The duality of spatial death-birth and birth-death processes and limitations of the isothermal theorem

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1 The duality of spatial death-bith and bith-death pocesses and limitations of the isothemal theoem axiv: v1 [q-bio.qm] 17 Nov 2014 Kaman Kaveh 1, Natalia L. Komaova 2, Mohammad Kohandel 1 1 Depatment of Applied Mathematics, Univesity of Wateloo, Wateloo, Ontaio, Canada N2L 3G1 2 Depatment of Mathematics and Depatment of Ecology & Evolutionay Biology, Univesity of Califonia Ivine, Ivine, CA 92697, United States May 7, 2018 Abstact Evolutionay models on gaphs, as an extension of the Moan pocess, have two majo implementations: bith-death (BD) models (o the invasion pocess) and deathbith (DB) models (o vote models). The isothemal theoem states that the fixation pobability of mutants in a lage goup of gaph stuctues (known as isothemal gaphs, which include egula gaphs) coincides with that fo the mixed population. This esult has been poven by Liebeman et al (Natue 433: , 2005) in the case of BD pocesses, whee mutants diffe fom the wild types by thei bith ate (and not by thei death ate). In this pape we discuss to what extent the isothemal theoem can be fomulated fo DB pocesses, poving that it only holds fo mutants that diffe fom the wild type by thei death ate (and not by thei bith ate). Fo moe geneal BD and DB pocesses with abitay bith and death ates of mutants, we show that the fixation pobabilities of mutants ae diffeent fom those obtained in the mass-action populations. We focus on spatial lattices and show that the diffeence between BD and DB pocesses on 1D and 2D lattices ae non-small even fo lage population sizes. We suppot these esults with a geneating function appoach that can be genealized to abitay gaph stuctues. Finally, we discuss seveal biological applications of the esults. 1 Intoduction Exploing the effect of spatial stuctue of an evolutionay system and its impotance on the dynamics of selection has long been of inteest in population genetics [1]. The oiginal stepping stone model developed by Weiss and Kimua is the backbone of the subsequent implementations of spatial stuctue into evolutionay modeling [2]. Mauyama analyzed the fixation behavio of a Moan pocess in a geogaphically stuctued population, and was 1

2 able to show that egula spatial stuctues do not enhance o suppess selection [3,4]. Moe ecently, Libeman et al. evisited the poblem and extended pevious models to an abitay gaph, whee at each node of the gaph a single species can eside [5]. They wee able to show that some categoies of gaphs (such as a sta gaph, whee a cental vetex is connected to all leaf vetices) enhance the selection, while othe gaphs suppess the selection (see also [6,7]). Othe notable ecent woks have focused on the effect of spatial stuctue on evolution of coopeation [8] and on the eplicato dynamics on a gaph stuctue [9]. Selection dynamics on vaious types of andom gaphs has been the subject of a lot of inteest ecently in the context of applications to social netwoks [10], infectious disease and epidemiology [11, 12], and cance modeling [13, 14] among othes. Multi-hit pocesses and tumou suppesso gene inactivation [15], spatial patten fomation in evolutionay models [16, 17], and the effect of spatial distibution of fitness on selection [18] ae some othe diections of cuent eseach. Fo a geneal eview of evolutionay dynamics on gaphs and some ecent tends see [19]. In the Moan pocess, we assume the existence of N individuals. To keep the population constant it is assumed that each elementay update consists of a bith and a death event, whee individuals ae chosen fo bith and death fom the pool consisting of the entie population. It is of little consequence whethe the fist event in each elementay update is a bith o a death event. Howeve, the ode of these events can become impotant if we conside a spatial genealization of the Moan pocess. To this end, we conside a egula spatial lattice o an unstuctued mesh, and again impose the condition of a constant population. Fo each update, an independent bith (o death) event can occu at a andomly chosen site of the lattice, but the subsequent death (o bith) event should now happen at a neighboing site of the lattice to epesent the spatial stuctue and the fact that only daughtes of individuals that ae connected to a paticula site can be placed thee. In a bith-death (BD) pocess, a bith event is followed by a death event, and in a death-bith (DB) pocess, a death event happens fist. These ae the two main implementations of the genealized Moan pocess that have been consideed in the liteatue. Aside fom death-bith o bith-death models thee have been othe implementations of evolutionay pocesses on a gaph stuctue. Sood et al. [10, 20] discuss a model of link dynamics, whee instead of andom sampling of nodes, one consides andom samples edges (links) of a gaph as a fist elementay event and then updates the values of the two nodes on the chosen link based on the fitness of individuals thee (simila models ae discussed in [9,21]). Evolutionay games in the context of linking dynamics ae also studied in [22 24]. Anothe vaiation of bith-death models on a gaph is consideed by [25] whee each of the elementay events can be eithe a bith o a death o a migation (see also [14, 26]). It has been obseved by [3, 4] and late genealized by [5] that in the case of the BD pocess, selection dynamics ae not affected by egula and symmetic stuctues (o moe geneally isothemal gaphs). This elegant esult is efeed to as the Isothemal theoem. In this pape we exploe the applicability of this theoem to othe types of pocesses. The motivation fo this question is the esult of [13] whee the Moan pocess was studied on a 1D spatial lattice, and the DB fomulation was used. It was shown that the pobability of mutant fixation in this case was diffeent fom the one obtained in the space-fee Moan pocess. Theefoe, in this pape we study the connection between DB and BD pocesses on spatial lattices and exploe the extent to which the ode of bith and death events influences 2

