A leader cell triggers end of lag phase in populations of Pseudomonas fluorescens

Abstract The relationship between the number of cells colonizing a new environment and time for resumption of growth is a subject of long-standing interest. In microbiology this is known as the “inoculum effect.” Its mechanistic basis is unclear with possible explanations ranging from the independent actions of individual cells, to collective actions of populations of cells. Here, we use a millifluidic droplet device in which the growth dynamics of hundreds of populations founded by controlled numbers of Pseudomonas fluorescens cells, ranging from a single cell, to one thousand cells, were followed in real time. Our data show that lag phase decreases with inoculum size. The decrease of average lag time and its variance across droplets, as well as lag time distribution shapes, follow predictions of extreme value theory, where the inoculum lag time is determined by the minimum value sampled from the single-cell distribution. Our experimental results show that exit from lag phase depends on strong interactions among cells, consistent with a “leader cell” triggering end of lag phase for the entire population.


Introduction
When bacteria encounter new environmental conditions, growth typically follows four phases: a lag phase, during which bacteria acclimate, but do not divide; an exponential phase, during which cells multiply; a stationary phase, where nutrient exhaustion causes cessation of growth; and finally a death phase, during which cells may lyse.In a fluctuating envir onment, eac h phase can play an important role in population persistence .T he lag phase has particular significance because of both benefits (enhanced growth) and eventual costs (sensitivity to external stressors) associated with the resumption of gr owth (Mor eno-Gámez et al .2020 , Ş im ş ek and Kim 2019 ).Mor eov er, the time to resumption of gr owth-and contr olling factors-has implications for the entire field of microbiology (Monod 1949 ), but especially for infection caused by pathogens and for food safety (Bertrand and Margolin 2019 , Swinnen et al . 2004, Pérez-Rodríguez 2014 ).
Despite its discov ery mor e than 100 years ago (Müller 1895 ), cellular and molecular details defining the lag phase, factors triggering resumption of growth, and contributions to fitness, are not well-understood.This is lar gel y a consequence of the difficulties associated with experimental quantification of the dynamics of populations founded by small numbers of cells .Nonetheless , adv ances ov er the last decade hav e shown that bacteria in la g phase ar e tr anscriptionall y and metabolicall y activ e (Rolfe et al . 2012 ), that lag phase is a dynamic state, that single cells are heterogeneous in time to resume division (Julou et al . 2020, Moreno-Gámez et al . 2020, Ş im ş ek and Kim 2019 ), and that numerous factors affect lag phase duration (Nikel et al . 2015, Bertrand and Margolin 2019, Basan et al . 2020 ).
Ar guabl y the most intriguing aspect of lag phase biology is the a ppar ent inv erse r elationship between the number of cells in the founding population and duration of lag phase-often r eferr ed to as the "inoculum effect."First reported in 1906 (Rahn 1906 ), the relationship has been shown to hold for a number of different bacteria (Penfold 1914, Lodge and Hinshelwood 1943, Lankford et al . 1966, Ka pr el yants and Kell 1996 ), although there exist fe w r ecent quantitativ e inv estigations.In certain instances, the inoculum effect is observed only under specific culture conditions (Ka pr el yants and Kell 1996 , Augustin et al . 2000 ).
Factors controlling the inoculum effect are of special interest (Dagley et al . 1950, Lankford et al . 1966, Halmann and Mager 1967, Augustin et al . 2000, Swinnen et al . 2004, Bertrand and Margolin 2019 ).Given that bacterial cells are typically variable in many of their properties, the simplest explanation (Explanation I) posits that population lag time is determined by the set of cells with the shortest time to first division.Accordingly, the larger the founding population, the more likely it is that the inoculum contains cells on the v er ge of division, with these cells contributing dispr oportionately to the resumption of population growth.
An alternate explanation is that resumption of growth depends on interaction among founding cells (Explanations II and III), e.g.via production of an endogenous growth factor: once a critical threshold concentration is achieved growth resumes, and the larger the inoculum the sooner this happens.Evidence in support of suc h contr ol deriv es fr om anal ysis of Bacillus (Lankford et al . 1966 ), Francisella (formerly Pasturella ) tularensis (Halmann and Mager 1967 ), Micrococcus luteus (Votyakova et al . 1994 ), and Aerobacter aerogenes (Dagley et al . 1950 ).
In instances where exit from lag-phase is determined by interactions among founding cells , models ha ve assumed that all cells are equal contributors to the production of growth activating factors (Explanation II; Lankford et al . 1966 , Ka pr el yants andKell 1996 ).Ho w ever, an alternative possibility exists, namely, that population lag time is set by the activity of a single "leader cell" that triggers resumption of growth for the entire population of cells (Explanation III).Distinguishing among competing hypotheses r equir es pr ecision measur ements of population growth, high le v els of replication, ability to control inoculum size, and crucially, knowledge of the distribution of lag times for single cells.
Here, we use a millifluidic droplet device in which the growth dynamics of hundreds of populations founded by different numbers of Pseudomonas fluorescens cells were followed in real time.Our data confirm that lag phase shortens with inoculum size increase and provide a quantitative characterization of the effect on: avera ge, v ariance and shape distribution of lag times values for various controlled inoculum size.We demonstrate that these statistical properties follow extreme value theory (EVT), where population lag time is determined by the minimum value sampled from the single cell distribution.Additionally, w e sho w that the inoculum effect cannot be explained by a sweep initiated from a small number of cells, but rather involves the parallel growth of many lineages .T hese results suggest that exit from lag phase depends on str ong inter actions among cells, consistent with a leader cell triggering end of lag phase for the entire population.And we derive the scaling laws that allows prediction of bacterial population lag time as a function of inoculum size.

