Derivation of the space horopter.

The horopter--the locus of those points in space that would stimulate corresponding points on the retinae of the two eyes--has been previously considered to be a plane curve lying in the horizontal plane. The two-dimensional character of this curve arises as a consequence of limiting all considerations to two dimensions only. However, by considering the retina as a two-dimensional surface in 3-space, geometric analysis reveals the horopter to be a non-planar curve: a twisted cubic curve in space. The classical horopter experiments can then be seen to be plotting self-corresponding lines rather than self-corresponding points, and these lines are found to be the chords of this cubic curve. The equations determining the horopter curve in parametric form have been found expressing each point of the curve as a function of the coordinates of the point of fixation.     

A note on the horopter

By assuming the fixity (but not the symmetry) of corresponding points on the two retinae, it is possible to derive the equation of any horopter when one is known. In particular when, as experiment shows, one horopter is linear, then all horopters must be conics. These have the form given by Ogle, but whereas Ogle leaves one parameter undetermined at each fixation, on our assumption the only arbitrary parameter is determined by the position of the linear horopter.     

The probability of obtaining complex kinetic curves for enzyme mechanisms with cubic terms in the pseudo-steady state rate equations

A number of details required for the classification of 3 : 3 double reciprocal plots are provided. It is shown that the ν(S) plot for a 3 : 3 function can have at most four inflexions and at most two inflexions adjacent to a turning point. Using this information, a classification of 3 : 3 ν(S) plots into ten main varieties with several subclasses is reported. The problem of defining the probability with which a given mechanism can give rise to specific curve shape features is considered. Applying this technique, the probability with which four simple enzyme mechanisms can give rise to 3 : 3 curve shapes is computed. It is shown that a 3 : 3 saturation function can have no turning points, at most two inflexions and at most one inflexion in double reciprocal space. The probability with which the available 3 : 3 shapes can arise is also computed. It is concluded that, with realistic values for rate constants, chemically reasonable enzyme mechanisms leading to rate equations of degree n : n can generate most of the kinetic profiles available to a rational function of degree n : n with positive coefficients. The probability of obtaining specific curve shapes is not so characteristic of the particular mechanism for 3:3 rate equations as it is for 2:2 rate equations. The probability of obtaining highly complex curves with several turning points or inflexions is rather lower for the enzyme mechanisms than with general 3 : 3 rational functions. There is a high probability that 3 : 3 mechanisms will generate kinetic curves that are geometrically similar to those possible for degree 2 : 2 but this is not so for binding isotherms. Hence differentiating 3 : 3 from 2 : 2 rate equations from experimental kinetic data is more likely to be successful by non-linear regression to the whole data set than by demonstrating a specific 3 : 3 feature. Binding curves, on the other hand, for three or more sites should give Scatchard plots with inflexions, features not possible with second degree equations which are conic sections in this space.     

Distribution of mapping points of 20 amino acids in the tetrahedral space

Summary Based on the genetic codes and a simple theorem for the geometrical property of the regular tetrahedron, each amino acid is mapped onto a unique point in a 3-dimensional tetrahedral space. The distribution of the 20 mapping points for 20 amino acids is studied in detail. It is found that the mapping points for the hydrophobic and hydrophilic amino acids are distributed at distinct regions in the 3-dimensional space. A plane separating the two kinds of points satisfactorily based on the Fisher's algorithm has been calculated. It is shown that the codons coding for the hydrophobic amino acids are constituted dominantly by the bases of keto group, i.e., G and T. While the codons coding for the hydrophilic amino acids are constituted dominantly by the bases of amino group, i.e., A and C. The biological implication of the mapping points and the separating plane has been discussed in some details.     

A spatial property of the retino-cortical mapping

Striate cortex topography derives from a stretching of retinal space along the optic axis. At the retina, relative distances are preserved in a mapping of retinal space onto a spherical surface in the environment. At the cortex, relative distances along visual meridia in the cortical map are preserved in a mapping of striate cortex onto an environmental conic surface whose base is in the plane of the eye. This eco-cortical relationship can be considered a reference frame through which spatial relationships at the cortex might provide information about the environment. The present analysis provides an explanation of changes in cortical magnification with visual eccentricity in the primate and a detailed three-dimensional model of striate topography for the macaque monkey. In man, a conic environmental surface is shown to be uniformly resolvable along meridia in the visual field. Finally, the implications of this analysis of the structural properties of the retino-striate pathway and visual resolution are considered in relation to depth and distance perception.     

