ParB-GFP foci are spaced equally over the nucleoid

To study par2-mediated plasmid segregation, we investigated ParB-GFP localization, expressed from a par2-carrying mini-R1 test plasmid. The par2 locus containing the parB::sfGFP fusion is fully functional as judged by loss-frequency assays (S1A Fig.). As previously described, usage of ParA-GFP and the tetO-TetR-mCherry labeling system also does not affect plasmid stability, indicating full functionality [9], [12]. ParB-GFP forms foci that are regularly positioned along the long cell axis in vivo (Fig. 1A), consistent with ParB-binding to plasmid-encoded parC regions [10], [11]. Since plasmid dynamics occur primarily over the nucleoid, we reasoned that plasmid positioning with respect to the nucleoid rather than cell length is most informative. Therefore we measured ParB-GFP foci localization, together with Hoechst (DNA) stain to determine the nucleoid boundaries. As expected ParB-GFP foci colocalized exclusively with the Hoechst stain, and were equally spaced over the nucleoid (Fig. 1B,C,D for n_p = 1,2, S2A,B Fig. for n_p = 3,4).

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Figure 1. Plasmid foci are equally spaced over the nucleoid.

(A) Fluorescence localization of plasmid-binding protein ParB-GFP (green) and Hoechst DNA stain (blue) in representative WT E. coli cells. Scale bars: 1 µm; plasmid: pFS21 (mini-R1, parC1⁺, parA⁺, parB::sfGFP, parC2⁺). (B) Scatter plot of plasmid foci positions (blue, red) with respect to nucleoid edges (purple) and cell edges (black) for wild-type cells with n_p = 1,2 plasmid foci. (C) Histograms of plasmid foci positions shown in (B) relative to nucleoid length. (D) Scatter plot (blue) of the interplasmid focus distance as a function of nucleoid length in cells exhibiting two plasmid foci. A least square fit (black line) indicates a slope of 0.5.

https://doi.org/10.1371/journal.pcbi.1004009.g001

Mathematical analysis shows that dynamic, competitive ParA concentrations can generate equal plasmid spacing

Several studies have proposed that plasmid positioning is controlled by a concentration gradient of ParA over the nucleoid [8], [16]–[20]. Intuitively in this mechanism, ParB bound to plasmid parC (ParB-parC complex) interacts with nucleoid associated ParA-ATP, which effectively anchors the plasmid to the nucleoid. At the same time, the ParB-parC complex stimulates ParA-ATP hydrolysis causing a local ParA-ATP depletion. These processes could then generate a ParA-ATP gradient which a plasmid is able to follow. Reorganization of ParA gradients under the influence of multiple ParB-parC complexes might then lead to equal plasmid spacing. To rigorously understand if, and with what requirements, equal spacing can be achieved we develop here a minimal mathematical model based on the above principles.

We model the nucleoid as a 1d system of length (along the long axis of the cell) on which ParA-ATP and plasmids can interact. Let denote the nucleoid-associated ParA-ATP concentration at position relative to one nucleoid edge at time . Let be the positions of the n_p plasmids. ParA can bind to the nucleoid with flux . Once bound, ParA-ATP can diffuse along the nucleoid with diffusion constant . For simplicity, we first assume that the ParA-ATP concentration at each plasmid is zero due to a high ParA-ATP hydrolysis rate. Later on we will relax this assumption. This system can be described by the deterministic reaction-diffusion equations:

We now use separation of time scales to obtain the steady-state solution for : we assume that plasmid motion is much slower than the time for individual ParA-ATP molecules to diffuse over the nucleoid and generate a concentration profile. In this way, the plasmid positions are effectively time-independent and a priori unknown. The equation for then simplifies to:(1)

This equation can be solved by integrating twice using the boundary conditions. The solution is given by:(2)

Next we use these equations to compute the diffusive fluxes of ParA-ATP, , at a plasmid location , where the + and – superscripts below refer to the flux from the right (+) and left (-) respectively. We find:

Clearly, a symmetric ParA concentration profile, where fluxes from either side balance, is only possible for . The plasmids are then equally distributed with . We note that the predicted inter-plasmid spacing arising from this analysis is consistent with our experimental findings (Fig. 1D, S2B).

Importantly, the above analysis provides insight into the equal spacing mechanism. The key is that the above fluxes depend on the distances either between the plasmid and nucleoid end, or between neighboring plasmids. This feature is a consequence of ParA binding to the nucleoid anywhere, but with ParA release only occurring at a plasmid. In order for these on and off fluxes to balance at steady-state, the off-flux at a plasmid must scale with the inter-plasmid or plasmid-nucleoid-end distance. In this way, non-local information about lengths is converted into local spacing information encoded in the slope of ParA-ATP concentration. For non-equal plasmid spacing, the competing ParA concentrations on either side of a plasmid will be unequal, with one gradient steeper than the other. The steeper gradient corresponds to the side with the greater available space for ParA binding. If a plasmid can preferentially move (on the appropriate slow time scale) towards the side with the locally steepest ParA-ATP concentration, the plasmids are then progressively restored towards equal spacing. As this process occurs, the ParA-ATP concentrations will dynamically reorganize such that a symmetric configuration around a plasmid is reached only when the plasmids are equally spaced. In this state, where the competing ParA-ATP concentrations are symmetric, plasmid movement would no longer have a directional preference and would thus remain, on average, stationary.

So far, we have assumed that the ParA-ATP concentration vanishes at a plasmid, corresponding to very fast ParA-ATP hydrolysis. However, our results also hold true when we only assume that this hydrolysis occurs with a finite rate , leading to a non-zero concentration of ParA-ATP at a plasmid. This ParA-ATP can then anchor a plasmid to the nucleoid before being hydrolysed. This more general and realistic case is presented in the S1 Text, but our overall conclusions reached above remain unchanged.

From the above analysis, we see that the following conditions are required for equal plasmid spacing: (1) movement of a plasmid towards higher ParA-ATP concentrations. (2) diffusion of (at least a fraction of) ParA-ATP over the nucleoid to ensure formation of competitive concentration gradients. Single molecule tracking experiments in vitro support this assumption [17], [18]. (3) ParA-ATP hydrolysis must occur (predominantly) by plasmid-associated ParB-parC complexes, again to ensure gradient formation. (4) ParA-ATP must adopt a 1d-like configuration, as previously claimed [9], [13], [14]. If ParA were not organized in this fashion, it would be possible for ParA to diffuse around the sides of a plasmid without encountering the hydrolyzing effect of the ParB-parC complex. This would equalize the ParA concentrations on both sides even in the case of asymmetrically placed plasmids, leading to failure of the equal spacing mechanism. This assumption is in line with our subsequent experiments (see below). Due to this proposed 1d-like nature, we will from now on refer to the ParA distributions away from a plasmid as ParA structures. (5) There must be a separation of time scales between plasmid movement and ParA concentration reorganization, as discussed above.

Importantly, this overall mechanism is not reliant on a specific type of plasmid translocation. Any process that would allow a plasmid to move into regions of higher ParA concentration will suffice. In the following sections we therefore analyze different means of plasmid movement and compare them with our experimental data to determine which is used in our par2 segregation system.

Diffusion/immobilization model could space highly mobile plasmids equally over the nucleoid

In the previous section the mechanistic details of plasmid movement towards a higher ParA concentration were not specified. We now examine a specific implementation involving a diffusion-immobilization mechanism. Using a minimal modelling approach, we assume that nucleoid-associated ParA-ATP can immobilize freely diffusing plasmids through its interaction with the ParB-parC complex and that ParA-ATP does not polymerize (Fig. 2A). Since the plasmid will tend to become immobilized in regions of higher ParA-ATP concentration, this process allows for effective plasmid translocation up a ParA-ATP concentration gradient. We also incorporate ParB-parC-stimulated ParA-ATP hydrolysis at a plasmid, in accordance with prior experimental data. To further investigate this mechanism, given the known physiological and biochemical constraints, we developed stochastic simulations using a Gillespie algorithm [22]. Here we use standard diffusion for the plasmid movement; below we discuss the potential impact of subdiffusive motion.

