JSim v1.1
import nsrunit; unit conversion on;
// MODEL NUMBER: 0030
// MODEL NAME: HbCoop:
// SHORT DESCRIPTION: Oxygen binding to hemoglobin at 4 cooperative sites.
math Hbcoop { //alp > 1 for pos cooperativity, alp < 1 for neg coop.
realDomain t sec; t.min=0 ; t.max=200 ; t.delta= 0.1;// time
// static variables and parameters
real //change binding rate to produce close to SS relationships
alp = 4.17, //alpha: cooperativity factor
KpO = 26.8 mmHg, //KpO is p50 of Hb in mmHg
Keq = alp^1.5*KpO,//Keq is Equil dissoc const for sites, mmHg
Kp1 = Keq/4, //Kp1 is p50 of first site mmHg
Kp2 = 2*alp^(-1)*Keq/3, //Kp2 is p50 of second site mmHg
Kp3 = 3*alp^(-2)*Keq/2, //Kp3 is p50 of third site mmHg
Kp4 = 4*alp^(-3)*Keq, //Kp4 is p50 of fourth site mmHg
k_1 = 10 s^-1, //off rate for first site
k_2 = 10 s^-1, //off rate for second site
k_3 = 10 s^-1, //off rate for third site
k_4 = 10 s^-1, //off rate for fourth site
k1 = k_1/Kp1, //s^(-1)/mmHg,
k2 = k_2/Kp2, //s^(-1)/mmHg,
k3 = k_3/Kp3, //s^(-1)/mmHg,
k4 = k_4/Kp4, //s^(-1)/mmHg,
HbTot = 1 mmHg,
Kpref = 1 mmHg, // dummy to normalize pO2 and KpO in Hill Eq
// nH = 2.7, // Hill exponent (No non-integer variable power in JSim)
//binding 1 O2 reduces free energy of binding by A kCal and rate k2 to alp*k1;
//binding 2 O2 reduces free energy of binding by 2A kCal & rate k3 to alp^2*k1;
// Shalf = Keq/alp^1.5,
rate = 1 mmHg/s; // rate of increase of pO2 starting from zero
//alpho = 1.4e-6 M/mmHg, soly of O2 in water, not used here
real O2(t) mmHg, Hb(t) mmHg, HbO(t) mmHg, SHb(t), SHill(t);
real HbO2(t) mmHg, HbO3(t) mmHg, HbO4(t) mmHg; //2,3,4, refer to site #
real G(t), SatSS(t); //SatSS = Saturation in Steady State
// initial conditions
when(t=t.min) { O2 = 0; Hb= HbTot; HbO = 0; HbO2 = 0; HbO3 = 0; }
// ODEs
O2:t = rate;
Hb:t =-k1*O2*Hb + k_1*HbO;
HbO:t = k1*O2*Hb - k_1*HbO - k2*O2*HbO + k_2*HbO2;
HbO2:t = k2*O2*HbO - k_2*HbO2 - k3*O2*HbO2 + k_3*HbO3;
HbO3:t = k3*O2*HbO2- k_3*HbO3 - k4*O2*HbO3 + k_4*HbO4;
HbO4 = HbTot -Hb - HbO - HbO2 -HbO3;
SHb = (HbO + 2*HbO2 + 3*HbO3 + 4*HbO4)/(HbTot*4);
SHill = (O2/Kpref)^2.7/((KpO/Kpref)^2.7 + (O2/Kpref)^2.7); //Hill Eqn
G = O2/Keq;
SatSS = (G +3*alp*G^2 + 3*alp^3*G^3 + alp^6*G^4)/
(1+ 4*G + 6*alp*G^2 + 4*alp^3*G^3+ alp^6*G^4);
}
/*
DETAILED DESCRIPTION:
Hemoglobin, a protein with 4 interdependent binding sites, can become
saturated with oxygen, i.e all of its binding sites can be occupied at high
concentrations. The fractional saturation is calculated here by a cooperative
scheme by which there is a constant ratio of increases in affinity as each site
is filled in succession. The cooperativity factor "alp" is >1 for positive
cooperativity, and < 1 for anticooperativity.
