MULTISCALE MODEL OF CRISPR-INDUCED COEVOLUTIONARY DYNAMICS: DIVERSIFICATION AT THE INTERFACE OF LAMARCK AND DARWIN.

0.9 MB·English

by  Childs, Lauren M

ORIGINAL ARTICLE
doi:10.1111/j.1558-5646.2012.01595.x
MULTISCALE MODEL OF CRISPR-INDUCED
COEVOLUTIONARY DYNAMICS:
DIVERSIFICATION AT THE INTERFACE
OF LAMARCK AND DARWIN
LaurenM.Childs,1 NicoleL.Held,2 MarkJ.Young,3 RachelJ.Whitaker,4,5 andJoshuaS.Weitz6,7
1SchoolofBiologyandSchoolofMathematics,GeorgiaInstituteofTechnology,310FerstDr,Atlanta,Georgia30332
2DepartmentofMicrobiology,UniversityofIllinoisatUrbana-Champaign,601S.GoodwinAvenue,Urbana,Illinois61801
3ThermalBiologyInstituteandDepartmentofPlantSciencesandPlantPathology,MontanaStateUniversity,307Plant
BioscienceBuilding,Bozeman,Montana59717
4DepartmentofMicrobiologyandInstituteforGenomicBiology,UniversityofIllinoisatUrbana-Champaign,601
S.GoodwinAvenue,Urbana,Illinois61801
5E-mail:[email protected]
6SchoolofBiologyandSchoolofPhysics,GeorgiaInstituteofTechnology,310FerstDr,Atlanta,Georgia30332
7E-mail:[email protected]
ReceivedJuly27,2011
AcceptedJanuary19,2012
DataArchived:Dryaddoi:10.5061/dryad.gc2260qm
TheCRISPR(ClusteredRegularlyInterspacedShortPalindromicRepeats)systemisarecentlydiscoveredtypeofadaptiveimmune
defenseinbacteriaandarchaeathatfunctionsviadirectedincorporationofviralandplasmidDNAintohostgenomes.Here,we
introduceamultiscalemodelofdynamiccoevolutionbetweenhostsandvirusesinanecologicalcontextthatincorporatesCRISPR
immunityprinciples.WeanalyzethemodeltotestwhetherandhowCRISPRimmunityinduceshostandviraldiversificationand
themaintenanceofmanycoexistingstrains.Weshowthathostsandvirusescoevolvetoformhighlydiversecommunities.We
observethepunctuatedreplacementofexistentstrains,suchthatpopulationshaveverylowsimilaritycomparedoverthelong
term.However,intheshortterm,weobserveevolutionarydynamicsconsistentwithbothincompleteselectivesweepsofnovel
strains(assinglestrainsandcoalitions)andtherecurrenceofpreviouslyrarestrains.Coalitionsofmultipledominanthoststrains
arepredictedtoarisebecausehoststrainscanhavenearlyidenticalimmunephenotypesmediatedbyCRISPRdefensealbeitwith
differentgenotypes.Weclosebydiscussinghowourexpliciteco-evolutionarymodelofCRISPRimmunitycanhelpguideefforts
tounderstandthedriversofdiversityseeninmicrobialcommunitieswhereCRISPRsystemsareactive.
KEY WORDS: Evolutionarybiology, host–parasiteinteractions, immunedefense, microbialecology, viralevolution.
TheCRISPR(ClusteredRegularlyInterspacedShortPalindromic example, plasmids and viruses (Mojica et al. 2005; Barrangou
Repeats) system is a recently discovered type of adaptive im- etal.2007;Brounsetal.2008;Deveauetal.2008;Horvathand
munesystemwhichdefendsagainstforeigngeneticmaterial,for Barrangou2010).Importantly,theCRISPRsystemispurported
tobethemeansbywhichsomebacteriaandarchaeaevadeviral
infection and lysis in the environment (Andersson and Banfield
Re-useofthisarticleispermittedinaccordancewiththeTermsand
2008;Heldetal.2010;Heidelbergetal.2009).Aswedescribe
Conditions set out at http://wileyonlinelibrary.com/onlineopen#
OnlineOpenTerms below,themoleculardetailsofhowtheCRISPRsystemoperates
(cid:2)C 2012TheAuthor(s).Evolution(cid:2)C 2012TheSocietyfortheStudyofEvolution.
2015 Evolution66-7:2015–2029
LAUREN M. CHILDS ET AL.
andhowvirusesevadeitaretopicsofintensivestudy.Nonethe- rulesofgenomicchangeassociatedwiththeCRISPRsystem.We
less,thefactthathostswithanoperativeCRISPRsystemundergo dosotofurthertheoreticalunderstandingoftwoquestions.First,
directedchangestotheirgenomewithrespecttotheintroduction arethemolecularmechanismsassociatedwiththeCRISPRsys-
offoreigngeneticmaterialposesachallengetotheoreticalefforts temsufficienttoleadtoandmaintainviralandhostdiversityand
tounderstandthebasisforcoevolutionary-induceddiversification acomplexhost–viralcommunity(AnderssonandBanfield2008;
amonghostsandviruses.Nearlyalltheoriesofevolutionarydy- Heidelberg et al. 2009; Held et al. 2010)? Second, what are the
namicshaveincommontwotenetsofDarwinianevolution:first, evolutionarymechanismsbywhichdirectedandundirectedmuta-
changestoorganismalgenomes,forexample,mutations,areran- tionalmechanismsremaininbalance,incaseswherecoexistence
dom(LuriaandDelbruck1943;LederbergandLederberg1953); isobserved(Heidelbergetal.2009;Heldetal.2010)?