3 the pobability of mutant fixation. In the liteatue, diffeences between DB and BD updates have been descibed in somewhat diffeent contexts. In papes by [27, 28], evolutionay games wee studied on gaphs, and diffeent behavio unde DB and BD models has been emphasized. In [9], the eplicato equation on gaphs is studied in the context of DB, BD, and imitation dynamics; applications include evolutionay games such as Pisone s Dilemma, the Snow-Dift game, a coodination game, and the Rock-Scissos-Pape game. Pape [29] studied evolution of coopeation, and fomulates the coesponding Moan pocess unde the DB and BD implementations. It is shown that fo DB updates, coopeation may be favoed in stuctued populations, while with BD updates this neve is. The authos popose a mixed ule whee in each time step DB o BD updates ae used with fixed pobabilities, and they futhe deive the conditions fo selection favoing coopeation unde the mixed ule fo vaious social dilemmas. Ou wok adds to these investigations by studying the diffeence between DB and BD updates in the Moan pocess in the context of mutant fixation. Anothe natual genealization of the Moan pocess that we focus on hee, concens the definition of fitness. In the conventional Moan model, all individuals ae chaacteized by thei fitness paametes. Usually, an individual is chosen fo division with a pobability weighted by its fitness, and an individual is chosen fo death with the pobability 1/N (such that all individuals ae equally likely to be chosen fo death, see e.g. [5, 29]). In a moe geneal setting, howeve, fitness could be influenced by the death ate as much as it is by the division ate. Fo example, a cance cell in tissue could diffe fom the suounding cells by its division ate and/o by its death ate. Theefoe, in this pape we aim to exploe how the division and the death ates both influence the fixation dynamics of mutants in the DB and BD pocesses. The est of this pape is oganized as follow. In section 2 we fomulate DB and BD models on a gaph, with geneal death and bith ates. We show that DB and BD tansition pobabilities can be tansfomed into each othe by using a duality popety of the two models. To study the diffeences between the DB and BD pocesses, we stat with the case of a complete gaph (whee all vetices ae connected, section 3) and deive closed algebaic foms fo fixation pobability in eithe of cases. In paticula, we show that the diffeence between DB and BD implementation on a complete gaph vanishes as 1/N, as the system size goes to infinity. In section 4 we discuss exact solutions of DB and BD models on a 1D cicle and show that the diffeences ae no longe negligible even fo lage system sizes. In section 5 we find an appoximation fo the fixation pobability in highe dimensions and the highe connectivities. Based on the esults obtained, we conjectue that the diffeence between the two models has the ode of the invese of the degee of connectivity of the gaph. This is suppoted by the exact solutions in the complete gaph and 1D cases, and numeical simulations in the 2D case. We show that ou appoximation in 2D fo a egula lattice matches the exact stochastic simulations within a vey good eo magin. Finally, a discussion is pesented in section 6. 3

4 2 Death-Bith (DB) and Bith-Death (BD) pocesses on a gaph Assume two species, A (nomal, o wild type) and B (mutant), with coesponding polifeation ates B and A and death ates d B and d A. A geneal death-bith pocess (DB) is defined as follows: at each time step, one cell at site i (eithe A o B) is andomly chosen to die with weight d A o d B. One of the neighbouing sites to i (denoted by j) is chosen at andom, with weight A o B to polifeate, and pobability w ij fo the offsping to be placed in site i (to eplace the dead cell). Fo each site i, we set n i = 1 if the esident cell is mutant, and n i = 0 if the esident cell is wild type. Thus the distibution of mutant cells at sites i is n i and the distibution of nomal cells is 1 n i. Given the vecto n = (n 1,.., n N ), the tansition pobabilities can be witten as: W + DB ({n i}) = W DB ({n i}) = d A (1 n i ) B j w ijn j d B k n k + d A k (1 n k) ( B A ) l w iln l + A d B n i A j w ij(1 n j ) d B k n k + d A k (1 n k) ( B A ) l w. (1) iln l + A Hee, W + DB (W DB ) stands fo the pobability that in the death-bith pocess, a new mutant (a new wild-type cell) appeas at site i afte one elementay update. Similaly, in a bith-death pocess (BD) the sequence of the two death and bith events is switched and can fomally be obtained by switching s and d s in equation (1), and also w ij w ji. W + BD ({n B n i d A j i}) = w ji(1 n j ) B k n k + A k (1 n k) (d B d A ) l w lin l + d A W BD ({n A (1 n i ) d B j i}) = w jin j B k n k + A k (1 n k) (d B d A ) l w. (2) lin l + d A In the case of BD, W + BD ({n i}) indicates that a new mutant appeas at any site j neighbouing site i. Notice that in the case of a DB pocess, the death event is a global event and the individual is chosen fo death among all the individuals on the gaph with weight of d A o d B. The following bith event, howeve, takes places among only the individuals that ae connected to the peviously chosen individual, and thus constitutes a local event. Even though both of the events occu though andom sampling with weighted biases d i o i, with i = A, B, the subset of andom sampling is smalle fo the second event unless the gaph is a complete gaph. Similaly, in a BD pocess, the global event is a bith event, while the death event takes place fo the neighbous of the chosen individual on the gaph and thus is a local event. Thee ae specific cases of the above pocesses that have been often discussed in the liteatue. The case of d A = d B = 1 has been typically implemented as eithe a DB o BD pocess on a gaph. As the eade can easily see, the opposite limit whee A = B = 1 while d A,B ae abitay, also leads to equivalent models while the dynamics of the two models ae 4

5 evesed. In othe wods a DB model with A = B = 1 is equivalent to a BD model with d A = d B = 1 and vice vesa. In the case of BD models with d A = d B it has been agued that fo a wide ange of gaph stuctues, known as isothemal gaphs, whee j w ij = i w ij = 1, the fixation pobability of the model is the same as the mixed population one [5] (see also [3], [4] and [7]). The main examples of isothemal gaphs ae egula lattices with peiodic bounday conditions, and the intoduction of boundaies o andomness in connectivities emoves the isothemal popety (fo effect of boundaies see [13], [14] and [6]). 3 Fixation pobability fo DB and BD pocesses on a complete gaph Let us conside the simplest pocess which is on a complete gaph, whee evey two nodes i and j ae connected, i.e. w ij = 1/N. Let us use the following convenient notation: B / A =, d B /d A = d. The Kolmogoov backwad equation fo the absoption pobability π m, whee m is the mutant population, is witten as π m = W + m+1π m+1 + W m 1π m 1 + (1 W + m W m)π m, π 0 = 0, π N = 1, (3) which is valid fo both BD and DB pocesses. Tansition pobabilities W ± m ae defined as the pobabilities to gain o lose one mutant in a system with m mutants. The geneal solution fo the fixation pobability stating fom only one mutant can be found in closed fom. Denoting we obtain fo the pobability of fixation, γ m = W m, W m + π 1 = N 1 j=1 γ 1... γ j. (4) Notice that the above esult is tue when the tansition pobabilities in such a one step pocess only depend on the numbe of mutants in the system at evey time step, and not on othe degees of feedom of the system. This condition holds fo the complete gaph, and, as we will see late, fo 1D ings. Next, we will conside some implementations of DB and BD pocesses on a complete gaph. Fist, we will assume the update ules whee the second event (death in the BD pocess and division in the DB pocess) occu in a neighbohood of a given cell, which includes the cell itself. That is, fo example, if a cell is chosen fo epoduction fist, duing the death event this (mothe) cell will be in a pool of cells that have a pobability to die. 5