T he str ain
The ancestral strain of P. fluorescens SBW25 was isolated from the leaf of a sugar beet plant at the University of Oxford farm (Wytham, Oxfor d, UK;Silb y et al . 2009 ).The strain was modified to incor por ate, via c hr omosomall y integr ated Tn 7 , the gene GFPm ut3B contr olled by an inducible Ptac pr omoter.

Prepar a tion of cells
Pseudomonas fluorescens SWB25 was grown in casamino acid medium (C AA).C AA for 1 l: 5g of Bacto Casamino Acids Technical (BD ref 223120), 0.25 g MgSO 4 •7H 2 O (Sigma CAS 10034-99-8), and 0.9 g K 2 HPO 4 .Prior to generation of droplets, SBW25 was grown from a glycerol stock for 19 h in 5 ml of CAA incubated at 28 • C and shaken at 180 rpm.At 19 h, this stationary phase culture was centrifuged at 3743 RCF for 4 min and the supernatant r emov ed fr om the pellet.The pellet was then r esupended in 5 ml of sterile CAA.It was then centrifuged and resuspended one further time in order to pr e v ent an y interfer ence of a gr owth-activ ator that may come fr om the supernatant of the ov ernight cultur e.The washed cultur e was then adjusted to OD 0.8 with CAA and mixed 1:1 in volume with autoclaved 30% v/v gl ycer ol.A volume of 100 μl aliquots were pipetted in 1 ml eppendorf and frozen at −80 • C.After freezing, one aliquot was taken to measure viable cells by plating on agar.We found 1.62 × 10 8 cell ml −1 .

Gener a tion of droplets with a range of inoculum sizes
Each experiment, with a range of inoculum sizes was prepared as follows.One frozen aliquot was thawed and diluted in 6 Falcon © tubes with a final volume of 4 ml of sterile C AA. T he frozen aliquot was diluted, with a ppr opriate intermediate dilutions, in the tubes r espectiv el y by 7.04 × 10 4 , 1.76 × 10 4 , 4.4 × 10 3 , 1.1 × 10 3 , 2.75 × 10 2 , and 6.875 × 10 1 to obtain, r espectiv el y, 1, 4, 16, 64, 256, and 1024 cells per droplet (on average).We completed the dilutions fr om fr ozen stoc k by adding 29.1 μl, 29 μl, 28.6 μl, 27.3 μl, 21.8 μl, and 0 μl, r espectiv el y, of sterile 60% v/v gl ycer ol.This step is very important to balance the glycerol coming from the frozen stoc k and ensur e that all the tubes have precisely the same composition of medium (Figure S2, Supporting Information).We then added 50 μl of sterile IPTG (100 mM) to each sample.Each dilution was then pipetted in wells (250 μl per well) of a 96-well micr otitr e plate befor e pr oceeding to gener ate dr oplets using the Millidrop Azur de vice.Dr oplets hav e a volume of 0.4 μl, whic h yields, with our dilutions, a range of inoculum sizes as follows: 1, 4, 16, 64, 256, and 1024 cells per droplet.We generated 40 replicate droplets for each population of a given founding inoculum size, except for populations founded by 1024 cells, which for technical reasons were restricted to 30 replicates.

The inoculum of droplets follows a Poisson distribution
Importantly, the inoculum size in each droplet is controlled by the Poisson process intrinsic to the formation of droplets from the 96well plates .T he inoculum size that we report is, thus the av er a ge of the corresponding Poisson distribution.In particular, the variance of the number of cells between the droplets is equal to the av er a ge inoculum.

Gener a tion of droplets founded by a single cell
To generate droplets with an inoculum of a single cell per droplet, we diluted in sterile CAA a frozen aliquot by 7.04 × 10 4 , added 29.1 μl of sterile 60% v/v gl ycer ol, and 50 μl of sterile IPTG (100 mM).We generated 230 droplets in the Millidrop Azur device, which yielded 156 droplets that grew due to the Poisson process inherent to the sampling process.

Log-normality of single-cell lag times
The cum ulativ e distribution function (CDF) for single-cell la g times is displayed in Fig. 2 (B), and is described by a lognormal fit.To quantify the goodness of fit, we used the Sha pir o-Wilk test with the null hypothesis that a sample log θ 1 ,..., log θ n is derived from a normal distribution.The null hypothesis was tested with significance le v el alpha 5% and gav e a P -v alue of 0.840 indicating that we can not reject the lognormality of the single-cell lag time distribution.In addition, a quantile-to-quantile plot of single-cell lag time against a lognormal distribution is shown in Figure S11 (Supporting Information).

Calculation of the uncertainty of measurement for populations founded by a single cell
N th = 1.6 × 10 8 cells ml −1 is the y -coordinate of the point taken in the exponential phase of growth to calculate the population lag time (see Fig. 1 ).N th is uncertainty surrounding the y -coordinate, i.e. deriv ed fr om uncertainty on the calibration of cell concentration v ersus fluor escent signal (Figur e S8, Supporting Information).The uncertainty of the calibration Figure S8 (Supporting Information) gives N th = 0.7 × 10 8 cells ml −1 for this value as depicted by the gray area.t th is the uncertainty of the time when the population r eac hes beyond the thr eshold N th .We take its v alue as equal to the sampling frequency of the machine t th = 18 min.λ is the av er a ge gr owth r ate of populations in dr oplets and λ is the uncertainty.These quantities are estimated with the distribution of the growth rate shown in Figure S6 (Supporting Information): λ = 0.84 h −1 and λ = 0.02 h −1 .N 0 is the uncertainty on the inoculum size.In the experiment with populations founded by a single cell shown in Fig. 2 (B), 230 − 156 = 74 dr oplets wer e empty despite being gener ated fr om the same seed cultur e .T his is due to the randomness of the pipetting process that fills droplets of bacteria according to a P oisson process .T he randomness of the process gives intrinsically an uncertainty on N 0 .In the follo wing, w e explain ho w this w as estimated.Kno wing the number of empty droplets allows calculation of the precise average inoculum of the experiment which correspond to the α parameter of a Poisson distribution having its first value p (0) = exp( −α) = 74/230 : α = 1.134 cells per droplet.T his a verage takes into account the empty droplets with zero bacteria but we only measure the nonempty droplets.
To estimate the av er a ge inoculum of the nonempty droplets we dr aw numericall y a lar ge series of r andom numbers with a Poisson probability of parameter α = 1.134 and calculate the average and standard deviation (SD) of the nonzero values .T his yielded an av er a ge of 1.7 and a SD of 0.9.Ther efor e, we consider that for our experiment in an ideal case with an infinite number of droplets the uncertainty on the inoculum of the droplets will be intrinsically N 0 = 0.9 cells per droplet and that the average inoculum (of the filled droplets) is N 0 = 1.7 cells per droplet.All together these values allow calculation of the uncertainty of the lag time estimated by Equation ( 1).The expression of uncertainty is given by Equation ( 2) and numerical application gives θ = 0.88 h.

