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https://doi.org/10.1038/s42005-021-00739-5

OPEN

Scale invariance of cell size ﬂuctuations in
starving bacteria

1234567890():,;

Takuro Shimaya

1 ✉, Reiko Okura2, Yuichi Wakamoto

2 & Kazumasa A. Takeuchi

1,3 ✉In stable environments, cell size ﬂuctuations are thought to be governed by simple physicalprinciples, as suggested by recent ﬁndings of scaling properties. Here, by developing a
microﬂuidic device and using E. coli, we investigate the response of cell size ﬂuctuations
against starvation. By abruptly switching to non-nutritious medium, we ﬁnd that the cell size
distribution changes but satisﬁes scale invariance: the rescaled distribution is kept unchanged
and determined by the growth condition before starvation. These ﬁndings are underpinned bya model based on cell growth and cell cycle. Further, we numerically determine the range ofvalidity of the scale invariance over various characteristic times of the starvation process, andﬁnd the violation of the scale invariance for slow starvation. Our results, combined withtheoretical arguments, suggest the relevance of the multifork replication, which helps
retaining information of cell cycle states and may thus result in the scale invariance.

1 Department of Physics, Graduate School of Science, University of Tokyo, Tokyo 113-0033, Japan. 2 Department of Basic Science, Graduate School of Arts
and Sciences, University of Tokyo, Tokyo 153-8902, Japan. 3 Department of Physics, School of Science, Tokyo Institute of Technology, Tokyo 152-8551, Japan.
✉email: t.shimaya@noneq.phys.s.u-tokyo.ac.jp; kat@kaztake.org

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R

ecent studies on microbes in the steady growth phasesuggested that the cellular body size ﬂuctuations may be
governed by simple physical principles. For instance, Giometto et al.1 proposed that size ﬂuctuations of various eukaryotic
cells are governed by a common distribution function if the cell
sizes of a given species are normalized by their mean value (seealso ref. 2). In other words, the distribution of cell volumes v, p(v),can be described as follows:pðvÞ ¼ v1 Fðv=VÞ;ð1Þ

with a function F( ⋅ ) and V = 〈v〉 being the mean cell volume.
This property of distribution is often called scale invariance.
Interestingly, this ﬁnding can account for power laws of community size distributions, i.e., the size distribution of all individuals regardless of species, which were observed in various
natural ecosystems3,4. Scale invariance akin to Eq. (1) was also
found for bacteria5,6 for each cell age, and the function F( ⋅ ) was
shown to be robust against changes in growth conditions, such as
the temperature.
Those results, as well as theoretical models proposed in thiscontext1,7,8, have been obtained under steady environments, forwhich our understanding of single-cell growth statistics has also
been signiﬁcantly deepened recently9–11. By contrast, it is unclear
whether such a simple concept as scale invariance is valid undertime-dependent conditions, where different regulations of cell
cycle kinetics may come into play in response to environmental
variations. In particular, when bacterial cells enter the stationary
phase from the exponential growth phase, they undergo reductive
division, during which both the typical cell size and the amount of
DNA per cell decrease12–15. Although this behavior itself iscommonly observed in batch cultivation, little is known about
single-cell statistical properties during the transient. The bacterial
reductive division is therefore an important stepping stone forstudying cell size statistics under time-dependent environments
and understanding the robustness of the scale invariance against
environmental changes.
To investigate size distributions of large bacterial populations
under time-dependent growth conditions, we should care about
experimental methods. A microﬂuidic device called the mothermachine16 consists of many small separate chambers of cells
supplied with medium and thus allows for tracking of bacteria
trapped therein. Although this type of device has also been used
for time-dependent problems as well17–21, for our purpose
involving size ﬂuctuations of a large population of cells, it is not
obvious if they are equivalent to those of a collection of many
independent small populations. More precisely, since the size of acell depends strongly on its age, it is reasonable to use large
enough chambers so that the chamber size may not affect the age
distribution. This led us to develop another system that can deala

Bubble trap
Medium outlet

Medium inlet
Medium outlet

with large enough populations in each chamber and uniformly
control non-steady environments without delicate optimization.
In this study, we develop a microﬂuidic device, which we namethe “extensive microperfusion system” (EMPS). This device canculture cells uniformly by supplying fresh medium through a
porous membrane, similarly to previously reported systems22–24,
but here we realize wide quasi-two-dimensional traps of dense
bacteria in such a system. We conﬁrm that bacteria can freely
swim and grow inside and evaluate the uniformity and theswitching efﬁciency of the culture condition. Then we use thissystem for quantitative observations of bacterial reductive division processes, triggered by abrupt switching to non-nutritious
medium. We observe Escherichia coli cells and ﬁnd that the distribution of cell volumes, collected irrespective of cell ages,
maintain the scale invariance as in Eq. (1) at each time, with themean cell size that gradually decreases. On the other hand, therescaled distribution function F is found to depend on the growth
condition before starvation, slightly but signiﬁcantly. To obtaintheoretical insights on these experimental ﬁndings, we devise acell cycle model describing reductive division processes, byextending the Cooper–Helmstetter (CH) model and its
variants25–27 for steady growth environments. We numericallyﬁnd that this model indeed shows the scale invariance understarvation conditions, conﬁrming the robustness of this property.
We also provide theoretical descriptions on the time evolution of
the cell size distribution and propose a condition for the scale
invariance. Finally, we numerically show the range of validity ofthe scale invariance over various characteristic times of the
starvation process, revealing that the number of multifork replications may be important for the scale invariance.
Results
Development of the EMPS. To achieve uniformly controlledenvironments with dense bacterial suspensions, we adopt a perfusion system, which supplies fresh medium through a porous
membrane attached over the observation area. Among several
existing devices of this kind22–24, here we choose the one developed in ref. 22 as a prototype. In this device, bacteria are conﬁned
in microwells made on a coverslip, covered by a cellulose porous
membrane attached to the coverslip via biotin–streptavidin
bonding. Note that cellulose cannot be metabolized by E. coli
strains common for laboratory use28, which we conﬁrmed
explicitly with MG1655 (Supplementary Fig. 1g). The pore size ofthe membrane is chosen so that it can conﬁne bacteria and alsothat it can exchange nutrients and waste substances across themembrane. To continuously perfuse the system with fresh medium, a polydimethylsiloxane (PDMS) pad with a bubble trap is
attached above the membrane by a two-sided frame seal (Fig. 1aand Supplementary Fig. 1a). This set-up can maintain a spatiallybMedium inlet

PDMS pad

Frame
seal

PET membrane
Cellulose membrane

Coverslip

Objective lens

Bubble trap

PDMS pad
Frame seal

Biotin-decorated coverslip

Medium outlet

Observation area
Stage

Fig. 1 Sketch of the extensive microperfusion system (EMPS). a Entire view of the device. Microwells are created on a glass coverslip. We attach apolydimethylsiloxane (PDMS) pad on the coverslip with a square frame seal to ﬁll the system with liquid medium. b Cross-sectional view inside the PDMS
pad. A polyethylene terephthalate (PET)–cellulose bilayer porous membrane is attached via the biotin–streptavidin bonding. Note that there are two outlets
as in (a).
2

