Reaction Kinetics: Exercises, Programs and Theorems: Mathematica for Deterministic and Stochastic Kinetics

476 Pages·2018·8.11 MB·English

by  János Tóth

János Tóth · Attila László Nagy 
Dávid Papp
Reaction Kinetics:
Exercises,
Programs and
Theorems
Mathematica for Deterministic and
Stochastic Kinetics
Reaction Kinetics: Exercises, Programs
and Theorems
János Tóth • Attila László Nagy (cid:129) Dávid Papp
Reaction Kinetics:
Exercises, Programs
and Theorems
Mathematica for Deterministic
and Stochastic Kinetics
123
JánosTóth AttilaLászlóNagy
DepartmentofAnalysis DepartmentofStochastics
BudapestUniversityofTechnology BudapestUniversityofTechnology
andEconomics andEconomics
Budapest,Hungary Budapest,Hungary
ChemicalKineticsLaboratory
EötvösLorándUniversity
Budapest,Hungary
DávidPapp
DepartmentofMathematics
NorthCarolinaStateUniversity
Raleigh,USA
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https://doi.org/10.1007/978-1-4939-8643-9
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Dedicatedtoour families,
upwards anddownwards,
and, lastbut notleast,horizontally.
Preface
Origin More than a quarter of a century ago the senior author (JT) published a
book with Péter Érdi on the mathematical models of chemical reactions.1
Before and after writing that book (and having courses and seminars at
the Budapest University of Technology and Economics and at the Eötvös
LorándUniversity,Budapest)werealizedthatitisalmostimpossibleto do
anythingin the theory and applicationsof chemical kinetics without using
the computer, even if one is mainly interested in symbolic calculations.
Luckily, all three of the present authors learned and taught Mathematica
from relatively early on, cf. Szili and Tóth;2 therefore it was quite natural
to use this extremely powerful and providently designed language for our
purposes.From time to time we wrote codesto solve problemsin reaction
kinetics, for the simulation of the usual stochastic model (what we have
started well before, in the beginning of the seventies of the last century3),
orforthedecompositionofoverallreactions4,5orforbiologicalmodeling.6
Asourprogramsstartedtobeusedbyotherpeoplewerealizedthatitwould
beusefultomakeitavailabletoawideraudience.
However, according to our experience, recent developments in the theory
of reaction kinetics, what may also be called formal reaction kinetics,
are known in the chemists’ and chemical engineers’ community much
less than optimal. Mathematicians are also not familiar enough with this
area full of interesting unsolved problems. Therefore, we tried to do the
1Érdi P, Tóth J (1989) Mathematical models of chemical reactions. Theory and applications of
deterministicandstochasticmodels.PrincetonUniversityPress,Princeton,NJ.
2SziliL,TóthJ(1996)MathematicsandMathematica.ELTEEötvösKiadó,Budapest.http://www.
math.bme.hu/~jtoth/Mma/M_M_2008.pdf.
3ÉrdiP,SiposT,TóthJ(1973)Stochasticsimulationofcomplexchemicalreactionsbycomputer.
MagyKémFoly79(3):97–108.
4Papp D, Vizvári B (2006) Effective solution of linear Diophantine equation systems with an
applicationinchemistry.JMathChem39(1):15–31.
5KovácsK,VizváriB,RiedelM,TóthJ(2004)Computerassistedstudyofthemechanismofthe
permanganate/oxalicacidreaction.PhysChemChemPhys6(6):1236–1242.
6TóthJ,RosparsJP(2005)Dynamicmodellingofbiochemicalreactionswithapplicationstosignal
transduction:principlesandtoolsusingMathematica.Biosystems79:33–52.
vii
viii Preface
impossible:ourbookisaimedatbeginnersinthefieldwithsomeknowledge
in introductorycalculus, linear algebra, and stochastics who would like to
proceedveryfasttowardsreadingtheoriginalliteratureandalsocontribute
tothetheoryandapplyittomoreandmorepracticalproblems.Atthesame
time we are also providing a kind of user’s manual to our Mathematica
programpackagecalledReactionKinetics,Version1.0.Theprogram
downloadedfreely,andremarksandcriticismsleadingtoVersions2,3etc.
arewelcome.WealsoshowhowtheprogramsworkinaCDF(Computable
DocumentFormat)filesreadablebyanybodyusingthefreelydownloadable
programCDFPlayer. One should not forgetthat the program Mathematica
is freely available for anyone having a Raspberry Pi computer (even if it
does not work as effectively as a paid version on a computer with larger
capacity).Nodeepknowledgeofprogrammingisneeded;weareconvinced
thatwell-writtenMathematicaprogramsareeasytoread,andtheyareself-
explanatory.
