Effect of Graph Structures on Selection for a Model of a Population on an Undirected Graph

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1 Effect of Gah Stuctue o Selectio fo a Model of a Poulatio o a Udiected Gah Watig Che Advio: Jao Schweibeg May 0, 206 Abtact Thi eeach focue o aalyzig electio amlifie i oulatio geetic. Sice the tuctue of a oulatio gah ca ifluece electio, thi ae focue o fidig udiected gah that ca amlify electio. To clealy demotate the idea, the ae will biefly dicu the baic Moa model at fit. The it will aalyze the ta gah a a examle of electio amlifie that ca iceae the fixatio obability of mutat comaed to the imle Moa model. Fially, it will dicu the ecific kid of udiected gah that amlify electio ad have fixatio obability cloe to. Itoductio I oulatio geetic, the Moa model i the mot fudametal model to mimic how oulatio evolve ove time coideig the effect of electio. The Moa model aume fixed oulatio ize of N ad each idividual i the oulatio i claified a eithe a mutat o a o-mutat. Fite meaue the uvival obability of each idividual, ad the fite of mutat ad o-mutat ae alway diffeet. I diffeet ettig, the mutat may be moe likely o le likely to be elected to eoduce ad elace othe idividual. Sice oigially the oulatio ha o mutat, we omalize the fite of the o-mutat idividual i the oulatio to be ad let the fite of the mutat to be. I thi ae, ice we ae oly iteeted i the cae that mutat ae advatageou, the fite of mutat i alway lage tha, i.e. + ad 0. Whe the fite i high, it i moe likely fo the mutat to elace othe idividual. Each idividual chooe to eoduce ad elace a adomly choe othe idividual i the oulatio at ate equal to it fite. Fo uch a evet to hae, we have to wait fo a exoetial ditibutio of time with ate. Let tate eeet the umbe of mutat i the oulatio, o tate i i the cicumtace that thee ae i mutat i the oulatio. Suoe thee ae i mutat iitially i the oulatio, the ate fo the model to go fom tate i to tate i + ad the ate to go fom tate i to tate i ae a follow. i( i) i i +, at ate, i( i) i i, at ate. Thu, the ate at which the oce leave the tate i i q(i) ( + )i( i) (2 + )i( i). ()

2 I ode to bette eet ad aalyze oulatio tuctue, we will eeet diffeet oulatio model uig gah G (V, E). The V tad fo the et of vetice o ode i the gah ad the E tad fo the et of edge coectig thoe ode i the gah. Whe a idividual eoduce, it offig elace the idividual at a adomly choe eighboig vetex. Sice each ode ca be elaced by ay of the othe ode i the oulatio, the Moa model we dicued above ca be eeeted by a comlete gah with all ode togly coected by all oible edge. The mutat i the gah have ate to eoduce ad elace othe ode ad the o-mutat have ate of eoductio. The oce of the Moa model i eeated util evetually the oulatio ha eached homogeeity, the tate that all idividual ae o-mutat o mutat. I eithe cae we ay that the oulatio eache fixatio. I thi ae, the mutat fixatio efe to the obability that all idividual i the gah become mutat, if o futhe exlaatio. We ae iteeted i the effect of electio i uch model, ad i aticula, we ae iteeted i the cae that thee i oly a igle mutat i the oulatio iitially. Accodig to Liebema et al.[4], fo a comlete gah ude the Moa model with ize N, the obability that all ode will evetually become mutat with the iitial occuece of a igle mutat i the oulatio i ρ N. (2) Howeve, whe the oulatio gah i ot comlete, ot all ode ae coected by all oible edge, o the fixatio obability become difficult to tudy. Oe uch examle i the ta gah. 2 Sta Gah Coide a ta gah with ode i the cete ad leaf ode coected to the cete ad to o othe vetex. I total, uch a ta gah ha N + ode. I the followig dicuio, we will look at the ode i the cete ad ode o the oke eaately. I 2008, M. Boom ad J. Rychta[] have foud the exact fixatio obability of uch ta gah with leaf ode ad fite to be Q [ ( + ) j ( + (+) ) j ]. (3) I thi ae, howeve, we will look at the fixatio obability of the ta gah by fidig it ue ad lowe boud whe goe to ifiity, ad ituitively ee why it ha lage electio amlificatio effect tha the Moa model. Coide a advatageou mutat with fite + ad 0 aeaig o oe of the leaf ode. We wat to fid the fixatio obability of the ta gah with uch iitial mutat. Let i be the umbe of mutat o the leaf ode. Thee ae oly fou evet oible to hae: the oibility of havig i + mutat ad i mutat o the leaf ode i the ext geeatio whe the cete i a mutat; ad the oibility of havig i + mutat ad i mutat o the leaf ode i the ext geeatio whe the cete i ot a mutat. Howeve, whe the cete ode i a mutat, thee i o way to make ay mutat o the leaf ode diaea i the ext geeatio; whe the cete ode i a o-mutat, thee i o way fo it to give bith 2