second,successoforganismsdependsontheirecologicalfitness A few other models have already made inroads in charac-
(Lande1976;Geritzetal.1997;NowakandSigmund2004).The terizingtheeffectthatCRISPRdefensemayhaveonecological
CRISPR system suggests that a new class of models are neces- and evolutionary dynamics. First, He and Deem (2010) utilized
sarytodescribehost–viruscoevolutionthatliesattheinterfaceof an immunological-based approach in which viral production is
DarwinianandLamarckianevolution(KooninandWolf2009). uncoupled from host density (and hence is less concerned with
As noted above, the CRISPR system utilizes a form of ecologicallydrivendynamics).Thatmodelconcludedthatspacers
genome-level imitation that permits a microbial cell to direct should be more diverse in the leading edge and also, that coex-
genomicchangesthatmaybebeneficialtoitssurvivalagainstin- istence is possible among diverse strains. Second, Levin (2010)
vadingelements(e.g.,Soreketal.2008;HorvathandBarrangou largely avoided the issue of coevolution, to examine ecologi-
2010; Marraffini and Sontheimer 2010a; Vale and Little 2010). calcompetitionbetweenstrainsthatpossessCRISPRimmunity
CRISPRlocihavebeenidentifiedin40%ofbacteriaand90%of versus those that possess receptor-based immunity. The present
archaea(Grissaetal.2007).Inbrief,theCRISPRsystemworks model aims to unite these two perspectives: (1) by utilizing an
asfollows:bacteriaandarchaeamayhavemultipleCRISPRloci, explicit density-dependent ecological formalism for host–viral
containingasetofCRISPR-associated(Cas)genesandarepeat- interactions,suchasLevin(2010);(2)byexaminingthecoupling
spacerregion(Soreketal.2008;Jansenetal.2002;Makarovaetal. between the ecological dynamics of strains and the evolution-
2009). This region has “spacers,” that is, genetic subsequences arychangeofthegenomicstateofstrains,suchasHeandDeem
usually 20–50 nucleotides long that match to the protospacers (2010).Insodoing,thepresentmodeltracksthedynamicsofboth
found in extrachromosomal elements such as viruses, plasmids, hostandviralstrainstatesaswellasdensities.Athirdmodel,by
and transposons, which are separated by repeats (Bolotin et al. Haerteretal.(2011),alsopresentsasimilarapproach,albeitwith
2005; Pourcel et al. 2005; Marraffini and Sontheimer 2010a). a focus on spatially-mediated interactions between viruses and
Therepeat-spacerregionsaretranscribedasRNA.Mediatedby hosts.
theCasproteins,theseCRISPRRNAsconferimmunityagainst Here,weanalyzecoevolutionary-induceddynamicswherein
virusesandplasmidsbytargetinghomologousstretchesofDNA hostspossessmultiplespacersandvirusespossessmultipleproto-
and/or RNA. Successful recognition of foreign genetic material spacers.Weobservethathighlydiverseassemblagesemergefrom
(i.e., via Watson–Crick base pairing with specific subsequences lowdiversityinitialconditions.Theemergenceandmaintenance
known as protospacers) can lead to repression and/or digestion ofdiversityisduetoaseriesofinvasionsbyvirusesandhosts.Di-
oftheforeigngeneticmaterial(Haleetal.2009;Marraffiniand versityismaintainedoverthelongterm,butthisdiversityreflects
Sontheimer2010a).However,theprecisemolecularmechanisms thepunctuatedemergenceofnovelhostandviralstrainsthathave
for immunity or interference and acquisition of new spacers re- relativelyshortlifetimes.Hence,wefindthatpopulationsareof-
mainsafocusofcontinuedresearch(e.g.,Haurwitzetal.2010; tenhighlysimilaronshorttime-scalesbuthighlydissimilarover
MarraffiniandSontheimer2010b;Haleetal.2009;vanderOost longtime-scales.Wealsoobservethreetypesofevolutionarydy-
etal.2009).Thehostgenomeevolvesbypartialimitationofviral namicsthatdriveshort-termchangesinourmodel:(1)invasion
genomes (or plasmids) for which it has survived exposure (see by rare strains with fitness advantages; (2) recurrence of rare,
Fig.1B).Incontrast,virusesthatinfectahostcellandavoidde- older strains that gain fitness advantages due to changes in the
tectionbytheCRISPRsystemandotherviralimmunitysystems geneticstatesofotherstrains;(3)invasionbycoalitionsofstrains
evolveviaundirectedmutation(seeFig.1A). with identical immune phenotypes but distinct genotypes. We
Inthisarticle,weintroduceamodelintendedtocapturethe notethatcoevolutionarydrivendiversificationisnotinevitablein
principlesofhost–virusinteractionsandcoevolutionviaCRISPR suchmodels,andpointoutconditionsthatfavorCRISPR-induced
immunity in an explicit ecological context. The model utilizes eliminationofviruses,whichmaybeofinterestinbioengineering
a multiscale approach to combine density-dependent ecological applications.Additionally,weobservethatCRISPRimmunityis
dynamicswithevolutionarychangesinformedbythemolecular dominated by the most recently acquired spacers, an emergent
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Figure 1. SchematicoftheDarwinianandLamarckiancomponentsofevolutionintheCRISPRmodel.(A)Undirectedmutationofviruses
following successful infection leads to replacement with a novel protospacer within the viral genomes. New protospacers can occur
anywhereintheprotospacerset.(B)Directedmutationofhostsleadstoinclusionofanovelspacerwithinthehostgenome.Newspacers
areaddedattheleadingend.Note:Wesimplifythedynamicsofspacerstatechangebyassumingthemaximumnumberofspacersper
straintypeisconstant.Whenthemaximumnumberisreached,theadditionofaspacerattheleadingendisaccompaniedbydeletion
ofaspaceratthetrailingend.
feature of our simulations. Hence, we predict that only the first Table 1. Descriptionandvaluesofparametersinmultiscaleeco-
few spacers play a role in shaping the selective forces driving evolutionarysimulations.Detailsofhowtheseparametersarein-
host–viralcoevolutionevenwhenthespacerlocusiscomprised tegratedinthemodelareexplainedinthemaintext.
ofmanyspacers.Weshowthattheacquisitionrateofnewspacers
Model
isastrongerdeterminantofthecomplexityoftheresultingcom-
component Parameter Meaning Values
munitythanisthefailurerateofhoststoprotectagainstviruses
forwhichCRISPRimmunityisalreadypresent.Finally,wedis- Molecular p CRISPRfailure 10−5
probability
cusshowthiscoevolutionarymodelframeworkcanbeutilizedto
q Newspacer 10−5
helpidentifythosefactorsdrivingCRISPR-inducedcoevolution
acquisition
intheenvironment.
