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Browsing by Author "Putkaradze, Vakhtang, advisor"

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    Analysis and modeling of cells, cell behavior, and helical biological molecules
    (Colorado State University. Libraries, 2011) Benoit, Steven Richard, author; Putkaradze, Vakhtang, advisor; Shipman, Patrick, committee member; Estep, Don, committee member; Marconi, Mario, committee member; Tobet, Stuart, committee member
    Mathematical models of biological systems have evolved over time and through the introduction and growth of computer simulation and analysis. Models have increased in sophistication and power through the combination of multi-scale approaches, molecular and granular dynamics simulations, and advances in parallelization and processing speed. However, current cell models cannot accurately predict behaviors at the whole-cell scale, nor can molecular models predict accurately the complex shape assumed by large biological molecules including proteins, although significant progress is being made toward this goal. The present work introduces new models in three domains within biological systems modeling. We first discuss a phenomenological model of observed cell motions in developing tissue that characterizes cells according to a best-fit generalized diffusion model and combines this data with Voronoi diagrams to effectively visualize patterns of cell behavior in tissue. Next, we present a series of component models for cells and cell structure that support simulations involving tens to hundreds of cells in a way that captures behaviors ignored by existing models, including pseudopod formation, membrane mechanics, cytoskeletal polymerization / depolymerization, and chemical signal transduction. The resulting models exhibit many of the behaviors of real-world cells including polarization and chemotaxis. Finally, we present a method for analysis of biological molecules that form helical conformations that includes long-range electrostatic interactions as well as short-range interactions to prevent self-intersections. We consider the stability of molecules with repeating monomers that include off-axis charge concentrations and derive energy landscapes to identify stable conformations, then analyze helical stability using geometric methods.
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    Automated methods for quantifying the tortuosity of microvascular networks
    (Colorado State University. Libraries, 2012) Dodd, Melody, author; Putkaradze, Vakhtang, advisor; Tobet, Stuart, committee member; Shipman, Patrick, committee member
    Networks of microscopic blood vessels can be studied for changes in morphology that correlate with biological abnormalities. Tortuosity, or vessel twistiness, is one of these morphological properties, and it can be surprisingly difficult to quantify. The purpose of this thesis is to present the development, testing, and analysis of new automated methods to measure and quantify the tortuosity of microvascular networks. We will explain necessary automated image processing techniques and background information before presenting our new metrics for measuring network tortuosity. Experiments using the methods will be presented, including a full analysis of the results. We will use the results from these experiments to justify our final conclusions and recommendations regarding the performance of the methods.
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    Constrained dynamics of rolling balls and moving atoms
    (Colorado State University. Libraries, 2011) Kim, Byungsoo, author; Putkaradze, Vakhtang, advisor; Tavener, Simon, committee member; Shipman, Patrick, committee member; Marconi, Mario C., committee member
    This dissertation is devoted to the study of the dynamics, conservation laws and symmetries of rolling spheres, with special attention to applications to atomic and molecular systems. Previously known conservation laws of the rolling motion are associated with the nonholonomic version of Noether's theorem. Moreover, the conservation laws are related to the reduction by Lie symmetries of the dynamic equations of motion. Symmetries in the Noether's theorem and in the reduction by Lie symmetries are compared in their applications. In addition, we analyze the collective motion of the system of rolling particles for its statistical quantities under the constraint condition of rolling without slipping motion. The numerical simulations revealed some of qualitative characteristics in the statistical mechanics of the rolling-constrained system. As a separate topic, the study of the molecular dynamics is discussed in relation to the results of recent experimental achievements with the non-contact atomic force microscopy. We propose a novel scenario explaining the process of single-atom manipulation in terms of the classical resonance effect.
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    Homotopy continuation methods, intrinsic localized modes, and cooperative robotic workspaces
    (Colorado State University. Libraries, 2012) Brake, Daniel Abram, author; Putkaradze, Vakhtang, advisor; Maciejewski, Tony, advisor; Marconi, Mario, committee member; Bates, Dan, committee member; Shipman, Patrick, committee member
    This dissertation considers three topics that are united by the theme of application of geometric and nonlinear mechanics to practical problems. Firstly we consider the parallel implementation of numerical solution of nonlinear polynomial systems depending on parameters. The program written to do this is called Paramotopy, and uses the Message Passing Interface to distribute homotopy continuation solves in another program called Bertini across a supercomputer. Paramotopy manages writing of Bertini input files, allows automatic re-solution of the system at points at which paths failed, and makes data management easy. Furthermore, parameter homotopy nets huge performance gains over fresh homotopy continuation runs. Superlinear speedup was achieved, up to hard drive throughput capacity. Various internal settings are demonstrated and explored, and the User's Manual is included. Second, we apply nonlinear theory and simulation to nanomechanical sensor arrays. Using vibrating GaAs pillars, we model Intrinsic Localized Modes (ILMs), and investigate ILM-defect pinning, formation, lifetime, travel and movement, and parameter dependence. Intrinsic Localized Modes have been analyzed on arrays of nonlinear oscillators. So far, these oscillators have had a single direction of vibration. In current experiments for single molecule detection, arrays made of Gallium Arsenide will be innately bidirectional, forced, dissipative. We expand previous full models to bidirectionality, and simulate using ODE solvers. We show that small regions of a very large parameter space permit strong ILM formation. Additionally, we use Hamiltonian mechanics to derive new simplified models for the monodirectional ILM travel on an infinite array. This monodirectional ILMs of constant amplitude have unrealistic behavior. Permitting the amplitude of the ILM to vary in time produces much more realistic behavior, including wandering and intermittent pinning. The final set of problems concerns the application of numerical algebraic geometric methods to untangle the phase space of cooperating robots, and optimize configuration for fault tolerance. Given two robots in proximity to each other, if one experiences joint failure, the other may be able to assist, restoring lost workspace. We define a new multiplicity-weighted workspace measure, and use it to solve the optimization problem of finding the best location for an assistance socket and separation distance for the two robots, showing that the solution depends on robot geometry, which link is being grasped, and the choice of objective function.
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    Mathematical methods for fluid-solid interfaces: meandering streams and sand ripples
    (Colorado State University. Libraries, 2008) Mertens, Keith, author; Putkaradze, Vakhtang, advisor
    This thesis presents several mathematical methods for modeling free surfaces, interfaces, and fluid-solid interactions. This work is done in the context of two physical systems. In the first two sections, the focus will be to understand the the physics of streams flowing down inclined substrates. Models will be derived to investigate both steady state and dynamic meandering profiles. It will be shown that, through the right approximation techniques, many physical insights can be drawn about this system. These results include: a complete understanding of the steady states, transitions between steady states, mechanism of meandering, forces involved in meandering, and spectral scaling laws of long-time ensemble averaged meandering stream profiles. In the third section, the focus will shift to how one can model underlying physics when it becomes too complicated to address from first principles. Here, the power of symmetries and conservation laws are explored to derive an amplitude equation describing the interface between sand and water when the water is subjected to oscillatory flow. The thesis will then close by posing a novel way to study scaling laws with respect to parameters using Lie's prolongation algorithm. Through this work various tools will be combined from the fields of physics, engineering, applied and pure mathematics to develop approaches for reducing complex systems into tractable pieces which can be studied carefully.