3 to ay mutat at the leaf ode i the ext geeatio. Theefoe, the above two evet ca be elimiated ad the oly two oible evet that ca hae with thei ate ae a follow. i i +, at ate i i, at ate i ( i) if the cete i a mutat; if the cete i a o-mutat. If the cete i a mutat, it ha ate to ifect all the othe ode coected to it, ice thee ae oly i o-mutat i the gah, the ate fom tate i to tate i + i ( i). Similaly, whe the cete i a o-mutat, it ca oly ifect othe to become o-mutat with ate. Sice thee ae i out of mutat that could be ifected to become o-mutat, the ate fom tate i to tate i i i. I ode to udetad the dyamic of the ta gah, we divide the oblem ito a equece of fou-tate Makov Chai. Let tate be the tate that thee ae i mutat o the oke egadle of what the cete ode i. Let tate 2 be the tate that thee ae i mutat o the oke ad cete of the ta gah i mutat. Let tate 3 be the tate that thee ae i mutat o the oke ad the cete of the ta gah i o-mutat. Let tate 4 be the tate that thee ae i + mutat o the oke egadle of what the cete ode i. So the taitio ate ae State 3 State, i ; State 2 State 3, i; State 3 State 2, i; State 2 State 4, ( i). Let g(i) be the obability that the oulatio eache tate 4 befoe tate, tatig fom tate i. We ca obtai the followig equatio. Likewie, g(2) i ( i) + ( i) g(3) + + g(3) + + g(4). ( i) ( i) + ( i) g(4) g(3) i i + ig() + i i + i g(2) g() g(2). Solve the above two equatio with g() 0 ad g(4), we get g(2) , the ue boud fo eachig tate 4 befoe tate. g(3) , the lowe boud fo eachig tate 4 befoe tate. 3

4 Let f(i) be the obability that the oulatio eache tate befoe tate 4. The the lowe boud fo eachig tate befoe tate 4 i g(2), f(2) The ue boud fo eachig tate befoe tate 4 i g(3) f(3) The, we will ue thee fou boud fo the imle 4-tate Makov Chai to calculate the ue ad lowe boud fo the ta gah. Let T y mi{t : X t y} be the fit hittig time of tate y. Let h(i) P i (T < T 0 ) be the obability that the mutat fixate whe thee ae iitially i mutat i the gah. We will ue the followig tadad eult about aymmetic adom walk fom [6]. Lemma. Let be the obability that the umbe of mutat i the gah goe fom i to i +, ad aume that thi obability i the ame fo all i. The P i (T < T 0 ) ( )i ( ). (4) Poof. Sice i the obability that the umbe of mutat i the gah goe fom i to i+, the obability that the umbe of mutat i the gah goe fom i to i i. Accodig to the fit te coditioig method, the fixatio obability with iitial tate i ca be divided ito two cae, the obability of fixatio with iitial tate i+ ad obability of fixatio with iitial tate i, with coeodig obability that goe fom tate i to each of them eectively. Reaage the equatio, we could get h(i) h(i + ) + ( )h(i ). h(i) + ( )h(i) h(i + ) + ( )h(i ) ( )[h(i) h(i )] [h(i + ) h(i)] h(i + ) h(i) [h(i) h(i )]. Statig fom tate 0, it i imoible to get to tate, o h(0) 0. Let c h(). Sice h(), j ( h(j) c i0 )j c ( c ( ), ) i. 4

