q-bio.CB 2012-08-14 0

A Markovian growth dynamics on rooted binary trees evolving according to the Gompertz curve

Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields model. Fix $n\ge 1$ and $\beta>0$. We start at time 0 with the tree composed of a root only. At any …

Authors: C. L, im, R. D. Portugal

A MARK O VIAN GR O WTH D YNAMICS ON R OOTED BINAR Y TREES EV OL VING A CCORDING TO THE GOMPER TZ CUR VE C. LANDIM, R. D. POR TUGAL AND B. F. SV AITER Abstract. Inspired b y biological dynamics, we consider a gr o wth Marko v process taking v alues on the space of ro oted binary trees, similar to the Aldous- Shields model [1]. Fix n ≥ 1 and β > 0. W e start at time 0 with the tree composed of a ro ot only . At a ny time, eac h no de with no descendan ts, indepen- den tly from the other no des, pro duces tw o successors at rate β ( n − k ) /n , where k is the distance fr om the node to the r oot. Denote by Z n ( t ) the num ber of nodes with no descendan ts at time t and l et T n = β − 1 n ln( n/ ln 4)+ (ln 2) / (2 β ). W e pr o ve that 2 − n Z n ( T n + nτ ) , τ ∈ R , conv erges to the Gomp ertz curve exp( − (ln 2) e − β τ ). W e also prov e a central li mit theorem for the martingale associated to Z n ( t ). 1. Introduction Gomp e rtz mo del was o r iginally prop o sed as an a ctuarial curve [7] to mo del mortality of an aging populatio n, and this appro ach still has applications in curr e n t surviv al analys is [5]. One century after its creation, the mo del surpassed its o r iginal realm and w as used as a biological growth curve [16, 15]. Since then, this function has successfully describ ed animal growth [11], r egeneratio n [14], and tumor growth after the pioneering w ork of Laird [12]. The astonishing fact is no t only that Gompertz mo del fits successfully in many cases of biologic a l growth, but mainly that its biolog ical foundation is still not fully understo o d and that the Go mper tz c urve has not yet b een der ived as the scaling limit o f so me micros c o pic dynamics. Inspired by recent biolo gical exp eriments on cellular aging, w e present in this article a micros copic Marko vian gro wth dynamics whic h leads to the Gomp ertz curve. T elomeres ar e nucleo-proteins lo cated at the end ter minal of chromosomes which shor ten at ea ch somatic cell division. In cultured cells, growt h is not o b- served indefinitely , the division rate slows do wn and ultimately cea ses [10]. There are significant evidences that telomeres act as a mo le c ular counting devic e which regulates the num ber of cell divisio ns and limits further division after a critic length is ac hieved [9]. Recent experimental evidences suggest also that telomer e shorten- ing may also b e related with the decr ease of mitotic rate, the prop or tion o f cells in a tiss ue that are undergoing mitosis [2]. On the mathematical s ide, computer simulation of a very simple discr ete-time sto chastic model of telomere reg ula ted growth yielded gr owth curv es similar to the Gomp e rtzian mo del [13]. The basic assumption of this mo del was a line ar decr ease in mitosis probability with telomere shortening. W e take a step further in this article by studying a contin uous-time branching pr o cess. In stead of sim ulations, Key wor ds and phr ases. Aging; Rando m binary trees, Gomp ertz curve; Gro wth pro cesses. 