Lineage space and the propensity of bacterial cells to undergo growth transitions

The molecular makeup of the offspring of a dividing cell gradually becomes phenotypically decorrelated from the parent cell by noise and regulatory mechanisms that amplify phenotypic heterogeneity. Such regulatory mechanisms form networks that contain thresholds between phenotypes. Populations of cells can be poised near the threshold so that a subset of the population probabilistically undergoes the phenotypic transition. We sought to characterize the diversity of bacterial populations around a growth-modulating threshold via analysis of the effect of non-genetic inheritance, similar to conditions that create antibiotic-tolerant persister cells and other examples of bet hedging. Using simulations and experimental lineage data in Escherichia coli, we present evidence that regulation of growth amplifies the dependence of growth arrest on cellular lineage, causing clusters of related cells undergo growth arrest in certain conditions. Our simulations predict that lineage correlations and the sensitivity of growth to changes in toxin levels coincide in a critical regime. Below the critical regime, the sizes of related growth arrested clusters are distributed exponentially, while in the critical regime clusters sizes are more likely to become large. Furthermore, phenotypic diversity can be nearly as high as possible near the critical regime, but for most parameter values it falls far below the theoretical limit. We conclude that lineage information is indispensable for understanding regulation of cellular growth.


Abstract 22
The molecular makeup of the offspring of a dividing cell gradually becomes phenotypically 23 decorrelated from the parent cell by noise and regulatory mechanisms that amplify pheno-24 typic heterogeneity. Such regulatory mechanisms form networks that contain thresholds 25 between phenotypes. Populations of cells can be poised near the threshold so that a subset 26 of the population probabilistically undergoes the phenotypic transition. We sought to char-27 acterize the diversity of bacterial populations around a growth-modulating threshold via 28 analysis of the effect of non-genetic inheritance, similar to conditions that create antibiotic-29 tolerant persister cells and other examples of bet hedging. Using simulations and experi-30 mental lineage data in Escherichia coli, we present evidence that regulation of growth am-31 plifies the dependence of growth arrest on cellular lineage, causing clusters of related cells 32 undergo growth arrest in certain conditions. Our simulations predict that lineage correla-33 tions and the sensitivity of growth to changes in toxin levels coincide in a critical regime. 34 Below the critical regime, the sizes of related growth arrested clusters are distributed ex-35 ponentially, while in the critical regime clusters sizes are more likely to become large. 36 Furthermore, phenotypic diversity can be nearly as high as possible near the critical regime, 37 but for most parameter values it falls far below the theoretical limit. We conclude that 38 lineage information is indispensable for understanding regulation of cellular growth. 39 40 Introduction 59 The process of cellular growth is both the distinguishing feature of living matter and 60 central to the roles of regulatory networks from microbes to metazoa. Growth and division 61 is also a primary source of phenotypic diversification. For instance, when a bacterial cell 62 divides, and its cellular contents become partitioned into two daughter cells, diffusible cy-63 toplasmic components are often randomly distributed into the daughter cells in a binomial 64 distribution. Such phenotypic diversification permits populations to be robust to unpredict-65 ably changing environments, a phenomenon known as bet-hedging. A striking example of 66 this effect is the regulation of growth rate by toxins. 67 Figure 1. Simulated effects of a molecular network with an endogenous growth-regulating threshold in bacteria. a. Simplified toxin-antitoxin module, depicting its interaction with cellular growth rate. b. Deterministic steady state model predictions for a toxin with growth feedback. A regime with no deterministic molecular steady state (labeled "Growth Arrest") arises when toxin production sufficiently exceeds the growth feedback-imposed threshold. Growth rate is normalized to the maximum = 1. c. Binomial phenotypic inheritance at a constant molecule production rate. With no effect on cellular growth rate, the population exhibits regression to the mean within a few generations of division. d. With a discrete growth arrest threshold, the population becomes increasingly skewed over time. Box and whisker plots represent median, interquartile range, and range of a population started from a single simulated cell. Details on model implementation are presented in Supplemental Materials.
