FluctuationsandScalinginBiology
T.Vicsek
editor
2001
Contents
Introduction
2
1
Basicconcepts(T.Vicsek)
3
1.1
Fluctuations................................
5
1.1.1
Noiseversus
uctuations.....................
5
1.1.2
Molecularmotorsdrivenbynoiseand
uctuations.......
6
1.2
Scaling...................................
8
1.2.1
Criticalbehaviour.........................
8
1.2.2
Scalingofeventsizes:Avalanches................
9
1.2.3
Scalingofpatternsandsequences:Fractals...........
11
1.2.4
Scalingingroupmotion:Flocks.................
14
2
Introductiontocomplexpatterns,
uctuationsandscaling
17
2.1
Fractalgeometry(T.Vicsek).......................
18
2.1.1
Fractalsasmathematicalandbiologicalobjects........
19
2.1.2
De�nitions.............................
22
2.1.3
Usefulrules............................
23
2.1.4
Self-similarandself-aÆnefractals................
25
2.1.5
Multifractals
...........................
27
2.1.6
Methodsfordeterminingfractaldimensions
..........
28
2.2
Stochasticprocesses(I.Der�enyi)
....................
31
2.2.1
Thephysicsofmicroscopicobjects
...............
31
2.2.2
KramersformulaandArrheniuslaw
..............
33
2.3
Continuousphasetransitions(Z.Csah�ok)................
35
2.3.1
ThePottsmodel.........................
38
2.3.2
Mean-�eldapproximation
....................
38
Bibliography
..................................
41
1
2
CONTENTS
3
Self-organisedcriticality(SOC)(Z.Csah�ok)
45
3.1
SOCmodel................................
46
3.2
Applicationsinbiology..........................
47
3.2.1
SOCmodelofevolution
.....................
48
3.2.2
SOCinlungin
ation.......................
54
Bibliography
..................................
57
4
Patternsandcorrelations
59
4.1
Bacterialcolonies(A.Czir�ok)......................
59
4.1.1
Introduction............................
59
4.1.2
Bacteriaincolonies........................
60
4.1.3
Compactmorphology.......................
67
4.1.4
Branchingmorphology......................
78
4.1.5
Chiralandrotatingcolonies...................
96
5
Microscopicmechanismsofbiologicalmotion(I.Der�enyi,T.Vicsek)105
5.1
Characterisationofmotorproteins
...................105
5.1.1
Cytoskeleton...........................105
5.1.2
Musclecontraction........................109
5.1.3
Rotarymotors...........................113
5.1.4
Motilityassay...........................115
5.2
Fluctuationdriventransport.......................118
5.2.1
Basicratchetmodels.......................120
5.2.2
Briefoverviewofthemodels...................121
5.2.3
Illustrationofthesecondlawofthermodynamics
.......122
5.3
Realisticmodels..............................123
5.3.1
Kinesin
..............................123
Bibliography
..................................131
6
Collectivemotion
141
6.1
Flocking:collectivemotionofself-propelledparticles(A.Czir�ok,T.
Vicsek)...................................141
6.1.1
Modelsandsimulations......................142
6.1.2
Scalingproperties.........................143
6.1.3
FurthervariantsofSPPmodels.................152
6.1.4
Continuumequationsforthe1dsystem.............163
6.1.5
Hydrodynamicformulationfor2D................165
6.1.6
Theexistenceoflong-rangeorder................170
Bibliography
..................................172
Chapter1
Basicconcepts
Thecomplexityofbiologicalsystemsismanifestedinmanyways.Hereweshall
considerthoseaspectsoflifewhichinvolverandom
uctuationsandahierarchical
underlyingstructureresultinginapowerlawdependenceofthevariousquantities
characterisingthesesystems.Aswillbeshown,thesetwofeatures{
uctuationsand
scaling{arefrequentlyandintimatelyrelated,althoughinsomecasestheyappear
independentlyofeachother.
Are
uctuationsanimportant,inherentingredientoflife?Whatistheirorigin
andimpact?
Theseareimportantquestionswhichhavetobeaddressedbeforewe
leadthereadertothemoreadvancedpartsofthetext.
Itisalmostatrivialstatementthatnearlyallprocessesinbiologyinvolverandom-
nessbeyondanegligiblelevel.Ontheotherhand,exceptsomearti�cialsituations,
allphenomenainnaturehavetheelementsofstochasticityinthem.Thereisno
suchathingasanoiselessorabsolutelydeterministicsystemandthisistruefrom
thebehaviourofgalaxiesdowntoelementaryparticles.Intheintermediaterangethe
presenceoftemperature
uctuations,thevariousnonlinearities/instabilitiesrepresent
themainsourceofrandomness.Thisisknowntobesoandisunderstoodinmany
casesfornon-livingsystems.
Biologicalsystemsarenotexceptionsfrom
thesamerule.Perhapsthemost
typicalfeatureofabiologicalobjectorsignalisthattheysimultaneouslypossess
somespeci�cpatternanddeviationsaroundthesepatterns.Ifwetakeasanexample
ananimal,letussayadog,wecanconsiderthefollowingillustrationsoftheabove
statement:i)notwodogsareexactlythesame,buttheyaresimilarinmanyways,
ii)theyhavetypicalreactions,butneverreactinaperfectlyidenticalway,iii)their
heartbeatsalmostregularly,butacloseranalysisshowsspeci�c
uctuationsaround
theaverageheartbeatrate,iv)ifwestudytheelectricsignals(withthehelpof
anelectrode)oftheirbrainweseeanalmostrandomlylookingseriesofspikes,v)
3
4
CHAPTER
1.
BASIC
CONCEPTS
ifwehappenedtoseethemotionoftheindividualcellsorevenlargemoleculesin
theirbody(e.g.,bloodcells,sperms,RNAmolecules)wewouldobserveanerratic
behaviouraroundanaveragetendency.Thelistofsuchexamplescouldbecontinued
forlong.
Aswillbediscussedlater,inmanycasestheabovementionedstochasticchanges
or
uctuationsarenotcompletelyrandomandcanbeassociatedwithpowerlawsor
scaling.
Inshort,ifasystemismadeofmanyinteractingunits,speci�cstatisticalfeatures
involving
uctuationsandscalingemerge.
Thereareafewimportantaspectsof
uctuationswhentheoriginofanapparently
randombehaviourisrelativelywellunderstood.
I) Microscopicobjectsaresubjecttothesocalledthermal
uctuations.Itisafun-
damentalfeatureofallsystemsthatiftheyhaveawellde�nedtemperatureT,than
eachmicroscopicparticle(atomormolecule)inthemhasinaverageanamountof
kineticenergy
1 2kT
(kistheBoltzmannconstant)foreachdegreeoffreedom(modeof
motion).Amorecomplexmoleculehasmanydegreesoffreedom,andisalsosubject
to"kicks"fromtheneighbouringmoleculesmovingwithavelocitycorrespondingto
theirkineticenergy.Asaresult,anindividualmoleculeinteractingwithmanyothers
followsamoreorlessrandomtrajectoryduetothemanysubsequent,randomlyoccur-
ring"kicks"fromtheirneighbours.Itisabeautifulsubjecttounderstandhowsuch
uctuationsaretamedbythespeci�cprocesses(di�usion,enzymaticinteractions)
insidelivingcellsandresultinanorganised(withsome
uctuations)behaviour.
II)
Whenmanysimilar,butnotnecessarilymicroscopicobjects(biologicalornon-
living)arepresentinasystemtherearefurtherreasonstoconsidertherandomaspect
ofthebehaviour.
Ingeneral,ifmanymovingobjectsinteract,themotionoftheindividualobjects
isboundtoberandom-like(likeintheabovementionedmicroscopiccase),evenif
themotionortheinteractionisdeterministicandweconsidermacroscopicobjects.
Thetrajectoriesaresubjecttomanychangesofvaryingdegreeandinsteadoflooking
atsuchprocessesasdeterministicitisconceptuallymoreusefultoassumethatthe
motionisstochasticanditisthestatisticalfeaturesoftheassemblyofobjectswhich
shouldbedeterminedwhendescribingthesystem.Itisnotonlythedirectionof
motionwhichcanbechangedduringinteraction.Forobjectswithadirectedness
theirdirectioncanalsobemodi�edduetosomedirection-dependentinteraction.
Now,asaresultoftheseinteractions,variousspontaneousprocessesmaytake
inthesystem:forattractiveforces,aggregatesarelikelytoform,orgroupswith
thesamedirectednessoftheirmembersmayappear.Schoolsof�shisacommon
1.1.
FLUCTUATIONS
5
exampleforaggregation.However,duetothecomplexnatureofasystemwithmany
objectsmovingbothrandomly,andorderly(inagroup,butwithsomeperturbations)
anotherkindof
uctuations,arandomdistributionofgroupsizesisproduced.Many
timesthedistributionofgroupsizesfollowsapowerlaw,orinotherwords,ascaling
distribution.
III)
Non-linearitiesareknowntoleadtoaverycomplexbehaviourwhich{especially
inthepresenceofthermal
uctuations{canbeconsideredasrandom.Acommon
spatialexampleistheformationofbranchingpatternsunderconditionsleadingtothe
unstablegrowthofmoreadvancedbranches.Inthesecasesthesmallestperturbations
arelikelytoresultinanew,quicklygrowingside-branchandthestructureattains
thewellknownbranchingmorphologysocommoninbiology(e.g.,treesorblood
vesselnetworks).Althoughsuchnetworkspossesssomespeci�cfeaturestypicalfor
thegivenbiologicalobject,theyarealsoirregular.Inaddition,aswillbeshownthey
haveaspeci�chierarchicalstructurebestdescribedbyfractalgeometry.
InthefollowingpartsofthisintroductorychapterIbrieydiscussthebasiccon-
ceptsrelatedto
uctuationsandscalinginbiologyandsummarisethemostimportant
�ndingsobtainedintherelatedinvestigations.
1.1
Fluctuations
1.1.1
Noise
versus
uctuations
Theoriginof
uctuationscanwidelyvary.Inmostofthecases,however,theyare
duetotheabovementioned"thermalnoise"orerraticmotionofmicroscopicparti-
cles.Typically,noiseisnotcorrelatedwhichmeansthatthevalueofthe
uctuating
quantityF
atthelocationr
attimetdoesnotdependonitsvalueatadi�erent
locationandatanearliermomentoftime.Symbolicallywewritethatthecorrelation
functionhastheformofadeltafunction
c(r)=hF(r;t)F(r
0;t
0)i�hF(r;t)ihF(r;t)i=CÆ(r�r
0;t�t0)
wheretheÆfunctionisequaltozeroforanynon-zerovaluesofitsargumentsand
C
issomeconstant.Theaveraging(denotedbyh:::iismadeoverallvaluesofthe
arguments.Thisexpressionholdsfortheuncorrelatedwhiteorshotnoise,whilefor
correlated
uctuationsc(r)hasmorecomplexforms.
Fluctuationscanbemorecomplexthanjustwhitenoise.Manytimestheyrepre-
sentaninherent,characteristicfeature(reactiontowhitenoise)ofthesystemitself.
6
CHAPTER
1.
BASIC
CONCEPTS
Forexample,duetothethermal(white)noisethe
uctuationsinthetotalmagnetisa-
tionofaferromagnetnearitscriticalpointcanstronglyincreaseandexhibitspeci�c
correlations.
Noiseand
uctuationsplayacentralroleinorderingphenomena.Asystemof
manyinteractingunitswithaninteraction"trying"toforcetheunitstobehaveinthe
samewayinthepresenceofstrongexternal
uctuations(noise)maynotbeableto
order.Ontheotherhand,asthemagnitudeofnoisedecreases(e.g.,thetemperature
islowered),theobjectsinthesystemmayalreadyassumetheircommonorcollective
patternofbehaviour,forexample,theyspontaneouslydevelopacommondirection
ofmotion,or�ndtheirrightplaceforacrystallinestructure.Thesenoisedriven
transitionswillbediscussedbrieyinthescalingpartofthisintroductorychapter.
