The utility of an empirically derived co-activation ratio for muscle force prediction through optimization

Biomechanical optimization models that apply efficiency-based objective functions often underestimate or negate antagonist co-activation. Co-activation assists movement control, joint stabilization and limb stiffness and should be carefully incorporated into models. The purposes of this study were to mathematically describe co-activation relationships between elbow flexors and extensors during isometric exertions at varying intensity levels and postures, and secondly, to apply these co-activation relationships as constraints in an optimization muscle force prediction model of the elbow and assess changes in predictions made while including these constraints. Sixteen individuals performed 72 isometric exertions while holding a load in their right hand. Surface EMG was recorded from elbow flexors and extensors. A co-activation index provided a relative measure of flexor contribution to total activation about the elbow. Parsimonious models of co-activation during flexion and extension exertions were developed and added as constraints to a muscle force prediction model to enforce co-activation. Three different PCSA data sets were used. Elbow co-activation was sensitive to changes in posture and load. During flexion exertions the elbow flexors were activated about 75% MVC (this amount varied according to elbow angle, shoulder flexion and abduction angles, and load). During extension exertions the elbow flexors were activated about 11% MVC (this amount varied according to elbow angle, shoulder flexion angle and load). The larger PCSA values appeared to be more representative of the subject pool. Inclusion of these co-activation constraints improved the model predictions, bringing them closer to the empirically measured activation levels.     

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.     

Predictive Simulation Generates Human Adaptations during Loaded and Inclined Walking

Predictive simulation is a powerful approach for analyzing human locomotion. Unlike techniques that track experimental data, predictive simulations synthesize gaits by minimizing a high-level objective such as metabolic energy expenditure while satisfying task requirements like achieving a target velocity. The fidelity of predictive gait simulations has only been systematically evaluated for locomotion data on flat ground. In this study, we construct a predictive simulation framework based on energy minimization and use it to generate normal walking, along with walking with a range of carried loads and up a range of inclines. The simulation is muscle-driven and includes controllers based on muscle force and stretch reflexes and contact state of the legs. We demonstrate how human-like locomotor strategies emerge from adapting the model to a range of environmental changes. Our simulation dynamics not only show good agreement with experimental data for normal walking on flat ground (92% of joint angle trajectories and 78% of joint torque trajectories lie within 1 standard deviation of experimental data), but also reproduce many of the salient changes in joint angles, joint moments, muscle coordination, and metabolic energy expenditure observed in experimental studies of loaded and inclined walking.     

Predicting Growth Conditions from Internal Metabolic Fluxes in an In-Silico Model of E. coli

A widely studied problem in systems biology is to predict bacterial phenotype from growth conditions, using mechanistic models such as flux balance analysis (FBA). However, the inverse prediction of growth conditions from phenotype is rarely considered. Here we develop a computational framework to carry out this inverse prediction on a computational model of bacterial metabolism. We use FBA to calculate bacterial phenotypes from growth conditions in E. coli, and then we assess how accurately we can predict the original growth conditions from the phenotypes. Prediction is carried out via regularized multinomial regression. Our analysis provides several important physiological and statistical insights. First, we show that by analyzing metabolic end products we can consistently predict growth conditions. Second, prediction is reliable even in the presence of small amounts of impurities. Third, flux through a relatively small number of reactions per growth source (∼10) is sufficient for accurate prediction. Fourth, combining the predictions from two separate models, one trained only on carbon sources and one only on nitrogen sources, performs better than models trained to perform joint prediction. Finally, that separate predictions perform better than a more sophisticated joint prediction scheme suggests that carbon and nitrogen utilization pathways, despite jointly affecting cellular growth, may be fairly decoupled in terms of their dependence on specific assortments of molecular precursors.     

