New out-of-plane angle and bond angle internal coordinates and related potential energy functions for molecular mechanics and dynamics simulations

With currently used definitions of out-of-plane angle and bond angle internal coordinates, Cartesian derivatives have singularities, at ±Π/2 in the former case and Π in the latter. If either of these occur during molecular mechanics or dynamics simulations, the forces are not well defined. To avoid such difficulties, we provide new out-of-plane and bond angle coordinates and associated potential energy functions that inherently avoid these singularities. The application of these coordinates is illustrated by ab initio calculations on ammonia, water, and carbon dioxide. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1067–1084, 1999Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/34691/1/9_ftp.pd

An exact expression to calculate the derivatives of position-dependent observables in molecular simulations with flexible constraints

In this work, we introduce an algorithm to compute the derivatives of physical observables along the constrained subspace when flexible constraints are imposed on the system (i.e., constraints in which the hard coordinates are fixed to configuration-dependent values). The presented scheme is exact, it does not contain any tunable parameter, and it only requires the calculation and inversion of a sub-block of the Hessian matrix of second derivatives of the function through which the constraints are defined. We also present a practical application to the case in which the sought observables are the Euclidean coordinates of complex molecular systems, and the function whose minimization defines the constraints is the potential energy. Finally, and in order to validate the method, which, as far as we are aware, is the first of its kind in the literature, we compare it to the natural and straightforward finite-differences approach in three molecules of biological relevance: methanol, N-methyl-acetamide and a tri-glycine peptideComment: 13 pages, 8 figures, published versio

Explicit factorization of external coordinates in constrained Statistical Mechanics models

If a macromolecule is described by curvilinear coordinates or rigid constraints are imposed, the equilibrium probability density that must be sampled in Monte Carlo simulations includes the determinants of different mass-metric tensors. In this work, we explicitly write the determinant of the mass-metric tensor G and of the reduced mass-metric tensor g, for any molecule, general internal coordinates and arbitrary constraints, as a product of two functions; one depending only on the external coordinates that describe the overall translation and rotation of the system, and the other only on the internal coordinates. This work extends previous results in the literature, proving with full generality that one may integrate out the external coordinates and perform Monte Carlo simulations in the internal conformational space of macromolecules. In addition, we give a general mathematical argument showing that the factorization is a consequence of the symmetries of the metric tensors involved. Finally, the determinant of the mass-metric tensor G is computed explicitly in a set of curvilinear coordinates specially well-suited for general branched molecules.Comment: 22 pages, 2 figures, LaTeX, AMSTeX. v2: Introduccion slightly extended. Version in arXiv is slightly larger than the published on

Monte Carlo algorithm based on internal bridging moves for the atomistic simulation of thiophene oligomers and polymers

We introduce a powerful Monte Carlo (MC) algorithm for the atomistic simulation of bulk models of oligo- and poly-thiophenes by redesigning MC moves originally developed for considerably simpler polymer structures and architectures, such as linear and branched polyethylene, to account for the ring structure of the thiophene monomer. Elementary MC moves implemented include bias reptation of an end thiophene ring, flip of an internal thiophene ring, rotation of an end thiophene ring, concerted rotation of three thiophene rings, rigid translation of an entire molecule, rotation of an entire molecule and volume fluctuation. In the implementation of all moves we assume that thiophene ring atoms remain rigid and strictly co-planar; on the other hand, inter-ring torsion and bond bending angles remain fully flexible subject to suitable potential energy functions. Test simulations with the new algorithm of an important thiophene oligomer, {\alpha}-sexithiophene ({\alpha}-6T), at a high enough temperature (above its isotropic-to-nematic phase transition) using a new united atom model specifically developed for the purpose of this work provide predictions for the volumetric, conformational and structural properties that are remarkably close to those obtained from detailed atomistic Molecular Dynamics (MD) simulations using an all-atom model. The new algorithm is particularly promising for exploring the rich (and largely unexplored) phase behavior and nanoscale ordering of very long (also more complex) thiophene-based polymers which cannot be addressed by conventional MD methods due to the extremely long relaxation times characterizing chain dynamics in these systems

Quantum mechanical calculation of the effects of stiff and rigid constraints in the conformational equilibrium of the Alanine dipeptide