6 This is equivalent to having a gaph on which evey node is connected to itself by a loop. (Ohtsuki and Nowak [8] call this an imitation updating in the context of game theoy on gaphs.) Late in this section, we will include a pocess whee the neighbohood does not includes self as a candidate fo the second event to take place. The pefeence fo eithe of the two updatings is somewhat abitay and depends on the paticula natue of the modeling poblem at hand. 3.1 The complete netwok including self Fist, we will conside the pocess whee the netwok of neighbos of a given cell includes the cell itself. To explain the update pocedue in moe detail, note that at evey time step, we andomly label a cell fo death (bith) and then label a cell fo bith (death). We let the death and bith events happen afte both labels have been assigned. This way, fo the complete gaph (mass-action) scenaio, the numbe of neighbos is always N, and we have the following tansition pobabilities: W m + m = m + N m N m dm + N m Wm N m = m + N m dm dm + N m. (5) Inteestingly, these pobabilities ae the same fo the DB and BD pocesses, the two ae completely equivalent in this case. We theefoe have γ m = d, and the following well-known esult fo the fixation pobability holds: π 1 = 1 d/ 1 (d/) N. (6) Next, we will study netwoks that do not include self. Fo example, in the BD pocess descibed below, afte the initial bith event, the cell that has just divided is excluded fom the death event. In the following section, fo the DB pocess, the cell that has been chosen fo death will not be paticipating in the subsequent epoduction event. 3.2 The BD pocess Fo a BD pocess on a complete gaph excluding self, the tansition pobabilities W ± m ae witten as, W + BD,m = m m + N m N m dm + N m d W BD,m = N m m + N m dm dm + N m 1. (7) 6

7 Fo d = 1 the atio of W BD,m /W + BD,m = 1/ fo any m, howeve, this is not in geneal tue as upon choosing a mutant/nomal cell to divide (bith event) the next death event among the est of the N 1 cells will nomalize diffeently in W and in W +. In othe wods, when a mutant cell is chosen to divide (W + contibution) the successive death event happens among the est of the m 1 mutants (excluding the one aleady chosen to divide) and N m existing nomal cells. When a nomal cell is chosen to divide (W contibution) the successive death event occus among m mutant and N m 1 nomal cells. In geneal the atio of W BD,m /W + BD,m is, γ BD m W BD,m W + BD,m = d (1 ) d 1. (8) N m + dm 1 Fo d = 1 the atio will be 1/ and substituting into equation (4) give ise to well-known Moan esult, equation (6), π 1 = 1 1 ( ) N. (9) 1 1 Fo geneal values of d, howeve, a closed fom fo π 1 can be obtained, π BD 1 = d(n 1)Φ ( d, 1, N+d 2 d 1 whee the Lech tanscendent is defined as k=0 (d 1) ) ( + Φ(z, s, a) = d 1 (N 1) ( d k=0 z k (a + k) s. ) N Φ ( d ) ), N+d 2, 1, d 1 To evaluate the lage-n behavio of the fixation pobability fo advantageous mutants, we note that the function Φ ( ) d N+d 2, 1, d 1 decays as a powe law as N, and (d/) N decays exponentially fo d <. Theefoe, we can ignoe the tem multiplying (d/) N. Futhe, we notice the following asymptotic behavio of the Lech tanscendent: ( ) z k Φ(z, 1, a) = a + kzk + O(1/a 3 1 ) = a 2 a(1 z) + z a 2 (1 z) + 2 O(1/a3 ). Substituting this expession with z = d/ and a = N+d 2 into the equation fo π d 1 1, and expanding futhe in Taylo seies in 1/N, we obtain π1 BD = 1 d (d 1)d + N + O(1/N 2 ). (10) The same expansion can also be obtained by a diffeent method. If we expand the expession fo γm BD, equation (8), in tems of small 1/N, we obtain γm BD = d (d 1)d N + O(1/N 2 ), (11) 7

8 an expession that does not depend on m. This means that up to the ode O(1/N) we can simply use the geometic pogession fomula and obtain the esult pesented in equation (10). 3.3 The DB pocess Similaly, fo a DB pocess on a complete gaph, the tansition pobabilities W ± m ae given by, and thus, W + BD,m = N m dm + N m m m + N m 1 W DB,m = dm dm + N m N m m + N m, (12) γm DB = d ) ( (13) m + N m Compaing equations (8) and (13) one can see a duality between the two models which comes natually fom the definiton of the models, γm DB 1 (, d) = γm BD (d, ). (14) This esults in an inteesting connection between the DB and BD esults fo fixation pobability. Also, the fixation pobability is obtained by π DB 1 = (N 1)(d ) 2 ( ) N (15) d ( Nd 2 N 3 d 2 + 2) + ((N 1) + d( N)), Fo = 1 this educes to equation (6), The expansion of equation (15) is, d 1 d N 1. (16) π DB 1 = 1 d ( ) d ( ) 1 N + O. (17) N 2 Again, the same esult can be obtained by fist expanding the expession fo γm DB, equation (13), in tems of 1/N, and then pefoming the summation of a geometic seies. 8

