An application of beam theory to determine the stress and deformation of long bones

We present a generalized beam theory in which deformation and load are determined simultaneously, in order to analyze statically indeterminant problems involving long bones. We regard a long bone as a beam curved in three dimensions for which the cross-sectional properties vary continuously along its length. The theory is used to determine the force, moment, deflection and twist along the fifth metatarsal when it is subjected to both a pointwise and a distributed load.     

CURVATURE: software for the analysis of curved DNA

Software is presented to plot the sequence-dependent spatialtrajectory of the DNA double helix and/or distribution of curvaturealong the DNA molecule. The nearest-neighbor wedge model isimplemented to calculate overall DNA path using local helixparameters: helix twist angle, wedge (deflection) angle anddirection (of deflection) angle. The procedures described provedto be very convenient as tools for investigation of a relationshipbetween overall DNA curvature and its gel electrophoretic mobility.All parameters of the model had been estimated from experimentaldata. Using these wedge parameters the program takes, as input,any DNA sequence and calculates the likely degree of curvatureat each point along the molecule. This information is displayedboth graphically and in the form of simplified representationsof curved double helices. The Software, CURVATURE, can thusbe used to investigate possible roles of curvature in modulationof gene expression and for location of curved portions of DNA,which may play an important role in sequence-specific protein-DNAinteractions.     

Terminal twist induced continuous writhe of a circular rod with intrinsic curvature

It is well known that a large linking number induces an abrupt writhing of a circular rod with zero intrinsic curvature, i.e., the stress-free state of the rod is straight. We show here that for any rod with a uniform natural curvature, no matter how small the intrinsic curvature is, a twist will induce a continuous writhing from the circular configuration and the abrupt writhing is only the limiting case when the intrinsic curvature is absolutely zero. The implication of this result on elastic models of circular DNA is discussed.     

Intrinsic curvature of DNA influences LacR-mediated looping

Protein-mediated DNA looping is a common mechanism for regulating gene expression. Loops occur when a protein binds to two operators on the same DNA molecule. The probability of looping is controlled, in part, by the basepair sequence of inter-operator DNA, which influences its structural properties. One structural property is the intrinsic or stress-free curvature. In this article, we explore the influence of sequence-dependent intrinsic curvature by exercising a computational rod model for the inter-operator DNA as applied to looping of the LacR-DNA complex. Starting with known sequences for the inter-operator DNA, we first compute the intrinsic curvature of the helical axis as input to the rod model. The crystal structure of the LacR (with bound operators) then defines the requisite boundary conditions needed for the dynamic rod model that predicts the energetics and topology of the intervening DNA loop. A major contribution of this model is its ability to predict a broad range of published experimental data for highly bent (designed) sequences. The model successfully predicts the loop topologies known from fluorescence resonance energy transfer measurements, the linking number distribution known from cyclization assays with the LacR-DNA complex, the relative loop stability known from competition assays, and the relative loop size known from gel mobility assays. In addition, the computations reveal that highly curved sequences tend to lower the energetic cost of loop formation, widen the energy distribution among stable and meta-stable looped states, and substantially alter loop topology. The inclusion of sequence-dependent intrinsic curvature also leads to nonuniform twist and necessitates consideration of eight distinct binding topologies from the known crystal structure of the LacR-DNA complex.     

Link, twist, energy, and the stability of DNA minicircles

We describe how the stability properties of DNA minicircles can be directly read from plots of various biologically intuitive quantities along families of equilibrium configurations. Our conclusions follow from extensions of the mathematical theory of distinguished bifurcation diagrams that are applied within the specific context of an elastic rod model of minicircles. Families of equilibria arise as a twisting angle alpha is varied. This angle is intimately related to the continuously varying linking number Lk for nicked DNA configurations that is defined as the sum of Twist and Writhe. We present several examples of such distinguished bifurcation diagrams involving plots of the energy E, linking number Lk, and a twist moment m3, along families of cyclized equilibria of both intrinsically straight and intrinsically curved DNA fragments.     

