Solutions of burnt-bridge models for molecular motor transport

Transport of molecular motors, stimulated by interactions with specific links between consecutive binding sites (called “bridges”), is investigated theoretically by analyzing discrete-state stochastic “burnt-bridge” models. When an unbiased diffusing particle crosses the bridge, the link can be destroyed (“burned”) with a probability $p$, creating a biased directed motion for the particle. It is shown that for probability of burning $p=1$ the system can be mapped into a one-dimensional single-particle hopping model along the periodic infinite lattice that allows one to calculate exactly all dynamic properties. For the general case of $p<1$ a theoretical method is developed and dynamic properties are computed explicitly. Discrete-time and continuous-time dynamics for periodic distribution of bridges and different burning dynamics are analyzed and compared. Analytical predictions are supported by extensive Monte Carlo computer simulations. Theoretical results are applied for analysis of the experiments on collagenase motor proteins.

Figure 1

A schematic view of the motion of the particle in the system with burnt bridges. Thick solid lines are strong links, while thin solid lines are weak links, or “bridges” that can be burnt. Dotted lines are already burnt bridges. The particle can jump with equal rates to the right or to the left unless the link is broken.

Figure 2

Dynamic properties of a continuous-time burnt-bridge model with periodic distribution of bridges for $p=1$. Lines are from analytic calculations, while symbols are obtained from Monte Carlo computer simulations—see text for details.

Figure 3

Reduced kinetic schemes for systems with periodic distribution of bridges: (a) Continuous-time forward BBM (only transition rates not equal to unity are shown); (b) discrete-time forward-backward BBM (only transition rates not equal to $1∕2$ are shown). The origin corresponds to the right end of the last burnt bridge.

Figure 4

(a) Velocity as a function of concentration of bridges for different burning probabilities for the periodic forward continuous-time BBM. Solid lines are results of analytical calculations, and symbols are from Monte Carlo simulations. Curve $a$ corresponds to $p=0.9$, curve $b$ corresponds to $p=0.7$, curve $c$ corresponds to $p=0.5$, and curve $d$ corresponds to $p=0.1$. (b) Velocity as a function of burning probability for different concentrations of weak links for the periodic forward continuous-time BBM. Solid lines are results of analytical calculations and symbols are from Monte Carlo simulations. Curve $a$ corresponds to $c=1$, curve $b$ corresponds to $c=1∕2$, curve $c$ corresponds to $c=1∕3$, and curve $d$ corresponds to $c=1∕10$.

Figure 5

Velocity as a function of concentration for different burning realizations for the continuous-time periodic BBM. Solid lines are results of analytical calculations. Curve $a$ corresponds to forward-backward BBM with $p=0.8$, curve $b$ corresponds to forward BBM with $p=0.8$, curve $c$ corresponds to forward-backward BBM with $p=0.2$, and curve $d$ corresponds to forward BBM with $p=0.2$.

Figure 6

Dispersion in the forward continuous-time periodic BBM from Monte Carlo simulations (a) as a function of burning probability and (b) as a function of concentration of bridges.