# Analyzing and Resolving Infeasibility in Flux Balance Analysis of Metabolic Networks

^{*}

## Abstract

**:**

## 1. Introduction

^{13}C-MFA, which uses measurements of isotopic labeling patterns from

^{13}C-tracer experiments to resolve unknown metabolic rates [8].

## 2. Preliminaries

#### 2.1. Definitions

**,**can be easily embedded in an FBA problem, for example, to incorporate an upper limit for the total amount of enzymes that are needed to catalyze the fluxes contained in the rate vector [12,13]. The relations (1)–(3) give rise to a system of linear inequalities and its solution space forms a (flux) polyhedron. FBA problems contain, in addition, a linear objective function, such as the maximization of the growth rate, which is represented by a vector $c$ and optimized subject to constraints (1)–(3):

#### 2.2. Classical MFA

#### 2.3. Conservation Relations

#### 2.4. Correcting Measured Rates to Make Inconsistent Systems Feasible

## 3. Results

#### 3.1. Infeasibility in FBA Scenarios with Known Fluxes

#### 3.2. Weighted Least-Squares Solution via Quadratic Optimization

- W1: If the variances of measurements (${\sigma}_{i}^{2}$)are available, these could be used for the weights: ${w}_{i}=1/{\sigma}_{i}^{2}$.

- W2: If the measurements variances are unavailable, one could set ${w}_{i}=1/\left|{f}_{i}\right|$ so that a deviation from a flux of larger magnitude weighs less than the same deviation from a flux of lower magnitude. This assumes that the measurement noise is correlated with the magnitude of the measured value.

- W3: As the simplest approach, one could choose equal weights for all corrections: ${w}_{i}=1$.

#### 3.3. Using Linear Optimization to Correct Flux Measurements in Infeasible FBA Systems

#### 3.4. Allowing Minimal Corrections of Other Constraints to Make Infeasible FBA Systems Feasible

#### 3.5. A Practical Guide for How to Proceed with Infeasible FBA Problems

- (1)
- Detect (in)feasibility: infeasibility of a metabolic flux (balance) scenario can or will be detected by an associated error message of the LP solver when performing an FBA optimization or an FVA. If this infeasibility is not a consequence of fixing some reaction rates, i.e., if the base system (1)–(3) is already infeasible, use the methods explained in the previous section for resolving inconsistencies in these constraints.
- (2)
- If not yet performed, identify all reactions with a fixed rate (e.g., by searching for reactions in the model where lower and upper bound are identical).
- (3)
- It is recommended to check via Equation (15) whether there are algebraic redundancies in the defined scenario (solely related to the steady-state constraint) and, if so, to find out which of the fixed reaction rates induce this redundancy (via the redundancy matrix (12)). This can be very helpful to find possible sources of mutually inconsistent rates (but it does not provide a (complete) explanation if constraints (2) and (3) are also involved in the inconsistency).
- (4)
- Decide on which optimization approach (QP (21) or LP (24)) and which of the three weighting schemes, W1–W3, are to be applied. Recommendation: if the measurements variances ${\sigma}_{i}^{2}$ are known and if a suitable QP solver is available, then use a QP with ${w}_{i}=1/{\sigma}_{i}^{2}$; otherwise, choose the LP approach with ${w}_{i}=1/\left|{f}_{i}\right|$. In the latter case, the ${w}_{i}$ may be manually (re)adjusted to increase knowledge about the (un)certainty of the fixed rates. Generally, large weights (${w}_{i}$ > 1000) should be used for rates fixed at zero (${f}_{i}=0)$.

- (5)
- Compute the corrections with the respective optimization approach and analyze them to identify the given rates that were assigned the largest changes and have thus caused the largest inconsistencies.
- (6)
- Apply the corrections and compute with the balanced (now feasible) system the solutions for the original FBA/FVA problem. In particular, FVA can be used to identify unknown rates that are uniquely determined from the measured rates (FVA delivers identical lower and upper bounds for those rates). Generally, when performing FBA or FVA in the corrected system, one may face numerical problems: the precision of the calculated (and applied) corrections might not be sufficient enough for the solver used in the subsequent FBA/FVA optimizations, again resulting in an infeasibility message of the solver. If this happens to be the case, it is advised to use a slightly lower precision for the subsequent optimizations. In the example applications described in Section 3.7, we did not encounter such a problem.

#### 3.6. Implementation in CellNetAnalyzer and CNApy

#### 3.7. Relevant Examples from Core and Genome-Scale Models of Escherichia coli

_{2}), while the others can be assumed to be zero. Again, the resulting redundancy and the magnitudes of the computed (minimal) corrections needed to make the system feasible can then deliver important insights on the overall carbon balance and point to possible modeling (or measurement) errors.

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Example network with different MFA scenarios. Green values mark known (“measured”) reaction rates and red values indicate computed minimal corrections for the measured rates that make the scenario feasible. Black values correspond to reaction rates that can be uniquely determined from the given rates or, in case of scenario S9, they reflect identified ranges of feasible fluxes as determined by FVA. Each scenario is characterized by determinacy (D: determined; UD: underdetermined), redundancy (R: redundant; NR: non-redundant), and consistency/feasibility (C: consistent, I: inconsistent).

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**MDPI and ACS Style**

Klamt, S.; von Kamp, A.
Analyzing and Resolving Infeasibility in Flux Balance Analysis of Metabolic Networks. *Metabolites* **2022**, *12*, 585.
https://doi.org/10.3390/metabo12070585

**AMA Style**

Klamt S, von Kamp A.
Analyzing and Resolving Infeasibility in Flux Balance Analysis of Metabolic Networks. *Metabolites*. 2022; 12(7):585.
https://doi.org/10.3390/metabo12070585

**Chicago/Turabian Style**

Klamt, Steffen, and Axel von Kamp.
2022. "Analyzing and Resolving Infeasibility in Flux Balance Analysis of Metabolic Networks" *Metabolites* 12, no. 7: 585.
https://doi.org/10.3390/metabo12070585