Terminal Twist-Induced Writhe of DNA with Intrinsic Curvature

Supercoiling of a closed circular DNA rod may result from an application of terminal twist to the DNA rod by cutting the rod, rotating one of the cut faces as the other being fixed and then sealing the cut. According to White's formula, DNA supercoiling is probably accompanied by a writhe of the DNA axis. Deduced from the elastic rod model for DNA structure, an intrinsically straight closed circular DNA rod does not writhe as subject to a terminal twist, until the number of rotation exceeds a rod-dependent threshold. By contrast, a closed circular DNA rod with intrinsic curvature writhes instantly as subject to a terminal twist. This noteworthy character in fact belongs to many intrinsically curved DNA rods. By solving the dynamic equations, the linearization of the Euler–Lagrange equations governing intrinsically curved DNA rods, this paper shows that almost every clamped-end intrinsically curved DNA rod writhes instantly when subject to a terminal twist (clamped-end DNA rods include closed circular DNA rods and topological domains of open DNA rods). In terms of physical quantities, the exceptions are identified with points in ℝ⁶ whose projections onto ℝ⁵ (through ignoring the total energy density of a rod) form a subset of a quadratic hypersurface. This paper also suggests that the terminal twist induced writhe is due to the elasticity and the clamped-end boundary conditions of the DNA rods. To my sister for her 50th birthday.     

Computer simulation of flagellar movement VIII: coordination of dynein by local curvature control can generate helical bending waves

Computer simulations have been carried out with a model flagellum that can bend in three dimensions. A pattern of dynein activation in which regions of dynein activity propagate along each doublet, with a phase shift of approximately 1/9 wavelength between adjacent doublets, will produce a helical bending wave. This pattern can be termed "doublet metachronism." The simulations show that doublet metachronism can arise spontaneously in a model axoneme in which activation of dyneins is controlled locally by the curvature of each outer doublet microtubule. In this model, dyneins operate both as sensors of curvature and as motors. Doublet metachronism and the chirality of the resulting helical bending pattern are regulated by the angular difference between the direction of the moment and sliding produced by dyneins on a doublet and the direction of the controlling curvature for that doublet. A flagellum that is generating a helical bending wave experiences twisting moments when it moves against external viscous resistance. At high viscosities, helical bending will be significantly modified by twist unless the twist resistance is greater than previously estimated. Spontaneous doublet metachronism must be modified or overridden in order for a flagellum to generate the planar bending waves that are required for efficient propulsion of spermatozoa. Planar bending can be achieved with the three-dimensional flagellar model by appropriate specification of the direction of the controlling curvature for each doublet. However, experimental observations indicate that this "hard-wired" solution is not appropriate for real flagella.     

HELANAL: a program to characterize helix geometry in proteins

A detailed analysis of structural and position dependent characteristic features of helices will give a better understanding of the secondary structure formation in globular proteins. Here we describe an algorithm that quantifies the geometry of helices in proteins on the basis of their C alpha atoms alone. The Fortran program HELANAL can extract the helices from the PDB files and then characterises the overall geometry of each helix as being linear, curved or kinked, in terms of its local structural features, viz. local helical twist and rise, virtual torsion angle, local helix origins and bending angles between successive local helix axes. Even helices with large radius of curvature are unambiguously identified as being linear or curved. The program can also be used to differentiate a kinked helix and other motifs, such as helix-loop-helix or a helix-turn-helix (with a single residue linker) with the help of local bending angles. In addition to these, the program can also be used to characterise the helix start and end as well as other types of secondary structures.     

A quantitative measure of DNA curvature enabling the comparison of predicted structures

A growing body of data indicates that the equilibrium structures of some DNA fragments are curved and that curvature is sequence-directed. We describe a quantitative measure of DNA curvature that can be used for evaluating and comparing current proposed models for the molecular basis of DNA curvature. We demonstrate that this measure, in conjunction with any given prediction model, enables both the comparison of experimental data to predictions and the scanning of nucleotide sequence databases for potential curved regions.     

A Quantitative Measure of DNA Curvature Enabling the Comparison of Predicted Structures

Abstract

A growing body of data indicates that the equilibrium structures of some DNA fragments are curved and that curvature is sequence-directed. We describe a quantitative measure of DNA curvature that can be used for evaluating and comparing current proposed models for the molecular basis of DNA curvature. We demonstrate that this measure, in conjunction with any given prediction model, enables both the comparison of experimental data to predictions and the scanning of nucleotide sequence databases for potential curved regions.     

Haptic perception of object curvature in Parkinson's disease

Background

The haptic perception of the curvature of an object is essential for adequate object manipulation and critical for our guidance of actions. This study investigated how the ability to perceive the curvature of an object is altered by Parkinson''s disease (PD).