Acetabular morphology and resurfacing design

The bony surfaces of 18 archaeological hemipelves were scanned using a 3D laser surface scanner and CyDir™ software on a Silicon Graphics workstation. The acetabular area was selected and point data from the approximately spherical bone surface saved. These data were input to a MATLAB routine that calculated the radius and centre of the best-fit sphere. The goodness of fit was estimated using the mean and standard deviation of the distance of the bone surface points from the sphere surface. Eight points, at approximately equal distances around the acetabular rim, were selected with reference to bony landmarks. A plane containing three of these points served as an orientation reference plane. The vectors joining the eight rim points to the centre of the best-fit sphere were found. The angles between these vectors and the normal to the reference plane were calculated. Paired angles were summed to give the angle subtended by the acetabular rim in four directions. The overall mean angle was 158° (range of mean angles 145°–173°). The largest individual angles, some exceeding 180°, were in the superior–inferior direction, while the mean angle in the anterior–posterior direction, i.e. that controlling flexion-extension, was 152°. Males had larger subtended angles than females, although the difference was not statistically significant. Simulated reaming increased all angles by approximately 10°. The subtended angles are important parameters in the design of the acetabular component of a hip replacement and particularly important in resurfacing hip replacement when the volume available is tightly constrained.     

Target point selection during scanning eye movements

The process of selection of target points during voluntary eye movements when polygonal random shapes are observed was analysed in humans by means of an eye movement recording technique. A computer model was constructed with the aim to explain the empirical results. It has been found that the majority of fixation points were located at the angles. The marginal distribution over the x-axis of the individual angles depends on the angle's size: the maximal value of distribution was found for acute angles more distant from the vertex than in obtuse ones. The distribution of output activity of the ganglion cells in a computer model, reproducing some basic features of the retina, is in good agreement with the empirical results.     

Polarised absorption spectroscopy of chlorophyll-lipid membranes

A technique has been developed for measuring visible absorption spectra of chlorophyll in lipid membranes. An expression is derived which enables the directions of the transition moments of the different absorption bands to be determined from polarisation data. It is found that the transition moments of the principal blue and red absorption bands of chlorophyll a make angles of 26° and 36.5° respectively with the plane of the membrane. On the assumption that these two transitions lie in the plane of the porphyrin ring and are mutually perpendicular, it may be deduced that the plane of the porphyrin ring is tilted at approx. 48° to the membrane surface. For chlorophyll b the transition moments of the blue and red bands are found to make angles of 29.5° and 36.5° with the plane of the bilayer, giving an angle of tilt of the porphyrin ring of approx. 51°.

These results are compared with measurements of dichroism in vivo.     

Derivation of the angle of torsion of the eye.

The angle of torsion of the eye, that is the amount by which the eye rotates about its anterio-posterior axis, changes with the direction of the line of fixation. If the eye can be considered as a rigid body under an elaborate system for constraints its motion can be described by the laws of kinematics. The techniques of linear algebra can then be used to derive an explicit relationship between the angles defining the direction of fixation and the angle of torsion of the eye. The angle of torsion can then be expressed as a function of the angle of rotation of the eye and the direction cosines of the axis of rotation. However, under conditions in which Listing's Law holds these angles, are well defined functions of the angles defining the direction of fixation, and so the angle of torsion, in this case, can be expressed as an explicit function of this direction.     

Evidence for surface-based processing of binocular disparity

It is convenient to think of an object's location as a point within a Cartesian framework; the x axis corresponds to right and left, the y axis to up and down, and the z axis to forward or backward. When an observer is looking straight ahead, binocular disparities provide information about distance along the z axis from the fixation plane. In this coordinate system, changes in disparity are treated as independent of changes in location along the orthogonal x and y axes. Does the human visual system use this three-dimensional coordinate system, or does it specify feature location in a coordinate frame determined by other nearby visible features? Here we show that the sensitivity of the human stereo system is determined by the distance of points with respect to a local reference plane, rather than by the distance along the z axis with respect to the fixation plane. There is a distinct advantage to using a local frame of reference for specifying location. It obviates the need to construct a complex three-dimensional space in either eye-centered or head-centered coordinates that must be updated with every shift of the eyes and head.     

The plasticity of correspondence: After-effects,illusions and horopter shifts in depth perception

Retinal disparity is the cue for stereoscopic depth perception. Disparity detection begins with cortical single units driven binocularly from the two eyes. A previous paper (Nelson, 1975) has shown that inhibitory and facilitatory interactions are essential to insure successful disparity detection, particularly with repeating stimulus patterns, and that such a system will display all the appropriate properties of sensory fusion. This paper shows that most depth illusions occur as by-products of the same domain interactions. Such illusion effects fall into two classes: those caused by shifts in the distribution of activity along the appropriate sensory domain (here, the disparity domain) and those caused by changes in the average activity level within the domain. Profile shifts cause depth contrast illusions. The most important profile level change is an activity lowering due to disparity domain inhibition. This adversely affects fusional range (Panum's area). It is postulated that all domain interactions persist following cessation of stimulation. Persistent profile shifts cause depth after-effect illusions; persistent profile lowering is responsible for threshold elevation after-effects.Sensory fusion, the coding errors seen in illusions, the induced effect, and widespread failure to perceive depth from disparity input show that retinal correspondence is not stable in the normal individual. Yet horopter research has attempted to specify one set of retinal points as corresponding. Not surprisingly, horopter research shows systematic shifts in retinal correspondence linked to eye position. Small, simple, tonic modulations of the domain interactions responsible for so many other stereopsis system properties provide a satisfactory cortical mechanism for horopter changes.     

