pnasresearcharticle Wyse Jackson I.C., L.J.B., L.F., and M.D. designed
research; T.W.J. performed the experimental research; T.W.J. analyzed data;
J.M., P.L., and M.D. designed model and performed simulations; T.W.J., L.J.B.,
and I.C. wrote the paper with all authors contributing. The authors declare no
conflict of interest. 1To whom correspondence should be addressed. E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Structural Origins of Cartilage Shear Mechanics

Thomas Wyse Jackson  Department of Physics, Cornell University, Ithaca, NY
Equal first author contribution  Jonathan Michel  School of Physics &
Astronomy, Rochester Institute of Technology, Rochester NY  Equal first author
contribution  Pancy Lwin  School of Physics & Astronomy, Rochester Institute
of Technology, Rochester NY  Lisa A. Fortier  Department of Clinical Sciences,
Cornell University, Ithaca, NY  Moumita Das  School of Physics & Astronomy,
Rochester Institute of Technology, Rochester NY  Equal corresponding author
contribution  Lawrence J. Bonassar  Meinig School of Biomedical Engineering,
Cornell University, Ithaca, NY  Sibley School of Mechanical and Aerospace
Engineering, Cornell University, Ithaca, NY  Equal corresponding author
contribution  Itai Cohen  Department of Physics, Cornell University, Ithaca,
NY  Equal corresponding author contribution

###### Abstract

Cartilage, the soft tissue that coats the long bones in mammalian joints, is a
remarkable material that is able to sustain millions of loading cycles over
decades of use and outperforms any synthetic substitute. The bulk properties
of this tissue reflect the mechanics of an extracellular matrix that is
comprised of primarily two components: a collagen network and a reinforcing
secondary network of aggrecan molecules linked by hyaluronic acid. Diseases of
cartilage involve degradation of one or both of these networks either due to
mechanical overloading or various biochemical processes. Crucially, how the
loss of these constituents manifest in changes to the mechanical performance
of the tissue, particularly in shear, is not sufficiently understood to enable
predictive modeling of tissue performance in disease. Here, we present
experiments and theory in support of a rigidity percolation framework that
describes quantitatively the structural origins of cartilage’s shear
properties and how they arise from the mechanical interdependence of the
collagen and aggrecan networks. This framework predicts that sensitivity to
degradation is highly dependent on how close the collagen network is to its
rigidity percolation threshold. Specifically, it explains that near the
cartilage surface, where the collagen network is sparse and close to the
rigidity threshold, slight changes in either collagen or aggrecan
concentrations, common in early stages of cartilage disease, create dramatic
changes in modulus that can lead to tissue collapse. More broadly, this
rigidity percolation framework provides a map for understanding how changes in
composition throughout the tissue alter its shear properties and ultimate in
vivo function.

###### keywords:

Cartilage $|$ Shear $|$ Rigidity Percolation $|$ Osteoarthritis $|$

###### doi:

www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX

Healthy cartilage is a remarkable tissue, able to withstand tens of millions
of loading cycles with minimal damage. While much is known about the
structural origins of its compressive mechanics, how composition determines
its mechanical behavior in shear, a major mode of failure, is poorly
understood. Through a combination of microscale structure-function experiments
and modeling, we develop a rigidity percolation framework to explain the
structural origins of cartilage shear mechanics. This framework provides a new
quantitative understanding for how the known degradative events in
osteoarthritis determine the mechanical changes that are a hallmark of this
disease. As such, this work provides a road map for understanding disease
progression and in combination with non-invasive techniques such as MRI will
enable more effective diagnosis and treatment. This manuscript was compiled on

Affecting over 27 million people in the United States (1), and over 250
million people worldwide, Osteoarthritis is one of the leading causes of
disability. Osteoarthritis can arise from trauma, mechanical forces,
inflammation, biochemical reactions, and metabolic changes to cells (2, 3, 4).
As this disease progresses, inflammatory mediators can induce release of
enzymes that result in degradation of the extracellular collagen and aggrecan
networks (5), the two most important constituents responsible for the
mechanical properties of cartilage. The networks formed by these constituents
are quite distinct in their properties. The collagen molecules form a network
of fibrils with very high tensile strength. The aggrecan network is comprised
of highly charged chondroitin sulfate molecules attached to a core protein in
a bottle brush geometry. These aggrecan monomers combine with hyaluronic acid
to form 10-1000 MDa aggregates that gel (6). The high amount of charge on the
aggrecan network produces an osmotic stress that draws in water and swells the
tissue. Broadly, damage to the collagen network leads to loss of tissue
integrity (3) and a reduction in the capacity of the tissue to resist tensile
(7) and shear strains (8, 9). Loss of aggrecan reduces the osmotic swelling,
makes the tissue more susceptible to compression (10), and is also associated
with the loss of shear properties (11, 12). As these networks degrade, the
tissue mechanics degrade as well (13, 11, 12), though often in a non-
intuitive, non-linear, and depth dependent manner, until eventually the tissue
fails catastrophically. Importantly, understanding the path towards mechanical
failure in cartilage requires knowledge of how the collagen and aggrecan
networks contribute to function under compression and shear in both healthy
and damaged tissue.

A major step towards understanding how the collagen and aggrecan networks
contribute to the compressive properties of cartilage were the developments of
poroelastic (14) and mixture theories (15), which account for water movement
through the extracellular matrix, as well as additional theories that describe
electrostatic/osmotic contributions to compressive mechanics (16, 17). These
theories have been useful quantitative frameworks (14) to understand
experimental observations describing how damage to the charged aggrecan
network makes the cartilage more susceptible to compression. As the aggrecan
network degrades, the osmotic stress driving tissue hydration dissipates and
the drainage time-scale is significantly shortened due to larger effective
pores in the extracellular matrix. This prediction is confirmed by studies
that used enzymes such as chondroitinase abc and trypsin to model degradation
of the aggrecan network and found that the bulk compressive modulus of a
cartilage explant can decrease by up to 50$\%$ (18) and that the tissue
hydraulic permeability can increase up to 15 fold (10). Due to their
quantitative predictive power, such theories have been important tools for
understanding the compressive mechanics of healthy and damaged cartilage.

