Protein molecules in 3D: complex behavior from simple local geometry plus weak interactions

Protein structures in 3D are large. elegant, complex, and subtle enough that their prediction and understanding still largely eludes us. In contrast, their local building blocks are quite simple: to a good approximation their connectivity is a tree graph of backbone and sidechains, and the atoms are connected with specific, near-invariant covalent bond-length and bond-angle geometry. However, the connections are mostly tetrahedral, with essentially no right angles and thus no independent, orthogonal degrees of freedom. Also, the non-local interactions such as van der Waals, hydrophobicity, and hydrogen bonds are very much weaker than the covalent bonds and produce the compact, semi-rigid "native" structure only collectively by their great numbers. These two factors are major causes of the cooperativity of protein folding, allostery, and function, and help guide the evolution of a protein sequence to achieve just one or a few equilibrium 3D structures. We will explore several ways in which geometrical analysis - or sometimes even a geometrical metaphor - has illuminated some facet of protein folding, conformation, or mobility. A final message is that the currently most successful algorithms in structure prediction and design attack the complexity by global energetics evaluated using unabashed brute-force computation. There may yet be opportunity for developing more efficient solutions by hierarchical buildup of cleaner and more intuitive geometrical understanding.