Mapping single‐cell responses to population‐level dynamics during antibiotic treatment

Abstract Treatment of sensitive bacteria with beta‐lactam antibiotics often leads to two salient population‐level features: a transient increase in total population biomass before a subsequent decline, and a linear correlation between growth and killing rates. However, it remains unclear how these population‐level responses emerge from collective single‐cell responses. During beta‐lactam treatment, it is well‐recognized that individual cells often exhibit varying degrees of filamentation before lysis. We show that the cumulative probability of cell lysis increases sigmoidally with the extent of filamentation and that this dependence is characterized by unique parameters that are specific to bacterial strain, antibiotic dose, and growth condition. Modeling demonstrates how the single‐cell lysis probabilities can give rise to population‐level biomass dynamics, which were experimentally validated. This mapping provides insights into how the population biomass time‐kill curve emerges from single cells and allows the representation of both single‐ and population‐level responses with universal parameters.


Introduction
Beta-lactam antibiotics are widely used to treat bacterial infections (Hamad, 2010). Studies have shown that beta-lactams exhibit both time-and antibiotic concentration-dependent killing (Wong & Amir, 2019). More specifically, two salient dynamic features of time-dependent killing were demonstrated in time-kill curves at a lethal antibiotic concentration. First, the total population biomass will first increase before collapsing, leading to an apparent time delay in the effect of the antibiotic (Craig, 1995(Craig, , 1998. Second, the maximum lysis rate is linearly correlated to the growth rate of the population before antibiotic treatment (Tuomanen et al, 1986;Lee et al, 2018). Quantitative measurements of these dynamics have been shown to be important for designing effective antibiotic treatment protocols (Meredith et al, 2015;Lee et al, 2018). However, it is unclear how these population-level features could emerge from the collective responses of single bacterial cells undergoing antibiotic treatment (Wong & Amir, 2019).
At single-cell level, exposure to beta-lactam antibiotics often results in morphological changes in many rod-shaped Gramnegative and Gram-positive bacteria (Elliott & Greenwood, 1983;Eng et al, 1985;Mason et al, 1995;Chen et al, 2005;Buijs et al, 2008;Paulander et al, 2014;Kjeldsen et al, 2015;Cushnie et al, 2016;preprint: Cayron et al, 2020). One of the known mechanisms of beta-lactam-induced filamentation is cell wall synthesis inhibition, when a beta-lactam binds to penicillin-binding proteins (PBPs) and inhibits peptidoglycan cross-linking (Chung et al, 2009;Cho et al, 2016;Vigouroux et al, 2020), while the cells continue to accumulate biomass (Rolinson, 1980b;Choi et al, 2014). As a result, individual cells elongate exponentially without cell division and become filamented until the accumulation of cross-link defects creates a pore where a bulge forms . Due to the loss of cell-wall integrity, filamentation is eventually followed by rapid cell lysis, consisting of cell membrane bulging through the cell wall pore and subsequent bursting of the cell membrane Spratt, 1975;Daly et al, 2011;Yao et al, 2012;Cho et al, 2014;Wong & Amir, 2019;Zahir et al, 2020). The time for each cell to lyse varies under prolonged antibiotic exposure (Ş imşek & Kim, 2019). Additionally, beta-lactam-induced filamentation can be reversible: if the antibiotic is removed before lysis occurs, filamented cells can divide into multiple cells (Chen et al, 2005;El Meouche et al, 2016;Zahir et al, 2019Zahir et al, , 2020, with the resulting cell number being roughly proportional to the filament length (Wehrens et al, 2018). Therefore, bacterial filamentation affects population recovery (preprint: Cayron et al, 2020). These observations underscore the importance in quantifying key parameters of cellular lysis, such as single-cell filament lengths and at what length cells burst, to couple with temporal population biomass dynamics.
Despite the evident connection between single-cell filamentation and population dynamics, the lack of quantitative analysis prevents a clear understanding of how the population dynamics emerge from the collective elongation and lysis of the single cells. Here, we found that the lysis probability increases with the extent of filamentation by measuring E. coli single-cell filamentation and lysis dynamics using time-lapse microscopy. We found that key parameters of the lysis probability are unique to bacterial strain and antibiotic dose. We further show how the single-cell elongation and lysis parameters can explain different, experimentally measured population dynamics resulting from beta-lactam treatment ( Fig 1A).

The probability of antibiotic-induced lysis depends on filamentation length
We tracked beta-lactam-mediated filamentation of E. coli MG1655 cells, which are rod-shaped Gram-negative bacteria, using timelapse microscopy. Our measurements confirmed a constant rate exponential elongation before burst (Fig 1B; Appendix Fig S1), which are reported previously (Rolinson, 1980a;Yao et al, 2012;Lee et al, 2016). Cells in exponential growth phase were loaded onto a thin agarose gel containing growth medium and beta-lactam antibiotics. We then collected initial and final lengths over time ( Fig 1C). We defined the final length as the longest length of a cell in a given time interval for two reasons: (i) cells elongated only in the long-axis direction and shrunk when the cell wall burst (Yao et al, 2012;Shi et al, 2021), and (ii) once bursting was observed, lysis took place in a short time (Spratt, 1975;Yao et al, 2012;Zahir et al, 2020). For example, one of the cells we observed had filamented for 100 min, followed by bulging and lysis in the next 10 min: bulging to lysis duration was 10 times shorter than the duration of filamentation ( Fig 1B).