Inthefollowingwhenonlytheirstochasticnatureisrelevantweusenoise,
uctu-
ationsandrandomperturbationsassynonyms.However,randomchangesappearing
asaninherentbehaviour(response)ofthesystemitselfwillalwaysbecalled
uctu-
ations.
Afurtheraspectof
uctuationsinvolvestransportprocessesinthepresenceof
noise.Interestingly,evenuncorrelated
uctuations(thesearerandomchangeswithout
anytendencies)mayresultinabehaviourwithawellde�nedtendency.Thishappens
inthecaseofmolecular
motors,wherewhitenoiseassiststhemotorproteinsto
proceedalongspeci�cintracellulartracks.Ontheotherhand,whitenoisealone
cannotproducecurrents,thiswouldcontradictthesecondlawofthermodynamics.
Herewementionthebasic�ndingsaboutthisfascinatingnewdirectioninbiological
physics.
1.1.2
Molecularmotorsdrivenbynoiseand
uctuations
Intheinorganicworldtransportalwaystakesplacealongamacroscopicgradientof
apotential(orananalogousquantity).Thingsfalldownduetothegravitational
forcewhichcanbeobtainedfromthederivativeofthegravitationalpotential.Even
theglobaltransportofmicroscopicobjectssuchasmoleculestakesplacealongthe
extendedgradientofthesocalledchemicalpotential.Forexample,particlestendto
di�usefromadenserregiontoalessdenseonemakingamacroscopicoveralldistance
ifthegradientofthedensityextendsoverthatdistance.Electronsmoveinawire
followingthegradientoftheelectricpotentialwhichislargeratoneendofthewire
thanattheotherend.
Thisisnothowtransportisrealizedinmostofthebiologicalsystems.Theabove
mechanismtendstobringasystemintoamotionlessstate:astheobjectsmovealong
thepotentialgradienttheysimultaneouslydecreasethevalueoftheoverallgradient.
Forexample,di�erenceintheconcentration(drivingthedi�usionaltransport)de-
1.1.
FLUCTUATIONS
7
creasesintimeastheparticlesdi�usetothespotsofsmallerconcentrations(and
increasethedensityatthesespots).Instead,lifeisaboutgeneratingdi�erences,
buildingstructuresfromalesspatternedenvironment.
Onemechanismfordoingthisismotionalongperiodic,butlocallyasymmetric
structures.Imagineasawtooth-like(orratchet)potential:itispiece-wiselinear,
withtwodi�erentgradients(slopes).Wealsohaveaparticleinthispotential,most
ofthetime"sitting"inoneoftheminima("valleys").Now,ifwepullthisparticle
periodicallybackandforthinthissawtooth-likepotential,thefollowingcasesare
possible:i)theforceislargeenoughfortheparticletobepulledoutfromaminimum
bothtoleftandrightii)theforceisstrongenoughonlytopullouttheparticleinthe
directionofthesmallergradient(steepness),iii)theparticleisnotpulledoutfrom
thevalleybecausetheforceistooweak.
Obviously,caseii)establishesasituationinwhichatransportispossiblewithout
anyglobalpotentialdi�erence:theparticlemovesinthedirectionofthesmaller
slopealthoughtheforceactingonitiszeroinaverage(actsbackandforth).Inthis
wayparticlescanbecollectedatoneendofatrackwithsuchapotential,thus,a
concentrationdi�erence,i.e.,structurecanbebuiltup.
However,lifeisnotsosimple.Suchprocessesoccuronthemolecularlevel,where
uctuationsareverystrongfortworeasons:a)Theparticle(calledmotorprotein)
iskickedbytheothermoleculesinthesystem
randomly,inanoisy,uncorrelated
mannerallthetime,b)theperiodic,deterministicbackandforthdrivingcannotbe
establishedinamicroscopicenvironmentaswell:instead,theenergysuppliedbythe
socalledATPmoleculesandprovidingtheconformationalchangesofthemolecular
motorresultinginitstendencytomovebackandforthalsoarrivesstochastically(the
ATPmoleculesare"consumed"attimesfollowingaPoissondistribution).
Thepictureemergingisthefollowing:Apossible,simpli�edrepresentationof
biologicalmotionisthatofthemotionofaBrownianparticleinanasymmetric
periodicpotential.Thecorrespondingequationhavetoaccountforthestochastic
natureoftheprocess:bothforthewhitenoisecomingfromtheenvironmentandfor
theirregularnatureoftheenergyinput.TherelatedLangevinequationapproachis
discussedindetailsinchapter5.
Thereisoneinterestingpointhere:inthecaseofmotionalongasymmetric
periodicpotentialsnoisemayplayaroleenhancingthetransport.Thisisabit
paradoxical,wearemoreusedtothenotionthatrandom
perturbationstypically
destroytendencies.Inthecaseofmolecularmotors,however,itmayhappen,that
addingwhitenoiseresultsinstrongercurrent.Theeasiestwaytoshowthisisthe
following:Imaginecaseiii)whentheexternalforcepullingtheparticlebackandforth
isnotstrongenoughfortheparticletoescapeinanyofthetwodirections.Now,if
weaddnoise(arandomlychangingsmallamountofextraforce)insomecasesthe
8
CHAPTER
1.
BASIC
CONCEPTS
overallforcemayexceedthecriticalvaluenecessarytopullouttheparticlefroma
valley.Thiswillhappenmorefrequentlyinthedirectionofthesmallerslope,since
thecriticalamountofforceissmallerinthatdirection,andtherewillbeanoverall
currentinthedirectionofthesmallerslope.Thisiswhysuchsystemsarealsocalled
thermalratchets.
Anotherinterestingvariantofthissituationiswhenweconsideronesingleforce
(insteadofthesumofadeterministicperiodicbackandforthactingforceandan
uncorrelatedwhitenoiseone)changingstochastically.Now,ifthissingle
uctuating
forceiscompletelyuncorrelated,orinotherwords,thermalorwhitenoise,thenno
globaltransportispossible.Ifthe
uctuationsarethermalnoise-like,thanthesystem
isinequilibriumandnotransportispossibleinthermalequilibrium.
Ontheotherhand,ifthe
uctuatingforce(noise)iscorrelated,transportalready
becomespossible.Theprobabilitytoleaveinthelesssteepdirectionwillbestilllarger
thanintheoppositedirection.Inthisway,ourratchet"recti�es"the
uctuations,it
isabletomakeuseofitsnon-whitepart.
1.2
Scaling
AquantityF
scalesasafunctionofitsargumentx
ifchangingtheargumentbya
factor(e.g.,changingxforAx)doesnotchangetheformofthefunctionaldependence
ofF
onx(apartfromaconstantfactor).Thisistriviallysoforafunctionoftheform
ofapowerlaw,butisnot,asarule,trueforotherfunctions.Forexample,F
=x
2
scalesbecauseF
0
=F(Ax)=A
2x
2
=A
2F(x),whileF
=log(x+B)doesnotscale
becauseinthiscaseF
0
=F(Ax+B)cannotbereducedtoaformcontainingF(x)
andaconstantfactoronly.
Thescalingquantitiesweshallconsideraremostlyofstochasticnature.Thus,
thespeci�cfunctionaldependenceswillbevalidfortheaverageofthegivenquan-
titiesasafunctionoftheirarguments.Eachrealizationofsomestochasticprocess
hasa
uctuatingoutcome,butmakinganaverageoverseveralprocesses,orovera
singleprocesshavingsequences
uctuatingaroundanaveragecanprovidetheproper
estimateofthequantityofinterest.
1.2.1
Criticalbehaviour
Perhapsthemosttypicalcollectivephenomenaexhibitedbyanassemblyofmany
interactingparticlesarethesocalledphasetransitions,when,asafunctionofanex-
ternalparameter(liketemperature),theparticlescollectivelychangetheirbehaviour.
1.2.
SCALING
9
Forexample,duringfreezingallofthemoleculesofa
uidmovetoaspeci�cposition
sothattheresultingstructurebecomesacrystalwithregularmicroscopicstructure.
Inthevicinityofsuchtransitionsinterestingspatialandtemporalcorrelations
canbeobservedinthesystemsandthesefeatureswillberelevantfromthepointof
themajorityofthetopicsdiscussedinthisbook.
Inparticular,duringsecondorderphasetransitionsthesocalled"criticalstate"
(orcriticalphenomena)canbeobservedinwhichtheordinarilyexponentialfunc-
tionaldependencesarereplacedbyalgebraic(powerlaw)dependenceoftherelevant
quantitiesontheirparameters.Apowerlawdependenceofthequantityn(s)(e.g.,
thenumberofschoolscontainings�sh)isofthefollowingformn(s)�
s�
�
;where�
expressesproportionality,and�
issomeexponent.Thepowerlawdependenceisvery
special:forexample,apowerlawdecayofthenumberofclusters(schoolsof�sh)as
afunctionoftheirsize(numberof�shinaschool)meansthatverylargeclustersmay
occurwithaprobabilitywhichisnotnegligible(thisprobabilitywouldbeextremely
smallifthenumberofclusterswoulddecreaseexponentiallywithgrowingclustersize,
asitdoesforregularstates).Ifaquantitychangesaccordingtoapowerlawwhenthe
parameteritdependsonisgrowinglinearly,wesayitscales,andthecorresponding
exponentiscalledcriticalexponent.
Whyaresuchstatescalledcritical?Becausetheyareextremelysensitivetosmall
changesorperturbations.Ifahumanbeingisinacriticalstateitmeanshisorher
statecangetworseveryeasily.Inthecaseoflatticemodelsasmallchangeinthe
temperaturemayleadtothequickcollapseorbirthofverylargeclusters.Duringthe
lasttwodecadesstatisticalphysicistshaveworkedoutdelicatetheoriesandmethods
tointerpretthebehaviourofsuchcriticaltransitionsandstatesandinthefollowing
weshallconsidertheapplicationoftherelatedconceptstobiologicalphenomena
involvingmanysimilarunits.
Theimportantpointisthatscalingtypicallyinvolvesuniversality:insteadofpar-
ticleswecanimaginesimilarorganisms.Iftheinteractionamongtheseorganismsis
relativelysimple,andisanalogoustothosewhichproducescalingorphasetransitions
innon-livingsystems,thanwecanexpectthesametypeofbehaviourinsuchsystems
oflivingentitiesaswell.
1.2.2
Scalingofeventsizes:Avalanches
Asmentionedabove,scalingcanbeobservedduringequilibriumphasetransition,but
inthefollowingweshallarguethatapowerlawdependenceofthevariousimportant
quantitiescanemergeinthenon-equilibriumworld(oflife)aswell.Infact,itisinthe
non-equilibrium
statewhenstructurescanemergespontaneouslyfromanoriginally
homogeneousmedium.
10
CHAPTER
1.
BASIC
CONCEPTS
Aparticularandimportantdeparturefromequilibriumiswhenthesystem
is
"slowlydriven"toastationarystate.Slowdrivingmaymeanthegradualaddition
ofsomequantity(energy)toaasystem
whichmayalsoloosethisenergydueto
interactions.Iftheinteractionbetweentwopartsofthesystemissuchthatachange
exceedingacriticalvalueofthegivenquantityinoneunitresultsinasimilarex-
ceedingofthesamecriticalvalueintheneighbouringunit,thanlarge,avalanche-like
seriesofchangesmaytakeplaceinthesystemwhenitisclosetoa(critical,balanced)
state.Asthisstateisbothspontaneouslyemergingandcritical,theassociatedphe-
nomenoniscalledInthisstatethesystemisverysensitiveto
uctuations,sincea
smallperturbationmayleadtoalargeavalanche.Inthissensethesystemisina
criticalstate.Notalloftheavalanchesarelarge,themajorityofthemaresmall,
buttheprobabilityofhavingalargeavalanchedoesnotgotozeroveryquicklywith
theavalanchesize.Inmanysuchslowlydrivensystemsscaling(powerlaw)ofthe
avalanchesizedistributioncanbeobserved.Avalanchesaresometimesverylarge(as
weknowfromthenewsonskiingareas,butinthiscontextanearthquakeisalso
anavalanche)andtheyarethesocalledbigeventsinthetheoryofslowlydriven
systems.