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homogeneous environment for cell populations in each microwell, in particular if the microwells are sufﬁciently shallow so that
all cells remain near the membrane. However, because the soft
cellulose membrane may droop and adhere to the bottom forwide and shallow microwells, the horizontal size of such quasitwo-dimensional microwells has been limited up to a few tens of
micrometers, preventing from characterization of the instantaneous cell size distribution.
By the EMPS, we overcome this problem and realize quasitwo-dimensional wells sufﬁciently large for statistical characterization of cell populations. This is made possible by introducing abilayer membrane, where the cellulose membrane is sustained bya polyethylene terephthalate (PET) porous membrane via
biotin–streptavidin bonding (Fig. 1b, Supplementary Fig. 1b,
and “Methods”). Because the PET membrane is more rigid thanthe cellulose membrane, we can realize extended area without
bending of the membrane.
Here we conducted several experiments to evaluate how wellthe experimental condition inside the EMPS can be controlled(see also Supplementary Note 2). We ﬁrst examined the ﬂatnessof the observation area using motile bacteria. If a cellulose
membrane alone is used, it is bent and adheres to the bottom of
the well (Supplementary Fig. 1c, d and Supplementary Movie 1).
However, if it is replaced by our PET–cellulose bilayer membrane,
it keeps ﬂat enough so that bacteria can freely swim in the shallow
well (Supplementary Fig. 1e, f and Supplementary Movie 2). We
also show that, using non-motile bacteria, growth rate of thebacteria is spatially uniform (Supplementary Fig. 2 andSupplementary Movies 3 and 4). The doubling time of the cell
population was 59 ± 10 min, which is comparable to that in theprevious system without the PET membrane22,29,30. Furthermore,similarly to other microﬂuidic devices, we can also switch theculture condition by changing the medium to supply. We evaluate
how efﬁciently the medium in the well is exchanged, by using
ﬂuorescent dye and non-motile bacteria. We found that themedium exchange was almost completed within 5 min (Supplementary Fig. 3), much shorter than the length of the bacterial cell
cycle. On the other hand, since medium exchange in EMPS relies
on diffusion of molecules through the membrane, other devices
that can replace medium more directly, such as mother machines
in which medium is poured to a main channel connected toobserved growth channels17–21, may be advantageous in thisrespect. Since this difference may affect, e.g., the amount ofmolecules that can remain on the surface of the observation area,
cellular states may also change differently after an environmental
switch. In this respect, advantages of EMPS in environmental
changes may be in the fact that (i) we can control theenvironment without hydrodynamic perturbations and, simultaneously, (ii) we can observe cells in large space under a uniform
and time-dependent environment without mechanical trapping.
The absence of hydrodynamic perturbations can be seen from
Brownian motion of non-motile cells in Supplementary Movies 5
and 6, which is hardly affected by relatively strong medium ﬂow
above the membrane used to switch the medium. Therefore,
EMPS is indeed able to change the growth condition for cells inlarge space uniformly, without noticeable ﬂuid ﬂow perturbations, which is a unique strength of our device.
Characterization of bacterial reductive division by EMPS. Now
we observe the reductive division of E. coli MG1655 in the EMPS,
triggering starvation by switching from nutritious medium to
non-nutritious buffer. In the beginning, a few cells are trapped ina quasi-two-dimensional well (diameter 55 μm and depth 0.8 μm)
and grown in nutritious medium, until a microcolony composedof approximately 100 cells appear. We then quickly switch themedium to a non-nutritious buffer, which is continuously supplied until the end of the observation (see “Methods” for moredetails). By doing so, we not only trigger cell starvation but also
intend to remove various substances secreted by cells, such as
autoinducers for quorum sensing and waste products, to reduce
their effects on cell growth31–34. Note that 5 min required to
exchange the medium in the trap is sufﬁciently shorter than the
typical length of the cell cycle of E. coli, which is several tens ofminutes. This implies that starvation is triggered abruptly forcells. Throughout this experiment, the well is entirely recorded byphase contrast microscopy. We then measure the length and thewidth of all cells in the well, to obtain the volume v of each cell by
assuming the spherocylindrical shape, at different times before
and after the medium switch. Here we mainly show the results forthe case where the medium is switched from Luria-Bertani (LB)
broth to phosphate-buffered saline (PBS) (denoted by LB → PBS)
in Fig. 2, while the results for M9 medium with glucose (Glc) and
12 amino acids (a.a.) → PBS, M9 medium with glucose (Glc) →PBS, M9 medium with glycerol (Glyc) → PBS, and M9 medium
with glucose (Glc) → M9 medium with α-methyl-D-glucoside(αMG), a glucose analog that cannot be metabolized35, are alsopresented in Supplementary Fig. 4. We observe that, afterswitching to the non-nutritious buffer, the growth of the total
volume decelerates, and the mean cell volume rapidly decays
because of excessive cell divisions (Supplementary Movies 7–11,
Fig. 2a, b, and Supplementary Fig. 4), until cells eventually stop
growing and dividing. Note that, unlike other cases (Supplementary Fig. 4), cell growth did not stop completely in the case of
LB → PBS, but we consider that this will not affect our analysis
because the ultimately remaining growth rate  104 min1(Supplementary Fig. 9a) was sufﬁciently low compared to thetime scale of the volume reduction  102 min (Fig. 2c) and allother time scales relevant in this study. Concerning the volume
reduction, it is mostly due to the decrease of the cell length, while
we notice that the mean cell width may also change slightly(Supplementary Fig. 5). We consider that this is not due toosmotic shock36, because then the cell width would increase when
the osmotic pressure is decreased, which is contradictory to ourobservation for LB → PBS (Supplementary Table 1 and Supplementary Fig. 5). Such a change in cell widths was also reported fora transition between two different growth conditions37. In any
case, Fig. 2d shows how the distribution of the cell volumes v,
p(v, t), changes over time; as the mean volume decreases, thehistograms shift leftward and become sharper. However, when we
take the ratio v/V(t), with V(t) = 〈v〉 being the mean cell volume
at each time t, and plot vp(v, t) instead, we ﬁnd that all thosehistograms overlap onto a single curve (Fig. 2e). In other words,
we ﬁnd that the time-dependent cell size distribution during the
reductive division maintains the following scale-invariant form all
the time:
pðv; tÞ ¼ v1 Fðv=VðtÞÞ:ð2Þ

This is analogous to Eq. (1) previously reported for the steady
growth condition, but here importantly the mean volume V(t)changes over time signiﬁcantly (Fig. 2c). The scale invariance also
holds for the length distribution (Supplementary Fig. 6); this is
expected because the length changes are dominant in the studied
volume changes.
To further test the scale invariance of the distribution, we
evaluate the coefﬁcient of variation (CV) and the skewness (Sk)deﬁned bypﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
hδv2 ihδv3 ið3Þ
;; Sk 
CV hvi
hδv2 i3=2
with δv ≡ v − 〈v〉. Both quantities measure the shape of the

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t = 0 min

LB growth mediab

PBS (no nutrient)timed

LB → PBSe

LB → PBSfc
LB → PBS

Fig. 2 Results from the observations of reductive division. a Snapshots taken during the reductive division process of E. coli MG1655 in the EMPS. The
medium is switched from LB broth to phosphate-buffered saline (PBS) at t = 0. See also Supplementary Movie 7. b, c Experimental data (blue symbols) for
the total cell volume Vtot(t) (b), the growth rate λ(t) (b inset, see also Supplementary Fig. 9a showing the same data in logarithmic scale), the mean cellvolume V(t) (c), and the number of the cells n(t) (c inset) in the case of LB → PBS, compared with the simulation results (red curves). The error bars
indicate segmentation uncertainty in the image analysis (see “Methods”). t = 0 is the time at which PBS enters the device (black dashed line). The data
were collected from 15 wells recorded in a single experiment. d Time evolution of the cell size distributions during starvation in the case of LB → PBS at
t ¼ 0; 5; 30; 60; 90; 120; 180; 240; 300; 360; 420; 480 min from right to left. The sample size is n(t) for each distribution (see (c) inset). e Rescaling of
the data in (d). The overlapped curves indicate the function F(v/V(t)) in Eq. (2). The dashed line represents the ﬁtted log-normal distribution(σ = 0.34(2)). f The coefﬁcient of variation (CV) and the skewness (Sk) (Eq. (3)) against V(t) = 〈v〉. The error bars were estimated by the bootstrap
method with 1000 realizations.

