted from Fratzl et al.²⁸ but modified to reflect the present model). (b) Fraction of tropocollagen molecules that are straightened (solid lines), or began sliding (dashed lines) vs λ. Inset: Weibull PDF of the fitted model. The parameters α_s, β_s are provided in the table in (a).

The molecular strain, $e_m^{\left(i\right)}$, is related to the fibril strain, e = (λ² − 1)/2, through,⁴¹

$$e_m^{\left(i\right)}=\frac{\left(e-e_s^{\left(i\right)}\right)}{\left(1+2e_s^{\left(i\right)}\right)}.$$ (2)

Upon stretching a collagen fibril by λ, the stress σ⁽ⁱ⁾ developed in the ith tropocollagen molecule takes one of the following values:

$${\sigma }^{\left(i\right)}\left(\lambda ,{\lambda }_s^{\left(i\right)}\right)=\left\{\begin{array}{c}0,\hspace{1em}\left[\lambda <{\lambda }_s^{\left(i\right)}\right]\hfill \\ S_m^e·G\left(t\right)+{\int }_0^{t}G\left(t-\tau \right)·\frac{d{S}_m^e}{d{e}_m^{\left(i\right)}}·\frac{d{e}_m^{\left(i\right)}}{d\tau }d\tau ,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left[{\lambda \ge {\lambda }_s^{\left(i\right)};\sigma }^{\left(i\right)}<{\sigma }_0\right]\hfill \\ {\sigma }_0,\hspace{1em}\left[{\sigma }^{\left(i\right)}\ge {\sigma }_0\right].\hfill \end{array}\right.$$ (3)

Based on prior modeling^12,14,15 evidence, the tropocollagen molecules are modeled as quasi-linear viscoelastic (QLV),⁴² with an instantaneous elastic response that is assumed to follow:⁴¹

$$S_m^e=A·e_m^{\left(i\right)}+B·{\left(e_m^{\left(i\right)}\right)}^N,$$ (4)

where A, B, and N are material parameters. A simple reduced relaxation function, G(t), is assumed

$$G\left(t\right)=k_0+k_1·\exp \left(-\frac{t}{{\tau }_1}\right),$$ (5)

where τ₁ is the material time-constant (allowed to vary with strain rate) and $k_0+k_1=1\hspace{0.17em}\left(k_0,k_1>0\right).$

The derivative $d{e}_m^{\left(i\right)}/dt$ in Eq. (3) depends on the applied strain rate to the fibril, $de/dt$ (constant), and to simplify the fitting process it is assumed that $d{e}_s^{\left(i\right)}/dt=0$. Finally, σ₀ is the critical tensile stress at which weak intermolecular crosslinks break resulting in initiation of molecular sliding at a steady stress, which is consistent with computational molecular modeling⁴³ and x-ray diffraction results.⁴⁴ The overall stress in the fibril, σ, is computed as the average for N tropocollagen molecules,

$$\sigma =\frac{1}{N}\sum _{i=1}^N{\sigma }^{\left(i\right)}\left(\lambda ,{\lambda }_s^{\left(i\right)}\right).$$ (6)

A large number of tropocollagen molecules (N = 10⁵) was used to simulate the collagen fibril behavior. Larger N did not yield different results. The rate dependent model parameters (α_s, β_s, k₀, τ₁, σ₀) were determined by fitting the experimental fibril response, Fig. 1(a), by using the trust-region-reflective least squares algorithm.⁴⁵ Figure 3(a) shows the experimental and the fitted model σ–λ response for the two extreme and a mid-range strain rate. The elastic response parameters A, B, and N are the same for all tested fibrils. The derived viscoelastic time constants ranged from 10 ms (35 s⁻¹) to 100 s (0.0001 s⁻¹). The model captured well the entire nonlinear σ–λ response until failure, predicting an instantaneous modulus A = 5.9 GPa for infinitesimal strains. Previous molecular dynamics simulations have shown that tropocollagen molecules are viscoelastic, with modulus values in the range of 4–16 GPa.^12,13 Similarly, the critical tensile stress for the onset of sliding also increased with the strain rate, from 56 to 201 MPa [inset Table, Fig. 3(a)]. This can be attributed to the increase in intermolecular shear strength with loading rate, originating in rate-dependent breaking and reforming of hydrogen bonds.^12,46 A kinetic model proposed by Lieou et al.⁴⁷ predicted that the peak force for breaking and re-forming intra- or intermolecular hydrogen bonds increases roughly by a factor of two when the stretch rate increases from 100 to 10 000 nm/s. These loading velocities, corresponding to strain rates 3 × 10⁻³–3 × 10⁻¹ s⁻¹, are well within the experimental range of strain rates applied in this study. The model prediction for an increase in σ₀ by a factor of four between 10⁻⁴ and 35 s⁻¹ [table in Fig. 3(a)] is roughly in agreement with the peak force modeling results in Ref. 47.