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Figure 2. Diffusion/immobilization model can move and maintain plasmids at equally spaced positions.

(A) Schematic illustration of par2 diffusion/immobilization model. The clock indicates the slow conversion of cytoplasmic ParA-ADP into cytoplasmic ParA-ATP that is competent to bind to the nucleoid. (B) Typical simulation kymograph of diffusion/immobilization model for growing cell, where plasmid (red) diffusion influenced by the local ParA-ATP (green) concentration leads to immobilization initially at mid cell. After plasmid duplication, the system dynamically self-organizes to reacquire equal plasmid spacing. (C) Time-averaged plasmid position distributions for diffusion/immobilization model with n_p = 1–2 on a simulated nucleoid growing from 1.5 µm to 3 µm in 40 min without plasmid duplication. Plasmid distributions were obtained by sampling positions every 5 s in 36 independent simulations.

https://doi.org/10.1371/journal.pcbi.1004009.g002

In our simulation, a one dimensional lattice with sites of size dx = 5 nm represents the nucleoid. ParA-ATP and plasmids can diffuse on the lattice with diffusion coefficient D_A and D_P respectively. Up to 35 ParA-ATP can bind to a plasmid at the same site with reaction parameter k_AB reflecting the binding interaction of ParA-ATP and the ParB-parC complex [11]. More than one ParA-ATP bound to a plasmid reduces the plasmid diffusion constant to zero. Plasmid-bound ParA-ATP can be hydrolysed with reaction parameter k_B. Whenever a ParA-ATP hydrolysis event occurs, ParA unbinds from the nucleoid and becomes a cytoplasmic ParA-ADP. ParA-ADP can then be converted into a cytoplasmic ParA-ATP that is competent in DNA binding (cytoplasmic ParA-ATP for short) with a slow reaction parameter k_W [8]. Cytoplasmic ParA-ATP can then bind anywhere along the nucleoid with parameter k_on (see Materials and Methods and Tables 1,2 for details).

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Table 1. Reactions and propensities used in the diffusion/immobilization model.

https://doi.org/10.1371/journal.pcbi.1004009.t001

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Table 2. Parameter values used in the diffusion/immobilization model.

https://doi.org/10.1371/journal.pcbi.1004009.t002

Prior work has demonstrated plasmid displacement along the long cell axis of up to 3–4 µm within 10 min [9], [15]. With a diffusion/immobilization mechanism all plasmid movement in between immobilization events is generated by (unbiased) free diffusion, for which we have (in 1d) a mean square displacement (MSD) of . By inserting the above length and time scales into this equation, we conclude that a plasmid diffusivity of at least D_P∼10⁻² µm²s⁻¹ would be required to generate sufficiently rapid diffusive movement in accordance with previous experiments. We therefore chose D_P = 10⁻¹ µm²s⁻¹. In order to physically justify that ParA can immobilise the plasmids, we chose the nucleoid bound ParA-ATP diffusivity to be lower than D_P, with D_A = 10⁻² µm²s⁻¹ (Table 2). We experimentally constrained the overall copy number of ParA for pB171 par2 by semi-quantitative Western blots, which revealed that there were approximately 8×10³ ParA monomers per cell (S1B Fig.). This diffusion/immobilization model could produce equal plasmid spacing on simulated growing nucleoids with varying numbers of plasmids (Fig. 2B,C, S3A). This result demonstrates that using a sufficiently high (low) plasmid (ParA) diffusivity, respectively, the equal plasmid spacing seen in our experiments (Fig. 1B,C,D, S2A,B Fig.) and previously [9], could in principle be achieved using a diffusion/immobilization mechanism.

Free plasmid mobility is too low for a diffusion/immobilization mechanism

To test whether the requirement of a relatively high free plasmid mobility is met in vivo, we compared the movement of test-plasmids with and without par2. We analyzed trajectories of labeled plasmid foci using the tetO-TetR-mCherry labeling system, measuring the positions over time (Fig. 3A) and MSDs for each time lag τ. Plasmid motion will be biased by a functional par2⁺ partitioning system, in contrast to the random motion of par^-. Nevertheless comparing MSDs can still be informative in comparing relative overall mobilities. On time scales up to a minute we found that the par2⁺ MSD is higher than in par^- (Fig. 3B), showing that, on average, par2⁺ plasmids are more mobile than their par^- counterparts. Note that the number of data points for the short time lags far exceeds the number of trajectories (n_par- = 747, n_par2+ = 763), since every trajectory contains multiple short time lags. Consequently our estimates for the mean are relatively precise for short time lags. It is true that the error on the mean does not reflect inaccuracy due to experimental limitations in determining the actual plasmid position, for instance due to a finite pixel size. However, that error is the same for both par2⁺ and par^-. Moreover, since the error is also time lag independent, it is taken into account in our fitting procedure as a time lag independent term (for more details see below and Materials and Methods). Overall, these results are hard to reconcile with a diffusion/immobilization mechanism where the par2 system can only immobilize plasmids, and thus lower their MSD. These MSD values could in principle be limited due to cellular confinement. However, we found that MSD saturation only starts to occur at much larger length scales at times of up to 10 min (Fig. 3C). In the presence of par2, plasmids generally reside within the nucleoid region, while in its absence they tend to become somewhat more polar localized, although they can still sample the entire cell volume on long enough timescales [23]. Consistently we still find many par^- plasmids located within the nucleoid region (S3B Fig.). Restricting the mobility analysis to par^- plasmids within the nucleoid region did not alter the resulting MSD curves significantly (S3B Fig.). We conclude that the presence of par2 can increase plasmid mobility in the nucleoid region, which is inconsistent with a diffusion/immobilization mechanism. We emphasize that this conclusion can be made irrespective of the underlying (par^-) plasmid transport processes, which we now describe in more detail.

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Figure 3. The par2 segregation system increases plasmid mobility.

(A) Time lapses showing the localization of par^- pMH82tetO120 (mini-R1, par^-, tetO120) and par2⁺ pSR236 (mini-R1, parC1⁺, parA^-, parB⁺, parC2⁺, tetO120, P_lac::parA::eGFP) plasmids in E. coli cells harboring pSR124 (P_BAD::tetR::mCherry). The par2⁺ time lapse, with ParA-GFP localization, shows a segregation event where two foci segregate ≥0.8 µm further apart within 20 s. PC =  hase contrast, scale bar: 1 µm. (B) Mean square displacements after time lag τ were extracted from plasmid trajectories (n_par- = 747, n_par2+ = 763) using strains specified in (A), par^- (red) and par2⁺ (black), error bars: standard error of the mean. (C) Log-log plot of experimental mean square displacements after time lag τ (red) were extracted from plasmid trajectories over 1 min as in (A,B) and (inset, linear scales, n = 50) over 15 min from par^- pMH82tetO120 (mini-R1, par^-, tetO120) plasmids in E. coli cells harboring pSR124 (P_BAD::tetR::mCherry). At timescales on the order of 10 min saturation of the MSD occurs due to cellular confinement. A nonlinear least square fit (black line) using the function was used to estimate parameter values: α = 0.73±0.02, D = 9.7±1.3×10⁻⁴ µm²s^−α, β = 1.6±2.4×10⁻³ µm², (R² = 0.99, p-values: 8×10⁻¹⁵, 8×10⁻³ and 0.50 respectively). See Materials and Methods for details; error bars: standard error of the mean.

