A simple method to obtain consistent and clinically meaningful pelvic angles from euler angles during gait analysis

Clinical gait analysis usually describes joint kinematics using Euler angles, which depend on the sequence of rotation. Studies have shown that pelvic obliquity angles from the traditional tilt-obliquity-rotation (TOR) Euler angle sequence can deviate considerably from clinical expectations and have suggested that a rotation-obliquity-tilt (ROT) Euler angle sequence be used instead. We propose a simple alternate approach in which clinical joint angles are defined and exactly calculated in terms of Euler angles from any rotation sequence. Equations were derived to calculate clinical pelvic elevation, progression, and lean angles from TOR and ROT Euler angles. For the ROT Euler angles, obliquity was exactly the same as the clinical elevation angle, rotation was similar to the clinical progression angle, and tilt was similar to the clinical lean angle. Greater differences were observed for TOR. These results support previous findings that ROT is preferable to TOR for calculating pelvic Euler angles for clinical interpretation. However, we suggest that exact clinical angles can and should be obtained through a few extra calculations as demonstrated in this technical note.     

Characterizing the relative orientation and dynamics of RNA A-form helices using NMR residual dipolar couplings

We present a protocol for determining the relative orientation and dynamics of A-form helices in 13C/15N isotopically enriched RNA samples using NMR residual dipolar couplings (RDCs). Non-terminal Watson-Crick base pairs in helical stems are experimentally identified using NOE and trans-hydrogen bond connectivity and modeled using the idealized A-form helix geometry. RDCs measured in the partially aligned RNA are used to compute order tensors describing average alignment of each helix relative to the applied magnetic field. The order tensors are translated into Euler angles defining the average relative orientation of helices and order parameters describing the amplitude and asymmetry of interhelix motions. The protocol does not require complete resonance assignments and therefore can be implemented rapidly to RNAs much larger than those for which complete high-resolution NMR structure determination is feasible. The protocol is particularly valuable for exploring adaptive changes in RNA conformation that occur in response to biologically relevant signals. Following resonance assignments, the procedure is expected to take no more than 2 weeks of acquisition and data analysis time.     

A fully automatic 3D reconstruction method using simulated annealing enables accurate posterioric angular assignment of protein projections

Single-particle analysis is a structure determining method using electron microscopic (EM) images, which does not require protein crystal. In this method, projections are picked up and used to reconstruct a three-dimensional (3D) structure. When the conical tilting method is not available, the particle images are usually classified and averaged to improve the signal-to-noise ratio. The Euler angles of these average images must be posteriorically assigned to create a primary 3D model. We developed a new, fully automatic unsupervised Euler angle assignment method, which does not require an initial 3D reference and which is applicable to asymmetric molecules. In this method, the Euler angle of each average image is initially set randomly and then automatically corrected in relation to those of the other averages by iterated optimizations using the Simulated Annealing (SA) algorithm. At each iteration, the 3D structure is reconstructed based on the current Euler angles and reprojected back in the average-input directions. A modified cross-correlation between each reprojection and its corresponding original average is then calculated. The correlations are summed as a total 3D echo-correlation score to evaluate the Euler angles at this iteration. Then, one of the projections is selected, its Euler angle is changed randomly, and the score is also calculated. Based on the score change, judgment of whether to accept or reject the new angle is made using the SA algorithm, which is introduced to overcome the local minimums. After a certain number of iterations of this process, the angles of all averages converge so as to create a reliable primary 3D model. This echo-correlated 3D reconstruction with simulated annealing also has potential for wide application to general 3D reconstruction from various types of 2D images.     

Antibody elbow angles are influenced by their light chain class

We have examined the elbow angles for 365 different Fab fragments, and observe that Fabs with lambda light chains have adopted a wider range of elbow angles than their kappa chain counterparts, and that the lambda light chain Fabs are frequently found with very large (>195 degrees ) elbow angles. This apparent hyperflexibility of lambda chain Fabs may be due to an insertion in their switch region, which is one residue longer than in kappa chains, with glycine occurring most frequently at the insertion position. A new, web-based computer program that was used to calculate the Fab elbow angles is described.     

