On the mathematical theory of post-Darwinian mutations, selection, and evolution

This paper is devoted to the modeling, qualitative analysis and simulation of Darwinian selection phenomena and their evolution. The approach takes advantage of the mathematical tools of the kinetic theory of active particles which are applied to describe the selective dynamics of evolution processes. The first part of the paper focuses on a mathematical theory that has been developed to describe mutations and selection processes. The second part deals with different modeling strategies and looks ahead to research perspectives.

[1]  Nicola Bellomo,et al.  Modeling crowd dynamics from a complex system viewpoint , 2012 .

[2]  Juan Soler,et al.  ON THE ASYMPTOTIC THEORY FROM MICROSCOPIC TO MACROSCOPIC GROWING TISSUE MODELS: AN OVERVIEW WITH PERSPECTIVES , 2012 .

[3]  D. Hanahan,et al.  Hallmarks of Cancer: The Next Generation , 2011, Cell.

[4]  Carlo C. Maley,et al.  Clonal evolution in cancer , 2012, Nature.

[5]  Robert A Gatenby,et al.  The Critical Roles of Information and Nonequilibrium Thermodynamics in Evolution of Living Systems , 2013, Bulletin of mathematical biology.

[6]  V. Caselles,et al.  Finite Propagation Speed for Limited Flux Diffusion Equations , 2006 .

[7]  D. Knopoff.  ON THE MODELING OF MIGRATION PHENOMENA ON SMALL NETWORKS , 2013 .

[8]  Martin A Nowak,et al.  Why Are Phenotypic Mutation Rates Much Higher Than Genotypic Mutation Rates? , 2006, Genetics.

[9]  J. Tinsley Oden,et al.  SELECTION AND ASSESSMENT OF PHENOMENOLOGICAL MODELS OF TUMOR GROWTH , 2013 .

[10]  Nicola Bellomo,et al.  Dynamics of tumor interaction with the host immune system , 1994 .

[11]  Marco Ajmone Marsan,et al.  Towards a mathematical theory of complex socio-economical systems by functional subsystems representation , 2008 .

[12]  Martin Burger,et al.  The Keller-Segel Model for Chemotaxis with Prevention of Overcrowding: Linear vs. Nonlinear Diffusion , 2006, SIAM J. Math. Anal..

[13]  Avner Friedman,et al.  Asymptotic phases in a cell differentiation model , 2009 .

[14]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes , 2000, SIAM J. Appl. Math..

[15]  Federica Cavallo,et al.  2011: the immune hallmarks of cancer , 2011, Cancer Immunology, Immunotherapy.

[16]  F. Weissing,et al.  Genetic versus phenotypic models of selection: can genetics be neglected in a long-term perspective? , 1996, Journal of mathematical biology.

[17]  E. Regazzini,et al.  ABOUT THE GENE FAMILIES SIZE DISTRIBUTION IN A RECENT MODEL OF GENOME EVOLUTION , 2010 .

[18]  Alexander Lorz,et al.  Applying ecological and evolutionary theory to cancer: a long and winding road , 2012, Evolutionary applications.

[19]  Gonzalo Galiano,et al.  Evolutionary Distributions and Competition by Way of Reaction-Diffusion and by Way of Convolution , 2013, Bulletin of mathematical biology.

[20]  Juan Soler,et al.  ON THE DIFFICULT INTERPLAY BETWEEN LIFE, "COMPLEXITY", AND MATHEMATICAL SCIENCES , 2013 .

[21]  Thimo Rohlf,et al.  Receptor cross-talk in angiogenesis: mapping environmental cues to cell phenotype using a stochastic, Boolean signaling network model. , 2010, Journal of theoretical biology.

[22]  Juan Soler,et al.  MULTISCALE BIOLOGICAL TISSUE MODELS AND FLUX-LIMITED CHEMOTAXIS FOR MULTICELLULAR GROWING SYSTEMS , 2010 .

[23]  F. C. Santos,et al.  Evolutionary dynamics of social dilemmas in structured heterogeneous populations. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[24]  Luisa Fermo,et al.  TOWARDS THE MODELING OF VEHICULAR TRAFFIC AS A COMPLEX SYSTEM: A KINETIC THEORY APPROACH , 2012 .

[25]  J. Stiglitz.  Information and the Change in the Paradigm in Economics , 2002 .

[26]  R. May.  Uses and Abuses of Mathematics in Biology , 2004, Science.

[27]  Yann Brenier,et al.  Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations , 2008, J. Nonlinear Sci..

