Scale invariance of cell size fluctuations in starving bacteria
Takuro Shimaya, Reiko Okura, Yuichi Wakamoto, Kazumasa A. Takeuchi
SScale invariance during bacterial reductive division observed by an extensivemicroperfusion system
Takuro Shimaya, ∗ Reiko Okura, Yuichi Wakamoto, and Kazumasa A. Takeuchi
1, 3, † Department of Physics, Graduate School of Science, University of Tokyo, Tokyo 113-0033, Japan Department of Basic Science, Graduate School of Arts and Sciences, University of Tokyo, Tokyo 153-8902, Japan Department of Physics, School of Science, Tokyo Institute of Technology, Tokyo 152-8551, Japan (Dated: June 26, 2020)In stable environments, cell size fluctuations are thought to be governed by simple physical prin-ciples, as suggested by recent finding of scaling properties. Here we show, using
E. coli , that thescaling concept also rules cell size fluctuations under time-dependent conditions, even though thedistribution changes with time. We develop a microfluidic device for observing dense and largebacterial populations, under uniform and switchable conditions. Triggering bacterial reductive divi-sion by switching to non-nutritious medium, we find evidence that the cell size distribution changesin a specific manner that keeps its normalized form unchanged; in other words, scale invarianceholds. This finding is underpinned by simulations of a model based on cell growth and intracellularreplication. We also formulate the problem theoretically and propose a sufficient condition for thescale invariance. Our results emphasize the importance of intrinsic cellular replication processes inthis problem, suggesting different distribution trends for bacteria and eukaryotes.
INTRODUCTION
Recent studies on microbes in the steady growth phasesuggested that the cellular body size fluctuations maybe governed by simple physical principles. For instance,Giometto et al. [1] proposed that size fluctuations ofvarious eukaryotic cells are governed by a common dis-tribution function, if the cell sizes of a given species arenormalized by their mean value (see also [2]). In otherwords, the distribution of cell volumes v , p ( v ), can bedescribed as follows: p ( v ) = v − F ( v/V ) , (1)with a function F ( · ) and V = (cid:104) v (cid:105) being the mean cell vol-ume. This property of distribution is often called scale in-variance. Interestingly, this finding can account for powerlaws of community size distributions, i.e., the size distri-bution of all individuals regardless of species, which wereobserved in various natural ecosystems [3, 4]. Scale in-variance akin to Eq. (1) was also found for bacteria [5, 6]for each cell age, and the function F ( · ) was shown to berobust against changes in growth conditions such as thetemperature.Those results, as well as theoretical models proposedin this context [1, 7], have been obtained under steadyenvironments, for which our understanding of single-cellgrowth statistics has also been significantly deepened re-cently [8–10]. By contrast, it is unclear whether such asimple concept as scale invariance is valid under time-dependent conditions, where different regulations of cellcycle kinetics may come into play in response to envi-ronmental variations. In particular, when bacterial cellsenter the stationary phase from the exponential growthphase, they undergo reductive division, during whichboth the typical cell size and the amount of DNA per celldecrease [11–14]. Although this behavior itself is com- monly observed in test tube cultivation, little is knownabout single-cell statistical properties during the tran-sient. The bacterial reductive division is therefore an im-portant model case for studying cell size statistics undertime-dependent environments and testing the robustnessof the scale invariance against environmental changes.It has been, however, a challenge to observe large pop-ulations of bacteria under uniform yet time-dependentgrowth conditions. For steady conditions, the MotherMachine [15], which allows for tracking of bacteriatrapped in short narrow channels, was proved to be apowerful tool for measuring cell size statistics. In such ex-periments, the channel width needs to be adapted to cellwidths in a given condition, and this renders the applica-tion to time-dependent conditions difficult. If we enlargethe channels, depletion of nutrients in deeper regions ofthe channels induces spatial heterogeneity, as discussedin ref.[16, 17] and later in this article. Hence, it is alsorequired to develop a system that can uniformly controlnon-steady environments and give a sufficient amount ofsingle-cell statistics with high efficiency.In this study, we establish a novel microfluidic device,which we name the “extensive microperfusion system”(EMPS). This system can trap dense bacterial popula-tions in a wide quasi-two-dimensional region and uni-formly control the culture condition for a long time. Weconfirm that bacteria can freely swim and grow inside,and evaluate the uniformity and the switching efficiencyof the culture condition. Then we use this system forquantitative observations of bacterial reductive divisionprocesses. We observe Escherichia coli cells and find thatthe distribution of cell volumes, collected irrespective ofcell ages, maintain the scale invariance as in Eq. (1) ateach time, with the mean cell size that gradually de-creases. To obtain theoretical insights on this experi-mental finding, we devise a cell cycle model describing a r X i v : . [ q - b i o . CB ] J un Objective lensBiotin-decorated coverslipPDMS pad
Medium outlet
Frame seal
Bubble trapMedium outletMedium inlet
Medium outletMedium inlet
PDMS pad
PET membraneCellulose membrane
Coverslip
Stage
Observation area a b
Frameseal Bubble trap
FIG. 1. Sketch of the extensive microperfusion system(EMPS). a Entire view of the device. Microwells are cre-ated on a glass coverslip. We attach a PDMS pad on thecoverslip with a square frame seal to fill the system with liq-uid medium. b Cross-sectional view inside the PDMS pad.A PET-cellulose bilayer porous membrane is attached via thebiotin-streptavidin bonding. reductive division processes, by extending the single-cellCooper-Helmstetter model [18, 19] for steady growth en-vironments. We numerically find that this model indeedshows the scale invariance, confirming the robustness ofthis property. Further, we provide theoretical descrip-tions on the time evolution of the cell size distribution,and propose a condition for the scale invariance.
RESULTSDevelopment of the extensive microperfusion system
In order to achieve uniformly controlled environmentswith dense bacterial suspensions, we adopt a perfusionsystem as developed in ref. [20]. This system allows forsupplying fresh medium through a porous membrane at-tached over the entire observation area, instead of supply-ing from an open end as polydimethylsiloxane (PDMS)-based systems usually do. In its prototypical setup, bac-teria are confined in microwells made on a coverslip, cov-ered by a cellulose porous membrane attached to the cov-erslip via biotin-streptavidin bonding. The pore size ofthe membrane is chosen so that it can confine bacteriaand also that it can exchange nutrients and waste sub-stances across the membrane. To continuously perfusethe system with fresh medium, a PDMS pad with a bub-ble trap is attached above the membrane by a two-sidedframe seal (Fig. 1a and Supplementary Fig. 1a). Thissetup can maintain a spatially homogeneous environmentfor cell populations in each microwell, in particular if themicrowells are sufficiently shallow so that all cells remainnear the membrane. However, because the soft cellulosemembrane may droop and adhere to the bottom for wideand shallow microwells, the horizontal size of such quasi-two-dimensional microwells has been limited up to a fewtens of micrometers, preventing from characterization ofthe instantaneous cell size distribution.By the EMPS, we overcome this problem and realizequasi-two-dimensional wells sufficiently large for statis- tical characterization of cell populations. This is madepossible by introducing a bilayer membrane, where thecellulose membrane is sustained by a polyethylene tereph-thalate (PET) porous membrane via biotin-streptavidinbonding (Fig. 1b, Supplementary Fig. 1b and Methods).Because the PET membrane is more rigid than the cel-lulose membrane, we can realize extended area withoutbending of the membrane. Specifically, in our setup withcircular wells of 110 µ m diameter and 1 . µ m depth, al-though a cellulose membrane alone is bent and adheresto the bottom of the well (Supplementary Fig. 1c andSupplementary Movie 1), our PET-cellulose bilayer mem-brane keeps flat enough so that bacteria can freely swimin the shallow well (Supplementary Fig. 1d and Supple-mentary Movie 2). The EMPS can realize such obser-vations for a long time with little hydrodynamic pertur-bation by medium flow and no mechanical stress, whichmay exist in a PDMS-based device that holds cells me-chanically. We also check whether the additional PETmembrane may affect the growth condition of cells, byusing E. coli
MG1655 and M9 medium with glucose andamino acids (Glc+a.a.). We find that the doubling timeof the cell population is 59 ±
10 min, which is compa-rable to that in the previous system without the PETmembrane [20–22]. Therefore, our bilayer membrane canstill exchange medium efficiently.We then test the spatial uniformity of the culture con-dition. We design U-shape traps with an open end,for both the EMPS (Fig. 2a) and for the conventionalPDMS-based device (Supplementary Fig. 2a,b). Withthis geometry, nutrients are supplied via diffusion fromthe open end in the PDMS-based system, while nutritiousmedium is directly and uniformly delivered through themembrane above the trap in the EMPS. When we culture
E. coli