Most of the molecular content in the bacterial cytoplasm undergoes growth-mediated 68 dilution (in some cases, such as most proteins, as the primary mechanism of degradation). 69 Reduction in cellular growth rate by a cytoplasmic toxin, or other molecule with toxic ef-70 fect, creates an effective positive feedback loop, trapping some cells in a growth arrested 71 state until they can escape in changed conditions [1][2][3]. This mechanism is associated with 72 antibiotic-tolerant persister cells arising in the population, which cause difficulty in antibi-73 otic treatment [4]. Various feedback mechanisms are associated with growth bistability [5]. 74 Thus, understanding the processes that result in growth diversification is an important goal 75 on the path to solving the impending antibiotic resistance crisis. 76 Growth arrested cells typically represent a small subset of a bacterial population [6]. In 77 E. coli, growth arrested persister cells are associated with alterations in metabolic activity 78 via the stringent response [7,8], and with efflux of antibiotics [9]. Depending on the mech-79 anism of induction, persister cell fractions can be spontaneously produced or respond to 80 external stresses [6]. Persistence in E. coli is associated with toxin-antitoxin systems and 81 global metabolic regulation [10], with a core mechanism of toxins that are neutralized by 82 antitoxins [11] ( Fig. 1a-b). The competing effects of toxin and antitoxin create a threshold 83 in a stoichiometric effect via molecular titration that can cause conditional cooperativity of 84 TA gene regulation [12,13]. When accounting for gene expression noise and proteolysis 85 of antitoxins, free toxin levels will gain sufficient concentration to result in a growth feed-86 back mechanism that ultimately induces growth arrest in above-threshold cells. The result 87 is skewed phenotypic distributions, with a core fast-growing group of cells along with rarer, 88 growth arrested cells, as opposed to regression to mean levels observed in networks without 89 the growth arrest threshold (Fig. 1c- To explore this hypothesis, we used an established experimental model of threshold-97 based growth arrest in E. coli to experimentally confirm lineage dependence. We then cre-98 ated a minimal multiscale computational framework that allowed more extensive charac-99 terization of the various growth regimes than were possible with time-lapse microscopy. 100 Our computational model represents the processes of cellular growth and division, with 101 binomially distributed inheritance of a simplified toxin-antitoxin-like system subject to sto-102 chastic molecular kinetics in individual cells over time. We modeled a functional depend-103 ence of growth on toxin concentrations as an exponential function with a key parameter, α, 104 that quantifies how toxic the toxin is. We used various specific realizations of the frame-105 work to simulate growth of small bacterial populations from a single common ancestor and 106 growth regulation by the simulated toxin for various toxin:antitoxin production ratios. We first sought to establish an empirical basis for growth arrest kinetics and threshold-118 based amplification of lineage correlations. An established experimental model of thresh-119 old-based growth arrest [17] provided a simple way to track growth in a lactose-sensitive 120 strain of E. coli. In this model, lactose stimulates growth at sufficiently low concentrations, 121 but creates toxicity in a subset of cells at high concentration that results growth arrest or 122 death of those cells. Presently, the precise mechanism of toxicity is not known in this 123 model, but the competing effects of lactose import rate and processing rate are the most 124 likely culprit, and the threshold-based mechanism for growth arrest and persistence is es-125 tablished [17]. In the high-lactose condition, bacterial colonies have a slow net growth rate 126 and a high likelihood of any individual cell eventually undergoing growth arrest and/or 127