distribution function of v/V(t) and are not affected by variation ofV(t). The results in Fig. 2f indeed conﬁrm that both CV and Sk
remain essentially constant, so that the function F( ⋅ ) remains
unchanged and the scale invariance holds during the reductive
division. Remarkably, we reach the same conclusion for all
combinations of the growth and starving conditions that we test,
as shown in Fig. 3a–c and Supplementary Fig. 7 (see Fig. 2f and
Supplementary Fig. 4 for the results of CV and Sk). Our results
therefore indicate that the scale invariance as in Eq. (2), which has
been observed for steady conditions1,2, also holds in non-steady
reductive division processes of E. coli rather robustly.
In addition to the robustness of the scaling relation (2), thefunctional form of the scale-invariant distribution, i.e., that ofF(x), is of interest. We detect weak dependence of F(x) on the
choice of the medium in the growth phase (Fig. 3c). Morespeciﬁcally, we ﬁnd the trend that the ﬂuctuations of the rescaled
cell volumes are larger for richer growth conditions (Supplementary Fig. 7h, i), consistently with a past observation in ref. 38. The
lower the nutrient level of the growth medium is, the sharper thefunction F(x) becomes, and therefore, the smaller the variance is.
We have also conﬁrmed that the variation in F(x) among
different sets of media is more signiﬁcant than that among
biological replicates (Supplementary Fig. 7h, i). This is somewhat
unexpected in view of the past studies reporting the robustness ofcell size ﬂuctuations against varying temperatures and otherenvironmental factors1,5,7.
Moreover, we ﬁnd that our observations for E. coli are
signiﬁcantly different from those for unicellular eukaryotes
reported by Giometto et al.1 (Fig. 3d). More precisely, they
showed that the rescaled cell size distribution for unicellular
eukaryotes is well ﬁtted by the log-normal distribution, /ð1=xÞ expððlog x  mÞ2 =2σ 2 Þ with m = − σ2/2 (due to thenormalization 〈x〉 = 1) and obtained σ = 0.471(3). We ﬁnd thatour data for E. coli can also be ﬁtted by the log-normal
distribution (Figs. 2e and 3a, b, d and Supplementary Fig. 7), buthere the value of σ, evaluated by the standard deviation of log x, is4

found to be around σ = 0.3 (σ = 0.34(2) for LB → PBS,
σ = 0.30(2) for M9(Glc+a.a.) → PBS, and σ = 0.29(2) for
M9(Glc) → PBS), much lower than σ = 0.471(3) for the unicellular eukaryotes. In the literature, a previous study on Bacillus
subtilis39 reported values of σ from 0.24 to 0.26, which arecomparable to our results for E. coli. Compared to this substantial
difference between bacteria and unicellular eukaryotes, thedependence on the environmental factors seems to be much
weaker (Fig. 3d).
Modeling the reductive division process. To obtain theoreticalinsights on the experimentally observed scale invariance of thecell size distributions, we construct a simple cell cycle model forthe bacterial reductive division. For the steady growth conditions,a large number of studies on E. coli have been carried out toclarify what aspect of cells triggers the division event9,10. Signiﬁcant advances have been made recently to provide molecularlevel understanding9,27,37,40–44. Here we extend such a model todescribe the starvation process.
One of the most established models in this context is the CHmodel25,45, which consists of cellular volume growth andmultifork DNA replication. The multifork replication is thephenomenon that a cell replicates its DNA not only for its
daughters but also for its granddaughters, before the birth of thedaughter cells (Fig. 4a, b)—a phenomenon well known for fast
growing bacteria such as E. coli and B. subtilis25,45. In the CHmodel, completion of the DNA replication triggers the cell
division, and this gives a homeostatic balance between the DNA
amount and the cell volume. An unknown factor of the CHmodel is how DNA replication is initiated, and a few studies
attempted to ﬁll this gap to complement the CH model27,40. Hoand Amir27 assumed that replication is initiated when a criticalamount of “initiators” accumulate at the origin of replication. In
the presence of a constant concentration of autorepressors,
expressed together with the initiators, this assumption means thatthe cellular volume increases by a ﬁxed amount between two

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COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00739-5baM9(Glc+a.a.)→ PBS

M9(Glc)
→ PBS

d

c

Fig. 3 Rescaled cell size distributions. a The results for M9(glucose (Glc) + amino acids (a.a.)) → PBS. The dashed line represents the ﬁtted log-normal
distribution (σ = 0.31(2)). The data were taken from 17 wells recorded in a single experiment. The sample size ranges from n(0) = 685 to n(180) = 1260
(see Supplementary Fig. 4). (Inset) Time evolution of the non-rescaled cell size distributions at t ¼ 0; 10; 20; 30; 40; 50; 60; 90; 120; 180 min. b The
results for M9(glucose (Glc)) → PBS. The dashed line represents the ﬁtted log-normal distribution (σ = 0.29(2)). The data were taken from 26 wells
recorded in a single experiment. The sample size ranges from n(0) = 836 to n(200) = 2160 (see Supplementary Fig. 4). (Inset) Time evolution of the nonrescaled cell size distributions at t ¼ 0; 10; 20; 30; 40; 50; 60; 80; 100; 150; 200 min. c Experimental results of F(v/V(t)) = vp(v, t) for the three cases
studied in this work. The raw data obtained at different times are shown by thin lines with relatively light colors, and the time-averaged data are shown by
the bold lines. The time-averaged distributions (bold lines) are found to be slightly but signiﬁcantly different among the three cases. The difference can also
be seen in the instantaneous distributions (thin lines; see the inset for enlargement near the peak). d Fitting of the experimentally obtained F(x) (solid lines;
time-averaged data in (c) are shown) to the log-normal distribution (yellow dotted line). Also shown is the ﬁtting result by Giometto et al.1 for unicellular
eukaryotes (green dotted line). σ is the standard deviation parameter of the log-normal distribution (see text).

a

Initiation to initiation = adderc Cell volume

GrowthStarvation

Ori

Initiation

Initiation

Initiation

b

time

d

Initiation
Initiatione

Simulation(LB→PBS)

DivisionFig. 4 Model of reductive division and simulation results. a, b Single (a) and multifork (b, where #ori = 4) intracellular cycle processes. See Eq. (4) for theCD;thCD
criterion that triggers the initiation. Progress of each cycle is represented by a coordinate XCD
i ðtÞ, which increases at speed μi(t) and ends at Xi ðtÞ ¼ Xi
by triggering cell division. c Illustration of cell cycles in this model. Each colored arrow represents a single intracellular cycle process. d Overlapping of therescaled cell size distributions during starvation in the model for LB → PBS. The dashed line represents the ﬁtted log-normal distribution (σ = 0.25(2)).
(Inset) The non-rescaled cell size distributions at t ¼ 0; 5; 30; 60; 90; 120; 180; 240; 300; 360; 420; 480 min from right to left. e Numerically measured
division rate, B(v, t), in the model for LB → PBS. See Supplementary Note 4.B for the measurement method. (Inset) Test of the condition of Eq. (9). HereBt(0)/Bt(t) is evaluated by Bt(0)/Bt(t) = ∫B(xV(0), 0)dx/∫B(xV(t), t)dx, with x running in the range 0 ≤ x ≤ 1.8. Overlapping of the data demonstrates thatEq. (9) indeed holds in our model.

