Exponential growth of bifurcating processes with ancestral dependence

25 Pages·2014·0.29 MB·English

Exponential growth of bifurcating
processes with ancestral dependence
Sana Louhichi∗ and Bernard Ycart∗
3
1 Laboratoire Jean Kuntzmann
51 rue des Math´ematiques 38041 Grenoble cedex 9, France
0
2
Abstract: Branching processes are classical growth models in cell kinet-

l ics. In their construction, it is usually assumed that cell lifetimes are in-
u dependent random variables, which has been proved false in experiments.
J Modelsofdependentlifetimesareconsideredhere,inparticularbifurcating

1 Markovchains.Underhypothesesofstationarityandmultiplicativeergod-
icity,thecorrespondingbranchingprocessisprovedtohavethesametype
] ofasymptotics asitsclassiccounterpart inthei.i.d.supercriticalcase: the
E cell population grows exponentially, the growth rate being related to the
P exponent of multiplicative ergodicity, in a similar way as to the Laplace
transformoflifetimesinthei.i.d.case.Anidentifiablemodelforwhichthe
.
o multiplicative ergodicitycoefficients and the growthrate can beexplicitly
i computed isproposed.
b
- AMS 2000 subject classifications:Primary60J85; secondary92D25.
q
Keywords and phrases: branching process, bifurcating Markov chain,
[
multiplicativeergodicity,cellkinetics.

1
v
3 1. Introduction
4
3 Let T denote the infinite complete binary tree where each vertex has exactly
0
. 2 descendants. Let (Tv)v∈T be a bifurcating process, i.e. a family of positive
7 random variables indexed by T, defined on a probability space (Ω,F,P). The
0 vertices of T are interpreted as cells, and T as the lifetime of cell v. The root
3 v
(ancestor) of the tree being born at time 0, let N the number of individuals
1 t
: alive at time t: (Nt)t>0 is a continuous time branching process (precise defini-
v
tions will be given in section 2). If lifetimes are i.i.d., the population N grows
i t
X exponentially in t: this is a particular case of one of the most basic results of
r the theory (see Bellmann & Harris [3], Harris [17, Chap. VI], and Athreya &
a Ney [1, Chap. IV]).
Theorem1.1. Assumethatthelifetimes T arei.i.d. copies ofanalmostsurely
v
positive random variable T with non lattice distribution. Then:
lim e−νtN =W a.s., (1.1)
t
n→∞
where:
• W is a random variable with expectation C and finite variance,
∗ResearchsupportedbyLaboratoired’ExcellenceTOUCAN(ToulouseCancer)
1
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 2
• the growth rate (also called Malthusian parameter) ν is such that:
2E[e−νT]=1, (1.2)
• the proportionality constant C is:
C = 4νE[Te−νT] −1 . (1.3)
(cid:0) (cid:1)
The aim of this paper is to extend Theorem1.1 to models in which lifetimes
maybedependent,andinparticulartoBifurcatingMarkovChains(BMC).Our
main result (Theorem 3.1) generalizes Theorem 1.1 to the case where (Tv)v∈T
is a multiplicatively ergodic, stationary BMC. The growth rate ν and the pro-
portionality constant C in that case are related to the multiplicative ergodicity
coefficients of birth dates.
