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doi: 10.3934/dcdss.2022075
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A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient

Brittni Hall , Xiaoying Han , Peter E. Kloeden and  Hans-Werner van Wyk ,

221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

* Corresponding author: Hans-Werner van Wyk

Dedicated to Professor Georg Hetzer on the occasion of his 75th birthday

Received  August 2021 Early access March 2022

Fund Project: This work is partially supported by FEDER-Junta de Andalucia, Spain (project P18-FR-4509)

A mathematical model describing the growth of gut microbiome inside and on the wall of the gut is developed based on the chemostat model with wall growth. Both the concentration and flow rate of the nutrient input are time-dependent, which results in a system of non-autonomous differential equations. First the stability of each meaningful equilibrium is studied for the autonomous counterpart. Then the existence of pullback attractors and its detailed structures for the nonautonomous system are investigated using theory and techniques of nonautonomous dynamical systems. In particular, sufficient conditions under which the microbiome vanishes or persists are constructed. Numerical simulations are provided to illustrate the theoretical results.

Keywords: Microbiome, chemostat, wall growth, nonautonomous dynamical system, pullback attractor, stability, persistence.
Mathematics Subject Classification: Primary: 34D45, 35B41, 37B55, 92B05, 92D25; Secondary: 35B35, 45M10, 97M60.
Citation: Brittni Hall, Xiaoying Han, Peter E. Kloeden, Hans-Werner van Wyk. A nonautonomous chemostat model for the growth of gut microbiome with varying nutrient. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022075
References:
[1]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, SpringerBriefs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6.

[2]

T. Caraballo, X. Han and P. Kloeden, Chemostats with time-dependent input and wall growth, Appl. Math. Inf. Sci., 9 (2015), 2283-2296.  doi: 10.12785/amis.

[3]

T. Caraballo, X. Han and P. E. Kloeden, Non-autonomous chemostats with variable delays, SIAM Journal on Mathematical Analysis, 47 (2015), 2178-2199.  doi: 10.1137/14099930X.

[4]

T. Caraballo, X. Han, P. E. Kloeden and A. Rapaport, Dynamics of non-autonomous chemostat models, Continuous and Distributed Systems. Ⅱ, Stud. Syst. Decis. Control, Springer, Cham, 30 (2015), 103-120.  doi: 10.1007/978-3-319-19075-4_6.

[5]

A. Carry, Piles of ancient poop reavel 'extinction event' in human gut bacteria, Science AAAS, (2021).

[6]

I. Cho and M. J. Blaser, The human microbiome: At the interface of health and disease, Nat. Rev. Genet., 13 (2012), 260-270.  doi: 10.1038/nrg3182.

[7]

A. V. Contreras, B. Cocom-Chan, G. Hernandez-Montes, T. Portillo-Bobadilla and O. Resendis-Antonio, Host-microbiome interaction and cancer: Potential application in precision medicine, Front. Physiol., 7 (2016), 606.  doi: 10.3389/fphys.2016.00606.

[8]

J. J. Farrell, L. Zhang, H. Zhou, D. Chia, D. Elashoff, D. Akin, B. J. Paster, K. Joshipura and D. T. Wong, Variations of oral microbiota are associated with pancreatic diseases including pancreatic cancer, Gut, 61 (2011), 582-588.  doi: 10.1136/gutjnl-2011-300784.

[9]

T. Gibson and G. Gerber, Robust and scalable models of microbiome dynamics for bacteriotherapy design, (2018).

[10]

J. A. Gilbert, R. A. Quinn, J. Debelius, Z. Z. Xu, J. Morton, N. Garg, J. K. Jansson, P. C. Dorrestein and R. Knight, Microbiome-wide association studies link dynamic microbial consortia to disease, Nature, 535 (2016), 94-103.  doi: 10.1038/nature18850.

[11]

A. Gonzalez, J. Stombaugh, C. Lozupone, P. J. Turnbaugh, J. I. Gordon and R. Knight, The mind-body-microbial continuum, Dialogues Clin. Neurosci., 13 (2011), 55-62. 

[12]

A. L. Gould, V. Zhang, L. Lamberti, E. W. Jones, B. Obadia, N. Korasidis, A. Gavryushkin, J. M. Carlson, N. Beerenwinkel and W. B. Ludington, Microbiome interactions shape host fitness, PNAS, 115 (2018), E11951–E11960. doi: 10.1073/pnas. 1809349115.

