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

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