9 3.4 Conclusions fo the complete gaph scenaio In the case whee self is a pat of the neighbohood gaph, the BD and DB pocesses ae identical, and we obtain the exact fomula fo the pobability of fixation, equation (6). If self is not included, then the BD and DB pocesses ae diffeent fom each othe. The diffeence is of the ode of 1/N. If d = 1 fo the BD pocess then the pobability of fixation is given by expession (6). Similaly, if = 1 in the DB pocess, equation (6) holds. These esults can be intepeted as the isothemal theoem fo the mass action scenaio. Fo lage values of N, the pobabilities of fixation can be appoximated by expessions (11) and (17) fo the BD and DB pocesses, espectively. 4 Exact esults fo fixation pobability in 1D In this section we focus on the solutions of the DB and BD models on a cicle with abitay bith and death ates and d fo the mutant cells. Simila but moe esticted cases have been discussed in the liteatue. Papes [10], [20] and [7] consideed a epesentation of the model which is a weak-selection limit of the model we discussed in the pevious section and showed that the esult agees with the isothemal theoem. Pape [13] has appoached the poblem moe diectly by solving a backwad Kolmogoov equation with peiodic bounday conditions. We will conside a simila implementation to [13] and suggest an altenative and moe intuitive method to calculate the fixation pobability that can be extended to highe dimensions and diffeent connectivities. One dimension is a paticula case fo which the mutant clone keeps a constant numbe of nomal (o mutant) neighbous. Moeove, the death-bith events which lead to an incease o decease of the mutant population only occus at the two ends of the bounday between mutant and nomal clones. On the bounday, at any time, the numbe of mutant neighbous and nomal neighbous emain the same. The sequence of events is depicted in figue 1. Thee ae two exceptions to this condition and that is when thee is one mutant and N 1 nomal cells in the system, o N 1 mutant and 1 nomal cell, in which the mutant (nomal) cell has two nomal (mutant), adjacent neighbouing cells (see figue 2). This is not tue in highe dimensions. In highe dimensions, the clonal font geomety and even the topology of a mutant clone can fluctuate. In the following subsections we will conside the DB and BD pocesses whee the neighbohoods do not include self. The case whee self is included is consideed in Appendix A. 4.1 The DB model Fo the one-dimensional DB model, it tuns out that the condition W m/w + m = d/ holds tue fo 1 < m < N 1, since in this case, the tansition pobabilities W ± can be ewitten as, 9

10 W + : W : mutant cells nomal cells Figue 1: Sequence of events in a death-bith pocess on a line (cicle). pobabilities W ± ae indicated by the coesponding events. The tansition W + DB 1D,m = 1 2 m + (N m)d + 1 W DB 1D,m = d 2 m + (N m)d + 1. (18) The atio of tansition pobabilities is diffeent fo m = 1, i.e. whee thee is only one mutant in the population, and fo m = N 1, whee only one nomal cell is left in the population. This is due to the fact that the condition of having the same numbe of mutant o nomal cell neighbous on the bounday is not coect in the latte two cases. A single mutant cell in the beginning has no neighbouing mutant cells, and the last emaining nomal cell befoe a full takeove does not have any nomal neighbous left. Theefoe, we have W DB 1D,m W + DB 1D,m = Substituting these values into equation (4) gives, d/ 1 < m < N 1 d( + 1)/2 γ 1 m = 1 2d/( + 1) γ N 1 m = N 1 (19) π1 DB 1D 2d( d) = (d/) N 2 ( 1 2d) + d(d( 1) + 2) In the case whee d = 1, we obtain π DB 1D 1 = In the limit N and > 1, this educes to 2( 1) ( 3)(1/) N 2. π DB 1D 1 = 2( 1) 3 1, (20) 10

11 a esult obtained by [13]. Fo = 1, equation (20) gives π DB 1D 1 = 1 d 1 d N, which coincides with equation (6) with = 1. Equation (20) can be also obtained using a geneating function method (Appendix C). W + 1 : W + N : W 1 : W N : mutant cells nomal cells Figue 2: Tansition pobabilities fo the fist and last events of fixation. In the text γ 1 = W1 /W 1 + and γ N 1 = W N 1 /W + N 1. (Figue is depicted fo a DB pocess). 4.2 The BD pocess Due to duality between the fomulation of a DB pocess and its coesponding BD pocess (see equations (1) and (2)) the same calculation can be epeated fo a BD pocess by switching death and bith ates (d and W + W ). The tansition pobabilities fo the coesponding BD pocess ae, W BD 1D,m W + BD 1D,m = Substituting these values into equation (4) gives, π BD 1D 1 = d/ 1 < m < N 1 2d/(d + 1) γ 1 m = 1 (d + 1)/(2) γ N 1 m = N 1 (21) d 2 (1 + d)(d ) d 2 (d(d 1) (d + 1)) + (d/) N (d( + d + 1) ). (22) In the limit whee d = 1, the fixation pobability above takes the well-known fom fo a Moan pocess, π BD 1D = 1 1 ( ) N, (23)

12 see equation (6) with d = Compaison of one-dimensional DB and BD pocesses Below we analyze the fixation pobabilities fo the one-dimensional DB and BD pocesses obtained in this section, see equations (20) and (22). Compaison between the 1D spatial pocesses and the nonspatial Moan pocess We can compae the DB and DB pocesses with the nonspatial Moan pocess fo geneal values of and d. The esults ae as follows: If d > 1, the BD pocess has a highe pobability of fixation than the Moan pocess; if d < 1, then it has a lowe pobability of fixation, see figue 3(a). If > 1, the DB pocess has a lowe pobability of fixation than the Moan pocess; if < 1, then it has a highe pobability of fixation, see figue 3(b). In figue 3, we plot the atio of the pobability of fixation in the spatial pocess (BD is (a) and DB in (b)) and the nonspatial Moan pocess. The set of values that satisfy the isothemal theoem sepaates the egions whee the spatial pocess yields a highe and a lowe pobability of fixation, compaed with the Moan pocess. To pove the esults above, we conside the cases d < and < d sepaately, and take the limit N. Fo the case of the BD pocess, the expession π BD 1D /π Moan simplifies to + d d d d and d 2 (1 + d) d(1 + d + ) in the two cases. Both expessions ae geate than 1 if d > 1, and smalle than one othewise. Similaly, fo the DB pocess, in the cases whee d < and < d, the atio π DB 1D /π Moan simplifies to 2 d (2 + d) and 2d (1 + 2d ). Both expessions ae geate than one if < 1, and they ae smalle than one othewise. The isothemal theoem can be consideed a bodeline case, see the dashed lines in figue 3. This can be summaized by the following statements: The BD pocess satisfies the isothemal theoem in the paticula case whee d = 1. The DB pocess satisfies the isothemal theoem if = 1. Compaing the BD and DB pocesses to each othe Fo the same values of d and, the DB and BD pocess yield diffeent pobabilities of fixation. In the special case whee d = 1, the BD pocess has a highe fixation pobability than the DB pocess iff > 1. That is, in the BD pocess, advantageous mutants have a highe pobability of fixation, and disadvantageous mutants have a lowe pobability of fixation. 12

