The authors have declared that no competing interests exist.

Conceived and designed the experiments: NT TS. Performed the experiments: NT TS. Analyzed the data: NT TS. Wrote the paper: NT TS.

Metabolic flux analysis (MFA) is a widely used method for quantifying intracellular metabolic fluxes. It works by feeding cells with isotopic labeled nutrients, measuring metabolite isotopic labeling, and computationally interpreting the measured labeling data to estimate flux. Tandem mass-spectrometry (MS/MS) has been shown to be useful for MFA, providing positional isotopic labeling data. Specifically, MS/MS enables the measurement of a metabolite tandem mass-isotopomer distribution, representing the abundance in which certain parent and product fragments of a metabolite have different number of labeled atoms. However, a major limitation in using MFA with MS/MS data is the lack of a computationally efficient method for simulating such isotopic labeling data. Here, we describe the tandemer approach for efficiently computing metabolite tandem mass-isotopomer distributions in a metabolic network, given an estimation of metabolic fluxes. This approach can be used by MFA to find optimal metabolic fluxes, whose induced metabolite labeling patterns match tandem mass-isotopomer distributions measured by MS/MS. The tandemer approach is applied to simulate MS/MS data in a small-scale metabolic network model of mammalian methionine metabolism and in a large-scale metabolic network model of

Metabolic flux analysis (MFA) is a method for quantifying in vivo metabolic fluxes that is commonly used to address problems in biotechnology and medicine [^{13}C labelled substrates), measuring metabolite isotopic labeling, and applying computational methods to estimate fluxes [

MFA is based on the key observation that metabolite isotopic labeling patterns are uniquely determined by the distribtuion of metabolic flux in the network [

A distinct labeling pattern of a certain metabolite is called an ^{n} distinct isotopomers; for example, as shown in

Isotopomers | Tandemers |
---|---|

0000 | [ |

0001 | [ |

0010 | [ |

0011 | [ |

0100 | [ |

0101 | [ |

0110 | [ |

0111 | [ |

1000 | [ |

1001 | [ |

1010 | [ |

1011 | [ |

1100 | [ |

1101 | [ |

1110 | [ |

1111 | [ |

Isotopomers are represented by sequences of zeroes and ones, denoting non-labeled and labeled atoms, respectively.

Measuring the complete isotopomer distribution of metabolites is technically infeasible. Instead, mass-spectrometry is typically used to measure the relative abundance of a given metabolite having different number of labeled atoms (i.e. zero labeled atoms, one, two, etc). A set of isotopomers of a certain metabolite having the same mass is referred to as

Information on the positional labeling of metabolites can be obtained by tandem mass-spectrometry (i.e. MS/MS) and was previously shown to significantly improve quantification of metabolic fluxes via MFA [

Currently, there is no method for efficiently simulating tandemer distributions. Previous applications of MFA given MS/MS data have inefficiently computed the complete isotopomer distributions for all metabolites in the network (for example, via cumomers [

We denote a

(a) Metabolite A and its MFP

The entire tandemer distribution of A, with respect to the MFP

Under isotopic steady-state, for metabolite B that is produced solely through one biochemical reaction with a single substrate, A, the tandemer distribution matrix

For metabolite C that is produced solely through one reaction with two substrates, A and B, _{1} and N_{1} are mapped to atoms in M and N, respectively) and _{2} and N_{2} are similarly mapped to M and N;

For example, let us consider the bi-substrate reaction shown in

Under isotopic steady-state, a tandemer distribution matrix for the MFP _{i} is the flux through reaction

Our goal is to efficiently simulate the tandemer distribution for a pre-defined set of metabolites (for which corresponding experimental data might be available). We assume that a metabolic network model with reaction atom-mappings (describing the mapping of substrate to product metabolite atoms in each reaction) and candidate fluxes are given. To address this problem, we present the tandemers approach, whose outline is shown in

First, given MS/MS measurements and a metabolic model, a minimal set of MFPs is identified constructing an MFP graph. Second, MFPs are clustered and sorted and third, isotopic balance equations are formulated for each MFP cluster. Given a candidate flux vector, tandemer distributions are calculated by solving the set of isotopic balance equations.

The identification of a minimal set of MFP's in the metabolic network whose tandemer distributions would enable simulating the tandemer distribution of a given set of metabolites is done using a recursive procedure, in a similar manner to that presented in the EMU approach [

The total number of MFPs depends on many factors, including the structure of the metabolic network, number of atoms per metabolite, reaction atom mappings, and number of metabolites for which experimental MS/MS data is available as input. In theory, for each metabolite with n carbons, the worst case number of possible MFPs could go up to:
^{n}). In practice, as we show below, applying this method on various metabolic networks, the number of MFPs found per metabolite is markedly lower than the number of isotopomers, resulting in improved running time compared to existing methods.

