# Thermodynamic Limits and Optimality of Microbial Growth

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Historical Overview

**Y**), the quantitative description of the production capabilities in microbial cultures evolved into a whole theory. The introduction of a simple hyperbolic relationship that connects the amount of limiting resources in the environment to growth rates of organisms opened the door for many different modeling approaches. Even today, the Monod growth law together with modern mathematical tools forms the basis for sophisticated theories of microbial growth. Although the work of Jacques Monod is undoubtedly one of the major achievements in microbiology of the 20th century, it is worth mentioning that others have proposed growth laws with slightly different functional dependencies of the growth rate on substrate availability, see for instance the work of Blackman and Tessier [17,18]. As Esener et al. show, all three models (Monod, Blackman, Tessier) can realistically describe experimental data of batch cultures [19].

_{2}as nitrogen source), and the enthalpy of formation for biomass, ${\Delta}_{f}{H}_{b}$. By using the well-known relation of entropy, enthalpy and Gibbs free energy, $\Delta G=\Delta H-T\Delta S$, he even attempted to estimate the entropy of formation of biomass, but concluded that this method is too prone to errors, because it highly depends on the quality of the approximation of enthalpy and Gibbs free energy [31].

## 3. Recent Applications of Thermodynamics in Microbial Growth

## 4. Thermodynamic Approaches to Self-Replication

**I**(a single cell and the substrates in the surrounding medium), to a macrostate

**II**(two daughter cells and the substrates), where each macrostate corresponds to an extremely large number of microstates. England’s reasoning starts from the fact that particles obey classical mechanics at the microscopic scale, and therefore follow a reversible dynamic. This allows quantifying the reversibility of a microscopic transition by the associated change in entropy. Applying these microscopic considerations to the macroscopic scale, the author obtains a generalization of the second law of thermodynamics for macroscopic irreversible biological processes.

**I**to macrostate

**II**(respectively from

**II**to

**I**). When a macroscopic transition is irreversible ($\pi (\mathbf{II}\to \mathbf{I})\ll \pi (\mathbf{I}\to \mathbf{II})$), the logarithm becomes negative, increasing the lower bounds for heat dissipation and entropy increase.

**I**(a single bacterium and the substrate), and will evolve through cell division to macrostate

**II**(two bacteria and the substrate). Application of Equation (4) to this system provides a lower limit of heat dissipation for cell replication. This amount is six times lower than what was experimentally observed for E. coli, which is, according to England, surprisingly close to the thermodynamic limit.

## 5. Combining Black Box Techniques with Modern Genome-Scale Approaches

_{1.595}O

_{0.374}N

_{0.263}P

_{0.023}S

_{0.006}this reads

_{1.595}O

_{0.374}N

_{0.263}P

_{0.023}S

_{0.006}+ 1.251 O

_{2}+ 0.012 KOH → CO

_{2}+ 0.131N

_{2}+ 0.006 P

_{4}O

_{10}+ 0.006 K

_{2}SO

_{4}+ 0.803 H

_{2}O.

_{2}is assumed to be the nitrogen source,

## 6. Outlook and Conclusion

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Y | Biomass yield |

$\gamma $ | Degree of reduction |

$\mathsf{\Phi}$ | Vector of rates of flow into the system |

$\mu $ | Vector of specific molar enthalpies |

${\Delta}_{R}G$ | Gibbs free energy of reaction |

${\Delta}_{f}{G}_{b}$ | Gibbs free energy of formation for biomass |

${\Delta}_{f}{G}_{i}^{0}$ | Standard Gibbs free energy of formation for a metabolite i |

$\sigma $ | Entropy production |

$\tau $ | Time |

T | Temperature in Joule, ${k}_{B}$ set to 1 |

$\mathcal{P}(\pm \sigma )$ | Probability to observe an entropy production of $\pm \sigma $ |

$\pi (\mathbf{II}\to \mathbf{I})$ | Probability to observe a transition from macrostate $\mathbf{II}$ to $\mathbf{I}$ |

$\pi (\mathbf{I}\to \mathbf{II})$ | Probability to observe a transition from macrostate $\mathbf{I}$ to $\mathbf{II}$ |

g | Duplication rate |

$\delta $ | Decaying rate |

## Appendix A. A Black Box Model of Overflow Metabolism

**Assumption 1: Two Modes of Catabolism**

**Assumption 2: Thermodynamic Driving Force Depends on Catabolism**

**Assumption 3: Upper Bound of Gibbs Energy Dissipation Rate**

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**Figure 1.**Energy of formation ${\Delta}_{f}{G}_{b}$ for biomass as encoded in genome-scale models of the BiGG database. Each point represents a biomass composition as described in the models. 85 models of the BiGG database containing 165 biomass functions were analyzed. Mean ${\Delta}_{f}{G}_{b}$ = −70.079 kJ/C-mol, Mean $\gamma $ = 4.996.

**Figure 2.**Anabolic properties of genome-scale models of the BiGG database. The y-axis indicates the minimum required amount of ATP per biomass carbon. The x-axis displays the ratio of carbon dioxide released by anabolism to carbon dioxide released by overall metabolism (including anabolism and catabolism).

**Table 1.**Standard free energy of formation for various organic compounds of interest necessary for estimating the energy of formation for biomass (see [31]).

Substance | Formula | ${\Delta}_{\mathit{f}}{\mathit{G}}_{\mathit{i}}^{0}$ [kJ/mol] |
---|---|---|

Oxygen | O_{2} (g) | 0 |

Potassium hydroxide | KOH (c) | −379.11 |

Carbon dioxide | CO_{2} (g) | −394.36 |

Nitrogen | N_{2} (g) | 0 |

Phosphorous decoxide | P_{4}O_{10} (c) | −2697.84 |

Potassium sulfate | K_{2}SO_{4} (c) | −1321.43 |

Water | H_{2}O (lq) | −237.18 |

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Saadat, N.P.; Nies, T.; Rousset, Y.; Ebenhöh, O.
Thermodynamic Limits and Optimality of Microbial Growth. *Entropy* **2020**, *22*, 277.
https://doi.org/10.3390/e22030277

**AMA Style**

Saadat NP, Nies T, Rousset Y, Ebenhöh O.
Thermodynamic Limits and Optimality of Microbial Growth. *Entropy*. 2020; 22(3):277.
https://doi.org/10.3390/e22030277

**Chicago/Turabian Style**

Saadat, Nima P., Tim Nies, Yvan Rousset, and Oliver Ebenhöh.
2020. "Thermodynamic Limits and Optimality of Microbial Growth" *Entropy* 22, no. 3: 277.
https://doi.org/10.3390/e22030277