This model simulates a single enzyme reaction which occurs inside a vesicle where the substrate and product diffuse across the vesicle boundary on each end of the reaction.

Description

This model represents the enzymatic conversion of a single substrate, S, to a single product, P, inside a vesicle such as an endoplasmic reticulum, across a cell membrane or inside an organelle such as a lysosome. First the solute must permeate across the permeable boundary and then it binds to the enzyme forming a substrate-enzyme complex. This is followed by a reaction-release step which yields the product and the enzyme and then diffusion of the product out of the vesicle into the surrounding volume. The reaction inside the vesicle is given as follows:

where P and S are defined previously, E represents the enzyme, ES the substrate-enzyme complex, k₁ is the forward binding rate of S to E, k_-1 is the backwards reaction rate of ES dissociating to E and S, k₂ is the forward reaction rate of ES forming E and P and k_-2 the reverse reaction rate of E and P producing ES. The permeability across the vesicle boundary are represented by PVs and PVp for the substrate and product respectively. The diagram shown below gives the entire reaction including diffusion across the vesicle boundaries.

   o-----------------------------------------------------------------------o
   | VolOut                                                                |
   |             o-------------------------------------------o             |
   |             | VolIn                                     |             |
   |             |                                           |             |
   |         PVs |               k1-->       k2-->           | PVp         |
   |  Sout <-----|-----> S + E <------> ES <-------> P <-----|-----> Pout  |
   |             | PVs          <--k_1      <--k_2       PVp |             |
   |             |                                           |             |
   |             |                                           |             |
   |             o-------------------------------------------o             |
   |                                                                       |
   o-----------------------------------------------------------------------o

Equations

This reaction is governed by a system of five ODEs which describe the concentrations of the substrate and product both inside and outside the vesicle and the concentration of the complex, ES, inside the vesicle. The sixth equation to close the system is given by specifying the total amount of 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 K_s is the equilibrium dissociation rate of S binding to E and K_p of p binding to E.