### Control analysis

Let us return to Equation (

3), a generalized description of the temporal evolution of a metabolic network in differential equation format. Let us also assume that the reference state

*x* =

*x** corresponds to a steady state -

*i.e. N v** = 0, where

*v** =

*v*(

*x**). Writing

(for algebraic simplicity),

*N* = diag(

*c*)

^{-1}·

*N* and

, and dropping hats for convenience, we transform the system into the more recognisable form

where

*x* = 0 now corresponds to the steady state. Linearizing about this steady state

where *ε*' is the *n × m* unscaled elasticity matrix.

In general, the rank(*N ε*') = *m*
_{0} <*m* and the system defined above will display moiety conservations - certain metabolites can be expressed as linear combinations of other metabolites in the system. Note that the number of independent metabolites is not given simply by rank(*N*), as is generally (and erroneously) suggested; rather the local dynamics of the system must also be taken into account via the elasticity matrix. The conservations may be removed through matrix decomposition, using a *m × m*
_{0} link matrix *L* that relates the complete vector of internal metabolites to the vector of independent metabolites [28]. Writing *A* = *N ε*' and letting *A*
_{
r
}denote a *m*
_{0} × *m* matrix composed of linearly independent rows of *A*, the corresponding link matrix is defined as
, where '^{+}' denotes the Moore-Penrose pseudoinverse [29]; hence *A = L·A*
_{
r
}.

From Equation (

6), and noting that the rows of

*L* corresponding to the independent metabolites

*x*
_{
r
}form the identity matrix, we find

*x* =

*L x*
_{
r
}and hence

where the *m*
_{0} × *m*
_{0} matrix (*N*
_{r}·*ε*'. *L*) is invertible through introduction of the link matrix *L*.

Having transformed the system, we add a small perturbation to reaction

*j*
where

*δ* is our perturbation;

*e*
_{
j
}denotes the

*j*
^{th} standard basis vector and the notation

*N*
_{r, j}is used to denote the

*j*
^{th} column of

*N*
_{
r
}. The new steady state resulting from this perturbation is given by

Using Equation (

9), we may resolve the definition of (unscaled) flux control and concentration control coefficients as

respectively. If we compare our expressions to those given in Reder [22], we see that they are identical, save in her case *r'* is defined as the independent rows of *N*, leading to
. If *r* = *r'* (*i.e*. if rank(*N ε'*) = rank(*N*)), then *L* = *L'* and the two results are equivalent.

As such, we may see that we have extended Reder's work to encompass the possibility that rank(*N ε'*) < rank(*N*), as is the case for our model (rank(*N ε'*) = 205, whilst rank(*N*) = 616). From Equations (10) and (11), one may trivially deduce the summation and connectivity theorems.

Equation (

10) may be used to calculate flux control coefficients for our genome-scale model. These parameters may also be defined in their more usual scaled form