Local Interactions and Protein Folding: A 3D Off-Lattice Approach

The thermodynamic behavior of a three-dimensional off-lattice model for protein folding is probed. The model has only two types of residues, hydrophobic and hydrophilic. In absence of local interactions, native structure formation does not occur for the temperatures considered. By including sequence independent local interactions, which qualitatively reproduce local properties of functional proteins, the dominance of a native state for many sequences is observed. As in lattice model approaches, folding takes place by gradual compactification, followed by a sequence dependent folding transition. Our results differ from lattice approaches in that bimodal energy distributions are not observed and that high folding temperatures are accompanied by relatively low temperatures for the peak of the specific heat. Also, in contrast to earlier studies using lattice models, our results convincingly demonstrate that one does not need more than two types of residues to generate sequences with good thermodynamic folding properties in three dimensions.Comment: 18 pages, 11 Postscript figure

Scaling and Scale Breaking in Polyelectrolyte

We consider the thermodynamics of a uniformly charged polyelectrolyte with harmonic bonds. For such a system there is at high temperatures an approximate scaling of global properties like the end-to-end distance and the interaction energy with the chain-length divided by the temperature. This scaling is broken at low temperatures by the ultraviolet divergence of the Coulomb potential. By introducing a renormalization of the strength of the nearest- neighbour interaction the scaling is restored, making possible an efficient blocking method for emulating very large polyelectrolytes using small systems. The high temperature behaviour is well reproduced by the analytical high-$T$ expansions even for fairly low temperatures and system sizes. In addition, results from low-$T$ expansions, where the coefficients have been computed numerically, are presented. These results approximate well the corresponding Monte Carlo results at realistic temperatures. A corresponding analysis of screened chains is performed. The situation here is complicated by the appearance of an additional parameter, the screening length. A window is found in parameter space, where scaling holds for the end-to-end distance. This window corresponds to situations where the range of the potential interpolates between the bond length and the size of the chain. This scaling behaviour, which is verified by Monte Carlo results, is consistent with Flory scaling. Also for the screened chain a blocking approach can be devised, that performs well for low temperatures, whereas the low-$T$ expansion is inaccurate at realistic temperatures.Comment: 18 pages, latex, 6 figure

Titrating Polyelectrolytes - Variational Calculations and Monte Carlo Simulations

Variational methods are used to calculate structural and thermodynamical properties of a titrating polyelectrolyte in a discrete representation. The Coulomb interactions are emulated by harmonic repulsive forces, the force constants being used as variational parameters to minimize the free energy. For the titrating charges, a mean field approach is used. The accuracy is tested against Monte Carlo data for up to 1000 monomers. For an unscreened chain, excellent agreement is obtained for the end-to-end distance and the apparent dissociation constant. With screening, the thermodynamical properties are invariably well described, although the structural agreement deteriorates. A very simple rigid-rod approximation is also considered, giving surprisingly good results for certain properties.Comment: 22 pages, PostScript, 9 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

Thermodynamic Studies of Macromolecular Models

The thermodynamic properties of single chain polyelectrolyte and protein models are studied using Monte Carlo simulations and (for the polyelectrolyte models) variational calculations and high- and low-temperature expansions. A variational method is used for minimizing Lennard-Jones energies and for estimating end-to-end distances for a rigid Coulomb chain at finite temperatures. A polyelectrolyte chain is viewed as Gaussian chain augmented with a Coulomb or screened Coulomb (Debye-Huckel) interaction between all pairs of monomers. Variational calculations are also used together with Monte Carlo simulations to study the behavior of a titrating polyelectrolyte. Furthermore a method for mapping the original polymer to a smaller one by introducing a corrective nearest-neighbor interaction is presented. The underlying assumptions for this approach is examined using high- and low-temperature expansions. The effect of the screening length on the stiffness, defined by the persistence length, of a screened Coulomb chain is studied. Moreover a three dimensional off-lattice model for protein folding is presented. The model has two types of residues, hydrophobic and hydrophilic respectively, that interact via a sequence dependent Lennard-Jones potential. The influence of sequence independent local interactions is studied as well as the folding properties and the formation of the native state

Blocking technique for emulating very large polyelectrolytes

A new Monte Carlo method for computing thermodynamical properties of very large polyelectrolytes is presented. It is based on a renormalization group relating the original polymer to a smaller system, where, in addition to the naively rescaled forces, a corrective nearest-neighbor interaction originating from the short distance Coulomb cutoff is introduced. The method is derived for low T but is in the unscreened case valid for all T. Large polymers with N monomers are emulated by Monte Carlo calculations on smaller systems, K = N/Q. The computational gain of the method is Q3 and is explored with emphasis on room temperature

A Blocking Technique for Emulating Very Large Polyelectrolytes

: A new Monte Carlo method for computing thermodynamical properties of very large polyelectrolytes is presented. It is based on a renormalization group relating the original polymer to a smaller system, where in addition to the naively rescaled forces, a corrective nearest-neighbor interaction originating from the short distance Coulomb cutoff is introduced. The method is derived for low T but is in the unscreened case valid for all T . Large polymers with N monomers are emulated by Monte Carlo calculations on smaller systems, K = N=Q. The computational gain of the method is Q 3 . It is explored with emphasis on room temperature. Results for N=10000 are presented. 1 [email protected] 2 [email protected] 3 [email protected] The thermodynamics of polyelectrolytes consisting of linear chains of monomers, with covalent harmonic bonding forces and Coulomb interactions, have been extensively studied with Monte Carlo Methods. Recently, high statistics results have emerged for relatively lo..