[13]

J. Halfvarson, C. J. Brislawn, R. Lamendella, Y. Vázquez-Baeza, W. A. Walters, L. M. Bramer, M. D'Amato, F. Bonfiglio, D. McDonald, A. Gonzalez, E. E. McClure, M. F. Dunklebarger, R. Knight and J. K. Jansson, Dynamics of the human gut microbiome in inflammatory bowel disease, Nat. Microbiol., 2 (2017), 1-7.  doi: 10.1038/nmicrobiol.2017.4.

[14]

L. V. Hooper, D. R. Littman and A. J. Macpherson, Interactions between the microbiota and the immune system, Science, 336 (2012), 1268-1273.  doi: 10.1126/science.1223490.

[15]

L. V. Hooper, M. H. Wong, A. Thelin, L. Hansson, P. G. Falk and J. I. Gordon, Molecular analysis of commensal host-microbial relationships in the intestine, Science, 291 (2001), 881-884.  doi: 10.1126/science.291.5505.881.

[16]

S. Huitzil, S. Sandoval-Motta, A. Frank and M. Aldana, Modeling the role of the microbiome in evolution, Front. Physiol, (2018). doi: 10.3389/fphys. 2018.01836.

[17]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J.Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.

[18]

P. E. Kloeden and T. Lorenz, Pullback incremental attraction, Nonauton. Dyn. Syst., 1 (2013), 53-60.  doi: 10.2478/msds-2013-0004.

[19]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[20]

P. E. Kloeden and M. Yang, An Introduction to Nonautonomous Dynamical Systems and their Applications, World Scientific, Singapore, 2020.

[21]

R. E. Ley, Obesity and the human microbiome, Curr. Opin. Gastroenterol, 26 (2010), 5-11.  doi: 10.1097/MOG.0b013e328333d751.

[22]

R. E. Ley, P. J. Turnbaugh, S. Klein and J. I. Gordon, Microbial ecology: Human gut microbes associated with obesity, Nature, 444 (2006), 1022. 

[23]

X. C. Morgan, T. L. Tickle, H. Sokol, D. Gevers, K. L. Devaney, D. V. Ward, J. A. Reyes, S. A. Shah, N. LeLeiko, S. B. Snapper, A. Bousvaros, J. Korzenik, B. E. Sands, R. J. Xavier and C. Huttenhower, Dysfunction of the intestinal microbiome in inflammatory bowel disease and treatment, Genome Biol., 13 (2012), R79. doi: 10.1186/gb-2012-13-9-r79.

[24]

V. S. H. Rao and P. R. S. Rao, Dynamical Models and Control of Biological Systems, Springer-Verlag, Berlin, 2009.

[25]

G. Rogers, D. Keating, R. Young, M. Wong, J. Licinio and S. Wesselingh, From gut dysbiosis to altered brain function and mental illness: Mechanisms and pathways, Mole. Physchiatry, 21 (2016), 738-748.  doi: 10.1038/mp.2016.50.

[26] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, 13. Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
[27]

R. R. Stein, V. Bucci, N. C. Toussaint, C. G. Buffie, G. Rätsch, E. G. Pamer, C. Sander and J. B. Xavier, Ecological modeling from time-series inference: Insight into dynamics and stability of intestinal microbiota, PLOS Computational Biology, 9 (2013), 31003388.  doi: 10.1371/journal.pcbi.1003388.

show all references

References:
[1]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, SpringerBriefs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6.

[2]

T. Caraballo, X. Han and P. Kloeden, Chemostats with time-dependent input and wall growth, Appl. Math. Inf. Sci., 9 (2015), 2283-2296.  doi: 10.12785/amis.

[3]

T. Caraballo, X. Han and P. E. Kloeden, Non-autonomous chemostats with variable delays, SIAM Journal on Mathematical Analysis, 47 (2015), 2178-2199.  doi: 10.1137/14099930X.

[4]

T. Caraballo, X. Han, P. E. Kloeden and A. Rapaport, Dynamics of non-autonomous chemostat models, Continuous and Distributed Systems. Ⅱ, Stud. Syst. Decis. Control, Springer, Cham, 30 (2015), 103-120.  doi: 10.1007/978-3-319-19075-4_6.

[5]

A. Carry, Piles of ancient poop reavel 'extinction event' in human gut bacteria, Science AAAS, (2021).

[6]

I. Cho and M. J. Blaser, The human microbiome: At the interface of health and disease, Nat. Rev. Genet., 13 (2012), 260-270.  doi: 10.1038/nrg3182.