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initiation events, regardless of the absolute volume at theinitiation. This “adder” principle between initiations is now
supported by several observations42–44. By the initiation considered above, the cell starts the C period of the bacterial cell
cycle, which is followed by the D period, and eventually the cell
divides25,45. While Ho and Amir assumed that a constant time isneeded to complete the C+D period ("timer” principle)27, further
experimental investigations by Witz et al. clariﬁed that the model
assuming the adder principle for the C+D period captured singlecell behavior better26. Clarifying the mechanism of cell divisioncontrol is currently a target of intensive studies and different
models have also been proposed42–44. In the present work, wechoose to extend Witz et al.’s model26 to cope with the switch to
the non-nutritious condition and measure the cell size ﬂuctuations during the reductive division process. We also checked that
our main conclusions do not change if we use instead Ho andAmir’s model27 as the starting point.
The model consists of two processes that proceed simultaneously, namely, the volume growth and the intracellular cycle.
The volume of each cell (indexed by i), vi(t), grows asdvi
dt ¼ λðtÞvi ðtÞ, with a time-dependent growth rate λ(t). Following
Witz et al.’s model27, we assume that the volume growth iscoupled to the intracellular cycle as follows. To begin with thesimplest case, suppose that a newborn cell i has a single origin of
replication in its chromosome and that the replication starts atsome point in time (Fig. 4a, red star). By this initiation of
replication, the cell starts to have two origins of replication. Then,the next initiation is triggered when the cell volume vi(t) increases
by a ﬁxed amount δi,1 per origin, i.e., when vi(t) increases byδi,1 × 2, since the last initiation (Fig. 4a, blue stars). Note that thiscriterion does not change whether a cell divides or not before theinitiation; if a cell divides and produces daughter cells i1 and i2,
the initiation in the daughter cells occurs whenvi1 ðtÞ þ vi2 ðtÞ  vi ðt init Þ ¼ δ i;1 ´ 2, where tinit is the time at which
the last initiation occurred. Similarly, if multifork replication
takes place in a single cell (i.e., #ori = 2j with j ≥ 2), the thresholdfor the added volume is given by δi,j × #ori (see Fig. 4b for an
example with #ori = 4). The criterion therefore reads:!
∑

i0 2 offspring of ivi0 ðtÞ

 vi ðt init Þ ¼ δ i;j ´ #ori:

ð4Þ

Following the experimental results by Si et al.41, we assume that
δi,j does not depend on environmental conditions. On the other
hand, to take into account stochastic nature of division events, we
generate δi,j randomly from the Gaussian distribution with mean
〈δi,j〉 = δmean and standard deviation Std[δi,j] = δstd.
After an initiation, the cell undergoes the C+D period andﬁnally divides. Here, for the sake of simplicity, the progression of
the C and D period is collectively expressed by a coordinate
X CD
i ðtÞ, which starts form zero and increases at time-dependent
dX CD

CD;th
,
speed μi(t), dti ¼ μi ðtÞ. When X CDi ðtÞ reaches a threshold X ithe cell divides (Fig. 4b), leaving two daughter cells of volumes
vi1 ðtÞ ¼ xsep vi ðtÞ and vi2 ðtÞ ¼ ð1  xsep Þvi ðtÞ. Here xsep is randomly drawn from the Gaussian distribution with mean 0.5 andstandard deviation 0.0325, the latter value being deduced fromexperimental observations (see “Methods” and Supplementary
Fig. 8). To deal with the multifork replication, the index i of
X CD
i ðtÞ denotes the cell to divide by the considered cell cycle
progression. Therefore, if #ori = 2 when the initiation is triggered,a pair of cell cycles for the future daughter cells, represented byCD
X CD
i1 ðtÞ and X i2 ðtÞ, start and run simultaneously (Fig. 4c).

Similarly to δi,j, we also assume that X iCD;th is a Gaussian random
CD;thvariable, with hX CD;thi ¼ 1 and Std ½X CD;th ¼ X std.
ii
6

Now we are left to determine the two time-dependent rates,
λ(t) and μi(t). Here we consider the situation where growth
medium is switched to non-nutritious buffer at t = 0; therefore, tdenotes time passed since the switch to the non-nutritious
condition. First, we set the volume growth rate λ(t) on the basis of
the Monod equation46, assuming that substrates in each cell aresimply diluted by volume growth and consumed at a constantrate, without uptake because of the non-nutritious condition
considered here. As a result, we obtain
1A
ð5Þ
;
λðtÞ ¼ λ0 ct
e Awith constant parameters A and c and the growth rate λ0( = λ(0))
in the exponential growth phase (see Supplementary Note 3.A for
details).
For the cycle progression speed μi(t), we propose a functional
form that conforms with the type of the principle assumed in theoriginal model for the C+D period in steady conditions, i.e., theadder principle for Witz et al.’s model and the timer principle for
Ho and Amir’s model. We ﬁrst note that the C+D period mainly
consists of DNA replication, followed by its segregation and theseptum formation45. Most parts of those processes involve
biochemical reactions of substrates, such as deoxynucleotide
triphosphates for the DNA synthesis. Here we can consider
different molecular mechanisms for the cycle progression,
depending on the type of the principle to adopt. For the case of
the adder principle (à la Witz et al.), we can assume that divisionoccurs when a given amount of relevant molecules, such as DNA,is produced. We assume that such relevant molecules aresynthesized from substrates through enzyme catalyses, according
to the Michaelis–Menten equation. Considering dilution
due to the volume growth too, we obtain μi(t) ∝ [SC+D]vi(t)/(K + [SC+D]), with [SC+D] being the concentration of thecorresponding substrates and K an adjustable parameter (seeSupplementary Note 3 for details). For simplicity, here we assume
that [SC+D] is common to all cells. Note that, since [SC+D] isconstant in steady conditions, μi(t) ∝ vi and this results in theadder principle as considered in Witz et al.’s model. For the
starvation process, we consider that [SC+D] decreases by dilution
due to volume growth, degradation, and consumption. Those areassumed to be independent of #ori, based on the experimental
results that the duration of the C+D period is independent of #ori
in steady environments41. From those considerations, we ﬁnallyobtain the following equation for the cycle progression speed:μi ðtÞ ¼

μ0
kþ1
v ðtÞ;
v0 k expðt=τÞ þ 1 i

ð6Þ

with parameters k and τ, the mean cycle progression speed μ0,
and the mean cell size v0 in the exponential phase (seeSupplementary Note 3.A). In the case of the timer principle for
the C+D period in steady environments (à la Ho and Amir), we
consider instead that assembly processes of molecules such as
deoxynucleotide triphosphates control the cycle progression
speed. As a result, we obtain a formula of μi(t) without vi
dependence (see Supplementary Note 3). In the following,
however, we mostly present results from the model à la Witz
et al. unless otherwise stipulated, while we checked that the main
conclusions did not change if the model à la Ho and Amir wasused instead.
The parameter values are determined from the experimentallymeasured total cell volume and the cell number, which oursimulations turn out to reproduce very well (Fig. 2b, c andSupplementary Fig. 4a, b), with the aid of relations reported byWallden et al.40 for some of the parameters (see Table 1 for theparameter values used in the simulations and “Methods” for theestimation method). With the parameters ﬁxed thereby, we