1 2 C. LANDIM, R. D. POR TUGAL AND B. F . SV AITER we prove that the size o f the p opulation prop erly resca led in time conv erges to the Gomp e rtz c ur ve. W e also estimate the mea n telo mere size of the cell p opulatio n, a quantit y that can b e actually measured in cultured ce lls . Our model is simila r to the Aldous-Shields mo del [1, 6 , 3] for a ro o ted, gr owing random binar y tree. While in their mo del each active vertex beco mes inactive and activ ates its tw o descendant a t a ra te which decreas es expo nentially with the distance o f the vertex to the r o ot, in our mo del the ra te decreases linearly . 2. The stochastic model W e pres e nt in this a r ticle a ra ndom gr owth mo del which we will interpret as a cell division pro c e ss reg ulated by telomere shortening. Cell divisio n is clearly no t an insta nt aneous pro cess since a fter a divis ion each cell must synthesize a num ber of c e llular comp onents b efore it can divide ag ain. W e will assume, howev er, that these pro ces s es o ccur in a time scale muc h smaller than the time sca le in which cell division tak es place. Assume that the initial state is a single cell with telomere length L 0 , that a fixe d amount of basis, say δ , is lo st by each telomere at each cell divis ion, a nd that a c ell reaches mitotic senes cence, which means that it doe s not divide an ymore, when its telomere a ttains a critical length L min . Without loss of generality , we ma y supp os e that n = ( L 0 − L min ) /δ is a p os itive integer r epresenting the maximum num ber of divisions which a cell may undergo , the so called H ayflic k limit in the bio lo gical literature. The total time spent in a cell cycle and arrest for a cell w hich has undergone k divisions, 0 ≤ k ≤ n , is mo deled as an exp onential random v ariable with para meter β ( n − k ) / n , where β > 0 is a fixed parameter representing the rate at which a cell with telomere o f le ng th L 0 undergo es a division. These exp onential rando m v a riables a re of course a ssumed to b e m utually indep endent. The dynamics just de s crib ed is a Ma rko v pr o cess taking v alues on the space o f ro oted binary trees, similar to the Aldous-Shields mo de l [1 ]. W e sta rt a t time 0 with the tree comp osed of a r o ot o nly . At any time, each no de with no descendants, independently from the other no des, pro duces tw o succes sors a t a r ate equal to β ( n − k ) / n , where k is the dis ta nce from the no de to the ro ot. Even tually the pro cess re a ches the ro oted binar y tree with n genera tions, which is the absor bing state for the proc e s s. Let X ( k , t ) = X n ( k , t ), 0 ≤ k ≤ n , be the num ber of cells in the population whic h has under g one ex actly k mitosis at time t . In the tree formulation, X ( k , t ) repr esents the num be r of no des in the k th generation with no desc endants, called the active no des o f the k th genera tion. The cell p o pula tion consists therefor e of the a ctive no des. It follows fr o m the previo us assumptions that X ( t ) = ( X (0 , t ) , . . . , X ( n, t )) is a Marko v chain in the sta te spa c e Ω = { ( x 0 , . . . , x n ) ∈ Z n +1 + | n X i =0 2 n − i x i = 2 n } with generator Q giv en by ( Qg )( x ) := n − 1 X k =0 λ k x k  g ( T k x ) − g ( x )  , (2.1) GR OWTH D YNAMICS EVOL VING A CCORDING TO THE GOMPER TZ CUR VE 3 where x = ( x 0 , . . . , x n ), T k x = ( x 0 , . . . , x k − 1 , x k +1 + 2 , . . . , x n ), λ k = β ( n − k ) / n , 0 ≤ k ≤ n − 1, and g : Ω → R is a generic function. Denote b y X n ( t ) the size of the p o pulation at time t reno rmalized by 2 n : X n ( t ) = 2 − n n X k =0 X ( k , t ) , by x k ( t ), 0 ≤ k ≤ n , t ≥ 0, the exp ected num ber o f cells which undergone k mitosis at time t : x k ( t ) = E  X ( k , t )  and let x ( t ) =  x 0 ( t ) , · · · , x n ( t )  T . Denote by S n ( t ) the exp ected n umber o f cells: S n ( t ) := 2 − n n X i =0 x i ( t ) . (2.2) As befor e, in the tree formulation, x k ( t ) represents the exp ected n umber of active no des in the k -th g eneration and S n ( t ) the exp ected n umber of active nodes , which corres p o nds to the expected size of the inner boundary of the tree. Let T n = n β ln  n ln 4  + ln 2 2 β · (2.3) W e show in (3.10) that T n is at dista nce O ( n − 1 ) from the time at whic h the exp ected nu mber of active no des is equa l to one half of the final size. As in [1], consider the progress of the tree along the leftmost branch. Let H n k the time of the k -th division, so that H n k = P 0 ≤ i 0 and every τ ∈ R , S n ( t ) =  1 − e − β t/n 2  n , lim n →∞ W n ( τ ) = exp  − (ln 2) e − β τ  . The pro of o f this r e s ult provides in fact an expansio n in n − 1 of ln W n ( τ ). W e may als o compute the as y mptotic b ehavior of the time deriv ativ e of the rescale d growth cur ve W n . A simple computation with the generator giv es that d dt S n ( t ) = β 2 n n X i =0 ( n − i ) n x i ( t ) . (2.4) W e show in Section 4 that ( d dt ln S n )( t ) = β e − β t/n 2 − e − β t/n , n ≥ 1 , t > 0 . (2.5) The next result follo ws fr om this identit y and Prop osition 2.3. Prop ositi on 2.4 . F or every τ ∈ R , lim n →∞ ( d dτ ln W n )( τ ) = (ln 2) β e − β τ . This result ha s an interpretation in our biological mo del. In cultured cells, telomere length is not ev aluated individually . Instead, what is actually measure d is the me an telomer e length of a colony of cells [2]. Therefore, in order to verify the fit of the stochastic mo del to real da ta, we must obtain the exp ected mea n telomere length pre dicted b y the model. The a s ymptotic telo mere length may be estimated in tw o different regimes . The telomere length of a ce ll which has undergone k mitosis is L min + ( n − k ) δ . Hence, the total telomer e length a t time t is P 0 ≤ k ≤ n [ L min + ( n − k ) δ ] X ( k , t ). Let ℓ n ( t ) b e the expectation of the tota l telomer e leng th a nd let L n ( t ) be the a verage telo mere length: ℓ n ( t ) = E h n X k =0  L min + ( n − k ) δ  X ( k , t ) i , L n ( t ) = ℓ n ( t ) 2 n S n ( t ) · Prop ositio n 2.5 below follows fro m identit y (2.5) and Pr o p osition 2.4. Its pro of is presented at the end of Section 4. Prop ositi on 2.5 . F or every t ≥ 0 , τ ∈ R , L n ( tn ) = L min + ( L 0 − L min ) e − β