Efficient Monte-Carlo Simulation of Proteins
Monte Carlo simulation (MCS) is a common methodology to compute pathways and thermodynamic properties of proteins. A simulation run is a series of random steps in conformation space, each perturbing some degrees of freedom of the molecule. A step is accepted with a probability that depends on the change in value of an energy function.
Typical energy functions sum many terms. The most costly ones to compute are contributed by atom pairs closer than some cutoff distance. This software uses a new method that speeds up MCS by efficiently computing the energy at each step. The method exploits the facts that proteins are long kinematic chains and that few degrees of freedom are changed at each step. A novel data structure, called the ChainTree, captures both the kinematics and the shape of a protein at successive levels of detail. It is used to find all atom pairs contributing to the energy. It also makes it possible to identify partial energy sums left unchanged by a perturbation, thus allowing the energy value to be incrementally updated.
The method used in this software is based on an algorithm for efficiently computing self-collisions in a deformable kinematic chain . This algorithm was extended and adapted for the efficient computation of the internal energy of a protein as it undergoes deformations during an MCS .
MCS of proteins can be performed in many different ways. The methodology that is used is customized to match the goals of the specific simulation at hand. Some of the aspects that are often changed are:
Move set: The changes applied to the protein structure at each step of the simulation.
Acceptance criterion: The rule according to which new conformations are accepted or rejected.
Energy Function: The score that is given to each conformation. This is often the internal energy of the conformation, but it is not required to have a physical meaning.
While the software we provide implements only one such MCS methodology, the underlying data structures can be used efficiently with a large variety of move sets, acceptance criteria and energy functions. The only restrictions are:
Only backbone and side-chain dihedral angles can be changed, leaving the bond lengths and angles fixed throughout the simulation.
The efficiency of the software drops as the number of simultaneous backbone angles that are changed at each simulation step is increased.
The energy computation is most efficient when energy terms depend on the distances between pairs of atoms and a cutoff distance is used to limit the number of pairs that contribute to the total energy. Single parameter terms such as a dihedral angle potential can also be used efficiently.
Our implementation can be used to run an MCS using the choices we have made, which are described below. It can also be used as an example for anyone that wants to use his/her own move set / acceptance criterion. The energy function that is used can also be replaced, although that may turn out to be more difficult.
This software implements the following MCS methodology:
Move Set: There are two kinds of moves:
Backbone move: Three backbone angles are changed simultaneously. Each angle is changed by a random amount drawn from a zero-mean Gaussian distribution.
Rotamer move: A number of side-chain rotamers are changed simultaneously to different rotamer values drawn from a fixed library of rotamers (The backbone independent rotamer library of Dunbrack and Cohen  from August '99).
Each step of the simulation is comprised of one attempted backbone move followed by a number of attempted rotamer moves.
Acceptance criterion: The Metropolis criterion:
Energy function: EEF1 of Lazaridis and Karplus .
The different parameters of the simulation are determined by the following command line options:
|-I file-name||The input protein structure used
as the initial conformation of the simulation. The structure is described by
the backbone dihedral angles and the index of the rotamer of each amino
acid. For example:
The first entry is the three letter amino acid code, then come the Φ and Ψ backbone angles and finally the index of the rotamer to be used. The file must have a ".angs" extension.
|-T temperature||The temperature of the simulation. (Default is 298.15 K)|
|-S step-number||The number of steps in the simulation (Default is 100,000)|
|-N save-interval||The number of steps between save operations. Each save operation the current conformation is saved in a ".angs" file as well as in a PDB style ".pdb" file. Also, the energy is saved in the ".out" file. (Default is 1000)|
|-K backbone-angs||The number of backbone angles that are changed simultaneously during a backbone move. (Default is 1)|
|-A angle-stdev||The standard deviation (in degrees) of the zero-mean Gaussian distribution from which the change to each backbone angle is drawn. (Default is 10°)|
|-F rotamer-moves||The number of rotamer moves attempted during each simulation step. (Default is 5)|
|-R changed-rotamers||The number of rotamers that are changed each rotamer move. (Default is 1)|
|-O output-files||The name of the output files without any extension. The different extensions are added by the program. At each save interval the current conformation and its energy value are saved. The default is to save only intermediate energy values and the final conformations.|
|-U random-seed||The random seed that is used by the programs random number generator. (Default is 0)|
|-C reference-structure||A protein structure specified in PDB style file with a ".pdb" extension that is used as the reference structure to which all saved structures are compared using the cRMSD similarity measure. The similarity measure of the current conformation is saved together with its energy value. The default is to use the initial conformation as the reference structure.|
|-H home-dir||The directory where the executable together with the files "rotamers.h", "rotamer_coords" and "exclusions" reside.|
|-X output-dir||The directory where intermediate results will be saved.|
The software source-code can be downloaded from here.
To build the executable type make -k mcs in the directory where you installed the source files.
Lotan, I., Schwarzer, F., Halperin, D., Latombe, J.C.: Efficient maintenance and self-collision testing for kinematic chains. In: Symposium on Computational Geometry (2002) 43–52. [PS], [PDF]
Lotan, I., Schwarzer, F., Latombe, J.C.: Efficient Energy Computation for Monte-Carlo Simulation of Proteins. In: Workshop on Algorithms in Bioinformatics (2003), To appear. [PS], [PDF]
Dunbrack Jr., R. L. and Cohen, F. E. Bayesian Statistical Analysis of Protein Sidechain Rotamer Preferences ." Protein Science 6 (1997) 1661-1681.
Lazaridis, T., Karplus, M.: Effective energy function for proteins in solution. Proteins 35 (1999) 133–152.
Please send questions or comments to Itay Lotan.