Biomechanics and Modeling in Mechanobiology (2019) 18:1809–1819 https://doi.org/10.1007/s10237-019-01178-6 ORIGINAL PAPER Effect of collagenase–gelatinase ratio on the mechanical properties of a collagen fibril: a combined Monte Carlo–molecular dynamics study Bethany Powell1,2 · David C Malaspina3 · Igal Szleifer2,4,5 · Yasin Dhaher6,7 Received: 21 March 2019 / Accepted: 27 May 2019 / Published online: 3 June 2019 © The Author(s) 2019 Abstract Loading in cartilage is supported primarily by fibrillar collagen, and damage will impair the function of the tissue, leading to pathologies such as osteoarthritis. Damage is initiated by two types of matrix metalloproteinases, collagenase and gelatinase, that cleave and denature the collagen fibrils in the tissue. Experimental and modeling studies have revealed insights into the individual contributions of these two types of MMPs, as well as the mechanical response of intact fibrils and fibrils that have experienced random surface degradation. However, no research has comprehensively examined the combined influences of collagenases and gelatinases on collagen degradation nor studied the mechanical consequences of biological degradation of collagen fibrils. Such preclinical examinations are required to gain insights into understanding, treating, and preventing degradation-related cartilage pathology. To develop these insights, we use sequential Monte Carlo and molecular dynamics simulations to probe the effect of enzymatic degradation on the structure and mechanics of a single collagen fibril. We find that the mechanical response depends on the ratio of collagenase to gelatinase—not just the amount of lost fibril mass—and we provide a possible mechanism underlying this phenomenon. Overall, by characterizing the combined influences of col- lagenases and gelatinases on fibril degradation and mechanics at the preclinical research stage, we gain insights that may facilitate the development of targeted interventions to prevent the damage and loss of mechanical integrity that can lead to cartilage pathology. Keywords Collagen fibril · Mechanical properties · Collagenase · Gelatinase · Molecular dynamics · Monte Carlo 1 Introduction In healthy tissue, collagen production is typically balanced Bethany Powell and David Malaspina have contributed equally to by collagen degradation mediated by matrix metalloprotein- this work. ases (MMPs), enzymes that bind and cleave the triple helical Igal Szleifer and Yasin Dhaher have contributed equally to this collagen molecule. This balance is disrupted in many dis- work. ease states, including arthritis, cancer, and fibrosis (Cawston and Wilson 2006; Clark et al. 2008; Giannandrea and Parks Electronic supplementary material The online version of this 2014; Malemud 2006; Mancini and Di Battista 2006). When article (https​://doi.org/10.1007/s1023​7-019-01178​-6) contains supplementary material, which is available to authorized users. 4 * Yasin Dhaher Department of Chemistry, Northwestern University, yasin.dhaher@utsouthwestern.edu Evanston, IL, USA 5 1 Department of Chemical and Biological Engineering, Department of Mechanical Engineering and Bioengineering, Northwestern University, Evanston, IL, USA Valparaiso University, Valparaiso, IN, USA 6 2 Department of Physical Medicine and Rehabilitation, Department of Biomedical Engineering, Northwestern University of Texas Southwestern, Dallas, TX, USA University, Evanston, IL, USA 7 3 Department of Orthopedic Surgery, University of Texas Institut de Ciencia de Materials de Barcelona Southwestern, Dallas, TX, USA (ICMAB-CSIC), Campus UAB, Bellaterra, Barcelona, Spain 13 Vol.:(0123456789) 1810 B. Powell et al. the balance favors destruction, permanent structural changes Accordingly, our purpose is to examine the combined can occur, especially in tissues that are metabolically mostly effect of collagenase and gelatinase on collagen fibril struc- quiescent, such as articular cartilage (Maroudas et al. 1992; ture and obtain the associated changes in the mechanical Verzijl et al. 2000). Such damage can lead to changes in properties using molecular simulations. To do so, we devel- mechanical properties that impair the function of the tissue. oped a Metropolis Monte Carlo (MC) approach to examine At the molecular level, collagen degradation is mediated the process of biological degradation of a collagen fibril by by two MMPs subfamilies: collagenases and gelatinases. collagenase MMP-1 and gelatinase MMP-9. The structures Single-molecule tracking techniques reveal that collagenases obtained by the MC approach were then used as inputs to (MMP-1) adsorb on collagen and then spend the majority of the coarse-grained MD model (Malaspina et al. 2017) to their time in paused states where cleavage eventually occurs examine the changes in the mechanical response of the fibril (Sarkar et