Our measurements showed that lysis probability depends on the extent of elongation. To find single-cell lysis kinetics in cell length during elongation, we first measured the lysis probability density (ρ L ) from the final lengths, which represents the fraction of cells with a certain final length before they lysed. We observed that ρ L is small for both short cells and long cells and peaks at intermediate cell lengths. A small ρ L at a small L reflects a small lysis probability for a short cell. A small ρ L at a large L, however, does not imply a small lysis probability for a long cell, but rather the rarity of such cells due to lysis before reaching a long length. The shape of the dependence of ρ L on L suggests the extent of elongation is predictive of the likelihood of lysis.
We next constructed the cumulative lysis probability (P L ) from ρ L , that is, P L ¼ R L 0 ρ L dL, which is the probability for a cell to lyse before reaching length L (Fig 1C). We empirically chose to fit the sigmoidal dependence of P L on L using a Hill equation (corresponding to a log-logistic distribution): (1) We term L C the critical length; 50% of cells would lyse before reaching this length and ρ L peaks at L C . The Hill coefficient, H, represents the steepness of P L . This fitted equation is phenomenological; the sigmoidal curve can be fit to other equations. However, as we shall see, the choice of the Hill equation provides a simple analytical interpretation of the population dynamics during antibiotic treatment. The dependence of lysis probability on extent of elongation is not unique to cells treated with a beta-lactam: it also applies to E. coli cells treated with another cell-wall synthesis inhibitor, Dcycloserine, according to our analysis of the raw data provided in a study (Vigouroux et al, 2020) (Appendix Fig S2A-C).

The critical length depends on the antibiotic dose
The lysis probability exhibited the same dependence on filamentation length in different antibiotic and growth conditions. We exposed bacteria to three different carbenicillin concentrations (20, 50, and 100 μg/ml) and incubated at two different temperatures (27 and 37°C) (Fig 2A; Appendix Fig S3A-D). Temperature changed elongation rate but did not affect L C . However, L C decreased with increasing carbenicillin concentration, with the correlation coefficient c = -0.9103 and the fitted inverse correlation ( Fig 2B). This inverse correlation confirms that cells can better tolerate lower-dose antibiotics and elongate further. Moreover, the correlation between initial and final lengths was weak under all conditions; that is, the elongation capacity was not defined by initial length during antibiotic exposure (Appendix Fig S3A). H did not show a dependence on antibiotic dose or temperature changes. Figure 1. Mapping single-cell kinetics to population dynamics during beta-lactam treatment.

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A A schematic of single-cell versus population-level responses. Bacteria elongate in response to beta-lactam antibiotics. During prolonged exposure, lysis occurs when both cell wall and membrane integrity are broken. At the population level, the total biomass, which is a sum of survivors' biomass, experiences a transient increase before a decline due to single-cell filamentation and lysis. B Single-cell lengths were measured from time-lapse microscopy images upon beta-lactam antibiotic treatment. Initial cell length was measured in the first time frame.
Elongation ends with cell length shrinkage due to bulging. The longest length of each cell across the time frames was used as the final length. Tracked intermediate cell lengths were shown over time, while initial and final lengths were shown in larger circles. Linear regression of log-scaled lengths over time showed a constant rate of exponential elongation (solid gray line). Scale bar: 5 μm. C Single-cell lysis probabilities were plotted with the extent of filamentation. Red circle markers connected with lines show the measured initial and final lengths of 110 carbenicillin-treated individual cells over time, while gray lines show the length of 15 non-treated cells with division (left panel, log scaled in length). Probability density function (ρ L ) over final cell lengths was collected by normalizing the probability distribution (middle panel). Dot plotted cumulative lysis probability function (P L ) of the ρ L is shown, and was fitted to the log-logistic distribution in cell length (right panel).
We further tested if the length dependence of P L and inverse correlation of L C to antibiotic dose were maintained in various conditions: different beta-lactams, bacterial strains, and growth media. First, we observed the same dependence of P L on cell length when using two other beta-lactam antibiotics that exhibit the filamentation (cefotaxime at 20 and 100 μg/ml and amoxicillin at 6.25 and 25 μg/ml) at two temperatures (27 and 37°C) (Fig 2C; Appendix Fig S4A and B). For each antibiotic, the inverse correlation was consistently observed. Second, we found that the same trends were maintained for three clinical isolates of pathogenic E. coli strains ( Fig 2D; Appendix Fig S4C). These isolates are resistant to betalactams due to their ability to express extended-spectrum beta- lactamases (ESBL). We inhibited this resistance by using clavulanic acid, a well-established beta-lactamase (Bla) inhibitor, thus rendering the isolates susceptible to beta-lactams. Specifically, these isolates were exposed to two amoxicillin doses (6.25 and 25 μg/ml) in the presence of clavulanic acid (50 μg/ml) at 37°C. P L of the isolates were also sigmoidal with the extent of filamentation, and L C was shorter at higher antibiotic dose. Finally, we observed that the inverse correlation was maintained in different growth media (Appendix Fig S5). The final length decreased with increasing carbenicillin dose regardless of growth medium, though the absolute final lengths differed across media.
