{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T06:04:59Z","timestamp":1774332299417,"version":"3.50.1"},"reference-count":83,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,2,28]],"date-time":"2020-02-28T00:00:00Z","timestamp":1582848000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["390686111"],"award-info":[{"award-number":["390686111"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["391465903"],"award-info":[{"award-number":["391465903"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["812616"],"award-info":[{"award-number":["812616"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Understanding microbial growth with the use of mathematical models has a long history that dates back to the pioneering work of Jacques Monod in the 1940s. Monod\u2019s famous growth law expressed microbial growth rate as a simple function of the limiting nutrient concentration. However, to explain growth laws from underlying principles is extremely challenging. In the second half of the 20th century, numerous experimental approaches aimed at precisely measuring heat production during microbial growth to determine the entropy balance in a growing cell and to quantify the exported entropy. This has led to the development of thermodynamic theories of microbial growth, which have generated fundamental understanding and identified the principal limitations of the growth process. Although these approaches ignored metabolic details and instead considered microbial metabolism as a black box, modern theories heavily rely on genomic resources to describe and model metabolism in great detail to explain microbial growth. Interestingly, however, thermodynamic constraints are often included in modern modeling approaches only in a rather superficial fashion, and it appears that recent modeling approaches and classical theories are rather disconnected fields. To stimulate a closer interaction between these fields, we here review various theoretical approaches that aim at describing microbial growth based on thermodynamics and outline the resulting thermodynamic limits and optimality principles. We start with classical black box models of cellular growth, and continue with recent metabolic modeling approaches that include thermodynamics, before we place these models in the context of fundamental considerations based on non-equilibrium statistical mechanics. We conclude by identifying conceptual overlaps between the fields and suggest how the various types of theories and models can be integrated. We outline how concepts from one approach may help to inform or constrain another, and we demonstrate how genome-scale models can be used to infer key black box parameters, such as the energy of formation or the degree of reduction of biomass. Such integration will allow understanding to what extent microbes can be viewed as thermodynamic machines, and how close they operate to theoretical optima.<\/jats:p>","DOI":"10.3390\/e22030277","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T04:16:16Z","timestamp":1583122576000},"page":"277","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Thermodynamic Limits and Optimality of Microbial Growth"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1262-6591","authenticated-orcid":false,"given":"Nima","family":"Saadat","sequence":"first","affiliation":[{"name":"Institute of Quantitative and Theoretical Biology, Heinrich-Heine-Universit\u00e4t D\u00fcsseldorf, Universit\u00e4tsstra\u00dfe 1, 40225 D\u00fcsseldorf, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1587-2971","authenticated-orcid":false,"given":"Tim","family":"Nies","sequence":"additional","affiliation":[{"name":"Institute of Quantitative and Theoretical Biology, Heinrich-Heine-Universit\u00e4t D\u00fcsseldorf, Universit\u00e4tsstra\u00dfe 1, 40225 D\u00fcsseldorf, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9118-7178","authenticated-orcid":false,"given":"Yvan","family":"Rousset","sequence":"additional","affiliation":[{"name":"Institute of Quantitative and Theoretical Biology, Heinrich-Heine-Universit\u00e4t D\u00fcsseldorf, Universit\u00e4tsstra\u00dfe 1, 40225 D\u00fcsseldorf, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7229-7398","authenticated-orcid":false,"given":"Oliver","family":"Ebenh\u00f6h","sequence":"additional","affiliation":[{"name":"Institute of Quantitative and Theoretical Biology, Heinrich-Heine-Universit\u00e4t D\u00fcsseldorf, Universit\u00e4tsstra\u00dfe 1, 40225 D\u00fcsseldorf, Germany"},{"name":"Cluster of Excellence on Plant Sciences (CEPLAS), Heinrich-Heine University, Universit\u00e4tsstrasse 1, 40225 D\u00fcsseldorf, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.jtbi.2019.06.002","article-title":"Is the cell really a machine?","volume":"477","author":"Nicholson","year":"2019","journal-title":"J. Theor. Biol."