2011, 8(2): 475-502. doi: 10.3934/mbe.2011.8.475

Bacteria--phagocyte dynamics, axiomatic modelling and mass-action kinetics

1. 

Department of Computer Science and Applied Mathematics, The Weizmann Institute, Rehovot, 76100, Israel

2. 

The Estrin Family Chair of Computer Science and Applied Mathematics, Department of Computer Science and Applied Mathematics, The Weizmann Institute, Rehovot, 76100, Israel

Received  March 2010 Revised  October 2010 Published  April 2011

Axiomatic modeling is ensued to provide a family of models that describe bacterial growth in the presence of phagocytes, or, more generally, prey dynamics in a large spatially homogenous eco-system. A classification of the possible bifurcation diagrams that arise in such models is presented. It is shown that other commonly used models that do not belong to this class may miss important features that are associated with the limited growth curve of the bacteria (prey) and the saturation associated with the phagocytosis (predator kill) term. Notably, these features appear at relatively low concentrations, much below the saturation range. Finally, combining this model with a model of neutrophil dynamics in the blood after chemotherapy treatments we obtain new insights regarding the development of infections under neutropenic conditions.
Citation: Roy Malka, Vered Rom-Kedar. Bacteria--phagocyte dynamics, axiomatic modelling and mass-action kinetics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 475-502. doi: 10.3934/mbe.2011.8.475
References:
[1]

C. A. Janeway and R. Medzhitov, Innate immune recognition,, Annual Review of Immunology, 20 (2002), 197. doi: 10.1146/annurev.immunol.20.083001.084359. Google Scholar

[2]

O. Soehnlein and L. Lindbom, Phagocyte partnership during the onset and resolution of inflammation,, Nature Reviews Immunology, 10 (2010), 427. doi: 10.1038/nri2779. Google Scholar

[3]

C. Nathan, Neutrophils and immunity: Challenges and opportunities,, Nature Reviews Immunology, 6 (2006), 173. doi: 10.1038/nri1785. Google Scholar

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A. F. M. Marée, M. Komba, C. Dyck, M. Labecki, D. T. Finegood and L. Edelstein-Keshet, Quantifying macrophage defects in type 1 diabetes,, Journal of theoretical biology, 233 (2005), 533. doi: 10.1016/j.jtbi.2004.10.030. Google Scholar

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A. F. M. Maree, M. Komba, D. T. Finegood and L. Edelstein-Keshet, A quantitative comparison of rates of phagocytosis and digestion of apoptotic cells by macrophages from normal (BALB/c) and diabetes-prone (NOD) mice,, Journal of Applied Physiology, 104 (2008), 157. doi: 10.1152/japplphysiol.00514.2007. Google Scholar

[6]

R. Malka, E. Shochat and V. Rom-Kedar, Bistability and bacterial infections,, PLoS ONE, 5 (2010). doi: 10.1371/journal.pone.0010010. Google Scholar

[7]

Z. Rahman, L. Esparza-Guerra, H. Y. Yap, G. Fraschini, G. Bodey and G. Hortobagyi, Chemotherapy-induced neutropenia and fever in patients with metastatic breast carcinoma receiving salvage chemotherapy,, Cancer, 79 (1997), 1150. doi: 10.1002/(SICI)1097-0142(19970315)79:6<1150::AID-CNCR13>3.0.CO;2-Z. Google Scholar

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G. P. Bodey, M. Buckley, Y. S. Sathe and E. J. Freireich, Quantitative relationships between circulating leukocytes and infection in patients with acute leukemia,, Annals of Internal Medicine, 64 (1966), 328. Google Scholar

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J. M. van den Berg, E. van Koppen, A. Åhlin, B. H. Belohradsky, E. Bernatowska, L. Corbeel, T. Español, A. Fischer, M. Kurenko-Deptuch, R. Mouy, T. Petropoulou, J. Roesler, R. Seger, M. J. Stasia, N. H. Valerius, R. S. Weening, B. Wolach, D. Roos and T. W. Kuijpers, Chronic granulomatous disease: The european experience,, PLoS ONE, 4 (2009). doi: 10.1371/journal.pone.0005234. Google Scholar

