Computational and Systems Biology Course 186— Modeling …Modeling of Biological Systems by...
Transcript of Computational and Systems Biology Course 186— Modeling …Modeling of Biological Systems by...
Instructor: Van Savage Winter 2017 Quarter
Monday and Wednesday, 2-3:50pm 3/15/2017
Computational and Systems Biology Course 186—
Modeling of Biological Systems by Connecting Biological
Knowledge and Intuition with Mathematics and Computing
Funproblem—PiDay—Buffon’sneedle
What is the probability that a needle of length, L, will cross a line is thrown randomly at a set lines spaced a length, d, apart?
P=2Lπd
Socialorganisms
Socialorganisms
Us
Cancercellsarelikecheatersandhealthycellsinmul?cellularorganisms
arelikecooperators
How do we understand these behaviors, and the complex interplay or altruistic/
generous behaviors and selfish behaviors?
PlatoandAristotle
ForAristotle,altruismshouldalwaysbeaccompaniedbyself-interestedmo?ves.Hissystemofprac?calthoughtcouldbedismissedoutofhandifonebeginswiththeassump?onthatmoralmo?va?onmustbepurelyaltruis?c,freefromalltaintofself-regard.Otherwise,itwouldnotcountasmoral.Thatideahassomecurrency,anditisoJenaKributed(rightlyorwrongly)toKant.Butonreflec?on,itisopentoques?on.Ifitisthecasethatwheneveronehasagoodreasontobenefitsomeoneelseforthatperson’ssake,thereisalsoasecondgoodreasonaswell—namely,thatindoingsoonewillalsobenefitoneself—itwouldbeimplausibletosupposethatoneshouldnotletthatsecondreasonhaveanyinfluenceonone’smo?va?on.RichardKraut,StanfordEncyclopediaofPhilosophy
General Framework— Game Theory as Evolutionary Theory
through Replicator Equations
represents payoff of player i to player j. Benefits are like increases or growth/births and costs like decreases/deaths in biology
Given rules of game, “players” can have different strategies and when different strategies interact, they have different benefits and costs. These interactions and payoffs are described through a payoff matrix that is very similar to our interaction matrix.
π i→ j
Payoffmatrixforinterac?onsamongindividualswithtwodifferent
strategiesAandB
Similartomatriceswe’veseenbefore!Almostallofbiologyorgameisencapsulateinherebycombiningwithevolu?on/growthequa?ons.
Π=π A→A π B→A
π A→B π B→B
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Selfinterac?ons Cross-strategyinterac?ons
Trajectoriesin?me
Needgrowthequa?onsandexpressthoseintermsofpopula?onsizesofindividualswithstrategyAandthosewithstrategyB.Iftotalpopula?onsizeisN=A+B,thenpA+pB=1wherepA=A/NandpB=B/N.
dpAdt
= pA(?)dpBdt
= pB(?)
Ourfrequencywillincreaseifwearegrowingfasterthanotherstrategyormoregenerallyfasterthanaverage
individualinpopula?onReplicatorEqua?ons
dpAdt
= pA(wA −w)dpBdt
= pB(wB −w)
Mean/averagefitnessacrosspopula?on
meanfitnessofindividualwithstrategyAorB
CalculatemeanfitnessesMeanfitnessofindividualwithstrategyAisprobabilityofinterac?ngwithanotherindividualWithstrategyAandge`ngthatassociatepayoffplustheprobabilityofinterac?ngwithindividualofstrategyBandge`ngthatpayoff.
wA = pAπ A→A + pBπ B→A
SimilarlyforindividualwithstrategyB
wB = pAπ A→B + pBπ B→B
Meanfitnessofen?repopula?on
w = pAwA + pBwb =
pA(pAπ A→A + pBπ B→A)+ pB(pAπ A→B + pBπ B→B )= pA
2π A→A + pB2π B→B + pApB(π B→A +π A→B )
Selfinterac?ons Cross-strategyinterac?ons
Moresimilartohowwereadoffinterac?ontermsandmatricesearlierinclass.
matrixformforreplicatorequa?ons
w =Πp
w =wTp
dpAdtdpAdt
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
= wA −w( ) wB −w( )( ) pApB
⎛
⎝⎜⎜
⎞
⎠⎟⎟
p=pApB
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Simplifyequa?onsRightnowequa?onforgrowthofpAappearstodependonpAandpBandwAand.Butisthereawaytosimplify?Anyconstraints?BeKertohavepAgrowthjustintermsofpAandpBgrowthjustintermsofpBandfitnessesjustintermsofdifferencesandra?os.Equa?onsforbuildingbiologicalra?onalearedifferentthanformathema?callysolving.
w
wA −w( ) = (1− pA)(wA −wB )≡ (1− pA)Δw
dpAdt
= pA(1− pA)ΔwpBissameequa?onwithnega?vesignbecauseincreaseForonemeansdecreaseforother.Cangetthisbysubs?tu?ngpA=1-pBorrealizingsignofflips.Reallyonlyoneequa?on!Δw
Biological model for cooperation, mutualism, kin altruism, and Hamilton’s
rule
Aisancooperator/altruistandBisacheaterorfreeloaderoffaltruis?csociety
b–benefitfrommutuallybeneficialinterac?onc–costofgivingbenefitPayoffmatrixis
Π= b− c −c
b 0⎛
⎝⎜⎞
⎠⎟
Δw =wA −wB = −c
Backtoreplicatorequa?ons
dpAdt
= −cpA(1− pA)
Thisisjustlogis?cequa?onagain!
pA(t)=
11+(ect /A0)
Whathappensassystemsevolves
?me
pA(t) pB(t)
Cheaterpopula?ongoestofixa?onandaltruistsgoex?nct,butaltruistshadtoestablishfirstbeforecheaterscouldinvade.Altruistscannotpersists.