Determining the three-dimensional fold of a protein from approximate constraints

We propose a new approach for calculating the three-dimensional (3D) structure of a protein from distance and dihedral angle constraints derived from experimental data. We suggest that such constraints can be obtained from experiments such as tritium planigraphy, chemical or enzymatic cleavage of the polypeptide chain, paramagnetic perturbation of nuclear magnetic resonance (NMR) spectra, measurement of hydrogen-exchange rates, mutational studies, mass spectrometry, and electron paramagnetic resonance. These can be supplemented with constraints from theoretical prediction of secondary structures and of buried/exposed residues. We report here distance geometry calculations to generate the structures of a test protein Staphylococcal nuclease (STN), and the HIV-1 rev protein (REV) of unknown structure. From the available 3D atomic coordinates of STN, we set up simulated data sets consisting of varying number and quality of constraints, and used our group's Self Correcting Distance Geometry (SECODG) program DIAMOD to generate structures. We could generate the correct tertiary fold from qualitative (approximate) as well as precise distance constraints. The root mean square deviations of backbone atoms from the native structure were in the range of 2.0 A to 8.3 A, depending on the number of constraints used. We could also generate the correct fold starting from a subset of atoms that are on the surface and those that are buried. When we used data sets containing a small fraction of incorrect distance constraints, the SECODG technique was able to detect and correct them. In the case of REV, we used a combination of constraints obtained from mutagenic data and structure predictions. DIAMOD generated helix-loop-helix models, which, after four self-correcting cycles, populated one family exclusively. The features of the energy-minimized model are consistent with the available data on REV-RNA interaction. Our method could thus be an attractive alternative for calculating protein 3D structures, especially in cases where the traditional methods of X-ray crystallography and multidimensional NMR spectroscopy have been unsuccessful.     

A novel method for predicting and using distance constraints of high accuracy for refining protein structure prediction

The principal bottleneck in protein structure prediction is the refinement of models from lower accuracies to the resolution observed by experiment. We developed a novel constraints‐based refinement method that identifies a high number of accurate input constraints from initial models and rebuilds them using restrained torsion angle dynamics (rTAD). We previously created a Bayesian statistics‐based residue‐specific all‐atom probability discriminatory function (RAPDF) to discriminate native‐like models by measuring the probability of accuracy for atom type distances within a given model. Here, we exploit RAPDF to score (i.e., filter) constraints from initial predictions that may or may not be close to a native‐like state, obtain consensus of top scoring constraints amongst five initial models, and compile sets with no redundant residue pair constraints. We find that this method consistently produces a large and highly accurate set of distance constraints from which to build refinement models. We further optimize the balance between accuracy and coverage of constraints by producing multiple structure sets using different constraint distance cutoffs, and note that the cutoff governs spatially near versus distant effects in model generation. This complete procedure of deriving distance constraints for rTAD simulations improves the quality of initial predictions significantly in all cases evaluated by us. Our procedure represents a significant step in solving the protein structure prediction and refinement problem, by enabling the use of consensus constraints, RAPDF, and rTAD for protein structure modeling and refinement. Proteins 2009. © 2009 Wiley‐Liss, Inc.     

Postural kinematics of trumpet playing

This paper examines the effects of anthropometry on body posture of trumpeters playing in standing position. Sixteen virtuosi trumpeters were photographed while hitting three notes (low C, high F and high F sustained) during performance of musical tasks. Initial standing posture and anthropometric data were recorded. Six body segment angles were computed and a vectorial sum was obtained to describe whole body posture in neutral and playing conditions. Horn angle and dental overbite were also computed. Earlier results showed that the musical task has no effect on playing posture. One-way ANOVA showed notable differences between the neutral posture and the note-related playing postures. A multiple regression model showed that in addition to the note effect, anthropometric variables, mainly neck length, explain the changes in playing posture. Horn angle is determined by the dental overbite. The importance of the anthropometric variables in playing the more demanding notes indicate that anthropometry may act to constrain trumpeters' performance.     

A three-dimensional model of the rat hindlimb: musculoskeletal geometry and muscle moment arms

As a first step towards developing a dynamic model of the rat hindlimb, we measured muscle attachment and joint center coordinates relative to bony landmarks using stereophotogrammetry. Using these measurements, we analyzed muscle moment arms as functions of joint angle for most hindlimb muscles, and tested the hypothesis that postural change alone is sufficient to alter the function of selected muscles of the leg. We described muscle attachment sites as second-order curves. The length of the fit parabola and residual errors in the orthogonal directions give an estimate of muscle attachment sizes, which are consistent with observations made during dissection. We modeled each joint as a moving point dependent on joint angle; relative endpoint errors less than 7% indicate this method as accurate. Most muscles have moment arms with a large range across the physiological domain of joint angles, but their moment arms peak and vary little within the locomotion domain. The small variation in moment arms during locomotion potentially simplifies the neural control requirements during this phase. The moment arms of a number of muscles cross zero as angle varies within the quadrupedal locomotion domain, indicating they are intrinsically stabilizing. However, in the bipedal locomotion domain, the moment arms of these muscles do not cross zero and thus are no longer intrinsically stabilizing. We found that muscle function is largely determined by the change in moment arm with joint angle, particularly the transition from quadrupedal to bipedal posture, which may alter an intrinsically stabilizing arrangement or change the control burden.     

