Molecular Motors Interacting with Their Own Tracks

Dynamics of molecular motors that move along linear lattices and interact with them via reversible destruction of specific lattice bonds is investigated theoretically by analyzing exactly solvable discrete-state ``burnt-bridge'' models. Molecular motors are viewed as diffusing particles that can asymmetrically break or rebuild periodically distributed weak links when passing over them. Our explicit calculations of dynamic properties show that coupling the transport of the unbiased molecular motor with the bridge-burning mechanism leads to a directed motion that lowers fluctuations and produces a dynamic transition in the limit of low concentration of weak links. Interaction between the backward biased molecular motor and the bridge-burning mechanism yields a complex dynamic behavior. For the reversible dissociation the backward motion of the molecular motor is slowed down. There is a change in the direction of the molecular motor's motion for some range of parameters. The molecular motor also experiences non-monotonic fluctuations due to the action of two opposing mechanisms: the reduced activity after the burned sites and locking of large fluctuations. Large spatial fluctuations are observed when two mechanisms are comparable. The properties of the molecular motor are different for the irreversible burning of bridges where the velocity and fluctuations are suppressed for some concentration range, and the dynamic transition is also observed. Dynamics of the system is discussed in terms of the effective driving forces and transitions between different diffusional regimes

New and Old Results in Resultant Theory

Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than three-hundred-year research, resultants are of course rather well studied: a lot of explicit formulas, beautiful properties and intriguing relationships are known in this field. We present a brief overview of these results, including both recent and already classical. Emphasis is made on explicit formulas for resultants, which could be practically useful in a future physics research.Comment: 50 pages, 15 figure

Transport of Molecular Motor Dimers in Burnt-Bridge Models

Dynamics of molecular motor dimers, consisting of rigidly bound particles that move along two parallel lattices and interact with underlying molecular tracks, is investigated theoretically by analyzing discrete-state stochastic continuous-time burnt-bridge models. In these models the motion of molecular motors is viewed as a random walk along the lattices with periodically distributed weak links (bridges). When the particle crosses the weak link it can be destroyed with a probability $p$, driving the molecular motor motion in one direction. Dynamic properties and effective generated forces of dimer molecular motors are calculated exactly as a function of a concentration of bridges $c$ and burning probability $p$ and compared with properties of the monomer motors. It is found that the ratio of the velocities of the dimer and the monomer can never exceed 2, while the dispersions of the dimer and the monomer are not very different. The relative effective generated force of the dimer (as compared to the monomer) also cannot be larger than 2 for most sets of parameters. However, a very large force can be produced by the dimer in the special case of $c=1/2$ for non-zero shift between the lattices. Our calculations do not show the significant increase in the force generated by collagenase motor proteins in real biological systems as predicted by previous computational studies. The observed behavior of dimer molecular motors is discussed by considering in detail the particle dynamics near burnt bridges.Comment: 21 pages and 11 figure

Dynamic Properties of Molecular Motors in Burnt-Bridge Models

Dynamic properties of molecular motors that fuel their motion by actively interacting with underlying molecular tracks are studied theoretically via discrete-state stochastic ``burnt-bridge'' models. The transport of the particles is viewed as an effective diffusion along one-dimensional lattices with periodically distributed weak links. When an unbiased random walker passes the weak link it can be destroyed (``burned'') with probability p, providing a bias in the motion of the molecular motor. A new theoretical approach that allows one to calculate exactly all dynamic properties of motor proteins, such as velocity and dispersion, at general conditions is presented. It is found that dispersion is a decreasing function of the concentration of bridges, while the dependence of dispersion on the burning probability is more complex. Our calculations also show a gap in dispersion for very low concentrations of weak links which indicates a dynamic phase transition between unbiased and biased diffusion regimes. Theoretical findings are supported by Monte Carlo computer simulations.Comment: 14 pages. Submitted to J. Stat. Mec