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It has been reported that chromosomal loci and RNA-protein particles exhibit subdiffusive, rather than diffusive, behavior in the cytoplasm [24], [25]. Therefore it is possible that plasmids without a segregation mechanism could also exhibit subdiffusive motion. Further analysis is required to fully distinguish subdiffusion from the additional effects of cellular confinement or glass-like properties of the bacterial cytoplasm [23], [25]. Nevertheless such additional analysis is not required for the conclusions on par^- plasmid mobility relevant to this study, as we now explain. Subdiffusion results in an expected MSD displacement of the form , with α<1 and D the apparent diffusion constant (in units of µm²s^-α). We find that our MSD displacements on both short and long timescales are well described by subdiffusion with α = 0.7–0.8 and an apparent diffusion constant D = 5–10×10⁻⁴ µm²s^−α (Fig. 3C and Materials and Methods for details). This is consistent with other recent reports on par^- plasmid mobility [23], [26]. Importantly the experimental MSD is lower on all observed timescales than a hypothetical particle that would perform free diffusion inside a cell with a diffusion constant D_f = 10×10⁻⁴ µm²s⁻¹. This upper limit is already much lower than that needed to be consistent with the previously reported plasmid displacement data discussed above. We will further exploit this upper limit in our analysis below.

To further investigate the effect of par2 on plasmid positioning, we also studied rapid plasmid segregation events. We defined these as cases where two plasmid foci whose separation is initially ≤0.3 µm, move within 20 s at least another 0.8 µm apart (Fig. 3A, S3C). We also allowed for the two foci to be initially merged. Using these criteria, despite equally large data sets, we found 13 such events in par2⁺ and only one such case in par^-. Furthermore, we only retrieved 2 further par2⁺ segregation events when we relaxed the criterion to separation within 60 s instead of 20 s. This analysis shows that most segregation events occur rapidly. When we investigated the 26 plasmid trajectories involved they showed larger maximal MSDs compared to sets of 26 trajectories that were repeatedly randomly sampled from the whole par2⁺ dataset (p<10⁻⁶). This finding indicates that the par2 system can particularly enhance the mobility of plasmids when they are in close proximity. We then simulated 300 plasmid duplication events with our diffusion/immobilization model to determine the magnitude of diffusion constant required to generate the experimentally observed segregation. Note that we used diffusion rather than subdiffusion here because we have already determined that par^- plasmid movement is slower on all observed timescales than free diffusion with a diffusion constant D_f = 10×10⁻⁴ µm²s⁻¹. Hence, if the required diffusion constant is larger than D_f then we have also ruled out a subdiffusion/immobilization model. We required that 5% (15 out of 300) of segregated distances within 20 s were at least 0.8 µm (a very conservative requirement, since the criterion was satisfied by 13 of our 15 experimental segregation events). This requirement necessitated a free plasmid diffusion constant on the order of 10⁻¹ µm²s⁻¹, about two orders of magnitude higher than our experimentally observed upper bound D_f on the experimental par^- plasmid mobility. Hence, we conclude that the plasmids are generally too immobile for a diffusion/immobilization (or subdiffusion/immobilization) mechanism to explain these segregation events. Also the qualitative behaviour of segregation events in the diffusion/immobilization model appears different, since experimental segregation events (Fig. 3A, S3C) show more directionally biased motion, while the diffusion/immobilization model generates more sustained random, diffusive motion during segregation, prior to immobilization at equally spaced positions (Fig. 2B). Nevertheless, these segregation events were sufficiently rare not to significantly alter the overall MSD behaviour of the entire dataset shown in Fig. 3B. Thus the increased average mobility in the presence of par2⁺ cannot only be ascribed to these segregation events.

It is possible that the tetO-TetR-mCherry labeling system caused reduced plasmid mobility as compared to unlabelled plasmids. However, as we used the same labeling method for both par2⁺ and par^- cases, our above conclusions on relative mobility are unaffected. Moreover, our tetO-TetR-mCherry labeled plasmids still exhibited rapid segregation events (such as in Fig. 3A), underscoring the ability of par2 to overcome low plasmid mobility. Overall, we find that diffusion/immobilization cannot explain our data on par2⁺ versus par^- plasmid mobility, as well as on rapid par2⁺ plasmid segregation.

ParA structures competing to direct plasmid motion can space plasmids equally over the nucleoid

Given the shortcomings of the diffusion/immobilization model, we next tested models based on directed motion, allowing more rapid directed rather than unbiased diffusive plasmid movement. More specifically, we tested models based on the formation of competing ParA polymers, with ParB-parC-stimulated ParA-ATP hydrolysis directing plasmid movement. By modulating the length of these polymers, we thereby tested the robustness of directed motion models to generate equal plasmid positioning.

We again used a Gillespie algorithm to simulate ParA dynamics on the nucleoid (see Fig. 4A, Materials and Methods and Tables 3,4 for details). The nucleoid was represented as a rectangular lattice (dx = 5 nm in both dimensions), with a much shorter width (30 nm) than length (several µm). Similar reactions as in the diffusion/immobilization model described the cytoplasmic dynamics of ParA-ADP and ParA-ATP. Nucleoid-associated ParA-ATP could also still diffuse across the nucleoid in a mobile state in all four directions to neighbouring sites with diffusion constant D_A. However, two of these molecules at sites neighboring each other along the long nucleoid axis could interact to form a ParA polymer of two subunits, with reaction parameter k_p. Further ParA-ATP polymerization could occur by attachment of mobile ParA-ATP, located at a site immediately next to the tip of an existing ParA polymer, but only along the long axis. ParA-ATP polymers were assumed to be immobile. A ParA-ATP polymeric subunit could depolymerize spontaneously with reaction parameter k_dp, i.e. be converted into a mobile ParA-ATP at the same site. Given that its size is similar to the width of the lattice, we only took into account the plasmid position along the long axis and we assumed that it occupied all sites along the short axis simultaneously. The plasmid could diffuse with our experimentally estimated diffusion coefficient D_P along the long axis when polymeric ParA-ATP was not present either at any of the sites that the plasmid occupied or sites neighbouring the plasmid. In the presence of polymeric ParA-ATP, the plasmid was assumed to be tethered to such a polymer (via a ParB-parC complex), which prevented plasmid diffusion. At sites with a plasmid present, polymeric ParA-ATP could be converted into cytoplasmic ParA-ADP with reaction parameter k_B. Reflecting directed motion, at sites neighbouring a plasmid occupied by polymeric ParA-ATP, a plasmid could with reaction parameter k_dm move to the coordinate along the long axis of that ParA-ATP subunit, coinciding with conversion of that ParA-ATP into cytoplasmic ParA-ADP. For wild-type simulations, any plasmid in the system formed a hard wall to mobile ParA-ATP diffusion so that diffusing ParA-ATP molecules could not diffuse past a plasmid.

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Figure 4. The directed motion model can move and maintain plasmids at equally spaced positions.

