Quaternionic representation of the genetic code

A heuristic diagram of the evolution of the standard genetic code is presented. It incorporates, in a way that resembles the energy levels of an atom, the physical notion of broken symmetry and it is consistent with original ideas by Crick on the origin and evolution of the code as well as with the chronological order of appearence of the amino acids along the evolution as inferred from work that mixtures known experimental results with theoretical speculations. Suggested by the diagram we propose a Hamilton quaternions based mathematical representation of the code as it stands now-a-days. The central object in the description is a codon function that assigns to each amino acid an integer quaternion in such a way that the observed code degeneration is preserved. We emphasize the advantages of a quaternionic representation of amino acids taking as an example the folding of proteins. With this aim we propose an algorithm to go from the quaternions sequence to the protein three dimensional structure which can be compared with the corresponding experimental one stored at the Protein Data Bank. In our criterion the mathematical representation of the genetic code in terms of quaternions merits to be taken into account because it describes not only most of the known properties of the genetic code but also opens new perspectives that are mainly derived from the close relationship between quaternions and rotations.Comment: 19 pages, 11 figure

CRANKITE: a fast polypeptide backbone conformation sampler

Background: CRANKITE is a suite of programs for simulating backbone conformations of polypeptides and proteins. The core of the suite is an efficient Metropolis Monte Carlo sampler of backbone conformations in continuous three-dimensional space in atomic details. Methods: In contrast to other programs relying on local Metropolis moves in the space of dihedral angles, our sampler utilizes local crankshaft rotations of rigid peptide bonds in Cartesian space. Results: The sampler allows fast simulation and analysis of secondary structure formation and conformational changes for proteins of average length

SABBAC: online Structural Alphabet-based protein BackBone reconstruction from Alpha-Carbon trace

SABBAC is an on-line service devoted to protein backbone reconstruction from alpha-carbon trace. It is based on the assembly of fragments taken from a library of reduced size, selected from the encoding of the protein trace in a hidden Markov model-derived structural alphabet. The assembly of the fragments is achieved by a greedy algorithm, using an energy-based scoring. Alpha-carbon coordinates remain unaffected. SABBAC simply positions the missing backbone atoms, no further refinement is performed. From our tests, SABBAC performs equal or better than other similar on-line approach and is robust to deviations on the alpha-carbon coordinates. It can be accessed at

Chirality in a quaternionic representation of the genetic code

A quaternionic representation of the genetic code, previously reported by the authors, is updated in order to incorporate chirality of nucleotide bases and amino acids. The original representation assigns to each nucleotide base a prime integer quaternion of norm 7 and involves a function that associates with each codon, represented by three of these quaternions, another integer quaternion (amino acid type quaternion) in such a way that the essentials of the standard genetic code (particulaty its degeneration) are preserved. To show the advantages of such a quaternionic representation we have, in turn, associated with each amino acid of a given protein, besides of the type quaternion, another real one according to its order along the protein (order quaternion) and have designed an algorithm to go from the primary to the tertiary structure of the protein by using type and order quaternions. In this context, we incorporate chirality in our representation by observing that the set of eight integer quaternions of norm 7 can be partitioned into a pair of subsets of cardinality four each with their elements mutually conjugates and by putting they in correspondence one to one with the two sets of enantiomers (D and L) of the four nucleotide bases adenine, cytosine, guanine and uracil, respectively. Thus, guided by two diagrams proposed for the codes evolution, we define functions that in each case assign a L- (D-) amino acid type integer quaternion to the triplets of D- (L-) bases. The assignation is such that for a given D-amino acid, the associated integer quaternion is the conjugate of that one corresponding to the enantiomer L. The chiral type quaternions obtained for the amino acids are used, together with a common set of order quaternions, to describe the folding of the two classes, L and D, of homochiral proteins.Comment: 17 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1505.0465

Combining Coarse-Grained Protein Models with Replica-Exchange All-Atom Molecular Dynamics

