Folding small proteins via annealing stochastic approximation Monte Carlo

Recently, the stochastic approximation Monte Carlo algorithm has been proposed by Liang et al. (2007) as a general-purpose stochastic optimization and simulation algorithm. An annealing version of this algorithm was developed for real small protein folding problems. The numerical results indicate that it outperforms simulated annealing and conventional Monte Carlo algorithms as a stochastic optimization algorithm. We also propose one method for the use of secondary structures in protein folding. The predicted protein structures are rather close to the true structures.     

Prediction of protein conformation in water and on surfaces by Monte Carlo simulations using united-atom method

The united-atom method has been used to model an avian pancreatic polypeptide (APP) in water and the adsorption process of an albumin subdomain (AS) onto graphite surface to observe the capability of this lumped modelling approach to generate structures observed in protein data bank (PDB) and from atomistic modelling. The subdomain structure of a protein is simplified by the united-atom approximation where the side chains and peptide groups are represented by lumped spheres. The total potential energy of the adsorption process involves the interaction between these lumped spheres by means of virtual bond chain interaction and the interaction of the spheres with the graphite surface by means of Lennard-Jones potential. The protein/polypeptide structure has been perturbed by Monte Carlo with energy minimisation to obtain the global minimum. Results on the APP in water showed a near-to-experimental PDB conformation revealing the two α-helix structures of this small protein molecule with the root mean square deviation among carbon backbone atoms of 5.9 Å. Protein adsorption on biosurfaces has been made by modelling AS, which has 60 amino acids. The surface is graphite, which is characterised by its hydrophobicity. Graphite was chosen because of its widely used applications in certain implants that interact with blood. Our simulation results showed final conformation close to that obtained by atomistic modelling. It also proved that the whole pattern of intramolecular hydrogen bonds was distorted. The model also demonstrated the random conformation of the original α-helix secondary structures of AS consistent with experimental and atomistic results. While atomistic simulation works well for simulating individual small proteins, the united-atom model is more efficient when simulating macromolecular and multiple protein adsorption where time and limiting computer capacity are key factors.     

Application of kinetic Monte Carlo method to the vapour–liquid equilibria of associating fluids and their mixtures

Canonical kinetic Monte Carlo (C-kMC) simulations have been carried out to assess their feasibility and potential for calculating the vapour–liquid equilibria of various pure components with increasingly strong electrostatic interactions (carbon dioxide, methanol, ammonia and water) over a wide range of temperatures and for methanol/water mixtures at 298 K. The simulation results show that C-kMC is successful as a method for studying phase equilibria and thermodynamic properties. For all the examples investigated, the performance of the C-kMC method is at least as good as that of the conventional Monte Carlo (MC) methods and is efficient at low temperature where these fail. It also provides a route that is superior to the Widom method for the calculation of chemical potential. We recommend this method for this purpose and as an alternative to conventional MC for simulations of strongly associating fluids and at low temperatures.     

Monte Carlo Simulation of Nonionic Surfactants at the Oil-Water Interface

Abstract

A Monte Carlo simulation method has been developed for modelling amphiphiles at an oil-water interface. Properties are calculated for the mixture water, benzene and tetraoxyethylene glycol dodecyl ether.     

Basic statistics and variational concepts behind the reverse Monte Carlo technique

We revise the statistical foundations of the reverse Monte Carlo (RMC) technique by constructing the associated functional of a variational principle which incorporates, without any ad hoc assumptions, the inherent errors accompanying the simulation and the experimental data. We propose a Bayesian criteria for acceptance/rejection of configurations, in terms of the variations of the functional. The loss function and variational functional minimization approaches are compared.     

