# Asymptotic decoupling of population growth rate and cell size distribution

Yaïr Hein  Institute for Theoretical Physics, Utrecht University, 3584 CC
Utrecht, Netherlands  Farshid Jafarpour  Institute for Theoretical Physics,
Utrecht University, 3584 CC Utrecht, Netherlands

###### Abstract

The rate at which individual bacterial cells grow depends on the
concentrations of cellular components such as ribosomes and proteins. These
concentrations continuously fluctuate over time and are inherited from mother
to daughter cells leading to correlations between the growth rates of cells
across generations. Division sizes of cells are also stochastic and correlated
across generations due to a phenomenon known as cell size regulation.
Fluctuations and correlations from both growth and division processes affect
the population dynamics of an exponentially growing culture. Here, we provide
analytic solutions for the population dynamics of cells with continuously
fluctuating growth rates coupled with a generic model of cell-size regulation.
We show that in balanced growth, the effects of growth and division processes
decouple; the population growth rate only depends on the single-cell growth
rate process, and the population cell size distribution only depends on the
model of division and cell size regulation. The population growth rate is
always higher than the average single-cell growth rate, and the difference
increases with growth rate variability and its correlation time. This
difference also sets the timescale for the population to reach its steady
state. We provide analytical solutions for oscillations in population growth
rate and traveling waves in size distribution during this approach to the
steady state.

For over a century, bacteria have been studied in population experiments in
which the number of cells grows exponentially. In the past decade, however,
numerous experiments have attempted to uncover models of bacterial growth,
division, and cell size control at the single-cell level Stewart  _et al._
(2005); Wang  _et al._ (2010); Campos  _et al._ (2014); Robert  _et al._
(2014); Taheri-Araghi  _et al._ (2015); Tanouchi  _et al._ (2017); Kar  _et
al._ (2021); Vashistha  _et al._ (2021); Golubev (2016); Nordholt  _et al._
(2020). The details of these single-cell models ultimately determine the
dynamics of an exponentially growing population and its population-level
observables, such as cell size distribution and population growth rate, the
latter of which is an important indicator of population fitness Powell (1956,
1964); Amir (2014); Hashimoto  _et al._ (2016); Lin and Amir (2017); Thomas
(2018); Thomas  _et al._ (2018); Jafarpour  _et al._ (2018); Jafarpour (2019);
García-García  _et al._ (2019); Vittadello  _et al._ (2019); Tănase-Nicola and
Ten Wolde (2008); Thomas (2017); Levien  _et al._ (2020a); Lin and Amir
(2020); van Boxtel  _et al._ (2017); Ho  _et al._ (2018); Nozoe and Kussell
(2020).

In constant environments, single bacterial cells grow approximately
exponentially in size and divide when they roughly double their sizes.
However, the division sizes and single-cell growth rates fluctuate and vary
from cell to cell Wang  _et al._ (2010); Campos  _et al._ (2014); Taheri-
Araghi  _et al._ (2015); Pirjol  _et al._ (2017). Moreover, the time it takes
for a cell to divide is correlated between a mother cell and its daughter
cells. There are two distinct sources of correlations. On one hand, if a cells
grows for too long before dividing, its daughter cells would be larger than
average and would have to divide earlier to compensate for their size; this
process is known as cell size regulation, and it induces negative correlations
in inter-division times of the mother and daughter cells. On the other hand,
cells inherit the concentrations of proteins and ribosomes (which set their
instantaneous growth rate) from their mother cells at the moment of division
Kiviet  _et al._ (2014); Lin and Amir (2021); Kleijn  _et al._ (2018); Thomas
_et al._ (2018). This leads to fast-growing cells having fast-growing daughter
cells, which induces positive correlations in inter-division times. These two
competing correlations both decay with a comparable timescale of about a
couple of generation times Wang  _et al._ (2010); Taheri-Araghi  _et al._
(2015). Despite numerous theoretical and experimental attempts, the effects of
the fluctuations in growth and division and the two competing sources of
correlations in inter-division times on population-level observables are not
well understood (the mathematical difficulty of the relationship between
single-cell and population variables is illustrated in Fig. 1).

Figure 1: Relationship between population properties and single-cell
statistics: (A) Lineage tree of a population, two lineages are highlighted.
(B) Cell sizes along each lineage grow approximately exponentially and divide
by two at the division. (C) Along each lineage, the growth rate fluctuates
over time. (D) The state of the population at time $t$ is characterized by the
number of cells and the distributions of their growth rates and sizes. Both
the number of cells and their sizes at time $t$ depend on the history of the
stochastically branching tree of stochastic differential equations of growth
rates in (C). The growth processes of every two lineages in the tree share a
random portion of their history before they branch. The positions of these
branches (division times) are random and depend on the size of the cell and
also the size at the time of the previous branch (birth size) both of which
again depend on the history of the growth rate and the number of earlier
divisions. To relate the properties of the population to the single-cell
statistics using this highly interconnected and growing set of both continuous
and discrete stochastic processes is a daunting task that has consistently
forced modelers to make unphysical simplifying assumptions such as ignoring
fluctuations or correlations in the growth or in the division process Powell
(1956, 1964); Amir (2014); Hashimoto  _et al._ (2016); Lin and Amir (2017);
Thomas (2018); Thomas  _et al._ (2018); Jafarpour  _et al._ (2018); Jafarpour
(2019); García-García  _et al._ (2019); Vittadello  _et al._ (2019); Tănase-
Nicola and Ten Wolde (2008); Thomas (2017); Levien  _et al._ (2020a).

In this letter, we introduce a model in which the fluctuations in single cell
growth rates are described by a continuous stochastic differential equation
and accurately incorporates noise and correlations from both growth and
division. We demonstrate a surprising decoupling of population growth rate and
the distribution of cell sizes in the population at the steady state:
population cell size distribution is determined by the model of division and
contains no information about the single-cell growth process, whereas
population growth rate is determined solely by the model of single-cell growth
dynamics, and is completely unaffected by details of cell division and cell-
size regulation. We show that the population growth rate is always larger than
the the average single-cell growth rate. This difference is simply the product
of two single-cell variables, namely growth rate variability and its
correlation time. It follows that both variability and correlations in growth
are beneficial to the population fitness, when fixing the time-average growth
rate.

In addition to the steady-state results, we have exact solutions for the
population dynamics. Numerical simulations based on a more simplistic
bacterial growth model Jafarpour (2019) as well as experiments data on various
cancer cell lines Chiorino  _et al._ (2001); Vittadello  _et al._ (2019);
Gavagnin  _et al._ (2021) have demonstrated that population properties can
oscillate for a long time before reaching a steady state. Our analytical
solutions provide an exact form for the oscillations in the growth rate and
the accompanying traveling waves in the cell size distribution. We show that
these oscillations decay at a rate proportional to the difference between
population growth rate and average single-cell growth rate. These oscillations
can therefore last for an unexpectedly long time when the correlation time and
growth rate variability are small.

_Model_.—We assume cell sizes grow exponentially with some growth rate
$\lambda_{t}$,

$\frac{d}{dt}v(t)=\lambda_{t}v(t),$ (1)

at all $t$ where no division is taking place. The growth rate itself depends
on internal protein and ribosome concentrations which fluctuate in time and
are inherited by the daughter cells at division Kiviet  _et al._ (2014);
Kleijn  _et al._ (2018). We therefore model the growth rate $\lambda_{t}$ by a
continuously fluctuating process with a well-defined mean that remains
continuous at division. Continuous fluctuations around a well-defined mean are
described by the Ornstein-Uhlenbeck process, characterized by the differential
equation

$\frac{d\lambda_{t}}{dt}=-\theta(\lambda_{t}-\bar{\lambda})+\sigma_{\lambda}\sqrt{2\theta}\,\xi(t),$
(2)

where $\bar{\lambda}$ is the mean growth rate, $\sigma^{2}_{\lambda}$ is the
growth rate variance, $\xi(t)$ is standard Gaussian white noise and $\theta$
sets the auto-correlation decay rate ($1/\theta$ is the correlation time). It
follows that the growth rate is Gaussian distributed with the covariance
$\text{Cov}(\lambda_{t},\lambda_{s})=\sigma_{\lambda}^{2}e^{-\theta|t-s|}$. An
example of this process is given in Fig. 1(C).

In general, the division size $v_{d,n}$ of an $n$th-generation cell depends on
its birth size and possibly those of its ancestors through a stochastic
function $h$,

$v_{d,n}=h(v_{b,n},v_{b,n-1},\dots).$ (3)

The function $h$ defines the model of division and cell size regulation and
determines the steady-state birth size distribution $f_{v_{b}}$ (see Refs.
Amir (2014); Ho  _et al._ (2018) for examples of such models). To keep the
argument general, we will not specify a model of cell size regulation $h$. We
will show that the population dynamics is only affected by the birth size
distribution $f_{v_{b}}$ and is independent of other intricacies of how the
birth sizes may be correlated.