As a plausible interpretation of the dependence of P L on L, we hypothesized that cell wall integrity remains until the antibioitic-induced damage accumulates to a threshold number in association with elongation ( Fig EV1A; see Appendix Supplementary Methods for details of the model). The model formulation is based on the current understanding of the beta-lactam-induced cross-linking failure which causes cell wall bursting at varying lengths (Daly et al, 2011;Yao et al, 2012;Cho et al, 2016;Şimşek & Kim, 2019). Our model incorporates the notion that accumulation of cell-wall defects would lead to cell-wall crack formation before bursting (Huang et al, 2008). Briefly, our model assumes that damage (cross-linking failure) accumulation on a cell follows the Poisson distribution, while cell wall bursting happens when the number of damage reaches a certain threshold. Then, the model generates a gamma distribution of final length with a given threshold number (α) and a A Probability density functions of final lengths of MG1655 cells exposed to carbenicillin. Individual cell lengths were measured from populations exposed to three doses of carbenicillin (20, 50, and 100 μg/ml) at two temperatures (27°C in blue and 37°C in red). Line plots show the measured initial and final lengths over time in log scale, and the first 15 cells were picked in each dataset for presentation among n ≥ 100 of the tracked cells in each condition (top panels, see Appendix Table S1 for n). Probability density functions (ρ L ) of final lengths were collected with the same number of bins within the same final cell length range (bottom panels). The peak of ρ L shifted to the left with increasing carbenicillin dose. B The probability of lysis increased with the filamentation length at each condition. P L was calculated from ρ L in panel A (filled circle: 100 μg/ml, star: 50 μg/ml, and lined circle: 20 μg/ml). Solid lines represent fits using a log-logistic distribution function (R 2 > 0.99. See Appendix Table S1 for fitted parameters). L C was insensitive to temperature (27°C in blue and 37°C in red, carbenicillin doses in gray text), but was inversely correlated with carbenicillin (CB) dose (panel B inset: correlation coefficient c = -0.9103 and L C ¼ 389:7 1 CB þ 3:2, unit shown in the figure, R 2 > 0.9991). C The probability of lysis increased with the filamentation length of bacteria treated with other beta-lactams. MG1655 cells were treated with cefotaxime (20 and 100 μ g/ml) and amoxicillin (6.25 and 25 μg/ml, shown in gray text) at two temperatures (27°C in blue and 37°C in red). D The probability of lysis increased with the filamentation length of E. coli pathogens treated with a beta-lactam. E. coli pathogens expressing extended beta-lactam resistance were treated with amoxicillin (6.25 or 25 μg/ml, shown in gray text) and clavulanic acid (50 μg/ml) simultaneously at 37°C. Cells were sensitized by clavulanate acid, which inhibits beta-lactamases.
damage rate (β), which stands for the damage capacity per unit length and may increase with the increase in antibiotic concentration. The sigmoidal cumulative distribution of the gamma distribution may explain the experimentally observed sigmoidal P L ( Fig EV1B) The parameters of each distribution-α and β of gamma distribution and L C and H of Hill equation-are correlated: H is well approximated by ffiffiffi α p and L c by α β (Fig EV1C; see Appendix Supplementary Methods for detailed derivations). Therefore, L C and H may provide simple proxies for the mechanical aspects of single-cell lysis.

Predicting temporal dynamics of population-level responses
To map single-cell and population-level responses, we simulated and tracked the elongation and lysis of single cells in a population using a simple stochastic model (Appendix Fig S6A-C, see Materials and Methods for model details). We simulated cell elongation with a constant (but cell-specific) exponential growth rate (μ) (Rolinson, 1980b) and calculated P L (using Equation 1) and ρ L (¼ PL L ) at each cell length. For each cell, we generated a random number between 0 to 1. The cell is set to be lysed (with no biomass contribution to the population) when its instantaneous lysis rate becomes larger than the number. The instantaneous lysis rate at length L is called the hazard function (Gompertz, 1825;Collett, 1994), which is determined by P L and ρ L : We present the formal definition of the hazard function and the derivation of Equation 2 in the Appendix Supplementary Methods. Briefly, Equation 2 captures the following relationship: the probability of observed lyses between lengths L and L þ dL is equal to the probability of cells having reached length L multiplied by the probability for these cells to lyse between L and L þ dL. That is, Þ also allows us to compute how a population size changes as the function of the average cell length during antibiotic treatment (see below).
In this model, we assumed cell division stops in the presence of antibiotics but elongation continues (Wehrens et al, 2018;preprint: Cayron et al, 2020). Therefore, our stochastic single-cell length simulations show that the total biomass of the population exhibits a characteristic transient increase before a decline, while the cell number monotonically decreases over time (Appendix Fig S6B). These results have been observed in in vitro assays (Eng et al, 1985;Zahir et al, 2020). We define the characteristic time point when the total biomass decreases to that of the initial point as the effective treatment duration τ E ð Þ, as it represents the duration of antibiotic treatment sufficient to suppress the target population, despite the continued elongation of surviving cells.
