Differential geometry of proteins: a structural and dynamical representation of patterns.

Abstract This is an essay on the applications of the differential geometry of curves and surfaces to the analysis of the spatial patterns of proteins. Differential geometry is the natural mathematical tool when a protein molecule is represented as a space curve passing through its α-carbons. We suggest a unifying and natural description of the three-dimensional conformation of proteins. In particular, we argue that the regular secondary structures correspond to geodesics on minimal surfaces, and that the tertiary structures result from the energetically best packing of these minimal surfaces. The a-helices and the strands of the β-barrels lie, respectively, on the conjugate minimal surfaces of the helicoid and the catenoid. These two surfaces can be transformed into each other isometrically, and the intermediate stages in the transformation model the various β-twisted sheets found in proteins. The geometry of these protein curves and surfaces is studied in detail, and biological interpretations are given along with analytic expressions. Attention is paid to the relationship between local and global properties in both mathematical and biological terms. Implications for the morphogenetic process of protein folding are outlined. The problem of the prediction of three-dimensional structures from amino acid sequence is also addressed. These turn out to be best formulated in terms of a dynamical analysis of vector fields on proteins. Modern differential geometry centres its attention on manifolds, which are generalizations of curves and surfaces, and which behave locally like Euclidean spaces. From the theory of manifolds arises the notion of a fibre space. Fibre spaces are the mathematical objects used in Robert Rosen's unified approach to pattern generation. Since the spontaneous folding of proteins into native conformations is a prototype biological example of pattern generation, the abstract formalism of fibre spaces provides an appropriate general setting in which to further the study of the mathematical biology of proteins.