Skip to main content
SpringerLink
Log in

Multiscale Modeling of Uranium Bioreduction in Porous Media by One-Dimensional Biofilms

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

We formulate a multiscale mathematical model that describes the bioreduction of uranium in porous media. On the mesoscale we describe the bioreduction of uranium [VI] to uranium [IV] using a multispecies one-dimensional biofilm model with suspended bacteria and thermodynamic growth inhibition. We upscale the mesoscopic (colony scale) model to the macroscale (reactor scale) and investigate the behavior of substrate utilization and production, attachment and detachment processes, and thermodynamic effects not usually considered in biofilm growth models. Simulation results of the reactor model indicate that thermodynamic inhibition quantitatively alters the dynamics of the model and neglecting thermodynamic effects may over- or underestimate chemical concentrations in the system. Furthermore, we numerically investigate uncertainties related to the specific choice of attachment and detachment rate coefficients and find that while increasing the attachment rate coefficient or decreasing the detachment rate coefficient leads to thicker biofilms, performance of the reactor remains largely unaffected.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Abbas F, Eberl HJ (2011) Analytical substrate flux approximation for the monod boundary value problem. Appl Math Comput 218(4):1484–1494

    MathSciNet  MATH  Google Scholar 

  • 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 Biostat 2(1):123–143

    MATH  Google Scholar 

  • Abbas F, Sudarsan R, Eberl HJ (2012) Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates. Math Biosci Eng 9(2):215–239

    MathSciNet  MATH  Google Scholar 

  • Aspa Y, Debenest G, Quintard M (2007) Effective transport properties of porous biofilms. In: Eurotherm seminar \(\text{N}^o\) 81 reactive heat transfer in porous media

  • Ballyk MM, Jones DA, Smith HL (2001) Microbial competition in reactors with wall attachment: a mathematical comparison of chemostat and flug flow models. Microb Ecol 41:210–221

    Google Scholar 

  • 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

    Google Scholar 

  • Battiato I, Tartakovsky DM, Tartakovsky AM, Scheibe TD (2011) Hybrind models of reactive transport in porous and fractured media. Adv Water Resour 34:1140–1150

    Google Scholar 

  • Battiato I, Ferrero PT, O’Malley D, Miller CT, Takhar PS, Valdés-Parada FJ, Wood BD (2019) Theory and applications of macroscale models in porous media. Transp Porous Med 130:5–76

  • Baumgärtner M (2009) The use of reprocessed uranium in light water reactors: Problem identification and solution finding. In: Reprocessed Uranium (Pro Tech Mgt Vienna, 2007)

  • 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

    Google Scholar 

  • Burden RL, Faires JD (2010) Numerical analysis. Brooks/Cole Publishing, Boston

    MATH  Google Scholar 

  • Carl S, Heikkilä S (2000) Nonlinear differential equations in ordered spaces. Chapman & Hall

  • 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

    Google Scholar 

  • Corapcioglu MY (ed) (1993) Advances in porous media, vol 3. Elsevier, Amsterdam

    Google Scholar 

  • Coyte KZ, Tabuteau H, Gaffney EA, Foster KR, Durham WM (2016) Microbial competition in porous environments can select against rapid biofilm growth. PNAS pp 161–170. www.pnas.org/cgi/doi/10.1073/pnas.1525228113

  • Coyte RM, Jain RC, Srivastava SK, Sharma KC, Khalil A, Ma L, Vengosh A (2018) Large-scale uranium contamination of groundwater resources in India. Environ Sci Technol Lett 5(6):341–347

    Google Scholar 

  • 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

    Google Scholar 

  • Deygout C, Lesne A, Rapaport A (2011) A multiscale approach for biofilm modelling: application for growth in a plug flow reactor. http://hal.archives-ouvertes.fr/hal-00604377/en/

  • Donlan RM (2002) Biofilms: microbial life on surfaces. Emerg Infect Dis 82(1):881–890

    Google Scholar 

  • 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, pp 359–383

  • Epperson JF (2002) An introduction to numerical methods and analysis. Wiley, Hoboken

    MATH  Google Scholar 

  • Gaebler HJ, Eberl HJ (2018a) First order versus Monod kinetics in numerical simulation of biofilms in porous media. In: Kilgour DM, Kunze H, Makarov R, Melnik M, Wang X (eds) Recent advances in mathematical and statistical methods. Springer International Publishing, Cham, pp 351–361

    MATH  Google Scholar 

  • Gaebler HJ, Eberl HJ (2018b) 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(6):1110–1140

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Gaebler HJ, Hughes JM, Eberl HJ (2021) Thermodynamic inhibition in a biofilm reactor model with suspended bacteria. Bull Math Biol. https://doi.org/10.1007/s11538-020-00840-w