Methodology/Principal Findings

Eight healthy subjects and 11 patients with mild to moderate PD had to judge, without vision, the curvature of a virtual “box” created by a robotic manipulandum. Their hands were either moved passively along a defined curved path or they actively explored the curved curvature of a virtual wall. The curvature was either concave or convex (bulging to the left or right) and was judged in two locations of the hand workspace–a left workspace location, where the curved hand path was associated with curved shoulder and elbow joint paths, and a right workspace location in which these joint paths were nearly linear. After exploring the curvature of the virtual object, subjects had to judge whether the curvature was concave or convex. Based on these data, thresholds for curvature sensitivity were established. The main findings of the study are: First, 9 out 11 PD patients (82%) showed elevated thresholds for detecting convex curvatures in at least one test condition. The respective median threshold for the PD group was increased by 343% when compared to the control group. Second, when distal hand paths became less associated with proximal joint paths (right workspace), haptic acuity was reduced substantially in both groups. Third, sensitivity to hand trajectory curvature was not improved during active exploration in either group.

Conclusion/Significance

Our data demonstrate that PD is associated with a decreased acuity of the haptic sense, which may occur already at an early stage of the disease.     

The role actin filaments play in providing the characteristic curved form of Drosophila bristles

Drosophila bristles display a precise orientation and curvature. An asymmetric extension of the socket cell overlies the newly emerging bristle rudiment to provide direction for bristle elongation, a process thought to be orchestrated by the nerve dendrite lying between these cells. Scanning electron microscopic analysis of individual bristles showed that curvature is planar and far greater near the bristle base. Correlated with this, as development proceeds the pupa gradually recedes from the inner pupal case (an extracellular layer that encloses the pupa) leading to less bristle curvature along the shaft. We propose that the inner pupal case induces elongating bristles to bend when they contact this barrier. During elongation the actin cytoskeleton locks in this curvature by grafting together the overlapping modules that comprise the long filament bundles. Because the bristle is curved, the actin bundles on the superior side must be longer than those on the inferior side. This is accomplished during grafting by greater elongation of superior side modules. Poor actin cross-bridging in mutant bristles results in altered curvature. Thus, the pattern of bristle curvature is a product of both extrinsic factors-the socket cell and the inner pupal case--and intrinsic factors--actin cytoskeleton assembly.     

A Unified Model of Heading and Path Perception in Primate MSTd

Self-motion, steering, and obstacle avoidance during navigation in the real world require humans to travel along curved paths. Many perceptual models have been proposed that focus on heading, which specifies the direction of travel along straight paths, but not on path curvature, which humans accurately perceive and is critical to everyday locomotion. In primates, including humans, dorsal medial superior temporal area (MSTd) has been implicated in heading perception. However, the majority of MSTd neurons respond optimally to spiral patterns, rather than to the radial expansion patterns associated with heading. No existing theory of curved path perception explains the neural mechanisms by which humans accurately assess path and no functional role for spiral-tuned cells has yet been proposed. Here we present a computational model that demonstrates how the continuum of observed cells (radial to circular) in MSTd can simultaneously code curvature and heading across the neural population. Curvature is encoded through the spirality of the most active cell, and heading is encoded through the visuotopic location of the center of the most active cell''s receptive field. Model curvature and heading errors fit those made by humans. Our model challenges the view that the function of MSTd is heading estimation, based on our analysis we claim that it is primarily concerned with trajectory estimation and the simultaneous representation of both curvature and heading. In our model, temporal dynamics afford time-history in the neural representation of optic flow, which may modulate its structure. This has far-reaching implications for the interpretation of studies that assume that optic flow is, and should be, represented as an instantaneous vector field. Our results suggest that spiral motion patterns that emerge in spatio-temporal optic flow are essential for guiding self-motion along complex trajectories, and that cells in MSTd are specifically tuned to extract complex trajectory estimation from flow.     

On the contribution of moment-bearing links to bending and twisting in a three-dimensional sliding filament model

Previously (Hines, M., and J.J. Blum 1983, Biophys. J., 41:67-79), a method was developed that allowed one to compute curvature and twist for a three-dimensional sliding filament model. In that formalism it was difficult to specify the shear and bending moments arising from moment-bearing interfilament links such as fixed 5-6 bridges or dyneins. Euler's equation offers a straightforward method for computing these bending and shear moments when the potential energy stored in the links as a function of axonemal shape is specified. We used this approach to examine the effect of 5-6 bridges on curvature and twist for several distributions of internal shear moments. Twist changes the angle that a link makes with a doublet and thus in some circumstances may reduce the potential energy stored in those links. Twist is a second-order effect proportional to the square of the distance between an outer doublet and the neutral axis. Fixed links will not generate twist if they are symmetrically located around the axoneme.     