Using a coarse-grained deterministic model, population biomass dynamics can be approximated by accounting for the average cell length as a function of t, L t ð Þ, and the total cell number as a function of L, N L ð Þ: Note that the derivative in Equation 4 is with respect to length, to follow the definition of P H L ð Þ. From Equations 3 and 4, we can derive the temporal dynamics of total surviving biomass, which should be proportional to LN. d LN ð Þ dt thus follows (see Materials and Methods for derivation): which is defined by H; P L , and μ. We note that P L , over time, increases and converges to 1 due to the definition of cumulative probability and the further elongation of cells. This deterministic model allows us to compute how the total biomass changes as a function of time with the average parameters of single cells (Fig 3A; Appendix Fig S7). Our coarse-grained model analytically demonstrates the time-and dose-dependent population biomass dynamics (Fig 3A). At the boundary condition, where total biomass is equal to the initial biomass ( , τ E increases with an increasing L C or a decreasing μ ( Fig 3B). Precisely, τ E satisfies the following: where L 0 is the initial cell length. Intuitively, LC L0 sets the limit to which cells can elongate before lysis. e μt indicates the extent of elongation. If t > τ E , the time duration of antibiotic treatment has exceeded the limit set by LC L0 . The larger μ is, the smaller τ E would be for the same LC L0 , and thus, the sooner the antibiotic will suppress the total biomass. Indeed, the immediate cell division and regrowth from elongated cells after few hours of antibiotic exposure has been reported (preprint: Cayron et al, 2020), which supports that the antibiotic did not thoroughly suppress the population in a short time. More importantly, filament length was roughly proportional to the number of daughter cells, following the conservation of biomass between filamented cells and their daughters (Wehrens et al, 2018).
We experimentally assessed the dependence of τ E on L C and μ, using different concentrations of carbenicillin and casamino acids to modulate L C and μ, respectively ( Fig 3C). Indeed, longer τ E was observed in lower antibiotic concentrations and in more slowly elongating populations. Our results establish a quantitative correlation between antibiotic dose and minimum killing time of the population.
Predicting the linear correlation between the population-level growth and lysis rates Equation 5 recapitulates the two salient features of population dynamics exposed to beta-lactams mentioned above. First, according to Equation 5, the total population biomass will increase until 1ÀHP L ð Þ< 0, which explains time-delayed lysis. As long as H > 1, 1ÀHP L will eventually become negative, as P L approaches 1. Second, the term μHP L in Equation 5 is the effective lysis rate of the population; it approaches μH as P L approaches 1. Thus, Equation 5 predicts a proportionality between the maximum growth rate (G ¼ μ) and the maximum lysis rate (D ¼ μH), with H being the coefficient. That is, D ¼ HG. This proportionality is exact if there is no cell-cell variability in initial cell lengths and growth rate of each cell. However, even when such variability is considered, our numerical simulations (Fig 4A and B) indicate that the proportionality is maintained. A caveat of these simulations and of Equation 5 is the assumption of H being constant at different growth rates. Our experimental measurements (Fig 2; Appendix Table S1) suggest moderately variable H. To test the effect of this variability, we conducted numerical simulations by using normally distributed H values (with a mean of 5 and a variance of 1). We then collected maximum growth (G) and lysis rates (D) of each population and conducted linear regression. Despite the variability in H, the proportionality between lysis rate and growth rate was approximately maintained; the slope (5.1) of the linear correlation was close to the mean H value (Fig 4C). Therefore, our model provides a simple, single-cell-based explanation for the emergence of a linear correlation between maximum population growth rate and lysis rate from previous studies (Tuomanen et al, 1986;Lee et al, 2018). We noticed that the fitted H values of the P L in Fig 2 were larger than those from previous experiments (Lee et al, 2018). On the one hand, this discrepancy reflects a potential limitation of our simplified model in quantitively matching the experimental data. On the other, it could also reflect a limitation in the resolution of the experimental data. For example, the residual biomass of lysed cells contributes to the optical density measurements, which could lead to an underestimation of the maximum lysis rates. Altogether, our results suggest that H poses a theoretical cap of lysis rates for different growth rates despite the variability. These results are along with our damage accumulation model, where the H approximates the threshold damage number (α) that may originate from biological mechanisms. A bigger α leads to a sharper response with a higher H (Fig EV1C).

Predicting the non-monotonic dependence of survivor cell lengths on antibiotic doses
Our measurements show that L C decreases with increasing antibiotic dose (Fig 2B, inset), which is likely due to the need to accumulate sufficient defects in the cell wall before it collapses. However, if the antibiotic dose is sub-lethal and the lysis probability therefore remains low, antibiotic exposure results in cells elongating longer on average before division or lysis relative to untreated cells. As such, for increasing antibiotic doses, we predicted that the average length of survivors would first increase and then decrease, that is, survivor length is biphasic (Fig 5A). Previous studies (Rolinson, 1980b;Chung et al, 2012;Fredborg et al, 2015;McLaughlin & Sue, 2018;Oh et al, 2020) have reported this nonmonotonic length dependence in antibiotic susceptibility testing but do not offer a mechanistic explanation. Figure 3. Single-cell lysis profiles predict temporal dynamics of population growth and lysis.

