Coordinate-Independent Computations on Differential Equations

This project investigates the computational representation of differentiable manifolds, with the primary goal of solving partial differential equations using multiple coordinate systems on general n- dimensional spaces. In the process, this abstraction is used to perform accurate integrations of ordinary differential equations using multiple coordinate systems. In the case of linear partial differential equations, however, unexpected difficulties arise even with the simplest equations.

A Modern Differential Geometric Approach to Shape from Shading

How the visual system extracts shape information from a single grey-level image can be approached by examining how the information about shape is contained in the image. This technical report considers the characteristic equations derived by Horn as a dynamical system. Certain image critical points generate dynamical system critical points. The stable and unstable manifolds of these critical points correspond to convex and concave solution surfaces, giving more general existence and uniqueness results. A new kind of highly parallel, robust shape from shading algorithm is suggested on neighborhoods of these critical points. The information at bounding contours in the image is also analyzed.

Automatic Qualitative Analysis of Ordinary Differential Equations Using Piecewise Linear Approximations

This paper explores automating the qualitative analysis of physical systems. It describes a program, called PLR, that takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values. PLR approximates intractable nonlinear systems with piecewise linear ones, analyzes the approximations, and draws conclusions about the original systems. It chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes, yet simple enough to be analyzed completely and efficiently. It derives additional properties, such as boundedness or periodicity, by theoretical methods. I demonstrate PLR on several common nonlinear systems and on published examples from mechanical engineering.

Differential effects of insulin signaling on individual carbon fluxes for fatty acid synthesis in brown adipocytes

Considering the major role of insulin signaling on fatty acid synthesis via stimulation of lipogenic enzymes, differential effects of insulin signaling on individual carbon fluxes for fatty acid synthesis have been investigated by comparing the individual lipogenic fluxes in WT and IRS-1 knockout (IRS-1 KO) brown adipocytes. Results from experiments on WT and IRS-1 KO cells incubated with [5-¹³C] glutamine were consistent with the existence of reductive carboxylation pathway. Analysis of isotopomer distribution of nine metabolites related to the lipogenic routes from glucose and glutamine in IRS-1 KO cells using [U-¹³C] glutamine as compared to that in WT cells indicated that flux through reductive carboxylation pathway was diminished while flux through conventional TCA cycle was stimulated due to absence of insulin signaling in IRS-1 KO cells. This observation was confirmed by quantitative estimation of individual lipogenic fluxes in IRS-1 KO cells and their comparison with fluxes in WT cells. Thus, these results suggest that glutamine’s substantial contribution to fatty acid synthesis can be directly manipulated by controlling the flux through reductive carboxylation of alpha-ketoglutarate to citrate using hormone (insulin).

SCAPA (Spacially Addressable Protein Array) – a Novel Protein Array for the Differential Profiling of Ligand-receptor Induced Signalling

While protein microarray technology has been successful in demonstrating its usefulness for large scale high-throughput proteome profiling, performance of antibody/antigen microarrays has been only moderately productive. Immobilization of either the capture antibodies or the protein samples on solid supports has severe drawbacks. Denaturation of the immobilized proteins as well as inconsistent orientation of antibodies/ligands on the arrays can lead to erroneous results. This has prompted a number of studies to address these challenges by immobilizing proteins on biocompatible surfaces, which has met with limited success. Our strategy relates to a multiplexed, sensitive and high-throughput method for the screening quantification of intracellular signalling proteins from a complex mixture of proteins. Each signalling protein to be monitored has its capture moiety linked to a specific oligo ‘tag’. The array involves the oligonucleotide hybridization-directed localization and identification of different signalling proteins simultaneously, in a rapid and easy manner. Antibodies have been used as the capture moieties for specific identification of each signaling protein. The method involves covalently partnering each antibody/protein molecule with a unique DNA or DNA derivatives oligonucleotide tag that directs the antibody to a unique site on the microarray due to specific hybridization with a complementary tag-probe on the array. Particular surface modifications and optimal conditions allowed high signal to noise ratio which is essential to the success of this approach.