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L. Edelstein-Keshet, J. Watmough and D. Grunbaum, Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts,, Journal of Mathematical Biology, 36 (1998), 515. doi: 10.1007/s002850050112. Google Scholar

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Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system,, Biophysical Journal, 94 (2008), 3684. doi: 10.1529/biophysj.107.120824. Google Scholar

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J. R. Pomerening, E. D. Sontag and J. E. Ferrell, Building a cell cycle oscillator: Hysteresis and bistability in the activation of cdc2,, Nature Cell Biology, 5 (2003), 346. doi: 10.1038/ncb954. Google Scholar

[25]

D. Angeli, J. E. Ferrell and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems,, Proceedings of the National Academy of Sciences, 101 (2004), 1822. doi: 10.1073/pnas.0308265100. Google Scholar

[26]

E. D. Sontag, Monotone and near-monotone biochemical networks,, Systems and Synthetic Biology, 1 (2007), 59. doi: 10.1007/s11693-007-9005-9. Google Scholar

[27]

V. S. Afraimovich, V. P. Zhigulin and M. I. Rabinovich, On the origin of reproducible sequential activity in neural circuits,, Chaos, 14 (2004), 1123. doi: 10.1063/1.1819625. Google Scholar

[28]

T. Gross and U. Feudel, Generalized models as a universal approach to the analysis of nonlinear dynamical systems,, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.016205. Google Scholar

[29]

E. Shochat, V. Rom-Kedar and L. A. Segel, G-CSF control of neutrophil dynamics in the blood,, Bull. Math. Biology, 69 (2007), 2299. doi: 10.1007/s11538-007-9221-1. Google Scholar

[30]

M. Chromek, Z. Slamova, P. Bergman, L. Kovacs, L. Podracka, I. Ehren, T. Hokfelt, G. H. Gudmundsson, R. L. Gallo, B. Agerberth and A. Brauner, The antimicrobial peptide cathelicidin protects the urinary tract against invasive bacterial infection,, Nature Medicine, 12 (2006), 636. doi: 10.1038/nm1407. Google Scholar

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M. H. Zwietering, I. Jongenburger, F. M. Rombouts and K. VAN 'T Riet, Modeling of the bacterial growth curve,, Application of Environmental Microbiology, 56 (1990), 1875. Google Scholar

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P. C. J. Leijh, M. T. van den Barselaar, T. L. van Zwet, I. Dubbeldeman-Rempt and R. van Furth, Kinetics of phagocytosis of Staphylococcus aureus and Escherichia coli by human granulocytes,, Immunology, 37 (1979), 453. Google Scholar

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C. C. Clawson and J. E. Repine, Quantitation of maximal bactericidal capability in human neutrophils,, Journal of Laboratory and Clinical Medicine, 88 (1976), 316. Google Scholar

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P. K. Peterson, J. Verhoef, D. Schmeling and P. G. Quie, Kinetics of phagocytosis and bacterial killing by human polymorphonuclear leukocytes and monocytes,, Journal of Infectious Diseases, 136 (1977), 502. doi: 10.1093/infdis/136.4.502. Google Scholar

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S. Budhu, J. D. Loike, A. Pandolfi, S. Han, G. Catalano, A. Constantinescu, R. Clynes and S. C. Silverstein, CD8+ T cell concentration determines their efficiency in killing cognate antigen-expressing syngeneic mammalian cells in vitro and in mouse tissues,, The Journal of Experimental Medicine, 207 (2010), 223. doi: 10.1084/jem.20091279. Google Scholar

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A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations,", World Scientific Pub Co Inc, (1998). Google Scholar

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show all references

References:
[1]

C. A. Janeway and R. Medzhitov, Innate immune recognition,, Annual Review of Immunology, 20 (2002), 197. doi: 10.1146/annurev.immunol.20.083001.084359. Google Scholar

