Modeling and Simulation in Science, Engineering and Technology Nicola Bellomo Abdelghani Bellouquid Livio Gibelli Nisrine Outada A Quest Towards a Mathematical Theory of Living Systems Modeling and Simulation in Science, Engineering and Technology Series Editors NicolaBellomo TayfunE.Tezduyar DepartmentofMathematics DepartmentofMechanicalEngineering FacultyofSciences RiceUniversity KingAbdulazizUniversity Houston,TX,USA Jeddah,SaudiArabia Editorial Advisory Board K.Aoki M.A.Herrero NationalTaiwanUniversity DepartamentodeMatematicaAplicada Taipei,Taiwan UniversidadComplutensedeMadrid Madrid,Spain K.J.Bathe DepartmentofMechanicalEngineering P.Koumoutsakos MassachusettsInstituteofTechnology ComputationalScience&Engineering Cambridge,MA,USA Laboratory,ETHZürich Zürich,Switzerland Y.Bazilevs DepartmentofStructuralEngineering H.G.Othmer UniversityofCalifornia,SanDiego DepartmentofMathematics LaJolla,CA,USA UniversityofMinnesota Minneapolis,MN,USA M.Chaplain DivisionofMathematics K.R.Rajagopal UniversityofDundee DepartmentofMechanicalEngineering Dundee,Scotland,UK TexasA&MUniversity CollegeStation,TX,USA P.Degond DepartmentofMathematics K.Takizawa ImperialCollegeLondon DepartmentofModernMechanical London,UK Engineering WasedaUniversity A.Deutsch Tokyo,Japan CenterforInformationServices andHigh-PerformanceComputing Y.Tao TechnischeUniversitätDresden DongHuaUniversity Dresden,Germany Shanghai,China More information about this series at http://www.springer.com/series/4960 Nicola Bellomo Abdelghani Bellouquid (cid:129) Livio Gibelli Nisrine Outada (cid:129) A Quest Towards a Mathematical Theory of Living Systems NicolaBellomo Livio Gibelli Department ofMathematics, Faculty of Schoolof Engineering Sciences University of Warwick KingAbdulaziz University Coventry Jeddah UK SaudiArabia Nisrine Outada and Mathematics andPopulation Dynamics Laboratory,UMMISCO,Faculty of Department ofMathematical Sciences Sciencesof Semlalia of Marrakech (DISMA) CadiAyyad University Politecnico di Torino Marrakech Turin Morocco Italy and AbdelghaniBellouquid Jacques-Louis LionsLaboratory EcoleNationaledesSciencesAppliquéesde Pierre et Marie CurieUniversity Marrakech Paris 6 Académie Hassan IIDes SciencesEt France Techniques,CadiAyyad Unviersity Marrakech Morocco ISSN 2164-3679 ISSN 2164-3725 (electronic) Modeling andSimulationin Science, Engineering andTechnology ISBN978-3-319-57435-6 ISBN978-3-319-57436-3 (eBook) DOI 10.1007/978-3-319-57436-3 LibraryofCongressControlNumber:2017941474 MathematicsSubjectClassification(2010): 35Q20,35Q82,35Q91,35Q92,91C99 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisbookispublishedunderthetradenameBirkhäuser TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland (www.birkhauser-science.com) To the memory of Abdelghani Bellouquid Preface This book is devoted to the design of a unified mathematical approach to the modeling and analysis of large systems constituted by several interacting living entities. It is a challenging objective that needs new ideas and mathematical tools based on a deep understanding of the interplay between mathematical and life sciences.Theauthorsdonotnaivelyclaimthatthisisfullyachieved,butsimplythat ausefulinsightandsomesignificantresultsareobtainedtowardthesaidobjective. The source ofthecontentsofthis bookistheresearchactivitydevelopedinthe last20years,whichinvolvedseveralyoungandexperiencedresearchers.Thisstory started with a book edited, at the beginning of this century, by N.B. with Mario Pulvirenti[51],wherethechaptersofthebookpresentedavarietyofmodelsoflife sciencesystemswhichwerederivedbykinetictheorymethodsandtheoreticaltools of probability theory. The contents of [51] were motivated by the belief that an important new research frontier of applied mathematics had to be launched. The basic idea was that methods of the mathematical kinetic theory and statistical mechanics ought to be developed toward the modeling of large systems in life science differently from the traditional application to the fluid dynamics of large systems of classical particles. Here, particles are living entities, from genes, cells, up to human beings. These entitiesarecalled,withintheframeworkofmathematics,activeparticles.Thisterm encompasses the idea that these particles have the ability to express special strategiesgenerallyaddressing totheirwell-being andhencedonotfollow lawsof classical mechanics as they can think, namely possess