# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 3 Results
### Sequence Optimization  
For the staggered hexagonal packings \(P_{1}\)-\(P_{8}\) that we consider here, each chain has the same environment. Hence, as the objective function for the sequence design we can use the interaction energy \(E_{rad}(S)\) of the central rod with lattice indices (_i,j,k_) = (0,0,0), with all the other chains in the crystalline arrangement:  
\[E_{rad}(S)=\sum_{(i,j)\
eq(0,0),0}V_{rad-rad}^{(ijk)}(S) \tag{12}\]  
The potential of the interaction of the central chain at (0,0,0) with another chain at lattice position (_i,j,k_) is a sum of bead-bead interactions:  
\[V_{rad-rad}^{(ijk)}(S)=\sum_{n,n=1,N}V_{s_{n}s_{m}}(r_{nm}^{ijk}) \tag{13}\]  
\[r_{nm}^{ijk}=\mathbf{r}_{ijk}+(\mathbf{m}-\mathbf{n})l_{b}\hat{z} \tag{14}\]  
where \(V_{s_{n}s_{m}}(r_{nm}^{ijk})\) is the interaction potential for two beads: bead \(n\) on the central rod and bead \(m\) on the rod at lattice position (_i,j,k_), with bead types \(s_{n}\) and \(s_{m}\), and for a bead-bead distance \(r_{nm}^{ijk}\) (see Section 2). For sequence optimization, we minimize the rod interaction energy \(E_{rad}(S)\) with respect to the sequence \(S\), for each of the packings \(P_{1}\)-\(P_{8}\). We do so for a fixed composition in terms of bead types. Bead compositions we choose do not reflect natural collagen-like sequences, but rather reflect our expectations as to a typical bead composition that would readily form fibrils. We expect fibril assembly will be opposed by a net charge, while both charged and hydrophobic beads are necessary to drive assembly into staggered hexagonal packings. Too many hydrophobic beads could lead to kinetically trapped assembly into off-target packings. These considerations lead us to choose, for the study here, a composition of 30% of hydrophobic beads, 30% uncharged hydrophilic beads, and 40% charged beads (half of which are positively and half of which are negatively charged). We search for sequences \(S\) with a minimal rod interaction energy \(E_{rad}(S)\) using a Metropolis Monte Carlo search algorithm that generates random sequences \(S\) with probability:  
\[p(S)\sim\exp(-E_{rad}(S)/\Delta E) \tag{15}\]  
For a value of \(\Delta E\) that we set to \(\Delta E=2\). We run the Metropolis Monte Carlo search for a large number of steps, and save the list of generated sequences and associated rod interaction energies, for later analysis. The Monte Carlo move that is used to generate a new sequence from a previous \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Packing** & \(\boldsymbol{L}\)/\(l_{b}\) & \(\boldsymbol{D}\)/\(l_{b}\) & \(\boldsymbol{N}_{\parallel}\) & \(\boldsymbol{g}\)/\(l_{b}\) \\ \hline \(P_{1}\) & 40 & 14 & 3 & 2 \\ \(P_{2}\) & 40 & 15 & 3 & 5 \\ \(P_{3}\) & 40 & 16 & 3 & 8 \\ \(P_{4}\) & 40 & 11 & 4 & 4 \\ \(P_{5}\) & 40 & 12 & 4 & 8 \\ \(P_{6}\) & 40 & 9 & 5 & 5 \\ \(P_{7}\) & 40 & 8 & 6 & 8 \\ \(P_{8}\) & 40 & 7 & 6 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters for the target-staggered hexagonal packings \(P_{1}\)–\(P_{8}\) of (straight) chains considered in the numerical calculations. sequence is a simple swap of the two beads in the sequence, provided the two bead types are different. For simplicity, we restrict ourselves to palindromic sequences, such that the orientation of the rods in the lattice does not matter. First, for each of the crystal packings \(P_{1}\)-\(P_{8}\) we search for sequences \(S\) with minimal interaction energy \(E_{rad}(S)\) in runs of \(10^{5}\) Monte Carlo steps (per packing). Results are shown in Figure 3. The lowest energy sequences are found for packings \(P_{1}\)-\(P_{3}\) with the lowest periodicity, \(N_{\parallel}=3\). Differences in results between these lattices are very small. We choose the \(P_{1}\) lattice for a more detailed investigation. Next, to get a complete histogram of the distribution of \(E_{rad}(S)\) for all possible sequences \(S\) arranged in a \(P_{1}\) lattice, we performed a simple random sampling search of \(10^{6}\) steps. The resulting energy distribution is shown in Figure 4a.