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BIOMD0000000006 - Tyson1991 - Cell Cycle 2 var

 

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Reference Publication
Publication ID: 1831270
Tyson JJ.
Modeling the cell division cycle: cdc2 and cyclin interactions.
Proc. Natl. Acad. Sci. U.S.A. 1991 Aug; 88(16): 7328-7332
Department of Biology, Virginia Polytechnic Institute and State University, Blacksburg 24061.  [more]
Model
Original Model: BIOMD0000000006.xml.origin
Submitter: Nicolas Le Novère
Submission ID: MODEL6614715255
Submission Date: 13 Sep 2005 20:32:12 UTC
Last Modification Date: 16 May 2013 22:38:56 UTC
Creation Date: 09 Feb 2005 02:36:17 UTC
Encoders:  Bruce Shapiro
   Lukas Endler
set #1
bqbiol:hasVersion Reactome REACT_152
bqbiol:is KEGG Pathway sce04111
bqbiol:occursIn Taxonomy Opisthokonta
bqbiol:isVersionOf Gene Ontology mitotic cell cycle
Notes
Tyson1991 - Cell Cycle 2 var

Mathematical model of the interactions of cdc2 and cyclin.

Description taken from the original Cellerator version of the model ( Tyson (1991, 2 variables) at http://www.cellerator.org ).

This model is described in the article:

Tyson JJ.
Proc. Natl. Acad. Sci. U.S.A. 1991; 88(16); 7328-32

Abstract:

The proteins cdc2 and cyclin form a heterodimer (maturation promoting factor) that controls the major events of the cell cycle. A mathematical model for the interactions of cdc2 and cyclin is constructed. Simulation and analysis of the model show that the control system can operate in three modes: as a steady state with high maturation promoting factor activity, as a spontaneous oscillator, or as an excitable switch. We associate the steady state with metaphase arrest in unfertilized eggs, the spontaneous oscillations with rapid division cycles in early embryos, and the excitable switch with growth-controlled division cycles typical of nonembryonic cells.

This is a two variable reduction of the larger 6-variable model published in the same paper. The equations are:

u'= k4(v-u)(alpha+u^2)-k6*u
v'=kappa-k6*u
z= v-u
with kappa = k1[aa]/[CT]

In the present implementation, an additional variable z is introduced with z = v-u is made, so that the different variables be interpreted as follows:

u=[activeMPF]/[CT]
v=([cyclin]+[preMPF]+[activeMPF])/[CT]
z=([ cyclin]+[preMPF])/[CT]
with [CT]=[CDC2]+{CDC2P]+[preMPF]+[aMPF].

The reactions included are only to show the flows between z and u, and do not influence the species, as they all are set to boundaryCondition=True , meaning, that they are only determined by the rate rules (explicit differential equations) and assignment rules.

If you set boundaryCondition=False and remove the rate rules for v, u and the the assignment rule for z, you get the more symmetrical, but equivalent, version from the Cellerator repository:

u'= k4*z*(alpha+u^2)-k6*u
z'=kappa-z*(alpha+u^2)

To the extent possible under law, all copyright and related or neighbouring rights to this encoded model have been dedicated to the public domain worldwide. Please refer to CC0 Public Domain Dedication for more information.

Model
Publication ID: 1831270 Submission Date: 13 Sep 2005 20:32:12 UTC Last Modification Date: 16 May 2013 22:38:56 UTC Creation Date: 09 Feb 2005 02:36:17 UTC
Mathematical expressions
Reactions
Reaction1 Reaction2 Reaction3  
Rules
Rate Rule (variable: u) Assignment Rule (variable: z) Rate Rule (variable: v) Assignment Rule (variable: alpha)
Physical entities
Compartments Species
cell EmptySet u z
v    
Global parameters
kappa k6 k4 k4prime
alpha      
Reactions (3)
 
 Reaction1 [EmptySet] → [z];  
 
 Reaction2 [u] → [EmptySet];  
 
 Reaction3 [z] → [u];  
 
Rules (4)
 
 Rate Rule (name: u) d [ u] / d t= k4*(v-u)*(alpha+u^2)-k6*u
 
 Assignment Rule (name: z) z = v-u
 
 Rate Rule (name: v) d [ v] / d t= kappa-k6*u
 
 Assignment Rule (name: alpha) alpha = k4prime/k4
 
  Spatial dimensions: 3.0  Compartment size: 1.0
 
 EmptySet
Compartment: cell
Initial amount: 1.0
 
 u
Compartment: cell
Initial amount: 0.0
 
  z
Compartment: cell
Initial amount: 0.0
 
 v
Compartment: cell
Initial amount: 0.0
 
Global Parameters (5)
 
   kappa
Value: 0.015
Constant
 
   k6
Value: 1.0
Constant
 
   k4
Value: 180.0
Constant
 
   k4prime
Value: 0.018
Constant
 
   alpha
Value: NaN
 
Representative curation result(s)
Representative curation result(s) of BIOMD0000000006

Curator's comment: (updated: 08 Apr 2011 02:56:21 PDT)

Nullclines (left) and time course trajectories (right) of the 2 variable system for various values of kappa in the u-v plane. The values of kappa were chosen to give similar results to figure 4 in the original publication, but as the exact values used are not known, they only show approximately the same results. The right graph shows time courses trajectories (200 time units) for the various values of kappa, and the u-nullcline (orange). the high and low values of kappa (kappa = 0.13, blue and kappa = 0.0075, green)lead to stable steady states, the intermediate one (kappa=0.018, yellow) approaches a limit cycle.
The results were calculated and plotted using XPPaut and the Biomodels.net sbml2xpp converter.

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