Subdiffusive movement of chromosomal loci in bacteria explained by DNA bridging

Chromosomal loci in bacterial cells show a robust subdiffusive scaling of the mean square displacement, $\text{MSD}\left(\tau \right)\sim {\tau }^{\alpha }$, with $\alpha <0.5$. On the other hand, recent experiments have also shown that DNA-bridging nucleoid associated proteins (NAPs) play an important role in chromosome organization and compaction. Here, using polymer simulations we investigate the role of DNA bridging in determining the dynamics of chromosomal loci. We find that bridging compacts the polymer and reproduces the subdiffusive elastic dynamics of monomers at timescales shorter than the bridge lifetime. Consistent with this prediction, we measure a higher exponent in a NAP mutant compared to the wild type. Furthermore, bridging can reproduce the rare but ubiquitous rapid movements of chromosomal loci that have been observed in experiments. In our model the scaling exponent defines a relationship between the abundance of bridges and their lifetime. Using this and the observed mobility of chromosomal loci, we predict a lower bound on the average bridge lifetime of around five seconds.

Figure 1

(a) The simulated polymer. Bridges form between spatially proximal monomers with a probability $p$ and have a mean lifetime $\lambda$. (b) Ensemble-averaged MSD curves with (orange) and without bridging (blue). Bridging parameters: $p=1.5\times {10}^{-3},\lambda =4\times {10}^4\mathrm{MCS}$. (c) Phase diagram from Fig. 9 with $\mu$ remapped to the percentage of monomers bridged. Contours indicate interpolated curves of the given exponent. (d) Velocity auto-correlation function (VAC) is negative at short lags and collapses for different windows of $\tau /\delta ,$ indicative of a subdiffusive process.

Figure 2

(a) The scaling exponent $\alpha$ is plotted as a function of percentage bridged for different bridge lifetimes (color). (b) $\alpha$ as function of bridge lifetime for different values of $\mu$ (color). Note that the curves come closer together because of the relationship in Fig. S1(e). (c), (e), (g) The two-point correlation function $G(s,\tau )$ is calculated from our simulations for different parameters. (d), (f), (h) Shows the collapse of curves in (c), (e), (g) upon rescaling time by $\tau \to s^{-b}\tau$. In the absence of bridging (d) we found that a slightly different exponent $b=1.9$ fits better than the expected $b=2.1$. In (f), (g) the calculated values of $b=3.03,b=2.59$ resulted in good collapse. In (d), (f), (h) $s_0=45$ is a reference distance for plotting.

Figure 3

(a) Snapshot of microscopy experiment of GFP-ParB/parS labeled ori loci in E. coli. (b) Sample MSD curves of individual foci obtained from WT strain (set 1). (c) Sample MSD curves of mutant $\mathrm{\Delta }\mathrm{HNS}$ (set 1). Ensemble averaged MSD is represented by the black lines (b), (c). (d) Ensemble-averaged MSD curves from experiments of wild-type E. coli and a strain lacking the NAP H-NS (see also Fig. 6). MSD curves are fitted up to a $2\mathrm{s}$ delay (black lines). $\mathrm{\Delta }\mathrm{H}$-NS: $\alpha =0.38$ 18089 tracks, WT: $\alpha =0.32$ 6717 tracks. Inset: Exponent $\alpha$ from individual replicates (see Table 1).

Figure 4

(a) Ensemble-averaged MSD from WT cells (blue circles) is fit to an fBm model (orange squares, Fit $0.0008{\tau }^{0.36}\mu {\text{m}}^2$, 6717 tracks). (b) Drift velocity of wild type tracks (blue circles) have a wider tail than from simulations of the parameter-matched fBm (orange squares). (c) Ensemble-averaged MSD from polymer simulations to an fBm model (orange squares, $\mu =80,\lambda ={10}^5$ MCS, Fit $0.0009{\tau }^{0.39}\mu {\text{m}}^2$, 5200 tracks). (d) Drift velocity distribution from bridging simulations (blue circles) has a similar broad tail compared to the parameter-matched fBm model (orange square). (e) Same as (a), (c) but for polymers simulations without bridging. (f) The distributions of $v_d$ overlap.

Figure 5

(a) Ensemble averaged MSD from FBM simulations overlaid with $\mathrm{\Delta }\mathrm{H}$-NS data (Fit, $0.0009{\tau }^{0.38}$). (b) Drift velocity distributions $v_d$ from FBM simulations have a smaller tail than $\mathrm{\Delta }\mathrm{H}$-NS data. (c) $\mathrm{\Delta }\mathrm{H}$-NS has a slightly higher number, faster tracks than the wild type. (d) Drift velocity distribution comparisons as in Figs. 2 and 2, but at shorter elasped time of 5000 MCS the differences between bridging simulations and fBm model is amplified. The elasped time here is much shorter than the bridge lifetime leading to more heterogeneous populations than in Fig. 2. Parameters: $\mu =80,\lambda ={10}^5$ MCS.

