A measure of folding complexity for d-dimensional polymers

A measure of folding characterizes aspects of the instantaneous organization of a polymer chain in space. For three-dimensional polymers (D = 3), one such measure is the mean overcrossing number. An intuitively similar property, the radial intersection number, has been proposed as a tool to characterize "folding features" in two-dimensional polymers (D = 2). In this work, we show rigorously that these measures are indeed related and that they can be derived as particular cases within a single, unified formulation. The present approach provides an analytical expression for a measure of folding complexity that can be applied to generic D-dimensional polymers. In the case D = 2, we show results for models derived from experimental structures by using optimized multidimensional scaling transformations for data compression.