TEMA (São Carlos) vol.13 número3

Uma Publiação daSoiedade BrasileiradeMatemátiaApliadaeComputaional.

Reombination and Geneti Diversity

1

T.C.COUTINHO 2

,Campus AltoParaopeba,UFSJ -Univ. Fed. deSãoJoão Del

Rei,36420-000OuroBrano,MG,Brazil.

T.T. DA SILVA 3

, Departamento de Físia e Matemátia, UFSJ - Univ. Fed.

deSãoJoãoDel Rei,36420-000OuroBrano,MG,Brasil.

G.L. TOLEDO 4

, Departamento deTenologiaem EngenhariaCivil,Computaçãoe

Humanidades, UFSJ -Univ. Fed. de São João Del Rei, 36420-000Ouro Brano,

MG,Brazil.

Abstrat. Inthispaperwepresentaspatialstohastimodelforgeneti

reombi-nation,thatanswersifdiversityispreservedinaninnitepopulationof

reombinat-ingindividualsdistributedspatially. Weshowthat,fornitetimes,reombination

may maintain all the various potential dierent types, but whentime grows

in-nitely,thediversityofindividualsextinguisheso. Sounderthemodelpremisses,

reombination and spatial loalization alone are not enough to explain diversity

inapopulation. Furtherwe disussanappliationofthe modelto aontroversy

regardingthediversityofMajor HistoompatibilityComplex (MHC).

Keywords. Genetireombination,spatialstohastimodel,Major

Histoompat-ibilityComplex(MHC).

1. Introdution

Mendelianlawsofinheritane,whenapplied toinnitepopulationsunderrandom

mating, lead to Hardy-Weinberg laws, whih state that gene and genotype

pro-portions donothange after therst generation [2℄. Whenonsidered overnite

populations without mutation, random geneti drift leads the population to

ho-mozigosity,evenin thepreseneofreombination. Ouraimis,then,to investigate

howtheproportionofdierentgenotypesvariesinaninnitepopulationthatis

dis-tributedspatially,tryingtoverifytheroleofreombinationinthissetting, mainly

itsimpliationforpopulationdiversity.

Inorderto buildthe model,weonsider somehypotheseswhih weexpliit in

thesequel:

1

ThisworkwaspartiallysupportedbytheBrazilianNationalResearhCounil-CNPq,under

Grants 314729/2009-7 and 503851/2009-4 and byFAPEMIG, under Grant 253-10 and

04719-10(PRONEM).PartofthepaperwaspresentedatCMAC2011-Sudeste[5℄.

2

thamaraoutinhogmail.om

3

timoteoufsj.edu.br

4

i)Thepopulationonsistsofhaploidindividuals;

ii)Thereareaninnitenumberofindividuals,eahoupyingaposition in

Z

;

iii) A newborn individual isalwaysformed bythe ontributionof genes from two

distintindividuals;

iv) We donot onsider anybiohemial ormetaboliinuene onthe geneti

in-heritane,i.e., mutationsdonotour,noranykindoferrorduringtheproessof

genetiinheritane;besidestherearenoseletiveforesatingoverthepopulation.

The model realls the Voter Model, a stohasti model originally developed

to study the interation of twodistint populationsompeting for aterritory[3℄.

Stohasti modelstreatnaturallyrandom utuations that usually happenin the

environment. In population genetis, e.g., it is natural to assume that allele

fre-quenyvariationisinuenedbyprobabilistifators. Then,throughtheknowledge

ofthe populationstatein ageneration,andgiven areprodutionshemefor

indi-viduals, we an determine the probability of reapearane of asample of genes in

thenextgeneration [6℄.

Themodelling proedure anbedesribedbriey asfollows. We disposeeah

individual indierentpositions foreah timestep. An individual's genesonestep

aheadare inheritedfromthereombinationofitsneighbours'genesin theurrent

stepwithequalprobability. This originatesastohastiproess thatwill be

anal-ysedbyadualproess. Thebuildingofthisdualproessallowsusto lookbakon

the evolution of the population and retrieveinformation about whih individuals

at time step

0

donated thegenes that onstitute someindividual at time step

n

.

