Influence of flanking residues on tilt and rotation angles of transmembrane peptides in lipid bilayers. A solid-state 2H NMR study

To gain insight into the parameters that determine the arrangement of proteins in membranes, (2)H NMR experiments were performed to analyze tilt and rotation angles of membrane-spanning alpha-helical model peptides upon incorporation in diacylphosphatidylcholine bilayers with varying thickness. The peptides consisted of the sequence acetyl-GW(2)(LA)(8)LW(2)A-NH(2) (WALP23) and analogues thereof, in which the interfacial Trp residues were replaced by Lys (KALP23) and/or the hydrophobic sequence was replaced by Leu (WLP23 and KLP23). The peptides were synthesized with a single deuterium-labeled alanine at four different positions along the hydrophobic segment. For all peptides a small but systematic increase in tilt angle was observed upon decreasing the bilayer thickness. However, significantly larger tilt angles were obtained for the Lys-flanked KALP23 than for the Trp-flanked WALP23, suggesting that interfacial anchoring interactions of Trp may inhibit tilting. Increasing the hydrophobicity resulted in an increase in tilt angle for the Trp-flanked analogue only. For all peptides the maximum tilt angle obtained was remarkably small (less than 12 degrees ), suggesting that further tilting is inhibited, most likely due to unfavorable packing of lipids around a tilted helix. The results furthermore showed that the direction of tilt is determined almost exclusively by the flanking residues: Trp- and Lys-flanked peptides were found to have very different rotation angles, which were influenced significantly neither by hydrophobicity of the peptides nor by the extent of hydrophobic mismatch. Finally, very small changes in the side chain angles of the deuterated alanine probes were observed in Trp-flanked peptides, suggesting that these peptides may decrease their hydrophobic length to help them to adapt to thin membranes.     

The effect of anisotropic systematic errors in estimating helical angles

A common question in movement studies is how the results should be interpreted with respect to systematic and random errors. In this study, simulations are made in order to see how a rigid body's orientation in space (i.e. helical angle between two orientations) is affected by (1) a systematic error added to a single marker (2) a combination of this systematic error and Gaussian white noise. The orientation was estimated after adding a systematic error to one marker within the rigid body. This procedure was repeated with Gaussian noise added to each marker.

In conclusion, results show that the systematic error's effect on estimated orientation depends on number of markers in the rigid body and also on which direction the systematic error is added. The systematic error has no effect if the error is added along the radial axis (i.e. the line connecting centre of mass and the affected marker).     

The effect of anisotropic systematic errors in estimating helical angles

A common question in movement studies is how the results should be interpreted with respect to systematic and random errors. In this study, simulations are made in order to see how a rigid body's orientation in space (i.e. helical angle between two orientations) is affected by (1) a systematic error added to a single marker (2) a combination of this systematic error and Gaussian white noise. The orientation was estimated after adding a systematic error to one marker within the rigid body. This procedure was repeated with Gaussian noise added to each marker. In conclusion, results show that the systematic error's effect on estimated orientation depends on number of markers in the rigid body and also on which direction the systematic error is added. The systematic error has no effect if the error is added along the radial axis (i.e. the line connecting centre of mass and the affected marker).     

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.     

A survey of formal methods for determining the centre of rotation of ball joints

The determination of an accurate centre of rotation (CoR) from segment marker positions is of interest across a wide range of applications, but particularly for clinical gait analysis and for estimating the hip joint centre during surgical intervention of the knee, for limb alignment purposes. For the first time in this survey of formal methods, we classify, analyse and compare different methods (geometric, algebraic, bias compensated algebraic, and Pratt sphere fit methods, as well as the centre transformation technique, the Holzreiter approach, the helical pivot technique, the Schwartz transformation techniques, the minimal amplitude point method and the Stoddart approach) for the determination of spherical joint centres from marker position data. In addition, we propose a new method, the symmetrical CoR estimation or SCoRE, in which the coordinates of the joint centre must only remain constant relative to each segment, thus not requiring the assumption that one segment should remain at rest. For each method, 1000 CoR estimations were analysed with the application of isotropic, independent and identically distributed Gaussian noise (standard deviation 0.1cm) to each of the marker positions, to all markers on the segment simultaneously and the two in combination. For the test conditions used here, most techniques were capable of determining the CoR to within 0.3 cm, as long as the spherical range of motion (RoM) of the joint was 45 degrees or more. Under the most stringent conditions tested, however, the SCoRE was capable of best determining the CoR, to within approximately 1.2mm with a RoM of 20 degrees . The correct selection and application of these methodologies should help improve the accuracy of surgical navigation and clinical kinematic measurement.     

