Systems Biology and Mathematical Modeling Group, Max-Planck-Institute of Molecular Plant Physiology, 14476 Potsdam, Germany

ETH Zurich, Institute of Molecular Systems Biology, 8093 Zurich, Switzerland

Abstract

Background

Changes in environmental conditions require temporal effectuation of different metabolic pathways in order to maintain the organisms’ viability but also to enable the settling into newly arising conditions. While analyses of robustness in biological systems have resulted in the characterization of reactions that facilitate homeostasis, temporal adaptation-related processes and the role of cellular pathways in the metabolic response to changing conditions remain elusive.

Results

Here we develop a flux-based approach that allows the integration of time-resolved transcriptomics data with genome-scale metabolic networks. Our framework uses bilevel optimization to extract temporal minimal operating networks from a given large-scale metabolic model. The minimality of the extracted networks enables the computation of elementary flux modes for each time point, which are in turn used to characterize the transitional behavior of the network as well as of individual reactions. Application of the approach to the metabolic network of

Conclusions

Altogether, our findings pinpoint the inherent relation between the systemic properties of robustness and adaptability arising from the interplay of metabolic network structure and changing environment.

Background

The steady-state metabolism of microorganisms has evolved to optimize growth under ambient conditions

While robustness has been widely studied

Several approaches that integrate data with graph-theoretic methods have been applied to obtain subnetworks engaged under different conditions. For instance,

With the increasing availability and quality of genome-scale metabolic models and high-throughput data, constraint-based methods that integrate these data have found broad applications. For instance, a genome-scale metabolic model has been coupled with transcriptomics data, based on Boolean logic, to improve flux predictions

While constraint-based methods usually provide solutions that optimize a certain objective, elementary flux modes (EFMs) capture the whole spectrum of metabolic steady states of a given network. An EFM is defined as a minimal set of reactions that can operate at steady state

Flux-based,

Here we present a novel method, which we term Adaptation of Metabolism (AdaM), to identify reactions and pathways that enable system adaptation upon external perturbation. AdaM integrates time-series transcriptomics data with flux-based bilevel optimization to extract minimal operating networks from a given large-scale metabolic model. The minimality of the extracted networks enables the computation of EFMs for each time point. These sets of EFMs are in turn used to characterize the transitional behavior of the network as well as of individual reactions (see Figure

Schematic depiction of the computational approach

**Schematic depiction of the computational approach.** A genome-scale network and time-series transcriptomics data are used to extract time- and condition-specific minimal networks. Data for different environmental conditions are analyzed to weight genes based on differential expression and bimodal distribution analyses. The gene-reaction annotation of the network reconstruction is used to map the weights onto the metabolic model. A minimization approach is applied to extract minimal networks. EFM analysis is conducted on the minimal networks, and the resulting sets of EFMs and the derived fractional appearance profiles are employed to characterize the transitional behavior of the network and of individual reactions, respectively.

Methods

Weighting of reactions

Transcriptomics data are used to weight reactions that are catalyzed by the enzymes encoded in the respective genes. To determine pathways that contribute to the metabolic state of the organism, we consider reactions, that are not only temporally activated upon the changed environmental condition, but also reactions that are constantly active. Therefore, we combine information obtained from the analysis of differential expression as well as the gene expression levels themselves. The significance of differential behavior, is captured by the

where

**Supplementary Information.**

Click here for file

We determine weights for transcriptomics data from the cold and heat stress and the control conditions spanning seven time points (0 - 90 min) and map them onto the genome-scale metabolic network reconstruction of

Data-driven network reduction—the min-max problem

In the following, we develop a formulation of the problem whose solution yields the minimal network of largest weight, quantifying the compliance with the data. More formally, we determine the minimal number of reactions that maximize
_{
j
} and _{
j
} are the weight and the flux of reaction _{
j
}∈ {0,1}. If _{
j
}= 0, reaction _{
j
}= 1, the reaction carries flux in the range determined by the flux boundaries (Equation (6)). The outer optimization level seeks to minimize the number of reactions in the network (Equation (2)), while fulfilling the inner constraints. The inner problem definition is a modification of the standard flux balance analysis (FBA)

