Evolution Equations Arising in the Modelling of Life Sciences PDF
Preview Evolution Equations Arising in the Modelling of Life Sciences
ISNM International Series of Numerical Mathematics Volume 163 ManagingEditors: K.-H.Hoffmann, München, Germany G.Leugering, Erlangen-Nürnberg, Germany AssociateEditors: Z.Chen, Beijing,China R.H.W.Hoppe, Augsburg, Germany/Houston, USA N.Kenmochi, Chiba,Japan V.Starovoitov, Novosibirsk,Russia HonoraryEditor: J.Todd, Pasadena, USA† Forfurthervolumes: www.birkhauser-science.com/series/4819 Messoud Efendiev Evolution Equations Arising in the Modelling of Life Sciences MessoudEfendiev InstituteofBiomathematicsandBiometry HelmholtzCenterMunich Neuherberg,Germany ISBN978-3-0348-0614-5 ISBN978-3-0348-0615-2(eBook) DOI10.1007/978-3-0348-0615-2 SpringerBaselHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013932776 MathematicsSubjectClassification: 92Bxx,92Cxx,92Dxx,35-XX,35Q92 ©SpringerBasel2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerBaselispartofSpringerScience+BusinessMedia(www.birkhauser-science.com) DedicatedtothememoryofmyauntNagiba Preface Thisbookdealswithmodelling,analysisandsimulationofproblemsarisinginthe lifesciences,inparticular,biologicalprocesses.Suchproblemsplayaveryimpor- tant role in many scientific and technological areas and is a truly interdisciplinary research topic. Indeed the models and results obtained in this book are a result of veryusefulandintensivediscussionswithmicrobiologists,doctorsandmedicalper- sonnel,physicists,chemists,aswellasindustrialengineers.Ourresultsgiveabetter understandingoftheproblemsoflifesciencesconsideredinthisbookandexplain theirexperimentalfindings.Weespeciallyemphasizethat,duetostronginteractions withcolleaguesinthetopicsmentionedabove,thegoverningequationsoflifesci- encemodelswehavederivedarebasedonexperimentaldataandleadtoanewclass ofdegeneratedensity-dependentnonlinearreaction-diffusion-convectiveequations comprising two kinds of degeneracy: porous-medium and fast-diffusion type de- generacy. It is worth mentioning that such a class of degenerate evolution partial differentialequations(PDEs),comprisingsimultaneouslytwokindsofdegeneracy, is so far inadequately understood in the mathematical literature. This shows how usefulsuchinterdisciplinaryresearchtopicsareforbothmathematiciansandrepre- sentativesofotherfields. Inthisbookwenotonlyderiverealisticlifesciencemodelsandtheirgoverning equationsofthedegeneratetypesmentionedabove,butatthesametimesystemat- ically study these classes of equations. In each concrete case we study their well- posedness, dependence of solutions on boundary conditions, which reflects some properties of our environment and the large-time behaviour of solutions; we also includesomenumericalanalysis. Thebookconsistsofsixchapters.Chapter1hasmoreofateachingaidcharacter andisdedicatedtosomebasicconceptsoflinearellipticboundaryvalueproblems (BVPs),propertiesofNemytskiioperatorsinSobolevspaces(fractionalaswell)and Hölderspaceswhichweuseintheanalysisofderivingmodelsinthenextchapters. Chapter2isconcernedwithlargetimebehaviourofsolutionsofevolutionequa- tionsintermsoftheglobalattractor,itsexistenceandproperties. In deriving evolution models of the problems of life sciences we are usually dealingwiththetemporalevolutionsofnon-negativequantitieslikeconcentrations vii viii Preface of nutrients and chemicals, population densities, temperature, pressure etc. Each modelissomeapproximationofarealsituationandthereforeitisnotclearbefore- hand, that the obtained models will obey the following crucial properties for the modellers: the solutions of continuous evolution equations remain non-negative if the initial data are non-negative. We emphasize that numerical simulations cannot answerthequestionwhetherthemodelinthissenseisvalidornot;eventhoughnu- mericalsimulationscanprovideempiricalevidencewecanneverbesure.Henceit