al. 2012). Once collagenase has cleaved a tropoco- that resulted from MMP-induced structural changes. We llagen molecule, the tropocollagen begins to denature and hypothesized that a higher ratio of collagenases to gelati- becomes susceptible to gelatinase cleavage. Employing a nases would lead to reduced toughness of the fibril at a fixed single-molecule tracking paradigm, Rosenblum et al. (2010) amount of degradation, since an increase in the number of reported that gelatinase (MMP-9) does not efficiently bind collagenases would increase the number of cleaved bonds to the native triple helical structure, but tends to bind and that cannot bear load. Further, we examined the time depend- cleave the partially denatured fragments that collagenases ence of degradation, hypothesizing that the time course of create. While the isolated interactions of MMPs with their degradation would differ based on the absolute number of substrates have been experimentally characterized, the com- collagenases and gelatinases on the fibril as well as the ratio bined influences of collagenase and gelatinase on the struc- of collagenases to gelatinases. To our knowledge, this is tural and mechanical properties of an isolated collagen fibril the first study to examine the relative influences of colla- have not been established. genases and gelatinases acting simultaneously on a collagen Several experimental examinations have shown that col- fibril and the first study to quantify their relative effects on lagen degradation mediated by MMP subtypes produces a fibril mechanical integrity. We argue that this computational dramatic decrease in the mechanical strength and toughness construct will help inform future experimental designs and of collagen fibers (Laasanen et al. 2003; Panwar et al. 2013, consequently serve as a bridge between experimental obser- 2015; Park et al. 2008). But, while connections between the vations and molecular mechanisms. Our results demonstrate molecular structure and the strength and toughness of colla- that the ratio of collagenase to gelatinase dictates fibril deg- gen have been drawn in some of these studies (Panwar et al. radation and mechanical integrity. The findings presented 2013, 2015), the precise nature of the relationship between herein could eventually inform future experiments that may MMP-induced structural changes and the mechanical prop- provide targeted clinical treatment strategies, which would erties of collagen fibrils has not been comprehensively stud- likely hinge on the relative effects of local collagenase and ied. In our recent work (Malaspina et al. 2017), we employed gelatinase inhibition on degradation and mechanics. a coarse-grained molecular dynamics (MD) model (Depalle et al. 2015) to examine how random surface degradation influenced the mechanical response of a collagen fibril. We 2 Methods showed that the surface degradation (determined using a sol- vent-accessible surface area constraint) produced dramatic 2.1 Dynamic Metropolis Monte Carlo Degradation changes in the toughness of the collagen fibril consistent Model with experimental observations (Laasanen et al. 2003; Pan- war et al. 2013, 2015; Park et al. 2008). These findings align A coarse-grained lattice model was used to represent the with the hypothesis that the molecular organization of the structural hierarchy of collagen, which is depicted in Fig. 1. fibril exerts a strong influence over the mechanical strength Enzymes were randomly selected and placed on empty lat- (Fratzl 2008) and a small disruption of the structure leads to tice sites using random selection of position indices on the a large reduction of the fibril toughness. However, although surface of the fibril. A detailed description of the structure the model yielded important insights into the relationship and the initial placement of the enzymes can be found in between degraded fibril structure and mechanical proper- S1 Text. Briefly, the lattice was organized according to ties, our previous work did not account for the complexity of experimental reports of collagen fibril organization, which structural changes that may stem from the underlying biol- includes tropocollagen molecules arranged into microfi- ogy and the thermodynamics of degradation, which includes brils and microfibrils arranged into a fibril. This structure the combined action of the two MMP subtypes, collagenase served as the basis for a dynamic Metropolis MC approach, and gelatinase. depicted in Fig. 1a–d. 13 Effect of collagenase–gelatinase ratio on the mechanical properties of a collagen fibril: a… 1811 Fig. 1  Illustrations of the components of the fibril MC lattice structure and resulting MD structure. a Illustration of the individual tropocollagen molecule. An intact molecule is shown on the far left, a molecule that has been cleaved at the collagenase