[2]

O. Soehnlein and L. Lindbom, Phagocyte partnership during the onset and resolution of inflammation,, Nature Reviews Immunology, 10 (2010), 427. doi: 10.1038/nri2779. Google Scholar

[3]

C. Nathan, Neutrophils and immunity: Challenges and opportunities,, Nature Reviews Immunology, 6 (2006), 173. doi: 10.1038/nri1785. Google Scholar

[4]

A. F. M. Marée, M. Komba, C. Dyck, M. Labecki, D. T. Finegood and L. Edelstein-Keshet, Quantifying macrophage defects in type 1 diabetes,, Journal of theoretical biology, 233 (2005), 533. doi: 10.1016/j.jtbi.2004.10.030. Google Scholar

[5]

A. F. M. Maree, M. Komba, D. T. Finegood and L. Edelstein-Keshet, A quantitative comparison of rates of phagocytosis and digestion of apoptotic cells by macrophages from normal (BALB/c) and diabetes-prone (NOD) mice,, Journal of Applied Physiology, 104 (2008), 157. doi: 10.1152/japplphysiol.00514.2007. Google Scholar

[6]

R. Malka, E. Shochat and V. Rom-Kedar, Bistability and bacterial infections,, PLoS ONE, 5 (2010). doi: 10.1371/journal.pone.0010010. Google Scholar

[7]

Z. Rahman, L. Esparza-Guerra, H. Y. Yap, G. Fraschini, G. Bodey and G. Hortobagyi, Chemotherapy-induced neutropenia and fever in patients with metastatic breast carcinoma receiving salvage chemotherapy,, Cancer, 79 (1997), 1150. doi: 10.1002/(SICI)1097-0142(19970315)79:6<1150::AID-CNCR13>3.0.CO;2-Z. Google Scholar

[8]

G. P. Bodey, M. Buckley, Y. S. Sathe and E. J. Freireich, Quantitative relationships between circulating leukocytes and infection in patients with acute leukemia,, Annals of Internal Medicine, 64 (1966), 328. Google Scholar

[9]

J. M. van den Berg, E. van Koppen, A. Åhlin, B. H. Belohradsky, E. Bernatowska, L. Corbeel, T. Español, A. Fischer, M. Kurenko-Deptuch, R. Mouy, T. Petropoulou, J. Roesler, R. Seger, M. J. Stasia, N. H. Valerius, R. S. Weening, B. Wolach, D. Roos and T. W. Kuijpers, Chronic granulomatous disease: The european experience,, PLoS ONE, 4 (2009). doi: 10.1371/journal.pone.0005234. Google Scholar

[10]

A. Reynolds, J. Rubin, G. Clermont, J. Day, Y. Vodovotz and B. G. Ermentrout, A reduced mathematical model of acute inflammation response: I. derivation of model and analysis of anti-inflammation,, Journal of Theoretical Biology, 242 (2006), 220. doi: 10.1016/j.jtbi.2006.02.016. Google Scholar

[11]

M. C. Herald, General model of inflammation,, Bulletin of Mathematical Biology, 72 (2010), 765. doi: 10.1007/s11538-009-9468-9. Google Scholar

[12]

E. M. C. D'Agata, M. Dupont-Rouzeyrol, P. Magal, D. Olivier and S. Ruan, The impact of different antibiotic regimens on the emergence of antibiotic-resistant bacteria,, PLoS ONE, 3 (2008). Google Scholar

[13]

Y. Li, A. Karlin, D. J. Loike and C. S. Silverstein, A critical concentration of neutrophils is required for effective bacterial killing in suspension,, Proceedings of the National Academy of Sciences, 99 (2002), 8289. doi: 10.1073/pnas.122244799. Google Scholar

[14]

Y. Li, A. Karlin, D. J. Loike and C. S. Silverstein, Determination of the critical concentration of neutrophils required to block bacterial growth in tissues,, The Journal of Experimental Medicine, 200 (2004), 613. doi: 10.1084/jem.20040725. Google Scholar

