The "practical mathematics" of recording three-dimensional eye position using scleral coils.

The "gold standard" for recording the three-dimensional rotation of the eye involves placing two coils of wire, embedded in a soft plastic ring, on the sclera of the eye, then placing the subject inside a set of orthogonal oscillating magnetic fields, and using the currents induced in the eye coils to deduce the position of the coil, and hence of the eye, in space. Eye movements are actually eye rotations, which can be described mathematically by a special class of matrices, rotation matrices, or, alternatively, by a rotation vector related to the axis of the rotation. This article deals with the mathematical tools needed to implement the signal processing from such a multifield, dual-coil system and compute the precise rotational movement of the eye. One reason for making such careful measurements is to study an interesting constraint on eye movements, called Listing's law, which expresses ocular torsion, or rotation of the eye about its line of sight, in terms of the direction of gaze. Techniques for experimentally quantitating these constraints are also presented. Following a treatment of the "ideal" case, with coils and eye in perfect alignment, the additional techniques for dealing with various departures from ideality that are almost always encountered experimentally are examined. A final section deals with developing a validation protocol for eye movement analysis techniques using mechanical and computer simulations of eye movements.     

Effects of linear acceleration on fish behavior and eye movements.

Behavioral responses and eye movements of fish during linear acceleration were reviewed. It is known that displacement of otoliths in the inner ear leads to body movements and/or eye movements. On the ground, the utriculus of the vestibular system is stimulated by otolith displacement caused by gravitational and inertial forces during horizontal acceleration of whole body. When the acceleration is imposed on the fish's longitudinal axis, the fish showed nose-down and nose-up posture for tailward and noseward displacement of otolith respectively. These responses were understood that the fish aligned his longitudinal body axis in a plane perpendicular to the direction of resultant force vector acting on the otoliths. When the acceleration was sideward, the fish rolled around his longitudinal body axis so that his back was tilted against the direction in which the inertial force acted on the otoliths. Linear acceleration applied to fish's longitudinal body axis evoked torsional eye movement. Direction of torsion coincided with the direction of acceleration, which compensate the change of resultant force vector produced by linear acceleration and gravity. Torsional movement of left and right eye coordinated with each other. In normal fish, both sinusoidal and rectangular acceleration of 0.1G could evoke clear eye torsion. Though the amplitude of response increased with increasing magnitude of acceleration up to 0.5 G, the torsion angle did not fully compensate the angle calculated from gravity and linear acceleration. Removal of the otolith on one side reduced the response amplitude of both eyes. The torsion angle evoked by rectangular acceleration was smaller than that evoked by sinusoidal acceleration in both normal and unilaterally labyrinthectomized fish. These results suggest that eye torsion of fish include both static and dynamic components.     

3D maps of RNA interhelical junctions

More than 50% of RNA secondary structure is estimated to be A-form helices, which are linked together by various junctions. Here we describe a protocol for computing three interhelical Euler angles describing the relative orientation of helices across RNA junctions. 5' and 3' helices, H1 and H2, respectively, are assigned based on the junction topology. A reference canonical helix is constructed using an appropriate molecular builder software consisting of two continuous idealized A-form helices (iH1 and iH2) with helix axis oriented along the molecular Z-direction running toward the positive direction from iH1 to iH2. The phosphate groups and the carbon and oxygen atoms of the sugars are used to superimpose helix H1 of a target interhelical junction onto the corresponding iH1 of the reference helix. A copy of iH2 is then superimposed onto the resulting H2 helix to generate iH2'. A rotation matrix R is computed, which rotates iH2' into iH2 and expresses the rotation parameters in terms of three Euler angles α(h), β(h) and γ(h). The angles are processed to resolve a twofold degeneracy and to select an overall rotation around the axis of the reference helix. The three interhelical Euler angles define clockwise rotations around the 5' (-γ(h)) and 3' (α(h)) helices and an interhelical bend angle (β(h)). The angles can be depicted graphically to provide a 'Ramachandran'-type view of RNA global structure that can be used to identify unusual conformations as well as to understand variations due to changes in sequence, junction topology and other parameters.     

On the estimation of joint kinematics during gait.

