Notes on Progress3.MM
Figure 1: Progress Curves, A->B->C : Default
parameter set.
The three concentration curves are plotted as functions
of time. Note the similarity to the progress curves in
Progress3.React.proj plotted here as dashed lines. The
solutions from Progress3.React can be obtained by this model
by setting the values of VmaxA, VmaxB, KmA, and KmB all to
the same large value, e.g. 100.
To see the effect of changing the initial concentration
of A on the final metabolite C, go to the 'Loops' page.
Confirm parameter 'A__init' has a check in the 'OK' column
and then hit the 'Run' button. This should run the simulation
at initial values of 1, 10, and 50 mM for A. Observe the
Progress curves for A and its metabolites: B and C. It is
necessary to change Vmax and/or Km for the reactions to
raise the production of C within the given time period.
Figure 2: Locating KmA : Fig2 parameter set
The flux of A to B, -dA/dt is plotted as a
function of A, where A is plotted logarithmically.
The curve has a maximum at VmaxA, the maximum flux,
when A>>KmA. This occurs at the beginning of the
simulation.
When A=KmA, the flux is -VmaxA/2
dA/dt = -VmaxA*KmA/(KmA+KmA)= -VmaxA/2.
Use the vertical slider to find the value of KmA.
Figure 3: Reaction Rate: Fig2 parameter set
The reaction rate is plotted as a function
of the concentration of A and is slowest
when A<<KmA and has the maximum value VmaxA/KmA.
/*MODEL NUMBER: 0270
MODEL NAME: Progress3.MM
SHORT DESCRIPTION:Sequence of two catalyzed 1st order
Michaelis Menten-type irreversible reactions.
*/
import nsrunit; unit conversion on;
math Progress3 {
// INDEPENDENT VARIABLE
realDomain t s; t.min=0.0; t.max=10.0; t.delta=0.1;
// PARAMETERS
real KmA = 1 mM, // Apparent Km for A->B
KmB = 1 mM, // Apparent Km for B->C
VmaxA = 1 mM/s, // Maximum reaction flux for A->B
VmaxB = 1 mM/s; // Maximum reaction flux for B->C
// VARIABLES
real A(t) mM, // Concentration of A
B(t) mM, // Concentration of B
C(t) mM; // Concentration of C
// INITIAL CONDITIONS
when(t=t.min) {A =1; B=0; C = 0;}
// ORDINARY DIFFERENTIAL EQUATIONS
A:t = -VmaxA * A / (KmA + A);
B:t = VmaxA * A / (KmA + A) - VmaxB * B / (KmB + B);
C:t = VmaxB * B / (KmB + B);
} // END
/*
DIAGRAM:
VmaxA/(KmA+A) VmaxB/(KmB+B)
A---------------->B---------------->C
DETAILED DESCRIPTION:
Two irreversible reactions are governed by Michaelis-Menten
kinetics. Consider the following two cases for the first
reaction, dA/dt = -VmaxA*A/(KmA+A) :
Case 1: A<<KmA, The reaction rate is ~VmaxA/Kma and the flux is
VmaxA*(A/KmA)<<VmaxA.
Case 2: A>>KmA. The reaction rate is ~VmaxA/A and the flux is
VmaxA.
The reaction rate is slowest when A<<KmA. The flux is greatest when
A>>KmA.
SHORTCOMINGS/GENERAL COMMENTS:
1. Having irreversible reactions is most uncommon and it is generally
more sensible to write reversble reactions.
2. With the particular setting of this model, starting with
A(t=t.min) = 1 mM, which is the same as the KmA.
3. Compare the concentration time curves with those produced by the model
Progress3.React.proj.
4. See 'Notes' page for description of the plot pages and parameter sets.
5. RELATED MODELS: Progress3, Progress3.enz
KEY WORDS: Progress curves, irreversible reactions,PK_PD, pharmacokinetics,
first order kinetics, Michaelis-Menten, tutorial, Progress3.MM
REFERENCES:
REVISION HISTORY:
Original Author : JBB Date: 21/Dec/09
Revised by : BEJ Date: 22/Dec/09:Update format of comments
Revised by : BEJ: 11/aug/15: Update and add comments for PK-PD tutorial
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.
*/