Figure 1.
Schematic of a continuous morbidostat. There is no drug injection when the total microbes are less than threshold $ U $ . There is continuous drug injection once the total microbes reach the threshold $ U $
-
Previous Article
Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions
- CPAA Home
- This Issue
-
Next Article
Droplet phase in a nonlocal isoperimetric problem under confinement
January
2020, 19(1): 203-220.
doi: 10.3934/cpaa.2020011
The continuous morbidostat: A chemostat with controlled drug application to select for drug resistance mutants
| 1. | Department of Mathematics, Swinburne University of Technology, Melbourne VIC 3122, Australia |
| 2. | Department of Mathematics and National Center of Theoretical Science, National Tsing Hua University, Hsinchu, Taiwan |
| 3. | Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan |
The morbidostat is a bacteria culture device that progressively increases antibiotic drug concentration and maintains a constant challenge for study of evolutionary pathway. The operation of a morbidostat under serial transfer has been analyzed previously. In this work, the global dynamics for the operation of a morbidostat under continuous dilution is analyzed. The device switches between drug on and drug off modes according to a simple threshold algorithm. We prove the extinction and uniform persistence of all species with both forward and backward mutations. Numerical simulations for the case of logistic growth and the Hill function for drug inhibition are also presented.
Mathematics Subject Classification:
Primary: 34C11, 92B; Secondary: 94D.
Citation:
Zhenzhen Chen, Sze-Bi Hsu, Ya-Tang Yang. The continuous morbidostat: A chemostat with controlled drug application to select for drug resistance mutants. Communications on Pure & Applied Analysis,
2020, 19
(1)
: 203-220.
doi: 10.3934/cpaa.2020011
![]()
References:
| [1] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Am. Nat., 115 (1980), 151-170.
doi: 10.1086/283553. |
| [2] |
M. Barber, Infection by penicillin resistant Staphylococci, Lancet, 2 (1948), 641-644. Google Scholar |
| [3] |
Z. Chen, S. B. Hsu and Y. T. Yang,
The Morbidostat: a bio-reactor that promotes selection for drug resistance in bacteria, SIAM J. Appl. Math., 77 (2017), 470-499.
doi: 10.1137/16M105695X. |
| [4] |
K. S. Cheng, S. B. Hsu and S. S. Lin,
Some results on global stability of a predator-prey system, J. Math. Biology., 12 (1981), 115-126.
doi: 10.1007/BF00275207. |
| [5] |
W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Health. Math. Monograph, 1965. |
| [6] |
J. B. Deris, M. Kim, Z. Zhang, H. Okano, R. Hermsen, A. Groisman and T. Hwa, The innate growth bistability and fitness landscapes of antibiotic resistant bacteria, Science, 342 (2013), 1237435. Google Scholar |
| [7] |
M. Dragosits and D. Mattanovich, Adaptive laboratory evolution - principles and applications for biotechnology, Microb. Cell Fact., 12 (2013), 64. Google Scholar |
| [8] |
R. Hermsen, J. B. Deris and T. Hwa, On the rapidity of antibiotic resistance evolution facilitated by a concentration gradient, Proc. Natl. Acad. Sci., 109 (2012), 10775-10780. Google Scholar |
| [9] |
R. Hermsen and T. Hwa, Sources and Sinks: A stochastic model of evolution in heterogeneous environments, Phys. Rev. Lett., 105 (2015), 248104. Google Scholar |
| [10] |
W. M. Hirsch, H. L. Smith and X. Q. Zhao,
Chain transitivity, attractivity and strong repellers for semidynamical systems, J. Dynam. Differential. Equations, 13 (2001), 107-131.
doi: 10.1023/A:1009044515567. |
| [11] |
S. B. Hsu,
Limiting behavior for competing species system, SIAM J. Applied Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
| [12] |
S. B. Hsu, Ordinary Differential Equations with Applications, World Scientific Press, 2013.
doi: 10.1142/8744.