_Derivation of Population Properties._ —We start by defining a dimensionless
stochastic time as

$\tau_{t}\equiv\int_{0}^{t}\lambda_{s}ds.$ (4)

Since $\lambda_{t}$ is Gaussian, $\tau_{t}$ is Gaussian too. Its mean and
variance are given by

$\langle\tau_{t}\rangle=\bar{\lambda}t,\text{
and}\quad\sigma_{\tau_{t}}^{2}=2\frac{\sigma_{\lambda}^{2}}{\theta}t-2\frac{\sigma_{\lambda}^{2}}{\theta^{2}}(1-e^{-\theta
t}).$ (5)

Dynamics of $v(\tau)$ in terms of stochastic time simplify to

$\frac{d}{d\tau}v(\tau)=v(\tau).$ (6)

This follows from the chain rule of differentiation and the fact that
$d\tau_{t}/dt=\lambda_{t}$. By doing this, we absorb all growth fluctuations
into the stochastic time, and as such, we get cell size dynamics where the
only stochasticity comes from the division noise. Equation (6) now implies
that

$v(\tau)=\frac{v_{0}\,e^{\tau}}{2^{\Delta(\tau)}},$ (7)

where $\Delta(\tau)$ is the number of divisions before $\tau$. Here
$v_{0}e^{\tau}$ corresponds to the size the cell would have at $\tau$ if no
divisions had taken place, and the factor of $1/2^{\Delta(\tau)}$corresponds
to the cell having halved its size for a total of $\Delta(\tau)$ times. It can
be shown that the distribution of $v(\tau)$ becomes periodic in $\tau$ with a
period of $\ln(2)$ (see the Supplemental Material, SM SM )

We can now relate the lineage behaviour to the population behaviour. Our first
focus is to find the expected population size $N(t)$ at some time $t$. There
are $2^{n}$ distinct $n$-th generation cells in the population tree, each of
which can be associated with a unique lineage. These lineages each have the
same probability $P(\Delta(\tau_{t})=n)$ of being in the $n$-th generation at
time $t$. Hence, the expected number of $n$-th generation cells present at
time $t$ is given by $2^{n}P(\Delta(\tau_{t})=n)$. The total expected number
of cells is the sum of this quantity over all generations Levien  _et al._
(2020b); Pigolotti (2021); Jafarpour (2019)

$N(t)=\sum_{n=0}^{\infty}2^{n}P(\Delta(\tau_{t})=n)=\left\langle
2^{\Delta(\tau_{t})}\right\rangle.$ (8)

Since divisions are triggered purely by cell size, $\Delta(\tau_{t})$ depends
only on $\tau_{t}$ and not directly on $t$. We may therefore average over
$\Delta(\tau_{t})$ and $\tau_{t}$ independently,

$N(t)=\left\langle\left\langle\left.2^{\Delta(\tau_{t})}\right|\tau_{t}\right\rangle\right\rangle$
(9)

By applying Eq. (S18), we can rewrite the inner expected value conditioned to
$\tau_{t}=\tau$ as

$\left\langle
2^{\Delta(\tau)}\right\rangle=\left\langle\frac{v_{0}}{v(\tau)}\right\rangle
e^{\tau}.$ (10)

This is where we can use the periodicity of the distribution of $v(\tau)$,
which implies that $\langle v_{0}/v(\tau)\rangle$ is periodic in $\tau$ with
period $\ln(2)$. We can therefore expand this function as a Fourier series

$\left\langle\frac{v_{0}}{v(\tau)}\right\rangle=\sum_{k}c_{k}e^{ik\omega\tau},$
(11)

with $\omega=2\pi/\ln(2)$, where $c_{k}$s are constants that depend on the
birth size distribution. By combining Eqs. (9), (10) and (11), we obtain

$\displaystyle N(t)=$ $\displaystyle\sum_{k\in\mathbb{Z}}c_{k}\left\langle
e^{\left(1+ik\omega\right)\tau_{t}}\right\rangle$ $\displaystyle=$
$\displaystyle
e^{\langle\tau_{t}\rangle+\frac{1}{2}\sigma_{\tau_{t}}^{2}}\sum_{k\in\mathbb{Z}}c_{k}e^{ik\omega\left(\langle\tau_{t}\rangle+\sigma_{\tau_{t}}^{2}\right)}e^{-\frac{1}{2}k^{2}\omega^{2}\sigma_{\tau_{t}}^{2}},$
(12)

where $\langle\tau_{t}\rangle$ and $\sigma_{\tau_{t}}^{2}$ are given by Eq.
(5). At large $t$ all components become negligible compared to the fastest
growing term, which is $k=0$. Hence asymptotically the population grows as

$N(t)\approx c_{0}\,e^{\langle\tau_{t}\rangle+\frac{1}{2}\sigma_{\tau}^{2}}.$
(13)

We may also use this method to find other quantities averaged over a
population. If $x$ is some cell property, its value averaged over all cells in
a population snapshot is

$\langle x\rangle_{p}=\frac{1}{N(t)}\left\langle
x\frac{v_{0}}{v(\tau_{t})}e^{\tau_{t}}\right\rangle,$ (14)

where the subscript p indicates population average. Here $x$ can be any cell
variable, such as instantaneous growth rate $\lambda_{t}$ or size
$v(\tau_{t})$. By applying Eq. (14) to Dirac-delta functions
$\delta(\lambda-\lambda_{t})$ and $\delta(v-v(\tau_{t}))$ we can calculate
expected population probability density distributions of growth rates and
sizes respectively.

Steady-State Population Properties.—Using Eqs. (5) and (13), we find a
surprisingly simple relationship for the asymptotic population growth rate

$\Lambda_{\infty}\equiv\lim_{t\to\infty}\frac{1}{N(t)}\frac{dN(t)}{dt}=\bar{\lambda}+\frac{\sigma_{\lambda}^{2}}{\theta}.$
(15)

Population growth rate is remarkably unaffected by the details of the division
process. When fixing the lineage time-average cell growth rate
$\bar{\lambda}$, increasing growth rate variability increases the population
growth rate. The difference between population growth rate and single-cell
growth rate, given by $\sigma_{\lambda}^{2}/\theta$, can be thought of as the
diffusion coefficient of accumulated lineage cell growth, since for large $t$
we have $\sigma_{\tau_{t}}^{2}\approx 2(\sigma_{\lambda}^{2}/\theta)t$. The
result that growth rate variability is always beneficial from the perspective
of population growth rate seems to contradict the results of previous work Lin
and Amir (2017, 2020). This difference can be explained by how growth rate
variability is defined differently in the previous work. See SM SM  for an in-
depth comparison.

The distribution of instantaneous single-cell growth rates in a population
snapshot can be obtained by calculating
$\langle\delta(\lambda-\lambda_{t})\rangle_{p}$. Asymptotically, this
distribution converges to a Gaussian distribution with mean $\Lambda_{\infty}$
and variance $\sigma_{\lambda}^{2}$. Measurements of instantaneous growth
rates of cells sampled from populations will thus be higher than those of
cells sampled from single lineage experiments. Given that this result is again
independent of the details of cell division, it is consistent with Ref. Levien
_et al._ (2021) where a similar model of growth is studied neglecting noise
and correlations in the division process.

Next, we can calculate the time-dependent size distribution
$G(t,v)=\langle\delta(v-v(\tau_{t}))\rangle_{p}$ using Eq. (14). By taking the
asymptotic time limit, we find the stationary distribution of cell sizes in a
population snapshot

$G(v)=\frac{2}{v^{2}\left\langle
1/v_{b}\right\rangle}\left(F_{v_{b}}(v)-F_{v_{b}}(v/2)\right),$ (16)

where $F_{v_{b}}$ is the cumulative distribution function of the birth size
$v_{b}$ (see SM SM  for derivation). When the birth size variability
approaches zero, $F_{v_{b}}$ becomes a step function and Eq. (16) turns into
the well-known “inverse square law” of size distribution for constant birth
size Koch and Schaechter (1962), characterized by $G(v)=2v_{b}/v^{2}$ for
$v_{b}\leq v<2v_{b}$ as shown in Fig. 2. Remarkably, Eq. (16) is independent
of the growth process (growth rate, its fluctuations, and its correlation
time), and as such, it agrees with the size distribution predicted by Powell
in 1964 Powell (1964), where it was assumed that all cells grow
deterministically with the same constant growth rate.

The asymptotic independence of the size distribution from the growth process
as well as the independence of population growth rate from the division model
hold more generally than the case of Ornstein-Uhlenbeck growth process. They
hold when growth along a lineage is any continuous stationary process that is
unaffected by division (see SM SM  for derivation)

The birth-size distribution in Eq. (16) is determined by the model of division
and cell-size control. We introduce a variation of the model from Ref. Amir
(2014), where cells division size $v_{d}$ is determined based on its birth
size $v_{b}$

$\ln(v_{d})=\alpha\ln(2\bar{v})+(1-\alpha)\ln(2v_{b})+\eta,$ (17)

where $\eta$ is a Gaussian random variable with mean zero and variance
$\sigma^{2}_{\eta}$, $\bar{v}$ is the constant equilibrium birth size, and
$\alpha$ determines the degree of cell size regulation ($\alpha=0$ is the
timer model, $\alpha=1$ is the sizer model, and $\alpha=0.5$ approximates the
adder model; see Ref. Amir (2014); Ho  _et al._ (2018) for more details). This
model predicts a log-normal distribution $f_{v_{b}}$ for the birth sizes.
Figure 2 shows reasonable agreement between the best fit of Eq. (16) with a
log-normal $f_{v_{b}}$ to experimental cell sizes from Ref. Gray  _et al._
(2019). The slight disagreement in the tail of the size distribution hints at
potential non-Gaussianity of the noise term $\eta$ in cell division.

Figure 2: Comparison of the size distribution from Eq. (16) (blue solid curve,
with birth-size distribution derived from the model of cell-size regulation in
Eq. (17)) to histograms of cell sizes from Ref. Gray  _et al._ (2019) and the
inverse square law (dashed red curve, assuming deterministic division).

