Browsing by Author "Sharon R. Lubkin, Committee Co-Chair"

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    Analysis and Computation for a Fluid Mixture Model of Tissue Deformations
    (2008-06-17) Jiang, Qunlei; Xiaobiao Lin, Committee Member; Kazufumi Ito, Committee Member; Hien Tran, Committee Member; Zhilin Li, Committee Chair; Sharon R. Lubkin, Committee Co-Chair
    A fluid mixture model of tissue deformations in one and two dimensions has been studied in this dissertation. The model is a mixed system of nonlinear hyperbolic and elliptic partial differential equations with interfaces. Both theoretical and numerical analysis are presented. We found the relationship between physical parameters and the resulting pattern of tissue deformations via linear stability analysis. Several numerical experiments support our theoretical analysis. The solution of the system exhibits non-smoothness and discontinuities at the interfaces. The conventional high order finite difference methods (FDM), such as the WENO scheme and TVD Runge Kutta method, for the hyperbolic equation, coupled with the central FDM for the elliptic equation, give spurious oscillations near the interfaces in our problem. By enforcing the jump conditions across the interfaces, our approach, the immersed interface method (IIM), eliminates non-physical oscillations, improves the accuracy of the solution, and maintains the sharp interface as time evolves. The IIM has been applied to solve a one dimensional linear advection equation with discontinuous initial conditions. By building the jump conditions into a conventional finite difference method, the Lax-Wendroff method, solutions of second order accuracy are observed. The IIM showed its robustness in solving the linear advection equation with nonhomogeneous jump conditions across the moving interface. The two dimensional fluid mixture model has been derived asymptotically from the three dimensional model so that the thickness of the gel is taken into account. Many numerical examples have been completed using Clawpack and qualitatively reasonable numerical solutions have been obtained.
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    Stochastic Modeling of the Behavior of Dynein
    (2005-04-11) Goedecke, David Michael; John F. Monahan, Committee Member; Kevin Gross, Committee Member; Carla Mattos, Committee Member; Sharon R. Lubkin, Committee Co-Chair; Timothy C. Elston, Committee Co-Chair
    Molecular motors are proteins that convert stored energy into physical work inside cells, and thus are the engines that drive many cellular functions. An individual motor can be studied using a laser trap to measure its response to working against an external force. Axonemal dynein is the molecular motor responsible for the rhythmic beating of eukaryotic cilia and flagella. An individual axonemal dynein molecule is capable of both unidirectional, processive motion and bidirectional motion when placed under a load (Shingyoji et al., 1998). This capability may be an important underlying factor in the mechanism for flagellar and ciliary motion. A detailed stochastic model is proposed which links the physical motion of a two-headed dynein molecule to the biochemical steps of its ATP hydrolysis cycle. Forward motion is driven by ATP hydrolysis, while backward motion is due to a passive process of biased diffusion. The model exhibits both processive and bidirectional behaviors. A simplified model which can be more easily analyzed is derived, as is an alternate version which steps backward actively, rather than sliding passively. The simplified models are then used to predict motor characteristics such as the load-velocity profile, the stall force, and the effective diffusion coefficient, which can be determined experimentally and used to distinguish among competing mechanisms.