both an individual and a collective intelligence [84]. Due to this specific feature, interactions between par- ticles are nonlinearly additive. In fact, the strategy developed by each particle depends on that expressed by the other particles, and in some cases develops a collectiveintelligenceofthewholeviewedasaswarm.Moreover,itoftenhappens that all these events occur in a nonlinear manner. An important conceptual contribution to describing interactions within an evolutive mathematical framework is offered by the theory of evolutive games [186, 189]. Once suitable models of the dynamics at the scale of individuals have been derived, methods of the kinetic theory suggest to describe the overall system vii viii Preface by a probability distribution over the microstate of the particles, while a balance of the number of particles within the elementary volume of the space of the microstatesprovidesthetimeandspacedynamicsofthesaiddistribution,viewedas adependentvariable.Quantitiesatthemacroscaleareusefulinseveralapplications, and these can be obtained from averaged moments of the dependent variable. The hallmarks that havebeen presentedabove are somehow analogoustothose proposed in the book [26], where the approach, however, was limited to linear interactions. Therefore, this present monograph provides, in the authors’ belief, a far more advanced approach, definitely closer to physical reality. Moreover, an additional feature is the search of a link between mutations and selections from post-Darwinist theories [173, 174] to game theory and evolution. Indeed, appli- cations, based on very recent papers proposed by several researchers, have been selectedforphysicalsystems,wherenonlinearitiesappeartoplayanimportantrole in the dynamics. Special attention is paid to the onset of a rare, not predictable event, called black swan according to the definition offered by Taleb [230]. The contents ofthe bookarepresentedattheend ofthefirst chapter aftersome generalspeculationsonthecomplexityofliving systems andonconceivablepaths that mathematicscanlookforaneffectiveinterplay with theirinterpretation.Some statements can possibly contribute to understanding the conceptual approach and the personal style of presentation: (cid:129) The study of models, corresponding to a number of case studies developed in the research activity of the author and coworkers, motivated the derivation of mathematical structures, which have the ability to capture the most important complexity features. This formal framework can play the role of paradigms in the derivation of specific models, where the lack of a background field theory creates a huge conceptual difficulty very hard to tackle. (cid:129) Each chapter is concluded by a critical analysis, proposed with two goals: focusingonthedevelopmentsneededforimprovingtheefficacyoftheproposed methods and envisaging further applications, possibly in fields different from those treatedin this book. Applications cover a broad range offields, including biology,socialsciences,andappliedsciencesingeneral.Thecommonfeatureof allthese applicationsisamathematical approach,whereallofthemareviewed as living, hence complex, systems. (cid:129) The authors of this book do not naively claim that the final objective of pro- viding a mathematical theory of living systems has been fully achieved. It is simply claimed that a contribution to this challenging and fascinating research field is proposed and brought to the attention of future generations of applied mathematicians. Finally,Iwishtomentionthatthisbookrepresentswhathasbeenachieveduntil now.Hopefully,newresultscanbeobtainedinfutureactivities.However,Idecided to write a book, in collaboration with Abdelghani Bellouquid, Livio Gibelli, and Nisrine Outada, according to the feeling that defining the state of the art at this stageisanecessarysteptolookforward.Abdelghani,Livio,andNisrinewerekind Preface ix enough to allow me to write this Preface, as my experience in the field was developedinalonger(notdeeper)lapseoftime.Therefore,Ihavestoriestotell,but mainly persons to thank. I mentioned that many results have been achieved by various authors. Among them the coworkers are very many and I will not mention them explicitly, as they appearinthebibliography.However,Iwouldliketoacknowledgethecontribution of some scientists who have motivated the activity developed in this book. ThefirsthintisfromHelmutNeunzert,whoisarguablythefirsttounderstandthat