F-1-ATPase (F-1) is an ATP-driven rotary motor protein ubiquitously found in many species as the catalytic portion of FoF1-ATP synthase. Despite the highly conserved amino acid sequence of the catalytic core subunits: alpha and beta, F-1 shows diversity in the maximum catalytic turnover rate V-max and the number of rotary steps per turn. To study the design principle of F-1, we prepared eight hybrid F(1)s composed of subunits from two of three genuine (F)1s: thermophilic Bacillus PS3 (TF1), bovine mitochondria (bMF(1)), and Paracoccus denitrificans (PdF1), differing in the V-max and the number of rotary steps. The V-max of the hybrids can be well fitted by a quadratic model highlighting the dominant roles of 0 and the couplings between alpha-beta. Although there exist no simple rules on which subunit dominantly determines the number of steps, our findings show that the stepping behavior is characterized by the combination of all subunits.

Density-based clustering methods are prominent clustering approaches to discover discrete structures buried in high-dimensional (HD) data in terms of density variations. Among them is the well-known Density Peak Clustering (DPC) proposed by Rodriguez and Laio (2014) that performs fairly well in detecting clusters with nonlinear shapes and varying densities. However, it has several shortcomings that it does not learn about the nonlinear shapes of the underlying HD data, is lack of a probabilistic framework to handle overlapping clusters, and is not fully automated.

Here we develop comprehensive generalizations of DPC, termed Graph-based Fuzzy Density Peak Clustering (GF-DPC), to circumvent these limitations. In GF-DPC, graph-based methods are employed to robustly estimate densities and capture nonlinearities in the HD data that enhances its power in detecting clusters with arbitrary shapes. Furthermore, a fuzzy extension is introduced that returns a probabilistic assignment of data points to the detected clusters. Finally, the identification of cluster centers and the number of clusters are automated and generalized in terms of fuzzy clustering validation index. The superior performances of GF-DPC compared to other well-known fuzzy clustering methods in discovering clusters with arbitrary shapes, densities, separations and overlapping are demonstrated using both intuitive examples and real datasets.

Validation is an essential part of clustering analysis to assess the quality of the detected patterns. One of the most well-known validation methods is the silhouette index that is only applicable to hard clustering results. In this paper, we develop a fuzzy clustering validation framework based on the silhouette index, termed Shape-aware Generalized Silhouette Analysis (SAGSA), which allows for an extensive evaluation and diagnoses of possible problems in the clustering results at the point-wise, cluster-wise and global levels.

In particular, a probabilistic framework to quantify the cohesion (compactness) and separation of the detected clusters is formulated to handle fuzzy clustering results. Furthermore, graph-based (shape-aware) distances are employed to faithfully capture nonlinear structures enabling an accurate validation of curved clusters. Finally, a graphical tool, cohesion-separation (CS) plot, is introduced that allows us to visually assess clustering results at different levels regardless of the dimensionality of the dataset. To show its effectiveness in diagnosing problems in clustering results, SAGSA is compared with other fuzzy clustering validation methods on test cases with different types of clustering challenges, namely, clusters with arbitrary shapes, imbalance sizes, overlapping, hierarchical structures, mixed with noises, etc.