Clustering Algorithms to Analyze Molecular Dynamics Simulation Trajectories for Complex Chemical and Biological Systems?

Molecular dynamics (MD) simulation has become a powerful tool to investigate the structurefunction relationship of proteins and other biological macromolecules at atomic resolution and biologically relevant timescales. MD simulations often produce massive datasets containing millions of snapshots describing proteins in motion. Therefore, clustering algorithms have been in high demand to be developed and applied to classify these MD snapshots and gain biological insights. There mainly exist two categories of clustering algorithms that aim to group protein conformations into clusters based on the similarity of their shape (geometric clustering) and kinetics (kinetic clustering). In this paper, we review a series of frequently used clustering algorithms applied in MD simulations, including divisive algorithms, agglomerative algorithms (single-linkage, complete-linkage, average-linkage, centroid-linkage and ward-linkage), center-based algorithms (K-Means, K-Medoids, K-Centers, and APM), density-based algorithms (neighbor-based, DBSCAN, density-peaks, and Robust-DB), and spectral-based algorithms (PCCA and PCCA+). In particular, differences between geometric and kinetic clustering metrics will be discussed along with the performances of different clustering algorithms. We note that there does not exist a one-size-fits-all algorithm in the classification of MD datasets. For a specific application, the right choice of clustering algorithm should be based on the purpose of clustering, and the intrinsic properties of the MD conformational ensembles. Therefore, a main focus of our review is to describe the merits and limitations of each clustering algorithm. We expect that this review would be helpful to guide researchers to choose appropriate clustering algorithms for their own MD datasets.     

Evaluation of pattern recognition methods by criteria based on information theory and euclidean geometry

Euclidean geometry and information and fuzzy-set theory are used to develop general criteria for the evaluation of clustering methods. A separation function, describing the geometric clustering in a feature space for a given separation state, is introduced. Suitable clustering algorithms for given data can be selected by using the measure derived. The criteria developed are used in studies of the homogeneity of solids.     

Computer method for the selection of a suitable stationary phase in gas chromatography

Summary A selection of stationary phases is carried out for a gas chromatographic separation of certain hydrocarbon mixtures with help of a computer. For this purpose a suitable catalogue including the relative retention data of substances investigated on a large number of stationary phases and an algorithm of sequence of operations are proposed. The program written in ALGOL algorithmic language points out three stationary phases being the best for the separation of a certain n-component mixture by the criterion R_{r, k}. Besides the three suitable phases the conditions for a given separation are also pointed out.     

Optimal separation times for electrical field flow fractionation with Couette flows

The prediction of optimal times of separation as a function of the applied electrical field and cation valence have been studied for the case of field flow fractionation [Martin M., Giddings J. C., J. Phys. Chem. 1981, 85, 727] with charged solutes. These predictions can be very useful to a priori design or identify optimal operating conditions for a Couette-based device for field flow fractionation when the orthogonal field is an electrical field. Mathematically friendly relationships are obtained by applying the method of spatial averaging to the solute species continuity equation; this is accomplished after the role of the capillary geometrical dimensions on the applied electrical field equations has been assessed [Oyanader M. A., Arce P., Electrophoresis 2005; 26, 2857]. Moreover, explicit analytical expressions are derived for the effective parameters, i.e. diffusivity and convective velocity as functions of the applied (orthogonal) electrical field. These effective transport parameters are used to study the effect of the cation valence of the solutes and of the magnitude of the applied orthogonal electrical field on the values of the optimal time of separation. These parameters play a significant role in controlling the optimal separation time, leading to a family of minimum values, for particular magnitudes of the applied orthogonal electrical field.     

Computer-assisted method development in liquid chromatography-mass spectrometry: new proposals

A new approach is proposed for computer-assisted method development in LC-MS. The procedure consists of three stages. Firstly, an accurate retention model is developed for the peaks in the mixture to be separated by use of an iterative approach with isocratic priming data, which is calibrated and validated by means of a few gradient runs. Secondly, a specially developed LC-MS objective function, based on selectivity targets (the selectivity matrix), is calculated and used to evaluate the simulated chromatograms and drive the optimization process. Thirdly, the retention model and the selectivity matrix objective function are used with an evolutionary algorithm in which the concepts of constrained Pareto optimality are applied, to carry out the unattended optimization process. The system was applied to real data for a complex separation and compared with the results provided by a commercial tool for computer-assisted method development.     