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"von Stockar, U., and van der Wielen, L.A. (2013). Biothermodynamics: The Role of Thermodynamics in Biochemical Engineering, EPFL Press.","DOI":"10.1201\/b15428"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2457","DOI":"10.1002\/bit.260221202","article-title":"Application of macroscopic principles to microbial metabolism","volume":"22","author":"Roels","year":"1980","journal-title":"Biotechnol. Bioeng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/BF00430368","article-title":"A new thermodynamically based correlation of chemotrophic biomass yields","volume":"60","author":"Heijnen","year":"1991","journal-title":"Antonie van Leeuwenhoek"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1002\/bit.260401003","article-title":"A black box mathematical model to calculate auto- and heterotrophic biomass yields based on Gibbs energy dissipation","volume":"40","author":"Heijnen","year":"1992","journal-title":"Biotechnol. Bioeng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"833","DOI":"10.1002\/bit.260390806","article-title":"In search of a thermodynamic description of biomass yields for the chemotrophic growth of microorganisms","volume":"39","author":"Heijnen","year":"1992","journal-title":"Biotechnol. Bioeng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/0167-7799(94)90056-6","article-title":"Thermodynamics of microbial growth and its implications for process design","volume":"12","author":"Heijnen","year":"1994","journal-title":"Trends Biotechnol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1780","DOI":"10.1002\/bit.260281205","article-title":"The application of a novel heat flux calorimeter for studying growth of Escherichia coli W in aerobic batch culture","volume":"28","author":"Marison","year":"1986","journal-title":"Biotechnol. Bioeng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/S0040-6031(97)00424-3","article-title":"Biological reaction calorimetry: development of high sensitivity bio-calorimeters","volume":"309","author":"Marison","year":"1998","journal-title":"Thermochim. Acta"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/0005-2728(93)90225-5","article-title":"Thermodynamic considerations in constructing energy balances for cellular growth","volume":"1183","author":"Gustafsson","year":"1993","journal-title":"BBA-Bioenergetics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S0005-2728(99)00065-1","article-title":"Does microbial life always feed on negative entropy? Thermodynamic analysis of microbial growth","volume":"1412","author":"Liu","year":"1999","journal-title":"BBA-Bioenergetics"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"von Stockar, U. (2010). Biothermodynamics of live cells: a tool for biotechnology and biochemical engineering. J. Non-Equilib. Thermodyn., 35.","DOI":"10.1515\/jnetdy.2010.024"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1042\/BST0381197","article-title":"Building and analysing genome-scale metabolic models","volume":"38","author":"Fell","year":"2010","journal-title":"Biochem. Soc. Trans."},{"key":"ref_14","unstructured":"Monod, J. (1942). Recherches Sur La Croissance Des Cultures Bacteriennes, Hermann."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1146\/annurev.mi.03.100149.002103","article-title":"The growth of bacterial cultures","volume":"3","author":"Monod","year":"1949","journal-title":"Annu. Rev. Microbiol."},{"key":"ref_16","unstructured":"Monod, J. (1950). La Technique De Culture Continue: Theorie Et Applications, Masson."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1093\/oxfordjournals.aob.a089000","article-title":"Optima and Limiting Factors","volume":"19","author":"Blackman","year":"1905","journal-title":"Ann. Bot."},{"key":"ref_18","first-page":"209","article-title":"Croissance des populations bact\u00e9riennes et quantit\u00e9 d\u2019aliment disponible","volume":"80","author":"Tessier","year":"1942","journal-title":"Rev. Sci. Paris"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2803","DOI":"10.1002\/bit.260251202","article-title":"Theory and applications of unstructured growth models: Kinetic and energetic aspects","volume":"25","author":"Esener","year":"1983","journal-title":"Biotechnol. Bioeng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1099\/00221287-14-3-601","article-title":"The Continuous Culture of Bacteria: A Theoretical and Experimental Study","volume":"14","author":"Herbert","year":"1956","journal-title":"J. Gen. Microbiol."},{"key":"ref_21","unstructured":"Herbert, D. (1958). VII Int. Congr. Microbiology. Recent Progress in Microbiology, Almqvist & Wiksell."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1098\/rspb.1965.0069","article-title":"The maintenance energy of bacteria in growing cultures","volume":"163","author":"Pirt","year":"1965","journal-title":"Proc. R Soc. Lond. B Biol. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1007\/s00248-006-9049-5","article-title":"Microbial Maintenance: A Critical Review on Its Quantification","volume":"53","year":"2007","journal-title":"Microb. Ecol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1002\/bit.260210405","article-title":"Utilization of mass-energy balance regularities in the analysis of continuous-culture data","volume":"21","author":"Erickson","year":"1979","journal-title":"Biotechnol. Bioeng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1002\/bit.260220104","article-title":"Simple model for the energetics of growth on substrates with different degrees of reduction","volume":"22","author":"Roels","year":"1980","journal-title":"Biotechnol. Bioeng."