Analysis of a Model for Transfer Phenomena in Biological Populations

We study the problem of transfer in a population structured by a continuum variable corresponding to the quantity being transferred. The transfer of the quantity occurs between individuals according to specified rules. The model is of Boltzmann type with kernels corresponding to the transfer process. We prove that the transfer process preserves total mass of the transferred quantity and the solutions of the simple model converge weakly to Radon measures. We generalize the model by introducing proliferation of individuals and production and diffusion of the transferable quantity. It is shown that the generalized model admits a globally asymptotically stable steady state, provided that transfer is sufficiently small. We discuss an application of our model to cancer cell populations, in which individual cells exchange the surface protein P-glycoprotein, an important factor in acquired multidrug resistance against cancer chemotherapy.

[1]  M. Gottesman,et al.  Multidrug resistance in cancer: role of ATP–dependent transporters , 2002, Nature Reviews Cancer.

[2]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[3]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[4]  I. Roninson,et al.  Induction of multidrug resistance in human cells by transient exposure to different chemotherapeutic drugs. , 1993, Journal of the National Cancer Institute.

[5]  Nicola Bellomo,et al.  On the foundations of cancer modelling: Selected topics, speculations, and perspectives , 2008 .

[6]  A. L. Jones,et al.  Mutation , 1967, Quantitative Genetics.

[7]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[8]  Andre Levchenko,et al.  Intercellular transfer of P-glycoprotein mediates acquired multidrug resistance in tumor cells. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[9]  F. Leboulenger,et al.  Regulation of volume-sensitive Cl- channels in multi-drug resistant MCF7 cells. , 2005, Biochemical and biophysical research communications.

[10]  Pierre Magal.  Global stability for differential equations with homogeneous nonlinearity and application to population dynamics , 2002 .

[11]  J. Zalcberg,et al.  Induction of MDR1 gene expression by anthracycline analogues in a human drug resistant leukaemia cell line , 1999, British Journal of Cancer.

[12]  A. Bellouquid,et al.  Mathematical Modeling of Complex Biological Systems: A Kinetic Theory Approach , 2006 .

[13]  R. Nagel,et al.  One-parameter Semigroups of Positive Operators , 1986 .

[14]  C. Villani.  Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory , 2002 .

[15]  M. Gottesman,et al.  A novel way to spread drug resistance in tumor cells: functional intercellular transfer of P-glycoprotein (ABCB1). , 2005, Trends in pharmacological sciences.

[16]  Eugene V Koonin,et al.  Mathematical modeling of evolution of horizontally transferred genes. , 2005, Molecular biology and evolution.

[17]  Nicola Bellomo,et al.  From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells ✩ , 2008 .

[18]  G. Webb,et al.  Mutation, selection, and recombination in a model of phenotype evolution , 1999 .

[19]  Anirvan M. Sengupta,et al.  Mutation-selection networks of cancer initiation: tumor suppressor genes and chromosomal instability. , 2003, Journal of theoretical biology.

[20]  Pierre Magal.  Mutation and recombination in a model of phenotype evolution , 2002 .

[21]  G. Webb.  An operator-theoretic formulation of asynchronous exponential growth , 1987 .

[22]  N. Komarova.  STOCHASTIC MODELING OF LOSS- AND GAIN-OF-FUNCTION MUTATIONS IN CANCER , 2007 .

[23]  Pierre Magal.  Perturbation of a Globally Stable Steady State and Uniform Persistence , 2009 .

[24]  C. Batty,et al.  ONE-PARAMETER SEMIGROUPS OF POSITIVE OPERATORS (Lecture Notes in Mathematics 1184) , 1987 .

[25]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[26]  Glenn F Webb,et al.  Asymptotic behaviour of solutions to abstract logistic equations. , 2007, Mathematical biosciences.

[27]  Geoffrey I. Webb,et al.  Dynamics of bacterial phenotype selection in a colonized host , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[28]  H. Amann.  Dual semigroups and second order linear elliptic boundary value problems , 1983 .