During fibril loading, a cascade of intramolecular kink straightening and stretching of tropocollagen molecules takes place, along with intermolecular sliding once σ₀ is reached. As shown in Fig. 3(b), for a given stretch ratio, molecular straightening is most effective at slow rates, yet, even at the slowest strain rate of 10⁻⁴ s⁻¹, only ∼85% of all molecules were straightened at the point of failure. Furthermore, the model results indicate that at all strain rates, the fractions of straightened and sliding molecules are almost identical, Fig. 3(b). Molecular straightening takes place continuously from the beginning of the loading process, whereas intermolecular sliding initiates at small (1–2)% strains and proceeds with increasing molecular fraction during fibril stretching.

The inset in Fig. 3(b) shows the PDF of the calibrated Weibull distribution. Although theoretically speaking the Weibull PDF stretches out to infinity, in practice the PDF of molecular kinks is limited to the finite interval 1 ≤ λ_s < 3 based on the fitting of the experimental data. According to the model results, the fraction of kinked molecules that become straightened is reduced at higher strain rates. Furthermore, the model predicts that the fraction of molecules sliding after reaching σ₀ is reduced by increasing the strain rate, Fig. 3(b). The calculated Weibull PDF, inset in Fig. 3(b), is broader at high strain rates, thus indicating that, statistically speaking, molecules must stretch to larger λ_s to begin carrying load, or alternatively, a smaller fraction of molecules has reached λ_s for a given macroscale fibril stress. This result must be put in perspective of the collagen fibril D-band structure: As mentioned, the present model is simplified by assuming that molecular straightening [physically taking place mainly in the gap regions shown as the less dense bands in the inset schematic in Fig. 3(a) and the inset TEM image in Fig. 1(a)] is distributed along the entire length of continuous tropocollagen molecules. Therefore, as smaller fractions of molecules reach λ_s at high strain rates for a given macroscale fibril stress, it is also deduced that a smaller fraction of gap regions in the collagen fibril D-band structure are straightened at a given fibril stress. X-ray diffraction studies by Karunaratne et al.⁷ have also pointed out that the change in the D-band gap length-to-fibril extension ratio significantly decreases with the increasing strain rate, which is in agreement with the present model prediction that, for a given stretch ratio, the fraction of straightened molecules decreases with the increasing strain rate. Furthermore, the parallel tropocollagen molecules in the present model have the same stretch ratio but experience different microscopic stress, Eq. (3), depending on the value of ${\lambda }_s^{\left(i\right)}$. X-ray diffraction studies by Sasaki et al.⁴⁴ led to the conclusion that the relative slip of adjacent molecules is proportional to an increase in D-band gap length, which is reduced at high strain rates. This proportionality between relative slip (molecular sliding) and increase in D-band gap length (molecular straightening) is also captured by the consistent tracking of the dashed and solid line curves, respectively, at all strain rates, as shown in Fig. 3(b).

In conclusion, microscale in vitro experiments demonstrated pronounced stiffening and strengthening of the mechanical behavior of reconstituted mammalian collagen fibrils with an increase in the strain rate. Notably, the increased mechanical stiffness and strength were not accompanied by a reduction in the ultimate tensile strain, which led to strain rate induced toughening of collagen fibrils. Consistent with results reported by molecular simulations, the nonlinear, strain rate dependent constitutive behavior of collagen fibrils was captured well by a microstructure-based model that incorporated the gradual recruitment of kinked viscoelastic tropocollagen molecules, followed by stretching of the straightened tropocollagen molecules and subsequent intermolecular slip beyond a threshold stress. The increasing fraction of tropocollagen molecules undergoing sliding leads to fibril softening, thus capturing the entire σ–λ response of reconstituted collagen fibrils.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.