Applications of branching process to cell lineage studies have a long his-
tory (see e. g. [25] and references therein). Independence of lifetimes was ques-
tioned very early: see [24]. Indeed, actual data show two types of correlation
[47]: between the lifetimes of a mother and its two daughters, and between
the two sisters conditioning on the mother; they will be referred to as mother-
correlationand sister-correlation.It was remarked long ago by Powell[42] that
sister-correlationsdo not influence exponential growth (see also [7, 18] and [17,
section 28.2 p. 158]). The effect of mother-correlation on growth rates was
discussed by Harvey in [18]. Since then, many models have been proposed to
account for ancestry dependence, in particular by Smith & Martin [45], Lu¨ck
& Lu¨ck [32], Brook [5], or Murphy et al. [36]: see [33, 38]. Here, lifetimes are
seen as a stochastic process indexed by the binary tree; see Pemantle [40] and
Benjamini & Peres [4] as general references on tree-indexed processes. Under
a minimal hypothesis of stationarity, exponential growth for the mean popula-
tion size E[N ] is proved and the growth rate ν as well as the proportionality
t
constant C are expressed in terms of the Laplace transforms of cell birth dates
(Theorem2.1).AsymptoticsofLaplacetransformsforpartialsumsofaMarkov
chain are usually described by multiplicative ergodicity properties, which have
beenthoroughlystudiedbyMeynandhisco-workers[2,27,28,34];seealso[35,
p.519]forashortintroduction.ItisthereforenaturaltouseaBMCasamodel
oflifetimes:seeBenjamini&Peres[4]fortree-indexedMarkovchains,Hwang&
Basawa[19]formoreasymptoticresults,andGuyon[16]forapplicationstocell
lineage data. Under a multiplicative ergodicity condition, Theorem 1.1 is gen-
eralized: e−νtN is shown to converge almost surely; moreover,the growth rate
t
ν andthe proportionalityconstantC areexplicitly relatedtothe multiplicative
ergodicity coefficients (Theorem 3.1). The proof follows a classical scheme, al-
ready used by Bellman & Harris for the i.i.d. case in [3]. It consists of studying
the firstandsecondmoments ofN ,thenproveconvergenceinquadraticmean,
t
and finally deduce almost sure convergence. This is related to what Pemantle
calls the “second-moment method” [40, section 2.3]. In applying it, we have
triedto give the weakestpossible conditions at eachstep, startingwith the sta-
tionarity hypothesis of Theorem 2.1. Proposition 6.1 gives sufficient conditions
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 3
that ensure quadratic convergence of e−νtN , Proposition 6.2 gives conditions
t
for almost sure convergence. These conditions will be shown to hold under the
hypotheses of Theorem 3.1.
An obvious drawback for applications is that the growth rate ν and the pro-
portionality constant C cannot be computed in general. Therefore an explicit
model, potentially adjustable to observed data and for which ν and C can be
computed in terms of the transition kernel, had to be proposed. It was con-
structed as a quadratic transformation of a bifurcating autoregressive process
[6,16,10,9].Itdependson5identifiableparameters,(location,scale,andshape
for lifetime distribution plus mother- and sister-correlations) and can be fitted
to actual data.
Having in mind the application to cell lineage studies, it was natural to write
theresultsforthebinarytree.Nevertheless,theyextendquitestraightforwardly
toprocessesontheinfinite completek-arytreefork >2,atthe onlyexpenseof
heaviernotations.Remarksinthetextwillmakethegeneralizationmoreprecise.
Further extensions are possible, firstly to the case where T is a supercritical
Galton-Watsontreeandthelifetimesofdaughtersareindependentconditionally
on their common mother, secondly to the case where cell deaths are modelled
by a binary process as in [10]. They will be the object of future work.
The paper is organized as follows. In section 2 the branching process (N )
t t>0
associated to a bifurcating lifetime process (Tv)v∈T is defined. Two notions of
stationarity along lineages are introduced and the exponential growth of E[N ]
t
is proved. Section 3 is devoted to the definition of a BMC, and the statement
of Theorem 3.1. The explicit example of a BMC for which the multiplicative
ergodicity coefficients can be computed, is presented in section 4. The relation
between mother-correlation and growth rate for a fixed marginal distribution
of lifetimes is discussed in section 5. Section 6 is devoted to conditions under
which e−νtN converges in L2 and almost surely. These conditions are verified
t
for a multiplicatively ergodic BMC in section 7.
2. Stationary bifurcating processes
Inthissection,notationsonbifurcatingprocessesareintroduced.Thebirthdate
process (Sv)v∈T and the branching process (Nt)t>0 associated to a bifurcating
process (Tv)v∈T are defined, and related by Lemma 2.1. Two notions of sta-
tionarityareintroduced:birth-stationarity(Definition2.2)isthestationarityof
birth dates in a given generation; fork-stationarity (Definition 2.3) is the sta-
tionarityofcouplesofbirthdateswhenthegenerationsofthetwocellsandtheir
most recent common ancestor are fixed. Under birth-stationarity the expecta-
tion of N is proved to grow exponentially, and the parameters of exponential
t
growth ν and C are related to the Laplace transforms of birth dates (Theorem
2.1).
Someclassicalnotationsforinfinitetreeswillberecalledfirst:seePemantle[40].
The infinite rooted complete binary tree is denoted by T and its root by 0. If v
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 4
is a vertex ofT, the number of edges connecting v to the rootis denotedby |v|.
