We present a model of coupled transcriptional-translational ultradian oscillators (TTOs) as a possible mechanism for the circadian rhythm observed at the cellular level. It includes nonstationary Poisson interactions between the transcriptional proteins and their affined gene sites. The associated reaction-rate equations are nonlinear ordinary differential equations of stochastic switching type. We compute the deterministic limit of this system, or the limit as the number of gene-proteins interactions per unit of time becomes large. In this limit, the random variables of the model are simply replaced by their limiting expected value. We derive the Kolmogorov equations — a set of partial differential equations —, and we obtain the associated moment equations for a simple instance of the model. In the stationary case, the Kolmogorov equations are linear and the moment equations are a closed set of equations. In the nonstationary case, the Kolmogorov equations are nonlinear and the moment equations are an open-ended set of equations. In both cases, the deterministic limit of the moment equations is in agreement with the deterministic state equations.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:BVIV.1828/933 |
| Date | 01 May 2008 |
| Creators | De La Chevrotière, Michèle |
| Contributors | Reinhard, Illner |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Rights | Available to the World Wide Web |
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