Effect of the Min System on Timing of Cell Division in Escherichia coli

In Escherichia coli the Min protein system plays an important role in positioning the division site. We show that this system also has an effect on timing of cell division. We do this in a quantitative way by measuring the cell division waiting time (defined as time difference between appearance of a division site and the division event) and the Z-ring existence time. Both quantities are found to be different in WT and cells without functional Min system. We develop a series of theoretical models whose predictions are compared with the experimental findings. Continuous improvement leads to a final model that is able to explain all relevant experimental observations. In particular, it shows that the chromosome segregation defect caused by the absence of Min proteins has an important influence on timing of cell division. Our results indicate that the Min system affects the septum formation rate. In the absence of the Min proteins this rate is reduced, leading to the observed strongly randomized cell division events and the longer division waiting times.


I. SI SUPPORTING FIGURES
We take total cell length as measure of OD of the culture. This quantity is calculated every minute in the simulations. As one can see the increase in total cell length is clearly linear on a log scale and the slope corresponds to a doubling time of 75 minutes.  Thus, chromosome aggregation occurs (on average) only at 5 out of the 7 potential division sites. These 5 sites are marked with green arrows in (c).  Table 1.

II. SI THEORETICAL BACKGROUND: MODEL PARAMETERS
Here, we derive the details of cell growth and the parameter values that enter into the theoretical models.

A. Cell growth
To be able to implement cell growth in the simulations we needed to figure out if E. coli cells increase their mass (and thus length) exponentially or linearly with time (or in a more complicated way [1]). To do so we took pictures of the cells every 5 minutes and measured their lengths. We analyzed the data in two different ways to determine the increment in cell length per time.

Rescaled cell length
If cell length L increases exponentially with time t, then where L 0 is the newborn cell length (that can be different for different cells). T is the doubling time, and t 0 the time at which the cell was born. The last equation can be written as where ln 2/T on the right hand side is a constant for the cells grown under the same condition. In particular, one obtains for the division length L d ln By combining Eq.
(3) we can eliminate the difficulties caused by the differences in newborn cell length, i.e.
Upon introducing the rescaled length increment and the rescaled time Eq. (4) then implies l resc = t resc , i.e. a linear correlation between cell length increment and time. Thus, for an exponential increase in cell mass all experimental data points should lie on a straight line from (0,0) to (1,1), see S7.
As can be seen, the rescaled experimental data (represented by blue dots) clearly lie on the straight line from (0,0) to (1,1). The red line shows how the curve would look like if the cells were increasing cell length linearly in time. In this way the differences between the two growth modes becomes apparent, indicating that the cells indeed increase their mass exponentially.

Cell length increment rate
Another way to distinguish exponential from linear mass increase is to calculate the cell length increment rate. For exponential time dependence, Eq. (1) implies while for a linear increase one has and Eq. (7) shows that for an exponential mass increase the length increment rate is proportional to its length, while for a linear time dependence the length increment rate is constant.
From Fig. S8 we can see that for both strains the length increment rate is indeed proportional to the cell length, and the ratio is about the same for both strains. This is also consistent with our conclusion that these two strains have similar growth rates.
From the combination of these two methods, we conclude that the cells increase their length exponentially with time. In particular, from our experimental data we can also exclude the possibility of a linear or bilinear length (or mass) increase.
In this case, in Fig. S8a all data points would line on or two horizontal lines.

B. Doubling time and division waiting time
In the simulations each cell gets a doubling time assigned that is drawn from the experimentally determined distribution of WT. We use the same distribution for minB − cells as the Min system does not affect cell growth, see OD plots in

C. Cell lengths parameters
As mentioned above, in the simulations a starting length and an ending length are assigned to each new compartment. The ending length is twice as long as the starting length, but neither of them can be measured directly from the experimental data. What we can measure is the new born cell length of WT cells (Fig. S11).
From the average new born cell length of WT L 0 we can calculate the starting length by Here, T is the average doubling time in Fig. S9 and T w is the waiting time in In the simulation the starting length is drawn from a normal distribution with average L s and standard deviation 0.1L s . This is set according to the distribution of new born cell lengths (Fig. S11). Again, we used the length distribution of WT cells as a parameter for the minB − strain in the simulations.                   Figure S11