Folding and Design in Coarse-Grained Protein Models

Recent advances in coarse-grained lattice and off-lattice protein models are reviewed. The sequence dependence of thermodynamical folding properties are investigated and evidence for non-randomness of the binary sequences of good folders are discussed. Similar patterns for non-randomness are found for real proteins. Dynamical parameter MC methods, such as the tempering and multisequence algorithms, are essential in order to obtain these results. Also, a new MC method for design, the inverse of folding, is presented. Here, one maximizes conditional probabilities rather than minimizing energies. By construction, this method ensures that the designed sequences represent good folders thermodynamically.Comment: LATTICE 99 (Spin Models), 3 pages, 1 figure, espcrc2.st

Hybrid Monte Carlo simulation of polymer chains

We develop the hybrid Monte Carlo method for simulations of single off-lattice polymer chains. We discuss implementation and choice of simulation parameters in some detail. The performance of the algorithm is tested on models for homopolymers with short- or long-range self-repulsion, using chains with $16\le N\le 512$ monomers. Without excessive fine tuning, we find that the computational cost grows as $N^{2+z^\prime}$ with $0.64<z^\prime<0.84$. In addition, we report results for the scaling of the end-to-end distance, $r_{1N}\sim N^\nu(\ln N)^{-\alpha}$.Comment: 13 pages, single postscript file, uuencoded form, Lund Preprint LU-TP 93-2

Thermodynamics of amyloid formation and the role of intersheet interactions

The self-assembly of proteins into $\beta$-sheet-rich amyloid fibrils has been observed to occur with sigmoidal kinetics, indicating that the system initially is trapped in a metastable state. Here, we use a minimal lattice-based model to explore the thermodynamic forces driving amyloid formation in a finite canonical ($NVT$) system. By means of generalized-ensemble Monte Carlo techniques and a semi-analytical method, the thermodynamic properties of this model are investigated for different sets of intersheet interaction parameters. When the interactions support lateral growth into multi-layered fibrillar structures, an evaporation/condensation transition is observed, between a supersaturated solution state and a thermodynamically distinct state where small and large fibril-like species exist in equilibrium. Intermediate-size aggregates are statistically suppressed. These properties do not hold if aggregate growth is one-dimensional.Comment: 22 page

Design of Sequences with Good Folding Properties in Coarse-Grained Protein Models

Background: Designing amino acid sequences that are stable in a given target structure amounts to maximizing a conditional probability. A straightforward approach to accomplish this is a nested Monte Carlo where the conformation space is explored over and over again for different fixed sequences, which requires excessive computational demand. Several approximate attempts to remedy this situation, based on energy minimization for fixed structure or high-$T$ expansions, have been proposed. These methods are fast but often not accurate since folding occurs at low $T$. Results: We develop a multisequence Monte Carlo procedure, where both sequence and conformation space are simultaneously probed with efficient prescriptions for pruning sequence space. The method is explored on hydrophobic/polar models. We first discuss short lattice chains, in order to compare with exact data and with other methods. The method is then successfully applied to lattice chains with up to 50 monomers, and to off-lattice 20-mers. Conclusions: The multisequence Monte Carlo method offers a new approach to sequence design in coarse-grained models. It is much more efficient than previous Monte Carlo methods, and is, as it stands, applicable to a fairly wide range of two-letter models.Comment: 23 pages, 7 figure

Local Interactions and Protein Folding: A Model Study on the Square and Triangular Lattices

We study a simple heteropolymer model containing sequence-independent local interactions on both square and triangular lattices. Sticking to a two-letter code, we investigate the model for varying strength $\kappa$ of the local interactions; $\kappa=0$ corresponds to the well-known HP model [K.F. Lau and K.A. Dill, Macromolecules 22, 3986 (1989)]. By exhaustive enumerations for short chains, we obtain all structures which act as a unique and pronounced energy minimum for at least one sequence. We find that the number of such designable structures depends strongly on $\kappa$. Also, we find that the number of designable structures can differ widely for the two lattices at a given $\kappa$. This is the case, for example, at $\kappa=0$, which implies that the HP model exhibits different behavior on the two lattices. Our findings clearly show that sequence-independent local properties of the chains can play an important role in the formation of unique minimum energy structures.Comment: 10 pages LaTeX, 3 Postscript figures. Figure and references adde

Enumerating Designing Sequences in the HP Model

The hydrophobic/polar HP model on the square lattice has been widely used to investigate basics of protein folding. In the cases where all designing sequences (sequences with unique ground states) were enumerated without restrictions on the number of contacts, the upper limit on the chain length N has been 18-20 because of the rapid exponential growth of the numbers of conformations and sequences. We show how a few optimizations push this limit by about 5 units. Based on these calculations, we study the statistical distribution of hydrophobicity along designing sequences. We find that the average number of hydrophobic and polar clumps along the chains is larger for designing sequences than for random ones, which is in agreement with earlier findings for N up to 18 and with results for real enzymes. We also show that this deviation from randomness disappears if the calculations are restricted to maximally compact structures.Comment: 18 pages, 4 figure

Binary Assignments of Amino Acids from Pattern Conservation

We develop a simple optimization procedure for assigning binary values to the amino acids. The binary values are determined by a maximization of the degree of pattern conservation in groups of closely related protein sequences. The maximization is carried out at fixed composition. For compositions approximately corresponding to an equipartition of the residues, the optimal encoding is found to be strongly correlated with hydrophobicity. The stability of the procedure is demonstrated. Our calculations are based upon sequences in the SWISS-PROT database.Comment: 9 pages, 4 Postscript figures. References and figure adde

Coupled folding-binding versus docking: A lattice model study

Using a simple hydrophobic/polar protein model, we perform a Monte Carlo study of the thermodynamics and kinetics of binding to a target structure for two closely related sequences, one of which has a unique folded state while the other is unstructured. We obtain significant differences in their binding behavior. The stable sequence has rigid docking as its preferred binding mode, while the unstructured chain tends to first attach to the target and then fold. The free-energy profiles associated with these two binding modes are compared.Comment: 17 pages, 7 figures (to appear in J. Chem. Phys.

Folding thermodynamics of three beta-sheet peptides: A model study

We study the folding thermodynamics of a beta-hairpin and two three-stranded beta-sheet peptides using a simplified sequence-based all-atom model, in which folding is driven mainly by backbone hydrogen bonding and effective hydrophobic attraction. The native populations obtained for these three sequences are in good agreement with experimental data. We also show that the apparent native population depends on which observable is studied; the hydrophobicity energy and the number of native hydrogen bonds give different results. The magnitude of this dependence matches well with the results obtained in two different experiments on the beta-hairpin.Comment: 17 pages, 7 figures, to appear in Protein

Studies of an Off-Lattice Model for Protein Folding: Sequence Dependence and Improved Sampling at Finite Temperature

We study the thermodynamic behavior of a simple off-lattice model for protein folding. The model is two-dimensional and has two different ``amino acids''. Using numerical simulations of all chains containing eight or ten monomers, we examine the sequence dependence at a fixed temperature. It is shown that only a few of the chains exist in unique folded state at this temperature, and the energy level spectra of chains with different types of behavior are compared. Furthermore, we use this model as a testbed for two improved Monte Carlo algorithms. Both algorithms are based on letting some parameter of the model become a dynamical variable; one of the algorithms uses a fluctuating temperature and the other a fluctuating monomer sequence. We find that by these algorithms one gains large factors in efficiency in comparison with conventional methods.Comment: 17 pages, 9 Postscript figures. Combined with chem-ph/950500