5 the Thu, h() c ( ). P i (T < T 0 ) ( ) ( )i (. ) ( )i We ca ue thi eult to calculate the ue boud ad lowe boud of the fixatio obability of a ta gah. The ue boud ca be calculated uig the equatio g(2) ad f(2) that we + eviouly obtaied. So the obability of gettig a additioal mutat o the oke i ad the obability of emovig a exitig mutat o the oke i The Let u(i) be the ue boud of h(i), the by Lemma, the ue boud of the fixatio obability i u() +2 ( ). +2 Whe goe to ifiity, ted to 0. The the ue boud of the fixatio obability u() ted to u() 2. Similaly, we ca calculate the lowe boud of the fixatio obability, l(), by equatio g(3) ad f(3). By Lemma, the lowe boud of the fixatio obability i Whe goe to ifiity, l() + 2 ( + 2 ). ted to 0, the lowe boud of the fixatio obability ted to l() 2. Sice whe goe to ifiity, both of the ue boud ad lowe boud of the fixatio obability goe to 2, 5

6 whe ted to ifiity, the fixatio obability of the ta gah ted to 2. (5) Comaig fomula (2) ad (5), we ca ee that the fixatio obability of the ta gah i highe tha the Moa model. Theefoe, we ca coclude that comaed to the Moa model, the ta gah ha geate electio amlificatio effect. 3 Gah Decitio I the liteatue, eeache have aleady foud diected gah uch a ueta that have a lage electio amlificatio effect which geatly iceae fixatio obability[3]. The iitial mutatio could hae adomly i ay lace i the gah to obtai a fixatio obability that ted to. Howeve, i thi ae we ae iteeted i fidig udiected gah that could amlify electio ate. We come u with a ecific gah G m, which ha a ode i the cete with m comlete gah of ize coected to it. I ode to each mutat fixatio, the oigial mutat ha to tat fom the cete of the gah. The mutat i the cete ca quickly ifect all m ode aoud it, ad each of thoe m ode ca the ifect all the othe ode i each comlete gah. Ou tudy how that thi udiected gah G m, ha fixatio obability covegig to whe the ize of each comlete gah goe to ifiity ad m ted to ifiity at a aoiate ate. Theoem 2. The udiected gah G m, ha fixatio obability covegig to whe goe to ifiity ad m log. 4 Poof of Fixatio Fitly, we would like to comute the exected time to fixatio of a comlete gah ude Moa model followig the agumet fom Duett book [5]. Let τ be the time oe comlete gah fixate, which iclude both the cae that all idividual ae mutat ad all ae o-mutat. Let tate umbe be the umbe of mutat i the gah at a aticula time. Let S j be the amout of time et at tate j befoe time τ. The exected fixatio time with iitial tatig tate i, deoted a E i, i the um of time et at each tate j. So E i [τ] E i [S j ]. (6) j Lemma 3. The exected time fo a comlete gah to fixate with iitially oe mutat i the gah ca be ue bouded by 4 ( + log ), i.e. E [τ] 4 ( + log ). Poof. Let N j be the umbe of viit to tate j. Let q(j) (2+)j(N j) N be the ate the chai leave tate j. Sice each viit to j lat a exoetial amout of time with mea q(j), we have E i [S j ] q(j) E i[n j ]. (7) 6