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1424","DOI":"10.1128\/jb.96.4.1424-1426.1968","article-title":"Factors derived from studies of aerobic growth in minimal media","volume":"96","author":"Mayberry","year":"1968","journal-title":"J. Bacteriol."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Gunsalus, I.C., and Steiner, R.Y. (1961). Energy-yielding metabolism in bacteria. The Bacteria, Academic Press, Inc.","DOI":"10.1016\/B978-0-12-395627-9.50009-8"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1007\/BF02875932","article-title":"Productivity and heat generation of fermentation under oxygen limitation","volume":"18","author":"Minkevich","year":"1973","journal-title":"Folia Microbiol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1080\/14786440208635627","article-title":"The relation of oxygen to the heat of combustion of organic compounds","volume":"33","author":"Thornton","year":"1917","journal-title":"Lond. Edinb. Dublin Philos. Mag. J. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1002\/bit.260230910","article-title":"Estimation of heats of combustion of biomass from elemental analysis using available electron concepts","volume":"23","author":"Patel","year":"1981","journal-title":"Biotechnol. Bioeng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1002\/bit.260410405","article-title":"Calculation of entropy change accompanying growth ofEscherichia coli K-12 on succinic acid","volume":"41","author":"Battley","year":"1993","journal-title":"Biotechnol. Bioeng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1749","DOI":"10.1002\/bit.260240803","article-title":"Modeling of bacterial growth; Formulation and evaluation of a structured model","volume":"24","author":"Esener","year":"1982","journal-title":"Biotechnol. Bioeng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"747","DOI":"10.15252\/msb.20145379","article-title":"Emergence of robust growth laws from optimal regulation of ribosome synthesis","volume":"10","author":"Scott","year":"2014","journal-title":"Mol. Syst. Biol."},{"key":"ref_34","first-page":"E1038","article-title":"Mechanistic links between cellular trade-offs, gene expression, and growth","volume":"112","author":"Danos","year":"2015","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1002\/bit.260370407","article-title":"Calculation of the heat of growth ofEscherichia coli K-12 on succinic acid","volume":"37","author":"Battley","year":"1991","journal-title":"Biotechnol. Bioeng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/S0040-6031(98)00584-X","article-title":"An empirical method for estimating the entropy of formation and the absolute entropy of dried microbial biomass for use in studies on the thermodynamics of microbial growth","volume":"326","author":"Battley","year":"1999","journal-title":"Thermochim. Acta"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1016\/j.cell.2010.05.034","article-title":"Whole-genome sequencing breaks the cost barrier","volume":"141","author":"Bonetta","year":"2010","journal-title":"Cell"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1093\/nar\/28.1.27","article-title":"KEGG: Kyoto encyclopedia of genes and genomes","volume":"28","author":"Kanehisa","year":"2000","journal-title":"Nucleic Acids Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1093\/nar\/30.1.59","article-title":"The metacyc database","volume":"30","author":"Karp","year":"2002","journal-title":"Nucleic Acids Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1038\/nrmicro1023","article-title":"Genome-scale models of microbial cells: Evaluating the consequences of constraints","volume":"2","author":"Price","year":"2004","journal-title":"Nat. Rev. Microbiol."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/j.copbio.2003.08.001","article-title":"Advances in flux balance analysis","volume":"14","author":"Kauffman","year":"2003","journal-title":"Curr. Opin. Biotechnol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1142\/S0218339094000131","article-title":"On elementary Flux modes in biochemical reaction systems at steady state","volume":"2","author":"Schuster","year":"1994","journal-title":"J. Biol. Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1093\/bioinformatics\/15.3.251","article-title":"METATOOL: For studying metabolic networks","volume":"15","author":"Pfeiffer","year":"1999","journal-title":"Bioinformatics"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Edwards, J.S., and Palsson, B.O. (2000). Metabolic flux balance analysis and the in silico analysis of Escherichia coli K-12 gene deletions. BMC Bioinformatics, 1.","DOI":"10.1186\/1471-2105-1-1"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Oliveira, A.P., Nielsen, J., and F\u00f6rster, J. (2005). Modeling Lactococcus lactis using a genome-scale flux model. BMC Microbiol., 5.","DOI":"10.1186\/1471-2180-5-39"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2790","DOI":"10.1128\/JB.01583-07","article-title":"Genome-scale metabolic network analysis of the opportunistic pathogen Pseudomonas aeruginosa PAO1","volume":"190","author":"Oberhardt","year":"2008","journal-title":"J. Bacteriol."