High-throughput quantification of bacterial population dynamics with millifluidic technology
To investigate the relationship between inoculum size and duration of lag phase, we used a millifluidic device to quantify the dynamics of bacterial population growth across time (Baraban et al . 2011, Boitard et al . 2015, Cottinet et al . 2016 ).The device allows the monitoring of 230 bacterial populations compartmentalized in droplets contained within a tube.Figure 1 (A) shows a portion of the tube with two droplets filled with cells .T he statistical po w er of the experiment comes fr om pr ecise contr ol of lar ge numbers of droplets, in terms of both inoculum size and homogeneity of culture conditions: Fig. 1 (B) shows the growth dynamics of 40 r eplicate populations.Exponential gr owth and stationary phase ar e clearl y seen, while la g time is concealed behind the detection thr eshold (gr ay ar ea).Figur e 1 (C) shows the gr owth dynamics of a population contained within a single droplet.Population density in the droplet is reported by fluorescence intensity from GFP-labeled bacteria with parameters describing the phases of growth being estimated from these time series (see Fig. 1 C).Exponential gr owth r ate, λ, is the maxim um slope of the time series on a y-semi-logscale (we use a Gaussian processes method that makes no a priori assumption about the shape of the gr owth curv e (Swain et al . 2016 )).Final population size is estimated dir ectl y fr om measur ements; death phase is not significant in our experiment and is ignor ed.La g phase τ , is the time cells spend in a nondividing phase prior to onset of exponential gro wth.Har dw are limitations mean that fluorescence data are unobtainable for cell densities below 1600 cells per droplet (4 × 10 6 cells ml −1 ) and, thus τ must be estimated indir ectl y.This indir ect measurements allows also to circumvent to classical caveat for lag time measurement related to variation of the fluorescence per cell during lag phase .T his is done by firstly taking an arbitrary point ( t th , N th ) in exponential phase where cell density is N th = 1.6 × 10 8  S8, Supporting Information).The range of detection extends from 4 × 10 6 to 5 × 10 9 cells ml −1 (1.6 × 10 3 to 2 × 10 6 cells per droplet).T he gra y ar ea in subfigur es (B) and (C) denotes the r egion wher e bacterial density is below the threshold of detection.(B) Fluorescent signal across time from 40 replicate populations (in semi-logscale) in dr oplets pr epar ed fr om the same seed cultur e .T he a v er a ge inoculum in eac h dr oplet is N 0 = 1.6 × 10 5 cell ml −1 , or 64 cells per droplet (this concentration is marked by the purple dashed line that goes across (B) and (C)).In this example, the signal exceeds the detection threshold at ∼7 h, by which populations are in exponential growth phase.At ∼20 h, stationary phase is r eac hed, marked by cessation of growth.(C) A single time series showing population growth within a single droplet coming from the set of replicates shown in (B).The left y -axis is shared between these two plots .T he blue line depicts cell density derived using DropSignal (Doulcier 2019 ) and the shaded area represents the standard deviation (SD).Population lag time is inferred as described in text.The purple dotted line crossing N th = 1.6 × 10 8 cells ml −1 (64 000 cells per dr oplet) extr a polates the exponential gr owth bac k to its intersection with the inoculum density (purple horizontal dotted line), giving τ ≈ 5 h.The red line gives the deri vati ve of the time series, with shaded SD, and corresponding right y -axis in red.cells ml −1 .By r earr anging the equation for exponential growth: N th = N 0 e λ(t th −τ ) , and making τ the subject (1) A geometrical counterpart of Equation ( 1 ) gives the population lag time as the time point at which exponential growth (line in semi-log scale) intersects the horizontal line, which depicts inoculum density (see Fig. 1 C).These measurements of lag-times provide a wealth of quantitative information on the inoculum effect, as described and inter pr eted in the following.

