Coordination of gene expression noise with cell size: analytical results for
agent-based models of growing cell populations
Philipp Thomas
a)
and Vahid Shahrezaei
Department of Mathematics, Imperial College London, UK
The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside
living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions
and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based
framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify
the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division
noise. We find that the solution of the chemical master equation – including static extrinsic noise – exactly agrees
with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis, a
novel condition that generalises concentration homeostasis in deterministic systems to higher order moments and
distributions. We illustrate stochastic concentration homeostasis for a range of common gene expression networks.
When this condition is not met, we demonstrate by extending the linear noise approximation to agent-based
models that the dependence of gene expression noise on cell size can qualitatively deviate from the chemical master
equation. Surprisingly, the total noise of the agent-based approach can still be well approximated by extrinsic noise
models.
I. INTRODUCTION
Cells must continuously synthesise molecules to grow and
divide. At a single cell level, gene expression and cell size
are coordinated but heterogeneous which can drive pheno-
typic variability and decision making in cell populations
1–5
.
The interplay between these sources of cell-to-cell variabil-
ity is not well understood since they have traditionally been
studied separately. A general stochastic theory integrating
size-dependent biochemical reactions with the dynamics of
growing and dividing cells is hence still missing.
Many models of noisy gene expression and its regulation
are based on the chemical master equation that describes
the stochastic dynamics of biochemical reactions in a fixed
reaction volume
6–8
. The small scale of compartmental sizes
of cells implies that only a small number of molecules is
present at any time leading to large variability of reaction
rates from cell to cell, commonly referred to as gene ex-
pression noise
9–11
. Another factor contributing to gene ex-
pression noise is the fact that cells are continuously growing
and dividing causing molecule numbers to (approximately)
double over the course of a growth-division cycle. A com-
mon approach to account for cell growth is to include extra
degradation reactions that describe dilution of gene expres-
sion levels due to cell growth
9–13
akin to what is done in de-
terministic rate equation models
14,15
. We will refer to this
approach as the effective dilution model (EDM, see Fig. 1a).
However, little is known of how well this approach repre-
sents the dependence of gene expression noise on cell size
observed in a growing population.
Cells achieve concentration homeostasis through coupling
reaction rates to cell size via highly abundant upstream fac-
a)
Electronic mail: p.thomas@imperial.ac.uk
tors like cell cycle regulators, polymerases or ribosomes that
approximately double over the division cycle
3,16,17
. Cell size
fluctuates in single cells, however, providing a source of ex-
trinsic noise in reaction rates that can be identified via noise
decompositions
18,19
. A few studies combined EDMs with
static cell size variations as an explanatory source of ex-
trinsic noise
20–22
. In brief, the total noise in these models
amounts to intrinsic fluctuations due to gene expression and
dilution, and extrinsic variation across cell sizes in the pop-
ulation. We refer to this class of models as extrinsic noise
models (ENMs, see Fig. 1b). Yet it remains unclear how re-
liably these effective models describe cells that continuously
synthesise molecules, grow and divide.
An increasing number of studies are investing efforts
towards quantifying the dependence of gene expression
noise on cell cycle progression and growth, either exper-
imentally via ergodic principles or pseudo-time
23,24
and
time-lapse imaging
22,25,26
or theoretically through noise
decomposition
27–29
, master equations including cell cycle
dynamics
4,17,30–35
and agent-based approaches including
age-structure of growing populations
35–40
. The essence of
agent-based models (ABMs) is that each cell in a popula-
tion is represented by an agent whose physiological state is
tracked along with their molecular reaction networks. In
principle, these models are able to predict gene expression
distributions of cells progressing through well-defined cell
cycle states as measured by time-lapse microscopy and snap-
shots of heterogeneous populations. The unprecedented de-
tail of these models must cast doubt on the predictions of
master equation models (EDMs and ENMs) in which growth
and division are modelled by effective dilution reactions.
Yet it is presently unclear why these effective models have
fared reasonably well in predicting gene expression noise re-
ported by single-cell experiments
10,17,41
.
Nevertheless, most ABMs still ignore cell size, a ma-
jor physiological factor affecting both intracellular reactions
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and cell division dynamics alike. Since cell size varies at
least two-fold as required by size homeostasis in a growing
population, and it scales some reaction rates as required by
concentration homeostasis, it is expected that cell size must
significantly contribute to gene expression variation across
a population. In this article, we bridge the gap between the
chemical master equation and agent-based approaches by
integrating cell size dynamics with the stochastic kinetics of
molecular reaction networks.