[7]

A. V. Contreras, B. Cocom-Chan, G. Hernandez-Montes, T. Portillo-Bobadilla and O. Resendis-Antonio, Host-microbiome interaction and cancer: Potential application in precision medicine, Front. Physiol., 7 (2016), 606.  doi: 10.3389/fphys.2016.00606.

[8]

J. J. Farrell, L. Zhang, H. Zhou, D. Chia, D. Elashoff, D. Akin, B. J. Paster, K. Joshipura and D. T. Wong, Variations of oral microbiota are associated with pancreatic diseases including pancreatic cancer, Gut, 61 (2011), 582-588.  doi: 10.1136/gutjnl-2011-300784.

[9]

T. Gibson and G. Gerber, Robust and scalable models of microbiome dynamics for bacteriotherapy design, (2018).

[10]

J. A. Gilbert, R. A. Quinn, J. Debelius, Z. Z. Xu, J. Morton, N. Garg, J. K. Jansson, P. C. Dorrestein and R. Knight, Microbiome-wide association studies link dynamic microbial consortia to disease, Nature, 535 (2016), 94-103.  doi: 10.1038/nature18850.

[11]

A. Gonzalez, J. Stombaugh, C. Lozupone, P. J. Turnbaugh, J. I. Gordon and R. Knight, The mind-body-microbial continuum, Dialogues Clin. Neurosci., 13 (2011), 55-62. 

[12]

A. L. Gould, V. Zhang, L. Lamberti, E. W. Jones, B. Obadia, N. Korasidis, A. Gavryushkin, J. M. Carlson, N. Beerenwinkel and W. B. Ludington, Microbiome interactions shape host fitness, PNAS, 115 (2018), E11951–E11960. doi: 10.1073/pnas. 1809349115.

[13]

J. Halfvarson, C. J. Brislawn, R. Lamendella, Y. Vázquez-Baeza, W. A. Walters, L. M. Bramer, M. D'Amato, F. Bonfiglio, D. McDonald, A. Gonzalez, E. E. McClure, M. F. Dunklebarger, R. Knight and J. K. Jansson, Dynamics of the human gut microbiome in inflammatory bowel disease, Nat. Microbiol., 2 (2017), 1-7.  doi: 10.1038/nmicrobiol.2017.4.

[14]

L. V. Hooper, D. R. Littman and A. J. Macpherson, Interactions between the microbiota and the immune system, Science, 336 (2012), 1268-1273.  doi: 10.1126/science.1223490.

[15]

L. V. Hooper, M. H. Wong, A. Thelin, L. Hansson, P. G. Falk and J. I. Gordon, Molecular analysis of commensal host-microbial relationships in the intestine, Science, 291 (2001), 881-884.  doi: 10.1126/science.291.5505.881.

[16]

S. Huitzil, S. Sandoval-Motta, A. Frank and M. Aldana, Modeling the role of the microbiome in evolution, Front. Physiol, (2018). doi: 10.3389/fphys. 2018.01836.

[17]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J.Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.

[18]

P. E. Kloeden and T. Lorenz, Pullback incremental attraction, Nonauton. Dyn. Syst., 1 (2013), 53-60.  doi: 10.2478/msds-2013-0004.

[19]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[20]

P. E. Kloeden and M. Yang, An Introduction to Nonautonomous Dynamical Systems and their Applications, World Scientific, Singapore, 2020.

[21]

R. E. Ley, Obesity and the human microbiome, Curr. Opin. Gastroenterol, 26 (2010), 5-11.  doi: 10.1097/MOG.0b013e328333d751.

[22]

R. E. Ley, P. J. Turnbaugh, S. Klein and J. I. Gordon, Microbial ecology: Human gut microbes associated with obesity, Nature, 444 (2006), 1022. 

[23]

X. C. Morgan, T. L. Tickle, H. Sokol, D. Gevers, K. L. Devaney, D. V. Ward, J. A. Reyes, S. A. Shah, N. LeLeiko, S. B. Snapper, A. Bousvaros, J. Korzenik, B. E. Sands, R. J. Xavier and C. Huttenhower, Dysfunction of the intestinal microbiome in inflammatory bowel disease and treatment, Genome Biol., 13 (2012), R79. doi: 10.1186/gb-2012-13-9-r79.