Kin altruism and Hamilton
Aisancooperator/altruistandBisacheaterorfreeloaderoffaltruis?csociety
b–benefitfrommutuallybeneficialinterac?onc–costofgivingbenefitr—percentrelatednessofindividualyouinteractwithcanbenefitasdirectindividualorbyindirectbenefitandcosttoageneyoushare.
Π=
b+ r(bi − ci )− c −c + rbib− rci 0
⎛
⎝⎜⎜
⎞
⎠⎟⎟≡ b'− c ' −c '
b' 0⎛
⎝⎜⎞
⎠⎟
Δw =wA −wB = −c '= −c + rbi
Backtoreplicatorequa?ons
dpAdt
= −c 'pA(1− pA)
Thisisjustlogis?cequa?onagain!
pA(t)=
11+(e(rbi−c )t /A0)
c > rbi
Dynamicsdependonsignofargumentofexponen?al
Cheaterstakeoverwhendirectindividualcostisgreaterthanindirectbenefittorela?on
c > rbiAltruiststakeoverwhendirectindividualcostislessthanindirectbenefittorela?on
c = rbi Altruistsandcheaterscancoexist.
ThisisHamilton’sInequality.FamousresultfromBillHamiltoninthe60’s.
Defaultassump?onofgametheory
Pureselfishnessisassumedtobebasicstate.Makesithardtoexplainpurealtruism.Butwhenflippingitaround,italsohardtoexplainandraretofindpureselfishness.Couldfliparoundaltruismtobedefaultstatebychangingassump?onsandchangingfocus.
Relevantquote"Soonerorlaterinlifewealldiscoverthatperfecthappinessisunrealizable,butfewofuspausetoconsidertheopposite:thatso,too,isperfectunhappiness.Theobstaclespreven?ngtherealiza?onofbothoftheseextremestatesareofthesamenature:theyderivefromourhumancondi?on,whichishos?letoeverythinginfinite.Oureverinadequateknowledgeofthefutureopposesit,andthisiscalled,intheoneinstance,hopeand,intheother,uncertaintyabouttomorrow.Thecertaintyofdeathopposesit,fordeathplacesalimitoneveryjoy,butalsooneverysorrow.Ourinevitablematerialcaresopposeit,for,astheypoisoneverylas?nghappiness,theyjustasassiduouslydistractusfromourmisfortunes,makingourawarenessofthemintermiKentandhencebearable.”--PrimoLevi
Nashequilibriumandevolu?on
JohnNashdevelopedbasicideasofnon-coopera?vegametheory,andwonNobelPriceinEconomics.MademovieABeau?fulMindabouthim.Replicatorequa?onsshownheregivesameresultsthroughdynamicsofevolu?onandsurvivorshipandpersistencewithoutexplicitlyevokingselfishnessorconsciousnessofthat.
Goodreviewandfurtherexplana?onofaltruismandgametheoryin
lecturesbyJeffreySachs
hKp://www.lse.ac.uk/Events/2017/02/20170213t1830vOT/Economics-and-the-Cul?va?on-of-Virtue
Whatismissingfromthismodel?1. Abilitytocatchandpunishcheaters.Nodirectcosthereto
cheaters.Bigomission.2. Punishmentmightbe?medelayedfrommostbenefits.
Timedelaysand?mescalesandreallymaKer.3. Don’thaveabilitytolearn.Ifsomeonecheatsmeonce,I
won’tplaythemagain.“Foolmeonce,shameonyou.Foolmetwice,shameonme.”
4. Noabilitytochangeandmixstrategiesin?me.5. Morethantwotypesofstrategies/individuals.6. No?medelaybetweencostandbenefit.Again,?me
delayscanmaKer.7. No3-wayorhigher-orderinterac?ons.Justpairwise
effects.
Othertypesofbiological“games”
1. ChickenorHawk–Dovegame—s?lltwostrategygame.Hawkanddovehaveasharedresource.Dovessplitsresource.Hawksalwaysfightoverresourcesatcosttofightandfullbenefitiftheywin.
2. Rock-Paper-ScissorsorRochambeau—Non-transi?vegame.Aggression,coopera?on,anddecep?onamonglizardsforwinningmates.Whatdoyouthinkhappensfordynamics?
DynamicsofRochambeau
Flipsbetweenstatessoleadstooscillatorybehaviors.Howwouldwefindthismathema?cally?Itisnowa3equa?onmodel,soevenusingprobabilityconserva?onequa?on,wes?llhave2x2matrix.Needtofindeigenvaluesandusestabilityanalysislikewelearnedearlierintheclass.
Weendclosetowherewebegan,muchlikeanoscilla?onitself.
Talkandreview- Differen?alequa?onsandgrowth- Complexity-Stabilityandlinearalgebra(don’tknowif
interac?onmatrixisgenerallyknownoverstandardJacobian)
- IndirectInterac?onsandhigher-ordereffects- Modelsforbrain,biochemicalreac?ons,disease
transmission,predator-prey,- Stochas?csandapplica?ontogenes,cells,species,cancer- Selfsimilarity,fractals,scaling,andapplica?ontoallometric
biologicalrela?onships,cancer,andsleep- Networksinbiologyandhowtocluster,iden?fymo?fs,
understanddegreedistribu?onsandpowerlawsandpreferen?alaKachment.
- Evolu?onofcoopera?on,altruism,andkinselec?onthroughgametheory,replicatorequa?ons,andNashequilibria
Studyguide
Feedbackforfuture
Classevalua?onsthroughmyucla!