We describe a combination of all-atom simulations with CABS, a well-established coarse-grained protein modeling tool, into a single multiscale protocol. The simulation method has been tested on the C-terminal beta hairpin of protein G, a model system of protein folding. After reconstructing atomistic details, conformations derived from the CABS simulation were subjected to replica-exchange molecular dynamics simulations with OPLS-AA and AMBER99sb force fields in explicit solvent. Such a combination accelerates system convergence several times in comparison with all-atom simulations starting from the extended chain conformation, demonstrated by the analysis of melting curves, the number of native-like conformations as a function of time and secondary structure propagation. The results strongly suggest that the proposed multiscale method could be an efficient and accurate tool for high-resolution studies of protein folding dynamics in larger systems.Comment: 12 pages, 4 figure

Structural alphabets derived from attractors in conformational space

Background: The hierarchical and partially redundant nature of protein structures justifies the definition of frequently occurring conformations of short fragments as 'states'. Collections of selected representatives for these states define Structural Alphabets, describing the most typical local conformations within protein structures. These alphabets form a bridge between the string-oriented methods of sequence analysis and the coordinate-oriented methods of protein structure analysis.Results: A Structural Alphabet has been derived by clustering all four-residue fragments of a high-resolution subset of the protein data bank and extracting the high-density states as representative conformational states. Each fragment is uniquely defined by a set of three independent angles corresponding to its degrees of freedom, capturing in simple and intuitive terms the properties of the conformational space. The fragments of the Structural Alphabet are equivalent to the conformational attractors and therefore yield a most informative encoding of proteins. Proteins can be reconstructed within the experimental uncertainty in structure determination and ensembles of structures can be encoded with accuracy and robustness.Conclusions: The density-based Structural Alphabet provides a novel tool to describe local conformations and it is specifically suitable for application in studies of protein dynamics. © 2010 Pandini et al; licensee BioMed Central Ltd

Assessing Protein Conformational Sampling Methods Based on Bivariate Lag-Distributions of Backbone Angles

Despite considerable progress in the past decades, protein structure prediction remains one of the major unsolved problems in computational biology. Angular-sampling-based methods have been extensively studied recently due to their ability to capture the continuous conformational space of protein structures. The literature has focused on using a variety of parametric models of the sequential dependencies between angle pairs along the protein chains. In this article, we present a thorough review of angular-sampling-based methods by assessing three main questions: What is the best distribution type to model the protein angles? What is a reasonable number of components in a mixture model that should be considered to accurately parameterize the joint distribution of the angles? and What is the order of the local sequence–structure dependency that should be considered by a prediction method? We assess the model fits for different methods using bivariate lag-distributions of the dihedral/planar angles. Moreover, the main information across the lags can be extracted using a technique called Lag singular value decomposition (LagSVD), which considers the joint distribution of the dihedral/planar angles over different lags using a nonparametric approach and monitors the behavior of the lag-distribution of the angles using singular value decomposition. As a result, we developed graphical tools and numerical measurements to compare and evaluate the performance of different model fits. Furthermore, we developed a web-tool (http://www.stat.tamu.edu/∼madoliat/LagSVD) that can be used to produce informative animations

Hsc70-induced changes in clathrin-auxilin cage structure suggest a role for clathrin light chains in cage disassembly

The molecular chaperone, Hsc70, together with its co-factor, auxilin, facilitates the ATP-dependent removal of clathrin during clathrin-mediated endocytosis in cells. We have used cryo-electron microscopy to determine the 3D structure of a complex of clathrin, auxilin401-910 and Hsc70 at pH 6 in the presence of ATP, frozen within 20 seconds of adding Hsc70 in order to visualize events that follow the binding of Hsc70 to clathrin and auxilin before clathrin disassembly. In this map, we observe density beneath the vertex of the cage that we attribute to bound Hsc70. This density emerges asymmetrically from the clathrin vertex, suggesting preferential binding by Hsc70 for one of the three possible sites at the vertex. Statistical comparison with a map of whole auxilin and clathrin previously published by us reveals the location of statistically significant differences which implicate involvement of clathrin light chains in structural rearrangements which occur after Hsc70 is recruited. Clathrin disassembly assays using light scattering suggest that loss of clathrin light chains reduces the efficiency with which auxilin facilitates this reaction. These data support a regulatory role for clathrin light chains in clathrin disassembly in addition to their established role in regulating clathrin assembly