Kosiuk2015-Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle

  public model
Model Identifier
BIOMD0000000933
Short description
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.
Format
SBML (L2V4)
Related Publication
  • Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle.
  • Kosiuk I, Szmolyan P
  • Journal of mathematical biology , 4/ 2016 , Volume 72 , Issue 5 , pages: 1337-1368 , PubMed ID: 26100376
  • Max Planck Institute for Mathematics in the Sciences, Inselstra├če 22, 04103, Leipzig, Germany. ilona.kosiuk@mis.mpg.de.
  • 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.
Contributors
Submitter of the first revision: Ahmad Zyoud
Submitter of this revision: Ahmad Zyoud
Modellers: Ahmad Zyoud

Metadata information

is (2 statements)
BioModels Database MODEL2004240001
BioModels Database BIOMD0000000933

isDescribedBy (1 statement)
PubMed 26100376

hasTaxon (1 statement)
hasProperty (4 statements)
Mathematical Modelling Ontology Ordinary differential equation model
Gene Ontology cell cycle
NCIt Enzyme Kinetics
NCIt Mitosis

isDerivedFrom (1 statement)
occursIn (1 statement)
Brenda Tissue Ontology embryo


Curation status
Curated


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Model files

Kosiuk2015.xml SBML L2V4 Kosiuk2015-Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle 33.66 KB Preview | Download

Additional files

Kosiuk2015.cps COPASI version 4.27 (Build 217) Kosiuk2015-Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle_Fig3_Curated 58.55 KB Preview | Download
Kosiuk2015.sedml sed-ml L1V2 Kosiuk2015-Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle_Fig3_Curated 2.66 KB Preview | Download

  • Model originally submitted by : Ahmad Zyoud
  • Submitted: Apr 24, 2020 1:09:29 PM
  • Last Modified: Apr 24, 2020 1:09:29 PM
Revisions
  • Version: 2 public model Download this version
    • Submitted on: Apr 24, 2020 1:09:29 PM
    • Submitted by: Ahmad Zyoud
    • With comment: Automatically added model identifier BIOMD0000000933
Legends
: Variable used inside SBML models


Species
Species Initial Concentration/Amount
M

0016746
0.0 mmol
X

0000652
0.0 mmol
C

Guanidine
0.3 mmol
Reactions
Reactions Rate Parameters
=> M; C compartment*(6*C/(1+2*C)*(1-M)/((epislon+1)-M)-3/2*M/(epislon+M)) epislon = 0.001
=> X; M compartment*(M*(1-X)/((epislon+1)-X)-7/10*X/(epislon+X)) epislon = 0.001
=> C; X compartment*1/4*((1-X)-C) []
Curator's comment:
(added: 24 Apr 2020, 13:09:05, updated: 24 Apr 2020, 13:09:05)
Figure 3 has been reproduced using Copasi 4.27 ( Build 217)