# A Statistical Thermodynamical Interpretation of Metabolism

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

^{th}elementary mode. R

_{j}is then the j

^{th}reaction rate of the metabolic network, the reaction rates r

_{i,j}are defined by the n elementary modes, and w

_{i}is the fractional contribution of the i

^{th}elementary mode to the reaction j. Due to the normalization step, the unit of R

_{j}and of r

_{i,j}is the glucose uptake rate. The reactions leading to and from external metabolites are normally the transport reactions through the cell envelope. They define the overall reaction equation, and the ratios between their rates and the glucose uptake rate represent the yields or stoichiometry coefficients in the overall reaction equation.

_{m}[17]. Multiplication of the entropy of reaction with the rate of glucose consumption results in the rate of entropy production. However, the glucose consumption rate represents only a multiplication factor and effects related to entropy formation can be analyzed based on reaction entropies.

_{TOT}can be computed based on the macroscopic reaction stoichiometry between external metabolites using Equation (2), or from:

_{i}, computed using Equation (2), contribute to the overall value with weighting factors w

_{i}. Since the reaction entropies are expressed per mole of glucose consumed, the total rate of entropy production is obtained when ΔS

_{i}is multiplied with the rate of glucose consumption.

_{i,j}= v

_{i}u

_{j}), i.e., it is defined by the product of the sum of column elements and the sum of row elements.

_{TOT}to have a positive value, b must be positive. In that case the extreme value is a maximum. The result can be rewritten as:

_{i}, and the probabilities to the weighting factors. Thus, the reaction entropies ΔS

_{i}are linearly correlated with the natural log of the probability of usage of the corresponding elementary modes. This provides the theoretical justification for the relationship that has been previously observed.

**Figure 1.**Entropy generation as a function of weighting factors of elementary modes for E. coli under anaerobic non-growth conditions. (a) Total entropy production without constraint (black) and with the applied constraint of weighting factor w

_{1}= exp(–ΔS

_{1}/b) = 0.549 (blue plane). The system shows a global maximum entropy production of 0.376 kJ/K-mole when entropy generation is uniformly distributed and a local maximum entropy production of 0.352 kJ/K-mole located at the intersection between the cone and the plane when entropies are constrained to the value of the reaction entropies of individual elementary modes. (b) Comparison of predicted, maximum entropy production (●)with experimentally determined entropy generation (${\u25cf}$) based on data reported by Aristidou et al. (1999) [24]. The weighted average of entropy production of any combination of existing elementary modes is located on the gray plane (Equation (3)) while the blue surface represents combinations of elementary modes distributed according to the Gibbs measure. The blue surface touches the gray plane at the location of the constrained maximum entropy production point. The experimentally determined point is located very close to the predicted maximum entropy generation point reflecting the highly evolved metabolism of wildtype E. coli cells. (see Appendix 1 for detailed data).

**Figure 2.**Total entropy production as a function of weighting factors for E. coli under anaerobic non-growth conditions. (a) Comparison of predicted maximum total entropy production with experimental entropy generation (${\u25cf}$) reported in Wlaschin et al. (2006) [19]. The experimental point is on the gray plane and is expected to evolve in time towards the predicted maximum entropy generation point (${\u25cf}$) (b) Time course of the entropy generation in an evolving system determined from data by Hua et al. (2006) [25]. The experimentally determined reaction entropies are located on the gray plane and move with time during adaptation towards the predicted maximum entropy production point: (●), unevolved system; (${\u25cf}$), after 30 days of adaptation; (${\u25cf}$), after 60 days of adaptation. (c) Total entropy generation as function of evolution time. With time the system is expected to reach the maximum entropy generation where the elementary mode weighting factors are distributed according to Equation (16). (see Appendix 1 for detailed data analysis).