Determining the three-dimensional relation between the skeletal elements of the human shoulder complex

In this paper, we present an inverse kinematics method to determining human shoulder joint motion coupling relationship based on experimental data in the literature. This work focuses on transferring Euler-angle-based coupling equations into a relationship based on the Denavit–Hartenberg (DH) method. We use analytical inverse kinematics to achieve the transferring. For a specific posture, we can choose points on clavicle, scapula, and humerus and represent the end-effector positions based on Euler angles or DH method. For both Euler and DH systems, the end-effectors have the same Cartesian positions. Solving these equations related to end-effector positions yields DH joint angles for that posture. The new joint motion coupling relationship is obtained by polynomial and cosine fitting of the DH joint angles for all different postures.     

Use of dual Euler angles to quantify the three-dimensional joint motion and its application to the ankle joint complex

This paper presents a modified Euler angles method, dual Euler angles approach, to describe general spatial human joint motions. In dual Euler angles approach, the three-dimensional joint motion is considered as three successive screw motions with respect to the axes of the moving segment coordinate system; accordingly, the screw motion displacements are represented by dual Euler angles. The algorithm for calculating dual Euler angles from coordinates of markers on the moving segment is also provided in this study. As an example, the proposed method is applied to describe motions of ankle joint complex during dorsiflexion–plantarflexion. A Flock of Birds electromagnetic tracking device (FOB) was used to measure joint motion in vivo. Preliminary accuracy tests on a gimbal structure demonstrate that the mean errors of dual Euler angles evaluated by using source data from FOB are less than 1° for rotations and 1 mm for translations, respectively. Based on the pilot study, FOB is feasible for quantifying human joint motions using dual Euler angles approach.     

Phosphorus-31 nuclear magnetic resonance of highly oriented DNA fibers. 1. Static geometry of DNA double helices

The static geometry of the phosphodiesters in oriented fibers of DNA and a variety of polynucleotides was investigated by solid-state 31P nuclear magnetic resonance (NMR) spectroscopy. The structural parameters of the phosphodiester backbone expressed by two Euler angles beta and gamma were estimated on the basis of the NMR spectra of natural DNA, poly(dA).poly(dT), poly(rA).poly(dT), and poly-(rA).poly(rU). The Euler angles were calculated by using the known single crystal structures of a decamer, r(GCG)d(TATACGC), and a dodecamer, d(CGCGAATTCGCG). The distribution pattern of the Euler angles was quite different between these two oligonucleotides due to the different types of conformation, and it was fully consistent with the 31P NMR results, showing that the conformation of the B form DNA is very heterogeneous while that of the A or A' form is much more invariable with regard to the base composition. The structural parameters were also calculated by using various structures determined by the X-ray fiber diffraction studies, and they were evaluated on the basis of the 31P NMR data. Notably, poly(dA).poly(dT) fibers exhibited abnormal 31P NMR spectra which were very broad in line width and were not appreciably perturbed by hydration; a coiled double-helical structure is proposed as the most plausible model for this polymer.     

The analysis of golf swing as a kinematic chain using dual Euler angle algorithm

The manner in which anatomical rotation from an individual segment contributes to the position and velocity of the endpoint can be informative in the arena of many athletic events whose goals are to attain the maximal velocity of the most distal segment. This study presents a new method of velocity analysis using dual Euler angles and its application in studying rotational contribution from upper extremity segments to club head speed during a golf swing. Dual Euler angle describes 3D movement as a series of ordered screw motions about each orthogonal axis in a streamlined matrix form-the dual transformation matrix- and allows the translation and rotation component to be described in the same moving frame. Applying this method in biomechanics is a novel idea and the authors have previously applied the methodology to clinical studies on its use in displacement analysis. The focus of this paper is velocity analysis and applications in sports biomechanics. In this study, electrogoniometers (Biometrics, UK) with a frequency of 1000 Hz were attached to a subject during the execution of the swing to obtain the joint angles throughout the motion. The velocity of the club head was then analyzed using the dual velocity which specifies the velocity distribution of a rigid body in screw motion at any point in time as the dual vector. The contributions of each segment to the club-head velocity were also compared. In order to evaluate this method, the calculated position and velocity of the club head were compared to the values obtained from video image analysis. The results indicated that there is good agreement between calculated values and video data, suggesting the suitability of using the Dual Euler method in analyzing a kinematic chain motion.     