death. 128
We used time-lapse fluorescence microscopy to track individual microcolonies in a 129 microfluidic device with constant perfusion of fresh minimal medium containing defined 130 concentrations of a single sugar as the sole carbon source. We used two carbon sources: a 131 growth-arrest-prone condition with a high lactose concentration (50 g/l), and a condition 132 that does not induce a growth arrest threshold, with a moderate glucose concentration 133 (2 g/l) ( Fig. 2; Movies S1 and S2). As inferred from extension of cellular major axis length, 134 cells grow exponentially at heterogeneous rates (Figs. 2a-b, 2e-f, S1) and are capable of 135 quickly shifting between growth rates, e.g., from fast to slower or non-growing (Fig. 2b,  136 2f). To identify cases of mid-cell cycle shifts in growth rate, we fit each cell cycle to an 137 exponential growth model, applied Bonferroni correction to the resulting fit significance 138 levels, and selected the non-significant cases (Fig. S3). A constitutive fluorescent reporter 139 provides clear visual evidence of mother-daughter cell correlations only in the growth ar-140 rest-prone condition (Fig. 2c, 2g). 141 We reconstructed the microcolony lineage in both conditions to quantify the effects of 142 non-genetic inheritance in this experiment (Fig 2d, 2h). The result of the growth arrest 143 threshold is a striking effect on the structure of the lineages. The growth arrest-prone line-144 age shows distinct clusters of growth arrested or dead cells, and clusters of faster growing 145 cells, resulting in an asymmetric tree (Fig. 2d). On the other hand, absent the growth arrest 146 threshold, the tree is nearly symmetric (Fig. 2h). In the growth arrest prone condition, we 147 classified cells into being growth arrested or dead (apparent growth rate = 0) or actively 148 growing. Of the 63 total cells in the final lineage, 16 (25.4%) were determined to be com-149 pletely growth arrested or dead at the final time point. We determined the pairwise lineage 150 distance, defined as the time since the most recent common ancestor, for three subsets: all 151 cells, only growing cells, and only growth arrested cells (Fig. S2). The all-growing and all-152 growth arrested subsets both had significantly closer lineage distances compared to the all 153 cells set (p < 0.05, Mann-Whitney U). From these results, we conclude that lineage has a 154 strong effect on phenotypic heterogeneity during colony development around a growth-155 modulating threshold. 156 Figure 2. Growth rate of E. coli B REL606 GFP+ cells prone to stochastic growth arrest in high lactose reveals lineage dependence. Numbers indicate time in hours. a -d. Colony grown in a commercial microfluidic device with continuous perfusion of minimal medium containing 50 mg/ml lactose as described in Methods. e -h. Colony grown with continuous perfusion of minimal medium containing 2.5 mg/ml glucose, which does not predispose cells to growth arrest. a, e. Growth kinetics of a selection of cells. Individual trajectories are divided by cell division or different growth rates by a leastsquares fit of the data to the model L(t) = L0e gt . b, f. Growth rates from exponential model fit. Vertical lines indicate cell division times for the corresponding trajectory color. c, g. Selected frames of the time-lapse microscopy experiment. d, h. Lineages derived from time-lapse microscopy. Colors indicate growth rate. Lack of color indicates insufficient data for a significant fit. Note asymmetry in d and symmetry in h.

Lineage Dependence is Reproduced in a Simple Computational Branching Process Model 158
To determine the minimal set of mechanisms necessary to reproduce the interactions 159 between threshold-based molecular regulation of growth rate and population dynamics, we 160 created a computational model containing cell agents growing and dividing at a typical rate 161 for enteric bacteria (30 minute doubling time), each with a cell volume and division upon 162 doubling of the volume. Each cell agent has embedded stochastic kinetics of a growth-163 inhibiting molecule (analogous to a toxin) and a neutralizing molecule that binds and pre-164 vents toxicity (analogous to an antitoxin). As discussed in more detail in Methods, we as-165 sume toxin and antitoxin production, growth-mediated dilution, and binding-unbinding ki-166 netics of the molecules. We used a phenomenological exponential function layer that trans-167 lates between concentrations of toxin and resultant growth rate, with a single parameter, α, 168 that determines the level of toxicity. 169 The key similarity between our experimental and computational approaches is the ex-170 istence of a threshold in the molecular network that determines the growth rate of the cell. 171 There are many potential mechanisms for such a threshold to arise, as discussed in the 172 Introduction. We do not claim that the mechanism implemented in the computational 173 model is the same as the experimental model. Rather, there is an underlying fundamental 174 interplay between growth regulation and lineage structure that we will show is conserved. 175 To determine the effect of the growth threshold on microcolony dynamics, we scanned 176 the rate of toxin production, keeping antitoxin production constant. (In most natural toxin-177 antitoxin systems, the antitoxin is unstable. We simulated this case as well, below). The 178 simulations were seeded with a single cell growing with excess antitoxin and permitted to 179 grow for 100 simulation minutes before changing the toxin production rate to a positive 180 value. After several generations of growth, we found three qualitative regimes across dif-181 ferent toxin production rates: symmetrical growth with no or little growth arrest (toxin 182 production rate 0-2.5 /min), a critical regime with clusters of growing and growth arrested 183 cells (toxin production rate 3-4.5 /min), and a regime of nearly instantaneous growth arrest 184 (toxin production rate >4.5 /min) with the colony trapped in its near-initial state. Figure 3  185 shows representative cases with growth rate (Fig. 3a) or toxin concentration (Fig. 3b) de-186 picted with coloring of each cell. 187 Sub-lineages of fast-growing and slow-growing cells are evident in the critical regime 188 (with toxin production rate 5-6 /min; Fig. 3a). Lineage effects are also evident from toxin 189 levels, where there are sublineages escaping from entry into high toxin concentrations (blue 190 clusters in Fig. 3b). The precise time of entry into growth arrest can have a large effect on 191 toxin levels, suggesting that growth rate is a more precise phenotype to follow for the study 192 of lineage effects in this system. 193 Figure 3. Simulated lineages over a range of toxin production rates. Time proceeds downward in each lineage and begins at the onset of toxin production (t = 100 h). a. Lineage growth rate superimposed on the lineages. b. Free toxin concentration superimposed on the lineage. Lineages for production rates 3.5 /min and higher are plotted with wider trajectories for visibility.