(A) Schematic illustration of par2 directed motion model. The clock indicates the slow conversion of cytoplasmic ParA-ADP into cytoplasmic ParA-ATP that is competent to bind to the nucleoid. (B) Typical simulation kymograph of directed motion model with short polymers for a simulated growing cell where a plasmid (red) is initially directed from a nucleoid edge to mid-cell by ParA (green) filament competition. After plasmid duplication, the system dynamically self-organizes to attain equal spacing. (C) Time-averaged plasmid position distributions for directed motion model with short polymers for n_p = 1,2 plasmids on a simulated nucleoid growing from 1.5 µm to 3 µm in 40 min without plasmid duplication. Plasmid distributions were obtained by sampling positions every 5 s in 36 independent simulations.

https://doi.org/10.1371/journal.pcbi.1004009.g004

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Table 3. Reactions and propensities used in the directed motion models.

https://doi.org/10.1371/journal.pcbi.1004009.t003

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Table 4. Parameter values used in the directed motion models.

https://doi.org/10.1371/journal.pcbi.1004009.t004

We first adjusted the ParA-ATP polymerization rate to generate short filaments, of approximately 10 subunits in length (Table 4 for parameters). Simulations again faithfully reproduced the equal spacing of plasmids along simulated growing nucleoids with varying numbers n_p of plasmids (Fig. 4B, n_p = 1,2 in Fig. 4C, n_p = 3,4 in S4A Fig.) in good agreement with our experiments (Fig. 1, S2). By adjusting the ParA-ATP polymerization rate (Table 4), long continuous ParA polymer bundles could also be generated. In that case equal spacing could also be achieved (S4B,C Fig.). Intuitively, in both short and long filament cases, this occurs because in an irregularly spaced plasmid configuration, the unequal ParA concentrations on either side of a plasmid result in an unequal degree of ParA polymerization. This in turn results in an unequal amount of competitive directed motion events to each side, resulting in effective directed translocation over longer length scales back towards an equally positioned state. Plasmid separation occurs when two nearby plasmids encounter two ParA-ATP structures extending in opposite directions away from the plasmids. The two ParA-ATP structures will then necessarily mediate a segregation event. The effect of directed movement in this model is clearest in the case of plasmid segregation events (Fig. 4B, S4B), where we see rapid segregation consistent with the fast segregation events observed experimentally (see Fig. 3A).

ParA-GFP oscillations are not continuously required for equal plasmid spacing

Intriguingly, simulations of the directed motion model did not generally produce sustained spatiotemporal oscillations of ParA across the nucleoid (short polymers: Fig. 4B, long polymers: S4B Fig.). A lack of sustained oscillations would therefore appear to be a common feature of models where competitive ParA structures generate equal plasmid spacing. This absence was unexpected, as prior experimental work had emphasized the oscillatory aspect of the ParA dynamics [12]–[14]. To experimentally test this key model prediction in an unbiased fashion, we experimentally measured the degree of ParA asymmetry in the par2 system in a large dataset (n = 134) of snapshots of ParA-GFP across the nucleoid. We examined only cases with a single plasmid tetO-TetR-mCherry focus, where sustained oscillations should be easiest to infer. The ParA-GFP fluorescence signal from pole to plasmid position was summed and divided by the respective pole-to-plasmid distance. This generated two ParA-GFP fluorescence densities I_L and I_R for either side extending to the two cell poles. This allowed us to compute the normalized asymmetry measure |I_L-I_R|/|I_L+I_R| [27] for ParA (see Materials and Methods for details). Asymmetric ParA-GFP distributions, arising for example from oscillations, where for example I_L≈0, I_R≈1, will give asymmetry values closer to one, whereas symmetric ParA-GFP distributions, where I_L≈I_R, will give values closer to zero. Note that the ParA-GFP exposure time used here was 1.5 s; clearly, we cannot measure asymmetries that occur on a timescale faster than this exposure time. However, the timescales of the plasmid and ParA-GFP dynamics are on the order of tens of seconds or longer and it is therefore unlikely that any significant asymmetry is being missed by our measurements.

When we examined our whole distribution of cells exhibiting single plasmid tetO-TetR-mCherry foci, we found that the degree of ParA-GFP asymmetry (Fig. 5A,B) was low in comparison with the well-established MinD spatiotemporal oscillator [27]. Furthermore, the ParA-GFP asymmetry did not correlate with cell length (S5A Fig., R² = 0.08), unlike the case of MinD-YFP [27]. We also compared the ParA-GFP asymmetry to the Hoechst signal. This DNA stain labels the nucleoid itself, which is relatively uniform along the long cell axis [28]–[30]. Here, any asymmetry is not expected to depend on the plasmid foci positions. The Hoechst asymmetry distribution was indeed concentrated around relatively small values, but was apparently measurable within our approach (Fig. 5B, S5B). Importantly, we found that the ParA-GFP asymmetry measure had a similarly low value as for the Hoechst case (Fig. 5B, S5B, no significant difference, Kolmogorov-Smirnov test), and that for both the asymmetry is uncorrelated to the plasmid focus position (S5C Fig.). We therefore conclude that for a single plasmid focus, ParA-GFP typically resides on both sides of a plasmid, with relatively little asymmetry or oscillation, as predicted by the directed motion model, irrespective of a weak (Fig. 5B) or strong (S5B Fig.) degree of polymerization.

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Figure 5. As predicted by the directed motion model, ParA-GFP distribution is relatively symmetric.

(A) Localization of plasmids and summed Z-stack of ParA-GFP distributions in a field of cells. Scale bar: 1 µm; plasmid: pSR233 (mini-R1, par2⁺, P_lac::parA::eGFP, tetO120) in E. coli cells harboring pSR124 (P_BAD::tetR::mCherry). (B) ParA-GFP (n = 134) asymmetry measure |I_L-I_R |/|I_L+I_R| using fluorescence densities I_L, I_R on left, right sides of a plasmid focus along long cell axis (see Materials and Methods). Comparison shown to the prediction of directed motion model with short polymers, Hoechst (n = 134) and MinD-YFP case [27]. Box plots represent minimal, first quartile, median, third quartile and maximal values of asymmetries in all cases. (C) Time lapses showing localization of par2⁺ pSR236 (mini-R1, parC1⁺, parA^-, parB⁺, parC2⁺, P_lac::parA::eGFP, tetO120) plasmids in E. coli cells harboring pSR124 (P_BAD::tetR::mCherry).

https://doi.org/10.1371/journal.pcbi.1004009.g005

Previous analyses had focused on plasmids migrating in the wake of retracting ParA-GFP structures [9]. Such events can transiently give rise to relatively high ParA-GFP asymmetries (see, for example, Fig. 3A, 5C). Accordingly, we conclude that ParA asymmetry or oscillations are not continuously required for par2 mediated plasmid positioning. Transient asymmetry, including oscillations, instead likely arises from the dynamics needed to bring about equal plasmid spacing following a spatial perturbation or plasmid duplication event (Fig. 5C). Once the ParA distribution has returned to being relatively symmetric, this coincides with an equally spaced plasmid configuration (Fig. 5C). Such dynamics can be seen in our model simulations (Figs. 4B, S4B): asymmetric during plasmid segregation events, but relatively symmetric otherwise. This analysis can therefore accommodate both our findings of a relatively symmetric ParA distribution with previous reports emphasizing asymmetry and oscillations. Overall, our finding of predominantly symmetric, non-oscillatory ParA dynamics may help to reconcile similar findings for ParA in other plasmid partitioning systems, such as for plasmid P1 [15], [16].

