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Flux Balances

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Matrix Structure

We consider two diffusion situations.

1. The substrate is an exchange flux with the first bin only, but is allowed to freely diffuse to adjacent bins, imposing a directionality on the system.

We denote the "source" substrate to be a point 0. For n points in the domain, we have substrate, enzyme, complex, and product at each of these points.

The reaction-diffusion matrix is organized as follows:

{\textbf  {v}}=(k_{{-1}}C_{1},...,k_{{-1}}C_{n},k_{1}S_{1}E_{1},...,k_{1}S_{n}E_{n},k_{2}P_{1},...,k_{2}P_{n},D_{{S_{0}\leftrightarrow S_{1}}},...,D_{{S_{{n-1}}\leftrightarrow S_{n}}},
D_{{E_{1}\leftrightarrow E_{2}}},...,D_{{E_{{n-1}}\leftrightarrow E_{n}}},D_{{C_{1}\leftrightarrow C_{2}}},...,D_{{C_{{n-1}}\leftrightarrow C_{n}}},D_{{P_{1}\leftrightarrow P_{{n+1}}}},...,D_{{P_{n}\leftrightarrow P_{{n+1}}}},)
{\textbf  {x}}=(S_{0},...,S_{n},E_{0},...,E_{n},C_{0},..,C_{n},P_{0},...,P_{n})

Then for the two-dimensional case, the matrix appears as follows:

S_{{2D}}=\left[{\begin{array}{cccccccccccc}1&0&-1&0&0&0&1&-1&0&0&0&0\\0&1&0&-1&0&0&0&1&0&0&0&0\\1&0&-1&0&1&0&0&0&-1&0&0&0\\0&1&0&-1&0&1&0&0&1&0&0&0\\-1&0&1&0&-1&0&0&0&0&-1&0&0\\0&-1&0&1&0&-1&0&0&0&1&0&0\\0&0&0&0&1&0&0&0&0&0&-1&0\\0&0&0&0&0&1&0&0&0&0&0&-1\\\end{array}}\right]

with reversibilities as follows, for their respective fluxes:

rev_{{2D}}=\left[{\begin{array}{cccccccccccc}0&0&0&0&0&0&1&1&1&1&1&1\\\end{array}}\right]

and for the three-dimensional case:

S_{{3D}}=\left[{\begin{array}{ccccccccccccccccccc}1&0&0&-1&0&0&0&0&0&1&-1&0&0&0&0&0&0&0&0\\0&1&0&0&-1&0&0&0&0&0&1&-1&0&0&0&0&0&0&0\\0&0&1&0&0&-1&0&0&0&0&0&1&0&0&0&0&0&0&0\\1&0&0&-1&0&0&1&0&0&0&0&0&-1&0&0&0&0&0&0\\0&1&0&0&-1&0&0&1&0&0&0&0&1&-1&0&0&0&0&0\\0&0&1&0&0&-1&0&0&1&0&0&0&0&1&0&0&0&0&0\\-1&0&0&1&0&0&-1&0&0&0&0&0&0&0&-1&0&0&0&0\\0&-1&0&0&1&0&0&-1&0&0&0&0&0&0&1&-1&0&0&0\\0&0&-1&0&0&1&0&0&-1&0&0&0&0&0&0&1&0&0&0\\0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&-1&0&0\\0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&-1&0\\0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&-1\\\end{array}}\right]

with reversibilities as follows:

rev_{{3D}}=\left[{\begin{array}{ccccccccccccccccccc}0&0&0&0&0&0&0&0&0&1&1&1&1&1&1&1&1&1&1\\\end{array}}\right]

Results

Consider the two-bin case:

The elementary flux modes that give rise to a steady state among the internal fluxes for situation 1:

  1. k_{{-1}}C_{1}=k_{1}S_{1}E_{1}
  2. k_{{-1}}C_{2}=k_{2}S_{2}E_{2}
  3. k_{{-1}}C_{1}=k_{1}S_{2}E_{2}=D_{{S_{1}\rightarrow S_{2}}}=D_{{E_{1}\rightarrow E_{2}}}=D_{{C_{1}\leftarrow C_{2}}}
  4. k_{{-1}}C_{2}=k_{1}E_{1}S_{1}=D_{{S_{1}\leftarrow S_{2}}}=D_{{E_{1}\leftarrow E_{2}}}=D_{{C_{1}\rightarrow C_{2}}}
  5. k_{1}E_{1}S_{1}=k_{2}P_{1}=D_{{S_{0}\rightarrow S_{1}}}=D_{{P_{1}\rightarrow P_{{bin}}}}
  6. k_{1}S_{2}E_{2}=k_{2}P_{2}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{1}\rightarrow S_{2}}}=D_{{P_{2}\rightarrow P_{{bin}}}}
  7. k_{1}S_{1}E_{1}=k_{2}P_{2}=D_{{S_{0}\rightarrow S_{1}}}=D_{{E_{1}\leftarrow E_{2}}}=D_{{C_{1}\rightarrow C_{2}}}=D_{{P_{2}\rightarrow P_{{bin}}}}
  8. k_{1}S_{2}E_{2}=k_{2}P_{1}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{1}\rightarrow S_{2}}}=D_{{E_{1}\rightarrow E_{2}}}=D_{{C_{1}\leftarrow C_{2}}}=D_{{P_{1}\rightarrow P_{{bin}}}}


We are of course interested only in fluxes in which an inflow of substrate from S_{0} leads to a buildup of product. Flux balances 1, 2, 3, and 4 correspond to internal cycles, and therefore are of no interest (the first two are "stationary" cycles, the second two involve diffusion). Flux 5 corresponds simply to the one-bin situation, and is of no spatial interest. Flux 6 involves substrate flowing to point 2, such that all of the chemical action occurs in that second bin. Fluxes 7 and 8 are of greatest interest, involving the occurrence of the two forward reactions in different bins with a steady-state maintained by diffusion.


Now, we consider the three-bin case: There are too many flux modes to list independently, so we present the flux mode matrix as computed by efmtool:

modes_{{3D}}=\left[{\begin{array}{ccccccccccccccccccc}1&1&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0\\0&0&0&0&1&0&1&0&0&0&0&1&0&1&0&0&0&0&0\\0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&1&1\\0&0&0&0&1&0&0&0&0&1&0&0&0&1&1&1&1&1&0\\0&1&0&0&0&1&0&1&1&0&0&1&1&0&0&0&0&0&0\\1&0&1&1&1&0&1&0&0&0&1&0&0&0&0&0&0&0&1\\0&0&0&1&0&0&0&0&0&0&0&0&1&0&0&0&1&0&0\\0&0&1&0&0&1&0&0&0&0&0&0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0&1&0&1&0&0&0&0&1&0&0&0\\0&0&1&1&1&1&0&0&1&0&1&0&1&0&1&1&1&0&0\\1&1&1&1&0&1&0&0&1&0&1&0&1&-1&0&0&0&-1&0\\1&0&1&1&1&0&1&-1&0&0&1&0&0&0&0&0&0&-1&0\\1&1&0&1&-1&0&0&0&0&0&0&0&1&-1&-1&-1&0&-1&0\\1&0&1&1&0&0&1&-1&-1&0&0&0&0&0&0&-1&0&-1&0\\-1&-1&0&-1&1&0&0&0&0&0&0&0&-1&1&1&1&0&1&0\\-1&0&-1&-1&0&0&-1&1&1&0&0&0&0&0&0&1&0&1&0\\0&0&0&1&0&0&0&0&0&0&0&0&1&0&0&0&1&0&0\\0&0&1&0&0&1&0&0&0&0&0&0&0&0&1&0&0&0&0\\0&0&0&0&1&0&0&0&1&0&1&0&0&0&0&1&0&0&0\\\end{array}}\right]

