The calculation to determine elementary modes

We calculated based on (Schuster et al. 2000). We combined two reactions to cancel internal substances. To make calculation easy, we use the matrixes. For example, reactions which catalyzed by OTC and CPS are shown below.
L-ornithine + carbamoyl phosphate → L-citrulline + Pi L-glutamine + HCO₃^- + H₂O + 2ATP → carbamoyl phosphate + L-glutamate + Pi + 2ADP

If we combine them, we get total reaction formula as below.

L-ornithine + L-glutamine + HCO₃^- + H₂O + 2ATP →L-citrulline + L-glutamate + 2ADP + Pi

This operation means we canceled carbamoyl phosphate (internal substance) by combining two reactions. If we do this operation in matrixes like, the operation is following.




The row represents one reaction. The column in the left-hand side submatrix represents the internal and external substances’ consumption or production in the reaction. The column in the right-hand side submatrix represents what reaction is take place to get left-hand side matrix. The first column represents carbamoyl phosphate consumption or production. Minus means consumption and plus means production. Therefore, the addition of two rows means the total reaction of the two reactions. We can get the reaction formula which doesn’t contain internal substances by repeating this operation. We show the initial tableau.



At first, we combined two rows to make the first column zeros. The rows whose first column is already zero are copied to the next tableau in the same part (reversible or irreversible) because they don’t need to be combined with other rows. The rows which are obtained by combining the same part (reversible or irreversible) of two rows go into the part of the same part of the previous part. The rows which are obtained by combining different parts go into irreversible part. Irreversible reactions don’t allow subtracting from other rows. Combination of two rows is limited by three conditions. We checked them through calculation. First, a pair of rows is combined only if it fulfills the condition

for all row indices l belonging to the respective part (reversible or irreversible) of the new tableau as it has been compiled until that stage. mi(j) represents for the ith row in the right-hand side submatrix (Internal and external substances part) of the tableau T(j) and S(mi(j)) is the positions of zeroes in this row. It is allow increasing or decreasing the number of the rows because making next tableau by combination.
The second condition is that “irreversible” rows can only be added not subtracted like mentioned above. The third condition is shown below.

It is sufficient to apply it only upon calculation of the final tableau.
We calculated based on this limit. We show the process of the calculation.

















Finally, we obtained the tableau below.




The rows in the right-hand side submatrix of the final tableau represent the elementary modes.