Thesimplestexampleisthatofasandpile.Imaginethatweaddgrainsofsand
toagrowingpile:asthegrainsaredropped,mostofthetimethesurfaceofthe
pilebecomesonlyslightlyrearranged,however,timetotimethenewgraintriggers
alongseriesofevents:grainsrollingdowntheslopedragmanymoregrainswith
them.Asimplemodelalongalinesegmentwouldcontaincolumnsofparticles.
Anewparticleisdroppedatarandomposition.Iftheheightdi�erencebetween
twocolumnsbecomeslargerthan2,thanfromthehighercolumntwoparticlesare
removedandaddedtothetwoneighbouringcolumn.Particlesattheedgeofthe
segmentaredroppedoutfromthesystemcompletely.
InthisbooktwoexamplesforbiologicalSOCarediscussed.Thestructureof
thelungissuchthatitcanbebroughtintoanalogywiththesandpilemodel.The
airenteringthelunghastogothroughasequenceofairwayseachopeningifthe
pressureexceedsacriticalvalue.Byforcingtheairgraduallyentertheexperimentally
investigatedlung,largejumpsintheterminalairwayresistancehavebeenobserved.
Thesejumpscorrespondedtoavalanches:tothesubsequentopeningofalargeset
ofairwaysinashorttime.Thedistributionofjumpsfollowedapowerlaw.The
observationofstrongly
uctuatingextinctionratesandthecorrespondingSOCrelated
theoryisalsodiscussedinchapter3.
1.2.
SCALING
11
1.2.3
Scalingofpatternsandsequences:Fractals
Natureisfullofbeautifulcomplexshapeswhicharefarmoreintricatethanthe
idealisedformsproposedbyEuclidmorethantwothousandyearsago.Thisispar-
ticularlytrueforthelivingworldwherecomplicatedstructuresaregeneratedduring
embriogenesis.Manyofthesepatternsarerandombranchingnetworks;examplesin-
cludetrees,thenetworkofbloodvessels,airwaysinthelung,neuralnets,etc.These
highlyhierarchicalpatternscanbebestinterpretedintermsoffractalgeometry.
Fractalsarefascinatinggeometricalobjectscharacterisedbyanon-trivialfrac-
tionaldimension.ImagineagrowingpatternwhosemassM
(thenumberofparticles
itcontains)increasesslowerthanthed-thpowerofitsradiusR,wheredisthedi-
mensionofthespaceinwhichthepatternisdeveloping.Thisisclearlydi�erentfrom
thecaseofhomogeneousstructuresthatweareusedto.ForfractalsM(R)�
RD
;
whereD
iscalledthefractaldimensionsinceinmanycasesitisnotaninteger,buta
fractionalnumberlessthand.Iftheaboverelationistrueforapattern,itisbound
tobeself-similarinthesensethatasmallpartofitlooksthesameasthewhole
structureafteritisexpandedisotropically.Fordeterministicmathematicalfractals
theblownuppiecelooksexactlythesameasthewholeobject.Forrandompatterns
self-similarityissatis�edinastochasticmanner.Thefractaldimensioncanalsobe
de�nedthroughtheexpressionc(r)=
1 N
Pr
0
�(r)�(r+r
0)�
rD
�
d
wherec(r)describes
thedensity-densitycorrelationswithinthepatternand�(r)isequaltounityifthereis
aparticleatthepositionr,anditisequaltozerootherwise.Forisotropicstructures
thecorrelationfunctionc(r)isequivalenttotheprobabilitythatone�ndsaparticle
belongingtotheclusteratadistancer=jr�r0jfroma�xedpointonthecluster.
Inthiscaseanaveragingcanbemadeoverthedirectionsaswell.
Themeaningoftheabovestatementsisthatfractalscanbelookedatasstructures
exhibitingscalinginspacesincetheirmassasthefunctionofsizeortheirdensityas
afunctionofdistancebehaveasapowerlaw.
Self-aÆnestructuresrepresentanothertypeoffractals.Forsuchobjectsasmall
partofthefractalmustbeenlargedinananisotropicwaytomatchtheentirepattern.
Forexample,ifthefractalisembeddedintotwodimensions,forself-aÆnefractals
oneachievesmatchingbyrescalingthesizehorizontallyandverticallybydi�erent
factors.
FractalBacterialColonies
Perhapsthebestde�nedbiologicalsystemsexhibitingfractalgrowtharebacterial
colonies.Byacarefulcontroloftheexperimentalconditionsithasbeenpossibleto
obtainwellreproducibleresultsonthedevelopmentofcomplexbranchingpatterns
madeofmanymillionsofbacteriaastheymultiplyonthesurfaceofanagar(gel)
layerinaPetridish.Therelatedbeautifulresultsarediscussedinchapter4.
12
CHAPTER
1.
BASIC
CONCEPTS
Typically,bacterialcoloniesaregrownonsubstrateswithahighnutrientlevel
andintermediateagarconcentration.Undersuch"friendly"conditions,thecolonies
developsimple(almoststructureless)compactpatternswithrelativelysmoothen-
velope.Thisbehaviour�tswellthecontemporaryviewofthebacterialcolonies
asacollectionofindependentunicellularorganisms.However,innature,bacterial
coloniesregularlymustcopewithhostileenvironmentalconditions.Whathappens
ifwecreatehostileconditionsinaPetridishbyusing,forexample,averylowlevel
ofnutrientsorahardsurface(highconcentrationofagar),orboth?Thebacterial
reproductionrate,whichdeterminesthegrowthrateofthecolony,islimitedbythe
levelofnutrientsconcentrationavailableforthebacteria.Thelatterislimitedbythe
di�usionofnutrientstowardsthecolony.Hence,thegrowthofthecolonyresembles
di�usion-limitedgrowthininorganicsystemsleadingtofractalpatterns.
Di�usion-limitedgrowthleadstorandombranchingpatternsbecauseofthefol-
lowinginstability:ifagivenpartofthegrowingsurfaceisabitmoreadvancedthan
thesurroundingregion,thispartwilladvancefasterthantheneighbouringpartsof
thecolony,becauseitwillbeclosertothesourceofthenutrientdi�usingfromthe
outerregionsofthePetridish.Inturn,partslaggingbehindtendnottogrowany
moresinceinthoseregionsnonutrientwillbeavailableasthenutrientdi�using
towardsthecolonywillbeconsumedbytheadvancedpartsofthecolony.Thisis
positive(negative)feedback:aprotrusiongrowsfaster(andproducesabranch)the
screenedfjordsstoptogrowcompletely.Theresultingpatternhasaradiallygrowing
tree-likestructure.
Inrealitythesituationissomewhatmorecomplexsincethebacteriacancom-
municatethroughchemotaxis.Theyareabletopassoninformationabouttheiren-
vironmentandincrease/decreasethegrowthrateatotherpointsinthecolony.The
communicationenableseachbacteriumtobebothactorandspectator(usingBohr's
expressions)duringthecomplexpatterning.Thebacteriadevelopedaparticle-�eld
duality:eachofthebacteriumisalocalised(moving)particlewhichcanproducea
chemicalandphysical�eldarounditself.Forresearchersinthepatternformation
�eld,theabovecommunicationregulationandcontrolmechanismopensanewclass
oftantalisingcomplexmodelsexhibitingamuchricherspectrumofpatternsthanthe
modelsforinorganicsystems.
Allthiscanbeinvestigatedbyconstructingsuitablecomputermodels.Inthe
correspondingcalculationsanumberoffactorsaretakenintoaccount(seechapter
4):goodagreementwiththeexperimentalobservationscanbeachievedbyassuming
anutrientdependentmultiplicationrate,di�usionalmotionofthebacteriaonthe
agarsurface,chemotacticsignalling,etc.Thesesimulationsaredi�erentfromthe
commonapproachesinphysicsandbiology.Physicistsprefertobuildsimplemodels
ignoringmanyofthedetailsandlookforuniversalbehaviour.Biologistsmostly
1.2.
SCALING
13
usespeci�cmodelsre
ectingthebiologicaldetailsofthesystemunderinvestigation.
Themodelsusedtomimicbacterialcolonydevelopmentinthecomputerinterpolate
betweentheseapproachesandareaimingat�ndinguniversalbehaviourtakinginto
accountmostofthebiologicallyrelevantdetails.
Correlationsinthegeneticcode
Onepossiblerepresentationofthevastinformationstoredintheextremelylong
sequencesofDNAdataisarandomwalkbuilttocorrespondtosuchsequences.In
thisapproachDNAismappedontoaprocesswhichcanberegardedasawalk:each
ofthefour\letters"ofaDNAsequenceisidenti�edwithastepinagivendirection.
Then,thespeci�cfeaturesofthiswalkcanbeanalysedusingmethodsborrowedfrom
statisticalphysics.
Giventhewalkonecanlookforcorrelations.Twoseriesofdata(XandY)are
correlatedifthereisarelationshipbetweenthecorrespondingelementsoftheseries.
Whensearchingforcorrelationswithinasinglesequenceofdatawecanaskhowthe
valueXiisrelatedtothevalueXi+j.Bycomparingthetwovalueswiththeaverage
ofX
onecangetinformationaboutthequestionwhethertwovaluesinthedataset
separatedbyjelementsarecorrelated.
Anordinaryrandomwalkhasnolong-rangecorrelations.Oneofthemostrelevant
questionsonecanraiseinthecontextofDNAsequencesisthelocationofcoding
andnon-codingpartsinthegenome.Inthecasethesetwokindsofsub-sequences
havedi�erentkindsofcorrelationswemaybeabletodi�erentiatebetweencoding
andnon-codingpartwithoutanypriorknowledgeaboutthesequences.Indeed,it
hasbeenshownthattherandomwalkscorrespondingtonon-codingpartshavelong-
rangecorrelationsincontrasttothecodingparts(whichhaveshort-rangecorrelations
only). A
sanalternativetotheDNAwalk,thesymbolsequencecorrespondingtoaDNA
moleculecanberegardedasawrittentextcomposedbyusingfourletters.Sincewe
donotknowthe"language"ofthetextwehavetotoapplymethodsdevelopedfor
analysingwritten(natural)textsofunknownorigin.Inparticular,onecanaskthe
questionwhethertwotextswerewritteninthesamelanguageornot.Weexpect
largercorrelationsbetweentextofthesameorigin(language).Heretwosequences
arecorrelatedifthescalarproductofthetwoappropriatelyde�nedvectors(corre-
spondingtothem)hasavaluedi�erentfromthatitwouldhavefortwouncorrelated
sequences.Inthe�rstapproximationhismethodislanguageinsensitive.
WhenapplyingthevectorspacetechniquetoDNAsequencesinawaywelookat
DNAasanencodedtextwritteninanunknownlanguage,still,weexpecttolocate
correlationsbetweenpartsofthesequencesduetosimilaritiesintheirunderlying
structure.Incasethelanguageofthecodingpartsisdi�erentfromthenon-coding
oneswegetahighervalueforthecorrespondingscalarproduct(asithasbeendemon-
14
CHAPTER
1.
BASIC
CONCEPTS
stratedinsomecases).
1.2.4
Scalingingroupmotion:Flocks
Groupmotion(ocking)isabeautifulphenomenonmanytimescapturingoureyes.
Here
ockingisunderstoodinageneralsenseoftheword,includingherdingof
quadrupeds,schoolingof�sh,etc).Inthelastchapterofthisbookweaddress
thequestionwhethertherearesomeglobal,perhapsuniversaltransitionsin
ocking
whenmanyorganismsareinvolvedandsuchparametersasthelevelofperturbations
orthemeandistanceoftheorganismsischanged.