Normal modes of vibration in bovine pancreatic trypsin inhibitor and its mechanical property

The normal mode analysis of conformational fluctuation is carried out for a small globular protein, bovine pancreatic trypsin inhibitor. Results are analyzed mainly to reveal the mechanical construction of the protein molecule. We take dihedral angles, including peptide omega angles, as independent variables for the normal mode analysis. There are 306 such angles in this molecule. Motions in modes with frequencies lower than 120 cm-1 are shown to involve atoms in the whole protein molecule, and spatial change of displacement vectors is continuous, i.e., those of atoms near in space are similar. To quantitate the observation of the continuity, a correlation function of direction vectors of atomic displacements is calculated. From this function we define a quantity that is interpreted as the wave length of an equivalent elastic plane wave. From this quantity we deduce effective Young's modulus for each mode. For the mode with the lowest frequency 4.4 cm-1, it turned out to be 0.8 x 10(9) dyn cm-2, the value two orders of magnitude softer than, for instance, alpha-helices. Prompted by this observation, the four lowest frequency modes and also the harmonic motions in the thermal equilibrium are analyzed further mainly to detect relatively rigid structural elements in the molecule. From this analysis emerges a mechanical picture of the protein molecule that is made up of relatively rigid elements held together by very soft parts.     

Effect of screw position on load transfer in lumbar pedicle screws: a non-idealized finite element analysis

Angled screw insertion has been advocated to enhance fixation strength during posterior spine fixation. Stresses on a pedicle screw and surrounding vertebral bone with different screw angles were studied by finite element analysis during simulated multidirectional loading. Correlations between screw-specific vertebral geometric parameters and stresses were studied. Angulations in both the sagittal and axial planes affected stresses on the cortical and cancellous bones and the screw. Pedicle screws pointing laterally (vs. straight or medially) in the axial plane during superior screw angulation may be advantageous in terms of reducing the risk of both screw loosening and screw breakage.     

Comparison of normal mode analyses on a small globular protein in dihedral angle space and Cartesian coordinate space

Normal mode analyses on the protein, bovine pancreatic trypsin inhibitor, in dihedral angle space and Cartesian coordinate space are compared. In Cartesian coordinate space it is found that modes of frequencies lower than 30 cm(-1) contribute 80% of the total mean-square fluctuation and are represented almost completely by motions in the dihedral angles. Bond angle and length fluctuations dominate in modes above 200 cm(-1), but contribute less than 2% to the total mean-square fluctuation. In the low-frequency modes a good correspondence between patterns of atomic displacements was found, but on average the root-mean-square fluctuations of the Cartesian coordinate modes are 13% greater than their dihedral angle counterparts. The main effect of fluctuations in the bond angles and lengths, therefore, is to allow the dihedral angles to become more flexible. As the important subspaces determined from the two methods overlap considerably, dihedral angle space analysis can be applied to proteins too large for Cartesian coordinate space analysis.     

Dynamic coordinate data for describing muscle-tendon paths: a mathematical approach

When modelling the musculoskeletal system over a range of joint angles the use of fixed points to describe muscle-tendon paths has inherent limitations. These result in fewer deflection points and the use of effective insertions to accommodate both relative marker movement and avoid muscle paths contacting bony structures. Model performance is dependent on the joint angle relative to the anatomical position where the muscle-tendon paths were defined. The present study proposes a scheme for the implementation of dynamic coordinates for describing muscle-tendon paths. For each muscle-tendon element a plane is defined in which the muscle-tendon complex acts when crossing a given joint. The muscle-tendon plane is dependent on 3D segment orientations and describes one degree of freedom, while the remaining two degrees of freedom are described by polar coordinates and locate the dynamic point in the muscle-tendon plane. The dynamic approach is implemented on four muscles of the lower limb in modelled and simulated joint movements and offers a significant improvement on previous approaches based on fixed deflection points. The scheme accommodates compound 3D rotations about joint axes, is not computationally difficult or require large data sets, and does not impose limitations on the number of points that may be defined along a muscle-tendon path.     

