Synthesis and degradation of FtsZ quantitatively predict the first cell division in starved bacteria

Abstract In natural environments, microbes are typically non‐dividing and gauge when nutrients permit division. Current models are phenomenological and specific to nutrient‐rich, exponentially growing cells, thus cannot predict the first division under limiting nutrient availability. To assess this regime, we supplied starving Escherichia coli with glucose pulses at increasing frequencies. Real‐time metabolomics and microfluidic single‐cell microscopy revealed unexpected, rapid protein, and nucleic acid synthesis already from minuscule glucose pulses in non‐dividing cells. Additionally, the lag time to first division shortened as pulsing frequency increased. We pinpointed division timing and dependence on nutrient frequency to the changing abundance of the division protein FtsZ. A dynamic, mechanistic model quantitatively relates lag time to FtsZ synthesis from nutrient pulses and FtsZ protease‐dependent degradation. Lag time changed in model‐congruent manners, when we experimentally modulated the synthesis or degradation of FtsZ. Thus, limiting abundance of FtsZ can quantitatively predict timing of the first cell division.


Supplemental Figures
Appendix Figure S1:

18.5%
For all flow cytometry experiments, cells were stained with SYBR Green I dye and incubated for at least 10 minutes before measurement (see Methods). 10 µL of events were measured (scattering and fluorescence) at the slow rate (14 µL/min) and then analyzed with MATLAB R2015B. A scatter plot of the measured events are shown for a blank (left) and cell sample (right). We only focused on the forward scattering and green fluorescence dimensions. The gating used for all samples is shown by the red region. In the sample shown, the gate captured 18.5% of the events, which were taken to be the bacterial cells.
Appendix Figure   B Accumulated valine, tyrosine, (iso)leucine, and guanine depleted and recovered after pulse occurrence suggesting protein and nucleic synthesis.
C Other amino acids and hypoxanthine were affected by corresponding antibiotics (f = 0.18 mmol/g/h). Antibiotics were added one minute after the second pulse (yellow region). Chloramphenicol (blue) inhibits protein biosynthesis, rifamycin (orange) inhibits RNA polymerase, and azidothymidine (AZT; red) inhibits DNA synthesis. The ion for tyrosine could not be annotated for the f = 0 mmol/g/h measurement.
Appendix Figure S4: Antibiotic metabolomics data with no antibiotic control.  Figure 4C is plotted against the no antibiotic control condition (f = 0.18 mmol/g/h from Figure 4A). Black dots indicate the ion intensity for the no antibiotic condition and the solid lines indicate the moving average filter of the ion intensity. Induction of FtsZ from a plasmid allowed for division at a feedrate (f = 0.17-0.18 mmol/g/h), which is below the critical rate in the control strain (f = 0.2 mmol/g/h). For induction, max, half, and zero correspond to addition of 50, 10, and 0 ng/µL of doxycycline respectively. The TI feedrate is abbreviated as f.

Data from
Appendix Figure S8: Division is limited by FtsZ also under "natural" starvation. In contrast to the previous experiments with a sudden, artificially induced starvation of cells harvested from the mid exponential growth phase, here we allowed cells to enter a more natural stationary phase upon depletion of the supplied limiting glucose. After optical density stabilization, cells remained for 2 to 6 h in starvation, before adding the inducer and the initiating he glucose pulsing experiment at the indicated TI feedrates. FtsZ was titrated via plasmid-based, inducible expression. For induction, max and zero correspond to addition of 50 and 0 ng/µL of doxycycline respectively. The TI feedrate is abbreviated as f. The parent plasmid pJKR-L-tetR [9] was used to determine the appropriate induction level for all titration experiments. GFP expression is driven by the titratable pLtetO promoter. Normalized GFP levels are shown for different levels of inducer doxycycline over time for microplate cultivations. Time at 0 indicates the addition of doxycycline and start of plate reader measurement. For titration experiments, 50 ng/µL was selected as high, 10 ng/µL as half, and 0 ng/µL as zero expression.
2 Supplemental Information -Development and parametrization of the FtsZ model

Introduction
This section discusses how the FtsZ model was developed and parameterized with values from previous literature. We started with the most basic model possible for the FtsZ threshold activation of division and used a differential equation to describe how FtsZ abundance changes. Starting with the equation from [12], we considered just the synthesis and degradation of FtsZ toward its changing abundance. We assumed dilution effects to be neglible because the cells are not dividing. Furthermore, we expect the number of FtsZ per cell to dictate division occurrence, not the intracellular concentration. For synthesis, we assumed that the yield of FtsZ on carbon is constant and, therefore, can be approximated as a TI feedrate dependent synthesis (αf ): Considering the abundance of FtsZ throughout stationary phase [10], we found that the level remains roughly constant (700 copies/cell) between 1 day and 3 days in stationary phase. This suggests that FtsZ never fully depletes during stationary phase, and that our model was oversimplified. To account for this, we posed an additional term for the basal synthesis of FtsZ (α 0 ):

Closed Form Solution for the Michaelis-Menten Equation
A closed form solution to time-dependent enzyme kinetics, based on the Michaelis-Menten rate law, was first described in 1997 by Schnell and Mendoza [11]. Using the quasi-steady-state approximation, the differential equation that describes the concentration of a substrate S that is degraded by and enzyme with a maximum velocity of V max is given by: Solving this equation for [S](t) yields: Where [S] 0 is the value of [S] at time t = 0, and W [·] is the Lambert-W function (also known as the omega function or the product logarithm) and is defined as the inverse of the function z → z · e z (see [3]).
In our work, however, this solution is not sufficient, since we are dealing with constant positive production on top of the enzymatic degradation, i.e. α 0 + α 1 f . Therefore, we modified equation 3 to be: The solution in this more general case is: e. the value of [S] at steady-state (or t → ∞). Note that if v in > V max , the value of [S] ∞ will be negative, indicating that the system has no steady-state (and [S] continues to grow indefinitely). Nevertheless, equation 6 holds even in such cases, except that we need to use the lower branch of the Lambert-W function (usually denoted W −1 ).
It is sometimes useful to consider the inverse function of [S](t), as we will soon see for predicting the lag time using [FtsZ] levels. Note that since the differential equation (5) is time invariant, there is a unique solution for the time difference ∆t = t 1 − t 0 , given the initial and final concentrations [S](t 0 ) and [S](t 1 ). The solution is given by:

Applying the Closed Form Solution
In the case of [FtsZ], we could now apply equation 6 which will be of the following form: where In order to answer the question what is the time t at which [FtsZ] accumulate from an initial concentration (F I ) to the threshold concentration required for division (F T ), we could use the inverse formula, i.e. equation 7: As one would expect, this function diverges (t lag → ∞) when [FtsZ] ∞ → F T . Since [FtsZ] ∞ is a function of the TI feedrate, we could solve for f from equation 2 and find that this happens when For any f < f critical , the production rate of FtsZ is too low and its level would never reach F T . Before using this closed form solution to predict lag times, we had to first find the values for the system parameters, namely α 0 , α 1 , K M , V max , F I , and F T .

Parametrization
We first parametrized the degradation term using in vitro measurements from previous work [2]. We adapted data from Figure 1F within [2] to find the turnover number of FtsZ into the ClpXP protease complex. The data shows the degradation of FtsZ over 90 minutes depending on the concentration of ClpX in 25 µL total volume. We considered two points seemingly within the linear range of degradation: Roughly, we have a turnover number of 0.05 min −1 . For the actual V max , we need to consider the number of active ClpXP within E. coli. From [4], we know that ClpX is stochiometrically limiting toward formation of the ClpXP complex. Therefore, to calculate V max , we can simply consider the number of ClpX in E. coli. In starved conditions, there is approximately 200 copies of ClpX per cell [10]. Our V max is approximated to be 200 · 0.05 = 10 [cell −1 min −1 ].
Unfortunately, K M is difficult to resolve using data from [2]. We, therefore, just used the same value from an earlier, non-specific protein degradation kinetics study [12] where K M = 600 [cell −1 ].
We could now use the final steady-state condition to calculate the basal synthesis term (α 0 ). Per the observation that FtsZ maintains a steady-state abundance of 700 copies per cell in stationary phase, which means that [FtsZ] ∞ = 700 [cell −1 ], when f = 0.
We then had enough information to predict how FtsZ depletes during starvation without feeding. Per [10], we see that number of FtsZ under growing conditions is about 2000 copies/cell. We took this value to be the FtsZ concentration both at the onset of starvation as well as the threshold F T needed to induce cell division. Using the above parameters from literature and plugging them into equation (6), we calculated that FtsZ levels after 2 hours of starvation would deplete to 1740 copies/cell. This value was taken to be the starting FtsZ at the onset of pulsing, i.e. F I = 1740 [cell −1 ]. The values for all the model parameters are summarized in Supplementary Table S5.

Fitting α 1
Now, the only missing parameter was α 1 , i.e. the yield of FtsZ from fed carbon (in units of cell −1 min −1 mmol glc/g DCW/h ). Since there is no available data for this relationship, we fitted the value of α 1 based on our TI feedrate versus lag time data and the results from equation (10), as illustrated in Supplementary Fig S10. Since the lag time spreads for lower feedrates, we applied a log transform on the lag times prior to fitting. We did not consider data point where no visible lag time was detected because our FtsZ model assumes t lag > 0. We found that α 1 = 12.9 We could now rewrite the lag time as a direct function of the TI feedrate f : and calculate the critical TI feedrate: This is very close to our experimentally measured threshold of non-division (over the first 6 hours), which was at 0.2 mmol glc/g DCW/h.
Note that although t lag (f ) seems to have a singularity point at f = Vmax−α 0 α 1 (in our case it is equal to 0.355 mmol glc/g DCW/h), the function is continuous finite at the entire range of (f critical , ∞).

Taking the pulsing into account
It is important to remember that our solution for the estimated lag time (equation 10) assumes that after the starvation phase, there is a steady production of [FtsZ] (α 0 +α 1 f ). However, our experimental system provides a pulse of glucose every 2-10 minutes, and the cells consume that glucose within ≈ 0.2 minutes (assuming a maximal glucose uptake rate of 10 mmol glc/g DCW/h [7]).
In order to check whether the smooth simplification (denoted the smooth model) alters the predicted lag phases, we redid our calculations without this assumption, by precisely tracking the changes in FtsZ production rates during the pulse (i.e. while glucose is present) and between the pulses (i.e. while the glucose level is 0). The results are presented in Supplementary Figure S11, and show that the effect is negligible. Finally, we compared the lag time predictions of the smooth and pulsing models, assuming the same average TI feedrate. In addition, we solved the ODE system numerically by integrating over time and finding the point where [FtsZ] crosses the threshold F T . We conclude that the differences between all three models are negligible (Supplementary Figure S12). Appendix Figure S12: All three models give very similar predictions for the lag time.
All code used for parametrization and model generation is available in Supplementary Material or at https://github.com/karsekar/pulsefeeding-analysis.

Supplemental Tables
Appendix Table S1: Summary information for wild-type pulse feed experiments. For the units, mmol is mmol of glucose, and g is grams dry cell weight of E. coli.
Appendix Table S2: C 13 Labeled fraction of measured amino acids and glycogen in washed, hydrolyzed extract.
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