This paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner.

Establishing the general and promoter-specific mechanistic features of gene transcription initiation requires improved understanding of the sequence-dependent structural/dynamic features of promoter DNA. Experimental data suggest that a spontaneous dsDNA strand separation at the transcriptional start site is likely to be a requirement for transcription initiation in several promoters. Here, we use Langevin molecular dynamic simulations based on the Peyrard-Bishop-Dauxois nonlinear model of DNA (PBD LMD) to analyze the strand separation (bubble) dynamics of 80-bp-long promoter DNA sequences. We derive three dynamic criteria, bubble probability, bubble lifetime, and average strand separation, to characterize bubble formation at the transcriptional start sites of eight mammalian gene promoters. We observe that the most stable dsDNA openings do not necessarily coincide with the most probable openings and the highest average strand displacement, underscoring the advantages of proper molecular dynamic simulations. The dynamic profiles of the tested mammalian promoters differ significantly in overall profile and bubble probability, but the transcriptional start site is often distinguished by large (longer than 10 bp) and long-lived transient openings in the double helix. In support of these results are our experimental transcription data demonstrating that an artificial bubble-containing DNA template is transcribed bidirectionally by human RNA polymerase alone in the absence of any other transcription factors.