## 2 Materials and Methods  
### Collagen Fibril Coarse-Grained Model  
The purpose of the molecular dynamic simulation was to obtain stress-strain curves for collagen fibrils in ligaments with different crosslink densities. The simulations were based on a mesoscopic model first developed by [PERSON] [17]. The collagen molecule is a triple-helical protein structure that consists of three chains of amino acids. Type I collagen is the most abundant type and is present specifically in human ligaments. It has a diameter of about 1.6 nm and a length of about 300 nm (Protein Data Bank entry 3 HR2 experimentally measured by X-ray crystallography [35]). The concept of the mesoscopic model was to abbreviate the full molecular geometry into a single chain of beads (or super atoms), where each bead represents several atoms in the full atomistic model. This concept is fully explained in previous publications [17; 36; 37]. This approach allows molecular dynamics to reach time and length scales otherwise inaccessible by simulating full molecular structures. In fact, one to two orders of magnitude on the size of the fibril or the strain rate used could be gained. We used tropocollagen molecules of 218 beads [36], where the coordinates of the beads were obtained by averaging the position of adjacent atoms. The resulting structure was a chain of quasi-equidistant beads with equilibrium angles ranging from 164\({}^{\circ}\) to 180\({}^{\circ}\). Details on the molecular, mesoscopic model parameters are given in Table 1. The fibril was then built by replicating the above-described molecule orthogonally to its principal axis. We assumed that molecules were hexagonally packed (with a lattice constant of 16.52 A), forming a fibril of 21.5 nm diameter and containing 151 molecules (32918 Beads). A gap length of 36 nm, and an overlap length of 28.2 nm, were used [38; 39; 40; 41]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Molecular Properties**} & \multicolumn{3}{c}{**Fiber Properties**} \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Molecule number of atoms & 3134 & Gap [Å] & 400 \\ Molecule total mass [g/mol] & 287,000 & Overlap [Å] & \(\sim\)282 \\ Number of beads per molecule & 218 & D-period [Å] & \(\sim\)682 \\ Mass of each bead [g/mol] & 1316 & Length of fibril [Å] & 3410 \\ Length along principal axis [Å] & 3011 & Hex. lattice constant [Å] & 16.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mesoscopic structural properties. The formulation of the coarse-grained model of collagen molecules was implemented in several studies [19; 36; 42] and was proven to mimic the actual behavior of the fibril accurately. Three main energies govern the force field. \[E=E_{inter}+E_{bond}+E_{angle} \tag{1}\]  
The interatomic energy (\(E_{inter}\)) is a pairwise Lennard-Jones (LJ) interaction between beads from different molecules responsible for keeping the fibril together in the radial direction. The LJ potential is given by:  
\[E_{inter}=4\epsilon\bigg{(}\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{ \sigma}{r}\right)^{6}\bigg{)} \tag{2}\]  
where \(\sigma\) and \(\epsilon\) represent, respectively, the characteristic distance and the minimum energy of the LJ potential. Parameters for pairwise potential are given in Table 3. The bond energy is a hyperelastic interaction between two adjacent beads from the same molecule. It is represented by three regimes of potential energy that are given by:  
\[F_{bond}=\frac{\partial E_{bond}}{\partial r}=\left\{ \begin{array}{ll}K_{ T0}(r-r_{0})&r<r_{1}\\ K_{T1}(r-\overline{r_{1}})&r_{1}<r<r_{b}\\ 0&r>r_{b}\end{array} \right. \tag{3}\]  
where \(K_{T0}\) and \(K_{T1}\) are spring constants, \(r_{1}\) is the distance at which the hyperelastic behavior of the bond is triggered, \(r_{b}\) is the bond-breaking distance, and \(\overline{r_{1}}\) it is a constant calculated to ensure the continuity of the force field.