Dur a tion of lag phase depends on the number of founding cells
In Fig. 2 (A), the av er a ge la g time fr om thr ee independent experiments (colors) is shown as a function of inoculum size (diamonds).gives the probability that cell lag times assume a value less than or equal to the x -value .T he measured distribution is fitted to a log-normal distribution (red dotted line) with a mean of 6.8 h and a SD of 1.3 h.A Gaussian "de-blurring" applied to these data generates the true distribution of cell lag times (blue dotted line).Both models in (A) simulate populations founded by bacteria with lag times drawn at r andom fr om this corr ected distribution: cells gr owing independentl y in droplets (dashed blue) and cells dividing according to a signal from the leader cell (solid blue).
Lag time decreases monotonically from 6.4 ± 1.1 h for droplets inoculated with a single cell, to 4.4 ± 0.3 h for an inoculum size of 1024 cells .T he SD, r epr esented by the err or-bars in the figure, decr eases monotonicall y and slowl y with incr easing inoculum size.
We also examined the dependence of other growth parameters on inoculum size.Initial experiments sho w ed an effect on final cell density, ho w e v er, this was found to be a consequence of subtle differences in glycerol concentrations arising from dilutions of fr ozen gl ycer ol-saline stoc k cultur es used to pr epar e founding inocula.When corrected, no effect of inoculum size on final cell density was observed.This technical, but important experimental observation is explained in Figure S2 (Supporting Information).Additionally, no effect of inoculum size on mean growth rate was detected, although the variance across droplets decreased.Details ar e pr ovided in Figur e S3 (Supporting Information).
What might be the basis of the decrease in mean lag time with inoculum size?There are three mutually exclusive explanations, all recognize that populations of cells are heterogeneous with regard to individual cell lag times as a consequence of innate phenotypic variability.Explanation I posits no interaction among cells, with population lag time being set by an event equivalent to a selecti ve swee p, i.e. initiated by the cell (or cells) with the shortest cell lag time.
Explanations II and III involve interactions among cells and can be thought of in terms of two extremes of a continuum.Explanation II posits that all cells contribute equally to the production of some gr owth-stim ulatory factor.Explanation III r ecognizes that population lag time could be set by the cell with the shortest lag time and whose activity triggers division of other cells.We demonstrate below that distinguishing between these alternate explanations is possible via quantitative data obtained from the millifluidic droplet device.Making this distinction requires knowledge of the lag time distribution of populations founded by single cells.

Precise estimation of the distribution of cell lag times from inocula containing a single bacterium
To quantify the lag time of individual bacteria, 230 droplets w ere inoculated b y-on av er a ge-a single bacterium, r esulting in growth in 156 droplets (the inoculation of droplets follows a Poisson pr ocess).For eac h dr oplet, the la g time was estimated as in Fig. 1 (C).The resulting distribution of lag times is shown in Fig. 2 (B) (blue dots).In this case, the lag time of each population is clearly equal to that of the founding cell.We denote the single-cell lag by θ to distinguish it from τ of larger inoculum size that may be affected by cooper ativ e effects .T he heterogeneity of cell lag times is br oad, r anging fr om ∼4 to ∼12 h, with a mean value of m = 6.8 h and SD σ = 1.3 h.A Sha pir o-Wilk test applied on the logarithm of the data r e v eals the underl ying distribution can be log-normal (see also the quantile-to-quantile plot shown in Figure S7, Supporting Information).Fitting a log-normal function (green dashed line in Fig. 2 ) yields parameters μ = 1.9 and s = 0.2.
Although the fit is good, there is uncertainty in the estimation of lag times due to measur ement err ors that pr opa gate to the extr a polation of Equation ( 1).This equation expresses the dependence of lag time on parameters t th , N 0 , N th , and λ for droplet populations, including the special case of a population being founded by a single cell.Expanding it to a Taylor series and assuming independent variables allows the uncertainty θ to be calculated as where t th , N th , N 0 , and λ correspond to the uncertainty of t th , N th , N 0 , and λ, r espectiv el y.Giv en the values of these uncertainties, we estimate θ = 0.88 h (see "Materials and Methods" for details of calculations).
The uncertainty associated with dir ect measur ements blurs the "true" distribution of single-cell lag times, which is less dispersed, i.e. has a smaller SD.We assume a Gaussian noise of zero mean and a SD equal to the measurement uncertainty σ noise = θ.Deconvolution of the Gaussian noise from the measured distribution (Blackwood 1995 ) amounts to subtracting its mean and v ariance fr om that of the measur ements The corrected distribution remains lognormal, with parameters Expression of the true probability density of lag time is thus The red dotted line in Fig. 2 (B) depicts the corresponding CDF after correction for measurement noise; it is narro w er than that obtained by dir ect measur ement.This distribution can now be used to examine the pr e viousl y pr oposed explanations for the dependence of population lag time on inoculum size.

A s w eep initia ted b y cells with the shortest lag time is inconsistent with the data
Intuitiv el y, one might ima gine that the lar ge v ariability in singlecell lag-times is sufficient to account for the observed inoculum effect, e v en for independently growing cells: larger inocula contain outlier cells that are fast to resume growth; these could in principle take over the population and reach maximal cell number faster, causing the observed inoculum effect.Having a precise estimate of the single-cell lag time distribution, it is now possible to put this hypothesis (Explanation I) to quantitative testing.
To this end, growth of populations within droplets established fr om differ ent numbers of founding cells wer e sim ulated and the match with experimental data determined.Virtual droplets were founded by cells with lag times drawn from the true distribution (shown in Fig. 2 B) and with exponential gr owth r ates dr awn from the measured distribution (see Figure S6, Supporting Information).Note that addition or not of the weak correlation between lag time and growth rate seen Figure S6 (Supporting Information) does not affect the conclusion of the simulation (code provided in Appendix, Supporting Information).Cells were then allowed to replicate within droplets.To mimic the experimental protocol, the time t th at which populations reach N th = 64 000 cells (equal to a density of 1.6 × 10 8 cells ml −1 ) was determined.Equation ( 1 ) was then used to calculate the lag time of each population with known N 0 and with known mean growth rate λ.The blue dotted line in Fig. 2 (A) shows the results of these simulations.
In marked contrast to the experimental results, these simulations of independent (non-interacting) cells show almost no dependence of the mean population lag time on inoculum size.In addition, the decrease in variation across droplets, represented by the SD of lag time (shaded blue region around dotted line), decr eases r a pidl y, wher eas in the experimental data the SD decr eases m uc h slo w er.
Failure of Explanation I to account for the data can also be understood by a simple calculation based on the growth c har acteristics .T he corrected CDF of the single-cell lag times, Fig. 2 (B), has a value of 0.025 for lag time 5 h.In other or ds, in a droplet inoculated by 1024 cells, ≈25 cells (0.025 × 1024) have a lag time equal to or shorter than 5 h.Similarly, the number of cells exiting lag phase between 5.8 and 7.8 h (around the mean 6.8 ± 1 h) is: (0.86 − 0.14) × 1024 ≈ 737 cells.Given the single-cell growth rate λ = 0.84 h −1 (Figure S6, Supporting Information), the generation time is g = 0.83 h.T hus , the 25 cells that start dividing before 5 h go thr ough r oughl y (6.8 − 5.0)/ g ∼ 2 gener ations befor e the 737 cells around the mean start dividing.A total of two generations of 25 cells yields 100 offspring; ther efor e, clearl y cells with a short cell lag time do not exert a dominant sweep effect on the population.
T he abo ve estimate implies that, during the time of the measur ed gr owth, man y linea ges within dr oplets gr ow sim ultaneousl y and produce offspring.T herefore , if cells are independent, the population lag time is expected to be r oughl y equal to the mean of the single-cell lag time distribution, which is independent of inoculum size as the simulation shows .Moreo ver, as an a verage ov er man y cells, the lag time variability across droplets should decrease as 1 / √ N 0 with inoculum size.Indeed, the SD of the simulation shown in Fig. 2 (A) decreases rapidly with inoculum size, in marked contrast to the experimental data-which decrease slowl y.This discr epancy of the variance behavior with inoculum size indicates that population lag time does not arise as an avera ge ov er independent cells in the inoculum.Alternative scenarios where cells are not independent are considered below.

