# Inspecting the Solution Space of Genome-Scale Metabolic Models

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Effect of Constraints on the Solution Space in the Network

#### 2.2. Inspecting the Solution Space Using Random Perturbations

^{−6}are considered to be “variable”. All other reaction fluxes are seen as “stable”. It is also interesting to note that the majority of reactions in all models, except for wt + med + met + pro, had an interval size larger than 10

^{−3}. For each reaction showing a flux interval determined by FVA, the respective flux value was fixed to 10 different randomly selected values within the given FVA interval. We will refer to this as a “perturbation”. Then, for each of the 10 randomly fixed flux values, FVA and FBA were recalculated and the effect on the flux distribution in terms of taking an alternative optimal distribution was investigated. In addition, we were interested in the extent to which reactions show a variable flux and therefore, we will refer to reactions that change their flux value in response to a perturbation by at least 5 percent as “sensitive”, and reactions that change their flux by less than that and therefore always carry approximately the same flux value irrespective of any perturbation as “robust”. It has to be pointed out that the value of ±5% resulted in an acceptable distinction between different characteristics among models within our set-up, but this can be adjusted to a higher/lower value if necessary. Please note that the concept of “stable” and “robust” are not the same, as “stable” refers to reactions for which FVA denotes no flux variability, while “robust” refers to variable reactions with a very small flux interval. We found the width of the permissible flux to be an important indicator of the size of the solution space.

^{−3}showed a similar correlation.

#### 2.3. Investigating Biological Phenotypes in FBA Results

#### 2.4. The Influence of Specific Quantitative Constraints on the Solution Space

#### 2.5. Analysing the Solution Space Using CoPE-FBA

#### 2.6. The Influence of ATP Maintenance on the Solution Space

#### 2.7. Validating the Results Using Models of Other Species

## 3. Discussion

- To decrease the number of biologically inconsistent results, it is vital to integrate biological constraints. In our analysis, generally the integration of proteome data was the most effective in reducing the solution space. However, metabolic flux data on exchange reactions (metabolic uptake and production rates) can also already significantly reduce the solution space (decrease in sensitivity), e.g., at branching points. The fact that the degree of reduction is really case-specific underlines the second point [29].
- The analysis of the solution space should be taken into account in any study using FBA. As the above cases demonstrate, constraints like metabolite exchange rates, which are arguably one of the more commonly used constraints, can effectively reduce the solution space such that biologically relevant results for certain questions (e.g., specific kind of fermentation) are achieved, but this is not the case for every model/data-set. There are different ways to investigate the solution space. In our study, sampling by perturbation was an easy and informative way to investigate the different optimal flux distributions. We suggest that the functional analysis of the solution space using our perturbation method gives an explicit account for the robustness as well as reliability of genome-scale models. This also enables us to understand which data sets and which biological phenotypes can effectively shrink the solution space and increase the predictability of models.However, there are alternative methods for the analysis of the solution space not investigated here, e.g., Monte-Carlo sampling. This approach is mostly used to calculate the probability distribution of individual fluxes as well as to determine correlated reaction sets which can be further used for experimental design [12,14]. While this aims at the probability distribution of individual fluxes, our method is focused on uncovering the uncertainty in the interplay between different metabolic fluxes. As mentioned above, Monte-Carlo sampling enables the calculation of correlated reaction sets, which can be used to select candidate reactions for flux measurements, helping to estimate the flux value of its correlated ones. Nevertheless, our method showed, while correlated, the integration of metabolic fluxes of fermentation products for which the internal reactions were reported to be correlated (e.g., LDH, PFK [12]), does not necessarily result in eliminating the physiologically inconsistent result (see analysis of the pyruvate branching point in the case of the E. faecalis mutant). Therefore, the analysis of the solution space using the perturbation procedure helped to yield more information regarding the behaviour of the network as a whole. Another difference between the method presented in this article and Monte-Carlo sampling is a far smaller sample size needed to capture the network response to different metabolic states. While Monte-Carlo sampling needs a large sample size to reveal a comprehensive overview (250,000 data points in [12]), our method uncovers different aspects of the solution space using a far smaller sample size (~10 times the number of variable reactions). However, this also implies that our method does a less complete sampling of the solution space and certain alternative solutions might be overlooked. Moreover, although it is hard to compare the computational performance since our method has a different purpose, we would like to state that our method is fast compared to the Monte-Carlo sampling methods with respect to computational time. The comparison of different Monte-Carlo sampling methods reported that the sampling time spans from 7.64 to 10.67 min, for models of comparable size to our models (especifically the model of E. faecalis) using the CHRR method (the most efficient Monte-Carlo method available right now ) on an intel Core i7 at 2.5 GHz as reported by Fallahi and colleagues [15]. In this study, a reduced version of the metabolic models was used, meaning that the reactions carrying no flux were discarded. Therefore, the number of reactions in the case of the four models, iLJ478, iSB619, iHN637 and iJN746 were reduced from 652, 743, 785 and 1054 to 380, 450, 522 and 652, respectively [15]. The perturbation process of our method took between 122 to 175 seconds depending on the model (wildtype or mutant) and how constrained a model was, in the case of the E. faecalis model on an Intel Core i5 2.3 GHz, 16 MB memory and HDD hard drive, when the flux distribution profiles were obtained using FBA (on MATLAB). Our method also allows the acquisition of flux distributions using FVA, which takes more time—in this case between 31 to 51 min for the same models on the same hardware setup. Comprehensive information regarding the run time of different models used in the above-mentioned study using CHRR and the perturbation process in this study, as well as the number of metabolites and reactions of each model used can be found in the Supplementary Tables S2a and S2b (Supplementary file 2, sheet: run time comparison). Of course; any additional statistical analysis takes further time.

