Random Walks on a Fluctuating Lattice: A Renormalization Group Approach Applied in One Dimension

We study the problem of a random walk on a lattice in which bonds connecting nearest neighbor sites open and close randomly in time, a situation often encountered in fluctuating media. We present a simple renormalization group technique to solve for the effective diffusive behavior at long times. For one-dimensional lattices we obtain better quantitative agreement with simulation data than earlier effective medium results. Our technique works in principle in any dimension, although the amount of computation required rises with dimensionality of the lattice.Comment: PostScript file including 2 figures, total 15 pages, 8 other figures obtainable by mail from D.L. Stei

Solvent viscosity dependence for enzymatic reactions

A mechanism for relationship of solvent viscosity with reaction rate constant at enzyme action is suggested. It is based on fluctuations of electric field in enzyme active site produced by thermally equilibrium rocking (cranckshaft motion) of the rigid plane (in which the dipole moment $\approx 3.6 D$ lies) of a favourably located and oriented peptide group (or may be a few of them). Thus the rocking of the plane leads to fluctuations of the electric field of the dipole moment. These fluctuations can interact with the reaction coordinate because the latter in its turn has transition dipole moment due to separation of charges at movement of the reacting system along it. The rocking of the plane of the peptide group is sensitive to the microviscosity of its environment in protein interior and the latter is a function of the solvent viscosity. Thus we obtain an additional factor of interrelationship for these characteristics with the reaction rate constant. We argue that due to the properties of the cranckshaft motion the frequency spectrum of the electric field fluctuations has a sharp resonance peak at some frequency and the corresponding Fourier mode can be approximated as oscillations. We employ a known result from the theory of thermally activated escape with periodic driving to obtain the reaction rate constant and argue that it yields reliable description of the preexponent where the dependence on solvent viscosity manifests itself. The suggested mechanism is shown to grasp the main feature of this dependence known from the experiment and satisfactorily yields the upper limit of the fractional index of a power in it.Comment: 36 LaTex pages, 9 Eps figures, final versio

Model for solvent viscosity effect on enzymatic reactions

Why reaction rate constants for enzymatic reactions are typically inversely proportional to fractional power exponents of solvent viscosity remains to be already a thirty years old puzzle. Available interpretations of the phenomenon invoke to either a modification of 1. the conventional Kramers' theory or that of 2. the Stokes law. We show that there is an alternative interpretation of the phenomenon at which neither of these modifications is in fact indispensable. We reconcile 1. and 2. with the experimentally observable dependence. We assume that an enzyme solution in solvent with or without cosolvent molecules is an ensemble of samples with different values of the viscosity for the movement of the system along the reaction coordinate. We assume that this viscosity consists of the contribution with the weight $q$ from cosolvent molecules and that with the weight $1-q$ from protein matrix and solvent molecules. We introduce heterogeneity in our system with the help of a distribution over the weight $q$. We verify the obtained solution of the integral equation for the unknown function of the distribution by direct substitution. All parameters of the model are related to experimentally observable values. General formalism is exemplified by the analysis of literature experimental data for oxygen escape from hemerythin.Comment: 16 LaTex pages, 5 eps figure

Temperature-dependent macromolecular X-ray crystallography

The dynamical behaviour of crystalline macromolecules and their surrounding solvent as a function of cryo-temperature is reviewed

A migration well model for the binding of ligands to heme proteins

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