Transient Dynamics of Population Properties.—The dynamics of the number of
cells in the population is extremely well approximated by only the zeroth and
first order terms from Eq. (12) and the large time behaviour of
$\sigma_{\tau_{t}}^{2}$, which results in

$\displaystyle N(t)\approx Ce^{\Lambda_{\infty}t}\left(1+ACos\left(\Omega
t+\phi\right)e^{-qt}\right),$ (18)

where $\Omega=\omega\left(\bar{\lambda}+2\sigma_{\lambda}^{2}/\theta\right)$,
$q=\omega^{2}\sigma_{\lambda}^{2}/\theta$ with $\omega=2\pi/\ln(2)$ (the
constants $C$, $A$, and $\phi$ can be calculated explicitly as functions of
the initial state of the population; see SM SM  for the exact solution).
Equation (18) tells us that the population growth rate (defined as
$\Lambda(t)\equiv d\ln(N)/dt$) oscillates, and its amplitude decays
exponentially with rate $q$. The uncorrelated limit ($\theta\to\infty$ keeping
$\sigma^{2}/\theta$ constant) of these oscillations is studied in Ref.
Jafarpour (2019). The numerically inferred relationship for the decay rate of
these oscillations in Ref. Jafarpour (2019) is in perfect agreement with our
analytical prediction for $q$.

The time-dependent population cell size distribution can be calculated
explicitly by taking $\langle\delta(v-v(\tau_{t}))\rangle_{p}$. The first
order large time approximation becomes

$\displaystyle G(t,v)\approx G(v)\frac{1+BCos\left(\Omega
t-\omega\ln(v)+\zeta\right)e^{-qt}}{1+ACos(\Omega t+\phi)e^{-qt}},$ (19)

where $B$ and $\zeta$ depend on the initial conditions of the population, and
$G(v)$ is the asymptotic size distribution given by Eq. (16) (see SM SM  for
the exact solution). A plot of Eq. (19) is given in Fig. 3, where we compare
it to the exact time dependent size distribution and the asymptotic case. We
see that the deviation from the asymptotic distribution decays at the rate
$q$. The parameter $q$ therefore provides us with a timescale for the internal
population properties to reach steady state. In Eq. (19) we also encounter
traveling waves in logarithmic cell size densities. These waves propagate at a
speed of $\Omega/\omega=\bar{\lambda}+2\sigma_{\lambda}^{2}/\theta$.
Interestingly, this is faster than the population growth rate
$\Lambda_{\infty}$ from Eq. (15).

Figure 3: Traveling waves in the dynamics of size distributions: analytically
calculated population size distributions are shown at three consecutive times
(red solid curves) compared with the first-order approximations (blue dashed
curves) and the asymptotic size distribution (green solid curve). Times are
given in units of average division times $\ln(2)/\bar{\lambda}$. Model
parameters: $\theta=\bar{\lambda}=\ln(2)$,
$\sigma_{\lambda}^{2}/\theta=0.006\bar{\lambda}$,
$\sigma_{\ln(v_{b})}^{2}=0.2$.

Discussion.—Previous works studying the role of growth variability in
population dynamics consider the growth rate as a constant random variable
associated with each cell and ignore its continuous fluctuations over time Lin
and Amir (2017, 2020); Jafarpour (2019); Taheri-Araghi  _et al._ (2015); Ho
_et al._ (2018); Marantan and Amir (2016). Here, we have shown that if instead
of simplified models, we consider a less coarse-grained more accurate model of
single-cell growth, the roles of growth and division statistics on population
dynamics decouple, and all aspects of population dynamics become analytically
solvable.

Our results on size distribution show that a snapshot image of a population of
cells cannot be used to extract any information about either the growth
process or the mechanism of cell size regulation. However, our time-dependent
solution Eq. (19) predicts traveling waves in population size distribution
that can be used to infer single-cell growth rate variability.

Equation (15) connects the statistics of growth fluctuations (and therefore
fluctuations in concentrations of the underlying molecular components) to the
fitness of the organism. This allows for studying (independently of any model
of cell division) how these fluctuations at the molecular level are shaped by
evolution Jafarpour  _et al._ (2022).

###### Acknowledgements.

We gratefully acknowledge Philipp Thomas and Ethan Levien for their comments
on the manuscript.

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Supplemental Material for:

Asymptotic decoupling of growth rate and cell-size distribution

Yaïr Hein and Farshid Jafarpour

##  I Birth size distribution

Here we show how a model like the one introduced in Ref. Amir (2014) leads to
identically distributed birth sizes. Consider a cell born with size $v_{b,n}$,
with $n$ denoting the generation of the cell considered within some fixed
lineage. It will attempt to divide at some size $v_{d,n}=h(v_{b,n})$, that
depends only on the cell’s birth size. This function is defined such that

$\ln(v_{d,n})=\alpha\ln(2\bar{v})+(1-\alpha)\ln(2v_{b,n})+\eta_{n}$ (S1)

where $\eta_{n}$ is an independent Gaussian noise parameter parameter with
mean $0$ and variance $\sigma_{\eta}^{2}$, $\bar{v}$ is the equilibrium birth
size and $\alpha$ is some constant that interpolates between different models
of cell size control. For $\alpha=1$, there is no dependency on birth size and
we obtain what is known as the sizer model. For $\alpha=0$, one can show that
the ratio $v_{d,n}/v_{b,n}$ becomes independent of $v_{b,n}$ and we obtain a
model that is equivalent to the timer model. For $\alpha=1/2$, we obtain a
model that is an approximation for the model where cells roughly add a
constant size $\bar{v}$ to their birth size upon division, known as the adder
model. In our simulations we will assume that $\alpha=1/2$, since the adder
model seems to most closely correspond to experimental findings Taheri-Araghi
_et al._ (2015).

Under the assumption of symmetric division, birth sizes are half the sizes of
their mother cells at division, meaning $v_{b,n+1}=v_{d,n}/2$, so we obtain a
recursive formula for the birth sizes

$\ln\left(\frac{v_{b,n+1}}{\bar{v}}\right)=(1-\alpha)\ln\left(\frac{v_{b,n}}{\bar{v}}\right)+\eta_{n}.$
(S2)

Now suppose the birth size of our starting cell $v_{b,0}$ is fixed, then the
$n$-th cell in our lineage will have its birth size given by

$\ln\left(\frac{v_{b,n}}{\bar{v}}\right)=(1-\alpha)^{n}\ln\left(\frac{v_{b,0}}{\bar{v}}\right)+\sum_{i=0}^{n-1}(1-\alpha)^{n-1-i}\eta_{i},$
(S3)

hence $\ln(v_{b,n})$ will be Gaussian with mean

$\left\langle\ln\left(v_{b,n}\right)\right\rangle=\ln(\bar{v})+(1-\alpha)^{n}\ln\left(\frac{v_{b,0}}{\bar{v}}\right)\quad\overset{n\to\infty}{\rightarrow}\quad\ln(\bar{v}),$
(S4)

and variance

$\sigma_{\ln(v_{b,n})}^{2}=\sum_{i=0}^{n-1}(1-\alpha)^{2i}\sigma_{\eta}^{2}=\frac{1-(1-\alpha)^{2n}}{\alpha(2-\alpha)}\sigma_{\eta}^{2}\quad\overset{n\to\infty}{\rightarrow}\quad\frac{1}{\alpha(2-\alpha)}\sigma_{\eta}^{2}.$
(S5)

In distribution, the birth sizes $v_{b,n}$ now quickly converge to some
asymptotic log-normal distribution $v_{b}$ such that $\ln(v_{b})$ has mean
$\ln(\bar{v})$ and variance
$\sigma_{\ln(v_{b})}^{2}=\sigma_{\eta}^{2}/\alpha(2-\alpha)$. One can check
that this is a steady-state distribution of equation (S2). Thus if we assume
$v_{b,0}$ is unknown and starts off in the steady state distribution, then all
subsequent cells will also have this distribution. This is what we used in the
main text. Note that even if the initial birth size is known and fixed, the
subsequent birth size distributions would converge back to the steady-state
distribution so rapidly, that after a few generations it hardly matters
whether the birth sizes follow equation (S2) or its corresponding steady state
distribution.

##  II Growth rate process

In the main text we defined our growth rate $\lambda_{t}$ as the stochastic
process satisfying the Langevin-equation

$\frac{d\lambda_{t}}{dt}=-\theta(\lambda_{t}-\bar{\lambda})+\sigma_{\lambda}\sqrt{2\theta}\xi(t),$
(S6)

with time-average growth rate $\bar{\lambda}$, instantaneous growth rate
variance $\sigma_{\lambda}^{2}$, auto-correlation decay rate $\theta$ and a
Gaussian white noise process $\xi(t)$ that satisfies $\langle\xi(t)\rangle=0$
and $\langle\xi(t)\xi(s)\rangle=\delta(t-s)$. The solution to equation (S6) is

$\lambda_{t}=\bar{\lambda}+(\lambda_{0}-\bar{\lambda})e^{-\theta
t}+\sigma_{\lambda}\sqrt{2\theta}e^{-\theta t}\int_{0}^{t}e^{\theta
s}\xi(s)ds$ (S7)

Due to the properties of $\xi(s)$, growth rate $\lambda_{t}$ is a gaussian
with mean

$\langle\lambda_{t}\rangle=\bar{\lambda}+(\lambda_{0}-\bar{\lambda})e^{-\theta
t}\qquad
Var(\lambda_{t})=\langle(\lambda_{t}-\langle\lambda_{t}\rangle)^{2}\rangle=\sigma_{\lambda}^{2}\left(1-e^{-2\theta
t}\right)$ (S8)

and auto-correlation

$Cov(\lambda_{t},\lambda_{s})=\langle(\lambda_{t}-\langle\lambda_{t}\rangle)(\lambda_{s}-\langle\lambda_{s}\rangle)\rangle=\sigma_{\lambda}^{2}\left(e^{-\theta|t-s|}-e^{-\theta(t+s)}\right)$
(S9)

When $t,s\gg 1/\theta$, $\lambda_{t}$ reaches a steady state with mean,
variance and auto-correlation

$\langle\lambda_{t}\rangle=\bar{\lambda},\qquad
Var(\lambda_{t})=\sigma_{\lambda}^{2},\qquad
Cov(\lambda_{t},\lambda_{s})=\sigma_{\lambda}^{2}e^{-\theta|t-s|},$ (S10)

Alternatively if $\lambda_{0}$ is unknown and assumed to be in the steady
state distribution characterized by equation (S10), then $\lambda_{t}$ will
stay in the steady-state distribution satisfy (S10) for all times. We could
have easily fixed $\lambda_{0}$ and used equations (S8) (S9) in the rest of
our theory, but for the sake of simplicity, we decided to opt for the steady-
state process satisfying equation (S9) in the main text.

