The authors have declared that no competing interests exist.

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

One of the most important characteristics of a cell is whether it is growing. Actively growing cells can multiply exponentially. In the case of infections and cancer, growth causes problems for the host organism. On the other hand, cells that have stopped growing can allocate cellular resources toward different activities, such as bacteria surviving antibiotics and tissues in multicellular organisms performing their physiological roles. Observing small bacterial colonies in a microscope over time, we have found that cells closely related to each other often have similar growth state. We were curious if lineage dependence was an intrinsic property of growth regulation or if other factors were needed to explain this effect. We therefore built a computational model of a growing and dividing cellular colony with an encoded growth regulation network. We found that regulation of growth is sufficient for lineage dependence to emerge. We next asked if lineage dependence constrains how diverse the cellular population can become. We found that cellular diversity can reach a peak that is nearly as high as possible near the conditions that have the highest lineage dependence, but that most conditions do not permit such high diversity. We conclude that lineage is an important constraint and discuss how the growth arrest transition is in some ways like a phase transition from physics, and in some ways strikingly different, making it a unique phenomenon.

The process of cellular growth is both the distinguishing feature of living matter and central to the roles of regulatory networks from microbes to metazoa. Growth and division is also a primary source of phenotypic diversification. For instance, when a bacterial cell divides, and its cellular contents become partitioned into two daughter cells, diffusible cytoplasmic components are often randomly distributed into the daughter cells in a binomial distribution. Such phenotypic diversification permits populations to be robust to unpredictably changing environments, a phenomenon known as bet-hedging. A striking example of this effect is the regulation of growth rate by toxins.

Most of the molecular content in the bacterial cytoplasm undergoes growth-mediated dilution (in some cases, such as most proteins, as the primary mechanism of degradation). Reduction in cellular growth rate by a cytoplasmic toxin, or other molecule with toxic effect, creates an effective positive feedback loop, trapping some cells in a growth arrested state until they can escape in changed conditions [

Growth arrested cells typically represent a small subset of a bacterial population [

Motivated by observations on phenotypic inheritance [

To explore this hypothesis, we used an established experimental model of threshold-based growth arrest in

These results show, for the first time, how important lineage is to growth regulation and bet-hedging phenotypes involving growth. Consideration of lineage is now indispensable for studies on phenotypic heterogeneity, phenotypic memory, and regulation of the growth arrest transition. Finally, our results suggest that lineage space used in evolutionary [

We first sought to establish an empirical basis for growth arrest kinetics and threshold-based amplification of lineage correlations. An established experimental model of threshold-based growth arrest [

We used time-lapse fluorescence microscopy to track individual microcolonies in a microfluidic device with constant perfusion of fresh minimal medium containing defined concentrations of a single sugar as the sole carbon source. We used two carbon sources: a growth-arrest-prone condition with a high lactose concentration (50 g/l), and a condition that does not induce a growth arrest threshold, with a moderate glucose concentration (2 g/l) (

Numbers indicate time in hours. _{0}^{gt}.

We reconstructed the microcolony lineage in both conditions to quantify the effects of non-genetic inheritance in this experiment (

To determine the minimal set of mechanisms necessary to reproduce the interactions between threshold-based molecular regulation of growth rate and population dynamics, we created a computational model containing cell agents growing and dividing at a typical rate for enteric bacteria (30 minute doubling time), each with a cell volume and division upon doubling of the volume. Each cell agent has embedded stochastic kinetics of a growth-inhibiting molecule (analogous to a toxin) and a neutralizing molecule that binds and prevents toxicity (analogous to an antitoxin). As discussed in more detail in Methods, we assume toxin and antitoxin production, growth-mediated dilution, and binding-unbinding kinetics of the molecules. We used a phenomenological exponential function layer that translates between concentrations of toxin and resultant growth rate, with a single parameter, α, that determines the level of toxicity.

The key similarity between our experimental and computational approaches is the existence of a threshold in the molecular network that determines the growth rate of the cell. There are many potential mechanisms for such a threshold to arise, as discussed in the Introduction. We do not claim that the mechanism implemented in the computational model is the same as the experimental model. Rather, there is an underlying fundamental interplay between growth regulation and lineage structure that we will show is conserved.

To determine the effect of the growth threshold on microcolony dynamics, we scanned the rate of toxin production, keeping antitoxin production constant. (In most natural toxin-antitoxin systems, the antitoxin is unstable. We simulated this case as well, below). The simulations were seeded with a single cell growing with excess antitoxin and permitted to grow for 100 simulation minutes before changing the toxin production rate to a positive value. After several generations of growth, we found three qualitative regimes across different toxin production rates: symmetrical growth with no or little growth arrest (toxin production rate 0–2.5 /min), a critical regime with clusters of growing and growth arrested cells (toxin production rate 3–4.5 /min), and a regime of nearly instantaneous growth arrest (toxin production rate >4.5 /min) with the colony trapped in its near-initial state.

