6B Effects of the cell cycle on stochastic gene expression
In Chap. 6, we mainly focused on the effects of intrinsic molecular noise on gene
expression. Here we consider the effects of a major source of extrinsic noise, which
arises from the stochastic nature of cell growth and division during the cell cycle. In
balanced population growth, the quantities of cellular components double on average during each cell cycle and then halve at cell division. However, individual cells
can deviate significantly from the average due to a combination of molecular noise
and stochastic growth and cell division. A variety of experimental and theoretical
studies have investigated the distribution of proteins across a population of cells under well-controlled conditions, with the goal of understanding the relative contributions of intracellular noise, the random partitioning of proteins at cell division, and
the stochastic exponential growth of cells [3, 2, 5, 6, 9, 16, 12, 13, 10, 17, 23, 4, 7].
6B.1 Non-equilibrium model of dividing cell populations
We begin by considering a minimal model of dividing cell populations due to Brenner and Shokef [6], which describes the joint effects of protein synthesis noise and
proliferation dynamics. The model uses a separation of time-scales between protein
production, which is continuous throughout the cell cycle, and cell division, which
occurs during a short fraction of the cell cycle. Another major simplification is that
the entire population of cells divides synchronously. The total protein content in a
cell is thus taken to evolve according to a discrete map
xn+1 = M xn = qn (xn + λn ),
(6B.1)
where xn is the protein content of a cell in the nth generation just after cell division,
λn is the amount of protein produced and accumulated in the cell during the nth
cell cycle, and qn is the fraction of protein inherited by one of the daughter cells at
the end of the nth cell cycle - the other inherits the fraction 1 − qn . Suppose that λn
and qn are i.i.d.’s drawn from the probability densities ξ (λ ) and η(q), respectively.
The discrete-time Liouville equation for the evolution of the distribution of proteins
across the population of cells at generation n is then given by
Pn+1 (x) =
Z 1
0
dq η(q)
Z ∞
dλ ξ (λ )
0
Z ∞
0
dx0 δ (M (x0 ) − x)Pn (x0 ).
(6B.2)
Note that the nature of cell division implies the symmetry η(q) = η(1 − q). This
means that hqi = 1/2, since
1
hqi =
=
Z 1
qη(q)dq =
0
Z 1
0
Z 1
0
qη(1 − q)dq
(1 − q)η(q)dq = 1 − hqi.
Suppose that a steady-state solution exists so that
lim Pn (x) = P(x),
n→∞
where P(x) is the (unique) fixed point solution of the Liouville equation. Introducing
the moment generating function
Z ∞
G(s) =
e−sx P(x)dx,
0
and using the Liouville equation we have
G(s) =
Z 1
dq η(q)
0
=
Z 1
0
=
Z 1
Z ∞
dλ ξ (λ )
0
dq η(q)
Z ∞
Z ∞
0
dx e−sx
Z ∞
0
dx0 δ (q(x0 + λ ) − x)P(x0 )
Z ∞
dx0 e−sq(x +λ ) P(x0 )
Z ∞
e−sλ ξ (λ )dλ
dλ ξ (λ )
0
0
0
dq η(q)G(qs)H(qs),
0
where
H(s) =
0
is the moment generating function of the random variable λ . If q and λ have finite
moments then so does x. Moreover, the statistical independence of q and λ means
that in the steady-state
k
∑ j hxk− j ihλ j i.
j=0
k
k
hxn+1
i = hqkn (xn + λn )k i = hqkn ih(xn + λn )k i = hqk i
In particular,
hxi = hqi(hxi + hλ i) = hλ i,
since hqi = 1/2.
In the case of constant λ and a uniform distribution η(q) = 1, the generating
function can be solved explicitly. The Liouville equation reduces to
G(s) =
Z 1
0
G(qs)e−sqλ dq,
which can be differentiated with respect to s to give
2
dG(s)
= −λ
ds
Z 1
0
qG(qs)e−sqλ dq +
Z 1
0
qG0 (qs)e−sqλ dq
Evaluating the second integral using integration by parts yields
Z 1
0
0
qG (qs)e
−sqλ
iq=1 1 Z 1
1h
d −sqλ −sqλ
dq =
qG(qs)e
−
G(qs)
qe
dq
s
s 0
dq
q=0
Z 1
h
i
1
1
G(qs) e−sqλ − sλ qe−sqλ dq.