Google Scholar
|
| [13] |
S. B. Hsu and P. E. Waltman,
Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM J. Applied Math., 52 (1992), 528-540.
doi: 10.1137/0152029. |
| [14] |
A. H. Melnyk, A. Wong and R. Kassen, The fitness costs of antibiotic resistance mutations, Evol. Appl., 8 (2015), 273-283. Google Scholar |
| [15] |
S. B. Levy and B. Marshall, Antibiotic resistance worldwide: causes, challenges and responses, Nat. Med., 10 (2004), s122–s129. Google Scholar |
| [16] |
P. Liu, Y. T. Lee, C. Y. Wang and Y.-T. Yang, Design and use of a low cost, automated Morbidostat for adaptive evolution of bacteria under antibiotic drug selection, J. Vis. Exp., 115 (2016), e54426. Google Scholar |
| [17] |
M. Mwangi, S. W. Wu and Y. Zhou, Tracking the in vivo evolution of jultidrug resistance in staphylococus aureus by whole genome sequencing, Pro. Natl. Acad. Sci., 104 (2007), 9451-9456. Google Scholar |
| [18] |
H. L. Smith,
Bacterial competition in serial transfer culture, Math. Biosci., 229 (2011), 149-159.
doi: 10.1016/j.mbs.2010.12.001. |
| [19] |
H. L. Smith and P. E. Waltman, The Theory of The Chemostat, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043.
Google Scholar
|
| [20] |
E. Toprak, A. Veres, J. B. Mitchel, D. L. Hartl and R. Kishony, Evolutionary paths to antibiotic resistance under dynamically sustained drug selection, Nat. Genet., 44 (2012), 101-106. Google Scholar |
| [21] |
A. Uri, An Introduction to System Biology Design Principles of Biological Circuits, Chapman and Hall Taylor and Francis Group, London, 2007. |
| [22] |
X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2015, 2nd Edition.
doi: 10.1007/978-0-387-21761-1. |
| [23] |
Q. Zhang, G. Lambert, D. Liao, H. Kim, K. Robin, C. Tung, N. Pourmand and R. H. Austin, Acceleration of emergence of bacterial antibiotic resistance in connected microenvironment, Science, 333 (2011), 1764-1767. Google Scholar |
show all references
References:
| [1] |
R. A. Armstrong and R. McGehee,
Competitive exclusion, Am. Nat., 115 (1980), 151-170.
doi: 10.1086/283553. |
| [2] |
M. Barber, Infection by penicillin resistant Staphylococci, Lancet, 2 (1948), 641-644. Google Scholar |
| [3] |
Z. Chen, S. B. Hsu and Y. T. Yang,
The Morbidostat: a bio-reactor that promotes selection for drug resistance in bacteria, SIAM J. Appl. Math., 77 (2017), 470-499.
doi: 10.1137/16M105695X. |
| [4] |
K. S. Cheng, S. B. Hsu and S. S. Lin,
Some results on global stability of a predator-prey system, J. Math. Biology., 12 (1981), 115-126.
doi: 10.1007/BF00275207. |
| [5] |
W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Health. Math. Monograph, 1965. |
| [6] |
J. B. Deris, M. Kim, Z. Zhang, H. Okano, R. Hermsen, A. Groisman and T. Hwa, The innate growth bistability and fitness landscapes of antibiotic resistant bacteria, Science, 342 (2013), 1237435. Google Scholar |
| [7] |
M. Dragosits and D. Mattanovich, Adaptive laboratory evolution - principles and applications for biotechnology, Microb. Cell Fact., 12 (2013), 64. Google Scholar |
| [8] |
R. Hermsen, J. B. Deris and T. Hwa, On the rapidity of antibiotic resistance evolution facilitated by a concentration gradient, Proc. Natl. Acad. Sci., 109 (2012), 10775-10780. Google Scholar |
| [9] |
R. Hermsen and T. Hwa, Sources and Sinks: A stochastic model of evolution in heterogeneous environments, Phys. Rev. Lett., 105 (2015), 248104. Google Scholar |
| [10] |
W. M. Hirsch, H. L. Smith and X. Q. Zhao,
Chain transitivity, attractivity and strong repellers for semidynamical systems, J. Dynam. Differential. Equations, 13 (2001), 107-131.
doi: 10.1023/A:1009044515567. |
| [11] |
S. B. Hsu,
Limiting behavior for competing species system, SIAM J. Applied Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
| [12] |
S. B. Hsu, Ordinary Differential Equations with Applications, World Scientific Press, 2013.
doi: 10.1142/8744.