Lineage Dependence is Strongest in the Critical Regime 194
To quantitatively characterize the properties of growth transitions in our simple com-195 putational framework, we considered the fate of simulated microcolonies at 250 minutes 196 of growth, which is shortly before the fastest growing cases begin to become computation-197 ally intractable, but after the population size is beyond the minimal requirement to be con-198 sidered a microcolony. Mean population growth rates and toxin concentrations across mul-199 tiple (N = 100) replicates reveal a growth-regulatable region flanked by regions of almost 200 full growth and almost complete growth arrest (Fig. 4a). In the region where population 201 growth is low but positive, toxin concentrations increase monotonically but non-linearly 202 with increases in toxin production (Fig. 4a). 203 To quantify the amount of lineage information shared by pairs of cells in their pheno-204 types, we calculated mutual information between phenotypic differences between pairs of 205 cells and pairwise lineage distance. From each simulation, we sampled one pair of cells 206 randomly to ensure independent, identically distributed samples and performed a 207 resampling procedure 100 times to increase the confidence in our estimate. This was done 208 for absolute growth rate differences and absolute toxin concentration differences (Fig. 4b). 209 Various studies of have found mutual information between different points on a lattice to 210 be indicative of a phase transition [24,25]. While our model may not exhibit a true phase 211 transition, our mutual information estimator reveals a similar peak for both growth rate and 212 Figure 4. Growth, lineage information, and diversity of simulated cellular lineages at various rates of toxin production at 4 h. a. Average cellular growth rates (red) and toxin concentrations (blue) 150 minutes after onset of stress are proportional to toxin production rate, with distinct growth regulation regimes. Error bars indicate standard deviation. b. Mutual information between cell pair growth rate differences, in red (or toxin concentration difference, in blue) and their lineage distance reveals a lineage-dependent effect on cellular phenotypes near the regulatable region. c. Dispersion of average growth rate for low toxin production rates. Vertical bar represents the peak mutual information depicted in panel b. d. Growth rate distributions in the population at various toxin production rates as indicated. Red represents the mean frequency at a given growth rate; blue, standard deviation in the frequency. toxin concentrations in the critical regime, where the population growth rate is most sensi-213 tive to changes in toxin production rates. 214 Distributions of growth rates reveal the underlying population structure not evident 215 from mean growth rates shown in Figure 4a. Distributions that emerge from the model 216 include uniformly fast (Fig. 4c, top left in Fig. 4d) or slow growing (Fig. 4c, bottom right  217 in Fig. 4d), bimodal between fast and slow growing (top right in Fig. 4d), and long-tailed 218 with a peak at either fast (at toxin production rate 3 /min, not shown) or slow growing 219 (bottom left in Fig. 4d). 220