A leader cell triggering end of lag time for the population is consistent with the data
We now turn to test Explanations II and III that involve interactions among cells within the founding inoculum.At one extreme case (Explanation III), collective growth is governed by a single e v ent that sync hr onizes population gr owth to the cell with the shortest lag time .T his would happen if the cell that divides first signals this e v ent to other cells, suc h that the entire population exits lag phase almost sim ultaneousl y.We first examine the consequences of this assumption and compare it to the data, and then consider the alternative scenario, namely, Explanation II, in which interactions among cells involve all cells contributing equally to the production of a growth-stimulating factor.
In statistical terms, we assume that an inoculum of N 0 cells is a random sample from the single-cell lag time distribution f ( θ).If there is a leader cell that triggers growth for all other cells, the measur ed population la g time will be equal to the shortest cell lag time in the sample , θ min .EVT pro vides a framework for statistical analysis of the extreme value of a sample, such as the shortest lag time θ min among N 0 cells (Embr ec hts et al .2013 ).In the limit of lar ge samples, EVT pr edicts the dependence of the mean and variance of a collection of θ min coming from samples of size N 0 .It also predicts that the distribution of θ min from populations founded by cells of a given inoculum size will approach a limiting fixed shape after a ppr opriate normalization as the sample size increases; the pr ecise sha pe is determined by global properties of the single-cell distribution f ( θ).
The unique features of our experiment create an ensemble of droplets with controlled inoculum size, and a measurement of the population lag time for each, labeled τ .These data provide the statistical properties required to test the hypothesis that τ = θ min ( N 0 ), namely that the population lag time is equal to the minimum cell lag time among the N 0 single cells of the inoculum.For this, we use predictions given by EVT on the distribution of θ min and ask whether they are consistent with the statistical properties of τ as measured in the droplets.
The first prediction is that both the av er a ge and the SD of θ min fr om populations decr ease slowl y with inoculum size N 0 .The precise scaling is derived from the single-cell distribution f ( θ) (see Appendix, Supporting Information); for a lognormal distribution we find the scaling to be where A and B are constants.Both predictions agree well with the population lag time τ .Fitting the curve of Equation ( 6) to the experimental relationship between mean population lag time ( τ ) and inoculum size, r e v eals a close match (Fig. 3 A).The same holds for the SD fitted to Equation ( 7 ) (Fig. 3 B).We note that although testing this prediction involves fitting constants, the dependence on sample size N 0 through ln ( N 0 ) is nontrivial and unique to the predictions of EVT.
A further prediction is that the distribution of minimal values ( θ min ), drawn from different sample sizes, tends to a universal shape in the limit of lar ge samples.Although, strictl y speaking, this holds in the limit N 0 → ∞ , in practice it may be expected to hold also for finite samples-e v en as small as se v er al dozen.For each sample size N 0 , our experiment provides a distribution of population lag times ( τ ), estimated over all droplets with the same inoculum size .T hese CDFs are depicted in Fig. 3 (C) for all inoculum sizes of at least four cells.To test whether the prediction on θ min holds for the population lag time τ , we normalize each CDF of τ by subtracting its mean and dividing its SD. Figure 3 (D) shows the result of this normalization and demonstrates that the distributions of τ collapse on one another, consistent with our hypothesis τ = θ min .The lightest shaded curv e, corr esponding to inoculum N 0 = 4, deviates from the rest-possibly indicating that this sample size is too small to acquire the limiting shape.The universal distribution itself is also predicted by EVT (Wikipédia 2018 ).Its CDF has the form with location and scale parameters θ 0 , γ , and a shape parameter k that reflects properties of the parent distribution f ( θ ), specifically the decay at its tails.Fitting the normalized data with this formula r e v eals an excellent match between the universal distribution formula (white dashed line in Fig. 3 B) and the normalized measured distributions of τ (green lines), at least for inoculum sizes above four cells per dr oplet.The anal ytical form ula for the distribution justifies the empirical pr ocedur e of normalizing by sample mean and SD used above as a test for the universal shape (see Appendix, Supporting Information).As a corollary of the predictions in Equations ( 6) and ( 7), the variance and mean of the distributions of extreme values drawn fr om differ ent sample sizes follo w a w ell-defined relationship (see Appendix).The a gr eement of this r elationship with the population lag time data is shown in the inset of Fig. 3 (B).
Taken together, the scaling of the mean and SD of τ according to the inoculum size [Equations ( 6) and ( 7)], the resulting relationship between variance and mean of τ , the collapse of normalized distributions of τ at different sample (inoculum) sizes, and the fit of the normalized distribution to the theoretical formula Equation ( 8 ), are consistent with population lag time being equal to the shortest cell lag time in the inoculum: τ = θ min .With our understanding that gro wth inv olves multiple simultaneous lineages, we conclude that, at the time of the shortest lag time in the inoculum, many cells must start growing in parallel.

