The role of directional interactions in the designability of generalized heteropolymers

Heteropolymers are important examples of self-assembling systems. However, in the design of artificial heteropolymers the control over the single chain self-assembling properties does not reach that of the natural bio-polymers, and in particular proteins. Here, we introduce a sufficiency criterion to identify polymers that can be designed to adopt a predetermined structure and show that it is fulfilled by polymers made of monomers interacting through directional (anisotropic) interactions. The criterion is based on the appearance of a particular peak in the radial distribution function, that we show being a universal feature of all designable heteropolymers, as it is present also in natural proteins. Our criterion can be used to engineer new self-assembling modular polymers that will open new avenues for applications in materials science.


Supplementary theory
A guiding theory for heteropolymer designability is given by the Random Energy Model (REM) [1][2][3]. A clear review can be found in the seminal works of Pande et al. [3], where it is shown how the designability of a heteropolymer increases with the total number of possible bonds for each bead (valence) and decreases with the conformational entropy per bead. Hence, it is reasonable to assume that directionality (the patches) combined with isotropic interactions would increase designability, because the valence (i.e. the total number of possible bonds) remains constant, while the conformational entropy per bead decreases. In fact, the introduction of the patches decreases the entropy by favouring the system to populate more specific structures with the patches along particular directions.
On the other hand, if the number of patches increases to much, the interactions become again close to isotropic and the designability decreases again. Hence, a model is needed that explicitly brings about the designability from a basic heteropolymer model by controlling the alphabet size and the conformational entropy per particle.

Symmetry of patches on the surface
Considering n as the number of patches on the surface, for freely jointed chains (FJC) with n = 3 the anchoring points are places perpendicular to the patch, while for n = 4 they are placed on a tetrahedron completed by the anchoring points. The freely rotating chains (FRC) are fixed on the vertices of a platonic solid or in the most symmetric way: equispaced on the equator and placed on the vertices of a equilateral triangle, of a tetrahedron and of a octahedron for the n = 3 n = 4 n = 6 cases, respectively. For n = 10 there is no platonic solid, so the patches are placed on the surface in the most symmetric way by using the following numerical procedure: 1. n patches are randomly placed on a sphere, their positions given by the set of vectors { r 1 , . . . , r n } 2. we assign a fictitious energy to the system, defined as U = 1 2 i =j | r i − r j | −1 3. we minimise U by attempting to move a randomly chosen patch, accepting the move if the total energy of the system consequently decreases. Formally, this can be regarded as a Monte Carlo (MC) simulation performed at temperature zero.
4. We iterate this procedure until convergence of U .
For completeness, we note that the above method produces a patch distribution which is independent of the definition of distance between two patches, being it the Euclidean distance or a spherical distance, for all the values of n considered here. In addition, the patch distribution makes sure that two particles cannot be involved in more than one bond. Free energy landscape sampled by SEEK for one patch and alphabet 3 in the freely rotating chain model. The free energy is in function of the total number of contacts between the spheres (distance below 6 R bead ) and the total number of contacts between the patches (distance below 1.25 R bead and angles θ1 and θ2 > 0.8 π). The target structure is chosen in the global minimum of this landscape. Following the above definition of patches contacts, in the target structure the 80% of the patches are maximally oriented between each others. However, we observe by looking closer at the structure, that all the patches are interacting. Nevertheless, the close packing peak in the radial distribution function dominates on the directional interaction peak. Hence, the close packing is not suppressed even when all the directional interactions are fulfilled. Thus, increasing the relative strength of the directional interactions will not make the first peak disappear.    For the polymer without patches we observe in Figure S.4 that the prevalent angle is 60 • corresponding to a random packing. For 10 patches we observe that the high number of patches decreases the random packing at distances close to 2R bead . In fact, at small distances (2.00 − 2.45R bead ) there is a component of directionality that cannot be attributed to the patch-patch bond, but it is instead due to the inter-penetration of the corona formed by the 10 patches, which can occur only with discrete angles. While at intermediate distances (2.45 − 2.90R bead ) be observe that the peak at 60 • reappears. In fact, this correspond to the new random packing, which is shifted due to the hindrance of the corona formed by the many patches. In the last part we observe only at distances 3R bead the reappearing of some directionality due to the patch-patch bond.