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Table 1 Parameters used for the simulations.
LB → PBS

Parameters
Parameters on the
exponential
growth phase

Time-dependent rates

M9(Glc+a.a.) → PBS
1

λ0

0.029 min

0.010 min1μ1
01:3λ0:84
þ 42 ’ 67min
0

1:3λ0:84
þ 42 ’ 104min
0v0

4.9 μm3

1.8 μm3

δmean or vthmeanδstd or vth
std
XCD;thstd
xsep
std
λðtÞ ¼ λ0 e1Act Aμμi ðtÞ ¼ v0 kekþ1t=τ þ1 vi0δmean

0.1 × δmean

0.1 × δmean = 0.0225 μm3
0:05 ´ hXCD;thi ¼ 0:05i

0.03250.0325
1

Theoretical conditions for the scale invariance. To seek for a
possible mechanism leading to the scale invariance, here we
describe, theoretically, the time dependence of the cell size distribution in a time-dependent process. Suppose N(v, t)dv is thenumber of the cells whose volume is larger than v and smaller
than v + dv. If we assume, for simplicity, that a cell of volume v
can divide to two cells of volume v/2, at probability B(v, t), we
obtain the following time evolution equation:
ð7Þ

Note that this equation has been studied by numerous paststudies for understanding stable distributions in steadyconditions1,47–52, but here we explicitly include the timedependence of the division rate, B(v, t), for describing thetransient dynamics. To clarify a condition for this equation tohave a scale-invariant solution, here we assume the scaleinvariant form, Eq. (2), where p(v, t) = N(v, t)/n(t) and n(t) is
the total number of the cells, and obtain the following selfconsistent equation (see Supplementary Note 4.A for derivation):∂FðxÞ Bðv; tÞBð2v; tÞFðxÞ þ 2 
Fð2xÞ:
 
ð8Þ
BðtÞ
BðtÞ
∂x
R
 ¼ dvBðv; tÞpðv; tÞ. For the scale
Here x = v/V(t) and BðtÞinvariance, Eq. (8) should hold at any time t. This is fulﬁlled if
B(v, t) can be expressed in the following form (see Supplementary
Note 4.A):FðxÞ ¼ xBðv; tÞ ¼ Bv ðv=VðtÞÞBt ðtÞ:δmean = 0.25 μm3= 0.025 μm3

0:05 ´ hXCD;thi ¼ 0:05
imeasure the cell size ﬂuctuations at different times and ﬁnd thescale invariance similar to that revealed experimentally (Fig. 4d
and Supplementary Fig. 9b, e). The constancy of CV and Sk is
also conﬁrmed (Supplementary Fig. 9c, f). Interestingly, the scale
invariance emerges despite the existence of characteristic scales in
the model deﬁnition, such as the typical volume added between
initiations, δmean. To check the robustness of those results, we also
extended Ho and Amir’s model along the same line (seeSupplementary Note 3.B for details) and conﬁrmed the scale
invariance of similar quality (Supplementary Fig. 9g, h). Theseﬁndings suggest the existence of a statistical principle underlying
the scale invariance, which is not inﬂuenced by details of themodel.

∂Nðv; tÞ∂
¼  ½λðtÞvNðv; tÞ∂t∂v
Bðv; tÞNðv; tÞ þ 4Bð2v; tÞNð2v; tÞ:= 0.275 μm3

ð9ÞThis is a sufﬁcient condition for the cell size distribution tomaintain the scale-invariant form, Eq. (2), during the reductive
division. Note that ref. 6 proposed a similar scale-invariant formof the division rate for the steady environment. It is also
important to remark that, as opposed to Eq. (7), Eq. (8) does not

A = 0.93, c = 0.0059min

A = 0.84, c = 0.011min1

k = 0.01, τ = 40min

k = 0.01, τ = 16mininclude the growth rate λ(t) explicitly. The scale-invariant
distribution F(x) is therefore completely characterized by thedivision rate B(v, t) in this framework.
To test whether the condition of Eq. (9) is satisﬁed in ourmodel, we measure the division rate B(v, t) in our simulations(Fig. 4e). The data overlap if B(v, t)Bt(0)/Bt(t) is plotted againstv/V(t), demonstrating that Eq. (9) indeed holds here. On theother hand, our theory does not seem to account for thefunctional form F(x) of the scale-invariant distribution; the righthand side of Eq. (8) differs signiﬁcantly from the left hand side, if
the numerically obtained B(v, t) is used together with the function
F(x) from the simulations or the experiments (Supplementary
Fig. 10). The disagreement did not improve by taking into
account the effect of septum ﬂuctuations (see Supplementary
Note 4.C). The lack of quantitative precision is probably not
surprising given the simplicity of the theoretical description,
which incorporates all effects of intracellular cycles into thesimple division rate function B(v, t). The virtue of this theory isthat it clariﬁes that the intracellular cycle seems to have important
relevance in the scale invariance and the functional form of thecell size distribution. The signiﬁcant difference in F(x) identiﬁed
between bacteria and unicellular eukaryotes (Fig. 3d) may beoriginated from the different replication mechanisms that the two
taxonomic domains adopt.
Violation of the scale invariance. Here we investigate therobustness of the scale invariance during the reductive division.
In particular, we aim to clarify whether it breaks down for otherstarvation conditions, and if so, what the condition is for the scale
invariance to hold. As shown in Fig. 3c and Supplementary Fig. 7,
the form of F(x) obtained by our experiments depends on thegrowth environment before starvation. This suggests that F(x)may change if one switches between two growth media in aquasistatic manner, i.e., the scale invariance may break down inthis case. Motivated by this hypothesis, we numerically investigate
whether there is a lower bound on the relaxation speed of thecellular state, below which the scale invariance breaks down. For
simplicity, we consider that the environment starts to change at
t = 0, and the volume growth rate λ(t) and the cycle progression
speed μi(t) decrease as follows:λðtÞ ¼ λ0 expðt=τ λ Þ;
μi ðtÞ ¼

μ0expðt=τ μ Þvi ðtÞ:
v0

ð10Þ
ð11Þ

We regard τλ, τμ, and λ0 as free parameters, while theparameters μ0 and v0 were set as follows (see also Supplementary
Note 3.C.2 for details). For μ0, we determined it from λ0 using
empirical relation reported by Wallden et al.40 for steadyenvironments. For v0, we set its value self-consistently, so that

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a

b

c

d

Fig. 5 Numerical results on the range of validity of the scale invariance. The initial growth rate is ﬁxed at λð0Þ ¼ λ0 ¼ 0:03 min1 unless otherwise
stipulated. a Trajectories in the 〈v〉–CV space for different τμ (from 50 to 150 min), with τ λ ¼ 40 min ﬁxed. The endpoint of each trajectory is indicated bya colored open circle. The black squares represent the states in steady growth conditions with the growth rate λ0 ranging from 0.01 to 0:03 min1 . The
dashed plateau indicates the initial CV at λ0 ¼ 0:03 min1 . b The master curves of the 〈v〉-CV trajectories for different τλ. Those are obtained by taking the
average of the CV values at each 〈v〉 over different τμ(>τλ). τλ ranges from 10 to 90 min. c Phase diagram. ×: the scale invariance breaks down. Blue ○: thescale invariance holds. Green △: near the boundary. Black dots: the scale invariance holds but the mean volume increases. d Pseudocolor plot of ρ fordifferent τλ and τμ. See the main text for the deﬁnition of ρ. The black region indicates ρ = 0. The white line represents the transition line obtained from c.
The boundaries (△) are not included in the region where the scale invariance breaks down.