ScopeandRelatedBooks Some of the readers might have noted a slight resem-
blancebetweenthetitleofourbookandthatofanearlySpringerbestseller,
aclassic:Pólya,G.andSzego˝,G.AufgabenundLehrsätzeausderAnalysis,
(4th edition), Heidelberger Taschenbücher, Springer, (1970) (it would be
inappropriate to add this item to the reference list). This is not by pure
chance.ThismaybetheonlysimilarityoftheauthorswithPólyaandSzego˝:
we also wanted to collect as much material as possible from the research
literature and present it in unified form to teach the elements and to be
availablefortheresearchersaswell.Thisturnedouttobeataskharderthan
anticipated in a period of explosive developmentof this branch of science
(after decades of sleeping at the end of last century). It may happen that
wedonotgiveanexactreferencewhenposingaproblem;however,allthe
paperswehavetakenproblemsfromareincludedinthelistofReferences.
Weshallalmostnevergivetheproofofatheorembutwedotrytogivethe
exactreferencespointingtothefirstoccurrence.
Érdi and Tóth7 has been used by some readers as a textbook and has also
been the starting point of many research papers. Related to that book the
present one coversa narrowerarea, is deeper, is more based on the use of
computer,andcontainsmuchmorefigures.Thatis whywe hopeit willbe
suitableforstudentsevenmoreandwillgiveideasforevenmoreresearchers.
Letustrytocharacterizeafewrelatedbookstohelpthereaderinorientation
andfurtherreading.(Astotheprograms,theywillbecomparedinChap.12
at the end of our book because by then the reader will have a certain
experiencewithourprogramandwillalsopossessthenecessaryvocabulary.)
7Érdi P, Tóth J (1989) Mathematical models of chemical reactions. Theory and applications of
deterministicandstochasticmodels.PrincetonUniversityPress,Princeton,NJ.
Preface ix
If you need more chemistry—including recipes of exotic reactions—you
might consult the books by Pilling and Seakins,8 Lente,9 Epstein and
Pojman,10 Scott,11 Temkin,12 orEspenson13 whicharealmostcomplemen-
tary to our book. Although the main interest of the author of Póta14 is
reaction kinetics, his book contains problems from many other areas of
mathematicalchemistryaswell.
The whole (application oriented) book by Érdi and Lente15 has been
dedicated to stochastic kinetics. The best source for numerical methods
needed to deal with problems of kinetics is Turányi and Tomlin.16 The
book by Marin and Yablonsky17 contains a very large number of concrete
examples fully analyzed, but its main goal is to try to rescue the “sinking
AtlantisofSovietScience,”astheauthorsputit:theycollectmanyvaluable
results which can only be found in hardly accessible (and untranslated)
journalsinRussian.
Purpose Thebookisaimedatchemists,engineers,andmathematiciansinterestedin
theinterdisciplinaryfieldofformalreactionkinetics,animportantsubfieldof
mathematicalchemistry.Reactionkineticsstudiesthe evolutionofreacting
species; it can predict time-dependent and stationary behavior, and it can
providea frameworkforcollectingdataeconomically.Hence,itisrelevant
for everyone working in or studying chemistry, chemical engineering,
biochemistry,environmentalchemistry,combustion,etc.
Let us say a few words about our package, which is used throughout and
taughtalsotothereader.Inalltheareasofchemistrywherereactionkinetics
playsanessentialrole(biochemistry,combustion,pharmacokinetics,organic
andinorganicchemistry)theusualsituationisthatonehasmechanisms,or
8Pilling MJ, Seakins PW (1995) Reaction kinetics. Oxford Science Publications, Oxford/New
York/Tokyo.
9Lente G (2015) Deterministic kinetics in chemistry and systems biology. The dynamics of
complexreactionnetworks.SpringerBriefsinmolecularscience.Springer,NewYork.
10EpsteinI,PojmanJ(1998)Anintroductiontononlinearchemicaldynamics:oscillations,waves,
patterns,andchaos.Topicsinphysicalchemistryseries.OxfordUniversityPress,NewYork.http://
books.google.com/books?id=ci4MNrwSlo4C.
11ScottSK(1991,1993,1994)Chemicalchaos.Internationalseriesofmonographsonchemistry,
vol24.OxfordUniversityPress,Oxford.