Monte Carlo simulation of intercalated carbon nanotubes

Monte Carlo simulations of the single- and double-walled carbon nanotubes (CNT) intercalated with different metals have been carried out. The interrelation between the length of a CNT, the number and type of metal atoms has also been established. This research is aimed at studying intercalated systems based on CNTs and d-metals such as Fe and Co. Factors influencing the stability of these composites have been determined theoretically by the Monte Carlo method with the Tersoff potential. The modeling of CNTs intercalated with metals by the Monte Carlo method has proved that there is a correlation between the length of a CNT and the number of endo-atoms of specific type. Thus, in the case of a metallic CNT (9,0) with length 17 bands (3.60 nm), in contrast to Co atoms, Fe atoms are extruded out of the CNT if the number of atoms in the CNT is not less than eight. Thus, this paper shows that a CNT of a certain size can be intercalated with no more than eight Fe atoms. The systems investigated are stabilized by coordination of 3d-atoms close to the CNT wall with a radius-vector of (0.18–0.20) nm. Another characteristic feature is that, within the temperature range of (400–700) K, small systems exhibit ground-state stabilization which is not characteristic of the higher ones. The behavior of Fe and Co endo-atoms between the walls of a double-walled carbon nanotube (DW CNT) is explained by a dominating van der Waals interaction between the Co atoms themselves, which is not true for the Fe atoms.     

Reverse Monte Carlo Simulation: A New Technique for the Determination of Disordered Structures

Abstract

We have developed a new technique, based on the standard Monte Carlo simulation method with Markov chain sampling, in which a set of three dimensional particle configurations are generated that are consistent with the experimentally measured structure factor. A(Q), and radial distribution function, g(r), of a liquid or other disordered system. Consistency is determined by a standard χ² test using the experimental errors. No input potential is required, we present initial results for liquid argon. Since the technique can work directly from the structure factor it promises to be useful for modelling the structures of glasses or amorphous materials. It also has other advantages in multicomponent systems and as a tool for experimental data analysis.     

Sequence specific resonance assignment via Multicanonical Monte Carlo search using an ABACUS approach

ABACUS [Grishaev et al. (2005) Proteins 61:36-43] is a novel protocol for automated protein structure determination via NMR. ABACUS starts from molecular fragments defined by unassigned J-coupled spin-systems and involves a Monte Carlo stochastic search in assignment space, probabilistic sequence selection, and assembly of fragments into structures that are used to guide the stochastic search. Here, we report further development of the two main algorithms that increase the flexibility and robustness of the method. Performance of the BACUS [Grishaev and Llinás (2004) J Biomol NMR 28:1-101] algorithm was significantly improved through use of sequential connectivities available from through-bond correlated 3D-NMR experiments, and a new set of likelihood probabilities derived from a database of 56 ultra high resolution X-ray structures. A Multicanonical Monte Carlo procedure, Fragment Monte Carlo (FMC), was developed for sequence-specific assignment of spin-systems. It relies on an enhanced assignment sampling and provides the uncertainty of assignments in a quantitative manner. The efficiency of the protocol was validated on data from four proteins of between 68-116 residues, yielding 100% accuracy in sequence specific assignment of backbone and side chain resonances.     

Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

Phylogenetic tree construction using sequential stochastic approximation Monte Carlo

Monte Carlo methods have received much attention recently in the literature of phylogenetic tree construction. However, they often suffer from two difficulties, the curse of dimensionality and the local-trap problem. The former one is due to that the number of possible phylogenetic trees increases at a super-exponential rate as the number of taxa increases. The latter one is due to that the phylogenetic tree has often a rugged energy landscape. In this paper, we propose a new phylogenetic tree construction method, which attempts to alleviate these two difficulties simultaneously by making use of the sequential structure of phylogenetic trees in conjunction with stochastic approximation Monte Carlo (SAMC) simulations. The use of the sequential structure of the problem provides substantial help to reduce the curse of dimensionality in simulations, and SAMC effectively prevents the system from getting trapped in local energy minima. The new method is compared with a variety of existing Bayesian and non-Bayesian methods on simulated and real datasets. Numerical results are in favor of the new method in terms of quality of the resulting phylogenetic trees.     