the mean cell volume 〈vi〉 obtained numerically in the exponential
phase with μi = (μ0/v0)vi falls within 1% error from the given
value of v0. The number of cells is set to be approximately 50,000
at t = 0 and kept constant afterward during the starvation
process, by eliminating one of the daughter cells produced bydivision (see “Methods” for details).
First we evaluate p
theﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
mean cell volume 〈v〉 and the coefﬁcientof variation, CV ¼ hδv2 i=hvi, in the exponential growth phase
under steady conditions, by varying the growth rate λ0 from 0.01
to 0:03 min1 . As shown by the black squares in Fig. 5a, b, lower
growth rates (smaller mean volumes) lead to lower CVs. This isconsistent with our experimental results (Fig. 3c and Supplementary Fig. 7). We then investigate how the mean volume 〈v〉and CV change during the starvation process, starting at t = 0
from the growth phase with λ0 ¼ 0:03 min1 (Fig. 5a, the top
right black square). As expected, our data showed that the meancell volume decreases if τμ > τλ and increases otherwise; therefore,
in the following, we deal with the case of τμ > τλ, whichcorresponds to the reductive division. The color curves in Fig. 5
show trajectories in the 〈v〉–CV space during the starvation
process, each curve corresponding to a different τμ( > τλ) with τλﬁxed at τ λ ¼ 40 min. Remarkably, these trajectories overlap to asingle curve with an extended plateau region, which indicates that
CV is kept constant, i.e., the scale invariance. Each curve stops in
the middle of the master curve, the location of the endpoint (att → ∞, shown by the open circles) being determined by τμ.
Importantly, for small τμ, the trajectories stop in the plateau
region, so that the scale invariance holds during the entire
process. By contrast, for large τμ, the trajectories go over theplateau and CV decreases abruptly; in other words, the scale
invariance breaks down. Next, Fig. 5b shows the master curves fordifferent τλ, each constructed by using the trajectories for all τμgreater than τλ. We ﬁnd that the smaller τλ is, the more extended8

the plateau region is. Finally, we show the phase diagram forvarious combinations of τμ and τλ in Fig. 5c. This clearly shows aregion in which the scale invariance is maintained during theentire starvation process, bordered by a transition line over which
the scale invariance breaks down. Note that the scale-invariant
region becomes narrower for larger τλ and τμ and seem todisappear eventually; this is consistent with our expectation
described at the beginning, that the scale invariance does not holdfor quasistatic changes. All those results were also conﬁrmed
when the extension of Ho and Amir’s model was used instead(Supplementary Fig. 11; see Supplementary Note 2.B for themodel deﬁnition).
To understand what triggers the violation of the scale
invariance, we focus on the state of the multifork replications,since our theory suggested the importance of the division rate,
which is controlled by the state of the cell cycle. As illustrated inFig. 4c, the ﬁrst few divisions after the onset of starvation are tied
to the initiation that occurred in the exponential growth phase.
We may expect that these division events retaining “memories”
from the growth phase are less affected by the starvation andtherefore may not violate the scale invariance. Based on thisexpectation, we investigate the state of the cell cycle as follows.
First, we observe that the change in the number of origins of
replication (#ori) during the growth phase is rather stable,
doubling (by initiation) and decreasing (by division) between#ori = 2j−1 and 2j with a ﬁxed j for the majority of cells (j = 3 inthe case of Fig. 4c; note that j − 1 and j correspond to thenumbers of parallel arrows therein). This number is maintained
for a while in the starvation process, but eventually it may
decrease, because a cell may divide without initiating a new
replication during the life. We therefore measure the fraction ofsuch cells, ρ. To be precise, with J being j of each cell in thegrowth phase, ρ is the fraction of cells such that the C+D period

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with j < J is initiated during the lifetime, and that this C+D period
ends and triggers a cell division afterward, before the cell cycle
progression completely stops (note that, since μi(t) → 0, not allcell cycles complete). It is measured at the ﬁnal time point of thesimulations (speciﬁcally t ¼ 600 min) and shown in Fig. 5d for
λ0 ¼ 0:03 min1 . Intuitively, ρ corresponds to the fraction of cells
that lost memories from the growth phase. Here we ﬁnd ρ = 0
indeed in most part of the scale-invariant region, while ρ > 0
when the scale invariance breaks down. We therefore consider
that the state of the multifork replications may be a key factor
that determines whether the scale invariance holds or not during
the reductive division. Note that, for gradual environmental
changes, actual cells are known to emit signals such as ppGpp53,54
that control growth and cycle progression, which are not taken
into account in our model. Investigating the effect of such signals
in this problem is an interesting problem left for future studies.
Discussion
In this work, we developed a membrane-based microﬂuidic
device that we named the extensive microperfusion system(EMPS). Advantages of this device are that we can realize a
uniformly controlled environment for wide-area observations of
microbes and can switch it without hydrodynamic perturbations.
Those advantages may be useful for applications in a wide rangeof problems with dense cellular populations, including living
active matter systems55,56 and bioﬁlm growth57–59. In this work,
we focused on statistical characterizations of single-cell morphology during the reductive division of E. coli. Thanks to the
EMPS, we recorded the time-dependent distribution of cell size
ﬂuctuations and revealed that the rescaled distribution is scale
invariant and robust against the abrupt environmental change,
despite the decrease of the mean cell size (e.g., Fig. 2). We conﬁrmed the robustness of the result against different combinationsof the growth and non-nutritious media, while we also found thatthe shape of the rescaled distribution does depend on the choiceof the growth medium before the switch (Fig. 3). Moreover, thoseﬁndings were successfully reproduced by simulations of a model
based on the CH model26,27, which we propose as an extension
for dealing with time-dependent environments (Fig. 4). We further inspected theoretical mechanism behind this scale invarianceand found the signiﬁcance of the division rate function B(v, t).
We obtained a sufﬁcient condition for the scale invariance, Eq.
(9), which was indeed conﬁrmed in our numerical data. Finally,
we numerically clariﬁed the range of validity of the scale invariance during the reductive division, showing that the state of themultifork replications may play a crucial role (Fig. 5).
Notably, our experiments (on the growth condition dependence) and simulations suggest that the scale invariance breaks
down for slow starvation. Further investigations of cell size
ﬂuctuations in such cases, both experimentally and theoretically,
will be an important step toward clarifying what determines thecritical time scale of environmental changes for the violation of
the scale invariance. Elucidating the (τλ, τμ) phase diagram isparticularly important, because it may serve as a further test of
the two cell cycle models used here, which predicted signiﬁcantly
different diagrams (Fig. 5c and Supplementary Fig. 11b). It is achallenge experimentally, but may also be possible with EMPS, bycombining a technique to control the progression speed of the C+D period, such as the one developed in ref. 41.
It is also worth noting that the cell size distribution we measured is that of the entire population, which is given by p(v) =∫p(v∣a)page(a)da with the size distribution p(v∣a) of cells at agiven age a and the age distribution page(a) of the population.
Since those distributions have also been studied in the past forsteady conditions (e.g., refs. 5,6 for p(v∣a), ref. 30 for page(a)), it is

