Table Of ContentLecture Notes on Mathematical Modelling
in the Life Sciences
Michael C. Mackey
Moisés Santillán
Marta Tyran-Kamińska
Eduardo S. Zeron
Simple
Mathematical Models
of Gene Regulatory
Dynamics
Lecture Notes on Mathematical Modelling
in the Life Sciences
Editors-in-Chief:
MichaelC.Mackey
AngelaStevens
Serieseditors
MartinBurger
MauriceChacron
OdoDiekmann
AnitaLayton
JinzhiLei
MarkLewis
LakshminarayananMahadevan
PhilipMaini
MasayasuMimura
ClaudiaNeuhauser
HansG.Othmer
MarkPeletier
AlanSPerelson
CharlesS.Peskin
LuigiPreziosi
JonathanRubin
MoisésSantillán
ChristophSchütte
The rapid pace and development of the research in mathematics, biology and
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andcomputationinthelife-sciences,atahighlevel,inbothprintedandelectronic
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Theycanserveasanintroductiontorecentandemergingsubjectareasand/orasan
advancedteachingaidatcolleges,institutesanduniversities.Besidesmonographs,
we envision that this series will also provide an outlet for material less formally
presentedandmoreanticipatoryoffutureneeds,yetofimmediateinterestbecause
of the novelty of its treatment of an application, or of the mathematics being
developed in the context of exciting applications. It is important to note that
the LMML focuses on books by one or more authors, not on edited volumes.
The topics in LMML range from the molecular through the organismal to the
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biology, neuroscience, organ, tissue and whole body science, immunology and
disease, bioengineering and biofluids, population biology and systems biology.
Mathematicalmethodsincludedynamicalsystems,ergodictheory,partialdifferen-
tial equations,calculus of variations,numericalanalysisand scientific computing,
differentialgeometry,topology,optimalcontrol,probability,stochastics,statistical
mechanics, combinatorics, algebra, number theory, etc., which contribute to a
deeperunderstandingofbiomedicalproblems.
Moreinformationaboutthisseriesathttp://www.springer.com/series/10049
Michael C. Mackey (cid:129) Moisés Santillán (cid:129)
Marta Tyran-Kamin´ska (cid:129) Eduardo S. Zeron
Simple Mathematical Models
of Gene Regulatory
Dynamics
123
MichaelC.Mackey MoisésSantillán
DepartmentofPhysiology UnidadMonterrey
McGillUniversity CinvestavdelIPN
Montreal,QC Apodaca,NL
Canada Mexico
MartaTyran-Kamin´ska EduardoS.Zeron
InstituteofMathematics DepartamentodeMatemáticas
UniversityofSilesia CinvestavdelIPN
Katowice CiudaddeMéxico
Poland Mexico
ISSN2193-4789 ISSN2193-4797 (electronic)
LectureNotesonMathematicalModellingintheLifeSciences
ISBN978-3-319-45317-0 ISBN978-3-319-45318-7 (eBook)
DOI10.1007/978-3-319-45318-7
LibraryofCongressControlNumber:2016956565
©SpringerInternationalPublishingSwitzerland2016
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To studentseverywhere: past,present,
andfuture.
Preface
We survey work that has been carried out in the attempts of biomathematicians
to understand the dynamic behavior of simple bacterial operons starting with the
initialworkofthe 1960s.We concentrateonthe simplestofsituations, discussing
both repressible and inducible systems as well as the bistable switch and then
turningtoadiscussionoftheroleofbothextrinsicnoiseandtheso-calledintrinsic
noiseintheformoftranslationaland/ortranscriptionalbursting.Weconcludewith
a consideration of the messier concrete examples of the lactose and tryptophan
operons and the lysis-lysogeny switch of phage (cid:2). This survey has grown out of
our work over the past 20 years and is an enlarged version of our review paper
(Mackeyetal.2015).