The outline of the paper is as follows. First, we explain
the analytical framework for EDMs, ENMs and ABMs (II).
Then we introduce the concept of stochastic concentration
homeostasis, a rigorous condition under which the chemi-
cal master equations of the EDM and ENM agree exactly
with the ABM (Sec. III A). This new condition is met by
some but not all common models of gene expression. We
show that when these conditions are not met, the effective
models agree with the ABM only on average (Sec. III B).
To address this problem, we propose a comprehensive theo-
retical framework extending the linear noise approximation
to agent-based dynamics with which we quantify cell size
scaling of gene expression in growing cells (Sec. III C). Our
findings indicate that the EDM can qualitatively fail to pre-
dict this dependence but our novel approximation method
accurately describes gene expression noise in the presence of
cell size control variations and division errors. We further
show that ENMs present surprisingly accurate approxima-
tions for the total noise statistics (Sec. III D).
II. METHODS
We consider a biochemical reaction network of N molecu-
lar species S = (S
1
, S
2
, . . . , S
N
)
T
embedded in a cell of size
s. The network then has the general form:
N
X
i=1
ν
−
ir
S
i
k
r
−→
N
X
i=1
ν
+
ir
S
i
, r = 1, . . . , R, (1)
where ν
±
r
= (ν
±
1r
, ν
±
2r
, . . . , ν
±
Nr
)
T
are the stoichiometric co-
efficients and k
r
is the reaction rate constant of the r
th
re-
action. In the following, we outline deterministic, effective
dilution and extrinsic noise models and develop a new agent-
based approach coupling stochastic reaction dynamics to
cell size in growing and dividing cells (Fig. 1).
A. Effective dilution models, extrinsic noise models and the
chemical master equation
1. Rate equation models and concentration homeostasis
Deterministically, the vector of molecular concentrations
¯
X = (
¯
X
1
,
¯
X
2
, . . . ,
¯
X
N
)
T
is governed by rate equation mod-
els in balanced growth conditions. The balanced growth
condition states that there exists a steady state between
reaction and dilution rates
α
¯
X =
R
X
r=1
(ν
+
r
− ν
−
r
)f
r
(
¯
X). (2)
Here, f
r
(
¯
X) are macroscopic reaction-rate functions and α
is the exponential growth rate of cells determining the dilu-
tion rate due to growth. Since these quantities are indepen-
dent of cell size, the balanced growth condition (2) implies
concentration homeostasis in rate equation models.
2. Effective dilution model
The chemical master equation
6
and equivalently the
stochastic simulation algorithm
7
are state-of-the-art
stochastic models of reaction kinetics inside cells. Al-
though well-established, they are strictly valid only when
describing cellular fluctuations at constant cell size s. A
straight-forward approach to circumvent this limitation is
to supplement (1) by additional degradation reactions of
rate α that model dilution of molecules due to cell growth:
S
i
α
−→ ∅, i = 1, 2, . . . , N, (3)
akin to what is traditionally for reaction rate equations (2).
The chemical master equation of this effective dilution model
(EDM) then takes the familiar form
0 =
∂Π
EDM
(x|s)
∂t
= [Q(s) + αD]Π
EDM
(x|s), (4)
governing the conditional probability of molecule numbers
x = (x
1
, x
2
, . . . , x
N
)
T
of the species S in a cell of size s and
where
Q
x,x
0
(s) =
R
X
r=1
w
r
(x
0
, s)(δ
x,x
0
+ν
+
r
−ν
−
r
− δ
x,x
0
), (5)
are the elements of the transition matrix of the molecu-
lar reactions (1) and we included the extra dilution reac-
tions (3) via D
x,x
0
(s) =
P
N
i=1
x
0
i
(δ
x
i
,x
0
i
−1
− δ
x
i
,x
0
i
). We are
here interested in the stationary solution and hence set the
time-derivative in Eq. (4) to zero. Such effective models are
motivated through the fact
6,42
that when the microscopic
propensities w
r
are linked to the macroscopic rate functions
f
r
of the rate equation models via mass-action kinetics
w
r
(x, s) ≈ sf
r
(X), (6)
where X = x/s is the concentration, the mean concentra-
tions of EDMs follow the concentrations
¯
X of the rate equa-
tions (2) (see Sec. II A 4).