Interpolation of segment Euler angles can provide a robust estimation of segment angular trajectories during asymmetric lifting tasks

Video-based field methods that estimate L5/S1 net joint moments from kinematics based on interpolation in the sagittal plane of joint angles alone can introduce a significant error on the interpolated joint angular trajectory when applied to asymmetric dynamic lifts. Our goal was to evaluate interpolation of segment Euler angles for a wide range of dynamic asymmetric lifting tasks using cubic spline methods by comparing the interpolated values with the continuous measured ones. For most body segments, the estimated trajectories of segment Euler angles have less than 5° RMSE (in each dimension) with 5-point cubic spline interpolation when there is no measurement error of interpolation points. Sensitivity analysis indicates that when the measurement error exists, the root mean square error (RMSE) of estimated trajectories increases. Comparison among different lifting conditions showed that lifting a load from a high initial position yielded a smaller RMSE than lifting from a low initial position. In conclusion, interpolation of segment Euler angles can provide a robust estimation of segment angular trajectories during asymmetric lifting when measurement error of interpolation points can be controlled at a low level.     

Determining dual Euler angles of the ankle complex in vivo using "flock of birds" electromagnetic tracking device

The dual Euler angles method has been proposed as an alternative approach to describe the general spatial human joint motion. In this study, the dual Euler angles method was applied to study the three-dimensional motion of the ankle complex. The methodology for obtaining dual Euler angles of the ankle complex was developed by using a "Flock of Birds" electromagnetic tracking device. The repeatability of the methodology was studied based on the intertester and intratester variability analysis. Finally kinematic coupling characteristics of the ankle complex during dorsiflexion-plantarflexion, eversion-inversion, and abduction-adduction were analyzed according to the parameters of the dual Euler angles.     

Angle determination for side views in single particle electron microscopy

In order to build a first model in single particle electron microscopy the relative angular orientation of each image of a protein complex must be determined. These orientations can be described by three Eulerian angles. Images of complexes that present the same view can be aligned in two-dimensions and averaged in order to increase their signal-to-noise ratio. Based on these averaged images, several standard approaches exist for determining Euler angles for randomly oriented projection images. The common lines and angular reconstitution methods work well for particles with symmetry while the random conical tilting and related orthogonal tilt reconstruction methods work in most cases but require the acquisition of tilt pairs of images. For the situation where views of particles can be identified that are rotations about a single axis parallel to the grid, an alternative algorithm to determine the orientations of class averages without the need to acquire tilt pairs can be applied. This type of view of a complex is usually called a side view. This paper describes the detailed workings and characterization of an algorithm, named rotational analysis, which uses real-space fiducial markers derived from the averages themselves to determine the Euler angles for side views. We demonstrate how this algorithm works in practice by applying it to a data set of images of affinity-purified bovine mitochondrial ATP synthase.     

Globographic visualisation of three dimensional joint angles

Three different methods for describing three dimensional joint angles are commonly used in biomechanics. The joint coordinate system and Cardan/Euler angles are conceptually quite different but are known to represent the same underlying mathematics. More recently the globographic method has been suggested as an alternative and this has proved particularly attractive for the shoulder joint. All three methods can be implemented in a number of ways leading to a choice of angle definitions. Very recently Rab has demonstrated that the globographic method is equivalent to one implementation of the joint coordinate system. This paper presents a rigorous analysis of the three different methods and proves their mathematical equivalence. The well known sequence dependence of Cardan/Euler is presented as equivalent to configuration dependence of the joint coordinate system and orientation dependence of globographic angles. The precise definition of different angle sets can be easily visualised using the globographic method using analogues of longitude, latitude and surface bearings with which most users will already be familiar. The method implicitly requires one axis of the moving segment to be identified as its principal axis and this can be extremely useful in helping define the most appropriate angle set to describe the orientation of any particular joint. Using this technique different angle sets are considered to be most appropriate for different joints and examples of this for the hip, knee, ankle, pelvis and axial skeleton are outlined.     

