Nöh, K.

Weitzel, M.

Wiechert, W.

Juelich Shared Electronic Resources

John von Neumann Institute for ComputingFrom Isotope Labeling Patterns toMetabolic Flux RatesK. No¨h, M. Weitzel, W. Wiechertpublished inFrom Computational Biophysics to Systems Biology (CBSB08),Proceedings of the NIC Workshop 2008,Ulrich H. E. Hansmann, Jan H. Meinke, Sandipan Mohanty,Walter Nadler, Olav Zimmermann (Editors),John von Neumann Institute for Computing, Ju¨lich,NIC Series, Vol. 40, ISBN 978-3-9810843-6-8, pp. 345-348, 2008.c© 2008 by John von Neumann Institute for ComputingPermission to make digital or hard copies of portions of this work forpersonal or classroom use is granted provided that the copies are notmade or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwiserequires prior specific permission by the publisher mentioned above.http://www.fz-juelich.de/nic-series/volume40From Isotope Labeling Patterns to Metabolic Flux RatesKatharina No¨h1, Michael Weitzel2, and Wolfgang Wiechert21 Institute of Biotechnology 2, Research Centre Ju¨lich, 52425 Ju¨lich, GermanyE-mail: k.noeh@fz-juelich.de2 Institute of Systems Engineering, University of Siegen, 57068 Siegen, GermanyE-mail: {michael.weitzel, wolfgang.wiechert}@uni-siegen.deIntracellular metabolic flux rates are the manifestation of metabolic activities within organisms.Aiming at a precise quantification of the in-vivo fluxes, Metabolic Flux Analysis based on label-ing experiments has become an valuable key technology in Systems Biology. Topology-basedcomputational algorithms are derived to facilitate informative high-throughput analyses as wellas predictive computational modeling and simulation approaches to generate new knowledgeand a robust experimental design of labeling experiments.1 IntroductionIntracellular metabolic flux rates are the most important indicators of the metabolic activ-ities in organisms. Depending on external conditions, the fluxome determines the cells’physiological phenotype and, thus, their metabolic capabilities. The in-vivo reaction rates,however, cannot be directly accessed from measurements. Aiming at a precise quantifi-cation, Metabolic Flux Analysis (MFA) based on Isotope Labeling Experiments (ILE) be-came an invaluable tool for Systems Biology.2 General Procedure: Isotope Labeling ExperimentsOver the past decade two types of isotope-based MFA emerged and are being successivelyrefined. Both methods rely on measurements of either labeled biomass components (withGC-MS, NMR) or labeled primary intermediates (LC-MS). The classical, well establishedstationary isotope MFA characterizes a cell’s fluxome in a metabolic and isotopic station-ary state. Typically, in a continuous culture the feed is switched from naturally labeled toisotopically labeled medium which propagates though the network and progressively re-places the unlabeled intermediates. When the labeling distribution is approximately time-invariant, samples are taken. In this field, recent experimental progress strengthened thedevelopment of high-throughput MFA1, the investigation of extensive metabolic networksas well as the utilization of elaborated nonlinear statistical methods for flux estimation2.A current area of research is isotopically non-stationary MFA which represents apromising generalization of the classical approach3. Here, the cells are likewise kept undermetabolic steady state conditions, however, now the time profiles of the labeling patternsare measured upon start of the labeling period in order to monitor the time-dissolved label-ing propagation of the isotopic tracer though the network. Compared to classical isotopeMFA, non-stationary ILEs are typically more informative and facilitate a cross check be-tween metabolome and fluxome data. In particular for comprehensive networks, however,the required computational effort is prohibitive3. Altogether, the new experimental tech-niques result in an increased demand for more efficient algorithms.3453 Modeling and Computational MachineryBesides the measurement of labeling patterns in key metabolites, extracellular rates, andpool sizes (non-stationary case only), a biochemical reaction network has to be provided.Additionally, atom transitions of all reactions have to be specified which essentially de-scribe how the flow of labeled material through the cell’s metabolic pathways is organized.The analytic determination of the fluxes as a function of the emerging labeling patterns,however, is impossible for realistic networks. Thus, for unraveling intracellular fluxes, aniterative parameter fitting procedure has to be applied. Summarizing, mathematical mod-eling and computational algorithms build the foundation of isotope-based MFA. In thiscontext, the computational bottleneck is the forward simulation step, i.e. the calculation ofemerging labeling patterns from given parameters. Because all possible labeling combina-tions of each metabolite have to be considered, at the end, the solution procedure involvesthe solution of large, cascaded systems of Labeling Balance Equations (LBEs)4, 5.The requirement for the computational evaluation of the model is in each case high:system dimensions range from 700 with 45 parameters (central metabolism, CM) and 5600with 65 parameters (CM with biosynthesis pathways) up to 355.000 with 580 parametersfor genome-scale models. The