A further constraint on the fluxes is imposed by demanding that a fraction _{min} of the maximum biomass production _{max }of the complete network is achieved (Equation (5) and

To reduce the computational complexity, we seek to reduce the number of integer variables. To this end, we distinguish between

where _{
ij
} is the stoichiometric coefficient of metabolite _{
j
}is the contribution of _{
j
} to the objective function and
_{
j
}, while

Although we investigate time-series data, the program formulation employs a the quasi-steady-state assumption (Equation (4)). We assume a separation of the time-constants at which transcriptional and metabolic regulations take place. This is justified by the evidence that changes taking place on the metabolic level are generally much faster (seconds) compared to those taking place on the transcriptional level (minutes)

To solve the min-max MILP, it is transformed from a bi-level to a single-level MILP. This procedure employs two steps: (1) finding the dual for the inner linear program

Fractional appearance of reactions in EFMs

The reduced size of the networks allows the computation of sets of EFMs for the time- and condition specific minimal networks. It has already been shown that the importance of a reaction for network functionality can be characterized by the number of EFMs in which it is involved
_{
ij
} of a reaction

Results

Time- and condition-specific minimal operating networks

For both, cold and heat stress, the minimal networks include 416 to 427 metabolites interconnected by 480 to 486 reactions. The biomass production, as a result of the constraint from the minimization approach, ranges between 0^{−5} and

Comparison to networks extracted from MADE

We compare the sets of reactions included in the time- and condition-specific networks extracted by both AdaM and MADE. We find that on average 66.6% of reactions are shared between the extracted networks over all time points and conditions, with larger average overlap for the heat condition of 67.2% (percentages with respect to the smaller network,

**Metabolites**

**Reactions**

**EFMs**

Given are the number of metabolites and reactions for the networks extracted by our approach and by MADE compared to the original network.

AdaM

416-427

480-486

1060 - 9582

MADE

545

658-806

†

original network

761

1075

†

**Time in min**

**Cold**

**Heat**

**Random**

Given is the overlap in % of reactions included in the networks extracted by our approach and by MADE with respect to the smaller network. The average value for all time-dependent cold- and heat-shock specific networks are significantly higher than the values for a randomly drawn set of reactions of the same size.

10

62.8

62.6

20

66.7

68.5

30

67.8

69.8

40

63.8

68.3

50

62.0

67.9

60

69.0

63.2

70

70.6

70.2

average

66.1

67.2

52.3

EFM-based characterization of adaptability

The number of EFMs in the minimal networks ranges between 1060 and 9582, which is small compared to other metabolic network models of similar size

Robustness of the optimization approach

We investigate the robustness of the approach with respect to: (1) threshold variations for the optimization approach and (2) slightly suboptimal networks. To address the first aspect, we repeat the network minimization for three different thresholds to distinguish between dispensable and indispensable reactions. In addition, we examine three different values of minimal biomass production (

To test the behavior of slightly suboptimal networks, we add noise to the weights of 100 randomly selected reactions. As a maximum noise level, we set 1% of the total range of weights. We repeat the analysis for different time points and threshold values, resulting in more than 1000 network perturbations. Comparing the resulting sets of EFMs, we find an average overlap of 82.0% over pairs of EFM sets. Therefore, the considered network perturbations together with variations in the used threshold values confirm the robustness of the extracted networks with respect to EFMs. Furthermore, this suggests that EFMs can be used to develop time-resolved descriptors of reactions’ contribution to network functionality.

Cold and heat stress response show distinct temporal behavior

To investigate the global properties of the transition, we determine the similarity for consecutive sets of EFMs and sets of dispensable reactions by using the Jaccard index. Changes in the usage of EFMs as well as dispensable reactions over time suggest adaptation-relation processes. The results are illustrated in Figure

Global characterization of temporal adaptation-related networks

**Global characterization of temporal adaptation-related networks.** Transitional behavior of the metabolic network after heat and cold stress perturbation. Heatmap of the Jaccard index from **(A)** sets of EFMs and **(B)** sets of dispensable reactions included in the minimal networks from consecutive time-points. A low Jaccard index denotes dissimilarity.