Time proceeds downward in each lineage and begins at the onset of toxin production (

Sub-lineages of fast-growing and slow-growing cells are evident in the critical regime (with toxin production rate 5–6 /min;

To quantitatively characterize the properties of growth transitions in our simple computational framework, we considered the fate of simulated microcolonies at 250 minutes of growth, which is shortly before the fastest growing cases begin to become computationally intractable, but after the population size is beyond the minimal requirement to be considered a microcolony. Mean population growth rates and toxin concentrations across multiple (N = 100) replicates reveal a growth-regulatable region flanked by regions of almost full growth and almost complete growth arrest (

To quantify the amount of lineage information shared by pairs of cells in their phenotypes, we calculated mutual information between phenotypic differences between pairs of cells and pairwise lineage distance. From each simulation, we sampled one pair of cells randomly to ensure independent, identically distributed samples and performed a resampling procedure 100 times to increase the confidence in our estimate. This was done for absolute growth rate differences and absolute toxin concentration differences (

Distributions of growth rates reveal the underlying population structure not evident from mean growth rates shown in

To examine a further indicator of criticality in this system, we calculated the dynamics of growing cell numbers below (toxin production rate 0–2.5 /min), near (toxin production rate 3–4.5 /min), and above the regulatable region (toxin production rate >4.5 /min) of growth rate. With toxin production well below the regulatable region, the predicted cell growth becomes equivalent to an uncoupled case where toxin has no effect on growth.

Growing cell numbers show variability between simulation replicates near the critical region (

If lineage is capable of constraining the attainable phenotypes of offspring cells, it stands to reason that the amount of phenotypic heterogeneity attainable in a microcolony is lowered by lineage dependence in systems that generate heterogeneity by diversifying growth rates. It is difficult to generalize what constitutes meaningful diversity in growth rates; small changes may or may not be important to fitness in the long run, but the importance of the distinction between growth arrested and fast-growing cells is clear. Therefore, we used two possible definitions of meaningful diversity: in one, arbitrarily small changes in growth rate or toxin concentration are meaningful. In the other extreme, we assumed that only growing versus non-growing cells (or high versus low toxin) is a meaningful distinction.

We quantified the phenotypic heterogeneity as information entropy (base 2), binning the simulated cells according to the two definitions of diversity (

Vertical line indicates the point of highest lineage-dependent mutual information between growth rate and lineage distance.

To explore the generality of our results, we created models with variations on the original, and tested for lineage dependence.

The first set of variations test two simplifications in the primary model: stability of the antitoxin, and bursty production of the molecular species. While we regard the model to be a general threshold-based growth control mechanism, it is worthwhile to determine if a toxin-antitoxin module with unstable antitoxin qualitatively reproduces our main results. Varying the stability of the antitoxin, we indeed found the same qualitative results (

Our next model variation was to vary the effect of growth regulation, increasing it (α = 0.3 in

Large clusters of growth arrested cells could have effects on the spatial development of bacterial colonies, as daughter cells tend to be correlated in space as well. We therefore asked what growth arrested cluster size distributions arise in the region where there is high mutual information between growth rate and lineage distance. We performed 10,000 simulations each and clustered the end-point populations according to lineage neighbors having similar growth rate (with a cutoff of 0.01 /h to be considered growth arrested). Resulting clusters were pooled across simulations of the same parameter set. We present distributions of raw absolute cluster size, not normalized.

Below the critical regime, the absolute cluster size distribution is nearly exponential (

Clusters are exponentially distributed below the critical region (red line, simulation; gray dashed line, exponential fit ^{–bc} for cluster sizes

Regulation of growth is a central part of phenotypic control. Many factors can control growth rate, including extrinsic conditions such as starvation, and intrinsic regulators of growth that often operate with a threshold-based mechanism. Using an experimental model of threshold-based growth arrest arising from metabolic toxicity, we tracked cell growth in a bacterial microcolony with a high probability of undergoing the growth arrest transition, and a colony grown in a condition that does not display the threshold-based growth arrest. We found several large, discrete shifts in growth rate to occur at a faster timescale than our 5-minute recording intervals (

We therefore sought the simplest possible model of microcolony growth dynamics that reproduces the effect. Our basic model captures single-cell biochemical kinetics on one scale (microscopic) interfacing population growth dynamics on another scale (macroscopic). We found striking phenotypic lineage dependence to emerge with the following criteria: (

As the probability of cellular transition to growth arrest increases, the mutual information between growth rate and lineage distance increases to a peak, then decreases as the simulated microcolony reaches the condition of immediate growth arrest. This transition bears a resemblance to a phase transition, with correlation of microscopic length scales peaking at the critical boundary. Here, the correlation length is in lineage space: we have assumed no traditional spatial information about the cells in the simulation.

