# Current State, Challenges, and Opportunities in Genome-Scale Resource Allocation Models: A Mathematical Perspective

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Stoichiometric Models of Metabolism (SMMs) and Flux Balance Analysis (FBA)

^{−1}h

^{−1}). FBA is a linear programming (LP) problem formulated so that it seeks to solve Objective (1) as follows:

## 3. Precursor Frameworks

#### 3.1. Flux Balance Analysis with Molecular Crowding (FBAwMC)

^{−1}), and $C$ is the cytoplasmic density, in g mL

^{−1}. Taken together, these terms define the parameter ${a}_{j}$ which is defined as the crowding coefficient of reaction $j$ (gDW h

^{−1}mmol

^{−1}). FBAwMC has been used to predict metabolic switching between high and low yield pathways and its relation to redox metabolism [33], predict growth rate and substrate utilization of mutant E. coli strains [30], and identify regulatory mechanisms controlling metabolic switches between states [34]. Given the nature of the new bounding constraint introduced, FBAwMC can be run without needing additional data beyond a typical SMM. However, to elucidate meaningful information on the kinetics of various fluxes, enzyme volume and cytoplasmic density data are necessary (see Figure 1b for a comparison of data needs between different modeling frameworks).

#### 3.2. FBA with Solvent Capacity Constraints (FBAwSCC)

^{−1}), ${k}_{cat,j}$ is the enzyme turnover number (in h

^{−1}) of the enzyme associated with reaction $j$, and $C$ is the limit on metabolic enzyme concentration in g gDW

^{−1}. This constraint was used to model the Warburg effect in proliferating cancer cells, which is a metabolic phenotype with high glycolytic flux and lactic acid fermentation [31]. Outside the typical SMM data needs, kinetic and molecular weight data are necessary to introduce the new constraint, although much of these data can be found in online databases, such as BRENDA [35]. However, an accurate limit on the metabolic enzyme concentration may need to be acquired from explicitly performed proteomics experiments (see Figure 1b for a comparison of data needs between different modeling frameworks). To our knowledge, FBAwSCC has only been applied once; however, this framework is influential in that some of the RAM frameworks utilize a very similar solute capacity constraint.

## 4. Resource Allocation Model (RAM) Frameworks

#### 4.1. Coarse-Grained RAMs (cgRAMs)

^{−1}, often ${k}_{cat}$ for cgRAMs [39,40,41]), and ${e}_{j}$ is the concentration of enzyme $e$ catalyzing reaction $j$ in mmol gDW

^{−1}. Generally, ${k}_{cat}$ values are extracted from a database such as BRENDA [35], though conceivably an apparent kinetic parameter (${k}_{app}$) could be used to correct for kinetics overestimation where substrate saturation is low. Depending on the framework, this constraint is modified to account for scenarios where an enzyme complex catalyzes a reaction (“and” GPR logic, Constraint (7) below) or where multiple isozymes catalyze a single reaction (“or” GPR logic, Constraint (8) below):

#### 4.1.1. Metabolic Modeling with Enzyme Kinetics (MOMENT) Framework and Successors

^{−1}in the cell. A follow up of MOMENT, referred to as short MOMENT (sMOMENT), reformulates Constraints (6) and (9) to reduce the number of model constraints as shown below [44]:

#### 4.1.2. GEM with Enzymatic Constraints Using Kinetics and Omics (GECKO) Framework and Its Progeny

#### 4.1.3. Automated Reconstruction of MOMENT and GECKO Models

#### 4.2. Fine-Grained RAMs (fgRAMs)

#### 4.2.1. Resource Balance Analysis (RBA)

^{−1}) on rRNA abundance (the same base symbol denotes its identical role to $C$, except applied to rRNA). Second, protein synthesis is limited by the capacity of ribosomes in the protein-ribosome coupling constraint, shown below:

^{−1}) and ${N}_{p}^{aa}$ is the number of amino acids in protein $p$. In total then, RBA models use Constraints (1) through (3), and (15) through (18).

#### 4.2.2. Model of Metabolism and Macromolecular Expression (ME-Models)

#### 4.2.3. Expression and Thermodynamics Flux (ETFL) Framework

^{−10}to 10

^{1}) can cause optimal solutions to occur outside of a solver’s accuracy limit (typically 10

^{−9}).