The values for the Jaccard index for consecutive sets of EFMs for the minimal cold stress networks are slightly lower (0.009-0.252) compared to those from the heat stress networks (0.012-0.271). These changes in the usage of EFMs, resulting from data-driven network extraction, can be regarded as changes in the distribution of fluxes through the networks. Moreover, the values for the Jaccard index for dispensable reactions remaining in the minimal networks are similar for cold (0.697-0.759) and heat stress (0.618-0.771), suggesting similar changes in activation patterns of reactions.

Considering the patterns of change, for cold shock we observe the first strong dissimilarity between sets of EFMs between 0 and 20 min after stress application, indicating that the main metabolic response takes place in this time range. For heat shock, the onset of the response is slightly shifted in time. These findings suggest that the stress response for the two conditions takes place on two slightly different time regimes. Such tendency can also be observed when comparing the time course of the similarities for the sets of dispensable reactions. In addition, for both conditions, the peak in dissimilarity between sets of EFMs occurs between 30 and 50 minutes. Between 50 and 90 minutes, both condition-specific networks exhibit similarity with respect to their sets of EFMs and sets of dispensable reactions, demonstrating that the system has started settling in the new condition.

Fractional appearance of reactions in EFMs discriminates two types of reactions

For ease of interpretation, we focus our analysis on the 50 reactions of highest fractional appearance for each time-point. The union of these selected reactions over all time points contains 71 and 76 reactions for the cold and heat shock, respectively. Out of these, 43 are conserved between the two stresses.

To gain general insights into the patterns of the fractional appearance profiles of these selected reactions, we determined the Kendall correlation

The working hypothesis is twofold: Reactions that are grouped together are expected to belong to the same pathways, or are regulated in a similar manner (

To further investigate this hypothesis we cluster the fractional appearance profiles of the previously selected reactions (

Clustering of time-resolved fractional appearance profiles

**Clustering of time-resolved fractional appearance profiles.** Shown are the fractional appearance profiles of reactions over time. Under both **(A)** cold and **(B)** heat shock, the selected reactions group into 9 clusters. Enzyme names discussed in the text are given. A complete list of all enzyme names is given in the Additional file

Flat patterns represent indispensable metabolic reactions

We first focus our analysis on reactions whose occurrence in EFMs does not change as a result of perturbation,

In total, these clusters consist of 36 and 47 reactions for cold and heat stress, respectively, of which 19 of these reactions are conserved between the two stresses. These reactions appear in 5 to 40% of all EFMs, which indicates a major role for network functionality. To gain a general overview of the higher biological processes of this set of reactions, we perform GO term over-representation analysis on their annotated genes (Additional file

Closer inspection reveals a group of 3 constantly used reactions - aconitase, trehalose transport via PEP:Pyr (PTS) and enolase, which belong to the central carbon metabolism. The constant utilization of glucose uptake system (PTS) is not surprising, since it is crucial for culture grown on this nutrient, as well as for the formation of G6P and flux from PEP to pyruvate. Furthermore, this reaction has also been shown to be constitutively active under different nutritional perturbations in

The second group contains reactions involved in amino acid metabolism, a process crucial for protein synthesis. Those reactions include acetylornithine transaminase, adenylsuccinate lyase, aspartate kinase, and L-serine transport. Another group contains reactions involved in nucleotides biosynthesis and degradation, processes essential for transcription and replication, including: purine-nucleoside phosphorylase (guanosine), pyrimidine-nucleoside phosphorylase (uracil), and thymidine phosphorylase, as well as thymidine kinase (ATP:thymidine). This group extends to three reactions involved in amino- and nucleotide-sugar metabolism, namely, UDP-N-acetylglucosamine 2-epimerase, UTP-glucose-1-phosphate uridylyltransferase, and UDP-glucose-hexose-1-phosphate uridylyltransferase.