Integration of the M6 Cyberknife in the Moderato Monte Carlo platform and prediction of beam parameters using machine learning

PurposeThis work describes the integration of the M6 Cyberknife in the Moderato Monte Carlo platform, and introduces a machine learning method to accelerate the modelling of a linac.MethodsThe MLC-equipped M6 Cyberknife was modelled and integrated in Moderato, our in-house platform offering independent verification of radiotherapy dose distributions. The model was validated by comparing TPS dose distributions with Moderato and by film measurements. Using this model, a machine learning algorithm was trained to find electron beam parameters for other M6 devices, by simulating dose curves with varying spot size and energy. The algorithm was optimized using cross-validation and tested with measurements from other institutions equipped with a M6 Cyberknife.ResultsOptimal agreement in the Monte Carlo model was reached for a monoenergetic electron beam of 6.75 MeV with Gaussian spatial distribution of 2.4 mm FWHM. Clinical plan dose distributions from Moderato agreed within 2% with the TPS, and film measurements confirmed the accuracy of the model. Cross-validation of the prediction algorithm produced mean absolute errors of 0.1 MeV and 0.3 mm for beam energy and spot size respectively. Prediction-based simulated dose curves for other centres agreed within 3% with measurements, except for one device where differences up to 6% were detected.ConclusionsThe M6 Cyberknife was integrated in Moderato and validated through dose re-calculations and film measurements. The prediction algorithm was successfully applied to obtain electron beam parameters for other M6 devices. This method would prove useful to speed up modelling of new machines in Monte Carlo systems.     

Bayesian phylogeny analysis via stochastic approximation Monte Carlo

Monte Carlo methods have received much attention in the recent literature of phylogeny analysis. However, the conventional Markov chain Monte Carlo algorithms, such as the Metropolis–Hastings algorithm, tend to get trapped in a local mode in simulating from the posterior distribution of phylogenetic trees, rendering the inference ineffective. In this paper, we apply an advanced Monte Carlo algorithm, the stochastic approximation Monte Carlo algorithm, to Bayesian phylogeny analysis. Our method is compared with two popular Bayesian phylogeny software, BAMBE and MrBayes, on simulated and real datasets. The numerical results indicate that our method outperforms BAMBE and MrBayes. Among the three methods, SAMC produces the consensus trees which have the highest similarity to the true trees, and the model parameter estimates which have the smallest mean square errors, but costs the least CPU time.     

Monte Carlo simulations in the isothermal-isobaric ensemble: use of a ‘shell’ particle for simulating polyatomic fluids

The recent reformulation of the isothermal-isobaric ensemble requires the use of a ‘shell’ particle to define uniquely the volume of the system, thereby avoiding the redundant counting of configurations. A previous modification of the Monte Carlo method, in which trial moves are generated and accepted consistent with the correct constant pressure partition function, is extended here to the case of polyatomic fluids. With a ‘shell’ molecule, either the centre of mass of the molecule or the location of any one of the atoms within the molecule can be chosen to define the system volume. Ensemble averages obtained with the use of the shell molecule differ from ensemble averages determined with the old (i.e. no shell particle) Monte Carlo algorithm, specifically for small system sizes, although both sets of averages become equal, as they must, in the thermodynamic limit. Monte Carlo simulations in the constant pressure ensemble for various Lennard-Jones polyatomic fluids, both for pure component and binary mixtures, demonstrate these differences for small systems. For mixtures, Monte Carlo simulations may include attempted identity swaps for the shell molecule, as the choice of which component serves as the shell molecule is arbitrary when periodic boundary conditions are applied.     

A new kinetic Monte Carlo scheme with Gibbs ensemble to determine vapour–liquid equilibria

We present a new kinetic Monte Carlo scheme, as an alternative to the Gibbs ensemble Monte Carlo (GEMC) method, to determine vapour–liquid equilibria using a canonical ensemble in a system composed of two boxes. To illustrate the method, we have tested it with two systems: (1) argon over a range of temperatures from below the triple point to close to the critical point; (2) methane and ethane mixtures of various compositions at 180 K. The advantage of the new scheme is that chemical potentials of all components are accurately determined in both boxes. In particular, the chemical potential in the liquid box is determined much more accurately than with the Widom method employed in conventional GEMC simulations.     