While these theories are effective at predicting the compressive behavior of
the tissue, much less attention has been paid to the shear behavior and its
dependence on tissue composition. Pioneering work has demonstrated that
electrostatic contributions from the aggrecan network account for a
significant portion of the shear modulus of healthy cartilage (19).
Additionally, it has been shown that the concentration-dependent shear
properties of the collagen network are well described by a rigidity
percolation model. Notably, however, a quantitative framework for the combined
contributions of the collagen and aggrecan networks to the shear modulus of
healthy and degraded cartilage does not exist. Because the shear mechanics of
cartilage are of critical importance to its function in joints and because
shear is the most common mode of failure for this tissue, developing such a
framework to understand the shear mechanics of cartilage and its dependence on
both the collagen and aggrecan constituents has been a major goal of the
biomechanics community.

Here, we build on previous work (9) to develop a complete rigidity percolation
framework for understanding the structural origins of cartilage shear
mechanics (Fig. 1). This framework reveals that describing the shear modulus
of cartilage requires accounting for the critical interplay between the
primary collagen and reinforcing aggrecan networks. It predicts that the shear
modulus of cartilage is governed by how close the composite network is to the
rigidity percolation threshold. Near this threshold, small compositional
changes in either aggrecan or collagen drive large changes in shear mechanics,
giving rise to a phase transition between a healthy, percolated, and
mechanically competent network and a degraded, sparse, and mechanically
incompetent network that can no longer sustain any shear loads. Such
predictions yield particularly important insights for the mechanical behavior
of the tissue near the cartilage surface where the collagen network is near
the rigidity percolation threshold (9) and minor changes to the aggrecan
network translate into large changes in the shear modulus (20).

This paper combines novel experiments measuring local composition and shear
mechanics with simulations of rigidity percolation phase transitions to
develop and validate this model. Ultimately, this framework will enable
predictions of how alterations in tissue structure and composition drive
changes in mechanics that occur over the course of disease with the potential
to inform diagnosis and therapy.

Figure 1: Shear Modulus Dependence on Collagen and Aggrecan Concentrations:
Experiments and Rigidity Percolation Framework Prediction. The data points are
experimental measurements of the modulus, measured via confocal elastography,
as a function of collagen and aggrecan concentrations, measured via Fourier
Transform Infrared Imaging. The surface depicts the theoretical prediction
from the rigidity percolation model.

##  1 Depth Dependent Composition and Mechanics

The complexity of the behavior predicted by the rigidity percolation model
requires a large amount of data relating matched measurements of collagen
concentration, aggrecan concentration, and shear modulus. To gather this data
we apply techniques that we developed to measure and register composition and
shear properties of cartilage on the microscale (9, 21, 22). Specifically, we
use a combination of Fourier Transform Infrared Imaging and confocal
elastography on matched samples of bovine cartilage as schematized in figure
2. Importantly, the extracellular matrix of cartilage shows significant
variation in collagen concentration primarily near the tissue surface (9). By
degrading the aggrecan molecules in this region we can obtain structure
function relationships for the extracellular matrix that span concentrations
in both the collagen and aggrecan networks. Towards this end, we harvested
cartilage tissue plugs from femoral condyles and subjected a subset of the
samples to tissue degradation with trypsin, which degrades the aggrecan core
protein so that aggrecan leaves the tissue (10). We then bisected all the
tissue plugs. We measured the local tissue composition in one hemi-cylinder
using histology and FTIR-I (9). We measured the depth dependent shear modulus
for the matching hemi-cylinder using confocal elastography (23, 24). Using
these measurements on healthy and degraded tissue we were able to quantify,
the dependence of the modulus on a wide range of collagen and aggrecan
compositions.

Figure 2: Experimental Protocols. Samples are extracted from medial and
lateral condyles of neonatal bovid. A subset of the samples are covered in an
epoxy coating leaving the surface exposed and submerged in a bath of trypsin
for 5 hours at 37∘C. This subset constitutes the degraded samples. All samples
are then bisected, with half being used for local compositional analysis with
FTIR-I, and the other half being used for local mechanical analysis with
confocal elastography. After mechanical testing samples are prepared for
biochemical assays used to measure the absolute concentration of the
constituents. These measurements are used to calibrate the FTIR data to
determine the absolute concentrations of collagen and aggrecan. Figure 3:
Matched Structural and Mechanical Measurements. A) Composition measurements.
top: Sarafin-O stained histology slides for the healthy and degraded tissue.
The areas stained red show regions of proteoglycan content. The yellow box
indicates one of three sample spanning sections where the FTIR spectra were
taken from, as well as three representative regions at depths $z$=100 µm,
$z$=1000 µm, and $z$=2000 µm for which the FTIR spectra are shown. middle:
Measured absorbance spectra along with the fitted aggrecan and collagen
contributions. The best fit spectra is also shown in dashed lines. bottom:
Collagen and aggrecan concentrations versus depth. B) Mechanical Measurements.
top: Lines are photobleached perpendicular to the tissue surface and shear
oscillations are applied parallel to the surface (Methods). middle: By
tracking the photobleached lines we extract the local strain within the
tissue. bottom: These data are combined with measurements of the sample
surface area and the total force needed to deform the tissue to extract the
shear modulus as a function of depth. The same procedures are repeated for the
degraded data to extract C) the collagen and aggrecan concentrations and D)
the shear modulus for the degraded tissue as a function of depth.