cleavage site is shown in the middle, and a molecule with a removed lattice site is shown on the far right. The blue site denotes the posi- tion where collagenase binds with high affinity and eventually cleaves. b Illustration of the microfibril structure (length- wise). The pattern of gaps and staggering of the tropocollagen molecules is shown in this flattened depiction. The index j indicates the vertical position of an individual tropocollagen within a microfibril. c Illustra- tion of the microfibril cross section. Bmf is the diameter of a single microfibril. d Illustra- tion of the fibril cross section when m = 3 (in the simulations presented in the results, m = 4). The index m is the number of microfibrils per edge on a fibril, and n and q are indices of microfibril position within the fibril. Refer to S1 Text for more details on the indices used to describe the structure. e Snap- shot of a section of the fibril equilibrated by MD simulation. The cleaved and removed sites were obtained from the results of the MC simulations 2.1.1 MMP–Collagen Interaction Energies collagen, k is the Boltzmann constant, T is the temperature, and i is the index of vertical position along a tropocollagen. When an MMP occupied a lattice site on a given tropoco- These interaction energies were estimated by running the llagen, it bound to the site with an energy that depended MC model with a single collagenase on the fibril, calculat- on the type of MMP, its position on the tropocollagen, and ing the percent of steps where the collagenase was paused whether the tropocollagen molecule had been cleaved by a at its characteristic cleave site (i = 5). The energy was collagenase. The interaction energy between collagenase and adjusted until the collagenase remained at the characteristic the collagen lattice sites was described by cleave site during approximately 18% of the total simula- tion steps, since Sarkar et al. (2012) observed that colla- ⎧ 0, i≤4 genases adsorbed to a fibril spend 18% of their time paused ⎪ EC,colla = ⎨ −2kT, i = 5 (1) at exposed collagenase cleave sites, which they denoted as ⎪ 0, i>5 “class II pauses.” Further, experiments indicate that colla- ⎩ genases are unable to diffuse over cleaved sites and have lower affinity for gelatin (Sarkar et al. 2012; Saffarian et al. where the first subscript C denotes the enzyme type, col- 2004); therefore, lagenase, the second subscript colla denotes the substrate, 13 1812 B. Powell et al. site 66 nm from the C terminus of an individual tropocol- EC,Ccl = ∞ lagen (i = 4), and gelatinase cleavage only occurred when (2) gelatinases were positioned on a tropocollagen that had EC,gela = 0.5 kT, 1 ≤ i ≤ 22 already been cleaved by a collagenase (Rosenblum et al. where C denotes collagenase, Ccl denotes a cleaved lattice 2010; Fields 2013). When the enzymes were in the posi- site, and the subscript gela denotes gelatin (a tropocollagen tions where cleavage could occur, cleavage succeeded at an molecule with at least one cleaved site). While the exact overall rate of 0.35 s−1 (Sarkar et al. 2012). When cleavage change in collagenase affinity for collagen after cleavage occurred, a bond was broken between the N terminus of the is unknown, we incorporated an increase in the interaction occupied site and the C terminus of the site directly above energy to phenomenologically incorporate qualitatively the occupied site (Fig. 1a). observed collagenase behavior. Once a lattice site and its lower neighbor had been Gelatinase interactions with collagen and gelatin, respec- cleaved, that site became disconnected from the rest of tively, were described by the equations the lattice and was removed from the system immediately (Fig. 1a). As the removal progressed, some sets of lattice EG,colla = 0, 1 ≤ i ≤ 22 sites became detached from neighboring lattice sites and (3) were removed from the system. A set of lattice sites was EG,gela = −0.5 kT, 1 ≤ i ≤ 22 considered detached when it had no vertical neighbors, where the subscript G indicates gelatinase, colla denote an and when it was not attached to at least two other beads uncleaved tropocollagen, and the subscript gela indicates a horizontally. cleaved tropocollagen or gelatin. Again, the exact change in A semi-quantitative verification of the MC simulations the gelatinase affinity for collagen after cleavage is unknown, of the degradation process against experimental observa- but we incorporated a decrease in the interaction energy to tions are shown in S1 Fig. Furthermore, S1 Text includes a phenomenologically take experimental observations into detailed description of the MC model formulation. account (Rosenblum et al. 2010). 