A central result of the discussion in the last chapter was the strong influence of finite-size effects on the freezing behavior of flexible polymers constrained to regular lattices. Thus, (unphysical) lattice effects interfere with (physical) finite-size effects and the question remains what polymer crystals of small size could look like. Since all effects in the freezing regime are sensitive to system or model details, this question cannot be answered in general. Nonetheless, it is obvious that the surface exposed to a different environment, e.g., a solvent, is relevant for the formation of the whole crystalline or amorphous structure. This is true for any physical system. If a system tries to avoid contact with the environment (a polymer in bad solvent or a set of mutually attracting particles in vacuum), it will form a shape with a minimal surface. A system that can be considered as a continuum object in an isotropic environment, like a water droplet in the air, will preferably form a spherical shape.

But what if the system is “small” and discrete? Small crystals consisting of a few hundred cold atoms, e.g., argon [154], but also as different systems as spherical virus hulls enclosing the coaxially wound genetic material [155, 156] exhibit an icosahedral or icosahedral-like shape. But why is just the icosahedral assembly naturally favored?

The capsid of spherical viruses is formed by protein assemblies, the protomers, and the highly symmetric morphological arrangement of the protomers in icosahedral capsids reduces the number of genes that are necessary to encode the capsid proteins. Furthermore, the formation of crystalline facets decreases the surface energy, which is particularly relevant for small atomic clusters.

Evolutionary aspects

The number of different functional proteins encoded in human DNA is of the order of about 100 000, which is an extremely small number compared to the total number of possibilities: Recalling that 20 types of amino acids occur in natural proteins and typical proteins consist of N ∼ O(102−103) amino acid residues, the number of possible primary structures, 20N, lies somewhere far, far above 20100 ∼ 10130. Assuming all proteins were of size N = 100 and a single folding event would take 1 ms, a sequential enumeration process would need about 10119 years to generate structures of all sequences, irrespective of the decision about their “fitness,” i.e., the functionality and ability to efficiently cooperate with other proteins in a biological system. Of course, one might argue that the evolution is a highly parallelized process that drastically increases the generation rate. So, we can ask the question, how many processes can maximally run in parallel. The visible universe contains of the order of 1080 protons. Assuming that an average amino acid consists of at least 50 protons, a chain with N = 100 amino acids has of the order O(103) protons, i.e., 1077 sequences could be generated in each millisecond (forgetting for the moment that some proton-containing machinery is necessary for the generation process and only a small fraction of protons is assembled in Earth-bound organic matter).