Exploring cell surface-nanopillar interactions with 3D super-resolution microscopy

Plasma membrane topography has been shown to strongly influence the behavior of many cellular processes such as clathrin-mediated endocytosis, actin rearrangements, and others. Recent studies have used 3D nanostructures such as nanopillars to imprint well-defined membrane curvatures (the "nano-bio interface"). In these studies, proteins and their interactions were probed by 2D fluorescence microscopy. However, the low resolution and limited axial detail of such methods are not optimal to determine the relative spatial position and distribution of proteins along a 100 nm-diameter object, which is below the optical diffraction limit. Here, we introduce a general method to explore the nanoscale distribution of proteins at the nano-bio interface with 10-20 nm precision using 3D single-molecule super-resolution (SR) localization microscopy. This is achieved by combining a silicone oil immersion objective and 3D double-helix point-spread function microscopy. We carefully optimize the objective to minimize spherical aberrations between quartz nanopillars and the cell. To validate the 3D SR method, we imaged the 3D shape of surface-labeled nanopillars and compared the results with electron microscopy measurements. Turning to transmembrane-anchored labels in cells, the high quality 3D SR reconstructions reveal the membrane tightly wrapping around the nanopillars. Interestingly, the cytoplasmic protein AP-2 involved in clathrin-mediated endocytosis accumulates along the nanopillar above a specific threshold of 1/R membrane curvature. Finally, we observe that AP-2 and actin preferentially accumulate at positive Gaussian curvature near the pillar caps. Our results establish a general method to investigate the nanoscale distribution of proteins at the nano-bio interface using 3D SR microscopy.

Fourier Frames for Surface-Carried Measures

In this paper, we show that the surface measure on the boundary of a convex body of everywhere positive Gaussian curvature does not admit a Fourier frame. This answers a question proposed by Lev and provides the 1st example of a uniformly distributed measure supported on a set of Lebesgue measure zero that does not admit a Fourier frame. In contrast, we show that the surface measure on the boundary of a polytope always admits a Fourier frame. We also explore orthogonal bases and frames adopted to sets under consideration. More precisely, given a compact manifold $M$ without a boundary and $D \subset M$, we ask whether $L^2(D)$ possesses an orthogonal basis of eigenfunctions. The non-abelian nature of this problem, in general, puts it outside the realm of the previously explored questions about the existence of bases of characters for subsets of locally compact abelian groups. This paper is dedicated to Alexander Olevskii on the occasion of his birthday. Olevskii’s mathematical depth and personal kindness serve as a major source of inspiration for us and many others in the field of mathematics.

Distinct cytoskeletal proteins define zones of enhanced cell wall synthesis in Helicobacter pylori

Helical cell shape is necessary for efficient stomach colonization by Helicobacter pylori, but the molecular mechanisms for generating helical shape remain unclear. The helical centerline pitch and radius of wild-type H. pylori cells dictate surface curvatures of considerably higher positive and negative Gaussian curvatures than those present in straight- or curved-rod H. pylori. Quantitative 3D microscopy analysis of short pulses with either N-acetylmuramic acid or D-alanine metabolic probes showed that cell wall growth is enhanced at both sidewall curvature extremes. Immunofluorescence revealed MreB is most abundant at negative Gaussian curvature, while the bactofilin CcmA is most abundant at positive Gaussian curvature. Strains expressing CcmA variants with altered polymerization properties lose helical shape and associated positive Gaussian curvatures. We thus propose a model where CcmA and MreB promote PG synthesis at positive and negative Gaussian curvatures, respectively, and that this patterning is one mechanism necessary for maintaining helical shape.

Endpoint Detection of Partially Overlapping Straight Fibers using High Positive Gaussian Curvature in 3D images

This paper introduces a method for detecting endpoints of partially overlapping straight fibers in three-dimensional voxel image data. The novel approach directly determines fiber endpoints without the need for more expansive single-fiber segmentation. In the context of fiber-reinforced polymers, endpoint information is of practical significance as it can indicate potential damage in endless fiber systems, or can serve as input for estimating statistical fiber length distribution. We tackle this challenge by exploiting Gaussian curvature of the surface of the fibers. Fiber endpoints have high positive curvature, allowing one to distinguish them from the rest of a structure. Accuracy data of the proposed method are presented for various data sets. For simulated fiber systems with fiber volume fractions of less than 20 %, true positive rates above 94 % and false positive rates below 5 % are observed. Two well-resolved real data sets show a reduction of the first rate to 90.3 % and an increase of the second rate to 13.1 %.

Distinct cytoskeletal proteins define zones of enhanced cell wall synthesis inHelicobacter pylori

Helical cell shape is necessary for efficient stomach colonization byHelicobacter pylori, but the molecular mechanisms for generating helical shape remain unclear. We show that the helical centerline pitch and radius of wild-typeH. pyloricells dictate surface curvatures of considerably higher positive and negative Gaussian curvatures than those present in straight- or curved-rod bacteria. Quantitative 3D microscopy analysis of short pulses with eitherN-acetylmuramic acid or D-alanine metabolic probes showed that cell wall growth is enhanced at both sidewall curvature extremes. Immunofluorescence revealed MreB is most abundant at negative Gaussian curvature, while the bactofilin CcmA is most abundant at positive Gaussian curvature. Strains expressing CcmA variants with altered polymerization properties lose helical shape and associated positive Gaussian curvatures. We thus propose a model where CcmA and MreB promote PG synthesis at positive and negative Gaussian curvatures, respectively, and that this patterning is one mechanism necessary for maintaining helical shape.

Examples of surfaces of constant mean curvature

A surface in E3 is called parallel to the surface M if it consists of the ends of constant length segments, laid on the normals to the surfaces M at points of this surface. The tangent planes at the corresponding points will be parallel. For surfaces in E3 the theorem of Bonnet holds: for any surface M that has constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature. Using Bonnet's theorem for a surfaces of revolution of constant positive Gaussian curvature, surfaces of constant mean curvature are constructed. It is proved that they are also surfaces of revolution. A family of plane curvature lines (meridians) is described by means of elliptic integrals. The surfaces of constant Gaussian curvature are also described by means of elliptic integrals. Using the mathematical software package, the surfaces under consideration are constructed.

On Bäcklund and Ribaucour transformations for hyperbolic linear Weingarten surfaces

We consider Bäcklund transformations for hyperbolic linear Weingarten surfaces in Euclidean 3-space. The composition of these transformations is obtained in the Permutability Theorem that generates a 4-parameter family of surfaces of the same type. Since a Ribaucour transformation of a hyperbolic linear Weingarten surface also gives a 4-parameter family of such surfaces, one has the following natural question. Are these two methods equivalent, as it occurs with surfaces of constant positive Gaussian curvature or constant mean curvature? By obtaining necessary and sucient conditions for the surfaces given by the two procedures to be congruent.The analytic interpretation of the geometric results is given in terms of solutions of the sine-Gordon equation.