A single leader cell determines population lag time
The a gr eement of statistical pr operties with pr edictions fr om EVT suggest that exit from lag phase is triggered by a single eventpossibly a single leader cell-that signals the exit from lag to all other cells.To test this hypothesis, we performed a set of simulations where cells are not independent.As for previous simulations, at each inoculum size thousands of virtual populations wer e gener ated (see sim ulation code in the Appendix).For each inoculum, the cell lag time of each founder cell was drawn at r andom fr om the experimental single-cell la g time distribution (Fig. 2 B) and then set to the shortest cell lag time in the sample.This means that all founder cells begin to pr olifer ate at the same time as the leader cell with population lag time τ = θ min .As in the pr e vious sim ulation, the numerical population w as allo w ed to grow exponentially with population lag time being estimated as per the experiments.
The results are depicted by the solid blue line in Fig. 2 (A) and are a close match to the experimental data over three orders of magnitude in inoculum size.Note that this is not a fit: the only input is the true single-cell lag distribution measured for populations founded by a single cell (Fig. 2 B).Additionall y, r esults fr om the sim ulations matc h the slow decr ease of la g time v ariability among dr oplets observ ed in the experiments (shaded area Fig. 2 A).Our r esults, thus pr ovide an explanation for the relationship between size of the founding inoculum and population la g time, whic h ar e fully consistent with resumption of growth of all cells in the inoculum being triggered by a leader cell.
Thus far, there remains uncertainty as to whether population gr owth is trigger ed by a single cell, or a small group of cells.To in vestigate , we performed further simulations (see code in Appendix) in which cells produce a growth activator after exit from lag phase .T he activ ator triggers end of la g phase for all cells in the inoculum upon passing a threshold.To have an effect, leader cells must activate growth of neighboring cells .T he time to reach the threshold is driven by two parameters related to the activator: the concentration threshold and rate of production.The ratio of the concentration threshold over the rate of production scales with time to r eac h the concentration threshold.For an effect to be evident, time to reach the threshold must be less than the lag time of neighboring cells (here 6.8 h on a verage).T hus , to study the influence of the time to r eac h the threshold it is necessary to fix the production rate and vary the concentration threshold (or the other way around).In the following sim ulation, we c hose to fix the production rate and vary the concentration threshold.
Giv en lac k of knowledge concerning the natur e of the activ ator, we assume that the rate of production is equal to the population gr owth r ate.Her e, the concentr ation thr eshold determines the fraction of cells in the inoculum that affects exit fr om la g phase.If the threshold is low, then production of growth activator by a single cell is sufficient to trigger end of lag phase for the entire population.If the threshold is high, then it is likely that multiple cells contribute to production of the gr owth activ ator.The dur ation of lag phase for e v ery cell was derived from experimental data as abo ve , and set by dra wing a random value from the single-cell lag distribution Fig. 2 (B).
We performed simulations for a range of activator thresholds at different inoculum sizes .T he results are depicted in Fig. 4 .First, it is seen in Fig. 4 (A) that a strong dependence of population lag F igure 4. Ho w many leader cells?Results of simulation in which each cell produces a growth activator as it exits lag phase , dra wn from the experimental distribution.The activator accumulates to a critical threshold and triggers end of lag phase for the entire population.(A) Population lag time as a function of inoculum size ( x -axis) and threshold of growth activator ( y -axis).(B) Number of leader cells that have exited lag phase before the critical activating threshold was reached.Note that the color-bar corresponds to a narrow range of between 1 and 5 cells.time on inoculum size appears only when the threshold concentr ation of gr owth activ ator is low.Giv en the significant inoculum effect observed in our experiments, we conclude that this result is consistent with our data only in the region of a low activation threshold.Second, Fig. 4 (B) shows the effecti ve n umber of cells that "lead" the population to exit from lag phase .T his number is defined by those cells, which had already reached the end of their individual lag time as dr awn fr om the single-cell distribution, before the threshold was reached.Strikingly, we see that for a large range of activator concentrations, only a single cell has time to exit lag phase before the critical concentration of activator is reached.These simulations support the conclusion that our experimental observ ations ar e consistent with Explanation III, where a single leader cell ends lag phase for the entire population.

