Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization

We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \mathbb {R}^{3}, as hyperelliptic curves, and as \mathbb {CP}^{1} modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller

Calculation of the relative metastabilities of proteins in subcellular compartments of Saccharomyces cerevisiae

[abridged] Background: The distribution of chemical species in an open system at metastable equilibrium can be expressed as a function of environmental variables which can include temperature, oxidation-reduction potential and others. Calculations of metastable equilibrium for various model systems were used to characterize chemical transformations among proteins and groups of proteins found in different compartments of yeast cells. Results: With increasing oxygen fugacity, the relative metastability fields of model proteins for major subcellular compartments go as mitochondrion, endoplasmic reticulum, cytoplasm, nucleus. In a metastable equilibrium setting at relatively high oxygen fugacity, proteins making up actin are predominant, but those constituting the microtubule occur with a low chemical activity. A reaction sequence involving the microtubule and spindle pole proteins was predicted by combining the known intercompartmental interactions with a hypothetical program of oxygen fugacity changes in the local environment. In further calculations, the most-abundant proteins within compartments generally occur in relative abundances that only weakly correspond to a metastable equilibrium distribution. However, physiological populations of proteins that form complexes often show an overall positive or negative correlation with the relative abundances of proteins in metastable assemblages. Conclusions: This study explored the outlines of a thermodynamic description of chemical transformations among interacting proteins in yeast cells. The results suggest that these methods can be used to measure the degree of departure of a natural biochemical process or population from a local minimum in Gibbs energy.Comment: 32 pages, 7 figures; supporting information is available at http://www.chnosz.net/yeas

Geometric rigidity for sequences of W2,2 conformal immersions

We analyse sequences of discs conformally immersed in ℝ with energy ∫{pipe}A{pipe} ≤ γ, where γ = 8π if n = 3 and γ = 4π when n ≥ 4. We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper's minimal surface if n = 3 or Chen's minimal graph if n ≥ 4. In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313-340, 2012; Rivière, Adv Calculus Variations 6(1), 1-31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then lim inf W(f) ≥ 8π. We apply the above analysis to show that in ℝ if the sequence diverges so that lim W(f) = 8π then there exists a sequence of Möbius transforms σ such that σ ○ f converges weakly to a catenoid