// MODEL NUMBER: 0322
// MODEL NAME: Vfnet_MM2substrate_reversible
// SHORT DESCRIPTION: Single enzyme reversible Michaelis-Menten Eqs for Hx->Xa->Ua, that is, two reactions on one enzyme.
// Data are progress curves for xanthine oxidase reactions to oxidize hypoxanthine, Hx, to xanthine, Xa,
// to uric acid, Ua, without inhibition by product.(Data from Escribano 1988).
import nsrunit; unit conversion on;
math Vfnet_MM2substrate_reversible {
realDomain t sec; t.min=0; t.max=5000.0; t.delta=1.00;
//PARAMETERS
real H2zero = 47 uM; // init condn for hypoxanthine
real X2zero = 0 uM; // init condn for xanthine
real U2zero = 0 uM; // init condn for uric acid
real VfHmax = 1.79 uM/s; // M-M Vmaxforward Hx -> Xa
real VbHmax = 0.0605 uM/s; // M-M Vmaxreverse Xa -> Hx
real Kmh = 2.47 uM; // Apparent Km for XO reaction with Hx -> Xa
real Kph = 19.1e4 uM; // Apparent Km for XO reaction with Xa -> Hx
real VfXmax = 1.95 uM/s; // M-M Vmaxforward Xa -> UA
real VbXmax = 0.865 uM/s; // M-M Vmaxreverse UA -> Xa
real Kmx = 4.06 uM; // Apparent Km for XO reaction with Xa -> UA
real Kpx = 244 uM; // Apparent Km for XO reaction with UA -> Xa
//VARIABLES
real H2(t) uM; // Hypoxanthine
real X2(t) uM; // Xanthine
real U2(t) uM; // Uric Acid
real U2cons(t) uM; // Uric acid calculated by conservation, as a check
//INITIAL CONDITIONS
when (t=t.min){ H2 = H2zero; X2 = X2zero; U2 = U2zero;}
//SYSTEM OF EQUATIONS
H2:t = -(VfHmax*H2/Kmh - VbHmax*X2/Kph) / (1 + H2/Kmh +X2/Kmx + U2/Kpx);
X2:t = (VfHmax*H2/Kmh - VbHmax*X2/Kph + VbXmax*U2/Kpx - VfXmax*X2/Kmx) /
(1 + H2/Kmh +X2/Kmx + U2/Kpx);
U2:t = (VfXmax*X2/Kmx - VbXmax*U2/Kpx ) / (1 + H2/Kmh +X2/Kmx + U2/Kpx);
U2cons = H2zero + X2zero + U2zero - H2 - X2;
// SECOND SET OF EQUATIONS TO SET UP FOR SIMULTANEOUS OPTIMIZATION for Fig 5 Data
//VARIABLES
real H5zero = 0 uM; // init condn for hypoxanthine
real X5zero = 93 uM; // init condn for xanthine
real U5zero = 0 uM; // init condn for uric acid
real H5(t) uM; // Hypoxanthine
real X5(t) uM; // Xanthine
real U5(t) uM; // Uric Acid
real U5cons(t) uM; // Uric acid calculated by conservation, as a check
//INITIAL CONDITIONS
when (t=t.min){ H5 = H5zero; X5 = X5zero; U5 = U5zero;}
//SYSTEM OF EQUATIONS
H5:t = -(VfHmax*H5/Kmh - VbHmax*X5/Kph) / (1 + H5/Kmh +X5/Kmx + U5/Kpx);
X5:t = (VfHmax*H5/Kmh - VbHmax*X5/Kph + VbXmax*U5/Kpx - VfXmax*X5/Kmx) /
(1 + H5/Kmh +X5/Kmx + U5/Kpx);
U5:t = (VfXmax*X5/Kmx - VbXmax*U5/Kpx ) / (1 + H5/Kmh +X5/Kmx + U5/Kpx);
U5cons = H5zero + X5zero + U5zero - H5 - X5;
}
//**************************************************************************************************************
/* FIGURE:
This model represents a series of single enzyme irreversible reactions for
the sequence: Hx -> Xa <-> UA for Hx, Xa, and Ua in vitor or in endothelial cells
Hx + E -----> EHx <--> EXa ---> E + Xa ------>EXa' <--> Eua ----> E + Ua
DETAILED DESCRIPTION: Bidirectional fluxes Hx <--> Xa <--> Ua facilitated
by a single enzyme, Xanthine Oxidase (EC# 1.7.3.2), in a hyperoxic medium at pH 8,
so it is oxidative. The equations are Michaelis-Menten, forward and backward, so
the concentration changes are driven by the NET flux through each reaction.
The optimization strategy is to have two models
operating simultaneously, the first one to fit the data of Fig 4 (Hx->xa->Ua) of
Escribano88, and the second to fit the data of Fig 5(Xa->Ua). Both models
use the identical parameters, The optimizer minimizes the RMS error for five (5)
data curves at once, thereby providing an overall best estimate of the parameters.
