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
* corresponds to a steady state - i.e. N v
* = 0, where v
* = v
(for algebraic simplicity), N
, and dropping hats for convenience, we transform the system into the more recognisable form
= 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') = m0 <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 × m0 link matrix L that relates the complete vector of internal metabolites to the vector of independent metabolites . Writing A = N ε' and letting A
denote a m0 × m matrix composed of linearly independent rows of A, the corresponding link matrix is defined as
, where '+' denotes the Moore-Penrose pseudoinverse ; hence A = L·A
From Equation (6
), and noting that the rows of L
corresponding to the independent metabolites x
form the identity matrix, we find x
= L x
where the m0 × m0 matrix (Nr·ε'. L) is invertible through introduction of the link matrix L.
Having transformed the system, we add a small perturbation to reaction j
is our perturbation; e
denotes the jth
standard basis vector and the notation Nr, j
is used to denote the jth
column of N
. 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 , 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.
) 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