Using stochastic cell division and death to probe minimal units of cellular replication

Abstract The invariant cell initiation mass measured in bacterial growth experiments has been interpreted as a minimal unit of cellular replication. Here we argue that the existence of such minimal units induces a coupling between the rates of stochastic cell division and death. To probe this coupling we tracked live and dead cells in Escherichia coli populations treated with a ribosome-targeting antibiotic. We find that the growth exponent from macroscopic cell growth or decay measurements can be represented as the difference of microscopic first-order cell division and death rates. The boundary between cell growth and decay, at which the number of live cells remains constant over time, occurs at the minimal inhibitory concentration (MIC) of the antibiotic. This state appears macroscopically static but is microscopically dynamic: division and death rates exactly cancel at MIC but each is remarkably high, reaching 60% of the antibiotic-free division rate. A stochastic model of cells as collections of minimal replicating units we term ‘widgets’ reproduces both steady-state and transient features of our experiments. Sub-cellular fluctuations of widget numbers stochastically drive each new daughter cell to one of two alternate fates, division or death. First-order division or death rates emerge as eigenvalues of a stationary Markov process, and can be expressed in terms of the widget’s molecular properties. High division and death rates at MIC arise due to low mean and high relative fluctuations of widget number. Isolating cells at the threshold of irreversible death might allow molecular characterization of this minimal replication unit.


Introduction
There are many approaches to define a 'minimal cell'. Some attempt to construct protocells out of elementary molecules and chemical processes [1,2]. Others start with a complex cell and reduce it to an essential core [3]. A third fruitful approach uses natural patterns of cell growth to infer basic requirements for cellular replication [4][5][6]. Campbell [5] realized that exponentially growing cellular populations were in a state of 'balanced growth': the chemical composition of a daughter cell immediately after division was invariant from one generation to the next, leading to a well-defined and constant doubling time. The specific dependence of the exponential growth rate (the exponent of the cell density versus time curve) on nutrient or antibiotic concentrations can be summarized as 'growth laws' [7]. Models of bacteria as autocatalytic chemical reactors accurately capture many mathematical features of these growth laws [8][9][10][11]. Combining such models with bacterial growth measurements, Jun and colleagues [11] have demonstrated an invariant cell initiation mass which they interpret as a minimal unit of cellular replication.
Studies of bacterial growth laws have focused mainly on exponential growth. However, bacterial populations in the presence of high antibiotic levels can also undergo sustained exponential decay over several orders of magnitude [12,13]. This is surprising: exponential growth can arise from deterministic cell doubling, but exponential decay with first-order kinetics typically occurs when individuals in a population die at random, like radioactive nuclei. If some cells die early while others die later, this must be due to some underlying cell-to-cell variability. The growth of single cells is known to be a stochastic, fluctuating process [14][15][16]. We should therefore consider the possibility that the choice between cell division and cell death could be a stochastic event. This contrasts with models that ascribe changes in exponential growth rates to changes in division rates alone [8,9]. No analysis has so far attempted to simultaneously account for molecular fluctuations, cell division, and cell death within a common framework.
Here we show that observed macroscopic features of cell growth and decay are consistent with the hypothesis that single cells make a stochastic choice between division and death. We also show that this type of stochastic choice naturally arises from a microscopic phenomenological model of cells as collections of subcellular replicating units. A replicating unit is an autocatalytic set of molecules and reactions, which might include ribosomes, DNA replication initiation complexes, and metabolic loops [8,9,11,17]. Such a unit is termed minimal if the removal of any of its components results in the loss of autocatalytic activity. Here we sidestep the issue of the precise composition of the minimal unit, grouping the entire replicating set of molecules and reactions into a single abstract 'widget'.
We model the synthesis, degradation and partitioning of widgets as stochastic biochemical processes. Cell division or death occurs when a cell hits high or low thresholds of these widgets. When the average number of widgets is small, sub-cellular fluctuations in their number drive a stochastic choice between cell division and death, thus coupling molecular dynamics with cellular dynamics. Remarkably, the predictions of this basic model match observed qualitative features of cell growth and decay. These observations suggest a fourth operational definition of a minimal cell: one at the threshold of death due to irreversible loss of its last functional replicating unit.

