13CFLUX2 Modeling Plug-In

The 13CFLUX2 Modeling Plug-In extends a network model by certain properties specific for 13CFLUX2. 13CFLUX2 is a high performance simulator framework for isotope-labeling-based metabolic flux analysis developed at the public research institute IBG-1, Forschungszentrum Jülich.


Metabolic flux analysis aims at identification and quantification of intracellular fluxes (i.e. reaction velocities). The flux rates in a cell cannot directly be measured. A widely established tool to quantify fluxes is the isotope labeling experiment. By labeling the input substrates with isotopes and by measuring the labeling enrichment of the metabolites all over the network with highly sensitive mass spectrometry devices the intracellular flux distribution can be estimated. Therefore, computer simulation is necessary.

13CFLUX2 is a simulator framework for the quantification of metabolic fluxes from experimental measurement data with high-performance computational methods. The framework consists of about twenty different command-line applications for simulation, exploration, parameter fitting, statistical analysis and other purposes.

Simulation with 13CFLUX2 bases on models provided in the simulator specific FluxML format, an XML-based model format that stores network stoichiometry, atom mappings, constraints, start conditions, configurations of the network, measurement specification and data, simulation settings and others.

Basically, the entire modeling process for 13CFLUX2 can take place in Omix. The entire 13CFLUX2 model consists of different model parts that are realized by a number of different plug-ins:
  • network stoichiometry - the stoichiometry of the network model is a built-in feature of Omix.
  • atom mappings - the Atomic Layer Plug-In provides modeling features to define carbon atom mappings for each reaction of the network. Optionally the Chemical Structure Plug-In allows to equip the network with chemical structures that make the atom mappings unique and well interpretable. If a metabolite provides a structural formula its InChi code is exported as annotation of the pool in FluxML.
  • stoichiometric constraints - the Network Constraints Plug-In equips the Omix document with the capability to define constraints for net and exchange fluxes as well as pool sizes. (The global bounds parameter available in the Network Constraints Plug-In is skipped in FluxML.)
  • model configuration - the 13CFLUX2 Modeling Plug-In provides extended modeling features for the 13CFLUX2 model prepared in Omix. This includes:
    • isotope enrichment of the network's input substrates
    • measurement data e.g. MS or NMR measured isotope enrichments of the network's intermediates
    • model variables and properties
    • simulation settings
The 13CFLUX2 FluxML Plug-In allows to export the entire model to FluxML which serves as input for the 13CFLUX2 simulator framework. Finally, the 13CFLUX2 Launcher Plug-In allows to start 13CFLUX2 directly from within Omix, observe the simulator logs at runtime and create comprehensive scientific workflows of MFA simulation.

Subsequently, the different modeling features of the 13CFLUX2 Modeling Plug-In are introduced in detail.

1.2. Using the Plug-In

After installation, the 13CFLUX2 Modeling Plug-In is available in the list of active plug-ins. Activate the plug-in within a network document.

Activating plug-in
Activating plug-in

After activation the list of network properties contains another entry "13CFLUX2 Model Properties".

Extended property list
Extended property list

1.2.1. Model Setup

Before starting to edit the 13FLUX2 model configuration, it is necessary to setup the network stoichiometry and define constraints and atom mappings for each reaction. The network stoichiometry is prepared, for instance, by sucessively adding metabolite and reaction symbols to the diagram and interconnecting them with each other.

If your network is compartmentalized, the same metabolite in two different compartments (including the non-compartment area) result in different pools in the exported 13CFLUX2 model (FluxML). Note, that cofactor edges, effector edges and additional graphical elements in the diagram are fully ignored by the FluxML exporter.

Finally, the completed network stoichiometry must have input pools and effluxes in order to be accepted by 13CFLUX2. An input pool is a metabolite that is only connected on the educt side of unidirectional reactions, i.e. no connection goes into the metabolite (cf. image a). An efflux is a unidirectional reaction without products, i.e. no connection goes out of the reaction (b).

Stoichiometric configuration of input pools (a) and network effluxes (b)
Stoichiometric configuration of input pools (a) and network effluxes (b)

Activate the Network Constraints Plug-In on the document in order to define further stoichiometric constraints for the network model.