We defined the dimensionless stochastic time by

$\tau_{t}=\int_{0}^{t}\lambda_{s}ds$ (S11)

Since $\lambda_{t}$ is always Gaussian, $\tau_{t}$ must be Gaussian too. We
can fully characterize its distribution by its mean and variance. The mean is

$\langle\tau_{t}\rangle=\int_{0}^{t}\bar{\lambda}ds=\bar{\lambda}t$ (S12)

and its variance is

$\sigma_{\tau_{t}}^{2}=\int_{0}^{t}\int_{0}^{t}Cov(\lambda_{s},\lambda_{u})duds=\sigma_{\lambda}^{2}\int_{0}^{t}\int_{0}^{t}e^{-\theta|u-s|}dsdu.$
(S13)

Due to the symmetry of the integrand in $s-u$, the integrals over the domain
halves $s-u\leq 0$ and $s-u\geq 0$ will yield the same value, so the integral
over either domain half will be half the value of the full integral. We find

$\begin{split}\sigma_{\tau_{t}}^{2}=&2\sigma_{\lambda}^{2}\int_{0}^{t}\int_{0}^{s}e^{\theta(u-s)}duds=2\sigma_{\lambda}^{2}\int^{t}_{0}\frac{1}{\theta}(1-e^{-\theta
s})ds=2\frac{\sigma_{\lambda}^{2}}{\theta}\left(t-\frac{1}{\theta}(1-e^{-\theta
t})\right)\end{split}$ (S14)

##  III Cell size trajectory function

The cell size function of time was initially defined as

$\frac{d}{dt}v(t)=\lambda_{t}v(t)$ (S15)

in all $t$ where no division is taking place. Though the mechanism described
in section I, we obtain an array of division sizes $(v_{d,0},v_{d,1},\dots)$.
A division happens when $v(t)$ reaches the next division size from this array.
Upon a division, $v(t)$ discontinuously jumps to half its value $v(t)/2$.

We also impose the initial condition $v(0)=v_{0}$. One may raise the point
that this initial condition affects the information we have about the birth
size of the initial cell, which contradicts the steady-state assumption of the
birth sizes made in section I. We can work around this by assuming $t$ is
large enough for the steady-state approximation of $v_{d,n}$ to hold for all
cells in the population.

Let us now define the cell size function of stochastic time as the function
satisfying

$\frac{d}{d\tau}v(\tau)=v(\tau)$ (S16)

for $\tau$ where no division is taking place. This function also halves its
value whenever it reaches the next division size taken from the same array
$v_{d,0},v_{d,1},v_{d,2},\dots$. We connect these cell size functions by
evaluating the stochastic time cell size in $\tau_{t}$, as defined in equation
(S11). Note that

$\frac{d}{dt}v(\tau_{t})=\frac{d\tau_{t}}{dt}\frac{d}{d\tau_{t}}v(\tau_{t})=\lambda_{t}\frac{d}{d\tau_{t}}v(\tau_{t})=\lambda_{t}v(\tau_{t}),$
(S17)

where we used equation (S16) and the fact that $d\tau_{t}/dt=\lambda_{t}$. It
follows that $v(\tau_{t})$ as a function of $t$ satisfies equation (S15). It
also has the same division behaviour as the cell size function of time, hence
$v(\tau_{t})=v(t)$.

As argued in the main text, we can express the cell size at any point as

$v(\tau)=v_{0}\frac{1}{2^{\Delta(\tau)}}e^{\tau},$ (S18)

where $\Delta(\tau)$ is the total number of divisions that have taken place
before stochastic time $\tau$. The right-hand side of equation (S18) simply
satisfies the definition of $v(\tau)$, which proves the equation.

Now we will show the periodicity of $v(\tau)$. Let
$\tau_{b,1},\tau_{b,2},\dots$ be the birth times such that
$\Delta(\tau_{b,n})=n$. Since $\Delta(\tau)$ does not increase outside of
division events, we have that for any $\tau$

$\Delta(\tau)=\sup\\{n\geq 0:\tau_{b,n}\leq\tau\\}$ (S19)

note that equation (S18) can be inverted to express the stochastic time in
terms of the other variables as

$\tau=\ln(2)\Delta(\tau)-\ln\left(\frac{v_{0}}{v(\tau)}\right).$ (S20)

This now allows us to express the birth times in terms of the other variables
as

$\tau_{b,n}=n\ln(2)-\ln\left(\frac{v_{0}}{v_{b,n}}\right),$ (S21)

hence we can rewrite equation (S19) as

$\Delta(\tau)=\sup\left\\{n\geq
0:n\ln(2)-\ln\left(\frac{v_{0}}{v_{b,n}}\right)\leq\tau\right\\}$ (S22)

Note that

$\begin{split}\Delta(\tau+\ln(2))-1=&\sup\left\\{n\geq
0:n\ln(2)-\ln\left(\frac{v_{0}}{v_{b,n}}\right)\leq\tau+\ln(2)\right\\}-1\\\
=&\sup\left\\{n\geq-1:n\ln(2)-\ln\left(\frac{v_{0}}{v_{b,n+1}}\right)\leq\tau\right\\},\end{split}$
(S23)

which has the same distribution as simply $\Delta(\tau)$. It follows that

$v(\tau+\ln(2))=v_{0}\frac{1}{2^{\Delta(\tau+ln(2))}}e^{\tau+\ln(2)}=\frac{1}{2^{\Delta(\tau+\ln(2))-1}}e^{\tau}$
(S24)

has the same distribution as $v(\tau)$, therefore the distribution of
$v(\tau)$ must be periodic in $\tau$ with a period $\ln(2)$.

##  IV Total population

Here we will more rigorously link the expected total population $N(t)$ to the
cell statistics along a single lineage. For each generation $n$, starting from
$n=0$, there are $2^{n}$ distinct cells in the population tree. Every cell can
therefore be characterized by its generation $n$ and some index $1\leq j\leq
2^{n}$. By $\Delta^{j}$ and $\tau^{j}_{t}$ we denote the the number of
divisions and stochastic time at time $t$ respectively for some lineage
containing cell $j$. A cell $j$ is present in the population at time $t$ if
the total number of divisions at that time equals the generation of the cell.
The total number of cells $\tilde{N}(t)$ can thus be obtained be counting the
number of cells satisfying the condition $\Delta^{j}(\tau^{j}_{t})=n$, which
is

$\tilde{N}(t)=\sum_{n=0}^{\infty}\sum_{j=1}^{2^{n}}\delta_{\Delta^{j}(\tau^{j}_{t}),n}$
(S25)

where $\delta$ is a Kronecker delta. The expected number of cells at time $t$
is

$N(t)=\left\langle\tilde{N}(t)\right\rangle=\sum_{n=0}^{\infty}\sum_{j=1}^{2^{n}}\left\langle\delta_{\Delta^{j}(\tau^{j}_{t}),n}\right\rangle$
(S26)

The expectation value of a Kronecker delta is always equal to the probability
of it being one, which is the probability that $\Delta^{j}(\tau^{j}_{t})=n$.
Since this probability is independent of the lineage considered, we can drop
the dependency on $j$, hence

$N(t)=\sum_{n=0}^{\infty}\sum_{j=1}^{2^{n}}\mathbb{P}(\Delta(\tau_{t})=n)=\sum_{n=0}^{\infty}2^{n}\mathbb{P}(\Delta(\tau_{t})=n)=\left\langle
2^{\Delta(\tau_{t})}\right\rangle$ (S27)

Let us now consider this expectation value conditioned to stochastic time
$\tau_{t}=\tau$, denoted by $\hat{N}(\tau)=\left\langle
2^{\Delta(\tau)}\right\rangle$. Using equation (S18), we rewrite this as

$\hat{N}(\tau)=\left\langle\frac{v_{0}}{v(\tau)}\right\rangle e^{\tau}$ (S28)

In section III we argued why the distribution of $v(\tau)$ is periodic with a
period of $\ln(2)$. It follows that $\langle v_{0}/v(\tau)\rangle$ is a
periodic function of $\tau$ with a period of $\ln(2)$. We may therefore
Fourier expand this expression as

$\left\langle\frac{v_{0}}{v(\tau)}\right\rangle=\sum_{k=-\infty}^{\infty}c_{k}e^{i\omega
k\tau}$ (S29)

with $\omega=2\pi/\ln(2)$ and where $c_{k}$ are complex numbers that depend on
the behaviour of $\langle v_{0}/v(\tau)\rangle$. We can plug equation (S29)
into equation (S28) and obtain

$\hat{N}(\tau)=\sum_{k=-\infty}^{\infty}c_{k}e^{(1+i\omega k)\tau}.$ (S30)

Since $\tau_{t}$ is a normal distribution, we can now exactly derive the
expected total population at time $t$ as

$N(t)=\left\langle\hat{N}(\tau_{t})\right\rangle=\sum_{k=-\infty}^{\infty}c_{k}\left\langle
e^{(1+i\omega
k)\tau_{t}}\right\rangle=\sum_{k=-\infty}^{\infty}c_{k}e^{(1+i\omega
k)\langle\tau_{t}\rangle+\frac{1}{2}(1+i\omega k)^{2}\sigma_{\tau_{t}}^{2}},$
(S31)

where $\langle\tau_{t}\rangle$ and $\sigma_{\tau_{t}}^{2}$ follow from section
II. By factoring out the exponential of the fastest growing term $k=0$, we
obtain the expression from the main text

