SYSTEM CLASSIFICATION

In order to carry out a proper classification o f a proposed stoichiometric reaction network the reactions need to be divided into reaction fluxes which w ill be measured and reaction fluxes which must be calculated. The classification method presented below was developed by van der Heijden et al. (1994a).

M athem atical M odelling o f Biochemical Pathw ays Classification and Sensitivity A nalysis

The purpose o f the classification procedure is to subdivide the reaction fluxes into four categories:

measured reaction fluxes which are balanceable, measured reaction fluxes which are not balanceable, non-measured reaction fluxes which are calculable, and non-measured reaction fluxes which are not calculable.

A balanceable reaction flux is a flux which can also be calculated from a combination o f the other measured reaction fluxes. Hence, a balanceable reaction flux need not necessarily be measured if the remaining reaction fluxes proposed as “measured” are.

If a calculated reaction flux is classified as not calculable this means that given the set o f measured fluxes, no meaningful flux value can be calculated for this particular reaction. To ensure that the system is completely solvable all reactions that are non-measured must be calculable.

The actual classification procedure starts with presenting the reaction network in an appropriate mathematical form. This is done by the matrix equation below:

E r =

0

Eq. 8-1

where E is the stoichiometric matrix o f dimensions (e x n) and r the vector containing all reaction fluxes o f dimensions (n x 1). A pseudo steady state is assumed for all the internal metabolites, hence the accumulation o f each o f the metabolites is set to zero (right hand side o f Eq. 8-1). A pseudo steady state may be asumed when the accumulation o f each o f the internal metabolites is negligible compared to the flux passing through the respective metabolite pools (Vallino and Stephanopoulos, 1993). Vector r can be partitioned into two vectors, one containing the measured reaction rates and one containing the reaction rates to be calculated. Hence:

M atnem aticai M odelling of tsiocnemicai ra tn w a ys Classification ana sen sitivity A nalysis

E c - r c + E m rm = 0

Eq. 8-2

Vector rm consist o f the m measured rates and vector rc consist of c non-measured rates. (Hence, c + m = n). Em and Ec are the columns of E corresponding to rm and rc

respectively. From Eq. 8-2, it can be seen that the calculated reaction fluxes may be obtained as follows:

r = _c

- E ~ x -

Eq. 8-3

However, matrix Ec is often non-square, in which case the inverse, Ec'1, does not exist. A generally valid expression using the pseudo inverse, Ec#, which is defined for singular and non-square matrices can be substituted.

rc = - E * c - E m -rm

Eq. 8-4

The use o f the pseudo inverse, E f , will always yield a value for all elements o f vector

rc, even if som e elements of rc principally can not be calculated from rm. The use of the pseudo inverse only guarantees that the calculated elements o f rc satisfy Eq. 8-1. A separate criterion may be required to determine which o f the calculated values for the elements o f rc that are meaningful. This will be discussed further below.

Substituting Eq. 8-4 into Eq. 8-2 yields:

R - r m = 0

Eq. 8-5

Where the redundancy matrix, R, is defined as:

M athem atical M odelling o j biochem ical ra th w a ys C lassification and sensitivity A nalysis

R = F - F_m • F # • _{c c} F_m

Eq. 8-6

The redundancy matrix has a central role in the classification procedure. The rank of the redundancy matrix, k, equals the number of independent equations in R. If this is larger than zero, the system o f equations is redundant and at least one o f the measured rates are balanceable. Balanceable reaction fluxes may be identified by inspecting the columns of matrix R. The presence of zero-columns in R indicate non-balanceable reaction fluxes in the corresponding rows o f rm.

Having determined the redundancy matrix and calculated the rank o f this, an assessment o f the calculability o f the remaining non-measured reaction fluxes may also be carried out. If e-c equals k, then all the non-measured rates are calculable. However, if e-c is greater than k, then some or all o f the rates may not be calculable. In the event o f e-c> k

matrix Ec should be analysed in further detail, by means of singular value decomposition, to determine which of the reaction rates that can and cannot be calculated. This procedure will not be discussed here, but is described in full by van der Heijden et a l (1994a). In the event that some or all the remaining reaction rates cannot be calculated, a new set o f “measured” reaction rates must be chosen. This may be an expansion o f the proposed set, or a completely new set chosen from the complete set of independent reactions.