Topological analysis denoted by K

properties in such situations. In the top-down approach Brown et al., 1990; Brand, 1996 and modular approach Westerhoff and Van Dam, 1987; Schuster et al., 1993, enzymes are grouped into blocks so that it is no longer necessary to know all the details within the blocks. The kinetic properties of the blocks and the control properties of the system are described by the overall elastic- ities and control coefficients, respectively. More- over, the modular approach is a suitable tool for describing the control properties of those enzymes that couple exergonic processes to endergonic processes, such as the various ATPases Schuster and Westerhoff, 1999. Appropriate linear combi- nation of the overall control coefficients gives the coefficients quantifying the control exerted by the enzymes as a whole and the control exerted by the slip. Kinetic parameters are often difficult to mea- sure for very fast reactions. In this situation, it is often justified to assume these reactions to attain quasi-equilibrium. We were able to show in a general way that quasi-equilibrium enzymes can be eliminated from the analysis of the control properties, allowing the calculation of the control coefficients of the slow reactions even if the ki- netic properties of the fast reactions are unknown Kholodenko et al., 1998.

3. Topological analysis

A number of interesting conclusions can be derived from knowledge of the stoichiometric structure of a reaction system alone. For example, one can analyze the so-called null-space of the stoichiometry matrix. This is the region in which all flux vectors J in steady-state must be situated Reder, 1988; Schuster and Schuster, 1991. A set of basis vectors to this space can be compiled as column vectors in a matrix K, which then fulfills the equation N K = 0. 4 It can now be tested whether this null-space matrix K can be block-diagonalized K = Á Ã Ã Ã Ã Ã Ä K 1 ··· K 2 ··· — — · · · — ··· K r ···  à à à à à Š, 5 where the null rows of K if any have been transferred to the bottom of K. The steady-state flux through the reactions corresponding to such null rows is always zero. The blocks of K in Eq.

5, denoted by K

i , correspond to subsystems of the reaction system, the fluxes of which are com- pletely independent. This means that the fluxes within one subsystem can be changed by suitable parameter changes without affecting the fluxes in the other subsystems. As any basis in linear algebra is non-unique, neither is the choice of the null-space matrix. A new basis can be formed by linear combination of basis vectors. Such a linear combination can, however, diminish the number of diagonal blocks. In the matrix given in Eq. 5, for example, replac- ing the first column which then includes part of K 1 by a linear combination of this column and the first column that includes part of K 2 would lead to a matrix that can be decomposed into r-1 diagonal blocks only. The question arises as to how to find that representation of K that can be partitioned into the maximum number of diago- nal blocks. It was proven Schuster and Schuster, 1991 that such a representation can be obtained by rearranging the rows and columns of the matrix K = K I , 6 which can be found by the Gaussian elimination method as applied to equation system Eq. 4. A block-diagonal structure of K can alterna- tively be obtained by determining the elementary modes in the system Schuster and Hilgetag, 1994; Heinrich and Schuster, 1996. These are the sim- plest flux vectors that comply with the orientation of the irreversible reactions. ‘Simplest’ here means that the flux vectors cannot be decomposed into two other flux vectors each of which invokes fewer enzymes. Important examples of matrices K with a block- diagonal structure are provided by the ubiquitous hierarchic systems in biology. Fig. 1A depicts such a system, for which the null-space matrix can be chosen so as to involve two diagonal blocks K = Á Ã Ã Ã Ä 1 1 1 1 Â Ã Ã Ã Å . 7 In addition to intermediary metabolism and protein metabolism, more levels such as those of DNA and mRNA can be included. If reactions 1 enzyme synthesis and 2 enzyme degradation are both activated in such a way that the enzyme level remains constant, the flux through the upper level increases without any change in the flux through the lower level. Conversely, by changing the corresponding parameters in reactions 3 or 4 or both, the flux through the lower level can be changed without any change in the upper level. Interestingly, the property of matrix K to be block-diagonal is retained even if the system is depicted on a more detailed level, as in Fig. 1B. In this representation, the stoichiometry matrix can no longer be block-diagonalized. The substances in Fig. 1B cannot be grouped in such a way that the concentration changes in each group are due to distinct chemical reactions, whereas the steady state fluxes can be arranged in two groups of distinct fluxes. This example shows that the null- space matrix K rather than the stoichiometry ma- trix N is the more accurate mathematical concept to analyze the subsystem structure.

4. Zero flux control