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FIG. 1. Modelling approaches for cell size dependence of gene expression. (a) The effective dilution model describes cells at
constant size with intracellular reactions coupled to effective dilution reactions. (b) The extrinsic noise model incorporates static cell
size variability as a source of extrinsic noise coupled with effective dilution models (c) The agent-based approach models intracellular
reactions occurring across a growing and dividing cell population without the need for effective dilution reactions.
3. Extrinsic noise model
A common way to incorporate static size variability be-
tween cells in the model is to consider cell size s to be
distributed across cells according to a cell size distribution
Π(s). We will refer to this approach as the extrinsic noise
model (ENM), which leads to a mixture model of concen-
trations X = x/s,
Π
ENM
(X) =
Z
∞
0
ds Π
EDM
(x = Xs|s)Π(s), (7)
and analogous expressions for the molecule number distri-
butions.
4. Analytical solutions and noise decomposition
The advantage of the EDM and ENM is that its noise
statistics can be approximated in closed-form using the lin-
ear noise approximation
6,43,44
. In this approximation, the
mean concentrations are approximated by the solution
¯
X
of the rate equations (2) and the probability distribution
Π
EDM
(x|s) is approximated by a Gaussian. In the same
limit, the covariance matrix Σ
Y
can be decomposed into in-
trinsic and extrinsic components, Σ
int
Y
and Σ
ext
Y
, using the
law of total variance
18,19
Σ
Y
= Σ
int
Y
|{z}
gene expression
+ Σ
ext
Y
|{z}
cell size variation
, (8)
which correspond to molecular fluctuations due to gene
expression and cell size variation, respectively, for Y ∈
{EDM, ENM}. Specifically, for molecule numbers x, we
have Σ
int
Y
= E
Π
[Cov
Π
Y
[x|s]] and Σ
ext
Y
= Cov
Π
[E
Π
Y
[x|s]],
where E
Π
denotes the expectation value with respect to the
distribution Π, and analogously for concentrations. The in-
trinsic components Σ
int
Y
satisfy a Lyapunov equation called
the linear noise approximation:
0 = J
d
Σ
int
Y
+ Σ
int
Y
J
T
d
+ Ω
−1
Y
D
d
(
¯
X), (9)
where Ω
Y
has to be chosen depending on whether concen-
tration or number covariances are of interest:
Ω
Y
concentration numbers
EDM s s
−1
ENM E
Π
[s
−1
]
−1
E
Π
[s]
−1
. (10)
The matrix J
d
is the Jacobian of the rate equations (2) and
D
d
denotes the diffusion matrix obeying
J
d
(
¯
X) = J(
¯
X) − α1, D
d
(
¯
X) = D(
¯
X) + α diag(
¯
X), (11)
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where J(
¯
X) =
P
R
r=1
(ν
+
r
− ν
−
r
)∇
T
¯
X
f
r
(
¯
X) and D(
¯
X) =
P
R
r=1
f
r
(
¯
X)(ν
+
r
− ν
−
r
)(ν
+
r
− ν
−
r
)
T
. The extrinsic compo-
nents Σ
ext
Y
follow from the dependence of the mean on cell
size, which features only in the molecule number variance
of the ENM:
Σ
ext
Y
concentration numbers
EDM 0 0
ENM 0 Var
Π
(s)
¯
X
¯
X
T
, (12)
where the last cell follows from Σ
ext
ENM
= Cov
Π
[E
Π
[x|s]] with
E
Π
[x|s] = s
¯
X.
As a concrete example, we consider transcription of mR-
NAs with a size-dependent transcription rate that are trans-
lated into stable proteins:
∅
k
0
s
−−→ M
k
dm
−−→ ∅, M
k
tl
−−→ M + P. (13)
We then account for dilution through the additional reac-
tions
M
α
−→ ∅, P
α
−→ ∅. (14)
The mean protein concentration is given by
¯
P = k
0
b/α and
the coefficient of variation predicted by the EDM and ENM
models follow the familiar expression
10
CV
2
Y
=
1
Ω
Y
¯
P
1 + b
δ
1 + δ
+
Σ
ext
Y
¯
P
2
, (15)
where we account for size-variability via Ω
Y
and Σ
ext
Y
given
by Eqs. (10) and (12), respectively, and the parameters
δ = 1 +
k
dm
α
, b =
k
tl
k
dm
+ α
, (16)
correspond to the ratio of mRNA and protein degrada-
tion/dilution rates and the translational burst size, respec-
tively. From Eq. (15), (10) and (12), it is clear that size
variation acts on the intrinsic noise component of molecule
concentrations (via E
Π
[s
−1
] ≈ E
Π
[s]
−1
(1 + CV
2
Π
[s])) but
the extrinsic noise component of molecule numbers (via Σ
ext
Y
(12)).