B A C
A Sample simulations of the temporal dynamics of total biomass. Three population simulations of different L C (marked near each plot, arbitrary unit) with slow (left panel) and fast (right panel) elongation rates are shown in time. The effective elongation duration point (τ E ), the point at which the total biomass is equal to the initial biomass, is depicted by a dot. The τ E occurs earlier in the population either with the faster elongation rate or shorter critical length. B The boundary of effective elongation condition depends on L C and τ E . Increased biomass is shown as the color of the heatmap. The boundary condition of τ E with respect to μ and L C is plotted in red, where the increased biomass is zero after initial points. C Experimental measurements confirm that τ E decreases with the increase of antibiotic dose and growth rates. For growth rate modulation, casamino acid at 0.02% (left panel) and 0.2% (right panel) were added to minimal media. Cells were first cultured without carbenicillin for 3 h (dashed line) and then exposed to carbenicillin (antibiotic doses are marked near each plot, μg/ml). The average of four technical repeats (gray lines) was plotted in a colored line. τ E points were marked by finding the first time point at which the averaged OD became equal to or less than the OD at the start of carbenicillin exposure.
To examine this hypothesis, we extrapolated L C of sublethal doses from the inverse correlation in Fig 2B and simulated length distributions of populations using our stochastic model mentioned above. We also ran separate cell length simulations for untreated cells, where cell division was enabled, using a previously established model (Tanouchi et al, 2015) (Fig 5B and C). After a short exposure (2 h), simulated cell lengths were similar across antibiotic doses because of low lysis probability for those short filaments (Fig 5B, left). After a prolonged exposure (6 h), the length of the survivors (cells with an intact cell wall) decreased with increasing drug concentration (Fig 5C, left). This is mainly because cell lysis is more probable at higher doses due to the higher cumulative lysis probability at a given length. Therefore, our simulation results recapitulated the biphasic cell length trend when increasing antibiotic concentration in the sublethal range and may explain the possible kinetics that causes the trend.
To experimentally verify the model predictions, we treated cells in liquid culture with low doses of carbenicillin (0, 5, 10 μg/ml for sublethal and 20 μg/ml for lethal, Appendix Fig S8) and imaged every 2 h starting from the time point when the antibiotic was added (Appendix Fig S9). Indeed, our single-cell simulations captured the biphasic trend of survivor lengths only in longer antibiotic treatment: after 2 h, the average lengths of surviving cells were similar for different antibiotic doses, and they were longer than in the absence of the antibiotic (Fig 5B, right); after 6 h, the biphasic dependence of average length of surviving cells on the antibiotic dose emerged (Fig 5C, right). We note that the quantitative aspects of experimental data, the absolute length of survivors, may differ from the simulated results due to extrapolation or different incubation conditions (liquid vs solid agarose gel). A Population biomass simulations in modulating elongation rates. Elongation rate modulation simulations were done with a controlled initial cell number (10,000 cells), initial length (4), and P L (L c = 16, H = 5). Temporal dynamics of normalized total biomass with constant elongation rates (0.1, 0.2, and 0.5) show a decrease in population biomass at different lysis rates. B The maximum lysis rate increases by H for each maximum growth rate. Linear regression of the maximum growth rate (G) and lysis rate (D) of each simulation in panel A shows the slope as 5.0, which perfectly matched the input H. C Lysis rate is linearly correlated to the growth rate. The linearity stays with the normalized distribution of H (inset histogram in panel c with a mean of 5), which was randomly introduced to the population simulations. The slope (5.1) of the linear regression was similar to the mean of randomized H. Khan et al, 2022). Our single-cell measurements reveal a robust dependence of lysis probability on cell length during antibiotic treatment. In particular, the cumulative lysis probability over length (P L ) showed a sigmoidal curve. We have shown that two coarse-grained empirical parameters, L C and H, characterize the P L , which was unique to cell strains, growth conditions, and the type and dose of antibiotics. These two parameters serve as a quantitative basis for interpreting well-documented population-level responses to betalactams. L C represents the critical length for single-cell filamentation. At a given antibiotic dose, L C does not change with elongation rate. It thus provides a simple explanation for why faster-growing cells are more susceptible to antibiotics: they reach L C earlier. Moreover, L C serves as the single-cell basis for the time-delayed killing by betalactams at the population level. L C and μ define the effective treatment duration (τ E ), which demonstrates the minimal duration needed for an antibiotic dose to effectively suppress bacterial growth (Fig 3).