[28]  Marcello Edoardo Delitala,et al.  MODELLING EPIDEMICS AND VIRUS MUTATIONS BY METHODS OF THE MATHEMATICAL KINETIC THEORY FOR ACTIVE PARTICLES , 2009 .

[29]  M. A. Herrero,et al.  A mathematical model for a T cell fate decision algorithm during immune response. , 2014, Journal of theoretical biology.

[30]  L. Segel,et al.  Traveling bands of chemotactic bacteria: a theoretical analysis. , 1971, Journal of theoretical biology.

[31]  Natalia L. Komarova,et al.  Targeted Cancer Treatment in Silico , 2014 .

[32]  Nicola Bellomo,et al.  On the dynamics of social conflicts: looking for the Black Swan , 2012, ArXiv.

[33]  Marcello Edoardo Delitala,et al.  FROM METHODS OF THE MATHEMATICAL KINETIC THEORY FOR ACTIVE PARTICLES TO MODELING VIRUS MUTATIONS , 2011 .

[34]  T. Vincent,et al.  EVOLUTIONARY DYNAMICS IN CARCINOGENESIS , 2005 .

[35]  Jerzy Tiuryn,et al.  SIZE DISTRIBUTION OF GENE FAMILIES IN A GENOME , 2014 .

[36]  B. Perthame,et al.  Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations , 2010, 1011.1768.

[37]  Robert L. Perlman,et al.  Evolutionary Biology: A Basic Science for Medicine in the 21st Century , 2011, Perspectives in biology and medicine.

[38]  Abdelghani Bellouquid,et al.  From kinetic models of multicellular growing systems to macroscopic biological tissue models , 2011 .

[39]  Robert A. Gatenby,et al.  Life history trade-offs in cancer evolution , 2013, Nature Reviews Cancer.

[40]  Jack T. Trevors,et al.  Self-organization vs. self-ordering events in life-origin models , 2006 .

[41]  Jorge Duarte,et al.  Avoiding healthy cells extinction in a cancer model. , 2014, Journal of theoretical biology.

[42]  N Bellomo,et al.  Toward a mathematical theory of living systems focusing on developmental biology and evolution: a review and perspectives. , 2011, Physics of life reviews.

[43]  Dominik Wodarz,et al.  Tumor growth dynamics: insights into evolutionary processes. , 2013, Trends in ecology & evolution.

[44]  Martin A. Nowak,et al.  Games on graphs , 2014 .

[45]  S. Carpenter,et al.  Early-warning signals for critical transitions , 2009, Nature.

[46]  H. Othmer,et al.  Models of dispersal in biological systems , 1988, Journal of mathematical biology.

[47]  N. Bellomo,et al.  From a class of kinetic models to the macroscopic equations for multicellular systems in biology , 2003 .

[48]  Francisco C. Santos,et al.  EVOLUTIONARY DYNAMICS OF CLIMATE CHANGE UNDER COLLECTIVE-RISK DILEMMAS , 2012 .

[49]  Juan Soler,et al.  ON A DISPERSIVE MODEL FOR THE UNZIPPING OF DOUBLE-STRANDED DNA MOLECULES , 2014 .

[50]  J. Soler,et al.  Morphogenetic action through flux-limited spreading. , 2013, Physics of life reviews.

[51]  K. Rajagopal,et al.  ON MODELS FOR VISCOELASTIC MATERIALS THAT ARE MECHANICALLY INCOMPRESSIBLE AND THERMALLY COMPRESSIBLE OR EXPANSIBLE AND THEIR OBERBECK–BOUSSINESQ TYPE APPROXIMATIONS , 2013 .

[52]  P. Pucci,et al.  On an initial value problem modeling evolution and selection in living systems , 2014 .

[53]  D. Knopoff,et al.  FROM THE MODELING OF THE IMMUNE HALLMARKS OF CANCER TO A BLACK SWAN IN BIOLOGY , 2013 .

[54]  Mirosław Lachowicz,et al.  Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions , 2014 .

[55]  D. Knopoff,et al.  ON A MATHEMATICAL THEORY OF COMPLEX SYSTEMS ON NETWORKS WITH APPLICATION TO OPINION FORMATION , 2014 .

[56]  Carlo C. Maley,et al.  Overlooking Evolution: A Systematic Analysis of Cancer Relapse and Therapeutic Resistance Research , 2011, PloS one.

[57]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations II: Chemotaxis Equations , 2002, SIAM J. Appl. Math..

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

[59]  Avner Friedman,et al.  Asymptotic limit in a cell differentiation model with consideration of transcription , 2012 .