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1186\/1475-2859-8-43","article-title":"Development and experimental verification of a genome-scale metabolic model for Corynebacterium glutamicum","volume":"8","author":"Shinfuku","year":"2009","journal-title":"Microb. Cell. Fact."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Fritzemeier, C.J., Hartleb, D., Szappanos, B., Papp, B., and Lercher, M.J. (2017). Erroneous energy-generating cycles in published genome scale metabolic networks: Identification and removal. PLoS Comput. Biol., 13.","DOI":"10.1371\/journal.pcbi.1005494"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"14440","DOI":"10.1016\/S0021-9258(18)98705-3","article-title":"Estimation of standard Gibbs energy changes of biotransformations","volume":"266","author":"Mavrovouniotis","year":"1991","journal-title":"J. Biol. Chem."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2037","DOI":"10.1093\/bioinformatics\/bts317","article-title":"An integrated open framework for thermodynamics of reactions that combines accuracy and coverage","volume":"28","author":"Noor","year":"2012","journal-title":"Bioinformatics"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"D770","DOI":"10.1093\/nar\/gkr874","article-title":"eQuilibrator\u2013the biochemical thermodynamics calculator","volume":"40","author":"Flamholz","year":"2012","journal-title":"Nucleic. Acids Res."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"K\u00fcmmel, A., Panke, S., and Heinemann, M. (2006). Systematic assignment of thermodynamic constraints in metabolic network models. BMC Bioinformatics, 7.","DOI":"10.1186\/1471-2105-7-512"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Jol, S.J., K\u00fcmmel, A., Terzer, M., Stelling, J., and Heinemann, M. (2012). System-level insights into yeast metabolism by thermodynamic analysis of elementary flux modes. PLoS Comput. Biol., 8.","DOI":"10.1371\/journal.pcbi.1002415"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Gudmundsson, S., and Thiele, I. (2010). Computationally efficient flux variability analysis. BMC Bioinform., 11.","DOI":"10.1186\/1471-2105-11-489"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"2006.0034","DOI":"10.1038\/msb4100074","article-title":"Putative regulatory sites unraveled by network-embedded thermodynamic analysis of metabolome data","volume":"2","author":"Panke","year":"2006","journal-title":"Mol. Syst. Biol."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Peres, S., Jolicoeur, M., Moulin, C., Dague, P., and Schuster, S. (2017). How important is thermodynamics for identifying elementary flux modes?. PLoS ONE, 12.","DOI":"10.1371\/journal.pone.0171440"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1792","DOI":"10.1529\/biophysj.106.093138","article-title":"Thermodynamics-based metabolic flux analysis","volume":"92","author":"Henry","year":"2007","journal-title":"Biophys. J."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1038\/msb4100155","article-title":"A genome-scale metabolic reconstruction for Escherichia coli K-12 MG1655 that accounts for 1260 ORFs and thermodynamic information","volume":"3","author":"Feist","year":"2007","journal-title":"Mol. Syst. Biol."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1002\/bit.22285","article-title":"Thermodynamic analysis of biodegradation pathways","volume":"103","author":"Finley","year":"2009","journal-title":"Biotechnol. Bioeng."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/S0006-3495(02)75150-3","article-title":"Energy balance for analysis of complex metabolic networks","volume":"83","author":"Beard","year":"2002","journal-title":"Biophys. J."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Hoppe, A., Hoffmann, S., and Holzh\u00fctter, H.G. (2007). Including metabolite concentrations into flux balance analysis: thermodynamic realizability as a constraint on flux distributions in metabolic networks. BMC Syst. Biol., 1.","DOI":"10.1186\/1752-0509-1-23"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1038\/nature15765","article-title":"Overflow metabolism in Escherichia coli results from efficient proteome allocation","volume":"528","author":"Basan","year":"2015","journal-title":"Nature"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"12663","DOI":"10.1073\/pnas.0609845104","article-title":"Intracellular crowding defines the mode and sequence of substrate uptake by Escherichia coli and constrains its metabolic activity","volume":"104","author":"Beg","year":"2007","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-019-47935-6","article-title":"Flux balance analysis with or without molecular crowding fails to predict two thirds of experimentally observed epistasis in yeast","volume":"9","author":"Alzoubi","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_65","first-page":"592","article-title":"Dependency on medium and temperature of cell size and chemical composition during balanced growth of Salmonella typhimurium","volume":"19","author":"Schaechter","year":"1958","journal-title":"Microbiology"},{"key":"ref_66","first-page":"607","article-title":"The transition between different physiological states during balanced growth of Salmonella typhimurium","volume":"19","author":"Kjeldgaard","year":"1958","journal-title":"Microbiology"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1038\/msb.2009.82","article-title":"Shifts in growth strategies reflect tradeoffs in cellular economics","volume":"5","author":"Molenaar","year":"2009","journal-title":"Mol. Syst. Biol."