Article
MSC:
34A30,
34A34,
34C14,
47B65,
47N20,
92C45 | MR 2133728 | Zbl 1099.34006 | DOI: 10.1007/s10492-005-0015-1
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Keywords:
dynamical system; input-output system; chemical network; boundary layer
dynamical system; input-output system; chemical network; boundary layer
Summary:
Our aim is to show a class of mathematical models in application to some problems of cell biology. Typically, our models are described via classical chemical networks. This property is visualized by a conservation law. Mathematically, this conservation law guarantees most of the mathematical properties of the models such as global existence and uniqueness of solutions as well as positivity of the solutions for positive data. These properties are consequences of the fact that the infinitesimal generators forming the underlying dynamical systems are (nonlinear) negative $M$-operators.
References:
[1] B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, and J. D. Watson: Molecular Biology of the Cell. Garland Publishing, New York-London, 1989.
[2] A. Berman, M. Neumann, R. J. Stern: Nonnegative Matrices in Dynamic Systems. J. Wiley, New York, 1989. MR 1019319
[3] A. Berman, R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York-San Francisco-London, 1979. MR 0544666
[4] E. Bohl: A boundary layer phenomenon for linear systems with a rank deficient matrix. Z. Angew. Math. Mech. 7/8 (1991), 223–231. MR 1121486 | Zbl 0792.65057
[5] E. Bohl: Constructing amplification via chemical circuits. In: Biomedical Modeling Simulation, J. Eisarfeld, D. S. Leonis, M. Witken (eds.), Elsevier Science Publ., Amsterdam, 1992, pp. 331–334.
[6] E. Bohl: Structural amplification in chemical networks. In: Complexity, Chaos and Biological Evolution, E. Mosekilde, L. Mosekilde (eds.), Plenum Press, New York, 1991, pp. 119–128.
[7] E. Bohl, W. Boos: Quantitative analysis of binding protein-mediated ABC transport systems. J. Theor. Biology 186 (1997), 65–74. DOI 10.1006/jtbi.1996.0342
[8] E. Bohl, W. Boos: Binding protein-dependent transporters: An answer of mathematics to biology. J. Comput. Appl. Math. 63 (1995), 11–25. DOI 10.1016/0377-0427(95)00055-0 | MR 1365548
[9] E. Bohl, P. Lancaster: Perturbation of spectral inverses applied to a boundary layer phenomenon arising in chemical networks. Linear Algebra Appl. 180 (1993), 35–59. DOI 10.1016/0024-3795(93)90524-R | MR 1206409
[10] E. Bohl, I. Marek: A model of amplification. J. Comput. Appl. Math. 63 (1995), 27–47. DOI 10.1016/0377-0427(95)00052-6 | MR 1365549
[11] E. Bohl, I. Marek: A nonlinear model involving $M$-operators. An amplification effect measured in the cascade of vision. J. Comput. Appl. Math. 60 (1995), 13–28. DOI 10.1016/0377-0427(94)00081-B | MR 1354645
[12] E. Bohl, I. Marek: A stability theorem for a class of linear evolution systems. Integral Equations Oper. Theory 34 (1999), 251–269. DOI 10.1007/BF01300579 | MR 1689389
[13] E. Bohl, I. Marek: Existence and uniqueness results for nonlinear cooperative systems. Oper. Theory, Adv. Appl. 130 (2002), 153–170. MR 1902006
[14] E. Bohl, H. A. Shuman, W. Boos: A mathematical treatment of the kinetics of binding protein-dependent transport systems reveals that both loaded and the unloaded binding proteins iteract with the membrane components. Theoret. Biol. 172 (1995), 83–94. DOI 10.1006/jtbi.1995.0006
[15] W. Boos, J. M. Lucht: Periplasmic binding-protein-dependent ABC transporters. In: Escherichia coli and Salmonella typhimurium. Cellular and Molecular Biology, F. C. Neidhardt, R. Curtiss, J. L. Ingraham, E. C. C. Lin, K. B. Low, B. Magasanik, W. S. Reznikoff, M. Riley, M. Schaechter, H. E. Umbarger (eds.), American Society of Microbiology, Washington, DC, 1996, pp. 1175–1209.
