Scale invariance during bacterial reductive division observed by an extensive microperfusion system

In stable environments, cell size fluctuations are thought to be governed by simple physical principles, as suggested by recent finding of scaling properties. Here we show, using E. coli, that the scaling concept also rules cell size fluctuations under time-dependent conditions, even though the distribution changes with time. We develop a microfluidic device for observing dense and large bacterial populations, under uniform and switchable conditions. Triggering bacterial reductive division by switching to non-nutritious medium, we find that the cell size distribution changes in a specific manner that keeps its normalized form unchanged; in other words, scale invariance holds. This finding is underpinned by simulations of a model based on cell growth and intracellular replication. We also formulate the problem theoretically and propose a sufficient condition for the scale invariance. Our results emphasize the importance of intrinsic cellular replication processes in this problem, suggesting different distribution trends for bacteria and eukaryotes.

A snapshot of the device filled with PBS without rhodamine. The surface of the coverslip and the cells still exhibit fluorescence because of adsorption of rhodamine. b,c Time evolution of the spatial profile of the fluorescent intensity, when the medium is switched to the rhodamine solution (b, see also Video 5) and to the PBS without rhodamine (c, see also Video 6). The intensity averaged over 5 pixels (0.6 m) from the substrate bottom is shown. Note that the location of the substrate bottom was detected by image analysis in each frame, in order to avoid the influence of vibrations (see Movies S5 and S6) caused by the high flow rate used here. The peaks seen in the profiles are due to bacterial cells, walls or dust. d,e Time series of the spatially averaged fluorescence 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. = 0 is the time at which the rhodamine solution entered the device (black dashed line). The spatial average of intensity in the well (blue curves) was taken in a square ROI of height 5 pixels (0.6 m) from the substrate bottom, and width 200 pixels (24 m) along the y-axis, around the center of the well. The spatial average of intensity in the membrane (red curves) was taken in a linear ROI of length 200 pixels (24 m) along the y-axis, located at 4.8 m above the substrate bottom, around the center. our results indicate that the EMPS can indeed realize a uniform and stable culture condition while 138 the same medium is kept supplied. 139 Another advantage of the EMPS is that we can also switch the culture condition, by changing 140 the medium to supply. Here we evaluate how efficiently the medium in the well is exchanged. 141 In the presence of non-motile E. coli W3110 ΔfliC Δflu ΔfimA, we switch the medium to supply  Figure 3b) and that it is almost completed within 2-4 min (Figure 3d). We also change the medium 146 from PBS with rhodamine to that without rhodamine (Figure 3a, from left to right). The exchange 147 then took longer time, ≳ 5 min, presumably because of adsorption of rhodamine on the substrate 148 and membrane (see Figure 3a). In any case, the time to take for exchanging medium is much shorter 149 than the timescale of the bacterial cell cycle. Our observations also indicate that the membrane 150 is indeed kept flat above the well (Figure 3a) and that the Brownian motion of non-motile cells 151 is hardly affected by relatively strong medium flow above the membrane (estimated at roughly 152 6 mm∕sec ) induced when switching the medium (Video 5 and Video 6). We therefore conclude that 153 the EMPS is indeed able to change the growth condition for cells under observation uniformly,   Table 1 for the slope of each line). The black solid lines are guides for eyes indicating unit slope, i.e., proportional relation.
with constant parameters and , and the growth rate 0 (= (0)) in the exponential growth phase with parameters and , and the replication speed 0 (= (0)) in the growth phase (see Appendix 1). 296 The parameter values are determined from the experimentally measured total cell volume 297 and the cell number, which our simulations turn out to reproduce very well (Figure 4bc and

