# A Topological Characterization of Medium-Dependent Essential Metabolic Reactions

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## Abstract

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## 1. Introduction

**Figure 1.**Network context of topological reaction categories. (

**a**) Simple scheme of a small fictitious metabolic reaction system with examples of UPUC and SA reactions. (

**b**) Wild-type network. (

**c**) Knockout of SA reaction R1. Fluxes are rerouted over R4 leading to an increase in the systems SA. (

**d**) Knockout of UPUC reaction R5. (

**e**) Knockout of reaction R4. R1 (SA) and R5 (UPUC) are now correct essentiality predictors. Edge thickness indicates flux magnitude.

Reactions | UPUC | SA | MC | No category | |
---|---|---|---|---|---|

Overall/Cytosol | 1284/707 | 296/193 | 238/230 | 231/197 | 820/375 |

Non-essential (Overall/Cytosol) | 789/402 | 137/57 | 27/25 | 25/16 | 608/312 |

Conditional lethal (Overall/Cytosol) | 326/162 | 90/74 | 73/73 | 85/79 | 198/55 |

Essential (Overall/Cytosol) | 169/143 | 69/62 | 138/132 | 121/102 | 14/8 |

## 2. Results

#### 2.1. Relative Essentiality Analysis

^{4}combinatorial minimal media conditions. Furthermore, all subsequent single reaction knockouts of active (non-zero flux carrying) reactions are performed to identify for each medium condition the set of essential reactions (see Methods for details). An illustrative example of this concept, involving E. coli central carbon metabolism [31], is provided in the supplementary materials.

**Figure 2.**Outcome of combinatorial minimal media simulations. (

**a**) Sorted relative essentiality profile determined by the simulation of reaction deletions under 72468 combinatorial minimal media conditions. The three different essentiality classes are indicated by dashed lines. Inset (

**b**) shows the same profile using semilog plot. (

**c**) No relation is observed between essentiality and activity (non-essential reactions have been removed in inset (

**d**); relative activity is the number of simulations that a reaction was active normalized by the total number of simulations). (

**e**) A reaction-centric network diagram illustrating the relative essentiality on the nodes (707 reactions) and co-occurring activity on the edges of the cytoplasmic part of the iAF1260 model (transport, periplasmic, and blocked reactions have been discarded; currency metabolite have been removed manually, see Methods).

#### 2.2. Topological Categories as Markers of Essential Reactions

**Figure 3.**(

**a**) Bipartite network representation of the two step conversion of L-glutamate (glu-l) into L-glutamate-1-semialdehyde (glu1sa) involving glutamyl-tRNAsynthetase (GLUTRS; 6.1.1.17) and glutamyl-tRNA reductase (GLUTRR; 1.2.1.-). Additionally, two reaction sources for L-glutamate, Δ

^{1}-pyrroline-5-carboxylate dehydrogenase (P5CD; 1.5.1.12) and the reverse direction of glutamate dehydrogenase (GLUDy_Rev; 1.4.1.4), and one sink for L-glutamate-1-semialdehyde, glutamate-1-semialdehyde aminotransferase (G1SAT; 5.4.3.8), are shown, among other reactions consuming L-glutamate. Node colors indicate relative essentiality (legend provided in Figure 2). (

**b**) A reaction centric projection of the bipartite network in (

**a**). Reaction categories (UPUC, SA, and MC) are shown for each reaction node and gray boxes indicate the occurrence of one subgraph of type id14 and two subgraphs of type id74 (see

**c**). (

**c**) All (13) combinatorial three-node subgraphs and corresponding identifiers (id110 and id238 have not been encountered in any effective network).

**Figure 4.**Reaction categories and essentiality classes. The proportions of the three different essentiality classes determined for UPUC, SA and MC component (for absolute numbers see Supplementary Figure S7).

^{c}∩MC

^{c}where X

^{c}denotes the absolute complement. On the other hand, the majority of combinatorial sets is significantly enriched for essential reactions (Figure 5c). More importantly, more than half of the combinatorial sets exhibit a clear separation of essential from conditional lethal and non-essential classes. Comparing Figure 5a and Figure 5c reveals that the sequence of combinatorial sets in the sorted non-essential enrichment resembles the essential sequence in reverse order (e.g., the exclusive UPUC set being visually absent for high essential reaction enrichment). This observation provides evidence for a strong negative association between these two essentiality classes in the context of the UPUC, SA and MC categories.

**Figure 5.**Enrichment of combinatorial reaction category sets for essentiality classes. Combinatorial sets sorted on the basis of (

**a**) non-essential, (

**b**) conditional lethal and (

**c**) essential class enrichment. Venn diagrams [34] on the abscissa indicate each of the 127 nonempty unions and intersections of UPUC (upper circle), SA (lower left circle) and MC (lower right circle), respectively (no set includes less than 13 reactions, the largest combinatorial set includes 332 reactions). For the computation of Z-scores, random distribution of essentiality classes among the Venn intersections and unions was used as the underlying null hypothesis.

#### 2.3. Distribution of Essentiality Classes Across Three-Node Subgraphs

**Figure 6.**Enrichment on three-node subgraphs. The statistical over- and under-representation of (

**a**) reaction categories and (

**b**) essentiality classes on all occurring three-node subgraphs (two motifs have been omitted as they were not detected in any effective network). For the computation of Z-scores, random distribution of reaction categories and essentiality classes among subgraphs was used as underlying null hypothesis.

## 3. Methods

#### 3.1. Metabolic Model and Network Representations

#### 3.2. Flux-Balance Analysis

**c**, the stoichiometric matrix

**S**, the flux vector

**v**and the constraint vectors

**v**and

_{min}**v**. As we are considering reversible reactions as two independent unidirectional reactions, we set

_{max}**v**to zero. Problems like Equation 1 can be efficiently solved using linear programming. In order to avoid thermodynamically infeasible loops, we utilized pFBA [39], effectively using the solution of Equation 1, to fix the objective to its maximum value and minimize the L1-norm of all other fluxes in a second optimization.

_{min}#### 3.3. Combinatorial Minimal Media and Reaction Essentiality

#### 3.4. Blocked Reactions

_{max}) maximal uptake and secretion rate was assigned to all available transporters in the system and then blocked reactions were confirmed by flux variability analysis [32]. These globally blocked reactions cannot carry a flux under any environmental conditions and consequently are not available to methods that use FBA.

#### 3.5. Metabolic Core

#### 3.6. Synthetic Accessibility Reactions

#### 3.7. UPUC Reactions

#### 3.8. Enumeration of Three-Node Subgraphs

## 4. Conclusions

## Acknowledgments

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## Supplementary Files

**Supplementary File 1:**

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Sonnenschein, N.; Marr, C.; Hütt, M.-T.
A Topological Characterization of Medium-Dependent Essential Metabolic Reactions. *Metabolites* **2012**, *2*, 632-647.
https://doi.org/10.3390/metabo2030632

**AMA Style**

Sonnenschein N, Marr C, Hütt M-T.
A Topological Characterization of Medium-Dependent Essential Metabolic Reactions. *Metabolites*. 2012; 2(3):632-647.
https://doi.org/10.3390/metabo2030632

**Chicago/Turabian Style**

Sonnenschein, Nikolaus, Carsten Marr, and Marc-Thorsten Hütt.
2012. "A Topological Characterization of Medium-Dependent Essential Metabolic Reactions" *Metabolites* 2, no. 3: 632-647.
https://doi.org/10.3390/metabo2030632