MG1655 in M9(Glc+a.a.), the trap is eventuallyfilled with cells, and they exhibit coherent flow toward theopen end due to the cell growth and proliferation (Fig. 2band Supplementary Movie 3,4). To evaluate the unifor-mity of the cell growth, we measure the velocity field ofthe cell flow by particle image velocimetry (PIV) (Sup-plementary Fig. 2c,d). The velocity component along thestream-wise direction (the y axis in Fig. 2b) averagedover the span-wise direction, u ( y ), clearly shows that thevelocity profile is stable for the EMPS over long time pe-riods, while it gradually decreases for the PDMS-baseddevice (Fig. 2c Main Panel and Inset, respectively). Thecell growth rate is then obtained by λ ( y ) = d u d y , whichis shown in Fig. 2d. The result shows that the growthrate λ is indeed uniform and kept constant for the EMPS(Fig. 2d, Main Panel), while for the PDMS device it isheterogeneous, being higher near the open end (Fig. 2d,Inset), and it decreases as time elapses. The growth ratedecays at the distance of roughly 30 µ m from the openend, which is located near y ≈ µ m for the PDMSdevice. This observation is consistent with the nutrientdepletion length we evaluate by following the calculation Drain ObservationareaPET Cellulose a bc d
PDMSEMPS EMPS PDMS
FIG. 2. Cell growth measurements in U-shape traps inEMPS. a Sketch of the design of microchannels. Non-motilecells are trapped in the shallow observation area. Cells in thetrap can escape to the deep drain channel. b Top view of thetrap (30 µ m × µ m, 1 . µ m depth) filled with E. coli
W3110∆fliC ∆flu ∆fimA (see also Supplementary Movie 4). Thescale bar is 25 µ m. Coherent flow of cells driven by cell pro-liferation is directed toward the drain (15 µ m depth). c Thestream-wise ( y ) component of the velocity field averaged overthe span-wise direction (the two-dimensional velocity field isshown in Supplementary Fig. 2c), u , measured in differenttime periods. The data were taken from a single trap. t = 0is the time at which the trap is filled with cells. Error barsrepresent the standard deviation of the ensemble. The openend is located near y ≈ µ m. (Inset) The same quantitymeasured for a PDMS-based device with a similar trap (Sup-plementary Fig. 2a,b,c). The open end is at y ≈ µ m. Timedependence is clearly observed. d The growth rate profiles λ ,evaluated by λ = d u d y , for the EMPS (main panel) and thePDMS-based device (inset). by Mather et al. [17], 32 µ m, for which we used the dif-fusion constant of glucose [23] and the division time of60 min. Such heterogeneity is not seen in the EMPS.While cell growth regulation pathways may also be in-fluenced by such factors as mechanical pressure causedby cell elongation [24–27], quorum sensing [28, 29], etc.,our results indicate that the EMPS can indeed realizea uniform and stable culture condition while the samemedium is kept supplied.Another advantage of the EMPS is that we can alsoswitch the culture condition, by changing the mediumto supply. Here we evaluate how efficiently the mediumin the well is exchanged. In the presence of non-motile E. coli
W3110 ∆fliC ∆flu ∆fimA, we switch the mediumto supply from phosphate buffered saline (PBS) to PBSsolution of rhodamine fluorescent dye, and monitor thefluorescent signal in a cross-section of the device by aconfocal microscope (Supplementary Fig. 3a, from rightto left). The result shows that the medium inside the well is exchanged uniformly in space (Supplementary Fig. 3b)and that it is almost completed within 2-4 min (Supple-mentary Fig. 3d). We also change the medium from PBSwith rhodamine to that without rhodamine (Supplemen-tary Fig. 3a, from left to right). The exchange then tooklonger time, (cid:38) / sec ) induced when switching the medium(Supplementary Movie 5,6). We therefore conclude thatthe EMPS is indeed able to change the growth conditionfor cells under observation uniformly, without noticeablefluid flow perturbations. Characterization of bacterial reductive division byEMPS
Now we observe the reductive division of
E. coli
MG1655 in the EMPS, triggering starvation by switch-ing from nutritious medium to non-nutritious buffer. Inthe beginning, a few cells are trapped in a quasi-two-dimensional well (diameter 55 µ m and depth 0 . µ m) andgrown in nutritious medium, until a microcolony com-posed of approximately 100 cells appear. We then quicklyswitch the medium to a non-nutritious buffer, which iscontinuously supplied until the end of the observation(see Methods for more details). By doing so, we intendto remove various substances secreted by cells, such asautoinducers for quorum sensing and waste products, toreduce their effects on cell growth [28–31]. Throughoutthis experiment, the well is entirely recorded by phasecontrast microscopy. We then measure the length andthe width of all cells in the well, to obtain the volume v ofeach cell by assuming the spherocylindrical shape, at dif-ferent times before and after the medium switch. Here wemainly show the results for the case where the medium isswitched from LB broth to PBS (denoted by LB → PBS)in Fig. 3, while the results for M9(Glc+a.a.) → PBS, andM9 medium with glucose (Glc) → M9 medium with α -methyl-D-glucoside ( α MG), a glucose analog which can-not be metabolized [32], are also presented in Supple-mentary Fig. 4 and S5. We observe that, after switch-ing to the non-nutritious buffer, the growth of the totalvolume decelerates (Supplementary Movie 7-9, Fig. 3b,Supplementary Fig. 4b and Supplementary Fig. 5b), andthe mean cell volume rapidly decays because of excessivecell divisions (Supplementary Movie 7-9, Fig. 3a,c). Thevolume change is mostly due to the decrease of the celllength, while we notice that the mean cell width may also t = 0 min time
LB growth media PBS (no nutrient) ad e f bc
LB → PBS LB → PBS LB → PBS
FIG. 3. 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 PBS at t = 0. See also Supplementary Movie 7. b , c Experimental data (blue symbols) for the total cell volume V tot ( t ) ( b ), the growth rate λ ( t ) ( b , Inset), the mean cell volume 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 entersthe device (black dashed line). The data were collected from 15 wells. d Time evolution of the cell size distributions duringstarvation in the case of LB → PBS at t = 0 , , , , , , , , , , ,
480 min from right to left. The sample sizeis 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 time average of the datasets. f The moment ratio (cid:104) v l (cid:105) / (cid:104) v l − (cid:105) against V ( t ) = (cid:104) v (cid:105) .The error bars were estimated by the bootstrap method with 1000 realizations. The colored lines represent the results of linearregression in the log-log plots (see Supplementary Table 3 for the slope of each line). The black solid lines are guides for eyesindicating unit slope, i.e., proportional relation. a b Glc+a.a. → PBS Glc → αMG
FIG. 4. Rescaled cell size distributions. a The re-sults for M9(Glc+a.a.) → PBS. The dashed line repre-sents the time average of the datasets. The data were takenfrom 17 wells. The sample size ranges from n (0) = 685to n (180) = 1260 (see Supplementary Fig. 4c). (Inset)Time evolution of the non-rescaled cell size distributions at t = 0 , , , , , , , , ,
180 min. b The resultsfor M9(Glc) → M9( α MG). The dashed line represents thetime average of the datasets. The data were taken from26 wells. The sample size ranges from n ( −
5) = 1029 to n (65) = 1591 (see Supplementary Fig. 5c). (Inset) Timeevolution of the non-rescaled cell size distributions at t = − , , , , , , , ,
65 min. change slightly (Supplementary Fig. 7). We consider thatthis is not due to osmotic shock [33], because then thecell width would increase when the osmotic pressure isdecreased, which is contradictory to our observation forLB → PBS (Supplementary Table 1 and SupplementaryFig. 7). Such a change in cell widths was also reported for a transition between two different growth conditions[34]. In any case, Fig. 3d shows how the distribution ofthe cell volumes v , p ( v, t ), changes over time; as the meanvolume decreases, the histograms shift leftward and be-come sharper. However, when we take the ratio v/V ( t ),with V ( t ) = (cid:104) v (cid:105) being the mean cell volume at each time t , and plot vp ( v, t ) instead, we find that all those his-tograms overlap onto a single curve (Fig. 3e). In otherwords, we find that the time-dependent cell size distribu-tion during the reductive division maintains the followingscale-invariant form all the time: p ( v, t ) = v − F ( v/V ( t )) . (2)This is analogous to Eq. (1) previously reported for thesteady growth condition, but here importantly the meanvolume V ( t ) changes over time significantly (Fig. 3c). Tofurther test the scale invariance of the distribution, weplot the moment ratios (cid:104) v j (cid:105) / (cid:104) v j − (cid:105) with j = 2 , , (cid:104) v (cid:105) . This is indeed the rela-tion expected if the distribution satisfies Eq. (2) [1]. Re-markably, for all starving conditions that we test, we findthat the scale invariance robustly appears (Fig. 3f, Sup-plementary Fig. 4d, Supplementary Fig. 5d, and Fig. 4).Our results therefore indicate that the scale invarianceas in Eq. (2), which has been observed for steady con-ditions [1, 2], also holds in non-steady reductive divisionprocesses of E. coli rather robustly.In addition to the robustness of the scaling relation(2), the functional form of the scale-invariant distribu-tion, i.e., that of F ( x ), is of central interest. In fact, wefind that our observations for E. coli are significantly dif-ferent from those for unicellular eukaryotes reported byGiometto et al. [1] (Supplementary Fig. 6a). More pre-cisely, they showed that the rescaled cell size distributionfor unicellular eukaryotes is well fitted by the log-normaldistribution, which corresponds to F ( x ) = 1 √ πσ e − (log x − m ) / σ (3)with σ = 0 . m = − σ / (cid:104) x (cid:105) = 1. We find that our data for E. coli can alsobe fitted by a single log-normal distribution (Supplemen-tary Fig. 6a, yellow dotted line), but here the value of σ , evaluated by the standard deviation of log x , is foundto be σ = 0 . σ = 0 . B. subtilis [35] reported values of σ from 0 .
24 to0 .
26, which are comparable to our results shown in Sup-plementary Table 3. Compared to this substantial differ-ence between bacteria and unicellular eukaryotes, the de-pendence on the environmental factors seems to be muchweaker, as suggested by our observations under three dif-ferent growth conditions (Supplementary Fig. 6a). Note,however, that we also noticed indications of weak depen-dence on the growth condition (Supplementary Fig. 6b).Clarifying the scope of the universality, or specifically,what factors affect the scale-invariant distribution andhow strongly they do, is therefore an important openproblem left for future studies.