Methods
Cell growth and cell density measurement protocols We grew Escherichia coli MG1655 cells from a single colony overnight in Luria Bertani medium at 37°C. We transferred 50 μl of this culture to 25 ml glucose M9 minimal medium in a 100 ml flask at 37°C. The OD 600 of this culture was monitored until it reached 0.1. At this point, we added the appropriate concentration of kanamycin, and this was defined as the t=0 time point of our measurement. Every 40 min up to a maximum of 280 min, 600 μl of this culture was collected for OD 600 measurements, and 100 μl was collected for colony forming unit counts (CFU ml −1 ). We use a (non-standard) stringent definition of the minimal inhibitory concentration (MIC) as the lowest kanamycin level at which CFU ml −1 is non-increasing. By serially increasing [Kan] we found MIC to be between 4.2 and 4.3 μg ml −1 . Pipetting errors cause variations beyond this level of precision. The numerical value [Kan] MIC =4.21 μg ml −1 represents the smallest increment above 4.2 μg ml −1 at which CFU ml −1 was stable over 280 min. We determined OD 600 and colony counts using multiple dilutions. Colony counts were measured for four technical replicates; at MIC we used two biological replicates, each with four technical replicates. Minimal medium (100 ml): water, 76.8 ml; 10X M9 salts 10 ml; 20% glucose, 2 ml; 1 M CaCl 2 , 10 μl; 100 mM thiamine, 1 ml; 4% casamino acids, 10 ml; 1 M MgSO 4 200 μl.

Stochastic model of cell division and death
The transition system shown in figure 2(D) defines a Markov process with the following Master equation: Here, each c w represents the number of cells (or the normalized probability of cells, depending on the context) with precisely w widgets for w 1, , 1, = ¼ Wwith the stipulation that c 0. = W The first line corresponds to cells gaining or losing individual widgets. The second line corresponds to the creation of two new daughter cells by the instantaneous division of a cell that hits W widgets, which happens at rate c 1 .
The resulting daughters are defined by w¢ and w such that w w . ¢ +  = W The first factor of 2 accounts for two ways of achieving any given w , ¢ in the left or right daughter. The binomial coefficient arises since each widget has an equal chance of being inherited by either daughter cell. A cell divides instantaneously when it hits W widgets. The usual normalizing factor of 1 2 W / is replaced by 1 2 2 : - } are ignored since cells repeatedly divide until some other partition occurs.
We assume a large number of total cells and widgets, so the branching process never goes extinct. If c c c ] is a column vector, the system of equations equation (1) can be written using a transition matrix A and solved by matrix exponentiation: L º ( ) ( ) ( ) At long times this distribution approaches the eigenvector of A corresponding to its largest eigenvalue: We can see by direct substitution that a g is an eigenvalue of A. Since the number of live cells cannot increase any faster than the number of widgets, we also know this is its largest eigenvalue. Once f t ( ) is determined we calculate the specific division and death rates t f + ( ) and t f -( ) as the rates at which cells cross the right boundary w 1 = Wand the left boundary w 1.
= By measuring time in units of , 1 awe can see that the values f a  / depend only on the ratio g a / and on W ( figure 4(B)).