probability
Ecological r Growthrate(1/h) 1
Models
K Carryingcapacity 105.5
The coevolutionary model presented here is comprised of three (1/mL)
parts:(1)ecological;(2)molecular;and(3)evolutionary.Thefull β Burstsize 50
φ Adsorptionrate 10−7
modelintegratesthesethreecomponentstogethertosimulatethe
(mL/h)
dynamicinteractionsbetweendiversehostsandviruses.Inbrief,
m Viraldecayrate(1/h) 0.1
hostandviraldensitiesaredeterminedbyecologicalrulesofin-
Evolutionary μ Mutationrate 5×10−7
teractionthatincludehostreproductionanddeath,viralinfection ρ Densitycutoff(1/mL) 0.1
c
of hosts, and viral deactivation outside of hosts. The molecular
componentdetermineswhetherviralinfectionleadstohostlysis,
viraldeactivation,orspacerintegration.Theevolutionarycompo- et al. 2005). More broadly, multiscale eco-evolutionary models
nentintroducesnewhostandviralstrainsandtheirgeneticstates ofthiskindhavebeenutilizedelsewhere,forexample,foodweb
(see Fig. 1). The dynamical steady state in the model is shaped dynamics(LoeuilleandLoreau2005)andinfluenza-hostdisease
bytheassumptionsbuiltintothemodelaswellasthequantitative dynamics(Koelleetal.2006).
valuesofmodelparameters(seebelowandTable1).Thedetails
ofthesecomponentsandofthecomputationalschemeusedtoim- ECOLOGICAL COMPONENT
plementthemaredescribedbelow.Themodelframeworkbuilds Weconsideracommunitycomprisedofhosts(eitherbacteriaor
uponanearlierefforttostudycoevolutionarydynamicsbetween archaea), viruses, and implicitly modeled resources. The densi-
bacteriaandphagesbasedupontheevolutionofenveloperecep- tiesofhostsandviruseschangebasedonthefollowingecolog-
torstateswithinbacteriaandtailfiberstateswithinphages(Weitz icalevents.First,hostscandividegivensufficientresourcesand
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LAUREN M. CHILDS ET AL.
theycanalsodie.Here,wefocusonarathersimplifiedecolog- MOLECULAR COMPONENT OF CRISPR IMMUNITY
icalcontext,whereresourcesareconsideredimplicitlyandhost Theimmunedefenseofahosttoviralinfectionisbasedonse-
populations would increase to their carrying capacity in the ab- quence matches between host and viral genomes. The immune
senceofviruses.Viralpopulationsincrease(andhostpopulations state of a host is denoted as S =(s ,s ,...,s ) where s is
1 2 u i
decrease) due to infection and lysis of hosts. Viral populations theithspacerofu spacersintheCRISPRlocus.Inreality,mul-
decrease due to spontaneous deactivation in the environment, a tiple CRISPR locimayexist within agivenhost,however,here
process thought to be characterized by a single time-scale (De we only analyze a single locus. We simplify the dynamics of
PaepeandTaddei2006).Viruspopulationsalsodecreasedueto spacerstatechangebyassumingthemaximumnumberofspac-
unsuccessfulinfections.Wedenote N asthedensityofhostsof ersperstrainisconstantandthatspacersarealwaysaddedtothe
i
strainiandV asthedensityofvirusesofstrain j.Eachhoststrain leading end. When the maximum is reached, the addition of a
j
hasauniquegenomicstatewhichwedenotebyS,corresponding spacertotheleadingendisaccompaniedbydeletionofaspacer
i
tothesetofspacersitcontainsthatconferitwithCRISPR-derived atthetrailingend.Spacersaredrawnfromprotospacers,thatis,
immunity.Eachviralstrainhasauniquegenomicstatewhichwe smallsubsequenceswithintheviralgenome.Assuch,wedenote
denotebyG ,correspondingtothesetofprotospacersitcontains the genomic state of the virus relevant to CRISPR immunity as
j
forwhichhostsmayormaynotbeimmune.Hostsreproduceat G =(g1,g2,...,gv), where gj is the jth protospacer of v pro-
a maximum per-capita rate ofr with a carrying capacity of K. tospacersinthevirus.Throughoutthisanalysis,weconsiderall
i
Virusesinfecthostsatarateφ .Here,weonlyconsidertwopos- undirectedmutationstobedrawnfromaninfinitenumberofal-
ij
sibleoutcomesforaviralinfection:(1)thehostdiesandnewviral leles. This assumption is supported by studies that suggest that
particlesareproduced;(2)thehostdisablestheviralgenomeand only a single base pair mismatch undermines CRISPR immu-
(possibly)modifiesitsowngenomeinadirectedfashion.Finally, nity(Barrangouetal.2007).Otherimmunitymodelsmayfollow
virusesdecayatadensity-independentrateofm.Togetherthese fromrelaxingthiscondition.Here,CRISPRimmunityisdefined
rulesleadtothefollowingdynamicalequations: asfollows:
Hostdivisionandmortality
(cid:2) (cid:3)(cid:4) (cid:5) (cid:15)
⎛ (cid:9) ⎞ 1 if s =g ∈G,
⎜ Ni⎟ M(S,G)= i j (3)
dNi = riNi⎜⎝1− i ⎟⎠ 0 if otherwise.
dt K
Inwords, M(S,G)=1ifthespacersetinthehostincludes
Virallysiswhennotimmune
(cid:2) (cid:9)(cid:13) (cid:3)(cid:4) (cid:14) (cid:5) (1) at least one perfect match to a protospacer in the virus, oth-
−(1−q) 1−M(Si,Gj) φijNiVj erwise M(S,G)=0. The CRISPR immune mechanism is not
j perfect(Barrangouetal.2007).Weexpecterrorswillbeoftwo
V(cid:2)irallysisw(cid:3)(cid:4)henimmun(cid:5)e types,falsenegativesandfalsepositives.Forfalsenegatives,the
(cid:9)
−p M(S,G )φ N V , CRISPRsystemmaynotidentifyaviralgenomeeventhoughit
i j ij i j
possessesaspacerwhichmatchesaprotospacerinthevirus.For
j
false positives, the host may randomly acquire a spacer match-
Virionreleasefromnonimmunehosts ing a viral protospacer during an interaction (see below) which
(cid:2) (cid:3)(cid:4) (cid:5)
dV (cid:9)(cid:13) (cid:14) brings immunity along with it. Both CRISPR and non-CRISPR
j =(1−q)β 1−M(S,G ) φ N V
dt i j ij i j mechanisms may be involved. We model these types of errors
i
quantitatively as follows. If M(S,G)=1 (the host is immune
Virionreleasefromimmunehosts
(cid:2) (cid:3)(cid:4) (cid:5)
(cid:9) to the virus via CRISPR defense), then two events can happen:
+ pβ M(Si,Gj)φijNiVj (2) (1) immune defense with probability 1− p, via which the host
i
survives and the virus is eliminated; (2) stochastic failure with
Viralinfection
(cid:9)(cid:2) (cid:3)(cid:4) (cid:5) Virald(cid:2)e(cid:3)a(cid:4)c(cid:5)tivation probability p, via which the host is lysed by the virus lead-
− φijNiVj − mVj . ing to a burst of progeny viruses. If M(S,G)=0 (the host is
i not immune to the virus via CRISPR defense), then two events
In these equations, M(S,G ) denotes whether a host with canhappen:(1)hostlysiswithprobability1−q andsubsequent
i j
spacerstate S isimmune(M =1)ornotimmune(M =0)toa burst of progeny viruses; (2) host survival with probability q
i
viruswithprotospacerstateG .Thedetailsofhowsuchimmunity in which the virus is eliminated. We assume that both p and q
j
is determined is explained in the following section, along with are small, that is, p,q (cid:4)1, as described in Model parameters
discussionsofthemeaningoftheimmunityparameters pandq. below.
