Abstract
We formulate a mathematical model of bacterial populations in a chemostat setting that also accounts for thermodynamic growth inhibition as a consequence of chemical reactions. Using only elementary mathematical and chemical arguments, we carry this out for two systems: a simple toy model with a single species, a single substrate, and a single reaction product, and a more involved model that describes bioreduction of uranium[VI] into uranium[IV]. We find that in contrast to most traditional chemostat models, as a consequence of thermodynamic inhibition the equilibria concentrations of nutrient substrates might depend on their inflow concentration and not only on reaction parameters and the reactor’s dilution rate. Simulation results of the uranium degradation model indicate that thermodynamic growth inhibition quantitatively alters the solution of the model. This suggests that neglecting thermodynamic inhibition effects in systems where they play a role might lead to wrong model predictions and under- or over-estimate the efficacy of the process under investigation.
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Notes
Note: In Reactions 1-4, U[VI] is equivalent to \(\hbox {U}^{6+}\) and U[IV] is equivalent to \(\hbox {U}^{4+}\).
References
Batstone DJ, Keller J, Angelidaki I, Kalyuzhnyi SV, Pavlostathis SG, Rozzi A, Sanders WTM, Siegrist H, Vavilin VA (2002) Anaerobic digestion model No. 1. IWA Publishing, London
Brown DG, Komlos J, Jaffe PR (2005) Simultaneous utilization of acetate and hydrogen by geobacter sulfurreducens and implications for use of hydrogen as an indicator of redox conditions. Environ Sci Technol 39(9):3069–3076
Cao B, Ahmed B, Beyenal H (2010) Immobilization of uranium in groundwater using biofilms. In: Shah V (ed) Emerging environmental technologies. Springer, Dordrecht, pp 1–37
de Leenheer P, Cogan NG (2009) Failure of antibiotic treatment in microbial populations. J Math Biol 59:563–579
de Leenheer P, Dockery J, Gedeon T, Pilyugin SS (2010) The chemostat with lateral gene transfer. J Biol Dyn 6:607–620
de Leenheer P, Schuster M, Smith H (2019) Strong cooperation or tragedy of the commons in the chemostat. Math Biosci Eng 16:139–149
Delattre H, Chen J, Wade MJ, Soyer OS (2020) Thermodynamic modelling of synthetic communities predicts minimum free energy requirements for sulfate reduction and methanogenesis. J R Soc Interface 17:1–11
Fekih-Salem R, Harmand J, Lobry C, Rapaport A, Sari T (2013) Extensions of the chemostat model with flocculation. J Math Anal Appl 397:292–306
Fgaier H, Eberl HJ (2010) A competition model between Pseudomonas fluorescens and pathogens via iron chelation. J Theor Biol 263(4):566–578
Fgaier H, Kalmokoff M, Ellis T, Eberl HJ (2014) An allelopathy based model for the Listeria overgrowth phenomenon. Math Biosci 247:13–26
Hajji ME, Mazenc F, Harmand J (2018) A mathematical study of a syntrophic relationship of a model of anaerobic digestion process. Math Biosci Eng 7:641–656
Harmand J, Lobry C, Rapaport A, Sari T (2017) The chemostat: mathematical theory of microorganism cultures. ISTE Ltd and Wiley, London
Haynes WM (2010) CRC handbook of chemistry and physics. CRC Press, Boca Raton
Henze M, Gujer W, Mino T, van Loosdrecht MCM (2000) Activated sludge models ASM1, AMS2, ASM2D, ASM3. IWA Publishing, London
Hoh CY, Cord-Ruwisch R (1996) A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant. Biotechnol Bioeng 51:597–604
Istok JD, Park M, Michalsen M, Spain AM, Krumholz LR, Liu C, McKinley J, Long P, Roden E, Peacock AD, Baldwin B (2010) A thermodynamically-based model for predicting microbial growth and community composition coupled to system geochemistry: application to uranium bioreduction. J Contam Hydrol 112:1–14
Jin Q, Bethke CM (2003) A new rate law describing microbial respiration. Appl Environ Microbiol 69:2340–2348
Jin Q, Bethke CM (2007) The thermodynamics and kinetics of microbial metabolism. Am J Sci 307:643–677
Jin Q, Roden EE (2011) Microbial physiology-based model of ethanol metabolism in subsurface sediments. J Contam Hydrol 115:1–12
Khassehkhan H, Eberl HJ (2016) A computational study of amensalistic control of listeria monocytogenes by lactococcus lactis under nutrient rich conditions. Foods 5(3):61