Fluctuating Cell Growth Dynamics in the Critical Regime 221
To examine a further indicator of criticality in this system, we calculated the dynamics 222 of growing cell numbers below (toxin production rate 0-2.5 /min), near (toxin production 223 rate 3-4.5 /min), and above the regulatable region (toxin production rate >4.5 /min) of 224 growth rate. With toxin production well below the regulatable region, the predicted cell 225 growth becomes equivalent to an uncoupled case where toxin has no effect on growth. 226 Growing cell numbers show variability between simulation replicates near the critical 227 region (Fig. 5a). Over time, the dynamics of the mean number of growing cells approaches 228 exponential growth at low toxin production rates, critical growth at intermediate toxin pro-229 duction rates (as shown in Fig. 5a), and extinction (elimination of all growth) at high toxin 230 production rates. Mean cell numbers in critical growth show persistent oscillations that 231 dampen as the simulated growth rates become decorrelated by noise (Fig. 5a). As toxin 232 production approaches the critical regime, some cells accumulate high toxin and, depend-233 ing on individual cellular toxin accumulation, subsets of the population will enter the ex-234 ponential or extinction phase. Thus, the time required to conform to the exponential or 235 extinction regimes is high in the critical regime, reminiscent longer relaxation times ob-236 served near critical points in other models [e.g. 26]. Autocorrelations of growing cell num-237 bers at lag times after the onset of toxin production reveal this effect. For example, high 238 autocorrelation around lag time 100 min in critical regime (vertical dotted line) signifies 239 growth remaining correlated for a longer time compared to the autocorrelation at toxin 240 production rate 3.0 /min. The presence of more than two zeroes in the absolute autocorre-241 lations indicates the oscillatory regime (Fig. 5b).

If lineage is capable of constraining the attainable phenotypes of offspring cells, it 244
stands to reason that the amount of phenotypic heterogeneity attainable in a microcolony 245 is lowered by lineage dependence in systems that generate heterogeneity by diversifying 246 growth rates. It is difficult to generalize what constitutes meaningful diversity in growth 247 rates; small changes may or may not be important to fitness in the long run, but the im-248 portance of the distinction between growth arrested and fast-growing cells is clear. There-249 fore, we used two possible definitions of meaningful diversity: in one, arbitrarily small 250 changes in growth rate or toxin concentration are meaningful. In the other extreme, we 251 assumed that only growing versus non-growing cells (or high versus low toxin) is a mean-252 ingful distinction. 253 We quantified the phenotypic heterogeneity as information entropy (base 2), binning 254 the simulated cells according to the two definitions of diversity (Fig. 6). We calculated the 255 maximum entropy in the fine-grained binning case by assuming each cell had a unique 256 value. Note that the maximum entropy is extensive, decreasing with lower total cell count 257 (Fig. 6a). In the binary case, the maximum entropy is simply 1 bit. Regardless of the 258 Figure 6. Entropy of growth rates and toxin concentrations at 250 h. Vertical line indicates the point of highest lineage-dependent mutual information between growth rate and lineage distance. a. Fine-grained binning. b. Binary binning into growing-non growing or high-low toxin concentration. Error bars indicate standard deviation. definition used, the peak entropy of the population can get surprisingly close to the maxi-259 mum entropy. Note that peak entropy of growth rate nearly coincides with peak mutual 260 information between growth rate differences and lineage distance (Fig. 6, vertical line). 261 However, entropy away from this peak sharply decreases from the maximum. In the critical 262 regime, population heterogeneity is affected by two key factors: sensitivity of growth rate 263 to toxin and lineage dependence. Given that we observed higher lineage dependence in the 264 critical regime, the key question here is whether this dependence reduces the possible at-265 tainable heterogeneity in bet-hedging. The entropy plot (Fig. 6) shows that sensitivity of 266 growth rate to toxin dominates and thus phenotypic heterogeneity is maximal at when the 267 lineage is most structured. 268