###  1.1 Healthy Tissue

Results of our depth dependent histology and Fourier Transform InfraRed
Imaging (FTIR-I) measurements for healthy tissue are shown in figure 3A.
Safranin-O sections showed healthy cartilage morphology with increasing
staining with tissue depth suggesting higher aggrecan concentrations deeper in
the tissue (Methods). For each sample, the FTIR spectra as a function of depth
was measured as described in the Methods section. Sample FTIR spectra at three
different tissue depths ($z$=100 µm, $z$=1000 µm, and $z$=2000 µm) are shown
in the spectrograms in figure 3A. Plotted are the absorption of infrared light
as a function of wave number. The measured spectrum (red) is fit (dashed) by a
sum of the contributions from the collagen spectrum (yellow), the aggrecan
spectrum (orange), and a linear background (not shown) as described in the
Methods section. We obtain excellent fits for all the spectra with the
aggrecan spectrum contributing most notably to the peak at 1050 cm-1.
Consistent with the histology results, we find that the aggrecan contribution
to the spectra is negligible at the tissue surface and increases with depth.

From the best fit coefficients and whole-tissue biochemical assays performed
on the samples (see Methods Section F for details of the assays), we
determined the collagen and aggrecan concentrations with depth (bottom of Fig.
3A). We find that the collagen concentration shows a typical dip just below
the cartilage surface (9). Specifically, we find that the concentration is 150
mg/ml at the surface, decreases by nearly 30$\%$ in the first 100 µm and then
increases again plateauing at 175 mg/ml beyond 800 µm. We find that the
aggrecan concentration monotonically increases from having a concentration of
0 mg/ml at the surface to a concentration of nearly 50 mg/ml in the deep zone.

We used confocal elastography to determine the depth dependent shear modulus
of the matching hemi-cylinder (Fig. 3B). Briefly, we photobleached lines
perpendicular to the articular surface and tracked their displacements with
depth as described in the Methods section. The change in displacement over a
given depth is used to determine the depth dependent strain. Using the
extracted strain, surface cross section of the sample, and measurement of the
total shear force, we determined the shear modulus $\it{G^{*}}$ (Methods). The
measured modulus shows a typical response with a value of about 20 KPa at the
surface that increases by almost two orders of magnitude over the first 500 µm
and remains roughly constant at greater depths (23, 25, 26).

###  1.2 Degraded Tissue

Repeating this analysis on degraded tissue (Fig. 3C), we find that the
concentration profile of collagen remains unchanged while the aggrecan is
completely removed throughout the degraded region. From histology, we observe
that aggrecan removal proceeds as a front that is parallel to the tissue
surface. The degraded region appears white or light blue while the region of
tissue below the degradation front appears deep red. These results are
consistent with previous histological measurements of trypsin degraded
cartilage samples (20).

For each sample, the FTIR spectra as a function of depth was measured as
described in the Methods section. Sample FTIR spectra at three different
tissue depths ($z$=100 µm, $z$=1000 µm, and $z$=2000 µm) are shown in Fig. 3C.
Once again, we obtain excellent fits for all the spectra. Consistent with the
histology results, we find that the aggrecan contribution to the spectra is
negligible in the degraded region as indicated by the absence of the peak at
1050 cm-1. From these data and the whole tissue biochemical assay we
determined the collagen and aggrecan concentrations with depth. We find that
the depth dependent collagen concentration remains similar to that of healthy
tissue with a slight dip in concentration just below the surface and higher
concentrations in the deeper regions. We find that the aggrecan concentration
is close to zero in the degraded region and rises monotonically to levels
similar to those in healthy tissues over a 200 µm region at a depth of 1300
µm.

As a result of this degradation we observe distinct changes to the depth
dependent shear modulus of the matching hemi-cylinder (Fig. 3D). We observe
that the shear modulus at the surface is lower than in the healthy sample. For
depths 400 µm $<z<$ 1000 µm we observe an intermediate value of the modulus
that is about an order of magnitude larger than that at the surface and an
order of magnitude lower than the modulus in the deep zone. The modulus in the
deep zone is very similar to that of healthy tissue, because the degradation
front never reaches this region.

###  1.3 Average Depth Dependent Composition and Mechanics

These trends in the tissue composition and mechanics for the healthy and
degraded tissues hold when averaged across multiple samples (N=13 healthy, N=8
degraded). We plot the average collagen composition versus depth for both
healthy and degraded samples in Fig. 4A. We find no statistical difference
between the healthy and degraded conditions. We plot the average aggrecan
concentration versus depth for both healthy and degraded samples in Fig. 4B.
As in the single sample data in Fig. 3, we find that aggrecan is completely
removed from the fully degraded region for depths $z<$ 1200 µm, indicating
that our degradation protocol (exposure to trypsin at 0.25% for 5 hours at 37
∘C) produces consistent results. Following this fully degraded region, we
observe a transition region where the aggrecan rapidly increases in
concentration until it reaches that of the healthy tissue and the two samples
are statistically indistinguishable. We plot the shear moduli for the healthy
and degraded samples in Fig. 4C. We observe a consistent downward shift of
roughly half an order of magnitude between the averages of the healthy and
degraded samples. Moreover, the averages for the degraded samples continue to
show an intermediate plateau region with a value between those of the surface
and deep zones. While the modulus reduction is roughly constant across the
degraded region, the difference in aggrecan concentration between the healthy
and degraded tissues varies substantially. These data indicate that the
dependence of the modulus on aggrecan varies with depth and depends on
collagen concentration.

