We study the thermodynamic behavior of a model protein with 54 amino acids that forms a three-helix bundle in its native state. The model contains three types of amino acids and five to six atoms per amino acid and has the Ramachandran torsional angles i, i as its degrees of freedom. The force field is based on hydrogen bonds and effective hydrophobicity forces. For a suitable choice of the relative strength of these interactions, we find that the three-helixbundle protein undergoes an abrupt folding transition from an expanded state to the native state. Also shown is that the corresponding one-and two-helix segments are less stable than the three-helix sequence. It is not yet possible to simulate the formation of proteins' native structures on the computer in a controlled way. This goal has been achieved in the context of simple lattice and off-lattice models, where typically each amino acid is represented by a single interaction site corresponding to the C ␣ atom, and such studies have provided valuable insights into the physical principles of protein folding (1-5) and the statistical properties of functional protein sequences (6, 7). However, these models have their obvious limitations. Therefore, the search for computationally feasible models with a more realistic chain geometry remains a highly relevant task.In this paper, we discuss a model based on the well-known fact that the main degrees of freedom of the protein backbone are the Ramachandran torsional angles i , i (8). Each amino acid is represented by five or six atoms, which makes this model computationally slightly more demanding than C ␣ models. On the other hand, it also makes interactions such as hydrogen bonds easier to define. The formation of native structure is, in this model, driven by hydrogen-bond formation and effective hydrophobicity forces; hydrophobicity is widely held as the most important stability factor in proteins (9, 10), and hydrogen bonds are essential to properly model the formation of secondary structure.In this model, we study in particular a three-helix-bundle protein with 54 amino acids, which represents a truncated and simplified version of the four-helix-bundle protein de novo designed by Regan and DeGrado (11). This example was chosen partly because there have been earlier studies of similar-sized helical proteins using models at comparable levels of resolution (12-18). The behavior of small fast-folding proteins is a current topic in both theoretical and experimental research, and a three-helix-bundle protein that has been extensively studied both experimentally (19,20) and theoretically (14,17,21,22) is fragment B of staphylococcal protein A.In addition to the three-helix protein, to study size dependence, we also look at the behavior of the corresponding oneand two-helix segments. By using the method of simulated tempering (23-25), a careful study of the thermodynamic properties of these different chains is performed.Not unexpectedly, it turns out that the behavior of the model strongly depends on the relative streng...

Abstract:Random Boolean networks, the Kauffman model, are revisited by means of a novel decimation algorithm, which reduces the networks to their dynamical cores. The average size of the removed part, the stable core, grows approximately linearly with N, the number of nodes in the original networks. We show that this can be understood as the percolation of the stability signal in the network. The stability of the dynamical core is investigated and it is shown that this core lacks the well known stability observed in full Kauffman networks. We conclude that, somewhat counter-intuitive, the remarkable stability of Kauffman networks is generated by the dynamics of the stable core. The decimation method is also used to simulate large critical Kauffman networks. For networks up to N = 32 we perform full enumeration studies. Strong evidence is provided for that the number of limit cycles grows linearly with N. This result is in sharp contrast to the often cited √ N behavior.

Abstract:We develop a new elementary move for simulations of polymer chains in torsion angle space. The method is flexible and easy to implement. Tentative updates are drawn from a (conformation-dependent) Gaussian distribution that favors approximately local deformations of the chain. The degree of bias is controlled by a parameter b. The method is tested on a reduced model protein with 54 amino acids and the Ramachandran torsion angles as its only degrees of freedom, for different b. Without excessive fine tuning, we find that the effective step size can be increased by a factor of three compared to the unbiased b = 0 case. The method may be useful for kinetic studies, too. *

An atomic protein model with a minimalistic potential is developed and then tested on an alpha-helix and a beta-hairpin, using exactly the same parameters for both peptides. We find that melting curves for these sequences to a good approximation can be described by a simple two-state model, with parameters that are in reasonable quantitative agreement with experimental data. Despite the apparent two-state character of the melting curves, the energy distributions are found to lack a clear bimodal shape, which is discussed in some detail. We also perform a Monte Carlo-based kinetic study and find, in accord with experimental data, that the alpha-helix forms faster than the beta-hairpin.

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.