classical method involves the solution of an algebraic equa-tion system while for the non-stationary method a system of ordinary differential equations,with possibly stiff characteristic, has to be solved.4 Topological Techniques for Dimension ReductionEfficient non-standard solution algorithms basically all rely on the structure of the IsotopeLabeling Network (ILN) graph associated with the metabolic network. Although the so-lution algorithms for the stationary and non-stationary approaches are quite different, bothapproaches certainly benefit from a reduction of the problem size. Dimension reductionis performed by a careful analysis of the ILN graph followed by a removal of specifiednodes and edges or a decomposition into smaller subsystems. The two basic approachesfor dimension reduction are:Path Tracing. Typically, measurement data describe only a small subset of metabolites.The forward simulation step can be restricted to a relevant sub-network which sufficientlydescribes the transport of labeling from the substrates to the measured metabolites5. Thenecessary path tracing procedure relies on the computation of the transitive closure of thenetwork graphs and is performed in two directions – a forward tracing, which determinesthe fate of the isotopic labeling found in the substrates, and a backward tracing which de-termines the topological predecessors of a labeling pattern or fragment.Network Graph Decomposition. In a divide-and-conquer approach the network graphsare decomposed into disconnected, unilaterally connected (acyclic) and cyclic subnetworkcomponents (so called CCs, DAGs, and SCCs, respectively). This decomposition heavilyuses the unidirectionality of reactions. Once labeling leaves a cyclically connected subnet-work (by taking an unidirectional reaction route), it is impossible for it to return. This es-sential information can be used for decomposition of a network into smaller subnetworks.The decomposition results in subproblems with lower dimension and, thus, dramaticallyreduces the running time of the solution algorithms5. Depending on the network connec-tivity the speed-up is at least in the order of two to three magnitudes.346Although the potential of these two methods is high, further reduction is possible usingan even more fine-grained approach: due to their combinatorial origin, the ILNs containisomorphic subgraphs in form of parallel paths. Since the generated LBEs are likewise iso-morphic, this property results in redundant computations which are usually hard to elim-inate. Fortunately, this problem is solved by the generation of analytical solutions whichfacilitate the elimination of common subexpressions. As a by-product analytical solutionsenable exact evaluation of sensitivities and the generation of highly efficient machine code.In case of non-stationary ILEs, typically with growing label exchange between neigh-boring metabolites the differential equation systems tend to be stiff. Application of e.g.a s-stage implicit Runge-Kutta scheme involves a sN × sN -dimensional linear equationsystem, where N denotes the number of differential equations to be solved. By imposingspecialized structures on the Runge-Kutta matrix, e.g. by choosing a SDIRK scheme, thecomputational cost for its solution can be reduced. However, the complexity of performingat least one Newton step remains. Clearly, both topological approaches presented abovecan be directly applied. Moreover, because the sparsitivity pattern of the Runge-Kutta ma-trix remains the same for all time steps, the network decomposition has to be performedonly once.5 Conclusion and OutlookThe computational routines used in stationary, and in particular non-stationary isotopicMFA suffer from the inherent computational complexity of the approach. However, ex-ploiting the nature of the underlying (algebraic and differential) equation systems improvesthe efficiency of the solution methods. A careful study of the labeling network topologyleads to a significant increase in performance. New algorithms emerge, having their rootsin Graph Theory, Linear Algebra, and Compiler Theory. However, particularly for the clas-sical approach the new techniques providing analytical and fast numerical solutions openthe perspective to simulate even genome-scale metabolic models.References1. U. Sauer, Metabolic networks in motion: 13C-based flux analysis. Mol. Syst. Biol. 2,62, 2005.2. V. Kadirkamanathan, J. Yang, S.A. Billings, P.C. Wright PC, Markov Chain MonteCarlo Algorithm based metabolic flux distribution analysis on Corynebacterium glu-tamicum. Bioinf. 22(21), 2681–2687, 2006.3. K. No¨h, K. Gro¨nke, B. Luo, R. Takors, M. Oldiges, W. Wiechert, Metabolic flux anal-ysis at ultra short time scale: isotopically non-stationary 13C labeling experiments.J. Biotechnol. 129(2), 249–267, 2007.4. K. No¨h, A. Wahl, W. Wiechert, Computational tools for isotopically instationary 13Clabeling experiments under metabolic steady state conditions. Metab. Eng. 8(6),554–577, 2006.5. M. Weitzel, W. Wiechert, K. No¨h, The topology of metabolic isotope labeling net-works. BMC Bioinformatics 8, 315, 2007.347

From Isotope Labeling Patterns to Metabolic Flux Rates

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From Isotope Labeling Patterns to Metabolic Flux Rates

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