Modeling the reductive division process
To obtain theoretical insights on the experimentallyobserved scale invariance of the cell size distributions,we construct a simple cell cycle model for the bacterialreductive division. For the steady growth conditions, alarge number of studies on
E. coli have been carried outto clarify what aspect of cells triggers the division event[8, 9]. Significant advances have been made recently toprovide molecular-level understanding [8, 19, 34, 36–38].Here we extend such a model to describe the starvationprocess.One of the most established models in this contextis the Cooper-Helmstetter (CH) model [18, 39], whichconsists of cellular volume growth and multifork DNAreplication (Fig. 5a). In this model, completion of theDNA replication triggers the cell division, and this givesa homeostatic balance between the DNA amount and thecell volume. An unknown factor of the CH model is howDNA replication is initiated, and a few studies attemptedto fill this gap to complement the CH model [19, 36]. Hoand Amir [36] assumed that replication is initiated when a critical amount of ”initiators” accumulate at the originof replication. In the presence of a constant concentra-tion of autorepressors, expressed together with the initia-tors, this assumption means that the cellular volume in-creases by a fixed amount between two initiation events.In contrast, Wallden et al. [19] proposed another modelnamed the single-cell CH (sCH) model, in which theyconsidered, based on experimental observations, that theinitiation occurs when the cell volume exceeds a giventhreshold independent of the growth rate and the initialvolume of the cell. Albeit not exactly [37], both modelssuccessfully reproduced major characteristics of cell cy-cles in the exponential growth phase, such as the ”adder”principle [9, 40, 41]. For modeling the reductive divisionprocess, we choose to extend the sCH model by Wallden et al. , which requires a smaller number of assumptions toadd to describe time-dependent processes studied here.The model consists of two processes that proceed si-multaneously, namely the volume growth and the intra-cellular replication. The volume of each cell (indexed by i ), v i ( t ), grows as d v i d t = λ ( t ) v i ( t ), with a time-dependentgrowth rate λ ( t ). Following Wallden et al. ’s sCH model[19], we assume that, when the cell volume v i ( t ) exceedsa threshold v th i, given below, DNA replication starts.This initiates the C+D period in the bacterial cell cy-cle [18, 19, 39], in which all processes regarding the celldivision take place. Its progression is represented by a co-ordinate X CD i (t), which starts form zero and increases attime-dependent speed µ ( t ), d X CD i d t = µ ( t ). When X CD i ( t )reaches a threshold X CD , th i , the cell divides (Fig. 5a, left),leaving two daughter cells of volumes v i ( t ) = x sep v i ( t )and v i ( t ) = (1 − x sep ) v i ( t ). Here, x sep is randomly drawnfrom the Gaussian distribution with mean 0 . v i ( t ) reaches the second threshold v th i, beforethis cell divides. Then the replications for the futuredaughter cells, X CD i ( t ) and X CD i ( t ), start and run simul-taneously (Fig. 5a, right and b). This, called the mul-tifork replication [18, 39], is well-known for fast grow-ing bacteria such as E. coli and
B. subtilis . Similarly,at v i ( t ) = v th i,j , the replication for the j th generationis triggered. Following Wallden et al. [19], we assumethat the initiation volume v th i,j takes a constant valueper chromosome, so that typically v th i,j ≈ j − v th i, . Totake into account stochastic nature of division events, v th i,j is generated randomly from the Gaussian distribu-tion with mean (cid:104) v th i,j (cid:105) = 2 j − v thmean and standard de-viation Std[ v th i,j ] = 2 j − v thstd . Similarly, X CD , th i is alsoa Gaussian random variable with (cid:104) X CD , th i (cid:105) = 1 andStd[ X CD , th i ] = X CD , thstd . Here, v thmean , v thstd , X CD , thstd as wellas x sepstd = Std[ x sep ] are considered to be parameters.Now we are left to determine the two time-dependentrates, λ ( t ) and µ ( t ). Here we consider the situation wheregrowth medium is switched to non-nutritious buffer at Ori
Growth StarvationCell volume time abc d
Simulation(LB→PBS)
FIG. 5. Model of reductive division and simulation re-sults. a Single and multifork intracellular replication pro-cesses. Progress of each replication is represented by a co-ordinate X CD i ( t ), which ends at X CD i ( t ) = X CD , th i by trig-gering cell division. In the multifork process, all replica-tions proceed simultaneously at the same rate µ ( t ). b Il-lustration of cell cycles in this model. Each colored arrowrepresents a single intracellular replication process, whichstarts when the cell volume v i ( t ) exceeds an initiation vol-ume threshold v th i,j . c Overlapping of the rescaled cellsize distributions during starvation in the model for LB → PBS. The dashed line represents the time average of thedatasets. (Inset) The non-rescaled cell size distributions at t = 0 , , , , , , , , , , ,
480 min fromright to left. d Numerically measured division rate, B ( v, t ),in the model for LB → PBS. See Supplementary Theoryfor the detailed measurement method. (Inset) Test of thecondition of Eq. (S22). Here B t (0) /B t ( t ) is evaluated by B t (0) /B t ( t ) = (cid:82) B ( xV (0) , dx/ (cid:82) B ( xV ( t ) , t ) dx , with x run-ning in the range 0 ≤ x ≤ .
8. Overlapping of the datademonstrates that Eq. (S22) indeed holds in our model. t = 0; therefore, t denotes time passed since the switchto the non-nutritious condition. First, we set the vol-ume growth rate λ ( t ) on the basis of the Monod equation[42], assuming that substrates in each cell are simply di-luted by volume growth and consumed at a constant rate,without uptake because of the non-nutritious conditionconsidered here. As a result, we obtain λ ( t ) = λ − Ae Ct − A , (4)with constant parameters A and C , and the growth rate λ (= λ (0)) in the exponential growth phase (see Sup-plementary Simulations for details). For the replication speed µ ( t ), or more precisely the progression speed of theC+D period, we first note that the C+D period mainlyconsists of DNA replication, followed by its segregationand the septum formation [39]. Most parts of those pro-cesses involve biochemical reactions of substrates, suchas deoxynucleotide triphosphates for the DNA synthe-sis, and assembly of macromolecules such as FtsZ pro-teins for the septum formation. We therefore considerthat the replication speed is determined by the intra-cellular concentration of relevant substrates and macro-molecules, which is known to decrease in the stationaryphase [43, 44]. By assuming an exponential decay of thesubstrate concentration, together with the Hill equationfor the binding probability of ligands to receptors, weobtain the following equation for the replication speed: µ ( t ) = µ k + 1 k exp( t/τ ) + 1 , (5)with parameters k and τ , and the replication speed µ (= µ (0)) in the growth phase (see Supplementary Sim-ulations).The parameter values are determined from the experi-mentally measured total cell volume and the cell number,which our simulations turn out to reproduce very well(Fig. 3b,c and Supplementary Fig. 4b,c), with the aid ofrelations reported by Wallden et al. [19] for some of theparameters (see Supplementary Table 2 for the parame-ter values used in the simulations, and Methods for theestimation method). With the parameters fixed thereby,we measure the cell size fluctuations at different timesand find the scale invariance similar to that revealedexperimentally (Fig. 5c and Supplementary Fig. 9b,c).The proportionality of the moment ratios is also con-firmed (Supplementary Fig. 9a,d and Supplementary Ta-ble 3). Interestingly, the numerically obtained distribu-tion is found to roughly reproduce the experimental one(Supplementary Fig. 6a), even though no information onthe distribution is used for the parameter adjustment.Note that the scale invariance emerges despite the exis-tence of characteristic scales in the model definition, suchas the typical initiation volume v thmean . This suggests theexistence of a statistical principle underlying the scale in-variance, which is not influenced by details of the modeland the experimental conditions. Theoretical conditions for the scale invariance
To seek for a possible mechanism leading to the scaleinvariance, here we describe, theoretically, the time de-pendence of the cell size distribution in a time-dependentprocess. Suppose N ( v, t ) dv is the number of the cellswhose volume is larger than v and smaller than v + dv .If we assume, for simplicity, that a cell of volume v candivide to two cells of volume v/
2, at probability B ( v, t ),we obtain the following time evolution equation: ∂N ( v, t ) ∂t = − ∂∂v [ λ ( t ) vN ( v, t )] − B ( v, t ) N ( v, t ) + 4 B (2 v, t ) N (2 v, t ) . (6)Note that this equation has been studied by numerouspast studies for understanding stable distributions insteady conditions [1, 45–49], but here we explicitly in-clude the time dependence of the division rate, B ( v, t ),for describing the transient dynamics. To clarify a condi-tion for this equation to have a scale-invariant solution,here we assume the scale invariant form, Eq. (2), where p ( v, t ) = N ( v, t ) /n ( t ) and n ( t ) is the total number of thecells, and obtain the following self-consistent equation(see Supplementary Theory for derivation): F ( x ) = − x ∂F ( x ) ∂x − B ( v, t )¯ B ( t ) F ( x ) + 2 B (2 v, t )¯ B ( t ) F (2 x ) . (7)Here, x = v/V ( t ) and ¯ B ( t ) = (cid:82) dvB ( v, t ) p ( v, t ). For thescale invariance, Eq. (7) should hold at any time t . Thisis fulfilled if B ( v, t ) can be expressed in the following form(see Supplementary Theory): B ( v, t ) = B v ( v/V ( t )) B t ( t ) . (8)This is a sufficient condition for the cell size distributionto maintain the scale invariant form, Eq. (2), during thereductive division. It is important to remark that, asopposed to Eq. (6), Eq. (7) does not include the growthrate λ ( t ) explicitly. The scale-invariant distribution F ( x )is therefore completely characterized by the division rate B ( v, t ) in this framework.To test whether the condition of Eq. (S22) is satis-fied in our model, we measure the division rate B ( v, t )in our simulations (Fig. 5d). The data overlap if B ( v, t ) B t (0) /B t ( t ) is plotted against v/V ( t ), demon-strating that Eq. (S22) indeed holds here. In our model,the replication speed µ ( t ) is assumed to be given com-pletely by the concentration of replication-related sub-strates, which takes the same value for all cells. Thismay be why the separation of variable, Eq. (S22), effec-tively holds, and the scale invariance follows. On theother hand, our theory does not seem to account for thefunctional form F ( x ) of the scale-invariant distribution;the right hand side of Eq. (7) differs significantly fromthe left hand side, if the numerically obtained B ( v, t ) isused together with the function F ( x ) from the simula-tions or the experiments (Supplementary Fig. 10). Thedisagreement did not improve by taking into account theeffect of septum fluctuations. The lack of quantitativeprecision is probably not surprising given the simplicityof the theoretical description, which incorporates all ef-fects of intracellular replication processes into the simpledivision rate function B ( v, t ). The virtue of this theoryis that it clarifies it is the replication process, not the cell body growth rate, that seems to have direct relevance inthe scale invariance and the functional form of the cellsize distribution. The significant difference in F ( x ) iden-tified between bacteria and unicellular eukaryotes (Sup-plementary Fig. 6a) may be originated from the differentreplication mechanisms that the two taxonomic domainsadopt. CONCLUDING REMARKS
In this work, we developed a novel membrane-basedmicrofluidic device that we named the extensive microp-erfusion system (EMPS), which can realize a uniformlycontrolled environment for wide-area observations of mi-crobes. We believe that the EMPS has potential applica-tions in a wide range of problems with dense cellular pop-ulations, including living active matter systems [50, 51]and biofilm growth [52–54]. Here we focused on statis-tical characterizations of single cell morphology duringthe reductive division of
E. coli . Thanks to the EMPS,we recorded the time-dependent distribution of cell sizefluctuations and revealed that the rescaled distributionis scale-invariant and robust against the environmentalchange, despite the decrease of the mean cell size. Thisfinding was successfully reproduced by simulations of amodel based on the sCH model [19], which we proposeas an extension for dealing with time-dependent envi-ronments. We further inspected theoretical mechanismbehind this scale invariance and found the significance ofthe division rate function B ( v, t ). We obtained a suffi-cient condition for the scale invariance, Eq. (S22), whichwas indeed confirmed in our numerical data.After all, our theory suggests that mechanism of in-tracellular replication processes may have direct impacton the scale-invariant distribution, which may accountfor the significant difference we identified between bac-teria and eukaryotes (Supplementary Fig. 6a). Since thenumber of species studied in each taxonomic domain israther limited ( E. coli (this work) and
B. subtilis [35]for bacteria, 13 protist species for eukaryotes [1]), it isof crucial importance to test the distribution trend fur-ther in each taxonomic domain, and to clarify how andto what extent the cell size distribution is determinedby the intracellular replication dynamics. We also notethat the scale-invariant distribution F ( x ) might dependweakly on the culture condition in the exponential growthphase (Supplementary Fig. 6b). If this dependence on thegrowth condition is significant, we expect that conditionsfor the scale invariance are not met when switching be-tween different growth environments. Investigations ofcell size fluctuations in such cases, both experimentallyand theoretically, will be an important step toward clar-ifying the scope of the universality of the scale-invariantcell size distributions. Combining with other theoreticalmethods, such as models considering the cellular age [55]and renormalization group approaches for living cell tis-sues [56], may be useful in this context. The influenceof cell-to-cell interactions, e.g., quorum sensing [28, 29],may also be important. We hope that our understandingof the population-level response against nutrient starva-tion will be further refined by future experimental andtheoretical investigations. METHODSStrains and culture media
We used wild-type
E. coli strains (MG1655 and RP437)and a mutant strain (W3110 ∆fliC ∆flu ∆fimA) in thisstudy. Culture media and buffer are listed in Supplemen-tary Table 1. The osmotic pressure of each medium wasmeasured by the freezing-point depression method by theOSMOMAT 030 (Genotec, Berlin Germany). Details onthe strains and culture conditions in each experiment areprovided below.