###  1.4 Structure Function Relationships

To determine the dependence of the modulus on both the aggrecan and collagen
concentrations, we plot the local shear modulus as a function of collagen
concentration for both the healthy (blue) and aggrecan degraded (red) tissues
in Fig. 4D. For the degraded tissue, we only include data with aggrecan
concentration below 2 mg/ml since above this value the tissue has not been
degraded and has the properties of healthy tissue. We find a uniform factor of
5 decrease in the modulus for the degraded tissue.

Figure 4: Depth Dependent Results and and Shear Modulus Dependence on Collagen
Concentration. A) Collagen concentration with depth. Near the tissue surface
there is significant variation in collagen concentration, until it plateaus
after a depth of 1000 µm. There is no significant difference between the
healthy and degraded samples. B) Aggrecan concentration with depth. In the
healthy samples, there is a natural increase in aggrecan concentration
throughout the depth of the tissue. In the degraded samples, the aggrecan
concentration is zero until the depth of the degradation front, where it
returns to a similar concentration as the healthy samples. C) Shear Modulus
with depth. The shear modulus at the surface of the degraded tissue is lower
than in the healthy tissue. In the region 400 µm $<z<$ 1000 µm, there is an
intermediate value of the modulus that is about an order of magnitude larger
than that at the surface and half an order of magnitude lower than the modulus
in the deep zone. The modulus in the deep zone is very similar to that of
healthy tissue, because the degradation front never reaches this region. D)
Shear modulus as a function of collagen concentration. The depth dependent
measurements of the shear modulus, collagen concentration, and aggrecan
concentration are combined to create a plot of shear modulus versus collagen
concentration for both the healthy and the degraded tissue. The degraded
tissue modulus is lower by close to a factor of 5. E) The modulus
amplification factor as a function of collagen concentration:
(Log$G^{*}_{Healthy}$ \- Log$G^{*}_{Degraded}$)/($\Delta$ Aggrecan
Concentration). The contribution of the aggrecan to the shear modulus is
highest at low collagen concentrations and rapidly decreases with increasing
collagen concentration.

Importantly, to ascertain the sensitivity of the modulus to aggrecan, we must
account for the fact that the aggrecan concentration varies by almost an order
of magnitude across these data sets. For example, at high collagen
concentrations associated with the deep zone, reducing the aggrecan
concentration from 30 mg/ml to zero reduces the modulus by a factor of 5.
Remarkably, the same factor of 5 reduction in modulus results from reducing
the aggrecan concentration from 3 mg/ml to zero near the surface region. To
understand this dependence we define the aggrecan amplification factor as the
difference in the Log of the healthy shear modulus minus Log of the degraded
shear modulus divided by the change in aggrecan concentration. When the
amplification factor is high, aggrecan plays a more important role in
contributing to the shear modulus of the tissue. Conversely, when the
amplification factor is low, the aggrecan concentration has a diminished role
in determining the modulus.

We plot the modulus amplification factor as a function of collagen
concentrations in Fig. 4E. We find that the contribution of aggrecan to the
shear modulus is highest at low collagen concentrations and decreases by over
an order of magnitude over a less than two fold change in collagen
concentration. These measurements are consistent with a rigidity percolation
framework to explain the shear mechanics of cartilage. At low collagen
concentrations the network is less likely to percolate on its own and addition
of aggrecan increases the likelihood of percolation and hence dramatically
increases the shear modulus. At higher concentrations of collagen where the
network is more likely to be percolated the presence of aggrecan makes a much
smaller contribution to the shear modulus of the tissue.

###  1.5 Rigidity Percolation model

To quantitatively assess the degree to which our framework can describe both
the healthy tissue and aggrecan depleted tissue moduli, we fit the data in
Figs. 4D,E to a rigidity percolation model. The model consists of a disordered
kagome lattice representing the stiff primary collagen network embedded in a
continuum elastic background gel representing the reinforcing aggrecan and
hyaluronic acid networks (Fig. 5). The fibers in the collagen network are
randomly removed according to given probabilities $p$, where $0<p<1$. Each
bond is characterized by a stretching modulus $\alpha$ and a bending modulus
$\kappa$. In a manner similar to (9) we include a background gel of aggrecan
and other matrix components that resists bond deformations in the transverse
direction and has an elastic modulus $\mu$. Here, however, to account for the
fact that trypsin primarily affects mechanics through degradation of the
aggrecan, we take $\mu$ to linearly depend on the concentrations of aggrecan,
hyaluronic acid, and collagen (See Methods). The energy cost of deforming this
composite network in the linear response regime is given by:

$\displaystyle E$ $\displaystyle=$ $\displaystyle\frac{\alpha}{2}\sum_{\langle
ij\rangle}p_{ij}\left(\textbf{u}_{ij}\cdot\hat{\textbf{r}}_{ij}\right)^{2}$
$\displaystyle+\frac{\kappa}{2}\sum_{\langle
ijk\rangle}p_{ij}p_{jk}\left[\left(\textbf{u}_{ji}+\textbf{u}_{jk}\right)\times\hat{\textbf{r}}_{ji}\right]^{2}$
$\displaystyle+\frac{\mu}{2}\sum_{\langle
ij\rangle}p_{ij}\left[\textbf{u}_{ij}^{2}-\left(\textbf{u}_{ij}\cdot\hat{\textbf{r}}_{ij}\right)^{2}\right]+\frac{\mu}{2}\epsilon_{s}^{2}A,$

where the terms correspond to the energy penalty for fiber stretching, fiber
bending, the coupling of the network to the background gel, and the
deformation of the background gel, respectively. In the final term,
$\epsilon_{s}$ denotes an affine background shear strain, while $A$ is the
area of the network. The indices $i,j,k$ refer to sites (nodes) in the
lattice-based network, such that $p_{ij}$ is 1 or 0 when a bond between those
lattice sites is or is not present. The quantities $\hat{\textbf{r}}_{ij}$ and
$\textbf{u}_{ij}=\textbf{u}_{i}-\textbf{u}_{j}$ are respectively the unit
vector along a bond $ij$ and the corresponding relative displacement.

