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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.27 n.1 San José Jan./Jun. 2020

http://dx.doi.org/10.15517/rmta.v27i1.39965 

Artículo

Backward bifurcation in neutrophil-pathogen interaction

Bifurcación hacia atrás en interacción neutrófilo-patógeno

Christopher M. Kribs1 

Omomayowa Olawoyin2 

1University of Texas at Arlington, Department of Mathematics, Arlington TX 76012, United States; kribs@uta.edu

2University of Texas at Arlington, Department of Mathematics, Arlington TX 76012, United States; omomayowa.olawoyin@mavs.uta.edu

Abstract

Bacterial infections elicit immune responses including neutrophils, whose recruitment is stimulated by the bacteria’s presence but which die after eliminating those bacteria. This dual interaction between bacteria and neutrophil concentrations, more complicated than the simple predator-prey relationship that describes macrophage-bacteria interactions, creates an environment in which neutrophils may only be able to clear sufficiently small infections. This study describes this relationship using a simple nonlinear dynamical system which exhibits bistability behavior known as a backward bifurcation. Bacterial growth is assumed limited by a key nutrient. In contrast to a previous study which held neutrophil and nutrient levels constant and required saturation terms to produce bistability, our model shows that simple bilinear terms support bistability when nutrient and neutrophil densities are allowed to vary in response to bacterial density. An example application involving Borrelia burgdorferi, which feeds on manganese, illustrates why neutrophils’ rapid response is key to their ability to contain bacterial infections.

Keywords: bistability; resource limitations; phagocytosis; hysteresis.

Resumen

Las infecciones bacterianas provocan respuestas inmunitarias, incluyendo neutrófilos, cuyo reclutamiento es estimulado por la presencia de la bacteria pero que muere después de eliminar esas bacterias. Esta doble interacción entre las concentraciones de bacterias y neutrófilos, más complicada que la simple relación depredador-presa que describe las interacciones entre bacterias y macrófagos, crea un ambiente en el que los neutrófilos tal vez sólo puedan despejar infecciones suficientemente pequeñas. Este estudio describe esta relación utilizando un sistema dinámico no lineal sencillo que exhibe un comportamiento de biestabilidad conocido como una bifurcación hacia atrás. El crecimiento bacteriano se supone limitado por un nutriente clave. En contraste con un estudio anterior que mantuvo los niveles de neutrófilos y nutrientes constantes y requería términos de saturación para producir la biestabilidad, nuestro modelo muestra que los términos bilineales sencillos fomentan la biestabilidad cuando las densidades de nutrientes y neutrófilos pueden variar en respuesta a la densidad bacteriana. Un ejemplo aplicado a la bacteria Borrelia burgdorferi, que se alimenta de manganeso, ilustra por qué la respuesta rápida de los neutrófilos es clave para su capacidad de contener las infecciones bacterianas.

Palabras clave: biestabilidad; recursos limitados; fagocitosis; histéresis.

Mathematics Subject Classification: 92C37.

Ver contenido complete en PDF.

Acknowledgements

This project was inspired by 4, for which the authors thank their colleagues. CK dedicates this project to the memory of Fred and Carolyn Kribs.

References

R, Antia; J,C, Koella. A model of non-specific immunity, Journal of Theoretical Biology 168(1994), no. 2, 141-150. Doi: 10.1006/jtbi.1994.1094 [ Links ]

M,G, Baker; C,D, Simpson; B, Stover; L, Sheppard; H, Checkoway; B,A, Racette; N,S, Seixas. Blood manganese as an exposure biomarker: state of the evidence, Journal of Occupational and Environmental Hygiene 11(2014), no. 4, 210-217. Doi: 10.1080/15459624.2013.852280 [ Links ]

V, Bane; M, Lehane; M, Dikshit; A, O’Riordan; A, Furey. Tetrodotoxin: chemistry, toxicity, source, distribution and detection, Toxins 6(2014), no. 2, 693-755. Doi: 10.3390/toxins6020693 [ Links ]

A, Carter; A, Toja; O, Olawoyin; J, Gonzalez; J, Grover; H, Kojouharov; C, Kribs Zaleta. A theoretical model of coinfection dynamics: modeling competition dynamics between Borrelia burgdorferi and Anaplasma phagocytophilum within a human host, Mathematics Department, Preprint Series 2014-06, University of Texas at Arlington, 2014. [ Links ]

V,G, Daniels; P,R, Wheater; H,G, Burkitt. Functional histology: A text and colour atlas, Churchill Livingstone, Edinburgh, 1979. [ Links ]

A,N, Datar; N, Kaur; S, Patel; D,F, Luecke; E, Sapi. In vitro effectiveness of samento and banderol herbal extracts on the different morphological forms of Borrelia burgdorferi, Ph.D. Thesis, Univ. of New Haven, West Haven CT, 2010. [ Links ]