ARTICLEan important future work to understand how these distributions
change under time-dependent conditions and how they contribute to the scale invariance. It is also important to understandthe dependence on the population size, which will be a crucial
point to consider an analogous experiment in the mothermachine.
In the context of possible follow-up experiments using themother machine, another aspect that deserves attention is the way
nutrients are delivered to cells and removed. As we have mentioned, EMPS relies on diffusion of molecules and is therefore
prone to have a slight amount of residual nutrients in theobservation area. While this may better correspond to natural
conditions, in which the surrounding medium is not necessarily
replaced by a strong ﬂow as in the mother machine17–21, wecannot exclude the possibility that the slightly remaining nutrients might affect the cellular state after starvation in EMPS.
Therefore, it would be interesting to investigate whether the scale
invariance, which our study has shown to be robust in various
starvation conditions in EMPS and in models, can also be veriﬁedin the mother machine.
After all, our results backed by the cell cycle model suggest thatmechanism of intracellular replication processes may have direct
impact on the scale-invariant distribution, which may account forthe signiﬁcant difference we identiﬁed between bacteria andeukaryotes (Fig. 3d). Since the number of species studied in eachtaxonomic domain is rather limited (E. coli (this work) and B.
subtilis39 for bacteria, 13 protist species for eukaryotes1), it is of
crucial importance to test the distribution trend further in eachtaxonomic domain and to clarify how and to what extent the cellsize distribution is determined by the intracellular replication
dynamics. The inﬂuence of cell-to-cell interactions, e.g., quorum
sensing32,33, may also exist. Theoretical approaches, such asmodels considering the cellular age60, knowledge from the universal protein number ﬂuctuations61–63, and renormalizationgroup approaches for living cell tissues64, may also be useful. We
hope that our understanding of the population-level response
against nutrient starvation will be further reﬁned by future
experimental and theoretical investigations.
Methods
Strains and culture media. We used wild-type E. coli strains (MG1655 andRP437) and a mutant strain (W3110 ΔﬂiC Δﬂu ΔﬁmA) in this study. Culture
media and buffer are listed in Supplementary Table 1. The osmotic pressure of eachmedium was measured by the freezing-point depression method by the OSMOMAT 030 (Genotec, Berlin, Germany). Details on the strains and culture conditions in each experiment are provided below (see also Supplementary Note 1).

Fabrication of the EMPS. The EMPS consists of a microfabricated glass coverslip,
a bilayer porous membrane, and a PDMS pad. The microfabricated coverslip andthe PDMS pad were prepared according to refs. 22,30. We fabricated the bilayer
porous membrane by combining a streptavidin-decorated cellulose membrane anda biotin-decorated PET membrane. The streptavidin decoration of the cellulose
membrane (Spectra/Por 7, Repligen, Waltham, MA, molecular weight cut-off
25,000) was realized by the method described in refs. 22,30. The PET membrane(Transwell 3450, Corning, Corning, NY, nominal pore size 0.4 μm) was decorated
with biotin as follows. We soaked a PET membrane in 1 wt% solution of 3-(2aminoethyl aminopropyl) trimethoxysilane (Shinetsu Kagaku Kogyo, Tokyo,
Japan) for 45 min, dried it at 125 °C for 25 min and washed it by ultrasoniccleaning in Milli-Q water for 5 min. This preprocessed PET membrane was stored
in a desiccator at room temperature, until it was used to assemble the EMPS.
The EMPS was assembled as follows. The preprocessed PET membrane was cut
into 5 mm × 5 mm squares, soaked in the biotin solution for 4 h, and dried on ﬁlter
paper. The biotin-decorated PET membrane was attached with a streptavidindecorated cellulose membrane, cut to the size of the PET membrane, bysandwiching them between agar pads (M9 medium with 2wt% agarose). In themeantime, a 1 μl droplet of bacterial suspension was inoculated on a biotindecorated coverslip (see also details below). We then took the bilayer membranefrom the agar pad, air-dried for tens of seconds, and carefully put on the coverslip
on top of the bacterial suspension. The bilayer membrane was then attached to thecoverslip via streptavidin–biotin binding as shown in Supplementary Fig. 1b. We

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then air-dried the membrane for a minute and attached a PDMS pad on the
coverslip by a double-sided tape.
Observation of the bacterial reductive division. We used a wild-type strain
MG1655. Before the time-lapse observation, we inoculated the strain from a glycerol stock into 2 ml growth medium in a test tube. The same medium as for themain observation was used (LB broth, M9(Glc+a.a.) or M9(Glc)). After shaking it
overnight at 37 °C, we transferred 20 μl of the incubated suspension to 2 ml fresh
medium and cultured it until the optical density (OD) at 600 nm wavelength
reached 0.1–0.5. The bacterial suspension was ﬁnally diluted to OD = 0.05 before it
was inoculated on the coverslip.
For this experiment, we used a substrate with wells of 55 μm diameter and0.8 μm depth. The well diameter was chosen so that all cells in the well can berecorded. The device was placed on the microscope stage, in the incubation box
maintained at 37 °C. The microscope we used was Leica DMi8, equipped with a×100 (N.A. 1.30) oil immersion objective and operated by Leica LasX. To ﬁll thedevice with growth medium, we injected fresh medium stored at 37 °C from theinlet (Supplementary Fig. 1), at the rate of 60 ml h−1 for 5 min by a syringe pump(NE-1000, New Era Pump Systems).
In the beginning of the observation, growth medium was constantly supplied at
the rate of 2 ml h−1 (ﬂow speed approximately 0.2 mm s−1 above the membrane).
When a microcolony composed of approximately 100 cells appeared, we quickly
switched the medium to a non-nutritious buffer (PBS or M9 medium with αMG,see Supplementary Table 1) stored at 37 °C, by exchanging the syringe. The ﬂow
rate was set to be 60 ml h−1 for the ﬁrst 5 min, then returned to 2 ml h−1.
Throughout the experiment, the device and the media were always in the
microscope incubation box, maintained at 37 °C. Cells were observed by phase
contrast microscopy and recorded at the time interval of 5 min. The data forobtaining each distribution are taken from several wells (stated in the ﬁgure
captions) in a single experiment.
The cell volumes were evaluated as follows. We determined the major axis andthe minor axis of each cell, manually, by using a painting software. By measuring
the axis lengths, we obtained the set of the length Li and the width wi for all cells(indexed by i). We estimated the uncertainty in manual segmentation at ±0.15 μm.
However, the measurement of the individual cell widths is less accurate than that of
the lengths, essentially because the width depends on the choice of the section of
the cell. To estimate the cell volume, therefore, we neglected the ﬂuctuation of thewidth among the cells as follows. We measured the width wi at the center of eachcell and took the ensemble average 〈wi〉. Together with the cell length Li, we
 3
 2
hwi i
obtained the volume of this cell, vi, by vi ¼ 4π
þ π hw2i i ðLi  hwi iÞ. Note32that the scale invariance holds for the length distribution as well (Supplementary
Fig. 6), which suggests that neglecting the width ﬂuctuation does not affect themain ﬁnding of the paper.
Finally, let us note that there may be some technical limitation speciﬁc to the
combination of the LB medium and the EMPS. When we used LB medium in thegrowth phase, the bacteria continued growing, albeit very slowly, even long timeafter the medium was switched to PBS (see Supplementary Movie 7), while they
stopped growing completely in all cases where we used chemically deﬁned medium
before the switch (Supplementary Fig. 4 and Supplementary Movies 8–11). Thismay be because some nutrient molecules speciﬁc to LB might remain on the wellsurface or inside the membrane. However, since we are focusing on the earlier stageof the starvation process, in which the typical cell sizes change most signiﬁcantly,
we believe that this remaining slow growth in the case LB → PBS does not affectour main results. More quantitatively, from Supplementary Fig. 9a, we can evaluate
the rate of this remaining cell growth observed in the case LB → PBS to be nearly104 min1 or eventually even less. Because the corresponding time scale≳104 min is much longer than the time scales relevant to the scale invariance wefound, which are around 100 min, there is a clear scale separation, from which we
can expect that the remaining slow cell growth will not affect our main ﬁnding. We
also noticed relatively poor reproducibility of experiments using LB medium(Supplementary Fig. 7), which may be attributed to its chemical undeﬁnedness65,
since the experiments using deﬁned media recorded good reproducibility (see thesame ﬁgure).