isextremelyimportantformodellerstohavesuchcriteria,thatisnecessaryandsuf- ficientconditionsforsolutionsofparabolicsystemscontainingdiffusion,transport (convection)andinteractionofspecies(nonlinearterm)topreservepositivecones. Itisworthnoting,thatsofarsuchacriterion,infullgenerality,hasonlybeenfound forscalarequations,whichcriterionisbasedonthemaximumprinciple.Itiswell- knownthatforasystemofparabolicPDEsthemaximumprinciplefails.Atpresent in the mathematical literature there exist many sufficient conditions for parabolic systems preserving positive cones. However this is not satisfactory for modellers. Theyneedtohaveanalgorithmtoknowwhetherornotthederivedrealisticmod- elssatisfythispropertybeforestartingtodoanalysisandcarryingoutsimulations. ThereforeinChap.3(whichinturnconsistsofthreesections)wederiveacriterion forpositivityofsolutionsforquitelargeclassesofdeterministicparabolicsystems containingdiffusion,transport(advectionorconvection)andinteractionofspecies (nonlinearterm). Chapter 4 is devoted to the positivity of solutions of a class of semilinear parabolicsystemsofstochasticpartialdifferentialequationsbyconsideringrandom approximations. For the family of random approximations we derive explicit nec- essary and sufficient conditions such that solutions preserve positivity. These con- ditions imply the positivity of the solutions of the stochastic system for both Ito’s andStratonovich’sinterpretationofstochasticdifferentialequations.Weemphasize thatthisresultisalsoofindependentinterestforthemathematicalcommunity,since inthisgeneralitysuchacriteriaforthesystemsmentionedinChap.3andChap.4 werenotpreviouslyknown. Chapter5isdevotedtobiofilmmodelling(meso-scalelevel),analysisandsimu- lationwhichisoneofthemostactiveareasinmodernmicrobiology.Tothisenditis enoughtoreferto:“Itisthebestoftimesforbiofilmresearch”(Nature76,vol.15, pp. 76–81, 2007). In contrast to existing biofilm models, which are based mostly ondiscreterulesorhybridmodels,wearemainlyinterestedinadeterministicand continuousmodelwhichisdescribedbyPDEs. Chapter 5 consists of five sections. The first two sections, 5.1 and 5.2 are con- cernedwiththesinglespecies/singlesubstratemodels.InSect.5.1wederivegov- erningequationswhichdescribespatialspreadingmechanismsofbiomass.Thefea- tureoftheseequationsisthattheyarehighlynonlineardensity-dependentdegener- atereaction-diffusionsystemscomprisingtwokindofdegeneracy:porousmedium andfastdiffusion.Weprovethewell-posednessoftheobtainedequationsandstudy the long-time dynamics of their solutions in terms of a global attractor. Moreover weanalyzedependenceofsolutionsonboundaryconditions.Ournumericalsimu- lationsofderivedequationsleadtomushroompatternswhichwereobservedinthe experimentalstudies. Preface ix Section 5.2 is concerned with biofilm growth in a porous medium (e.g. soil) which changes the hydraulic conductivity of the medium (bioclogging), which in turn changes substrate transport, and as a result, the food supply of bacteria. This naturally occurring nonlinear phenomenon is used by engineers to devise microbially based technologies for groundwater protection and soil remediation. We derive bioclogging models that account for the spatial expansion of the bac- terial population in the soil and make a numerical analysis of the corresponding reaction-diffusion-convective parabolic systems describing the behaviour of this model. More precisely, it consists of a density-dependent, doubly degenerate dif- fusion equation that is coupled with the Darcy equations and a transport-reaction equation for growth limiting substrates. It is worth mentioning that, in contrast to previousexistingbiocloggingmodels,ourmodelallowsbacteriatomoveintoneigh- bouringvoidregionswhenthespacebecomeslocallylimitedandtheenvironmental