Estimating Z-ring radius and contraction in dividing Escherichia coli

We present a fluorescence recovery after photobleaching-based method for monitoring the progression of septal Z-ring contraction in dividing Escherichia coli cells. In a large number of cells undergoing division, we irreversibly bleached cytosolically expressed Enhanced Green Fluorescent Protein on one side of the septal invagination and followed the fluorescence relaxation on both sides of the septum. Since the relaxation time depends on the cross-sectional area of the septum, it can be used to determine the septal radius r. Assuming that the fraction of the observed cells with r-values in a given interval reflects the duration of that interval in the division process we could derive an approximate time-course for the contraction event, as a population average. By applying the method repeatedly on individual cells, the contraction process was also followed in real time. On a population average level, our data are best described by a linear contraction process in time. However, on the single cell level the contraction processes display a complex behaviour, with varying levels of activity. The proposed approach provides a simple yet versatile method for studying Z-ring contraction in vivo, and will help to elucidate its underlying mechanisms.


Models used to determine the cytosolic diffusion coefficient of EGFP from FRAP and FCS
measurements. The analysis of FRAP data on cells, with length L and without any septum, was based on the one-dimensional diffusion equation: Eq. S1, with the boundary conditions of S2 and the initial condition of S3 yields the following solutions in the center of each half of the cell: The detected fluorescence is proportional to the concentration; hence the relative fluorescence intensity in the center of each half of the cell is given by: and where K is a proportionality constant and ) ( 1 t F and ) ( 2 t F describes the detected fluorescence in the center of the first and second half of the cell, respectively. In the experimental FRAP curves, the measured ) ( 1 t F and ) ( 2 t F were fitted simultaneously to Eq. S5A and S5B, respectively, including only the first two terms in each infinite sum ) (t S i (since the magnitude of the remaining terms are negligible compared to the first two).
For the FCS measurements, the experimental autocorrelation curves were fitted to the following model: Here, 1 N and 2 N are the number of molecules, within the detection volume, with diffusion coefficients 1 D and 2 D , respectively. T is the average fraction of molecules that temporarily resides in a dark state, and with a relaxation time T  related to the corresponding blinking (Widengren et al., 1999, Haupts et al., 1998). In the model, a two-dimensional diffusion of the molecules is assumed, since the height of the detection volume is larger than the diameter of the bacterium. In the fitting of Eq. S6 to the experimental FCS curves, T  was fixed to 300 s. This is a typical relaxation time found in FCS measurements for GFP under the excitation conditions used (see Materials and Methods) and the expected pH of the E. coli cytoplasm -3 -(pH ~ 7). The origin of this relaxation component was not investigated further, but may be either due to a photo-induced non-or weakly fluorescent state of EGFP, or follow from proton exchange. In the experimental FCS curves, the fraction of the faster component ) was 0.93 ± 0.03, indicating that most of the dynamics can be attributed to free diffusion of EGFP.
Model to determine the septal radius from FRAP curves The model is based on a geometry, consisting of two compartments with volumes V 1 and V 2 , interconnected by a cylindrical septal region of length l and radius r. According to Fick's first law the net number of molecules n passing through the septum connecting the two compartments, satisfy: Here, c is the concentration, . Now, based on Eq. S7, the number of molecules entering compartment 1 per time unit is given by: is the constant total number of molecules, and i N and i V are the number of molecules and the volume of compartment i. The general solution to Eq. S5 is: where  is an arbitrary constant.
The fluorescence ) (t F i is proportional to the concentration of fluorophores, i i V N / , hence: where  is an arbitrarily constant and is a function that corrects for bleach and drift during the measurement, see Eq. 4 in Materials and Methods. To determine ) , ( l r  , FEM-based (Strang & Fix, 1973) simulations of the FRAP experiment were performed. The geometry used in the simulations, specified to resemble a dividing cell, included two cylindrical compartments of length 2 m and radius R=500 nm. A cylindrical 50 nm long septum joined the two compartments. The concentration of the fluorescent molecule (EGFP) inside the cell obeyed the three dimensional diffusion equation with a negative source term generated by photobleaching: with boundary condition: on the surface of the cell, with surface normal n . * D is the diffusion coefficient of EGFP, taking steric and hydrodynamic hindrance into account, which is equal to Eq. 2 in the septal region and equal to D in the compartments. The rate of degradation, ) , ( t r k D , is only nonzero during the 40 ms of bleaching in one of the compartments, and depends on the excitation power and photophysical properties of the diffusing molecule (EGFP) (Widengren & Thyberg, 2005): Here 0  is the fluorescence lifetime, D  the photobleaching quantum yield and 1 S the (normalized) steady state population of the (first) excited state of the EGFP. The values for 0  , D  and 1 S were specified to 4 . 2 ns (Jakobs et al., 2000), 6 10 8   (Peterman et al., 1999) and 7%, respectively. The 7% value for 1 S was calibrated from experimental normalized fluorescence profiles immediately after the bleaching on cells lacking a visible septum. In principle it is possible to derive a value for 1 S based on the excitation intensity and various photophysical properties of EGFP. However, at intensities in the order of mW, estimations of D  become uncertain. Moreover, uncertainties in ) , ( t r k D mainly affects the amplitude  in Eq. S10, and not significantly the other parameters, and  is always kept free in the fitting of the simulated FRAP data to Eq. S10 (see below).
Simulated FRAP relaxation curves were generated for different septal radii by solving Eq. S11 together with border condition Eq. S12 numerically. These relaxation curves were then fitted to Eq. S10. Except for ) , ( l r  and  , which were allowed to vary freely, all the other parameters were in the fitting procedure fixed to the same values as used as input values in the simulations. Thereby, a relation was empirically found between r and ) , ( l r  with D = 4.5 µm 2 /s and with r varying from 6 to 500 nm (Fig. S5B). The generated relation could nicely be fitted to a sum of two exponentials (red curve in Fig. S5B):  (r,l  50 nm)  -2.5797 10 -4 +0.84068  e -r10 9 /36.1956 +0.2745  e -r10 9 /222.75349 (S14) Simulations were also performed to test how well Eq. S10 fits to the simulated FRAP data and how well the fitted parameters agree with the input parameters, for different sets of parameters and different cell geometries (data not shown). For cylindrical compartments, Eq. S10 was tested within the parameter range (6<r<400 nm, 3<D<6 µm 2 /s, 1< 1 L , 2 L <3.5 um, l = 50 nm). Eq. S10 was then found to fit well with the simulated FRAP relaxation curves and to yield fitting parameters well in perfect agreement with the input parameters.
If the compartments are ellipsoids instead of cylinders, (and their axes given by R and L1, L2) then the error in using a cylindrical model to determine r for ellipsoid compartments is -6 -negligible for radii less than 20 and larger than 100 nm, but can be as high as 20% for r = 50 nm.
The influence of the septum length, l, was also tested in the same way. The correction factor ) , ( l r  depends on l. However, neglecting this dependence, and assuming l to be as long as 200 nm instead of the assumed 50 nm, the estimated error will not exceed 25% for small septal radii (6 < r < 10 nm), 40% for middle sized radii (10 < r < 100 nm) and 20% for larger radii (r > 100 nm).