Fabrication of the PDMS-based device
We prepared PDMS-based microfluidic devices by fol-lowing the method reported in Ref.[57]. We adopted amicrochannel geometry similar to that in Ref.[17], whichconsists of deep drain channels and shallow U-shape traps(Supplementary Fig. 2a,b). In our setup, the drain chan-nels are 25 µ m deep, and the U-shape traps are 30 µ mwidth, 70-90 µ m long, and 1 . µ m deep. Tygon tubesof 1/16” outer diameter were connected to the inlets andthe outlets (Supplementary Fig. 2b) via steel tubes (Elve-flow, Paris France). The width of the drains (see Sup-plementary Fig. 2b) and the length of tygon tubes wereadjusted to realize the desired flow rate in the drain nearthe U-shape traps. The tube length was 55 cm for themedium inlet, 40 cm for the medium outlet, and 30 cmfor the waste outlet. The cell inlet was connected to asyringe filled with bacterial suspension via a 10 cm tube,to inject cells at the beginning of observation. After theintrusion of cells, the syringe with the suspension wasfixed, and no flow was generated around the cell inlet. Fabrication of the EMPS
The EMPS consists of a microfabricated glass cov-erslip, a bilayer porous membrane and a PDMS pad.The microfabricated coverslip and the PDMS pad wereprepared according to ref. [20, 22]. We fabricatedthe bilayer porous membrane by combining a strepta-vidin decorated cellulose membrane and a biotin deco-rated polyethylene-terephthalate (PET) membrane. The streptavidin decoration of the cellulose membrane (Spec-tra/Por 7, Repligen, Waltham Massachusetts, molecu-lar weight cut-off 25000) was realized by the method de-scribed in ref. [20, 22]. The PET membrane (Transwell3450, Corning, Corning New York, nominal pore size0 . µ m) was decorated with biotin as follows. We soakeda PET membrane in 1 wt% solution of 3-(2-aminoethylaminopropyl) trimethoxysilane (Shinetsu Kagaku Kogyo,Tokyo Japan) for 45 min, dried it at 125 ◦ C for 25 minand washed it by ultrasonic cleaning in Milli-Q water for5 min. This preprocessed PET membrane was stored ina desiccator at room temperature, until it was used toassemble the EMPS.The EMPS was assembled as follows. The prepro-cessed PET membrane was cut into 5 mm × µ l droplet of bacterialsuspension was inoculated on a biotin decorated cover-slip (see also details below). We then took the bilayermembrane from the agar pad, air-dried for tens of sec-onds, and carefully put on the coverslip on top of thebacterial suspension. The bilayer membrane was thenattached to the coverslip via streptavidin-biotin bindingas shown in Supplementary Fig. 1b. We then air-driedthe membrane for a minute and attached a PDMS padon the coverslip by a double-sided tape. Observation of motile
E. coli in the EMPS
We used a wild-type motile strain of
E. coli , RP437.First, we inoculated the strain from a glycerol stock into2 ml TB medium (see Supplementary Table 1 for com-ponents) in a test tube. After shaking it overnight at37 ◦ C, we transferred 20 µ l of the incubated suspensionto 2 ml fresh TB medium and cultured it until the opti-cal density (OD) at 600 nm wavelength reached 0 . . . µ m diameter and1 . µ m depth. After the assembly of the device withthe bacterial suspension, it was fixed on the microscopestage inside an incubation box maintained at 37 ◦ C. Themicroscope we used was Leica DMi8, equipped with a63x (N.A. 1.30) oil immersion objective and operated byLeica LasX. To fill the device with medium, we injectedfresh TB medium stored at 37 ◦ C from the inlet (Sup-plementary Fig. 1), at the rate of 60 ml / hr for 5 minby a syringe pump (NE-1000, New Era Pump Systems,Farmingdale New York).During the observation, TB medium was constantlysupplied from the inlet at the rate of 2 ml / hr (flow speedabove the membrane was approximately 0 . / sec) bythe syringe pump. Cells were observed by phase con-trast microscopy and recorded at the time interval of30 sec for the cellulose-membrane device (SupplementaryFig. 1c) and 118 msec for the EMPS (SupplementaryFig. 1d). The time interval for the former was long, be-cause cells hardly moved in this case (see SupplementaryMovie 1). We checked that the EMPS can realize quasi-two-dimensional space in which bacteria can freely swim,even for large wells, at least up to 210 µ m in diameter,while we have not investigated the reachable largest di-ameter. Cell growth measurement in U-shape traps in thePDMS-based device
We used a non-motile mutant strain W3110 withoutflagella and pili (∆fliC ∆flu ∆fimA) to prevent cell ad-hesion to the surface of a coverslip. Before the time-lapse observation, we inoculated the strain from a glyc-erol stock into 2 ml M9 medium with glucose and aminoacids (Glc+a.a.) (see also Supplementary Table 1) in atest tube. After shaking it overnight at 37 ◦ C, we trans-ferred 20 µ l of the incubated suspension to 2 ml freshM9(Glc+a.a.) medium and cultured it until the OD at600 nm wavelength reached 0 . .
5. We then injectedthe bacterial suspension into the device from the cell in-let (Supplementary Fig. 2b) and left it until a few cellsentered the U-shape traps. The device was placed onthe microscope stage, in the incubation box maintainedat 37 ◦ C. The microscope we used was Leica DMi8,equipped with a 63x (N.A. 1.30) oil immersion objectiveand operated by Leica LasX.During the observation, we constantly suppliedM9(Glc+a.a.) medium and 0 . . / hr (flow speed in the drainwas approximately 1 mm / sec). Cells were observed byphase contrast microscopy and recorded at the time in-terval of 3 min. The velocity field of the coherent flowwas obtained by particle image velocimetry, using Mat-PIV (MATLAB toolbox). The stream-wise componentof the velocity field (Fig. 2c) was averaged over the span-wise direction, and also over the time period of 150 min. Cell growth measurement in U-shape traps in theEMPS
Here the choice of the strain, the medium, the cul-ture condition and the instruments was the same as those for the measurement with the PDMS-based device, un-less otherwise stipulated. Here, the cultured bacterialsuspension was diluted to OD = 0 .
04 before it was in-oculated on the coverslip of the EMPS. As sketchedin Fig. 2A, the substrate consisted of drain channels(100 µ m wide, 7 mm long, 13 µ m deep) and U-shapetraps (30 µ m wide, 80 µ m long, 1 . µ m deep), which wereprepared by the methods described in ref. [22]. When thebilayer membrane was attached to the substrate, care wastaken not to cover the two ends of the drain channel touse, so that cells in the drain could escape from it. Afterthe assembly of the device with the bacterial suspension,it was fixed on the microscope stage inside the incubationbox maintained at 37 ◦ C. To fill the device with medium,we injected fresh medium stored at 37 ◦ C from the inlet(Supplementary Fig. 1), at the rate of 60 ml / hr for 5 minby a syringe pump (NE-1000, New Era Pump Systems).During the observation, the flow rate of theM9(Glc+a.a. and BSA) medium was set to be 2 ml / hr(approximately 0 . / sec above the membrane), ex-cept that it was increased to 60 ml / hr (approximately6 mm / sec above the membrane) at a constant interval,in order to remove the cells expelled from the trap effi-ciently. The time interval of this flush was 60 min beforethe observed trap was filled with cells, and 30 min there-after. Evaluation of the rhodamine exchange efficiency inthe EMPS
We used a non-motile mutant strain W3110 ∆fliC ∆flu∆fimA. Before the time-lapse observation, we inoculatedthe strain from a glycerol stock into 2 ml M9(Glc+a.a.)medium in a test tube. After shaking it overnight at37 ◦ C, we transferred 20 µ l of the incubated suspensionto 2 ml fresh M9(Glc+a.a.) medium and cultured it untilthe OD at 600 nm wavelength reached 0 . .
