Alex Rodriguez

Clustering by fast search and find of density peaks

Discerning clusters of data points Cluster analysis is used in many disciplines to group objects according to a defined measure of distance. Numerous algorithms exist, some based on the analysis of the local density of data points, and others on predefined probability distributions. Rodriguez and Laio devised a method in which the cluster centers are recognized as local density maxima that are far away from any points of higher density. The algorithm depends only on the relative densities rather than their absolute values. The authors tested the method on a series of data sets, and its performance compared favorably to that of established techniques. Science, this issue p. 1492 Local density of points is ranked and analyzed to categorize data. Cluster analysis is aimed at classifying elements into categories on the basis of their similarity. Its applications range from astronomy to bioinformatics, bibliometrics, and pattern recognition. We propose an approach based on the idea that cluster centers are characterized by a higher density than their neighbors and by a relatively large distance from points with higher densities. This idea forms the basis of a clustering procedure in which the number of clusters arises intuitively, outliers are automatically spotted and excluded from the analysis, and clusters are recognized regardless of their shape and of the dimensionality of the space in which they are embedded. We demonstrate the power of the algorithm on several test cases.

Metadynamics Simulations Reveal a Na+ Independent Exiting Path of Galactose for the Inward-Facing Conformation of vSGLT

Sodium-Galactose Transporter (SGLT) is a secondary active symporter which accumulates sugars into cells by using the electrochemical gradient of Na+ across the membrane. Previous computational studies provided insights into the release process of the two ligands (galactose and sodium ion) into the cytoplasm from the inward-facing conformation of Vibrio parahaemolyticus sodium/galactose transporter (vSGLT). Several aspects of the transport mechanism of this symporter remain to be clarified: (i) a detailed kinetic and thermodynamic characterization of the exit path of the two ligands is still lacking; (ii) contradictory conclusions have been drawn concerning the gating role of Y263; (iii) the role of Na+ in modulating the release path of galactose is not clear. In this work, we use bias-exchange metadynamics simulations to characterize the free energy profile of the galactose and Na+ release processes toward the intracellular side. Surprisingly, we find that the exit of Na+ and galactose is non-concerted as the cooperativity between the two ligands is associated to a transition that is not rate limiting. The dissociation barriers are of the order of 11–12 kcal/mol for both the ion and the substrate, in line with kinetic information concerning this type of transporters. On the basis of these results we propose a branched six-state alternating access mechanism, which may be shared also by other members of the LeuT-fold transporters.

An efficient algorithm to perform local concerted movements of a chain molecule.

The devising of efficient concerted rotation moves that modify only selected local portions of chain molecules is a long studied problem. Possible applications range from speeding the uncorrelated sampling of polymeric dense systems to loop reconstruction and structure refinement in protein modeling. Here, we propose and validate, on a few pedagogical examples, a novel numerical strategy that generalizes the notion of concerted rotation. The usage of the Denavit-Hartenberg parameters for chain description allows all possible choices for the subset of degrees of freedom to be modified in the move. They can be arbitrarily distributed along the chain and can be distanced between consecutive monomers as well. The efficiency of the methodology capitalizes on the inherent geometrical structure of the manifold defined by all chain configurations compatible with the fixed degrees of freedom. The chain portion to be moved is first opened along a direction chosen in the tangent space to the manifold, and then closed in the orthogonal space. As a consequence, in Monte Carlo simulations detailed balance is easily enforced without the need of using Jacobian reweighting. Moreover, the relative fluctuations of the degrees of freedom involved in the move can be easily tuned. We show different applications: the manifold of possible configurations is explored in a very efficient way for a protein fragment and for a cyclic molecule; the “local backbone volume”, related to the volume spanned by the manifold, reproduces the mobility profile of all-α helical proteins; the refinement of small protein fragments with different secondary structures is addressed. The presented results suggest our methodology as a valuable exploration and sampling tool in the context of bio-molecular simulations.

METAGUI 3: A graphical user interface for choosing the collective variables in molecular dynamics simulations☆

Abstract Molecular dynamics (MD) simulations allow the exploration of the phase space of biopolymers through the integration of equations of motion of their constituent atoms. The analysis of MD trajectories often relies on the choice of collective variables (CVs) along which the dynamics of the system is projected. We developed a graphical user interface (GUI) for facilitating the interactive choice of the appropriate CVs. The GUI allows: defining interactively new CVs; partitioning the configurations into microstates characterized by similar values of the CVs; calculating the free energies of the microstates for both unbiased and biased (metadynamics) simulations; clustering the microstates in kinetic basins; visualizing the free energy landscape as a function of a subset of the CVs used for the analysis. A simple mouse click allows one to quickly inspect structures corresponding to specific points in the landscape. Program summary Program Title: METAGUI 3 Program Files doi: http://dx.doi.org/10.17632/wyxjndwkbp.1 Licensing provisions: GPLv3 Programming language: Tcl/Tk, Fortran Journal reference of previous version: METAGUI [1] Does the new version supersede the previous version?: No Nature of problem: Choose the appropriate collective variables for describing the thermodynamics and kinetics of a biomolecular system through biased and unbiased molecular dynamics. Solution method: Provide an environment to compute and visualize free energy surfaces as a function of collective variables, interactively defined. Additional comments: METAGUI 3 is not a standalone program but a plugin that provides analysis features within VMD (version 1.9.2 or higher). [1] X. Biarnes, F. Pietrucci, F. Marinelli, A. Laio, METAGUI. A VMD interface for analyzing metadynamics and molecular dynamics simulations, Computer Physics Communications 183 (2012) 203–211.

