Joseph Rudnick

The Shapes of Random Walks

A theoretical description of the shape of a random object is presented that is analytically simple in application but quantitatively accurate. The asymmetry of the object is characterized in terms of the invariants of a tensor, analogous to the moment-of-inertia tensor, whose eigenvalues are the squares of the principal radii of gyration. The complications accompanying ensemble averaging because of random processes are greatly reduced when the object is embedded in a space of high dimensionality, d. Exact analytical expressions are presented in the case of infinite spatial dimensions, and a procedure for developing an expansion in powers of l/d is discussed for linear chain and ring-type random walks. The first two terms in such an expansion lead to results for various shape parameters that agree remarkably well with those calculated by computer simulation. The method can be extended to yield an approximate, but extremely accurate, expression for the probability distribution function directly. The theoretical approach discussed here can, in principle, be used to describe the shape of other random fractal objects as well.

Elements of the Random Walk: An Introduction for Advanced Students and Researchers

Preface 1. Introduction to techniques 2. Generating functions I 3. Generating functions II: recurrence, sites visited, and the role of dimensionality 4. Boundary conditions, steady state, and the electrostatic analogy 5. Variations on the random walk 6. The shape of a random walk 7. Path integrals and self-avoidance 8. Properties of the random walk: introduction to scaling 9. Scaling of walks and critical phenomena 10. Walks and the O(n) model: mean field theory and spin waves 11. Scaling, fractals, and renormalization 12. More on the renormalization group References Index.

Growth and Erosion of Thin Solid Films

Thin films that are grown by the process of sputtering are, by and large, quite unlike the smooth, featureless structures that one might expect. In general, these films have a complicated surface morphology and an extended network of grooves and voids in their interiors. Such features can have a profound effect on the physical properties of a thin film. The surface irregularities and the bulk defects are the result of a growth instability due to competitive shadowing, an effect that also plays a role in geological processes such as erosion. For amorphous thin films, the shadow instability can be described by a remarkably simple model, which can be shown to reproduce many important observed characteristics of thin film morphology.

Fluctuation-facilitated charge migration along DNA.

We propose a model Hamiltonian for charge transfer along the DNA double helix with temperature-driven fluctuations in the base pair positions acting as the rate limiting factor for charge transfer between neighboring base pairs. We compare the predictions of the model with the recent work of Barton and Zewail on the unusual two-stage charge transfer of DNA.

CONFORMATIONS OF LINEAR DNA

We examine the conformations of a model for under- and overwound DNA. The molecule is represented as a cylindrically symmetric elastic string subjected to a stretching force and to constraints corresponding to a specification of the link number. We derive a fundamental relation between the Euler angles that describe the curve and the topological linking number. Analytical expressions for the spatial configurations of the molecule in the infinite- length limit were obtained. A unique configuraion minimizes the energy for a given set of physical conditions. An elastic model incorporating thermal fluctuations provides excellent agreement with experimental results on the plectonemic transition.

Elasticity Theory of the B-DNA to S-DNA Transition

We propose in this note a simple model--the two-state Worm Like Chain--to describe the elasticity of the recently discovered stress-induced transformation from B-DNA to S-DNA. The model reduces for low tractions to the well-known Worm Like chain theory, which is used to describe the elastic properties of B-DNA, while in the limit of high chain-bending moduli it reduces to the two-state Ising model proposed by Cluzel et al. for the B-S transition [Cluzel, P., A. Lebrun, C. Heller, R. Lavery, J-L. Viovy, D. Chatenay, and F. Caron. 1996. DNA: an extensible molecule. Science. 271:792-794]. Our model can be treated analytically to produce an explicit form of the force-extension relationship which agrees reasonably with the observations. We use the model to show that conformational fluctuations of the chain play a role also for the B to S transformation.

Finite-size scaling and the renormalization group

Renormalization group calculations ind = 4 andd = 4 −ɛ are performed for a system of finite size. A form of mean-field theory is used which yields a rounded transition for a finite system, and this allows a sensible expansion in fluctuations. A combination of Ewald and Poisson sum techniques is used to produce explicit numerical results for the specific heat ind = 4 which, with the setting of two nonuniversal metrical factors and the fourth-order coupling constant may be compared with simulations. The numerical visibility of logarithmic corrections is investigated. The universal scaling function for the specific heat to relativeO(ɛ) is also evaluated numerically.

What drives the translocation of stiff chains

We study the dynamics of the passage of a stiff chain through a pore into a cell containing particles that bind reversibly to it. Using Brownian molecular dynamics simulations we investigate the mean first-passage time as a function of the length of the chain inside for different concentrations of binding particles. As a consequence of the interactions with these particles, the chain experiences a net force along its length whose calculated value from the simulations accounts for the velocity at which it enters the cell. This force can in turn be obtained from the solution of a generalized diffusion equation incorporating an effective Langmuir adsorption free energy for the chain plus binding particles. These results suggest a role of binding particles in the translocation process that is in general quite different from that of a Brownian ratchet. Furthermore, nonequilibrium effects contribute significantly to the dynamics; e.g., the chain often enters the cell faster than particle binding can be saturated, resulting in a force several times smaller than the equilibrium value.

Finite-Size Scaling and the Renormalization Group

Renormalization group calculations ind = 4 andd = 4 −ɛ are performed for a system of finite size. A form of mean-field theory is used which yields a rounded transition for a finite system, and this allows a sensible expansion in fluctuations. A combination of Ewald and Poisson sum techniques is used to produce explicit numerical results for the specific heat ind = 4 which, with the setting of two nonuniversal metrical factors and the fourth-order coupling constant may be compared with simulations. The numerical visibility of logarithmic corrections is investigated. The universal scaling function for the specific heat to relativeO(ɛ) is also evaluated numerically.

DNA-protein cooperative binding through variable-range elastic coupling.

Cooperativity plays an important role in the action of proteins bound to DNA. A simple mechanism for cooperativity, in the form of a tension-mediated interaction between proteins bound to DNA at two different locations, is proposed. These proteins are not in direct physical contact. DNA segments intercalating bound proteins are modeled as a worm-like chain, which is free to deform in two dimensions. The tension-controlled protein-protein interaction is the consequence of two effects produced by the protein binding. The first is the introduction of a bend in the host DNA and the second is the modification of the bending modulus of the DNA in the immediate vicinity of the bound protein. The interaction between two bound proteins may be either attractive or repulsive, depending on their relative orientation on the DNA. Applied tension controls both the strength and the range of protein-protein interactions in this model. Properties of the cooperative interaction are discussed, along with experimental implications.