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Dynamical Systems in Cell Division Cycle, Winnerless Competition Models, and Tensor Approximations

Gong, Xue
http://orcid.org/0000-0003-1994-8216
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1458303716

Abstract Details

2016, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
This dissertation discusses the application of dynamical systems in three different fields. We study the cell cycle dynamics in a cell-cell coupling system, heteroclinic binding networks modeling sequential working memory and the tensor approximation problems using the gradient flow. We consider the cell cycle coupling model for budding yeast (Saccharomyces cerevisiae), where the rates of progression for cells in a certain phase (responsive region R) of the cell division cycle are influenced by cells in another phase (signaling region S). We first study a model to mimic the situation where there is a delay between cells in S producing the chemical agents and cells in R experiencing the feedback from the agents by including a gap between the two regions. We analyze the dynamics of this system for the k-periodic solutions and compare the dynamics with the system without a gap. Second, we study a model where two biologically motivated sources of noise are introduced into the cell cycle system and compare them with the model that has Gaussian white noise perturbations. In simulations, we explore how an ordered two-cluster periodic state of cells becomes disordered by the increase of noise and a uniform distribution emerges for large noise. As a corollary to the results, we can estimate the coupling strength of the cell cycle in yeast autonomous oscillation experiments where clustering is observed. We also study high-dimensional dynamical systems with the existence of a heteroclinic network in the phase space under certain assumptions. This kind of dynamical systems can be used to describe the phenomenon of winnerless competition. We study the heteroclinic network that is considered as a heteroclinic chain of heteroclinic cycles: Starting from each saddle point that is not in the last heteroclinic cycle, there are two heteroclinic orbits, one stays in its heteroclinic cycle and the other one goes to the next heteroclinic cycle; there is only one heteroclinic orbit starting from each saddle point in the last heteroclinic cycle. We prove under certain technical assumptions that for each collection of successive heteroclinic orbits inside this network, there is an open set of initial points such that any trajectory starting from each of them follows the prescribed collection of heteroclinic orbits and stays in a small neighborhood of it. We show also that the symbolic complexity function of the system restricted to this small neighborhood is a polynomial of degree L-1, where L is the number of heteroclinic cycles in the network. Approximations of a high-dimensional tensor by sums of separable tensors can be used to avoid the curse of dimensionality in numerical computations. We study the approximation of a rank-2 symmetric tensor using a rank-1 tensor according to the gradient flow of the regularized error function. It had been previously known that at least one local minimum is attained in the symmetric set. We prove that the non-symmetric equilibria to the gradient flow are either saddle points or the maximum points. This implies that outside of the symmetric set there do not exist local minimum points. Furthermore, for these saddle points, we study the possibility of the so-called swamp behavior, where the trajectory stays in a neighborhood of the saddle point for a relatively long time before it leaves the neighborhood.
Todd Young (Advisor)
145 p.
Mathematics
yeast metabolic oscillations; order-disorder transition; heteroclinic networks; non-symmetric saddle points

Recommended Citations

Citations

  • Gong, X. (2016). Dynamical Systems in Cell Division Cycle, Winnerless Competition Models, and Tensor Approximations [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1458303716

    APA Style (7th edition)

  • Gong, Xue. Dynamical Systems in Cell Division Cycle, Winnerless Competition Models, and Tensor Approximations. 2016. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1458303716.

    MLA Style (8th edition)

  • Gong, Xue. "Dynamical Systems in Cell Division Cycle, Winnerless Competition Models, and Tensor Approximations." Doctoral dissertation, Ohio University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1458303716

    Chicago Manual of Style (17th edition)

ohiou1458303716
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This open access ETD is published by Ohio University and OhioLINK.