Negative Gaussian curvature distribution in physical and biophysical systems—Curved nanocarbons and ion-channel membrane proteins

Abstract

Soft, polymeric and biological systems are self-assembled and hierarchical that involves a multitude of length scales, geometrical shapes and topological variation besides being elastically soft and easily deformable unlike their inorganic solid counterparts. Within the framework of topology and geometry applied to nanocarbons in our recent work [Gupta and Saxena, J. Appl. Phys. 109, 074316 (2011)], we invoke a similar approach to understanding soft/bio-macromolecular systems having structural diversity specifically within the context of minimal surfaces (i.e., mean curvature H = 0 and Gaussian curvature K < 0 everywhere). The systems of interest include non-periodic and periodic minimal surfaces such as catenoids (synthetic or natural ion-channel membrane proteins), helicoids (β-sheet proteins), and Schwarzites, respectively, which are analyzed within the framework of differential geometry to obtain the information about Gaussian curvature variation, Gaussian bending rigidity, elastic bending energy, and corresponding topological features. Specifically, we study the negative Gaussian curvature distribution providing surface structure of membrane proteins and Schwarzites and corresponding bending energy cost. We focus on ion-channel membrane proteins approximated as a symmetric catenoid, biological sheets as a helicoid and negatively curved carbons and certain mixed di- or triblock copolymers as periodic minimal surfaces, e.g., gyroids. Through these analyses, we identify the role of geometry (shape) and topology in energy storage and catalysis, nanomedicine and drug delivery applications and derive an overarching geometry/topology → property → functionality relationship paradigm.