If v and w are two vertices of T, v 4 w is the order relation that holds if v is
in the path from 0 to w; v∧w is the most recent common ancestor of v and w,
i.e. the vertex at which the paths from 0 to v and w diverge. If v 6= 0, v˜ is the
vertex such that v˜ 4 v and |v˜| = |v|−1 (referred to as the mother of v). For
n>0, the n-th generationΓ is the set of vertices v such that |v|=n (vertices
n
at distance n from the root). One simple way to explicitly construct T is to
identify Γ to the set of binary vectors of length n+1, with first coordinate 0.
n
With that identification, v 4w iff v coincides with the |v|+1 first coordinates
of w. The mother of v, v˜ is deduced from v by removing its last coordinate.
The two daughters of v are obtained by appending to v a new coordinate 0
or 1: they will be denoted by v0 and v1. The concatenation of n zeros will be
denotedby0n ∈Γ .Besidesalgorithmicconsiderations,oneadvantageofthis
n−1
construction is to naturally endow T with the alphabetical order.
A bifurcating process is a set of almost surely positive random variables
(Tv)v∈T indexedbythe binarytreeT: Tv isthe lifetime ofcell v.Thebirthdate
process(Sv)v∈T isalsoabifurcatingprocess:Sv is thesumofcelllifetimes from
0 to v˜. The branching process (N ) is the counting process of living cells at
t t>0
time t.
Definition 2.1. Let (Tv)v∈T be a bifurcating process.
1. For v ∈T, The birth date of cell v is defined by S =0 and for |v|>0:
0
S =S +T . (2.1)
v v˜ v˜
2. For t>0, the number of living cells at time t is defined by:
N = I − I , (2.2)
t Sv6t Sv06t
v∈T v∈T
X X
where I denotes the indicator of event A.
A
If S is the birth date of cell v, the common birth date of its two daughters
v
S =S is also the death date of v. So (2.2) expressesthe fact that cells alive
v0 v1
at time t are the set difference of cells born no later than t with cells dead no
later than t. A simpler expression will be used.
Lemma 2.1. With the notations above,
1 1
N = + I . (2.3)
t 2 2 Sv6t
v∈T
X
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 5
Proof. From (2.2), and using the relation S =S ,
v0 v1
N = I − I
t Sv6t Sv06t
v∈T v∈T
X X
1
= I − I
Sv6t 2 Sw6t
v∈T w∈T
X X
w6=0
1
= 1+ I ,
2 Sv6t
v∈T
X
v6=0
hence (2.3).
Remark 2.1. On the k-ary tree, (2.3) becomes:
1 k−1
N = + I .
t k k Sv6t
v∈T
X
Consider the particular case where lifetimes in a given generation are con-
stant:
∀v ∈Γ , T =T ,
n v 0n+1
Denote by S the common birth date of all cells in generation Γ , and assume
n n
a law of large numbers is satisfied.
S
lim n =t¯>0 a.s.
n→∞ n
The rank of the generation alive at time t, denoted by G , is the counting
t
process associated to the sequence (Sn)n∈N, and Nt =2Gt. Since Nt doubles at
S , e−νtN never converges,although
n t
log(N ) log(2)
t
lim = a.s.
t→∞ t t¯
Consider now
log(E[Nt]) log(E[eGtlog2])
= .
t t
The convergence of log(E[eθGt]) is a G¨artner-Ellis condition on G . Glynn and
t t
Whitt [15] have proved that it is equivalent to the analogous condition on S .
n
But the convergence of log(E[Nt]) does not imply that of e−νtE[N ] (cf. the case
t t
where all lifetimes are equal to some constant). A law of large numbers, even
strengthened by large deviations inequalities, does not suffice to prove our re-
sults: additional hypotheses are needed. We begin with stationarity require-
ments.
The notionof stationaritythat seems the mostnaturalis invariancethrough
automorphisms of the tree, as in Pemantle [39]. It will be satisfied by the BMC
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 6
models of the next two sections. Weaker hypotheses will suffice for our prelimi-
nary convergence results. The first one says that birth dates of cells in a given
generation have the same distribution. For n > 0, we shall denote by S the
n
birth date of the first cell in generation Γ , by alphabetical order.
n
S =S =T +T +···+T .
n 0n 0 02 0n
Definition 2.2. The bifurcating process (Tv)v∈T is birth-stationary if for all
n∈N and for all v ∈Γ :
n
D
S = S .