7 Let T j mi{t : X t j} be the time the fit viit to tate j ad R j mi {t : X t j ad X j fo ome < t} be the time of the fit etu to tate j, the accodig to [5], Whe j, 0 i j, accodig to Lemma, Likewie, whe j i, Ude ou Moa model, Uig equatio (9) ad (0), Sice E i [N j ] P i(t j < ) P j (R j ). P i (T j < T 0 ) ( )i (, (8) )j P i (T 0 < T j ) P i (T j < T 0 ) ( )i ( )j (. (9) )j P i (T < T j ) ( )i j (, (0) ) j P i (T j < T ) P i (T < T j ) ( ( j) j j( j) +. )i j ( ) j ( ) j. P j (R j ) P j+(t < T j ) P j (T 0 < T j ) )j+ j ( ( + ) j 2 + ( (+) j (+) j ad 0, (+) j )j ( )j ( )j ( + ) ( + + ) j 2 + ( + )j ( + )j ( + [ )j + ( + + ] + ) j ( + ) j [ ] ( + + ) j ( + ) j [ ] ( + ) j ( + ) j + ( + ) j. ( + ) j ( + ) j + ( + ) j. 7

8 Theefoe, Accodig to equatio (8), P j (R j ) 2 +. () P (T j < T 0 ) ( ) ( ( + ) )j ( ( + )j ( + )j + )j ( + ) j Sice P (T j < T 0 ) P (T j < ), ( + )j ( + ) j. Accodig to equatio () ad (2), Sice q(j) (2+)j( j), The uig (7) ad (3), P (T j < ) E [N j ] P (T j < ) P j (R j ) E [S j ] q(j) E [N j ] ( + )j ( + ) j. (2) ( + )j ( + ) j 2 +. ( + ) j j( j)[( + ) j ]. (3) E [τ] j 2 j ( + ) j j( j)[( + ) j ] ( + ) j j( j) [( + ) j ] + j 2 + ( + ) j j( j) [( + ) j ]. Amog j 2, j ca be ue bouded whe j 2 ad amog 2 + j, j ca be ue bouded whe j (+)j 2 +. Alo (+) j ca be ue bouded (+) (+) j+ by, 2 E [τ] 2 2 j [ 2 j j + 2 j + j 2 + j j 2 + j ] 4 ( + log ), uig the fact that j j + log. 8

9 The above ue boud we calculated i the ue boud fo the exected time fo a aticula comlete gah to fixate with iitially mutat. Fo ou gah G m,, the cete ode i the oly iitial mutat ad it take time fo it to ifect all the comlete gah coected to it. O aveage, the exected time fo the cete to ifect a aticula comlete gah i m, ice thee ae m uch comlete gah ad the ate of ifectio of the cete ode i. Fo a comlete gah, if all mutat die out, thee i o way fo it to each mutat fixatio. Howeve, oe beefit of ou gah G m, i that it ovide multile chace o attemt fo each comlete gah to fixate. If oe comlete gah ha o mutat, a log a the cete i till a mutat, the comlete gah ca be eifected. The we ca begi ou oof fo Theoem 2. Poof of Theoem 2. Accodig to Lemma 3, fo each attemt, the ue boud fo the exected time each comlete gah take to fixatio i 4 ( + log ), which coide both the cae that all mutat die out ad all mutat fixate, ad the exected time fo the cete to ifect a aticula comlete gah i m, o the exected time fo each attemt a aticula gah take i ue bouded by m + 4 ( + log ). Sice the obability of fixatio fo each comlete gah i, o aveage, the exected umbe of attemt eeded fo each comlete gah to fixate i obability of fixatio. Let T i be the time fo a comlete gah i to fixate. With m uch comlete gah, the exected time fo a aticula comlete gah with ize to fixate i E[T i ] [ m + 4 ] ( + log ) The by Makov iequality, the obability that oe gah take loge tha time t to fixate i P (T i > t) E[T i] t {[ m t + 4 ] ( + log ). }. The by Boole iequality, the obability that at leat oe gah take loge tha time t to fixate i ( m ) m {[ m P {T i > t} t + 4 ] ( + log ) } i i m {[ m t + 4 ] ( + log ) }. 9

