Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow

It is well known that the Poiseuille flow of a visco-elastic polymer fluid between plates or through a tube is linearly stable in the zero Reynolds number limit, although the stability is weak for large Weissenberg numbers. In this paper we argue that recent experimental and theoretical work on the instability of visco-elastic fluids in Taylor-Couette cells and numerical work on channel flows suggest a scenario in which Poiseuille flow of visco-elastic polymer fluids exhibits a nonlinear "subcritical" instability due to normal stress effects, with a threshold which decreases for increasing Weissenberg number. This proposal is confirmed by an explicit weakly nonlinear stability analysis for Poiseuille flow of an UCM fluid. Our analysis yields explicit predictions for the critical amplitude of velocity perturbations beyond which the flow is nonlinearly unstable, and for the wavelength of the mode whose critical amplitude is smallest. The nonlinear instability sets in quite abruptly at Weissenberg numbers around 4 in the planar case and about 5.2 in the cylindrical case, so that for Weissenberg numbers somewhat larger than these values perturbations of the order of a few percent in the wall shear stress suffice to make the flow unstable. We have suggested elsewhere that this nonlinear instability could be an important intrinsic route to melt fracture and that preliminary experiments are both qualitatively and quantitatively in good agreement with these predictions.Comment: 20 pages, 16 figures. Accepted for publication in J. of Non-Newtonian Fluid Mechanic

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

How the Proximal Pocket May Influence the Enantiospecificities of Chloroperoxidase-Catalyzed Epoxidations of Olefins

Chloroperoxidase-catalyzed enantiospecific epoxidations of olefins are of significant biotechnological interest. Typical enantiomeric excesses are in the range of 66%–97% and translate into free energy differences on the order of 1 kcal/mol. These differences are generally attributed to the effect of the distal pocket. In this paper, we show that the influence of the proximal pocket on the electron transfer mechanism in the rate-limiting event may be just as significant for a quantitatively accurate account of the experimentally-measured enantiospecificities

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

Faces of matrix models

Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as tau-functions of integrable hierarchies and as special-geometry prepotentials, as result of the action of W-operators and of various recursions on elementary input data, as gluing of certain elementary building blocks. All this explains the central role of such matrix models in modern mathematical physics: they provide the basic "special functions" to express the answers and relations between them, and they serve as a dream model of what one should try to achieve in any other field.Comment: 10 page