## Acknowledgements

## Authors' Contributions

## References

- Carlson, R.; Srienc, F. Fundamental Escherichia coli biochemical pathways for biomass and energy production: identification of reactions. Biotech. Bioeng.
**2004**, 85, 1–18. [Google Scholar] [CrossRef] [PubMed] - Carlson, R.; Srienc, F. Fundamental Escherichia coli biochemical pathays for biomass and energy production: creation of overall flux states. Biotech. Bioeng.
**2004**, 86, 149–161. [Google Scholar] [CrossRef] [PubMed] - Klamt, S.; Stelling, J. Two approaches for metabolic pathway analysis? Trends Biotech.
**2003**, 21, 64–69. [Google Scholar] [CrossRef] - Schuster, S.; Dandekar, T.; Fell, D.A. Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering. Trends Biotech.
**1999**, 17, 53–60. [Google Scholar] [CrossRef] - Schuster, S.; Hilgetag, C.; Fell, D. Detecting elementary modes of functioning in metabolic networks. Mode. Tren. Bio. Thermo. Kinetics
**1994**, 3, 103–105. [Google Scholar] - Trinh, C.T.; Wlaschin, A.; Srienc, F. Elementary mode analysis: a useful metabolic pathway analysis tool for characterizing cellular metabolism. Appl. Microbiol. Biotech.
**2009**, 81, 813–826. [Google Scholar] [CrossRef] [PubMed] - Pfeiffer, T.; Sanchez-Valdenebro, I.; Nuno, J.C.; Montero, F.; Schuster, S. METATOOL: for studying metabolic networks. Bioinformatics
**1999**, 15, 251–257. [Google Scholar] [CrossRef] [PubMed] - Von Kamp, A.; Schuster, S. Metatool 5.0: fast and flexible elementary modes analysis. Bioinformatics
**2006**, 22, 1930–1931. [Google Scholar] [CrossRef] [PubMed] - Wagner, C. Nullspace approach to determine the elementary modes of chemical reaction systems. J. Phys. Chem. B
**2004**, 108, 2425–2431. [Google Scholar] [CrossRef] - Terzer, M.; Stelling, J. Large-scale computation of elementary flux modes with bit pattern trees. Bioinformatics
**2008**, 24, 2229–2235. [Google Scholar] [CrossRef] [PubMed] - Stelling, J.; Klamt, S.; Bettenbrock, K.; Schuster, S.; Gilles, E.D. Metabolic network structure determines key aspects of functionality and regulation. Nature
**2002**, 420, 190–193. [Google Scholar] [CrossRef] [PubMed] - Urbanczik, R.; Wagner, C. Functional stoichiometric analysis of metabolic networks. Bioinformatics
**2005**, 21, 4176–4180. [Google Scholar] [CrossRef] [PubMed] - Trinh, C.T.; Unrean, P.; Srienc, F. Minimal Escherichia coli cell for the most efficient production of ethanol from hexoses and pentoses. Appl. Env. Microbiol.
**2008**, 74, 3634–3643. [Google Scholar] [CrossRef] [PubMed] - Schuster, S.; Hilgetag, C.; Woods, J.H.; Fell, D.A. Reaction routes in biochemical reaction systems: Algebraic properties, validated calculation procedure and example from nucleotide metabolism. J. Math. Biol.
**2002**, 45, 153–181. [Google Scholar] [CrossRef] [PubMed] - Poolman, M.G.; Venkatesh, K.V.; Pidcock, M.K.; Fell, D.A. A method for the determination of flux in elementary modes, and its application to Lactobacillus rhamnosus. Biotech. Bioeng.
**2004**, 88, 601–612. [Google Scholar] [CrossRef] [PubMed] - Wiback, S.J.; Mahadevan, R.; Palsson, B.O. Reconstructing metabolic flux vectors from extreme pathways: defining the alpha-spectrum. J. Theor. Biol.
**2003**, 224, 313–324. [Google Scholar] [CrossRef] - Sandler, S.I.; Orbey, H. On the thermodynamics of microbial-growth processes. Biotech. Bioeng.
**1991**, 38, 697–718. [Google Scholar] [CrossRef] [PubMed] - Roels, J.A. Energetics and Kinetics in Biotechnology; Elsevier Biomedical Press: Amsterdam, the Netherlands/New York, NY, USA, 1983. [Google Scholar]
- Wlaschin, A.P.; Trinh, C.T.; Carlson, R.; Srienc, F. The fractional contributions of elementary modes to the metabolism of escherichia coli and their estimation from reaction entropies. Metab. Eng.
**2006**, 8, 338–352. [Google Scholar] [CrossRef] [PubMed] - Prigogine, I. Modération et transformations irréversibles des systèmes ouverts. Bull. Acad. Roy. Belg. Cl. Sci.
**1945**, 31, 600–606. [Google Scholar] - De Groot, S.R.; Mazur, P. Non-equilibrium thermodynamics; North-Holland Pub. Co.: Amsterdam, The Netherlands, 1962. [Google Scholar]
- Martyushev, L.M.; Seleznev, V.D. Maximum entropy production principle in physics, chemistry and biology. Phys. Rep-Rev. Sect. Phys. Lett.
**2006**, 426, 1–45. [Google Scholar] [CrossRef] - Dill, K.A.; Bromberg, S. Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology; Garland Science Press: New York, NY, USA, 2003. [Google Scholar]
- Aristidou, A.A.; San, K.Y.; Bennett, G.N. Metabolic flux analysis of Escherichia coli expressing the Bacillus subtilis acetolactate synthase in batch and continuous cultures. Biotech. Bioeng.
**1999**, 63, 737–749. [Google Scholar] [CrossRef] - Hua, Q.; Joyce, A.R.; Fong, S.S.; Palsson, B.O. Metabolic analysis of adaptive evolution for in silico-designed lactate-producing strains. Biotech. Bioeng.
**2006**, 95, 992–1002. [Google Scholar] [CrossRef] [PubMed] - Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Tarbell, J.M. Thermodynamic liapunov-function for near equilibrium CSTR. Chem. Eng. Sci.
**1977**, 32, 1471–1476. [Google Scholar] [CrossRef] - Zhao, Q.Y.; Kurata, H. Maximum entropy decomposition of flux distribution at steady state to elementary modes. J. Biosci. Bioeng.
**2009**, 107, 84–89. [Google Scholar] [CrossRef] [PubMed] - Zhao, Q.Y.; Kurata, H. Use of maximum entropy principle with Lagrange multipliers extends the feasibility of elementary mode analysis. J. Biosci. Bioeng.
**2010**, 110, 254–261. [Google Scholar] [CrossRef] [PubMed]