Cyclic coordinate descent: A robotics algorithm for protein loop closure

In protein structure prediction, it is often the case that a protein segment must be adjusted to connect two fixed segments. This occurs during loop structure prediction in homology modeling as well as in ab initio structure prediction. Several algorithms for this purpose are based on the inverse Jacobian of the distance constraints with respect to dihedral angle degrees of freedom. These algorithms are sometimes unstable and fail to converge. We present an algorithm developed originally for inverse kinematics applications in robotics. In robotics, an end effector in the form of a robot hand must reach for an object in space by altering adjustable joint angles and arm lengths. In loop prediction, dihedral angles must be adjusted to move the C-terminal residue of a segment to superimpose on a fixed anchor residue in the protein structure. The algorithm, referred to as cyclic coordinate descent or CCD, involves adjusting one dihedral angle at a time to minimize the sum of the squared distances between three backbone atoms of the moving C-terminal anchor and the corresponding atoms in the fixed C-terminal anchor. The result is an equation in one variable for the proposed change in each dihedral. The algorithm proceeds iteratively through all of the adjustable dihedral angles from the N-terminal to the C-terminal end of the loop. CCD is suitable as a component of loop prediction methods that generate large numbers of trial structures. It succeeds in closing loops in a large test set 99.79% of the time, and fails occasionally only for short, highly extended loops. It is very fast, closing loops of length 8 in 0.037 sec on average.     

An inertial and magnetic sensor based technique for joint angle measurement

This paper describes the design and evaluation of a miniature kinematic sensor based three dimensional (3D) joint angle measurement technique. The technique uses a combination of rate gyroscope, accelerometer and magnetometer sensor signals. The technique enables 3D inter-segment joint angle measurement and could be of benefit in a variety of applications which require monitoring of joint angles. The technique is not dependent on a fixed reference coordinate system and thus may be suitable for use in a dynamic system such as a moving vehicle. The technique was evaluated by applying it to joint angle measurement of the ankle joint. Experimental results show that accurate measurement of ankle joint angles is achieved by the technique during a variety of lower leg exercises including walking.     

Conformational studies of nucleic acids. I. A rapid and direct method for generating furanose coordinates from the pseudorotation angle

The greatest difficulty in modeling a nucleic acid is generating the coordinates of its furanoses. This difficulty arises from constraints imposed by the closed ring geometries of these sugars. We have developed a new method for modeling these furanose rings. Using this method, the coordinates of a sugar can be obtained quickly and unambiguously for any point on the pseudorotational pathway from one parameter: the phase angle of pseudorotation P. The significant difference between this and previous sugar modeling schemes is that here the endocyclic bond lengths of the five-membered sugar ring are allowed to vary a small amount according to simple, explicit, and experimentally reasonable analytic functions of P. The coefficients of these functions follow from the empirical behavior of the endocyclic bond angles and from geometrical constraints due to ring closure. The ability to model the sugars directly from one parameter greatly facilitates carrying out the global conformational studies on nucleic acid constituents which will be attempted in subsequent papers of this series.     

The use of side-chain packing methods in modeling bacteriophage repressor and cro proteins.

In recent years, it has been repeatedly demonstrated that the coordinates of the main-chain atoms alone are sufficient to determine the side-chain conformations of buried residues of compact proteins. Given a perfect backbone, the side-chain packing method can predict the side-chain conformations to an accuracy as high as 1.2 Å RMS deviation (RMSD) with greater than 80% of the χ angles correct. However, similarly rigorous studies have not been conducted to determine how well these apply, if at all, to the more important problem of homology modeling per se. Specifically, if the available backbone is imperfect, as expected for practical application of homology modeling, can packing constraints alone achieve sufficiently accurate predictions to be useful? Here, by systematically applying such methods to the pairwise modeling of two repressor and two cro proteins from the closely related bacteriophages 434 and P22, we find that when the backbone RMSD is 0.8 Å, the prediction on buried side chain is accurate with an RMS error of 1.8 Å and approximately 70% of the χ angles correctly predicted. When the backbone RMSD is larger, in the range of 1.6–1.8 Å, the prediction quality is still significantly better than random, with RMS error at 2.2 Å on the buried side chains and 60% accuracy on χ angles. Together these results suggest the following rules-of-thumb for homology modeling of buried side chains. When the sequence identity between the modeled sequence and the template sequence is >50% (or, equivalently, the expected backbone RMSD is <1 Å), side-chain packing methods work well. When sequence identity is between 30–50%, reflecting a backbone RMS error of 1–2 Å, it is still valid to use side-chain packing methods to predict the buried residues, albeit with care. When sequence identity is below 30% (or backbone RMS error greater than 2 Å), the backbone constraint alone is unlikely to produce useful models. Other methods, such as those involving the use of database fragments to reconstruct a template backbone, may be necessary as a complementary guide for modeling.     

