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

* Corresponding author

Received  October 2018 Revised  March 2019 Published  July 2019

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.

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 and 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. 

[3]

Z. ChenS. 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. ChengS. 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.

[7]

M. Dragosits and D. Mattanovich, Adaptive laboratory evolution - principles and applications for biotechnology, Microb. Cell Fact., 12 (2013), 64.

[8]

R. HermsenJ. 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. 

[9]

R. Hermsen and T. Hwa, Sources and Sinks: A stochastic model of evolution in heterogeneous environments, Phys. Rev. Lett., 105 (2015), 248104.

[10]

W. M. HirschH. 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.
[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. MelnykA. Wong and R. Kassen, The fitness costs of antibiotic resistance mutations, Evol. Appl., 8 (2015), 273-283. 

[15]

S. B. Levy and B. Marshall, Antibiotic resistance worldwide: causes, challenges and responses, Nat. Med., 10 (2004), s122–s129.

[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.

[17]

M. MwangiS. 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. 

[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.
[20]

E. ToprakA. VeresJ. B. MitchelD. L. Hartl and R. Kishony, Evolutionary paths to antibiotic resistance under dynamically sustained drug selection, Nat. Genet., 44 (2012), 101-106. 

[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. ZhangG. LambertD. LiaoH. KimK. RobinC. TungN. Pourmand and R. H. Austin, Acceleration of emergence of bacterial antibiotic resistance in connected microenvironment, Science, 333 (2011), 1764-1767. 

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. 

[3]

Z. ChenS. 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. ChengS. 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.

[7]

M. Dragosits and D. Mattanovich, Adaptive laboratory evolution - principles and applications for biotechnology, Microb. Cell Fact., 12 (2013), 64.

[8]

R. HermsenJ. 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. 

[9]

R. Hermsen and T. Hwa, Sources and Sinks: A stochastic model of evolution in heterogeneous environments, Phys. Rev. Lett., 105 (2015), 248104.

[10]

W. M. HirschH. 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.
[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. MelnykA. Wong and R. Kassen, The fitness costs of antibiotic resistance mutations, Evol. Appl., 8 (2015), 273-283. 

[15]

S. B. Levy and B. Marshall, Antibiotic resistance worldwide: causes, challenges and responses, Nat. Med., 10 (2004), s122–s129.

[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.

[17]

M. MwangiS. 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. 

[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.
[20]

E. ToprakA. VeresJ. B. MitchelD. L. Hartl and R. Kishony, Evolutionary paths to antibiotic resistance under dynamically sustained drug selection, Nat. Genet., 44 (2012), 101-106. 

[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. ZhangG. LambertD. LiaoH. KimK. RobinC. TungN. Pourmand and R. H. Austin, Acceleration of emergence of bacterial antibiotic resistance in connected microenvironment, Science, 333 (2011), 1764-1767. 

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 $
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 $
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