t 2 − e − β t , n ≥ 1 , lim n →∞ n { L n ( T n + τ n ) − L min } = (ln 2) ( L 0 − L min ) e − β τ . GR OWTH D YNAMICS EVOL VING A CCORDING TO THE GOMPER TZ CUR VE 5 3. The dynamics of the stochastic model W e prov e in this s e ction P rop osition 2.3. By Kolmogo rov’s forward equation d dt x ( t ) = M x ( t ) , (3.1) where M is the squar e matr ix with en tries m i,j , 0 ≤ i, j ≤ n , given by m i,j =      − λ i , j = i, 2 λ i − 1 , j = i − 1 , 0 otherwise (3.2) The so lutio n of the pr evious linear ordinary differential equatio n with initial condition x (0) =  1 , 0 , · · · , 0  T is x ( t ) = exp( t M ) x (0), t ≥ 0. Moreov er, the exp ected nu mber of cells at time t ca n be ca lculated as the matrix pro duct of the vector [1 , 1 , . . . , 1] with x ( t ): n X i =0 x i ( t ) = [1 , 1 , . . . , 1] x ( t ) = [1 , 1 , . . . , 1] exp( t M ) x (0) . The pro of of Pro po sition 2.4 relies on the ana lysis of the matr ix M , presented in Lemma 3 .1 and in Coro llary 3 .3 b elow. Since M is low er-tria ngular, the spec tr um of M is − λ 0 , − λ 1 , · · · , − λ n . F or γ ∈ R and 0 ≤ i, k ≤ n , let a ( i, k , γ ) = a ( γ ) i,k =    0 i < k , γ i − k  n − k i − k  i ≥ k . (3.3) Lemma 3.1. The ve ctor w k ∈ R n +1 , 0 ≤ k ≤ n , define d by w k =  w 0 ,k , w 1 ,k , · · · , w n,k  T , w i,k = a ( − 2) i,k , (3.4) is a right-side eigenve ctor of M c o rr esp onding to the eigenvalues − λ k . Pr o of. Fix 0 ≤ k ≤ n . T o pr ov e this lemma we must ev aluate b = ( M + λ k I ) w k where I is the identit y matrix. Using the fact that M is lower-triangular we ge t from the definition of w k that b i = 0 for i < k . F or i = k , since m k,k = − λ k , b k = ( m k,k + λ k ) w k,k = 0. This proves the lemma for k = n . F or 0 ≤ k < n and i > k , b i = m i,i − 1 w i − 1 ,k + ( m i,i + λ k ) w i,k By (3.2) and by the definition of λ i , we have b i = 2 λ i − 1 w i − 1 ,k + ( − λ i + λ k ) w i,k = β n  2( n + 1 − i ) w i − 1 ,k + ( i − k ) w i,k  = β n 2( n + 1 − i )  w i − 1 ,k + i − k 2( n + 1 − i ) w i,k  . According to (3.3), for k < i ≤ n , w i − 1 ,k w i,k = a ( − 2) i − 1 ,k a ( − 2) i,k = i − k − 2 ( n + 1 − i ) · This pr oves the lemma.  6 C. LANDIM, R. D. POR TUGAL AND B. F . SV AITER Define the squar e matr ices A ( γ ), D b y A ( γ ) = { a ( γ ) i,j } , D = diag {− λ 0 , . . . , − λ n − 1 , − λ n } . (3.5) In view of the previous lemma, M = A ( − 2) D [ A ( − 2 )] − 1 so that exp( t M ) = A ( − 2) exp( t D ) [ A ( − 2)] − 1 , t ≥ 0 . (3.6) T o ev a luate this express ion we will need tw o auxiliary results. Lemma 3.2. F or any η ∈ R , [ η n , η n − 1 , . . . , η , 1] A ( γ ) = [( η + γ ) n , ( η + γ ) n − 1 , . . . , ( η + γ ) , 1] . Pr o of. Le t  u 0 , · · · , u n  = [ η n , η n − 1 , . . . , η , 1] A ( γ ) . A direct calculation, together with (3.3) yields u k = n X i =0 η n − i a i,k ( γ ) = n X i = k η n − i γ i − k  n − k i − k  = ( η + γ ) n − k . This co ncludes the pro of of the lemma.  Corollary 3.3. F or any γ , µ ∈ R , A ( γ ) A ( µ ) = A ( γ + µ ) , and A (0) = I . In p a rticular [ A ( γ )] − 1 = A ( − γ ) . Pr o of. B y Lemma 3 .2, fo r any η ∈ R , [ η n , η n − 1 , . . . , η , 1] A ( γ ) A ( µ ) = [ η n , η n − 1 , . . . , η , 1] A ( γ + µ ) . The first a ssertion of the Corollary follows from this identit y and from the fact that the s pan of { [ η n , η n − 1 , . . . , η , 1] : η ∈ R } is R n +1 . The seco nd claim o f the corolla r y follows fro m the definition of A ( γ ).  