[16] A. Cornish-Bowden: Fundamentals of Enzyme Kinetics. Portland Press, London, 1995.
[17] I. Hendekovic: Konstanz University PhD. Thesis, to be completed in 2005.
[18] E. Hille, R. S. Phillips: Functional Analysis and Semi-groups. Amer. Math. Soc. Coll. Publ. Vol. XXXI, third printing of Revised Edition, Providence, 1974. MR 0423094
[19] J. Keener, J. Sneyd: Mathematical Physiology. Springer-Verlag, New York, 1998. MR 1673204
[20] M. G. Krein, M. A. Rutman: Linear operators leaving invariant a cone in a Banach space. Uspekhi mat. nauk III (1948), 3–95 (Russian). MR 0027128
[21] J. R. Manson, W. Boos, P. J. Bassfort, B. A. Rasmussen: Dependence of maltose and chemotaxis on the amount of maltose-binding protein. J. Biol. Chem. 260 (1985), 9727–9733.
[22] I. Marek: Schwarz-like methods for approximate solving cooperative systems. Kybernetika 40 (2004), 611–624. MR 2121000 | Zbl 1151.65341
[23] I. Marek, D. Szyld: Pseudoirreducible and pseudoprimitive operators. Linear Algebra Appl. 154–156 (1991), 779–791. MR 1113169
[24] I. Marek, K. Žitný: Analytic Theory of Matrices for Applied Sciences, Vol. 1. Teubner Texte zur Mathematik, Band 60. Leipzig, 1983. MR 0731071
[25] G. Merino, W. Boos, H. A. Shuman, E. Bohl: The inhibition of maltose transport by unliganded form of the maltose-binding protein of Escherichia coli: Experimental findings and mathematical treatment. J. Theor. Biology 177 (1995), 171–179. DOI 10.1006/jtbi.1995.0236
[26] L. Michaelis, M. L. Menten: Die Kinetic der Invertierung. Biochem. Z. 49 (1913), 333–369.
[27] J. Monod, J. Wyman, J. P. Changeux: On the nature of allosteric transitions. A plausible model. J. Mol. Biol. 12 (1965), 88–118. DOI 10.1016/S0022-2836(65)80285-6
[28] I. Sawashima: On spectral properties of some positive operators. Natur. Sci. Rep. Ochanomizu Univ. 15 (1964), 53–64. MR 0187096 | Zbl 0138.07801
[29] H. H. Schaefer: Banach Lattices and Positive Operators. Springer-Verlag, Berlin-Heidelberg-New York, 1974. MR 0423039 | Zbl 0296.47023
[30] H. H. Schaefer: Topological Vector Spaces. Springer Verlag, New York-Heidelberg-Berlin, 1971. MR 0342978 | Zbl 0217.16002
[31] H. Schneider, M. Vidysagar: Cross-positive matrices. SIAM J. Numer. Anal. 7 (1970), 508–519. DOI 10.1137/0707041 | MR 0277550
[32] S. Szmelcman, M. Schwartz, T. J. Silhavy, W. Boos: Maltose transport in Escherichia coli K12. A comparison of transport kinetics in wild and $\lambda $-resistant mutants with the dissociation constant of the maltose-binding protein as measured by fluorescence quenching. Eur. J. Biochem. 65 (1976), 13–19. DOI 10.1111/j.1432-1033.1976.tb10383.x
[33] A. E. Taylor, D. C. Lay: Introduction to Functional Analysis, second edition. J. Wiley, New York, 1980. MR 0564653
[34] C. Tralau: Das Upg-Transportsystem in Escherichia coli: Mathematische Modellierung und experimentelle Befunde. Berichte aus der Biologie, Shaker Verlag, Aachen, 2003, Dissertation der Universität Konstanz.
[35] C. Tralau, G. Greller, M. Pajatsch, W. Boos, E. Bohl: Mathematical treatment of transport data of bacterial transport systems to estimate limitation in diffusion through the outer membrane. J. Theor. Biol. 207 (2000), 1–14. DOI 10.1006/jtbi.2000.2140
[36] A. B. Vasil’eva: Asymptotic behaviour of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives. Russ. Math. Surv. 18 (1963), 13–84. DOI 10.1070/RM1963v018n03ABEH001137 | MR 0158137
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