Growth Regulation as a Criterion for Lineage Dependence 269
To explore the generality of our results, we created models with variations on the orig-270 inal, and tested for lineage dependence. 271 The first set of variations test two simplifications in the primary model: stability of the 272 antitoxin, and bursty production of the molecular species. While we regard the model to be 273 a general threshold-based growth control mechanism, it is worthwhile to determine if a 274 toxin-antitoxin module with unstable antitoxin qualitatively reproduces our main results. 275 Varying the stability of the antitoxin, we indeed found the same qualitative results (Fig.  276 S4a). Similarly, simulating bursts of gene expression producing toxin and antitoxin pro-277 duced the same qualitative results (Fig. S4b). 278 Our next model variation was to vary the effect of growth regulation, increasing it 279 (α=0.3 in ( , ); see Methods below) and abolishing it completely (α=0 in ( , )). As 280 expected, a larger quantitative effect of toxin preserved the main results, but shifted the 281 toxin concentration necessary to see the lineage dependence (Fig. S4c). Abolishing growth 282 regulation eliminated the peak in mutual information, and thus lineage dependence (Fig.  283 S4d). 284

Distributions of Growth Arrested Cluster Sizes 285
Large clusters of growth arrested cells could have effects on the spatial development 286 of bacterial colonies, as daughter cells tend to be correlated in space as well. We therefore 287 asked what growth arrested cluster size distributions arise in the region where there is high 288 mutual information between growth rate and lineage distance. We performed 10,000 sim-289 ulations each and clustered the end-point populations according to lineage neighbors hav-290 ing similar growth rate (with a cutoff of 0.01 /h to be considered growth arrested). Resulting 291 clusters were pooled across simulations of the same parameter set. We present distributions 292 of raw absolute cluster size, not normalized. 293 Below the critical regime, the absolute cluster size distribution is nearly exponential 294 (Fig. 7, red line with exponential fit as gray dashed line). As the probability of growth arrest 295 increases (with high toxin production rate), the distributions diverge from exponential to 296 make large clusters of growth arrested cells more likely (Fig. 7). At higher toxin production 297 rates, the distribution is bimodal between large clusters and single growth arrested cells.   Figure 7. Distribution of growth arrested cluster sizes in simulated lineages. Clusters are exponentially distributed below the critical region (red line, simulation; gray dashed line, exponential fit ae -bc for cluster sizes c) but diverge from an exponential distribution near the critical region, eventually becoming bimodal (purple, blue, green, and orange lines). Each parameter set was simulated 10,000 times. a. Raw probability distributions. b. Probability distributions normalized to the probability of cluster size 1.