= G(s)e−sλ −
s
s 0
Hence,
s
dG(s)
= G(s)e−λ s − G(s).
ds
The latter equation can be solved using an integrating factor to give
Z s
i
1h
0
0
φ (s )ds , φ (s) = − 1 − e−λ s .
G(s) = exp
s
0
(6B.3)
(6B.4)
We have used the identity G(0) = 1. Taylor expanding φ (s) in powers of λ s and
integrating shows that
ψ(s) ≡
Z s
0
φ (s0 )ds0 =
(−λ s)k
.
k=1 k!k
∞
∑
Finally, the cumulants of the protein distribution P(x) are
n
λn
n d ln G(s) =
κn = (−1)
.
n
ds
n
s=0
(6B.5)
Note for s → ∞ we have φ (s) ∼ −1/s, ψ(s) ∼ − ln s and G(s) ∼ 1/s.
Given the generating G(s), one can obtain P(x) by expressing the inverse Laplace
transform using a Bromwich integral and steepest descents for large x:
P(x) =
Z c+i∞
esx G(s)ds =
c−i∞
Z c+i∞
esx eψ(s) ds
c−i∞
∗
∗
1
≈√
es x+ψ(s ) .
2π|ψ 00 (S∗ )|1/2
Here s∗ is the unique solution to the equation x + ψ 0 (s∗ ) = 0, that is,
x=
i
∗
1h
1 − e−λ s ,
∗
s
with s∗ < 0. It follows that, for this example, the tail of the probability density P(x)
is non-exponential (since s∗ = s∗ (x)) with P(x) ∼ x−x . It turns out that the shape
3
100
101
(a)
(b)
10-1
P(x)
G(s)
100
10-1
10-2
10-3
10-4
10-2 -1
10
100
s
0
101
1
2
3
x
4
5
6
Fig. 6B.1: Protein distribution in the case of constant protein synthesis. (a) Generating function
G(s) for a uniform distribution η(q) = 1 of the partition fraction q. Asymptotically G(s) ∼ 1/x.
(b) Corresponding distribution function P(x) (black curve). Also shown is the distribution for a
non-uniform η(q) (gray curve), illustrating non-universality. [Redrawn from Brenner and Shokef
[6].]
of the tail is also sensitive to the choice of the division density η(q) [6]. On the
other hand, if significant protein production noise is included, with λ drawn from
an exponential distribution, then one finds that the the density P(x) exhibits a similar
shape for different choices of η(q) and has an exponential tail. This suggests that
it might be possible to distinguish between the two types of noise in experimental
data.
6B.2 Contributions of cell growth to stochastic gene expression
One major simplification of the above model is that all cells divide synchronously.
Asynchronous cell division means that individual cells across the population may
be at different stages of the cell cycle when their molecular content is measured,
which adds another level of stochasticity. In order to take this into account one must
track cell growth. Moreover, if one wants to determine fluctuations in protein concentration rather than copy number, then one has to take into account changes in cell
volume during cell growth, which can also be stochastic. A schematic illustration of
the various contributions of cell growth to cell-to-cell variability is illustrated in Fig.
6B.2. In this section we carry out a decomposition of the variance in protein copy
number into the various sources of stochasticity, following the analysis of Schwabe
and Bruggeman [23], see also [13].
4
protein inherited from
mother cell
newly synthesized
protein
R0 = 2, S0 = 0
stochatic protein synthesis
and degradation
Ra = 1, Ra = 4
RT = 0, ST = 6
stochatic partitioning
q
1-q
variations in cell division
times
Fig. 6B.2: Schematic diagram illustrating the chemical kinetic and cell-growth processes contributing to cell-to-cell variability. Sources of stochasticity include (i) fluctuations in the rate of protein
synthesis and its regulation, (ii) binomial partitioning of molecules at cell division with partition
fraction q, (iii) variability in molecular content of mother cells, and (iv) variability in cell-cycle
stage across the population. For a given stage a of a cell cycle, Ra denotes the number of molecules
inherited from the mother cell that have not yet degraded, and Sa is the number of newly synthesized proteins since cell division.
Law of total variance
In the following, we will make continued use of the law of total variance, also known
as Eve’s law: if X and Y are random variables on the same probability space, then
Var[Y ] = EX [Var[Y |X]] + VarX [E[Y |X]].
