Internal Rotational Degrees Of Freedom Research Articles

A computational toolbox for the assembly yield of complex and heterogeneous structures

The self-assembly of complex structures from a set of non-identical building blocks is a hallmark of soft matter and biological systems, including protein complexes, colloidal clusters, and DNA-based assemblies. Predicting the dependence of the equilibrium assembly yield on the concentrations and interaction energies of building blocks is highly challenging, owing to the difficulty of computing the entropic contributions to the free energy of the many structures that compete with the ground state configuration. While these calculations yield well known results for spherically symmetric building blocks, they do not hold when the building blocks have internal rotational degrees of freedom. Here we present an approach for solving this problem that works with arbitrary building blocks, including proteins with known structure and complex colloidal building blocks. Our algorithm combines classical statistical mechanics with recently developed computational tools for automatic differentiation. Automatic differentiation allows efficient evaluation of equilibrium averages over configurations that would otherwise be intractable. We demonstrate the validity of our framework by comparison to molecular dynamics simulations of simple examples, and apply it to calculate the yield curves for known protein complexes and for the assembly of colloidal shells.

An energy‐momentum scheme for extended continuum models with rotational degrees of freedom

AbstractIn mechanical engineering, the finite element method is well‐known in the simulation of homogeneous materials without microstructures. But, in particular, microstructures can improve materials with respect to their applicability in engineering. Examples are 2D‐ and 3D‐fiber‐reinforced materials, which are manufactured in the microscale or mesoscale, respectively. An efficient finite element simulation of these materials provides an anisotropic continuum formulation. However, 3D‐fiber‐reinforced materials demand continua with extended kinematics. The introduction of such extended continua is possible by mixed finite element methods. In this paper, we introduce internal rotational degrees of freedom to model also a stiffness with respect to fiber flexure and twist. The kinematic formulation can be explained by using a covariant continuum formulation based on curvilinear convected coordinates. In this context, new objective deformation measures and power‐conjugate stress tensors are derived. Therefore, a corresponding energy‐momentum scheme is obtained and simulates numerical examples stable and efficient in accordance with discrete balance laws. These examples demonstrate the additional material stiffness with fiber curvature‐twist strain energy functions of Kauderer‐type by parameter studies with respect to different fiber directions.

Aggregation-Induced Emission and Temperature-Dependent Luminescence of Potassium Perylenetetracarboxylate.

Investigation of temperature-dependent photoluminescent properties of potassium perylene-3,4,9,10-tetracarboxylate (K4PTC), a molecule with no internal rotational degrees of freedom, shows aggregation-induced enhanced emission at room temperature. The different excitonic emission processes are dependent of temperature, some of which quenches in an intermediate temperature range (from 50 to 150K). The exciton excited states switching phenomenon from "dark" to "bright" states is observed and its explained using Herzberg-Teller selection rule. K4PTC is a molecule comparable to the size of its precursor, perylene-3,4,9,10-tetracarboxylic anhydride (PTCDA) and is highly soluble in water, contrary to PTCDA, which is poorly soluble in most solvents. Powder x-ray diffraction measurements corroborate a lesser degree of ordering of bulk K4PTC compared to bulk PTCDA. The green luminescent molecule could, in principle, be used as a biomarker, or in photodynamic therapy, if further studies show relatively low toxicity.

Synthetic spin-orbit coupling mediated by a bosonic environment

We study a mobile quantum impurity, possessing internal rotational degrees of freedom, confined to a ring in the presence of a many-particle bosonic bath. By considering the recently introduced rotating polaron problem, we define the Hamiltonian and examine the energy spectrum. The weak-coupling regime is studied by means of a variational ansatz in the truncated Fock space. The corresponding spectrum indicates that there emerges a coupling between the internal and orbital angular momenta of the impurity as a consequence of the phonon exchange. We interpret the coupling as a phonon-mediated spin-orbit coupling and quantify it by using a correlation function between the internal and orbital angular momentum operators. The strong-coupling regime is investigated within the Pekar approach and it is shown that the correlation function of the ground state shows a kink at a critical coupling, that is explained by a sharp transition from the non-interacting state to the states that exhibit strong interaction with the surroundings. The results might find applications in such fields as spintronics or topological insulators, where spin-orbit coupling is of crucial importance.

Docking of oligopeptides

The paper presents the results concerning the use of the supercomputer docking program SOL-P, which performs a search for low-energy minima of protein—ligand complexes in the MMFF94 force field without using a grid of precalculated potentials for protein—ligand interactions. The SOL-P docking program can be applied to the generalized docking of ligands with a large number of internal rotational degrees of freedom (torsions). This program is based on a tensor train global optimization algorithm. The SOL-P program was successfully applied to the docking of oligopeptide ligands consisting of 3–6 amino acid residues and having 18–25 internal rotational degrees of freedom.