ParA-GFP forms structures within the nucleoid region

One required feature to achieve equal plasmid spacing is that the ParA-ATP should be organized in a 1d-like structure along the nucleoid as concluded above. However, it is unclear why ParA-ATP on either side of a plasmid would align in a coherent 1d-like structure with their ends coinciding with a plasmid. One potential explanation for this 1d-like behavior is that the ParA-ATP structures are sensitive to the overall nucleoid architecture. To test these features, we examined the localization of ParA-GFP and Hoechst signal simultaneously using optical sectioning in WT cells (n = 678) without par2-carrying plasmids to prevent dynamic ParA-GFP structure disassembly. ParA-GFP intensity correlated well with the DNA stain (Fig. 6A,B, S6A, Pearson's correlation coefficient r_P = 0.81), indicating that ParA-GFP localization was indeed dependent on the underlying nucleoid. Importantly, ParA-GFP overlaid more with Hoechst than the reverse (Fig. 6C), indicating that ParA forms structures within the nucleoid region rather than uniformly covering the nucleoid. Although the resolution of our techniques does not allow identification of potential individual ParA polymers, in many cases we did observe extended 1d-like ParA-GFP structures on the nucleoid (Fig. 6B, S6A). Care must be taken in interpreting fluorescent localization studies due to potential artifacts, for example GFP-induced polymerization [31]. However, wild-type plasmid loss rates and plasmid foci positioning in cells expressing ParA-GFP argue against localization or polymerization artifacts in our case [9], [12].

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Figure 6. ParA forms structures within the nucleoid region.

(A) Fluorescence localization of ParA-GFP (green), Hoechst DNA stain (red) and overlay, at mid-height through cell, taken from deconvolved Z-stacks showing structures that are disrupted with 50 µg/ml nalidixic acid treatment (Nal) compared to WT. Scale bar: 1 µm; plasmid: pGE230 (mini-R1, par^-, P_lac::parA::eGFP). (B) Normalized fluorescence intensity profiles along the long cell axis for 9 in focus z heights (dz = 0.1 µm) resulting from deconvolved Z-stacks in representative WT and Nal-treated strains. (C) Manders overlap coefficients in WT cells (error bars: standard error of the mean, n = 678) showing the fraction of ParA-GFP fluorescence intensity that overlaps with Hoechst DNA stain when the latter is above a threshold T_Manders (ParA-GFP, green) and the reverse (Hoechst, red). ParA-GFP overlaps more with Hoechst DNA stain (p-values ranging from 10⁻¹² to 10⁻¹³², see Materials and Methods) than the reverse.

https://doi.org/10.1371/journal.pcbi.1004009.g006

Equal plasmid spacing is compromised in cells with a perturbed nucleoid

We reasoned that if ParA structures are reliant on the nucleoid morphology for their organization, then mutants/treatments that perturb the overall nucleoid structure should also exhibit alterations in ParA localization and therefore plasmid focus positioning (Fig. 7A). We measured plasmid focus positioning in mukE, mukF and matP mutant strains, as well as in cells treated with the DNA gyrase inhibitor nalidixic acid (Nal), all of which exhibit defects in nucleoid organization [32]–[34]. Nucleoid length distributions were altered in all of these cases (S7A Fig.) and, consistent with our hypothesis, there was in each case a similar deterioration in the fidelity of plasmid focus positioning (n_p = 1,2 in Fig. 7B, S7B,C, n_p = 3,4 in S7B,C,D Fig.) towards a random distribution (S7E Fig.). This deterioration may not have been large enough to detect in stability assays [35], [36]. Similarly, in E. coli mukB mutants, perturbed plasmid positioning without compromising plasmid stability has also been observed, although for the segregation mechanism mediated by ParM [37]. The deteriorations in plasmid positioning could have resulted from other effects, such as an induction of the SOS response in Nal-treated cells. However, the similarity of the altered plasmid positioning in all four cases instead suggests a common positioning defect based on nucleoid perturbation. This deterioration could also be due to an altered plasmid structure. However, at least for the case of matP we do not favor this hypothesis, due to the absence of MatP target sites (matS) on our test-plasmid.

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Figure 7. Nucleoid morphology disruption causes aberrant plasmid focus positioning.

(A) Fluorescence localization of ParB-GFP (green) and Hoechst DNA stain (blue) in mukE, mukF, matP mutants and wild-type cells treated with 50 µg/ml nalidixic acid (Nal). Scale bar: 1 µm; plasmid: pFS21 (mini-R1, parC1⁺, parA⁺, parB::sfGFP, parC2⁺). (B) Histograms of plasmid foci positions (n_p = 1,2) for mutants/treatments described in (A) relative to nucleoid size. According to Kolmogorov-Smirnov tests, all distributions are broader than WT (Fig. 1C) with p<10⁻² except Nal n_p = 1: p<0.05. (C) Manders overlap coefficients (error bars: standard error of the mean) of ParA-GFP comparing WT (n = 678) and Nal-treated cells (n = 862). Consistent with a decrease in the Pearson's correlation coefficient r_P (p<10⁻³⁸), ParA-GFP overlaps less with Hoechst in Nal-treated cells as compared to WT (p-values ranging from 10⁻⁵¹ to 10⁻¹⁴⁴). (D) Time-averaged plasmid position distributions for directed motion model with short polymers obtained as in Fig. 2C from 124 independent simulations. Here, mobile DNA-bound ParA-ATP was now able to diffuse past a plasmid.

https://doi.org/10.1371/journal.pcbi.1004009.g007

To provide evidence that the above deterioration in plasmid positioning arose from an altered ParA distribution, we systematically examined localization of ParA-GFP and Hoechst stain simultaneously in Nal-treated cells (n = 862), which had the largest visible perturbations. We were able to quantify (Fig. 7C, S6B, p<10⁻¹⁴⁹, see Materials and Methods) perturbations in nucleoid structure that were detectable by eye (Fig. 6A, S6A). Moreover, visual inspection showed that the ParA-GFP distribution followed the nucleoid structure less closely than in the WT (Fig 6A, S6A). This finding was quantitatively confirmed by a correlation coefficient of r_P = 0.68, decreased from its WT value of 0.81 (p<10⁻³⁴), and also by a decrease in the ParA-GFP overlap coefficient (Fig. 7C). Altogether, these findings support our hypothesis that the nucleoid provides a template for 1d-like ParA-ATP structure formation, which is partially compromised when the nucleoid structure is perturbed.

To reproduce this behavior in the directed motion model, we assumed that mobile DNA-bound ParA-ATP could now diffuse past a plasmid (see Materials and Methods for details). This could be due to the disordered nucleoid structure resulting in a deteriorated ParA-ATP structure organization, thereby allowing ParA-ATP to spatially bypass ParB-parC complexes and compromise the ParA concentration differences between either side of a plasmid. The directed motion model with a weak (Fig. 7D, S7F Fig.) or strong (S7F Fig.) degree of polymerization could then reproduce the observed plasmid focus distributions (Fig. 7B).

Diffusion/immobilization model

On the one dimensional lattice with sites of size dx = 5 nm, sites are numbered 0… (L-1). Reactants are : ParA-ATP at site i with number (≥0), : plasmids with j ParA-ATP bound to it at site i with number (≥0), : cytoplasmic ParA-ADP with number AADP (≥0), : cytoplasmic ParA-ATP with number ACYTO (≥0) The reactions and corresponding propensities are described in Table 1. Parameter values used are listed in Table 2.

We varied the exact number of ParA-ATP molecules forming a complex that are required to completely immobilize the plasmid and this variation does not alter the qualitative behavior of the system. Introduction of a low spontaneous ParA-ATP hydrolysis parameter k_off also does not alter the behaviour of the system. We do not keep track of the spatial positions of ParA-ADP and ParA-ATP in the cytoplasm. Instead we merely keep track of their number.

The ParA concentration is assumed to be constant throughout the cell cycle, consistent with the total ParA-GFP fluorescence as a function of cell volume when expressed from an inducible promoter (S1C Fig.). In accordance with estimates for average ParA copy numbers obtained by semi-quantitative Western blots (S1B Fig.), the ParA concentration is assumed to be 2400 ParA (dimers) per µm of nucleoid. Simulations start at time t = 0 and run until time t, updated according to the Gillespie algorithm, exceeds a predefined time T. To simulate nucleoid growth during the cell cycle the nucleoid lattice is extended by two sites of size dx (not containing any ParA or plasmids), at one randomly chosen position along the nucleoid length. Such a growth event occurs at regular time intervals. Reaction propensities are then updated in accordance with the new state.