(6B.6)
The proof follows from the law of expectation (see also Chap. 11). First, from the
definition of variance,
Var[Y ] = E[Y 2 ] − (E[Y ])2 .
5
Conditioning on the random variable X and applying the law of total expectation to
each term gives
Var[Y ] = E E[Y 2 | X] − (E[E[Y | X]])2 .
Now we rewrite the conditional second moment of Y in terms of its variance and
first moment:
Var[Y ] = E Var[Y | X] + (E[Y | X])2 − (E[E[Y | X]])2 .
Since the expectation of a sum is the sum of expectations, the terms can now be
regrouped as
Var[Y ] = E[Var[Y | X]] + E[(E[Y | X])2 ] − (E[E[Y | X]])2
The result follows from observing that the terms in parentheses on the right-hand
side give the variance of the conditional expectation E[Y |X].
Variance at a fixed cell cycle stage
Let a, 0 ≤ a ≤ T , denote the cell-cycle stage measured with respect to the last cell
division. Let Xa denote the protein number at stage a, Ra the number of molecules
inherited from the mother cell that have not yet degraded, and Sa the number of
newly synthesized proteins since cell division:
Xa = Ra + Sa ,
R0 = X0 ,
S0 = 0.
(6B.7)
We assume that the distribution of non-degraded proteins at stage a is given by a
Binomial distribution:
P[Ra = r|R0 = X0 ] =
X0 !
p(a)r (1 − p(a))X0 −r ,
(X0 − r)!r!
where p(a) is the corresponding survival probability for an inherited molecule. For
independent, first-order degradation we have p(a) = e−kd a where kd is the degradation rate. It follows that
hRa |X0 i = p(a)X0 ,
Var[Ra |X0 ] = p(a)(1 − p(a))X0 ,
where we have taken expectation with respect to the Binomial distribution for fixed
X0 . Given that there is also variation in X0 ,
hRa i = E[hRa |X0 i] = p(a)hX0 i,
and from the law of total variance
Var[Ra ] = E[p(a)(1− p(a))X0 ]+Var[p(a)X0 ] = p(a)(1− p(a))hX0 i+ p(a)2 Var[X0 ].
6
Combining these various results, we see that
hXa i = p(a)hX0 i + hSa i
(6B.8)
and
Var[Xa ] = p(a)(1 − p(a))hX0 i + p(a)2 Var[X0 ] + 2p(a)Cov[X0 , Sa ] + Var[Sa ].
(6B.9)
The next step is to determine the variation in the number of proteins X0 in a
daughter cell just after cell division. This will depend on two factors: (i) the variation in the number of proteins XT in a mother cell just before cell division, and
fluctuations due to randomly partitioning the molecules between the two daughter
cells with partition ratio q. Conditioning on XT and q, which are independent random variables, we apply the law of total variation twice. First,
Var[X0 | q] = E[Var[X0 | q, XT ]] + Var[E[X0 | q, XT ]],
with expectation taken with respect to XT . Since
hX0 | q, XT i = qXT ,
Var[X0 | q, XT ] = q(1 − q)XT ,
it follows that
Var[X0 | q] = q(1 − q)hXT i + q2 Var[XT ].
Applying the law of total variation a second time, with expectation now taken with
respect to q,
Var[X0 ] = E[Var[X0 | q]] + Var[E[X0 | q]]
2
(6B.10)
2
= hq(1 − q)ihXT i + hq iVar[XT ] + Var[q]hXT i
1
1
2
− Var[q] hXT i + Var[q]hXT i +
+ Var[q] Var[XT ].
=
4
4
We have used the fact that hqi = 1/2 (see Sec. 6A.1) and E[X0 | q] = qhXT i. Equation (6B.10) establishes that fluctuations in the partitioning probability q enhances
Var[X0 ].
We can express the population level variance Var[Xa ] given by equations (6B.9)
and (6B.10) solely in terms of the variance in protein synthesis Var[Sa ], its correlations with X0 , the variance in the partition distribution Var[q], and the survival
probability p(a). First, setting a = T in equation (6B.8) we have
hXT i = p(T )hX0 i + hST i.
Since hXT i = 2hX0 i (doubling at cell division), we see that
hX0 i =
hST i
= hXT i/2.