[15]

E. Shochat and V. Rom-Kedar, Novel strategies for g-csf treatment of high-risk severe neutropenia suggested by mathematical modeling,, Clinical Cancer Research, 14 (2008), 6354. doi: 10.1158/1078-0432.CCR-08-0807. Google Scholar

[16]

R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence,, Journal of Theoretical Biology, 139 (1989), 311. doi: 10.1016/S0022-5193(89)80211-5. Google Scholar

[17]

A. D. Bazykin, F. S. Berezovskaya, G. A. Denisov and Y. A. Kuznetzov, The influence of predator saturation effect and competition among predators on predator-prey system dynamics,, Ecological Modelling, 14 (1981), 39. doi: 10.1016/0304-3800(81)90013-2. Google Scholar

[18]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response,, SIAM Journal of Applied Mathematics, 16 (2001), 1445. Google Scholar

[19]

L. Edelstein-Keshet, J. Watmough and D. Grunbaum, Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts,, Journal of Mathematical Biology, 36 (1998), 515. doi: 10.1007/s002850050112. Google Scholar

[20]

Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system,, Biophysical Journal, 94 (2008), 3684. doi: 10.1529/biophysj.107.120824. Google Scholar

[21]

M. Golubitsky, I. Stewart, P. L. Buono and J. J. Collins, A modular network for legged locomotion,, Physica D, 115 (1998), 56. doi: 10.1016/S0167-2789(97)00222-4. Google Scholar

[22]

M. W. Hirsch and H. Smith, Monotone dynamical systems,, in, (2005), 239. Google Scholar

[23]

D. Angeli and E. D. Sontag, Monotone control systems,, IEEE Transactions on Automatic Control, 48 (2003), 1684. doi: 10.1109/TAC.2003.817920. Google Scholar

[24]

J. R. Pomerening, E. D. Sontag and J. E. Ferrell, Building a cell cycle oscillator: Hysteresis and bistability in the activation of cdc2,, Nature Cell Biology, 5 (2003), 346. doi: 10.1038/ncb954. Google Scholar

[25]

D. Angeli, J. E. Ferrell and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems,, Proceedings of the National Academy of Sciences, 101 (2004), 1822. doi: 10.1073/pnas.0308265100. Google Scholar

[26]

E. D. Sontag, Monotone and near-monotone biochemical networks,, Systems and Synthetic Biology, 1 (2007), 59. doi: 10.1007/s11693-007-9005-9. Google Scholar

[27]

V. S. Afraimovich, V. P. Zhigulin and M. I. Rabinovich, On the origin of reproducible sequential activity in neural circuits,, Chaos, 14 (2004), 1123. doi: 10.1063/1.1819625. Google Scholar

[28]

T. Gross and U. Feudel, Generalized models as a universal approach to the analysis of nonlinear dynamical systems,, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.016205. Google Scholar

[29]

E. Shochat, V. Rom-Kedar and L. A. Segel, G-CSF control of neutrophil dynamics in the blood,, Bull. Math. Biology, 69 (2007), 2299. doi: 10.1007/s11538-007-9221-1. Google Scholar

[30]

M. Chromek, Z. Slamova, P. Bergman, L. Kovacs, L. Podracka, I. Ehren, T. Hokfelt, G. H. Gudmundsson, R. L. Gallo, B. Agerberth and A. Brauner, The antimicrobial peptide cathelicidin protects the urinary tract against invasive bacterial infection,, Nature Medicine, 12 (2006), 636. doi: 10.1038/nm1407. Google Scholar

[31]

M. H. Zwietering, I. Jongenburger, F. M. Rombouts and K. VAN 'T Riet, Modeling of the bacterial growth curve,, Application of Environmental Microbiology, 56 (1990), 1875. Google Scholar

[32]

P. C. J. Leijh, M. T. van den Barselaar, T. L. van Zwet, I. Dubbeldeman-Rempt and R. van Furth, Kinetics of phagocytosis of Staphylococcus aureus and Escherichia coli by human granulocytes,, Immunology, 37 (1979), 453. Google Scholar

