A Variational Approach to the Structure and Thermodynamics of Linear Polyelectrolytes with Coulomb and Screened Coulomb Interactions

A variational approach, based on a discrete representation of the chain, is used to calculate free energy and conformational properties in polyelectrolytes. The true bond and Coulomb potentials are approximated by a trial isotropic harmonic energy containing force constants between {\em all}monomer-pairs as variational parameters. By a judicious choice of representation and the use of incremental matrix inversion, an efficient and fast-convergent iterative algorithm is constructed, that optimizes the free energy. The computational demand scales as $N^3$ rather than $N^4$ as expected in a more naive approach. The method has the additional advantage that in contrast to Monte Carlo calculations the entropy is easily computed. An analysis of the high and low temperature limits is given. Also, the variational formulation is shown to respect the appropriate virial identities.The accuracy of the approximations introduced are tested against Monte Carlo simulations for problem sizes ranging from $N=20$ to 1024. Very good accuracy is obtained for chains with unscreened Coulomb interactions. The addition of salt is described through a screened Coulomb interaction, for which the accuracy in a certain parameter range turns out to be inferior to the unscreened case. The reason is that the harmonic variational Ansatz becomes less efficient with shorter range interactions. As a by-product a very efficient Monte Carlo algorithm was developed for comparisons, providing high statistics data for very large sizes -- 2048 monomers. The Monte Carlo results are also used to examine scaling properties, based on low-$T$ approximations to end-end and monomer-monomer separations. It is argued that the former increases faster than linearly with the number of bonds.Comment: 40 pages LaTeX, 13 postscript figure

The Electrostatic Persistence Length Calculated from Monte Carlo, Variational and Perturbation Methods

Monte Carlo simulations and variational calculations using a Gaussian ansatz are applied to a model consisting of a flexible linear polyelectrolyte chain as well as to an intrinsically stiff chain with up to 1000 charged monomers. Addition of salt is treated implicitly through a screened Coulomb potential for the electrostatic interactions. For the flexible model the electrostatic persistence length shows roughly three regimes in its dependence on the Debye-H\"{u}ckel screening length, $\kappa^{-1}$.As long as the salt content is low and $\kappa^{-1}$ is longer than the end-to-end distance, the electrostatic persistence length varies only slowly with $\kappa^{-1}$. Decreasing the screening length, a controversial region is entered. We find that the electrostatic persistence length scales as $sqrt{\xi_p}/\kappa$, in agreement with experiment on flexible polyelectrolytes, where $\xi_p$ is a strength parameter measuring the electrostatic interactions within the polyelectrolyte. For screening lengths much shorter than the bond length, the $\kappa^{-1}$ dependence becomes quadratic in the variational calculation. The simulations suffer from numerical problems in this regime, but seem to give a relationship half-way between linear and quadratic. A low temperature expansion only reproduces the first regime and a high temperature expansion, which treats the electrostatic interactions as a perturbation to a Gaussian chain, gives a quadratic dependence on the Debye length. For a sufficiently stiff chain, the persistence length varies quadratically with $\kappa^{-1}$ in agreement with earlier theories.Comment: 20 pages LaTeX, 9 postscript figure

A Potts Neuron Approach to Communication Routing

A feedback neural network approach to communication routing problems is developed with emphasis on Multiple Shortest Path problems, with several requests for transmissions between distinct start- and endnodes. The basic ingredients are a set of Potts neurons for each request, with interactions designed to minimize path lengths and to prevent overloading of network arcs. The topological nature of the problem is conveniently handled using a propagator matrix approach. Although the constraints are global, the algorithmic steps are based entirely on local information, facilitating distributed implementations. In the polynomially solvable single-request case the approach reduces to a fuzzy version of the Bellman-Ford algorithm. The approach is evaluated for synthetic problems of varying sizes and load levels, by comparing with exact solutions from a branch-and-bound method. With very few exceptions, the Potts approach gives legal solutions of very high quality. The computational demand scales merely as the product of the numbers of requests, nodes, and arcs.Comment: 10 pages LaTe

Random Boolean Network Models and the Yeast Transcriptional Network

The recently measured yeast transcriptional network is analyzed in terms of simplified Boolean network models, with the aim of determining feasible rule structures, given the requirement of stable solutions of the generated Boolean networks. We find that for ensembles of generated models, those with canalyzing Boolean rules are remarkably stable, whereas those with random Boolean rules are only marginally stable. Furthermore, substantial parts of the generated networks are frozen, in the sense that they reach the same state regardless of initial state. Thus, our ensemble approach suggests that the yeast network shows highly ordered dynamics.Comment: 23 pages, 5 figure

A Variational Approach for Minimizing Lennard-Jones Energies

A variational method for computing conformational properties of molecules with Lennard-Jones potentials for the monomer-monomer interactions is presented. The approach is tailored to deal with angular degrees of freedom, {\it rotors}, and consists in the iterative solution of a set of deterministic equations with annealing in temperature. The singular short-distance behaviour of the Lennard-Jones potential is adiabatically switched on in order to obtain stable convergence. As testbeds for the approach two distinct ensembles of molecules are used, characterized by a roughly dense-packed ore a more elongated ground state. For the latter, problems are generated from natural frequencies of occurrence of amino acids and phenomenologically determined potential parameters; they seem to represent less disorder than was previously assumed in synthetic protein studies. For the dense-packed problems in particular, the variational algorithm clearly outperforms a gradient descent method in terms of minimal energies. Although it cannot compete with a careful simulating annealing algorithm, the variational approach requires only a tiny fraction of the computer time. Issues and results when applying the method to polyelectrolytes at a finite temperature are also briefly discussed.Comment: 14 pages, uuencoded compressed postscript fil

Diagnostics of the structure of AGN's broad line regions with reverberation mapping data: confirmation of the two-component broad line region model

We re-examine the ten Reverberation Mapping (RM) sources with public data based on the two-component model of the Broad Line Region (BLR). In fitting their broad H-beta lines, six of them only need one Gaussian component, one of them has a double-peak profile, one has an irregular profile, and only two of them need two components, i.e., a Very Broad Gaussian Component (VBGC) and an Inter-Mediate Gaussian Component (IMGC). The Gaussian components are assumed to come from two distinct regions in the two-component model; they are Very Broad Line Region (VBLR) and Inter-Mediate Line region (IMLR). The two sources with a two-component profile are Mrk 509 and NGC 4051. The time lags of the two components of both sources satisfy $t_{IMLR}/t_{VBLR}=V^2_{VBLR}/V^2_{IMLR}$, where $t_{IMLR}$ and $t_{VBLR}$ are the lags of the two components while $V_{IMLR}$ and $V_{VBLR}$ represent the mean gas velocities of the two regions, supporting the two-component model of the BLR of Active Galactic Nuclei (AGN). The fact that most of these ten sources only have the VBGC confirms the assumption that RM mainly measures the radius of the VBLR; consequently, the radius obtained from the R-L relationship mainly represent the radius of VBLR. Moreover, NGC 4051, with a lag of about 5 days in the one component model, is an outlier on the R-L relationship as shown in Kaspi et al. (2005); however this problem disappears in our two-component model with lags of about 2 and 6 days for the VBGC and IMGC, respectively.Comment: 7 pages, 5 figures. Accepted for publication in the Special Issue of Science in China (G) "Astrophysics of Black holes and Related Compact Objects