    Article  MATH  Google Scholar 

  • Golfier F, Wood BD, Orgogozo L, Quintard M, Bues M (2009) Biofilms in porous media: development of macroscopic transport equations via volume averaging with closure for local mass equilibrium conditions. Adv Water Resour 32(3):463–485

    Google Scholar 

  • Hallatschek O, Hersen P, Ramanathan S, Nelson DR (2007) Genetic drift at expanding frontiers promotes gene segregation. Proc Natl Acad Sci USA 104(50):19926–19930

    Google Scholar 

  • Haynes WM (2010) CRC handbook of chemistry and physics. CRC Press, Boca Raton

    Google Scholar 

  • Health Canada (2019) Guidelines for Canadian drinking water quality: uranium. Technical report, Government of Canada, catalogue No—144-13/12-2019E-PDF

  • Henze M, Gujer W, Mino T, van Loosdrecht MCM (2000) Activated sludge models ASM1, AMS2, ASM2D, ASM3. IWA Publishing, London

    Google Scholar 

  • Horn H, Lackner S (2014) Modeling of biofilm systems: a review. Adv Biochem Eng Biotechnol 146:53–76

    Google Scholar 

  • 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

    Google Scholar 

  • Jin Q, Bethke CM (2003) A new rate law describing microbial respiration. Appl Environ Microbiol 69:2340–2348

    Google Scholar 

  • Jin Q, Bethke CM (2007) The thermodynamics and kinetics of microbial metabolism. Am J Sci 307:643–677

    Google Scholar 

  • Jin Q, Roden EE (2011) Microbial physiology-based model of ethanol metabolism in subsurface sediments. J Contam Hydrol 115:1–12

    Google Scholar 

  • Jones D, Kojouharov HV, Le D, Smith H (2003) The Freter model: a simple model of biofilm formation. J Math Biol 47:137–152

    MathSciNet  MATH  Google Scholar 

  • Kissel JC, McCarty PL, Street RL (1984) Numerical simulation of a mixed-culture biofilm. J Environ Eng 2:393–411

    Google Scholar 

  • Kleerebezem R, van Loosdrecht MCM (2010) Generalized method for thermodynamic state analysis of environmental systems. Crit Rev Environ Sci Technol 40:1–54

    Google Scholar 

  • Kurttio P, Auvinen A, Salonen L, Saha H, Pekkanen J, Mäkeläinen I, Väisänen SB, Penttilä IM, Komulainen H (2002) Renal effects of uranium in drinking water. Environ Health Perspect 110:337–342

    Google Scholar 

  • Kus F, Wiesmann U (1995) Degradation kinetics of acetate and propionate by immobilized anaerobic mixed cultures. Water Res 29(6):1437–1443

    Google Scholar 

  • Liotta SF, Romano V, Russo G (2000) Central schemes for balance laws of reaction type. SIAM J Numer Anal 38(4):1337–1356

    MathSciNet  MATH  Google Scholar 

  • 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

    Google Scholar 

  • Logan JD (2000) Transport modeling in hydrogeochemical systems. Springer-Verlag, New York

    MATH  Google Scholar 

  • 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

    Google Scholar 

  • Manobala T, Shukla SK, Rao TS, Kumar MD (2019a) A new uranium bioremediation approach using radio-tolerant Deinococcus radiodurans biofilm. J Biosci 44(122):1–9

    Google Scholar 

  • Manobala T, Shukla SK, Rao TS, Kumar MD (2019b) Uranium sequestration by biofilm-forming bacteria isolated from marine sediment collected from southern coastal region of india. Int Biodeter Biodegr 145:1–10

    Google Scholar 

  • MaÅ¡ić 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

    MathSciNet  MATH  Google Scholar 

  • MaÅ¡ić 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 International Publishing, Cham, pp 267–276. https://doi.org/10.1007/978-3-319-30379-6_25

  • McCarty PL, Bae J (2011) Model to couple anaerobic process kinetics with biological growth equilibrium thermodynamics. Environ Sci Technol 45:6838–6844

    Google Scholar 

  • Millet YA, Alvarez D, Ringgaard S, von Andrian UH, Davis BM, Waldor MK (2014) Insights into Vibrio cholerae intestinal colonization from monitoring fluorescently labeled bacteria. PLoS Pathog 10(10):e1004405

    Google Scholar 

  • Mitri S, Foster KR (2013) A genotypic view of social interactions in microbial communities. Annu Rev Genet 47:265–291