Intrinsic linear heterogeneity of amyloid β protein fibrils revealed by higher resolution mass-per-length determinations

Amyloid β proteins spontaneously form fibrils in vitro that vary in their thermodynamic stability and in morphological characteristics such as length, width, shape, longitudinal twist, and the number of component filaments. It is vitally important to determine which variant best represents the type of fibril that accumulates in Alzheimer disease. In the present study, the nature of morphological variation was examined by dark-field and transmission electron microscopy in a preparation of seeded amyloid β protein fibrils that formed at relatively low protein concentrations and exhibited remarkably high thermodynamic stability. The number of filaments comprising these fibrils changed frequently from two to six along their length, and these changes only became apparent when mass-per-length (MPL) determinations are made with sufficient resolution. The MPL results could be reproduced by a simple stochastic model with a single adjustable parameter. The presence of more than two primary filaments could not be discerned by transmission electron microscopy, and they had no apparent relationship to the longitudinal twist of the fibrils. However, the pitch of the twist was strongly affected by the pH of the negative stain. We conclude that highly stable amyloid fibrils may form in which a surprising amount of intrinsic linear heterogeneity may be obscured by MPL measurements of insufficient resolution, and by the negative stains used for imaging fibrils by electron microscopy.     

HELANAL: A Program to Characterize Helix Geometry in Proteins

Abstract

A detailed analysis of structural and position dependent characteristic features of helices will give a better understanding of the secondary structure formation in globular proteins. Here we describe an algorithm that quantifies the geometry of helices in proteins on the basis of their C^α atoms alone. The Fortran program HELANAL can extract the helices from the PDB files and then characterises the overall geometry of each helix as being linear, curved or kinked, in terms of its local structural features, viz. local helical twist and rise, virtual torsion angle, local helix origins and bending angles between successive local helix axes. Even helices with large radius of curvature are unambiguously identified as being linear or curved. The program can also be used to differentiate a kinked helix and other motifs, such as helix-loop-helix or a helix-turn-helix (with a single residue linker) with the help of local bending angles. In addition to these, the program can also be used to characterise the helix start and end as well as other types of secondary structures.     

Oblique circle method for measuring the curvature and twist of mitotic spindle microtubule bundles

The highly ordered spatial organization of microtubule bundles in the mitotic spindle is crucial for its proper functioning. The recent discovery of twisted shapes of microtubule bundles and spindle chirality suggests that the bundles extend along curved paths in three dimensions, rather than being confined to a plane. This, in turn, implies that rotational forces, i.e., torques, exist in the spindle in addition to the widely studied linear forces. However, studies of spindle architecture and forces are impeded by a lack of a robust method for the geometric quantification of microtubule bundles in the spindle. In this work, we describe a simple method for measuring and evaluating the shapes of microtubule bundles by characterizing them in terms of their curvature and twist. By using confocal microscopy, we obtain three-dimensional images of spindles, which allows us to trace the entire microtubule bundle. For each traced bundle, we first fit a plane and then fit a circle lying in that plane. With this robust method, we extract the curvature and twist, which represent the geometric information characteristic for each bundle. As the bundle shapes reflect the forces within them, this method is valuable for the understanding of forces that act on chromosomes during mitosis.     

A-tract clusters may facilitate DNA packaging in bacterial nucleoid

Molecular mechanisms of bacterial chromosome packaging are still unclear, as bacteria lack nucleosomes or other apparent basic elements of DNA compaction. Among the factors facilitating DNA condensation may be a propensity of the DNA molecule for folding due to its intrinsic curvature. As suggested previously, the sequence correlations in genome reflect such a propensity [Trifonov and Sussman (1980) Proc. Natl Acad. Sci. USA, 77, 3816–3820]. To further elaborate this concept, we analyzed positioning of A-tracts (the sequence motifs introducing the most pronounced DNA curvature) in the Escherichia coli genome. First, we observed that the A-tracts are over-represented and distributed ‘quasi-regularly’ throughout the genome, including both the coding and intergenic sequences. Second, there is a 10–12 bp periodicity in the A-tract positioning indicating that the A-tracts are phased with respect to the DNA helical repeat. Third, the phased A-tracts are organized in ~100 bp long clusters. The latter feature was revealed with the help of a novel approach based on the Fourier series expansion of the A-tract distance autocorrelation function. Since the A-tracts introduce local bends of the DNA duplex and these bends accumulate when properly phased, the observed clusters would facilitate DNA looping. Also, such clusters may serve as binding sites for the nucleoid-associated proteins that have affinities for curved DNA (such as HU, H-NS, Hfq and CbpA). Therefore, we suggest that the ~100 bp long clusters of the phased A-tracts constitute the ‘structural code’ for DNA compaction by providing the long-range intrinsic curvature and increasing stability of the DNA complexes with architectural proteins.