# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 3 Results
### Langevin Dynamics Simulations
Both this cross-section and the other cross-section, through the hydrophobic part of the chain, should show hexagonal order, but again, the numbers of chains in the cross-sections are too low to be absolutely sure. As an additional test to establish whether the preferred cross-sectional order for the \(S_{0}\) sequence is indeed the hexagonal \(P_{1}\) packing, we performed Langevin-Dynamics simulations for the subsequences of the \(S_{0}\) sequence, which together form the non-gapped part of the fibril. These were arranged in a \(10\times 10\) hexagonal lattice, and Langevin-Dynamics at \(\widetilde{T}~{}=~{}1\) was used to establish the stability of this lattice against the thermal motion. Results are shown in Figure 7. The \(10\times 10\) hexagonal starting lattice is shown in Figure 6a. The final configuration, at \(t=23\tau_{r}\) (or \(10^{6}\) simulation steps) is shown in Figure 7b. The top view shown in Figure 6c more clearly shows the ordering that remains in the presence of thermal fluctuations. The interaction energy per bead during the simulation run is shown in Figure 7d. During the simulation, the initial hexagonal lattice contracts somewhat, lowering the interaction energy per bead, and then remains constant. Especially from Figure 7d, it is evident that at \(\widetilde{T}~{}=~{}1\), thermal motion does not destroy the initial hexagonal ordering for these larger \(10\times 10\) cross-sectional lattices. Some disorder is observed at the edges, but not in the central part of the lattice. We conclude therefore that at to \(\widetilde{T}~{}=~{}1\), the \(S_{0}\) sequence indeed spontaneously assembles into the target-staggered hexagonal \(P_{1}\) structure, as designed. Figure 6: Langevin Dynamics simulation of assembly of chains with sequences \(S_{0}\) and \(S_{nand}\) after quench from \(\widetilde{T}=1.4\) to \(\widetilde{T}=1\). (**a**) Initial (\(t=0\)) and final (\(t=75\tau_{r}\)) configurations for sequence \(S_{nand}\), where \(\tau_{r}\) is the rotational diffusion time of the corresponding rigid rods. (**b**) Initial (\(t=0\)) and final (\(t=75\tau_{r}\)) configurations for sequence \(S_{0}\), where \(\tau_{r}\) is the rotational diffusion time of the corresponding rigid rods. (**c**) Chain–chain interaction energy \(E_{int}\) (per bead, in Lennard–Jones units, \(\epsilon\)) during the assembly, for chains with sequences \(S_{nand}\) (red symbols) and \(S_{0}\) (blue symbols), as a function of the simulation time in Lennard Jones units, \(t\), scaled by the diffusional rotation time \(\tau_{r}\). (**d**) Zoom-in of the final structure for the chains with sequence \(S_{0}\), with cross sections at the two different non-gapped regions. long-range as in our model. Hence, we expect that the patterns of electrostatic interactions that we found will certainly be helpful in driving staggered fibrillar assembly also in experimental systems such as helical peptides or proteins. It should be noted again that the translation of the coarse-grained model to real systems such as collagen-like helices is non-trivial. For example, the contact energy of pairs of oppositely charged ions is a very sensitive function of the distance of the charged groups and the number of intervening water molecules. Here we simply took a single average value for the contact energy of ion-pairs whereas in reality there will most likely be quite a broad spread of the contact energies of different ion-pairs. The pattern of the \(S_{0}\) sequence can also be considered as a more general pattern for staggered assembly into hexagonal (quasi) crystalline fibrils, independent of the specific type of bead interactions. This is illustrated in Figure 8. The \(S_{0}\) sequence is palindromic (non-polar) and consists of two elements: a polar \(\alpha\) domain, and a non-polar \(\beta\) domain, as shown in Figure 8a,b. Favorable interactions for the domains are antiparallel interactions of \(\alpha\)-domains, and interactions of \(\beta\)-domains with \(\alpha\) domains in any orientation.