5. The bac-terial suspension was finally diluted to OD = 0 . µ M rhodamine to adsorb fluorescentdye on the surface of the coverslip, then removed non-adsorbed rhodamine molecules by injecting a pure PBSbuffer. We repeated this medium exchange several times.For the observation, we used a laser-scanning confo-cal microscope, Nikon Ti2, equipped with a 100x (N.A.1.49) oil immersion objective and operated by Nikon NIS-Elements. The resolution along the Z-axis was 0 . µ m,and cross-sectional images were taken over a height of10 µ m, at the time interval of 1 . µ M rhodamine,at the flow rate of 6 mm / sec (approximately 60 ml / hrabove the membrane) (Supplementary Fig. 3b,d). The0medium switch was performed by exchanging the syringeconnected to the device. After this observation, followedby a time interval of a few minutes without flow of thesolution in the device, we started to monitor the fluores-cent intensity while we switched the medium from thePBS with rhodamine to that without rhodamine, at theflow rate of 6 mm / sec (Supplementary Fig. 3c,e). Observation of the bacterial reductive division
We used a wild-type strain MG1655. Before the time-lapse observation, we inoculated the strain from a glyc-erol stock into 2 ml growth medium in a test tube. Thesame medium as for the main observation was used (LBbroth, M9(Glc+a.a.) or M9(Glc)). After shaking itovernight at 37 ◦ C, we transferred 20 µ l of the incubatedsuspension to 2 ml fresh medium and cultured it until theOD at 600 nm wavelength reached 0 . .
5. The bacterialsuspension was finally diluted to OD = 0 .
05 before it wasinoculated on the coverslip.For this experiment, we used a substrate with wells of55 µ m diameter and 0 . µ m depth. The well diameterwas chosen so that all cells in the well can be recorded.The device was placed on the microscope stage, in theincubation box maintained at 37 ◦ C. The microscope weused was Leica DMi8, equipped with a 100x (N.A. 1.30)oil immersion objective and operated by Leica LasX. Tofill the device with growth medium, we injected freshmedium stored at 37 ◦ C from the inlet (SupplementaryFig. 1), at the rate of 60 ml / hr for 5 min by a syringepump (NE-1000, New Era Pump Systems).In the beginning of the observation, growth mediumwas constantly supplied at the rate of 2 ml / hr (flowspeed approximately 0 . / sec above the membrane).When a microcolony composed of approximately 100 cellsappeared, we quickly switched the medium to a non-nutritious buffer (PBS or M9 medium with α -methyl-D-glucoside ( α MG), see Supplementary Table 3) storedat 37 ◦ C, by exchanging the syringe. The flow rate wasset to be 60 ml / hr for the first 5 minutes, then returnedto 2 ml / hr. Throughout the experiment, the device andthe media were always in the microscope incubation box,maintained at 37 ◦ C. Cells were observed by phase con-trast microscopy and recorded at the time interval of5 min.The cell volumes were evaluated as follows. We de-termined the major axis and the minor axis of each cell,manually, by using a painting software. By measuring theaxis lengths, we obtained the set of the lengths L and thewidths w for all cells. We then estimated the volume v of each cell, by assuming the spherocylindrical shape ofthe cell: v = π (cid:0) w (cid:1) + π (cid:0) w (cid:1) ( L − w ). We estimatedthe uncertainty in manual segmentation at ± . µ m. Simulations
Details on the derivation of the functional forms of λ ( t ) and µ ( t ) are provided in Supplementary Simulations.The parameters were evaluated as follows. First, fromthe observations of the exponential growth phase, we de-termined the growth rate λ directly. This allowed usto set the replication speed µ too, by using the rela-tion µ − (cid:39) (1 . λ − . + 42) proposed by Wallden et al. [19] (the units of λ and µ are min − ). Concerning thevolume threshold for initiating the replication, we foundsuch a value of v thmean that reproduced the experimentallyobserved mean cell volume in the growth phase. Thestandard deviation v thstd was set to be 10% of the mean v thmean , based on the relation found by Wallden et al. [19].They also measured the fluctuations of the time length ofthe C+D period; this led us to estimate X CD , thstd at 5% of (cid:104) X CD , th (cid:105) , i.e., X CD , thstd = 0 .
05. On the septum positions,we measured their fluctuations and found little differ-ence in x sepstd among the different growth conditions weused, and also in the non-nutritious case (SupplementaryFig. 8). We therefore used a single value x sepstd = 0 . λ ( t ) can be de-termined independently of the cell divisions, becausethe total volume V tot ( t ) = (cid:80) i v i ( t ) grows as V tot ( t ) = V tot (0) exp( (cid:82) t λ ( t ) dt ). With λ ( t ) given by Eq. (4), wecompared V tot ( t ) with experimental data and determinedthe values of A and C (Fig. 4c). Finally, only k and τ inEq. (5) remained as free parameters. We tuned them sothat the mean cell volume V ( t ) and the number of thecells n ( t ) observed in the simulations reproduced thosefrom the experiments (Fig. 4d).The parameter values determined thereby are summa-rized in Supplementary Table 2, for the simulations forLS → PBS and M9(Glc+a.a.) → PBS. Note that we didnot perform simulations for M9(Glc) → M9( α MG), be-cause we found it difficult to fit the experimental datafor V tot ( t ) and n ( t ) in this case (Supplementary Fig. 5).This suggests that the consequence of the replacementof glucose by glucose analog α MG may not be simplestarvation.We started the simulations from 10 cells with volumesin the range of 0 . . µ m , randomly generated fromthe uniform distribution. The cells grew in the expo-nential phase (with the constant growth rate λ and thereplication speed µ ) until the number of cells reached100,000. We then randomly picked up 10 cells from this“precultured” sample and used them as the initial pop-ulation of each simulation. Thus, the cell cycles of thecells were sufficiently mixed.1 ACKNOWLEDGMENTS
We acknowledge discussions with H. Chat´e, Y. Fu-ruta, H. Nakaoka, T. Hiraiwa, Y. Kitahara and D.Nishiguchi. We thank I. Naguro for letting us use theOSMOMAT 030. This work is supported by KAKENHIfrom Japan Society for the Promotion of Science (JSPS)(No. 16H04033, No. 19H05800), a Grant-in-Aid forJSPS Fellows (No. 20J10682) and by the grants asso-ciated with the “Planting Seeds for Research” programand Suematsu Award from Tokyo Tech.
COMPETING INTERESTS
The authors declare that no competing interests exist.
AUTHOR CONTRIBUTIONS
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I. SUPPLEMENTARY SIMULATIONS:DERIVATION OF THE FUNCTIONAL FORMS OF λ ( t ) AND µ ( t ) Here we describe how we obtained the functional forms of λ ( t ) and µ ( t ), Eqs. (4) and (5), which were used to defineour sCH model for the bacterial reductive division. We consider the situation where growth medium is switched tonon-nutritious buffer at t = 0; therefore, t denotes time passed since the switch to the non-nutritious condition. First,on the basis of the Monod equation [42], we assume that the growth rate λ ( t ) can be expressed as a function of theconcentration of substrates, S ( t ), inside each cell: λ ( t ) = A (cid:48) S ( t ) B (cid:48) + S ( t ) , (S1)where A (cid:48) and B (cid:48) are constant coefficients. We consider that substrates in each cell are simply diluted by volumegrowth and consumed at a constant rate C . Therefore, dS ( t ) dt = − λ ( t ) S ( t ) − CS ( t ) . (S2)Note that there is no uptake of substrates because of the non-nutritious condition considered here. By combiningEq. (S1) and Eq. (S2), we obtain the following differential equation for S ( t ): dS ( t ) dt = − S ( t )( A (cid:48) S ( t ) + CB (cid:48) + CS ( t )) B (cid:48) + S ( t ) . (S3)We can solve this equation and obtain S ( t ) = DCB (cid:48) e Ct − D ( A (cid:48) + B (cid:48) ) , (S4)where D is a constant of integration. Substituting it to Eq. (S1), we obtain λ ( t ) = DCA (cid:48) e Ct − D ( C − A (cid:48) − B (cid:48) )= λ − Ae Ct − A , (S5)with A = D ( C − A (cid:48) − B (cid:48) ) and λ = λ (0) = DCA (cid:48) / (1 − D ( C − A (cid:48) − B (cid:48) )) being the growth rate in the growth phase,before the onset of starvation.Next, we determine the functional form of the replication speed µ ( t ), more precisely the progression speed of theC+D period. We first note that the C+D period mainly consists of DNA replication, followed by its segregationand the septum formation [39]. Most parts of those processes involve biochemical reactions of substrates, such asdeoxynucleotide triphosphates for the DNA synthesis, and assembly of macromolecules such as FtsZ proteins for theseptum formation. We therefore consider that the replication speed is determined by the intracellular concentrationof relevant substrates and macromolecules. Here we simply assume that the progression of the C+D period can berepresented by assembly-like processes of relevant molecules, represented collectively by RM C+D . We then considerthat the progression speed µ ( t ) is given through the Hill equation, which usually describes the binding probability ofa receptor and a ligand, with cooperative effect taken into account. Specifically, µ ( t ) ∝ [RM C+D ] n K n + [RM C+D ] n , (S6)where n is the Hill coefficient, and K is the equilibrium constant of the (collective) assembly process. One can evaluatethe time evolution of the concentration [RM C+D ] in the same way as for S ( t ). With a constant consumption rate E ,the differential equation can be written as d [RM C+D ] dt = − λ ( t )[RM C+D ] − E [RM C+D ]= − λ − Ae Ct − A [RM C+D ] − E [RM C+D ] . (S7)4The solution to this equation is [RM C+D ] = c (cid:48) e − Ht e Ct − A , (S8)with a constant c (cid:48) and H = E − λ (1 − A ) /A . To reduce the number of the parameters in the model, we simplyassume that [RM C+D ] exponentially decreases during starvation, i.e. ,[RM
