# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 2 Materials and Methods
For a rod consisting of \(N\) beads of diameter \(\sigma\), spaced at a distance \(l_{b}\), in the absence of bead-bead hydrodynamic interactions, \(\tau_{r}\ =\ \zeta_{0}N^{3}l_{b}{}^{2}/12\ k_{B}T\), where \(\zeta_{0}\) is the hydrodynamic friction of a single bead. For our parameters this gives \(\tau_{r}=8.67\times 10^{4}\tau\) (at \(\widetilde{T}=1\)). The cut-off for both the Lennard-Jones and Debye-Huckel interactions was set to \(5\sigma\). Non-bonded interactions were only taken into account between beads on different chains. For assembly simulations a rectangular simulation box was used with dimensions \((l_{x}l_{y}l_{z})=(20\sigma,20\sigma,120\sigma)\). Reflective rather than periodic boundary conditions were used in order to enforce the formation of finite rather than infinite fibrils. Initially, \(M=40\) rods were placed regularly in the box, with their long axis pointing in the \(z\)-direction. Particles were given random initial velocities \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(\kappa\) & 1.5 & \(\sigma^{-1}\) \\ \(l_{B}\) & 0.47 & \(\sigma\) \\ \(q\) & 2.0 & – \\ \(\epsilon_{h}\) & 2.0 & \(k_{B}T\) \\ \(\epsilon_{0}\) & 0.1 & \(k_{B}T\) \\ \(l_{b}\) & 0.57 & \(\sigma\) \\ \(k_{s}\) & 100 & \(k_{B}T/\sigma^{2}\) \\ \(k_{b}\) & 10 & \(k_{B}T\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values for the coarse-grained patterned semiflexible chain model. corresponding to a LJ temperature \(\widetilde{T}=5\) and a simulation was run for \(58\tau_{r}\) (or \(10^{8}\) simulation steps). Next, particles were given random initial velocities corresponding to a LJ temperature \(\widetilde{T}=2.5\) and a simulation was again run for \(58\tau_{r}\) (or \(10^{8}\) simulation steps). Finally, particles were given random initial velocities corresponding to a LJ temperature \(\widetilde{T}=1.5\) and a simulation was run for \(46\tau_{r}\) (or \(8\times 10^{7}\) simulation steps). During equilibration, temperatures for all Langevin dynamics runs, decreases somewhat. For the run with \(\widetilde{T}=1.5\), the final equilibrium temperature was \(\widetilde{T}=1.0\). By running the simulations for a large number of rotational diffusion times, we guarantee that thermal equilibrium can indeed be reached during the simulations. For cross-sectional assembly simulations, only those parts of the sequence \(S_{0}\) are used that together form the non-gapped region. These are:  
\(S_{0,A}=\{4,A,4,4,2,2,2,2,1,1,1,1\}\)  
\(S_{0,B}=\{3,3,3,3,2,2,2,2,3,3,3,3\}\),  
\(S_{0,C}=\{1,1,1,1,2,2,2,2,4,A,4,4\}\). A \(10\times 10\) hexagonal array was prepared of A, B, and C chains, corresponding to the \(P_{1}\) packing of the \(S_{0}\) sequence, with a lattice spacing \(l_{c}=1.3\sigma\). For these simulations, a rectangular simulation box was used with dimensions (\(l_{x}l_{y}l_{z}\)) = (24\(\sigma\),18\(\sigma\),11.84\(\sigma\)). Again, reflective boundary conditions were used. Particles were given random initial velocities corresponding to an LJ temperature \(\widetilde{T}=1.4\). After equilibration, this leads to an average temperature during the simulations of \(\widetilde{T}=1\). The rotational diffusion time for these much shorter rods with \(N=12\) is \(\tau_{r}=2.34\times 10^{3}\tau\). Simulations were run for \(21\tau_{r}\) (or \(10^{6}\) simulation steps).