Abstract
We formulate a biofilm reactor model with suspended bacteria that accounts for thermodynamic growth inhibition. The reactor model is a chemostat style model consisting of a single replenished growth promoting substrate, a single reaction product, suspended bacteria, and wall attached bacteria in the form of a bacterial biofilm. We present stability conditions for the washout equilibrium using standard techniques, demonstrating that analytical results are attainable even with the added complexity from thermodynamic inhibition. Furthermore, we numerically investigate the longterm behaviour. In the computational study, we investigate model behaviour for select parameters and two commonly used detachment functions. We investigate the effects of thermodynamic inhibition on the model and find that thermodynamic inhibition limits substrate utilization/production both inside the biofilm and inside the aqueous phase, resulting in less suspended bacteria and a thinner biofilm.
Similar content being viewed by others
References
Abbas F, Eberl H (2013) Investigation of the role of mesoscale detachment rate expressions in a macroscale model of a porous medium biofilm reactor. Int J Biomath Biostats 2(1):123–143
Abbas F, Sudarsan R, Eberl HJ (2012) Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates. Math Biosc Eng 9(2):215–239
Anderson RT, Vrionis HA, Ortiz-Bernard I, Resch CT, Long PE, Dayvault RD, Karp K, Marutzky S, Metzler DR, Peacock AD, White DC, Lowe M, Lovely DR (2003) Stimulating the in situ activity of Geobacter species to remove uranium from the groundwater of a uranium-contaminated aquifer. Appl Environ Microbiol 69(10):5884–5891
Carl S, Heikkilä S (2000) Nonlinear differential equations in ordered spaces. Chapman & Hall, Baco Raton, FL
Characklis WG (1983) Process analysis in microbial systems:biofilms as a case study. In: Bazin M (ed) Mathematics in microbiology. Academic Press Inc., New York, pp 171–234
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
Donlan RM (2002) Biofilms: Microbial life on surfaces. Emerg Infect Dis 82(1):881–890
Eberl HJ, Wade MJ (2020) Challenges and perspectives in reactor scale modeling of biofilm processes. In: Simões M, Borges A, Simões LC (eds) Recent trends in biofilm science and technology. Elsevier, Amsterdam, pp 359–383
Fenchel T, King G, Blackburn T (2012) Bacterial metabolism. In: Fenchel T, King G, Blackburn T (eds) Bacterial biogeochemistry, 3rd edn. Academic Press, Boston, pp 1–34
Gaebler HJ, Eberl HJ (2018) A simple model of biofilm growth in a porous medium that accounts for detachment and attachment of suspended biomass and their contribution to substrate degradation. Eur J Appl Math 29:1110–1140
Gaebler HJ, Eberl HJ (2020) Thermodynamic inhibition in chemostat models: with an application to bioreduction of uranium. Bull Math Biol. https://doi.org/10.1007/s11538-020-00758-3
Gaebler HJ, Hughes JM (2020) Contaminant removal in ceramic water filter by bacterial biofilms. In: Kilgour DM, Kunze H, Makarov R, Melnik R, Wang X, Ebert HJ (eds) Recent developments in mathematical, statistical and computational sciences. Springer, Cham
Haynes WM (2010) CRC handbook of chemistry and physics. CRC Press, Boca Raton, FL
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
Hughes JM (2020) A mathematical model for biofilm growth on degradable substratum. SIURO 13:94–115. https://doi.org/10.1137/19S1308475
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
Kissel JC, McCarty PL, Street RL (1984) Numerical simulation of mixed-culture biofilm. J Environ Eng 110(2):393–411
Kleerebezem R, van Loosdrecht MCM (2010) Generalized method for thermodynamic state analysis of environmental systems. Crit Rev Env Sci Tec 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
Lewandowski Z, Beyenal H (2003) Fundamentals of biofilm research. CRC Press, Baco Raton, FL