This strategy maximizes the ratio of data to parameters and narrows the confidence
limits on the parameters.
The inhibitory action of Ua was found by Escribano et al (1988) in a set of
inital velocity experiments, showing an apparent Ki, they report, of 178 uM,
but no data were provided. As an exercise, set up this model to show a set of
initial consumptions of Xa at varied background levels of Ua. Alternatively,
add a new variable for tracer Ua to be produced from tracer Xa and show initial
rates of production of tracer Ua.
SHORTCOMINGS/GENERAL COMMENTS:
The rates of binding of substrate to enzyme are assumed instantaneous (M-M 1913), and the
substrate concentration are assumed high compared to that of the enzyme. In the
Escibano experiments the enzyme concentration is 2.7 uM, over 5% of Hx(t=0).
The model assumes high rates of reversibility ES <--> EP, considering them as
the same complex and does not define any mechanism for their interconversion.
There is a small approximation made with respect to the binding for the
reverse reactions: it can be argued that, even if there is only one binding site,
that the term X/Kmx is a little too large since, when Kmx for the forward reaction
of X is smaller than Kph for the backward reaction, there is not as much E occupied
as when there is no E bound with affinity KpH. This idea is based on the premise
that there is a change in conformational state between X binding E and then forming
H by a backward reaction that is different from the state where X is binding E
and then reacting in the forward direct to form U.
The alternative view is that Kmx is the only affinity for the site. Then the
rate of the backward reaction X->H is governed solely by VbHmax. By putting Kph
equal to Kmh, the number of free parametera is reduced by one, and the estimate of
VbHmax must be similarly reduced so as to keep the ratio VbHmax/Kph more or less
the same in order to get a good fit. Try it.
RELATED MODELS:
One enzyme - two substrate models:
M-M2Substrate_irreversible (model #0320)
MM2Substrate_product_inhibited (Model #0321)
Vfnet_MM2substrate_reversible (Model #0322)
FullXO (Model #0323)
FourXO, 4 xanthine oxidase models together (Model #0324)
Comp1EnzReact4 (Model #275)
One enzyme - one substrste models:
Aspirin Clearance (model #280)
One_Enzymatic_Reversible (Model #130)
PG_Isomerase (Model #271)
Progress3.MM (Model #270)
KEY WORDS: purine nucleosides, xanthine oxidase, simple Michaelis-Menten equations,
double reaction, serial reactions, progress curves, unidirectional, bovine milk, XO,
Hx, Xa, Ua, hypoxanthine, xanthine, uric acid, saturable binding, data, publication
REFERENCES:
Bassingthwaighte James B., Chinn Tamara Meiko,
Re-examining Michaelis-Menten enzyme kinetics for xanthine oxidase,
Adv Physiol Educ 37: 37-48, 2013
Escribano, J., Garcia-Canovas, F., and Garcia-Carmona,F.
A kinetic study of hypoxanthine oxidation by milk xanthine oxidase.
Biochem. J. 254: 829-833, 1988.
Hofmeyr J-HS and Cornish-Bowden A. The reversible Hill equation:
How to incorporate cooperative enzymes into metabolic models.
Comput Appl Biosci 13: 377-385, 1997.
Houston M, Estevez M, Chumley P, Aslan M, Marklund S, Parks D, and Freeman BA.
Binding of Xanthine Oxidase to vascular endothelium. Kinetic characterization
and oxidative impairment of nitric oxide-dependent signaling.
J Biol Chem 274: 4985-4994, 1999.
Michaelis L and Menten ML. Die Kinetik der Invertinwirkung.
Biochem Z 49: 333-369, 1913.
REVISION HISTORY:
Written 18 Aug 2011 by JBB (Revised from Comp1EnzReact4 (model#275)
and Progress3.MM (Model # 270)
19dec2012 JBB revised General Comments.
14aug2013 BEJ revised: add keywords
20jan2015 BEJ revised copyright and main reference, update t.max for simulation.
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2015 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Academic use is unrestricted. Software may be copied so long as this copyright notice is included.
When citing JSim please use this reference: Butterworth E, Jardine BE, Raymond GM, Neal ML, Bassingthwaighte JB.
JSim, an open-source modeling system for data analysis [v3; ref status: indexed, http://f1000r.es/3n0]
F1000Research 2014, 2:288 (doi: 10.12688/f1000research.2-288.v3)
This software was developed with support from NIH grants HL088516 and HL073598, NIBIB grant BE08417
and the Virtual Physiological Rat program GM094503 (PI: D.A.Beard). Please cite this grant in any
publication for which this software is used and send an email with the citation and, if possible,
a PDF file of the paper to: staff@physiome.org.
*/