Google Scholar
|
| [13] |
S. B. Hsu and P. E. Waltman,
Analysis of a model of two competitors in a chemostat with an external inhibitor, SIAM J. Applied Math., 52 (1992), 528-540.
doi: 10.1137/0152029. |
| [14] |
A. H. Melnyk, A. Wong and R. Kassen, The fitness costs of antibiotic resistance mutations, Evol. Appl., 8 (2015), 273-283. Google Scholar |
| [15] |
S. B. Levy and B. Marshall, Antibiotic resistance worldwide: causes, challenges and responses, Nat. Med., 10 (2004), s122–s129. Google Scholar |
| [16] |
P. Liu, Y. T. Lee, C. Y. Wang and Y.-T. Yang, Design and use of a low cost, automated Morbidostat for adaptive evolution of bacteria under antibiotic drug selection, J. Vis. Exp., 115 (2016), e54426. Google Scholar |
| [17] |
M. Mwangi, S. W. Wu and Y. Zhou, Tracking the in vivo evolution of jultidrug resistance in staphylococus aureus by whole genome sequencing, Pro. Natl. Acad. Sci., 104 (2007), 9451-9456. Google Scholar |
| [18] |
H. L. Smith,
Bacterial competition in serial transfer culture, Math. Biosci., 229 (2011), 149-159.
doi: 10.1016/j.mbs.2010.12.001. |
| [19] |
H. L. Smith and P. E. Waltman, The Theory of The Chemostat, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043.
Google Scholar
|
| [20] |
E. Toprak, A. Veres, J. B. Mitchel, D. L. Hartl and R. Kishony, Evolutionary paths to antibiotic resistance under dynamically sustained drug selection, Nat. Genet., 44 (2012), 101-106. Google Scholar |
| [21] |
A. Uri, An Introduction to System Biology Design Principles of Biological Circuits, Chapman and Hall Taylor and Francis Group, London, 2007. |
| [22] |
X. Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2015, 2nd Edition.
doi: 10.1007/978-0-387-21761-1. |
| [23] |
Q. Zhang, G. Lambert, D. Liao, H. Kim, K. Robin, C. Tung, N. Pourmand and R. H. Austin, Acceleration of emergence of bacterial antibiotic resistance in connected microenvironment, Science, 333 (2011), 1764-1767. Google Scholar |
Figure 2.
Forward mutations between species. Mutant $ v_{i} $ mutates to mutant $ v_{i+1} $ with a forward mutation rate $ q_{ii+1} $ , and there is no backward mutations. We have $ v_{0} = u $ and $ i = 0, 1,2,\cdots, N-1. $
Figure 3.
Forward-backward mutations between species. Mutant $ v_{i} $ mutates to mutant $ v_{i+1} $ with a forward mutation rate $ q_{ii+1} $ , while mutant $ v_{i+1} $ mutates to mutant $ v_{i} $ with a backward mutation rate $ \tilde{q}_{ii+1} $ . We have $ v_{0} = u $ and $ i = 0, 1,2,\cdots, N-1. $
Figure 4.
Cell, substrate, and inhibitor densities of system (3) when $ U = 8 $ . The wild type $ u $ , mutants $ v_{1} $ , $ v_{2} $ and inhibitor $ P $ go extinction in the drug on drug off model, while mutant $ v_{3} $ and substrate $ S $ persist at fixed values in the long-term. In this figure, we take $ S^{0} = 10 $ , $ D = 0.9 $ , $ P^{0} = 10, $ $ q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4}, $ $ m = 0.3 $ , $ r = 0.5 $ , $ a = 0.5 $ , $ L = 1 $ , $ K_{1} = 1 $ , $ K_{2} = 3 $ , $ K_{3} = 10 $ and $ K_{4} = 30 $
Figure 5.