v n
Observe that birth-stationarity does not imply that lifetimes T are identi-
v
callydistributed,eveninagivengeneration.Itwillbeusedtoprovethe Ces`aro
convergenceofE[e−νtN ]inTheorem2.1below.Fortheconvergenceinquadratic
t
mean and almost sure, a stronger notion will be used: the joint distribution of
the birth dates of two cells in generations n+i and n+j with most recent
common ancestor in generation n, should depend only on n, i, and j. The first
such couple in alphabetical order is (0n+i,0n10j−1). The corresponding birth
dates will be denoted by S(0) and S(1) (Figure 1):
n,i n,j
Sn(0,i) =Sn+i =S0n+i and Sn(1,j) =S0n10j−1 . (2.4)
S(0): n + i lifetimes
i,n
S : n lifetimes i lifetimes
n
T
n
0 0
T
0n 1 j lifetimes
T
(1)
0 S : n + j lifetimes
j,n
time
Fig 1. Birth dates S(0) and S(1) of the first two cells of generations n+i and n+j, whose
n,i n,j
most recent common ancestor is ingeneration n.
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 7
Definition 2.3. The bifurcating process (Tv)v∈T is fork-stationary if for all
(n,i,j)∈N×N×N∗ and for all (v,w)∈Γ ×Γ such that v∧w∈Γ :
n+i n+j n
(S ,S )=D (S(0),S(1)).
v w n,i n,j
For i=0 the definition includes the case v 4w: for all n,k∈N×N∗, for all
(v,w)∈Γ ×Γ such that v 4w,
n n+j
D
(S ,S )=(S ,S ).
v w n n+j
In particular all couples (S ,S ) are identically distributed for v ∈ Γ , hence
v v0 n
lifetimes in a given generation have the same distribution.
Remark 2.2. On thek-arytree,onemight thinkthat forks should have k teeth.
Yet fork-stationarity is used in Proposition 6.1 to express E[N N ] in terms
t t+τ
of the joint distribution of (S(0),S(1)). This remains the same on the k-ary tree.
n,i n,j
The main result of this section concerns the exponential growth of E[N ]; it
t
relates the growth rate ν and the proportionality constant C to the Laplace
transform of S . In order to enhance the link with Theorem 1.1, we chose to
n
express our results in terms of Laplace transforms, instead of characteristic
functions or logarithmic moment generating functions, as is customary in large
deviations theory (see e.g. [11]). Throughoutthe paper, the Laplacetransforms
evaluated at γ >0 of S , and of S conditioned on T =u, will be denoted by
n n 0
L (γ) and L (γ,u):
n n
L (γ)=E[e−γSn] and L (γ,u)=E[e−γSn|T =u].
n n 0
Theorem 2.1. Let (Tv)v∈T be a birth-stationary bifurcating process. Assume
that ν and C given below are well defined, positive, and finite.
∞
ν :=inf{γ >0, 2nL (γ)<∞}, (2.5)
n
n=1
X
∞
γ
C := lim 2n−1L (γ+ν). (2.6)
n
γց0γ+ν
n=1
X
Then for all t>0, E[N ]<∞ and:
t
1 t
lim e−νsE[N ]ds=C . (2.7)
s
t→∞ t
Z0
In the particular case where the lifetimes T are i.i.d. random variables,
v
Ln(γ) = (E[e−γT0])n; it can be easily checked that (2.5) and (2.6) reduce to
(1.2) and (1.3).
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 8
Proof. From (2.3):
1 1
N = + I
t 2 2 Sv6t
v∈T
X
∞
1
= 1+ I .
2 Sv6t
nX=1vX∈Γn
By birth-stationarity:
∞
E[N ]=1+ 2n−1P[S 6t]. (2.8)
t n
n=1
X
By Markov’s inequality, for all γ >0 and t>0,
P[S 6t]6eγtE[e−γSn].
n
The hypotheses of Theorem 2.1 imply that there exists γ >ν such that
∞
2n−1E[e−γSn]<∞.
n=1
X
Hence E[N ]<∞ for all t>0. Consider now:
t
∞
A (t)=e−νt 2n−1P[S 6t].
ν n
n=1
X
Let A˜ (γ) be the Laplace transform of A . The Laplace transform of P[S 6
ν ν n
t], evaluated at γ > 0 is 1L (γ). The Laplace transform of e−νtP[S 6 t] is
γ n n
1 L (γ+ν). Therefore:
γ+ν n
∞
1
A˜ (γ)= 2n−1L (γ+ν).
ν n
γ+ν
n=1
X
By (2.6):
limγA˜ (γ)=C .
