Metabolic isotopomer labeling systems - Part I: global dynamic behavior

In the last few years metabolic flux analysis (MFA) using carbon labeling e xperiments (CLE) has become a major diagnostic tool in metabolic engineerin g. The mathematical centerpiece of MFA is the solution of isotopomer labeli ng systems (ILS). An ILS is a high-dimensional nonlinear differential equat ion system that describes the distribution of isotopomers over a metabolic network during a carbon labeling experiment. This contribution presents a g lobal analysis of the dynamic behavior of general ILSs. It is proven that a n ILS is globally stable under very weak conditions that are always satisfi ed in practice. In particular it is shown that in some sense ILSs are a non linear extension to the classical theory of compartmental systems. The cent ral stability condition for compartmental systems, i.e., the non-existence of traps in linear compartmental networks, is also the major stability cond ition for ILSs. As an important side result of the proof, it is shown that ILSs can be transformed to a cascade of linear systems with time-dependent inhomogeneous terms. This cascade structure has considerable consequences f or the development of efficient numerical algorithms for the solution of IL Ss and thus for MFA. (C) 2001 Elsevier Science Inc. All rights reserved.