Physicochemical characterization of dilute n-alcohol/biodiesel mixtures by inverse gas chromatography

Inverse gas chromatography (IGC) has been used to determine the physicochemical parameters that characterize solution thermodynamic interactions in biodiesel-n-alcohol solute systems. Such data is of value to chemical engineers and separation scientists in optimizing separation processes to separate alcoholic solutes at low concentrations in soybean oil methyl ester mixtures (biodiesel). The derived activity and Henry's Law coefficient data can be used to rationalize the interaction of four members of an n-alcoholic homologous series and the soya-based methyl ester solvent in terms of such esters as "green" renewable solvents. Sorption isotherm data confirm linear behavior in most cases between the solute (alcohol) vapor state concentrations and their uptake into the biodiesel phase. Overall, the experimentally determined activity coefficients agree well with those predicted by solution thermodynamic theories as well as correlative chemical engineering equations.     

Separation of ions by alteration of the concentration of the ligand in a discontinuous countercurrent extraction system

A new technique in Craig-type liquid-liquid, discontinuous, countercurrent solvent extraction was developed. In this, the ligand concentration in the extraction train was altered at several points, causing non-equivalent changes in the extractability of the ions being separated. This allowed quantitative separation of a mixture with a minimum number of transfers, thereby keeping the solvent volume to a minimum. The technique was applied to achieve total or partial separation of a number of elements, using methyl isobutyl ketone as the solvent and hydrochloric acid as the ligand. A computer programme was developed for calculating the distribution of a solute in an extraction train operating under these conditions.     

Reconciliation of Solute Concentration Data with Water Contents and Densities of Multi-Component Electrolyte Solutions

Density and chemical masses are two of the most important parameters tracked in chemical plant flowsheets. Unfortunately, chemical plant laboratories commonly avoid density and solvent concentration measurements. Without these data, it is difficult to reconcile solute concentrations reported by the laboratories with the total mass and volume tracked in flowsheets. In this paper, the Laliberté-Cooper density model is used in conjunction with a numerical algorithm to simultaneously estimate both density and water content from measured solute concentrations for aqueous electrolyte solutions. The algorithm numerically optimizes the water content until the sum of the water and solute concentrations (in mass per volume units) equals the density predicted by the Laliberté-Cooper model for that composition. The algorithm was tested against an experimental dataset of simulated nuclear waste supernatant solutions containing mixtures of ten different electrolytes with total ionic strengths up to 8 mol⋅L⁻¹. The algorithm was able to predict the measured densities with an R² of 0.9912 and an average relative percent error of just 0.05%. The model error was not correlated to the estimated water content or any of the electrolyte concentrations. Thus, the algorithm can be successfully used to simultaneously predict density and water content of aqueous electrolyte solutions containing many electrolytes at high concentrations from analytical data reported in moles or mass of solute per volume.     

K-means Find Density Peaks in Molecular Conformation Clustering

Performing cluster analysis on molecular conformation is an important way to find the representative conformation in the molecular dynamics trajectories. Usually, it is a critical step for interpreting complex conformational changes or interaction mechanisms. As one of the density-based clustering algorithms, find density peaks (FDP) is an accurate and reasonable candidate for the molecular conformation clustering. However, facing the rapidly increasing simulation length due to the increase in computing power, the low computing efficiency of FDP limits its application potential. Here we propose a marginal extension to FDP named K-means find density peaks (KFDP) to solve the mass source consuming problem. In KFDP, the points are initially clustered by a high efficiency clustering algorithm, such as K-means. Cluster centers are defined as typical points with a weight which represents the cluster size. Then, the weighted typical points are clustered again by FDP, and then are refined as core, boundary, and redefined halo points. In this way, KFDP has comparable accuracy as FDP but its computational complexity is reduced from O\begin{document}$(n^2)$\end{document} to O\begin{document}$(n)$\end{document}. We apply and test our KFDP method to the trajectory data of multiple small proteins in terms of torsion angle, secondary structure or contact map. The comparing results with K-means and density-based spatial clustering of applications with noise show the validation of the proposed KFDP.     

Further developments in the modelling of a sequential chromatographic refiner unit

Summary A classical plate-type model developed to simulate the separation of binary mixtures by sequential semicontinuous chromatography is modified by the inclusion of a separate expression to account for the effects of solute concentration on the feed plate. The improved model is tested on three systems of different separation difficulty, for which experimental data are available. Comparison of experimental and simulated separations is achieved by consideration of the solute concentration profiles. In determining the success of the simulation, particular note is made of profile shape, feed point location and concentration levels. Best agreement between experimental and simulated profiles is shown to exist for high separation factor systems at low flowrates and low temperatures. Relatively poor agreement for difficult systems at higher flowrates is attributed to phase changes, column temperature effects and the influence of the anti-Langmuir shape of the absorption isotherm.     