If constraints are imposed on a macromolecule, two inequivalent classical models may be used: the stiff and the rigid one. This work studies the effects of such constraints on the Conformational Equilibrium Distribution (CED) of the model dipeptide HCO-L-Ala-NH2 without any simplifying assumption. We use ab initio Quantum Mechanics calculations including electron correlation at the MP2 level to describe the system, and we measure the conformational dependence of all the correcting terms to the naive CED based in the Potential Energy Surface (PES) that appear when the constraints are considered. These terms are related to mass-metric tensors determinants and also occur in the Fixman's compensating potential. We show that some of the corrections are non-negligible if one is interested in the whole Ramachandran space. On the other hand, if only the energetically lower region, containing the principal secondary structure elements, is assumed to be relevant, then, all correcting terms may be neglected up to peptides of considerable length. This is the first time, as far as we know, that the analysis of the conformational dependence of these correcting terms is performed in a relevant biomolecule with a realistic potential energy function.Comment: 37 pages, 4 figures, LaTeX, BibTeX, AMSTe

Developing a triangular tessellation method for the analysis of medium ring pucker conformations

The main focus of this thesis is to investigate the relative conformational flexibilities of α-, β- and γ-cyclodextrins in water by analysing their macrocyclic ring puckering motion from Molecular Dynamics (MD) simulations. In particular, the puckering of the CDs is investigated through a coarse grained analysis of full atomistic simulations, where the CD conformational motions are studied on the macrocyclic scale rather than the atomistic scale. The flexibilities of the cyclodextrins (CDs) are then compared to their experimentally-observed aqueous solubility trend in order to try explain the anomalously flow solubility of β-cyclodextrin. β-CD has important applications in industry, such as the pharmaceutical industry, thus exploring the conformational reasons for its low solubility can help to design more effective cyclodextrin-based products in future. The ring puckering of the CDs is measured quantitatively using a reduced system of puckering coordinates based on the method of triangular tessellation. The triangular tessellation definition for monocylic 6-membered rings is first extended to 7- and 8-membered rings, and the corresponding puckering coordinates are derived mathematically. The macrocyclic CD rings are then simplified to monocyclic representations through an appropriate coarse graining of the molecules (specifically, α-, β- and γ-cyclodextrins are simplified to 6-, 7- and 8-sided rings, respectively), and the corresponding triangular tessellation definition is then used to measure their macrocyclic puckering. The rates of decay of the puckering motion are then calculated using time correlation functions, from which the relative flexibilities of the CDs is determined. Probability distributions are also used to investigate the ranges of the CD puckering. In addition, the horizontal contraction and expansion of the macrocyclic rings (termed """"breathing"""" herein) is analysed to supplement the puckering analysis. Puckering coordinates based on the triangular tessellation of 6-membered rings have been used previously to characterise all 38 canonical states of cyclohexane. In this thesis, a systematic procedure is developed to generate the triangular tessellation puckering coordinates of all the canonical states of 6-, 7- and 8-membered rings, and the coordinates for all canonical states of cycloheptane and cyclooctane are subsequently generated. These puckering coordinates can be useful not only in the conformational analysis of cyclohexane, cycloheptane and cyclooctane, but also to quantitatively characterise the conformations of 6-, 7- and 8-membered rings in general, both from experimental and computational studies

Relating chaos to deterministic diffusion of a molecule adsorbed on a surface

Chaotic internal degrees of freedom of a molecule can act as noise and affect the diffusion of the molecule on a substrate. A separation of time scales between the fast internal dynamics and the slow motion of the centre of mass on the substrate makes it possible to directly link chaos to diffusion. We discuss the conditions under which this is possible, and show that in simple atomistic models with pair-wise harmonic potentials, strong chaos can arise through the geometry. Using molecular-dynamics simulations, we demonstrate that a realistic model of benzene is indeed chaotic, and that the internal chaos affects the diffusion on a graphite substrate

Introduction to protein folding for physicists

The prediction of the three-dimensional native structure of proteins from the knowledge of their amino acid sequence, known as the protein folding problem, is one of the most important yet unsolved issues of modern science. Since the conformational behaviour of flexible molecules is nothing more than a complex physical problem, increasingly more physicists are moving into the study of protein systems, bringing with them powerful mathematical and computational tools, as well as the sharp intuition and deep images inherent to the physics discipline. This work attempts to facilitate the first steps of such a transition. In order to achieve this goal, we provide an exhaustive account of the reasons underlying the protein folding problem enormous relevance and summarize the present-day status of the methods aimed to solving it. We also provide an introduction to the particular structure of these biological heteropolymers, and we physically define the problem stating the assumptions behind this (commonly implicit) definition. Finally, we review the 'special flavor' of statistical mechanics that is typically used to study the astronomically large phase spaces of macromolecules. Throughout the whole work, much material that is found scattered in the literature has been put together here to improve comprehension and to serve as a handy reference.Comment: 53 pages, 18 figures, the figures are at a low resolution due to arXiv restrictions, for high-res figures, go to http://www.pabloechenique.co