Goingbackstillfurther,eachofyourancestorsinturnhadtwoparents,indicated - PDF document

Goingbackstillfurther,eachofyourancestorsinturnhadtwoparents,indicatedbyprepend-inganextra\great"eachtime.Forexample,yourmaternallineageis: great-great-great-grandmother(level5ancestor) great-great-grandmother(level4ancestor) _ great-grandmother(level3ancestor) _ grandmother(level2ancestor) _ mother(level1ancestor) _ you _ andsoon(andsimilarlyfor\fathers"insteadof\mothers"atanylevel).Sinceeachancestorhastwoparents(onemotherandonefather),youhaveatotalof2nancestorsatleveln:twoparents,fourgrandparents,eightgreat-grandparents,sixteengreat-great-grandparents,etc.Summingup,youhaveatotalof2+22+23+:::+2n=2n+1�2ancestorsoflevelnorlower;forexample,yourtotalnumberofparentsandgrandparentsandgreat-grandparentscombinedis23+1�2=16�2=14.Inshort,yourancestorsformaperfectbinarytree{simplicityitself.DescendantLegacyIfyouhavechildrenyourself,thentheirchildrenareyourgrandchildren,andyourgrand-children'schildrenareyourgreat-grandchildren,andsoon:2 mother father you _ siblings sister _ � (Here,andthroughout,relationshipsto\you"arewrittenwithintheboxes,andrelationshipsbetweenotherpairsofindividualsareindicatedbyconnectinglines.)Ifyouandyoursiblingseachhavechildren,thenthosechildrenarerst-cousinsofeachother.Then,ifthetworst-cousinseachhavechildren,thenthosechildrenaresecond-cousinsofeachother;andtheirchildrenarethird-cousins,andsoon: mother father you _ siblings sister _ � daughter _ rstcousins sister'sdaughter _ granddaughter _ secondcousins sister'sgranddaughter _ great-granddaughter _ thirdcousins sister'sgreat-granddaughter _ great-great-granddaughter _ fourthcousins sister'sgreat-great-granddaughter _ (andsimilarlyfor\son"insteadof\daughter"atanylevel).Ingeneral,n-levelcousinssharetwo(n+1)-levelancestors(butnon-levelancestors).Thus,rst-cousinssharetwograndparents(butnoparents),andsecond-cousinssharetwogreat-grandparents(butnograndparents),andsoon.ItfollowsthatifAandBaren-levelcousins,thenA'schildandB'schildare(n+1)-levelcousins.Thus,childrenofrst-cousinsaresecond-cousins,andchildrenofsecond-cousinsarethird-cousins,andsoon.Infact,ifweregardsiblingsas0-levelcousins,thenthisreasoningappliestosiblingstoo:childrenof0-levelcousins(i.e.,siblings)arethemselvesrst-cousins.Finally,yoursibling'schildisyourniece(ornephew,ifmale),andtheirchildisyourgreat-niece(orgreat-nephew),andsoon:4 grandmother grandfather mother _ aunt _ � you _ rst-cousin _ rst-cousin-once-removed _ rst-cousin-twice-removed _ rst-cousin-three-times-removed _ Toseewhereyoursecond-cousinscomefrom,wehavetomoveonemorelevelup.Yourgrandparents'siblingsareyourgreat-auntsandgreat-uncles.Sotheirchildren,i.e.yourparents'cousins,areyourrst-cousins-once-removed.Andtheirchildrenareyoursecond-cousins: great-grandmother great-grandfather grandmother _ great-aunt _ � mother _ rst-cousin-once-removed _ you _ second-cousin _ Thesamepatterncontinuesupwardsforallearliergenerations.Onceagain,yournthcousinsshareyour(n+1)-levelancestors,butnotyournth-levelancestors.Siblingsofyournth-levelancestorsareyourgreat-...-greatauntsandgreat-...