10 To imlify the iequality, thee exit cotat c ad d uch that ( m P i ) {T i > t} c [ ] m 2 + m( + log ) t d t [ ] m 2 + m log. The above comutatio aume that the cete i alway mutat. Howeve, the cetal mutat could alo die ad be elaced by a o-mutat. So we would like to ivetigate the obability that the cete die befoe time t ad comae it with the obability that thee i at leat oe comlete gah ha ot fixated by that time t, which we obtaied above. The ate of a o-mutat ifectio i ad the obability fo the cete to be ifected by a o-mutat i a uit time, the obability the cete die, i m, ice thee ae m comlete gah coected to the cete, each with obability to ifect the cete ode. Let D be the time the cete die. The obability that the cete die befoe time t i P (D t) m t. I ode to make both equatio above to be mall, ick m log, t 2(log ) 3. Let T be the time that all comlete gah fixate. The ad P (D t) 2(log )4 2(log )2 P (T > t) d 2(log ) 3 d log. We ca ee that both of the equatio go to 0 whe goe to ifiity. Thu, whe goe to ifiity, by time t, the cete of the gah die with obability covegig to 0, ad all gah fixate with obability covegig to, which mea that ou gah G m, will fixate with obability tedig to. Thu, the udiected gah G m, amlifie electio ad ha fixatio obability cloe to whe the ize of each ub-gah,, goe to ifiity. Whe 0,, thee i o electio i the oulatio. We ae iteeted i the fixatio obability of ou gah G m, fo that cae to guaatee that ou gah ca tuly amlify electio ate. Pooitio 4. If thee i o electio, i.e. 0 ad, the fixatio obability of the gah G m, goe to 0 whe goe to ifiity. Poof. Give a udiected gah G (V, E), let V deote the et of all vetice ad E deote the et of all edge i the gah. Alo, let C be the et of all mutat ode ad C V. Let P C be the obability of mutat fixatio, ad d m be the degee fo each ode m, which i the 0

11 umbe of ode coected to m. Accodig to the fomula give by Boom et al.[2], fo fite, P C i C d i k V d k. (4) I ou gah G m,, the cete ode ha m ode coected to it o the degee of the cete ode i m. Sice iitially thee i oly oe mutat i the gah G m, which i the cete ode, the ummatio of the ivee of the degee of all mutat i i C d i m. I additio, thee ae m comlete gah coected with the cete ode ad the degee fo each ode i each comlete gah i. Meawhile, thee ae ode i each comlete gah ad thee ae m uch comlete gah. Fo all ode that ae ot coected to the cete i a comlete gah, thei degee i ; fo the ode that coected to the cete, it degee i. So the ummatio of the ivee of the degee of all ode i gah G m, i d k [ m + m m ] + ( m), Accodig to fomula (4), k V which ca be bouded above by P C i C d i k V d k [ m + m m m [ ] + m 2 m + m P C + ( m) ] + m 2. (5) Whe goe to ifiity, m log goe to ifiity, ad the fixatio obability P C alo goe to 0, which mea that G m, caot each mutat fixatio without electio. Theefoe, ou gah G m, i ideed a electio amlifie. Refeece [] Boom, M., Rychta, J. (2008): A aalyi of the fixatio obability of a mutat o ecial clae of o-diected gah. Poceedig of the Royal Society A 464, doi:0.098/a

12 [2] Boom, M., Hadjichyathou, C., Rychta, J. ad Stadle, B.T. (200): Two eult o evolutioay ocee o geeal o-diected gah. Poceedig of the Royal Society A. doi:0.098/a [3] Jamieo-Lae, A., Hauet, C. (205): Fixatio obabilitie o ueta, eviited ad evied. Joual of Theoetical Biology Volume 382, doi:0.06/j.jtbi [4] Liebema, E., Hauet, C., Nowak, M.A.(2005): Evolutioay Dyamic o Gah. Natue 433, 3236 [5] Rick Duett(2008): Pobability Model fo DNA Sequece Evolutio. Sige [6] Rick Duett(999): Eetial of Stochatic Pocee. Sige 2