Origin of sperm cell association in the “male germ unit” ofBrassica pollen

The association of the two sperm cells inBrassica napus pollen following the generative cell division was investigated. The generative cell during division is located in the center of the pollen grain, within the vegetative cell. The space present between the two cells is slightly irregular as seen following standard glutaraldehyde fixation. After completion of mitosis vesicles appear in the equatorial plane, coalescing centripetally to form a cell plate which fuses with the membrane of the generative cell, dividing it in two sperm cells. They are isolated from the vegetative cell by the space between the two cell membranes and are separated from each other by a similar space resulting from the cell plate formed during cytokinesis.     

Solid state conformational study of complexes containing the (o-P-C6H4-CN-κN,P) palladium six-membered chelated ring

The Cambridge Structural Database (CSD) V. 5.23 has been searched for all the structures containing the fragment Pd{o-P-C₆H₄-CN-κ²N,P}. Bond lengths, bond angles and the conformations adopted by the PdPCCCN chelated ring have been studied statistically. It has been found Pd-P and Pd-N distances in the mentioned compounds are, respectively, shorter and longer than the mean value found for these parameters when the CSD was searched irrespective of the ligands. The conformation of the chelated ring has been characterized by means of two torsion angles, and can be described in most cases as a plane containing the P and carbon atoms with Pd and N atoms situated at the same side out of this plane. Molecular mechanics calculations have been employed to justify the conformational preferences found. The calculated strain energy suggests a path for the movement of Pd and N atoms from one side of the PCCC plane to the other through a planar conformation. MM calculations in complex (η³-allyl)-(N-(2-(diphenylphosphino)benzylidene)-4-methoxyphenylamine)-palladium(II) indicate that planar conformation is energetically accessible, and the extent of preorganization for the free ligand in this compound has been evaluated of 74%.     

Efficiency as a predictor of human jaw design in the sagittal plane

The effects of changing the direction of the bite force and of the mandibular joint reaction have been studied with a mathematical model assisted by a computer using the technique of linear programming. We conclude the following: In the sagittal plane the long axes of lower molars are each tilted in the direction that most efficiently converts muscle force into work at the bite point rather than in the direction that would maximize static bite force. These genetically determined angles are referred to as the most 'work efficient' angles. Collectively they lead to the appearance of the curve of Spee associated with the postcanines. Given the most work efficient angle of the first molar, the model indicates for bite forces generated in this direction the joint reaction is least when tilted forward from the vertical at between 20 degrees and 30 degrees. The joint reaction is normal to the articular surface of the condyle which is itself tilted forward 20-30 degrees from the occlusal plane. We conclude the condyle and articular eminence are remodelled to the angle that minimizes the joint reaction. The direction of the bite force may be controlled via neuronal circuitry connecting mechanoreceptors of the periodontal ligament with motor nerves supplying the jaw-closing muscles. The height of the occlusal plane in the molar region has little effect on jaw efficiency.     

Optimization of ski jumper's posture considering lift-to-drag ratio and stability

An optimization analysis of a ski jumper's posture has been performed to improve the lift-to-drag ratio, and to examine aerodynamic stability to ensure flight control and safety. Three-dimensional Reynolds-averaged Navier-Stokes equations were discretized using finite volume approximations for the flow analysis, and the shear stress transport k-ω turbulence model was used for a turbulence closure. The Airfoil theory and principles of aircraft stability were used to examine the stability mechanism. Two ski jumper posture angles were chosen as design variables through a preliminary test, and the lift-to-drag ratio was used as an objective function for the optimization problem. Thirteen design points within design spaces are selected by Latin hypercube sampling. In order to predict the objective function values in the design space, the Kriging model was constructed using the numerical results on the design points. By the sequential quadratic programming, the optimal point was found from the constructed the Kriging model. The Kriging model predicted the objective function value at the optimum point with a 1.1% error compared to the value obtained by numerical analysis. The optimum design showed a considerable lift-to-drag ratio improvement compared to the reference design.     

An electrophile-nucleophile interaction in metalloprotein structures.

A new attractive interaction in metalloprotein structures, between the thiolate anion of a metal-bound cysteine (acting as a nucleophile) and a carbonyl carbon of a peptide group (an electrophile), has been identified. From 82 cases extracted from 23 metalloprotein structures, the interacting S and C atoms are found to be at a distance of 3.2 (+/-2) A, such that the angle S ... C-O is 109 degrees (+/-15 degrees). Usually, the interacting atoms are from the same Cys residue, and to allow the S to interact with the carbonyl group the side-chain and the main-chain torsion angles deviate from those found in cysteines not bound by metals. There is a good correlation between the S...C distance and the angular deviation of the S...C vector from the normal to the peptide plane. Various data points may be envisaged to represent "snapshots" along the reaction coordinate for the intra-residue attack of Cys S on the CO group.