Everyonehasexperiencedhowaninitiallyrandomlydirectedgroupofbirdsfeed-
ingonthegroundisspontaneouslyorderedintoawellorganised
ockwhentheyleave
becauseofsomeexternalperturbation.Thisorderingisahighlynon-trivialquestion
sinceinahuge
ockofseveralhundredorthousandbirdsusuallythereisno"leader"
bird(wedonotconsiderheretheV
shapedorotherstructured
ightofsomelarge
bodiedbirds)andnotevenallbirdscanvisuallyinteract.Still,thewhole
ockselects
awellde�neddirection.Suchorderingisfamiliarfromequilibriumphasetransition
ofmagneticsystemsandthecorresponding�ndingsmayprovidecluestotheunder-
standingofthemorecomplexfar-fromequilibriumorderingofmovingorganisms.
Self-propulsionisanessentialfeatureofmostlivingsystems.Inaddition,the
motionoftheorganismsisusuallycontrolledbyinteractionswithotherorganismsin
theirneighbourhoodandrandomnessplaysanimportantroleaswell.Itispossible
todesignsimplecomputermodelswhichsimulatethecollectivemotionandtakeinto
accountthemostrelevantingredientsofthephenomenon.
Asimplemodelofcollectivemotionconsistsofparticlesmovinginone,twoor
threedimensions.Theparticlesarecharacterisedbytheir(o�-lattice)locationx
iand
velocityv
i
pointinginthedirection#i.Toaccountfortheself-propellednatureof
theparticlesthemagnitudeofthevelocityis�xedtov0.Asimplelocalinteraction
isde�nedinthemodel:ateachtimestepagivenparticleassumestheaverage
directionofmotionoftheparticlesinitslocalneighbourhoodwithsomeuncertainty.
Suchamodelisatransportrelated,non-equilibriumanalogueoftheferromagnetic
models.Theanalogyisasfollows:thefunctiontendingtoalignthespinsinthesame
directioninthecaseofequilibriumferromagnetsisreplacedbytheruleofaligning
thedirectionofmotionofparticles,andtheamplitudeoftherandomperturbations
canbeconsideredproportionaltothetemperature.
Inaddition,collectivemotioncanbedescribedbycontinuumequationsaswell.
Thecollectionof"birds"isthenlookedatasparticlesina
uidsubjectto
uctuations
andsatisfyingtheconditionoftryingtomovewithagivenvelocity.
Boththeoreticalapproachesledtotheconclusionthatthereareinteresting,in
1.2.
SCALING
15
casesunexpected(comparedtoequilibriumsystems)transitionsincollectivemotion.
Forexample,ifthenoise(levelofperturbations,correspondingtotemperatureinthe
caseofferromagnets)isdecreased,theoriginallydisordered
ockbecomesorderedin
analogywithsecondorderphasetransitions.Thelevelofglobalorder,the
uctuations
aroundthisorderandseveralrelatedquantitiesallscale,i.e.,behaveaccordingto
powerlawasafunctionofthedistancefromacriticallevelofperturbations.
Pedestriansimulations
Aspecialkindof
ocksisagroupofpeople.Inthelastchapterinterestingappli-
cationsofpedestriansimulationsarediscussed.Justasinthecaseofotherorganisms,
peoplecanberepresentedbyparticles"dressed"bytheappropriateinteractions.Sim-
ulationsofhumansmovingincon�nedplacesleadstoanumberofinterestinge�ects
reproducingrelatedobservations.
Freezingbyheatingisane�ectobservedwhenaparticlesaredriveninopposite
directions.Therelatedsimulationsdemonstratedthatmorenervousorhecticchanges
(heating)ofthedirectionofmotioncancauseabreakdownofaneÆcientpatternof
cooperativeinteractionsand�nallyproduceadeadlock(freezing).Inparticular,this
mayberelevantforpanickingpedestriansinasmokyenvironment,whotendtobuild
upfatalblockings.Thesystemdescribedinthelastchapterconsistsofamesoscopic
numberofdrivenparticleswithrepulsivehard-coreinteractionsmovingintoopposite
directionsunderthein
uenceof
uctuations.Exampleforsuchsystemsispedestrians
walkinginapassage.
Inshort,\freezingbyheating"meansatransitionfroma
uidstate(withself-
organisedlanesofuniformdirectionofmotiontoasolid,crystallised(\frozen")state
justbyincreasingthenoiseamplitude(\temperature").Thisisincontrastto,for
example,melting(whereincreasingthetemperatureincreasestheenergyandorder
isdestroyed)andtonoise-inducedorderinginglassesorgranularmedia,wherein-
creasingthetemperaturedrivesthesystemfromadisorderedmetastablestate(cor-
respondingtoalocalenergyminimum)toanorderedstablestate(corresponding
totheglobalenergyminimum).Instead,\freezingbyheating"showsanincrease
intheorderatincreasingtemperature,althoughthetotalenergyincreasesatthe
sametime.Thecrystallisedstatecanalsobedestroyedbyongoing
uctuationswith
extremenoiseamplitudesgivingrisetoathird,disordered(\gaseous")statewith
randomlydistributedparticles.Thus,withincreasing\temperature"�,wehavethe
atypicalsequenceoftransitions
uid�!
solid�!
gaseous.
Furthervariantsofpedestriansimulationsallowthequantitativeinvestigationof
trailformation,optimisationofpassagegeometries,etc.
InthischapterIhaveattemptedtopresentinasimpli�edmanneraselectionof
concepts,topicsandresultsdiscussedinmuchmoredetailinthemainbodyofthe
book.Fordetailsnecessaryforadeeperunderstandingoftheconceptsand�ndings
16
CHAPTER
1.
BASIC
CONCEPTS
relatedto
uctuationsandscalinginbiologyIadvisethereadertoreadotherparts
ofourbookaswell(wheretherelatedreferencesarealsogiven).
Chapter2
Introductiontocomplexpatterns,
uctuationsandscaling
Inthisbookwemostlyconsidermodelsofreality,since"reality"inthecaseofbiology,
isfartoocomplextoallow
completetheoreticaltreatment.
Ontheotherhand,
wheneveritispossible,wearetryingtopresentmodelsasrealisticaspossiblein
ordertore
ecttheessentialfeaturesofthespeci�cphenomenonoccurringinnature.
Inmanycasesweareconcernedwithsystemsconsistingofmanysimilarobjectsand
thisfeaturehasparticularimplicationsonthekindsofmodelsweconsider.
Variousmodelsallowingexactornumericaltreatmenthavebeenplayinganim-
portantroleinthestudiesofbiologicalprocesses.Becauseofthecomplexityofthe
phenomenaitisusuallyadiÆculttasktodecidewhichofthemanyfactorsin
uenc-
ingtheprocessesarethemostsigni�cant.Inarealsystemthenumberofallpossible
factorscanbetoolarge;thisnumberisdecreasedtoafew
byappropriatemodel
systems.Thus,theinvestigationofthesemodelsprovidesapossibilitytodetectthe
mostrelevantfactors,anddemonstratetheire�ectsintheabsenceofanydisturbance.
Systemsconsistingofmanysimilarunitscanbesuccessfullydescribedintermsof
particles,wherethewordparticlecanstandforamoleculeaswellasformorecomplex
objects,includingorganisms.Then,theparticularnatureofthemodelisgivenby
thefeaturesoftheindividualparticlesandbythewaystheseparticlesinteractwith
eachother.
Thespatialarrangementoftheseparticlesisfrequentlyofmajorinterest.Struc-
turesconsistingofconnectedparticlesareusuallycalledclustersoraggregates.In
mostofthecasestheparticlesareassumedto"exist"onalatticeforcomputational
convenience,andtwoparticlesareregardedasconnectediftheyoccupynearestneigh-
boursitesofthelattice.However,forstudyinguniversalityandrelatedquestions,
o�-latticeorfurtherneighbourversionsofclusteringprocessescanalsobeinvesti-
17
18
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
gated.Alatticesitewithaparticleassignedtoitiscalledoccupiedor�lled.An
importantadditionalfeatureincludedintothemajorityofmodelstobedescribedis
stochasticitywhichistypicalforbiologicalphenomena.
Inthischapterwediscussthebasicfeaturesofthecomplexpattersproduced
byawidevarietyofbiologicalgrowthprocesses.Inmanycasesbiologicalgrowth
leadstorandomfractalstructurescharacterisedbyanon-integerdimensionde�ned
below.Inthesecondsectionofthischaptertheprinciplesbehindthemotioninthe
presenceof
uctuationswillbepresented.Sincebiologicalmotionisproducedby
motormoleculesactingonamicroscopicscale,thermalnoiseandotherstochastic
perturbationsareessential.Finally,wediscusscontinuousphasetransitionswhere
theconceptofscalingplaysacentralrole.Scalingmeansapowerlawdependenceofa
quantityonitsargumentand,aswillbedemonstrated,isafeatureshowingupinan
unexpectedlylargeselectionofbiologicalsystems.Weknowfromtheearlystudiesof
scalinginphysicsthatitisafundamentalcharacteristicsofasystem.Thepowerlaw
dependenceofaquantityusuallyinvolvesasimilarbehaviouroftheotherquantities
inasystem;inaddition,theexponent(correspondingtothepowerlaw)istypically
notsensitivetothedetailsuniversal)oftheprocessesconsidered.
2.1
Fractalgeometry
Duringthelastdecadeithaswidelybeenrecognisedbyresearchersworkingindiverse
areasofsciencethatmanyofthestructurescommonlyobservedpossessarather
specialkindofgeometricalcomplexity.Thisawarenessislargelyduetotheactivity
ofBenoitMandelbrot[1]whocalledattentiontotheparticulargeometricalproperties
ofsuchobjectsastheshoreofcontinents,thebranchesoftrees,orthesurfaceof
clouds.Hecoinedthenamefractalforthesecomplexshapestoexpressthattheycan
becharacterisedbyanon-integer(fractal)dimensionality.Withthedevelopmentof
researchinthisdirectionthelistofexamplesoffractalshasbecomeverylong,and
includesstructuresfrommicroscopicaggregatestotheclustersofgalaxies.Objects
ofbiologicaloriginaremanytimesfractal-like.
Beforestartingamoredetaileddescriptionoffractalgeometryletus�rstcon-
siderasimpleexample.Fig.2.1showsaclusterofparticleswhichcanbeusedfor
demonstratingthemainfeaturesoffractals.Thisobjectwasproposedtodescribe
di�usion-limitedgrowth[2]andhasalooplessbranchingstructurereminiscentof
manyshapesofbiologicalorigin.ImagineconcentriccirclesofradiiRcenteredatthe
middleofthecluster.Forsuchanobjectitcanbeshownthatthenumberofparticles
inacircleofradiusRscalesas
N(R)�
RD
;
(2.1)
2.1.
FRACTAL
GEOMETRY
19
Figure2.1:Atypicalstochasticfractalgeneratedinacomputerusingthedi�usion-
limitedaggregationmodel.
whereD
<
disanon-integernumbercalledthefractaldimension.Naturally,fora
realobjecttheabovescalingholdsonlyforlengthscalesbetweenalowerandanupper
cuto�.Obviously,foraregularobjectembeddedintoaddimensionalEuclideanspace
Eq.2.1wouldhavetheformN(R)�
Rd
expressingthefactthatthevolumeofad
dimensionalobjectgrowswithitslinearsizeR
asR
d.Clustershavinganon-trivialD
aretypicallyself-similar.Thispropertymeansthatalargerpartoftheclusterafter
beingreduced\looksthesame"asasmallerpartoftheclusterbeforereduction.This
remarkablefeatureoffractalscanbevisuallyexaminedonFig.2.1,wherepartsof
di�erentsizes(includedintorectangularboxes)canbecomparedfromthispointof
view.