Discussion 325
Regulation of growth is a central part of phenotypic control. Many factors can control 326 growth rate, including extrinsic conditions such as starvation, and intrinsic regulators of 327 growth that often operate with a threshold-based mechanism. Using an experimental model 328 of threshold-based growth arrest arising from metabolic toxicity, we tracked cell growth in 329 a bacterial microcolony with a high probability of undergoing the growth arrest transition, 330 and a colony grown in a condition that does not display the threshold-based growth arrest. 331 We found several large, discrete shifts in growth rate to occur at a faster timescale than our 332 5-minute recording intervals (Fig. 2). Quantifying the lineage dependence of cellular 333 growth phenotype, we found that growth arrested or dead cells tend to be clustered in the 334 lineage, as do fast-growing cells. The difference in lineage shapes between the growth ar-335 rest-prone and and non-growth arrest prone conditions is striking (Fig. 2d,h). 336 We therefore sought the simplest possible model of microcolony growth dynamics that 337 reproduces the effect. Our basic model captures single-cell biochemical kinetics on one 338 scale (microscopic) interfacing population growth dynamics on another scale (macro-339 scopic). We found striking phenotypic lineage dependence to emerge with the following 340 criteria: (i) growth rate dependence on a toxin; (ii) stochastic dynamics around a cellular 341 threshold embedded within the network; (iii) kinetic parameters calibrated so that the pop-342 ulation average growth rate is near the regulatable region. 343 As the probability of cellular transition to growth arrest increases, the mutual infor-344 mation between growth rate and lineage distance increases to a peak, then decreases as the 345 simulated microcolony reaches the condition of immediate growth arrest. This transition 346 bears a resemblance to a phase transition, with correlation of microscopic length scales 347 peaking at the critical boundary. Here, the correlation length is in lineage space: we have 348 assumed no traditional spatial information about the cells in the simulation. 349 Lineage space is a binary tree growing with extinction probability based on micro-350 scopic dynamics. Distances are modified by dynamical growth rates, which explains why 351 a higher probability of heterogeneous growth results in structured trees. Thus, relating per-352 sister and other threshold-based growth arrest mechanisms to the established mathematics 353 of branching processes [27,28] is an important direction for microbial physiology. 354 After 100 simulated minutes we imposed a continuous rate of increased toxin produc-355 tion (or antitoxin degradation, in one derived model) on the developing microcolony. The 356 constant input of more toxin created an irreversible threshold. Once a cell crosses the 357 growth arrest threshold, there is an irreversible stoppage of growth that arises from toxin 358 growth feedback. The growth arrest condition can then be considered an absorbing state. 359 Continuous transitions from active to absorbing states are generically characterized by the 360 scaling properties of critical directed percolation [29][30][31]. Our model qualitatively repro-361 duces characteristics of directed percolation, including longer relaxation times near the 362 critical region (Fig. 5) and different regimes of growth arrested cluster size distribution 363 (Fig. 7). However, the dimensionality of the space is unclear, and may be shaped by the 364 probability of growth arrest. Thus, we are doubtful that bet hedging quantitatively con-365 forms to the classic criteria for directed percolation. 366 If lineages impart spatial structure onto growth phenotypes, then do they impose an 367 upper limit to the level of phenotypic heterogeneity that can be attained by a microcolony? 368 The population is most sensitive to fluctuations directly in the region with the highest lin-369 eage dependence, the latter of which appears to imply a dampening of phenotypic 370 heterogeneity. However, multiple methods of measuring total population entropy suggest 371 that the population can still approach the maximum total entropy in cases where growth 372 rates are both finely-binned and binned into only two phenotypes -growing and growth 373 arrested (Fig. 6). Heterogeneity is reduced as the population reaches either extreme of high 374 or low average toxin level. Thus, counterintuitively, a more highly structured lineage yields 375 a higher level of heterogeneity. Lineage plays an interesting role in determining the phe-376 notypes of extant growing cells, but it does not appear to restrict what phenotypes can be 377

attained. 378
The purely intracellular phenomena considered here allow lineage to be the only type 379 of space considered. However, closely related cells in many conditions, such as surface-380 attached conditions or channels, will be physically closer together as well. In many bacte-381 rial colonies with a substantial chance of endogenous and exogenous conditions interacting 382 to determine the growth arrest transition (such as quorum sensing), an information metric 383 that includes components of both real space and lineage space will need to be considered. 384 385 386

Methods 387
Cell Culture Conditions 388 E. coli B REL606 lacI -PlacO1-GFP was grown from -80º C cryogenic culture for 18 h 389 in LB medium in a shaking incubator (37º C), acclimatized by incubating in Davis minimal 390 medium containing either 50 mg/ml lactose (DMlac50) or 2 mg/ml glucose (DMglc2) for 391 24 h, and resuspended either in fresh DMlac50 or DMglc2 culture, respectively, for 3 hours 392 before beginning time-lapse microscopy. 393

Microscopy and Image Analysis 394
We used an Olympus IX81 inverted fluorescence microscope with an incubated imag-395 ing chamber (Olympus, Tokyo, Japan). The chamber with objective was pre-heated, bac-396 terial cultures were added to a pre-heated CellAsic ONIX microfluidic plate (Millipore, 397 Billerica, Massachusetts) at an approximate OD450 of 0.005, and a continuous media flow 398 of 1 psi DMlac50 or DMglc2 was maintained for the duration of the experiment. Images 399 in brightfield and green fluorescence (488 nm stimulation / 509 nm emission) channels 400 were captured every 5 minutes with a 4k CMOS camera, followed by ZDC autofocus. For 401 the DMlac50 experiment, we used a 100x oil immersion objective. Due to technical issues 402 with the objective, we used a 60x air objective for the DMglc2 experiment. Thus, the pixel 403 lengths of the cells between the two experiments should not be directly compared. 404 Images were cropped after identifying a stable microcolony originating from a single 405 cell. We developed a semi-supervised cell tracking algorithm in Mathematica (Wolfram 406 Research, Champaign, Illinois) with manually input cell division times and cell lengths. 407 From this information, we reconstructed the lineage and approximated growth rates with 408 exponential growth models. When mapping the growth rates to the lineages in Fig. 2, we 409 approximated growth rates of cells with non-significant exponential fits using piecewise 410 linear regression as reviewed in [32]. 411