13 (a) (b) Bith- death Isothemal theoem Death- bith Isothemal theoem d d Figue 3: 1D pocess: Compaison of the BD and DB pocesses with the nonspatial pocess. (a) The atio π BD 1D /π Moan is plotted as a function of vaiables d and. The contous coespond to equal values of this quantity, and lighte colos mak highe values. (b) The same, fo the function π DB 1D /π Moan. The paamete N = 100 is used. If > d and N, the pobabilities of fixation have a simple fom: π BD 1D = (1 + d)(d ) d(d 1) (d + 1), πdb 1D = 2( d) d( 1) + 2. In this limit, the two pocesses have the same pobability of fixation if = 3 d 1 + d. In geneal, thee is a cetain theshold value, c, such that fo > c ( < c ), the BD pocess (the DB pocess) has a highe pobability of fixation. This can be intepeted as follows: fo mutants that ae sufficiently advantageous, the BD pocess is moe successful, and fo less advantageous mutants, the DB pocess is moe successful. This is illustated in figue 4(a), whee we plot the atio π BD 1D /π DB 1D as a function of the vaiables d and. The contous coespond to equal values of this quantity, and lighte colos mak highe values. Above the line = c, the pobability of fixation is lage fo the BD pocess. Anothe inteesting limit is fo lage-n and lage-. Even in the simple case of d = 1, while the BD pocess leads to π1 BD = 1 1/ 1, the DB esult is π1 DB = 2( 1)/(3 1) 2/3. The fact that in a DB pocess, the fixation pobability of an abitaily advantageous mutant neve appoaches unity might sound supising but in fact is vey intuitive. In a DB updating scheme, no matte how high the chances ae fo a single advantageous mutant to polifeate, the chance of extinction is always contibuted by the ealy death events. In the case of a 1D cycle, thee is a 1/N chance fo the mutant to be picked and eplaced by one of its neighbous in the fist time step, while thee is 2/N chance fo the mutants neighbous 13

14 to be chosen to die and be eplaced by a mutant offsping. The atio of the total numbe of events that the population changes to the ones that the mutant suvives is 2/3, which is exactly the fixation pobability fo limit. Neutality of the mutants The spatial pocesses equie a diffeent definition of neutality fo the mutant. In the nonspatial Moan pocess, neutal mutants satisfy /d = 1, and the pobability of fixation of such mutants is 1/N. Fo 1D spatial models, the sets of neutality in the space (d, ) ae pesented in figue 4(b). The ed line coesponds to the DB pocess, the blue line - to the BD pocess, and the black line to the nonspatial Moan model. These lines become vey close to the /d = 1 line as N inceases. (a) 2.5 (b) = c Advantageous DB BD Disadvantageous d d Figue 4: 1D pocess: popeties of the fixation pobabilities. (a) The atio π BD 1D /π DB 1D is plotted as a function of vaiables d and. The contous coespond to equal values of this quantity, and lighte colos mak highe values. Above the line = c, the pobability of fixation is lage fo the BD pocess. The paamete N = 100 is used. (b) Neutality of mutants. Plotted ae he lines in the (d, ) space along which the pobability of mutant fixation is equal to 1/N (the line π BD 1D = 1/N is blue, and the line π DB 1D = 1/N is ed; the line /d = 1 is black). Paamete N = 5 was used. 5 Appoximate esults fo fixation pobability in 2D In the following, we apply some of the undestanding gained fom the pevious analysis on DB models and genealize it to find an appoximate analytical esult in highe dimensions. The 1D case was athe special due to the following popeties: The numbe of mutant/nomal neighbous at evey time duing the evolution of the clone emains constant on the two fonts of the mutant clone. The only two exceptions 14

15 ae when thee is only one mutant in the system (N 1 nomal cell) o one nomal (N 1 mutant). Due to the above popety, the atio of tansition pobabilities fo any clone size emains the same as fo the mixed population Moan model othe than the two exception of n = 1 and n = N 1. Including these two new tansition pobabilities into the solutions of the Kolmogoov equation lead to an exact esult fo the fixation pobability of the 1D model. The topology of the mutant clone does not change with time. If we begin with a single mutant the domain of the mutant clone always emains a simply connected egion. In highe dimensions, none of the above popeties ae coect in a stict mathematical sense. The tansition pobabilities in two- o highe dimensions ae not only functions of the clone size but of the geomety of the clonal font. Also thee ae elementay events that can lead to splitting of a clone into two and thus the topology of a clone might change with time. As in the 1D case, we assume peiodic bounday conditions and without loss of geneality we conside a egula squae lattice with degee of connectivity k. The sequence of events fo a DB pocess on a squae lattice is depicted in figues 5 and 6. The atio of tansition pobabilities in highe dimensions fo loss o gain of a mutant cell diffes fo lage clone sizes and small clone sizes. Hee, we assume that the dominant contibution comes fom m = 1 (in 1D, m = 1 was the only diffeent tem), and use the appoximation W m/w + m d/ fo m 2 (see Appendix B). We have W DB 2D,m W + DB 2D,m Thus, we obtain, = d/ 2 < m < N 2 d( + k 1)/(k) = γ1 DB 2D m = 1 = d/ m = 2 (24) π DB 2D 1 = 1 + N 1 j=2 ( j W k W + k=2 k 1 ) N 2 ( ( 1 d ( ) d ) ) γ DB 2D 1 γ DB 2D 1,, (25) upon substitution fom equation (24) we end up with an algebaic expession fo π DB 2D 1, fo lage N, π DB 2D 1 k( d) k + d( 1). (26) 15

