Theses and Dissertations - Department of Mathematics

Permanent URI for this collection

https://ir.ua.edu/handle/123456789/120

Browse

Browsing Theses and Dissertations - Department of Mathematics by Subject "Bioinformatics"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Fast alternating direction implicit schemes for geometric flow equations and nonlinear poisson equation in biomolecular solvation analysis
    (University of Alabama Libraries, 2014) Tian, Wufeng; Zhao, Shan; University of Alabama Tuscaloosa
    The present work introduces new alternating direction implicit (ADI) methods to solve potential driven geometric flow partial differential equations (PDEs) for biomolecular surface generation and the nonlinear Poisson equations for electrostatic analysis. For solving potential driven geometric flow PDEs, an extra factor is usually added to stabilize the explicit time integration. However, there are two existing ADI schemes based on a scaled form, which involves nonlinear cross derivative terms that have to be evaluated explicitly. This affects the stability and accuracy of these ADI schemes. To overcome these difficulties, we propose a new ADI algorithm based on the unscaled form so that cross derivatives are not involved. Central finite differences are employed to discretize the nonhomogenous diffusion process of the geometric flow. The proposed ADI algorithm is validated through benchmark examples with analytical solutions, reference solutions, or literature results. Moreover, quantitative indicators of a biomolecular surface, including surface area, surface-enclosed volume, and solvation free energy, are analyzed for various proteins. The proposed ADI method is found to be unconditionally stable and more accurate than the existing ADI schemes in all tests of biomolecular surface generation. The proposed ADI schemes have also been applied in solving the nonlinear Poisson equation for electrostatic solvation analysis. Compared with the existing biconjugate gradient iterative solver, the ADI scheme is more efficient. The CPU time cost is validated through the solvation analysis of an one atom Kirkwood model and a set of 17 small molecules whose experimental measurements are available. Additionally, application of the proposed ADI scheme is considered for electrostatic solvation analysis of a set of 19 proteins. The proposed ADI scheme enables the use of a large time increment in the steady state simulation so that the proposed ADI algorithm is efficient for biomolecular surface generation and solvation analysis.
  • Thumbnail Image
    Item
    Metabolic network inference with the graphical lasso
    (University of Alabama Libraries, 2015) Aicher, Joseph Krittameth; Song, Song; Reed, Laura K.; University of Alabama Tuscaloosa
    Metabolic networks describe the interactions and reactions between different metabolites (e.g. sugars, fatty acids, amino acids) in a biological system, which together give rise to the chemical processes which make life possible. Efforts to further knowledge of the structure of metabolic networks have taken place for well over a century through the efforts of numerous biochemists and have revolutionized our understanding of biology and the capabilities of modern medicine. The introduction in the recent past of metabolomics technologies, which allow for the simultaneous measurement of the concentrations of a significant number of metabolites, has led to the development of mathematical and statistical algorithms that aim to use the information and data that these technologies have made available to make inferences about the structure of metabolic networks. In this thesis, I investigate the application of the graphical lasso algorithm to metabolomics data for the purposes of metabolic network inference. I use the graphical lasso on a metabolomics dataset collected by gas chromatography-mass spectrometry from Drosophila melanogaster to estimate graphical models of varying levels of sparsity that describe the conditional dependence structure of the observed metabolite concentrations. With these estimated models, I describe how they can be chosen from and interpreted in the context of both the data and the underlying biology to inform our knowledge of metabolic network structure.