In gait analysis, the concepts of Euler and helical (screw) angles are used to define the three-dimensional relative joint angular motion of lower extremities. Reliable estimation of joint angular motion depends on the accurate definition and construction of embedded axes within each body segment. In this paper, using sensitivity analysis, we quantify the effects of uncertainties in the definition and construction of embedded axes on the estimation of joint angular motion during gait. Using representative hip and knee motion data from normal subjects and cerebral palsy patients, the flexion-extension axis is analytically perturbed +/- 15 degrees in 5 degrees steps from a reference position, and the joint angles are recomputed for both Euler and helical angle definitions. For the Euler model, hip and knee flexion angles are relatively unaffected while the ab/adduction and rotation angles are significantly affected throughout the gait cycle. An error of 15 degrees in the definition of flexion-extension axis gives rise to maximum errors of 8 and 12 degrees for the ab/adduction angle, and 10-15 degrees for the rotation angles at the hip and knee, respectively. Furthermore, the magnitude of errors in ab/adduction and rotation angles are a function of the flexion angle. The errors for the ab/adduction angles increase with increasing flexion angle and for the rotation angle, decrease with increasing flexion angle. In cerebral palsy patients with flexed knee pattern of gait, this will result in distorted estimation of ab/adduction and rotation. For the helical model, similar results are obtained for the helical angle and associated direction cosines.(ABSTRACT TRUNCATED AT 250 WORDS)     

Moiré patterns: an accurate technique for determination of the locus of the centres of rotation

This study describes an accurate technique for the determination of the centre of rotation of small angles. The moiré fringe method localizes the centre of rotation by defining two primary fringes, each of which is found by the intersection of three lines. The primary fringes intersect at the centre of rotation at 90 degrees to each other, the angle least likely to produce an error in measurement. By utilizing joints with known centres of rotation, we have found that the method is extremely accurate and reproducible to within 2 mm of the real centre for angular changes as small as 3 degrees. This technique is useful in evaluating whether a joint is a simple hinge, i.e. rotating about a single axis of rotation or whether the joint moves about a changing axis of rotation referred to as a locus or centrode.     

The Opening of a Single Base Without Perturbations of Neighboring Nucleotides: A Study on Crystal B-DNA duplex d(CGCGAATTCGCG)2

Abstract

In this work we explore the possibility of the opening of a single base without perturbation of its neighboring nucleotides. Low energy base opening into the grooves can be accomplished by rotation of the relevant backbone and glycosidic bond torsion angles. The pathway has been determined by identifying ζ torsion angle as the reaction coordinate together with the accompanying geometric requirement that guides the displacement of other torsion angles. Our study on Dickerson dodecamer duplex d(CGCGAATTCGCG)₂ showed that all bases with normal equilibrium ζ can be rotated by ~ 30°, corresponding to ~ 3.5Å base displacement, towards the major groove. Such an opening extent is comparable with estimated amplitudes of local angular motions in DNA bases from NMR experiments, which might facilitate proton exchange. The computed base opening energy barrier is also comparable with measured base pair opening enthalpy. These results indicate possible relevance of the pathway studied in this work with experimentally observed base pair opening process. Our analysis also showed a preference for base opening along the major groove and an abnormal behavior for bases with unusual equilibrium ζ torsion angle.     

Geometry of the five-membered ring mathematical demonstration of the pseudorotation formulae

The geometrical relations between the 15 typical parameters (bond lengths and angles, torsion angles) of a five-membered ring are derived for any ring then for a regular one. It is demonstrated that for the case of the 20 symmetrical C ₂ and C _sconformations, only geometrical considerations are needed to obtain the pseudorotation formulae for the torsion angles. However, the puckering intensity as well as the bond angle values cannot be expressed from geometrical constraints alone but would require energetical considerations.     

Conformational analysis of protein and nucleic acid fragments with the new grid search algorithm FOUND

The new computer algorithm FOUND, which is implemented as an integrated module of the DYANA structure calculation program, is capable of performing systematic local conformation analyses by exhaustive grid searches for arbitrary contiguous fragments of proteins and nucleic acids. It uses torsion angles as the only degrees of freedom to identify all conformations that fulfill the steric and NMR-derived conformational restraints within a contiguous molecular fragment, as defined either by limits on the maximal restraint violations or by the fragment-based DYANA target function value. Sets of mutually dependent torsion angles, for example in ribose rings, are treated as a single degree of freedom. The results of the local conformation analysis include allowed torsion angle ranges and stereospecific assignments for diastereotopic substituents, which are then included in the input of a subsequent structure calculation. FOUND can be used for grid searches comprising up to 13 torsion angles, such as the backbone of a complete -helical turn or dinucleotide fragments in nucleic acids, and yields a significantly higher number of stereospecific assignments than the precursor grid search algorithm HABAS.     