B. Agent-based modelling
Little is known about the accuracy of EDMs and ENMs
in predicting cellular noise in growing populations. In the
following, we introduce an agent-based modelling approach
that serves as a gold standard to assess the validity of these
effective models. The ABM represents cells as agents that
progressively synthesise molecules via intracellular reactions
(1), grow in size and undergo cell division. Every division
gives rise to two daughter cells of varying birth sizes, each
of which inherits a proportion of molecules from the mother
cell via stochastic size-dependent partitioning at division.
The ABM simulation algorithm is given in Box 1, which
combines the First-Division algorithm, previously intro-
duced for agent-based cell populations
38
, with the Extrande
method adapted to simulate reaction networks embedded in
a growing cell
45
. In the following, we describe the exact an-
alytical framework with which we characterise the snapshot
distributions that underlie such a population of agents.
Master equation for agent-based populations
We consider the number of cells n(τ, s, x, t) with age τ
(time since the last division), cell size s and molecule counts
x in a snapshot at time t, which evolves as
∂
∂t
+
∂
∂τ
+
∂
∂s
αs + ¯γ(s, τ)
| {z }
growth
n(τ, s, x, t) = Q(s)n(τ, s, x, t)
| {z }
stochastic reactions
,
n(0, s, x, t)
| {z }
# newborn cells
= 2
∞
∫
0
dτ
0
∞
∫
0
ds
0
B(s|s
0
)
| {z }
division error
×
X
x
0
B(x|x
0
, s/s
0
)
| {z }
partitioning of molecules
¯γ(s
0
, τ
0
)n(τ
0
, s
0
, x
0
, t)
| {z }
# dividing cells
, (17)
and describes cell growth, stochastic reaction kinetics and
a boundary condition for cell division that ensures that the
number of newborn cells is twice the number of dividing
cells after partitioning their size and molecular contents.
These evolution equations have been derived in
38,39
for age-
dependent snapshots but here we extend such agent-based
models to include also cell size dynamics and size-dependent
reaction dynamics. We allow for the following generalisa-
tions: (i) size increases exponentially in single cells, (ii) cells
divide with rate ¯γ(s, τ) that is both size- and age-dependent,
(iii) the transition matrix Q(s) of the molecular reactions
depends on cell size s via the propensities (see definition
after Eq. (4)), and (iv) the molecular partitioning kernel
B(x|x
0
, s/s
0
) depends on the inherited size fraction s/s
0
of
a daughter cell. We now describe in detail how we model
the individual noise sources associated with cell size control,
division errors, and molecule partitioning.
a. Cell size control fluctuations. Recent studies
46,47
have shown that the distribution of sizes with which cells
divide does not explicitly depend on cell age but on the
birth size s
0
. Assuming that ¯γ(s, τ ) = αsγ(s, s
−ατ
), where
γ(s, s
0
) is the division rate per unit size (see also
48,49
), the
division-size distribution is given by
ϕ(s
d
|s
0
) = γ(s
d
, s
0
)e
−
R
s
d
s
0
dsγ(s,s
0
)
. (18)
As a concrete example of (18) we consider a model where
the division size is linearly related to birth size
49–51
s
d
= as
0
+ ∆. (19)
The division rate can be calculated from the distribu-
tion ˜ϕ(∆) of the noise term ∆ in (19) via ˜γ(∆) =
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˜ϕ(∆)/(
R
∞
∆
du ˜ϕ(u)) and setting γ(s, s
0
) = ˜γ(s −as
0
), which
gives the correct division-size distribution ϕ(s
d
|s
0
) = ˜ϕ(s
d
−
as
0
) as expected. The model generalises the sizer (a = 0)
to concerted cell size controls such as the adder (a = 1) and
timer-like (2 > a > 1) models
46,47,52
. In the following, we
will refer to CV
ϕ
[∆] as the size-control noise.