That is, the dual proess retrievesthe genealogy of genes in the population. We

willproposethemodellingfor

2−

and

3−

loiindividuals,notingthatthelastgives

opportunityformorereombinationto our.

Inthe next setion we propose the models and obtain someonlusions from

them. InSetion3.wedisussanappliationtoaontroversyregardingthe

diver-sity of Major Histoompatibility Complex (MHC). MHCmoleules playa key

roleinmanyimmunefuntions,onsequently,thesemoleulesarisespeialmedial

interest,sinetheyarediretlyrelatedtoorganandtissuerejetion,topathogeni

suseptibility,aswellastoindividualvariabilityregardingthesuseptibilityto

dis-ordersofself-immuneaetiology.

2. Mathematial modelling

2.1. The

2

-loi model

Consideraninnitepopulationonsistingofhaploidindividuals,forwhihwe

anal-ysetwodistint loi

A

and

B

. Eah individual is at apoint of

Z

and eah lous

admits onlytwoalleles. Inthe rstgeneration,the individualsat oddpointswill

reprodue, their newgenes will be areombination of their neighbours'genes, in

suhawaythat,iftheindividualatposition

i

−

1

is,e.g.,

A1B1

andtheindividual

at position

i

+ 1

is

A2B2

, then theindividual at position

i

in therst generation

SeetheDiagram

1

foranexampleofhowthemodelevolves.

a

₂

b

2

a

1

b

2

a

₁

b

2

a

2

b

2

n

= 5

ր

տ

ր

տ

ր

a

1

b

1

a

1

b

₂

a

2

b

2

a

2

b

₂

n

= 4

տ

ր

տ

ր

տ

a

1

b

1

a

1

b

2

a

2

b

2

a

2

b

1

n

= 3

ր

տ

ր

տ

ր

a

2

b

2

a

₁

b

2

a

1

b

1

a

₂

b

1

n

= 2

տ

ր

տ

ր

տ

a

₂

b

₂

a

2

b

1

a

₁

b

₁

a

1

b

2

n

= 1

ր

տ

ր

տ

ր

a

1

b

1

a

2

b

1

a

2

b

2

a

1

b

2

n

= 0

· · ·

−1

0

1

2

· · ·

Z

Diagram

1

: Votermodelmodied,adaptedtogenetis.

Letusdevelopmathematialythemodel: Let

{

V

(

i, n

)

, i

∈

Z, n

∈

IN}

be aset

ofrandomvariablesuniformlydistributedontheinterval

[0

,

1]

. Denetheintervals

I1

= [0

,

1

/

2[

and

I2

= [1

/

2

,

1]

. For eah

n

∈

IN

and

i

∈

Z

onsider the random

vetor

X

(

i, n

) = [

x1

(

i, n

)

x2

(

i, n

)]

, whih for

k

= 1

,

2

,

x

k

(

i, n

)

has either the

value

0

or

1

(only two distint alleles per loum). So,

X

(

n

) :

Z

→ {0

,

1}

2

. We

dene, then,thedynamisof themodelin thefollowingway: if

i

+

n

isodd,then

X

(

i, n

) =

X

(

i, n

−1)

;if

i

+

n

iseven,then

X

(

i, n

) =

2

X

α

=1

[

x1

(

i

+ (−1)

α

_{, n}

₋₁₎

_x2

₍

_i

₋

₍₋₁₎

α

_{, n}

_{−1)] 1I}

[

V

(

i,n

)

∈

I

α

]

.

The initial distributionis given by

P

(

X

(

i,

0) = [

a b

]) =

π

ab

,

for all

i

∈

Z

, with

P

a,b

=0

,

1

π

ab

= 1

and

π

ab

>

0

,

for

a, b

∈ {0

,

1}

. Thefuntion

1I

A

istheharateristi funtion oftheset

A

. Theinitialdistribution of

a

's intherst oordinateand

b

's

intheseondare, respetively,

P

(

x1

(·

,

0) =

a

) =

1

X

j

=0

π

aj

,

and

P

(

x2

(·

,

0) =

b

) =

1

X

i

=0

π

ib

,

with

a, b

= 0

,

1

.