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(2016)) for some of the parameters (see Table 3 for the parameter values used in the simulations,  )), which we propose as an extension for dealing with time-dependent environments. 358 We further inspected theoretical mechanism behind this scale invariance and found the significance 359 of the division rate function ( , ). We obtained a sufficient condition for the scale invariance, 360 Equation 8, which was indeed confirmed in our numerical data. 361 After all, our theory suggests that mechanism of intracellular replication processes may have 362 direct impact on the scale-invariant distribution, which may account for the significant difference we 363 identified between bacteria and eukaryotes (Figure 5-Figure Supplement 1a to what extent the cell size distribution is determined by the intracellular replication dynamics. We 368 also note that the scale-invariant distribution ( ) might depend weakly on the culture condition 369 in the exponential growth phase (Figure 5-Figure Supplement 1b)   Observation of motile E. coli in the EMPS 422 We used a wild-type motile strain of E. coli, RP437. First, we inoculated the strain from a glycerol 423 stock into 2 ml TB medium (see Table 2 for components) in a test tube. After shaking it overnight at 424 37 • C, we transferred 20 l of the incubated suspension to 2 ml fresh TB medium and cultured it 425 until the optical density (OD) at 600 nm wavelength reached 0.1-0.5. The bacterial suspension was 426 finally diluted to OD = 0.1 before it was inoculated on the coverslip of the EMPS. 427 Regarding the device, here we compared the EMPS with the previous system developed in 428 ref. (Inoue et al. (2001)), whose membrane was composed of a cellulose membrane alone instead  (Figure 1-Figure Supplement 1c) and 118 msec for the EMPS (Figure 1-Figure   440 Supplement 1d Cell growth measurement in U-shape traps in the PDMS-based device 445 We used a non-motile mutant strain W3110 without flagella and pili (ΔfliC Δflu ΔfimA) to prevent cell 446 adhesion to the surface of a coverslip. Before the time-lapse observation, we inoculated the strain 447 from a glycerol stock into 2 ml M9 medium with glucose and amino acids (Glc+a.a.) (see also

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Here the choice of the strain, the medium, the culture condition and the instruments was the same 463 as those for the measurement with the PDMS-based device, unless otherwise stipulated. Here, the 464 cultured bacterial suspension was diluted to OD = 0.04 before it was inoculated on the coverslip of 465 the EMPS. As sketched in Figure 2A, the substrate consisted of drain channels (100 m wide, 7 mm 466 long, 13 m deep) and U-shape traps (30 m wide, 80 m long, 1.0 m deep), which were prepared by 467 the methods described in ref. (Hashimoto et al. (2016)). When the bilayer membrane was attached 468 to the substrate, care was taken not to cover the two ends of the drain channel to use, so that cells in 469 the drain could escape from it. After the assembly of the device with the bacterial suspension, it was 470 fixed on the microscope stage inside the incubation box maintained at 37 • C. To fill the device with 471 medium, we injected fresh medium stored at 37 • C from the inlet (Figure 1-Figure Supplement 1), 472 at the rate of 60 ml∕hr for 5 min by a syringe pump (NE-1000, New Era Pump Systems). 473 During the observation, the flow rate of the M9(Glc+a.a. and BSA) medium was set to be 474 2 ml∕hr (approximately 0.2 mm∕sec above the membrane), except that it was increased to 60 ml∕hr To fill the device with growth medium, we injected fresh medium stored at 37 • C from the inlet 509 (Figure 1-Figure Supplement 1), at the rate of 60 ml∕hr for 5 min by a syringe pump (NE-1000, New 510 Era Pump Systems). 511 In the beginning of the observation, growth medium was constantly supplied at the rate of 512 2 ml∕hr (flow speed approximately 0.2 mm∕sec above the membrane). When a microcolony com-513 posed of approximately 100 cells appeared, we quickly switched the medium to a non-nutritious 514 buffer (PBS or M9 medium with -methyl-D-glucoside ( MG), see Table 1) stored at 37 • C, by ex-515 changing the syringe. The flow rate was set to be 60 ml∕hr for the first 5 minutes, then returned to 516 2 ml∕hr. Throughout the experiment, the device and the media were always in the microscope incu-517 bation box, maintained at 37 • C. Cells were observed by phase contrast microscopy and recorded 518 at the time interval of 5 min.