b. Division errors. After division, size is partitioned
between cells and the birth size of the two daughter cells is
obtained from s
0
0
= θs
d
and s
00
0
= (1 − θ)s
d
where θ is the
inherited size fraction, a random variable between 0 and 1
with distribution ¯π(θ) (see Box 1). This can be modelled
using the division kernel
B(s
0
|s
0
) =
Z
1
0
dθ π(θ)δ(θ − s
0
/s
0
),
where π(θ) =
1
2
¯π(θ) +
1
2
¯π(1 −θ) including the case of asym-
metric division. We will refer to CV
π
[θ] as the division error
about the centre E
π
[θ] =
1
2
.
c. Molecule partitioning at cell division. The partition-
ing kernel B(x|x
0
, θ) denotes the probability that a cell in-
herits x molecules from a total of x
0
molecules from its
mother and this probability depends on the daughter’s in-
herited size fraction θ. We assume that cells are sufficiently
well mixed and each molecule is partitioned independently
with probability θ such that the division kernel is binomial
B(x|x
0
, θ) =
N
Y
i=1
x
0
i
x
i
θ
x
i
(1 − θ)
x
0
i
−x
i
. (20)
To make analytical progress, we assume that the pop-
ulation establishes a long-term stationary distribution
Π(s, s
0
, x) characterising the fraction of cells with molecule
numbers x, cell size s and birth size s
0
that is invariant in
time. To this end, we let n(τ, s, x, t) ∝ e
αt
Π(s, τ, x) and
change variables from cell age τ to birth size s
0
such that
Π(s, s
0
, x) = (αs)
−1
Π(s, τ = ln(s/s
0
)/α, x). We find that
this transformation reduces the PDE (17) to an integro-
ODE:
α +
∂
∂s
αs + αsγ(s, s
0
)
Π(s, s
0
, x) = Q(s)Π(s, s
0
, x)
(21a)
s
0
Π(s
0
, s
0
, x) =
2
X
x
0
R
∞
0
ds
0
R
s
0
0
ds
0
0
B(x|x
0
, s
0
/s
0
)B(s
0
|s
0
)s
0
γ(s
0
, s
0
0
)Π(s
0
, s
0
0
, x
0
).
(21b)
We finally characterise the marginal cell size distribution
Π(s, s
0
) and the conditional molecule number distribution
Π(x|s, s
0
) via Bayes’ formula
Π(s, s
0
) =
P
x
Π(s, s
0
, x), Π(x|s, s
0
) =
Π(s, s
0
, x)
Π(s, s
0
)
, (22)
which together provide the full information about the pop-
ulation snapshot.
Cell size distribution
The evolution of the size distribution Π(s, s
0
) is obtained
by summing Eqs. (21) over all possible x, which yields:
α +
∂
∂s
αs + αsγ(s, s
0
)
Π(s, s
0
) = 0 (23a)
s
0
Π(s
0
, s
0
) =
2
R
∞
0
ds
0
R
s
0
0
ds
0
0
B(s
0
|s
0
)s
0
γ(s
0
, s
0
0
)Π(s
0
, s
0
0
). (23b)
Eqs. (23) can be solved analytically
Π(s, s
0
) =
2
Z
ψ
bw
(s
0
)Φ(s|s
0
)
1
s
2
, (24)
where ψ
bw
(s
0
) is the birth size distribution in a backward
lineage (see
48
for details), Φ(s|s
0
) = exp(−
R
s
s
0
ds
0
γ(s
0
, s
0
))
is the probability that a cell born at size s
0
has not divided
before reaching size s, and Z = E
ψ
bw
[s
−1
0
] is a normalising
constant.
Molecule number distributions for cells of a certain size
The conditional molecule number distribution Π(x|s, s
0
)
gives the probability to find the molecule numbers x in a
cell of size s that was born at size s
0
and satisfies
αs
∂
∂s
Π(x|s, s
0
) = Q(s)Π(x|s, s
0
), (25a)
Π(x|s
0
, s
0
) =
X
x
0
R
∞
0
ds
0
R
s
0
0
ds
0
0
B(x|x
0
, s
0
/s
0
)ρ(s
0
, s
0
0
|s
0
)Π(x
0
|s
0
, s
0
0
).
