# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 3 Results
### Langevin Dynamics Simulations  
Sequence optimized sequences \(S\) are those which have the lowest rod interaction energy \(E_{rad}(S)\) for a given target packing \(P\). This \"ground-state\" approximation for sequence design is shown to be very powerful in other cases, such as the de-novo computational design of protein sequences [2], but it cannot guarantee that the target structure will be adopted by the optimized sequence. First of all, entropy competes with ordering and may preclude the ordered packing from being actually realized at  
Figure 5: (a) Lowest rod interaction energy sequences \(S_{0}\)–\(S_{10}\) from Metropolis Monte Carlo sequence search. Additionally shown is the representative random sequence \(S_{rand}\) drawn from sequences with the rod interaction energy for which the probability distribution peaks, see Figure 4a. (**b,c**) Packing of optimized sequence \(S_{0}\) into the staggered hexagonal \(P_{1}\) target lattice. (b) Cylindrical subsection of crystal lattice. (c) slice in \(x\)-\(z\) plane: single layer of the lattice of chains, with staggered arrangement and perpendicular slices, in the \(x\)-\(y\) plane, demonstrating the three different perpendicular packings, one gapped and two non-gapped. Color code for interaction beads: green = hydrophobic (type 1), white = hydrophilic uncharged (type 2), red = positively charged (type 3), blue = negatively charged (type 4). finite temperatures. Furthermore, the optimized sequence may have yet lower rod interaction energies in arrangements other than the target packing. Finally, kinetic barriers may prevent the target structure from being adopted. Therefore, we check using Langevin Dynamics simulations, whether semiflexible chains with the optimized sequences indeed spontaneously assemble in the target packing or not. We do so for the \(S_{0}\) sequence, which was computationally optimized to assemble into the \(P_{1}\) structure, and compare with a typical random sequence, \(S_{rand}\). Random non-assembled initial states for \(M=40\) semiflexible chains were prepared at a temperature \(\widetilde{T}~{}=~{}1.4\) by stepwise cooling from still higher temperatures. A rectangular, closed simulation box was employed. To start assembly, the system was cooled to \(\widetilde{T}~{}=~{}1\), and simulated for \(t=46\tau_{r}\) (or \(8\times 10^{7}\) simulation steps). Results are shown in Figure 5. For the random sequence \(S_{rand}\) we observe clustering of the chains into a very loosely associated anisotropic aggregate, but no clear fibril formation (Figure 6a). This is associated with just a small decrease in the total interaction energy \(E_{int}\) per interaction bead (Figure 6c). On the other hand, for the \(S_{0}\) sequence, we observe rapid staggered assembly into fibrils (Figure 6b), associated with a large decrease in the interaction energy \(E_{int}\) per interaction bead (Figure 6c). A zoom of the final fibril structure for the optimized \(S_{0}\) sequence is shown in Figure 6d. How close is this to the target \(P_{1}\) packing it was designed to assemble into, and which is shown in Figure 5b,c? The staggered assembly in the parallel direction clearly matches that of the \(P_{1}\) target structure. For the cross-sectional order, in the plane perpendicular to the long axis of the fibril, it is more difficult to establish whether this matches the \(P_{1}\) packing or not, because of the relatively low number of chains in the cross-sections. Representative cross-sections are shown in Figure 6d. The left cross-section, through the charged part of the chains, is expected to show the structure indicated in Figure 5c (top): positively charged beads should have a surrounding consisting of alternating negatively charged and neutral beads, etc. This can indeed be qualitatively recognized, but the numbers of chains in the cross-section are too low to be sure about the regularity of the pattern.