Multiscale Growth Simulation Framework 412
To capture the minimal mechanisms necessary that recapitulate non-genetic inheritance 413 and effects of cellular lineage, we created a multiscale growth simulation framework with 414 individual cell agents, each containing a molecular network of interacting proteins, referred 415 to as toxin and antitoxin, with toxin affecting cellular growth rate. 416 We track the simulated number of toxin and antitoxin molecules as well as cell volumes 417 for each cell agent across time. In the next time step, t+δt, the number of toxin and antitoxin 418 molecules are determined by stochastic simulation (below) and are updated for that cell. 419 Cellular growth rates are set by a deterministic function of the toxin concentration (#/vol). 420 The change in the volume (δv) in δt is determined by the amount of toxin present at that 421 time. When cell volume doubles, the number of each molecule is distributed binomially 422 into the two daughter cells. From that time on, the two daughter cells are labeled as differ-423 ent cells and are iterated in the same way. We initiate each simulation as a single cell with 424 no toxin and allow growth for a few generations (100 minutes) before applying toxin pro-425 duction rate (or antitoxin degradation rate) of a given quantity. The primary purpose of this 426 model is to capture the qualitative effect of the growth arrest threshold, so several important 427 details about the biophysics of kinetics in growing cells were omitted, such as the effects 428 of chromosome replication and the volume dependence of bimolecular stochastic reaction 429 propensities. 430

Estimation of Mutual Information from Simulated Lineages 431
We sought to develop a sampling methodology to ensure independent, identically dis-432 tributed samples from lineage simulations to estimate the mutual information between lin-433 eage distance d and phenotypic differences between pairs of cells φ. Phenotypic differences 434 (φ) could be growth rate or intracellular toxin concentration. To do so, we performed 100 435 independent simulations in each condition, and randomly drew a single pair of cells from 436 each lineage. Our estimate of mutual information was calculated from the resulting distri-437 bution of i.i.d. samples: ( , ) = ∑ ∑ ( , ) log 2 ( 3(4,5) 3(4)3 (5) ) 4∈7 5∈8 . A more accurate es-438 timate of absolute mutual information may extrapolate to an infinite sample size. In our 439 case, the relative mutual information between different locations in parameter space suf-440 fices to demonstrate the existence of a strong lineage dependence for certain parameter 441 ranges. To estimate the uncertainty of our relative mutual information estimate, we 442 resampled 100 cell pairs with replacement and present the resulting mean ± standard devi- To illustrate the effects of growth arrest on distributions of growth-modulating cyto-475 plasmic contents (Fig. 1), we created a simplified computational model with constant pro-476 duction, constant sub-threshold generation times, and binomially distributed molecular 477 contents between two daughter cells. One simulation for each initial condition was run for 478 12 generations, with 10 molecules produced per generation, and a growth arrest threshold 479 of 20 molecules. Initial conditions were 0, 10, 20, or 30 molecules. A second case with no 480 threshold was simulated with the same parameters and initial conditions. The Mathematica 481 code is given in S4 Model. 482

483
Deterministic Molecular-Scale Model as a Basis for Growth Feedback 484 The exact functional dependency of growth on toxin is unknown. In our stochastic 485 simulation framework, we considered an exponential dependence of growth on toxin. Fig.  486 1b depicts a deterministic model of toxin growth feedback by a free toxin as follows: 487 , where kt is the toxin production rate, γ is the maximum growth rate, 488 and θ determines the toxicity level of the toxin. We chose the Hill form for the determin-