2.1.3 Degradation Conditions 2.1.2 Dynamic Metropolis Monte Carlo Moves The number of enzymes on the fibril was selected in a way A dynamic Metropolis MC approach was used to simulate that facilitated study of the relative effects of collagenases the interactions between the MMPs and the collagen to cap- and gelatinases, since the patterns of cleavage differ for the ture the stochastic, time-dependent nature of the collagen two classes of MMPs. The amount of each type enzyme degradation process. can be found in Table 1. For each enzyme surface coverage For each step of the simulation, an enzyme was randomly condition, the changes in fibril structure were recorded as selected from the ensemble of collagenases and gelatinases, the simulation progressed. and the selected enzyme attempted to move vertically or horizontally or attempted to cleave. The movement attempts 2.2 Mechanical Analysis with Coarse‑Grained were accepted based on the Boltzmann factor of the change Molecular Dynamics in energy between one lattice site and the next. For vertical motion, trial moves in the positive and nega- The degraded structures from the MC simulations were tive z-directions were chosen with the same probability. In then used to perform mechanical analysis using MD. The a similar way, trial moves were chosen with the same prob- mapping of sites from the lattice MC to the MD model ability for “left or right” movements for horizontal motion (Malaspina et al. 2017) is such that one site on the MC but constrained to remain on the outermost edge of the fibril. structure is equivalent to ten sites in MD structure. The In accordance with experimental observations (Sarkar et al. 2012), the ratio of vertical to horizontal trial moves was Table 1  Simulated systems and relative amounts of collagenases and selected in the model such that vertical movements were gelatinases much more probable. Furthermore, the probabilities of dis- Simulated system Number of Number of Ratio of col- placement attempts were selected such that the enzymes collagenases gelatinases lagenase to would move with the diffusion coefficients reported in the gelatinase literature (Collier et al. 2011). System 1 (c 4: g 4) 4 4 1 For cleave attempts, the model presented here was param- System 2 (c 8: g 8) 8 8 1 eterized to mimic experimental observations (Sarkar et al. System 3 (c 6: g 2) 6 2 3 2012; Rosenblum et al. 2010). Collagenase cleavage only System 4 (c 2: g 6) 2 6 0.33 occurred when collagenases were positioned on the lattice 13 Effect of collagenase–gelatinase ratio on the mechanical properties of a collagen fibril: a… 1813 simulation protocol is identical to the one used in our pre- changes to the mechanical properties of collagen fibrils. To vious work (Malaspina et al. 2017) with a fibril that is ten do so, we used a combined MC/MD approach and observed times longer. The length was increased so that the size of the complex changes to the fibril structure when collagenases MD structure would be equal to the MC structure. In the MD and gelatinases degrade a collagen fibril simultaneously. structure, which is depicted in Fig. 1e, each tropocollagen is The MC model revealed distinct distributions of cleaved represented by a polymeric chain that contains 220 bonded and removed beads along the fibril for structures that had beads. A total of 925 tropocollagen chains interact through all experienced 1.1% degradation (Fig. 2). A snapshot of a a Lennard–Jones potential, in conjunction with a harmonic section of an equilibrated fibril (Fig. 2a, b gray), displaying potential and a hyper-elastic bond energy. The force field removed beads (Fig. 2a, b blue lines) and the beads with for the coarse-grained MD model was developed by Depalle adjacent cleaved bonds (Fig. 2a, b red spheres), served as et al. (2015) and is described in more detail in S2 Text. an illustration of the general distribution of cleaved and The stress–strain response was assessed at 1.1% degra- removed bonds obtained from the MC simulation. Note that dation for all treatment conditions, where percent degrada- in the current study, one removed site in the MC simulation tion was defined as number of removed lattice sites divided was equivalent to 10 sites in the MD simulation, which is by the initial number of lattice sites × 100. This amount of why the removed sites were represented by blue lines instead degradation (1.1%) was chosen based on a previous work of individual beads in the MD snapshot (Fig. 2b). The spatial (Malaspina et al. 2017) and represents the beginning of a distribution of cleaved sites from the cross-sectional view plateau in the change in toughness of a collagen fibril as of the fibril for all the systems can be observed in the heat function of surface degradation. Since the MC degradation maps of Fig. 2c. In a similar way, the spatial distribution of simulation progressed at different rates for each condition, removed sites from the cross-sectional view of the fibril for the time required to reach 1.1% degradation differed for each all the