[24]

V. S. H. Rao and P. R. S. Rao, Dynamical Models and Control of Biological Systems, Springer-Verlag, Berlin, 2009.

[25]

G. Rogers, D. Keating, R. Young, M. Wong, J. Licinio and S. Wesselingh, From gut dysbiosis to altered brain function and mental illness: Mechanisms and pathways, Mole. Physchiatry, 21 (2016), 738-748.  doi: 10.1038/mp.2016.50.

[26] H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge Studies in Mathematical Biology, 13. Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
[27]

R. R. Stein, V. Bucci, N. C. Toussaint, C. G. Buffie, G. Rätsch, E. G. Pamer, C. Sander and J. B. Xavier, Ecological modeling from time-series inference: Insight into dynamics and stability of intestinal microbiota, PLOS Computational Biology, 9 (2013), 31003388.  doi: 10.1371/journal.pcbi.1003388.

Figure 1.  Plot of the fraction $ R(t) $ of bacteria in the gut for different initial conditions
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Figure 2.  Plot showing the joint dynamics of the nutrient $ x(t) $ and the total bacteria $ Y(t) $ over the time interval $ [0,2] $ under assumptions (A1) and (A2). The arrows indicate the trajectory directions
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Figure 3.  The time evolution of the fraction $ R(t) $ of bacteria in the gut and the total bacteria $ Y(t) $ under nonautonomous, periodic forcing with parameters satisfying Assumptions (A1), (A2), and (A3), but not (A4)
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Figure 4.  Plots showing the convergence of the total bacteria $ Y(t) $ to $ 0 $ and that of the nutrition level $ x(t) $ to $ x^*(t) $ under nonautonomous, periodic forcing and Assumptions (A1)–(A4)
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Figure 5.  Joint trajectories of the nutrition $ x $, the bacteria in the gut $ y_1 $, and the bacteria on the wall $ y_2 $ subject to nonautonomous, periodic forcing, with parameters safisfying (A1)–(A4)
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Figure 6.  Plot of the joint dynamics of the nutrition level $ x $ and the total bacteria population $ Y $ under nonautonomous, periodic forcing, with parameters satisfying Assumptions (A1)–(A3), (A5), and (A6)
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Figure 7.  Plot of the trajectories in Figure 6 on a logarithmic scale
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Figure 8.  Trajectory plot $ (x(t),Y(t)) $ in the time interval $ [0,50] $ of the nutrition $ x $ and the total bacteria population $ Y $ under constant nutrient injection for various initial conditions
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Figure 9.  Plot of the fraction of bacteria in the gut over time under autonomous forcing and Assumptions (A1)–(A3), (A5), and (A6)
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Table 1.  Model parameters
Parameter Meaning
$ D(t)> 0 $ input and output flow rate
$ I(t)> 0 $ input concentration of the nutrient
$ \mu> 0 $ ratio between output and input flow rates
$ a> 0 $ maximum consumption rate of the nutrient by the microorganisms
$ b \geq 0 $ growth rate of the microorganisms due to consumption
$ m> 0 $ half-saturation rate of the consumption function
$ \delta \in (0, 1) $ recycle rate from dead microorganisms to new microbe biomass
$ \nu \geq 0 $ collective death rate of microorganisms
$ r_1 \geq 0 $ rate at which microbe attaches to the wall
$ r_2 \geq 0 $ rate at which microbe detaches from wall
$ \alpha \geq 0 $ intra-specific competition rate of microbe population in reservoir
$ \gamma \geq 0 $ intra-specific competition rate of wall population of microbe
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Parameter Meaning
$ D(t)> 0 $ input and output flow rate
$ I(t)> 0 $ input concentration of the nutrient
$ \mu> 0 $ ratio between output and input flow rates
$ a> 0 $ maximum consumption rate of the nutrient by the microorganisms
$ b \geq 0 $ growth rate of the microorganisms due to consumption
$ m> 0 $ half-saturation rate of the consumption function
$ \delta \in (0, 1) $ recycle rate from dead microorganisms to new microbe biomass
$ \nu \geq 0 $ collective death rate of microorganisms
$ r_1 \geq 0 $ rate at which microbe attaches to the wall
$ r_2 \geq 0 $ rate at which microbe detaches from wall
$ \alpha \geq 0 $ intra-specific competition rate of microbe population in reservoir
$ \gamma \geq 0 $ intra-specific competition rate of wall population of microbe
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