A Priori Analysis of Metabolic Flux Identifiability from 13 C-Labeling Data Wouter A. van Winden, 1, * Joseph J. Heijnen, 1 Peter J. T. Verheijen, 2 Johan Grievink 2 1 Department of Bioprocess Technology, Faculty of Applied Sciences, Delft University of Technology, The Netherlands 2 Department of Process Systems Engineering, Faculty of Applied Sciences, Delft University of Technology, The Netherlands Received 3 March 2001; accepted 4 April 2001 Abstract: The 13 C-labeling technique was introduced in the field of metabolic engineering as a tool for determin- ing fluxes that could not be found using the ‘classical’ method of flux balancing. An a priori flux identifiability analysis is required in order to determine whether a 13 C- labeling experiment allows the identification of all the fluxes. In this article, we propose a method for identifi- ability analysis that is based on the recently introduced ‘cumomer’ concept. The method improves upon previ- ous identifiability methods in that it provides a way of systematically reducing the metabolic network on the ba- sis of structural elements that constitute a network and to use the implicit function theorem to analytically deter- mine whether the fluxes in the reduced network are theo- retically identifiable for various types of real measure- ment data. Application of the method to a realistic flux identification problem shows both the potential of the method in yielding new, interesting conclusions regard- ing the identifiability and its practical limitations that are caused by the fact that symbolic calculations grow fast with the dimension of the studied system. © 2001 John Wiley & Sons, Inc. Biotechnol Bioeng 74: 505–516, 2001. Keywords: 13 C-labeling; cumomer; flux analysis; identi- fiability; network reduction INTRODUCTION It has long been recognized that analysis of stationary meta- bolic fluxes is often impeded by the unobservability of fluxes due to parallel pathways, metabolic cycles, or bidi- rectional reactions. Over the last few years 13 C-labeling experiments have become a well-established tool in the analysis of these otherwise unobservable fluxes. Various measurement methods are available for determining the la- beling distribution of metabolic intermediates (for an over- view, see Mo ¨llney et al., 1999; Szyperski, 1998). Labeling balances are bi- or even trilinear expressions. This characteristic prohibits direct calculation of the un- known intracellular fluxes from the 13 C-labeling data and the measured extracellular fluxes. Therefore, the unknown fluxes are mostly determined using an iterative numerical procedure. This procedure consists of using a mathematical model to simulate labeling data as a function of the fluxes and varying the fluxes until the simulated data fit the mea- sured data. A disadvantage of such a flux-fitting algorithm is that a flux estimate cannot be proven to be unique (Schmidt, 1998; Wiechert, 1996). The possibility to uniquely identify the fluxes depends on the topology of the studied metabolic network, the measurable labeled com- pounds, the 13 C-measurement method used, and the sub- strate labeling applied. Although repeating the iterative flux estimation procedure for various starting values of the fluxes will increase the confidence in the uniqueness of a flux estimate, an analytical a priori (i.e., independent of the values of the actual fluxes) flux identifiability analysis is preferable. Recently, Wiechert et al. (1999) introduced the new ‘cu- momer’ concept for describing labeling distributions in me- tabolites. Cumomers represent sets of molecules that are 13 C-labeled at specific atomic positions and of which the remaining atomic positions may be either 13 C-labeled or not. As a consequence, cumomers are sums of isotopomers. In contrast to isotopomers, cumomers can be expressed as explicit functions of the metabolic fluxes and substrate la- beling in a metabolic network. This opens the way for ana- lytically calculating isotopomer distributions for a given set of fluxes. Moreover, the concept of cumomers can be used to tackle the inverse problem: the structural analysis of flux identifiability from known cumomer fractions. In their ar- ticle, Wiechert et al. (1999) illustrate the use of cumomer balances for identifiability purposes by means of two ex- ample networks. The authors mention that their identifiabil- ity analysis of the examples is rather intuitive and refer to computer algebraic algorithms developed by Wiechert (1995) for a more systematic method. In that article about identifiability and redundancy analysis, the author describes how to algebraically solve the flux identifiability problem for positional 13 C-enrichment data using so-called Gro ¨bner bases. However, he concludes that the computation of Gro ¨b- ner bases is strictly limited to small problems, because the computational effort required to find an answer increases dramatically with the complexity of the problem. In another Correspondence to: Wouter A. van Winden * Present address: Kluyver Laboratory for Biotechnology, Julianalaan 67, 2628 BC Delft, The Netherlands, Phone/fax: +31-15-278-5307/-2355, Email: W.A.VanWinden@tnw.tudelft.nl © 2001 John Wiley & Sons, Inc.