/*
 * A Minimal Cascade Model for the Mitotic Oscillator Involving
 * Cyclin and cdc2 Kinase
 * 
 * Model Status
 * 
 * The CellML model runs in OpenCell to reproduce figure 3 of the
 * published paper. The model also runs in COR but doe to the long
 * duration of the simulation (100 minutes) COR is not reall suitable
 * for such a simulation. The units have been checked and they
 * are consistent.
 * 
 * Model Structure
 * 
 * ABSTRACT: A minimal model for the mitotic oscillator is presented.
 * The model, built on recent experimental advances, is based on
 * the cascade of post-translational modification that modulates
 * the activity of cdc2 kinase during the cell cycle. The model
 * pertains to the situation encountered in early amphibian embryos,
 * where the accumulation of cyclin suffices to trigger the onset
 * of mitosis. In the first cycle of the bicyclic cascade model,
 * cyclin promotes the activation of cdc2 kinase through reversible
 * dephosphorylation, and in the second cycle, cdc2 kinase activates
 * a cyclin protease by reversible phosphorylation. That cyclin
 * activates cdc2 kinase while the kinase triggers the degradation
 * of cyclin has suggested that oscillations may originate from
 * such a negative feedback loop [Felix, M. A., Labbe, J. C., Doree,
 * M., Hunt, T. and Karsenti, E. (1990) Nature (London) 346, 379-382].
 * This conjecture is corroborated by the model, which indicates
 * that sustained oscillations of the limit cycle type can arise
 * in the cascade, provided that a threshold exists in the activation
 * of cdc2 kinase by cyclin and in the activation of cyclin proteolysis
 * by cdc2 kinase. The analysis shows how miototic oscillations
 * may readily arise from time lags associated with these thresholds
 * and from the delayed negative feedback provided by cdc2-induced
 * cyclin degradation. A mechanism for the origin of the thresholds
 * is proposed in terms of the phenomenon of zero-order ultrasensitivity
 * previously described for biochemical systems regulated by covalent
 * modification.
 * 
 * The original paper reference is cited below:
 * 
 * A minimal cascade model for the mitotic oscillator involving
 * cyclin and cdc2 kinase, Albert Goldbeter, 1991, Proceedings
 * of the National Academy of Sciences, 88, 9107-9111. PubMed ID:
 * 1833774
 * 
 * cell diagram
 * 
 * [[Image file: goldbeter_1991_diagram.png]]
 * 
 * Minimal cascade model for the mitotic oscillations between cyclin
 * and cdc2 kinase (M) during the cell cycle. X represents the
 * fraction of active (phosphorylated) cyclin protease. * represents
 * the fraction of inactive enzymes.
 */

import nsrunit;
unit conversion on;
unit minute=60 second^1;
unit first_order_rate_constant=.01666667 second^(-1);
unit flux=1.6666667E-5 meter^(-3)*second^(-1)*mole^1;
// unit micromolar predefined

math main {
	realDomain time minute;
	time.min=0;
	extern time.max;
	extern time.delta;
	real C(time) micromolar;
	when(time=time.min) C=0.01;
	real X(time) dimensionless;
	when(time=time.min) X=0.01;
	real kd first_order_rate_constant;
	kd=0.01;
	real Kd micromolar;
	Kd=0.02;
	real vi flux;
	vi=0.025;
	real vd flux;
	vd=0.25;
	real M(time) dimensionless;
	when(time=time.min) M=0.01;
	real M_star(time) dimensionless;
	real V1(time) first_order_rate_constant;
	real V2 first_order_rate_constant;
	V2=1.5;
	real K1 dimensionless;
	K1=0.005;
	real K2 dimensionless;
	K2=0.005;
	real X_star(time) dimensionless;
	real V3(time) first_order_rate_constant;
	real V4 first_order_rate_constant;
	V4=0.5;
	real K3 dimensionless;
	K3=0.005;
	real K4 dimensionless;
	K4=0.005;
	real Kc micromolar;
	Kc=0.5;
	real VM1 first_order_rate_constant;
	VM1=3;
	real VM3 first_order_rate_constant;
	VM3=1;

	// <component name="environment">

	// <component name="C">
	C:time=(vi-vd*X*C/(Kd+C)-kd*C);

	// <component name="M">
	M:time=(V1*M_star/(K1+M_star)-V2*M/(K2+M));

	// <component name="M_star">
	M_star=(1-M);

	// <component name="X">
	X:time=(V3*X_star/(K3+X_star)-V4*X/(K4+X));

	// <component name="X_star">
	X_star=(1-X);

	// <component name="model_parameters">
	V1=(VM1*C/(Kc+C));
	V3=(M*VM3);
}