Finally, we observe constant utilization of glutathione reductase and trimethylamine N-oxide reductase. The first can be understood by the fact that the ratio of reduced to oxidized glutathione in

The results of the functional enrichment analysis and biological interpretation of the metabolic role of indispensable reactions, showing flat profiles of fractional appearance, supports our assumption that these reactions constitute the most crucial part of the metabolic network.

Fluctuating patterns capture condition-specific temporal response

Next we investigate reactions whose temporal appearance in EFMs changes as a result of the applied stress. The number of reactions showing such behavior is smaller compared to that of reactions which are constantly used. There are 35 reactions for cold stress and 29 for heat stress.

Over-representation analysis of biological processes reveals that under cold stress cluster 2 is enriched for catabolic processes, in particular, of amino acids, organic acids, and coenzymes, followed by acetyl-CoA biosynthetic process from pyruvate, and glycolysis. The reactions in this cluster are excluded from the networks at 10 min and peak with respect to their fractional appearances at 30 min. A prominent representative in this group of reactions is the glycine cleavage system, which has been found to be slightly affected by cold stress

Cluster 3 is enriched for biosynthesis of the aspartate family of amino acids (

The coupling between the pentose phosphate pathway and catabolic processes is also apparent in the enrichment of GO terms in cluster 4. Here, the two considered reactions are only present in the extracted networks for the last time points. One of these reactions, phosphoglycerate mutase, takes part in glycolysis, which together with glucose consumption is reduced under low temperatures, especially in the early time points after stress

Under heat stress, cluster 2 consists of profiles where the reactions are initially used, then excluded from the network, and finally reintroduced. Over-representation analysis demonstrates that catabolic processes involving amino acids, glyoxylate and coenzymes are enriched. Interestingly, the reactions in this cluster are also grouped together in cluster 2 under cold stress. However, it appears that after initial usage of the glycine cleavage system under heat stress, it is transiently shut down in a manner opposite of that under cold stress.

Cluster 5 includes ammonium exchange which is down-regulated after application of heat stress. This is in line with the catabolic processes observed in cluster 2, suggesting that protein synthesis is present to support maintenance of cell vitality without the need to sustain growth. In addition, cluster 3 under heat stress has a high overlap with cluster 3 under cold stress. However, the patterns of fractional appearance, as already observed for cluster 2, show a different temporal behavior. We therefore suggest the hypothesis that although same biological processes are involved in adaptation to temperature stresses, the temporal usage in terms of (in)activation may slightly differ. The activation pattern of these processes may further amplify the effect of genes specific to cold/heat stress.

Discussion

Here we proposed a novel approach to investigate adaptation of metabolism upon external perturbation. Based on experimental data we determine time- and condition-specific minimal networks for which sets of EFMs can be calculated. These sets are used to determine the fractional appearance profiles of reactions. This integrative profile combines information from transcriptomics data, the underlying network structure, and biologically meaningful flux distributions in a quasi steady-state; thus it includes information which transcriptomics data would never be able to reveal on their own.

The fractional appearance of reactions has already been investigated with respect to the concept of robustness

It must be noted that our proposed approach extracts network for individual time points, without accounting for their dependency in the time domain. However, since the weighting of the reactions is conducted by using data which already embed the temporal dependency, this also extends to the extracted networks.

Since transcriptomics data do not necessarily reflect enzyme activities (due to post-transcriptional modification and regulatory effects), we use the results from the analysis of the expression data only as indicator for the activity of the respective reaction rather that definite values. Furthermore, the approach does not rely on a condition-specific objective function,

Conclusion

We applied our approach to time-resolved transcriptomics data from heat and cold shock experiments in

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

NT and ZN designed the study and wrote the manuscript. NT performed the study. SJ, NT and ZN interpreted the results. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the Max-Planck society for financial support.