Notes on Parameter Sets for One_Enzyme_Reversible
1. Michaelis_Menten
This parameter set shows the rapid equilibration which was assumed by Michaelis and Menten for their reaction kinetic approximation to be valid. This results in the reaction from ES complex to product P being the rate limiting step in the process. Under the conditions set, with KP >> KS the reaction is almost irreversible.
2. Briggs_Haldane
This parameter set allows time for the quantity of ES to fill up before reaching a steady state. This is a looser constraint than Michaelis and Menten applied. Here the reaction from ES complex to product P is the same order of magnitude as ES complex decomposing to enzyme E and substrate S and E and S forming the ES complex. The key here is that ES reaches a pseudo-equilibrium state after the initial fill time and before the quantity of S begins to approach that of the available E.
3. Reversible_MM
a. This parameter set shows what happens when the rate of E and P forming the complex ES is significant. The maximum amount of P produced is limited by reversibility.
b. Loops run: Show area of Progress curves where Michaelis-Menten estimate agrees with kinetic model:
- Confirm 'Reversible_MM' parameter set is loaded.
- Go to loops page and confirm paramters Etot, k1, and k2 are listed and checked with the following settings
Start Other Values
Etot 0.1 @*0.1
k1 100 @*10
k2 0.01 @*10
- Run two or three loops and look at the effects on the Progress_Curves plot page.
With Etot << S0, the progress curves, S and P, converge to Sm and Pm (Michaelis-Menten approximation) respectively for P less than S0/2. But we see that the Michaelis-Menten approximation for substrate and product concentration never reach the actual thermodynamic equilibrium (t >> 0) as defined by the ratio of Kp/Ks. The full kinetic calculation does go towards the thermodynamic equilibrium.
This example shows why the Michaelis-Menten approximation is fundamentally wrong. It can only describe reactions where the total enzyme concentration and the product concentration are much less then the substrate conentration.
Note: in all cases S >> E at t=0.
// MODEL NUMBER: 0096
// MODEL NAME: One_Enz_Reverse.MM
// SHORT DESCRIPTION:
// First-order reversible enzymatic reaction with binding of either substrate or product to
// enzyme and allows thermodynamic equilibrium. Uses two methods, the enzyme binding and
// also the Michaelis-Menten approximation.
JSim v1.1
import nsrunit; unit conversion on;
math One_Enz_Reverse.MM{
// *****************************************************************************
// I N P U T P A R A M E T E R S
// *****************************************************************************
realDomain t sec; t.min=0; t.max=5000.0; t.delta=1.00;
real k1 = 100 1/(uM*s); // Forward reaction binding S to E
real Ks = 0.1 uM; // Equil. dissoc. for binding S to E
real k2 = 0.01 1/s; // Forward reaction for ES to E + P
real Kp = 1e6 uM; // Equil. dissoc. for binding P to E
real Etot = 0.1 uM; // Total amount of enzyme
real k_1 1/s; // Backwards reaction ES to S + E
real k_2 1/(uM*sec); // Backwards reaction P + E to ES
real Km uM; // Michaelis Menten constant
real S0 = 1 uM, P0 = 0 uM, ES0 = 0 uM;
// *****************************************************************************
// M O D E L V A R I A B L E S
// *****************************************************************************
real S(t) uM; // Substrate
real P(t) uM; // Product
real E(t) uM; // Enzyme
real ES(t) uM; // Enzyme-substrate complex
// *****************************************************************************
// I N I T I A L C O N D I T I O N S
// *****************************************************************************
when (t=t.min){ S = S0; ES = ES0; P = P0; }
// *****************************************************************************
// P A R A M E T E R E Q U A T I O N S
// *****************************************************************************
k_1 = Ks * k1; // Backwards reaction ES to S + E; Ks = dissoc const
k_2 = k2 / Kp; // Backwards reaction P + E to ES; Kp = dissoc const
Km = (k2 + k_1) / k1; // Michaelix Menten constant
// *****************************************************************************
// S Y S T E M O F E Q U A T I O N S
// *****************************************************************************
E = Etot - ES;
S:t = k_1*ES - k1*S*E;
ES:t = k1*S*E - k_1*ES - k2*ES + k_2*E*P;
P:t = k2*ES - k_2*E*P;
// *****************************************************************************
// Reversible Reaction using Michaelis Menten-type equations
// M O D E L V A R I A B L E S for MM Equations
// The two methods use exactly the same kinetic constants but, unlike the full
// enzyme equations, the Michaelis-Menten expression assumes that enzyme is in
// negligible concentration compared to substrate and does not account for
// substrate binding. Note that Km .NE. Ks.