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
Mašić A, Eberl HJ (2012) Persistence in a single species CSTR model with suspended flocs and wall attached biofilms. Bull Math Biol 75(5):1001–1026
Mašić A, Eberl HJ (2016) A chemostat model with wall attachment: the effect of biofilm detachment rates on predicted reactor performance. In: Bélair J, Frigaard I, Kunze H, Makarov R, Melnik R, Spiteri R (eds) Mathematical and computational approaches in advancing modern science and engineering. Springer, Cham, pp 267–276
McCarty PL, Bae J (2011) Model to couple anaerobic process kinetics with biological growth equilibrium thermodynamics. Environ Sci Technol 45:6838–6844
Muffler K, Ulber R (eds) (2014) Productive biofilms. Springer, Cham
Quemener EDL, Bouchez T (2014) A thermodynamic theory of microbial growth. ISME J 8:1747–1751
Rittmann BE, McCarty PL (2001) Environmental biotechnology: principles and applications. McGraw-Hill, Boston, MA
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
Stewart PS (2003) Diffusion in biofilms. J Bacteriol 185(5):1485–1491
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
Wade MJ (2020) Not just numbers: mathematical modelling of anaerobic digestion: past, present and future. Processes 8(8):888. https://doi.org/10.3390/pr8080888
Walter W (1998) Ordinary differential equations. Springer, New York, NY
Wanner O, Gujer W (1986) A multispecies biofilm model. Biotechnol Bioeng 28:314–386
Wanner O, Eberl H, Morgenroth E, Noguera D, Piciroeanu C, Rittmann B, van Loosdrecht M (2006) Mathematical modeling of biofilms. IWA Publishing, London
Yabusaki SB, Fang Y, Long PE, Resch CT, Peacock AD, Komlos J, Jaffe PR, Morrison SJ, Dayvault RD, White DC, Anderson RT (2007) Uranium removal from groundwater via in situ biostimulation:Field-scale modeling of transport and biological processes. J Contam Hydrol 93:216–235
Yabusaki SB, Sengőr SS, Fang Y (2015) A uranium bioremediation reactive transport benchmark. Comput Geosci 19:551–567
Acknowledgements
This study was financially supported through an Ontario Graduate Scholarship and by a Highdale Farms - Arthur and Rosmarie Spoerri Scholarship in Natural Sciences awarded to HJG, an NSERC (CGS-M) Scholarship awarded to JMH, and an NSERC Discovery Grant (RGPIN-2019-05003) awarded to HJE.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
A Additional Information: Jacobian Entries
B Numerical Investigation of a Unique Asymptotically Stable Non-Trivial Equilibrium
To investigate the uniqueness and stability of a non-trivial equilibrium, we select a parameter set, given in Table 1, such that the washout equilibrium \(E^0=(S_1^\text {in},0,0,0)\) is unstable as per the conditions in Corollary 1. Initial conditions are fixed in the range
and we randomly sample 2000 sets of initial conditions and run the model (8) to steady state. Results are presented in Table 4. We find that for each state variable, the standard deviation is orders of magnitude smaller than the calculated mean, suggesting a unique non-trivial asymptotically stable equilibrium exists when the trivial equilibrium is unstable.
C A Biofilm Reactor Model with Suspended Bacteria and Monod Growth
The biofilm reactor model with suspended bacteria and without thermodynamic inhibition is given by
where
and the concentration of substrates \(c_1=c_1(z)\) and \(c_2=c_2(z)\) inside the biofilm are given by the solution to
Rights and permissions
About this article
Cite this article
Gaebler, H.J., Hughes, J.M. & Eberl, H.J. Thermodynamic Inhibition in a Biofilm Reactor with Suspended Bacteria. Bull Math Biol 83, 10 (2021). https://doi.org/10.1007/s11538-020-00840-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11538-020-00840-w
Keywords
- Biofilm reactor
- Chemostat
- Gibbs free energy
- Mathematical model
- Productive biofilm
- Suspended bacteria
- Thermodynamic inhibition