systems can be observed in Fig. 2d. In both Fig. 2c system described in Table 1. Regardless of the time required and Fig. 2d, the heat maps are purposely arranged the lowest to reach 1.1% degradation, the analysis was performed at a amount of collagenase (top panels) to the highest amount of single instant in time, and no further enzyme-mediated deg- collagenase (bottom panels) to better illustrate the behavior radation occurred as the tensile loading was applied. of the systems. For the quantification of the fibril stress–strain response The density and location of cleaved bonds were highly to axial loading, the strain was computed by tracking the dependent on the ratio of collagenase to gelatinase, as indi- change in the size of the simulation box and the stress ten- cated by the heat maps in Fig. 2c. System 4 (c 2: 6 g, top sor was computed using the virial stress in the MD package panel Fig. 2c) had the smallest ratio collagenase to gelati- LAMMPS (Plimpton 1995). The strain rate used in these nase and the most localization of cleaved bonds. As the ratio simulations was 107 s−1. In the current model, the abrupt of collagenase to gelatinase increased, the distribution of fracture of the fibril does not occur because the fracture cleaved bonds became more uniform across the outer sur- energy dissipation was constrained with a maximum velocity face of the fibril cross section, as shown in System 1 (4 limit to avoid the collapse of the simulation at large strains. c: 4 g, second panel Fig. 2c). As the ratio of collagenase We analyzed the strain distribution inside the fibril by to gelatinase continued to increase, the density of cleaved calculating the tropocollagen bond length distributions. The bonds increased around the outer surface of the fibril cross distribution is calculated based on the length of bonds in section, as shown in System 3 (6 c: 2 g, third panel Fig. 2c). the fibril structure at the different fibril bulk strain values. Finally, System 2 (8 c: 8 g, bottom panel Fig. 2c) showed a This distribution allows for the quantification of sliding and similar distribution and density of cleaved bonds compared stretching of tropocollagen chains during the deformation to System 1 (4 c: 4 g, second panel Fig. 2c), which had the of the fibril. We defined bonds as sliding bonds when their same ratio of collagenase to gelatinase as System 2 but half length during fibril deformation was comparable to their the total number of enzymes. This observation indicates that, length in the undeformed fibril, and we defined bonds as for the same amount of degradation, the fibril had less con- stretching bonds when their length during fibril deformation nectivity (more cleaved bonds) on the surface as the ratio of was large compared to their length in the unstretched fibril. collagenase to gelatinase increased. Moreover, for a small ratio of collagenase to gelatinase, the distribution of cleaved sites tended to be more localized. 3 Results The density map of removed beads (Fig. 2d) appeared to be related to the density map of cleaved bonds (Fig. 2c), 3.1 Fibril Morphology After Degradation since gelatinase could remove a bead in the MC model only if a collagenase had previously cleaved a neighboring We sought to study the combined effects of collagenases bond. Thus, the location of removed bonds (Fig. 2d) was and gelatinases on fibril degradation, and the subsequent similar to the density map of cleaved bonds (Fig. 2c) but 13 1814 B. Powell et al. Fig. 2  Representation of cleaved and removed sites for different con- of cleaved sites from the fibril cross-sectional view for system 4, 1, ditions. a Snapshot of a cross-sectional view of the fibril and b longi- 3 and 2, ordered from top to bottom, respectively. d Density map of tudinal view of the fibril. In a, b removed sites are represented as blue removed sites from the fibril cross-sectional view for system 4, 1, 3 tubes and cleaved sites are represented as red spheres. c Density map and 2, ordered from top to bottom, respectively much more localized in certain regions. Note that the num- 3.2 Toughness of the Fibril ber of removed beads was the same for each of the panels in Fig. 2d, but the distribution of the removed beads dif- To analyze the simultaneous effect of degradation by col- fered for each system. Moreover, the distribution appeared lagenase and gelatinase on the mechanical properties, we to differ for a small ratio of collagenase to gelatinase (see studied the stress–strain response by using MD simula- system 4; c 2: 6 g, top panel Fig. 2d), but the effect of the tions. The stress–strain curves are shown in Fig. 3 for the collagenase to gelatinase ratio was less evident in the other simulated systems in Table 1. As a reference, we included systems (Fig. 2d, bottom 3 panels). the stress–strain behavior of an intact fibril with its