Spherical reconstruction: a method for structure determination of membrane proteins from cryo-EM images

We propose a new method for single-particle reconstruction, which should be generally applicable to structure determination for membrane proteins. After reconstitution into a small spherical vesicle, a membrane protein takes a particular orientation relative to the membrane normal, and its position in the projected image of the vesicle directly defines two of its three Euler angles of orientation. The spherical constraint imposed by the vesicle effectively reduces the dimensionality of the alignment search from 5 to 3 and simplifies the detection of the particle. Projection images of particles in vesicles collectively take all possible orientations and therefore cover the whole Fourier space. Analysis of images of vesicles in ice showed that the vesicle density is well described by a simple model for membrane electron scattering density. In fitting this model we found that osmotically swollen vesicles remain nearly spherical through the freezing process. These results satisfy the basic experimental requirements for spherical reconstruction. A computer simulation of particles in vesicles showed that this method provides good estimates of the two Euler angles and thus may improve single-particle reconstruction and extend it to smaller membrane proteins.     

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.     

New mathematical definition and calculation of axial rotation of anatomical joints

In the field of joint kinematics, clinical terms such as internal-external, or medical-lateral, rotations are commonly used to express the rotation of a body segment about its own long axis. However, these terms are not defined in a strict mathematical sense. In this paper, a new mathematical definition of axial rotation is proposed and methods to calculate it from the measured Euler angles are given. The definition and methods to calculate it from the measured Euler angles are given. The definition is based on the integration of the component of the angular velocity vector projected onto the long axis of the body segment. First, the absolute axial rotation of a body segment with respect to the stationary coordinate system is defined. This definition is then generalized to give the relative axial rotation of one body segment with respect to the other body segment where the two segments are moving in the three-dimensional space. The well-known Codman's paradox is cited as an example to make clear the difference between the definition so far proposed by other researchers and the new one.     

Nucleic Acid Structure Analysis: A Users Guide to a Collection of New Analysis Programs

Abstract

Common nomenclature describing the geometry of nucleic acid structures was established at a 1988 EMBO Workshop on DNA Curvature and Bending (Diekmann, S. (1988) J. Mol. Biol. 208, 787–791; Diekmann, S. (1989) The EMBO Journal 8, 1–4; Sarma, RH. (1988) J. Biomol. Structure & Dynamics 6, 391–395; Dickerson, R.E. (1989) J. Biomol. Structured Dynamics 6, 627–634; Dickerson, RE. et al. (1989)Nuc. Acids Res. 17, 1979–1803). We have subsequently developed and incorporated sophisticated mathematics in a computer program to calculate the parameters described by the guidelines. The program calculates all the local parameters relating complementary bases and neighboring base and base pairs in both Cartesian and helical coordinate frames. In addition, the main mathematical property requested by the EMBO guidelines—that the magnitude of the parameters be independent of strand or direction of measurement—is accomplished without the use of a midway coordinate frame for the rotational parameters. The mathematics preserve the physical intuition used in defining the parameters; in particular, the rotational parameters are true rotations based on a simple physical model (rotation at constant angular velocity for a unit amount of time), not Euler angles or angles between vectors and planes as is the case with other approaches. As a result, the mathematical equations are symmetric with the property that a 5° tilt is the same as a 5° roll or a 5° twist, except that the rotations take place about different axes. In other approaches, a 5° tilt can mean a different amount of net rotation from a 5° roll or a 5° twist. In addition, a great deal of flexibility is built into the program so that the user has control over the analysis, including the input format, the coordinate frame used for the base pairing relationship, the point about which the rotations are performed, and which geometric relationships are analyzed. While there is a great deal of flexibility, the program is easy to use. Interactive queries and user accessible files make the options in the program very convenient to tailor to individual needs. In addition, there is also a program that calculates bond lengths, valence angles, and torsion angles along the nucleic acid backbone, and within the sugar and base rings. Another program ‘learns’ the identities of the bond lengths, valence angles, and torsion angles that the user would like to determine. This last program is especially useful for calculating the hydrogen bonds between atoms in complementary strands as well as the unusual hydrogen bonds found in recently determined nucleic acid NMR structures or within protein/nucleic acid complexes.