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.

Description

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.

   o----------------------------------------------------------------------o
   | 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             |
   |                                                                      |
   o----------------------------------------------------------------------o

Equations

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 K_s1 is the equilibrium dissociation rate of S binding to E₁, K_p1 of P₁ binding to E₁, K_s2 is the equilibrium dissociation rate of S binding to E₂ and K_p2 of P₂ binding to E₂.