char- The time course of degradation appeared to depend on acteristic deformation regions (Malaspina et al. 2017; the ratio of collagenases to gelatinases, as shown in S1 Fig. Depalle et al. 2015). The fibril exhibited an abrupt change Degradation progressed most quickly when more gelatinases in the mechanical properties as a result of small amount were present (system 4) and progressed most slowly when of degradation and showed a shift in the yield strain for more collagenases were present (system 3) when the total the degraded curves, characteristics consistent with our number of enzymes on the surface was constant. However, earlier report (Malaspina et al. 2017). One key observation when the absolute number of enzymes on the fibril was is that the fibril toughness was progressively reduced as increased, the degradation progressed more quickly (sys- the collagenase-to-gelatinase ratio increased. Moreover, tem 2). the stress–strain curves for a ratio of 1 (systems 1 and 2) were nearly identical, even though the absolute num- bers of enzymes differed by a factor of two, indicating 13 Effect of collagenase–gelatinase ratio on the mechanical properties of a collagen fibril: a… 1815 Fig. 3  Stress–strain curves for different collagenase and gelati- nase amounts at 1.1% degrada- tion. Simulated systems 1, 3 and 4 are represented in solid lines and system 2 is represented in blue dashed line to denote difference in total amount of enzymes. The stress–strain curve for the intact fibril is included in gray dashed lines the stress–strain curves for a specified level of degrada- 3.3 Bond Length Distribution at Fixed Strain tion appeared to depend on the ratio of collagenases to gelatinases and not the total number of enzymes. The yield Bond length distribution was examined to investigate the stress was reduced by ~ 52% for the system with the small- molecular origin of the change in toughness for the different est ratio of collagenase to gelatinase (system 4) and ~ 43% stress–strain curves in Fig. 3. These distributions provided for the system with the largest ratio collagenase to gelati- the probabilities of tropocollagen stretching and tropocol- nase (system 3; Fig. 3 inset). lagen sliding (Fig. 4 inset schematic) and were related to the stress distribution inside the fibril. Two populations of bond lengths could be observed for each structure when the strain Fig. 4  Distributions of tropocol- lagen bond lengths for systems 1, 3 and 4 described in Table 1. The distribution is evaluated at the same strain (19%) and same degradation (1.1 wt%). A gray dashed line represents the mean value where bonds transition from sliding to stretching. An inset scheme is included to show the difference between sliding and stretching bonds 13 1816 B. Powell et al. was equal to 19%, a pre-yield state (Fig. 4). The probability 4 Discussion peak associated with the lower lengths was centered slightly above the equilibrium bond length of 14.7 Angstroms, indi- 4.1 Lessons Learned cating that minimal stretching had occurred and that the beads connected by those bonds were sliding with respect Using sequential MC and MD simulations, we probed to the rest of the structure. The peak centered around higher the effect of enzymatic degradation on the structure and lengths was associated with stretching of the bonds, indicat- mechanics of a single collagen fibril. These models inves- ing that the bonds connecting these beads could bear load tigated the combined degradation behavior of collagenase (Malaspina et al. 2017). As the collagenase-to-gelatinase and gelatinase and the molecular origin of the mechanical ratio increased, the proportion of unstretched (sliding) bonds integrity of a degraded collagen fibril. Our MD simula- increased and the proportion of stretched bonds decreased. tions revealed that the loss of fibril mechanical integrity This trend suggested that fewer tropocollagen molecules may depend not only on the number of removed beads, but bore load and more molecules passively slid apart as the also on the relative abundance of collagenase and gelati- ratio increased, which may explain why fibril toughness nase. An increase in the amount of collagenase increased decreased as ratio increased. the total number of cleaved bonds at a fixed amount of degradation. Further, the ratio of collagenase to gelati- 3.4 Bond Length Distribution Versus Strain nase appeared to regulate the spatial distribution of the cleaved sites on the fibril surface, with a more diffuse pat- The analysis of the bond length distribution was extended to tern of cleaved bonds for higher ratios and a more local- the whole range of strains calculated in Fig. 3. The distribu- ized pattern for low ratios. Finally, our examination of tion of bond lengths is displayed in Fig. 5 as a contour plot to the bond length distribution demonstrated that