Mehran Kardar

Intricate Knots in Proteins: Function and Evolution

Our investigation of knotted structures in the Protein Data Bank reveals the most complicated knot discovered to date. We suggest that the occurrence of this knot in a human ubiquitin hydrolase might be related to the role of the enzyme in protein degradation. While knots are usually preserved among homologues, we also identify an exception in a transcarbamylase. This allows us to exemplify the function of knots in proteins and to suggest how they may have been created.

Anomalous dynamics of translocation

We study the dynamics of the passage of a polymer through a membrane pore (translocation), focusing on the scaling properties with the number of monomers N. The natural coordinate for translocation is the number of monomers on one side of the hole at a given time. Commonly used models that assume Brownian dynamics for this variable predict a mean (unforced) passage time tau that scales as N2, even in the presence of an entropic barrier. In particular, however, the time it takes for a free polymer to diffuse a distance of the order of its radius by Rouse dynamics scales with an exponent larger than two, and this should provide a lower bound to the translocation time. To resolve this discrepancy, we perform numerical simulations with Rouse dynamics for both phantom (in space dimensions d=1 and 2), and self-avoiding (in d=2) chains. The results indicate that for large N, translocation times scale in the same manner as diffusion times, but with a larger prefactor that depends on the size of the hole. Such scaling implies anomalous dynamics for the translocation process. In particular, the fluctuations in the monomer number at the hole are predicted to be nondiffusive at short times, while the average pulling velocity of the polymer in the presence of a chemical-potential difference is predicted to depend on N.

Anomalous dynamics of forced translocation

We consider the passage of long polymers of length N through a hole in a membrane. If the process is slow, it is in principle possible to focus on the dynamics of the number of monomers s on one side of the membrane, assuming that the two segments are in equilibrium. The dynamics of s(t) in such a limit would be diffusive, with a mean translocation time scaling as N2 in the absence of a force, and proportional to N when a force is applied. We demonstrate that the assumption of equilibrium must break down for sufficiently long polymers (more easily when forced), and provide lower bounds for the translocation time by comparison to unimpeded motion of the polymer. These lower bounds exceed the time scales calculated on the basis of equilibrium, and point to anomalous (subdiffusive) character of translocation dynamics. This is explicitly verified by numerical simulations of the unforced translocation of a self-avoiding polymer. Forced translocation times are shown to strongly depend on the method by which the force is applied. In particular, pulling the polymer by the end leads to much longer times than when a chemical potential difference is applied across the membrane. The bounds in these cases grow as N2 and N1+nu, respectively, where nu is the exponent that relates the scaling of the radius of gyration to N. Our simulations demonstrate that the actual translocation times scale in the same manner as the bounds, although influenced by strong finite size effects which persist even for the longest polymers that we considered (N=512).

Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities

The statistical mechanics of interfaces subject to quenched impurities is studied in two dimensions. The presence of randomness changes the scaling of domain wall fluctuations, and modifies critical behavior at interface-driven depinning (wetting), and commensurate-to-incommensurate phase transitions. All these problems are examined by combining the replica method with Bethe ansatz calculations. Results include expressions for quench-averaged free energies, their cumulants, expectation values, distribution functions, in addition to a number of new critical exponents. The intermediate results include some novel Bethe ansatz solutions, such as the ground state energy of a system of n attracting fermion species.

Scattering theory approach to electrodynamic Casimir forces

We give a comprehensive presentation of methods for calculating the Casimir force to arbitrary accuracy, for any number of objects, arbitrary shapes, susceptibility functions, and separations. The technique is applicable to objects immersed in media other than vacuum, nonzero temperatures, and spatial arrangements in which one object is enclosed in another. Our method combines each objects classical electromagnetic scattering amplitude with universal translation matrices, which convert between the bases used to calculate scattering for each object, but are otherwise independent of the details of the individual objects. The method is illustrated by rederiving the Lifshitz formula for infinite half-spaces, by demonstrating the Casimir-Polder to van der Waals crossover, and by computing the Casimir interaction energy of two infinite, parallel, perfect metal cylinders either inside or outside one another. Furthermore, it is used to obtain new results, namely, the Casimir energies of a sphere or a cylinder opposite a plate, all with finite permittivity and permeability, to leading order at large separation.

Fractional Laplacian in bounded domains.

The fractional Laplacian operator -(-delta)(alpha/2) appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.

Nonequilibrium dynamics of interfaces and lines

Abstract The lectures examine several problems related to non-equilibrium fluctuations of interfaces and flux lines. We start by introducing the phenomenology of depinning, with particular emphasis on interfaces and contact lines. The role of the anisotropy of the medium in producing different universality classes is elucidated. We then focus on the dynamics of lines, where transverse fluctuations are also important. We shall demonstrate how various non-linearities appear in the dynamics of driven flux lines. The universality classes of depinning, and also dynamic roughening, are illustrated in the contexts of moving flux lines, advancing crack fronts, and drifting polymers.

Casimir interaction between a plate and a cylinder

We find the exact Casimir force between a plate and a cylinder, a geometry intermediate between parallel plates, where the force is known exactly, and the plate sphere, where it is known at large separations. The force has an unexpectedly weak decay approximately L/[H3 ln(H/R)] at large plate-cylinder separations H (L and R are the cylinder length and radius), due to transverse magnetic modes. Path integral quantization with a partial wave expansion additionally gives a qualitative difference for the density of states of electric and magnetic modes, and corrections at finite temperatures.

Instabilities of Charged Polyampholytes

We consider polymers formed from a (quenched) random sequence of charged monomers of opposite signs. Such polymers, known as polyampholytes (PAs), are compact when completely neutral and expanded when highly charged. We examine the transition between the two regimes by Monte Carlo simulations, and by analogies to charged drops. We find that the overall excess charge Q is the main determinant of the size of the PAs. A polymer composed of N charges of \ifmmode\pm\else\textpm\fi{}

Pressure and Phase Equilibria in Interacting Active Brownian Spheres

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