# Concentration Control Coefficients

The function ConcentrationControlMatrix in the MetabolicControl-Analysis add-on package can be used to calculate the concentration control coefficients for the erythrocyte model. These coefficient values express the importance of a particular reaction in controlling the concentration of a particular metabolite (Section 5.2).

Q: How is the concentration control coefficient matrix computed?

A: This is done by executing the following function with the three arguments that define the whole erythrocyte model, namely, S, VN, and v:

The whole matrix is not printed here because of space restrictions, but it can be inspected by simply deleting the semicolon after the function and then re-evaluating the

### Mathematica Cell.

Also, note that each entry in row i and column j of the control matrix gives the concentration control coefficient of reaction j with respect to metabolite i. The concentration control coefficient matrix has dimensions of 56 x 53 because there are 56 internal substrate variables and 53 reactions in the erythrocyte model.

Q: Which reactions are important in controlling the concentration of 2,3-BPG?

A: Before answering this question directly, it turns out to be useful to define first some replacement rules and functions which make the overall analysis much simpler.

First define a replacement rule of the form substrate name 0 row, and one of the form row 0 substrate name.

substrateToNum = Table [Spj 0 i, {i, Length[S]}] ; numToSubstrate = Table [i 0 Spj , {i, Length[S]} ] ;

Then, as was done for the flux control coefficients, we define a function, CC[x,y],which gives the concentration control coefficients for concentration x which are > y.

CC[x_, y_] : = Module[{bigCccPosition, names, values}, bigCccPosition =

Position[ccmjx/.substrateToNumi , z_? (Abs [#1] >= y&)] // Flatten; names = bigCccPosition / . numToRxn ; values = Part[ccmIx/.substrateToNumi , bigCccPosition] ; Transpose[{names,values}]]

Now check whether the summation theorem (Eqn [5.7]) holds for the control of 2,3-BPG concentration.

Clearly, this output indicates a value that is very nearly 0, as expected, thus indicating a successful analysis. And finally, the important reactions for controlling free 2,3-BPG concentration are determined with the function CC, as follows:

{{hk, 1.57071}, {pfk, 0.345435}, {eno, -0.183782}, {pk, -0.269142}, {bpgsp2, 0.127866}, {bpgsp9, -0.410397}, {atpase, -1.16271}, {ox, -0.0190557}}

The list indicates that there are several enzymes that exert significant control over the concentration of 2,3-BPG. The most important of these are hexokinase, phosphofructokinase, enolase, pyruvate kinase, the two irreversible steps of the 2,3-BPG synthase-phosphatase, and the ATPase.