In order to define atom mappings activate the Atomic Layer Plug-In and follow the instructions of the plug-in documentation. For 13CFLUX2 models it is absolutely recommended to use chemical structures as basis for the atom mappings. By using chemical structural, the atom mappings are based on well-defined atom positions in the reactant molecules. Furthermore, if chemical structures are available, they are used when defining input labeling and labeling measurements. This warrants consistency between the atom mappings and the labeling model because it is based on the same unique and reproducible atom identification algorithm.

The KEGG and BioCyc plug-ins allow to load the metabolic pathways of entire organisms from corresponding databases. The imported model also contains atom mappings and chemical structures.

1.2.2. 13CFLUX2 Model Properties

After preparing the stoichiometric and atomic part of the model, the "13CFLUX2 Model Properties" can be defined. By clicking on the "Edit..." label, the 13CFLUX2 model properties can be edited in the following dialog window:

13CFLUX2 model properties dialog
13CFLUX2 model properties dialog Simulated Element

On the left side in the "Simulated Element" box, the chemical element has to be chosen, the isotope labeling is based on. The carbon isotope 13C is widely used for isotope labeling experiments and the original motivation for the name of 13CFLUX2. Thus, carbon is chosen by default. However, you can alternatively select oxygen or nitrogenium as element base for the isotope labeling.

In 13CFLUX2, it is not important for simulation of isotope labeling experiment what chemical element is actually used for isotope labeling. But important is, that the weight difference between the main isotope and the labeling isotope is 1:
  • 12C, natural abundance 98.9%
    13C for labeling
  • 16O, natural abundance 99.8%
    17O for labeling
  • 14N, natural abundance 99.6%
    15N for labeling

The Atomic Layer Plug-In supports the definition of atom mappings for all element types simultaneously. The "Simulated Element" restricts the considered atom mappings to the selected atom type. Hence, only the atom mappings are exported to FluxML that correspond to the "Simulated Element". Comment

Beyond the "Simulated Element" field, a comment section is given, where you can enter arbitrary context information about the model or modeling process. The comment will later be part of the exported FluxML model. Configurations

On the right side of the model property dialog window different model configurations can be created. A 13CFLUX model configuration represents a parametrization of a static network structure with isotopic labeling for the substrates, additional constraints, measurement values, variables and settings for the simulation. Each configuration has to specify a unique name.

Each model should provide a model configuration named "default". However, arbitrary different configurations can be specified which must be explicitely referenced to be used when starting the 13CFLUX2 tools (see simulator documentation). By default, 13CFLUX2 uses the "default" configuration.

By selecting the configuration in the list and clicking the "Edit" button you can start editing the model configuration. You can also add a new empty configuration, duplicate the selected one and import model configurations from FluxML files.

1.2.3. Editing Configurations

The following image shows a screenshot of the model configuration editor. On the left side of the editor, all different editor pages are listed:
  • Basic Properties
  • Constraints
  • Network Input
  • Measurement
  • Variables
By selecting between the list entries, you can switch between the different pages.

Screenshot of the 13CFLUX2 model configuration editor
Screenshot of the 13CFLUX2 model configuration editor Basic Properties

On the "Basic Properties" page, you can specify the name of the configuration (e.g. "default") and optionally write a describing comment about the current configuration. On bottom, you find an "Advanced Settings" button and a "Measurement" box containing four options:
  • Include measurement model (checked by default)
  • No measurement data
  • Experimental design
  • Non-stationary
These options are advanced settings required for different simulation tasks in 13CFLUX2 (cf. sections Advanced Configuration and Special Application Cases). Constraints

The "Constraints" page provides an editor for linear stoichiometric constraints for net and exchange fluxes. Each model configuration can provide further stoichiometric constraints in addition to the general network constraints provided by the network property "Constraints" (Network Constraints Plug-In). How to use the editor to define stoichiometric constraints is described here. Network Input

Each configuration is required to provide a specification of the isotopic labeling sources. This is done on the "Network Input" editor page:

Screenshot of the 'Network Input' editor page
Screenshot of the 'Network Input' editor page

If the "Network Input" section does not contain any input pool entries 13CFLUX2 automatically initializes the isotopomer distribution of each input source according to the natural abundance of 12C (98.945%) and 13C (1.055%) isotopes (even when the simulated element is not carbon).