-greatuncles,where\great"isrepeatedn�1times.Furthermore,thenthcousinsofyourmth-levelancestors,andalsothemth-leveldescendantsofyournthcousins,areyournthcousinsmtimesremoved.6 ThickerThanWaterOneofthereasonswecareaboutfamilytreesisbecauseofasensethatcertainfamilyrelationsare\morerelated"tous,andshouldbeassistedandprotectedandlovedonthatbasis.Thisattitudepresumablyhasanevolutionarybasis:ourgenessurvivedthroughtheagesbecauseourancestorsmadeeortstohelpthemsurvivebycaringnotonlyforthemselves,butfortheircloserelativestoo.Indeed,thereisanancientBedouinArabsaying,\Iagainstmybrother,mybrothersandmeagainstmycousins,thenmycousinsandIagainststrangers",whichnicelyillustratesthephilosophyofcaringmostforthosewhoaregeneticallyclosesttous.Thisraisesthequestionofjusthowsimilarourrelatives'genesaretoourown.Well,rstofall,about99.9%ofourgeneticmaterialiscommontoallhumans(yes,evenyourin-laws),andindeediswhatmakesushuman.Furthermore,somepeoplemayshareothergeneswithusjustbychance;forexample,ifImeetastrangerwhoseeyesarebrownjustlikemineare,thatdoesnotnecessarilyestablishthatwearecloserelatives.Inaddition,thereislotsofrandomnessinhowgenesarepassedon(eachindividualgetshalfoftheirgeneticmaterialfromtheirmotherandhalffromtheirfather,butwhichbitscomefromwhichparentischosenatrandomandcannotbepredicted),sowecannotdrawpreciseconclusionswithcertainty.Todealwithallofthis,weassigntoeachpairofindividualsarelatednesscoecientwhichrepresentstheexpectedfraction(i.e.,thefractiononaverage)oftheirgeneswhichareforcedtobeidenticalbyvirtueoftheirfamilyrelationship.Thisapproachaveragesoutalloftherandomness,whilefocusingongeneticsimilaritiesspecicallyduetofamilyconnections.Accordingtothisdenition,strangershavearelatednessofzero(thesmallestpossiblevalue).Bycontrast,yourrelatednesswithyourselfisone(thelargestpossiblevalue).Otherrelatednesscoecientsfallbetweenthesetwoextremes.Forexample,yourrelatednesswithyourmotheris1/2,sinceyouobtainhalfofyourgeneticmaterialfromher.Andyourrelatednesswithyourfatherisalso1/2.Bythesamereasoning,yourrelatednesswithyourchildisagain1/2.Sofarsogood: mother father you you 1=2_ you 1=2_ daughter 1=2_ Nextconsideryourmaternalgrandmother.Shegavehalfofhergenestoyourmother,andthenyourmothergavehalfofhergenestoyou.Itispossiblethatthehalfyoutookisexactlythesameasthehalfyourgrandmothergave.Itisalsopossiblethatthehalfyoutookhasnooverlapatallwiththehalfyourgrandmothergave.Butonaverage,i.e.inexpectation,exactlyhalfofthegeneticmaterialyoutookfromyourmotheroriginatedfromyourmaternalgrandmother.So,yourrelatednesscoecientwithyourgrandmotherisone-halfofone-half,i.e.(1=2)(1=2),or1/4:8 mother you 1=2 1/4* half-sister 1=2 � Regular(full)siblingssimilarlyshare1/4oftheirgeneticmaterialthroughtheirmother,butalsoshare1/4oftheirgeneticmaterialthroughtheirfather.Thisgivesatotalrelatednesscoecientof1=4+1=4=1=2: mother father you _ 1/4+1/4=1/2* sister _ � (Onespecialcaseisidenticaltwins,whohaveidenticalgenesandthusarelatednesscoecientofone.Butfraternaltwinshaverelatednesscoecient1/2,justlikeothersiblings.)Continuingonward,sinceyourmotherandauntaresiblings,theyhaverelatednesscoef-cient1/2.Meanwhile,youandyourmotherhaverelatednesscoecient1/2.Puttingthistogether,youandyouraunt(oruncle)haverelatednesscoecient(1=2)(1=2)=1=4: mother 1=2 aunt you 1=2_ 1/4(andsimilarlywith\aunt"replacedby\uncle").And,yourrelatednesscoecientwithyournieceornephewisalso1/4.Takingittothenextlevel,yourgrandmotherandyourgreat-auntarealsosiblingsandhencealsohaverelatednesscoecient1/2.Sinceyouhaverelatednesscoecient1/4withyourgrandmother,itfollowsthatyouhaverelatednesscoecient1/8withyourgreat-aunt:10 