Dynamics Of Gene Networks With Time Delays
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The study of time delays and their influence on gene network dynamics is the central topic of this thesis. Three different gene models with delay are studied: (1) a single gene-mRNA-protein model (2) a multiple gene network model (3) a continuous gene network model Chapters 2 and 3 present a brief introduction to the fields of gene regulatory networks and delay differential equations, respectively. Previous results relevant for this thesis are presented and a friendly overview with figures and examples is given. The purpose of these two chapters is to present a self-contained introduction of the main ideas to the rest of the thesis. Chapter 4 presents a linear and nonlinear analysis of (1) a single gene-mRNAprotein system given by M = alpha (sub m)X (1/(1+(P(sub d)/P (sub 0) exp(n))) - Mu (sub m)X M; P = alpha (sub p)X M - Mu (sub p)X P. The study of this model is divided into three cases: Case 1: Perturbation Methods with Constant Delay Case 2: Center Manifold Reduction with Constant Delay Case 3: Perturbation Methods with State-Dependent Delay Theoretical proof that the model exhibits oscillatory behavior of mRNA and protein expressions was found. The final outcome results in closed form expressions for the limit cycle amplitude and frequency of oscillation. An important result of these findings is the theoretical evidence that delays can drive oscillations in gene activity. Chapter 5 presents a study of a network model of N coupled gene units. This analysis is the natural extension of a single-gene model by considering multiple gene-mRNA-protein units interconnected. Two different cases are studied and theoretical and numerical results are presented. Chapter 6 presents a study of a continuous network model. The model takes the form m = -Mu X m + intrgral (K X (x-(bar x)) X H X (P (sub d) X (bar x)) X D X (bar x); p = m - Mu X p where m = m X (x,t), p = p X (x,t), p (sub d) X (bar x) = p X (bar x,t - T), H is a Hill function, and where K(x - (bar x)) is a weighting function. We choose K(x - (bar x)) in two different ways K(x - (bar x)) = 1 and K(x - (bar x)) = e (exp -|x - (bar x)| which we name uniform and exponential weighting, respectively. Both of these cases are studied by either theoretical or numerical analysis, and a detailed stability study of the steady states is given. Closed form expressions for the critical delay T sub(cr) and associated frequency "w" are found. Finally, confirmation of our results are presented by discretizing the continuous system into an N-dimensional system and showing that the discrete results in Chapter 5 approach the continuous results as N->infinity.
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