# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 3 Results
### Target-Staggered Crystalline Arrangements  
While for some collagen-like peptides is was found that simple patterns of just electrostatic interactions can lead to cubic arrangements of the helices inside fibrils and two-dimensional membranes [17], we anticipate that for engineering stable fibrils using weak interactions, it is helpful to allow for as many contacts as possible. For these reasons, we here focus on the staggered assembly of semiflexible chains into hexagonal (para)crystalline arrays. For pattern design we focus on minimizing the interaction energy \(E_{rod}(S)\) of patterned rods arranged in staggered hexagonal lattices, with respect to their sequence \(S\). If the ground state that is found in this manner is sufficiently deep, we anticipate it will be stable against the thermally excited shape fluctuations of the semiflexible chains. This will be checked explicitly by verifying that semiflexible chains with the optimized sequences do indeed assemble into the designed crystal structure, using Langevin Dynamics simulations. Similar ground state design approaches are very successful for designing, e.g., de-novo designed self-assembling proteins [2]. Target structures that we consider are a family of staggered crystalline arrangements of straight chains with length \(L=Nl_{b}\), aligned with their long axis in the \(z\)-direction, with a hexagonal arrangement in the \(x\)-\(y\) plane. The hexagonal lattice spacing is \(l_{c}\). In the \(x\)-\(z\) plane, neighbouring rods are shifted  
Figure 1: Illustration of coarse-graining of microscopic collagen triple-helical structure, to the coarse-grained representation in terms of interaction beads. Shown is the ideal triple helical structure for a (GPP)\({}_{4}\)-GPR-(GPP)\({}_{4}\) homotrimer, with three GPP triplets (one in each strand) being represented by a single hydrophilic (white) bead, and the three central GPR triplets (one in each strand) being represented by a positively charged (red) bead. in the \(z\)-direction by the stagger \(D\). The geometry of the family of lattices is illustrated in Figure 2. The ratio \(L/D\) determines both the periodicity \(N_{\parallel}\) of the staggered arrangement and the gap width \(g\):  
\[N_{\parallel}=\text{int}(L/D)+1 \tag{5}\]  
\[g=N_{\parallel}D-L \tag{6}\]  
Positions \(\mathbf{r}_{ijk}\) of the origins of the rods in the lattices are:  
\[\mathbf{r}_{ijk}=\text{i}\cdot\mathbf{e}_{i}+\text{j}\cdot\mathbf{e}_{j}+\text{k}\cdot \mathbf{e}_{k} \tag{7}\]  
where \(i,j\) and \(k\) are the lattice indices. The lattice vectors are:  
\[\mathbf{e}_{\text{i}}=l_{c}\;\hat{x} \tag{8}\]  
\[\mathbf{e}_{\text{j}}=-\frac{3}{2}l_{c}\hat{x}\;+\frac{1}{2}\sqrt{3}l_{c}\hat{y} \tag{9}\]  
\[\mathbf{e}_{\text{k}}=D\hat{z} \tag{10}\]  
For a given value of \(k\), only certain values of \(i\) and \(j\) correspond to the origin of a rod:  
\[(i+j)\text{mod}\;N_{\parallel}=k\;\text{mod}\;N_{\parallel} \tag{11}\]  
Different lattices in the family have different values for the stagger \(D\), the periodicity \(N_{\parallel}\) and the gap width \(g\). For the numerical calculations, we choose the stagger \(D\) to be an integer number of times  
Figure 2: Representative packing chosen from the family of simple hexagonal staggered packings of (straight) chains considered. Shown is the packing \(P_{5}\), for (straight) chains of \(N=40\) interaction beads and length \(L=40l_{b}\), arranged with a stagger \(D=12l_{b}\) such that the periodicity of the lattice is \(N_{\parallel}=4\) and the gap is \(g=8l_{b}\). (**a**) Projection onto \(x\)-\(z\) plane, showing a single staggered layer of rods. The rod length \(L\), stagger \(D\) and gap width \(g\) are indicated, as well as the crystal spacing \(l_{c}\) and the bond length \(l_{b}\) (separation between beads in the chains). (**b**) Three-dimensional representation of cylindrical subsection of the crystalline lattice. Gapped and non-gapped regions of the fibril for which \(y\)-\(x\) cross-sections are shown in (**c**), are indicated by (\(g\)) and (\(n\)).