2.1.1. Dual proess and genealogy

Consider,foreah

(

i, n

)

∈

Z

×

IN

,theproess

Y

i,n

₍

_k

_{) = (}

_y

i,n

1

(

k

)

, y

i,n

2

(

k

))

,suhthat

Y

i,n

₍

_k

_{) :}

_Z

_→

_Z

2

isgivenby

Y

i,n

_{(0) = (}

_{i, i}

₎

;

Y

i,n

_{(1) = (}

_{i, i}

₎

,if

i

+

n

isodd;and,

if

k

= 1

and

i

+

n

iseven,orif

k >

1

Y

i,n

₍

_k

_{) =}

_Y

i,n

₍

_k

₋

_{1) +}

X

α,β

=1

,

2

(−1)

α

₍

_δ

where

δ

βk

isequalto

1

if

β

=

k

,andequaltozerootherwise. Thisproessrepresents

thegenealogyfortheindividualatposition

i

,in generation

n

.

The following Diagram

2

represents apossible genealogy for an individual at

generation

5

.

a

1

b

2

n

= 5

ւ

ց

a

1

b

2

n

= 4

ւ

ւ

a

1

b

2

n

= 3

ց

ւ

a

1

b

2

n

= 2

ւ

ց

b

2

a

1

n

= 1

ւ

ց

a

1

n

= 0

· · ·

−1

0

1

2

· · ·

Z

Diagram

2

: Genealogyfortheindividualatposition

1

,generation

5

.

Byonstrutionoftheproess

Y

i,n

wehavethefollowing

Lemma 2.1 (Duality relation,gene phylogeny). The following duality identity is

valid:

X

(

i, n

) =

h

x1

(

y

i,n

1

(

n

)

,

0)

x2

(

y

2

i,n

(

n

)

,

0)

i

.

(2.1)

Thegenotypeoftheindividual atposition

i

,generation

n

, onsistsofthegene

attherstlousoftheindividualattherandomposition

y

i,n

1

(

n

)

andofthegeneat theseond lousoftheindividual attherandom position

y

i,n

2

(

n

)

, bothpertaining totheinitialgeneration.

Theorem 2.1. The proportion of genotypes, from the rst generation on, keeps

onstant.

TheproofanbefoundinAppendix A.

2.1.2. Diversityloss

Considerthefollowingequality,whosevalidityisshowninAppendixB:

P

(

X

(

i, n

)

6=

X

(

j, n

)) =





1

−

X

a,b

=0

,

1

π

2

ab





P Y

i,n

(

n

)

6=

Y

j,n

(

n

)

.

(2.2)

Thisequalitytranslatesmathematialyananestralityrelationbetweentwo

individ-ualshosenatrandomfromgeneration

n

. Thatis,weaninferthattheprobability

of two individuals having distint genotypes is assoiated with the probability of

theirgeneshavingome fromdistintanestorsintheinitialgeneration. Applying

Theorem 2.2 (Geneti diversity loss). The probability of

X

(

i, n

)

being dierent

from

X

(

j, n

)

goes tozerofor large

n

. Itfollows that the geneti diversity does not

keep itselfon thepopulation.

