Advantages and disadvantages of aggregating fluxes into synthetic and degradative fluxes when modelling metabolic pathways

It is now widely accepted that mathematical models are needed to predict the behaviour of complex metabolic networks in the cell, in order to have a rational basis for planning metabolic engineering with biotechnological or therapeutical purposes. The great complexity of metabolic networks makes it crucial to simplify them for analysis, but without violating key principles of stoichiometry or thermodynamics. We show here, however, that models for branched complex systems are sometimes obtained that violate the stoichiometry of fluxes at branch points and as a result give unrealistic metabolite concentrations at the steady state. This problem is especially important when models are constructed with the S-system form of biochemical systems theory. However, the same violation of stoichiometry can occur in metabolic control analysis if control coefficients are assumed to be constant when trying to predict the effects of large changes. We derive the appropriate matrix equations to analyse this type of problem systematically and to assess its extent in any given model. The metabolic activities of living cells are accomplished by a regulated and highly coupled network of enzyme-catalysed reactions and selective transport systems. The complexity of the networks makes it necessary to construct models to understand and predict their behaviour, but these are always simplifications of reality. Several approaches for modelling biochemical pathways have appeared over the last 30 years, and with the help of computers it is not difficult to simulate behaviour of simple metabolic pathways with a set of equations representing kinetics of the enzymes in the pathway. However, dealing with complex systems and progressing further in the field requires a theoretical basis for formulating different strategies for simplify- ing the complexity and relating local properties of enzymes with global properties of the pathway. To fulfil this purpose two approaches to the analysis of complex biochemical systems have emerged in the past two decades, biochemical systems theory (1-3) and metabolic control analysis (4-6). The need for these derives from the large numbers of components in such systems and the nonlinear interactions between them. Both are based on sensitivity theory, but whereas biochemical systems theory uses explicit methods metabolic control analysis uses implicit methods (7-10). Equivalent matrix equations have been developed in both formalisms to relate the response of a whole system to a perturbation (logarithmic gains in biochemical systems theory, or control coefficients in metabolic control analysis) and the local responses of individual enzymes to changes in their substrate concentrations (kinetic orders or elasticity coefficients). Two variants of biochemical systems theory have long been distinguished, generalized mass action and S-systems. In both variants each rate law is simplified by writing it as a product of power-law functions of all metabolites involved. The exponent associated with each metabolite is called a kinetic order and is regarded as a constant. At the operating point it is equivalent to an elasticity coefficient defined in metabolic control analysis, with the difference that in control analysis there is no impli- cation that it is a constant. In generalized mass action each individual reaction is a unit of representation, but in S-systems