Discussion
An inverse relationship between the number of bacterial cells founding a new environment and the time to exit lag phase was first noted more than 100 years ago (Rahn 1906, Penfold 1914 ).Despite its significance, rigorous validation has been lacking, and understanding of the causes and controlling factors remains incomplete.P aucity of pr ogr ess has stemmed lar gel y fr om difficulties associated with experimental analysis of populations founded by few cells.
Her e, taking adv anta ge of ne w opportunities presented by millifluidic technologies (Baraban et al . 2011, Boitard et al . 2015, Cottinet et al . 2016 ) we have obtained quantitative evidence from highl y r eplicated populations founded by contr olled numbers of cells, that in populations of P. fluorescens SBW25, the time to resume growth after transfer to a new environment is strongly influenced by size of the founding inoculum.Mor eov er, the same technology has allo w ed determination of the duration of lag phase for a sample of individual cells .T he a v er a ge decr ease in time to gr owth r esumption, v ariance acr oss dr oplets, and distribution sha pes, follow pr edictions of EVT, consistent with the inoculum lag-time being determined by the minimum value sampled from the single-cell distribution.At the same time, within droplets, growth of multiple cell lineages in parallel contribute to population expansion with no single lineage providing a disproportionate effect on duration of lag time.Our experimental results, thus show that exit from lag phase depends on str ong inter actions among cells, suggesting that a "leader cell" triggers end of lag phase for the entire population.
This finding builds on recent work in which the time to first division of single bacteria maintained in isolated cavities of microfluidic devices has been measured (Julou et al . 2020, Moreno-Gámez et al . 2020, Ş im ş ek and Kim 2019 ).From such studies, it is clear that there is substantial variation in cell-level lag time with evidence that this variance can have profound fitness consequences for population growth.For example, in fluctuating environments, heterogeneity in the time for individual cells to resume growth, can facilitate survival in the face of environmental change (Julou et al . 2020 ), especially that wrought by periodic antibiotic stress (Fridman et al . 2014, Ş im ş ek and Kim 2019, Moreno-Gámez et al . 2020 ).
Although microfluidic chambers used for analysis of isolated cells allow precision measures of the distribution of lag times for single cells, such experimental devices do not allow for interactions among cells, thus making problematic any attempt to connect the distribution of single-cell lag times to population lag times.In fact, extr a polation of population la g times fr om knowledge of the distribution of single-cell lag times would be justified only in the case of independent cells.
Linking cell and population le v el behaviors necessarily requires measures of lag times both for individual cells and for populations in pr ecisel y the same envir onment.Mor eov er, the envir onment should be well-mixed (spatially homogeneous and devoid of surface effects) so that should emer gent population-le v el behaviors be r ele v ant, mediated via, e.g.pr oduction of diffusible gr owth factors (Lankford et al . 1966 , Ka pr el yants andK ell 1996 ), then effects can be observed.In this regard, the millifluidic device has pr ov en fit for purpose.
In seeking an explanation for the observed inoculum effect, we consider ed thr ee m utuall y exclusiv e explanations.Centr al to Ex-planation I was absence of interactions among cells with the inoculum effect being explained by disproportionate growth of a subset of cells with the shortest time to first cell division.Both simulations and simple calculations led to unequivocal rejection of this hypothesis.
Explanations II and III recognized the possibility of interactions among cells.Because of the po w er of EVT, combined with wellunderstood statistical properties, we chose to focus on whether population lag times were determined by the minimum value sampled from the single-cell distribution (Explanation III).EVT makes predictions as to the distribution of minimal cell lag times acr oss dr oplets, whic h sur prisingl y, hold for the distribution of population lag times measured in experiments, leading to the conclusion that population lag time is equal to the minimal cell lag time present in the inoculum.Simulations of population growth in droplets based on this evidence delivered an almost perfect match to experimental data.While conformity to Explanation III means that Explanation II in a strict sense (in which all cells contribute equally to exit from lag phase) cannot hold, the fact that there is a continuum of possibilities led us to perform additional simulations to address whether our data are consistent with resumption of growth being triggered by just a single leader cell.In these simulations, a gr owth activ ator was assumed to be produced by all cells as they exited lag phase, but was r equir ed to accum ulate to a threshold before all other cells started growing; this interpolates between a single leader cell and a contribution from all cells, depending on the threshold level.We found that a strong decrease of population lag with increasing inoculum size was r epr oduced in sim ulations, but onl y when the thr eshold w as lo w.In the rele v ant par ameter r egion wher e the inoculum effect matched the experiment, we found that only a single cell contributes to the production of growth activator.
In our simulations, it was assumed that droplets are well-mixed (see also Figure S13, Supporting Information) so that the time for transport of growth activating molecules is negligible.In reality, three times scale are relevant: the time it takes for single cells to exit lag phase (reported in Fig. 2 B at 6.8 h on av er a ge), the r ate of production of the growth activating molecule, and the time it takes for the activator to pr opa gate thr ough the population.Transport is dominated either by diffusion or convection, with the former gener all y slo w er than the latter.In our droplet-based system, continual movement of droplets (and thus stirring of contents) points to convection as the primary mechanism of signal pr opa gation; thus, as soon as the signal is produced, it is likely to spread in the droplet and be sensed by all other cells .Con vection is r ele v ant in man y cases, including stirr ed r eactors, moistur e dr oplets on food, local envir onments within eukaryotic hosts, mo ving water bodies , and e v en undisturbed culture flasks (Ardré et al . 2019 ).In the absence of convection-and in environments where the time to transport effector molecules is longer than the intrinsic lag time of cells-the inoculum effect may be of less significance.
The rate at which the signaling molecule is produced is a k e y factor especially in environments of large volume.For instance, if the rate of production is low, then the leader cell will fail to produce sufficient signaling molecule to trigger neighboring cells.Under such circumstances, no inoculum effect will be observed and resumption of population growth will be determined by the intrinsic lag time of each cell.In our simulations (Fig. 4 ), we assumed that the pr oduction r ate is equal to the population gr owth r ate (Figur e S3, Supporting Information), which while a r easonable assumption, lac ks, at the pr esent time, a mec hanistic basis.
An obvious next question concerns identity of the growth activ ator.While detailed inv estigations ar e beyond the scope of this study, we nonetheless, considered the possibility that iron chelation might trigger the effect.Such a possibility has been pr e viously suggested (Kaprelyants and Kell 1996 ).To this end, we repeated the initial experiment in which the time to resumption of growth was determined in replicate populations founded by different numbers of cells as in Fig. 2 (A).Instead of SBW25, a mutant deficient in production of the ir on-c helating compound p y over din w as used: P. fluorescens SBW25 pvdS G229A (Zhang and Rainey 2013 ).No change in the inoculum effect was observed (see Figure S4, Supporting Information), thus ruling out p y over din as the activating molecule.We next asked whether the inoculum effect might be eliminated by addition of culture supernatant, deriv ed fr om an ov ernight cultur e of SBW25 grown in CAA, to populations of cells in droplets .T he ensuing data show that indeed cultur e supernatant significantl y dampens the inoculum effect (Figure S11, Supporting Information).This is consistent with Lankford et al . ( 1966 ) and Ka pr el yants and K ell ( 1996 ), who observ ed an inoculum effect in flasks that could be abolished by addition of culture supernatant.
A further set of factors that stand to influence the inoculum effect, stem from the en vironment.T he effect of differences in chemical composition of gro wth media, gro wth stage of cells, and external str essors ar e pr esentl y unknown, but the subject of curr ent inv estigation.
The relationship between the number of cells founding growth in a new environment and duration of lag phase has profound implications for microbiology.While much remains to be understood, including generality and molecular bases, the rigorous quantification ac hie v ed her e pr ovides unequivocal e vidence of an inoculum effect in P. fluorescens SBW25.Mor eov er, w e sho w that the effect is best understood in terms of interactions among cells with statistical analysis of the distribution of population lag times indicating that a single leader cell is sufficient to trigger simultaneous growth of all cells in the founding population.