In a uni-substrate reaction producing MFP

Following a method proposed by [

Tandemer distributions for MFPs in a cluster are linearly dependent on each other, given the corresponding tandemer distributions for MFPs in previous clusters. Notably, the set of balance equations for tandemer distribution matrices for MFPs in the i’th cluster can be formulated as following:
_{i} is a matrix whose rows represent the tandemer distributions for MFPs in the i’th cluster (i.e. each tandemer distribution represented as a row vector; removing infeasible tandemers), _{i} is a matrix whose rows represent tandemer distributions and Cauchy product of tandemer distributions computed for previous clusters, and _{i} and _{i} consist of corresponding fluxes.

Solving a set of balance equations for MFPs in the i’th cluster requires calculating the inverse of _{i}, whose number of rows (and columns) is equal to the number of MFPs in the cluster. As the cumomers approach also involves grouping cumomers in clusters and inverting flux matrices (similar to _{i}, here) whose size depends on the number of cumomers in each cluster, the running time of cumomers and tandemers approaches can be compared in terms of cumomers and MFPs cluster sizes, and the time needed to invert the corresponding flux matrices. Theoretically, an ^{3} improvement in running time. Notably, the number of non-feasible tandemers in each tandemer distribution matrix (represented by the number of column of _{i} has a negligible effect on the time require to solve ^{3} time required to calculate the inverse of _{i}).

In this section we describe the application of the tandemers approach on a toy metabolic network shown in

(a) A toy metabolic network, where the labeling pattern of A that is supplied in the media is assumed to be known, and the tandemer distribution of

For example, the isotopic balance equation for tandemer distribution matrices in cluster (I) is formulated as following, according to

To demonstrate the applicability of the tandemers method for efficiently computing experimental MS/MS data in ^{13}C labeling experiments, we applied it on a simplified metabolic network model of mammalian cellular metabolism of methionine (_{2} prior to the propylamine transfer). Considering that the number of isotopomers of a metabolite with n carbons is 2^{n}, explicitly modeling the entire isotopomer distribution of all five metabolites in this network (as done in the cumomers method) would require 52,306 variables.

Metabolites abbreviations: SAM: S-Adenosylmethionine; SAH: S-Adenosylhomocysteine; HCyc: L-Homocysteine; MTA: Methylthioadenosine.

To apply the tandemers method, we utilized experimentally determined fluxes in this network as input [

Next, we ran both the tandemers and cumomers methods multiple times, choosing a different subset of metabolites to calculate their tandemer distribution in each run. The parent fragment was assumed to be the intact metabolite and the product fragment was chosen to be the ribose, adenine, propylamine, or the four methionine carbons other than the methyl group. We find an average number of only 33 MFPs per run of the tandemers method, while the cumomers method requiring of 52,306 cumomer variables regardless of assumed input. The average running time of the tandemers method was found to be 0.00046 seconds, while that of the cumomers method was 0.68 seconds, ~1500-fold higher (

Mammalian methionine metabolism model | ||||||
---|---|---|---|---|---|---|

Variable count | Maximal cluster size | Running time | Variable count | Maximal cluster size | Running time | |

Cumomers | 52,306 | 10,197 | 0.68 | 19,404 | 4,016 | 3.3 |

Tandemers | 33 | 4 | 0.00046 | 695 | 32 | 0.01 |

To further demonstrate the applicability of the tandemers approach, we applied it on a large-scale metabolic network model of

We applied the tandemers method 1000 times to compute the tandemer distribution of randomly chosen sets of metabolites (having between 1 to 20 metabolites). The average number of resulting MFPs was 695, with a maximal MFP cluster size of 32. In comparison, applying the cumomers approach resulted in 19,404 cumomers, and a maximal cluster size of 4,016. The average running time of the tandemers and cumomers methods are 0.01 and 3.3 seconds, respectively, representing a ~300-fold improvement by the tandemers method (

For the tandemers methods, the number of variables represents the number of MFP whose tandemer distribution is calculated; for the cumomers approach, it represents the number of cumomers. The number of these variables corresponds to the size of the flux matrices whose inverse is calculated by each method, and is hence proportional to overall running time (see Section 2.3.3). We report the average variable count, clustersize, and running time for the cumomers and tandemers methods in multiple simulations given different sets of metabolites and collisional fragments, as described above. Notably, considering that MFA applications and especially experimental design of isotope tracing experiments require thousands of repeated simulations of metabolite isotopic labeling, the ~1500-fold and 300-fold improvement in running time observed in the mammalian methionine network and on the

Tandem MS holds great promise for metabolic flux analysis as it provides information on metabolite positional labeling [

In our application of the tandemers method on a metabolic network of mammalian methionine metabolism and for

In a recent study, we described a method, Metabolic Flux Analysis/Unknown Fragments (MFA/UF), capable of using MS/MS data to improve flux inference even when the positional origin of fragments is unknown [

Considering that a major current complication in utilizing MS/MS data in metabolic flux analysis involves the lack of computationally efficient methods for simulating such experimental measurements, we expect the tandemers approach to promote broader usage of this technology.

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