Discussion
From the bacterial perspective, τ E represents the effective duration for the population to survive the antibiotic treatment, before the total biomass starts to decline. Previous studies have shown that an antibiotic-treated population can recover upon the removal of A Illustration of single cells and total biomass of populations during antibiotic treatment. At the population level, total biomass measurements of the sublethal dose antibiotic-treated population often match those of a non-treated population. Single-cell measurements, however, can distinguish the presence of antibiotics due to the elongation of individuals. Therefore, time-point single-cell measurements offer more detailed information about the lethality of antibiotics than populationlevel measurements. B, C Stochastic simulations and measurements of survivor length. Simulated survivor lengths were plotted using the predicted L C of lower doses of carbenicillin (left panels) at 2 h (B) and 6 h (C). The corresponding single-cell length measurements in vitro were plotted (right panels). In the simulation, only non-antibiotic treated cells were set to divide, when the cell length reaches the division length in the linear model (L d ¼ aL i þ b, a = 0.871, b = 2.7. Parameters were extracted from the original paper; Tanouchi et al, 2015). In both simulations and measurements, cells in carbenicillin-treated conditions were further elongated than control cells at all time points. At 2 h exposure, antibiotic-exposed cells showed similar lengths across antibiotic concentrations since lysis is not highly probable for short filaments. At 6 h exposure, survivor lengths decreased with increasing carbenicillin dose. Independent t-tests between non-treated and treated group showed extreme significance, ****P < 0.0001. The box represents the middle 50%, the central line represents the median, and the whiskers represent the lower and upper quartile of the data points (n > 300). , if the antibiotic is removed before τ E , filamentation (despite the associated risk of lysis) can allow a population to recover biomass more quickly than one consisting of cells that do not grow or die. L C also represents the lethality of the drug or susceptibility of bacteria in a given condition. Since L C was inversely correlated to antibiotic dose, our results suggest that 1 LC can directly report the killing capacity in a quantitative manner (Fig 2). The specificity of L C allows comparisons of antibiotic efficacy across different antibiotics and bacterial strains. For example, if all conditions except the antibiotic agent remain identical, the agent with a shorter L C is more detrimental to bacteria; if all conditions except bacterial strain remain the same, the strain with a shorter L C is more vulnerable to the given antibiotic treatment condition.
The Hill coefficient (H) of P L reflects the cell-to-cell variations in antibiotic killing kinetics. It is remarkable how the population-level metric (the linear correlation between growth and lysis rates) (Lee et al, 2018) quantitatively connects with a single-cell metric, H (Fig 5). The Hill coefficient measures the steepness of the response curve, and in P L , H shows how tightly the lysis is determined by cell length. Therefore, H can report the amount of cell-to-cell length variations in antibiotic-induced lysis: a smaller H corresponds a wider distribution of cell lengths before lysis. Such phenotypic heterogeneity in single-cell lysis dynamics has been shown to affect the emergence of antibiotic resistance (Andrews, 2001;Corona & Martinez, 2013;Artemova et al, 2015).
Additionally, our damage accumulation model suggests that H can emerge from underlying molecular events: the threshold number of defects (α) in the cell wall that triggers the collapse of the cell wall ( Fig EV1A). Our analysis showed that L C is inversely correlated to damage capacity per length (β), which increases with the antibiotic dose. That is, our work reveals the fundamental constraint of possible molecular mechanisms that underlie single-cell responses, as illustrated by our proposed damage accumulation model.
Together, our work provides the single-cell lysis kinetics and the analytical relationship between the kinetics and population biomass dynamics using coarse-grained modeling. The choice of Hill equation (corresponding to log-logistic distribution) to describe the cumulative distribution function of single-cell lysis is empirical and does not imply a molecular mechanism. However, the Hill equation provides a foundation for establishing a quantitative mapping between single-cell filamentation and lysis dynamics (Equations 1-4) and dynamics of collective biomass accumulation and decline (Equations 5 and 6).
Our study opens the question if the same equations can be applicable to other bacteria. Our analysis has focused on E. coli, a rodshaped bacterium, while many other rod-shaped bacteria also undergo filamentation when exposed to antibiotics. Moreover, certain bacteria enlarge in all directions when treated with betalactams (Choi et al, 2014). For these cells, it is possible that an equation relating the lysis probability to cell size may be applicable. Further study is needed to examine if the size-mediated single-cell lysis kinetics and mapping it to population biomass dynamics are available.
Growth media, chemicals, and OD measurement Unless otherwise noted, we picked a single colony from an LB plate and cultured it in 3 ml of M9CA media with 0.4% glucose overnight ( ∼ 16-18 h). For imaging, we diluted the overnight culture in fresh media (1:10), incubated it for 2 h, and used 1 μl of the diluted culture for time-lapse microscopy. All cultures were incubated in test tubes and placed in a 37°C shaker with 225 rpm. For time-course population-level measurements, we washed and resuspended overnight cultures in PBS and diluted to make 0.1 of OD 600 in fresh M9 media containing either 0.2% or 0.02% casamino acids at the same rate, using 96-well plates (Corning) with a sterile transparent seal (ThermalSeal RTS, Excel Scientific) to prevent evaporation. All population dynamics were measured using Tecan Infinite 200 Pro, where OD 600 was measured every 10 min with 10-s orbital shaking before each measurement. All single-cell experiments and most population measurements were done at 27 or 37°C. Some population-level dynamics were also measured at 30°C (Fig 3C; Appendix Fig S8), which was used to further tune overall population growth rates.
When applicable, kanamycin at 50 μg/ml (Sigma) to select for plasmid-containing cells and carbenicillin (Genesee Scientific), amoxicillin (Sigma), cefotaxime (Sigma), and/or clavulanic acid (Sigma) with appropriate concentrations were added to the growth media.