Figure 1 .
Figure 1.Bacterial population growth in dr oplets.Subfigur e (A) shows two droplets of 0.4 μl are separated by an air spacer (to prevent droplet coalescence) inside the tube of a millifluidic mac hine.Dr oplets ar e pr epar ed by "sipping" samples from a 96-well plate.Typically, 230 dr oplets ar e pr oduced fr om six seed cultur es that differ solel y in the number of founding cells (the inoculum ).Each seed culture delivers 40 r eplicate dr oplets, but for tec hnical r easons that last deliv ers 30 r eplicates.Dr oplets mov e bac k-and-forth, via c hanges in pr essur e, passing in front of a fluorescence detector every ∼18 min.Pseudomonas fluorescens SBW25 cells express GFP from a chromosomally integrated r eporter, allowing c hanges in biomass to be determined based on intensity of the fluorescent signal (excitation at 497 nm emission at 527 nm).Signal intensity is calibrated to cell density by plate counting (FigureS8, Supporting Information).The range of detection extends from 4 × 10 6 to 5 × 10 9 cells ml −1 (1.6 × 10 3 to 2 × 10 6 cells per droplet).T he gra y ar ea in subfigur es (B) and (C) denotes the r egion wher e bacterial density is below the threshold of detection.(B) Fluorescent signal across time from 40 replicate populations (in semi-logscale) in dr oplets pr epar ed fr om the same seed cultur e .T he a v er a ge inoculum in eac h dr oplet is N 0 = 1.6 × 10 5 cell ml −1 , or 64 cells per droplet (this concentration is marked by the purple dashed line that goes across (B) and (C)).In this example, the signal exceeds the detection threshold at ∼7 h, by which populations are in exponential growth phase.At ∼20 h, stationary phase is r eac hed, marked by cessation of growth.(C) A single time series showing population growth within a single droplet coming from the set of replicates shown in (B).The left y -axis is shared between these two plots .T he blue line depicts cell density derived using DropSignal(Doulcier 2019 ) and the shaded area represents the standard deviation (SD).Population lag time is inferred as described in text.The purple dotted line crossing N th = 1.6 × 10 8 cells ml −1 (64 000 cells per dr oplet) extr a polates the exponential gr owth bac k to its intersection with the inoculum density (purple horizontal dotted line), giving τ ≈ 5 h.The red line gives the deri vati ve of the time series, with shaded SD, and corresponding right y -axis in red.

Figure 2 .
Figure2.Quantitative data on lag-times are consistent with a strongly cooper ativ e exit from lag phase.(A) Population lag time τ as a function of inoculum size for three independent experiments (colors).Symbols are the mean lag times over droplets with a given inoculum size, with error bars denoting the standard deviation (SD).The data are compared to two models (blue lines-av er a ge, shaded blue-SD).(B) Cum ulativ e distribution (CDF) of single-cell lag times ( θ) from 156 droplets inoculated with a single bacterium on av er a ge (dark line).The y -value gives the probability that cell lag times assume a value less than or equal to the x -value .T he measured distribution is fitted to a log-normal distribution (red dotted line) with a mean of 6.8 h and a SD of 1.3 h.A Gaussian "de-blurring" applied to these data generates the true distribution of cell lag times (blue dotted line).Both models in (A) simulate populations founded by bacteria with lag times drawn at r andom fr om this corr ected distribution: cells gr owing independentl y in droplets (dashed blue) and cells dividing according to a signal from the leader cell (solid blue).

Figure 3 .
Figure 3. Statistical properties of lag times.For three independent experiments (colored symbols), mean lag times (A) over populations and their SD (B) are depicted as a function of inoculum size.Each point is calculated over 40 replicate populations (droplets).Inset: variance as a function of mean.The scaling relations predicted by EVT are shown in dashed lines: y = 6 .84 −0 .86 ln N 0 for the mean and y = 1 / ln N 0 for the SD.(C) Cum ulativ e Distributions of population la g times for differ ent inoculum sizes [ N 0 , colors; legend in (D)].The curves derive from the pooled data of three independent experiments yielding at least 120 population lag times for each.(D) Same data as in (C), scaled by subtraction of empirical mean and division by SD.The white dashed line depicts the fit by the universal distribution predicted by the EVT.The y -axis is shared between (C) and (D).