2 − p(T )
7
(6B.11)
Second, setting a = T in equation (6B.9) gives
Var[XT ] = p(T )(1 − p(T ))hX0 i + p(T )2 Var[X0 ] + 2p(T )Cov[X0 , ST ] + Var[ST ],
and substituting this into the second line of (6B.10) yields
Var[X0 ] =
hq(1 − q)ihXT i + Var[q]hXT i2
1 − hq2 ip(T )2
+
(6B.12)
hq2 i (p(T )(1 − p(T ))hX0 i + 2p(T )Cov[X0 , ST ] + Var[ST ])
.
1 − hq2 ip(T )2
Equations (6B.9), (6B.11) and (6B.12) yield the desired result.
In the above analysis, we have not specified the particular mechanism underlying
the variation in the partition ratio q. This issue is explored by Huh and Paulsson [13],
who consider some kinetic models of fluctuations in q due to various mechanisms
such as differences in the available volumes of the daughter cells, and molecular
clustering. In order to calculate the variance due to cell division, they introduce
an effective birth death process that captures the exact segregation statistics even
though it need not describe the precise physical process underlying partitioning.
Here we will illustrate the basic approach by considering the simpler case of unbiased independent partitioning, where each protein has 0.5 probability of being
assigned to either daughter cell, that is, q = 1/2. This could be physically realized
by several mechanisms: molecules rapidly diffusing between the two halves during cell division; proteins being synthesized at the same rate in either half of the
mother cell but not diffusing or degrading; molecules binding to either mitotic spindle at cell division (see Sec. 8.2) with equal probability. Irrespective of the particular
model, the statistical partitioning error can be realized by Ehrenfest’s urn model in
the stationary state. The latter is described by a birth-death process for the number
of molecules L in one daughter cell given a total of X molecules in the mother cell.
The reactions are
X−L
L
L → L + 1, L → L − 1,
that is, each molecule (L on one side and X − L on the other) switches sides at a
constant rate taken to be unity. The associated birth-death master equation for the
probability distribution P(L,t) is, for fixed X,
dP(L,t)
= (X − L + 1)P(L − 1,t) + (L + 1)P(L + 1,t) − XP(L,t),
dt
which is identical in form to the master equation for a two-state ion channel (see
Sec. 3.3). The stationary solution is thus the Binomial distribution
Ps (L) = qL (1 − q)X−L
X!
,
(X − L)!L!
with q = 1/2 (deterministic partition ratio) and E[L|X] = qX, Var[L|X] = q(1 − q)X.
It follows from the law of total variance that
8
1
1
Var[L] = hXi + Var[X].
4
4
This agrees with equation (6B.10), since Var[q] = 0.
Variance due to the distribution of cell cycle stages
So far we have determined the population variance at a specific cell cycle stage a,
given some unspecified model of stochastic protein synthesis Sa . However, there is
an additional source of variance due to the fact that, across the population, cells are
at different stages of the cell cycle. Making the identifications
hXa i = E[X | a],
Var[Xa ] = Var[X | a],
and again applying the law of total variation, we have that the total variance in
protein synthesis across the population of cells is
Var[X] = E[Var[Xa ]] + Var[hXa i],
(6B.13)
where expectation is now taken with respect to the distribution u(a) of cell cycle
stages. The latter can be specified in terms of the distribution f (τ) of interdivision
times τ, under the assumptions that the cells divide asynchronously and population
growth has reached a stationary state [19]. In particular, all extensive quantities grow
exponentially so that the number of cells N(t) is given by
N(t) = eµt N0 ,
where µ is the growth rate and N0 is the number of cells at time t = 0. In the case of
binary division, the rate at which new cells are generated is then given by 2µN0 eµt .
We can relate the age distribution u(a) to f (τ) as follows. The number of cells
younger than some fixed age a is by definition
N(a) = N0
Z a
u(y)dy.
0
An alternative way to calculate N(a) is to note that these cells are precisely those
that formed during the time-interval (−a, 0) and do not divide until a time t > 0.
The rate of formation of such cells at time t ∈ (−a, 0) is
γ(t) = 2µN0 eµt
It follows that
N(a) =
Z 0
Z ∞
−a
−t
f (y)dy.
γ(t)dt
Differentiating both sides with respect to a then gives
9
(6B.14)
u(a) = N0−1 γ(−a) = 2µe−µa
Z ∞
f (y)dy.