Bias in the gradient-sensing response of chemotactic cells

We apply linear stability theory and perform perturbation studies to better characterize, and to generate new experimental predictions from, a model of chemotactic gradient sensing in eukaryotic cells. The model uses reaction-diffusion equations to describe 3(') phosphoinositide signaling and its regulation at the plasma membrane. It demonstrates a range of possible gradient-sensing mechanisms and captures such characteristic behaviors as strong polarization in response to static gradients, adaptation to differing mean levels of stimulus, and plasticity in response to changing gradients. An analysis of the stability of polarized steady-state solutions indicates that the model is most sensitive to off-axis perturbations. This biased sensitivity is also reflected in responses to localized external stimuli, and leads to a clear experimental prediction, namely, that a cell which is polarized in a background gradient will be most sensitive to transient point-source stimuli lying within a range of angles that are oblique with respect to the polarization axis. Stimuli at angles below this range will elicit responses whose directions overshoot the stimulus angle, while responses to stimuli applied at larger angles will undershoot the stimulus angle. We argue that such a bias is likely to be a general feature of gradient sensing in highly motile cells, particularly if they are optimized to respond to small gradients. Finally, an angular bias in gradient sensing might lead to preferred turn angles and zigzag movements of cells moving up chemotactic gradients, as has been noted under certain experimental conditions.     

Probing a label-free local bend in DNA by single molecule tethered particle motion

Being capable of characterizing DNA local bending is essential to understand thoroughly many biological processes because they involve a local bending of the double helix axis, either intrinsic to the sequence or induced by the binding of proteins. Developing a method to measure DNA bend angles that does not perturb the conformation of the DNA itself or the DNA-protein complex is a challenging task. Here, we propose a joint theory-experiment high-throughput approach to rigorously measure such bend angles using the Tethered Particle Motion (TPM) technique. By carefully modeling the TPM geometry, we propose a simple formula based on a kinked Worm-Like Chain model to extract the bend angle from TPM measurements. Using constructs made of 575 base-pair DNAs with in-phase assemblies of one to seven 6A-tracts, we find that the sequence CA₆CGG induces a bend angle of 19° ± 4°. Our method is successfully compared to more theoretically complex or experimentally invasive ones such as cyclization, NMR, FRET or AFM. We further apply our procedure to TPM measurements from the literature and demonstrate that the angles of bends induced by proteins, such as Integration Host Factor (IHF) can be reliably evaluated as well.     

Conformational Studies of Nucleic Acids I. A Rapid and Direct Method for Generating Furanose Coordinates from the Pseudorotation Angle

Abstract

The greatest difficulty in modeling a nucleic acid is generating the coordinates of its furanoses. This difficulty arises from constraints imposed by the closed ring geometries of these sugars. We have developed a new method for modeling these furanose rings. Using this method, the coordinates of a sugar can be obtained quickly and unambiguously for any point on the pseudorotational pathway from one parameter: the phase angle of pseudorotation P. The significant difference between this and previous sugar modeling schemes is that here the endocyclic bond lengths of the five-membered sugar ring are allowed to vary a small amount according to simple, explicit, and experimentally reasonable analytic functions of P. The coefficients of these functions follow from the empirical behavior of the endocyclic bond angles and from geometrical constraints due to ring closure. The ability to model the sugars directly from one parameter greatly facilitates carrying out the global conformational studies on nucleic acid constituents which will be attempted in subsequent papers of this series.     