Figure 6

(a) MSD curve transitions to a higher exponent $\alpha \sim 0.56$. (b) Location of the kink depends linearly on the bridge lifetime ($\lambda$). Inset shows the same plot after conversion to seconds. (c) Phase diagram from Fig. 1 converted to seconds. The contours of $\alpha$ are overlaid. Dashed lines are estimated projections to higher values of $\lambda$. Shading indicates that the kink lies beyond $1\mathrm{s}$.

Figure 7

(a) Phase diagram of MSD at 25000 MCS for different parameters. MSD value reached at longer time delays decreases with increasing bridging. (b) Conversion between bridge lifetime $\lambda$ in MCS to seconds. $\lambda$(s) increases linearly with $\lambda$ (MCS) but the slope decreases sharply for higher levels of bridging. Hence, it is computationally challenging to access longer bridge lifetimes in seconds for higher levels of bridging.

Figure 8

(a) Phase diagram of the dynamic bridging model in the $\mu =p\lambda$ and $\lambda$ space. Note that $\mu$ is positively related to the percentage of monomers bridged. Contours indicate a fixed exponent $\alpha$. (b) Example polymer conformations with and without bridging. (c) Radius of Gyration $\langle R_g^2/N\rangle$ as function of Percentage Bridged. (d) The change in excluded volume relative to the nonbridging polymer $\mathrm{\Delta }V$ decreases linearly with bridging. (Inset) The relationship between $\mathrm{\Delta }V$ and the model parameter $\mu$. Black line indicates the average across bridge lifetimes. (e) Distributions of $\langle R_g^2/N\rangle$ are overlapping for different levels of bridging. (f) Distributions of $V_{\text{occupied}}$ are significantly separated.

Figure 9

(a) Mesh size ($\xi$) decreases with increasing bridges. (b) End to end distance for subsegments on the chain scale as $\langle R\rangle \sim s^{\nu }$. We observe an exponent $\nu =0.59$ in the absence of bridging as expected for a self-avoiding polymer. The scaling exponent decreases with increasing bridging. But, we do not observe a plateauing of the curves which is indicative of the globule state. (c) The contact probability $P\left(s\right)$ between monomers is plotted as function of monomer distance (s). In the absence of bridging (blue curve), we find $P\left(s\right)\sim s^{2.18}$ as expected for a self-avoiding chain in a good solvent [52, 53]. With increasing bridging the exponent increases, but for our level of bridging we are never below the compact globule regime. $\lambda =100000$ MCS in (b), (c). (d) Percentage of monomers bridged increases with $\mu$ and is independent of bridge lifetime.

Figure 10

(a)–(d) Confinement affects the scaling exponent of MSD even in the absence of bridging. Mesh size decreases with increasing confinement. Parameters: lattice size L, mesh size $\xi$, polymer length N = 400.

Figure 11

A circular polymer in a cuboid with hard walls matching the confinement of E. coli cells. In the presence of bridging, the polymer is compacted and has a density that decreases across the cross section. Lattice dimensions $88\times 22\times 22$.

Figure 12

(a) Drift velocity distribution comparisons between WT data and fBm measured over 28s (entire track length). (b) Ensemble averaged MSD of the wild type strain with tracks selected based on drift velocity $v_d$ (grey dashed line in (a) shows the $v_d$ threshold). Tracks with $v_d$ greater that fBm distribution display different dynamics to the rest of the population and transition to a higher exponent at longer time lags. (c) Same as in (a) for the bridging simulations with $v_d$ defined over the entire track length of $5\times {10}^5$ MCS. (d) We find a similar subset of tracks (split based $v_d$, grey dashed line in (c)) which transition to higher exponent at longer time lags. Parameters $\mu =80,\lambda ={10}^5$ MCS. (e) We select the indicated sub-population of faster moving tracks in the fBm simulations. The ensemble-averaged MSD of the faster tracks deviate from the rest of the tracks only at longer time lags and do not reproduce the behavior seen in the experimental data in (b).

Figure 13

(a) Mean intensity of WT tracks plotted versus scaling exponent $\alpha$ fitted to the entire track length. Shows weak positive correlation. (b) Same as in (a) but for the mutant $\mathrm{\Delta }\mathrm{H}$-NS. (c) Intensity distributions of WT and $\mathrm{\Delta }\mathrm{H}$-NS is plotted. WT distribution has a broader tail. (d) Ensemble averaged MSD of WT plotted for two different subpopulations with intensity $I>\text{median}\left(I\right)$ and $I<\text{median}\left(I\right)$. Scaling exponent $\alpha$ shows marginal change while $D_{\mathrm{app}}$ decreases with higher intensities [5]. (e) Same as in (d) but for the mutant. The subpopulations overlap. (f) Intensity of loci in the WT at a MSD of 10 s. We find a very weak positive correlation.

Figure 14

Mean logarithmic squared displacement (MLSD) also shows a transition to higher exponent at longer delays.