Proof. Wemay onsider

y

i,n

1

a symmetrirandom walk in

Z

, without any lossof generality. Ontheotherhand,

y

i,n

2

walksin

Z

independentlyof

y

i,n

1

, exeptwhen

y

1

i,n

=

y

2

i,n

,beausewhen theymeeteahother, if

y

i,n

1

(

k

+ 1) =

y

1

i,n

(

k

) + 1

, then we must have

y

i,n

2

(

k

+ 1) =

y

i,n

2

(

k

)

−

1

, but if

y

i,n

1

(

k

+ 1) =

y

i,n

1

(

k

)

−

1

, then

y

₂

i,n

(

k

+ 1) =

y

₂

i,n

(

k

) + 1

. Thebehaviourof

y

j,n

1

and

y

j,n

2

isanalogous. So,withprobabilityone,

y

i,n

1

willouplewith

y

j,n

1

when

n

growstoinnity,sine theyareunidimensionalsymmetrireurrentrandomwalks[8℄. Intheasethat

y

i,n

2

isalreadyequalto

y

j,n

2

,thenthereisnothingelsetoprove. Intheasethat

y

i,n

2

is dierentfrom

y

j,n

2

,weanhangethepointofviewandonsiderthat

y

i,n

2

and

y

j,n

2

areindependentproessesfrom

y1

=

y

i,n

1

=

y

j,n

1

. Thus

y

i,n

2

and

y

j,n

2

,unidimensional symmetrirandomwalks,willouplewith eahotherwithprobabilityonewhen

n

inreases.

2.2. The

3

-loi model

The model for three loi onstitutes an extension of the model for two loi. For

3

distint loi

A, B

and

C

, wewill have thefollowingreombination possibilities.

If the individual at position

i

−

1

is, for example,

A1B1C1

and the individual at

position

i

+ 1

is

A2B2C2

,thentheindividualatposition

i

willbeeither

A1B2C2

or

A1B1C2

or

A1B2C1

or

A2B1C2

or

A2B1C1

or

A2B2C1

withequalprobabilities.

Weonsider

{

V

(

i, n

)}

and

{

U

(

i, n

)}

twosetsof

[0

,

1]

-uniformlydistributed

ran-domvariables. Wedene

J

β

=

h

β

−

1

3

,

β

3

h

,

β

= 1

,

2

,

and

J3

=

2

3

,

1

. Foreah

n

∈

IN

and

i

∈

Z

let

X

(

i, n

) = [

x1

(

i, n

)

x2

(

i, n

)

x3

(

i, n

)]

be the random vetor where,

for

k

= 1

,

2

,

3

,

x

k

(

i, n

)

takesthevalues

0

or

1

. That is,

X

(

n

) :

Z

→ {0

,

1}

3

_.

The

initialdistributionisgivenby

P

(

X

(

i,

0) = [

a b c

]) =

π

abc

.

Themodeldynamisis:

if

i

+

n

isodd,then

X

(

i, n

) =

X

(

i, n

−

1)

;if

i

+

n

iseven,then

X

(

i, n

) =

2

X

α

=1

3

X

β

=1





x1

i

−

(−1)

α

+

δ

β

1

, n

−1

x2

i

−

(−1)

α

+

δ

β

2

, n

−1

x3

i

−

(−1)

α

+

δ

β

3

, n

−1





T

1I

[

V

(

i,n

)

∈

I

α

]

1I

[

U

(

i,n

)

∈

J

β

]

.

2.2.1. Dual proess and genealogy

Wealso build thedual proess forthis model. Consider,for eah

(

i, n

)

∈

Z

×

IN

,

theproess

Y

i,n

₍

_k

_{) = (}

_y

i,n

1

(

k

)

, y

i,n

2

(

k

)

, y

i,n

3

(

k

))

,suhthat

Y

i,n

₍

_k

_{) :}

_Z

_→

_Z

3

isgiven

by

Y

i,n

_{(0) = (}

_{i, i, i}

₎

;

Y

i,n

_{(1) = (}

_{i, i, i}

₎

,if

i

+

n

isodd;andif

k

= 1

and

i

+

n

iseven,

orif

k >

1

,then

Y

i,n

₍

_k

₎

₌

_Y

i,n

₍

_k

₋

₁₎₊

P

2

α

=1

P

3

β

=1

P

3

γ

=1

(−1)

α

+1

(−1)

δ

β

1

δ

γ

1

,

(−1)

δ

β

2

δ

γ

2

,

(−1)

δ

β

3

δ

γ

3

×

_1I[

_V

₍

_Y

i,n

Lemma2.2 (Dualityrelation). The following dualityidentity istrue:

X

(

i, n

) =

h

x1

(

y

1

i,n

(

n

)

,

0)

x2

(

y

i,n

2

(

n

)

,

0)

x3

(

y

i,n

3

(

n

)

,

0)

i

(2.3)

Theorem2.3. Theproportionofgenotypeskeepsonstantfromtherstgeneration

on.