a natural development of the mathematical kinetic theory needed to be addressed tosystemsfarfromthatofmolecularfluids.Namely,thepioneerideasonvehicular trafficbyPrigogineshouldhavebeenapplied,accordingtohishint,alsotobiology and applied sciences. He organized a fruitful, small workshop in Kaiserslautern, where discussions, critical analysis,and hintsleft a deep trace inmy mind. Subsequently I met Wolfgang Alt, who also had the feeling that methods of kinetic theory and statistical mechanics in general could find an interesting area of application in biology. He invited me to an Oberwolfach workshop devoted to mathematical biology, although I had never made, as a mathematician, a contri- butiontothespecificfieldofthemeeting.Iwasasortofaguestscientist,whowas lucky to have met, on that occasion, Lee Segel. His pragmatic way of developing research activity opened my eyes and convinced me to initiate a twenty-year activity, which is still going on and looks forward. However, I still wish to mention three more lucky events. The first one is the collaboration with Guido Forni, an outstanding immunologist who helped me to understand the complex and multiscale essence of biology and of the immune competition in particular. Indeed, my first contributions are on the applications of mathematics totheimmune competition. Recently,Imet ConstantineDafermosin Rome, who strongly encouraged me to write this book to leave a trace on the interplay between mathematics and life. Finally, I had the pleasure to listen the opening lecture of Giovanni Jona Lasinio at the 2012 meeting of the Italian Mathematical Union. I am proud to state that I do share with him the idea that evolution is a key feature of all living systems and that mathematics should take into account this specific feature. My special thanks go to Abdelghani Bellouquid. He has been a precious coworker for me and several colleagues. For my family, he has been one of us. Ihavetousethepasttense,ashepassedawaywhenthebookwasreachingtheend of the authors’ efforts. My family and I, the two other authors of this book, and many others will never forget him. The approach presented in this book was certainly challenging and certainly on theborderofmyknowledgeandability.Inmanycases,Ihavebeenalonewithmy thoughts and speculations. However, my scientific friends know that I have never been really alone, as my wife Fiorella was always close to me. Without her, this book would have simply been a wish. Turin, Italy Nicola Bellomo January 2017 Contents 1 On the “Complex” Interplay Between Mathematics and Living Systems.. .... ..... .... .... .... .... .... ..... .... 1 1.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 1 1.2 A Quest Through Three Scientific Contributions . .... ..... .... 3 1.3 Five Key Questions .. ..... .... .... .... .... .... ..... .... 5 1.4 Complexity Features of Living Systems.... .... .... ..... .... 5 1.5 Rationale Toward Modeling and Plan of the Book.... ..... .... 11 2 A Brief Introduction to the Mathematical Kinetic Theory of Classical Particles. .... ..... .... .... .... .... .... ..... .... 15 2.1 Plan of the Chapter... ..... .... .... .... .... .... ..... .... 15 2.2 Phenomenological Derivation of the Boltzmann Equation ... .... 16 2.2.1 Interaction dynamics. .... .... .... .... .... ..... .... 19 2.2.2 The Boltzmann equation.. .... .... .... .... ..... .... 20 2.2.3 Properties of the Boltzmann equation .... .... ..... .... 23 2.3 Some Generalized Models .. .... .... .... .... .... ..... .... 25 2.3.1 The BGK model.... .... .... .... .... .... ..... .... 26 2.3.2 The discrete Boltzmann equation ... .... .... ..... .... 26 2.3.3 Vlasov and Enskog equations.. .... .... .... ..... .... 28 2.4 Computational Methods .... .... .... .... .... .... ..... .... 30 2.5 Critical Analysis. .... ..... .... .... .... .... .... ..... .... 31 3 On the Search for a Structure: Toward a Mathematical Theory to Model Living Systems.... .... .... .... .... ..... .... 33 3.1 Plan of the Chapter... ..... .... .... .... .... .... ..... .... 33 3.2 A Representation of Large Living Systems . .... .... ..... .... 35 3.3 Mathematical Structures for Systems with Space Homogeneity ... 39 3.3.1 A phenomenological description of games .... ..... .... 39 3.3.2 Modeling interactions by tools of game theory. ..... .... 42 3.3.3 Mathematical structures for closed systems.... ..... .... 44 xi