simulations. Note that, without this stochastic asymmetric division, the cell size
distribution exhibited periodic oscillations, presumably because cellular states
between siblings were strongly correlated then.
In the following, we describe how the remaining parameters were evaluated andhow the simulations were carried out for each set of the simulations presented inthis work.
Methods for the results that reproduced the experimental observations. We evaluatedthe time-dependent rates λ(t) and μi(t) as follows. The growth rate λ(t) can be
determined independently of the cell divisions, because the total volume
Rt
Vtot(t) = ∑ivi(t) grows as V tot ðtÞ ¼ V tot ð0Þ expð 0 λðtÞdtÞ. With λ(t) given by Eq.
(5), we compared Vtot(t) with experimental data and determined the values of A
and c (Fig. 3c). Finally, only k and τ in Eq. (6) remained as free parameters. We
tuned them so that the mean cell volume V(t) and the number of the cells n(t)observed in the simulations reproduced those from the experiments (Fig. 3d). The
parameter values determined thereby are summarized in Table 1 for the simulations for LB → PBS and M9(Glc+a.a.) → PBS.
We started the simulations from 10 cells with volumes in the range of
0.07–1.13 μm3, randomly generated from the uniform distribution. The cells grewin the exponential phase (with the constant growth rate λ0 and the cycle
progression speed μi = (μ0/v0)vi) until the number of cells reached 100,000. We
then randomly picked up 10 cells from this “precultured” sample and grew them
until the number of cells exceeded 500. Those cells were kept growing for 1,000 min
to sufﬁciently mix cell cycle progressions in the population. During this process, we
kept the number of cells constant, by eliminating one of the daughter cells after
each division. We then used them as the initial population of each simulation. To
precisely compare the number of cells in simulations with the experimentallyobtained population nexp ðtÞ (Fig. 2c and Supplementary Fig. 4b), the numericallyobtained population nsim ðtÞ is rescaled by multiplying nexp ð0Þ=nsim ð0Þ.
Methods for the results on violation of the scale invariance. The functional forms of
λ(t) and μi(t) were given by Eqs. (10) and (11), with variable parameters τλ and τμ.
For μ0, we determined it from λ0 using the empirical relation reported by Wallden
et al.40. For v0, we set its value self-consistently, so that the mean cell volume 〈vi(0)〉obtained numerically in the exponential phase with μi = (μ0/v0)vi falls within 1%
error from the given value of v0. As a result, we obtained v0 = 1.9, 2.5, 3.2, 4.0,
5.0 μm3 for λ ¼ 0:01; 0:015; 0:02; 0:025; 0:03 min1 , respectively. These values
satisfy the growth law, i.e., the mean cell size increases exponentially with thegrowth rate52. The other parameters were ﬁxed at δmean = 0.25 μm3, δstd =sep¼ 0:05, and xstd ¼ 0:0325. We started the simulations from 50
0.025 μm3, X CD;thstdcells with volumes in the range of 0.07–1.13 μm3, randomly generated from theuniform distribution. The cells grew in the exponential phase until the number ofcells reached 500,000. We then randomly picked up 50 cells and grew them until
the number of cells exceeded 50,000. Those cells were kept growing for 1,000 min
to sufﬁciently mix cell cycle progressions in the population, with the number ofcells kept constant by eliminating one of the daughter cells produced by division.
Using them as the initial population (at t = 0), we started the simulations for t ≥ 0,with the number of cells still kept constant by the same method. Strictly, thissituation with a constant number of cells is different from the experimental setting,but we conﬁrmed that this change did not inﬂuence the validity of the scale
invariance and had only a minor effect on the value of CV, at least for the situation
shown in Fig. 4c. Therefore, for the results on the violation of the scale invariance(Fig. 5), we carried out simulations with a constant number of cells as describedabove, to reduce the computation time.
Reporting summary. Further information on research design is available in the Nature
Research Reporting Summary linked to this article.

Data availability
The data that support the ﬁndings of this study are available at https://github.com/
shimasaan/bacterial_rd.

Code availability
Simulation. The parameters used in the simulations were evaluated as follows.
First, from the observations of the exponential growth phase, we determined thegrowth rate λ0 and the mean cell size v0 directly. This allowed us to set the cycle
0:84
progression speed μ0 too, by using the relation μ1þ 42Þ proposed by
0 ’ ð1:3λ0
Wallden et al.40 (the values of λ0 and μ0 in the unit of min1 are used here).
Concerning the volume threshold for initiating the replication, we found such avalue of δmean (or vth
mean ) that reproduced the experimentally observed mean cellvolume in the growth phase. The standard deviation δstd (or vthstd ) was set to be 10%of the mean δmean (vthmean ), based on the relation on the initiation volume found by
Wallden et al.40. They also measured the ﬂuctuations of the time length of the C+D
at 5% of 〈XCD,th〉, i.e., X CD;th¼ 0:05. On the
period; this led us to estimate X CD;th
std
std
sep
septum positions, we measured their ﬂuctuations and found little difference in xstd
among the different growth conditions we used and also in the non-nutritious casesep
(Supplementary Fig. 8). We therefore used a single value xstd ¼ 0:0325 for all
10

The codes used in this study are available at https://github.com/shimasaan/bacterial_rd.

Received: 29 September 2020; Accepted: 15 October 2021;

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Acknowledgements
We are grateful to Y. Himeoka for motivating us to compare the abrupt and quasi-staticchanges, which led to the results presented in Fig. 5. We also acknowledge useful discussions with H. Chaté, Y. Furuta, T. Hiraiwa, Y. Kitahara, H. Nakaoka, and

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11

ARTICLE

COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00739-5

D. Nishiguchi. We thank I. Naguro for letting us use the OSMOMAT 030. This work is
supported by KAKENHI from Japan Society for the Promotion of Science (JSPS) (No.
16H04033, No. 19H05800), a Grant-in-Aid for JSPS Fellows (No. 20J10682), and by the
grants associated with the “Planting Seeds for Research” program and Suematsu Award
from Tokyo Tech.

Peer review information Communications Physics thanks Thomas Julou and the otheranonymous reviewer(s) for their contribution to the peer review of this work. Peerreviewer reports are available.

Author contributions

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional afﬁliations.

T.S. and K.A.T. designed research. T.S., R.O., Y.W., and K.A.T. developed the extensive
microperfusion system. T.S. performed all bacterial experiments and analyzed data. T.S.
and K.A.T. did the modeling, and T.S. wrote the codes for the simulations. T.S. performed the theoretical calculations. T.S. and K.A.T. wrote the manuscript, and all authors
worked for revision.

Competing interests
The authors declare no competing interests.

Additional information
Supplementary information The online version contains supplementary material
available at https://doi.org/10.1038/s42005-021-00739-5.
Correspondence and requests for materials should be addressed to Takuro Shimaya or
Kazumasa A. Takeuchi.

12

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