$N(t)=e^{\langle\tau_{t}\rangle+\frac{1}{2}\sigma_{\tau_{t}}^{2}}\sum_{k=-\infty}^{\infty}c_{k}e^{i\omega
k\left(\langle\tau_{t}\rangle+\sigma_{\tau_{t}}^{2}\right)}e^{-\frac{1}{2}\omega^{2}k^{2}\sigma_{\tau_{t}}^{2}}.$
(S32)

If there were no growth rate variability, we would have
$\sigma_{\tau_{t}}^{2}=0$ and the population would oscillate forever. This is
something that has also been concluded in Ref. Jafarpour (2019) in the case of
constant growth rate. The spread in stochastic time caused by growth rate
variability causes the relative amplitudes of all the oscillatory components
in equation (S32) to decrease with time. We can read off that
$e^{\langle\tau_{t}\rangle+\frac{1}{2}\sigma_{\tau_{t}}^{2}}$ is the leading
exponential behaviour of the population, $e^{i\omega
k\left(\langle\tau_{t}\rangle+\sigma_{\tau_{t}}^{2}\right)}$ capture the
oscillating behaviour of the $k\neq 0$ terms, $c_{k}$ are their initial
amplitudes and $e^{-\frac{1}{2}\omega^{2}k^{2}\sigma_{\tau_{t}}^{2}}$ capture
their decay relative to the leading term

Let us rewrite equation (S32) to get rid of all complex components. Since all
coefficients satisfy $c_{k}^{*}=c_{-k}$, we can take all positive and negative
$k$ terms together.

$N(t)=e^{\langle\tau_{t}\rangle+\frac{1}{2}\sigma_{\tau_{t}}^{2}}\left[c_{0}+\sum_{k=1}^{\infty}2|c_{k}|Cos\left(\omega
k\left(\langle\tau_{t}\rangle+\sigma_{\tau_{t}}^{2}\right)+Arg(c_{k})\right)e^{-\frac{1}{2}\omega^{2}k^{2}\sigma_{\tau_{t}}^{2}}.\right]$
(S33)

After a few generations we can easily approximate equation (S14) by

$\sigma_{\tau_{t}}^{2}=2\frac{\sigma_{\lambda}^{2}}{\theta}t-2\frac{\sigma_{\lambda}^{2}}{\theta^{2}}$
(S34)

Equation (S33) can now be rewritten as

$N(t)=Ce^{\Lambda_{\infty}t}\left[1+\sum_{k=1}^{\infty}A_{k}Cos\left(k\Omega
t+\phi_{k}\right)e^{-k^{2}qt}\right],$ (S35)

with fundamental factors of time given by

$\Lambda_{\infty}=\bar{\lambda}+\frac{\sigma_{\lambda}^{2}}{\theta},\qquad\Omega=\omega\left(\bar{\lambda}+2\frac{\sigma_{\lambda}^{2}}{\theta}\right),\qquad
q=\omega^{2}\frac{\sigma_{\lambda}^{2}}{\theta},$ (S36)

and initial condition dependent constants given by

$C=c_{0}e^{-2\frac{\sigma_{\lambda}^{2}}{\theta^{2}}},\qquad
A_{k}=2\frac{|c_{k}|}{c_{0}}e^{\omega^{2}k^{2}\frac{\sigma_{\lambda}^{2}}{\theta^{2}}},\qquad\phi_{k}=Arg(c_{k})-2\omega
k\frac{\sigma_{\lambda}^{2}}{\theta^{2}}$ (S37)

##  V Coefficients

In this section we will discuss how the coefficients $c_{k}$ are related to
the birth size distribution and initial conditions. From the Fourier expansion
in equation (S29), it follows that

$c_{k}=\frac{1}{\ln(2)}\int_{x}^{x+\ln(2)}\left\langle\frac{v_{0}}{v(\tau)}\right\rangle
e^{-i\omega k\tau}d\tau,$ (S38)

where $x$ can be any real number large enough for $v(\tau)$ to have reached
steady-state. Due to cell size homeostasis there must be some finite interval
on which all birth sizes lie with full certainty. We assume that all log birth
sizes lie within some interval of length $\ln(2)$. Though this is not always
true in nature, it does hold by approximation and it simplifies our
calculations a lot. Suppose $y$ is such that $\ln(v_{b,n})\in(y,y+\ln(2))$.
Choose $x=y+n\ln(2)-\ln(v_{0})$, with $n$ large enough for $v_{b,n}$ to be in
steady-state. Using equation (S21), we find that

$\tau_{b,n}\in(x,x+\ln(2))$ (S39)

and that all other times of birth fall outside of this interval. For all
$\tau$ considered in the integral in equation (S38), the division count can
only take on two values, depending on whether the one division occurred before
or after $\tau$.

$\Delta(\tau)=\begin{cases}n-1&\text{ if }\tau\leq\tau_{b,n}\\\ n&\text{ if
}\tau>\tau_{b,n}\end{cases}$ (S40)

It follows that

$\frac{v_{0}}{v(\tau)}=e^{-\tau}\left(2^{n-1}\mathds{1}\\{\tau\leq\tau_{b,n}\\}+2^{n}\mathds{1}\\{\tau>\tau_{b,n}\\}\right),$
(S41)

where $\mathds{1}$ is an indicator function that equals $1$ when the condition
in the brackets is satisfied and $0$ otherwise. We can evaluate this function
in a shifted position and find that

$\frac{v_{0}}{v(\tau+n\ln(2)-\ln(v_{0}))}=v_{0}e^{-\tau}\left(\frac{1}{2}\mathds{1}\\{\tau\leq\ln(v_{b,n})\\}+\mathds{1}\\{\tau>\ln(v_{b,n})\\}\right)$
(S42)

its expectation value can be obtained by integrating over the possible values
of the one birth size of interest

$\left\langle\frac{v_{0}}{v(\tau+n\ln(2)-\ln(v_{0}))}\right\rangle=v_{0}e^{-\tau}\int_{y}^{y+\ln(2)}f_{\ln(v_{b})}(u)\left(\frac{1}{2}\mathds{1}\\{\tau\leq
u\\}+\mathds{1}\\{\tau>u\\}\right)du$ (S43)

Now we perform a shift in the integral of equation (S38) of
$\tau\to\tau+n\ln(2)-\ln(v_{0})$, plug in equation (S43) and switch the order
of integration to obtain

$\begin{split}c_{k}=&\frac{1}{\ln(2)}e^{i\omega
k\ln(v_{0})}\int_{y}^{y+\ln(2)}\left\langle\frac{v_{0}}{v(\tau+n\ln(2)-\ln(v_{0}))}\right\rangle
e^{-i\omega k\tau}d\tau\\\ =&\frac{v_{0}}{\ln(2)}e^{i\omega
k\ln(v_{0})}\int_{y}^{y+\ln(2)}f_{\ln(v_{b})}(u)\int_{y}^{y+\ln(2)}e^{-(1+i\omega
k)\tau}\left(\frac{1}{2}\mathds{1}\\{\tau\leq
u\\}+\mathds{1}\\{\tau>u\\}\right)d\tau du\end{split}$ (S44)

The inner integral can be evaluated for a given $u$ as

$\begin{split}&\int_{y}^{y+\ln(2)}e^{-(1+i\omega
k)\tau}\left(\frac{1}{2}\mathds{1}\\{\tau\leq
u\\}+\mathds{1}\\{\tau>u\\}\right)d\tau=\frac{1}{2}\int_{y}^{u}e^{-(1+i\omega
k)\tau}d\tau+\int_{u}^{y+\ln(2)}e^{-(1+i\omega k)\tau}d\tau\\\
=&\frac{1}{1+i\omega k}\left(\frac{1}{2}e^{-(1+i\omega
k)y}-\frac{1}{2}e^{-(1+i\omega k)u}+e^{-(1+i\omega k)u}-e^{-(1+i\omega
k)(y+\ln(2))}\right)=\frac{1}{1+i\omega k}\frac{1}{2}e^{-(1+i\omega
k)u}.\end{split}$ (S45)

Plugging this back into equation (S44) results in

$c_{k}=\frac{v_{0}e^{i\omega k\ln(v_{0})}}{2\ln(2)(1+i\omega
k)}\int_{y}^{y+\ln(2)}f_{\ln(v_{b})}(u)e^{-(1+i\omega k)u}du.$ (S46)

We can calculate these more explicitly if we assume that $v_{b}$ is log-normal
with $\langle\ln(v_{b})\rangle=\ln(\bar{v})$ and variance
$\sigma_{\ln(v_{b})}^{2}$. For simplicity we think of
$\sigma_{\ln(v_{b})}^{2}$ as small enough to the point where it doesn’t matter
if we take the integral over all of space instead of the finite interval. We
obtain

$c_{k}=\frac{v_{0}}{\bar{v}}\frac{e^{i\omega
k\left(\ln(v_{0}/\bar{v})+\sigma_{\ln(v_{b})}^{2}\right)}}{2\ln(2)(1+i\omega
k)}e^{\frac{1-\omega^{2}k^{2}}{2}\sigma_{\ln(v_{b})}^{2}}.$ (S47)

This can be used to obtain more explicit forms of the constants from equation
(S37), namely

