Our ability to model the growth of microbes only relies on empirical laws, fundamentally restricting our understanding and predictive capacity in many environmental systems. In particular, the link between energy balances and growth dynamics is still not understood. Here we demonstrate a microbial growth equation relying on an explicit theoretical ground sustained by Boltzmann statistics, thus establishing a relationship between microbial growth rate and available energy. The validity of our equation was then questioned by analyzing the microbial isotopic fractionation phenomenon, which can be viewed as a kinetic consequence of the differences in energy contents of isotopic isomers used for growth. We illustrate how the associated theoretical predictions are actually consistent with recent experimental evidences. Our work links microbial population dynamics to the thermodynamic driving forces of the ecosystem, which opens the door to many biotechnological and ecological developments.
In his famous book ‘What is life?’, Erwin Schrödinger opened the debate on how life could be envisioned from the thermodynamic standpoint (Schrödinger, 1944). Ilyia Prigogine (Prigogine, 1955) then made an important contribution by pioneering the application of nonequilibrium thermodynamics to biology, underlying modern developments of biological flux-force models (Westerhoff et al., 1982, 1983). Today, thermodynamic state functions are widely applied to living systems at different organization levels (Jørgensen and Svirezhev, 2004). The study of microbes, the simplest form of life, however, led to a deeper physical conceptualization of the problem (McCarty, 1965; Roels, 1980; Heijnen and Vandijken, 1992; Rittmann and McCarty, 2001; Kleerebezem and Van Loosdrecht, 2010). In these contributions, microbial anabolism was linked to catabolism through energy dissipation, sometimes expressed as a universal efficiency factor. A relation between dissipated energy and growth stoichiometry was established, enabling the prediction and calculation of energy and matter balances of microbial growth. However, the key question of the link between microbial thermodynamics and growth kinetics remained unanswered.
At the beginning of the 20th century, chemistry was facing a similar problem that finally resulted in a thermochemical kinetic theory 80 years ago (Eyring, 1935). The existence of a high-energy transition state, resulting from the collision of reactants, was postulated. Statistical physics was invoked to estimate the probability of the colliding molecules to have enough energy to overcome the transition state energy. A link between reaction kinetics and thermodynamic state of the system was thus established. Lotka (1922a) suggested that a similar approach could be applied to biological units: ‘The similarity of the [biological] units invites statistical treatment [...], the units in the new statistical mechanics will be energy transformers subject to irreversible collisions of peculiar type-collisions in which trigger action is a dominant feature…’
Let us treat statistically a clonal population of N microbes consuming substrates, transducing energy and dividing. For each individual, we propose that harvesting a threshold level of energy from the environment fundamentally triggers the microbial division process. Each division can thus be described as a succession of two steps: (i) the reversible capture of energy by the microbe and (ii) its irreversible transduction leading to division. We indeed assume that the second irreversible step is a slow kinetically limiting process and that the first step can thus be considered as close to equilibrium. The elementary microbial division act can thus be symbolized as follows:
where M represents a microbe and X‡ the intermediate microbial activated state in which the microbe is able to divide (Figure 1a).
During the first step, each microbe has access to a volume (Vharv) in which it can harvest the chemical energy in the form of substrate molecules. The part of the chemical energy that is available for growth depends on the thermodynamic state of the environment surrounding the microbe. It can thus be transcribed using the meaningful concept of chemical catabolic exergy, which represents the maximum work available when bringing the substrate molecules in thermodynamic equilibrium with the microbe’s environment (see Supplementary Material). Consequently, exergy levels can be attributed to each elementary volume Vharv and thus to each microbe. Using statistical physics, we demonstrate that the occupancies of exergy levels by microbes follow Boltzmann statistics (see Supplementary Information for the detailed demonstration). Let us denote E‡ the threshold exergy level triggering division corresponding to both dissipated exergy Edis and stored exergy EM during growth (Figure 1a). These exergies can be evaluated using energy balances for growth established by different authors (see Supplementary Information and reference Kleerebezem and Van Loosdrecht, 2010 for a review). Let us now finally introduce the parameter μmax, representing the division rate of an activated microbe (see Supplementary Information). Then comes the expression of microbial growth rate as a function of microbial exergy balance:
S represents the energy-limiting substrate, Ecat being the catabolic exergy of one molecule of energy-limiting substrate.