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MULTISCALE MODEL OF CRISPR-INDUCED COEVOLUTIONARY DYNAMICS
EVOLUTIONARY COMPONENT Wesimulatetheecologicalandmolecularinteractionsofthehosts
Undirectedmutationsofviralprotospacerscanoccuruponsuc- andviruses(seeeqs.1–2)deterministicallyusingode45inMatlab.
cessfulinfectionofahost.VirusesthatevadeCRISPRdefenses Populationdensitieschangeuntil:(1)ahostorvirusstraingoes
andotherhostdefensesexploithostcellularmachineryandpro- extinct;(2)amutationeventoccurs,eitheroftheundirected(viral)
duceβvirions(seeeq.(2)).Errorsinreplicationcanleadtothe or directed (host) type; or (3) the simulation reaches a defined
modification of one of the protospacer alleles in a given virion timepointfordataoutputandrecalculationofmutationratesthat
withaper-alleleprobabilityμ(e.g.,thealleleindicatedbyanar- occuratperiodicintervals.Whenanyoftheseeventsoccur,the
rowinFig.1A).Theundirectedmutationofavirusprotospacerset simulationispausedandthestrainmutationrates,whichdepend
isdenotedasG →G(cid:6).Notethatinimplementingthemodel,we on the continually varying strain abundances, are recalculated
donotconsidersimultaneousmutationsorrecombinationmech- (see Supporting Information for details.). Note that we use the
anismsasmeansforintroducingvariationinprotospacerstates. term “mutation” here to denote the insertion of a sequence into
Hence, if viruses are produced at a rate b per unit time in the theCRISPRlocusofahostorthechangeinsequenceofaviral
entire system, then the expected number of mutations per unit protospacer. This process is repeated until one of the following
timeisbμv,wherevisthenumberofprotospacerspervirus.The occurs:allhoststrainsgoextinct,allviralstrainsgoextinct,orthe
stochastictimingoftheseeventsisdeterminedviarandomsam- simulationreachesthemaximumrunningtime(generally2500h
pling from an exponential distribution with mean time between in the model). Simulations are run with 100 replicates, unless
mutationsof1/(bμv).Duetochangesinhostandviralpopula- the computation per replicate is excessive in which case 75 or
tionsthataffecttheviralbirthrate,thisrateisrecalculatedduring 25replicatesareused.
thesimulation(seeSupportingInformationfordetails). Strain extinction occurs when the population density of a
Directed mutation can occurwhen a host identifies andin- strainfallsbelowourcriticalpopulationthreshold,ρ ;thisactsas
c
tegrates a new protospacer into its CRISPR locus, where it is anabsorbingstateforstrains.Throughaneventfunctioninode45,
denotedasaspacer(seeFig.1B).Duringeveryhost–viralinter- thesimulationispaused,andthesystemofordinarydifferential
action, there is a small probability q of acquiring a new spacer equations (ODEs) is reduced by removing the equation for the
throughuptakeofaprotospacer.Dependinguponthepreviousim- strainwhichhasfallenbelowthecutoff.Mutationeventscanoc-
munestateofthehost,theadditionofaprotospacermaychange cur upon replication by viruses (via an undirected mechanism)
the immune state of the host with respect to the identified and anduponvirusinfectionofahost(viaadirectedmechanism)(see
imitatedgenotype.Inotherwords,thehostwillbecomeimmune Fig.1).Mutationaleventscauseanadditionofanewstrainand
to all viruses which contain the protospacer that was integrated thustheadditionofanewODEtothesystem.Allmutantstrains
intoS(cid:6)asaspacer.Ifthehostalreadycontainedamatchingproto- aregivenaninitialdensity10%greaterthanρ .Thetimeuntilthe
c
spacer,thisnewspacerdoesnotprovideanyadditionalimmunity. nextmutationaleventiscalculatedusingtheGillespiealgorithm.
Ifahostdidnotpreviouslycontainamatchingspacer,thenewly Sincethetimetothenextmutationaleventisstochastic,replicate
addedprotospacerallowsthehosttosurvive.Thisoftenleadsto simulationswillnotgiveidenticalresults.Inthecaseofviralmu-
a selective advantage for this host strain. To summarize: proto- tation, agivenstrain ofvirusis randomlyselectedtoundergoa
spacer integration to the host can occur with rate q during any mutationeventwithprobabilityinproportiontotheinstantaneous
unsuccessfulattemptbyavirustoinfectahost,regardlessifthe growthrateofthatstrain.Similarly,inthecaseofhostacquisition
hostpreviouslyhadimmunitytotheattackingvirus.Recallthat ofaspacer,agivenhoststrainisrandomlyselectedtoacquirea
theadditionofaspacerattheleadingedgeisaccompaniedbythe new spacer in proportion to its instantaneous rate of successful
lossofasinglespaceratthetrailingedgeoftheCRISPRlocus defenseevents.Dataoutputoccursatregularintervalsthroughout
when the locus has a maximum number of spacers considered thesimulation.Aftereachevent—strainextinction,strainmuta-
(eightinoursimulations).Additionally,notethatahostneedonly tion, or data output—the strain mutation rates are recalculated.
differbyasinglespacerfromallotherhoststrainstobeconsid- Additional details of the simulation procedure are found in the
ereditsownstrain.Allnovelmutantstrainsareintroducedinthe SupportingInformation.
simulation,regardlessofwhethertheyhaveaselectiveadvantage
ornot. MODEL PARAMETERS
The choice of model parameters will vary depending on the
SIMULATION PROTOCOL CRISPR system of interest. In general, we consider ecological
Webeginoursimulationswithasinglehoststrainandasingleviral parameterstypicalofEscherichiacolianditsphages(DePaepe
strain along with their respective spacer and protospacer states. andTaddei2006)andmolecularparametersconsistentwithsmall
Ourinitialhoststrainissusceptibletotheinitialviralstrainand, errorratesinCRISPRimmunity, p(cid:4)1andq (cid:4)1.Thevalueof
thus,doesnotcontainaspacermatchingaprotospacerofthevirus. p is based on work in Streptococcus thermophilus for which p
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LAUREN M. CHILDS ET AL.
canbeconsideredanefficiencyofplatingofvirusesonCRISPR
immunehostsandcanrangefrom10−4to10−7(Barrangouetal.