Measurement of Scaled Residual Dipolar Couplings in Proteins Using Variable-angle Sample Spinning

NMR spectra of ubiquitin in the presence of bicelles at a concentration of 25% w/v have been recorded under sample spinning conditions for different angles of rotation. For an axis of rotation equal to the magic angle, the (1)H/(15)N HSQC recorded without any (1)H decoupling in the indirect dimension corresponds to the classical spectrum obtained on a protein in an isotropic solution and allows the measurement of scalar J-couplings (1) J (NH). For an angle of rotation smaller than the magic angle, the bicelles orient with their normal perpendicular to the spinning axis, whereas for an angle of rotation greater than the magic angle the bicelles orient with their normal along the spinning axis. This bicelle alignment creates anisotropic conditions that give rise to the observation of residual dipolar couplings in ubiquitin. The magnitude of these dipolar couplings depends directly on the angle that the rotor makes with the main magnetic field. By changing this angle in a controlled manner, residual dipolar couplings can be either scaled up or down thus offering the possibility to study simultaneously a wide range of dipolar couplings in the same sample.     

Conformational analysis of the 3'-5'-cyclic dinucleotide d less than pApA greater than by means of molecular mechanics

The conformational properties of the cyclic dinucleotide d less than pApA greater than were studied by means of molecular mechanics calculations in which a multiconformation analysis was combined with minimum energy calculations. In this approach models of possible conformers are built by varying the torsion angles of the molecule systematically. These models are then subjected to energy minimization; in the present investigation use was made of the AMBER Force field. It followed that the lowest energy conformer has a pseudo-two-fold axis of symmetry. In this conformer the deoxyribose sugars adopt a N-type conformation. The conformation of the sugar-phosphate backbone is determined by the following torsion angles: alpha +, beta t, gamma +, epsilon t and zeta +. The conformation of this ringsystem corresponds to the structure derived earlier by means of NMR spectroscopy and X-ray diffraction. The observation of a preference for N-type sugar conformations in this molecule can be explained by the steric hindrance induced between opposite H3' atoms when one sugar is switched from N- to S-type puckers. The sugars can in principle switch from N- to S-type conformations, but this requires at least the transition of gamma + to gamma -. In this process the molecule obtains an extended shape in which the bases switch from a pseudo-axial to a pseudo-equatorial position. The calculations demonstrate that, apart from the results obtained for the lowest energy conformation, the 180 degrees change in the propagation direction of the phosphate backbone can be achieved by several different combinations of the backbone torsion angles. It appeared that in the low energy conformers five higher order correlations are found. The combination of torsion angles which are involved in changes in the propagation direction of the sugar-phosphate backbone in DNA-hairpin loops and in tRNA, are found in the dataset obtained for cyclic d less than pApA greater than. It turns out, that in the available examples, 180 degrees changes in the backbone direction are localized between two adjacent nucleotides.     

Kinematics of Visually-Guided Eye Movements

One of the hallmarks of an eye movement that follows Listing’s law is the half-angle rule that says that the angular velocity of the eye tilts by half the angle of eccentricity of the line of sight relative to primary eye position. Since all visually-guided eye movements in the regime of far viewing follow Listing’s law (with the head still and upright), the question about its origin is of considerable importance. Here, we provide theoretical and experimental evidence that Listing’s law results from a unique motor strategy that allows minimizing ocular torsion while smoothly tracking objects of interest along any path in visual space. The strategy consists in compounding conventional ocular rotations in meridian planes, that is in horizontal, vertical and oblique directions (which are all torsion-free) with small linear displacements of the eye in the frontal plane. Such compound rotation-displacements of the eye can explain the kinematic paradox that the fixation point may rotate in one plane while the eye rotates in other planes. Its unique signature is the half-angle law in the position domain, which means that the rotation plane of the eye tilts by half-the angle of gaze eccentricity. We show that this law does not readily generalize to the velocity domain of visually-guided eye movements because the angular eye velocity is the sum of two terms, one associated with rotations in meridian planes and one associated with displacements of the eye in the frontal plane. While the first term does not depend on eye position the second term does depend on eye position. We show that compounded rotation - displacements perfectly predict the average smooth kinematics of the eye during steady- state pursuit in both the position and velocity domain.     