To determine the network properties we strain the network, use conjugate
gradient minimization of the deformation energy (Eq. 1.5) to mechanically
equilibrate the network, and extract the shear modulus (See Methods).
Specifically, for each set of parameters ($\alpha$,$\kappa$,$\mu$), a network
containing $\sim 10^{5}$ nodes was randomly generated with a fraction $1-p$ of
bonds missing, subjected to a compressive strain of $5\%$ and a shear strain
of $1\%$ applied via the top boundary, and then allowed to relax via fiber
deformations, with periodic boundary conditions imposed along the left and
right sides of the network. With these simulations we obtained the modulus as
a function of the bond occupation probability $p$ and the gel modulus $\mu$.

Figure 5: Rigidity Percolation Model. Shown is a schematic of a portion of the
simulated kagome network. Hash marks indicate continuous boundary conditions
in the lateral dimension. The black links represent the collagen fibers and
the gray represents the background aggrecan gel. The links or bonds in the
network are characterized by a stretching modulus $\alpha$ and a bending
stiffness $\kappa$. The background gel further couples the links in the
network through inhibition of transverse transport. This gel is characterized
by a modulus $\mu$.

To map the results of the simulation to experiments, we linearly scale the
network modulus, bond occupation probability $p$, and gel modulus $\mu$ to the
experimentally measured modulus, collagen concentration, and aggrecan
concentration, respectively. We then fit the model to the collagen dependent
modulus and amplification factor data in Figs. 4D and 4E and find excellent
agreement. In all, the rigidity percolation predictions for both the healthy
and degraded data sets are obtained using 5 fitting parameters. Moreover, the
fits for the bending to stretching ratio $\kappa/\alpha$ and the modulus of
the bare hyaluronic acid gel are in very close agreement with literature
values (27, 28, 29, 19). Importantly, once these parameters are fit to the
data, it is possible to use this rigidity percolation framework to
quantitatively predict the modulus for arbitrary values of the collagen and
aggrecan concentrations as shown in Fig. 1.

##  2 Discussion

By collecting a large amount of matched experimental data for local
composition and local shear mechanics and fitting them using simulations, we
have constructed a rigidity percolation framework that provides valuable
insights into the dependence of cartilage mechanical properties on the tissue
constituents (Fig. 1). Our measurements and simulations show that the
contribution of aggrecan to the shear modulus is highly dependent on the
concentration of collagen. When the collagen concentration is high, the
aggrecan provides a relatively minuscule contribution to the modulus.
Conversely, aggrecan plays a critical role in enhancing the shear properties
of cartilage in regions of the extracellular matrix where collagen
concentration is low (Fig. 4E). By capturing the interactions between the
supporting aggrecan gel and the collagen network, the rigidity percolation
framework provides important intuition for the origins of this non-linear and
unusual behavior. When the collagen network is sufficiently concentrated that
it percolates by itself, the relative contribution of aggrecan is small.
Conversely, when the collagen network is close to the rigidity percolation
threshold, even a small reinforcement of the network by aggrecan makes it
easier for the composite network to transmit stresses. In combination, the
model and experimental data provide a powerful framework for understanding how
the shear mechanics of cartilage arise from the interactions of the collagen
and aggrecan networks.

This rigidity percolation framework for describing how collagen and aggrecan
interact to determine the shear mechanics of articular cartilage is remarkably
effective despite its simplicity. The model uses a conventional 2D Kagome
lattice structure to represent the crosslinked collagen network that is
coupled to a background reinforcing network representing the contributions of
aggrecan and other matrix components. The contributions of the collagen
network concentration, connectivity, and crosslinking, are effectively
described by the bond occupation probability, $p$, and the lattice structure.
Future studies using different enzymes that target these structural aspects of
the collagen network would be helpful in elucidating their effective relative
contributions to $p$.

Importantly, a critical contribution of the reinforcing background gel is to
provide a coupling between the fibers in the network. As such, its
contribution is not only additive as has been suggested in prior literature
(19). As shown in the supplementary information (SI), when we exclude the
third term in Eq. 1.5, which couples the network deformations to the
background gel, we obtain inferior fits and unreasonable values for the
fitting parameters. These include a bending to stretching ratio that
approaches 1 and a value for the hyaluronic acid gel contribution to the
modulus $\sim 1$ Pa, which is three orders of magnitude too small (6). These
findings highlight that the aggrecan gel can dramatically alter the shear
properties of cartilage by helping to drive the composite network through its
rigidity percolation transition. This mechanism for controlling the tissue
properties is important biologically since the turnover rate for aggrecan is
orders of magnitude faster than the turnover rate for collagen. As such,
fabrication and degradation of aggrecan by chondrocytes, the cells in
cartilage, can be used to rapidly alter the tissue shear mechanics in response
to changes in shear loads.

A model that describes the dependence of cartilage shear mechanics on
composition is also a powerful tool for understanding the progression of
diseases of cartilage such as arthritis. Such diseases develop over many years
and are typically characterized by slow degradation of one or more components
of the extracellular matrix, but at late stages often result in rapid loss of
joint function due to compromised cartilage mechanics. In the context of the
proposed framework, this rapid loss of function may arise from the rapid
decrease in mechanical integrity associated with crossing the rigidity
percolation threshold. Recent advances in clinical MRI of cartilage tissue
enable mapping of collagen and aggrecan composition and organization in vivo
(30). Relating such measurements to tissue and joint scale mechanics would be
a major advance in the field. The rigidity percolation framework presented
here provides an opportunity to accomplish this important goal and may even
allow for identifying tipping points during disease progression where small
additional changes in composition lead to tissue failure. This understanding,
could be critical for informing treatment by identifying stages of disease
progression that are most in danger of compromising function due to loss of
collagen, aggrecan, or both.