2007).Thevalueforq isbasedonworkinthesamesystemfor
whichacquisitionofresistancetovirulentbacteriophagesoccurs
rarely, q ≈10−6 (Barrangou et al. 2007; Horvath et al. 2008;
Deveauetal.2008).Notethatinthismodel,thevalueq denotes
thesuccessfulintegrationofanovelspacer,andhence,directed
mutationofthehost.
Giventhevariationinherentinviralandhostdynamics,we
further restrict our attention to model parameterizations which
obey the following two conditions: (1) viruses eventually die
outwhen infecting immune hosts; (2) viruses coexist with non-
immune hosts. Given small error rates and large burst sizes
and the definitions of the ecological and molecular compo-
nents, these conditions can be written compactly as: Kφβ >1
m
and 1 <β< 1 (seetheSupportingInformation).Notethatac-
1−q p
tual ecological parameters remain poorly known for all but the
mostwellstudiedoflaboratoryhost–virussystems.Furthermore,
whenselectinghostsandvirusestomodel,wenotethatparame-
terchoicesarenotindependent.Forexample,viralmortalityrates
andtheirproductionrates(burstsizedividedbylatentperiod)are
positivelycorrelatedforsomephages(i.e.,phageswhichproduce
morevirionsdegradefasterandthosewhichproducefewervirions
aremorestable)(DePaepeandTaddei2006).Parametervalues
that are used as baselines for all simulations (unless otherwise
noted)arelistedinTable1.
Thoughthemodelframeworkcanhandlearbitrarynumbers
Figure 2. Dynamics and diversification of multiple spacer–
of spacers and protospacers, we focus computational efforts on
protospacermodel(eightspacers,10protospacers).(A)Viralpop-
caseswhenu ≤8spacersandv ≤10protospacers.Thechoiceof
ulation dynamics (green online) and host population dynamics
spacerandprotospacernumberensuresthatthenumberofspacers
(black) show that population densities undergo fluctuations. (B)
islessthanthenumberofprotospacers,asisthecasebiologically.
Viralstraincount(greenonline)andhoststraincount(black)show
Further,ourchoiceismadeasaconcessiontoefficientnumerical thediversificationintomultiplehostandviralstrains.Thesegraphs
simulation of the model, whose simulation time increases with showresultsfromasinglerepresentativesimulation(outof100
protospacer number. Computational time and power also limits replicates).
thenumberofreplicatesitisfeasibletoconsider.
spacer sites for which hosts have CRISPR immunity may gain
somefitnessadvantage.Inthismodel,hosttypesemergethatare
CRISPRimmunetomultiple(butnotnecessarilyall)virusesand,
Results
likewise,viraltypesemergethatinfectsome(butnotnecessarily
VIRAL AND HOST DIVERSIFICATION IN A MULTIPLE all)hosts(seeFig.S1).
SPACER, MULTIPLE PROTOSPACER MODEL Inthissystem,complexcoevolutionarydynamicsunfold(see
Here, we examine the dynamics of a host–virus community in Fig.2A). Notethatsincethehostgenerationtimeis∼1hinthis
whicheachhostpossessesmultiple spacersandeachviruspos- model(seeTable1),wewillrefertodynamicsintheequivalent
sesses multiple protospacers. We find that hosts rapidly acquire scale of generations. In the first ∼50 generations, the densities
CRISPR immunity through directed incorporation of spacers. of hosts and viruses oscillate around a dramatically increasing
Viruses mutate randomly at one of multiple protospacer sites average.Thenumberofhostandvirusstrainsinitiallyincreases,
so that not all viral mutations are immediately beneficial. Non- often exceeding dozens and sometimes hundreds of strains (see
beneficial viral mutations may arise if viral mutations occur at Fig. 2B). After the system passes transient dynamics, the virus
protospacersitesforwhichnohostpossessesCRISPRimmunity. populationexceedsthehostpopulationinbothdensityandstrain
In contrast, viral strains that have mutations of specific proto- count(seeFig.2). Typicallyby∼750generations,theCRISPR
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MULTISCALE MODEL OF CRISPR-INDUCED COEVOLUTIONARY DYNAMICS
finding is important as it points out that multiscale eco-
evolutionaryCRISPRmodelsmaybeappropriateforthestudyof
host–virusdynamicsinnaturalenvironments(wherecoexistence
maybeofinterest)orhost–virusdynamicsinindustrialcontexts
(whereviraleliminationmaybeagoal).
