162
Chapter 5
Note 5.5 Sensitivity analysis of stoichiometric matrices
When metabolic fluxes are estimated using matrix inversion it is important to pay attention to the system
matrix. If it is ill-conditioned even small errors in measured rates may propagate as large errors in the
estimated fluxes. One way to check this is through evaluation of the condition number, which is given by:
condition number =
T2
r
0 )
Here | | indicates any matrix norm and (T2r )
is the pseudo inverse of the stoichiometric matrix given by
eq. (5.28) (if a determined system is analyzed the pseudo inverse becomes identical to the inverse of the
matrix). If the condition number is high (above 100) the matrix is ill-conditioned. This is illustrated by a
very simple equation system:
P
'
1
[l
1.0001J
v,
(2)
The condition number of this matrix is 104, and the matrix is clearly ill-conditioned. Now let the rate vector
be given by:
r =
'2
(3)
whereby the solution is V|=2 and v2=0. However, if the second element in the r vector is changed to 2.0001,
a change of less then 0.1%, the solution is v,= 1 and v2=l. Thus, a very small change in the measured rates
results in a drastic change in the estimated flux vector, and in practice this means that even small
measurement errors propagate as large errors in the estimated fluxes. If the stoichiometric matrix is ill-
conditioned it is necessary to revise the model, e.g., to remove certain reactions or choosing a different set
of reaction rates for the flux estimation.
A different approach to evaluate whether the fluxes are sensitive to small variations in the measured rates is
through the sensitivity matrix, which is given as the inverse of T2:
d \
= - t 2
-'
(4)
Here the partial derivate specifies the sensitivity of all the elements in the flux vector to the individual
measured rates, i.e. the element in theyth row and the ;th column specifies the sensitivity of theyth flux with
respect to variations in the measurement of the ith rate. For the example we find the sensitivity matrix to be:
104
1.0001
(5)
All the elements are very large and it is quite clear that the two fluxes are very sensitive to even small
variations in the measured rates.______________________________________________________ ________