Adsorption model for retention in normal-phase liquid chromatography with ternary mobile phases

The review gives a link between the theory of adsorption from multicomponent solutions and liquid chromatography. The article surveys the methods developed to describe the retention in normal-phase chromatography with ternary mobile phases with emphasis on the results of the authors. In the model used the driving force for the separation is the difference in adsorption of a solute and all solvents onto the solid surface. The general equation generates a series of simple linear relationships to predict the retention factor in ternary mobile phase for which certain parameters remain fixed. Theoretical concepts are tested by comparison with experimental data. The correlations between parameters characterizing retention in ternary, binary and pure solvents are discussed.     

Metrics of separation performance in chromatography. Part 1. Definitions and application to static analyses

Earlier introduced metrics of separation performance are described in a systematic way. After providing the definitions of the metrics suitable for a broad variety of applications, the study focuses on static analyses (isothermal GC, isocratic LC, etc.) and their general separation performance. Statistically expected number of resolved (adequately separated) single-component peaks is treated as the ultimate metric of general separation performance of chromatographic analysis. That number depends on the peak capacity of the analysis and the number of components in a test mixture. The peak capacity, in turn, depends on the separation capacity of a column and the lowest separation required by the data-analysis system for resolving poorly separated peaks. The separation capacity is a special case of a broader metric of the separation measure which is a function of column efficiency, solute separability, and the level of the solute interaction with a column stationary phase. The formulae for theoretical prediction of all these metrics for arbitrary pairs of peaks in static analyses are derived. To provide a better insight into the basic metrics of the separation performance, additional metrics such as the solute discrimination (relative difference in solute velocities), utilization of separability (solute discrimination per unit of their separability), specific separation (the separation per unit of separability), and others are defined and found for static analyses.     

Study of traffic-emitted lead pollution of soil and plants using different fuzzy clustering algorithms

We discuss the clustering of 234 environmental samples resulting from an extensive monitoring program concerning soil lead content, plant lead content, traffic density, and distance from the road at different sampling locations in former East Germany. Considering the structure of data and the unsatisfactory results obtained applying classical clustering and principal component analysis, it appeared evident that fuzzy clustering could be one of the best solutions. In the following order we used different fuzzy clustering algorithms, namely, the fuzzy c-means (FCM) algorithm, the Gustafson–Kessel (GK) algorithm, which may detect clusters of ellipsoidal shapes in data by introducing an adaptive distance norm for each cluster, and the fuzzy c-varieties (FCV) algorithm, which was developed for recognition of r-dimensional linear varieties in high-dimensional data (lines, planes or hyperplanes). Fuzzy clustering with convex combination of point prototypes and different multidimensional linear prototypes is also discussed and applied for the first time in analytical chemistry (environmetrics). The results obtained in this study show the advantages of the FCV and GK algorithms over the FCM algorithm. The performance of each algorithm is illustrated by graphs and evaluated by the values of some conventional cluster validity indices. The values of the validity indices are in very good agreement with the quality of the clustering results. Figure Projection of all samples on the plane defined by the membership degrees to cluster A₂, and A₄ obtained using Fuzzy c-varieties (FCV) algorithm (expression of objective function and distance enclosed)     

Computer-aided determination of relative stereochemistry and 3D models of complex organic molecules from 2D NMR spectra

A method for elucidation of the relative stereoconfiguration of natural product molecular structures and their 3D models based on NOE data and the application of a genetic algorithm is described. The method is applicable mainly for rigid polycyclic structures commonly encountered in natural products. It is demonstrated that the technique of simulated annealing cannot be easily used when dealing with low-weight fused ring molecules but the application of a genetic algorithm is proven successful. Examples of a typical genetic algorithm workflow and the optimization of the algorithmic parameters are discussed. The efficiency of the approach developed here is demonstrated on the complex natural products of both Taxol^® (C₄₇H₅₁NO₁₄) and brevetoxin B (C₅₀H₇₀O₁₄).     