In Fig. 2B the nucleoid grows from 1.5 µm to 3 µm in T = 40 min, reflecting one cell cycle. Initially a quarter of the total ParA in the system is in the cytoplasmic ParA-ADP form, 11 ParA-ATP are bound to each plasmid to ensure initial anchoring, and the rest are bound randomly to the nucleoid. In Fig. 2B the plasmid is initially located at site 0. In the simulations used to generate the histograms shown in Fig. 2C, S3A, all plasmids are initially distributed randomly across the nucleoid. At regular time intervals of 5 s the simulation state is output along with the plasmid positions to generate a time-averaged probability distribution for the plasmid positions along the long axis of the cell. In cases where the total number of plasmids (n_p) is more than one, the plasmids are ordered and labeled 1…n_p according to their positions (by increasing site number) along the nucleoid. Their position is then used to generate distributions for every plasmid label 1…n_p for that particular overall number of plasmids n_p.

In the event of plasmid duplication at a particular site where an existing plasmid is located, a new plasmid without any bound ParA is added to the same site and the reaction propensities are updated accordingly. In case of two or more existing plasmids, one is chosen randomly for duplication. Plasmid duplication events in Fig. 2B occur at regular time intervals T/3, although the model behaves equally well with duplication at any time as it dynamically segregates the plasmids to equally spaced positions.

Directed motion model

The nucleoid was represented as a rectangular lattice divided into square sites of sides dx = 5 nm. The long axis could grow from 1.5 µm to 3 µm in length, while the short axis of the nucleoid lattice remained fixed. For wild-type directed motion model simulations the short axis length was 30 nm (directed motion model with short polymers) and 25 nm (directed motion model with long polymers). Thus every site had a coordinate along the long axis (labelled as 0… L-1) as well as a coordinate along the short axis (labelled 0… S-1). Reactants are: : mobile ParA-ATP at site (i,j) with number (≥0), : polymeric ParA-ATP at site (i,j) with number (0 or 1); : plasmids at site i with number (≥0); : cytoplasmic ParA-ADP with number (≥0) and : cytoplasmic ParA-ATP with number (≥0). The reactions and corresponding propensities are listed in Table 3. Parameter values used are listed in Table 4.

In the perturbed-nucleoid simulations, mobile ParA-ATP can diffuse past a plasmid with 10% (short) or 100% (long) of the normal diffusion rate and the short axis length of the nucleoid is altered to 10 nm in the long polymer model. Lastly, to allow for mobile ParA-ATP to move past the plasmid without being hydrolyzed, k_mB is reduced 10-fold compared to its standard value.

As for the diffusion/immobilization model, the total ParA concentration was constrained to be 2400 ParA (dimers) per µm of nucleoid (long axis) and the total length of simulated time was T = 40 min. Initially a quarter of the total number of ParA in the system was in the cytoplasmic ParA-ADP form, with the rest distributed randomly on the nucleoid in the mobile ParA-ATP form. Initial plasmid positioning, state output, plasmid position distribution generation and plasmid duplication rules were also as described previously. Nucleoid growth was implemented as described previously, with one generalization: a position along the long axis of the nucleoid was first chosen randomly. Then two nucleoid slices of 1 site (along the long axis) by S sites (along the short axis) were inserted.

Plasmids and strains

The ParA-GFP fusion and tetO-TetR-mCherry plasmid labeling system were described previously [9], [12]. To obtain the functional ParB-GFP fusion, the parB gene in the par2 locus was replaced by parB::sfGFP and inserted in a mini-R1 test-plasmid. See S2 Text for more details on the strains and plasmids construction, semi-quantitative ParA western blotting and supplemental figure data analysis.

Epifluorescence microscopy

E. coli strains carrying plasmids of interest (see Table S1, S2 in S2 Text for details on strains and plasmids) were grown to stationary phase while being shaken at 37°C in LB medium supplemented with appropriate antibiotics (30 or 50 µg/ml ampicillin, 25 or 50 µg/ml kanamycin, 15 µg/ml chloramphenicol), with the exception of the muk strains, which were grown at 24°C. Cultures were diluted to an OD₄₅₀ of 0.025 in antibiotic-free M9 minimal medium containing supplements (0.2% casamino acids, 0.2% glycerol, 1 µg/ml thiamine, 1 mM MgSO₄, 0.1 mM CaCl₂). Inoculated cultures were incubated until an OD₄₅₀ of ≈0.2 was reached, typically taking 3 h.

When nalidixic acid was used to condense the nucleoids, the antibiotic was added to a growing culture at a final concentration of 50 µg/ml two hours before imaging. Where appropriate, culture samples were mixed with Hoechst 33342 (Invitrogen) at a final concentration of 50 µg/ml for DNA staining immediately before microscopy.

For imaging, cells were immobilized on 1.5% agarose-M9 pads mounted on microscopy slides using Gene Frames (Thermo Scientific). All microscopy experiments, unless specified otherwise (see below), were carried out using an Olympus IX71 inverted microscope with a CoolSNAP HQ EMCCD digital camera (Photometrics, pixel size  = 0.066 µm). A temperature-controlled incubation chamber (Applied Scientific) fitted to a Weather Station (Precision Control) kept samples at a constant 30°C. Images were acquired using SoftWoRx version 5.5.0 with a Zeiss Plan-Neofluar 100X/1.30 NA oil objective and Olympus Mercury 100 W burner (U-LH100HG) fluorescent light source. Filter set specifics are given in Table S3 in S2 Text.

Optical sectioning of fluorescence signals from ParA-GFP and Hoechst-stained nucleoid DNA

Expression of ParA-GFP from plasmid pGE230 (mini-R1, par^-, P_lac::parA::eGFP) in E. coli strain KG22 or FS1 (KG22ΔmatP) was induced by adding 10 µM IPTG to the culture medium two hours before microscopy. A 31 image Z-stack with 0.1 µm section widths was taken for all projections (exposure times Phase Contrast (GFP channel): 0.05 s, ParA-GFP: 1.5 s, Phase Contrast (Hoechst channel): 0.1 s, Hoechst: 2 s). Image stacks were subsequently deconvolved using SoftWoRx v.5.5.0 with the following parameters: 10 iterations, medium noise reduction, conservative method.

Measuring asymmetry in ParA-GFP distributions using optical sectioning

Expression of ParA-GFP from plasmid pSR233 (mini-R1, par2⁺, P_lac::parA::eGFP, tetO120) in E. coli KG22 cells harboring pSR124 (P_BAD::tetR::mCherry) was induced by adding 10 µM IPTG to the culture medium one hour before microscopy. Samples were treated with Hoechst stain and imaged immediately thereafter. Expression of TetR-mCherry was not induced, as baseline activity of P_BAD produced sufficient amounts of TetR-mCherry to detect foci in a single image at mid Z-height. Similarly, a single Hoechst stain image was acquired. For ParA-GFP, a 21 image Z-stack with 0.2 µm section widths was taken (exposure times Phase Contrast: 0.1 s, TetR-mCherry 1.5 s, ParA-GFP: 1.5 s, Hoechst 0.15 s). Images were acquired using a Zeiss Axiovert 200 M inverted epifluorescence microscope with a Zeiss Plan-Neofluar 100X/1.3 NA oil objective in a temperature-controlled room at 22°C. The microscope was controlled using MetaMorph software version 7.7.5.0 (Molecular Devices, Inc.). Cells were illuminated using a Lambda LS xenon-arc lamp and images acquired using a CoolSnap HQ² EMCCD digital camera (Photometrics, pixel size  = 0.0625 µm). Filter set specifics are given in Table S3 in S2 Text.

