Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle

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• 作者：Ilona Kosiuk ; Peter Szmolyan
• 关键词：Cell cycle ; Mitotic oscillator ; Enzyme kinetics ; Geometric singular perturbation theory ; Relaxation oscillations ; Blow ; up method
• 刊名：Journal of Mathematical Biology
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：72
• 期：5
• 页码：1337-1368
• 全文大小：1,885 KB
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• 作者单位：Ilona Kosiuk (1)
  Peter Szmolyan (2)
  
  1. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany
  2. Institute for Analysis and Scientific Computing, Technische Universität Wien, Wiedner Hauptstraße 8-10, 1040, Vienna, Austria
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Biology
  Applications of Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1416

文摘

A minimal model describing the embryonic cell division cycle at the molecular level in eukaryotes is analyzed mathematically. It is known from numerical simulations that the corresponding three-dimensional system of ODEs has periodic solutions in certain parameter regimes. We prove the existence of a stable limit cycle and provide a detailed description on how the limit cycle is generated. The limit cycle corresponds to a relaxation oscillation of an auxiliary system, which is singularly perturbed and has the same orbits as the original model. The singular perturbation character of the auxiliary problem is caused by the occurrence of small Michaelis constants in the model. Essential pieces of the limit cycle of the auxiliary problem consist of segments of slow motion close to several branches of a two dimensional critical manifold which are connected by fast jumps. In addition, a new phenomenon of exchange of stability occurs at lines, where the branches of the two-dimensional critical manifold intersect. This novel type of relaxation oscillations is studied by combining standard results from geometric singular perturbation with several suitable blow-up transformations.