## 5. Discussion and Conclusions

^{13}C MFA) measurements. Despite the potential for inaccurate ${k}_{app}$ estimates, few RAM model investigations include parameter sensitivity analyses. Model sensitivity to kinetics has been investigated in multiple ways, including substituting known ${k}_{app}$ values for average values [21] and perturbing the effective kinetic parameter by an order of magnitude in either direction [65]. A robust analysis of a RAM model should use this gold standard as a baseline, then apply sensitivity analysis to determine if model conclusions are valid under different kinetic estimates.

## 6. Future Directions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Radar graphs depicting the compared abilities and constraints of a given model type for (

**a**) stoichiometric metabolic models, (

**b**) precursor frameworks and coarse-grained resource allocation models (cgRAM), and (

**c**) fine-grained resource allocation models (fgRAM). Each of the four categories is assigned an arbitrary value 1–6, designating a comparative “ranking” between the six modeling types within each category (e.g., a value of “1” designates the highest amount within a category, “2” designates second highest, and so on). When necessary, equivalent values are assigned to model types that have no meaningful distinction within a given category. Descriptions of MOMENT and GECKO frameworks can be found in Section 4.1.1 and Section 4.1.2, respectively. Descriptions of RBA, ME Modeling, and ETFL modeling frameworks can be found in Section 4.2.1, Section 4.2.2 and Section 4.2.3, respectively.

**Figure 2.**Pictorial representation of example research questions that can be answered with various model types. The trapezoidal shape showcases the relative number of model types that can answer a given question, with all models above a given question able to provide an answer, and all models below unable to do so.

**Figure 3.**Diagram of components in precursor, cgRAM, and fgRAM models. For each model group, any element included in at least one model (e.g., in RBA but no other fgRAMs) is included. Areas included in each model group are translucent, so aspects found in multiple are noted by overlapping colors.

**Table 1.**Constraints involved with resource analysis frameworks to compare and contrast what is modeled by different frameworks discussed here. Yellow cells indicate the constraint present in the SMM tool or RAM framework. This table also notes the type of problem for each tool or framework.

Framework Category | Constraints | |||||||
---|---|---|---|---|---|---|---|---|

Precursor | cgRAM | fgRAM | ||||||

FBAwMC | FBAwSCC | GECKO | MOMENT | RBA | ME Model | ETFL | Conceptual Description | Eqn. No. |

× | × | × | × | × | × | × | Objective function | (1) |

× | × | × | × | × | × | × | Mass balance | (2) |

× | × | × | × | × | × | × | Flux bounds | (3) |

× | Molecular crowding | (4) | ||||||

× | Solute capacity | (5) | ||||||

× | × | × | × | × | Linear enzyme kinetics limitation | (6) | ||

× | × | × | × | × | Enzyme capacity | (9) | ||

× | Enzyme pool determination | (10) | ||||||

× | Enzyme pool limit | (11) | ||||||

× | × | × | Macromolecule mass balance (pseudosteady-state) | (14) | ||||

× | × | rRNA capacity constraint | (17) | |||||

× | Protein-ribosome coupling constraint | (18) | ||||||

× | Transcription capacity constraint | (19) | ||||||

× | Macromolecular machinery capacity constraint | (20) | ||||||

× | Thermodynamic constraints on reaction direction | (21)–(26) | ||||||

× | Petersen linearization of growth-driven dilution | (29)–(31) | ||||||

LP | LP | LP | LP | Iterative LP | Iterative LP or NLP | MILP | Type of problem |

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## Share and Cite

**MDPI and ACS Style**

Schroeder, W.L.; Suthers, P.F.; Willis, T.C.; Mooney, E.J.; Maranas, C.D.
Current State, Challenges, and Opportunities in Genome-Scale Resource Allocation Models: A Mathematical Perspective. *Metabolites* **2024**, *14*, 365.
https://doi.org/10.3390/metabo14070365

**AMA Style**

Schroeder WL, Suthers PF, Willis TC, Mooney EJ, Maranas CD.
Current State, Challenges, and Opportunities in Genome-Scale Resource Allocation Models: A Mathematical Perspective. *Metabolites*. 2024; 14(7):365.
https://doi.org/10.3390/metabo14070365

**Chicago/Turabian Style**

Schroeder, Wheaton L., Patrick F. Suthers, Thomas C. Willis, Eric J. Mooney, and Costas D. Maranas.
2024. "Current State, Challenges, and Opportunities in Genome-Scale Resource Allocation Models: A Mathematical Perspective" *Metabolites* 14, no. 7: 365.
https://doi.org/10.3390/metabo14070365