Time-lapse microscopy
We prepared new 1.5% agarose growth media gel ahead of every imaging experiment. In the 15 ml conical tube, UltraPure Low Melting Point Agarose (Invitrogen) powder was dissolved into 3 ml growth media by putting it in a 70°C water bath for 3 min and mixed well by pipetting. We aliquoted 1 ml of the solution in microtubes, added antibiotics, gently mixed, and loaded into small wells made with adhesive isolators (either Geneframe [Thermo Fisher,25 μl] or Press-To-Seal Silicon Isolator [Grace Bio-Labs, Ø 8 mm × 0.8 mm depth]) on a clean glass slide. The solution was flattened with another glass slide and solidified at room temperature for 5 min. Pre-culture (1 μl) was loaded onto the gel, spread by tilting, allowed to sit for 3 min for cell setting, and covered with a coverslip.
A Keyence microscope (BZ-X710 and BZ-X800) with an incubation chamber (INU Tokai hit) for a microscope was set up for timelapse imaging. We used a 40X objective for phase-contrast images and when applicable, including a DsRed filter for fluorescent images. Images were taken at 5-min intervals, and 7 levels of focus at 0.7 μm z-stack intervals were investigated to determine focus with the focus tracking option of the software. Most focused images were automatically chosen by the microscope software (BZ-X analyzer) for image analysis. The incubation chamber was set up with temperature control only: to achieve x°C, the top was set to (x + 12)°C and the bottom to (x + 2)°C according to the manufacturer's instruction. We used either 27 or 37°C as x for time-lapse microscopy.
Image analysis of final length from time-lapse microscopy and lysis probability fitting Manually drawn line segment length was measured using Fiji (Schindelin et al, 2012) to capture the long-axis length of elongated and curved cells. Length measurements were mainly done with phase-contrast images, with additional red fluorescence channel images being used only when cell outlines were faint in phasecontrast images. We used MATLAB codes to convert units and generate probability distributions. The MATLAB R2021a curve fitting tool was used for ill equation fitting of cumulative probability distributions.
Mathematical model of mapping single-cell lysis to population biomass dynamics From Equations 2-4, we can derive the temporal dynamics of total cell number: Further derivation of total biomass then follows: Assuming an initial cell length of L 0 and an initial cell number of N 0 , L t ð Þ and N t ð Þ can be derived from Equations 1-3 and 7 by following: In stochastic simulations (Appendix Fig S6), we initialized and ran simulations of the model as follows: 1 For a population, set L c and H of P L and initialize 2,000 cells (N i ) with an average initial length (L i ) and an average single-cell elongation rate (μ). 2 Add Gaussian noise to L i and μ of each cell, but not to exceed 10% of the given average values, and ensure L i > 0 and μ > 0. 3 For cells without division, compute L t ð Þ ¼ L i e μt , ρ L L t ð Þ ð Þ, and P L L t ð Þ ð Þ. To integrate the hazard function lysis rate (P H ) within simulation time step size, we used P H L t ð Þ ð Þ L t ð ÞÀL tÀΔt ð Þ ð Þ as hazard function. By generating one random number (d), uniform between 0 and 1, for each cell, we set L t ≥ T ð Þ¼0 when P H L t ð Þ ð Þ> d, which implies lysis with no residual biomass. 4 For cells with division under non-treated conditions, force a cell to divide into two cells of equal length after reaching the designated division length (L d ¼ aL i þ b) according to the linear model (Tanouchi et al, 2015). We tracked up to a maximum of 10,000 cells. 5 Compute total biomass by summing up L t ð Þ of all cells.
In deterministic simulations, we used an averaged initial length (L 0 ) and elongation rate (μ). Equations 3 and 7, to yield L t ð Þ and N t ð Þ, were computed with ode45. Total biomass was then the product of those two as shown in Equation 5. In the exponential growth and lysis rate simulations, the net growth rate of a population was calculated as the log of total biomass divided by the time interval, lnODtþ1ÀlnODt Δt . In each simulation, the maximum growth rate was found from the maximum net growth rate, and the maximum lysis rate was found by subtracting the minimum from the maximum net growth rate.

Survivor length measurements
Overnight cultures were split into new culture tubes. We added carbenicillin to the culture in proper doses and incubated the tubes in the 37°C shakers with 225 rpm. We loaded 1 μl of the culture onto a glass slide and covered. Due to low cell number, cells treated with 20 μg/ml carbenicillin at the 6 h timepoint were spun down (100 μl, 2,000 g, 2 min) and resuspended in 20 μl of the same media for imaging. 40× phase contrast images from 3 locations of each experimental condition were taken to yield more than 300 cells to be analyzed. Cell lengths were extracted with customized single-cell segmentation Python code based on the scikit-image package (van der Walt et al, 2014;Bois, 2018).