//
// *****************************************************************************
real Sm(t) uM, // Substrate
Pm(t) uM, // Product
Etotm = Etot, // Use identical value for Etotm
Vmaxf = Etotm*k2, // Vmaxforward from Sm -> Pm
Vmaxb = Etotm*k_1; // Vmaxbackward from Pm -> Sm
// *****************************************************************************
// I N I T I A L C O N D I T I O N S
// *****************************************************************************
when (t=t.min){ Sm = S0; Pm = P0; }
// *****************************************************************************
// S Y S T E M O F E Q U A T I O N S
// *****************************************************************************
Sm:t = -(Vmaxf*Sm/Ks - Vmaxb*Pm/Kp) / (1 + Sm/Ks + Pm/Kp);
Pm:t = -Sm:t;
} // End of PROGRAM O N E E N Z Y M E R E V E R S I B L E
/* FIGURE:
This model represents the single enzyme reversible reaction:
k1 --> k2 -->
S + E <-----------> ES <-----------> P + E
<-- k_1 <-- k_2
DETAILED DESCRIPTION:
This model represents the enzymatic conversion of a single substrate, S, to a single product, P.
First is binding of the solute to the enzyme, E, forming the substrate-enzyme complex, ES.
The binding is followed by a reaction-release event, which yields the product and releases free
enzyme. The model parameters can be adjusted to fit Michaelis-Menten conditions (Substrate
concentrations high compared to enzyme; on/off reaction fast compared to forward
reaction rate) and Briggs-Haldane conditions where the on-rate is low but not over a wide range
of conditions.
A second part of the program is given for the same enzyme concentrations and kinetics
constants but now expressed as "reversible Michaelis Menten Equations.
Comparisons are odious, 'tis said. This one reveals the inaccuracy of the MM approximation
for a reversible reaction when Kp >> Ks: Run the loops; these are set up to decrease Etot
by 10-fold per loop, and to increase both k1 and k_2 by 10-fold so that the apparent
forward and backward Vmax's are unchanged. The result is that successive reductions of Etot
results in a better and better approximation of the true kinetics to the MM-kinetics, as expected,
because the fundamental assumption of the MM kinetics is that S >> Etot. However, as the reaction
proceeds toward equilibrium (here Kp/Ks = 5000 = P(equil)/S(equil), the MM expression
cannot be correct. The MM expression contains contradictory assumptions, namely that k1 and k_1
are fast compared to k2 (for the S -> P reaction), and that k2 and k_2 are fast compared to k_1.
Therefore the MM fails when P > 0.3*S0 in this progress experiment.
SHORTCOMINGS/GENERAL COMMENTS:
The model assumes high rates of reversibility ES <--> EP
and does not define any mechanism for the interconversion.
KEY WORDS: Transport Physiology, Chemical Reaction Enzymes, Enzymatic Reaction,
Single Enzyme, Reversible, Michaelis-Menten Kinetics, Briggs-Haldane Kinetics
REFERENCES:
Bassingthwaighte JB.: Enzymes and Metabolic Reactions, Chapter 10 in "Transport and Reactions
in Biological Systems", Pages 7-8
REVISION HISTORY:
Original Author : JBB 18jan2010 deriving it from and contrasting it to model 130
Revised format : BEJ 19jan2010
Revised description : JBB 17apr2018
RELATED MODELS: PGIsomerase, One_Enzyme_Reversible
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2018 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,
the Cardiac Energy Grid HL122199 (PI: J.B. Bassingthwaighte), and the Virtual Physiological Rat program
GM094503 (PI: D.A.Beard). Please cite these grants 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.
*/