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Effective Sampling of Protein Conformation Space: Identification of Independent Folding Units

A simplified off-lattice model of protein molecules has been developed which uses a physico-chemical force field based on experimental data. Employing a search strategy based on Simulated Annealing, we are able to predict a high percentage of correct folds for proteins up to 40 residues in length. We have developed a robust and efficient hybrid Monte Carlo algorithm (Density Guided Importance Sampling) that outperforms other methods designed to overcome sampling inefficiencies encountered at low temperatures (J-Walking and S-Walking). We have applied our model and sampling algorithm to the identification of independent folding units within the primary sequences of two homologous four-helix proteins, Im7 and Im9. The results are in agreement with experimental folding data collected for these two systems, and are consistent with a hierarchical folding mechanism. 1. RAFT: a Simplified Model of Protein Molecules We have developed a simplified off-lattice model for describing proteins called RAFT (Reduced Ramachandran Angle and Folding Force Field for Tertiary Structure Prediction) [1]. Central to the method are the observations by Wodak [2] and later by Levitt [3] that polypeptide conformations may be adequately described using only four or six pairs of φψ angles. Such representations reduce substantially the conformation space of a polypeptide from the enormous number that can populate unrestricted Ramachandran space, and can represent the backbones of known protein structures with overall RMS deviation from the experimental coordinates typically less than 2.0 Å. RAFT uses a set of six discrete φψ angles per residue optimised from the PDB; two cover the α-helical region of Ramachandran space, three cover the β-sheet region and one covers the reverse α-helical region. Each residue is represented as a sphere centred on its backbone atoms and the side chain by between zero (for Gly) and three (for Lys) spheres placed along the Cα-Cβ vector, with the total volume approximating that which the real side chain occupies in water. The side chains are modelled to approximate rotationally averaged real side chains, working on the premise that intimate side chain docking is not an absolute requirement for achieving a native-like fold. The structure of a protein in our model can thus be described by a ‘conformer descriptor’; a sequence of numbers taking values between 1 and 6, each denoting the φψ angle for a particular residue. This takes up far less memory than a series of xyz coordinates. Each sphere has an associated polar (P) or non-polar (NP) potential such that the sum of these for a particular residue corresponds to the free energy of transfer of the residue from water to the protein interior calculated from experimental data [4-6]. The energy of a particular conformation is given by the sum of these sphere-sphere potentials according to an approximate Lennard-Jones potential function, such that NP-NP interactions are favourable, NP-P are unfavourable, and P-P are null. A backbone hydrogen bond potential derived from experimental data [7] is also incorporated to stabilise α and β secondary structures. To locate the global energy minimum conformation we employ a search strategy based on Simulated Annealing [8]. Random changes are made to a given conformer descriptor and the resulting change in energy computed. The acceptance probability is then calculated using the standard Metropolis criterion [8]. When the search is complete we increase the temperature to that at which the lowest energy conformation was found and perform another, more refined, cooling schedule. Performed in this way, the minimisation is both more efficient and reliable. The algorithm predicts a high percentage of correct folds for peptides up to forty residues in length containing both α and β secondary structures, and is similarly effective as a more complex, allatom model at predicting the structures of proposed independent folding units [1,9] (Figure 1). Thus, the geometry and force field employed in RAFT provide a reasonably realistic model with which to explore fundamental questions related to protein folding. Figure 1. LHS: backbone structures of independent folding units predicted using RAFT (magenta) superimposed onto the native crystal structures (red). RHS: comparison of the accuracy of the predictions using RAFT with those using a full-atom description [9]. 2. Density Guided Importance Sampling: a Hybrid Monte Carlo Method Calculating protein folding thermodynamic parameters demands knowledge of the temperature dependent Boltzmann distribution of the conformation space in terms of the molecular partition function. Owing to the size of conformation space, sampling algorithms must be employed to estimate this. The standard Metropolis-Boltzmann importance sampling algorithm [8] is inefficient at sampling conformation space. At critical temperatures the simulations become quasi-ergodic as tunnelling problems are encountered. Moreover, there is no direct way of assessing equilibration. Hybrid Monte Carlo sampling techniques proposed to overcome sampling inefficiencies include methods such as Jump Walking (J-Walking) [10] and Smart Walking (S-Walking) [11]. These conduct walks at higher temperatures in parallel to the walk at the low sampling temperature. At appropriate intervals jumps are attempted from the conformation occupied at the low temperature to that occupied at the high temperature. In general, this overcomes quasi-ergodicity as the transition barriers are reduced, resulting in a larger area of the conformation space being thermally accessible. JWalking alone does not strictly obey detailed balance and thus is not guaranteed to generate the correct low temperature Boltzmann distribution. A further modification to J-Walking, parallel 1alc2132 1bgs1022 1bgs8898 1fkf2738 1lmb1526 1mbc617 1pga4354 1ubq315 2i1b6982 2i1b103112 2mhr6778 2sns2235 4pti22-33 1m bc 617