2.1.1
Fractalsasmathematicalandbiologicalobjects
Inadditiontoself-similaritymentionedabove,acharacteristicpropertyoffractals
isrelatedtotheirvolumewithrespecttotheirlinearsize.Todemonstratethiswe
�rstneedtointroduceafewnotions.WecallembeddingdimensiontheEuclidean
dimensiond
ofthespacethefractalcanbeembeddedin.Furthermore,d
hasto
bethesmallestsuchdimension.Obviously,thevolumeofafractal(oranyobject),
20
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
V(l),canbemeasuredbycoveringitwithddimensionalballsofradiusl.Thenthe
expression
V(l)=N(l)l
d
(2.2)
givesanestimateofthevolume,whereN(l)isthenumberofballsneededtocoverthe
objectcompletelyandlismuchsmallerthanthelinearsizeLofthewholestructure.
Thestructureisregardedtobecoverediftheregionoccupiedbytheballsincludesit
entirely.Thephrase\numberofballsneededtocover"correspondstotherequirement
thatN(l)shouldbethesmallestnumberofballswithwhichthecoveringcanbe
achieved.ForordinaryobjectsV(l)quicklyattainsaconstantvalue,whileforfractals
typicallyV(l)!
0asl!
0.Ontheotherhand,thesurfaceoffractalsmaybe
anomalouslylargewithrespecttoL.
ThereisanalternativewaytodetermineN(l)whichisequivalenttothede�nition
givenabove.Considerad-dimensionalhypercubiclatticeoflatticespacinglwhich
occupiesthesameregionofspacewheretheobjectislocated.Thenthenumberof
boxes(meshunits)ofvolumeldwhichoverlapwiththestructurecanbeusedasa
de�nitionforN(l)aswell.Thisapproachiscalledboxcounting.
ReturningtotheclustershowninFig.2.1wecansaythatitcanbeembedded
intoaplane(d=2).Measuringthetotallengthofitsbranches(correspondingto
thesurfaceinatwo-dimensionalspace)wewould�ndthatittendstogrowalmost
inde�nitelywiththedecreasinglengthlofthemeasuringsticks.Atthesametime,
themeasured\area"ofthecluster(volumeind=2)goestozeroifwedetermine
itbyusingdiscsofdecreasingradius.Thereasonforthisisrootedintheextremely
complicated,self-similarcharacterofthecluster.Therefore,suchacollectionof
branchestobede�nitelymuch"longer"thanalinebuthavingin�nitelysmallarea:
itisneitheraone-noratwo-dimensionalobject.
Thus,thevolumeofa�nitegeometricalstructuremeasuredaccordingtoEq.2.2
maygotozerowiththedecreasingsizeofthecoveringballswhile,simultaneously,its
measuredsurfacemaydivergefollowingapowerlawinsteadofthebetterbehaving
exponentialconvergence.Ingeneral,wecallaphysicalobjectfractal,ifmeasuringits
volume,surfaceorlengthwithd,d�
1etc.dimensionalhyperballsitisnotpossible
toobtainawellconverging�nitemeasureforthesequantitieswhenchanginglover
severalordersofmagnitude.
Itispossibletoconstructmathematicalobjectswhichsatisfythecriterionof
self-similarityexactly,andtheirmeasuredvolumedependsonleveniflor(l=L)
becomessmallerthanany�nitevalue.Fig.2.2givesexampleshowonecanconstruct
suchfractalsusinganiterationprocedure.Usuallyonestartswithasimpleinitial
con�gurationofunits(Fig.2.2a)orwithageometricalobject(Fig.2.2b).Then,in
thegrowingcasethissimpleseedcon�guration(Fig.2.2a,k=2)isrepeatedlyadded
2.1.
FRACTAL
GEOMETRY
21
k=3
k=0
k=1
k=2
Figure2.2:Fig.2.2ademonstrateshowonecangenerateagrowingfractalusingan
iterationprocedure.InFig.2.2bananalogousstructureisconstructedbysubsequent
divisionsoftheoriginalsquare.Bothproceduresleadtofractalsfork!
1
withthe
samedimensionD
'
1:465[3].
toitselfinsuchawaythattheseedcon�gurationisregardedasaunitandinthe
newstructuretheseunitsarearrangedwithrespecttoeachotheraccordingtothe
samesymmetryastheoriginalunitsintheseedcon�guration.Inthenextstage
thepreviouscon�gurationisalwayslookedatastheseed.TheconstructionofFig.
2.2bisbasedondivisionoftheoriginalobjectanditcanbewellfollowedhowthe
subsequentreplacementofthesquareswith�vesmallersquaresleadstoaself-similar,
scaleinvariantstructure.
Onecangeneratemanypossiblepatternsbythistechnique;thefractalshown
inFig.2.2waschosenjustbecauseithasanopenbranchingstructureanalogousto
manyobservedbiologicalfractals[3].Onlythe�rstcoupleofsteps(uptok=3)of
theconstructionareshown.Mathematicalfractalsareproducedafterin�nitenumber
ofsuchiterations.Inthisk
!
1
limitthefractaldisplayedinFig.2.2abecomes
in�nitelylarge,whilethedetailsofFig.2.2bbecomeso�nethatthepictureseems
to\evaporate"andcannotbeseenanymore.Ourexampleshowsaconnected
construction,butdisconnectedobjectsdistributedinanontrivialwayinspacecan
alsoformafractal.
Inanyrealsystemthereisalwaysalowercuto�ofthelengthscale;inourcase
thisisrepresentedbythesizeoftheparticles.Inaddition,arealobjecthasa�nite
22
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
linearsizewhichinevitablyintroducesanuppercuto�ofthescaleonwhichfractal
scalingcanbeobserved.Thisleadsustotheconclusionthat,incontrasttothe
mathematicalfractals,forfractalsobservedinnaturalphenomena(includingbiology)
theanomalousscalingofthevolumecanbeobservedonlybetweentwowellde�ned
lengthscales.
Then,apossiblede�nitionforabiologicalfractalcanbebasedontherequirement
thatapowerlawscalingofN(l)hastoholdoveratleasttwoordersofmagnitude.
2.1.2
De�nitions
BecauseofthetwomaintypesoffractalsdemonstratedinFig.2.2,tode�neand
determinethefractaldimensionDonetypicallyusestworelatedapproaches.
Forfractalshaving�xedLanddetailsonverysmalllengthscaleDisde�ned
throughthescalingofN(l)asafunctionofdecreasingl,whereN(l)isthenumber
ofddimensionalballsofdiameterlneededtocoverthestructure.
Inthecaseofgrowingfractals,wherethereexistsasmallesttypicalsizea,one
cutsoutd-dimensionalregionsoflinearsizeLfromtheobjectandthevolume,V(L),
ofthefractalwithintheseregionsisconsideredasafunctionofthelinearsizeLof
theobject.WhendeterminingV(L),thestructureiscoveredbyballsorboxesof
unitvolume(l=a=1isusuallyassumed),thereforeV(L)=N(L),whereN(L)is
thenumberofsuchballs.
ThefactthatanobjectisamathematicalfractalthenmeansthatN(l)diverges
asl!
0orL!
1
,respectively,accordingtoanon-integerexponent.
Correspondingly,forfractalshavinga�nitesizeandin�nitelysmallrami�cations
wehave
N(l)�
l�D
(2.3)
with
D=liml!
0
lnN(l)
ln(1=l)
;
(2.4)
while
N(L)�
LD
(2.5)
and
D=
limL!
1
lnN(L)
ln(L)
:
(2.6)
forthegrowingcase,wherel=1.Here,aswellasinthefollowingexpressionsthe
symbol�
meansthattheproportionalityfactor,notwrittenoutin2.3,isindependent
ofl.
2.1.
FRACTAL
GEOMETRY
23
Obviously,theabovede�nitionsfornon-fractalobjectsgiveatrivialvaluefor
D
coincidingwiththeembeddingEuclideandimensiond.Forexample,thearea
(correspondingtothevolumeV(L)ind=2)ofacirclegrowsasitssquaredradius
whichaccordingto2.6resultsinD=2.
Nowweareinthepositiontocalculatethedimensionoftheobjectsshownin
Fig.2.2.Itisevidentfromthe�gurethatforthegrowingcase
N(L)=5k
with
L=3k;
(2.7)
wherekisthenumberofiterationscompleted.Fromhereusing2.6wegetthevalue
D
=ln5=ln3=1:465:::whichisanumberbetweend=1andd=2justaswe
expected.
2.1.3
Usefulrules
Inthissectionwementionafewruleswhichcanbeusefulinpredictingvarious
propertiesrelatedtothefractalstructureofanobject.Ofcourse,becauseofthe
greatvarietyofself-similargeometriesthenumberofpossibleexceptionsisnotsmall
andtheruleslistedbelowshouldberegarded,atleastinpart,asstartingpointsfor
moreaccurateconclusions.
a)Manytimesitistheprojectionofafractalwhichisofinterestorcanbeexper-
imentallystudied(e.g.apictureofafractalembeddedintod=3).Ingeneral,
projectingaD<dsdimensionalfractalontoadsdimensionalsurfaceresultsin
astructurewiththesamefractaldimensionDp=D.ForD�
dstheprojection
�llsthesurface,Dp=ds.
b)Itfollowsfroma)thatforD<dsthedensitycorrelationsc(r)(seenextsection)
withintheprojectedimagedecayasapowerlawwithanexponentds�Dinstead
ofd�
Dwhichistheexponentcharacterisingthealgebraicdecayofc(r)ind.
c)Cuttingoutads
dimensionalslice(cross-section)ofaD
dimensionalfractal
embeddedintoaddimensionalspaceusuallyleadstoaD+ds�ddimensional
object.Thisseemstobetrueforself-aÆnefractals(nextsection)aswell,with
Dbeingtheirlocaldimension
d)ConsidertwosetsAandBhavingfractaldimensionsDA
andDB,respectively.
MultiplyingthemtogetherresultsinafractalwithD=DA
+DB.Asasimple
example,imagineafractalwhichismadeofparallelsticksarrangedinsucha
waythatitscross-sectionisthefractalshowninFig.2.2b.Thedimensionof
thisobjectisD=1+ln5=ln3.
24
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
e)TheunionoftwofractalsetsAandB
withDA
>
DB
hasthedimension
D=DA.
f)ThefractaldimensionoftheintersectionoftwofractalswithDA
andDB
isgiven
byDA\B
=DA
+DB
�
d.Toseethis,consideraboxoflinearsizeLwithin
theoverlappingregionoftwogrowingstochasticfractals.ThedensityofAand
BparticlesisrespectivelyproportionaltoLDA
=Ld
andLDB
=Ld.Thenumber
ofoverlappingsitesN
�
LDA
\
B
isproportionaltothesedensitiesandtothe
volumeoftheboxwhichleadstotheabovegivenrelation.Theruleconcerning
intersectionsoffractalswithsmoothhypersurfaces(rulec)isaspecialcaseof
thepresentone.
g)Thedistributionofemptyregions(holes)inafractalofdimensionDscalesas
afunctionoftheirlinearsizewithanexponent�
D�
1.
Self-similaritycanbedirectlycheckedforadeterministicfractalconstructedby
iteration,butinthecaseofrandomstructuresoneneedsothermethodstodetectthe
fractalcharacterofagivenobject.Infact,random
fractalsareself-similaronlyina
statisticalsense(notexactly)andtodescribethemitismoreappropriatetousethe
termscaleinvariancethanself-similarity.Naturally,fordemonstratingthepresence
offractalscalingonecanusethede�nitionbasedoncoveringthegivenstructurewith
ballsofvaryingradii,however,thiswouldbearathertroublesomeprocedure.Itis
moree�ectivetocalculatethesocalleddensity-densityorpaircorrelationfunction
c(~ r)=
1 VX ~ r
0
�(~ r+~ r0)�(~ r
0)
(2.8)
whichistheexpectationvalueoftheeventthattwopointsseparatedby~ rbelongto
thestructure.ForgrowingfractalsthevolumeoftheobjectisV=N,whereNisthe
numberofparticlesinthecluster,and2.9givestheprobabilityof�ndingaparticle
attheposition~ r+~ r0,ifthereisoneat~ r0.In2.9�isthelocaldensity,i.e.,�(~ r)=1if
thepoint~ rbelongstotheobject,otherwiseitisequaltozero.Ordinaryfractalsare
typicallyisotropic(thecorrelationsarenotdependentonthedirection)whichmeans
thatthedensitycorrelationsdependonlyonthedistancersothatc(~ r)=c(r).