You can add input labeling entries to the model by pressing the "Add" button. In the following dialog, you have to select which input metabolite you wish to specify (a). The selection dialog only lists actual input pools of the network because selecting an intracellular pool as input will be rejected by 132CFLUX2 as an invalid model.

a) 13CFLUX2-Modeling6.png b) 13CFLUX2-Modeling7.png

After selecting an input pool a second dialog window appears where you can define the fractional abundance of a certain labeling pattern in the input source (b). Here, you have to specify which atoms of the input substrate are labeled. Therefore, click on the corresponding atom in the scene. If a structural formula is available for the selected input pool, you define the labeling pattern based on the displayed chemical structure which warrants clear atom identification.

After confirming the dialog, the new input pool and labeling pattern is listed. Alternative, you can choose "Default Labeling Pattern". When no labeling pattern is listed, 13CFLUX2 assumes natural carbon isotope abundance for the input pool (see above).

You can add multiple labeling patterns for each input substrate ("Add Labeling Pattern" button) leading to a labeling mixture. In this case, specify the ratio of the individual labeling patterns in the mixture by editing the value column.

For each labeling pattern you can optionally specify the positional isotopic purity of labeled carbon atom positions as value between 0.0 and 1.0. If an atomic purity is entered, all unlabeled carbon atom positions are assumed to be naturally labeled (i.e. 98.945% 12C, 1.055% 13C). If no atomic purity is given, they are taken 100% 12C and 0% 13C.

Instead of specifying the labeling patterns as isotopomer (labeled and unlabeled positions) you can also use the cumomer notation (states: labeled, unlabeled and either/or). Therefore, select "Cumomer" in the "Type" combobox on top. Measurement

On the "Measurement" page, you can specify the measurement model and include labeling measurement and flux measurement data:

Screenshot of the 'Measurement' editor page
Screenshot of the 'Measurement' editor page

The "Measurement" page contains different tabs for labeling and flux measurements. In modeling non-stationary isotope labeling experiments, also poolsize measurements can be defined. Labeling Measurement
The labeling measurement table contains columns for measurement group, value, standard deviation and scale mode. At first, you ave to define the measurement group. Therefore, click on the cell in the first table column. The cell becomes editable. Here, you can enter the measurement group textually according to the measurement short notation of FluxML (cf. documentation).

However, a more convenient way to define a labeling measurement group is by using the graphical features of the modeling plug-in. Therefore, click on the "Add measurement group" button in the cell which opens the following wizard dialog:

Screenshot of the 'Measurement Group' editor dialog
Screenshot of the 'Measurement Group' editor dialog

Here, you can select the measured pool from a list of all metabolites in the network and choose which measurement method has been applied:
  • MS
  • MS/MS
  • 1H NMR
Alternatively, you can combine the measured isotopomer mixture by selecting "Use Generic Labeling Pattern". If a structural formula is available for the measured pool, you define the labeling measurement based on the displayed chemical structure which warrants clear atom identification.

After specifying a measurement group in the first column, insert measured value and standard deviation in the second and third column. The fourth column represents the "scale" attribute of the measurement group. This attribute controls whether the measurement values are automatically scaled (value "auto") or not (value "1"). MS Measurement Group
A simple mass spectrometry (MS) measurement describes a single metabolite or fragment in different "mass isotopomers". A mass isotopomers of M+n (M = mass of measured ion) indicates that exactly "n" of the labeling positions of the molecule or fragment are occupied by an isotopic labeling. The measurement value is simulated by summing up all isotopomer fractions which fit this pattern. In the real MS measurement, a specific mass-increment of a metabolite results in a peak in the spectrogram and the measurement value is obtained by an integration of the peak.

First, specify the measured molecule fragment. Therefore, click on the individual atoms in order to select which belongs to the fragment (green) and which does not (red). Then, select the mass isotopomers that have been measured by clicking on the rectangular buttons (M0..Mn) on bottom.

Specifying MS measurement group
Specifying MS measurement group

After finishing, the table is filled with the measurement group expression and additional rows for each mass isotopomer (M0..Mn). MS/MS Measurement Group
A MS/MS (Tandem-MS) measurement describes a MS measurement followed by a fragmentation of the molecule and a second MS measurement on the resulting fragments. Since the first MS measurement may also refer to a molecule fragment, a MS/MS measurement value describes a certain mass isotopomer observed in a fragment of a fragment.

In order to define the MS/MS group, you must first specify the measured molecule fragment in two fragmentation steps. Therefore, click on the individual atoms of the mother ion (M=mass of mother ion) and daughter ion (m=mass of daughter ion) in order to define the fragmentation. Green atoms belong to the fragment, red one do not. Then, select the mass isotopomers that have been measured by clicking on the rectangular buttons (M0..Mp × m0..mqM(0..p,0..q)) on bottom.