Role of geometrical dimensions in electrophoresis applications with orthogonal fields

The role of geometrical dimensions in electrophoresis applications with axial and orthogonal (secondary) electric fields is investigated using a rectangular capillary channel. In particular, the role of the applied orthogonal electrical field in controlling key parameters involved in the effective diffusivity and effective (axial) velocity of the solute is identified. Such mathematically friendly relationships are obtained by applying the method of spatial averaging to the solute species continuity equation; this is accomplished after the role of the capillary geometrical dimensions on the applied electrical field equations has been studied. Moreover, explicit analytical expressions are derived for the effective parameters, i.e., diffusivity and convective velocity as functions of the applied (orthogonal) electric field. Previous attempts (see Sauer et al., 1995) have only led to equations for these parameters that require numerical solution and, therefore, limited the use of such results to practical applications. These may include, for example, the design of separation processes as well as environmental applications such as soil reclamation and wastewater treatment. An illustration of how a secondary electrical field can aid in reducing the optimal separation time is included.     

NMR determination of smectic ordering of probe molecules

The NMR spectra of the three solutes ortho-, meta-, and para-dichlorobenzene in the nematic and smectic A phases of the liquid crystals 8CB and 8OCB are analyzed to yield two orientational order parameters for each solute. Extrapolation of the asymmetry in the energy parameters that describe the orientational ordering in the nematic phase are used to provide estimates of the strength of the nematic potential in the smectic A phase. The experimentally determined asymmetry of the orientational order parameters in the smectic A phase is then used in conjunction with Kobayashi-McMillan theory applied to solutes to give information about the smectic A layering and the nematic/smectic A coupling. In both smectic A solvents, the solute smectic coupling constant, tau, is negative (with the origin fixed at the center of the smectic layer) for all solutes. The signs and relative values of tau indicate that the ortho and para solutes favor the interlayer region while the meta solute is more evenly distributed throughout the layers.     

Analysis of retrograde behavior and the cross-over effect in supercritical fluids

In this paper, results based upon thermodynamic stability theory are developed which lead to a set of both necessary and sufficient conditions for the existence of retrograde behavior in a multicomponent solute system dissolved in a pure supercritical fluid. While experimental evidence of retrograde behavior in single solute systems has been known for some time, recently data have been obtained showing the retrograde effect in binary solute systems dissolved in a pure supercritical fluid.In such systems, cross-over regions may be defined. These are pressure—temperature regions where the solubility of one solute increases while that of the other decreases with a change in temperature at constant pressure. The existence of cross-over regions in multicomponent mixtures can have implications for a new separation technique using pure supercritical fluids. In conjunction with an equation of state, the results derived here allow cross-over regions to be predicted, thus enabling one to identify candidate systems and thermodynamic conditions for the cross-over process. For this work a variation of a perturbed hard sphere model equation of state was used for the calculations.     

A kinetic modeling of particle formation by gas antisolvent process: Precipitation of aspirin

Population balance equations (PBEs) are often integro-partial differential in nature due to complexities involved in nonconventional crystallization processes, especially gas antisolvent (GAS). The reason is that they include phenomena such as primary nucleation, secondary nucleation, crystal growth, agglomeration, and/or breakage of crystals. Therefore, the solution to such models has become rather difficult. Considering these difficulties, a powerful numerical algorithm was adopted in this paper to treat the population balance model for the precipitation of aspirin by the GAS process. This method was the combination of Lax–Wendroff and Crank–Nicholson numerical methods. It was used to investigate the effect of significant operating parameters, that is, antisolvent addition rate, process temperature, and solute concentration, on the final product properties for two solvents. The antisolvent addition rate was varied between 8 and 40 bar/min, the process temperature was kept constant at levels 37°C and 42°C, the solute concentration was manipulated at two levels, namely, 0.2 and 0.27 g solute/g solution, and methanol and acetone were used as the organic solvents. The results indicated the successful performance of the applied method in treating PBE, since smooth particle size distributions were produced, which were in an acceptable agreement with the experimental data of the investigated system.     

Micellar gradients in size-exclusion simulated moving bed chromatography

The selectivity of size-exclusion chromatography (SEC) can be modified by adding non-ionic micelles to the mobile phase. Surfactant-aided size-exclusion chromatography (SASEC) can therefore very well be performed in a gradient mode on an SMB, as is reported in this paper. A method has been developed for correctly positioning a micellar gradient over an SMB. The method is applied for size-exclusion chromatography with the non-ionic surfactant C12E23 as gradient forming solute, and demonstrated by applying it to a relevant chromatographic protein separation problem.