/*
 * Protein phosphorylation driven by intracellular calcium oscillations:
 * A kinetic analysis
 * 
 * Model Status
 * 
 * This model has been built with the differential expressions
 * in Dupont and Goldbeter's 1992 paper. This file is known to
 * run in PCEnv and COR, and variables for constants (vMK) can
 * be altered to produce figure4b, c and figure5b and c (the ratio
 * vMK/vP is held constant at vMK/vP = 20/0.89). The current parameterization
 * is set to reproduce 4b with vMK=100 (note the erratum received
 * in 1995: wT=10 not 1, to replicate figure 5. Initial conditions
 * for Z, Y and Wstar were set by letting the model settle into
 * a steady state.
 * 
 * Model Structure
 * 
 * Abstract: Given the ubiquitous nature of signal-induced Ca2+
 * oscillations, the question arises as to how cellular responses
 * are affected by repetitive Ca2+ spikes. Among these responses,
 * we focus on those involving protein phosphorylation. We examine,
 * by numerical simulations of a theoretical model, the situation
 * where a protein is phosphorylated by a Ca2+-activated kinase
 * and dephosphorylated by a phosphatase. This reversible phosphorylation
 * system is coupled to a mechanism generating cytosolic Ca2+ oscillations;
 * for definiteness, this oscillatory mechanism is based on the
 * process of Ca2+-induced Ca2+ release. The analysis shows that
 * the average fraction of phosphorylated protein increases with
 * the frequency of repetitive Ca2+ spikes; the latter frequency
 * generally rises with the extent of external stimulation. Protein
 * phosphorylation therefore provides a mechanism for the encoding
 * of the external stimulation in terms of the frequency of signal-induced
 * Ca2+ oscillations. Such a frequency encoding requires precise
 * kinetic conditions on the Michaelis-Menten constants of the
 * kinase and phosphatase, their maximal rates, and the degree
 * of cooperativity in kinase activation by Ca2+. In particular,
 * the most efficient encoding of Ca2+ oscillations based on protein
 * phosphorylation occurs in conditions of zero-order ultrasensitivity,
 * when the kinase and phosphatase are saturated by their protein
 * substrate. The kinetic analysis uncovers a wide variety of temporal
 * patterns of phosphorylation that could be driven by signal-induced
 * Ca2+ oscillations.
 * 
 * model diagram
 * 
 * [[Image file: dupont_1992.png]]
 * 
 * Schematic diagram of the cell model.
 * 
 * The original paper reference is cited below:
 * 
 * Protein phosphorylation driven by intracellular calcium oscillations:
 * A kinetic analysis, Dupont G, Goldbeter A 1992, Biophysical
 * Chemistry 41, 257-270. PubMedID: 1316185
 */

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

math main {
	realDomain time minute;
	time.min=0;
	extern time.max;
	extern time.delta;
	real VM2 micromolar_min;
	VM2=65;
	real VM3 micromolar_min;
	VM3=500;
	real KR micromolar;
	KR=2;
	real KA micromolar;
	KA=0.9;
	real KP micromolar;
	KP=1;
//	Var below replaced by constant in model eqns to satisfy unit correction
//	real n dimensionless;
//	n=2;
//	Var below replaced by constant in model eqns to satisfy unit correction
//	real m dimensionless;
//	m=2;
//	Var below replaced by constant in model eqns to satisfy unit correction
//	real p dimensionless;
//	p=4;
	real kf per_minute;
	kf=1;
	real k per_minute;
	k=10;
	real Y(time) micromolar;
	when(time=time.min) Y=1.454;
	real Z(time) micromolar;
	when(time=time.min) Z=0.2281;
	real v2(time) micromolar_min;
	real v3(time) micromolar_min;
	real v0 micromolar_min;
	v0=1;
	real v1beta micromolar_min;
	v1beta=2.4;
	real K1 dimensionless;
	K1=0.01;
	real K2 dimensionless;
	K2=0.01;
	real WT micromolar;
	WT=10;
	real vP micromolar_min;
	real vK(time) micromolar_min;
	real vMK micromolar_min;
	vMK=100;
	real Wstar(time) dimensionless;
	when(time=time.min) Wstar=0.8916;
	real Ka micromolar;
	Ka=1;
//	Var below replaced by constant in model eqns to satisfy unit correction
//	real q dimensionless;
//	q=4;

	// <component name="environment">

	// <component name="parameters">
	v2=(VM2*Z^2/(KP^2+Z^2));
	v3=(VM3*(Y^2/(KR^2+Y^2))*(Z^4/(KA^4+Z^4)));

	// <component name="cytosol">
	Z:time=(v0+v1beta-v2+v3+kf*Y-k*Z);

	// <component name="insensitive_pool">
	Y:time=(v2-v3-kf*Y);

	// <component name="phosphorylation">
	vP=(vMK*(.89/20));
	Wstar:time=(vP/WT*(vK/vP*(1-Wstar)/(K1+1-Wstar)-Wstar/(K2+Wstar)));

	// <component name="kinase_reaction">
	vK=(vMK*(Z^4/(Ka^4+Z^4)));
}