Specifying MS/MS measurement group
Specifying MS/MS measurement group

After finishing, the table is filled with the measurement group expression and additional rows for each mass isotopomer (M(0..p,0..q)). 1H NMR Measurement Group
The proton and electron of a hydrogen atom have a non-zero spin with opposite signs. For an unpaired hydrogen atom these spins would normally cancel out since the electron’s spin shields the spin of the proton. However, for a hydrogen in a chemical bond and depending on the electronegativity of the bonding partners, the electron is pulled away from the proton an the proton shows up as a peak in the NMR spectrum. This chemical shift results in different resonance frequencies, and thus different hydrogens in the molecules can be discriminated.

More importantly it is possible to discriminate labeled from unlabeled atom positions in the neughborhood of hydrogens, and thus, this 1H-NMR allows the description of positional isotopic labeling enrichment. A 1H-NMR measurement group specification is essentially a list of labeling fractions at specified positions. It is assumed that all listed atom positions have at least one bond to a hydrogen atom.

Define a 1H-NMR measurement group by clicking on the individual atoms in order to select which position was measured (green) and which was not (red).

Specifying 1H NMR measurement group
Specifying 1H NMR measurement group

After finishing, the table is filled with the measurement group expression and additional rows for each labeling position (P1..Pn). Generic Measurement Group
After all, the NMR and MS measurement groups given above are nothing but macros allowing the convenient description of sums of subsets of the full set of a metabolite’s isotopomer fractions. Alternatively, a measurement group can also be defined in generic manner.

Therefore, select "Use Generic Labeling Pattern" and mark each atom of the metabolite to be measured (black), not measured (white) or "either/or" (shaded). After finishing, the content of the table cell remains editable and you can edit the generic formula. A generic measurement specification consists of a semicolon-separated list of formulas. Each of these formulas describes exactly one measurement value. The formulas contain isotopomer notations and support various operators ("+", "-", "*", "/", "yˆx") including some built-in functions (sqrt, abs, min, max, exp, log, log2, log10). Syntactical valid examples:
  • E4P#x1x1 + E4P#1xxx
  • E4P#0101; E4P#1000
  • 2 * E4P#0x11 * log(E4P#x1x0)

Specifying generic measurement group
Specifying generic measurement group

Finish editing the generic measurement formulas by pressing the ENTER key or clicking on a different table cell. After finishing, the table is filled with additional rows for each independent formula in the generic group (separated by ";"). Flux Measurement
The flux measurement page provides different tables for net and exchange fluxes. Each table row specifies one measurement. In the first column, you have to specify what exactly you have measured. Here, you can insert an arbitrary formula containing reaction names. The formula supports various operators ("+", "-", "*", "/", "yˆx") including some built-in functions (sqrt, abs, min, max, exp, log, log2, log10). Examples:
  • v1
  • v2+v5
  • 2*v3-v8
Insert measured value and standard deviation in the second and third column. Import & Copy-Paste Functionality
You can import measurement data from different file formats:
  • CSV spreadsheets - available by default
  • Excel spreadsheets - requires the Excel plug-in
  • FluxML - requires the FluxML plug-in
  • FWD - requires the FWDSIM plug-in
CSV and Excel must provide a table that is formatted in the same manner as the target measurement table. When importing from CSV, you can specify the delimiter character and quotation character used in the file.

When importing measurement data from FWD file, you must specify the standard deviation. This is done in the dialog shown in the next picture. Here, you can specify a constant value as standard deviation for all imported values. Alternatively, you can specify a mathematical expression how the standard deviation shall be computed from the imported value x.

Specifying standard deviation for measurements
Specifying standard deviation for measurements

Furthermore, it is possible to copy-paste data from Excel tables or from other model configurations to the measurement tables. Meta Information
By clicking "Edit Details >>>" in the lower right corner, you can specify additional background information about the measurement, e.g. date, version, start and end of the experiment, experimenter, strain, and units of the measurement values. Variables

In a metabolic network all fluxes can be computed when a certain number of fluxes are known. These known fluxes are called free fluxes while all others are called dependent. The dependency results from the network stoichiometry and constraints. The free fluxes are the variables of the equation system.

On the "Variables" page, you can specify the variables and specify their initial values. The page provides a table for net and exchange fluxes. In non-stationary mode, you can also define free poolsizes.

