Operational Principles for the Dynamics of the In Vitro ParA-ParB System

In many bacteria the ParA-ParB protein system is responsible for actively segregating DNA during replication. ParB proteins move by interacting with DNA bound ParA-ATP, stimulating their unbinding by catalyzing hydrolysis, that leads to rectified motion due to the creation of a wake of depleted ParA. Recent in vitro experiments have shown that a ParB covered magnetic bead can move with constant speed over a DNA covered substrate that is bound by ParA. It has been suggested that the formation of a gradient in ParA leads to diffusion-ratchet like motion of the ParB bead but how it forms and generates a force is still a matter of exploration. Here we develop a deterministic model for the in vitro ParA-ParB system and show that a ParA gradient can spontaneously form due to any amount of initial spatial noise in bound ParA. The speed of the bead is independent of this noise but depends on the ratio of the range of ParA-ParB force on the bead to that of removal of surface bound ParA by ParB. We find that at a particular ratio the speed attains a maximal value. We also consider ParA rebinding (including cooperativity) and ParA surface diffusion independently as mechanisms for ParA recovery on the surface. Depending on whether the DNA covered surface is undersaturated or saturated with ParA, we find that the bead can accelerate persistently or potentially stall. Our model highlights key requirements of the ParA-ParB driving force that are necessary for directed motion in the in vitro system that may provide insight into the in vivo dynamics of the ParA-ParB system.