[33]

C. C. Clawson and J. E. Repine, Quantitation of maximal bactericidal capability in human neutrophils,, Journal of Laboratory and Clinical Medicine, 88 (1976), 316. Google Scholar

[34]

M. C. Hammer, A. L. Baltch, N. T. Sutphen, R. P. Smith and J. V. Conroy, Pseudomonas aeruginosa: Quantitation of maximum phagocytic and bactericidal capabilities of normal human granulocytes,, Journal of Laboratory and Clinical Medicine, 98 (1981), 938. Google Scholar

[35]

P. K. Peterson, J. Verhoef, D. Schmeling and P. G. Quie, Kinetics of phagocytosis and bacterial killing by human polymorphonuclear leukocytes and monocytes,, Journal of Infectious Diseases, 136 (1977), 502. doi: 10.1093/infdis/136.4.502. Google Scholar

[36]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,", Cambridge University Press, (1995). Google Scholar

[37]

K. P. Hadeler and P. Van den Driessche, Backward bifurcation in epidemic control,, Mathematical Biosciences, 146 (1997), 15. doi: 10.1016/S0025-5564(97)00027-8. Google Scholar

[38]

J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases,, Journal of Mathematical Biology, 36 (1998), 227. doi: 10.1007/s002850050099. Google Scholar

[39]

F. Brauer, Backward bifurcations in simple vaccination models,, Journal of Mathematical Analysis and Applications, 298 (2004), 418. doi: 10.1016/j.jmaa.2004.05.045. Google Scholar

[40]

S. Budhu, J. D. Loike, A. Pandolfi, S. Han, G. Catalano, A. Constantinescu, R. Clynes and S. C. Silverstein, CD8+ T cell concentration determines their efficiency in killing cognate antigen-expressing syngeneic mammalian cells in vitro and in mouse tissues,, The Journal of Experimental Medicine, 207 (2010), 223. doi: 10.1084/jem.20091279. Google Scholar

[41]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction,, Ecology, 56 (1975), 881. doi: 10.2307/1936298. Google Scholar

[42]

A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations,", World Scientific Pub Co Inc, (1998). Google Scholar

[43]

R. Lindqvist, Estimation of Staphylococcus aureus growth parameters from turbidity data: Characterization of strain variation and comparison of methods,, Applied and Environmental Microbiology, 72 (2006), 4862. doi: 10.1128/AEM.00251-06. Google Scholar

[44]

R. J. Carroll, D. Ruppert, L. A. Stefanski and C. M. Crainiceanu, "Measurement Error in Nonlinear Models: A Modern Perspective,", Chapman and Hall/CRC, (2006). Google Scholar

[45]

W. Press, S. Teukolsky, W. Vetterling and B. Flannery, "Numerical Recipes in C,", Cambridge University Press, (1992). Google Scholar

[46]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the nelder-mead simplex method in low dimensions,, SIAM Journal of Optimization, 9 (1999), 112. doi: 10.1137/S1052623496303470. Google Scholar

[47]

B. Efron, R. Tibshirani and R. J. Tibshirani, "An Introduction to the Bootstrap,", Chapman & Hall/CRC, (1993). Google Scholar

[48]

M. A. Nowak and R. M. May, "Virus Dynamics,", Oxford University Press, (2000). Google Scholar

[49]

A. A. Berryman, The orgins and evolution of predator-prey theory,, Ecology, 73 (1992), 1530. doi: 10.2307/1940005. Google Scholar

[50]

V. Lee, C. K. Li, M. M. K. Shing, K. W. Chik, K. Li, K. S. Tsang, D. C. Zhao, D. H. Lai, A. Wong and P. M. P. Yuen, Single vs twice daily g-csf dose for peripheral blood stem cells harvest in normal donors and children with non-malignant diseases,, Bone Marrow Transplantation, 25 (2000), 931. doi: 10.1038/sj.bmt.1702338. Google Scholar

[51]

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