# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 2 Materials and Methods  
The coarse-grained model for the patterned semiflexible chains is formulated in terms of 4 types of interaction beads, hydrophobic, uncharged hydrophilic, positively charged, and negatively charged. Note that the use of interaction beads is an approximation of at least two levels. First, the molecular structures that we have in mind such as peptide alpha-helices and triple helices are only approximately semiflexible rods, and secondly, additional errors are introduced by approximating semiflexible rods in terms of bead-spring models. Type numbers s associated with the bead types are: \(s=1\) (uncharged hydrophilic), \(s=2\) (hydrophobic), \(s=3\) (positively charged) and \(s=4\) (negatively charged). We consider \(M\) patterned semiflexible chains, each chain consists of \(N=40\) interaction beads, such that the total number of interaction beads is \(N_{tot}=M.N\). All semiflexible chains have the same sequence of bead types, \(S=\{s_{1},s_{2},\ldots,s_{N}\}\), and all beads have the same diameter \(\sigma\). Electrostatic interactions are represented by a screened Coulomb potential, hydrophobic attraction and excluded volume interactions by a Lennard-Jones potential. Hence, the non-bonded interaction between two beads \(i\) and \(j\) with bead types \(s_{i}\) and \(s_{j}\) is:  
\[V_{s_{i}s_{j}}\Big{(}r_{ij}\Big{)}=4\epsilon_{s_{i}s_{j}}\Big{(}\big{(}\sigma/ r_{ij}\big{)}^{12}-\big{(}\sigma/r_{ij}\big{)}^{6}\Big{)}+q_{s_{i}}q_{s_{j}} \frac{I_{B}}{r_{ij}}\exp\Big{(}-\kappa r_{ij}\Big{)}k_{B}T \tag{1}\]  
where \(I_{B}=e^{2}/4\pi\epsilon_{0}e_{r}k_{B}T\) is the Bjerrum length, in terms of the elementary charge \(e\), the absolute and relative dielectric permittivities \(\epsilon_{0}\) and \(\epsilon_{r}\), and the thermal energy \(k_{B}T\). The Debye screening length is \(\kappa^{-1}=(8\pi l_{B}n_{s})^{-1/2}\), where \(n_{s}\) is the number density of (monovalent) ions. The effective bead charges \(q_{s_{i}}\) (number of elementary charges \(e\)) are:  
\[q_{s_{i}}= \begin{cases}0&s_{i}=1,2\\ +q&s_{i}=3\\ -q&s_{i}=4\end{cases}  \tag{2}\]  
and the Lennard-Jones interaction parameters are:  
\[\epsilon_{s_{i}s_{j}}= \begin{cases}\epsilon_{h}&s_{i}=s_{j}=2\\ \epsilon_{0}&\text{otherwise}\end{cases}  \tag{3}\]  
Bonded interactions used to describe the semiflexible chains are harmonic stretching- and bending potentials, such that the total potential energy function for the \(M\) chains is:  
\[U(r_{1},\ldots,r_{N_{tot}})=\sum_{pairs\ i,j}V_{s_{i}s_{j}}\Big{(}r_{ij}\Big{)} +\ \frac{1}{2}k_{s}\sum_{bonds\ i,j}\Big{(}r_{ij}-l_{b}\Big{)}^{2}+\frac{1}{2}k_{ b}\sum_{bonds\ i,j,k}\theta_{ijk}^{2} \tag{4}\]  
The bending angle \(\theta_{ijk}\) is the angle between the (\(i\))... (\(j\)) and (\(j\))... (\(k\)) virtual bond vectors. The linear chains are assumed to be straight at equilibrium hence the equilibrium bending angle is zero, and \(j=i+1,k=i+2\). Parameter values are chosen such that they represent one of the possible chemistries that can be used to implement patterned semiflexible chains, viz., the collagen triple helix. The bead diameter \(\sigma\), therefore, corresponds to the diameter of a collagen triple helix, of around 1.5 nm. Experimental values reported for the persistence length \(l_{p}\) of collagen triple helices cover a wide range of values [20; 21]. We assume thermal bending fluctuations have a destabilizing effect on fibril assembly, therefore we here use a value of \(l_{p}\approx 35\) nm which is on the lower side of the reported values. We do so since we would rather like to overestimate the effect of bending fluctuations on the fibril stability than to underestimate it. Collagen amino acid sequences are repeats of G-Xaa-Yaa triplets, where Xaa is often proline and Yaa often hydroxyproline. If we let each bead represent the helical rise 0.85 nm of a single G-Xaa-Yaa triplet of amino acids, we arrive at a bond distance \(l_{b}=0.57\sigma\). A single interaction bead then represents 9 amino acids (3 strands, a G-Xaa-Yaa triplet on each strand).