Combinatorial Surfaces Research Articles

Double‐Orthogonal Gradient‐Based High‐Throughput Screening Platform for Studying Cell Response Toward Combined Physicochemical Biomaterial Properties

Implant‐associated complications arise due to non‐optimized cell–biomaterial interactions. It is well known that cells respond to their physicochemical microenvironment on 2D interfaces and 3D networks. Attempts to manipulate this interaction target surface parameters such as wettability (W), stiffness (S), and topography (T) to influence cell differentiation, adhesion, and morphology, due to induction of gene activation and protein expression. Investigating the combinatorial influence of all three mentioned parameters simultaneously remains challenging, though most realistic, since all three parameters are inherently present on a surface. Herein, a novel high‐throughput screening technology, which allows investigating the cell response of human bone‐marrow‐derived mesenchymal stem cells toward three varying biomaterial surface parameters simultaneously, is presented. The platform provides efficient screening and cell response readout to a vast amount of combined biomaterial surface properties, in a single‐cell experiment. Surface gradients of aligned wrinkle T, S, and W are orthogonally combined giving four combinatorial surfaces. The screening outcome is validated by translating interesting regions to homogeneous surfaces. Cells are found to behave similar to the screening in terms of adhesion, spreading, and vimentin expression. The technology tremendously supports the identification of optimal surface parameter combinations and potentially addressing many of the current implant‐associated complications.

Combinatorial Minimal Surfaces in Pseudomanifolds and Other Complexes

We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds and their generalizations. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called \emph{thin position}. Thin position is defined using orderings of the $n$-dimensional simplices of an $n$-dimensional pseudomanifold. In addition to defining and finding combinatorial minimal surfaces, from thin orderings, we derive invariants of even-dimensional closed simplicial pseudomanifolds called \emph{width} and \emph{trunk}. We study additivity properties of these invariants under connected sum and prove theorems analogous to those in knot theory and 3-manifold theory.

Clean-up tool path generation for combinatorial surfaces based on cutter center point of fillet boundary

The fillet is a typical transitional form for a region with a common edge between combinatorial surfaces of blisk parts, and the processing quality of that has a strong influence on the aerodynamic performances of the parts. As a special type, the incomplete fillet is generated due to missing data of the combinatorial surfaces, which is not tangent to the combinatorial surfaces. For machining the clean-up region with incomplete fillet, the planned tool path using the conventional method may result in excess residual material, which will affect the part performance. To address this issue, a novel clean-up tool path planning method based on the cutter center point of the fillet boundary is proposed. In this paper, the method constructs a mapping model from a point on the fillet boundary to the cutter center point. The model is solved with a point-searching algorithm so that the tool path of the fillet boundary can be calculated. Then, the region between the adjacent tool path of the fillet boundary is equivalent to circular curve for planning the clean-up tool path. Finally, this method is applied to the clean-up machining between the blade and hub of an open blisk using a computational simulation and experiment to verify its effectiveness. The results show that the maximum allowance is 0.311 mm with the conventional method, whereas it is approximately 0 mm with the proposed method, which satisfies the design requirement. Thus, the correctness and effectiveness of the proposed method are validated.

Lineage‐Specific Commitment of Stem Cells with Organic and Graphene Oxide–Functionalized Nanofibers

AbstractControlling the differentiation to certain lineages is the main goal of current stem cell research, which might exploit new routes based on the interaction of cells with nanomaterials. Here it is shown that primary neurospheres from dental pulp stem cells grown on combinatorial surfaces with different fibrous morphology and graphene oxide functionalization exhibit different differentiation propensity. The developed materials strongly influence the stem cell fate, as highlighted by morphological, immunofluorescence, molecular biology, and functional analyses. Instructive cues lead to the increased expression of markers that are characteristic of selective differentiation into osteoblasts, glial cells, fibroblasts, and neurons even in basal medium conditions, and randomly oriented fibers are found to revert neuronal precommitment and to trigger osteoblastic differentiation. Graphene oxide coatings lead instead to the relatively enhanced expression of genes typical of either glial or neuronal commitment, depending on the underlying nanofibrous morphology. The mechanisms addressing cell fate are investigated, highlighting the correlation of wetting anisotropy and protein adsorption capacity of different surfaces, ultimate cell conformational changes reflected by skeletal and nuclear elongation, and directed cell commitment. Cues from the different surfaces are therefore lineage‐specific, unveiling remarkable potentialities for cellular programming by means of nanomaterials.

Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki (Graph structure theory. Contemporary Mathematics, vol. 147, 1993), a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov's systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length $$O(g^{3/2}n^{1/2})$$O(g3/2n1/2) for any triangulated combinatorial surface of genus $$g$$g with $$n$$n triangles, and describe an $$O(gn)$$O(gn)-time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.

Conformal nanopatterning of extracellular matrix proteins onto topographically complex surfaces.

We report a method for conformal nanopatterning of extracellular matrix proteins onto engineered surfaces independent of underlying microtopography. This enables fibronectin, laminin, and other proteins to be applied to biomaterial surfaces in complex geometries inaccessible using traditional soft lithography techniques. Engineering combinatorial surfaces that integrate topographical and biochemical micropatterns enhances control of the biotic-abiotic interface, used here to understand cardiomyocyte response to competing physical and chemical cues in the microenvironment.

Electronic tongue generating continuous recognition patterns for protein analysis.

In current protocol, a combinatorial approach has been developed to simplify the design and production of sensing materials for the construction of electronic tongues (eT) for protein analysis. By mixing a small number of simple and easily accessible molecules with different physicochemical properties, used as building blocks (BBs), in varying and controlled proportions and allowing the mixtures to self-assemble on the gold surface of a prism, an array of combinatorial surfaces featuring appropriate properties for protein sensing was created. In this way, a great number of cross-reactive receptors can be rapidly and efficiently obtained. By combining such an array of combinatorial cross-reactive receptors (CoCRRs) with an optical detection system such as surface plasmon resonance imaging (SPRi), the obtained eT can monitor the binding events in real-time and generate continuous recognition patterns including 2D continuous evolution profile (CEP) and 3D continuous evolution landscape (CEL) for samples in liquid. Such an eT system is efficient for discrimination of common purified proteins.

Formal specification and proofs for the topology and classification of combinatorial surfaces

We describe one of the first attempts at using modern specification techniques in the field of geometric modeling and computational geometry. Using the Coq system, we developed a formal multi-level specification of combinatorial maps, used to represent subdivisions of geometric manifolds, and then exploited it to formally prove fundamental theorems. In particular, we outline here an original and constructive proof of a combinatorial part of the famous Surface Classification Theorem, based on a set of so-called “conservative” elementary operations on subdivisions.

Schnyder woods for higher genus triangulated surfaces (abstract)

Abstract We study a well known characterization of planar graphs, also called Schnyder wood or Schnyder labelling, which yields a decomposition into vertex spanning trees. The goal is to extend previous algorithms and characterizations designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. We define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and colouration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how to characterize our edge coloration in terms of genus g maps.

Local configurations in discrete combinatorial surfaces

Representing discrete objects by polyhedral complexes, we can define all conceivable topological characteristics of points in discrete objects, namely those of vertices of polyhedral complexes. Such a topological characteristic is determined for each point by observing a configuration of object points in the 3 × 3 × 3 local point set of its neighbors. We study a topological characteristic such that the point is in the boundary of a 3D polyhedral complex and the boundary forms a 2D combinatorial surface. By using the topological characteristic, we present an algorithm which examines whether the central point of a local point set is in a combinatorial surface, and show how many local point configurations exist in combinatorial surfaces in a 3D discrete space.

Stellar Mesh Simplification Using Probabilistic Optimization

Abstract This paper proposes the stellar mesh simplification method, a fast implementation of the Four‐Face Cluster (FFC) algorithm. In this method, a probabilistic optimization heuristic substitutes the priority queue of the original method, which results in a 40% faster algorithm with the same order of distortion. It extends naturally to a progressive and/or multiresolution scheme for combinatorial surfaces. This work also presents a simple way to encode the hierarchy of the resulting multiresolution meshes. This work also focuses on important aspects for the development of a practical and robust implementation of this simplification technique, and on the analysis of the influence of the parameters.

Doubly Transitive Automorphism Groups of Combinatorial Surfaces

Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S 4 , the alternating group A 5 and the Frobenius group C 7 · C 6 . In each case the combinatorial surface is uniquely determined. The symmetric group S 4 acts doubly transitively on the tetrahedron surface, the alternating group A 5 on the triangulation of the projective plane with six vertices and the Frobenius group C 7 · C 6 on the Moebius torus with seven vertices.