(6B.15)
a
In the analysis of Ref. [23], it is assumed that the interdivision time is the same for
all cells, τ = T , that is, f (τ) = δ (τ − T ). It follows that
u(a) = 2µe−µa
0 ≤ a ≤ T,
(6B.16)
and is zero otherwise. Moreover, since the population doubles over a time interval
of size T , we have
ln 2
µ=
.
T
Example: zero degradation (kd = 0) and constant rate of synthesis.
In the case of zero degradation, p(a) = 1 for all 0 ≤ a ≤ T . Moreover, the synthesis
of proteins is given by a Poisson process with
hST i = Var[ST ] = ks T.
Hence
hXT i = 2hX0 i = 2hST i = 2ks T,
and from equation (6B.12) with Cov[X0 , Sa ] = 0,
hq(1 − q)ihXT i + Var[q]hXT i2 + Var[ST ]
1 − hq2 i
ks T (3 − 4Var[q] + 16Var[q]ks T )
=
.
3 − 4Var[q]
Var[X0 ] =
Similarly, equation (6B.9) reduces to
Var[Xa ] = Var[X0 ] + Var[Sa ] = Var[X0 ] + ks a.
Hence,
E[Var[Xa ]] =
Z T
0
u(a)[Var[X0 ] + ks a]da
= Var[X0 ] + ks (µ −1 − T ).
Finally,
hXa i = hX0 i + hSa i = ks (T + a),
so that
10
Var[hXa i] = ks2
Z T
0
u(a)[a − hai]2 da
= ks2 (µ −2 − 2T 2 ).
Hence, from equation (6B.13), the total variance is
Var[X] =
ks
(ks T )2 16Var[q]
+ ks2 (µ −2 − 2T 2 ) +
µ
3 − 4Var[q]
(6B.17)
6B.3 Stochastic exponential growth of single bacterial cells
The above two sections focused on growth and division at the population level,
for which all extensive quantities grow exponentially under balanced growth conditions. This is irrespective of how the size of an individual cell changes in time. That
is, the observation of population growth is not sufficient to determine the the growth
law at a single cell level. Both linear and exponential growth laws have previously
been considered [8], and linear protein synthesis was assumed in the above models.
However, recent advances in imaging techniques has allowed more precise measurements of the stochastic growth of individual Caulobacter cresentus bacterial cells
[15], revealing that mean sizes grow exponentially in time. Moreover, the size distributions collapse to a single curve when rescaled by their means. This universal
behavior of fluctuations during the growth of single bacterial cells has also been
accounted for theoretically by Iyer-Biswa et al. [14] using a minimal microscopic
model. In the following we describe this model in more detail, since it also provides
an illustrative example of a cyclic biochemical process.
Iyer-Biswas et al. consider a stochastic version of a simple kinetic model introduced by Hinshelwood [11]. The latter assumes that the cell components controlling
cell growth are linked via a cycle of autocatalytic reactions, whereby each chemical
X1
X1
XN
X2
X2
XN
k2X1
k3X2
k1XN
X1 + X2
X2 + X3
XN + X1
X3
Fig. 6B.3: Hinshelwood model for exponential growth, showing the autocatalytic cycle, in which
each species activates production of the next, and the corresponding reactions.
11
species catalyzes the production of the next, see Fig. 6B.3. The stochastic Hinshelwood cycle (SHC) consists of N species {X1 , X2 , . . . , XN }. The mean rate of production of X j is taken to be k j x j−1 , 1 ≤ j ≤ N, where x j is the copy number of X j , and
X0 ≡ XN . The reaction scheme is
k j x j−1
X j−1 → X j−1 + X j .
(6B.18)
(We will use Xi to denote the chemical species and the stochastic copy number of the
given reactant.) The corresponding master equation for the probability distribution
P(x,t), x = (x1 , x2 , . . . xN ) is
∂P
=
∂t
N
∑ k j x j−1 [P(x1 , . . . , x j − 1, . . . xN ) − P(x1 , . . . , x j , . . . xN )] .
(6B.19)
j=1
with P(x1 , . . . , x j − 1, . . . xN ) = 0 if x j = 0. Multiplying both sides by xi and integrating with respect to x, we see that all terms on the right-hand side vanish except
when j = i, for which
Z
dhXi i
= ki xi xi−1 [P(x1 , . . . , xi − 1, . . . xN ) − P(x1 , . . . , xi , . . . xN )] dx
dt
= ki hXi−1 i.