DIRECTED MUTATION OF HOSTS CHANGES HOST
POPULATION SIZE AND CONTENT OVER THE LONG
TERM
We find that the strain composition changes over the course of
asimulation,despitethemaintenanceofhighdiversitythrough-
out. In other words, strains arise, exist for some period of time
typicallybetween0and400h,andarelost(seeFig.S2).Toquan-
titatively compare the strain composition of the host population
overtime,weemploytheMorisita–Hornsimilarityindex(Wolda
Figure 3. Incorporationofspacerscauseschangesinhostpopu- 1981)whichdefinesthesimilarityofcommunitiesattimest and
1
lationsizeandhostpopulationcontent.Ecologicalsimilaritybe-
t takingintoaccountthetypesofstrainsandtheirabundances:
2
tween the whole host population at two time points using the
Morisita–Horn index which takes into account both abundance (cid:9)S
andtype(seeeqs.4–5).Timeintervalsof2hareused.Thecolor 2 n n
1j 2j
barindicatessimilarityfromblue(lowsimilarity)tored(highsim- (cid:2) = j=1 , (4)
ilarity).Thediagonalisthecomparisonofonecommunityagainst N N (ψ +ψ )
1 2 1 2
itselfandhencehasperfectsimilarity(darkred).Communitiessig-
where
nificantlyseparatedintimeareblueindicatingnosimilarity(see
bottomleftofthefigure).Theverticalbarsindicateanincreasein (cid:9)S
theaveragenumberofspacersperhost.Theaveragenumberof n2
ij
spacers,sismarkedabovethegraphandsaturatesatamaximum ψ = j=1 . (5)
ofs=8.Theinsetisanenlargedversionoft=1500tot=1700. i N2
i
Thisgraphshowsresultsfromasinglerepresentativesimulation
(outof100replicates). Inequations(4–5), Sisthetotalnumberofuniquestrainsfound
inoneorbothcommunities,N isthetotalnumberofindividuals
i
spacerlocusofallhoststrainshasacquiredafullarrayofeight incommunityi,andn isthenumberofindividualsofstrain jin
ij
spacers.Subsequentstrainsalsohaveafulllocus.Thereafter,the communityi (Wolda1981).Further,notethatψ isthesimilarity
i
averagenumberofhostandvirusstrains(from100replicates)re- of community i, and is equivalent to the probability that two
mainsrelativelyconstant,althoughthedensityandabundanceof randomlychosenindividualsinthatcommunityareofthesame
anyparticularstraininanyparticularsimulationchangesdramati- species. The index functions as follows: the index is zero when
cally.Cross-correlationanalysiswillbeconsideredinafollow-up thereisnooverlapofstraintypesbetweentimepoints;itislow
workgiventheinterestinphaselagsofconsumersandresources (nearzero)whenthereissomeoverlapofstraintypesbutatvastly
withineco-evolutionarydynamicalsystemssuchasthis(Yoshida differentabundances;itishigh(nearone)whenthereisoverlap
etal.2007). of many strains and these strains exist at similar abundances; it
Aviralstraindoesnotnecessarilysufferanextinctionwhen isonewhenthereisexactlythesamestrainsatexactlythesame
a host acquires immunity to that viral strain, because there are abundance(i.e.,comparisonofatimepointtoitself).
other host strains present which that viral strain may be able to Intheabsenceofmutation,ourchoiceofparametersallows
infect (see Fig. S1). However, evolutionarily induced extinction hostsandvirusestostablycoexist(seetheSupportingInforma-
ofvirusesdoesoccurwhenundirectedmutationdoesnotgenerate tion).Althoughsuchacaseisbiologicallyunfeasible,itisinstruc-
viralstrainsthatcanevadeCRISPRimmunitytoindividualhosts tivetoconsidertheoretically.Insuchacasewithoutmutation,the
or a coalition of hosts. For example, if μ, the mutation rate of content,orstraintypes,ofthepopulationwouldnotchange,but
ourviruses,istoosmall(e.g.,μ<10−8givenparametersutilized the population density of viruses and hosts would change over
here)thenthereisthepossibilitythattherearetoofewviralin- time, such that the Morisita–Horn index, (cid:2), would never reach
fections that lead to novel viral strains with the ability to evade zerosolongashostsandvirusespersist.
CRISPR immunity (see Table S1). Such viral extinctions can For our multiscale model that includes both directed and
also be ecologically induced even for larger values of μ. This undirected mutation, we calculate (cid:2) for all pairs of recorded
EVOLUTION JULY2012 2021
LAUREN M. CHILDS ET AL.
timepointsinoursimulation,leadingtothematrixinFigure3, Wefindthatthedominanthoststrainsdieoutandarereplaced
wherereddenoteshigh(cid:2) andbluedenoteslow(cid:2).Asshownin bynewstrains(asdefinedinourmodel)withdifferentcharacter-
Figure3,wefindrapidturnoverofhoststrainsovertime-scales isticevolutionarydynamics:(1)incompleteselectivesweeps;(2)
thatspanthesimulation.Directedmutationofahostcanprovide negativefrequency-dependentselection;and(3)clonalcompeti-
thenewhoststrainwithimmunitytoagreaterproportionofthe tionofstrainswiththesameimmunephenotype.Basedonthese
virus population, allowing the host strain to increase in relative dynamics,oneofthesethreetypesofstraincohortsdominateat
abundance.Weobserverepeatedinstancesinwhichthehostpop- any one time: (1) a newly evolved resistant strain, (2) an older
ulationissimilartoitselfoverashortperiodoftime(seeinsertto strainmaintainedinthepopulationatlowabundance,or(3)mul-
Fig.3).Instancesofshort-termsimilarityaretypicallycorrelated tiplestrainsresistanttosimilarviralsubsets,albeitwithdifferent
withoscillationsinhostdensitydefinedbyourecologicalmodel spacers.
parameters (the small red triangles near the diagonal in Fig. 3). Withtheexceptionofearlytimepointsinoursimulation,we
Weobservethatbothhostsandviruseshaveashortlifetime(see donotobservecompletesweepswithasinglestraineliminating
Fig.S2). allothers;thetime-scaleofevolutionandecologyismixedsuch
Similaritybetweenpopulationscanextendbeyondshortos- that multiple strains nearly always coexist in our simulations.
cillationtimescalesofinvasionwhenthereisnear-dominanceby Further, we find that in the different evolutionary dynamics de-
alimitednumberofhoststrainsasshownintheFigure3insert.We scribedabove,hostsrarelyincorporatethenewestprotospacer(see
donotobserveafixedtimescaleorintervalforhostpopulations Fig.S4).Rather,thehostspacerstatereflectsahistoryoftheeco-
tobemaintainedwithnonzerosimilarity.Eventually,thederived logicalsuccessofvirusesandtheirprotospacers,butnotneces-
strains out compete their ancestors, causing these older strains sarilyachronologicalhistoryofprotospacerappearance.Below,
tofallbelowthethresholdandberemovedfromthesimulation. wedescribeeachoftheseevolutionarydynamicsingreaterdetail.
Throughoutthesimulation,thefactthatpopulationsarenotsimi- First, incomplete selective sweeps are expected given the
lartopopulationsatmuchlatertimesimpliesthatthesubsequent fact that ecological and evolutionary time scales are mixed in
populationsaredominatedbynovelevolvedhostsratherthanre- this model. An incomplete selective sweep occurs when a host
currences of prior hosts over the simulation time scale of 2500 strainevolvesthatisresistanttoviraltypes,expandingitsrange
generations(seethelargebluesectioninthebottomleftofFig.3). ofresistance.Suchahoststrainusuallyevolvesfromahoststrain
thatwasabundant,asnotedabove.Ifthisstrainhasasignificant
INCOMPLETE SWEEPS AND BOUTS OF DIVERSIFYING advantage(i.e.,maintainsaspacerthatmatchesthedominantviral
EVOLUTION DRIVE CHANGES IN POPULATION variants),itcangrowtodominateinabundanceduringthenext
COMPOSITION IN THE SHORT TERM period of high density (see red, green, and dark blue curves in
Here,wefocusonevolutionarydynamicsoccurringontheshort Fig. S3). In response, a viral variant that can infect the newly
term.Wenotethatatanypointinthesimulation,thereareonly derived host has a competitive advantage and quickly increases
a few dominant host and viral strains whereas many diversified in frequency. This is consistent with the arms race dynamic of
strainsexistatlowabundance(seeFigs.S1andS3).Throughout coevolution through successive rises and falls of newly evolved
thesimulation,novelhostvariantsevolvefromtwotypesofances- hostandviralstrains(BucklingandRainey2002).