Developmental communalities of homologous and non-homologous body joints

Reducing 4970 center- and sex-specific correlations for age at appearance of postnatal ossification centers to 15 mean values of r representing homologous and non-hmologous inter-limb and intra-limb joint segment communalities, it was shown that homologous joint segments in different limbs (hand-foot, elbow-knee, shoulder-hip) showed systematically higher developmental communality than intra-limb non-homologous joints, and these in turn exceeded mean r values for inter-limb non-homologous joint segments.     

A non-linear model for visual-vestibular interaction during body rotation in man

A mathematical model for visual-vestibular interaction during body rotation in an illuminated visual surround is obtained by combining a previous model of the optokinetic reflex (OKR) with a simplified model of the vestibulo-ocular reflex (VOR). OKR is activated by the slip of the image of the external world on the retina, and represents a negative feedback loop around VOR. For large retinal slip velocities OKR behaves as a basically non-linear system. The validity of the model is proved via computer simulation by comparing predicted responses with the experimental results obtained in man by Koenig et al. (1978) in different situations of visual-vestibular interaction.Work supported by C.N.R. (Rome, Italy), Special Project on Piomedical Engineering, Grant No. 79.01255.86     

Evolution of body size: an optimization model

The growth of individuals is often described by bioenergetic equations which partition assimilated energy into maintenance, growth, reproduction, and so on. Such energy flows vary with body size. The bioenergetic equations in this paper, under the assumptions that organisms will adopt behavior that channels a maximum amount of energy into the production of surviving progeny, produce a model optimizing growth and reproduction. Using the Weierstrass theorem, we show that a solution of the optimization problem exists. The problem is further solved analytically using the Pontryagin maximum principle. The main conclusion of the paper is that in a given environment an optimal body size exists, one which maximizes the energy channeled to the production of progeny. This body size depends on the mortality rate, the maximum life span, and the derivative of the growth equation with respect to body size. The biological results predicted from the model are compared with ecological data for zooplankton and vertebrate species, which support the conclusions obtained.     

Multi-body optimization with subject-specific knee models: performance at high knee flexion angles

When estimating knee kinematics from skin markers and stereophotogrammetry, multi-body optimization (MBO) has provided promising results for reducing soft tissue artefacts (STA), but can still be improved. The goal of this study was to assess the performance of MBO with subject-specific knee models at high knee flexion angles (up to 110°) against knee joint kinematics measured by magnetic resonance imaging. Eight subjects were recruited. MBO with subject-specific knee models was more effective in compensating STA compared to no kinematic and spherical constraints, in particular for joint displacements. Moreover, it seems to be more reliable over large ranges of knee flexion angle. The ranges of root mean square errors for knee rotations/displacements were 3.0°–9.2°/1.3–3.5 mm for subject-specific knee models, 6.8°–8.7°/6.0–12.4 mm without kinematic constraint and 7.1°–9.8°/4.9–12.5 mm for spherical constraints.     