Wewill skip theproof sineit is, mutatismutandis, analogous to theproof of

Theorem

2

.

1

.

2.2.2. Loss ofdiversity

Thefollowingidentityisvalid:

P

(

X

(

i, n

)

6=

X

(

j, n

)) =





1

−

X

a,b,c

=0

,

1

π

abc

2





P Y

i,n

(

n

)

6=

Y

j,n

(

n

)

.

(2.4)

Thedemonstrationisanalogoustothedemonstrationofequality

(2

.

2)

.

Theorem 2.4. The probability of being

X

n

(

i

)

dierent from

X

n

(

j

)

goes to zero

when

n

inreases. Thereforeitfollows thatthegeneti diversitydoesnotkeep itselt

inthe population.

Proof. Wemayonsider,withoutlossofgenerality,that

y

i,n

1

and

y

i,n

2

aresymmetri randomwalksin

Z

,independentofeahother. Ontheotherhand,

y

i,n

3

movesin

Z

independentlyfrom

(

y

i,n

1

, y

i,n

2

)

, exeptwhen

y

i,n

1

=

y

i,n

2

=

y

i,n

3

, beausewhenthey meet,wehavethefollowingpossibleimpliations:

•

if

y

i,n

1

(

k

+ 1) =

y

i,n

1

(

k

) + 1

and

y

i,n

2

(

k

+ 1) =

y

i,n

2

(

k

) + 1

,then

y

i,n

3

(

k

+ 1) =

y

i,n

3

(

k

)

−

1

,

•

if

y

i,n

1

(

k

+ 1) =

y

i,n

1

(

k

)

−

1

and

y

i,n

2

(

k

+ 1) =

y2

(

k

)

−

1

, then

y

i,n

3

(

k

+ 1) =

y

i,n

₃

(

k

) + 1

,

•

otherwise,themovementof

y

i,n

3

totheleftortotherighthappenswithequal probabilities.

Weananalyseanalogouslythemovementof

y

j,n

1

, y

j,n

2

and

y

j,n

3

. So, theproess

y

i,n

1

will almost surely ouplewith

y

j,n

1

when

n

inreases, and in the sameway,

y

i,n

2

will a.s. ouple with

y

j,n

2

when

n

inreases, sinethey are unidimensionalreurrentsymmetrirandomwalks[8℄.

Intheasethat

y

i,n

3

isalreadyequalto

y

j,n

3

,thentheproofends. In the ase that

y

i,n

3

is dierent from

y

j,n

3

, we an hange the point of view and take, for example,

y

i,n

3

,

y

j,n

3

and

y1

as independent from eah other, where

y1

=

y

₁

i,n

=

y

₁

j,n

; besidesthemovementof

y2

=

y

i,n

2

=

y

j,n

2

will depend onthat of

y

3

i,n

and

y

j,n

3

,if

y

i,n

3

=

y

j,n

3

. Weonlude, therefore,that when

n

goesto innity, theprobabilityof

y

i,n

3

and

y

j,n

2.3. Disussion

Firstly, in eah model, weestablishaduality relationbetweenthe stohasti

pro-esses

X

and

Y

. It follows, bythestohasti proess ouplingtehnique,that, in

bothmodels,theprobabilityoftwoindividualsbeinggenetiallydistint,

P

(

X

(

i, n

)

6=

X

(

j, n

))

,goesto zerowhen

n

goestoinnity. Thatis,thegenetidiversity

disap-pearsfromthepopulationastimegoesby.