Theory of the rotating polaron: Spectrum and self-localization

We study a quantum impurity possessing both translational and internal rotational degrees of freedom interacting with a bosonic bath. Such a system corresponds to a `rotating polaron', which can be used to model, e.g., a rotating molecule immersed in an ultracold Bose gas or superfluid Helium. We derive the Hamiltonian of the rotating polaron and study its spectrum in the weak- and strong-coupling regimes using a combination of variational, diagrammatic, and mean-field approaches. We reveal how the coupling between linear and angular momenta affects stable quasiparticle states, and demonstrate that internal rotation leads to an enhanced self-localization in the translational degrees of freedom.

Crystal structures of 1,8‐bis(dimethylamino)naphthalene (DMAN) and dicarboxylic acids complexes determined from single‐crystal and powder diffraction data

New complexes of DMAN proton sponge with 3 dicarboxylic acids (dca): glutaric {1}, suberic {2} and dodecanedioic acid {3} were obtained. The structures of the complexes {1,2} were solved on the basis of laboratory powder data, while the structure of {3} was solved using single crystal methods. The additional goal of the study was to test some new approaches to organic crystal structure solution from powder diffraction data, particularly with presence of molecules with high number of internal rotational degrees of freedom (dca molecules). Performance of structure solution using the popular FOX program was tested with alternative definitions of flexible dca molecules.

Soft elasticity in solids composed of ellipse-shaped particles

We present a method for studying the influence of internal rotational degrees of freedom on the elastic properties of crystals composed of ellipsoidal particles. We derive the conditions under which a stretched-triangular lattice of ellipsoidal particles can exhibit a vanishing shear modulus. Analytical predictions are confirmed with numerical calculations. Numerical results also show that internal rotational modes can delay the proliferation of dislocations in the plastic regime.

Increasing Thermoelectric Efficiency: A Dynamical Systems Approach

Inspired by the kinetic theory of ergodic gases and chaotic billiards, we propose a simple microscopic mechanism for the increase of thermoelectric efficiency. We consider the cross transport of particles and energy in open classical ergodic billiards. We show that, in the linear response regime, where we find exact expressions for all transport coefficients, the thermoelectric efficiency of ideal ergodic gases can approach the Carnot efficiency for sufficiently complex charge carrier molecules. Our results are clearly demonstrated with a simple numerical simulation of a Lorentz gas of particles with internal rotational degrees of freedom.

Role of rotational entropy in benzyl isocyanide isomerization

We present here a detailed analysis of benzyl isocyanide isomerization as a test case for the ability of different theoretical methods to treat changes in internal rotational degrees of freedom during a reaction. In this reaction, the phenyl rotates freely in the reactant, but has hindered rotation at the transition state; this yields a negative contribution to the reaction entropy. Free energy of activation, internal energy of activation, and entropy of activation were quantified using both thermodynamic integration and harmonic frequency analysis. Compared with experiment, thermodynamic integration generated more accurate activation parameters than the harmonic frequency analysis. A hindered rotor model was also applied to retrieve thermodynamic properties of phenyl rotation at the reactant and at the transition state. The results from this hindered rotor calculation were used to improve the results from the harmonic frequency analysis as well as to verify that the decrease in entropy from the phenyl rotation hindered at the transition state is a major contributor to the entropy of activation seen experimentally in this reaction.

Theoretical determination of accurate rate constants: Application to the decomposition of a single-molecule precursor

The ring opening is proposed to be the initial step in the unimolecular decomposition of a single-molecule precursor bisazido(dimethylaminopropyl)gallium (BAZIGA) used in a CVD process of gallium nitride. An efficient Monte Carlo importance sampling technique is developed to estimate the canonical partition functions and absolute entropies therefrom by taking the internal rotor hindrance and all coupling arising from the external and internal rotational degrees of freedom into account. This allows the estimation of the free energy difference in the ring opening process of BAZIGA to be made. Canonical flexible transition state theory is applied to the radical Ga–C bond homolysis in the unimolecular decomposition of BAZIGA. The limiting high-pressure rate constant is calculated using a simple model for the potential energy surface based on density functional calculations.

Effect of Protonation on the Conformation of Cinchonidine

The protonation of cinchona alkaloids in solution leads to the severe restriction of their internal rotational degrees of freedom and to the locking of the molecule around a specific conformation held in place by a bridging counterion of the acid used for protonation. For HF, direct interactions were detected by NMR between the fluorine anion and not only with the acidic hydroxo group but also with non-acidic hydrogen atoms in the quinoline ring.

