Model of chromosomal loci dynamics in bacteria as fractional diffusion with intermittent transport

The short-time dynamics of bacterial chromosomal loci is a mixture of subdiffusive and active motion, in the form of rapid relocations with near-ballistic dynamics. While previous work has shown that such rapid motions are ubiquitous, we still have little grasp on their physical nature, and no positive model is available that describes them. Here, we propose a minimal theoretical model for loci movements as a fractional Brownian motion subject to a constant but intermittent driving force, and compare simulations and analytical calculations to data from high-resolution dynamic tracking in E. coli. This analysis yields the characteristic time scales for intermittency. Finally, we discuss the possible shortcomings of this model, and show that an increase in the effective local noise felt by the chromosome associates to the active relocations.

Figure 1

Illustration of the model. An intermittent process switches on and off a constant driving force, acting on top of a basal subdiffusive process with viscoelasticlike behavior, modeled as fractional Brownian motion. The resulting tracks are compared with experimental data from Ref. [8].

Figure 2

Model reproduces the mean-square displacement of experimental data allowing one to fix all model parameters. (a) Determination of the scaling exponent $2H$ of the background subdiffusion. Purple circles are the mean of the discrete derivative (1 s steps) of $D_{\mathrm{app}}:=〈r^2\left(\tau \right)/t^{2H}〉$, for values of lag time below 10 s, where the averages are not affected by the influence of ballistic stretches [8]. Since a well-defined $D_{\mathrm{app}}$ should not vary with time scale, the optimal estimated value for the scaling exponent $H\simeq 0.2$ is obtained when the plot crosses zero. (b) Large purple circles represent mean-square displacement vs lag time for the data on the Ori3 locus in Ref. [8] (error bars, SE, are smaller than symbols). Red line and small black circles are, respectively, a fit with the analytical formula, Eq. (4), and simulation results obtained with the parameter values fixed by the fits [reported in Table 1]. $2H=0.41$ is fixed from the analysis shown in panel (a). (c) Fits with $w=1/2$ cannot reproduce the data. The plot shows the ratio of the rms residual of fits imposing $w=1/2$ to the rms residual of the unconstrained fit shown in panel (b), for different values of the time scale. The ratio between the two values is in the range 50–70.

Figure 3

Subtraction procedure based on the model confirms the superdiffusive nature of the observed rapid chromosomal movements. Purple circles are mean-square displacement vs lag time from the same data as in Fig. 1. Indigo squares are obtained by subtracting the background fBm contribution from the mean-squared displacement [Eq. (3)]. The remaining contribution to the mean-square displacement is superlinear (the red dashed line is the linear scaling), as predicted by Eq. (4), consistent with the hypothesis of intermittent active relocations.

Figure 4

Model predicts correctly the qualitative behavior of single-track end-to-end distances and conditional mean-square displacements. (a) Distribution of the track end-to-end distance $R_{\mathrm{e}}$ in experimental data (purple circles), the intermittent active force model (black filled smaller circles), and fractional Brownian motion (open smaller circles). Simulations have been matched with experimental data, both in number and length of tracks (parameters shown in Table 1). (b) Conditional averages of mean-square displacements on tracks within a given percentile of $R_{\mathrm{e}}$ (diamonds, top $30\%$, and squares, bottom $70\%$, shown as example) show qualitative agreement and some quantitative discrepancies. Circles are the overall MSD average shown in Fig. 2.

Figure 5

Parameter values for different chromosomal loci. Top panel: typical velocity of excursions $V$. Middle panel: characteristic time ${\tau }_{\mathrm{on}}$. Bottom panel: characteristic time ${\tau }_{\mathrm{off}}$. The results of the less reliable fit of the Left1 locus are shown in faded colors. The strong differences observed for different chromosomal loci can be interpreted as local differences in the physical organization or in the action of the active drive [4, 8]. (Estimated errors are around the size of the points.)

Figure 6

Additional diffusivity of active excursions. (a) The model variant including active noise (black filled symbols) fully captures the conditional mean-square displacement at short time lags of experimental data (large filled symbols), for trajectories that fall in different percentiles of end-to-end distance $R_{\mathrm{e}}$ (top 30% $R_{\mathrm{e}}$, diamonds, vs bottom 70%, squares). (b) Conditional mean-square displacement at 1 s time lag computed as a function of end-to-end distance percentile, and compared with simulations. Model parameters in Table 1, $q=0.78$.