Time-lapse imaging of plasmid foci movement

Plasmid foci of the par^- mini-R1 plasmids pMH82tetO120 (par^-, tetO120⁺) or pSR236 (parC1⁺, ΔparA, parB⁺, parC2⁺, P_lac::parA::eGFP, tetO120⁺) in E. coli strain SR1 (KG22ΔpcnB) were visualised by labelling tetO arrays on the plasmid in trans with TetR-mCherry provided from the pSR124 vector (see [40] for the original method). TetR-mCherry expression was induced by adding L-arabinose to a final concentration of 0.02% to growing cultures for 15 minutes, followed by catabolite repression with 1% glucose for 10 minutes. In strains harbouring pSR236, expression of ParA-GFP was induced by the addition of 10 µM IPTG inducer 2 h before microscopy. Time-lapse image series were acquired for different total durations/time intervals: 1 min/4 s or 15 min/30 s for pMH82tetO120 (exposure times phase contrast: 0.1 s; TetR-mCherry: 1.5 s) and 1 min/5 s or 15 min/20 s for pSR236 respectively (exposure times phase contrast: 0.1 s; TetR-mCherry: 1.5 s; ParA-GFP: 1 s). The maximum rate of image acquisition possible with our imaging system was every 4 s and 5 s (without and with ParA-GFP channel) for pMH82tetO120 and pSR236 respectively. Sample focus was maintained in the mid-cell plane throughout the experiment using the UltimateFocus system (Applied Precision) sampling and refreshing before the acquisition of each individual frame.

Plasmid focus positioning microscopy

E. coli strains KG22, FS1 (KG22ΔmatP), FS2 (KG22: mukE::kan) or FS3 (KG22: mukF::kan) harbouring pFS21 (mini-R1, parC1⁺, parA⁺, parB::sfGFP, parC2⁺) were grown to an OD₄₅₀ of 0.3. Samples were treated with Hoechst stain and imaged immediately in the mid-cell plane (exposure times ParB-GFP: 1 s, Hoechst: 0.5 s). Images of muk strains were acquired using a Zeiss Axiovert 200 M inverted epifluorescence microscope with a Zeiss Plan-Neofluar 100X/1.3 NA oil objective in a temperature-controlled room at 22°C. The microscope was controlled using the MetaMorph software version 7.7.5.0 (Molecular Devices, Inc.). Cells were illuminated using a Lambda LS xenon-arc lamp and images acquired using a CoolSnap HQ² EMCCD digital camera (Photometrics, pixel size  = 0.0625 µm). Filter set specifics are given in Table S3. Other strains were imaged using both the Olympus IX71 and Zeiss Axiovert 200 M systems described above.

Plasmid foci mobility determination

Using the MATLAB-based software suite MicrobeTracker (MT) [27], we determined E. coli cell outlines from phase contrast (PC) images, as well as the distribution of tetO-TetR-mCherry-labeled plasmids along the long axis of cells. The cell outlines were used together with the MATLAB tools spotFinderZ and spotFinderM [27] to determine tetO-TetR-mCherry foci positions in par^- time-lapses of 1 min (short) or 15 min (long) in duration with images taken at intervals of 4 s or 30 s respectively. The linear tetO-TetR-mCherry distribution was used to control the peak detection method for false positives/negatives. For the short time-lapses we analysed cells with one or more foci, although all our results were unchanged if analysis was restricted to one focus cells to prevent potential foci labelling errors. For the long time-lapses, we only analysed cells exhibiting one focus. This was due to difficulties in distinguishing between multiple foci due to merging/splitting events, out of focus plane movement and photobleaching when acquiring images using a time interval of 30 s. These effects could have resulted in biases in the analysis due to labelling errors. We were unable to lower the time interval and simultaneously image for long time periods due to TetR-mCherry photobleaching.

At every time point the two-dimensional squared foci displacements r²(τ) after time lag τ were determined. All measured displacements for the same time lag were then averaged together to obtain Mean Square Displacements (MSD) with time lags from 4 s to 15 min (Fig. 3B,C,S3B). The measured plasmid displacement r_p(τ) can report the true plasmid displacement r_p(τ) at a resolution no greater than our measurement error, which can be up to 0.1 µm due to microscope drift. Our measurements are also limited by a finite pixel size of 0.066 µm. We therefore have: , where ε is the error due to both of the above effects. Squaring and averaging over many plasmid trajectories results in an MSD: . The last term vanishes due to averaging, but the second term remains and generates a small time independent value for τ>0. Even at short timescales of up to a minute, the MSD has a nonlinear shape, as has been reported before [26]. This is fully consistent with subdiffusive motion on these timescales. We thus expect the experimentally observed planar MSD for free particle subdiffusion in three dimensions to have the form: . We carefully measured the par^- MSD up to 1 min with short time intervals between measurements (Fig. 3B). We performed a nonlinear least squares fit (weighted by the standard error of the mean (SEM): 1/SEM(τ)) for resulting in the values α = 0.78±0.04, D = 6.8±1.2×10⁻⁴ µm²s^−α, β = 6±1×10⁻³ µm² (R² = 0.99, p-values: 4×10⁻¹⁰, 1×10⁻⁴ and 8×10⁻⁴ respectively). On longer timescales up to 15 min (Fig. 3C), plasmid mobility also showed subdiffusive behaviour with a similar analysis giving α = 0.78±0.05, D = 6.2±2.1×10⁻⁴ µm²s^−α, β = 4±1×10⁻² µm² (R² = 0.99, p-values: 8×10⁻¹⁵, 8×10⁻³ and 2×10⁻³ respectively). Analysing the two datasets combined (Fig. 3C) also generated consistent results, although the constant β was not significantly different from zero in this case: α = 0.73±0.02, D = 9.7±1.3×10⁻⁴ µm²s^−α, β = 1.6±2.4×10⁻³ µm², (R² = 0.99, p-values: 8×10⁻¹⁵, 8×10⁻³ and 0.50 respectively, fit shown in Fig. 3C). Fitting instead to this combined data set did not alter our estimates for α and D significantly. On all observable timescales (i.e. 4 s and longer) the experimentally found par^- MSD is bounded from above by the function, with D_f  =  10×10⁻⁴ µm²s⁻¹. Moreover, free diffusion with diffusion constant D_f inside a box of cellular dimensions still exceeds the experimental subdiffusive mobility.

Determining plasmid foci positions

Cell outlines and linear projections of ParB-GFP and Hoechst signal distributions along their long axis were determined as described above using MicrobeTracker (MT) [27]. ParB-GFP foci detection of snapshots was also performed using the methods described above. The positions of the half-maxima of the linear Hoechst signal distribution in every cell were then determined. We defined the nucleoid length as the length between the two half-maxima of the Hoechst stain. This analysis allowed us to determine the positions of plasmid foci with respect to the nucleoid.

ParA asymmetry analysis

Here, we summed 6 planes that are in focus from a Z-stack of ParA-GFP fluorescence signal images (dz = 0.2 µm), although the results are not different when using the ParA-GFP signal obtained from single confocal planes focused at mid-cell. Cell outlines, linear projections of ParA-GFP, tetO-TetR-mCherry and Hoechst stain fluorescence signal distributions, and tetO-TetR-mCherry foci positions were determined as described above. We confirmed that positioning of the tetO-TetR-mCherry foci from this dataset was similar to that measured previously [9]. In cells containing one plasmid focus (n = 134), the ParA-GFP fluorescence signal from pole to plasmid position was summed and divided by the respective pole-to-plasmid distance. This generates two ParA-GFP fluorescence densities I_L and I_R for either side extending to the two cell poles. This allows us to compute the normalized ParA asymmetry measure |I_L-I_R|/|I_L+I_R|. Irrespective of the plasmid position, a completely uniform fluorescence distribution would give an asymmetry value of zero. On the other hand, if all the ParA-GFP was located on one side of the plasmid the asymmetry measure would be one. Using a single confocal plane focused at mid-cell, we also computed the Hoechst asymmetry measure with respect to the plasmid position in the same manner.