The J-coupling Restrained Molecular Mechanics (JrMM) Protocol - An Efficient Alternative for Deriving DNA Endocyclic Torsion Angle Constraints Part II: Experimental Application of the JrMM Protocol

Abstract

The J-coupling restrained molecular mechanics (JrMM) protocol, which correlates deoxyribose endocyclic torsion angles and vicinal proton-proton torsion angle φ_{1′ 2′} in Part I of this study, is demonstrated to be a viable alternative to efficiently derive the endocyclic torsion angle constraints for the determination of the solution structures of DNA molecules. Extensive testing demonstrating the validity of the JrMM-derived torsion angle constraints in the restrained molecular dynamics and energy minimization structural refinement processes is performed theoretically using an energy-minimized B-DNA model and experimentally using a DNA hexamer d(CGTACG)₂. The results show that only a 0.2 Å difference exists between the RMSD values of the refined structures using the ideal and the JrMM-derived endocyclic torsion angle constraints. The JrMM-derived torsion angles are also determined to be in good agreement with the torsion angles derived through the use of the vicinal J-derived torsion angles. These results show that through the use of reliably measured J_{1′ 2′} values and computer simulation method, the endocyclic torsion angle constraints can be derived reliably and efficiently. Thus the JrMM method serves as an alternative strategy to generate endocyclic torsion angle constraints for the determination of the solution structures of DNA molecules.     

Effective Young's modulus of bacterial and microfibrillated cellulose fibrils in fibrous networks

The deformation micromechanics of bacterial cellulose (BC) and microfibrillated cellulose (MFC) networks have been investigated using Raman spectroscopy. The Raman spectra of both BC and MFC networks exhibit a band initially located at ≈ 1095 cm(-1). We have used the intensity of this band as a function of rotation angle of the specimens to study the cellulose fibril orientation in BC and MFC networks. We have also used the change in this peak's wavenumber position with applied tensile deformation to probe the stress-transfer behavior of these cellulosic materials. The intensity of this Raman band did not change significantly with rotation angle, indicating an in-plane 2D network of fibrils with uniform random orientation; conversely, a highly oriented flax fiber exhibited a marked change in intensity with rotation angle. Experimental data and theoretical analysis shows that the Raman band shift rate arising from deformation of networks under tension is dependent on the angles between the axis of fibrils, the strain axis, the incident laser polarization direction, and the back scattered polarization configurations. From this analysis, the effective moduli of single fibrils of BC and MFC in the networks were estimated to be in the ranges of 79-88 and 29-36 GPa, respectively. It is shown also that for the model to fit the data it is necessary to use a negative Poisson's ratio for MFC networks and BC networks. Discussion of this in-plane "auxetic" behavior is given.     

The effect of wire plane tilt and olive wires on proximal tibia fracture fragment stability and fracture site motion

To analyze the effect of the tilt angle relationship between the crossed wire plane and the bone axis on the stiffness of fine wire external fixation, load-deformation behavior was compared across different tilt angles (0 degree, 10 degrees, and 20 degrees) of the plane containing crossed smooth or olive wires under identical conditions of central axial compression, medial compression-bending, posterior compression-bending, posteromedial compression-bending, and torsion. Stiffness values were calculated from the load-deformation and torque-angle curves. A tilt angle of 20 degrees with olive wires provided significantly greater stiffness compared to smooth wires at any angle in any loading condition (p < 0.05). A tilt angle of 20 degrees with olive wires was also significantly more stiff than a tilt angle of 0 degree with olive wires in any loading condition. In torsion, olive wires with 10 degrees and 20 degrees tilt were not significantly different, while in posterior bending olive wires with 10 degrees tilt were significantly stiffer than olive wires with 0 degree or 20 degrees tilt. With smooth wires, tilting the wire plane caused a decrease in stiffness in posterior bending, posteromedial bending, and torsion. Overall, the use of olive wires in conjunction with tilting the wire plane enhances the fixation stiffness for proximal tibia fractures while allowing more options for wire configurations that avoid neurovascular and musculotendinous structures, and wounds.     

Eye Movements Induced by Changes in the Internal Representation of Body Posture

Oculomotor responses to body rotation were investigated in subjects standing with the eyes closed. A rotatable platform was used to provide body rotation relative to the space-stationary head or upper part of the body (fixation of the head; the head and the shoulders; and the head, the shoulders, and the pelvis). A slow rotation of the body about the longitudinal axis by ±6.5° within 10–150 s evoked an illusion of the upper part of the body turning in space, while the moving footplate was perceived as stationary in space. This illusion was accompanied by marked eye movements in the direction of the illusory rotation. In subjects grasping a rigid ground-based handle, the perception of body movements corresponded to the actual rotation of body parts. In this case, the amplitude of eye movements was substantially lower. It was concluded that the eye movement pattern depends not only on the actual relative movement of the body segments but also on the perception of this movement relative to the extrapersonal space.     

