Model number

This model represents a two enzyme reversible reaction where the two enzymes are sequestered inside a vesicle and substrate and product are allowed to diffuse across the vesicle boundary.


This model represents the enzymatic conversion of a single substrate, S, into two products, P1 and P2, inside a vesicle such as an endoplasmic reticulum. First the solute must permeate across the vesicle boundary where it binds to either one of two enzymes forming the substrate-enzyme complexes, ES1 or ES2. This is followed by a reaction-release step which yields product, P1 or P2, and the enzymes, E1 or E2 respectively. The diffusion of the products out of the vesicle into the surrounding volume follows. The constant flow rates of substrate and product across the boundary are allowed to be different depending on direction of flow. The reaction inside the vesicle for E1 binding to S and forming P1 is given as follows:


where P1 and S are defined previously, E1 represents the first enzyme, ES1 the corresponding substrate-enzyme complex, k11 is the forward binding rate of S to E1, k-11 is the backwards reaction rate of ES1 dissociating to E1 and S, k21 is the forward reaction rate of ES1 forming E1 and P1 and k-21 the reverse reaction rate of E1 and P1 producing ES1. We have the same reaction for the conversion of S to P2 inside the vesicle with the appropriate change of parameter indices:



The permeability across the vesicle boundary are represented by PVsin, PVsout PVp1in, PVp1out, PVp2in and PVp2out for the substrate and products respectively. The diagram shown below gives the entire reaction including diffusion across the vesicle boundaries.

   | VolOut                                                               |
   |             o------------------------------------------o             |
   |             | VolIn                                    |             |
   |             |                                          | PVp1out     |         |
   |             |           k11->       k21->            --|-------->    |
   |             |   * + E1 <------> ES1 <------> E1 + P1   |  P1out      |
   |      PVsin  |   |       <-k_11      <-k_21           <-|---------    |
   |     --------|-> |                                      | PVp1in      |
   |       Sout  |   S                                      |             |
   |     <-------|-- |                                      | PVp2in      |
   |      PVsout |   |        k12->       k22->           <-|---------    |
   |             |   * + E2 <------> ES2 <------> E2 + P2   |  P2out      |
   |             |          <-k_12       <-k_22           --|-------->    |
   |             |                                          | PVp2out     |
   |             |                                          |             |
   |             o------------------------------------------o             |
   |                                                                      |



This reaction is governed by a system of eight ODEs which describe the concentrations of the substrate and products both inside and outside the vesicle and the concentration of the two complexes, ES1 and ES2, inside the vesicle. The last two equations to close the system are given by specifying the total amount of each enzyme present which must be conserved. The initial conditions given are that all of the substrate is outside the vesicle and no complex or product are present at time t=0. The system of equations are:


The backward reaction rates in this model are determined from the equilibrium dissociation rates of S binding to E and P binding to E and are given by:


where Ks1 is the equilibrium dissociation rate of S binding to E1, Kp1 of P1 binding to E1, Ks2 is the equilibrium dissociation rate of S binding to E2 and Kp2 of P2 binding to E2.

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    Bassingthwaighte JB.: Enzymes and Metabolic Reactions, Chapter 10 in 
    "Transport and Reactions in Biological Systems"

Key terms
Two Enzyme
enzymatic reaction

Please cite in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.