We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci curvature can be employed in place of Ollivier-Ricci curvature for faster computation in larger real-world networks whenever coarse analysis suffices.
Perturbations in protein structure define the mechanism of allosteric regulation and biological information transfer. In cytochrome c (cyt c), Met80 ligation to the heme iron is critical for the protein’s electron-transfer (ET) function in oxidative phosphorylation and for suppressing its peroxidase activity in apoptosis. The hard base Lys is a better match for the hard ferric iron than the soft base Met, suggesting the key role of the protein scaffold in favoring Met ligation. To probe the role of the protein structure in maintenance of Met ligation, mutations T49V and Y67R/M80A were designed to disrupt hydrogen bonding and packing of the heme coordination loop, respectively. Electronic absorption, NMR, and EPR spectra reveal that ferric forms of both variants are Lys-ligated at neutral pH. A minor change in the tertiary contacts in T49V, away from the heme coordination loop, appears to be sufficient to execute a change in ligation, suggesting a cross-talk between the different regions of the protein structure and a possibility of built-in conformational switches in cyt c. Analyses of thermodynamic stability, kinetics of Lys binding and dissociation and the pH-dependent changes in ligation provide detailed characterization of the Lys coordination in these variants and relate these properties to the extent of structural perturbations. The findings emphasize the importance of the hydrogen-bonding network in controlling the native Met80 ligation to the heme iron.
We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian, which is easy to compute or to estimate numerically. It can therefore serve as a rough classification of large empirical graphs into families that share the same asymptotic behavior of the spectrum so that the distance of two graphs from the same family is bounded by O(1/n) in terms of size n of their vertex sets. Numerical experiments demonstrate that the spectral distance provides a practically useful measure of graph dissimilarity.
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