- 3.
- Caution has to be taken if outcomes of FBA are close to the edge of the feasible solution space w.r.t. some parameter, e.g., ATP maintenance. This is at least true when applying methods that are based on FVA, as shown in Section 2.6, since FVA often fails under these conditions and a solution space smaller than the actual space is reported.

## 4. Materials and Methods

#### 4.1. Models, Experimental Data and Constraints Integration

#### 4.2. Perturbation Procedure

_{max}and v

_{min}are the vectors of maximum and minimum allowable flux values, respectively, for each reaction. Using this characteristic of constraint-based models, the perturbation procedure we proposed is based on the idea that a change in the flux value of a reaction would result in a different combination of fluxes in the network, as shown in the Figure 10, in the example of two flux combinations:

^{−6}was applied to consider a reaction as variable. Next, for each variable reaction, 10 random values within the determined permissible interval were selected using the ‘rand’ function in MATLAB. The rand function yields a single uniformly distributed number within the given interval. The respective reaction was fixed at the given random value (lower and upper bounds having the same value) and the flux distribution profile was recalculated each time using FVA. For validation purposes, the analysis was repeated and the flux distribution was obtained with FBA (using CPLEX and GLPK 4.65 [36]) in COBRA. The analysis was also repeated using PySCeS-CBMPy 0.8.0 [22] on Python 2.7 [37] and the flux distribution profiles were again obtained using FVA (using CPLEX as solver). The obtained flux distribution for each reaction was then compared to the original flux distribution and a flux value was considered to change significantly, if it was altered beyond ±5% of the original flux value. In the cases where the original flux value was zero, the threshold was set to 10

^{−6}.

#### 4.3. Analysing the Solution Space Using CoPE-FBA

^{−6}as a threshold. Afterwards, flux modules, which are the sets of variable reactions that are linearly independent, were determined. The modules were then used to analyse the solution space.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The fraction of variable reactions in each model, separated into sensitive and robust reactions. The term “robust” in this figure refers only to the fraction of variable reactions with a variability interval of less than ±5%.

**Figure 2.**The distribution of alternative flux values across the flux intervals. Panel (

**A**–

**C**) show the frequency of flux values of sensitive reactions divided into 20 bins in the mt + med, mt + med + met and mt + med + met + pro models, respectively. Panel (

**D**–

**F**) show the intervasl in the respective models in which sensitive reactions responded to perturbations (red and blue lines, indicating lowest and highest flux values, respectively), and the interval given by FVA, indicated by red dots (lower bounds) and blue dots (upper bounds). For the sake of clarity, a few extreme points in panel (

**D**–

**F**) are excluded.

**Figure 3.**The pyruvate branching point. The distribution of flux at this point determines the fermentation profile of the cell.

**Figure 4.**The relative flux distribution in the branching point of carbohydrate fermentation (y-axis) in response to one perturbation in each of the variable reactions (x-axis) in the two studied genome-scale models of E. faecalis, resulting in homolactic or mixed acid fermentation in the two genome-scale models of E. faecalis.

**Figure 5.**The serine branching point by which serine is distributed. Pyr: pyruvate; sertrna: L-seryl-tRNA; acetyl-ser: Acetyl-serine.