Computing the Free Energy without Collective Variables

We introduce an approach for computing the free energy and the probability density in high-dimensional spaces, such as those explored in molecular dynamics simulations of biomolecules. The approach exploits the presence of correlations between the coordinates, induced, in molecular dynamics, by the chemical nature of the molecules. Due to these correlations, the data points lay on a manifold that can be highly curved and twisted, but whose dimension is normally small. We estimate the free energies by finding, with a statistical test, the largest neighborhood in which the free energy in the embedding manifold can be considered constant. Importantly, this procedure does not require defining explicitly the manifold and provides an estimate of the error that is approximately unbiased up to large dimensions. We test this approach on artificial and real data sets, demonstrating that the free energy estimates are reliable for data sets on manifolds of dimension up to ∼10, embedded in an arbitrarily large space. In practical applications our method permits the estimation of the free energy in a space of reduced dimensionality without specifying the collective variables defining this space.

Estimating the intrinsic dimension of datasets by a minimal neighborhood information

Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a manifold whose Intrinsic Dimension (ID) is much lower than the crude large number of coordinates. Such manifold is generally twisted and curved; in addition points on it will be non-uniformly distributed: two factors that make the identification of the ID and its exploitation really hard. Here we propose a new ID estimator using only the distance of the first and the second nearest neighbor of each point in the sample. This extreme minimality enables us to reduce the effects of curvature, of density variation, and the resulting computational cost. The ID estimator is theoretically exact in uniformly distributed datasets, and provides consistent measures in general. When used in combination with block analysis, it allows discriminating the relevant dimensions as a function of the block size. This allows estimating the ID even when the data lie on a manifold perturbed by a high-dimensional noise, a situation often encountered in real world data sets. We demonstrate the usefulness of the approach on molecular simulations and image analysis.

Non-Markovian effects on protein sequence evolution due to site dependent substitution rates

BackgroundMany models of protein sequence evolution, in particular those based on Point Accepted Mutation (PAM) matrices, assume that its dynamics is Markovian. Nevertheless, it has been observed that evolution seems to proceed differently at different time scales, questioning this assumption. In 2011 Kosiol and Goldman proved that, if evolution is Markovian at the codon level, it can not be Markovian at the amino acid level. However, it remains unclear up to which point the Markov assumption is verified at the codon level.ResultsHere we show how also the among-site variability of substitution rates makes the process of full protein sequence evolution effectively not Markovian even at the codon level. This may be the theoretical explanation behind the well known systematic underestimation of evolutionary distances observed when omitting rate variability. If the substitution rate variability is neglected the average amino acid and codon replacement probabilities are affected by systematic errors and those with the largest mismatches are the substitutions involving more than one nucleotide at a time. On the other hand, the instantaneous substitution matrices estimated from alignments with the Markov assumption tend to overestimate double and triple substitutions, even when learned from alignments at high sequence identity.ConclusionsThese results discourage the use of simple Markov models to describe full protein sequence evolution and encourage to employ, whenever possible, models that account for rate variability by construction (such as hidden Markov models or mixture models) or substitution models of the type of Le and Gascuel (2008) that account for it explicitly.

The permeation mechanism of organic cations through a CNG mimic channel

Several channels, ranging from TRP receptors to Gap junctions, allow the exchange of small organic solute across cell membrane. However, very little is known about the molecular mechanism of their permeation. Cyclic Nucleotide Gated (CNG) channels, despite their homology with K+ channels and in contrast with them, allow the passage of larger methylated and ethylated ammonium ions like dimethylammonium (DMA) and ethylammonium (EA). We combined electrophysiology and molecular dynamics simulations to examine how DMA interacts with the pore and permeates through it. Due to the presence of hydrophobic groups, DMA enters easily in the channel and, unlike the alkali cations, does not need to cross any barrier. We also show that while the crystal structure is consistent with the presence of a single DMA ion at full occupancy, the channel is able to conduct a sizable current of DMA ions only when two ions are present inside the channel. Moreover, the second DMA ion dramatically changes the free energy landscape, destabilizing the crystallographic binding site and lowering by almost 25 kJ/mol the binding affinity between DMA and the channel. Based on the results of the simulation the experimental electron density maps can be re-interpreted with the presence of a second ion at lower occupancy. In this mechanism the flexibility of the channel plays a key role, extending the classical multi-ion permeation paradigm in which conductance is enhanced by the plain interaction between the ions.

Predicting Amino Acid Substitution Probabilities Using Single Nucleotide Polymorphisms

Fast genome sequencing offers invaluable opportunities for building updated and improved models of protein sequence evolution. We here show that Single Nucleotide Polymorphisms (SNPs) can be used to build a model capable of predicting the probability of substitution between amino acids in variants of the same protein in different species. The model is based on a substitution matrix inferred from the frequency of codon interchanges observed in a suitably selected subset of human SNPs, and predicts the substitution probabilities observed in alignments between Homo sapiens and related species at 85–100% of sequence identity better than any other approach we are aware of. The model gradually loses its predictive power at lower sequence identity. Our results suggest that SNPs can be employed, together with multiple sequence alignment data, to model protein sequence evolution. The SNP-based substitution matrix developed in this work can be exploited to better align protein sequences of related organisms, to refine the estimate of the evolutionary distance between protein variants from related species in phylogenetic trees and, in perspective, might become a useful tool for population analysis.