In vector form with µ j (t) = hX j (t)i, we have
dµ(t)
= Kµ(t),
dt
where K is a cyclic matrix of period N, that is,
KN = k1 k2 . . . kN IN ,
and IN is the N × N identity matrix.
Consider the eigenvalue equation
Ku = λ u,
which can be iterated to give
KN u = κ N u = λ N u,
where κ is the geometric mean of the reaction rates k j :
κ N = k1 k2 . . . kN .
It follows that the m-th eigenvalue is
λm = κe2πim/N ,
12
(6B.20)
and the q-th component of the corresponding eigenvector um is
q
(q)
um = λm−q ∏ k p .
p=1
The eigenvalue with the largest real part is λN = κ, and this will dominate the
asymptotic dynamics with relaxation time-scale κ −1 . In particular, suppose that we
expand the first moment vectors in terms of the eigenvectors um :
N
µ(t) =
∑ cm (t)um ,
m=1
so that
N
N
∑ ċm (t)Uqn = ∑ λn cn (t)Unq ,
m=1
(q)
Uqm = um
n=1
Since the eigenvalues are distinct, the matrix of eigenvectors U is invertible so that
cm (t) = eλm t cm (0),
ċm (t) = λm cm (t),
and the eigenvalue expansion takes the form
N
µ j (t) =
N
∑ eλmt U jm cm (0) = ∑
m=1
Hence, in the asymptotic limit (t κ −1 ),
N
µ j (t) ∼
−1
eλm t U jmUmi
µi (0)
m,i=1
∑
!
−1
UiN
µi (0)
i=1
eκt U jN ,
(6B.21)
which implies that the mean copy number of all the reactants evolve asymptotically
as the single exponential eκt . It also follows that
µ j (t) U jN
=
,
µr (t) UrN
which is independent of initial conditions.
An interesting interpretation of the geometric mean κ can be obtained by assuming that each individual reaction rate is given by the Arrhenius form (see Sec.
3.3):
ki = Ai exp(−∆ Ei /kB T ),
where ∆ Ei is the activation energy of reaction i. Then
13
κ = (k1 k2 . . . kN )1/N
∆ E1 + . . . + ∆ EN
= (A1 A2 . . . AN )1/N exp −
NkB T
≡ A exp(−∆ E/kB T ),
where ∆ E is the arithmetic mean of the elementary activation energies. Hence, assuming that each reaction step has an energy barrier that is of the order of a typical
enzyme reaction, then so does the effective growth rate. This is consistent with experimental measurements of growth rates in single bacterial cells [15].
The next step is to determine the asymptotic behavior of the covariance matrix
C(t) with
Ci j = hXi X j i − hXi ihX j i.
A basic result from the theory of chemical master equations with transition rates
that are linear in the copy number is that the covariance matrix satisfies the same
Ricatti equation (2.2.22) as the corresponding OU process obtained using a system
size expansion. That is,
dC(t)
= KC(t) + C(t)KT +Θ T ,
dt
(6B.22)
where
Θi j = δi j µ j (t).
It can be shown that in the asymptotic limit [14],
N
Ci j (t) ∼ UiN U jN e2κt
∑ br µr (0),
(6B.23)
r=1
with the coefficients br dependent on the rates but not the initial conditions. Comparing equations (6B.21) and (6B.23), one finds that the rescaled covariance
b i j (t) = hYi (t)Y j (t)i − hYi (t)ihY j (t)i,
C
Yi (t) =
Xi (t)
,
µi (t)
takes the asymptotic form
!−2
N
Ci j (t) ∼
∑ UlN−1 µl (0)
l=1
N
∑ br µr (0),
(6B.24)
r=1
In particular, Var[Xi (t)/µi (t)] ∼ constant. Since the rescaled covariance is independent of i, j and time, it follows that the random variables Yi (t) are perfectly correlated. This only occurs if the random variables Yi (t) are linearly related. We conclude
that although the means µi (t) grow exponentially in time, the rescaled random variables Xi (t)/µi (t) have the same time-independent probability density in the asymp-
14
totic limit, which is also observed experimentally [15]. This, in turn, implies that the
n-th order moment of Xi varies as enκt .
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