conditionsaresuchthatfurthergrowthofthebacterialpopulationissustained. Sections 5.3–5.5 are devoted to multi-species/multi-substrate biofilm models. MorepreciselyinSect.5.3weareinterestedinantibioticdisinfectionofbiofilms. Inthiscasethebiofilmsystemconsistsoftheparticulatevolumefractionofactive andinertbiomassandthedissolvedsubstratesofnutrientsandantibiotics.Wede- rivegoverningequationsdescribingantibioticdisinfectionofbiofilmsandstudythe modelbothanalyticallyandnumerically.Weproveglobalintimeexistenceofso- lutions,but,incontrasttothesinglespecies/singlesubstratemodel,theuniqueness of solutions remains an open problem. This proof uses among other techniques a positivity criterion, which is formulated and proved in Chap. 3. Moreover, using the difference in characteristic time-scales of the components of our model in the one-dimensional case we derive a dimensionless parameter, the disinfection num- ber,thatprovidesinformationforthenetbiomassproductionpreventingantibiotic disinfection. Section5.4dealswithamathematicalmodelthatdescribeshowa“good”bacte- rialbiofilmcontrolsthegrowthofharmfulpathogenicbacterialbiofilm.Theunder- lyingmechanismisamodificationofthelocalprotonatedacidconcentration,which inturndecreasesthelocalpHand,thus,makesgrowthconditionsforthepathogens lessfavourable,whilethecontrol-agentitselfismoretoleranttothesechanges.We giveanexistenceprooffortheresultingdegeneratedmixed-culturebiofilmmodel. Neitherauniquenessnornon-uniquenessresultisobtained.Weillustrateinournu- merical simulations workings of these bio-control mechanisms. In particular, it is shownthatpathogensareeradicatedfirstinthedeeperlayersofthebiofilm,closeto thesubstratum,whereasintraditionalantibioticbiofilmcontrolfirstthecellsinthe outerlayersaredeactivated. Section 5.5 is concerned with the role of quorum sensing in biofilm formation. Quorum sensing is a cell-cell communication mechanism used by bacteria to co- ordinategeneexpressionandbehaviouringroupsbasedonthelocaldensityofthe bacterialpopulation.Wepresentamodelofquorumsensinginbiofilmcommunities, whichextendsthemono-speciesbiofilmgrowthmodelderivedinSect.5.1andcom- binesit witha mathematicalmodelof quorumsensingfor suspendedpopulations. Thedependentmodelvariablesarethevolumefractionoccupiedbydown-regulated x Preface andup-regulatedcells,concentrationofsignallingmolecules,andconcentrationof thegrowthlimitingnutrient.Weprovethewell-posednessofthemodel.Inparticu- lar,wepresentforthefirsttimeauniquenessresultforthistype(multi-component) ofproblem.Moreover,weillustratethebehaviourofmodelsolutionsinnumerical simulations. InthelastChap.6,weshallillustratetheuseofmathematicalmodellinginthe pharmaceuticalindustrybyanexamplefromthedevelopmentofabloodcoagula- tiontreatmentwithacoagulationfactor.Morespecificallywederiveamathematical modelforabloodcoagulationcascadesetupinaperfusionexperimentconducted atthepharmaceuticalcompanyNovoNordiskA/SinDenmark.Weinvestigatethe influenceofbloodflowanddiffusiononthebloodcoagulationpathwaybyderiving amodelconsistingofasystemofpartialdifferentialequations,takingintoaccount thespatialdistributionofthebiochemicalspecies.Thevalidityofthemodelises- tablishedviapositivitycriteriaprovedinChap.3.Themodelissolvedusingafinite elementcodeinordertoillustratetheinfluenceofdiffusionandconvectiononthe coagulationcascadewithdynamicboundaryconditionmodellingadhesionofblood plateletstoacollagencoatedsurface. Iwouldliketothankthemanyfriendsandcolleagueswhogavemesuggestions, adviceandsupport.Inparticular,IwishtothankH.Berestycki,H.Eberl,F.Hamel, C. Griebler, L. Roques, K.-H. Hoffmann, N. Kenmochi, R. Lasser, H. Matano, P. Maloszewskii, W. Meckenstock, M. Pedersen, L.A. Peletier, F. Rupp, S. Son- ner, M.P. Sorensen, A. Stevens, M. Otani, J. R. L. Webb, W. L. Wendland, J. Wu, A.Yagi.
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