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.

A topological twist on materials science

“…Surfaces with H = 0 are called minimal surfaces; some examples include a plane (trivial), a helicoid, a catenoid, and periodic negatively curved carbon (Schwarzite). 20 Next, we demonstrate and discuss nanocarbon allotropes as a prominent example of topological variation in greater detail.…”

“…Among the negative Gaussian curvature ( K < 0) periodic carbon allotropes, Schwarzite has a complex topology with g = 3, χ = -8 per unit cell ( Table I ). 20 Similarly, a helicoid -shaped graphene nanoribbon would have g = 0, χ = 1. 20 We summarize the global topology of nanocarbons in Table I in terms of g and χ .…”

“…20 Similarly, a helicoid -shaped graphene nanoribbon would have g = 0, χ = 1. 20 We summarize the global topology of nanocarbons in Table I in terms of g and χ . Note that the nanocones, nanodisks, and nanotubes closed at one end are topologically equivalent to the fl at graphene sheet ( g = 0).…”

“…Analogously, the deformation of negative curvature periodic minimal surfaces such as Schwarzites and double gyroids can be calculated, specifi cally under hydrostatic stress, assuming that only the lattice parameter changes under deformation. 20 Similarly, carbon and graphene nanoribbons can exist in helicoidal shape, and their axial deformation can be readily calculated by varying the pitch of the helicoid. 20 In both cases, the elastic energy is proportional to the Gaussian curvature and the material's bulk (or axial) modulus.…”

Mechanical and energy-absorption properties of schwarzites

“…Other works investigated the stability of some schwarzite structures [7][8][9][10] and possible synthesis routes [11][12][13][14]. Although the complex geometry of schwarzites poses significant challenges for their synthesis, they have been used to model realistic synthesized carbon nanofoams [15][16][17][18][19] and to clarify the role of negative Gaussian curvature in chemical structures [20] and biophysical systems [21,22]. With the recent synthesis of new porous carbon networks [14,23] there is a renewed interest in schwarzite-like structures due to their unique topological properties and many potential applications, such as catalysis, molecular sieving [24,25], gas storage [26], alkali ion batteries [27], as anode for lithium-ion batteries [28] and energy-absorbing materials [15][16][17][18][19].…”