$C=\frac{v_{0}}{\bar{v}}\frac{e^{\frac{1}{2}\sigma_{\ln(v_{b})}^{2}}}{2\ln(2)}e^{-2\frac{\sigma_{\lambda}^{2}}{\theta^{2}}},\qquad
A_{k}=\frac{2e^{-\frac{1}{2}\omega^{2}k^{2}\sigma_{\ln(v_{b})}^{2}}}{\sqrt{1+\omega^{2}k^{2}}}e^{\omega^{2}k^{2}\frac{\sigma_{\lambda}^{2}}{\theta^{2}}}$
(S48)

and

$\phi_{k}=\omega
k\left(\ln(v_{0}/\bar{v})+\sigma_{\ln(v_{b})}^{2}\right)-arctan(\omega
k)-2\omega k\frac{\sigma_{\lambda}^{2}}{\theta^{2}}$ (S49)

##  VI Population properties

Let $x^{j}_{t}$ be some value related to cell $j$, like cell size $v^{j}(t)$
or growth rate $\lambda^{j}_{t}$. We define the total value $\tilde{X}(t)$ as
the sum of $x^{j}_{t}$ over all cells in the population at time $t$. This sum
can be obtained by summing over $x_{t}^{j}$ over all cells within the
population tree but with $x^{j}_{t}$ multiplied by zero if $j$ is not in the
population and by one otherwise. We can thus write

$\tilde{X}(t)=\sum_{n=0}^{\infty}\sum_{j=1}^{2^{n}}x_{t}^{j}\delta_{\Delta^{j}(\tau^{j}_{t}),n}.$
(S50)

In analogy to equation (S26), we find that the expected total sum of
$x^{j}_{t}$ over all cells is given by

$X(t)=\left\langle\tilde{X}(t)\right\rangle=\sum_{n=0}^{\infty}\sum_{j=1}^{2^{n}}\left\langle
x_{t}^{j}\delta_{\Delta^{j}(\tau^{j}_{t}),n}\right\rangle=\sum_{n=0}^{\infty}2^{n}\left\langle
x_{t}\delta_{\Delta(\tau_{t}),n}\right\rangle$ (S51)

where in the last equality we again used the fact that the distributions of
cell properties are lineage independent. For each term we can now put the
factor two within the expectation brackets and replace $n$ by
$\Delta(\tau_{t})$ due to the presence of the Kronecker delta. We obtain

$X(t)=\sum_{n=0}^{\infty}\left\langle
x_{t}2^{\Delta(\tau_{t})}\delta_{\Delta(\tau_{t}),n}\right\rangle=\left\langle
x_{t}2^{\Delta(\tau_{t})}\right\rangle=\left\langle
x_{t}\frac{v_{0}}{v(\tau_{t})}e^{\tau_{t}}\right\rangle$ (S52)

The average value of $x_{t}$ per cell in the population can now simply
obtained by dividing this total by the expected number of cells, hence

$\langle x_{t}\rangle_{p}=\frac{X(t)}{N(t)}=\frac{1}{N(t)}\left\langle
x_{t}\frac{v_{0}}{v(\tau_{t})}e^{\tau_{t}}\right\rangle$ (S53)

By setting $x_{t}$ to $\delta(\lambda_{t}-\lambda)$ or $\delta(v(\tau_{t})-v)$
one can calculate the time-dependent distributions of $\lambda_{t}$ and
$v(\tau_{t})$ respectively.

##  VII Population growth rate distribution

Growth rate distribution in a population at time $t$ is given by

$\rho(t,\lambda)=\left\langle\delta(\lambda_{t}-\lambda)\right\rangle_{p}=\frac{1}{N(t)}\left\langle\delta(\lambda_{t}-\lambda)\frac{v_{0}}{v(\tau_{t})}e^{\tau_{t}}\right\rangle$
(S54)

The only dependency on the division process in this expression comes from
$v_{0}/v(\tau_{t})$ again, hence we may first average over the division
process independently from $\tau_{t}$ and $\lambda_{t}$

$\rho(t,\lambda)=\frac{1}{N(t)}\left\langle\left\langle\left.\frac{v_{0}}{v(\tau_{t})}\right|\tau_{t}\right\rangle\delta(\lambda_{t}-\lambda)e^{\tau_{t}}\right\rangle_{\tau_{t},\lambda_{t}}$
(S55)

We may now plug in the Fourier expansion from equation (S29) to obtain

$\rho(t,\lambda)=\frac{1}{N(t)}\sum_{k=-\infty}^{\infty}c_{k}\left\langle\delta(\lambda_{t}-\lambda)e^{\left(1+i\omega
k\right)\tau_{t}}\right\rangle.$ (S56)

Let us now calculate the expectation values

$\left\langle\delta(\lambda_{t}-\lambda)e^{\left(1+i\omega
k\right)\tau_{t}}\right\rangle=\left\langle\left\langle\left.\delta(\lambda_{t}-\lambda)e^{\left(1+i\omega
k\right)\tau_{t}}\right|\lambda_{t}\right\rangle\right\rangle=\left\langle\delta(\lambda_{t}-\lambda)\right\rangle\left\langle\left.e^{\left(1+i\omega
k\right)\tau_{t}}\right|\lambda_{t}=\lambda\right\rangle.$ (S57)

Note that $\langle\delta(\lambda_{t}-\lambda)\rangle$ must simply be equal to
the steady-state distribution of $\lambda_{t}$, which is Gaussian with mean
$\bar{\lambda}$ and variance $\sigma_{\lambda}^{2}$. In order to solve the
rightmost factor in equation (S57), we notice that the distribution of
$\tau_{t}$ is unaffected by a growth process time-reversal of
$\lambda_{s}\to\lambda_{t-s}$. Fixing the final growth rate as
$\lambda_{t}=\lambda$ will therefore have the same effect as fixing the
initial growth rate as $\lambda_{0}=\lambda$. Using equations (S8) and (S9),
we can calculate the mean and variance of $\tau_{t}$ given that
$\lambda_{0}=\lambda$ is fixed, so

$\langle\tau_{t}\rangle_{0}=\int_{0}^{t}\langle\lambda_{s}\rangle_{0}=\int_{0}^{t}\bar{\lambda}+(\lambda_{0}-\bar{\lambda})e^{-\theta
s}ds=\bar{\lambda}t+(\lambda_{0}-\bar{\lambda})\frac{1}{\theta}(1-e^{-\theta
t})$ (S58)

and

$\sigma_{\tau_{t},0}^{2}=\int_{0}^{t}\int_{0}^{t}Cov(\lambda_{s},\lambda_{u})duds=\sigma_{\lambda}^{2}\int_{0}^{t}\int_{0}^{t}e^{-\theta|u-s|}duds-\sigma_{\lambda}^{2}\int_{0}^{t}\int_{0}^{t}e^{-\theta(t+s)}duds.$
(S59)

Note that the left integral on the right-hand side is equal to the old
stochastic time variance. We therefore find

$\sigma_{\tau_{t},0}^{2}=\sigma_{\tau_{t}}^{2}-\sigma_{\lambda}^{2}\int_{0}^{t}\int_{0}^{t}e^{-\theta(u+s)}duds.=\sigma_{\tau_{t}}^{2}-\sigma_{\lambda}^{2}\frac{1}{\theta^{2}}\left(1-e^{-\theta
t}\right)^{2}=2\frac{\sigma_{\lambda}^{2}}{\theta}t-\frac{\sigma_{\lambda}^{2}}{\theta^{2}}\left(3-4e^{-\theta
t}+e^{-2\theta t}\right)$ (S60)

We can now use this to calculate the aforementioned conditional expectation

$\left\langle e^{\left(1+i\omega
k\right)\tau_{t}}|\lambda_{t}=\lambda\right\rangle=e^{(1+i\omega
k)\langle\tau_{t}\rangle_{0}+\frac{1}{2}(1+i\omega
k)^{2}\sigma_{\tau_{t},0}^{2}}$ (S61)

with $\langle\tau_{t}\rangle_{0}$ and $\sigma_{\tau_{t},0}^{2}$ given by
equations (S58) and (S60) respectively. We find

$\left\langle\delta(\lambda_{t}-\lambda)e^{\left(1+i\omega
k\right)\tau_{t}}\right\rangle=\frac{1}{\sqrt{2\pi\sigma_{\lambda}^{2}}}e^{-\frac{1}{2\sigma_{\lambda}^{2}}(\lambda-\bar{\lambda})^{2}}e^{(1+i\omega
k)\langle\tau_{t}\rangle_{0}+\frac{1}{2}(1+i\omega
k)^{2}\sigma_{\tau_{t},0}^{2}}$ (S62)

One could plug this expression into equation (S56) to obtain the full
analytical dynamic population cell growth distribution. To get the asymptotic
growth rate distribution we send $t\to\infty$. In this limit, all $k\neq 0$
terms as well as all $e^{-\theta t}$ vanish. We get

$\begin{split}\rho(\lambda)=&\frac{1}{N(t)}\frac{1}{\sqrt{2\pi\sigma_{\lambda}^{2}}}e^{-\frac{1}{2\sigma_{\lambda}^{2}}(\lambda-\bar{\lambda})^{2}}e^{\langle\tau_{t}\rangle_{0}+\frac{1}{2}\sigma_{\tau_{t},0}^{2}}\\\
=&\frac{1}{\sqrt{2\pi\sigma_{\lambda}^{2}}}e^{-\frac{1}{2\sigma_{\lambda}^{2}}(\lambda-\bar{\lambda})^{2}}e^{\frac{1}{\theta}(\lambda-\bar{\lambda})-\frac{1}{2}\frac{\sigma_{\lambda}^{2}}{\theta^{2}}}\\\
=&\frac{1}{\sqrt{2\pi\sigma_{\lambda}^{2}}}e^{-\frac{1}{2\sigma_{\lambda}^{2}}(\lambda-\Lambda_{\infty})^{2}}\end{split}$
(S63)

with $\Lambda_{\infty}=\bar{\lambda}+\sigma_{\lambda}^{2}/\theta$. Hence in a
steady-state population, instantaneous growth rate is normally distributed
with mean $\Lambda_{\infty}$ and variance $\sigma_{\lambda}^{2}$