2.3.1. A omparisonbetween the models

Whenweaugmentthenumberofloifromtwotothree,thediversityismaintained

longerwhenreombinationispresent. ToilustratethisbehaviourseeFigure

1

that

showsthesimulatedmeantimefor

Y

0

,n

and

Y

j,n

toouple,forvariousvaluesof

j

(

j

= 2

,

12

,

22

, . . . ,

102

). Themore

j

isdistantfrom

0

,ittakeslonger,in mean,for

Y

0

,n

and

Y

j,n

toassumethesamevaluein

Z

2

or

Z

3

. Buttheoalesenetimesin

Z

3

arelongerthanin

Z

2

.

Figure1: Variation oftheoalesenemeantimewith theinitialdistane

between individuals.

3. Appliation

3.1. Reombination and diversity of the MHC

responsi-generateimmuneresponses[10℄. Thisistheaseaboutrejetiontograftingandto

transplantationsperformedbetweentwopeopleimmunologiallyinompatible.

Therole playedby theimmune systemis to exhibit antigens against

miroor-ganismsthat invadethebodyto thelymphoytes thateliminatethese pathogens.

Speialized proteins, the Human Leukoyte Antigens (HLA), exeute this

fun-tion; they are odiedby a highly polygeni, polymorphi system, alled Major

HistoompatibilityComplex (MHC).

Thetermmajorhistoompatibilityomplex derivedfromresearhesinwhih

tissuesweretransplantedbetweenmembersofthesamespeies. Rejetionouring

inmanytransplantationswasthoughtofbeingdeterminedbyonegenesolely,that

was alled the major histoompatibility gene. Later, it was disovered that this

gene wasin fat aomplex, anensemble of genes inherited asone that sinehas

beenknown asthe major histoompatibility omplex (MHC). Today, it is known

thateahspeieshasanMHContainingmultiple genes.

MHCgenesappearin allvertebrates,in humans they aredesignated Human

Leukoyte Antigens(HLA),sinetheywereinitiallydetetedinleukoytes. The

humanMHC is odied mainly by aregion of the 6thhromosome that ontains

morethan

200

genes [9℄. Atleast sixpolymorphigeni loi,separatedand

orga-nized in lusters, were dened in auniquearea ofthe 6thhromosome[4℄. They

are themostpolymorphiof thehumangenome, havinghundredsof stable forms

(alleles)foreahgenein thepopulationalreadydesribed. Forexample,ageneof

the humanMHC that is polymorphiis HLA-B. Nowadaysit has morethan

150

alleles desribed. Nevertheless,this polymorphismisnotvalidfor allMHCgenes,

some of them havelittle polymorphism orare monomorphi. Approximately

224

genetiloiwereidentiedenglobing

3

.

5

megabases(Mgb)ofDNAinMHCregions. Possibly

180

genes areexpressed and around

40%

of them have somefuntion in

theimmunesystem. Thisregionwasoneoftherstmultimegabaseofthehuman

genomewhihwasompletely sequened[9℄.

MHCpolymorphism is aonsequene of vertebrates'evolutionaryresponse

againstinvasionbymiroorganisms;thusitreassurestheontinuityofthespeies,

evenin thepreseneof pandemis. Someindividualsmaysurviveapandemi due

to the protetive eet of MHC geneti polymorphism. The polymorphisms at

the binding region with the antigen determine the speiity of peptide binding.

ThereforetheMHCmoleulebindsonly withsomefewpeptidesamongthemany

atdisposalaroundtheellularmiro-environment[9℄. Beauseofpolymorphismit

isimprobabletheexisteneoftwoindividualsthatexpressidentialMHCmoleules.

Thishugediversityisthemainobstalefororganandtissuetransplantationsuess.

MHCmoleuleshaveanotheressentialharateristi: theyarepolygeni. Being

polygenimeansthatthesemoleulesareodiedbymultiple independentgenes.

Theyareinheritedinlustersalledhaplotypesandexpressedo-dominantlyin

eahindividual[10℄.