Screenshot of the 'Variables' editor page
Screenshot of the 'Variables' editor page

Add a variable by clicking on a cell in the first column and select an entry from the list of reactions. Thereafter, you can specify an initial value in the second column.

The selection of the system variables must be valid, i.e. no variable may be selected that stoichiometrically depends on other variables. If the selected set of variables is incomplete, the simulator chooses additional variables with initial value 0. The 13CFLUX2 utility sscanner can be used to automatically determine an optimal selection of free fluxes with optimal initial values. After running sscanner, import the variables from the provided FluxML file in order to carry the variables over into the model. Using Free Flux Plug-In
While specifying variables the 13CFLUX2 Modeling Plug-In does not check, whether the selected flux can be a variable or is dependent. Therefore, we recommend to use the Free Flux Plug-In to specify free fluxes within the network diagram. Every time you select a reaction as free, the Free Flux Plug-In recomputes the dependencies in the stoichiometry. By this, only those reactions can be selected that do not have any dependency.

You can import the free fluxes selection from the Free Flux Plug-In by selecting the "Import from Free Flux Plug-In" button on the toolbar. Import & Copy-Paste Functionality
You can import variables from different file formats:
  • CSV spreadsheets - available by default
  • Excel spreadsheets - requires the Excel plug-in
  • FluxML - requires the FluxML plug-in
  • FWD - requires the FWDSIM plug-in
  • HDF5 - requires the HDF5 plug-in
CSV and Excel must provide a table that is formatted in the same manner as the variables table. When importing from CSV, you can specify the delimiter character and quotation character used in the file.

It is also possible to copy-paste data from Excel tables or from other model configurations to the variables tables.

1.2.4. Validity Check

By pressing on the "Validity Check" button on the toolbar, the 13CFLUX2 model configuration is checked for consistency.

Validity check button
Validity check button

The consistency check includes:
  • Correct specification of input pools and network effluxes (see section Model Setup)
  • Specification of constraints
  • Specification of measurements and measurement data
  • Specification of variables

1.2.5. Advanced Configuration

By pressing the "Advanced Settings" button, additional options are available:

Advanced model configuration dialog
Advanced model configuration dialog

The attribute "Simulation Type" controls the behavior of network reduction:
  • "Auto" (the default) selects automatic network reduction. This results in choosing the optimal simulation model based on either a Cumomer or EMU network for the specified measurement model. This should be used by default to obtain the fastest possible simulation.
  • "Full" deactivates automatic network reduction. As a consequence a simulation of the full / unreduced Cumomer or EMU network is enforced - although this might not be necessary. Using this setting, simulations are usually much slower or even impossible for large networks comprising metabolites with many labeling positions. Use with care.
  • Checking "Explicit" results in evaluation of the measurement model, explicitly listing the measurement specifications that are to be simulated. This option does neither require nor allow actual measurement values. Therefore, specify "No measurement data" in the "Measurement" box on bottom of the "Basic Properties" page. In this mode, all measurement values are assumed to be zero and all standard deviations are assumed to have the value one.

Furthermore, you can you can influence the applied simulation method. The default value is "Auto", which selects the optimal simulation method for the specified measurement model. Other possible choices are "Cumomer" (for all NMR and MS/MS measurements) and "EMU" (for MS measurements). A non-automatic selection of the simulation method makes sense combination with simulation type "Full".

1.3. Special Application Cases

1.3.1. Experimental Design

The 13CFLUX2 framework provides the tools edscanner and edopt for experimental design.

In combination with a specification of a mixture of different available labeling substrates the program edscanner samples the possibly high dimensional mixing simplex at evenly spaced points. At each sampling point a linearized statistical analysis is performed and a chosen single-numbered measure (design criterion) is calculated providing information on the high dimensional ellipsoid given by the fluxes' covariance matrix is computed. The coordinates of the sampling points including the information criterion are written to a HDF5 file. A further analysis of this data may include the visualization of mixing triangles.