(25b)
Eqs. (25) follow directly from substituting Eq. (22) into (21)
and using (18) and (23). The solution of these equations
depends implicitly on the ancestral cell size distribution ρ,
ρ(s
0
, s
0
0
|s
0
) =
1
ψ
bw
(s
0
)
s
0
s
0
B(s
0
|s
0
)ϕ(s
0
|s
0
0
)ψ
bw
(s
0
0
), (25c)
that gives the probability of a cell born at size s
0
having an
ancestor with division size s
0
and birth size s
0
0
. The main
difference between the molecule number distributions of the
ABM and the EDM/ENM is the boundary condition at cell
division, which as we shall see can have a significant effect
on the reaction dynamics.
III. RESULTS
We here introduce the concept of stochastic concentra-
tion homeostasis (SCH) as a generalisation of concentra-
tion homeostasis in deterministic systems (see Sec. II A 1)
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![FIG. 2. Distributions of CME and agent-based models agree for reaction networks with stochastic concentration homeostasis. (a-c) The EDM (solid lines, analytical solution53,54) agrees with agent-based simulations (shaded areas) for a range of gene expression models. Panels show (a) bursty transcription53 with transcription rate proportional to cell size, (b) bursty translation54 , and (c) bursty transcription and translation54 with geometrically distributed bursts ms whose average is proportional to cell size s (see main text for details). (d) mRNA distributions simulated using the ABM (shaded areas) are shown for cells of sizes s = s0 (red), s = 1.5s0 (orange) and s = 2s0 (green), which agree with the effective dilution model (dots, Poisson distribution). (e) Simulated protein distributions (shaded areas) disagree with the effective dilution model (solid lines, solution in Ref.55). (f) Absolute error (`1) of the effective dilution model as a function of cell size for mRNA (teal) and protein (red) distributions. ABM simulations were obtained using the First-Division Algorithm (Box 1) assuming an adder model (a = 1) and parameters k0 = 10, kdm = 9, ktl = 100, α = 1. Cell cycle noise assumes gamma distribution ϕ̃(∆) with unit mean and CVϕ[∆] = 0.1, while division noise assumes symmetric beta distribution with CVπ[θ] = 0.01.](/figures/fig-2-distributions-of-cme-and-agent-based-models-agree-for-a65kgvb1.webp)
![FIG. 5. The extrinsic noise model approximates gene expression noise with size control and division errors. (a) Scaling of mRNA concentration noise with mean concentrations for various noise levels in added size CV[∆] and partition noise CV[θ] when the transcription rate k0 is varied (top). Corresponding scaling is shown for mRNA numbers (bottom). Analytical predictions of the ENM (solid lines, Eq. (8) with (10) and (12)) and ABM simulations using the First-Division Algorithm (dots, Box 1) are shown. The inset shows the relative error in CV of the EDM compared to ABM simulations [100%× (CVENM/CVABM − 1)]. (b) Scaling of protein noise with mean protein concentration (top) and numbers (bottom) when the translation rate ktl is varied. (c) Same as (b) but varying the transcription rate ktl. For comparison, the ABM predictions without cell cycle noise are shown (dashed black lines, Eq. (41)) and the error bounds of 2% predicted by the theory (solid grey). See caption of Fig. 3 for the remaining parameters.](/figures/fig-5-the-extrinsic-noise-model-approximates-gene-expression-2q0t28nn.webp)
![FIG. 6. Retrospective averages of birth size. Matched asymptotic expansions of EΠ[s0|s] (Eq. (38b), lines) for the adder size control (a = 1, s0 = 1) and agent-based simulations (dots) for (a) varying size control noise CV[∆] and (b) division errors CV[θ].](/figures/fig-6-retrospective-averages-of-birth-size-matched-gnjevqve.webp)
![FIG. 4. Effect of noise in cell size control and division on gene expression noise in single cells. (a-d) Protein noise as a function of cell size s for various noise levels in added size CV[∆]. For comparison, the prediction without cell cycle noise (dashed black line, Eq. (36)) and the cell size distributions (shaded grey) are shown. (e-h) Same as (a-d) but with division noise affecting the inherited size fraction CV[θ]. Analytical predictions (solid lines, Eq. (38a) with (36) and (38b)) and ABM simulations (dots) using the First-Division Algorithm (Box 1) are shown. Gene expression model and all other parameters are as given in Fig. 3. Added size ∆ assumes a gamma distribution with unit mean and CV[∆] = 0.01 (e-f) while division errors θ followed a symmetric beta distribution with CV[θ] = 0.01 (a-d).](/figures/fig-4-effect-of-noise-in-cell-size-control-and-division-on-2ussotmq.webp)