H, Drakesmith; A,M, Prentice. Hepcidin and the iron-infection axis, Science 338(2012), no. 6108, 768-772. Doi: 10.1126/science.1224577 [ Links ]

L, Geggel. How much blood is in the human body, LiveScience, 2016. Available at https://www.livescience.com/32213-how-much-blood-is-in-the-human-body.htmlLinks ]

R,B, Jain; Y,S, Choi. Normal reference ranges for and variability in the levels of blood manganese and selenium by gender, age, and race/ethnicity for general U.S. population, Journal of Trace Elements in Medicine and Biology 30(2015), 142-152. Doi: 10.1016/j.jtemb.2014.12.004 [ Links ]

Y, Li; A, Karlin; J,D, Loike; S,C, Silverstein. A critical concentration of neutrophils is required for effective bacterial killing in suspension, Proceedings of the National Academy of Sciences of the United States of America 99(2002), no. 12, 8289-8294. Doi: 10.1073/pnas.122244799 [ Links ]

R, Malka; V, Rom-Kedar. Bacteria-phagocyte dynamics, axiomatic modelling and mass-action kinetics, Mathematical Biosciences and Engineering 8(2011), no. 2, 475-502. Doi: 10.3934/mbe.2011.8.475 [ Links ]

R, Malka; E, Shochat; V, Rom-Kedar. Bistability and bacterial infections, PLoS ONE 5(2010), no. 5, 1-10. Doi: 10.1371/journal.pone.0010010 [ Links ]

R, Malka; B, Wolach; R, Gavrieli; E, Shochat; V, Rom-Kedar. Evidence for bistable bacteria-neutrophil interaction and its clinical implications, Journal of Clinical Investigation 122(2012), no. 8, 3002-3011. Doi: 10.1172/JCI59832 [ Links ]

H, Mayer; K, S, Zaenker; U, an der Heiden. A basic mathematical model of the immune response, Chaos: An Interdisciplinary Journal of Nonlinear Science 5(1995), no. 1, 155-161. Doi: 10.1063/1.166098 [ Links ]

M,J, Niemiec; B, De Samber; J, Garrevoet; E, Vergucht; B, Vekemans; R, De Rycke; E, Björn; L, Sandblad; G, Wellenreuther; G, Falkenberg; P, Cloetens; L, Vincze; C,F, Urban. Trace element landscape of resting and activated human neutrophils on sub-micrometer level, Metallomics 7(2015), no. 6, 996-1010. Doi: 10.1039/c4mt00346b [ Links ]

G, Petschenka; A,A, Agrawal. Milkweed butterfly resistance to plant toxins is linked to sequestration, not coping with a toxic diet, Proceedings of the Royal Society B 282(2015), no. 1818: 20151865, 1-9. Doi: 10.1098/rspb.2015.1865 [ Links ]

S,S, Pilyugin; R, Antia. Modeling immune responses with handling time, Bulletin of Mathematical Biology 62(2000), no. 5, 869-890. Doi: 10.1006/bulm.2000.0181 [ Links ]

J,E, Posey; F,C, Gherardini. Lack of a role for iron in the Lyme disease pathogen, Science 288(2000), no. 5471, 1651-1653. Doi: 10.1126/science.288.5471.1651 [ Links ]

A, B, Santamaria. Manganese exposure, essentiality & toxicity, Indian Journal of Medical Research 128(2008), no. 4, 484-500. [ Links ]

S,C, Silverstein; R, Rabadan. How many neutrophils are enough (redux, redux)?, Journal of Clinical Investigation 122(2012), no. 8, 2776-2779. Doi: 10.1172/JCI63939 [ Links ]

A M, Smith; J,A, McCullers; F,R, Adler. Mathematical model of a three-stage innate immune response to a pneumococcal lung infection, Journal of Theoretical Biology 276(2011), no. 1, 106-116. Doi: 10.1016/j.jtbi.2011.01.052 [ Links ]

T, Tak; K, Tesselaar; J, Pillay; J,A, Borghans; L, Koenderman. What’s your age again? Determination of human neutrophil half-lives revisited, Journal of Leukocyte Biology 94(2013), no. 4, 595-601. Doi: 10.1189/jlb.1112571 [ Links ]

B, Troxell; M, Ye; Y, Yang; S,E, Carrasco; Y, Lou; X,F, Yang. Manganese and zinc regulate virulence determinants in Borrelia burgdorferi, Infection and Immunity 81(2013), no. 8, 2743-2752. Doi: 10.1128/IAI.00507-13 [ Links ]

J,R, Zhang; S,J, Norris. Kinetics and in vivo induction of genetic variation of vlsE in Borrelia burgdorferi, Infection and Immunity 66(1998), no. 8, 3689-3697. [ Links ]

Received: May 12, 2019; Revised: August 31, 2019; Accepted: September 17, 2019

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