Cell, substrate, and inhibitor densities of system (3) when $ U = 2 $ . The wild type $ u $ and mutants $ v_{1} $ , $ v_{2} $ go extinction in the drug on drug off model, while mutant $ v_{3} $ , substrate $ S $ , and inhibitor $ P $ persist at fixed values in the long-term. In this figure, we take $ S^{0} = 10 $ , $ D = 0.9 $ , $ P^{0} = 10, $ $ q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4}, $ $ m = 0.3 $ , $ r = 0.5 $ , $ a = 0.5 $ , $ L = 1 $ , $ K_{1} = 1 $ , $ K_{2} = 3 $ , $ K_{3} = 10 $ and $ K_{4} = 30 $
Figure 6.
Cell, substrate, and inhibitor densities of system (3) when $ U = 6.5 $ . The wild type $ u $ and mutants $ v_{1} $ , $ v_{2} $ go extinction in the drug on drug off model, while mutant $ v_{3} $ , substrate $ S $ , and inhibitor $ P $ oscillate in the long-term. The inset figure shows the density of mutant $ v_{3} $ (green) and concentration of the Substrate $ S $ (blue) in the long-term. In this figure, we take $ S^{0} = 10 $ , $ D = 0.9 $ , $ P^{0} = 10, $ $ q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4}, $ $ m = 0.3 $ , $ r = 0.5 $ , $ a = 0.5 $ , $ L = 1 $ , $ K_{1} = 1 $ , $ K_{2} = 3 $ , $ K_{3} = 10 $ and $ K_{4} = 30 $
Figure 7.
Cell, substrate, and inhibitor densities of system (3) when $ U = 6.1 $ . The wild type $ u $ and mutants $ v_{1} $ , $ v_{2} $ go extinction in the drug on drug off model, while mutant $ v_{3} $ , substrate $ S $ , and inhibitor $ P $ persist at fixed values in the long-term. In this figure, we take $ S^{0} = 10 $ , $ D = 0.9 $ , $ P^{0} = 10, $ $ q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4}, $ $ m = 0.3 $ , $ r = 0.5 $ , $ a = 0.5 $ , $ L = 1 $ , $ K_{1} = 1 $ , $ K_{2} = 3 $ , $ K_{3} = 10 $ and $ K_{4} = 30 $
Figure 8.
Extinction of all the microbes of system (4). In this case, all the cells and inhibitor go to extinction in the drug on drug off model, while the substrate persists at a fixed level. In this figure, we take $ S^{0} = 10 $ , $ D = 0.9 $ , $ P^{0} = 10, $ $ U = 5 $ , $ q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4} $ , $ \tilde{q}_{01} = \tilde{q}_{02} = \tilde{q}_{03} = \tilde{q}_{12} = \tilde{q}_{13} = \tilde{q}_{23} = 10^{-4} $ , $ m = 0.08 $ , $ r = 0.5 $ , $ a = 0.5 $ , $ L = 1 $ , $ K_{1} = 1 $ , $ K_{2} = 3 $ , $ K_{3} = 10 $ and $ K_{4} = 30 $
Figure 9.
Persistence of the all the microbes of system (4). In this case, all the microbes persist in the drug on drug off model. However, the most resistant microbe dominates all the species. In this figure, we take $ S^{0} = 10 $ , $ D = 0.9 $ , $ P^{0} = 10, $ $ U = 6 $ , $ q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 0.005 $ , $ \tilde{q}_{01} = \tilde{q}_{02} = \tilde{q}_{03} = \tilde{q}_{12} = \tilde{q}_{13} = \tilde{q}_{23} = 0.005 $ , $ m = 0.3 $ , $ r = 0.5 $ , $ a = 0.5 $ , $ L = 1 $ , $ K_{1} = 1 $ , $ K_{2} = 3 $ , $ K_{3} = 10 $ and $ K_{4} = 30 $
Figure 10.