Probability of division
An immediate post-division daughter cell can have any widget number in the range w 1, , the probability of next division is a first-passage-time problem with absorbing boundaries at w 0 = and w . = W This corresponds to a new transition matrix Â where the binomial partition terms have been removed. We can find c c The integrated flux leaving the right and left boundaries, corresponding to probabilities of division or death, are:

Results
Measuring cell growth in the presence of antibiotics Different classes of antibiotics act through distinct mechanisms [18]. Since we focus on replicating units, here we use the aminoglycoside antibiotic kanamycin which irreversibly binds to and inhibits the ribosome [19].
The effect of an antibiotic is typically quantified in terms of its impact on growth rates. It is common to use turbidity measurements (OD 600 ) for this purpose, since these are easy to perform and automate [20]. However, cell growth is more accurately determined by measuring the density of viable colony-forming units (CFU ml −1 ) [13]. These two are not equivalent: colony-forming units measure the density of live cells, whereas turbidity measures the total density of all non-lysed cells, live or dead ( figure 1(A)). For the remainder of our analysis we always compare CFU ml −1 (live cells) with the rescaled value 8×10 7 ×OD 600 (total cells) as these coincide for exponentially growing E. coli. cells in the absence of antibiotics. Since we use a fixed volume of media we use the terms cell number and cell density interchangeably.
The MIC of an antibiotic is often defined as the concentration at which OD 600 no longer increases, but this depends on the duration and sensitivity of the measurement [21]. Here we rigorously define MIC as the antibiotic concentration at which CFU ml −1 is asymptotically constant over time.

Live and total cell counts under antibiotic treatment
We monitored the effect of kanamycin addition on E. coli cells grown in an initially antibiotic-free medium . This interpretation assumes a low rate of cell lysis. We have also not considered persister cells that slow their division under antibiotic treatment [22]; these are significant once nearly all cells in the original population have already died.
Exponential growth or decay arises from first-order cell division and death rates As a first attempt to understand these dynamics, we decomposed the separate contributions of cell division (f + ) and death (f -) rates using a first-order kinetic model ( figure 2(C)):  figure 1(B)). At high antibiotic we -This again matches the data: we see that live cell number decays exponentially, while total cell number approaches a flat asymptote ( figure 1(D)).
At MIC the situation is more interesting since by definition , That is, live cell numbers are constant because death and division rates balance, while dead cell numbers increase linearly because they arise from the continuing death of live cells. This is precisely what we observe: the slope of the linear portion of the total cell curve at MIC shows that 0.011 MIC f = min −1 ( figure 1(C), left panel). The total cell curve tracks the live cell curve for the first hour following antibiotic treatment, after which the live cell curve flattens while the total curve increases linearly. This suggests cell death only begins after a lag, while cell division is relatively unperturbed by the antibiotic. This is corroborated by the ratio 0.6 MIC max f f+ / being close to unity: cell division is nearly as rapid at MIC as at zero antibiotic. In summary, the following three observations support the idea that cell division and death operate at the single-cell level with apparent first-order kinetics, once transients die out. At low antibiotic the growth of total cell number and live cell number are both exponential. At high antibiotic live cell number decays exponentially, while total cell number approaches a constant. In between, at MIC, total cell number increases linearly, while live cell number is constant. Seeing first-order kinetics across the full range of antibiotic concentrations is surprising, since cells are not elementary chemical entities. In the following section we show how such kinetics emerge from a stochastic model of a cell as a collection of minimal replicating units.

Widgets: sub-cellular replicating units
The phenomenological model of equation (5) fails to predict the transient dynamics because it assumes a cell has no internal structure. If we wish to determine how f  depend on time, this must either be directly measured, or predicted from a more microscopic model. We therefore consider a cell as a collection of replicating units we term 'widgets' (figure 2; Methods: Stochastic model of cell division and death). The widgets themselves obey a birth-death dynamics analogous to equation (5), but with microscopic birth and death rate constants a and g ( figure 2(A)). For concreteness we imagine a to be constant (e.g. the catalytic efficiency of ribosomes) while g depends on the antibiotic concentration (e.g. the rate of irreversible ribosome inhibition by kanamycin), but these assumptions may be relaxed. We specify how cell division and death depend on the widgets as follows ( figure 2(B)). When the widget number hits w 0 = the cell dies since no new widgets can be made without an existing widget. When the widget number hits a threshold w = W the cell instantaneously divides, and the widgets are partitioned binomially between two daughter cells. This is arguably the simplest possible microscopic model of cell growth.
We consider a population of cells, binned according to the number of widgets they contain: c w is the number of cells with precisely w widgets, for w 1, , l If we define a normalized distribution f , That is, the shape of the distribution becomes constant, while the total number of cells increases or decreases exponentially.

