Theory of Active Intracellular Transport by DNA Relaying

The spatiotemporal organization of bacterial cells is crucial for the active segregation of replicating chromosomes. In several species, including Caulobacter crescentus, the ATPase ParA binds to DNA and forms a gradient along the long cell axis. The ParB partition complex on the newly replicated chromosome translocates up this ParA gradient, thereby contributing to chromosome segregation. A DNA-relay mechanism—deriving from the elasticity of the fluctuating chromosome—has been proposed as the driving force for this cargo translocation, but a mechanistic theoretical description remains elusive. Here, we propose a minimal model to describe force generation by the DNA-relay mechanism over a broad range of operational conditions. Conceptually, we identify four distinct force-generation regimes characterized by their dependence on chromosome fluctuations. These relay force regimes arise from an interplay of the imposed ParA gradient, chromosome fluctuations, and an emergent friction force due to chromosome-cargo interactions.

Figure 1

Minimal model for force generation by DNA relaying. (a) The relay force $F$ arises from the interactions of the cargo with ParA ATPases bound to the chromosome, represented by a set of chromosomal elements modeled as a bead-spring system with an associated ParA concentration (indicated by the green tone). We assume that the ParA gradient is comoving with the cargo. Chromosomal elements fluctuate due to thermal energy, with the magnitude of the fluctuations, $\sigma =\sqrt{k_{B}T/k}$ (red Gaussian). Our 1D model is presented in 2D for visual purposes. (b),(c) Cargo trajectory (b) and the corresponding DNA-relay force (c) obtained from the numerical solution of Eq. (1) using Brownian dynamics simulation. The horizontal line shows the time average of $F$.

Figure 2

Average relay force $F$ in the weak-binding limit ($c_0\ll 1$) for different values of the friction coefficient ${\gamma }_{\mathrm{c}}$ of the cargo in the cytoplasm and the magnitude of chromosome fluctuations $\sigma$. (a) We compare results from simulations (dots) with theory (lines), obtained from Eqs. (8) and (10) for a static (black) and moving cargo (blue), respectively. The dotted vertical line at $\sigma =1/2$ separates the different force generation regimes. (b) Phase diagram of force generation regimes.

Figure 3

The influence of the ParA concentration gradient $m$ and the cargo velocity $v$ on the force density $f(x)$. (a),(b) $f(x)$ for a static cargo given by Eq. (7) without ($m=0$) and with ($m=2$) a ParA gradient. (c),(d) $f(x)$ for a static ($v=0$) and a moving ($v=0.05$) cargo both with $m=2$. The force density for a moving cargo is obtained numerically. We compare results from simulations (dots) and our theoretical results (lines). Note that the dark green and the light blue curves in (a),(c) and (b),(d) show the same data.