Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 92-06, 92B99; Secondary: 92D25.

 Citation:

•  [1] C. A. Janeway and R. Medzhitov, Innate immune recognition, Annual Review of Immunology, 20 (2002), 197-216.doi: 10.1146/annurev.immunol.20.083001.084359. [2] O. Soehnlein and L. Lindbom, Phagocyte partnership during the onset and resolution of inflammation, Nature Reviews Immunology, 10 (2010), 427-439.doi: 10.1038/nri2779. [3] C. Nathan, Neutrophils and immunity: Challenges and opportunities, Nature Reviews Immunology, 6 (2006), 173-182.doi: 10.1038/nri1785. [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), 533-551.doi: 10.1016/j.jtbi.2004.10.030. [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), 157-169.doi: 10.1152/japplphysiol.00514.2007. [6] R. Malka, E. Shochat and V. Rom-Kedar, Bistability and bacterial infections, PLoS ONE, 5 (2010), e10010.doi: 10.1371/journal.pone.0010010. [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), 1150-1157.doi: 10.1002/(SICI)1097-0142(19970315)79:6<1150::AID-CNCR13>3.0.CO;2-Z. [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), 328-340. [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), e5234.doi: 10.1371/journal.pone.0005234. [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), 220-236.doi: 10.1016/j.jtbi.2006.02.016. [11] M. C. Herald, General model of inflammation, Bulletin of Mathematical Biology, 72 (2010), 765-779.doi: 10.1007/s11538-009-9468-9. [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), e4306. [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), 8289-8294.doi: 10.1073/pnas.122244799. [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), 613-622.doi: 10.1084/jem.20040725. [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), 6354-6363.doi: 10.1158/1078-0432.CCR-08-0807. [16] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, Journal of Theoretical Biology, 139 (1989), 311-326.doi: 10.1016/S0022-5193(89)80211-5. [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), 39-57.doi: 10.1016/0304-3800(81)90013-2. [18] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal of Applied Mathematics, 16 (2001), 1445-1472. [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), 515-549.doi: 10.1007/s002850050112. [20] Y. Mori, A. Jilkine and L. Edelstein-Keshet, Wave-pinning and cell polarity from a bistable reaction-diffusion system, Biophysical Journal, 94 (2008), 3684-3697.doi: 10.1529/biophysj.107.120824. [21] M. Golubitsky, I. Stewart, P. L. Buono and J. J. Collins, A modular network for legged locomotion, Physica D, 115 (1998), 56-72.doi: 10.1016/S0167-2789(97)00222-4. [22] M. W. Hirsch and H. Smith, Monotone dynamical systems, in "Handbook of Differential Equations: Ordinary Differential Equations 2," Elsevier, 2005, 239-357. [23] D. Angeli and E. D. Sontag, Monotone control systems, IEEE Transactions on Automatic Control, 48 (2003), 1684-1698.doi: 10.1109/TAC.2003.817920. [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), 346-351.doi: 10.1038/ncb954. [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), 1822-1827.doi: 10.1073/pnas.0308265100. [26] E. D. Sontag, Monotone and near-monotone biochemical networks, Systems and Synthetic Biology, 1 (2007), 59-87.doi: 10.1007/s11693-007-9005-9. [27] V. S. Afraimovich, V. P. Zhigulin and M. I. Rabinovich, On the origin of reproducible sequential activity in neural circuits, Chaos, 14 (2004), 1123-1129.doi: 10.1063/1.1819625. [28] T. Gross and U. Feudel, Generalized models as a universal approach to the analysis of nonlinear dynamical systems, Phys. Rev. E, 73 (2006), 016205.doi: 10.1103/PhysRevE.73.016205. [29] E. Shochat, V. Rom-Kedar and L. A. Segel, G-CSF control of neutrophil dynamics in the blood, Bull. Math. Biology, 69 (2007), 2299-2338.doi: 10.1007/s11538-007-9221-1. [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), 636-641.doi: 10.1038/nm1407. [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), 1875-1881. [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), 453-465. [33] C. C. Clawson and J. E. Repine, Quantitation of maximal bactericidal capability in human neutrophils, Journal of Laboratory and Clinical Medicine, 88 (1976), 316-327. [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), 938-948. [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), 502-509.doi: 10.1093/infdis/136.4.502. [36] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, 1995. [37] K. P. Hadeler and P. Van den Driessche, Backward bifurcation in epidemic control, Mathematical Biosciences, 146 (1997), 15-35.doi: 10.1016/S0025-5564(97)00027-8. [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), 227-248.doi: 10.1007/s002850050099. [39] F. Brauer, Backward bifurcations in simple vaccination models, Journal of Mathematical Analysis and Applications, 298 (2004), 418-431.doi: 10.1016/j.jmaa.2004.05.045. [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), 223-235.doi: 10.1084/jem.20091279. [41] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.doi: 10.2307/1936298. [42] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations," World Scientific Pub Co Inc, 1998. [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), 4862-4870.doi: 10.1128/AEM.00251-06. [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. [45] W. Press, S. Teukolsky, W. Vetterling and B. Flannery, "Numerical Recipes in C," Cambridge University Press, 2nd edition, 1992. [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), 112-147.doi: 10.1137/S1052623496303470. [47] B. Efron, R. Tibshirani and R. J. Tibshirani, "An Introduction to the Bootstrap," Chapman & Hall/CRC, 1993. [48] M. A. Nowak and R. M. May, "Virus Dynamics," Oxford University Press, 2000. [49] A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535.doi: 10.2307/1940005. [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), 931-935.doi: 10.1038/sj.bmt.1702338. [51] J. Crawford, D. C. Dale and G. H. Lyman, Chemotherapy-induced neutropenia: Risks, consequences, and new directions for its management, Cancer, 100 (2004), 228-237.doi: 10.1002/cncr.11882. [52] C. Cheretakis, R. Leung, C. X. Sun, Y. Dror and M. Glogauer, Timing of neutrophil tissue repopulation predicts restoration of innate immune protection in a murine bone marrow transplantation model, Blood, 108 (2006), 2821-2826.doi: 10.1182/blood-2006-04-018184. [53] K. Todar, Growth of Bacterial Populations, in "Online Textbook of Bacteriology," Accessed 5-24-2010, 2008. [54] M. L. Cohen, M. T. Murphy, G. W. Counts, C. D. Buckner, R. A. Clift and J. D. Meyers, Prediction by surveillance cultures of bacteremia among neutropenic patients treated in a protective environment, Journal of Infectious Diseases, 147 (1983), 789-793.doi: 10.1093/infdis/147.5.789. [55] P. A. Abrams, The fallacies of "ratio-dependent" predation, Ecology, 75 (1994), 1842-1850.doi: 10.2307/1939644. [56] A. A. Berryman, A. P. Gutierrez and R. Arditi, Credible, parsimonious and useful predator-prey models: A reply to Abrams, Gleeson, and Sarnelle, Ecology, 76 (1995), 1980-1985.doi: 10.2307/1940728. [57] P. A. Abrams and L. R. Ginzburg, The nature of predation: Prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15 (2000), 337-341.doi: 10.1016/S0169-5347(00)01908-X. [58] J. Sotomayor, Generic bifurcations of dynamical systems, In "em Proc. Sympos., Univ. Bahia, Salvador, 1971, Dynamical Systems," Academic Press, Berlin, 1973. [59] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, 1983.