# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 1 Introduction  
The design of soft materials that self-assemble into programmable structures with physical properties that can be precisely controlled, is now within reach thanks to technologies such as DNA nanotechnology [1], and the computational design of sequences for small globular proteins that self-assemble into precisely designed structures [2]. At larger length scales, and more coarse-grained levels of description, interaction potentials for soft matter components can be designed that assemble to stable target structures [3]. For example, isotropic interaction potentials have been designed that lead to the assembly of colloidal particles into quasi-crystals [4]. A large body of work, both experimental and theoretical, also deals with patchy colloidal particles, with anisotropic interaction potentials, which can be designed for assembly into, e.g., into monodisperse icosahedral clusters [5] or specific crystal structures [6]. Virtually no work has been carried out yet on patchy rods, or patchy semiflexible chains. No doubt this is related to the fact that as of yet there are not many experimental examples of such systems for which patterns on rods or semiflexible chains can be precisely controlled. However, in nature, patchy semiflexible chains are abundant, a particularly important example being those that exhibit staggered assembly into fibrils such as intermediate filaments and collagen-I. Both collagen fibrilsand intermediate filaments form by the staggered assembly of long, rod-shaped, semiflexible protein subunits. Presumably, some form of \"interaction code\" along the rod-shaped protein subunits directs the staggered assembly into fibrils. Indeed a regular pattern of electrostatic and other interactions for parallel triple helices has been implicated in the assembly of collagen-I triple helices [7; 8], but only at a qualitative level, leaving the possibility that very local and specific interactions may in fact dominate the total driving force for the staggered assembly. The same holds for the staggered assembly of intermediate filaments: patterns of electrostatic interactions have been implicated, but only at a qualitative level [9]. An additional complication is that these fibrils assemble in the context of a living organism, such that it is not clear whether indeed the assembly is fully encoded in the primary sequence of these proteins, or that it is assembled under the influence of external forces. For example, the staggered assembly of collagen-I triple helices into fibrils and fibers is a highly regulated process that is influenced by a multitude of cellular processes. Indeed, while isolated collagen-I triple helices have been shown to be able to assemble into staggered fibrils in-vitro, showing the familiar \"banding\" pattern, structurally these fibrils are different from those formed in-vivo [10]. Nevertheless, the fact that the staggered assembly is also found for collagen-I in-vitro (albeit not the same as in-vivo) does indicate that it is indeed encoded to some extent in the primary sequence. Further evidence in this direction also comes from studies on the assembly of engineered concatemers of (non-hydroxylated) fragments of collagen-I sequences, that form staggered fibrils, but with different gap sizes than natural collagen-I fibrils [11; 12; 13]. Designed fibrillar systems have been explored widely as mimics of collagen-I fibrils of the extracellular matrix. A number of systems have been developed that consist of sticky semiflexible chains that bundle into fibrils and form dilute hydrogels [14; 15; 16]. These have shown great promise as scaffolds for cell growth. However, semiflexible chains that exhibit precise staggered assembly into (quasi) crystalline arrays have not been developed yet. For short, helical peptides, it has been discovered that simple patterns of rather long-range interactions, such as electrostatic interactions, suffice to direct staggered assembly into fibrils, showing the characteristic \"banding\" when imaged using electron microscopy.