##  VIII Population cell size distribution

Cell size distribution in a population at time $t$ is given by

$G(t,v)=\left\langle\delta(v(\tau_{t})-v)\right\rangle_{p}=\frac{1}{N(t)}\left\langle\delta(v(\tau_{t})-v)\frac{v_{0}}{v(\tau_{t})}e^{\tau_{t}}\right\rangle=\frac{1}{N(t)}\frac{v_{0}}{v}\left\langle\delta(v(\tau_{t})-v)e^{\tau_{t}}\right\rangle$
(S64)

To solve this, we first average over values of $v(\tau_{t})$ for fixed
$\tau_{t}$ and then subsequently average over $\tau_{t}$. Since $v(\tau)$ is a
periodic function of $\tau$ we have that
$\langle\delta(v(\tau_{t}))-v)\rangle$ is periodic too with a period of
$\ln(2)$. A Fourier expansion of this function gives

$\langle\delta(v(\tau))-v)\rangle=\sum_{k=-\infty}^{\infty}b_{k}(v)e^{i\omega
k\tau},$ (S65)

hence in analogy to the derivation of equation (S32), we find a time-dependent
cell size distribution of

$\begin{split}G(t,v)=&\frac{1}{N(t)}\frac{v_{0}}{v}\sum_{k=-\infty}^{\infty}b_{k}(v)\left\langle
e^{(1+i\omega k)\tau_{t}}\right\rangle\\\
=&\frac{1}{N(t)}\frac{v_{0}}{v}e^{\langle\tau_{t}\rangle+\frac{1}{2}\sigma_{\tau_{t}}^{2}}\sum_{k=-\infty}^{\infty}b_{k}(v)e^{i\omega
k\left(\langle\tau_{t}\rangle+\sigma_{\tau_{t}}^{2}\right)}e^{-\frac{1}{2}\omega^{2}k^{2}\sigma_{\tau_{t}}^{2}}.\end{split}$
(S66)

Let us calculate the size-dependent coefficients

$b_{k}(v)=\frac{1}{\ln(2)}\int_{x}^{x+\ln(2)}\langle\delta(v(\tau))-v)\rangle
e^{-i\omega k\tau}d\tau.$ (S67)

We apply the same definitions as during the derivation of the expected
population coefficients. From equation (S42) it follows that for
$\tau\in(x,x+\ln(2))$ we have

$\begin{split}&\delta(v(\tau+n\ln(2)-\ln(v_{0}))-v)\\\
=&\delta\left(2e^{\tau}-v\right)\mathds{1}\\{\tau\leq\ln(v_{b,n})\\}+\delta\left(e^{\tau}-v\right)\mathds{1}\\{\tau>\ln(v_{b,n})\\}\\\
=&\frac{1}{v}\delta\left(\tau-\ln\left(v/2\right)\right)\mathds{1}\\{v/2\leq
v_{b,n}\\}+\frac{1}{v}\delta\left(\tau-\ln(v)\right)\mathds{1}\\{v>v_{b,n}\\}.\end{split}$
(S68)

We can now take the expectation over possible values of the birth size to find

$\left\langle\delta(v(\tau+n\ln(2)-\ln(v_{0}))-v)\right\rangle=\frac{1}{v}\left[\delta(\tau-\ln(v/2))(1-F_{v_{b}}(v/2))+\delta(\tau-\ln(v))F_{v_{b}}(v)\right],$
(S69)

where $F_{v_{b}}(v)$ is the cumulative distribution function of the birth size
$v_{b}$. Since the division size is just double the birth size in
distribution, we have that $F_{v_{b}}(v/2)$ is the cumulative distribution
function of division size. We can now solve the size-dependent coefficients by
shifting the integration region in equation (S67) and plugging in equation
(S69) to obtain

$\begin{split}b_{k}(v)=&\frac{1}{\ln(2)}e^{i\omega
k\ln(v_{0})}\int_{y}^{y+\ln(2)}\left\langle\delta(v(\tau+n\ln(2)-\ln(v_{0}))-v)\right\rangle
e^{-i\omega k\tau}d\tau\\\ =&\frac{1}{\ln(2)v}e^{-i\omega
k\ln(v/v_{0})}\left[(1-F_{v_{b}}(v/2))\mathds{1}\\{\ln(v)\in(y+\ln(2),y+2\ln(2))\\}+F_{v_{b}}(v)\mathds{1}\\{\ln(v)\in(y,y+\ln(2))\\}\right]\\\
=&e^{-i\omega k\ln(v/v_{0})}b_{0}(v).\end{split}$ (S70)

Thus, all the coefficients are just complex rotations of each other, with the
zeroth coefficient given by

$b_{0}(v)=\frac{1}{\ln(2)v}\left[(1-F_{v_{b}}(v/2))\mathds{1}\\{\ln(v)\in(y+\ln(2),y+2\ln(2)\\}+F_{v_{b}}(v)\mathds{1}\\{\ln(v)\in(y,y+\ln(2))\\}\right].$
(S71)

Recall that $y$ was defined to be a real number such that
$\ln(v_{b})\in(y+y+\ln(2))$. In other words, all births happen on the interval
$(y+y+\ln(2))$ and all divisions happen on the separate interval
$(y+\ln(2)+y+2\ln(2))$. Therefore, if the size considered lies in the birth
region given by $\ln(v)\in(y+y+\ln(2))$, we have that $F_{v_{b}}(v/2)=0$.
Conversely if the size lies in the division region given by
$\ln(v)\in(y+\ln(2),y+2\ln(2))$ we have $F_{v_{b}}(v)=1$. In either case,
equation (S71) matches with

$b_{0}(v)=\frac{1}{\ln(2)v}\left[F_{v_{b}}(v)-F_{v_{b}}(v/2)\right]$ (S72)

Combining this result with equations (S66) and (S33), we find the exact result

$G(t,v)=G(v)\frac{1+\sum_{k=1}^{\infty}2Cos\left(\omega
k\left(\langle\tau_{t}\rangle+\sigma_{\tau_{t}}^{2}\right)-\omega
k\ln(v/v_{0})\right)e^{-\frac{1}{2}\omega^{2}k^{2}\sigma_{\tau_{t}}^{2}}}{1+\sum_{k=1}^{\infty}2\frac{|c_{k}|}{c_{0}}Cos\left(\omega
k\left(\langle\tau_{t}\rangle+\sigma_{\tau_{t}}^{2}\right)+Arg(c_{k})\right)e^{-\frac{1}{2}\omega^{2}k^{2}\sigma_{\tau_{t}}^{2}}},$
(S73)

where

$G(v)=\frac{b_{0}(v)}{c_{0}}=\frac{2}{v^{2}\langle
1/v_{b}\rangle}\left[F_{v_{b}}(v)-F_{v_{b}}(v/2)\right].$ (S74)

It’s clear that equation (S74) is the time-limit of equation (S73). If we
again use the large time approximation of $\sigma_{\tau_{t}}^{2}$ from
equation (S34), we can rewrite equation (S73) as

$G(t,v)=G(v)\frac{1+\sum_{k=1}^{\infty}B_{k}Cos\left(k\Omega t-\omega
k\ln(v)+\zeta_{k}\right)e^{-k^{2}qt}}{1+\sum_{k=1}^{\infty}A_{k}Cos\left(k\Omega
t+\phi_{k}\right)e^{-k^{2}qt}}$ (S75)

where $\Omega$ and $q$ were given by equation (S36), $A_{k}$ and $\phi_{k}$ by
equation (S48) and (S49) respectively and where we introduce new constants

$B_{k}=2e^{-k^{2}q\frac{\sigma_{\lambda}^{2}}{\theta}},\qquad\zeta_{k}=\omega
k\ln(v_{0})-2\omega k\frac{\sigma_{\lambda}^{2}}{\theta}$ (S76)

By taking a finite number of terms from the denominator and numerator in
equation S73 or S75, one obtains great approximations for the population cell
size distribution that becomes progressively better for larger $t$. These
approximations are properly normalized.

##  IX Asymptotic results for a general growth rate process

In this section we will explain why population growth rate $\Lambda_{\infty}$
is independent cell division sizes and the population size distribution is
independent of growth rate, even when $\lambda_{t}$ is no longer necessarily
an Ornstein-Uhlenbeck process.

If we define $V(t)$ as the expected total volume of the population at time
$t$, then we can use equation (S52) to write

$V(t)=v_{0}\left\langle e^{\tau_{t}}\right\rangle.$ (S77)

The expected total volume is thus completely independent of the birth and
division size variables described in section I. This should however not come
at a surprise as we will illustrate in a little thought experiment. Suppose we
had one cell starting at volume $v_{0}$ that never divides and grows in
accordance with equation (S15) for a time $t$, where the growth rate
$\lambda_{t}$ is now some arbitrary stochastic process. This cell’s total
expected volume would then clearly be given by equation (S77). If the cells
were now split in two, say $v_{0,(1)}$ and $v_{0,(2)}$ such that
$v_{0,(1)}+v_{0,(2)}=v_{0}$ at time $t=0$, but their growth rate trajectories
$\lambda_{t}$ were still identical, then the total volume would naturally be
unaffected by this split. Note that we may thus write

$V(t)=\left\langle v_{0}e^{\tau_{t}}\right\rangle=\left\langle
v_{0,(1)}e^{\tau_{t}}\right\rangle+\left\langle
v_{0,(2)}e^{\tau_{t}}\right\rangle.$ (S78)

Due to the linearity of expectation values, the above result does not care
about how the growth rate trajectories of the two cells are correlated; they
only need to be identical in distribution. We may therefore split cells as we
please without affecting the expected total volume of the population.