Kleerebezem R, van Loosdrecht MCM (2010) Generalized method for thermodynamic state analysis of environmental systems. Crit Rev Environ Sci Technol 40:1–54
Kus F, Wiesmann U (1995) Degradation kinetics of acetate and propionate by immobilized anaerobic mixed cultures. Water Res 29(6):1437–1443
Liu C, Gorby YA, Zachara JM, Fredrickson JK, Brown CF (2002) Reduction kinetics of Fe(III), Co(III), U(VI), Cr(VI), and Tc(VII) in cultures of dissimilatory metal reducing bacteria. Biotechnol Bioeng 80(6):637–649
Lovely DR, Phillips EJP, Gorby YA, Landa ER (1991) Microbial reduction of urnaium. Nature 350:413–416
Malaguerra F, Chambon JC, Bjerg PL, Scheutz C, Binning PJ (2011) Development and sensitivity analysis of a fully kinetic model of sequential reductive dechlorination in groundwater. Environ Sci Technol 45:8395–8402
Masic A, Eberl HJ (2014) A modeling and simulation study of the role of suspended microbial populations in nitrification in a biofilm reactor. Bull Math Biol 76(1):27–58
McCarty PL, Bae J (2011) Model to couple anaerobic process kinetics with biological growth equilibrium thermodynamics. Environ Sci Technol 45:6838–6844
Monod J (1949) The growth of bacterial cultures. Annu Rev Microbiol 3:371–394
Natural Resources Canada (2016) Inventory of radioactive waste in Canada. Tech. rep., Government of Canada., Cat. No. M134-48/2016E-PDF (Online) ISBN 978-0-660-26339-7
Quemener EDL, Bouchez T (2014) A thermodynamic theory of microbial growth. ISME J 8:1747–1751
Rapaport A (2018) Properties of the chemostat model with aggregated biomass. Eur J Appl Math 29:972–990
Rittmann BE, McCarty PL (2001) Environmental biotechnology: principles and applications. McGraw-Hill, Boston
Smeaton CM, Cappellen PV (2018) Gibbs energy dynamic yield method (GEDYM): predicting microbial growth yields under energy-limiting conditions. Geochim Cosmochim Acta 241:1–16
Smith WR, Missen RW (1982) Chemical reaction equilibrium analysis. Wiley, Toronto
Smith HL, Waltman P (1995) The theory of the chemostat: dynamics of microbial competition. Cambridge University Press, New York
Tang Y, Liu H (2017) Modeling multidimensional and multispecies biofilms in porous media. Biotechnol Bioeng 114(8):1679–1687
Thauer RK, Jungermann K, Decker K (1977) Energy conservation in chemotrophic anaerobic bacteria. Bacteriol Rev 41(1):100–180
Wanner O, Eberl H, Morgenroth E, Noguera D, Piciroeanu C, Rittmann B, van Loosdrecht M (2006) Mathematical modeling of biofilms. IWA Publishing, London
Watson IA, Oswald SE, Mayer KU, Wu Y, Banwart SA (2003) Modeling kinetic processes controlling hydrogen and acetate concentrations in an aquifer-derived microcosm. Environ Sci Technol 37:3910–3919
Weedermann M, Seo G, Wolkowicz GSK (2013) Mathematical model of anaerobic digestion in a chemostat: effects of syntrophy and inhibition. J Biol Dyn 7:59–85
Weedermann M, Wolkowicz GSK, Sasara J (2015) Optimal biogas production in a model for anaerobic digestion. Nonlinear Dyn 81:1097–1112
Williams KH, Long PE, Davis JA, Wilkins MJ, N’Guessan AL, Steefel CI, Yang L, Newcomer D, Spane FA, Kerkhof LJ, McGuinness L, Dayvault R, Lovely DR (2011) Acetate availability and its influence on sustainable bioremediation of uranium-contaminated groundwater. Geomicrobiol J 28:519–539
Williams KH, Bargar JR, Lloyd JR, Lovely DR (2013) Bioremediation of uranium-contaminated groundwater: a systems approach to subsurface biogeochemistry. Biotechnology 24:489–497
Acknowledgements
The authors thank Professor William Smith (University of Guelph) for his help with the thermodynamic formulation and Heather M Gaebler (Wilfrid Laurier University) for her help with general chemistry background. HJG was supported by an Ontario Graduate Scholarship and by a Highdale Farms—Arthur and Rosmarie Spoerri Scholarship in Natural Sciences. HJE was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grants RGPIN-2019-05003 and RTI-2016-00080.
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Gaebler, H.J., Eberl, H.J. Thermodynamic Inhibition in Chemostat Models. Bull Math Biol 82, 76 (2020). https://doi.org/10.1007/s11538-020-00758-3
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DOI: https://doi.org/10.1007/s11538-020-00758-3