},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Noor, E., Bar-Even, A., Flamholz, A., Reznik, E., Liebermeister, W., and Milo, R. (2014). Pathway thermodynamics highlights kinetic obstacles in central metabolism. PLoS Comput. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003483"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1038\/s42255-018-0006-7","article-title":"An upper limit on Gibbs energy dissipation governs cellular metabolism","volume":"1","author":"Niebel","year":"2019","journal-title":"Nat. Metab."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"878","DOI":"10.1038\/nprot.2009.58","article-title":"13 C-based metabolic flux analysis","volume":"4","author":"Zamboni","year":"2009","journal-title":"Nat. Protoc."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1099\/00221287-114-2-267","article-title":"An Examination of the Crabtree Effect in Saccharomyces cerevisiae: the Role of Respiratory Adaptation","volume":"114","author":"Barford","year":"1979","journal-title":"J. Gen. Microbiol."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1099\/00221287-129-3-653","article-title":"The Role Of Limited Respiration In The Incomplete Oxidation Of Glucose By Saccharomyces Cerevisiae","volume":"129","author":"Rieger","year":"1983","journal-title":"Microbiology"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"2401","DOI":"10.1103\/PhysRevLett.71.2401","article-title":"Probability of second law violations in shearing steady states","volume":"71","author":"Evans","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1103\/PhysRevLett.78.2690","article-title":"Nonequilibrium Equality for Free Energy Differences","volume":"78","author":"Jarzynski","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"2721","DOI":"10.1103\/PhysRevE.60.2721","article-title":"Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences","volume":"60","author":"Crooks","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"1529","DOI":"10.1080\/00018730210155133","article-title":"The Fluctuation Theorem","volume":"51","author":"Evans","year":"2002","journal-title":"Adv. Phys."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"121923","DOI":"10.1063\/1.4818538","article-title":"Statistical physics of self-replication","volume":"139","author":"England","year":"2013","journal-title":"J. Chem. Phys."},{"key":"ref_78","doi-asserted-by":"crossref","unstructured":"Pi\u00f1ero, J., and Sole, R. (2017). Nonequilibrium Entropic Bounds for Darwinian Replicators. Entropy, 20.","DOI":"10.1101\/225011"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"D515","DOI":"10.1093\/nar\/gkv1049","article-title":"BiGG Models: A platform for integrating, standardizing and sharing genome-scale models","volume":"44","author":"King","year":"2015","journal-title":"Nucleic Acids Res."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Lane, N. (2018). Hot mitochondria?. PLoS Biol., 16.","DOI":"10.1371\/journal.pbio.2005113"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1002\/bit.10645","article-title":"Metabolic load of recombinant protein production: Inhibition of cellular capacities for glucose uptake and respiration after induction of a heterologous gene in Escherichia coli","volume":"83","author":"Neubauer","year":"2003","journal-title":"Biotechnol. Bioeng."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1002\/(SICI)1097-0290(19990705)64:1<74::AID-BIT8>3.0.CO;2-3","article-title":"Thermodynamic analysis of growth of Methanobacterium thermoautotrophicum","volume":"64","author":"Schill","year":"1999","journal-title":"Biotechnol. Bioeng."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1086\/670529","article-title":"A theoretical study of the thermodynamics of microbial growth using Saccharomyces cerevisiae and a different free energy equation","volume":"88","author":"Battley","year":"2013","journal-title":"Q. Rev. Biol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/277\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:02:41Z","timestamp":1760173361000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/277"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,28]]},"references-count":83,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["e22030277"],"URL":"https:\/\/doi.org\/10.3390\/e22030277","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,28]]}}}