Nowwecanusethepaircorrelationfunctionintroducedaboveasacriterionfor
fractalgeometry.Anobjectisnon-triviallyscaleinvariantifitscorrelationfunction
determinedaccordingto2.9isunchangeduptoaconstantunderrescalingoflengths
byanarbitraryfactorb:
c(br)�
b��c(r)
(2.9)
2.1.
FRACTAL
GEOMETRY
25
with�anon-integernumberlargerthanzeroandlessthand.Itcanbeshownthat
theonlyfunctionwhichsatis�es2.9isthepowerlawdependenceofc(r)onr
c(r)�
r�
�
(2.10)
correspondingtoanalgebraicdecayofthelocaldensitywithinarandom
fractal,
sincethepaircorrelationfunctionisproportionaltothedensitydistributionaround
agivenpoint.LetuscalculatethenumberofparticlesN(L)withinasphereofradius
Lfromtheirdensitydistribution
N(L)�
Z L 0c(r)ddr�
Ld�
�;
(2.11)
wherethesummationin2.8hasbeenreplacedbyintegration.Comparing2.11with
2.5wearriveattherelation
D
=d��
(2.12)
whichisaresultwidelyusedforthedeterminationofD
fromthedensitycorrelations
withinarandomfractal.
2.1.4
Self-sim
ilarand
self-aÆ
ne
fractals
Therearethreemajortypesoffractalsasconcerningtheirscalingbehaviour.Self-
similarfractalsareinvariantunderisotropicrescalingofthecoordinates,whilefor
self-aÆnefractalsscaleinvarianceholdsforaÆne(anisotropic)transformation.Until
thispointmainlytheformercasehasbeendiscussed.
Therandom
motionofaparticlerepresentsaparticularlysimpleexampleof
stochasticprocessesleadingtogrowingfractalstructures.Awidelystudiedcaseis
whentheparticleundergoesarandomwalk(Brownianordi�usionalmotion)making
stepsoflengthdistributedaccordingtoaGaussianinrandomlyselecteddirections.
SuchprocessescanbedescribedintermsofthemeansquareddistanceR
2
=hR
2(t)i
madebytheparticlesduringagiventimeintervalt.Forrandom
walksR
2
�
t
independentlyofdwhichmeansthattheBrowniantrajectoryisarandomfractal
inspaceswithd>
2.Indeed,measuringthevolumeofthetrajectorybythetotal
numberofplacesvisitedbytheparticlemakingtsteps,(N(R)�
t),theabove
expressionisequivalentto
N(R)�R
2
(2.13)
andcomparing2.13with2.5weconcludethatforrandomwalksD
=2<d
ifd>2.
Inthiscase,ratherunusually,thefractaldimensionisanintegernumber.However,
thefactthatitisde�nitelysmallerthantheembeddingdimensionindicatesthatthe
objectmustbenon-triviallyscaleinvariant.
26
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
Inmanyphysicallyrelevantcasesthestructureoftheobjectsissuchthatitis
invariantunderdilationtransformationonlyifthelengthsarerescaledbydirection
dependentfactors.Theseanisotropicfractalsarecalledself-aÆne[4,5,6].
Single-valued,nowhere-di�erentiablefunctionsrepresentasimpleandtypical
forminwhichself-aÆnefractalsappear.IfsuchafunctionF(x)hastheproperty
F(x)'b�H
F(bx)
(2.14)
itisself-aÆne,whereH>0issomeexponent.Eq.2.14expressesthefactthatthe
functionisinvariantunderthefollowingrescaling:shrinkingalongthexaxisbya
factor1=b,followedbyrescalingofvaluesofthefunction(measuredinadirection
perpendiculartothedirectioninwhichtheargumentumischanged)byadi�erent
factorequaltob�H
.Inotherwords,byshrinkingthefunctionusingtheappro-
priatedirection-dependentfactors,itisrescaledontoitself.Forsomedeterministic
self-aÆnefunctionsthiscanbedoneexactly,whileforrandomfunctionstheabove
considerationsarevalidinastochasticsense(expressedbyusingthesign').
Ade�nitionofself-aÆnityequivalentto2.14isgivenbytheexpressionforthe
heightcorrelationfunctionc(�x)
c(�x)=h[F(x+�x)�F(x)]
2i��x
2H
(2.15)
whichcanbeeasilyusedforthedeterminationoftheexponentH.Inadditionto
functionssatisfying2.14and2.15,therearealsoself-aÆnefractalsdi�erentfrom
single-valuedfunctions.
Letus�rstconstructadeterministicself-aÆnemodel,inordertohaveanobject
whichcanbetreatedexactly.
Anactualconstructionofsuchaboundedself-aÆnefunctionontheunitinterval
isdemonstratedinFig.2.3.Theobjectisgeneratedbyarecursiveprocedureby
replacingtheintervalsofthepreviouscon�gurationwiththegeneratorhavingthe
formofanasymmetricletterzmadeoffourintervals.However,thereplacement
thistimeshouldbedoneinamannerdi�erentfromtheearlierpractice.Hereevery
intervalisregardedasadiagonalofarectanglebecomingincreasinglyelongated
duringtheiteration.Thebasisoftherectangleisdividedintofourequalpartsand
thez-shapedgeneratorreplacesthediagonalinsuchawaythatitsturnoversare
alwaysatanalogouspositions(atthe�rstquarterandthemiddleofthebasis).The
functionbecomesself-aÆneinthek!
1
limit.
Sucharandomfunctionis,forexample,theplotofthedistancesX(t)measured
fromtheoriginasafunctionoftimet,ofaBrownianparticledi�usinginonedimen-
sion.ItisobviousthatasocalledfractionalBrownianplotforwhichhX
2 H
(t)i�t2H
satis�es2.15.
2.1.
FRACTAL
GEOMETRY
27
k=1
k=2
k=3
Figure2.3:Self-aÆnefunctionscanbegeneratedbyiterationprocedures.ThisThe
single-valuedcharacterofthefunctionispreservedbyanappropriatedistortionof
thez-shapedgenerator(k=1)ofthestructure[3].
NextwegiveafurtherbasicfeatureoffractionalBrownianmotion.
CalculatingtheFourierspectrumofafractionalBrownianfunctionone�ndsthat
thecoeÆcientsoftheseries,A(f),areindependentGaussianrandomvariablesand
theirabsolutevaluescaleswiththefrequencyfaccordingtoapowerlaw
jA(f)j�f�
H
�
1 2
:
(2.16)
2.1.5
M
ultifractals
Intheprevioussectionscomplexgeometricalstructureswerediscussedwhichcould
beinterpretedintermsofasinglefractaldimension.Thepresentsectionismainly
concernedwiththedevelopmentofaformalismforthedescriptionofthesituation
whenasingulardistributionisde�nedonafractal[7,8].
Itistypicalforalargeclassofphenomenainnaturethatthebehaviourofasystem
isdeterminedbythespatialdistributionofascalarquantity,e.g.,concentration,
electricpotential,probability,etc..Forsimplergeometriesthisdistributionfunction
anditsderivativesarerelativelysmooth,andtheyusuallycontainonlyafew(ornone)
singularities,wherethewordsingularcorrespondstoalocalpowerlawbehaviourof
thefunction.(Inotherwords,wecallafunctionsingularintheregionsurrounding
point~ xifitslocalintegraldivergesorvanisheswithanon-integerexponentwhen
28
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
theregionofintegrationgoestozero).Inthecaseoffractalsthesituationisquite
di�erent:aprocessinnatureinvolvingafractalmayleadtoaspatialdistributionof
therelevantquantitieswhichpossessesin�nitelymanysingularities.
Asanexample,consideranisolated,chargedobject.Ifthisobjecthassharptips,
theelectric�eldaroundthesetipsbecomesverylargeinaccordwiththebehaviour
ofthesolutionoftheLaplaceequationforthepotential.Inthecaseofchargingthe
branchingfractalsproducedinthek!
1
limitofconstructionsshowninFig.2.1or
2.2onehasin�nitenumberoftipsandcorrespondingsingularitiesoftheelectric�eld.
Moreover,tipsbeingatdi�erentpositions,ingeneralhavedi�erentlocalenvironments
(con�gurationoftheobjectintheregionsurroundingthegiventip)whicha�ectthe
strengthofsingularityassociatedwiththatposition.
Theabovediscussedtimeindependentdistributionsde�nedonafractalsubstrate
arecalledfractalmeasures.Ingeneral,afractalmeasurepossessesanin�nitenumber
ofsingularitiesofin�nitelymanytypes.Theterm
\multifractality"expressesthe
factthatpointscorrespondingtoagiventypeofsingularitytypicallyformafractal
subsetwhosedimensiondependsonthetypeofsingularity.Thedescriptionofthe
multifractalformalismgoesbeyondthescopeofthepresentsection,butcanbefound,
forexample,inRef.3.
2.1.6
Methodsfordeterminingfractaldimensions
Whenonetriestodeterminethefractaldimensionofbiologicalstructuresinpractice,
itusuallyturnsoutthatthedirectapplicationofde�nitionsforD
givenintheprevious
sectionsisine�ectiveorcannotbeaccomplished.Instead,oneisledtomeasureor
calculatequantitieswhichcanbeshowntoberelatedtothefractaldimensionofthe
objects.Threemainapproachesareusedforthedeterminationofthesequantities:
experimental,computerandtheoretical.
Experimentalmethodsformeasuringfractaldimensions
Anumberofexperimentaltechniqueshavebeenusedtomeasurethefractaldimension
ofscaleinvariantstructuresgrowninvariousexperiments.Themostwidelyapplied
methodscanbedividedintothefollowingcategories:(a)digitalimageprocessingof
two-dimensionalpictures,(b)scatteringexperimentsand(c)directmeasurementof
dimension-dependentphysicalproperties.
(a)Digitisingtheimageofafractalobjectisastandardwayofobtainingquan-
titativedataaboutgeometricalshapes.Theinformationispickedupbyascanneror
anordinaryvideocameraandtransmittedintothememoryofacomputer(typicallya
2.1.
FRACTAL
GEOMETRY
29
PC).Thedataarestoredintheformofatwo-dimensionalarrayofpixelswhosenon-
zero(equaltozero)elementscorrespondtoregionsoccupied(notoccupied)bythe
image.Oncetheyareinthecomputer,thedatacanbeevaluatedusingthemethods
describedinthenextsection,wherecalculationofD
forcomputergeneratedclusters
isdiscussed.
Theonlyprincipalquestionrelatedtoprocessingofpicturesarisesiftwo-dimensional
imagesofobjectsembeddedintothreedimensionsareconsidered.Ithasalreadybeen
mentionedthatthefractaldimensionoftheprojectionofanobjectontoa(d�
m)-
dimensionalplaneisthesameasitsoriginalfractaldimension,ifD
<
d�
m.
(b)Scatteringexperimentsrepresentapowerfulmethodtomeasurethefractal
dimensionofstructures.Dependingonthecharacteristiclengthscalesassociated
withtheobjecttobestudied,light,X-rayorneutronscatteringcanbeusedto
revealfractalproperties.Thereareanumberofpossibilitiestocarryoutascattering
experiment.Onecaninvestigatei)thestructurefactorofasinglefractalobject,ii)
scatteringbymanyclustersgrowingintime,iii)thescatteredbeamfromafractal
surface,etc.