3.1.1. Controversy

hy-lationsisstill disputed,sinefewomparativeresearheshaveomputedestimates

ofthisomplexreombinationrates[11℄. 5

Thespatialstohastimodelfor

reombi-nationpresentedaboveshowsthatreombinationisabletomaintainMHCdiversity

inapopulation through longtimeperiods,but whentimegoestoinnity,diversity

goestozeroalmostsurely.

The model puts in relevane the importane of polymorphism, polygeny and

reombinationto thediversityof MHCmoleules. Other issuessuh astheMHC

odominantpattern ortheexistene ofmorethan

3

alleles foronelous for most

ofMHCgenesarenotonsidered. 6

Ontheother hand,thevariabilityof MHCsystemevokesaseriesof questions

ofsientiinterestonitsown,relatedtoMHCunommonpolymorphism,natural

evolution, biologialfuntion of itsdiversegenesand theirationsontheimmune

system. DuetoMHCgenetipolymorphismitisimprobabletondtwoindividuals

that express idential MHC moleules. This suh great diversity is the main

ob-staleto suessful organandtissue transplantations[10℄. Nonetheless, aording

to the onlusionsof the mathematialmodel developedabove, thisdiversity will

extinguishointhelongrun. Therefore,theobserveddiversityofMHCmoleules

isnotlikelytodependontheirhighpolymorphism,highpolygeny,oronthegreat

numberofloiinvolvedinreombination;ifitisnotatransienteet,thisdiversity

maybeduetootherfatorssuhasmutation.

3.2. Other pratial issues

Reombinationisreognizedasanimportantfatorpotentiallyleadingtoevolution

advantageinpopulations[2℄,duetoitsroleonthemaintenaneofpopulation

diver-sity. But reombination solely, in spatially distributed innite populations, isnot

abletomaintaindiversityforlongertimes, in theontextproposed bythemodels

desribedinthispaper,foranitenumberofloi. However,furtherresearhshould

bedevelopedinordertoputinrelevaneotherharateristisnotonsideredsofar,

forexample,reprodutionofdiploidindividuals,seletivepressure,dominane

rela-tionbetweengenes,ornumberofallelesperloum. Itislikelythat,e.g.,inreasing

thenumberofpossibleallelesforeahlous,diversitywilltakelongertodisappear

fromthepopulation.

Anotherimportant aspet is the rate of reombination whih may not bethe

sameoronstant throughthe population. This isarelevantissue, e.g.,for

phylo-genetitreeestimation. IfhighratesofreombinationareommoninMHCgenes,

re-evaluation of manyinferene-basedphylogeneti analyses ofMHC loi,suh as

estimatesofthedivergenetimeofallelesandtrans-speipolymorphism,maybe

required[11℄.

5

Intheabseneofreombination,thegenesofHLAomplexareinheritedasanisolatedunity

of the

6

th hromosome,thehaplotype;the probabilityoftwobrothersbeingHLA-identialis

25%

,aordingtoMendellaws: thehildinheritsahaplotypefromthefatherandanotherfrom

themother.

6

4. Conlusion

Weproposedamathematialmodelapabletoverifytheinterfereneof

reombina-tioninthediversityofaspatiallydistributedinnitepopulation. Fromthemodel,

weonludethat,astimeinreases,theprobabilityof takingtwodistint

individ-uals with thesame genetiload, goes to one. Besides, the greaterthe numberof

reombiningloionsidered,thelongerthepopulationdiversityismaintained.

WhenthemodelwasappliedtothereombinationofMHCmoleules,wefound

thatreombinationwasnotasuientausetothemaintenaneofMHCdiversity.

Appendix

A Proof of Theorem 2.1

Bythedualityrelation

(2

.