As shown in [27] by using the same MT software package for analysis, the MinD-YFP asymmetry measure with respect to mid-cell follows an approximate sinusoidal oscillation over time, with a cell-length-dependent oscillation amplitude. In large cells the MinD-YFP oscillations are clearest with an amplitude of around 0.6. To generate an asymmetry measure appropriate for the MinD-YFP oscillations, we sampled 10³ time points t uniformly in [0,2π] (which constitutes one period). We then computed for every time point . The resulting asymmetry distribution (Fig. 5B) therefore reflects the experimental MinD-YFP asymmetry with respect to mid-cell in large cells [27]. In this way, we can directly compare the asymmetry present in the ParA-GFP and Hoechst signal distributions with that induced by the spatiotemporal oscillations of MinD-YFP. We also generated asymmetry measures using our directed motion model. In simulation outcomes shown in Fig. 4B (directed motion model with short polymers) and S4B (directed motion model with long polymers), the plasmid position, cytoplasmic ParA-ADP, cytoplasmic ParA-ATP, nucleoid-bound mobile ParA-ATP and polymeric ParA-ATP levels on either side of the plasmid were output at regular time intervals of dt = 5 s during a time period prior to plasmid duplication (first 2 min and 1.5 min of simulated time for directed motion model with short and long polymers respectively). Cytoplasmic ParA was assumed to be uniformly distributed throughout the cell (independently of the plasmid position), thus effectively only contributing to the denominator |I_L + I_R|. With this information we computed the ParA asymmetry using the same method as described for the experimental data. Results are shown in Fig. 5B (short polymers) and S5B Fig. (long polymers). It should be noted that according to both models, the ParA asymmetry remains very low once a plasmid is stably positioned at mid-cell, pushing the asymmetry distribution further towards zero over time. This is consistent with time lapses where stable equally spaced plasmid foci positioning correlates with ParA-GFP on either side of a plasmid focus (Fig. 5C and [9]).

Three dimensional nucleoid and ParA structure analysis

To compare the extent of overlay and 3D structure of Hoechst (nucleoid DNA) stain and ParA-GFP, we first had to align the Z-stack pairs in an unbiased manner. To achieve this, one phase contrast (PC) image (at mid z height) of the Hoechst signal sections was aligned with one GFP section PC image (at the same z position) using the TurboReg ImageJ plugin (option: translation) [41], after cropping both PC images to match the output size of the deconvolved Z-stacks. Using the same translation as for the Hoechst PC image, Hoechst Z-stacks were then translated in ImageJ to align them with the ParA-GFP Z-stacks.

We determined cell outlines in MT as described above using the PC image acquired with the GFP channel and excluded cells that did not show visible ParA-GFP and Hoechst stain simultaneously. We then computed the linear distributions (for every z height) along the long cell axis for the deconvolved Hoechst and ParA-GFP Z-stacks. We next determined for the ParA-GFP and Hoechst signals separately in every cell the maximal intensity value in the whole cell (I_maxcell) and the maximal values at every z height (I_max(z)). To find the 9 z planes from the Z-stacks (dz = 0.1 µm) that are in-focus for each cell in an automated fashion, we summed I_max(z) over 9 consecutive z positions including a given starting plane and determined the starting plane that gave the largest associated summed value. This starting plane and its 8 consecutive planes formed the in focus plane set. We verified that this method generated the right focus planes by inspecting the chosen planes visually for several cells. This method circumvents the problem of different focus planes for cells on the same image stack as well as alignment inaccuracies in the z direction between ParA-GFP and Hoechst signals which are difficult to control for manually.

Visual comparison of the nucleoid shape between WT and nalidixic acid (Nal) treated cells (Fig. 6A,B, S6A) revealed clear differences. In Nal-treated cells, the nucleoid signals, where present inside a cell, were more uniform along the long cell axis than in the WT (S6A,B Fig.). Shape differences were also visible in the raw Z-stacks suggesting they were not artefacts of the deconvolution method. To quantify these shape differences in an unbiased and systematic manner, we performed the following analysis (S6B Fig.).

We reasoned that a more uniform pattern would result in a profile along the long axis that resembled a first harmonic (first non-constant term of a Fourier expansion) between the nucleoid edges. Such a harmonic would not fit so well to a more spatially oscillating pattern that would arise, for example, from helical structures. Using the Hoechst stain I_maxcell and the I_max(z) arising from the 9 relevant focus planes we determined the half-maximum intensity locations along the long cell axis closest to the cell poles x_L and x_R at every z height. At every focus plane z height we could now define the ‘first harmonic’ function defined for x_L≤x≤x_R:

.

For every (x,z) we calculated the squared error SE(x,z) between the actual intensity value I(x,z) and H(x,z): . Lastly we summed over the SEs at every (x,z) and divided by the number of position points (x,z) to obtain a single measure of deviation SE_cell in a cell that is independent of the number of data points (and thus nucleoid size) and expression level variation between cells (because of normalization to I_maxcell). We then performed a Wilcoxon rank sum test on the set of SE_cell comparing a population of WT cells with nucleoid-perturbed cells (n_WT = 678 and n_Nal = 862). Nucleoid shapes in Nal-treated cells were indeed altered (p<10^-149). Note that this method did not detect a notable shape change in matP cells (n_matP = 579), potentially due to our techniques not being sufficiently sensitive.

To quantitate the colocalization of ParA-GFP and Hoechst signal in each cell, we also calculated, for every cell, the Pearson's correlation coefficient r_P using all the intensity values I_ParA-GFP(x,z) and I_Hoechst(x,z) [42].

To determine the fraction of ParA-GFP intensity signal that overlaps with Hoechst signal and vice versa we computed Manders overlap coefficients [42]. This method requires a choice of threshold T_Manders to distinguish between positions (x,z) that are considered to contain or lack sufficient intensity signal. We therefore performed our analyses for the complete range of threshold values to show that our qualitative conclusions are insensitive to the choice of a particular T_Manders (Fig. 6C, 7C). Manders overlap coefficients of ParA-GFP and Hoechst were calculated as follows:with

Likewise the Manders overlap coefficient of Hoechst onto ParA-GFP is defined as:with

Note that taking T_Manders = 0, will generate an overlap coefficient of one by construction. The normalization to I_maxcell in determining the colocalizing positions allows the overlap coefficients to be comparable between cells.

In a small fraction of cells the alignment procedure described above did not result in proper alignment. This is clearly reflected in the r_P values being considerably lower for these cases than for the cell population mean r_P value. However, without excluding these few, possibly false negative, cases the population mean r_P value is still high (0.81 and 0.68 for WT and Nal-treated cells respectively), indicating that ParA-GFP and Hoechst signals generally correlate strongly at a population level. Poor alignment affects Manders overlap coefficients for the ParA-GFP and Hoechst signals on average equally and is not biased towards a particular strain/treatment. Therefore the observed misalignment of a small fraction of cells does not affect the qualitative conclusions that we state in this study.

Note that in matP cells, we did not observe any significant alteration in intensity correlation (r_P = 0.80 for matP), nor ParA-GFP overlap coefficient, as compared to the WT. This result was expected given that we could not detect any significant nucleoid structure alteration, as described above.