Lineage space is a binary tree growing with extinction probability based on microscopic dynamics. Distances are modified by dynamical growth rates, which explains why a higher probability of heterogeneous growth results in structured trees. Thus, relating persister and other threshold-based growth arrest mechanisms to the established mathematics of branching processes [

After 100 simulated minutes we imposed a continuous rate of increased toxin production (or antitoxin degradation, in one derived model) on the developing microcolony. The constant input of more toxin created an irreversible threshold. Once a cell crosses the growth arrest threshold, there is an irreversible stoppage of growth that arises from toxin growth feedback. The growth arrest condition can then be considered an absorbing state. Continuous transitions from active to absorbing states are generically characterized by the scaling properties of critical directed percolation [

If lineages impart spatial structure onto growth phenotypes, then do they impose an upper limit to the level of phenotypic heterogeneity that can be attained by a microcolony? The population is most sensitive to fluctuations directly in the region with the highest lineage dependence, the latter of which appears to imply a dampening of phenotypic heterogeneity. However, multiple methods of measuring total population entropy suggest that the population can still approach the maximum total entropy in cases where growth rates are both finely-binned and binned into only two phenotypes–growing and growth arrested (

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

^{−}P_{lacO1}-GFP was grown from -80° C cryogenic culture for 18 h in LB medium in a shaking incubator (37° C), acclimatized by incubating in Davis minimal medium containing either 50 mg/ml lactose (DMlac50) or 2 mg/ml glucose (DMglc2) for 24 h, and resuspended either in fresh DMlac50 or DMglc2 culture, respectively, for 3 hours before beginning time-lapse microscopy.

We used an Olympus IX81 inverted fluorescence microscope with an incubated imaging chamber (Olympus, Tokyo, Japan). The chamber with objective was pre-heated, bacterial cultures were added to a pre-heated CellAsic ONIX microfluidic plate (Millipore, Billerica, Massachusetts) at an approximate OD450 of 0.005, and a continuous media flow of 1 psi DMlac50 or DMglc2 was maintained for the duration of the experiment. Images in brightfield and green fluorescence (488 nm stimulation / 509 nm emission) channels were captured every 5 minutes with a 4k CMOS camera, followed by ZDC autofocus. For the DMlac50 experiment, we used a 100x oil immersion objective. Due to technical issues with the objective, we used a 60x air objective for the DMglc2 experiment. Thus, the pixel lengths of the cells between the two experiments should not be directly compared.

Images were cropped after identifying a stable microcolony originating from a single cell. We developed a semi-supervised cell tracking algorithm in Mathematica (Wolfram Research, Champaign, Illinois) with manually input cell division times and cell lengths. From this information, we reconstructed the lineage and approximated growth rates with exponential growth models. When mapping the growth rates to the lineages in

To capture the minimal mechanisms necessary that recapitulate non-genetic inheritance and effects of cellular lineage, we created a multiscale growth simulation framework with individual cell agents, each containing a molecular network of interacting proteins, referred to as toxin and antitoxin, with toxin affecting cellular growth rate.

We track the simulated number of toxin and antitoxin molecules as well as cell volumes for each cell agent across time. In the next time step,

We sought to develop a sampling methodology to ensure independent, identically distributed samples from lineage simulations to estimate the mutual information between lineage distance _{i}

We considered a simple network consisting of three variables: toxin, antitoxin and toxin-antitoxin bound complex. Possible reaction events are synthesis of toxin and antitoxin, and binding and unbinding between toxin and antitoxin molecules. The reaction scheme for the basic model is:
_{t} is the toxin production rate varied in the simulations. Antitoxin production parameter, _{a}, is kept constant (_{a} = 4.2 /min) to allow the production ratio of toxin and antitoxin to be changed. Growth-mediated loss is implemented through _{b} and _{u} are binding and unbinding rates; _{b} = 0.1 and _{u} = 0.1 throughout. In the most basic model, each species is considered long-lived on the timescales of the simulation, so we do not consider any additional degradation processes. Variations on this model are discussed in Results.

The relationship between toxin concentration and cellular growth rate, the most phenomenological part of the framework, captures the interface between molecular and population dynamic scales. We reasoned that, while some random factors may reduce or increase the effect of toxin, the generality with which toxin affects global protein synthesis rates [^{-αT(t)/Ω(t)}

To illustrate the effects of growth arrest on distributions of growth-modulating cytoplasmic contents (

The exact functional dependency of growth on toxin is unknown. In our stochastic simulation framework, we considered an exponential dependence of growth on toxin. _{t} is the toxin production rate, _{t}/_{t} = 4.2 /min,

Time-lapse fluorescence microscopy of a cell lineage of ^{−}P_{lacO1}-GFP in DMlac50. Cells are tracked and measured as indicated. Numbers represent time (minutes) after the first frame. Experimental details are given in Methods.

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Time-lapse fluorescence microscopy of a cell lineage of ^{−}P_{lacO1}-GFP in DMglc2. Cells are tracked and measured as indicated. Numbers represent time (minutes) after the first frame. Experimental details are given in Methods.

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All cells (

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Lengths of growth arrest-prone cells between divisions were tested for a significant fit to an exponential growth model in the growth arrest-prone condition. These cases failed the significance test with a Bonferroni-adjusted

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We acknowledge Sheng Chen, Dominique Chu, Eric J. Deeds, and Uwe Täuber for useful conversations. Microscopy experiments were performed in the Microscopy and Analytical Imaging Laboratory at the University of Kansas.