    Google Scholar 

  • Muffler K, Ulber R (eds) (2014) Productive biofilms. Springer, Cham

    Google Scholar 

  • Muslu Y (1993) Mass transfer and substrate utilization in biofilm reactors. J Chem Technol Biotechnol 57:127–135

    Google Scholar 

  • Nadell CD, Foster KR, Xavier JB (2010) Emergence of spatial structure in cell groups and the evolution of cooperation. PLoS Comput Biol 6(3):e1000716

    Google Scholar 

  • Picioreanu C, Van Loosdrecht MCM, Heijnen JJ (2000) Effect of diffusive and convective substrate transport on biofilm structure formation: a two-dimensional modeling study. Biotechnol Bioeng 69(5):504–515. https://doi.org/10.1002/1097-0290(20000905)69:5<504::AID-BIT5>3.0.CO;2-S

  • Quemener EDL, Bouchez T (2014) A thermodynamic theory of microbial growth. ISME J 8:1747–1751

    Google Scholar 

  • Renslow RS, Ahmed B, Nuñez JR, Cao B, Majors PD, Fredrickson JK, Beyenal H (2017) Modeling substrate utilization, metabolie production, and uranium immobilization in Shewanella oneidensis biofilms. Front Environ Sci. https://doi.org/10.3389/fenvs.2017.00030

    Article  Google Scholar 

  • Rittmann BE (1982) The effect of shear stress on biofilm loss rate. Biotechnol Bioeng 24:501–506

    Google Scholar 

  • Rittmann BE, McCarty PL (2001) Environmental biotechnology: principles and applications. McGraw-Hill, Boston

    Google Scholar 

  • Singh R, Paul D, Jain RK (2006) Biofilms: implications in bioremediation. Trends Microbiol 14(9):388–397

    Google Scholar 

  • 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

    Google Scholar 

  • Tang Y, Liu H (2017) Modeling multidimensional and multispecies biofilms in porous media. Biotechnol Bioeng 114(8):1679–1687

    Google Scholar 

  • Thauer RK, Jungermann K, Decker K (1977) Energy conservation in chemotrophic anaerobic bacteria. Bacteriol Rev 41(1):100–180

    Google Scholar 

  • US Environmental Protection Agency (2020) National primary drinking water regulations. https://www.epa.gov/ground-water-and-drinking-water/national-primary-drinking-water-regulations#Radionuclides. Accessed 16 March 2020

  • van Gestel J, Weissing FJ, Kuipers OP, Kovács AT (2014) Density of founder cells affects spatial pattern formation and cooperation in Bacillus subtilis biofilms. ISME J 8(10):2069–2079

    Google Scholar 

  • Wanner O, Gujer W (1986) A multispecies biofilm model. Biotechnol Bioeng 28:314–386

    Google Scholar 

  • Wanner O, Cunningham AB, Lundman R (1995) Modeling biofilm accumulation and mass transport in a porous medium under high substrate loading. Biotechnol Bioeng 47:703–712

    Google Scholar 

  • Wanner O, Eberl HJ, Morgenroth E, Noguera DR, Picioreanu C, Rittmann BE, van Loosdrecht M (2006) Mathematical modeling of biofilms. IWA Publishing, London

    Google Scholar 

  • 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

    Google Scholar 

  • 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. Curr Opin Biotechnol 24:489–497

    Google Scholar 

  • World Health Organization (2017) Guidelines for drinking-water quality: fourth edition incorporating the first addendum. Technical report, licence: CC BY-NC-SA 3.0 IGO

  • 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

    Google Scholar 

  • Yabusaki SB, Sengör SS, Fang Y (2015) A uranium bioremediation reactive transport benchmark. Comput Geosci 19:551–567

    Google Scholar 

  • Zamora ML, Tracy BL, Zielinski JM, Meyerhof DP, Moss MA (1998) Chronic ingestion of uranium in drinking water: a study of kidney bioeffects in humans. Toxicol Sci 43:68–77

    Google Scholar 

Download references

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 and an NSERC Discovery Grant (RGPIN-2019-05003) awarded to HJE.

Author information

Authors and Affiliations

  1. University of Guelph, 50 Stone Rd. E., Guelph, ON, N1G 2W1, Canada

    Harry J. Gaebler & Hermann J. Eberl

Authors
  1. Harry J. Gaebler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hermann J. Eberl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Harry J. Gaebler.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 1819 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gaebler, H.J., Eberl, H.J. Multiscale Modeling of Uranium Bioreduction in Porous Media by One-Dimensional Biofilms. Bull Math Biol 83, 105 (2021). https://doi.org/10.1007/s11538-021-00938-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11538-021-00938-9

Keywords

Mathematics Subject Classification

Search

Navigation