C+D ] ≈ c exp( − t/T ) , (S9)with initial concentration c and the degradation time scale τ . As a result, we obtain µ ( t ) = µ k + 1 k exp( t/τ ) + 1 , (S10)with k := ( K/c ) n , τ = T /n , and µ (= µ (0)) being the replication speed in the growth phase before the onset ofstarvation.5 II. SUPPLEMENTARY THEORYTheoretical conditions for the scale invariance
Here, we describe the detailed derivation of the sufficient condition for the scale invariance of the cell size distribution.We start from the time evolution equation, Eq. (6): ∂N ( v, t ) ∂t = − ∂∂v [ λ ( t ) vN ( v, t )] − B ( v, t ) N ( v, t ) + 4 B (2 v, t ) N (2 v, t ) , (S11)where λ ( t ) is the growth rate and B ( v, t ) is the division rate function (see the main article for its definition). We assumethe scale invariance in the form of Eq. (2) for the cell size distribution p ( v, t ) = N ( v, t ) /n ( t ), where n ( t ) = (cid:82) ∞ N ( v, t ) dv is the total number of the cells. For N ( v, t ), this reads N ( v, t ) = n ( t ) v F (cid:18) vV ( t ) (cid:19) , (S12)where V ( t ) is the mean cell volume at time t , V ( t ) = (cid:104) v (cid:105) . One can then rewrite the time evolution equation (S11) interms of the function F ( x ) with x = v/V ( t ), as follows: ∂n ( t ) ∂t v F ( x ) − n ( t ) V ( t ) ∂V ( t ) ∂t ∂F ( x ) ∂x = − λ ( t ) n ( t ) 1 V ( t ) ∂F ( x ) ∂x − B ( v, t ) n ( t ) v F ( x ) + 4 B (2 v, t ) n ( t )2 v F (2 x ) . (S13)If we neglect cell-to-cell fluctuations of the growth rate, one can evaluate λ ( t ) from the total biomass growth, asfollows: λ ( t ) = d ( V ( t ) n ( t )) dt ( V ( t ) n ( t )) − = (cid:18) V ( t ) ∂n ( t ) ∂t + n ( t ) ∂V ( t ) ∂t (cid:19) ( n ( t ) V ( t )) − = 1 n ( t ) ∂n ( t ) ∂t + 1 V ( t ) ∂V ( t ) ∂t . (S14)From Eqs. (S13) and (S14), we obtain ∂n ( t ) ∂t v F ( x ) = − V ( t ) ∂n ( t ) ∂t ∂F ( x ) ∂x − B ( v, t ) v n ( t ) F ( x ) + 4 B (2 v, t )2 v n ( t ) F (2 x ) . (S15)Now, the time derivative of n ( t ) can be calculated as ∂n ( t ) ∂t = (cid:90) ∞ dv ∂N ( v, t ) ∂t = − (cid:90) ∞ dvB ( v, t ) N ( v, t ) + 4 (cid:90) ∞ dv (cid:48) B (2 v (cid:48) , t ) N (2 v (cid:48) , t )= n ( t ) (cid:90) ∞ dv B ( v, t ) v F (cid:18) vV ( t ) (cid:19) = n ( t ) B ( t ) , (S16)where B ( t ) = (cid:82) ∞ dvB ( v, t ) p ( v, t ) = (cid:82) ∞ dvB ( v, t ) v − F ( v/V ( t )). Substituting it to Eq. (S15), we finally obtain thefollowing self-consistent equation for F ( x ) (Eq. (7)): F ( x ) = − x ∂F ( x ) ∂x − B ( v, t ) B ( t ) F ( x ) + 2 B (2 v, t ) B ( t ) F (2 x ) . (S17)6For the scale invariance, Eq. (S17) should hold at any time t . In other words, the coefficient B ( v, t ) /B ( t ) shouldbe independent of both t and V ( t ). Note here that B ( v, t ) /B ( t ) can be rewritten as B ( t ) B ( v, t ) = (cid:90) ∞ dv (cid:48) B ( v (cid:48) , t ) B ( v, t ) 1 v (cid:48) F (cid:18) v (cid:48) V ( t ) (cid:19) = (cid:90) ∞ dx (cid:48) B ( x (cid:48) V ( t ) , t ) B ( xV ( t ) , t ) 1 x (cid:48) F ( x (cid:48) ) . (S18)Therefore, B ( v, t ) /B ( t ) is time-independent if B ( x (cid:48) V ( t ) ,t ) B ( xV ( t ) ,t ) does not depend on V ( t ), being a function of two dimen-sionless variables x and x (cid:48) only, as follows: B ( x (cid:48) V ( t ) , t ) B ( xV ( t ) , t ) = β ( x (cid:48) , x ) . (S19)This condition can be rewritten as follows. For a constant x , we can define B t ( t ) by B ( x V ( t ) , t ) = B t ( t ) (S20)and B v ( x ) by β ( x, x ) = B v ( x ) . (S21)Then, the division rate can be expressed as B ( xV ( t ) , t ) = B v ( x ) B t ( t ) for any x and t . This gives the sufficientcondition we presented in the main text, Eq. (8), B ( v, t ) = B v (cid:18) vV ( t ) (cid:19) B t ( t ) . (S22) Test of the derived conditions for B ( v, t ) and F ( x ) We tested the sufficient condition for B ( v, t ) (Eq. (S22)) and the resulting self-consistent equation for F ( x )(Eq. (S17)) with numerical data we obtained from our sCH model (Fig. 5d). We evaluated the division rate B ( v, t )in the simulations for LB → PBS by B ( v, t ) = v and v + ∆ v , between time t and t + ∆ t v and v + ∆ v at time t . (S23)Here, ∆ v was set to be approximately 0 . × V ( t ) and ∆ t to be approximately 20-30 min for each time point, respectively.The value of B ( v,
0) was determined by counting all division events in the exponential growth phase ( t < B t (0) /B t ( t ) can be evaluated by B t (0) /B t ( t ) = (cid:82) B ( xV (0) , dx/ (cid:82) B ( xV ( t ) , t ) dx . We found that the curves B ( v, t ) B t (0) /B t ( t ) taken at different t overlap reasonably well (Fig. 5d, Inset), which support the variable separabilitycondition of the division rate, Eq. (S22).Given the functional form of B ( v, t ) that we obtained numerically, we can also test the self-consistent equationfor F ( x ), Eq. (S17). Here we remind that the ratio B ( v, t ) /B ( t ) in Eq. (S17) can be expressed as Eq. (S18), and B ( x (cid:48) V ( t ) ,t ) B ( xV ( t ) ,t ) is time-independent (cf. Eq. (S19)). Therefore, B ( t ) B ( v, t ) = (cid:90) ∞ dx (cid:48) B ( x (cid:48) V (0) , B ( xV (0) ,
0) 1 x (cid:48) F ( x (cid:48) ) (S24)with B ( xV (0) ,
0) = B ( v,
0) = B ( v, t ) B t (0) /B t ( t ). Since we already confirmed the time independence of B ( v, t ) B t (0) /B t ( t ) (Fig. 5d, Inset), we took the average of this quantity obtained at t = 0 , , ,
90 min. Sincethe observed range of v is finite and F ( x ) almost vanishes for x (cid:38)
2, we evaluated the integral over x (cid:48) in the range0 ≤ x (cid:48) ≤ B ( t ) /B ( v, t ) evaluated thereby, we substituted F ( x ) obtained by the simulations for LB → PBS to Eq. (S17)(Supplementary Fig. 10a), using the time average of F ( x ) (the dashed line in Fig. 5c) and F ( x ) = 0 for x ≥
2. We alsotested Eq. (S17) with F ( x ) obtained in the experiment for LB → PBS, in the same way as for the model (SupplementaryFig. 10b). In both cases, the right-hand side (rhs) of Eq. (S17) differs significantly from the observed form of F ( x ).7 Theory with septum fluctuations
As a possible improvement of our theory, here we take into account septum fluctuations. We define the kernelfunction q ( v | ν ) which represents the probability that a mother cell of volume ν produces daughter cells of volume v and ν − v (therefore, q ( v | ν ) = q ( ν − v | ν )). The time evolution equation of N ( v, t ) is then written as ∂N ( v, t ) ∂t = − ∂ ( λ ( t ) vN ( v, t )) ∂v − B ( v, t ) N ( v, t ) + (cid:90) ∞ v q ( v | ν ) B ( ν, t ) N ( ν, t ) νv dν. (S25)Now, assuming that the fluctuations of the septum position are Gaussian for simplicity, we have q ( v | ν ) = 2 × (cid:115) π ( ξν ) exp (cid:18) − ( v − ν/ ξν ) (cid:19) , (S26)where the coefficient 2 corresponds to the two daughter cells produced from a single division event. Here, following ex-perimental observations (Supplementary Fig. 8, see also [41]), we assumed that the standard deviation is proportionalto the volume of the mother cell, ν , with coefficient ξ = 0 . F ( x ) as follows. One can again calculate thetime derivative of n ( t ) as ∂n ( t ) ∂t = (cid:90) ∞ dv ∂N ( v, t ) ∂t = − n ( t ) (cid:90) ∞ dvB ( v, t ) p ( v, t ) + n ( t ) (cid:114) πξ (cid:90) ∞ dv (cid:48) (cid:90) ∞ v (cid:48) dν B ( ν, t ) v (cid:48) p ( ν, t ) exp (cid:18) − ( v (cid:48) /ν − / ξ (cid:19) = n ( t ) B ( t ) , (S27)where B ( t ) is defined by B ( t ) = − (cid:90) ∞ dvB ( v, t ) p ( v, t ) + (cid:114) πξ (cid:90) ∞ dv (cid:48) (cid:90) ∞ v (cid:48) dν B ( ν, t ) v (cid:48) p ( ν, t ) exp (cid:18) − ( v (cid:48) /ν − / ξ (cid:19) . (S28)With x = v/V ( t ) and y = ν/V ( t ), one can obtain the self-consistent equation for F ( x ) as F ( x ) = − x ∂F ( x ) ∂x − B ( V ( t ) x, t ) B ( t ) F ( x ) + (cid:114) πξ (cid:90) ∞ x dy B ( V ( t ) y, t ) B ( t ) y − exp (cid:18) − ( x/y − / ξ (cid:19) F ( y ) . (S29)with B ( t ) B ( V ( t ) x, t ) = (cid:90) ∞ dx (cid:48) (cid:18) − B v ( x (cid:48) ) B v ( x ) F ( x (cid:48) ) x (cid:48) + (cid:114) πξ x (cid:48) (cid:90) ∞ x (cid:48) dy (cid:48) B v ( y (cid:48) ) B v ( x ) exp (cid:18) − ( x (cid:48) /y (cid:48) − / ξ (cid:19) F ( y (cid:48) ) y (cid:48) (cid:19) , (S30) B ( t ) B ( V ( t ) y, t ) = (cid:90) ∞ dx (cid:48) (cid:18) − B v ( x (cid:48) ) B v ( y ) F ( x (cid:48) ) x (cid:48) + (cid:114) πξ x (cid:48) (cid:90) ∞ x (cid:48) dy (cid:48) B v ( y (cid:48) ) B v ( y ) exp (cid:18) − ( x (cid:48) /y (cid:48) − / ξ (cid:19) F ( y (cid:48) ) y (cid:48) (cid:19) . (S31)We tested this for both experimentally and numerically observed F ( x ), but the modification did not improve theresults (Supplementary Fig. 10).8 III. SUPPLEMENTARY TABLES AND FIGURES