The results are compared with the result using a Hill equation with a
Hill coefficient of 2.7. The value is chosen because the Hill equation
with nH (Hill coefficient with nH = 2.7 fits oxyhemoglobin saturation curves well).
The math is straightforward, based on the equation for single site binding, modified
in recognition that there are, at varying concentrations, 4 sites available to fill. When
one is filled, only 3 remain, reducing the odds from 4 to 3, and so on. The actual O2
carriage depends on the relative abundances of HbO, HbO2, etc, and the fact that there
is twice as much O2 on HbO2 as on HbO, etc. The sum of the products of the relative
concentrations times the O2s being carried in each form is divided by 4*HbO4, the
maximum that can be carried.
This model serves as a basis of other cooperativity models wherein the filling of
the first and successive sites causes (by cooperativity = positive feedback through
molecular conformational rearrangement) successively higher affinities. The ratio, "alp"
is not necessarily constant. For example the Adair eqautions are equivalent to
having "alp as a variable.
SHORTCOMINGS/GENERAL COMMENTS:
Compare saturation curve with the more general formulation of Dash and Bassingthwaighte
which accounts for CO2, pH, DPG, and Temperature:
Dash RK and Bassingthwaighte JB. Blood HbO2 and HbCO2 dissociation curves at varied O2,
CO2, pH, 2,3-DPG and temperature levels. Ann Biomed Eng 32(12): 1676-1693, 2004.
This model is used in blood-tissue exchange models (convection-diffusion-reaction models
as described by: Dash RK and Bassingthwaighte JB.
Simultaneous blood-tissue exchange of oxygen, carbon dioxide, bicarbonate and hydrogen ion.
Ann Biomed Eng 34(7): 1129-1148. 2006.
KEY WORDS: hemoglobin, oxygen, carbon dioxide, saturation, Haldane, Bohr, acidity,
pH, blood gases, Hill equation, solubility, cooperativity, Data
REFERENCES:
Keener J and Sneyd J. Mathematical Physiology. New York, NY:
Springer-Verlag, 1998, 766 pp.
Dash RK and Bassingthwaighte JB. Erratum to: Blood HbO2 and HbCO2 dissociation curves at
varied O2, CO2, pH, 2,3-DPG and Temperature Levels. Ann Biomed Eng 38(4): 1683-1701, 2010.
Hill AV. The diffusion of oxygen and lactic acid through tissues.
Proc R Soc Lond (Biol) 104: 39-96, 1928.
Adair GS. The hemoglobin system. VI. The oxygen dissociation curve of
hemoglobin. J Biol Chem 63: 529-545, 1925.
Hill AV. The possible effects of the aggregation of the molecules of haemoglobin on its
dissociation curves. J Physiol 40: iv-vii, 1910
Hill R. Oxygen dissociation curves of muscle hemoglobin. Proc Roy Soc Lond B
120: 472-480, 1936.
Roughton FJW, Deland EC, Kernohan JC, and Severinghaus JW. Some recent studies of the
oxyhemoglobin dissociation curve of human blood under physiological conditions and the
fitting of the Adair equation to the standard curve. In: Oxygen Affinity of Hemoglobin and
Red Cell Acid Base Status. Proceedings of the Alfred Benzon Symposium IV Held at the
Premises of the Royal Danish Academy of Sciences and Letters, Copenhagen 17-22 May,
1971, edited by Rorth M and Astrup P. Copenhagen: Munksgaard, 1972, p. 73-81.
Winslow RM, Swenberg M-L, Berger RL, Shrager RI, Luzzana M, Samaja M,and
Rossi-Bernardi L. Oxygen equilibrium curve of normal human blood and its evaluation by
Adair's equation. J Biol Chem 252: 2331-2337, 1977.
REVISION HISTORY:
Original Author : JBB Date: 01/12/09
Revised by : BEJ Date: 09/JUN/2010
Revision: 1) Update comments and graph labels.
Revised by BEJ: Date: 26/Feb/10: add keyword Data
Revised by BEJ: Date: 27/may/15: updated reference/copyright
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.
*/