Montreal,QC,Canada MichaelC.Mackey
Apodaca,NL,Mexico MoisésSantillán
Katowice,Poland MartaTyran-Kamin´ska
CiudaddeMéxico,Mexico EduardoS.Zeron
June2016
vii
Acknowledgments
We have benefited from the comments, suggestions, and criticisms of many
colleaguesover the years (you will know who you are) and from the institutional
supportofourhomeuniversitiesaswellastheUniversityofOxford,theUniversity
of Bremen, Bergischen Universität Wuppertal, and the International Centre for
Theoretical Physics. MCM is especially grateful to a comment from Dr. Jérôme
Lossonmanyyearsagothatdirectedattentiontothesefascinatingproblems.
This work was supported by the Natural Sciences and Engineering Research
Council(NSERC)ofCanada,thePolishNCNgrantno.2014/13/B/ST1/00224,and
theConsejoNacionaldeCienciayTecnología(Conacyt)inMéxico.
ix
Contents
PartI DeterministicModelingTechniques
1 GenericDeterministicModelsofProkaryoticGeneRegulation........ 3
1.1 InducibleRegulation.................................................... 3
1.2 RepressibleRegulation ................................................. 5
2 GeneralDynamicConsiderations ......................................... 7
2.1 OperonDynamics....................................................... 7
2.1.1 NoControl ...................................................... 9
2.1.2 InducibleRegulation............................................ 9
2.1.3 RepressibleRegulation......................................... 13
2.1.4 BistableSwitches............................................... 13
2.2 TheAppearanceofCellGrowthEffectsandDelaysDue
toTranscriptionandTranslation........................................ 23
2.3 FastandSlowVariables................................................. 26
PartII DealingwithNoise
3 MasterEquationModelingApproaches .................................. 31
3.1 TheChemicalMasterEquation ........................................ 32
3.2 RelationtoDeterministicModels...................................... 34
3.2.1 TheChemicalLangevinEquation ............................. 36
3.3 StabilityoftheChemicalMasterEquation ............................ 37
3.3.1 AlgorithmstoFindSteadyStateDensityFunctions.......... 40
3.4 ApplicationtoaSimpleRepressibleOperon.......................... 43
4 NoiseEffectsinGeneRegulation:IntrinsicVersusExtrinsic .......... 49
4.1 DynamicswithBursting................................................ 50
4.1.1 Generalities ..................................................... 50
4.1.2 Distributions in the Presence of Bursting
forInducibleandRepressibleSystems ........................ 52
4.1.3 BurstinginaSwitch............................................ 57
xi
xii Contents
4.1.4 RecoveringtheDeterministicCase ............................ 61
4.1.5 ADiscreteSpaceBurstingModel ............................. 62
4.2 GaussianDistributedNoiseintheMolecularDegradationRate...... 64
4.3 TwoDominantSlowGeneswithBursting............................. 66
PartIII SpecificExamples
5 TheLactoseOperon ......................................................... 73
5.1 TheLactoseOperonRegulatoryPathway ............................. 73
5.2 MathematicalModelingoftheLactoseOperon....................... 77
5.3 QuantitativeStudiesoftheLactoseOperonDynamics ............... 83
6 TheTryptophanOperon .................................................... 87
6.1 TheTryptophanOperoninE.coli...................................... 87
6.2 MathematicalModelingofthetrpOperon ............................ 89
6.3 QuantitativeStudiesofthetrpOperon................................. 92
7 TheLysis-LysogenySwitch ................................................. 99
7.1 Phage(cid:2)Biology ........................................................ 102
7.2 TheLysis-LysogenySwitch ............................................ 104
7.3 MathematicalModelingofthePhage(cid:2)Switch ....................... 107
7.4 BriefReviewofQuantitativeStudiesonthePhage(cid:2)Switch......... 112
7.5 ClosingRemarks........................................................ 114
References......................................................................... 115
Index............................................................................... 123