Evaluationofnumericaldata
Throughoutthissectionweassumethattheinformationaboutthestochasticstruc-
turesisstoredintheformofd-dimensionalarrayswhichcorrespondtothevalues
ofafunctiongivenatthenodes(orsites)ofsomeunderlyinglattice.Inthecaseof
studyinggeometricalscalingonly,thevalueofthefunctionattributedtoapointwith
givencoordinates(thepointbeingde�nedthroughtheindexesofthearray)iseither
1(thepointbelongstothefractal)or0(thesiteisempty).Whenmultifractalprop-
ertiesareinvestigatedthesitefunctiontakesonarbitraryvalues.Ingeneral,such
discretesetsofnumbersareobtainedbytwomainmethods:i)bydigitisingpictures
takenfromobjectsproducedinexperiments,ii)bynumericalproceduresusedforthe
simulationofvariousbiologicalstructures.Forconvenience,inthefollowingweshall
frequentlyusetheterminology\particle"foralatticesitewhichbelongstothefractal
(is�lled)andclusterfortheobjectsmadeofconnectedparticles.
BelowwediscusshowtomeasureD
forasingleobject.Tomaketheestimates
moreaccurateoneusuallycalculatesthefractaldimensionformanyclustersand
averagesovertheresults.
PerhapsthemostpracticalmethodistodeterminethenumberofparticlesN(R)=
RD
withinaregionoflinearsizeR
andobtainthefractaldimensionD
fromtheslopes
oftheplotslnN(R)versuslnR.IfthecentersoftheregionsofradiusR
arethepar-
ticlesofthecluster,thanN(R)isequivalenttotheintegralofthedensitycorrelation
function.Inpracticeonechoosesasubsetofrandomlyselectedparticlesofthefractal
30
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
[N(R
)]ln
Rln
R=
1
Rm
ax
Slop
e=D
Figure2.4:Schematiclog-logplotofthenumericallydeterminednumberofparticles
N(R)belongingtoafractalandbeingwithinasphereofradiusR.IfR
issmaller
thantheparticlesizeorlargerthanthelinearsizeofthestructureatrivialbehaviour
isobserved.Thefractaldimensionisobtainedby�ttingastraightlinetothedata
inthescalingregion[3].
(asmanyasneededforareasonablestatistics)anddetermineshN(R)iforasequence
ofgrowingR(orcountsthenumberofparticlesinboxesoflinearsizeL).Inorderto
avoidundesirablee�ectscausedbyanomalouscontributionsappearingattheedgeof
theclusteroneshouldnotchooseparticlesascentresclosetotheboundaryregion.
ThesituationisshowninFig.2.4.Typicallythereisadeviationfromscalingfor
smallandlargescales.
TheroughnessexponentH
correspondingtoself-aÆnefractalsisusuallydeter-
minedfromthede�nition2.15.Analternativemethodistoinvestigatethescalingof
thestandarddeviation�(l)=[hF
2(x)ix�hF(x)i
2 x]1=2
oftheself-aÆnefunctionF
h�(l)i�lH;
(2.17)
wherethelefthandsideistheaverageofthestandarddeviationofthefunctionF
calculatedforregionsoflinearsizel.TheroughnessexponentH
canbecalculated
bydetermining�(l)forpartsoftheinterfacesforvariousl.Anaveragingshould
bemadeoverthesegmentsofthesamelengthandtheresultsplottedonadouble
2.2.
STOCHASTIC
PROCESSES
31
logarithmicplotasafunctionofl.
2.2
Stochasticprocesses
2.2.1
Thephysicsofmicroscopicobjects
Everybiologicalprocesseventuallytakesplaceatthemolecularlevel.Thephysicsof
thismicroscopicrealmisfundamentallydi�erentfromthephysicsofourmacroscopic
world,andrequiresacompletelydi�erentdescription.Firstofall,asthelength-scale
andvelocity-scalegodowntomolecularscales,theReynoldsnumbergoesdowntoo,
andweapproachtheoverdampedregimeinwhichinertiaplaysnoroleanymore[9]
andwherethevelocity(andnottheacceleration)oftheobjectsisproportionalto
theforcesactingonthem.Secondly,thereisBrownianmotion.Microscopicobjects
arebeingrandomlykickedaroundbymoleculesofthesurroundingmedium,andthe
processeshaveaninherentlystochasticnature.
Thetimescaleofmacroscopicprocessesissetbythevelocityandaccelerationof
massiveobjects,thethermal
uctuationsarenegligible,andwhenwedesignamacro-
scopicdevicewetrytosuppressanystochasticelementasmuchaspossible.Onthe
otherhand,formicroscopicobjectseverydegreeoffreedomhasinevitablyasignif-
icant
1 2kBTthermalenergyonaverage(whereTdenotestheabsolutetemperature
andkB
istheBoltzmanncoeÆcient),thetimingoftheprocessesissetbythermally
assistedevents(suchasdi�usionoractivatedtransitionsoverenergybarriers),and
forthedesignofmicroscopicdevicesthermal
uctuationsshouldbeexploitedrather
thansuppressed.
Ingeneral,themotionofanyobjectinathermalenvironmentcanbedescribed
bytheLangevinequation[10]:
mx(t)=�
_x(t)+
p 2D�(t)+F(x;t);
(2.18)
wherex,m,
,andD
denotetheposition,mass,viscousfrictioncoeÆcient,and
di�usioncoeÆcientoftheobject,respectively.Thethreeforcetermsontheright
handsideoftheequationaretheviscousfrictionbythemedium,
_x(t);thethermal
noisecomingfromthemoleculesofthemedium,
p 2D�(t);andalltheotherforces
unrelatedtothemedium,F(x;t).Sincethethermalnoisetermisastochasticfunction
theLangevinequationisreferredtoasastochasticdi�erentialequation.Thenoise
factor,�(t),isusuallymodelledbyaGaussianwhitenoisewithzerotimeaverage,
h�(t)i=0;
(2.19)
32
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
andautocorrelationfunction
h�(t)�(t
0)i=Æ(t�t0):
(2.20)
Boththeviscousfrictionandthethermalnoiseareexertedbythemedium,and
arenotindependent.Theirmagnitudesareconnectedbythe
uctuation-dissipation
theorem(orEinsteinrelation):
D=
kBT
:
(2.21)
ForbiomoleculesinwatersolutiontheLangevinequationcanbesimpli�ed.The
ratio� relax
=
m=isacharacteristictimescaleoftheLangevinequation(2.18),
andtellsushowlongittakesforaparticletolooseitsinitialvelocityviaviscous
frictionifthethermalnoiseandFareturnedo�.Multiplyingthisbytheparticle's
characteristicvelocityv,wegetthecharacteristicdistance�relax
=vm=onwhich
theparticlecomestoahalt.Comparingthisdistancetothecharacteristicsizeof
theparticlea,wegetsomeinformationaboutthestrengthoftheviscousdamping:
if�relax=a�
1thedampingisstrong,becausetheparticlestopsonamuchshorter
distancethanitssize;andif�relax=a�
1thedampingisweak.Supposingthatm
isproportionaltoa
3�and
isproportionaltoa�(cf.Stokeslaw),where�and�
arethedensityanddynamicviscosityofthemediumrespectively,�relax=abecomes
proportionalto
R=
va
�=�
=va �;
(2.22)
whichiscalledtheReynoldsnumber(�=�=�isthekinematicviscosity).Thus,itis
theReynoldsnumberthatcharacterisesthestrengthofthedamping.LowReynolds
numbermeansstrongdamping.
LetusnowestimatetheReynoldsnumberforbiologicalmolecules.Thetypical
sizeofaproteinisintheorderofnanometers(a�1nm),thedensityanddynamic
viscosityofwaterare��10
3
kg/m
3
and��10�
3
kg/s/m.Themaximalforcesacting
onaproteinareintheorderofpiconewtons(afewkBToverafewnanometers),thus
thecharacteristicvelocityofaproteincannotbemuchlargerthanv�1pN=(a�)�
1m/s.ThisshowsthattheReynoldsnumberforbiomoleculesisintheorderof10�
3
orevensmaller,i.e.,weareinthestronglydampedoroverdampedregime.R=10�
3
isasomewhatshockingresult.Itmeansthattheviscousfrictioncanstopaprotein
onadistance(�10
�
3
nm)muchshorterthanthesizeoftheatoms.
Inthisoverdampedregimewhentheforceschange,thevelocityofaparticle
relaxessoquickly(during� relax)andonsuchasmalldistance(�relax
�
a)thatthe
accelerationterm(thederivativeofthevelocitywithrespecttotime)onthelefthand
2.2.
STOCHASTIC
PROCESSES
33
sideoftheLangevinequation(2.18)canbeneglected:
_x(t)=F(x;t)=
+
p2D�(t):
(2.23)
Thiskindofreductioniscalledadiabaticeliminationofthefastvariables[11,10].
SincethemotionofbiomoleculescanbewelldescribedbytheoverdampedLangevin
equation,fromnowonwewilluseonlythisversionoftheequation,andalsotheterm
Langevinequationwillalwaysrefertoitsoverdampedversion(2.23).
Fromthisstochasticordinarydi�erentialequationonecanderiveadeterministic
partialdi�erentialequation,theFokker-Planckequation(orSmoluchowskiequation)
[10],whichdescribesthetimeevolutionoftheprobabilitydensityP(x;t)ofthepo-
sitionoftheparticle:
@tP(x;t)=�@xJ(x;t);
(2.24)
where
J(x;t)=
F(x;t)
P(x;t)�
kBT
@xP(x;t)
(2.25)
istheprobabilitycurrentoftheparticle.Iftheforce�eldF(x;t)isthenegative
gradientofapotential:F(x;t)=�@xV(x;t),theprobabilitycurrentcanbewritten
intheform
J(x;t)=�
kBT e�
V(x;t)=kB
T@x
� eV(x;t)=kB
TP(x;t)�:
(2.26)
2.2.2
Kram
ersform
ula
and
Arrheniuslaw
InmanysystemsBrownianparticlesarewigglingindeeppotentialwells(compared
tokBT)forlongperiodsoftime,rarelyinterruptedbyquickjumpsintooneofthe
neighbouringwells.Ifthepotentialisstaticorchangesinamuchlongertime-scale
thanthedurationofthesejumps(whichisusuallythecase),akineticapproachcan
beusedtodescribethemotionoftheparticles,withtransitionrateconstantsbetween
discretestates.
Thedisciplineofratetheory(forreviewseeRef.[12])wascreatedwhenArrhenius
[13]extensivelydiscussedvariousreaction-ratedataandshowedthattheyvaryona
logarithmicscalelinearlytotheinversetemperatureT�
1.Inotherwords,theescape
(orjumping)rateconstants,k,followtheArrheniuslaw
k=�e�
�E=kB
T;
(2.27)
where�Edenotesthethresholdenergyforactivationand�isafrequencyprefactor.
UsingKramers'method[14,10],theArrheniuslawcanbeeasilyderivedfrom
theFokker-PlanckequationforanoverdampedBrownianparticlemovinginaone-
dimensionalpotentialV(x)(depictedinFig.2.5),ifthepotentialwellfromwhich
34
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
theparticleistryingtoescapeismuchdeeperthankBT.Inthiscasetheprobability
densitynearthebottomcanbewellapproximatedbyitsequilibriumvalue
Peq(x)=P0e�
V(x)=kB
T;
(2.28)
whichcanbederivedfromEq.(2.26)bysettingitsright-handsidetozero.The
normalisationfactorisapproximately
P0=
1
Z c be�
V(x)=kB
Tdx
;
(2.29)
becausethevastmajorityoftheprobabilityfallsintotheinterval[b,c]wherethe
potentialdi�erencefromthebottomofthepotentialisnotlargerthanafew(�5)
kBT.Kramers'approachisbasedontheassumptionthattheprobabilitycurrentover
thepotentialbarrierbetweenBandCisaconstantJ.Indeed,thisconditionholds,
becausetheinterval[B,C]containsonlyaverysmallfractionofthetotalprobability.