16 W + : W : mutant cells nomal cells Figue 5: Sequence of death and bith events on the bounday of mutant clone (ed) that give ise to the incease o decease of mutant population. (Figue is depicted fo a DB pocess). W + 1 : W + 2 : W 1 : W 2 : Figue 6: Sequence of death and bith events in the ealy stages of one-mutant and twomutant clone (ed=mutant). (Figue is depicted fo a DB pocess). Fo d = 1 and k = 2, this educes to 2( 1)/(3 1). Fo lage k, we obtain π DB 2D 1 1 d/. Equation (26) can be compaed with the esult of the stochastic simulations esult on a squae lattice (k = 4, d = 1) as shown in figue 7. We can even test the validity of this appoximation fo 2D egula gaphs fo smalle lattice sizes. To do so we can use the finite-n contibution fom equation (25). In figue 8 16

17 fixation pobability moan DB 2D (eight neighbou) DB 2D (fou neighbou) DB 1D (cicle) fitness N=20x20 (2D) and N=20 (1D) compaison w lage-n theoetical esults (solid lines) Figue 7: Compaison of analytical esults fo the fixation pobability π 1 as a function of the division ate fo 1D DB (N=20) and 2D DB (N=20x20) cases with k=4 (4-neighbou) and k=8 (8-neighbou) egula lattices with peiodic bounday conditions. The solid lines ae analytical esult fom equation (9) (mixed population Moan model), equation (20) (1D DB with d = 1) and equation (26) (2D DB with d = 1) while cicles ae the esults of stochastic simulations with 40,000 iteations. we plotted the esult fo squae lattices (k = 4) with N = 3 3 and N = 5 5 and N = The match is good fo close to neutal limit cases while thee is slight deviation fom the analytical pediction as fitness inceases away fom the neutal limit. We can also futhe test validity of equations (25) and (26) by compaing the esults fo fixation pobability when one begins with two mutants, i.e. the tansition pobability γ 1 which is claimed to be the main cause of deviation fom isothemal behaviou in the DB case does not have any effect and we do expect the fixation pobability to be close to a mixed population fomula with the two-mutant initial condition. Notice that γ 1 is not the only cause of deviation fom the isothemal esult as othe tansition pobabilities γ i ae only appoximately l/. This is in fact the case fo the esults of one-mutant and two-mutant fixation pobabilities which ae depicted fo a squae lattice with N = and d = 1 in figue 9. Recalling the Moan fixation pobability with the two-mutant initial condition π 2, 17

18 fixation pobability DB 2D N=20x20 DB 2D N=5x5 DB 2D N=3x3 fitness DB2D esult (solid lines) in small sizes N=3x3(blue), N=5x5 (ed) also compaed with N=20x20 (magenta cicles). Figue 8: Compaison of analytical esults fo the fixation pobability π 1 as a function of the division ate fo a squae lattice 2D DB with diffeent lattice sizes cases with k=4 (4- neighbou). Solid lines ae analytical esults fom equation (25) while cicles ae the esults of stochastic simulations with 40,000-80,000 iteations (fo smalle sizes we used highe numbe of iteations to educe the statistical eo). π Moan 2 = ( ) ( ) N. (27) 1 1 The exact esult can be obtained (in eithe 1D cicle o 2D egula lattices) by using the ecusive elation, ( π 2 = ) π 1, (28) which is obtained fom solutions of Kolmogoov equation. This leads to the following esults fo π 2 in 1D and 2D cases in the lage-n limit, 18

19 2 1 fixation pobability 2 moan 2-mut i.c. DB 2D 2-mut i.c. DB 1D 2-mut i.c. moan 1-mut i.c. 1 DB 2D 1-mut i.c. DB 1D 1-mut i.c. fitness Figue 9: Compaison of fixation pobability fo two-mutant and one-mutant initial conditions fo N = 100 (1D DB) and N = squae lattice (2D DB) as a function of division ate. Solid lines ae coesponding theoetical esults. As can be seen the two-mutant esults ae close to Moan fomula, equation (28). π2 DB 1D = ( 3 1 π DB 2D 2 = ), ( 1 1 ). (29) The above appoximation fits a lage-n simulation esults well as shown in figue 9. Since equations (29) ae obtained using a mixed population fom of Kolmogoov equation, the fits with the simulations futhe suppot ou assumptions in deiving the fomula fo the 2D DB fixation pobability, given by equation (25). In the case of a BD updating, while the special case of d = 1 leads to the isothemal theoem fo egula gaphs, the fixation pobability fo abitay values of and d might not follow isothemal behaviou analogous to the case discussed ealie. Simila to the 2D DB case we obtain, 19

20 W BD 2D,m W + BD 2D,m = d/ 2 < m < N 2 kd/((d + k 1)) = γ1 BD 2D m = 1 = d/ m = 2 (30) This gives, fo lage N, π BD 2D 1 (d + k 1)( d) (d + k 1)( d) + kd. (31) Fo d = 1, equation (31) educes to 1 1/ (independent of k). π1 BD 2D 1 d/. Setting k = 2, this educes to the 1D case. Fo lage k, we have 6 Discussion In this wok, we discussed genealized vesions of death-bith and bith-death updating models fo evolutionay dynamics on gaphs and spatial stuctues. In contast to most pevious appoaches, we assume that the mutants can diffe fom the wild type not only by thei bith ate (), but also by thei death ate, (d), giving ise to a a two-paamete model whee and d ae independent abitay paametes affecting the selection pocess. This is in fact moe ealistic, as in many scenaios eithe death ate o bith ate o both can vay and affect the selection dynamics in tissues in situations involving invasion mechanisms such as cance. Focusing on egula lattices with peiodic bounday conditions, we show that the models fo both DB and BD updating ae exactly solvable in a 1D cicle. We futhe studied to what extent the isothemal theoem can be genealized. This theoem states that the selection dynamics ae the same on any isothemal gaph as they ae fo a mixed population/complete gaph. This theoem was poven in [5] to hold fo the d = 1 BD update, and hee we show that it also holds fo the = 1 DB update. We futhe demonstated that the moe geneal, two-paamete models deviate fom this theoem. The BD and DB systems exhibit fixation pobabilities that ae diffeent fom each othe and fom the canonical mass-action Moan esult. While fo the case of the complete gaph, the diffeence is vanishingly small fo lage gaphs, it emains finite fo 1D and 2D spatial stuctues. In geneal, we conjectue that the diffeence between the two models has the ode of the invese of the degee of connectivity of the gaph. Fo example, the connectivity of the complete gaphs (the mass-action system) is N, and thus the diffeence scales with 1/N. The connectivity of egula spatial lattices is given by the numbe of neighbos of each node. In the 1D ing, the numbe of neighbos is 2, and in typical 2D lattices, it is 4 o 8. Thus the diffeence in the fixation pobability between the BD and DB models is the lagest fo the 1D neaest neighbo ing, it is slightly smalle in the case of the von Neumann (4-cell) neighbohood in 2D, and still smalle in the case of the Mooe (8-cells) neighbohood. It decays with the the numbe neighbos. The eason fo the deviation fom the isothemal theoem is the diffeent tansition pobabilities fo the case when thee is only one mutant in the system, compaed to the est of the tansition pobabilities. When the quantity of inteest is the fixation pobability 20