Recalling the format of {\textbf  {v}} given above, we have the following flux balances: Columns 10, 12, and 19 have the stationary internal flux format:

k_{{-1}}C_{n}=k_{1}S_{n}E_{n}

Observe that there are necessarily n of these fluxes.
Meanwhile, Columns 1, 2, 7, 8, 14, and 18 all correspond to internal cycles, but through multiple bins via diffusion. Observe that there is no uptake of substrate and no production of product in these situations. There are necessarily n\times n-1 of these fluxes: the first reaction flux may occur at any of these locations, whereas the second reaction flux may occur at any of those locations except the one chosen for the first reaction flux.
It is worth examining the individual structures of the fluxes in which substrate is taken up and product is produced:

Column 3:k_{1}S_{3}E_{3}=k_{2}P_{2}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{1}\rightarrow S_{2}}}=D_{{S_{2}\rightarrow S_{3}}}=D_{{E_{2}\rightarrow E_{3}}}=D_{{C_{2}\leftarrow C_{3}}}=D_{{P_{2}\rightarrow P_{{bin}}}}
Column 4:k_{1}S_{3}E_{3}=k_{2}P_{1}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{1}\rightarrow S_{2}}}=D_{{S_{2}\rightarrow S_{3}}}=D_{{E_{1}\rightarrow E_{2}}}=D_{{E_{2}\rightarrow E_{3}}}=D_{{C_{1}\leftarrow C_{2}}}=D_{{C_{2}\leftarrow C_{3}}}=D_{{P_{1}\rightarrow P_{{bin}}}}
Column 5:k_{{-1}}C_{1}=k_{1}S_{1}E_{1}=k_{3}S_{3}E_{3}=k_{3}P_{3}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{2}\rightarrow S_{3}}}=D_{{E_{1}\leftarrow E_{2}}}=D_{{C_{1}\rightarrow C_{2}}}=D_{{P_{3}\rightarrow P_{{bin}}}}
Column 6:k_{1}S_{2}E_{2}=k_{2}P_{2}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{1}\rightarrow S_{2}}}=D_{{P_{2}\rightarrow P_{{bin}}}}
Column 9:k_{1}S_{2}E_{2}=k_{2}P_{3}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{1}\rightarrow S_{2}}}=D_{{E_{2}\leftarrow E_{3}}}=D_{{C_{2}\rightarrow C_{3}}}=D_{{P_{3}\rightarrow P_{{bin}}}}
Column 11:k_{1}S_{3}E_{3}=k_{2}P_{3}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{1}\rightarrow S_{2}}}=D_{{S_{2}\rightarrow S_{3}}}=D_{{P_{3}\rightarrow P_{{bin}}}}
Column 13:k_{1}S_{2}E_{2}=k_{2}P_{1}=D_{{S_{0}\rightarrow S_{1}}}=D_{{S_{1}\rightarrow S_{2}}}=D_{{E_{1}\rightarrow E_{2}}}=D_{{C_{1}\leftarrow C_{2}}}=D_{{P_{1}\rightarrow P_{{bin}}}}
Column 15:k_{1}S_{1}E_{1}=k_{2}P_{2}=D_{{S_{0}\rightarrow S_{1}}}=D_{{E_{1}\leftarrow E_{2}}}=D_{{C_{1}\rightarrow C_{2}}}=D_{{P_{2}\rightarrow P_{{bin}}}}
Column 16:k_{1}S_{1}E_{1}=k_{2}P_{3}=D_{{S_{0}\rightarrow S_{1}}}=D_{{E_{1}\leftarrow E_{2}}}=D_{{E_{2}\leftarrow E_{3}}}=D_{{C_{1}\rightarrow C_{2}}}=D_{{C_{2}\rightarrow C_{3}}}=D_{{P_{3}\rightarrow P_{{bin}}}}
Column 17:k_{1}S_{1}E_{1}=k_{2}P_{1}=D_{{S_{0}\rightarrow S_{1}}}=D_{{P_{1}\rightarrow P_{{bin}}}}