Equation (2) introduces a flux-force relationship between microbial growth rate (μ) and catabolic exergy density ([S]·Ecat). It correctly transcribes the well-known microbial growth rate dependence on substrate concentration enabling the modeling of any microbial experimental growth data as illustrated using Monod’s historical experiments (Figure 1b; Monod, 1942). In addition, our theory naturally accounts for the existence of an apparent threshold substrate concentration for growth (Figure 1b, detail), correcting a flaw of previous empirical equations (Kovarova-Kovar and Egli, 1998). However, the correct transcription of microbial growth rate dependence on substrate concentration is not per se sufficient to support the validity of our theory. This can indeed be achieved with any empirical equation exhibiting a sigmoid shape as outlined by Monod himself (Monod, 1942). Moreover, the simple process of fitting growth rate equations with measurements do not provide a sufficiently precise framework to compare different growth models as already demonstrated (Senn et al., 1994).
Hopefully, in our theory, the growth rate more exactly depends on the spatial distribution of exergy around microbes. This implies that, in addition to substrate concentration, the intrinsic thermodynamic properties of molecules involved in the metabolism also determine the growth rate. We thus realized that the study of microbial isotopic fractionation related to pure culture experiments could provide us with an adequate experimental opportunity to challenge the fundamental nature of the relationship between exergy and growth rate. Indeed, for a given microbe and in relation to a well-defined metabolic reaction (that is, μmax and Vharv are fixed), our theory predicts that the variation in catabolic exergy (Ecat) due to the differences in thermodynamic properties of isotopic isomers (isotopomers) would induce slight differences in substrate consumption rates between heavy or light molecules, thus giving a theoretical ground to the well-known microbial isotopic fractionation phenomenon.
Using our theory, we therefore derived a literal expression of the widely employed empirical kinetic fractionation factor αS/P of substrates toward products (Mariotti et al., 1981; see Supplementary Information for details):
α0 being the residual biochemical fractionation factor of the catabolic reaction, represents the difference between exergy of the catabolic reaction involving one molecule of the heavy isotopomer minus the exergy of its light counterpart.
Unexpectedly, two classes of isotopic fractionation behaviors are predicted from Equation (3) depending on the sign of , in relation to the thermodynamic properties of isotopomers (see Supplementary Material): if , then α should decrease when increases and this case could be named ‘microbial overfractionation’ (Figure 2a, upper curve).
Conversely, if , then α should increase when increases and this case could be named ‘microbial underfractionation’ (Figure 2a, lower curve).
For both cases, α should vary with substrate concentration and catabolic exergy, increasing values leading to the convergence of microbial fractionation toward an asymptotical value (α0). These theoretical predictions were then questioned with experimental microbial isotopic fractionation data from the litterature.
Although it has long been considered that microbial isotopic fractionation was only dependent on the type of metabolism (Mariotti et al., 1981; Hayes, 1993), recent evidences have suggested that it could vary with environmental conditions (Conrad, 1999, 2005). Strikingly, the dependency of microbial fractionation on substrate concentration (Valentine et al., 2004; Kampara et al., 2008; Goevert and Conrad, 2009) and on Gibbs energy (Penning et al., 2005) was recently proven. Figures 2b–d illustrate how these experimental data are actually in agreement with the predictions obtained from our model and thus supports the dependency of microbial kinetics to the spatial distribution of exergy. Moreover, it demonstrates how the reanalysis of these data under a consistent theoretical framework (see Supplementary Information) show the existence of over- and underfractionation as predicted from our theory, which had never been explicitly claimed in the literature to our knowledge. The fact that these two fractionation classes are related to the thermodynamic properties of molecules (the sign of ) is a strong argument in favor of a threshold microbial exergy level triggering division, as considered in our theory. Without denying the need for additional data on purposely designed and carefully controlled isotopic experiments, the overall consistency of the different mathematical predictions of our model with microbial isotopic fractionation data already strongly supports our theory of microbial growth sustained by Boltzmann statistics of exergy distribution.