torstrains:highlyabundantstrains,becausetheymakeupmore Rare, older host strains may also rise to dominance at pe-
of the population and thus have a greater chance of interacting riods of high host density in a manner consistent with negative
withvirusesandgainingaspacerviadirectedmutation;andless- frequency-dependent selection, albeit for mechanisms that are
abundantstrainsthatalreadyhaveimmunitytoaninfectingvirus predominantlyevolutionaryinnature(seeFig.S3lightblueand
becausetheyhavethepossibilitytoacquireanadditionalspacer magentacurvesandthetwoorangepeaksinFig.4,notethatcolors
duringsuccessfulCRISPRimmunedefensetoviralinfection.The shouldbeinterpretedseparatelyforthesetwofigures).Incontrast
secondmechanismforgeneratingnewstrainsoccursinfrequently totheselectivesweepmodel,thesestrainsarenotderiveddirectly
becausefewviral–hostinteractionsoccuramongrarestrains. from the previously highly abundant strain. Instead their fitness
Because viral mutation is undirected, the majority of new advantagearisesbecauseviralpopulationstowhichtheyareim-
viralmutantsdonothavesignificantlyincreasedfitnessbecause munegrowinnumberwhentargetingadifferenthigh-abundant
theyhavenotmutatedthespecificprotospacerthatdominanthost nonimmune host. Likewise, viral populations to which they are
strainshaveimmunityto.Onaverage,only1/vmutations,where notimmunemaydecreaseinnumberwhentargetedbyCRISPR
visthenumberofprotospacers,willproduceaviralmutantthat immunity of abundant hosts. Together, these mechanisms may
alters the dominant host immunity. In contrast, because of the haveasecondaryeffectofdecreasingviral-inducedmortalityof
CRISPRmechanismofdirectedmutation,hostsresponddirectly this rare, older strain (which possess a different array of spac-
toviralselection. ers).Decreaseofviral-inducedmortalityleadstothepopulation
2022 EVOLUTION JULY2012
MULTISCALE MODEL OF CRISPR-INDUCED COEVOLUTIONARY DYNAMICS
−5 −5
p=10 ,q=10
1
d
erre 0.8
nf
o
c
y 0.6
nit
u
m
m 0.4
e i
v
ati
el 0.2
R
0
Figure 4. Proportion of host strains in the population. Host 0 1 2 3 4 5 6 7 8
strains (independent colors—colors repeat when not directly Number of spacers
touching)arebornintothepopulationandincreaseinsizeover
time.Thetotalheightofthecoloredareaisproportionaltothe Figure 5. Mostrecentlyacquiredspacersprovidegreatestimmu-
population size and the vertical height of each color within the nity.Relativeimmunityconferredbythenewestnspacersinthe
coloredareaisproportionaltothepercentofthepopulationcom- locuscomparedtotheimmunityfromthefulllocusofeightspac-
prised by each strain. Strains first appear in the middle of the ers.Mean(circles)andstandarddeviation(errorbars)werecom-
colorthatistheirparentstrain.Somenovelstrains(i.e.,lightblue putedfor100replicatesaveragedoverthetimepointsafterthe
att≈1625denotedbyN)rapidlybecomethedominantstrain.At locusisfilledwithspacers.Immunityisdeterminedbycalculating
times,multiplehostsemergeascoalitionsandcomprisesignificant what percentage of the viruses the most recent nspacers from
portionsofthepopulation(i.e.,att≈1675denotedbyC).Finally, all hosts can match, where n=1,2,...,8. Relative immunity is
recurrenceofstrainscanbeobserved(orangepeaksat t≈1610 thepercentageofvirusesthatthemostrecentnspacersfromall
and t≈1640correspondtothesamestraindenotedbyR).Only hosts can match compared to the percentage of viruses the full
hoststrainscomprisingatleast1%ofthepopulationareincluded. spacerlocus(inourcaseeightspacers)matches.Themajorityof
Thetotalviralpopulationdensityisshowninthelowerpanel.This theimmunityisprovidedbythefirstspacerandmorethan80%
graphshowsresultsfromasinglerepresentativesimulation(out immunityisprovidedbythefirstthreespacers.
of100replicates).
expansionoftherare,olderstrain.Aspointedoutabove,wedo aCRISPRlocus(HeandDeem2010)inagreementwithobser-
not observe any host strains that persist over the time course of vation (Horvath et al. 2008; Held et al. 2010). This diversity is
theentiresimulation. consistentwiththemechanismbywhichspacersareinsertedat
Finally,wealsoobserveperiodsofhighhostpopulationden- theleadingpositionofthelocus.Ourmodeldemonstratesthatnot
sity in which there is not a single dominant host strain but a onlyaretheleadingspacersmorediverse,buttheyalsoemerge
coalitionofhoststrainsthatrisetogethertohighabundance.This asthemostimportantspacersforprovidingthehoststrainswith
can result from multiple host strains each gaining a new spacer CRISPRimmunity(seeFig.5). Foreachtimepoint,immunity
thatmatchesadifferentprotospacerinadominantvirusorviruses isdeterminedbycalculatingthepercentageofthetotalviralpop-
(i.e., around 1675 h in Fig. 4). These strains are phenotypically ulation to which hosts harbor matching spacers. For example,
nearlyidenticalbutdiffergenotypically.Thesecoalitionsfallin the immunity provided by the first two spacers measures what
abundanceduetotheriseofadivergentvirusthatdoesnotpossess percentage of all viruses the first two spacers of all host strains
anyprotospacersthatmatchanyofthenewlyaddedspacers.The match at a particular time point. Relative immunity is the im-
riseofacoalitionofhostscanalsoresultfromasetofrarestrains munitycalculatedforparticularsetsofspacerscomparedtothe
alreadyexistinginthepopulationinamannersimilartowhatis immunity calculated for the full spacer locus (eight spacers in
describedabove. ourcase).Averagevaluesarecomputedoverallhostsacrossall
time points after the locus is full of spacers. This measurement
IMMUNITY OF HOSTS IS CONTROLLED BY RECENTLY indicateswhichofthespacerpositionsaremostimportantforpro-
ACQUIRED SPACERS vidingCRISPRimmunity.Wefindthatthefirst(andmostrecent)
A recent coevolutionary CRISPR model of hosts and viruses spacerofthelocuscontributesthegreatesttotheimmunityofthat
found that spacers are more diverse in the leading position of locus,relativetotheimmunityprovidedfromtheentirelocus.The
EVOLUTION JULY2012 2023
LAUREN M. CHILDS ET AL.