1)

, for

n

≥

1

,wehave

P

(

X

(

i, n

) = [

a b

]) =

P

[

x1

(

y

₁

i,n

(

n

)

,

0)

x2

(

y

₂

i,n

(

n

)

,

0)] = [

a b

]

=

P

r,s

∈

Z

P

([

x1

(

r,

0)

x2

(

s,

0)] = [

a b

])

P

(

Y

i,n

(

n

) = (

r, s

))

=

P

r,s

∈

Z

P

1

j

=0

π

aj

P

1

_i

=0

π

ib

P

(

Y

i,n

₍

_n

_{) = (}

_{r, s}

₎₎

=

P

1

j

=0

π

aj

P

1

_i

=0

π

ib

P

r,s

∈

Z

P

(

Y

i,n

(

n

) = (

r, s

))

=

P

1

j

=0

π

aj

P

1

i

=0

π

ib

B Proof of Equality

(2

.

2)

P

(

X

(

i, n

)

6=

X

(

j, n

))

=

P

h

x1

(

y

i,n

1

(

n

)

,

0)

x2

(

y

i,n

2

(

n

)

,

0)

i

6=

h

x1

(

y

1

j,n

(

n

)

,

0)

x2

(

y

j,n

2

(

n

)

,

0)

i

=

X

r

i

6

=

r

j

or

s

i

6

=

s

j

P

([

x1

(

r

i

,

0)

x2

(

s

i

,

0)]

6= [

x1

(

r

j

,

0)

x2

(

s

j

,

0)])

×

P Y

i,n

₍

_n

_{) = (}

_r

i

, s

i

)

, Y

j,n

(

n

) = (

r

j

, s

j

)

=

X

r

i

6

=

r

j

or

s

i

6

=

s

j

[1−

P

([

x1

(

r

i

,

0)

x2

(

s

i

,

0)] = [

x1

(

r

j

,

0)

x2

(

s

j

,

0)])]

×

P Y

i,n

₍

_n

_{) = (}

_r

i

,s

i

)

, Y

j,n

(

n

) = (

r

j

,s

j

)

=

1

−

P

1

a,b

=0

π

2

ab

X

r

i

6

=

r

j

or

s

i

6

=

s

j

P Y

i,n

₍

_n

_{) = (}

_r

i

, s

i

)

, Y

j,n

(

n

) = (

r

j

, s

j

)

=

1

−

P

a,b

=0

,

1

π

ab

2

P Y

i,n

₍

_n

₎

₆₌

_Y

j,n

₍

_n

₎

Referenes

Else-[2℄ R.Bürger, Themathematialtheory ofseletion, reombination, and

muta-tion,JohnWiley&Sons,Chihester,2000.

[3℄ P. Cliord, A. Sudbury, A model for spatial onit, Biometrika Trust, 60

(1973),581588.

[4℄ R.Coio,G. Sunshine, Immunology: ashort ourse,6thed., JohnWiley &

Sons-Blakwell,NewJersey,2009.

[5℄ T.C.Coutinho, T.T. Da Silva, MHC, ReombinaçãoeDiversidade Genétia,

Congresso de Matemátia Apliada e Computaional - Sudeste, Uberlândia,

2011.

[6℄ W. Ewens, G.R. Grant, StatistialMethods in Bioinformatis: an

introdu-tion,2nded.,Springer,NewYork,2005.

[7℄ P. Ferrari, A. Galves, Aoplamento eproessos estoástios. IMPA, Riode

Janeiro,1997.

[8℄ O.Kallenberg,FoundationsofModernProbability,2nded.,Springer,Berlin,

2002.

[9℄ P.S.C.Magalhães,M.Böhlke,F.Neubarth,ComplexoPrinipalde

Histoom-patibilidade(MHC):odiaçãogenétia,basesestruturaiseimpliações

líni-as,Rev. Med. UCPEL (Pelotas),2(2004),5459.

[10℄ A.M. Miranda Vilela, A diversidade genétia do omplexo prinipal de

his-toompatibilidade (MHC) e sua relaçãoom asuseptibilidade para doenças

auto-imuneseâner. SBG,RibeirãoPreto,2007.

[11℄ H. Shashl, P. Wandeler, F. Suhentrunk, G. Obexer-Ru, S.J. Goodman,

SeletionandreombinationdrivetheevolutionofMHClassIIDRBdiversity