Mathematical modeling of bacterial track-altering motors: Track cleaving through burnt-bridge ratchets

The generation of directed movement of cellular components frequently requires the rectification of Brownian motion. Molecular motor enzymes that use ATP to walk on filamentous tracks are typically involved in cell transport, however, a track-altering motor can arise when an enzyme interacts with and alters its track. In Caulobacter crescentus and other bacteria, an active DNA partitioning (Par) apparatus is employed to segregate replicated chromosome regions to specific locations in dividing cells. The Par apparatus is composed of two proteins: ParA, an ATPase that can form polymeric structures on the nucleoid, and ParB, a protein that can bind and destabilize ParA structures. It has been proposed that the ParB-mediated alteration of ParA structures could be responsible for generating the directed movement of DNA during bacterial division. How precisely these actions are coordinated and translated into directed movement is not clear. In this paper we consider the C. crescentus segregation apparatus as an example of a track altering motor that operates using a so-called burnt-bridge mechanism. We develop and analyze mathematical models that examine how diffusion and ATP-hydrolysis-mediated monomer removal (or cleaving) can be combined to generate directed movement. Using a mean first passage approach, we analytically calculate the effective ParA track-cleaving velocities, effective diffusion coefficient, and other higher moments for the movement a ParB protein cluster that breaks monomers away at random locations on a single ParA track. Our model results indicate that cleaving velocities and effective diffusion constants are sensitive to ParB-induced ATP hydrolysis rates. Our analytical results are in excellent agreement with stochastic simulation results.

Figure 1

Diagram of the components of the ParA-ParB machinery. (a) A track of ParA dimers extends between the two cell poles. Each ParA dimer has one reactive site for the ParB complex, whereas the complex contains several binding sites that can react with a single ParA dimer. The ParB complex has affinity to bind anywhere on the surface of the ParA track, however a subset of the ParB binder sites (small spheres) are considered reactive proteins (dark spheres). Reactive ParB binder sites cleave the connections between consecutive ParA dimers. When a dimer $i$ reacts with the ParB complex, the connection between the $\left(i-1\right)\text{th}$ and $i\text{th}$ sites is broken, thus a cleaving reaction occurs. The ParB complex is assumed to stay attached and ahead of the $i\text{th}$ reactive site due to multivalent connections, in a burnt-bridge model fashion. If the $i\text{th}$ ParA dimer reacts, then all the ParA dimers with index $\le i-1$ are removed from the nucleoid. (b) Model diagram for the process. A one-dimensional semi-infinite axis represents the ParA track. The ParB complex and corresponding ParA sites are both represented by uniformly distributed locations of length $L$ (black rectangles) on the semi-infinite track axis. Only the center of the ParB complex is tracked through a position variable $x$ in the model. Here $x=0$ corresponds to the most recently cleaved ParA dimer site. (c) Diagram comparing a ParA dimer with our model track subunit. The black section of a model ParA track subunit is referred to as the reactive site for each dimer and it encapsulates both the ParA binding site and the ParB binders.

Figure 2

Comparison of discrete- vs continuous-space models. (a) In the discrete model, we assume that the CA is a random walker that hops between discrete-space points on a lattice with regular spacing $\Delta x$. Any point on the lattice $j$ can trigger a cleaving reaction, so the width of the reactive region length $L$ is not explicitly included in the model. (b) In the continuous model, the position of the CA $x$ is a continuous variable. Further, each reactive site has a prescribed nonzero width $L$ (black rectangles), corresponding to regions in space where a cleaving reaction can be triggered with the rate $g$. In this model, the locations $x_j+L<x<x_{j+1}$ are not reactive and the CA experiences simple diffusion between cleaving sites.

Figure 3

Discrete model results. (a) Comparison between nondimensional analytical velocity results in Eq. (2.16) and simulation results. Numerical simulations were performed using a Gillespie algorithm for a discrete lattice track. Numerical results are in excellent agreement with calculated velocities. (b) Effective diffusion coefficient $\frac{{\nu }^2}{{\mu }_{1}d\Delta x^2}$. The analytical effective diffusion is computed using Eq. (2.32) (solid line). The effective diffusion is compared with stochastic simulation diffusion $\frac{{\widehat{\sigma }}^2}{{\mu }_{1}d\Delta x^2}=\frac{1}{d{\mu }_1\Delta x}\text{var}(n-\frac{\overline{n}}{\overline{t_n}}{t}_n)$, where $n$ is the amount of cleaved track during $t_n$ interval, obtained from Gillespie simulations.

Figure 4

Continuous model results. (a) Nondimensional velocity $\widehat{v}$ plotted as a function of $\alpha$ with varying trap length measured by the nondimensional parameter $k$. For high cleaving rates $\alpha$ velocities limit to the same constant values independent of the size of the cleaving sites on the track. If cleaving does not occur, i.e., $\alpha =0$, then the cleaving velocity is zero and the CA is not reacting with the track. (b) Steady-state probability density solution $p_s\left(y\right)$ with $\Delta x=1$; $k=0.4$; and $\widehat{\alpha }=0.5,2,6$. The peaks on the distributions show that the CA is more likely to be found at the edge of the track if the cleaving rate increases, since there is less time to diffuse away from $x=0$ in each reaction cycle.

Figure 5

Model results for the point-cleaving rate limit. (a) Exit probabilities ${\pi }_i\left(x\right)$ with $\Delta x=1$; $i=2,4,6$; and three cases of cleaving rates: (a) $\alpha =0.25$, (b) $\alpha =4$, and (c) $\alpha =36$. The probabilities peak at the index site $i$ with higher probabilities occurring for higher cleaving rates $\alpha$. (b) Mean first passage times $T_i\left(x\right)$ for three exit sites indexed $i=2,4,6$; $\alpha =0.25,4,36$; and $\Delta x=1$. The mean exit times are the lowest at the exit site positions $x=i$. (c) The nondimensional track-cleaving velocity $\widehat{v}$ is shown as a function of the parameter $\alpha$. The velocity shows qualitatively similar saturating behavior to the discrete model velocities.

Figure 6

Comparison of all model results. (a) Plot comparing all the average velocity results for each model. (b) Comparison of the effective diffusion constant for the discrete model and the continuous model with point-cleaving rates $L\to 0$.