For clarity, our approach was exposed for a single substrate in the case of energy-limited microbial growth. However, we also demonstrate how it can easily be extended to multiple substrates or to the cases of stoichiometric growth limitation (see Supplementary Information). Equation (2) applied to mixed cultures links microbial population dynamics to the thermodynamic driving forces of the ecosystem, which has wide practical and fundamental implications. For example, implemented into engineering models for environmental bioprocesses such as ADM1 for anaerobic digestion (Batstone et al., 2002), it could thus naturally transcribe the well-known dependence of microbial activity to thermodynamic conditions (Jin and Bethke, 2007), correcting a major flaw of current kinetic equations (Kleerebezem and van Loosdrecht, 2006; Rodriguez et al., 2006). Moreover, our model offers the possibility to couple multiple biochemical reactions through the concept of exergy and to infer the fluxes generated by the ‘microbial engines that drive earth’s biogeochemical cycles’ (Falkowski et al., 2008). On a more fundamental point of view, our theory could constitute a mathematical framework to evaluate how microbial communities would evolve considering various thermodynamic goal functions such as the maximum power (Lotka, 1922a, 1922b; DeLong, 2008), maximum exergy (Jørgensen and Svirezhev, 2004) or minimum entropy production (Prigogine, 1955). Finally, and more generally, we also believe that the study of microbes, the simplest form of life, constitutes a fertile thinking ground for a deeper interlinking between physical and biological concepts.
Batstone DJ, Keller J, Angelidaki I, Kalyuzhnyi SV, Pavlostathis SG, Rozzi A et al. (2002). The IWA Anaerobic Digestion Model No 1 (ADM1). Water Sci Technol 45: 65–73.
Conrad R . (1999). Contribution of hydrogen to methane production and control of hydrogen concentrations in methanogenic soils and sediments. Fems Microbiol Ecol 28: 193–202.
Conrad R . (2005). Quantification of methanogenic pathways using stable carbon isotopic signatures: a review and a proposal. Org Geochem 36: 739–752.
DeLong JP . (2008). The maximum power principle predicts the outcomes of two-species competition experiments. Oikos 117: 1329–1336.
Eyring H . (1935). The activated complex in chemical reactions. J Chem Phys 3: 107–115.
Falkowski PG, Fenchel T, Delong EF . (2008). The microbial engines that drive Earth's biogeochemical cycles. Science 320: 1034–1039.
Goevert D, Conrad R . (2009). Effect of substrate concentration on carbon isotope fractionation during acetoclastic methanogenesis by Methanosarcina barkeri and M. acetivorans and in rice field soil. Appl Environ Microbiol 75: 2605–2612.
Hayes JM . (1993). Factors controlling C-13 contents of sedimentary organic-compounds—principles and evidence. Mar Geol 113: 111–125.
Heijnen JJ, Vandijken JP . (1992). In search of a thermodynamic description of biomass yields for the chemotropic growth of microorganisms. Biotechnol Bioeng 39: 833–858.
Jin Q, Bethke CM . (2007). The thermodynamics and kinetics of microbial metabolism. Am J Sci 307: 643–677.
Jørgensen SE, Svirezhev IM . (2004) Towards a Thermodynamic Theory for Ecological Systems. Elsevier: Amsterdam; Boston.
Kampara M, Thullner M, Richnow HH, Harms H, Wick LY . (2008). Impact of bioavailability restrictions on microbially induced stable isotope fractionation. 2. Experimental evidence. Environ Sci Technol 42: 6552–6558.
Kleerebezem R, van Loosdrecht MCM . (2006). Critical analysis of some concepts proposed in ADM1. Water Sci Technol 54: 51–57.
Kleerebezem R, Van Loosdrecht MCM . (2010). A generalized method for thermodynamic state analysis of environmental systems. Crit Rev Environ Sci Technol 40: 1–54.