Calculation of rotational partition functions by an efficient Monte Carlo importance sampling technique

The evaluation of the classical rotational partition function represented by a configuration integral over all external and internal rotational degrees of freedom of nonrigid chain polyatomic molecules is described. The method of Pitzer and Gwinn is used to correct the classical partition function for quantum mechanical effects at low temperatures. The internal rotor hindrance and all coupling arising from the external and internal rotational degrees of freedom are explicitly taken into account. Importance sampling Monte Carlo based on the adaptive VEGAS algorithm to perform multidimensional integration is implemented within the TINKER program package. A multidimensional potential energy hypersurface is calculated with the MM3(2000) molecular mechanics force field. Numerical tests are performed on a number of small n-alkanes (from ethane to octane), for which the absolute entropies calculated at three different temperatures are compared both with the experimental values and with the previous theoretical results. The application of a more efficient importance sampling technique developed here results in a substantial reduction of statistical errors in the evaluation of the configuration integral for a given number of Monte Carlo steps. Error estimates for the calculated entropies are given, and possible sources of systematic errors, and their importance for a reliable prediction of the absolute entropy, are discussed.

Cartan–Weyl Dirac and Laplacian Operators, Brownian Motions: The Quantum Potential and Scalar Curvature, Maxwell’s and Dirac-Hestenes Equations, and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincare–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q

Statistical thermodynamics of internal rotation in a hindering potential of mean force obtained from computer simulations.

A method of statistical estimation is applied to the problem of one-dimensional internal rotation in a hindering potential of mean force. The hindering potential, which may have a completely general shape, is expanded in a Fourier series, the coefficients of which are estimated by fitting an appropriate statistical-mechanical distribution to the random variable of internal rotation angle. The function of reduced moment of inertia of an internal rotation is averaged over the thermodynamic ensemble of atomic configurations of the molecule obtained in stochastic simulations. When quantum effects are not important, an accurate estimate of the absolute internal rotation entropy of a molecule with a single rotatable bond is obtained. When there is more than one rotatable bond, the "marginal" statistical-mechanical properties corresponding to a given internal rotational degree of freedom are reduced. The method is illustrated using Monte Carlo simulations of two public health relevant halocarbon molecules, each having a single internal-rotation degree of freedom, and a molecular dynamics simulation of an immunologically relevant polypeptide, in which several dihedral angles are analyzed.

Application of the stochastic tunneling method to high throughput database screening

The stochastic tunneling technique is applied to screen a database of chemical compounds to the active site of dihydrofolate reductase for lead candidates in the receptor–ligand docking problem. Using an atomistic force field we consider the ligand’s internal rotational degrees of freedom. It is shown that the natural ligand (methotrexate) scores best among 10 000 randomly chosen compounds. We analyze the top scoring compounds to identify hot-spots of the receptor. We mutate the amino acids that are responsible for the hot-spots of the receptor and verify that its specificity is lost upon modification.

Phase behavior of colloidal molecular crystals on triangular light lattices.

We investigate the melting process of a two-dimensional charge-stabilized colloidal system in the presence of a triangular substrate potential. This potential is formed by an optical interference pattern that allows the substrate strength to be varied continuously. By means of an additional scanned optical tweezer the particle density can be adjusted to different numbers m of colloidal particles per substrate minima; here we concentrate on the case of trimers, i.e., m = 3. Because trimers exhibit additional internal degrees of rotational freedom, the phase behavior of such a system is very different from homogeneous or one-dimensional periodic substrate potentials.

Constraint dynamics for quantum Monte Carlo calculations

We describe how to apply classical constraint dynamics to problems in diffusion Monte Carlo. We apply the method to rigid and nonrigid water molecules with an internal rotational degree of freedom. The method is applicable to a wide variety of problems.

A triangular thick plate finite element with an exact thin limit

We present a new formulation for a triangular finite element developed within the framework of a shear deformable plate theory. The element takes advantage of internal rotational degrees of freedom and a linked interpolation between the transverse displacement and the rotations. The element has excellent interpolating capacity and presents no locking effects; in fact, the shear energy may be set identically equal to zero without introducing any ill-conditioning, thus recovering a proper thin plate limit.

A shear deformable plate element with an exact thin limit

We present a quadrilateral finite element developed within the framework of a shear deformable plate theory. The interpolation for the rotation takes advantage of internal rotational degrees of freedom (through the use of bubble functions), while the interpolation for the transverse displacement is linked to the nodal rotations. A careful study of the element behavior is performed using an extensive set of mixed patch tests; results from several numerical examples are also presented. The element has proper rank and excellent interpolating capacity. Moreover, without using any ad-hoc assumption (e.g., energy balancing schemes) the element presents no locking effects at all; in fact, the shear energy may be set identically equal to zero without introducing any ill-conditioning in the problem, thus recovering a proper thin plate limit.