Additionally, this framework provides new insights for understanding and
designing cartilage therapies. Specifically, regenerative medicine approaches
frequently involve delivery of cells alone (31) or in combination with weak
scaffolds (32, 33) to promote cartilage regeneration. Critical to the success
of such approaches is the generation of a new mechanically competent
extracellular matrix. Cells alone or those embedded in sparse matrices are
likely far below the rigidity percolation threshold and, as such, will take a
significant amount of time to achieve mechanical competence. The framework
described here suggests that designing implants close to the rigidity
percolation threshold will maximize the impact of cellular matrix biosynthesis
on their mechanical performance.

##  3 Conclusions

In this work we developed a rigidity percolation framework to understand the
structural origins of cartilage shear mechanics based on the composition and
interactions of the collagen network and the reinforcing aggrecan gel that
together account for the majority of the tissue extracellular matrix
properties. Since these are also the main constituents of all extracellular
matrices in connective tissue in mammals, this framework and its extensions
into nonlinear deformation regimes may be a widely applicable tool for
understanding the mechanics of many if not all connective tissues in health,
disease, and repair (33). Similarly, this framework may also prove useful for
understanding artificial constructs with tissue-like properties (33). More
broadly, this work illustrates a striking example of how biology exploits
compositional perturbations driving physical processes at the proximity of
phase transitions to achieve remarkable function in tissue homeostasis,
disease, and repair.

###  3.1 Tissue Harvesting

One- to three-day old neonatal bovine knee joints were acquired from a local
abattoir (Gold Medal Packing, Rome, NY). Neonatal cartilage was chosen due to
its similarity in shear modulus profile to human cartilage (34, 24), as well
as ease of access, consistency of tissue samples, and prior use in similar
studies (23, 24, 25, 26, 20, 9). Cylindrical explants 3 mm in height and 6 mm
in diameter were dissected from the medial and lateral femoral condyles.

###  3.2 Enzymatic Degradation

Samples were enzymatically degraded by immersing them in a bath of trypsin-
EDTA 0.25% (Sigma) at 37°C for five hours (Fig. 2). Epoxy glue was used to
create a protective coating around the sample edges, with the top surface of
the tissue left exposed. This procedure allowed for creating a well-controlled
degradation front. Trypsin was chosen for its ability to primarily affect
mechanics through degradation of the aggrecan and minimal effect on the
collagen network (35). Post degradation, the samples were rinsed with PBS, the
glue was peeled off, and the samples were bisected into two hemi-cylinders.
One hemi-cylinder was placed in protease inhibitors for mechanical testing,
and the other was fixed in 10% PBS-buffered formalin for compositional
analysis via Fourier Transform InfraRed Imaging (FTIR-I).

###  3.3 Histology

Qualitative analysis of aggrecan removal was conducted via Safranin-O staining
as described previously (20). Tissue samples were fixed in neutral buffered
formalin, embedded in paraffin, cut into 4 µm thick sections, and placed onto
glass slides. Sections were dewaxed in three xlyene baths for 2 min each and
rehydrated in three baths of ethyl alcohol (100, 95, and 70% ethanol,
appropriately diluted with distilled water) for 2 min each. The nuclei were
stained with Weigert’s Iron hematoxylin for 10 min, then the samples were
rinsed for 10 min in running water. The samples were then stained with fast
green solution for 5 min, and rinsed with 1% acetic acid solution for 15
seconds. Finally the samples were stained with 0.1% Safranin-O, pH 2.0,
solution for 8 min.

###  3.4 Compositional Measurements

Quantitative measurements of collagen and aggrecan relative concentrations
were obtained via Fourier Transform InfraRed Imaging similarly to previous
studies (36, 37, 9, 21, 22). Sections, 4 µm thick from each tissue sample,
were placed on 2-mm-thick mid-infrared (IR) transparent BaF2 disks that were
25 mm in diameter (Spectral Systems, Hopewell Junction, NY). Although the
sectioning plane was fixed, the section cutting direction was randomized to
prevent systematic biases in section thickness due to cutting. Sections were
dewaxed in three xylene baths for 2 min each and rehydrated in three baths of
ethyl alcohol (100, 95, and 70% ethanol, appropriately diluted with distilled
water) for 2 min each.

Samples were loaded into a Hyperion 2000 Fourier Transform InfraRed Imaging
(FTIR-I) microscope (Bruker, Billerica, MA) in transmission mode and we
acquired data for wavenumbers between 600 and 4000 cm-1 with a resolution of 4
cm-1. A 15x objective was used with a slit aperture configured to acquire
spectra over a rectangular region 25 µm x 200 µm, where the long dimension was
parallel to the articular surface. Fifteen background-corrected scans were
repeated at a given measurement point and averaged to generate a single IR
spectrum. The acquisition window was scanned along the tissue sample’s depth
at 25 µm intervals by a computer-controlled stage to acquire measurements
throughout the depth of the tissue. This procedure was repeated three times
for each tissue sample with each scan separated laterally by roughly 2 mm
taking care to avoid blood vessels and other artifacts.