So,yourrelatednesscoecientwithyourrstcousinis1=8;withyoursecondcousinis1=32;withyourthirdcousinis1=128;andsoon.Whataboutrst-cousins-once-removed,andallofthat?Well,sinceyouandyourrst-cousinhaverelatedness1/8,andsinceyourrst-cousinandtheirchild(yourrst-cousin-once-removed)haverelatedness1/2,itfollowsthatyouandyourrst-cousin-once-removedhaverelatednesscoecient(1=8)(1=2)=1=16: you 1=8 rst-cousin 1/16- rst-cousin-once-removed 1=2_ Thepatterncontinues,witheachnew\removed"introducinganextrafactorof1/2intotheproduct.Itfollowsthatyourrelatednesscoecientwithyournthcousin,mtimesremoved,isequalto1=22n+m+1.Forexample,yourrelatednesscoecientwithyourthirdcousin(n=3)twiceremoved(m=2)isequalto1=26+2+1=1=29=1=512{notverycloseatall.Wecansummarisetherelatednesscoecientsofvariousrelationshipsinatable: Relationshiptoyou relatednesscoecient yourself 1 identicaltwin 1 parent,child 1/2 grandparent,grandchild 1/4 great-grandparent,great-grandchild 1/8 nthlevelancestorordescendant 1=2n sibling(sisterorbrother) 1/2 half-sibling 1/4 aunt,uncle 1/4 niece,nephew 1/4 great-aunt,great-uncle 1/8 great-niece,great-nephew 1/8 rst-cousin 1/8 rst-cousin-once-removed 1/16 second-cousin 1/32 second-cousin-once-removed 1/64 third-cousin 1/128 nthcousin 1=22n+1 nthcousin,mtimesremoved 1=22n+m+1 stranger 0 12 relatednesscoecientwithyourstep-relationsisalsozero,sinceagainyourrelationshipisthroughmarriagenotblood.Furthervariationsarisebyconsideringotherspecieswithothermechanismsforgeneticselection.Forexample,ifaspeciespropagatesbycloning,i.e.makingexactgeneticcopies,thenallrelationshaveidenticalgeneticmaterial,andhencearelatednesscoecientofone.Moresubtly,manyspeciesofbeesandantsarehaplodiploid.Thismeansthattheirmaleshavehalfasmuchgeneticmaterialastheirfemales.Infact,maleshavenofatheratall,butrathergetalltheirgenesfromtheirmother.Bycontrast,femalesgethalfoftheirgenesfromtheirmother(bytakinghalfofhergenes,chosenatrandom,justlikehumans)andhalfoftheirgenesfromtheirfather(bytakingallofhisgenes,withnorandomnessatall).Insuchasystem,therelatednesscoecientisnolongersymmetric.Mothersanddaughtersstillhavearelatednesscoecientof1/2,andmothersstillhavearelatednesscoecientof1/2withtheirsons.However,sonshavearelatednesscoecientof1withtheirmothers(whocontainalloftheirgeneticmaterial).Mostinterestingly,fullsistershavearelatednesscoecientof3/4,sincetheysharealloftheirfather'sgenesinadditiontohalfoftheirmother's.So,since3=4�1=2,femalebeesandantshaveageneticimperativetoassisttheirsistersevenaheadoftheirownchildren!Familyrelationscanleadtounexpectedsurprises.Atarecentlargefamilyreunion,ImetayoungmanwhomIdidnotknow.Aftersomediscussion,wedeterminedthatmygreat-grandfatherwasthebrotherofhisgreat-grandmother{makingusthirdcousins.Furthermore,mygreat-grandmotherwasthesisterofhisgreat-grandfather,too.Thatis,threegenerationsearlier,abrother-and-sisterpairhadmarriedowithasister-and-brotherpair.ThismeantthatheandIwerethird-cousinsbyeachoftwodierentpaths{wewere\doublethird-cousins"!Itfollowedthatourrelatednesscoecientwastwicethatofusualthird-cousins,i.e.equalto2(1=128)=1=64{stillnotveryclose,butinterestingnonetheless.IwishIhadhadthepresenceofmindtoimmediatelysaytohim,\Pleasedtomeetyou,double-third-cousin.Iamhonouredtoshareonesixty-fourthofyourgenes." JereyS.RosenthalisaprofessorintheDepartmentofStatisticsattheUni-versityofToronto,andtheauthorofthepopularbook\StruckbyLightning:TheCuriousWorldofProbabilities"(Granta,2008;www.probability.ca/sbl). 14