Supplementary Table 1. Culture conditions, ingredients and osmotic pressure.Medium(Osmotic pressure [Osm/kg]) Ingredients ConcentrationLB broth (Millar) Tryptone 1 wt%(0.38) Sodium Chloride 1 wt%Yeast extract 0.5 wt%M9(Glc+a.a.) medium Disodium Phosphate (Anhydrous) 0.68 wt%(0.21) Monopotassium Phosphate 0.3 wt%Sodium Chloride 0.05 wt%Ammonium Chloride 0.1 wt%Magnesium Phosphate 2 mMCalcium Chloride 0.1 mMGlucose 0.2 wt%MEM Amino Acids solution (M5550, Sigma) 1 wt%M9(Glc) medium Disodium Phosphate (Anhydrous) 0.68 wt%(0.22) Monopotassium Phosphate 0.3 wt%Sodium Chloride 0.05 wt%Ammonium Chloride 0.1 wt%Magnesium Phosphate 2 mMCalcium Chloride 0.1 mMGlucose 0.2 wt%M9( α MG) medium Disodium Phosphate (Anhydrous) 0.68 wt%(0.21) Monopotassium Phosphate 0.3 wt%Sodium Chloride 0.05 wt%Ammonium Chloride 0.1 wt%Magnesium Phosphate 2 mMCalcium Chloride 0.1 mMAlpha-methyl-D-glucopyranoside. 0.2 wt%TB medium Tryptone 1 wt%(0.21) Sodium Chloride 0.5 wt%PBS(-) Potassium Dihydrogenphosphate 0.02 wt%(0.27) Disodium phosphate (Anhydrous) 0.115 wt%Potassium Chloride 0.02 wt%Sodium Chloride 0.8 wt% Supplementary Table 2. Parameters used for the simulations. The method of parameter determination is described in Materialsand Methods in the main article. Parameters LB → PBS M9(Glc+a.a.) → PBSParameters on λ (0) = λ .
029 min − .
010 min − the exponential growth phase µ (0) − = µ − . λ − . + 42 (cid:39)
67 min 1 . λ − . + 42 (cid:39)
104 min v thmean . µ m . µ m v thstd . × v thmean = 0 . µ m . × v thmean = 0 . µ m X CD , thstd . × (cid:104) X CD , th i (cid:105) = 0 .
05 0 . × (cid:104) X CD , th i (cid:105) = 0 . x sepstd . . λ ( t ) = λ − Ae Ct − A A = 0 . C = 0 . − A = 0 . C = 0 .
011 min − µ ( t ) = µ k + 1 ke t/τ + 1 k = 10, τ = 75 min k = 0 . τ = 24 minSupplementary Table 3. Statistical data on the size distributions obtained by the experiments and the simulations. Theexponents α and the coefficients of determination R are obtained by fitting (cid:104) v j (cid:105) / (cid:104) v j − (cid:105) = c (cid:104) v (cid:105) α in the corresponding log-logplots (Fig. 3f, Supplementary Fig. 4d, Supplementary Fig. 5d and Supplementary Fig. 9a,d). The standard deviation parameter σ of the log-normal distribution is defined in Eq.(3) in the main text.Slope ( β ) R valuesMedia (cid:104) v (cid:105) / (cid:104) v (cid:105) (cid:104) v (cid:105) / (cid:104) v (cid:105) (cid:104) v (cid:105) / (cid:104) v (cid:105) (cid:104) v (cid:105) / (cid:104) v (cid:105) (cid:104) v (cid:105) / (cid:104) v (cid:105) (cid:104) v (cid:105) / (cid:104) v (cid:105) σ LB → PBS 0.99(4) 0.97(5) 0.88(8) 0.999 0.994 0.990 0.34(1)M9(Glc+a.a.) → PBS 0.97(3) 0.96(4) 0.95(5) 1.000 0.998 0.993 0.29(2)M9(Glc) → M9( α MG) 1.00(4) 1.02(4) 1.10(5) 1.000 0.999 0.996 0.28(1)Simulation (LB → PBS) 1.01(3) 1.02(7) 1.02(10) 1.000 1.000 0.999 0.25(1)Simulation (M9(Glc+a.a.) → PBS) 0.98(2) 0.96(3) 0.95(4) 1.000 1.000 0.999 0.22(1) a b CoverslipCellulosemembraneStreptavidinBiotinPET membrane c d
Supplementary Fig. 1. Supplementary figures on the setup of the EMPS. a Photograph of the device mounted on the microscopestage. The coverslip and tubes are fixed to the stage by mending tapes. b Illustration of the chemical bonding between abilayer membrane and a glass coverslip. A streptavidin-decorated cellulose membrane is sandwiched by a biotin-coated PETmembrane and coverslip. c (top) Sketch of growth of bacterial cells inside a circular well covered only by a cellulose membrane.Since the membrane bends and presses cell beneath, cells do not swim but form clusters, extending toward the wall. (bottom)Photograph of motile E. coli
RP437 in a well (diameter 110 µ m, depth 1 . µ m) covered only by a cellulose membrane. TBmedium was constantly supplied at 37 ◦ C (see Materials and Methods for details). Despite the motility, cells were confinednear the wall and unable to swim freely (see also Supplementary movie 1). The scale bar is 30 µ m. d (top) Sketch of growthof cells inside a well covered by a PET-cellulose bilayer membrane. The rigid bilayer membrane is sustained without bending,leaving a sufficient gap beneath for cells. (bottom) Photograph of motile E. coli
RP437 in a well (same diameter and depthas in ( c , bottom) covered by a bilayer membrane. Cells were able to swim freely (see also Supplementary movie 2). Growthconditions are same as in ( c , bottom). c d Flow D r a i n Observationarea a Medium InletMedium Outlet Waste OutletCell Inlet500 µm µm depth µm depth… … µm b Supplementary Fig. 2. Supplementary figures on the cell growth measurement. a Sketch of the design of microchannels in thePDMS-based device. Medium flow in the drain channel removes cells expelled from the observation area (trap). Nutrient issupplied to the cells inside the trap via diffusion from the drain channel. b The design of the PDMS-based device. The greenregion corresponds to the drain channel, and the blue regions are the U-shape traps. c Top view of the trap (30 µ m wide,88 µ m long, 1 . µ m deep) in the PDMS-based device. The trap is filled with E. coli
W3110 ∆fliC ∆flu ∆fimA. The yellowarrows represent the velocity field of flow driven by cell proliferation, measured by particle image velocimetry (PIV). The scalebar is 25 µ m. See also Supplementary movie 3. d Top view of the trap (30 µ m wide, 80 µ m long, 1 . µ m deep) in the EMPS.See also Supplementary movie 4. a c,eb,dw/ Rhodamine (10 µM ) w/o Rhodamine coverslipmembrane bacteria d e time time b c Supplementary Fig. 3. Direct observations of medium exchange in the EMPS. a Cross-sectional images of the device taken byconfocal microscopy. The scale bar is 5 µ m. Medium flows from the front side to the rear side of the images. Although theflow speed of the medium is relatively high above the membrane, the diffusive motion of cells is hardly affected (see MoviesS5 and S6). (left) A snapshot of the device filled with a PBS solution with 10 µ M rhodamine. Bacterial cells (W3110 ∆fliC∆flu ∆fimA) and the bilayer membrane are dyed and visualized. (right) A snapshot of the device filled with PBS withoutrhodamine. The surface of the coverslip and the cells still exhibit fluorescence because of adsorption of rhodamine. b , c Timeevolution of the spatial profile of the fluorescent intensity, when the medium is switched to the rhodamine solution ( b , see alsoSupplementary Movie 5) and to the PBS without rhodamine ( c , see also Supplementary Movie 6). The intensity averagedover 5 pixels (0 . µ m) from the substrate bottom is shown. Note that the location of the substrate bottom was detected byimage analysis in each frame, in order to avoid the influence of vibrations (see Movies S5 and S6) caused by the high flow rateused here. The peaks seen in the profiles are due to bacterial cells, walls or dust. d , e Time series of the spatially averagedfluorescence intensity when the medium is switched to the rhodamine solution ( d ) and to the PBS without rhodamine ( e ).During the experiment, medium flowed above the membrane at a constant speed of approximately 6 mm / sec. t = 0 is thetime at which the rhodamine solution entered the device (black dashed line). The spatial average of intensity in the well (bluecurves) was taken in a square ROI of height 5 pixels (0 . µ m) from the substrate bottom, and width 200 pixels (24 µ m) alongthe y-axis, around the center of the well. The spatial average of intensity in the membrane (red curves) was taken in a linearROI of length 200 pixels (24 µ m) along the y-axis, located at 4 . µ m above the substrate bottom, around the center. t = 0 min time Glucose + 12 a.a. (M9) PBS (no nutrient) ab c
Glc+a.a. → PBS d Supplementary Fig. 4. Results from the observations of reductive division in the case of M9(Glc+a.a.) → PBS. The datawere collected from 17 wells. a Snapshots taken during the reductive division process. See also Supplementary Movie 8. b , c Experimental data (blue symbols) for the total cell volume V tot ( t ) b , the growth rate λ ( t ) ( b , Inset), the number of the cells n ( t ) ( c ) and the mean cell volume V ( t ) ( c , Inset) in the case of M9(Glc) → PBS, compared with the simulation results (redcurves). The error bars indicate segmentation uncertainty in the image analysis (see Supplementary Materials and Methods). t = 0 is the time at which PBS entered the device (black dashed line). d The moment ratio (cid:104) v j (cid:105) / (cid:104) v j − (cid:105) against V ( t ) = (cid:104) v (cid:105) .The error bars were estimated by the bootstrap method with 1000 realizations. The colored lines represent the results of linearregression in the log-log plots (see Supplementary Table 3 for the slope of each line). The black solid lines are guides for eyesindicating unit slope, i.e., proportional relation. a t = 0 min time Glucose (M9) αMG (no nutrient) b c