Next, we require that the average volume of cells within a population reaches
some steady state, which should be true for any realistic cell division
process. We thus say that $\langle v(t)\rangle_{p}=V(t)/N(t)$ goes to some
constant for large $t$. We now see that for large $t$,

$\Lambda_{\infty}=\frac{d}{dt}\ln N(t)=\frac{d}{dt}\left[\ln V(t)-\ln\langle
v(t)\rangle_{p}\right]=\frac{d}{dt}\ln\left\langle e^{\tau_{t}}\right\rangle,$
(S79)

from which it is evident that the population growth rate only depends on the
single-cell growth rate process.

As for the asymptotic population cell size distribution, let’s assume that
some time-independent asymptotic distribution exists, and that the cell birth
sizes have identical distributions. Under these assumptions we can re-derive
equation (S66),

$G(t,v)=\frac{1}{N(t)}\frac{v_{0}}{v}\sum_{k=-\infty}^{\infty}b_{k}(v)\left\langle
e^{(1+i\omega k)\tau_{t}}\right\rangle.$ (S80)

We have previously argued that for large $t$, it holds that
$N(t)\propto\left\langle e^{\tau_{t}}\right\rangle$, hence

$G(t,v)\propto\frac{1}{v}\sum_{k=-\infty}^{\infty}b_{k}(v)\frac{\left\langle
e^{(1+i\omega k)\tau_{t}}\right\rangle}{\left\langle
e^{\tau_{t}}\right\rangle}.$ (S81)

The only way for this expression to converge to a time-independent
distribution is if all the transient terms disappear, in other words if
$\left\langle e^{(1+i\omega k)\tau_{t}}\right\rangle/\left\langle
e^{\tau_{t}}\right\rangle\to 0$ for all $k\neq 0$. What we are left with is

$G(t,v)\propto\frac{1}{v}b_{k}(v)\propto\frac{1}{v^{2}}\left[F_{v_{b}}(v)-F_{v_{b}}(v/2)\right],$
(S82)

which corresponds to the desired expression after finding the right
normalization constants.

##  X Comparison to the discretely varying growth rate model

In other literature Stewart  _et al._ (2005); Robert  _et al._ (2014); Taheri-
Araghi  _et al._ (2015); Lin and Amir (2017, 2020); Campos  _et al._ (2014),
cell growth rate is commonly defined the variable $\kappa$ satisfying

$v_{d}=v_{b}e^{\kappa(t_{d}-t_{b})},$ (S83)

where $v_{b}$ and $v_{d}$ are the sizes of birth and division of a cell and
$t_{b}$ and $t_{d}$ are the times of birth and division respectively.
Alternatively we can rewrite this as

$\kappa=\frac{1}{t_{d}-t_{b}}\ln\left(\frac{v_{d}}{v_{b}}\right)=\frac{1}{t_{d}-t_{b}}\left(\tau_{t_{d}}-\tau_{t_{b}}\right)=\frac{1}{t_{d}-t_{b}}\int_{t_{b}}^{t_{d}}\lambda_{s}ds,$
(S84)

hence $\kappa$ as defined in equation (S83) is also equal to the growth rate
process time-averaged over the lifetime of a cell. Since the lifetime of a
cell is tied to the growth rate process itself, the distribution of $\kappa$
may differ a lot from that of the instantaneous growth rate $\lambda_{t}$.
Most notably, their stochastic means are different. This may lead to some
seemingly contradictory claims such as in Ref. Lin and Amir (2020), where it
is said that in the absence of mother-daughter growth rate correlations, the
population number growth rate decreases with growth rate variability. In
contrast, our model predicts that population number growth rate always
increases with growth rate variability, regardless of any growth rate
correlations.

To resolve this, we will show how our models are related in a special limit of
vanishing auto-correlations where we let $\theta\gg\bar{\lambda}$ while fixing
$D:=\sigma_{\lambda}^{2}/\theta$. The stochastic time is now fully equivalent
to a Brownian motion with drift, characterized by the parameters

$\langle\tau_{t}\rangle=\bar{\lambda}t,\qquad\sigma_{\tau_{t}}^{2}=2Dt.$ (S85)

We define the first hitting time of $\tau_{t}$ reaching a fixed value $\tau$
by

$t_{\tau}=\inf\\{t:\tau_{t}\geq\tau\\}$ (S86)

The distribution density function of such a hitting is known as an inverse
Gaussian distribution. It is given by

$g_{t_{\tau}}(t)=\frac{\tau}{\sqrt{4\pi
Dt^{3}}}e^{-\frac{(\bar{\lambda}t-\tau)^{2}}{4Dt}},$ (S87)

for which we can find the following inverse moments

$\left\langle\frac{\tau}{t_{\tau}}\right\rangle=\bar{\lambda}+\frac{2D}{\tau},\qquad\left\langle\frac{\tau^{2}}{t_{\tau}^{2}}\right\rangle=\bar{\lambda}^{2}+\frac{6D\bar{\lambda}}{\tau}+\frac{12D^{2}}{\tau^{2}}$
(S88)

We define $u=\ln(v_{d}/v_{b})$ as the stochastic time needed for a cell to
reach its division size. Since divisions are triggered by cells attaining
certain sizes, the division times can be defined as hitting times of the
stochastic time reaching the right size, hence

$t_{d}-t_{b}=\inf\\{t:v(t+\tau_{b})\geq
v_{d}\\}=\inf\\{t:\tau_{t_{b}+t}-\tau_{t_{b}}\geq u\\}\sim t_{u}.$ (S89)

Due to size homeostasis we must have that $\langle u\rangle=\ln(2)$. We can
now calculate mean of the cell-average growth rate from equation (S84), as

$\bar{\kappa}=\left\langle\frac{u}{t_{u}}\right\rangle=\bar{\lambda}+\left\langle\frac{2D}{u}\right\rangle\approx\bar{\lambda}+\frac{2D}{\ln(2)}\left(1+\frac{\sigma_{u}^{2}}{\ln(2)^{2}}\right)$
(S90)

and its variance as

$\sigma_{\kappa}^{2}=\left\langle\frac{u^{2}}{t_{u}^{2}}\right\rangle-\left\langle\frac{u}{t_{u}}\right\rangle^{2}=\left\langle\frac{2D\bar{\lambda}}{u}\right\rangle+\left\langle\frac{12D^{2}}{u^{2}}\right\rangle-\left\langle\frac{2D}{u}\right\rangle^{2}\approx\frac{2D\bar{\lambda}}{\ln(2)}\left(1+\frac{\sigma_{u}^{2}}{\ln(2)^{2}}\right)+\frac{8D^{2}}{\ln(2)^{2}}\left(1-\frac{\sigma_{u}^{2}}{\ln(2)^{2}}\right)$
(S91)

A close look at equations (S90) and (S91) reveals a lot about the differences
between the conclusions from our model and the discretely varying growth rate
model from Ref. Lin and Amir (2020). Most importantly in the limit of small
growth rate variability $D\ll\bar{\lambda}$, we can express the population
growth rate in terms of the per-cell growth rate parameters

$\Lambda_{\infty}=\bar{\lambda}+D\approx\bar{\kappa}-\left(1-\frac{\ln(2)}{2}\left(1-\frac{\sigma_{u}^{2}}{\ln(2)^{2}}\right)\right)\frac{\sigma_{\kappa}^{2}}{\bar{\kappa}}.$
(S92)

This formula is the exact same as the known result for the discretely varying
growth rate model in the limit of vanishing mother-daughter growth rate
correlations and no division size variability. Equation (S92) directly
illustrates how their and our model lead to seemingly different conclusions
about the benefits of growth rate variability; fixing the average
instantaneous growth rate $\bar{\lambda}$ while increasing its variability
benefits the population growth rate $\Lambda_{\infty}$ whereas fixing the
average per-cell growth rate $\bar{\kappa}$ while increasing its variability
decreases the population growth rate $\Lambda_{\infty}$.

The size increase variability $\sigma_{u}^{2}$ can be linked to the parameters
from the model discussed in section I by noting that

$u=\ln(2)-\alpha\ln(v_{b}/\bar{v})+\eta_{n},$ (S93)

from which we derive

$\sigma_{u}^{2}=\alpha^{2}\sigma_{\ln(v_{b})}^{2}+\sigma_{\eta}^{2}=2\alpha\sigma_{\ln(v_{b})}^{2}=\frac{2}{2-\alpha}\sigma_{\eta}^{2},$
(S94)

thus the difference between $\bar{\lambda}$ and $\bar{\kappa}$ depends on the
size additive noise $\sigma_{\eta}^{2}$ and the cell size control parameter
$\alpha$. the population growth rate $\Lambda_{\infty}$ is therefore also
affected by $\sigma_{\eta}^{2}$ and $\alpha$ when fixing $\bar{\kappa}$. This
explains why the discretely varying growth rate model does not exhibit a full
decoupling of division size variability and cell size control from population
growth rate Lin and Amir (2020). This difference also acts as an example of
why the equivalence between the models breaks down when mother-daughter cell
correlations are introduced.

In Ref.Jafarpour (2019), the time-dependent behaviour of the discretely
varying growth rate model is studied through simulations. Mother-daughter
growth rate correlations are ignored, so their model is equivalent to our
model in the limit of $D=\sigma_{\lambda}^{2}/\theta\ll 1$. We may relate our
model parameters via equations (S90) and (S91). Ignoring cell division
variability, the oscillation amplitude decay rate from equation (S36) written
in terms of these alternative parameters is

$q\approx\frac{2\pi^{2}}{\ln(2)}\frac{\sigma_{\kappa}^{2}}{\bar{\kappa}}\approx
28.5\frac{\sigma_{\kappa}^{2}}{\bar{\kappa}}.$ (S95)

Which is in perfect agreement with their empirically inferred oscillation
amplitude decay rate of $q\approx 29\sigma_{\kappa}^{2}/\bar{\kappa}$