21 stating fom moe than 1 mutant cells, then the diffeence between DB and BD models (and the deviation fom the isothemal theoem) is smalle, compaed to the case when we stat with one mutant. Depending on the paametes, the spatial DB and DB pocesses could be chaacteized by a smalle o lage mutant fixation pobability compaed with the conventional non-spatial Moan esult. In paticula, fo the BD pocess, the fixation pobability is highe than the Moan value as long as d > 1. Fo the DB pocess, the fixation pobability is highe than the Moan value as long as < 1. The genealized two-paametic BD and DB models can be applied to investigate the evolutionay dynamics of tissue tunove. We would like to emphasize the impotance of consideing both bith and death ates of mutants (compaed to those of the suounding cells) in tems of the conceptual constuct of the hallmaks of cance [30]. Thee of them ae the most elevant fo ou study. They ae listed below and put in ou context by linking them with the cells bith and death ates [31 33]: 1. Self-sufficiency in gowth signals. While nomal cells cannot polifeate in the absence of stimulatoy signals, cance cells can do this with the help of oncogenes, which mimic nomal gowth signaling. This can be achieved by means of diffeent mechanisms. Fo example, cells can acquie the ability to synthesize thei own gowth factos, e.g. the poduction of PDGF (platelet deived gowth facto) and TGF-α (tumo gowth facto α) by glioblastomas and sacomas. Futhe, cell suface eceptos that tansduce gowth-stimulatoy signals into the cell, fo example EGFR and HER2/neu, may be ove expessed o stuctually alteed, leading to ligand-independent signaling. Finally, downsteam tagets of the signaling pathway can be alteed, e.g. the Ras oncogene, which is found mutated in about 25% of human tumos. In all these cases, the mutant cells ae chaacteized by an inceased bith ate compaed to the suounding cells. 2. Insensitivity to antigowth signals. Antigowth signals can block polifeation by (i) focing cells out of the active polifeative cycle into the quiescent (G0 ) state, until appopiate gowth signals put them back into the cell cycle; o (ii) inducing diffeentiation, which pemanently emoves thei polifeative potential. Cance cells evade these antipolifeative signals, by e.g. loss of TGFβ, loss of Smad4, o loss of CDK inhibitos such as p16, p21, o p53. The coesponding cells again ae chaacteized by an inceased bith ate compaed to the suounding cells. 3. Evading apoptosis. The ability of tumo cell populations to expand in numbe is detemined not only by the ate of cell polifeation but also by the ate of cell attition. Pogammed cell death (apoptosis) epesents a majo souce of this attition. Resistance to apoptosis as can be acquied by cance cells by e.g. loss of p53 (which nomally activates po-apoptotic poteins and epesents the most common loss of a poapoptotic egulato), o by activation o upegulation of anti-apoptotic Bcl2. In these cases, the mutants ae chaacteized by a deceased death ate compaed to the suounding cells. Othe applications ae the cases when selection dynamics ae affected by the intoduction of a dug, which inceases the mutant cell death while the bith ate is detemined by the cell 21

22 division ate and is independent of the dug concentation. This can be applied to poblems in both cance and infectious diseases, whee spatial stuctues affect selection dynamics. Anothe inteesting application is the stem cell dynamics in the case of the intestinal cypt. Recently, Vemuelen et al [34] investigated the dynamics of niche success in the intestinal cypt base by inducing diffeent types of mutations including, APC, p53 and Kas, and wee able to measue the faction of mutant stem cell clones at vaious time points fo a wide numbe of cypts (in mice). The authos fitted the esults to a simple bith-death model on a one-dimensional ing. They used Bayesian infeence to infe model paametes such as polifeation ate and total niche size N. It has been obseved that Kas oncogenic mutation infe a elatively high selection advantage in a newly intoduced mutant to the stem cell niche while APC /+ is weakly disadvantageous and APC / mutation in a backgound of APC /+ mutants is weakly advantageous. Similaly, it has been epoted that p53 mutations in a nomal intestinal base infe a vey small selection advantage while in the inflamed gut the advantageous p53 has the highe chance of succeeding in the niche. As discussed above, to compae with the expeimental data, the bith and death ates of nomal and mutant stem cells should be taken into account independently, paticulaly when both mechanisms ae contibuting to the selection dynamics at the same time. In the case of intestinal cypt stem cells, aside fom the fact that diffeent mutations can confe diffeent death and bith ates (o combination of both), othe mechanisms such as symmetic diffeentiation, and cell cycle and quiescent states can act as additional effective mechanisms fo a death event in such an evolutionay model. We have used the the epoted fixation pobability to estimate the possible set of death and bith ates. We also applied both DB and BD pocesses to see if thee is any significant diffeence between the two models. The esults ae depicted in figues 10a and 10b. In figue 10a, we have plotted the lines indicating possible sets of bith ate and death ate fo thee mutants, Kas, APC /+ and APC /. Dashed lines indicate the values fo death ates d = 1 and vaious bith ates epoted in [34]. Simila esults have been depicted fo the case of a DB pocess whee the death event occus fist. In the case of elatively advantageous mutations, such as Kas, the d = 1 case leads to an unusual high division ate. In fact highe diviso ates also point to much highe bith ates due to negative cuvatue of the (, d) gaphs. This is basically a special case of the duality epoted in the pevious sections. Ou finding also suppots the belief that the stem cell dynamics inside a cypt niche is dominated by bith (division) events, followed by death events due to geometical constaints in the systems. Such death events some times ae efeed to as a etaction in the biological context [35]. Acknowledgments: We would like to thank Ane Taulsen fo fuitful discussions. M. Kohandel is suppoted by the Natual Sciences and Engineeing Reseach Council of Canada (NSERC, discovey gants) as well as an NSERC/CIHR Collaboative Health Reseach gant. 22