Persistence of the all the microbes of system (4). In this case, all the microbes persist in the drug on drug off model. In this figure, we take $ S^{0} = 10 $ , $ D = 0.9 $ , $ P^{0} = 10, $ $ U = 6 $ , $ q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 0.05 $ , $ \tilde{q}_{01} = \tilde{q}_{02} = \tilde{q}_{03} = \tilde{q}_{12} = \tilde{q}_{13} = \tilde{q}_{23} = 0.05 $ , $ m = 0.3 $ , $ r = 0.5 $ , $ a = 0.5 $ , $ L = 1 $ , $ K_{1} = 1 $ , $ K_{2} = 3 $ , $ K_{3} = 10 $ and $ K_{4} = 30 $
| [1] |
Giuseppe Toscani. A kinetic description of mutation processes in bacteria. Kinetic & Related Models, 2013, 6 (4) : 1043-1055. doi: 10.3934/krm.2013.6.1043 |
| [2] |
Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129 |
| [3] |
Cristian Tomasetti, Doron Levy. An elementary approach to modeling drug resistance in cancer. Mathematical Biosciences & Engineering, 2010, 7 (4) : 905-918. doi: 10.3934/mbe.2010.7.905 |
| [4] |
Avner Friedman, Najat Ziyadi, Khalid Boushaba. A model of drug resistance with infection by health care workers. Mathematical Biosciences & Engineering, 2010, 7 (4) : 779-792. doi: 10.3934/mbe.2010.7.779 |
| [5] |
Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227 |
| [6] |
Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014 |
| [7] |
Ami B. Shah, Katarzyna A. Rejniak, Jana L. Gevertz. Limiting the development of anti-cancer drug resistance in a spatial model of micrometastases. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1185-1206. doi: 10.3934/mbe.2016038 |
| [8] |
Yu Wu, Xiaopeng Zhao, Mingjun Zhang. Dynamics of stochastic mutation to immunodominance. Mathematical Biosciences & Engineering, 2012, 9 (4) : 937-952. doi: 10.3934/mbe.2012.9.937 |
| [9] |
Roman Czapla, Vladimir V. Mityushev. A criterion of collective behavior of bacteria. Mathematical Biosciences & Engineering, 2017, 14 (1) : 277-287. doi: 10.3934/mbe.2017018 |
| [10] |
John Boscoh H. Njagarah, Farai Nyabadza. Modelling the role of drug barons on the prevalence of drug epidemics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 843-860. doi: 10.3934/mbe.2013.10.843 |
| [11] |
Timothy C. Reluga, Jan Medlock. Resistance mechanisms matter in SIR models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 553-563. doi: 10.3934/mbe.2007.4.553 |
| [12] |
Don A. Jones, Hal L. Smith, Horst R. Thieme. Spread of viral infection of immobilized bacteria. Networks & Heterogeneous Media, 2013, 8 (1) : 327-342. doi: 10.3934/nhm.2013.8.327 |
| [13] |
Maia Martcheva, Mimmo Iannelli, Xue-Zhi Li. Subthreshold coexistence of strains: the impact of vaccination and mutation. Mathematical Biosciences & Engineering, 2007, 4 (2) : 287-317. doi: 10.3934/mbe.2007.4.287 |
| [14] |
P. Magal, G. F. Webb. Mutation, selection, and recombination in a model of phenotype evolution. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 221-236. doi: 10.3934/dcds.2000.6.221 |
| [15] |
Pierre-Emmanuel Jabin. Small populations corrections for selection-mutation models. Networks & Heterogeneous Media, 2012, 7 (4) : 805-836. doi: 10.3934/nhm.2012.7.805 |
| [16] |
Jacek Banasiak, Aleksandra Falkiewicz. A singular limit for an age structured mutation problem. Mathematical Biosciences & Engineering, 2017, 14 (1) : 17-30. doi: 10.3934/mbe.2017002 |
| [17] |
Lambertus A. Peletier. Modeling drug-protein dynamics. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 191-207. doi: 10.3934/dcdss.2012.5.191 |
| [18] |
Sebastian Bonhoeffer, Pia Abel zur Wiesch, Roger D. Kouyos. Rotating antibiotics does not minimize selection for resistance. Mathematical Biosciences & Engineering, 2010, 7 (4) : 919-922. doi: 10.3934/mbe.2010.7.919 |
| [19] |
Donato Patrizia, Andrey Piatnitski. On the effective interfacial resistance through rough surfaces. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1295-1310. doi: 10.3934/cpaa.2010.9.1295 |
| [20] |
Piotr Bajger, Mariusz Bodzioch, Urszula Foryś. Singularity of controls in a simple model of acquired chemotherapy resistance. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2039-2052. doi: 10.3934/dcdsb.2019083 |
2018 Impact Factor: 0.925
Tools
Metrics
Other articles
by authors
[Back to Top]