Kovarova-Kovar K, Egli T . (1998). Growth kinetics of suspended microbial cells: From single-substrate-controlled growth to mixed-substrate kinetics. Microbiol Mol Biol Rev 62: 646–666.
Lotka AJ . (1922a). Natural selection as a physical principle. Proc Natl Acad Sci USA 8: 151–154.
Lotka AJ . (1922b). Contribution to the energetics of evolution. Proc Natl Acad Sci USA 8: 147–151.
Mariotti A, Germon JC, Hubert P, Kaiser P, Letolle R, Tardieux A et al. (1981). Experimental-determination of nitrogen kinetic isotope fractionation—some principles—illustration for the denitrification and nitrification processes. Plant Soil 62: 413–430.
McCarty PL . (1965). Thermodynamics of biological synthesis and growth. Air Water Pollut 9: 621–639.
Monod J . (1942) Recherches sur la croissance des cultures bactériennes. Paris.
Penning H, Plugge CM, Galand PE, Conrad R . (2005). Variation of carbon isotope fractionation in hydrogenotrophic methanogenic microbial cultures and environmental samples at different energy status. Glob Change Biol 11: 2103–2113.
Prigogine I . (1955) Introduction to Thermodynamics of Irreversible Processes. Thomas: Springfield, IL, USA.
Rittmann BE, McCarty PL . (2001) Environmental Biotechnology: Principles and Applications. McGraw-Hill: Boston, MA, USA.
Rodriguez J, Lema JM, van Loosdrecht MCM, Kleerebezem R . (2006). Variable stoichiometry with thermodynamic control in ADM1. Water Sci Technol 54: 101–110.
Roels JA . (1980). Application of macroscopic principles to microbial-metabolism. Biotechnol Bioeng 22: 2457–2514.
Schrödinger E . (1944) What is Life?: The Physical Aspect of the Living Cell. The University Press: Cambridge.
Senn H, Lendenmann U, Snozzi M, Hamer G, Egli T . (1994). The growth of Escherichia-coli in glucose-limited chemostat cultures—a reexamination of the kinetics. Biochim Biophys Acta 1201: 424–436.
Valentine DL, Chidthaisong A, Rice A, Reeburgh WS, Tyler SC . (2004). Carbon and hydrogen isotope fractionation by moderately thermophilic methanogens. Geochim Cosmochim Ac 68: 1571–1590.
Westerhoff HV, Hellingwerf KJ, Vandam K . (1983). Thermodynamic efficiency of microbial-growth is low but optimal for maximal growth-rate. Proc Natl Acad Sci USA 80: 305–309.
Westerhoff HV, Lolkema JS, Otto R, Hellingwerf KJ . (1982). Thermodynamics of growth—non-equilibrium thermodynamics of bacterial-growth—the phenomenological and the mosaic approach. Biochim Biophys Acta 683: 181–220.
We thank Dr Bart Haegman for his very useful comments on this manuscript. Tim Vogel, Christian Duquennoi, Ivan Delbende and Stéphane Ghozzi are also acknowledged for sharing their views with us and giving their feedbacks, Ariane Bize and Laurent Mazéas for helpful discussions. This work was completed thanks to the support of Irstea (we are especially grateful to Philippe Duchène who made this work possible) and Agence Nationale de la Recherche in the framework of the Programme Investissements d’Avenir ANR-10-BTBR-02. The two anonymous reviewers are also acknowledged for their useful comments.
The authors declare no conflict of interest.
Supplementary Information accompanies this paper on The ISME Journal website
About this article
Cite this article
Desmond-Le Quéméner, E., Bouchez, T. A thermodynamic theory of microbial growth. ISME J 8, 1747–1751 (2014). https://doi.org/10.1038/ismej.2014.7
- Gibbs energy
- growth rate
- microbial division
- microbial isotopic fractionation
- statistical physics
This article is cited by
Nature Reviews Earth & Environment (2021)
Nature Communications (2021)
Bulletin of Mathematical Biology (2021)
Bulletin of Mathematical Biology (2021)
Scientific Reports (2020)