contribution of subsequent spacers decreases such that only the is not a monotonically decreasing function of q. When q =0,
firstfivelociarerequiredtoprovide>90%ofimmunity,onav- thereexistsasinglesusceptiblehost,andtheviralpopulationhas
erage.Hence,theoldestspacerscontributeinsignificantlytothe the steady-state value V∗ = r(1−N∗/K) where N∗ = m (see
φ φ(β−1)
immunityofthelocus(seeFig.5).Althoughtheincreaseddiver- Supporting Information). When q =1, all viral strains will be
sityofleadingspacerswaspreviouslyknown,boththeoretically eliminated. However, when q is increased slightly above zero,
and empirically, until this model, it was not clear how impor- we find that viral density increases. Viral density increases re-
tant the recently acquired spacers were to CRISPR immunity. sultfromagreaterabilityofvirusestoreplicate,eitherbecause
Theemergentpropertyfromourmodelstronglysupportsthehy- there exist more host strains that lack immunity or because the
pothesisthatrecentlyacquiredspacerscontributesubstantiallyto hoststrainsthatlackimmunityhavelargerpopulations.Notethat
CRISPRimmunity,andmoreover,predictsthattheyaresufficient multistrain Lotka–Volterra models with fixed carrying capacity
for the CRISPR immune response, regardless of spacer identity predictthatviraldensityincreaseswiththenumberofhoststrains
atthetailendofthelocus. (see Supporting Information). In the case of increasing q, we
expectincreasesinthenumberofhoststrainsandthusthepos-
DEPENDENCE OF COEVOLUTIONARY sible secondary effect of increasing viral diversity and density
DIVERSIFICATION ON THE CRISPR IMMUNITY (seeSupportingInformation).Highervaluesofqalsoleadtohost
PARAMETERS populationsthatrecognizeagreaterproportionoftheviralpopu-
The molecular component determines whether viral infection lation(seeFig.6E).Additionally,athighervaluesofqandfaster
leadstohostlysisorviraldeactivation.Theerrorsassociatedwith acquisition,morethanjustthefirstfewspacersareimportantfor
CRISPRimmunityappearinthemodelthroughtheconstants p immunity (see Fig. 7). This is because the viral strains which
andq.Recallthatprepresentsthestochasticfailureofahostwith the older spacers recognize still exist in the population. See the
CRISPRimmunitytorecognizeaninvadingvirus.Further,recall SupportingInformationforanalysisofvariationinthespacerand
thatq representstheacquisitionrateofspacersbyhosts. protospacernumberswhileholding pandqconstant—wedonot
Here, we consider the effect of varying values of p and q observeanyqualitativedifferencesindynamics.
(aroundexperimentallyobservedvalues)ontheoutcomesofthe
multiscale coevolutionary model. In varying these parameters,
we considered values of p and q ranging from 10−6 to 10−4. Discussion
We ran simulations using all nine possible combinations of p Wehavepresentedamodelofcoevolutionarydynamicstoanalyze
and q values in this range separated by a factor of 10, that is, thediversificationofhostsandviruses.Themodeldemonstrates
p=10−6,10−5,10−4andq =10−6,10−5,10−4.Wefindthatal- how an initially small number of host and viral strains can di-
tering pwithinthisrangehasalmostnoeffectonthedynamicsin versifyintoadynamiccommunityofmanyhostsandvirusesas
thesimulations.Thereasonwhypisnotamajordriverofdynam- anticipatedfromempiricalstudies(Heidelbergetal.2009;Held
icsatsmallvaluescanbeunderstoodbyexaminingequations(1– etal.2010).Insodoing,weconfirmtheoreticallythehypothesis
2).Small valuesof p increaselysis rates,albeit multiple orders suggestedfromempiricaldatathatifCRISPRimmunityandviral
of magnitude less than lysis rates of hosts that are not CRISPR diversificationremaininbalance,arelativelystablevirusandhost
immune. Hence, viral lysis is driven by the interaction of hosts communitymayresult(AnderssonandBanfield2008;Heldetal.
andvirusesforwhichCRISPRimmunityisnotpresent,whereas 2010). Specifically, the model predicts that diversity over time
changing p onlyaffectsvirallysisinthoseinteractionsofhosts ismaintainedbythetemporallylimitedemergence,dominance,
and viruses for which CRISPR immunity is present. However, andreplacementofstrains(andcoalitionsofstrains).Weobserve
varying q significantly modifies the complexity of communi- incompletesweepsbysinglestrains,theoccasionalrecurrenceof
ties even at small values. Higher values of q, corresponding to rare,olderstrainsthatobtaintemporaryfitnessadvantages(sim-
morerapidacquisitionofspacers,leadtoahighnumberofhost ilar in form to negative frequency-dependent selection), as well
strainswithoutsignificantchangeinthehostpopulationsize(see astheemergenceofcoalitionswhopossessnearlyidenticalphe-
Fig.6A,C).Weexpectmorehoststrainswithhigherq because notypeswithdistinctspacergenotypesaspredictedinHeldetal.
hoststrainsaredistinguishedbytheirspacerstates,whichevolve (2010).Thebalanceofviralandhostcoevolutionoccursdespite
morerapidly.Infact,onlyonespacerneedstobedifferenttobe thefactthattheCRISPRsystemundergoesdirectedmutation.We
consideredadifferentstrain.Athighervaluesofq,viralstrains,on findthatthehostsgenerallycannotacquiresomanyspacerssuch
theotherhand,haveahighnumberofstrainsandhighpopulation thattheviralpopulationgoesextinct.Indeed,viralmutantsthat
size (see Fig. 6B, D). This trend of increasing viral population can target dominant hosts are under positive selection because
size as the hosts can more easily acquire spacers may at first theirreplicationwillbegreaterondominanthosts.Hence,evolu-
seemcounterintuitivebecauseitimpliesthatviralpopulationsize tionarychangesinviralstraincompositiondrivesthechangein
2024 EVOLUTION JULY2012