# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 3 Results  
### Coarse-Grained Model for Patterned Semiflexible Chains  
Driving forces for self-assembly that are taken into account in the coarse-grained model are electrostatic interactions, hydrophobic interactions, and excluded volume interactions. More specific, shorter-ranged and directional interactions such as hydrogen bonds, can possibly be implemented in later, more refined further sequence optimalizations, that start from patterns optimized at the most coarse-grained level, as obtained here. Semiflexible chains are represented using a simple bead-spring model (bead diameter \(\sigma\), equilibrium bond length \(l_{b}\)) with harmonic bond stretching energy and harmonic bending energy. Electrostatic interactions are represented using a screened Coulomb interaction potential, steric and hydrophobic interactions using a Lennard-Jones potential. We focus on sequence variation and therefore choose a single set of interaction parameter values representative for semiflexible collagen-like triple helices (see Section 2). Specifically, we assume a persistence length of the semiflexible chains of \(l_{p}=20\sigma\).  
A difference with the helical peptide systems for which staggered assembly has already been demonstrated [17; 18] is that we focus on longer semiflexible chains that can have more complicated interaction patterns. Specifically, for the numerical examples, we use chains of \(N=40\) interaction beads. Taking collagen triple helices as an example, this would translate to a length of 40 G-Xaa-Yaa triplets or 120 amino acids per strand, much longer than typical collagen-like peptides, but still much shorter than the triple-helical part of human collagen I, which would correspond to \(N\approx 350\). A shorter length was chosen to ensure that we can still adequately sample the interaction patterns and perform the Langevin Dynamics simulations of fibril assembly.  
Interaction patterns are represented in terms of four characteristic bead types: hydrophobic (type 1, represented as green beads), hydrophilic uncharged (type 2, represented as white beads), positively charged (type 3, represented as red beads) and negatively charged beads (type 4, represented as blue beads). Interaction patterns therefore correspond to coarse-grained sequences \(S=\{s_{1},s_{2},\ldots,s_{\rm N}\}\), where the bead type \(s_{i}=1\ldots 4\). For the values of the non-bonded interaction parameters that we use (Table 1), contact interactions between the different types of beads are on the order of the thermal energy \(k_{B}T\), such that multiple attractive contact interactions along the chains will be necessary to stabilize the fibrillar state.  
Numerical values of parameters are chosen to roughly match triple-helical collagen, with three G-Xaa-Yaa triplets (one for each strand), making up a single bead in the coarse-grained model. For collagen, a single bead would therefore correspond to nine amino acids. We do not attempt to make an explicit mapping of the bead types to underlying microscopic chemistries here. The present very coarse-grained approach is mainly meant to guide future more detailed computational design approaches for specific microscopic chemistries, such as triple helical collagen. A qualitative mapping however is still possible. For example, staying with the case of triple-helical collagen, GPO triplets (where O is hydroxylated proline) that occur frequently in the sequence of triple-helical human collagen-I would probably be best represented by hydrophilic uncharged beads, whereas GPR triplets, known to occur at quite high frequencies in collagen sequences due to its stabilizing effect on collagen triple helices [26], would be best represented by a positively charged bead. An example comparison of a microscopic structure of a collagen-like triple-helical peptide and its coarse-grained representation is shown in Figure 1.