We obtained the relative concentrations of aggrecan and collagen from the
measured spectra by fitting the spectral window between 900 to 1725cm-1 (9,
22) for each measurement point with a linear combination of previously
measured spectra for collagen and aggrecan (37), as well as a linear
background. This method is based on Beer-Lambert’s law, which states that IR
absorbance is proportional to molecular concentration, and two mixed species
of molecules have additive contributions. Including a linear background
correction accounts for the instrument-specific deviations and drift which can
occur in different labs for different environmental conditions (38). In
fitting the spectra, we constrained the relative concentration of aggrecan and
collagen to be greater than zero and assume that the linear background is the
same throughout each sample. We did not consider the trace amounts of type IX
collagen, type XI collagen, elastin, small nonaggregating proteoglycans, and
other matrix macromolecules which occur in cartilage as their limited
contributions to the absorption spectrum were negligible (39). The aggrecan
and collagen concentration coefficients of the scans were then averaged for
each depth, to determine the relative collagen and aggrecan concentrations as
a function of depth.

###  3.5 Mechanical Measurements

To prepare for mechanical testing, each hemi-cylindrical sample was soaked for
1h in PBS with 7 mg/mL 5-DTAF (5-dichlorotriazinylaminofluorescein) (Life
Technologies, Carlsbad, CA), an all-protein stain. Samples were rinsed in PBS
for 30 minutes to remove excess dye, then loaded into a Tissue Deformation
Imaging Stage (Harrick Scientific, Pleasantville, NY). In this apparatus, the
sample is gripped between two plates. Shear oscillations are applied to the
surface of the sample, and the displacement of the second plate is used to
measure shear stresses (24, 25). The sample was held in place between the
shearing plates using cyanoacrylate glue to minimize the need for compression.
Samples were immersed in PBS to maintain tissue hydration during mechanical
testing.

The Tissue Deformation Imaging Stage (Harrick Scientific) was mounted onto an
inverted LSM 5 Live confocal microscope (Carl Zeiss, Jena, Germany), where
five lines were photobleached onto the rectangular surface of the hemi-
cylinder perpendicular to the tissue surface. These lines caused no damage to
the tissue and were used to facilitate automated computer tracking of the
strain with a depthwise resolution of 10.4 µm. Imaging of 1 Hz oscillatory
shear at 1% peak strain amplitude was carried out with a 10x objective, and
movies of the tissue oscillations were acquired at 20 FPS throughout the
entire depth of the tissue (Fig 2). All mechanical testing was performed
within 48 h of tissue harvest. The samples were retrieved after mechanical
testing and frozen for biochemical assay.

Importantly, in order to isolate the effects of aggrecan on the tissue
properties, our protocol entailed conducting our shear experiments under no
compression. This procedure was necessary to avoid any effects due to buckling
of the collagen network (23), which would be much more localized in the
degraded tissue. The main consequence of this procedure was that we observed
much smaller changes in the moduli and energy dissipation for the degraded
tissue than those observed in (20, 9). Avoiding this complication was also
important for testing the rigidity percolation model since the model does not
yet account for buckling of collagen fibers under compression. Since the
normal function of cartilage tissue does entail compression, extending the
model to address this regime would also be important.

Automated tracking of the local shear strain was facilitated by the five
photobleached lines along the depth of the tissue. A custom Matlab code
tracked the amplitude of the oscillations of the tissue, and the local shear
strain was calculated by numerically differentiating the displacement with
respect to the depth (25). The strain curves were fit to a sinusoidal
function. The force applied to the tissue was able to be calculated by
observing the displacement of the stationary shearing plate of a known
stiffness. The depth-independent shear stress was able to be calculated by
dividing this force by the surface cross section of the tissue. The shear
modulus is this shear stress divided by the local shear strain. The difference
in phase of oscillations between the local region and the applied stress are
also recorded, which provides a measure of the viscous shear modulus relative
to the total shear modulus.

###  3.6 Biochemical Assay

To determine the absolute concentrations of collagen and aggrecan in a tissue
sample, we calibrated the coefficients obtained from the FTIR-I measurements
using biochemical analysis (40). Mechanically tested hemi-cylinders were
weighed for their wet weight, and frozen at -80∘C. The samples were dehydrated
in a lyophilizer for 48 hours and then weighed again to obtain the dry weight.
The samples were incubated in papain digest buffer (PDB) made from 125 mg/mL
papain (Sigma) and 10 mM N-acetyl cysteine (Sigma) in PBD buffer (100 mM
phosphate and 10 mM EDTA, pH 6.5) for 12 hours at 60 ∘C.

To measure the aggrecan concentration, GAG standards were created by
dissolving chondroitin sulfate in PBD to known concentrations from 2 µg/ml to
250 µg/ml. 50 µL of each standard and 50 µL of the digest solution were added
to a 96 well plate. 250 µL of pH3 DMMB dye (Sigma) were added to each well.
The plate was shaken for 30 seconds then the absorbance was read at 525 nm.
The absorbance of the samples were matched to the absorbance of the known
standards.

To measure the collagen concentration, hydroxyproline standards were created
by dissolving hydroxyproline in PBD to known concentrations from 2 µg/ml to
1000 µg/ml. 50 µL of 2M NaOH was added to 50 µL of each standard and samples.
The standards and samples were heated to 110 ∘C for 18 hours. 30.5 µL of HCl
solution, 100 µL 0.001 M CuSO4, 100 µL 2.5M NaOH, 100 µL 6$\%$ H2O2, were each
added to the standards and samples, ensuring to vortex after each addition.
The standards and samples were allowed to rest at room temperature for 2
hours. The samples were vortexed and heated to 80 ∘C for 5 minutes, then
frozen. 400 µL 3 M H2SO4 was added to each standard and sample, then they were
frozen again. 200 µL of DMAB (Sigma) were added to each standard and sample,
then they were heated at 70 ∘C for 15 minutes. 200 µL of each standard and
sample were added to a 96 well plate, and the absorbance was read at 540nm.
The absorbance of the sample was matched to the absorbance of the known
standards. Collagen is approximately 13.5$\%$ hydroxyproline, so the measure
of hydroxyproline was multiplied by 7.4 to find the concentration of collagen
in the sample.