Orientation and location of the finite helical axis of the equine forelimb joints

To reduce anatomically unrealistic limb postures in a virtual musculoskeletal model of a horse's forelimb, accurate knowledge on forelimb joint constraints is essential. The aim of this cadaver study is to report all orientation and position changes of the finite helical axes (FHA) as a function of joint angle for different equine forelimb joints. Five horse cadaver forelimbs with standardized cuts at the midlevel of each segment were used. Bone pins with reflective marker triads were drilled into the forelimb bones. Unless joint angles were anatomically coupled, each joint was manually moved independently in all three rotational degrees of freedom (flexion–extension, abduction–adduction, internal–external rotation). The 3D coordinates of the marker triads were recorded using a six infra-red camera system. The FHA and its orientational and positional properties were calculated and expressed against joint angle over the entire range of motion using a finite helical axis method. When coupled, joint angles and FHA were expressed in function of flexion–extension angle. Flexion–extension movement was substantial in all forelimb joints, the shoulder allowed additional considerable motion in all three rotational degrees of freedoms. The position of the FHA was constant in the fetlock and elbow and a constant orientation of the FHA was found in the shoulder. Orientation and position changes of the FHA over the entire range of motion were observed in the carpus and the interphalangeal joints. We report FHA position and orientation changes as a function of flexion–extension angle to allow for inclusion in a musculoskeletal model of a horse to minimize calculation errors caused by incorrect location of the FHA.     

A two-phase finite element model of the diastolic left ventricle

A porous medium finite element model of the passive left ventricle is presented. The model is axisymmetric and allows for finite deformation, including torsion about the axis of symmetry. An anisotropic quasi-linear viscoelastic constitutive relation is implemented in the model. The model accounts for changing fibre orientation across the myocardial wall. During passive filling, the apex rotates in a clockwise direction relative to the base for an observer looking from apex to base. Within an intraventricular pressure range of 0-3 kPa the rotation angle of all nodes remained below 0.1 rad. Diastolic viscoelasticity of myocardial tissue is shown to reduce transmural differences of preload-induced sarcomere stretch and to generate residual stresses in an unloaded ventricular wall, consistent with the observation of opening angles seen when the heart is slit open. It is shown that the ventricular model stiffens following an increase of the intracoronary blood volume. At a given left ventricular volume, left ventricular pressure increases from 1.5 to 2.0 kPa when raising the intracoronary blood volume from 9 to 14 ml (100 g)-1 left ventricle.     

TANGLE: two-level support vector regression approach for protein backbone torsion angle prediction from primary sequences

Protein backbone torsion angles (Phi) and (Psi) involve two rotation angles rotating around the C(α)-N bond (Phi) and the C(α)-C bond (Psi). Due to the planarity of the linked rigid peptide bonds, these two angles can essentially determine the backbone geometry of proteins. Accordingly, the accurate prediction of protein backbone torsion angle from sequence information can assist the prediction of protein structures. In this study, we develop a new approach called TANGLE (Torsion ANGLE predictor) to predict the protein backbone torsion angles from amino acid sequences. TANGLE uses a two-level support vector regression approach to perform real-value torsion angle prediction using a variety of features derived from amino acid sequences, including the evolutionary profiles in the form of position-specific scoring matrices, predicted secondary structure, solvent accessibility and natively disordered region as well as other global sequence features. When evaluated based on a large benchmark dataset of 1,526 non-homologous proteins, the mean absolute errors (MAEs) of the Phi and Psi angle prediction are 27.8° and 44.6°, respectively, which are 1% and 3% respectively lower than that using one of the state-of-the-art prediction tools ANGLOR. Moreover, the prediction of TANGLE is significantly better than a random predictor that was built on the amino acid-specific basis, with the p-value<1.46e-147 and 7.97e-150, respectively by the Wilcoxon signed rank test. As a complementary approach to the current torsion angle prediction algorithms, TANGLE should prove useful in predicting protein structural properties and assisting protein fold recognition by applying the predicted torsion angles as useful restraints. TANGLE is freely accessible at http://sunflower.kuicr.kyoto-u.ac.jp/~sjn/TANGLE/.