## Appendix 1.

**(1)**Entropy calculations in Figure 1b based on secretion data from Aristidou et al. [24]. The metabolic model used for the elementary mode analysis was the model for wildtype E. coli described by Carlson and Srienc (2004) [1] under anaerobic conditions. This model results in 21 elementary modes that can be grouped into four families of modes that have the same overall stoichiometry. The overall stoichiometries and the number of associated modes are shown in Table S1. The entropy of reaction has been computed from Equation (2) using the methods by Sandler and Orbey [17] for computing the standard entropies of formation for the individual components. With the entropies of reaction we computed the constant b from Equation (15) which resulted in a value of b = 0.1511 kJ/K-mole.

Family | Reaction stoichiometry | No. of EMs | △Si (kJ/K-mole) |
---|---|---|---|

1 | Glucose = Acetate + Succinate | 3 | 0.1934 |

2 | Glucose = 2 Lactate | 6 | 0.1604 |

3 | Glucose = Ethanol + Acetate + 2 Formate | 6 | 0.2326 |

4 | Glucose = Ethanol + Acetate + Formate + CO_{2} | 6 | 0.2752 |

Metabolic flux (mmole/g CDW h) | |
---|---|

Glucose | −1.000 |

Ethanol | 0.457 |

Acetate | 0.180 |

Lactate | 0.699 |

Succinate | 0.566 |

Formate | 0.637 |

^{1}The fluxes are extrapolated to a zero growth rate from measured fluxes from a series of chemostats at different dilution rates as described by Aristidou et al. [24].

