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:

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)

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

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

This model is hosted on BioModels Database
and identified by: BIOMD0000000006
. To cite BioModels Database, please use: BioModels Database: An enhanced, curated and annotated resource for published quantitative kinetic models
.

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.

*/
// unit definitions
import nsrunit;
unit conversion off;
// SBML property definitions
property sbmlRole=string;
property sbmlName=string;
property sbmlCompartment=string;
// SBML reactions
// Reaction1: EmptySet <=> z
// Reaction2: u <=> EmptySet
// Reaction3: z <=> u
math main {
realDomain time second;
time.min=0;
extern time.max;
extern time.delta;
// variable definitions
real cell = 1 L;
real kappa = .015;
real k6 = 1;
real k4 = 180;
real k4prime = .018;
real alpha(time);
real EmptySet = 1 mol;
real u(time) mol;
real z(time) mol;
real v(time) mol;
real Reaction1(time) katal;
real Reaction2(time) katal;
real Reaction3(time) katal;
// equations
alpha = k4prime/k4;
when (time=time.min) u = 0;
u:time = k4*(v-u)*(alpha+u^2)-k6*u;
z = v-u;
when (time=time.min) v = 0;
v:time = kappa-k6*u;
Reaction1 = kappa;
Reaction2 = k6*u;
Reaction3 = k4*z*(k4prime/k4+u^2);
// variable properties
cell.sbmlRole="compartment";
kappa.sbmlRole="parameter";
k6.sbmlRole="parameter";
k4.sbmlRole="parameter";
k4prime.sbmlRole="parameter";
alpha.sbmlRole="parameter";
EmptySet.sbmlRole="species";
EmptySet.sbmlCompartment="cell";
u.sbmlRole="species";
u.sbmlCompartment="cell";
z.sbmlRole="species";
z.sbmlCompartment="cell";
v.sbmlRole="species";
v.sbmlCompartment="cell";
Reaction1.sbmlRole="rate";
Reaction2.sbmlRole="rate";
Reaction3.sbmlRole="rate";
}