Pro of o f Prop osition 2.3. By (3.6) a nd by the previous co rollar y , x ( t ) = A ( − 2) exp( t D ) A (2) x (0) . (3.7) Since e − tλ k = ( e − tβ /n ) n − k , 0 ≤ k ≤ n , b y definition o f the diagonal matrix D , exp( t D ) = diag { ( e − tβ /n ) n , · · · , ( e − tβ /n ) 1 , 1 } . Hence, n X i =0 x i ( t ) = [1 , 1 , . . . , 1] A ( − 2) exp( t D ) A (2 ) x (0) . (3.8) Applying twice L emma 3 .2 we g et that [1 , 1 , . . . , 1] A ( − 2 ) exp( t D ) A (2) = [(2 − e − β t/n ) n , (2 − e − β t/n ) n − 1 , . . . , 2 − e − β t/n , 1] , so that n X i =0 x i ( t ) = (2 − e − β t/n ) n = 2 n  1 − e − β t/n 2  n . This co ncludes the pro of of the first assertion of the theorem. GR OWTH D YNAMICS EVOL VING A CCORDING TO THE GOMPER TZ CUR VE 7 Recall that we denote by W n ( τ ) the norma lized growth curve. It follows fro m the definition of S n ( t ) tha t W n ( τ ) =  1 − e − β τ θ n ln 2 n  n , (3.9) where θ n = 2 − (1 / 2 n ) . It rema ins to let n ↑ ∞ .  W e conclude this s ection showing that T n is at distance n − 1 from the time a t which the exp ected n um b er of active no des is equal to one half of the final size. Denote b y t ∗ the time at whic h the expected p opulation size is half of the final size: n ln  1 − e − β t ∗ /n 2  = − ln 2 . W e claim that t ∗ = n β ln  n ln 4  + ln 2 2 β + O ( n − 1 ) . (3.10) Indeed, dividing b oth sides of the p enultimate for m ula by n and tak ing ex po - nent ials on b oth sides o f this equation, we obtain that e − β t ∗ /n = 2  1 − e − ln(2) /n  . By T aylor expa nsion and setting t ∗ = ( n/β ) ln( n/ ln 4) + h ∗ we o btain ln 4 n e − β h ∗ /n = 2  ln 2 n − 1 2  ln 2 n  2 + O ( n − 3 )  . Hence e − β h ∗ /n = 1 − ln 2 2 n + O ( n − 2 ) . T a king loga rithms o n b oth sides, b y T aylor expansio n, − β n h ∗ = − ln 2 2 n + O ( n − 2 ) . This pr oves cla im (3.1 0). 4. A genera ting function W e prov e in this s e ction P rop osition 2.4 and iden tit y (2.5). Let Ψ( u, t ) = n X k =0 u n − k X ( k , t ) , u , t ∈ R + , (4.1) and deno te by ψ ( u, t ) be the expected v alue of Ψ( u, t ): ψ ( u, t ) = E [Ψ( u, t )] = n X k =0 u n − k E [ X ( k , t )] = n X k =0 u n − k x k ( t ) . (4.2) Hence, by (3.7), ψ ( u, t ) = [ u n , u n − 1 , · · · , u, 1] x ( t ) = [ u n , u n − 1 , · · · , u, 1] A ( − 2) exp( t D ) A (2) x (0) . By Lemma 3.2 and by the formula (3.5) for the diag onal matrix D , ψ ( u, t ) = (2 + ( u − 2) e − β t/n ) n . (4.3) 8 C. LANDIM, R. D. POR TUGAL AND B. F . SV AITER On the other hand, E h n X k =0 X ( k , t ) i = n X k =0 x k ( t ) = ψ (1 , t ) , E h n X k =0 ( n − k ) X ( k , t ) i = n X k =0 ( n − k ) x k ( t ) = ψ u (1 , t ) , (4.4) where ψ u stands fo r the partial deriv a tive o f ψ with respe c t to the first v ariable. It follo ws fro m (2.2), (2.4), the prev ious equatio ns and (4.3 ) that ( d dt ln S n )( t ) = β n ψ u (1 , t ) ψ (1 , t ) = β e − β t/n 2 − e − β t/n for n ≥ 1, t > 0. This is the identit y pres e nted in (2.5). Recall the definition of L n ( t ), the av erage telomere leng th a t time t , intro duced just b efor e the statement Pr op osition 2.5. It follows from (4.3) and (4.4) that ℓ n ( t ) = E h n X k =0  L min + ( n − k ) δ  X ( k , t ) i = L min (2 − e − β t/n ) n + δ n (2 − e − β t/n ) n − 1 e − β t/n , and L n ( t ) = L min + ( L 0 − L min ) e − β t/n 2 − e − β t/n bec ause δ n = L 0 − L min . The as sertions of P rop osition 2.5 are a straig ht forward consequence o f the previo us form ula. 5. Proof of Theorem 2.1 W e prov e in this sec tion Theorem 2.1 b y estimating the cov ar iances o f the pro cess X n ( t ). Let F j ( t ) = E [ X ( j, t )] , 0 ≤ j ≤ n , and let F j,k ( t ) b e the co v ariance b etw een X ( j, t ) and X ( k , t ): F j,k ( t ) = E  X ( j, t ) X ( k , t )  − E  X ( j, t )  E  X ( k , t )  , 0 ≤ j ≤ k ≤ n . Let F ( t ) b e the column v ector with m = ( n + 1)( n + 2) / 2 coo rdinates given by F ( t ) =  F 0 , 0 ( t ) , F 0 , 1 ( t ) , F 1 , 1 ( t ) , . . . , F 0 ,j ( t ) , F 1 ,j ( t ) , . . . , F j,j ( t ) , . . . , F n,n ( t )  T . An e le men tary c omputation shows that d dt F ( t ) = Σ F ( t ) + G ( t ) , (5.1) where Σ is the m × m matrix giv en by Σ =          M 0 0 0 0 0 0 D 1 M 1 0 0 0 0 0 D 2 M 2 0 0 0 . . . . . . 0 0 0 D n − 1 M n − 1 0 0 0 0 0 D n M n          , GR OWTH D YNAMICS EVOL VING A CCORDING TO THE GOMPER TZ CUR VE 9 M j is a squa re ( j + 1) × ( j + 1) matr ix with en tries M j ( a, b ), 0 ≤ a, b ≤ j , given by M j ( a, a ) = − ( λ a + λ j ) , 0 ≤ a ≤ j , M j ( j, j − 1) = 4 λ j − 1 , M j ( a + 1 , a ) = 2 λ a , 0 ≤ a ≤ j − 2 , M j ( a, b ) = 0 o therwise : M j =          − ( λ 0 + λ j ) 0 0 0 0 0 2 λ 0 − ( λ 1 + λ j ) 0 0 0 0 0 2 λ 1 − ( λ 2 + λ j ) 0 0 0 . . . . . . 0 0 0 2 λ j − 2 − ( λ j − 1 + λ j ) 0 0 0 0 0 4 λ j − 1 − 2 λ j          , and D j is the ( j + 1 ) × j matrix whose firs t j lines form the matrix 2 λ j − 1 I j , where I j is the j × j iden tit y , and whose last line has only zer os. Moreov er, G ( t ) is the vector [ G 0 ( t ) , G 1 ( t ) , . . . , G n ( t )] T and G j ( t ) is the column v ector with j + 1 en tries given by G 0 ( t ) =  λ 0 F 0 ( t )  T , G 1 ( t ) =  − 2 λ 0 F 0 ( t ) , 4 λ 0 F 0 ( t ) + λ 1 F 1 ( t )  T , G j ( t ) =  0 , . . . , 0 , − 2 λ j − 1 F j − 1 ( t ) , 4 λ j − 1 F j − 1 ( t ) + λ j F j ( t )  T , 2 ≤ j ≤ n − 1 , G n ( t ) =  0 , . . . , 0 , − 2 λ n − 1 F n − 1 ( t ) , 4 λ n − 1 F n − 1 ( t )  T . Lemma 5.1. The matrix Σ is diago nalizable. Pr o of. The eigenv alues of Σ ar e − ( λ 0 + λ j ), − ( λ n + λ j ), 0 ≤ j ≤ n . The eig env a lues − ( λ j + λ 0 ), 0 ≤ j ≤ n , ha ve multiplicit y u j = ⌊ j / 2 ⌋ + 1 , and the eigenv alues − ( λ j + λ n ), 0 ≤ j ≤ n , have m ultiplicit y v j = ⌊ ( n − j ) / 2 ⌋ + 1, where ⌊ a ⌋ stands for the in teger part of a . Fix 0 ≤ j ≤ n and a ssume without loss of gene r ality that j is e ven, j = 2 k . Consider the matrix Σ j = Σ + ( λ 0 + λ j ) I m , where I m is the m × m identit y . This matrix has u j zeros on the diag onal. Star ting from the b o ttom of the matr ix , the first zero appears at the p os ition (0 , 0) of the matrix M j , th e s econd one at the po sition (1 , 1) o f the matrix M j − 1 , and the la st one at the p osition ( k , k ) o f the Matrix M j − k . W e claim that we may reduce the matrix Σ j to obta in a matrix such that all lines where the matrix Σ j has an entry on the dia gonal equal to zero b ecome iden tically equal to zero. The assertion of the lemma follows fro m this claim. This reduction is p erfor med recursively . Let ( ℓ 1 , ℓ 1 ) be the p ositio n of the upmost zero in the diago nal a nd keep in mind that we star t coun ting from 0 which mea ns that the pos itio n ( ℓ 1 , ℓ 1 ) indicates in realit y the ( ℓ 1 + 1)-th line and column. This ent ry corresp onds to the en try ( k, k ) of the matrix M j − k . W e may first reduce the matr ix Σ j to obtain a matrix Σ (1) j which coincides with Σ j on the so uth-e ast square matrix co rresp onding to the entries { ℓ 1 , . . . , m − 1 } × { ℓ 1 , . . . , m − 1 } , whose restriction to the north- west sq uare matrix corr esp onding to the en tries { 0 , . . . , ℓ 1 − 1 } × { 0 , . . . , ℓ 1 − 1 } is the identit y , and whose ent ries v anish on the remaining t w o parts of the matrix. Notice that the ( ℓ 1 + 1)-th line of Σ (1) j v a nishes, a nd that the ent ries ( i, i ), 0 ≤ i ≤ k − 1 , o f the Matrix D j − k +1 also v anish. Denote by ( ℓ 2 , ℓ 2 ) the co or dinates of the s econd upmost zero in the diago na l of the matrix Σ j and b y ( m 2 , m 2 ) the co ordinates of the first entry of the matrix M j − k +1 , so that m 2 ≤ ℓ 2 . W e ma y reduce the matrix Σ (1) j using the diag onal en tries o f the 10 C. LANDIM, R. D. POR TUGAL AND B. F . SV AITER matrix M j − k +1 ab ov e the zero en try to obtain a ne w matrix Σ (2) j with the following prop erties. Only the e nt ries with coor dinates in { m 2 , . . . , m } × { m 2 , . . . , ℓ 2 − 1 } hav e been c hanged. The restr iction of M j − k +1 to the square { m 2 , . . . , ℓ 2 − 1 } × { m 2 , . . . , ℓ 2 − 1 } is the identit y and a ll entries below the diagonal o f this matr ix are zeros. Note tha t the line of Σ (2) j corres p o nding to the second upmost zer o in the diagonal of the ma trix Σ j v a nishes, and that the en tries ( i, i ), 0 ≤ i ≤ k − 2 , of the Matrix D j − k +2 also v anish. W e may therefo re re pea t the ar gument and conclude the proo f of the claim. Since the same argument applies to the eigenv alues − ( λ n + λ j ), 0 ≤ j ≤ n , the lemma is proved.  Prop ositi on 5.2 . F or al l τ ∈ R , lim n →∞ V ar h 1 2 n n X k =0 X ( k , T n + nτ ) i = 0 . Pr o of. Le t v b e the v ector [ v 0 , v 1 , . . . , v n ], where v 0 = [1], and v j is the vector with j + 1 co or dinates given b y v j = [2 , . . . , 2 , 1]. It is w ell known that the solution of (5.1) with initial condition F (0 ) = 0 is given by F ( t ) = Z t 0 e ( t − s )Σ G ( s ) ds . (5.2) Therefore, V ar h 1 2 n n X k =0 X ( k , t ) i = 1 4 n h v , F ( t ) i = 1 4 n Z t 0 h v , e ( t − s )Σ G ( s ) i ds , where h · , · i stands for the usual inner pro duct. Since, b y Lemma 5 .1, Σ is a diagonalizable matrix and all its eig env alues are negative, h v , e ( t − s )Σ G ( s ) i is absolutely bo unded b y | v | | G ( s ) | . Clearly , | v | 2 = ( n + 1)(2 n + 1). On the other hand, | G ( s ) | 2 =  λ 0 F 0 ( s )  2 +  4 λ n − 1 F n − 1 ( s )  2 + n − 1 X j =0  2 λ j F j ( s )  2 + n − 2 X j =0  4 λ j F j ( s ) + λ j +1 F j +1 ( s )  2 . Since λ j ≤ β and eac h F j is positive, all terms ins ide brack ets a re bounded by 4 β P 0 ≤ j ≤ n F j ( s ). Hence, | G ( s ) | ≤ 4 β √ 2 n + 1 S n ( s ) , and V ar h 1 2 n n X k =0 X ( k , T n + nτ ) i ≤ 4 β (2 n + 1 ) √ n + 1( T n + nτ ) 2 n S n ( T n + nτ ) 2 n , bec ause S n is an increasing f unction. The r esult follo ws from this estimate and Prop ositio n 2.3.  