Comparison of widget model to experimental growth curves
If we measure time in units of 1 athe model has two dimensionless parameters: the widget death/birth ratio , g a / and the threshold number of widgets at cell division . W The value of g a / is some monotonically increasing function of antibiotic concentration, not necessarily linear, with 1 g a = / at MIC. We are left with a single tunable parameter W which controls the number of widgets and therefore influences the scale of stochastic fluctuations: higher values of W correspond to lower fluctuations relative to the mean. We will return to this point in our discussion.
In our experiments we first grow cells in in the absence of antibiotic and then add kanamycin at the initial measurement point. To model this we first find the stationary distribution of widgets at zero antibiotic, and set this as the initial condition: )The addition of antibiotic is modeled by shifting g to some nonzero value, causing the cell population to evolve toward a new asymptotic distribution: )This corresponds to the transient phase of the experiment, as cells adapt to the presence of the antibiotic. As cells go through this transient we can use equation (8) to find c t w ( ) and equation (7) to find c t L ( ) and c t D ( ) for various ratios . Note that the assignment 10 W = is not a numerical fit, it is a representative parameter choice. It is not justifiable to fit the abstract widget model to quantitative measurements of cell growth, which are expected to depend on more complex aspects of metabolism and cell size control [11]. Nevertheless it is remarkable that such a basic model captures diverse qualitative aspects of cell growth and decay across a range of antibiotic concentrations.

Widget fluctuations drive a stochastic choice between cell division and death
A key aspect of the model is the choice of a post-division cell between two ultimate fates: division and death (Methods: Probability of division). Immediately following division the widgets partition binomially between daughter cells, leading to an initial post-division variation (histogram, figure 3(E)). At this point, the fluctuating birth-death widget dynamics take over. Cells with an initially low value of w are more likely to hit the left boundary and die, while cells with an initially high value are more likely to hit the right boundary and divide (curves, figure 3(E)). The addition of antibiotics (increasing g a / ) biases the choice against division. The squared coefficient of variation of the post-division binomial distribution 1 W / is a convenient measure of fluctuations. Note that increasing 1 W / increases fluctuations in both binomial widget partitioning and Poisson birth/death dynamics. The fewer the number of widgets, the larger the scale of fluctuations relative to the mean.
Deriving cellular parameters from widget properties Having validated the model against a specific set of experiments, we now use it to predict aspects of cell growth over a broader range of conditions. The dynamics of the widgets can be used to determine the effective parameters f  that appear in equation (5) (figure 2(D)). To do this we first find the number of cells in each widget bin, then track how many cells cross the right or left boundary. We can thus write an equation similar to equation (5), where f  are now time-dependent because the distribution of cells evolves from its initial state: L The inset shows the prediction for a cell that has no internal structure and divides as soon as 2; W = no transient is observed in this case. (E) Two sources of fluctuations: random partitioning and random birth/death of widgets. Immediately after cell division, the number of widgets w in a daughter cell is binomially distributed (gray histogram). Starting at any widget number, random birth/death dynamics can take a cell to either boundary. We show the probability that a cell will successfully divide again rather than die (curves; colors represent different values of g a / for 10 W = ).
Classically, exponentially growing populations are thought to arise when post-division daughter cells reach a time-invariant composition [5]. Our analysis suggests a broader pattern in which exponentially growing or decaying populations arise because the entire population of cells reaches a time-invariant distribution over compositions, due to which the per-cell rates of division and death appear to be first-order constants. However, these constants themselves obey certain constraints. Comparing equations (9) to (6) we have . l f f = -+ -On the other hand, it is easy to check that the largest eigenvalue of A is given by l a g = -(Methods: Stochastic model of cell division and death). This gives us two completely distinct ways to decompose the growth exponent l in the limit t :  ¥ .