Anotherassumptionisthattheprobabilitydistributionforx�
C
iszero,because
theparticlehasbasicallynochancetogetbacktothewellandtheescapecanbe
consideredtobecompleted.Thus,afterrearrangingandintegratingEq.(2.26)from
BtoCweget
J
kBT
Z C BeV
(x)=kB
Tdx=�
� eV(x)=kB
TPeq(x)� C x=B
;
(2.30)
kT
~5BkT
~5B
B
V
ba
cA
Cx
JFigure2.5:PotentialV(x)withadeepwellandabarrieroverwhichanoverdamped
Brownianparticletriestoescape.
2.3.
CONTINUOUSPHASE
TRANSITIONS
35
wheretheexpressionbetweenthesquarebracketsisP0
forx=Bandzeroforx=C.
From
thisthecurrentJoverthebarrier(whichisequivalenttotheescaperate
constantk)canbeexpressedas
J�
k=
D
Z c be�
V(x)=kB
Tdx
Z C BeV
(x)=kB
Tdx
=
De�
[V(A)�V(a)]=kB
T
Z c be�
[V(x)�V(a)]=kB
Tdx
Z C Be�
[V(A)�V(x)]=kB
Tdx
:
ThisexpressionhasindeedthesameformasthatoftheArrheniuslaw(2.27).Here
theactivationenergy�EistheheightofthebarrierV(A)�
V(a),andthefrequency
prefactor�dependsonlyontheshapeofthepotentialnearthebottomofthewell
andthetopofthebarrier.
ThisderivationholdsevenifthepotentialV(x)changesintimebutmuchslower
thentheintrawellrelaxationtimeoftheparticle[15].Inthiscasetheescaperate
constantbecomesalsotimedependent.
Mostchemicalreactionscanalsobedescribedintermsofkineticrateconstants,
soiftheyarepresent,theyrepresentanothersourceofstochasticityinmolecular
processesinadditiontothethermalnoise.Thewaitingtimeforanyescapeprocess
orchemicalreactioncharacterisedbyarateconstantkhasanexponentialdistribution
withmeanvalue1=k.
2.3
Continuousphasetransitions
Inthefollowingchapterwediscusstherelevanceofthesocalledself-organised
critically(SOC)forbiology.However,beforedescribingthismoreadvanced,non-
equilibriumconceptwegiveashortintroductiontothecloselyrelatedprecursor,the
secondorder(orcontinuous)phasetransitionoccurringinequilibrium.
Phasetransitionscanbeeasilyunderstoodonasimplethermodynamiclevel.Let
usconsiderasubstancewhichcanexistintwodi�erentphaseslikewater(liquidand
ice)oriron(paramagneticandferromagnetic).Usuallyoneofthephasesisdisordered
whiletheotherisordered.Thedistinctionisbasedonthesymmetryofthestate:
thesymmetric(orisotropic)stateisthedisorderedone.Tocharacterisethestrength
oforderingatgivenvaluesofthermodynamicparameters(likethetemperature)we
introduceanorderparameter.Itmeasureshowwellorderedthesubstanceis.By
conventioniftheorderparameteriszerowespeakofacompletelydisorderedstate
36
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
0.0
0.2
0.4
0.6
0.8
1.0
m
-10123
F(T,m)
T>
Tc
T=
Tc
T<
Tc
Tc
T
0.0
0.5
1.0
m
(a)
0.0
0.2
0.4
0.6
0.8
1.0
m
-10123
F(T,m)
T>
Tc
T=
Tc
T<
Tc
Tc
T
0.0
0.5
1.0
m
(b)
Figure2.6:Tworoutestoaphasetransition:(a)�rstorderand(b)secondorder.
Theinsetsshowtheorderparameterasthefunctionoftemperature.
(e.g.,liquidwater)andifitisnon-zerothenthesubstanceisinanorderedstate
(e.g.,icecrystal).Asitisknownfromthermodynamics,atconstanttemperaturethe
phasewiththelowestfreeenergyisstable.Sincethefreeenergygenerallycanalso
beafunctionoftheorderparameter,thisenergyminimumrequirementwillselect
whichphasecanbeobservedatagiventemperatureT.Inthiscasethetemperature
playstheroleofcontrolparameterwhichallowsfortuningthesystemtothephase
transition.
InFig.2.6wesketchtwopossiblewaysleadingtoaphasetransition.Whenthe
temperatureisabovethecriticaltemperatureTc
(wherethetransitionoccurs)theonly
globallystablesolutiontotheenergyminimumcriterionisthephasewithzeroorder
parameter(m=0).ThismeansthatforT
>
Tc
thesystemisinitsdisorderedstate.
Fig.2.6(a)showsacasewhenloweringthetemperature,atT
=Tc
�rstanon-trivial
(m>
0)minimumappearsandthenitshiftstolargervaluesasthetemperatureis
lowered(seeinsetinFig.2.6(a)).Thisisthescenarioofa�rstorderphasetransition.
Thecharacteristicfeatureofthistypeoftransitionisajumpdiscontinuityinthe
orderparameter(andcertainotherquantities).Incontrary,Fig.2.6(b)demonstrates
anothertypeoftransitionwheretheorderparameterchangescontinuouslyasthe
temperatureisloweredbelowTc.Thistypeoftransitionisreferredasasecondorder
phasetransitionandithasreceivedmuchlargerattentioninthepastdecades.A
motivationforthisinterestliesinthespecialpropertiesofsuchtransitions.
Withoutlossofgeneralityletusconsideramorespeci�cexampleforasecond
ordertransition:amagneticmaterialattemperatureT
andmagnetic�eldH
.If
thissystemshowsaparamagnetic{ferromagneticsecondordertransitionatTc
(for
2.3.
CONTINUOUSPHASE
TRANSITIONS
37
H
=0)thenitisconvenienttousethereducedtemperature
t=
T�
Tc
Tc
(2.31)
ascontrolparameterinsteadofT.
Measuringthephysicalpropertiesofthesamplerevealsdivergencesofvarious
physicalobservablesasthecriticaltemperatureisapproached.Theorderparameter
inthiscaseisthezero-�eld(H=0)magnetisationM0
sinceitiszeroifT>Tc,and
non-zerobelowTc.InthevicinityofTc
itbehavesas
M0(t)�jtj�;
(2.32)
where�isthecriticalexponentofthemagnetisation.Similarly,forthespeci�cheat
(whichgivesthechangeofenergyforasmallchangeoftemperature)
CH=0(t)�jtj��;
(2.33)
andthesusceptibility(whichisthesensitivityofthemagnetisationwithrespectto
theexternal�eldH)scalesas
�(T)�jtj�
:
(2.34)
Bothofthesequantitiesdescribearesponseofthesystemtosomeexternalperturba-
tion.ClosetoTc
theydivergeshowingthattherethesystemisextremelysensitive:
itisinacriticalstate.
Theabovede�nedthreecriticalexponents(�;�;)arenotindependentofeach
other.Amoredetailedanalysisshowsthattheexponentrelation
�=
2���
2
(2.35)
holds. N
earthetransitionpointthespontaneous
uctuationsinthesystem
become
largeduethehighsusceptibilities.Forthecaseofa
uidthesestrong
uctuations
areobservableasthedecreaseoflighttransmittance(criticalopalescence).Sincethe
lengthscale�associatedwiththese
uctuations,i.e.,thetypicalsizeof
uiddroplets,
alsohaspowerlawdivergence
��t��;
(2.36)
atthecriticalpointtherewillbenotypicallengthscaleexceptthetriviallower
(atomicsize)andupper(systemsize)scales.Thisfactismanifestedviathefractal[3]
structureofthe
uctuationsinthesystemanditiscloselyconnectedtootherpower
38
CHAPTER2.PATTERNS,FLUCTUATIONS,ANDSCALING
lawdivergencespresentatTc.Thefractalityfromtheexperimenter'spointofview
meansthatthe
uctuationsarestatisticallyinvariantunderthetransformation
(x;y;:::)7�!
(�x;�y;:::);
(2.37)
i.e.,notypicallengthscalecanbeidenti�ed.
2.3.1
ThePottsmodel
Usingthethermodynamicapproachitispossibletoderivethecriticalexponents
onlyheuristically,basedonsymmetryarguments,supposingsomeformofthe(coarse
grained)freeenergynearthecriticalpoint.ThismethodisusedbytheLandau
theoryofcriticalphenomena.Toovercomethelimitationsofthisapproachonehas
tointroducemodelswhichincludemoredetailsabouttheinteractionsleadingtothe
phasetransition.Anumberofdi�erentmodelscanbeconstructeddependingon
thelevelofabstractionatwhichinteractionsarehandled.Herewediscussarather
generallatticemodelintroducedbyPottsin1952[16].
Theq-statePottsmodelconsistsofasetof\spins"(orparticles)fsigeachof
whichmayhaveintegervaluessi
=0;1;:::(q�1).Thesespinssitonalatticeand
theHamiltonian(theenergyfunction)isde�nedas
H[fsig]=�X
JÆ Kr(si;sj);
(2.38)
whereÆ KristheKroneckerdeltafunctionandthesummationgoesovernearestneigh-
boursonly(shortrangeinteraction).ThemeaningoftheenergyfunctionEq.(2.38)
isthatonlyparticleswithsi=sj\like"eachother,onlysuchcombinationslowerthe
energyofthesystemasillustratedinFig.2.7.
Ifthetemperatureishighthenalltheqstateswillbeequallypopulated,so
anyquantitycanserveasanorderparameterwhichmeasuresthedi�erenceofthe
distributionofspinstatesfromuniform.Atlowtemperaturesthesystemorganises
itselfintoacon�gurationwheremostspinsareinarandomlyselectedstatewhilethe
otherstatesareweaklypopulated.
ThePottsmodelisrelatedtomanyotherlatticemodelsinstatisticalphysics[17].
Forq=2(twostates:`up'and`down')itisequivalenttothewellknownIsingmodel.
Theq=1limitreproducesthepercolationproblem,whileq=0canbemappedto
theresistornetworkproblem.
2.3.2
Mean-�eldapproximation
Althoughsomeofthespecialcasesmentionedabovecanbesolvedexactly,nogeneral
solutionexiststothePottsmodelitself.Herewepresentthesimplestapproachto
2.3.
CONTINUOUSPHASE
TRANSITIONS
39
��������
��������
����������
����������
��������
����������
����������
���
���
���
���
���
���
������
������
��������
��������
���
���
���
���
������
������ ���
���
���
���
���
���
��������
���
���
���
���
H 0 -J
Figure2.7:InteractionenergyofparticlesinthePottsmodel.
determinetheexponents[17].Considerthefollowingslightlymodi�edHamiltonian
H[fsig]=�
zJ N
X i<j
Æ Kr(si;sj);
(2.39)
whereziscoordinationnumberofthelattice(numberofneighbours,i.e.,z=4fora
planarsquarelattice)andN
isthetotalnumberofspins.IncontrasttoEq.(2.38)
thisHamiltonianallowslongrangeinteractionssinceinteractionofeveryspin-pair
contributestothetotalenergy.Inotherwords,aspinisa�ectednotonlybyits
neighbours,butratherbythemeanstateofthewholesystem.Forthisreasonsuch
anapproximationiscalledamean-�elddescription.Thisapproachhasseveralde�-
ciencies:byintroducinglongrangeinteractionitneglectsthe
uctuationswhichare
essentialpartsofphasetransitions,andthereforedoesnotgivecorrectresultsfor
thecriticaltemperatureandtheexponents.Nevertheless,sincegivesaqualitatively
correctpictureitisworthexamining.
Insteadofaccountingforthestateofeverys
Top Related