**Figure 6.**The relative flux distribution in the branching point of serine metabolism (y-axis) in response to one perturbation in each of the variable reactions (x-axis) in the two studied genome-scale models of E. faecalis, resulting in the production of acetyl serine, or seryl-tRNA or serine secretion.

**Figure 7.**The glutamine branching point distributing amino-groups via the amino acid L-glutamine. Gln_L: glutamine, pram: 5-Phospho-beta-D-ribosylamine, fpram: 2-(Formamido)-N1-(5-phospho-D-ribosyl) acetamidine, gmp: guanosine monophosphate, gam6p: glucoseamine 6 phosphate.

**Figure 8.**The relative flux distribution in the glutamine branching point(y-axis) in response to one perturbation in each of the variable reactions (x-axis)) in the two studied genome-scale models of E. faecalis, resulting in the distribution of glutamine in different pathways, namely amino acid, purine and pyrimidine metabolism.

**Figure 9.**Flux scan of the ATPm value over the feasible region of the wildtype model. The blue line shows the ratio between the number of variable reactions and the number of modules generated by CoPE-FBA.

**Figure 10.**Perturbation procedure to determine the robustness of FBA/FVA outcome. The figure shows how fixing one reaction at various random values results in a different range for flux combinations between two fluxes.

**Table 1.**Number of variable reactions in differently constrained genome-scale models of E. faecalis wildtype (wt) and ΔglnA mutant (mt). Here, “nc” indicates model version without any constraints, “med” indicates integration of medium composition, “met” the additional integration of data on metabolite uptake and release and “pro” the additional integration of proteome data.

Model Name | Number of Variable Reactions Variability > 10 ^{−6} | Number of Variable Reactions Variability > 10 ^{−3} | No of Reactions |
---|---|---|---|

mt + nc | 397 | 397 | 708 |

mt + med | 362 | 340 | 708 |

mt + med + met | 347 | 315 | 708 |

mt + med + met + pro | 298 | 289 | 708 |

wt + nc | 398 | 398 | 709 |

wt + med | 363 | 341 | 709 |

wt + med + met | 362 | 340 | 709 |

wt + med + met + pro | 307 | 85 | 709 |

**Table 2.**The number of reactions in the existing modules in each model when their solution space was analysed with CoPE-FBA.

Model Name | Number of Reactions in Each Module |
---|---|

mt + nc | 399 |

mt + med | 360, 4 |

mt + med + met | 345, 4 |

mt + med + met + pro | 286, 5, 4, 4 |

wt + nc | 400 |

wt + med | 361, 4 |

wt + med + met | 360, 4 |

wt + med + met + pro | 295, 5, 4, 4 |

**Table 3.**Number of variable reactions according to FVA (with the interval size larger than 10

^{−6}), number of sensitive reactions according to the solution space inspection procedure (perturbation analysis), and the number of reactions in existing modules in each model (CoPE-FBA). All three methods were used with two optimality tolerance values (100% and 99.9%) and the respective results were compared.

Wt + Med + Met + Pro-Edge | Wt + Med + Met + Pro | |||
---|---|---|---|---|

optimality tolerance | 100 | 99.9 | 100 | 99.9 |

FVA | 209 | 387 | 307 | 387 |

#reactions, sensitive to perturbation | 94 | 137 | 137 | 151 |

#modules according to CoPE-FBA | 4, 13, 7, 5, 4, 4, 12, 3 | 295, 5, 4, 4 | 295, 5, 4, 4 | 295, 5, 4, 4 |

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**MDPI and ACS Style**

Loghmani, S.B.; Veith, N.; Sahle, S.; Bergmann, F.T.; Olivier, B.G.; Kummer, U. Inspecting the Solution Space of Genome-Scale Metabolic Models. *Metabolites* **2022**, *12*, 43.
https://doi.org/10.3390/metabo12010043

**AMA Style**

Loghmani SB, Veith N, Sahle S, Bergmann FT, Olivier BG, Kummer U. Inspecting the Solution Space of Genome-Scale Metabolic Models. *Metabolites*. 2022; 12(1):43.
https://doi.org/10.3390/metabo12010043

**Chicago/Turabian Style**

Loghmani, Seyed Babak, Nadine Veith, Sven Sahle, Frank T. Bergmann, Brett G. Olivier, and Ursula Kummer. 2022. "Inspecting the Solution Space of Genome-Scale Metabolic Models" *Metabolites* 12, no. 1: 43.
https://doi.org/10.3390/metabo12010043