Monte Carlo Simulation of Tetrahedral Chains II: Properties of “First Self-avoiding Walks” and their Usability as Starting Configurations for Dynamic Relaxation Mechanisms

Very long model chains may be produced in a highly efficient manner using dynamic Monte Carlo methods. As any dynamic Monte Carlo procedure transforms one chain into another one, some starting configuration is necessary. This might be an unbiased self-avoiding walk (SAW) obtained by any static method, or an arbitrary configuration, e.g. a rodlike chain, equilibrated by a sufficiently large number of relaxations, the corresponding chains not being used for data sampling. An alternative method is to start with a non reversal random walk (NRRW) and to apply a dynamic Monte Carlo procedure under the constraint that the new chain must have a smaller (or at least an equal) number of double occupancies than the old one. The properties of those chains that are free of overlaps for the first time (FSAWs) are strongly dependent on the relaxation mechanism chosen. Whereas FSAWs obtained by local motions are very similar to the (initial) NRRWs on a macroscopic scale, pivot algorithms and reptation yield configurations with properties comparable to unbiased self-avoiding chains. When reptation is used and the relaxation is continued until each bond of the initial NRRW is replaced by a new bond (if the chain is self-avoiding earlier) no further equilibration is necessary prior to data sampling.     

Clinical implementation of a Monte Carlo based treatment plan QA platform for validation of Cyberknife and Tomotherapy treatments

PurposeThe main focus of the current paper is the clinical implementation of a Monte Carlo based platform for treatment plan validation for Tomotherapy and Cyberknife, without adding additional tasks to the dosimetry department.MethodsThe Monte Carlo platform consists of C++ classes for the actual functionality and a web based GUI that allows accessing the system using a web browser. Calculations are based on BEAMnrc/DOSXYZnrc and/or GATE and are performed automatically after exporting the dicom data from the treatment planning system. For Cyberknife treatments of moving targets, the log files saved during the treatment (position of robot, internal fiducials and external markers) can be used in combination with the 4D planning CT to reconstruct the actually delivered dose. The Monte Carlo platform is also used for calculation on MRI images, using pseudo-CT conversion.ResultsFor Tomotherapy treatments we obtain an excellent agreement (within 2%) for almost all cases. However, we have been able to detect a problem regarding the CT Hounsfield units definition of the Toshiba Large Bore CT when using a large reconstruction diameter. For Cyberknife treatments we obtain an excellent agreement with the Monte Carlo algorithm of the treatment planning system. For some extreme cases, when treating small lung lesions in low density lung tissue, small differences are obtained due to the different cut-off energy of the secondary electrons.ConclusionsA Monte Carlo based treatment plan validation tool has successfully been implemented in clinical routine and is used to systematically validate all Cyberknife and Tomotherapy plans.     

New version of Monte Carlo expanded ensemble method for precise calculations of free energy difference

A new version of Monte Carlo (MC) expanded ensemble (EE) method is proposed for the calculations of free energy difference (FED) between two different systems with close values of the free energy. In order to check the method the FED between simple model systems (fluid of hard spheres and freely jointed polymer chain of hard spheres) was calculated. The free energy of the mentioned above systems was also calculated by a standard MC EE method in order to compare the results of two simulations. It was shown that the accuracy of a new algorithm is the same as of a standard one. At the same time new version of EE allows us to obtain FED between two systems having quite different structures, but similar free energies, during one simulation run.     

The relationship between cell and population growth

A steady-state model of cell volume frequency distribution using the method of Williams (1971) is derived. Results are compared to a Monte Carlo simulation of cell growth and division. It is suggested that the Monte Carlo method might be of value for investigating cell and population properties for which analytic methods are not currently available.     

Two-step validation of a Monte Carlo dosimetry framework for general radiology

The Monte Carlo technique is considered gold standard when it comes to patient-specific dosimetry. Any newly developed Monte Carlo simulation framework, however, has to be carefully calibrated and validated prior to its use. For many researchers this is a tedious work. We propose a two-step validation procedure for our newly built Monte Carlo framework and provide all input data to make it feasible for future related application by the wider community. The validation was at first performed by benchmarking against simulation data available in literature. The American Association of Physicists in Medicine (AAPM) report of task group 195 (case 2) was considered most appropriate for our application. Secondly, the framework was calibrated and validated against experimental measurements for trunk X-ray imaging protocols using a water phantom. The dose results obtained from all simulations and measurements were compared. Our Monte Carlo framework proved to agree with literature data, by showing a maximal difference below 4% to the AAPM report. The mean difference with the water phantom measurements was around 7%. The statistical uncertainty for clinical applications of the dosimetry model is expected to be within 10%. This makes it reliable for clinical dose calculations in general radiology. Input data and the described procedure allow for the validation of other Monte Carlo frameworks.