ν
γց0
If both lim A (t) and limγA˜ (γ) exist, the fact that they are equal is a
ν ν
t→+∞ γց0
well known basic result of Laplace transform theory, known as the Final Value
Theorem.Deducing that the former limit exists fromthe existence ofthe latter
requires a Tauberian theorem: see Feller [14, section XIII.5] or Korevaar [29].
As a particular case of [14, Theorem 2 p. 445]:
1 t
limγA˜ (γ)=C ⇐⇒ lim A (s)ds=C ,
ν ν
γց0 t→+∞ t Z0
which is the announced result.
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 9
Remark 2.3. On the k-ary tree, (2.5) and (2.6) become:
∞
ν =inf{γ >0, knL (γ)<∞},
n
n=1
X
and
∞
γ(k−1)
C = lim kn−1L (γ+ν).
n
γց0 γ+ν
n=1
X
The proof is the same.
Without any further assumption, nothing more can be obtained than the
Ces`aro convergence (2.7), as the example of constant lifetimes shows. To con-
clude that limA (t)=C in the i.i.d. case, Bellman and Harris [3] use Ikehara’s
ν
Tauberian theorem. For the BMC case, we shall need not only a limit, but also
an exponential speed of convergence. Although we have not found Lemma 2.2
in the literature, it cannot be considered as new; it is closely related to a large
corpus of results going back to Haar,Wiener, and Ikehara:see [29] as a general
reference, [13, 37] for similar results.
Lemma 2.2. Let f be a function and f˜ its Laplace transform. Suppose that
there exist two positive reals δ and ǫ such that:
1. f˜is analytic in {z =x+iy, |x|<δ+ǫ}\{0},
2. f˜has a simple pole at 0, with residue C,
3.
∞
|f˜(δ+iy)|dy <∞,
Z−∞
4.
lim f˜(x+iy)=0,
y→±∞
uniformly in x∈[−δ,δ],
5.
∞
ψ := |f˜(−δ+iy)|dy <∞.
Z−∞
Then, for all t>0,
ψ
|f(t)−C|6 e−δt .
2π
Proof. By the inversion formula,
1 δ+iβ
f(t)= lim f˜(γ)eγtdγ .
2πiβ→∞Zδ−iβ
LetC betheclosedrectangularcontourlinkingthepointsδ−iβ,δ+iβ,−δ+iβ,
−δ−iβ.Thiscontourenclosesthesimplepoleatγ =0.Bytheresiduetheorem,
f˜(γ)eγtdγ =2πi res(f˜(γ)eγt),
ZC γ=0
S. Louhichi and B. Ycart/Bifurcating processes with ancestral dependence 10
where
res(f˜(γ)eγt)= lim γf˜(γ)eγt =C .
γ=0 γ→0
Consequently,
1 δ+iβ
f˜(γ)eγtdγ−C =I +I +I ,
1 2 3
2πi
Zδ−iβ
with:
1 −δ+iβ
I = − f˜(γ)eγtdγ ,
1
2πi
Zδ+iβ
1 −δ−iβ
I = − f˜(γ)eγtdγ ,
2
2πi
Z−δ+iβ
1 δ−iβ
I = − f˜(γ)eγtdγ .
3
2πi
Z−δ−iβ
By condition 4, I and I tend to 0 as β tends to infinity. Therefore:
1 3
1 +∞
|f(t)−C| 6 f˜(−δ+iy)e(−δ+iy)tdy
2π
(cid:12)Z−∞ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
+(cid:12)∞ (cid:12)
6 |f˜(−δ+iy)|e−δtdy
Z−∞
ψ
= e−δt ,
2π
by condition 5 .
3. Bifurcating Markov chains
Bifurcating Markov chains (BMC) were considered long ago, beginning with
Spitzer [46] in the binary valued case (see Benjamini & Peres [4] for further
reference). They were studied as cell lineage models by Guyon [16]. As in the
one-dimensional case, the probability distribution of a BMC is determined by
aninitialmeasureandatransitionkernel.Hereisthe definition,adaptedtoour
case (as usual, B denotes the Borel σ-algebra).
Definition3.1. AtransitionkernelP isamappingdefinedonR+×B(R+×R+)
such that:
• For all B ∈B(R+×R+), t7→P(t,B) is B(R+)-measurable,
• For all t∈R+, B 7→P(t,B) is a probability measure on B(R+×R+).
Definition 3.2. Let µ be a probability measure on R+, and P be a transition
kernel. The distribution of aBMC (Tv)v∈T with initial measureµ and transition
kernel P is inductively defined as follows.