12TemkinON(2012)Homogeneouscatalysiswithmetalcomplexes:kineticaspectsandmecha-
nisms.Wiley,Chicester.
13Espenson JH (2002) Chemical kinetics and reaction mechanisms, 2nd edn. McGraw-Hill,
Singapore.
14PótaG(2006)Mathematicalproblemsforchemistrystudents.Elsevier,Amsterdam.
15Érdi P, Lente G (2016) Stochastic chemical kinetics. Theory and (mostly) systems biological
applications.Springerseriesinsynergetics.Springer,NewYork.
16TurányiT,TomlinAS(2014)Analysisofkineticreactionmechanisms.Springer,NewYork.
17MarinGB,YablonskyGS(2012)Kineticsofchemicalreactions.Decodingcomplexity.Wiley,
Weinheim.
x Preface
complex chemical reactions consisting of a large number of (elementary)
reactionsteps,andofdifferentchemicalspecies.Thissituationgeneratesthe
needtodoasmuchcalculationsautomaticallyaspossible.Ontheotherside,
developments in formal reaction kinetics in the last three to four decades
madeitpossibletocreatetoolstogivethiskindofhelptothechemist.Taking
allthisintoconsiderationitisquiteastonishingthatsofewgeneralpackages
inreactionkineticsexist.Ourpackagehasthefollowingproperties:
1. Itisbasedonrecentdevelopmentsinformalreactionkinetics.
2. It has been written in a modern language, Mathematica (or Wolfram Lan-
guage,ifyouwish),usingallthenicefeaturesandefficienttoolsofit.
3. Itisofmodularstructure;thereforethepresentversioncan,should,andwill
befurtherdeveloped.
4. Itincludesstochasticsimulationmethodsaswell.
Plan After an introduction to time-independent, mainly combinatorial tools, the
book covers several mathematical and algorithmic aspects of continuous
time(deterministicandstochastic)modelsofchemicalreactions.Thebook
combines the formal presentation of mathematical models and algorithms
withthedemonstrationoftheapplicationsusingthesymbolicandnumerical
featuresofMathematica.Eachtopicisintroducedbyelementaryexamplesto
illustratetherelevantconcepts.Thenareviewoftheliteraturefollows,also
coveringthe state of the art. Finally, both simple and real-world examples
areanalyzedusingourMathematicaimplementation.
RelatedDisciplinesandIntendedAudienceWithinDifferentAreas
1. Mathematics: linear algebra, linear and integer programming, Diophantine
equations,polynomialordinarydifferentialequations,reactiondiffusionequa-
tions,graphtheory,continuoustimediscretestateMarkovprocesses.
2. Chemistry: reaction kinetics, physicalchemistry, chemical reaction network
theory.
3. Industryandtechnology:combustion,environmentalissues,pharmacology.
4. Biology:biochemistry,drugdesign,enzymology,populationbiology,systems
biology.
5. Government:environmentalissues,foodordrugsafety.
TheprimaryaudiencewillbeMScandPhDstudentsinappliedmathematics,
chemistry,biochemistry,chemicaland environmentalengineering.Itmight
alsobeatextbookforgraduate-levelcoursesonmathematicalandphysical
chemistry. We think that newcomers in the field might also learn from it,
playingwiththeprograms,andtheabundantliteraturemayalsohelpexperts.
We do not claim that the whole book will be a mandatory text for all the
studentsintheabovesciences;however,wearesurethatsomestudentsfrom
alloftheabovedisciplineswillbeinterested.
Preface xi
BeforethoughtsandAfterthoughts We are going to work with clear mathematical
concepts,precisedefinitions,andtheorems,neverthelesstrytokeepinmind
thepotentialapplications.Furthermore,wedonotwanttocoveralltheareas
of reaction kinetics, only those which we know relatively well. We collect
andreviewthecorrespondingmodernliterature.Ourprogramscanbeused
to treat almost all important problems of homogeneous reaction kinetics,
both deterministic and stochastic. The nontrivial algorithms we use come
partiallyfromourworkshop.
Theexercisesandproblemsfollowverycloselythematerialexposedinthe
chapters.Theycanbebestusedifthereadertriestosolvethem,andevenif
(s)heisabletosolveaproblem,(s)hemightlearnmethodsortricksfromthe
solutionpresentedbyus.
Budapest,Hungary JánosTóth
Budapest,Hungary AttilaLászlóNagy
Raleigh,USA DávidPapp
March25,2018