Joint axes of rotation and body segment parameters of pig limbs

To enable a quantification of net joint moments and joint reaction forces, indicators of joint loading, this study aimed to locate the mediolateral joint axes of rotation and establish the body segment parameters of the limbs of pigs (Sus scrofa). To locate the joint axes of rotation the scapulohumeral, humeroradial, carpal complex, metacarpophalangeal, coxofemoral, femorotibial, tarsal, and metatarsophalangeal joints from 12 carcasses were studied. The joints were photographed in three positions, bisecting lines drawn at fixed landmarks with their intersection marking the joint axes of rotation. The body segment parameters, i.e. the segment mass, center of mass and moment of inertia were measured on the humerus, radius/ulna, metacarpus, forepastern, foretoe, femur, tibia, metatarsus, hindpastern, and hindtoe segments from five carcasses. The segments were weighed, and their center of mass was found by balancing them. The moments of inertia of the humerus, radius/ulna, femur and tibia were found by rotating the segments. The moments of inertia of the remaining segments were calculated. Generally, the joint axes of rotation were near the attachment site of the lateral collateral ligaments. The forelimb, with segments taken as one, was significantly lighter and shorter than the hindlimb (P < 0.001). In all segments the center of mass was located 31 to 50% distal to the proximal segment end. The segment mass decreased with distance from the trunk, as did the segment moment of inertia. The results may serve as reference on the location of the joint axes of rotation and on the body segment parameters for inverse dynamic modeling of pigs.     

Variation of cellulose microfibril angles in softwoods and hardwoods-a possible strategy of mechanical optimization

Position-resolved small-angle X-ray scattering was used to investigate the nanostructure of the wood cell wall in two softwood species (Norwegian spruce and Scots pine) and two hardwood species (pedunculate oak and copper beech). The tilt angle of the cellulose fibrils in the wood cell wall versus the longitudinal cell axis (microfibril angle) was systematically studied over a wide range of annual rings in each tree. The measured angles were correlated with the distance from the pith and the results were compared. The microfibril angle was found to decrease from pith to bark in all four trees, but was generally higher in the softwood than in the hardwood. In Norwegian spruce, the microfibril angles were higher in late wood than in early wood; in Scots pine the opposite was observed. In pedunculate oak and copper beech, low angles were found in the major part of the stem, except for the very first annual rings in pedunculate oak. The results are interpreted in terms of mechanical optimization. An attempt was made to give a quantitative estimation for the mechanical constraints imposed on a tree of given dimensions and to establish a model that could explain the general decrease of microfibril angles from pith to bark.     

Constrained least-squares optimization for robust estimation of center of rotation

This paper presents a new direct method for estimating the average center of rotation (CoR). An existing least-squares (LS) solution has been shown by previous works to have reduced accuracy for data with small range of motion (RoM). Alternative methods proposed to improve the CoR estimation use iterative algorithms. However, in this paper we show that with a carefully chosen normalization scheme, constrained least-squares solutions can perform as well as iterative approaches, even for challenging problems with significant noise and small RoM. In particular, enforcing the normalization constraint avoids poor fits near plane singularities that can affect the existing LS method. Our formulation has an exact solution, accounts for multiple markers simultaneously, and does not depend on manually-adjusted parameters. Simulation tests compare the method to four published CoR estimation techniques. The results show that the new approach has the accuracy of the iterative methods as well as the short computation time and repeatability of a least-squares solution. In addition, application of the new method to experimental motion capture data of the thumb carpometacarpal (CMC) joint yielded a more plausible CoR location compared to the previously reported LS solution and required less time than all four alternative techniques.     

Calculating the axes of rotation for the subtalar and talocrural joints using 3D bone reconstructions

Orientation of the subtalar joint axis dictates inversion and eversion movements of the foot and has been the focus of evolutionary and clinical studies for a number of years. Previous studies have measured the subtalar joint axis against the axis of the whole foot, the talocrural joint axis and, recently, the principal axes of the talus. The present study introduces a new method for estimating average joint axes from 3D reconstructions of bones and applies the method to the talus to calculate the subtalar and talocrural joint axes. The study also assesses the validity of the principal axes as a reference coordinate system against which to measure the subtalar joint axis. In order to define the angle of the subtalar joint axis relative to that of another axis in the talus, we suggest measuring the subtalar joint axis against the talocrural joint axis. We present corresponding 3D vector angles calculated from a modern human skeletal sample. This method is applicable to virtual 3D models acquired through surface-scanning of disarticulated 'dry' osteological samples, as well as to 3D models created from CT or MRI scans.