the fibrils identify how the distribution of bond lengths evolved as the with reduced toughness had fewer stretched bonds and fibril stretched. At low strains, the bond length distributions more sliding bonds compared to the tougher fibrils. These showed that almost all of the bonds were sliding bonds. As results, which demonstrated a complex interplay between the strain increased, a bimodality in the distribution started collagenase and gelatinase effects on fibril degradation to develop, representing both stretching and sliding bonds. and mechanical integrity, could inform future experiments This bimodality can be identified in Fig. 5 as the region that would provide targeted clinical treatment strategies, containing a “hole” (dark blue) that represents the mini- which would likely hinge on the relative effects of local mum between the sliding and stretching peaks (also shown collagenase and gelatinase inhibition on degradation and in Fig. 4). At midrange strains, system 3 appeared to show mechanics. this bimodality, while systems 1 and 4 did not appear to have The toughness of the fibril appeared to be linked to the as many sliding bonds in this range of strains before yield. connectivity of the fibril, which we characterized using bond However, as the strain increased beyond yield, the distribu- length distributions (Fig. 5). These distributions revealed tions of bond lengths shifted back toward the equilibrium that increases in the relative amount of collagenase led to bond length of 14.7 Angstroms for all three systems. more sliding bonds and fewer stretched bonds, which may help explain the reduced load-bearing capacity of fibrils that were exposed to higher ratios of collagenase to gelatinase. Fig. 5  Contour plot of the distribution of bond lengths as a function of the strain for a collagen fibril at different collagenase/gelatinase ratios. The ratios are indicated above the plots, which are arranged in increasing ratio collagenase/gelatinase from left to right 13 Effect of collagenase–gelatinase ratio on the mechanical properties of a collagen fibril: a… 1817 The sliding bonds only provided a small amount of resist- between this surface coverage and MMP concentration in ance to fibril loading through non-bonded cohesive molecu- adjacent solution. That is, the model neglected the processes lar forces within the fibril (hydrogen bonds, electrostatic and of adsorption and desorption. Had the model included the van der Waals interactions). However, the stretched bonds adsorption and desorption processes, the total number of provided greater resistance to load because they were con- enzymes on the fibril would have remained constant on aver- nected by covalent bonds in addition to the cohesive molec- age but would have fluctuated around that average through- ular forces within the fibril. Thus, because sliding bonds out the simulation. However, it would have been difficult to indicate reduced load-bearing capacity, the presence of more include these processes in the model because the interaction sliding bonds (shown in Fig. 5) may help explain the overall energies associated with the adsorption and desorption pro- reduction in toughness that was observed for systems with cesses have not been experimentally reported in the context more collagenases (shown in Fig. 3). of MMP–tropocollagen interactions. Thus, it would not be Our simulation results indicated that the relative abun- possible to use information about the processes of adsorp- dance of collagenases and gelatinases affected the spatial tion and desorption to establish a functional relationship distribution of fibril degradation (Fig. 2c, d). It appeared between MMP surface coverage and MMP concentration in that a low ratio would lead to damage that was more local- solution at the single fibril level. At a coarser level, kinetic ized to one region of the fibril surface, while a higher ratio studies of collagenase and gelatinase have established such would lead to a more diffuse pattern of cleaved bonds and functional relationships for bulk solutions. However, the removed fragments. For lower ratios, the damage remained mean-field approximation underlying those approaches lim- confined to a particular region of the fibril, which may have its their utility at the nanoscale (Collier et al. 2001; Welgus allowed more bonds in the undamaged regions to stretch et al. 1980). Nonetheless, the MC model could be adapted to and bear the applied load. Conversely, the diffuse pattern examine adsorption/desorption if the necessary parameters of cleaved sites may have disrupted the continuity of the become available. molecular structure at more locations along the fibril surface In the present investigation, the computational frame- and prevented the uptake of load by the individual tropocol- work assumes that the entire fibril is