23 Figue 10: The plots of (d, ) sets fo diffeent mutations. The values of the fixation pobability ae extacted fom [34]. 23

24 A 1D pocesses whee self is included in the neighbohood netwok Let us conside a one-dimensional, neaest neighbo pocess, whee the cell itself is consideed to be one of its own neighbo. Theefoe, instead of two neighbos, each cell has thee neighbos. Pefoming calculations simila to those of Section 4, we obtain the following esults. Note that although qualitatively, the fixation pobabilities behave simila to those of Section 4, the numeical values diffe significantly in the two pocesses. Thus, whethe o not we include self in the neighbohood will significantly change the esulting fixation pobabilities, and the change does not disappea in the lage N case, like it did in the mass-action scenaio. A.1 The BD pocess We have the following expession fo tansition ate atios: γm BD = W d(1 + 2d)/((d + 2)) 1 < m < N 1 BD,m W + = d/ m = 1 BD,m d/ m = N 1 (32) Note that if d = 1, we have γm BD = 1/ fo all m. Using equation (4), the pobability of fixation can be easily calculated. Because the expession is athe cumbesome, hee we only pesent the appoximation fo the case when simultaneously, > d, > d(1 + 2d)/(2 + d), and N : π1 BD d(d + 2) = 1 (d + 2) d(d 1). The isothemal theoem (fo the geneal N) is satisfied fo d = 1. A.2 The DB pocess In this case, the tansition ate atios become: γm DB = W d( + 2)/(1 + 2) 1 < m < N 1 DB,m W + = d/ m = 1 BD,m d/ m = N 1 (33) If = 1, we have γm DB = d fo all m. Using equation (4), the pobability of fixation can be calculated; again, the expession is cumbesome. Hee we only pesent the appoximation fo the case when simultaneously, > d, d < (1 + 2)/( + 2), and N : The isothemal theoem is satisfied fo = 1. π1 BD d(1 + 2) = 1 d( 1) + (2 + 1). 24

25 B 2D gaph tansition pobability In this section we biefly discuss the atio of tansition pobabilities fo both DB and BD models when the mutant population is lage. Fo simplicity, we assume d A = d B = 1, A = 1, and B =. Using equations (2), the atio of tansition pobabilities ae given by W BD W + BD = i,j i,j w ji n j (1 n i ) ( 1) k n k + N. (34) w ji n i (1 n j ) ( 1) k n k + N Fo isothemal gaphs the sum ove indices i and j can be switched and the atio becomes 1/ fo any configuation. On the othe hand, this is not quite tue fo the DB pocess. The atio of tansition pobabilities, using equations (1) ae, W DB W + DB = i,j i,j w ij n i (1 n j ) N (( 1) l w iln l + k), (35) w ij n j (1 n i ) N (( 1) l w iln l + k) whee k = l w il is the numbe of neighbous fo each site. Notice that the denominato and numeato do not simplify as the tems in the denominato now depend on i as well. Howeve, fo lage mutant populations, we can appoximate ( 1) l w iln l + k and make it independent of the index i. In this case, one can simplify the atio and aive at the esult 1/. This is, howeve, incoect fo smalle populations, as discussed in section 5. The main contibution comes fom the m = 1 population. Fo m = 1, 2 (m is the mutant population), we have W1 = + 3 W + 1 W2 W 2 + Simila calculations fo m = 3 gives W 3 /W + 3 = /. = 1. (36) C Geneating function fomalism C.1 Mixed population in weak selection In the following we epeat the altenative deivation of the fixation pobability fo a nonspatial Moan model fom [36]. Beginning with the maste equation fo the Moan pocess in the weak-selection limit, p(n; t) t = W + (n 1)p(n 1; t) + W (n + 1)p(n + 1; t) (W + (n) + W (n))p(n, t), (37) 25

26 whee p(n; t) is the pobability of having n mutants at time t with a given initial condition. The tansition ates W ± (n) ae given by W + (n) = W (n) = n(n n) ((n + (N n))(dn + (N n)), d n(n n) ((n + (N n))(dn + (N n)), (38) whee the tansition pobabilities at the absobing states ae taken to be zeo, W ± (n = N) = W ± (n = 0) = 0. Pape [36] consideed the special case of the above model whee d = 1, 1 1. This way the denominato in tansition pobabilities W ± (n) can be appoximated by N 2. The pobability geneating function F (z, t): obeys the following equation (d = 1): F (z, t) t F (z, t) = m=0 z n p(m, t), (39) = (z 1)(z 1) z Notice that equation (40) is deived using the identity, ( ) F (z, t) N F (z, t) z. (40) z nz n = z z n. (41) z n Bounday and initial conditions fo the PDE in equation (40), ae, n F (1, t) = 1 (bounday condition) F (z, t = 0) = z n 0 (initial conidian), (42) whee n 0 is the numbe of mutant cells at t = 0. The steady state solution fo F (z, t) is, F (z, t ) = A + Bz N, (43) whee coefficients A and B ae extinction and fixation pobabilities. Applying the bounday and initial conditions and noticing that z = 1/ is a fixed point of equation (40), we can solve fo the coefficient B to obtain, ( ) n0 1 1 π n0 = B = 1 ) N. (44) The tivial fixed point z = 1 leads to the absoption pobability being unity. This coesponds to the initial condition of beginning with N initial mutants. Notice that in geneal we do not need to find a fixed point of the PDE fo the geneating function but athe a point z (o set of points) fo which the ight-hand-side of equation (40) is zeo. 26 ( 1