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The cell volumes were evaluated as follows. We determined the major axis and the minor axis 520 of each cell, manually, by using a painting software. By measuring the axis lengths, we obtained the 521 set of the lengths and the widths for all cells. We then estimated the volume of each cell, by 522   (Figure 5d). 544 The parameter values determined thereby are summarized in Table 3 Next, we determine the functional form of the replication speed ( ), more precisely the progression speed of the C+D period. We first note that the C+D period mainly consists of DNA replication, followed by its segregation and the septum formation (Wang and Levin (2009)). Most parts of those processes involve biochemical reactions of substrates, such as deoxynucleotide triphosphates for the DNA synthesis, and assembly of macromolecules such as FtsZ proteins for the septum formation. We therefore consider that the replication speed is determined by the intracellular concentration of relevant substrates and macromolecules.
Here we simply assume that the progression of the C+D period can be represented by assembly-like processes of relevant molecules, represented collectively by RM C+D . We then consider that the progression speed ( ) is given through the Hill equation, which usually describes the binding probability of a receptor and a ligand, with cooperative effect taken into account. Specifically, where is the Hill coefficient, and is the equilibrium constant of the (collective) assembly process. One can evaluate the time evolution of the concentration [RM C+D ] in the same way as for ( ). With a constant consumption rate , the differential equation can be written as The solution to this equation is with a constant ′ 0 and = − 0 (1 − )∕ . To reduce the number of the parameters in the model, we simply assume that [RM C+D ] exponentially decreases during starvation, i.e., with initial concentration 0 and the degradation time scale . As a result, we obtain with ∶= ( ∕ 0 ) , = ∕ , and 0 (= (0)) being the replication speed in the growth phase before the onset of starvation.
For the scale invariance, Equation 25 should hold at any time . In other words, the coefficient ( , )∕ ( ) should be independent of both and ( ). Note here that ( , )∕ ( ) can be rewritten as Therefore, ( , )∕ ( ) is time-independent if ( ′ ( ), ) ( ( ), ) does not depend on ( ), being a function of two dimensionless variables and ′ only, as follows: This condition can be rewritten as follows. For a constant 0 , we can define ( ) by and ( ) by Then, the division rate can be expressed as ( ( ), ) = ( ) ( ) for any and . This gives the sufficient condition we presented in the main text, Equation 8, Test of the derived conditions for ( , ) and ( ) 820 We tested the sufficient condition for ( , ) (Equation 8) and the resulting self-consistent equation for ( ) (Equation 25) with numerical data we obtained from our sCH model (Figure 6d). We evaluated the division rate ( , ) in the simulations for LB → PBS, by measuring the number of division events of cells between time and + Δ , divided by the number of cells at time , where only the cells of volume between and +Δ were considered in both the numerator and the denominator. Here, Δ was set to be approximately 0.2 × ( ) and Δ to be approximately 20-30 min for each time point, respectively. The value of ( , 0) was determined by counting all division events in the exponential growth phase ( < 0). The ratio (0)∕ ( ) can be evaluated by (0)∕ ( ) = ∫ ( (0), 0) ∕ ∫ ( ( ), ) . We found that the curves ( , ) (0)∕ ( ) taken at different overlap reasonably well (Figure 6d, Inset), which support the variable separability condition of the division rate, Equation 8.
with ( (0), 0) = ( , 0) = ( , ) (0)∕ ( ). Since we already confirmed the time independence of ( , ) (0)∕ ( ) (Figure 6d, Inset), we took the average of this quantity obtained at = 0, 30, 60, 90 min. Since the observed range of is finite and ( ) almost vanishes for ≳ 2, we evaluated the integral over ′ in the range 0 ≤ ′ ≤ 2. With the ( )∕ ( , ) evaluated thereby, we substituted ( ) obtained by the simulations for LB→PBS to Equation 25 (Figure 6-Figure Supplement 2a), using the time average of ( ) (the dashed line in Figure 6c) and ( ) = 0 for ≥ 2. We also tested Equation 25 with ( ) obtained in the experiment for LB→PBS, in the same way as for the model (Figure 6-Figure  Supplement 2b). In both cases, the right-hand side (rhs) of Equation 25 differs significantly from the observed form of ( ).   Table 1 for the slope of each line). The black solid lines are guides for eyes indicating unit slope, i.e., proportional relation. = 0 is the time at which MG entered the device (black dashed line). d The moment ratio ⟨ ⟩∕⟨ −1 ⟩ against ( ) = ⟨ ⟩. The error bars were estimated by the bootstrap method with 1000 realizations. The colored lines represent the results of linear regression in the log-log plots (see Table 1 for the slope of each line). The black solid lines are guides for eyes indicating unit slope, i.e., proportional relation.  Table 1