Family | △S_{i}(kJ/K-mole) | Weighting factors^{1} | |
---|---|---|---|

Experimental values | Predicted values | ||

1 | 0.1934 | 0.2681 | 0.2780 |

2 | 0.1604 | 0.3688 | 0.3458 |

3 | 0.2326 | 0.2865 | 0.2146 |

4 | 0.2752 | 0.0766 | 0.1618 |

△S_{TOT} | 0.1987 | 0.2037 |

^{1}The weighting factors are computed from the measured fluxes reported in Aristidou et al. [24]. The predicted weight factors are calculated from entropies of the family modes as described in Equation (14). The total entropies are calculated from the weighted sum of entropies of the family modes, Σwi△Si.

**(2)**Entropy calculations in Figure 2a of the main text contain the total entropy data as described by Wlaschin et al. [19]. The entropy calculations in Figure 2b are based on experimental data described by Hua et al. [25]. We used the metabolic model described by Carlson and Srienc [1] containing the two gene knockouts, △adhE△pta (Hua et al., [25]), for elementary mode calculations. The computation yielded 16 elementary modes that can be grouped into five families of modes with the same overall stoichiometry. They are shown in Table 4 together with the entropies of reaction. Table 5 summarizes the weighting factor for the three time points during the adaptation.

Family | Reaction stoichiometry | No. of EMs | △S_{i}(kJ/K-mole) |
---|---|---|---|

1 | Glucose = 2 Lactate | 8 | 0.2735 |

2 | Glucose = Lactate + 0.86 Succinate | 3 | 0.2927 |

3 | Glucose = 0.49 Lactate + 1.28 Succinate | 1 | 0.3022 |

4 | Glucose = 0.8 Lactate + 1 Succinate + 0.2 Formate | 3 | 0.3027 |

5 | Glucose = 1.67 Succinate + 0.33 Formate | 1 | 0.3222 |

**Table 5.**The weighting factors and entropies for each family of modes of E. coli containing the deletions △adhE△pta during adaptive evolution. The total reaction entropies per mole glucose consumed, ΔS

_{TOT}, are the sums of products of individual weighting factors and associated family entropies ΔS

_{i}.

Family | △S_{i}(kJ/K-mole) | Weighting factors^{1} | |||
---|---|---|---|---|---|

△adhE△pta | 30-day evolved △adhE△pta | 60-day evolved △adhE△pta | Predicted w _{i} | ||

1 | 0.2735 | 0.7280 | 0.7000 | 0.7150 | 0.2258 |

2 | 0.2927 | 0.1460 | 0.0710 | 0.0000 | 0.2054 |

3 | 0.3022 | 0.0000 | 0.1410 | 0.1300 | 0.1960 |

4 | 0.3027 | 0.1270 | 0.0880 | 0.1550 | 0.1955 |

5 | 0.3222 | 0.0000 | 0.0000 | 0.0000 | 0.1776 |

△S_{TOT} | 0.2818 | 0.2825 | 0.2848 | 0.2975 |

^{1}The experimental weighting factors are computed from the measured fluxes reported in Hua et al., [25] while the predicted weighting factors are calculated from entropies of the family modes.

## Appendix 2. Detailed Derivation of the Maximum Entropy Production

_{i}such that the macroscopic reaction entropy:

_{i}and over a column j is v

_{j}:

© 2010 by the authors; licensee MDPI, Basel, Switzerland. This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Srienc, F.; Unrean, P.
A Statistical Thermodynamical Interpretation of Metabolism. *Entropy* **2010**, *12*, 1921-1935.
https://doi.org/10.3390/e12081921

**AMA Style**

Srienc F, Unrean P.
A Statistical Thermodynamical Interpretation of Metabolism. *Entropy*. 2010; 12(8):1921-1935.
https://doi.org/10.3390/e12081921

**Chicago/Turabian Style**

Srienc, Friedrich, and Pornkamol Unrean.
2010. "A Statistical Thermodynamical Interpretation of Metabolism" *Entropy* 12, no. 8: 1921-1935.
https://doi.org/10.3390/e12081921