Data availability
Raw datasets and MATLAB model simulation codes are available at https://github.com/youlab/MappingFilamentation_KyeriKim.   C. Amoxicillin-treated ESBL-producing E. coli from the patient isolate library with Bla inhibition. Clavulanate acid (50 µg/ml) was added with amoxicillin (6.25 and 25 µg/ml) to the clinical isolates and incubated at 37 °C. Cell length, probability density, and elongation rate plots are derived by the same methods in panels A and B. At the later stage of low-dose amoxicillin treatment, these cells formed localized swelling Cushnie et al, 2016)a gradually inflated body from the middle of its cylindrical region-and the lab strain formed a bulb through the cell wall without changing the cell width. Therefore, we note that the final length of amoxicillin treated (6.25 µg/ml) cells could be slightly longer than the measured length, because the biomass that contributed to extending the width may have extended length if cells did not form local swell.
Appendix Figure S5. Dose-dependency of maintained in different growth media.
Exponentially growing cells were loaded on different media gel with the same absolute carbenicillin doses for time-lapse microscopy. Images taken after around 6 to 8 hours of antibiotic exposure were picked for displaying low-dose carbenicillin-treated cells. The inverse relationship of and carbenicillin dose was conserved in each medium. Compared to cells grown in M9CA medium, used in Figure 2 experiments, the lengths of the cells grown in LB and t-broth were longer; therefore, is sensitive to growth media. Scale bars: 20µm.
results show that the cell number only decreases because the cells cannot divide but are lysed (left panel). However, the sum of the biomass of survivors (total biomass) increases due to elongation and low lysis probability during early exposure and then decreases due to higher lysis probability (right panel). The time point at which total biomass crosses the initial biomass is the effective elongation duration, .
C. Time-course cell length distributions. Survivor length distributions at 0 to 6h elongation.

! . S6
If we assume that the lysis occurs when a cell attains damages, the final length distribution of cells with damages ( α ) should follow: Here, we note that Eq. S7 is the cumulative density function of final lengths and known as the Gamma distribution. Essentially, the model introduces the Poisson process on damage accumulation and the length to the α-th damage follows a Gamma distribution.
In accordance with the log-logistic fitting, the dependence of on in gamma distributions shifts leftward at a larger with constant ( Figure EV1A). This trend is consistent with the experimental observations: all else being equal, a higher antibiotic dose (indicates larger ) resulted in a smaller (the length at the mean of cumulative distributions) (Figure 2).
Our numerical analysis showed that was proportional to / , and increased approximately with the square root of ( Figure EV1B). These correlations emerge from the analytic solutions under the limiting case when ≫ 1.
Briefly, we assume the log-logistic distribution (with its CDF being the Eq 1 from direct data fitting) and the gamma distribution (arising from the damage-accumulation model) approximately describe the same data. Thus, the means and the variances of two-fitted distributions should be equal to each other: , S8 2 2 . S9 Further simplification of Eq. S8 shows that since 1 when ≫ 1.
Plugging into Eq. S9 gives: 2 2 1 . S10 Expanding the left-hand-side terms to the second order using 1 gives: 1 1 6 2 1 1 6 2 1 . S11 Further simplification of the left-hand-side terms to the second order of derives .
These results indicate that measured and are approximately determined by the maximum number of accumulated cross-linkage failures ( ) and the damage rate ( ) to retain the cell wall integrity.

Derivation of the hazard function formula
The hazard function, also known as the instantaneous death rate or instantaneous failure rate, was initially introduced by Gompertz in 1825 (Gompertz). It is often used in modeling the death of individuals in an age-distributed population, where the independent variable is time (age). However, it can be defined in terms of any monotonically increasing variable.
The Eq. 2, , is the hazard function formula, expressed as a function of cell length ( ), where length is monotonically increasing variable between initial to final lengths.
Below is a derivation of the hazard function formula from its definition (Collett, 1994).
In our study, we define in terms of cell length ( ): ≡ lim → | . S12 By definition, the hazard function describes the instantaneous lysis rate of an individual cell at cell length , that has survived to . The numerator of Eq. S12 is a conditional probability, which can be expanded and rearranged according to the Bayes theorem: | | , S13 as | is always 1, reflecting certainty.
The denominator of Eq. S13 is the survival function of the population: 1 1 . S14 Plugging Eqs. S13 and S14 into Eq. S12, we have: A Damage accumulates on the cell wall until lysed. A schematic diagram of the damage accumulation model for cell lysis. Upon addition of antibiotics, a cell is assumed to accumulate damage on its cell wall during elongation, with an antibiotic dose-dependent rate of β. When the total number of damages to a cell reaches a threshold value α, the cell formed a bulge and lysed in a short time. B Final length follows a gamma distribution. Under the damage accumulation model, the final length of a cell that has α damages follows a gamma distribution with the parameters of α and β (see Appendix Supplementary Methods). Probability density function (PDF, ρ L , left) and cumulative distribution function (CDF, P L , right) of the three different rates (β = 0.4, 0.8, and 1.2) recapitulate the experimental distributions shown in Fig 2B. All three plots used a constant threshold (α = 10). C Parameters of gamma and log-logistic distributions are correlated. Gamma distributions that were generated with different parameter sets of α and β were fitted to log-logistic distributions (P L ¼ L H L H þL H c ). H was not sensitive to β (left) but highly sensitive to α (bottom left). Black dashed line shows the approximation of H to α in 2 nd order. L C was proportional to α=β (right). R 2 of all log-logistic fits were larger than 0.997.