Glc → αMG d Supplementary Fig. 5. Results from the observations of reductive division in the case of M9(Glc) → M9( α MG). The datawere collected from 26 wells. a Snapshots taken during the reductive division process. See also Supplementary Movie 9. b , c Experimental data for the total cell volume V tot ( t ) ( b ), the number of the cells n ( t ) ( c ) and the mean cell volume V ( t )( c , Inset) in the case of M9(Glc) → M9( α MG). The error bars indicate segmentation uncertainty in the image analysis (seeSupplementary Materials and Methods). t = 0 is the time at which α MG entered the device (black dashed line). d The momentratio (cid:104) v j (cid:105) / (cid:104) v j − (cid:105) against V ( t ) = (cid:104) v (cid:105) . The error bars were estimated by the bootstrap method with 1000 realizations. Thecolored lines represent the results of linear regression in the log-log plots (see Supplementary Table 3 for the slope of each line).The black solid lines are guides for eyes indicating unit slope, i.e., proportional relation. a b Supplementary Fig. 6. Comparison of the function F ( v/V ( t )) = vp ( v, t ) among different cases. a F ( x ) obtained by ourexperiments and simulation, as well as that obtained by Giometto et al. [1] for unicellular eukaryotes. The solid lines representtime-averaged data. The yellow dashed line is obtained by fitting Eq. (3) for the log-normal distribution to our experimentaldata. The green dashed line is the fitting result by Giometto et al. [1] for unicellular eukaryotes. σ is the standard deviationparameter of the log-normal distribution (see Eq. (3)). b vp ( v, t ) for the three cases studied in this work, plotted in a linearscale. The raw data obtained at different times are shown by thin lines with relatively light colors, and the time-averaged dataare shown by the bold lines. Instantaneous distributions (thin lines) also seem to be slightly but significantly different amongthe three cases. d fe LB → PBS Glc → αMGGlc+a.a. → PBS a cb
LB → PBS Glc → αMGGlc+a.a. → PBS
Supplementary Fig. 7. Time series of the mean cell length and the mean cell width during the starvation process. t = 0 is thetime at which the non-nutritious buffer entered the device (black dashed line). a , b , c Time series of the mean cell length inthe case of LB → PBS ( a ), M9(Glc+a.a.) → PBS ( b ) and M9(Glc) → M9( α MG) ( c ). The error bars indicate segmentationuncertainty in the image analysis (see Supplementary Materials and Methods). d , e , f Time series of the mean cell width inthe case of LB → PBS ( d ), M9(Glc+a.a.) → PBS ( e ) and M9(Glc) → M9( α MG) ( f ). The error bars indicate the standarddeviation in each ensemble. ac bd Supplementary Fig. 8. Cell-to-cell fluctuations of the septum position. Scatter plots of the mother cell length against theseptum position x sep are shown for four different cases. The data were taken from more than 100 cell division events chosenrandomly for each case. The error bars indicate segmentation error in the image analysis. There is no visible correlationbetween the mother cell size and the standard deviation of the septum position. a Scatter plot for the exponential growthphase in LB broth. The standard deviation of the septum position is x sepstd = 0 . b Scatter plot during the starvation processfor LB → PBS. The color represents the time passed since PBS entered the device. The standard deviation of the septumposition is x sepstd = 0 . c Scatter plot for the exponential growth phase in M9(Glc+a.a) medium. The standard deviationof the septum position is x sepstd = 0 . d Scatter plot during the starvation process for M9(Glc+a.a.) → PBS. The standarddeviation of the septum position is x sepstd = 0 . ab c d Simulation(LB→PBS) Simulation(Glc+a.a.→PBS)Simulation(Glc+a.a.→PBS)Simulation(Glc+a.a.→PBS)
Supplementary Fig. 9. Supplementary figures on the simulation results. a The moment ratio (cid:104) v j (cid:105) / (cid:104) v j − (cid:105) against V ( t ) = (cid:104) v (cid:105) ,in the model for LB → PBS. The error bars estimated by the bootstrap method with 1000 realizations were smaller thanthe symbol size. The colored lines represent the results of linear regression in the log-log plots (see Supplementary Table 3for the slope of each line). The black solid lines are guides for eyes indicating unit slope, i.e., proportional relation. b Time evolution of the cell size distributions during starvation in the model for M9(Glc+a.a.) → PBS, obtained at t =0 , , , , , , , , ,
180 min from right to left. c Rescaling of the data in b . The overlapped curves indicate thefunction F ( v/V ( t )) in Eq.(2) in the main text. The dashed line represents the time average of the datasets. d The momentratio (cid:104) v j (cid:105) / (cid:104) v j − (cid:105) against V ( t ) = (cid:104) v (cid:105) , in the model for M9(Glc+a.a.) → PBS. The error bars estimated by the bootstrap methodwith 1000 realizations were smaller than the symbol size.
Experiment (LB→PBS) a b
Simulation(LB→PBS)
Supplementary Fig. 10. Test of the self-consistent equations derived in Supplementary Theory. The functional form of F ( x ),obtained numerically ( a ) or experimentally ( b ) for the case LB → PBS, is compared with the right-hand side (rhs) of theself-consistent equations derived in Supplementary Theory, Eqs. (S17) and (S29). The effect of septum fluctuations is neglectedin Eq. (S17) and considered in Eq. (S29). F ( x ) is obtained as the time average of the instantaneous data. IV. SUPPLEMENTARY MOVIE DESCRIPTIONS
Supplementary Movie 1:
Growth of motile
E. coli
RP437 inside a well covered only by a cellulose membrane. The diameter of the well is110 µ m, and the depth is 1 . µ m. Being pressed by the bent cellulose membrane, cells do not swim but form clusters,extending toward the wall. The movie is played at 600 × real-time speed. Supplementary Movie 2:
Growth of motile
E. coli
RP437 inside a well covered by a PET-cellulose bilayer membrane. The diameter of the wellis 110 µ m, and the depth is 1 . µ m. Cells freely swim inside the quasi-two-dimensional well. The movie is played atreal-time speed. Supplementary Movie 3:
Coherent flow of non-motile bacterial cells driven by self-replication in a U-shape trap of the PDMS-based device.The trap is 30 µ m wide, 88 µ m long, and 1 . µ m deep. E.coli strain W3110 ∆fliC ∆flu ∆fimA is used.
Supplementary Movie 4:
Coherent flow of non-motile bacterial cells driven by self-replication in a U-shape trap of the EMPS. The trap is30 µ m wide, 80 µ m long, and 1 . µ m deep. E.coli strain W3110 ∆fliC ∆flu ∆fimA is used.
Supplementary Movie 5:
A cross-sectional movie of the EMPS, recorded while medium is switched from transparent PBS to a PBS solution of10 µ M rhodamine. The diameter of the well is 45 µ m and the depth is 1 . µ m. A few non-motile E.coli (W3110 ∆fliC∆flu ∆fimA) are present in the well. The rhodamine solution flowed at a constant speed of approximately 6 mm / secabove the membrane (flow rate 60 ml / hr). The movie is played at 19 × real-time speed. Supplementary Movie 6:
A cross-sectional movie of the EMPS, recorded while medium is switched from a PBS solution of 10 µ M rhodamineto transparent PBS. The diameter of the well is 45 µ m and the depth is 1 . µ m. A few non-motile E.coli (W3110∆fliC ∆flu ∆fimA) are present in the well. The PBS without rhodamine flowed at a constant speed of approximately6 mm / sec above the membrane (flow rate 60 ml / hr). The movie is played at 19 × real-time speed. Supplementary Movie 7:
Reductive division of
E. coli
MG1655 for the case LB → PBS. The diameter of the well is 55 µ m and the depth is0 . µ m. Until t = 0, fresh LB broth was supplied at a constant flow speed of approximately 0 . / sec above themembrane (flow rate 2 ml / hr). PBS entered the device at t = 0 and quickly replaced the LB broth, by setting a highflow speed ∼ / sec (60 ml / hr) until t = 5 min. After flushing, we continued supplying PBS at the flow speed ofapproximately 0 . / sec (2 ml / hr). Supplementary Movie 8:
Reductive division of
E. coli
MG1655 for the case M9(Glc+a.a.) → PBS. The diameter of the well is 55 µ m and thedepth is 0 . µ m. PBS entered the device at t = 0. The flow rates were controlled and set in the same manner as forSupplementary Movie 7. Supplementary Movie 9:
Reductive division of
E. coli
MG1655 for the case M9(Glc) → M9( α MG). The diameter of the well is 55 µ m and thedepth is 0 . µ m. M9( α MG) entered the device at tt