The total concentrations of collagen and chondroitin sulfate were taken from
the biochemical assays and divided by the volumes of each sample, approximated
by the depth of the sample measured during mechanical testing multiplied by
its surface cross section. The concentrations were averaged for the healthy
samples, and the degraded samples, and the FTIR-I curves were calibrated to
the total average concentrations (22).

###  3.7 Mathematical Model

Rigidity percolation theories model biopolymer networks as disordered fiber
networks and provide a framework to connect their rigidity to the network
structure, composition, and single filament properties. These models have been
immensely successful in predicting the mechanical properties and phase
transitions of in vitro cytoskeletal and extracellular matrix networks as a
function of filament concentrations. Previously, we combined this framework
with a lattice-based disordered fiber network reinforced by a background gel
to explain the depth-dependent shear properties of AC(9). In this framework,
the shear modulus of the tissue was taken to be a sum of a background shear
modulus, $\mu$, and a simulated shear modulus,
$G_{sim}(\kappa/\alpha,\mu/\alpha,p)$, where $\kappa$ and $\alpha$ denote the
bending and the stretching moduli of the collagen fibers, and $p$ is the bond
occupation probability of the collagen network, respectively. Here we adapted
this model to predict the mechanical response of both healthy and enzyme
degraded articular cartilage.

Since degradation with trypsin removes the aggrecan leaving the background
hyaluronic acid network nearly intact, we extended the model to describe
separate contributions of aggrecan and the associated hyaluronic acid network.
Furthermore, the background modulus also includes mechanical reinforcement
from collagen fibers that do not participate in the formation of a percolated
network. Therefore, we expand $\mu$ about the bare HA value $\mu_{0}$, to
linear order in aggrecan and collagen concentrations, $\rho_{a}$ and
$\rho_{c}$, respectively, as $\mu=\mu_{0}+\beta\rho_{a}+\gamma\rho_{c}$, where
$\beta$ and $\gamma$ are expansion coefficients capturing the reinforcement of
the bare hyaluronic acid network by aggrecan and collagen. We computed the
full shear modulus as

$G=c\Big{[}\mu+G_{network}(\kappa/\alpha,\mu/\alpha,p)\Big{]},$ (2)

where $c$ is a scaling factor from simulation to experimental units, and the
network modulus $G_{sim}$ was obtained from the simulations by minimizing the
deformation energy density of the system, $\mathcal{E}$, under a given small
strain $\epsilon_{s}$ and calculating
$\frac{\partial^{2}\mathcal{E}}{\partial{\epsilon_{s}}^{2}}$ (See SI for model
construction and simulation details).

###  3.8 Determination of Model Parameters

We assumed the collagen fibers to be rods with a cross-sectional radius $r\sim
10$nm (27) and Young’s modulus $E\sim 1$ GPa (28). In the simulations, lengths
and displacements were scaled by the bond length $l_{c}$, and rigidities were
scaled by the fiber stretching modulus $\alpha$. This meant that for a fiber
of length $l_{c}$, $\alpha\sim Er^{2}/l_{c}$ and the scaled bending rigidity
$\kappa\sim Er^{4}/l_{c}^{3}$, the bending to stretch ratio was given by
$\kappa/\alpha\sim r^{2}/l_{c}^{2}\sim 10^{-4}$ for $l_{c}=1\mu$m. (9). The
elasticity of a densely connected network is stretching-dominated, and we
expected the shear modulus to scale as
$G\sim\frac{\alpha}{l_{c}}\sim\frac{Er^{2}}{l_{c}^{2}}\sim 10^{5}$ Pa, setting
the order of magnitude for $c$. To inform an estimate of the stiffness for the
HA gel, we used the data of Holmes, et al. (6), who found the concentration of
hyaluronic acid in articular cartilage to be of the order of 1 mg/ml, with a
molecular mass on the order of 1 MDa. The classical estimate for a shear
modulus of an aggregate of Gaussian chains, with a concentration of $c$ chains
per unit volume is $\sim k_{B}Tc$ (41), yielding a shear modulus of the order
of 1 KPa. These results suggested the normalized ratio of shear modulus to
stretching stiffness, $\mu_{0}l_{c}/\alpha$ should be $\sim 10^{-2}$. As the
product of $c$ and $\mu_{0}$ should yield a modulus of the order $10^{3}$ Pa,
this separately suggests $c$ should be of order $10^{5}$.

We fitted for $c$, $\beta$, $\gamma$, $\mu_{0}$ and $\kappa$, using the above
considerations to constrain our search. Parameters were chosen using
$\chi^{2}$ minimization via the Nelder-Mead simplex algorithm in Mathematica.
We found optimal values of $c=2.33\times 10^{6}$ Pa, $\mu_{0}=.001$,
$\beta=2.52\times 10^{-3}$ ml/mg, $\gamma=1.49\times 10^{-4}$ ml/mg,
$\kappa=.01$ and $\delta=.0052$, with $\chi^{2}/D.O.F=3.58$). The $\chi^{2}$
value for each data bin was computed using the log base 10 of the experimental
and model shear moduli. See SI for further details.

This work was supported by the National Science Foundation grant DMR-1807602,
DMR-1808026, CBET-1604712, CMMI 1927197, and BMMB-1536463. This work was also
supported by the NIH National Institute of Arthritis and Musculoskeletal and
Skin Diseases, Contract: 5R01AR071394-04. Finally, this work made use of the
Cornell Center for Materials Research Facilities supported by the National
Science Foundation under Award Number DMR-1719875.

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