Pro of o f Theo rem 2 .1. It follo ws from Pro p o sition 2.3 a nd Pr op osition 5 .2 that the finite dimensio nal distributions of X n ( t ) converge to the finite dimensional distri- butions of a Dira c measure co ncentrated on the Gomper tz curve exp( − (ln 2) e − β τ ). GR OWTH D YNAMICS EVOL VING A CCORDING TO THE GOMPER TZ CUR VE 11 Tigh tness. T o conclude the proof o f Theorem 2.1, it remains to sho w that the pro cess X n is tigh t on eac h finite interv al [0 , T ]. Since 0 ≤ X n ( t ) ≤ 1 a.s. for all t and n , it is enough [4] to show that fo r any T > 0 and ǫ > 0, lim sup δ → 0 lim sup n →∞ P h sup   X n ( t ) − X n ( s )   > ǫ i = 0 , where the supremum is carried o v er all 0 ≤ s, t ≤ T suc h that | t − s | ≤ δ . B y construction, the pro ces s X n is increasing. In par ticular, if t i = iδ , 0 ≤ i ≤ M δ = ⌊ δ − 1 ⌋ + 1, wher e ⌊ a ⌋ stands for the int eger part o f a , P h sup   X n ( t ) − X n ( s )   > ǫ i ≤ M δ max 0 ≤ i ≤ M δ P h   X n ( t i +1 ) − X n ( t i )   > ǫ/ 2 i . By Prop osition 2 .3, W n conv erges uniformly to the Gomper tz curv e, whic h is uni- formly con tin uous. It is therefore enough to sho w that lim sup δ → 0 lim sup n →∞ M δ max 0 ≤ i ≤ M δ P h   X n ( t i +1 ) − X n ( t i )   > ǫ/ 2 i , where X n ( t ) = X n ( t ) − W n ( t ). By Chebyc hev inequality and by Prop osition 5.2, for a ll t ∈ R , a > 0, lim sup n →∞ P h   X n ( t )   > a i ≤ lim sup n →∞ 1 a 2 E h X n ( t ) 2 i = 0 . This co ncludes the pro of of the tightness of X n ( t ). 6. Central Limit Theorem W e prov e in this last section Theor em 2.2. Reca ll the definition of the martingale M n and o f the pro ces s V n . W e re ly in this pro o f o n [8, Theor em VI I I.3 .11]. Let ∆ M n ( s ) = M n ( s ) − M n ( s − ). By definition o f the martinga le M n , ∆ M n ( s ) = 2 − n/ 2 { X n ( s ) − X n ( s − ) } . Since X n ( s ) − X n ( s − ) is either 0 or 1, for every t ≥ 0, ǫ > 0, lim n →∞ P h sup s ≤ t   ∆ M n ( s )   ≥ ǫ i = 0 . (6.1) Denote by h M n i t the predictable quadratic v a riation o f the ma rtingale M n ( t ). W e claim that for each τ ≥ 0 , h M n i τ conv erges in probability a s n ↑ ∞ to e − ln 2 e − β τ − 1 2 . (6.2) A straight forward computation shows tha t h M n i τ = 1 2 n Z T n + τ n T n n X j =0 λ j X ( j, s ) ds . By (4.4), uniformly in an y compact interv al of R , lim n →∞ E h n 2 n n X j =0 λ j X ( j, T n + τ n ) i = d dτ e − ln 2 e − β τ . Hence, for all τ ≥ 0, lim n →∞ E h h M n i τ i = e − ln 2 e − β τ − 1 2 · 12 C. LANDIM, R. D. POR TUGAL AND B. F . SV AITER On the o ther hand, b y Sch warz ineq uality a nd with the notation in tro duced in Section 5 , E h h M n i τ − E  h M n i τ   2 i = E h 1 2 n Z T n + τ n T n n X j =0 λ j { X ( j, s ) − x j ( s ) } ds  2 i ≤ τ n 4 n Z T n + τ n T n n 2 X j

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