Universal criterion for designability of heteropolymers

Proteins are an example of heteropolymers able to self-assemble in specific target structures. The self-assembly of designed artificial heteropolymers is still, to the best of our knowledge, not possible with control over the single chain self-assembling properties comparable to what natural proteins can achieve. What artificial heteropolymers lacks compared to bio-heteropolymers that grants the latter such a versatility? Is the geometry of the protein skeleton the only a particular choice to be designable? Here we introduce a general criteria to discriminate which polymer backbones can be designed to adopt a predetermined structure. With our approach we can explore different polymer backbones and different amino acids alphabets. By comparing the radial distribution functions of designable and not-designable scenarios we identify as designability criteria the presence of a particular peak in the radial distribution function that dominates over the random packing of the heteropolymer. We show that the peak is a universal feature of all designable heteropolymers, as it is dominating also the radial distribution function of natural proteins. Our finding can help in understanding the key features that make proteins a highly designable system. The criteria that we present can be applied to engineer new types of self-assembling modular polymers that will open new applications for polymer based material science.