Based on a 13CFLUX2 model with an optimized flux distribution the program edopt can be used to generate an optimal design for a subsequent isotope labeling experiment. For this purpose, edopt performs a linearized statistical analysis and determines the mixture of substrates which minimizes the criterion value for the flux distribution’s covariance ellipsoid, i.e. by using the A-criterion it allows the most accurate flux determination having the smallest sum of standard deviations (see technical documentation of 13CFLUX2 for futher information). In contrast to edscanner, edopt is not limited to choosing the equidistant points in mixture triangle, but tries to optimize the substrate mixture directly to obtain a minimal A-criterion. Labeling Mixture

You can create the labeling mixture file required by edscanner and edopt directly in Omix. Therefore, go to menu "Extras" -> "13CFLUX2 Modeling Options" -> "Create Mixture for Experimental Design" which opens the fillowing dialog. Here, you can define an input labeling mixture as known from the "Network Input" editor (see above).

In order to perform experimental design, make sure, that the network stoichiometry only contains one input pool. Specify a number of input labeling states for this input pool. Optimally, you specify three different labelings in order to use the mixing tiangle visualization.

Each of the labeling states should be represented by one item in the left list. Each state should have one labeling pattern entry in the right list. Do not insert the labeling mixture to be evaluated in experimental design under the same input state. Each labeling pattern has furthermore an optional "Cost" attribute. The actual costs of labeling patterns are considered in experimental design while finding an optimal labeling mixture.

Specifying labeling mixture for experimental design
Specifying labeling mixture for experimental design Error Model

Successful experimental design requires to specify an error model for extrapolating the standard deviations of the simulated measurement values. Therefore, select "Experimental design" in the "Measurement" box on bottom of the "Basic Properties" page. Now, the table for labeling measurements is equipped with another column "Error Model". Click on a cell in order to specify the error model. An error model is a formula composed of constant values, the predefined variables "meas_real" (provided measurement value), "std_real" (provided standard deviation), and "meas_sim" (simulated measurement value), standard mathematical operators ("+", "-", "*", "/", "yˆx") and the built-in functions "abs(x)", "exp(x)", "max(x,y)", "min(x,y)", "sqrt(x)", "log(x)", "log2(x)", "log10(x)", and "sqr(x)".

In case of a multi-value measurement group (e.g. multiple MS mass isotopomers) you can specify the error model in the row of the measurement group specification or in each value row. In the first case, the error model is used for all measurement values in the measurement group. In the second case, each measurement value requires an error model. That is, it is possible to specify different error models for individual measurement values.

In case the error model is completely omitted no extrapolation of the standard error is performed and the original provided standard error is used when computing the covariance matrix required for experimental design. The error model may be set to "1" for disabling the scaling of the covariance matrix completely. Weight of Free Fluxes

In "Experimental Design" mode, the variable tables of net, exchange fluxes is equipped with the additional column "Weight".

The attribute "Weight" takes a floating-point value between 0.0 and 1.0 (inclusive) and is used to adjust the influence of a free flux on the elements of the covariance matrix. Setting "Weight" to 0.0 elimates the influence of a free flux. Any value between 0.0 and 1.0 results in a scaling of the corresponding row and column of the covariance matrix. The default value of the attribute is 1.0.

1.3.2. Non-Stationary Isotope Labeling Experiments

The "Non-stationary" option on the "Basic Properties" page is for modeling isotopic non-stationary isotope labeling experiments. Here, labeling measurements can be specified for different time points. Furthermore, poolsizes have to be taken into account (pool measurements, pool variables).

Specifying different timesteps of the measurements
Specifying different timesteps of the measurements

Extended labeling measurement table including a time step column
Extended labeling measurement table including a time step column

The non-stationary model parts are experimental and subject to be changed or removed.

1.4. Further Reading

  • Wiechert W. (2001) 13C metabolic flux analysis. Metab Eng 3(3):195-206
  • Wiechert W, Möllney M, Isermann N, Wurzel M, de Graaf AA (1999) Bidirectional reaction steps in metabolic networks. Part III: Explicit solution and analysis of isotopomer labeling systems. Biotechnol Bioeng 66:69–85
  • Wiechert W, Wurzel M (2001) Metabolic isotopomer labeling systems. Part I: Global dynamic behaviour. Math Biosci 169:173–205
  • Weitzel, M., Wiechert, W., and Nöh, K. 2007. The topology of metabolic isotope labeling networks. BMC Bioinformatics, 8 (1), 315.
  • Nöh, K., Weitzel, M., and Wiechert, W. 2008. From Isotope Labeling Patterns to Metabolic Flux Rates. In U. H. E. Hansmann, J. H. Meinke, S. Mohanty, W. Nadler, and O. Zimmermann (Eds.) From Computational Biophysics to Systems Biology (CBSB08), vol. 40 of NIC Series, (pp. 345–348).