accessible to MMPs lagen molecules. Thus, multiple lines of evidence seem to throughout the MC simulations. We recognize, however, help explain the observed differences in load-bearing capa- that a single fibril in solution does not adequately repre- bilities for the systems with different ratios of collagenase sent the environment of a fibril that is incorporated into a to gelatinase (Fig. 3). tissue, such as cartilage, where degradation will proceed Although the mechanical response of a fibril appeared to much more slowly. Indeed, numerous reports indicate that depend on the ratio of collagenases to gelatinases and not collagen fibrils in cartilage are protected by the glycosami- on the absolute number of enzymes after a fixed amount of noglycan (GAG) molecules that surround them (Kar et al. degradation, the speed of degradation appeared to depend 2016; Li et al. 2015; Pratta et al. 2003) and that the GAGs on both the absolute number of enzymes on the fibril and the slow the degradation process such that it occurs over years ratio of collagenase to gelatinase. S1 Fig shows that the time and decades rather than minutes and hours. However, this course of degradation accelerated when the total number of slow degradation of GAG-protected tissues can be traced enzymes increased (system 2) and when the collagenase- back to the degradation of collagen fibrils. Thus, we argue to-gelatinase ratio decreased (system 4). These two factors, that it is necessary to gain a fundamental understanding of MMP ratio and surface coverage, are likely regulated in vivo single fibril degradation to fully understand the degradation by local cellular production rate and clearance by the lym- at larger scales. phatic system and may have important implications for the progression of damage in the joint. Thus, future investiga- 4.3 Regarding Verification tions should attempt to holistically examine the progression of damage to the collagen by combining information about Multiple lines of evidence suggest that the MD model in these two factors that influence degradation and mechanics this work adequately reflects experimental observations. In with our understanding of the biological processes (cellu- our previous work (Malaspina et al. 2017), we compared lar production and lymphatic clearance) that regulate the the results of simulated mechanical testing to experimental factors. data obtained using axial extension tests (Svensson et al. 2013). While we were only able to compare the results for 4.2 Model Assumptions and Limitations intact fibrils because axial extension tests have not been performed on degraded collagen fibrils, we observed rea- The MC model of degradation employed in this study sonable agreement between experimental and simulated included a constant number of collagenases and gelati- results for the intact fibril. Furthermore, the MD model has nases on the fibril surface and did include a relationship been successfully applied in different scenarios (Depalle 13 1818 B. Powell et al. et al. 2015, 2016) where it showed general agreement with of the relative roles of collagenases and gelatinases may experimental observations. Finally, recent work on collagen eventually help illuminate paths to targeted treatments that fascicles seems to corroborate our results that show dam- may prevent or reduce pathological collagen degradation age to the fibril will reduce its mechanical strength (Zitnay and loss of mechanical integrity, particularly in articular et al. 2017). That work shows that collagen fascicles sub- cartilage. jected to sub-yield loading conditions experience subtle damage that leads to reduced strength and toughness com- pared to intact specimens (Zitnay et al. 2017), though we Funding Funding was provided by National Institute of Biomedical Imaging and Bioengineering (US) (1U01 EB015410-01A1), National note that the mechanism of damage differs between their Institutes of Health (T32 HD07418) experiment and our model. Despite the similarities between the MD model and experimental observations, the present Open Access This article is distributed under the terms of the Crea- model, which incorporates biological mechanisms of deg- tive Commons Attribution 4.0 International License (http://creat​iveco​ radation, has not yet been validated thoroughly against mmons​.org/licen​ses/by/4.0/), which permits unrestricted use, distribu- tion, and reproduction in any medium, provided you give appropriate experimental data and further verification is needed. How- credit to the original author(s) and the source, provide a link to the ever, such experiments would require examinations at the Creative Commons license, and indicate if changes were made. single fibril level, which is experimentally challenging (Laasanen et al. 2003; Panwar et al. 2013, 2015). 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