Determining fall direction and impact location for various disturbances and gait speeds using the articulated total body model

Because fall experiments with volunteers can be both challenging and risky, especially with older volunteers, we wished to develop computer simulations of falls to provide a theoretical framework for understanding and extending experimental results. To perform a preliminary validation of the articulated total body (ATB) model for passive falls, we compared the model predictions of fall direction, impact location, and impact velocity as a function of disturbance type (faint, slip, step down, trip) and gait speed (fast, normal, slow) to experimental results with young adult volunteers. The three-dimensional ATB model had 17 segments and 16 joints. Its physical characteristics, environment definitions, contact functions, and initial conditions were representative of our experiment. For each combination of disturbance and gait speed, the ATB model was left to fall passively under gravity once disturbed, i.e., no joint torques were applied, until impact with the floor occurred. Finally, we also determined the sensitivity of the model predictions to changes in the model's parameters. Our model predictions of fall angles and impact angles were qualitatively in agreement with those observed experimentally for ten and seven of the 12 original simulations, respectively. Quantitatively, the model predictions of fall angles, impact angles, and impact velocities were within one experimental standard deviation for seven, three, and nine of the 12 original simulations, respectively, and within two experimental standard deviations for ten, nine, and 11 of the 12 original simulations, respectively. Finally, the fall angle and impact angle region did not change for 92% and 95% of the 74 input variation simulations, respectively, and the impact velocities were within the experimental standard deviations for 78% of the 74 input variation simulations. Based on our simulations and a sensitivity analysis, we conclude that our preliminary validation of the ATB model for passive falls was successful. In fact, these ATB model simulations represent a significant step forward in fall simulations. We believe that with additional work, the ATB model could be used to accurately simulate a variety of human falls and may be useful in further understanding the etiology and mechanisms of fall injuries such as hip fractures.     

ADAPTATION FROM RESTRICTED GEOMETRIES: THE SHELL INCLINATION OF TERRESTRIAL GASTROPODS

The adaptations that occur for support and protection can be studied with regard to the optimal structure that balances these objectives with any imposed constraints. The shell inclination of terrestrial gastropods is an appropriate model to address this problem. In this study, we examined how gastropods improve shell angles to well‐balanced ones from geometrically constrained shapes. Our geometric analysis and physical analysis showed that constantly coiled shells are constrained from adopting a well‐balanced angle; the shell angle of such basic shells tends to increase as the spire index (shell height/width) increases, although the optimum angle for stability is 90° for flat shells and 0° for tall shells. Furthermore, we estimated the influences of the geometric rule and the functional demands on actual shells by measuring the shell angles of both resting and active snails. We found that terrestrial gastropods have shell angles that are suited for balance. The growth lines of the shells indicated that this adaptation depends on the deflection of the last whorl: the apertures of flat shells are deflected downward, whereas those of tall shells are deflected upward. Our observations of active snails demonstrated that the animals hold their shells at better balanced angles than inactive snails.     

Olecranon orientation as an indicator of elbow joint angle in the stance phase,and estimation of forelimb posture in extinct quadruped animals

Reconstruction of limb posture is a challenging task in assessing functional morphology and biomechanics of extinct tetrapods, mainly because of the wide range of motions possible at each limb joint and because of our poor knowledge of the relationship between posture and musculoskeletal structure, even in the extant taxa. This is especially true for extinct mammals such as the desmostylian taxa Desmostylus and Paleoparadoxia. This study presents a procedure that how the elbow joint angles of extinct quadruped mammals can be inferred from osteological characteristics. A survey of 67 dried skeletons and 113 step cycles of 32 extant genera, representing 25 families and 13 orders, showed that the olecranon of the ulna and the shaft of the humerus were oriented approximately perpendicular to each other during the stance phase. At this angle, the major extensor muscles maximize their torque at the elbow joint. Based on this survey, I suggest that olecranon orientation can be used for inferring the elbow joint angles of quadruped mammals with prominent olecranons, regardless of taxon, body size, and locomotor guild. By estimating the elbow joint angle, it is inferred that Desmostylus would have had more upright forelimbs than Paleoparadoxia, because their elbow joint angles during the stance phase were approximately 165° and 130°, respectively. Difference in elbow joint angles between these two genera suggests possible differences in stance and gait of these two mammals. J. Morphol. 2009. © 2009 Wiley‐Liss, Inc.     

New Insights into the Interdependence between Amino Acid Stereochemistry and Protein Structure

To successfully design new proteins and understand the effects of mutations in natural proteins, we must understand the geometric and physicochemical principles underlying protein structure. The side chains of amino acids in peptides and proteins adopt specific dihedral angle combinations; however, we still do not have a fundamental quantitative understanding of why some side-chain dihedral angle combinations are highly populated and others are not. Here we employ a hard-sphere plus stereochemical constraint model of dipeptide mimetics to enumerate the side-chain dihedral angles of leucine (Leu) and isoleucine (Ile), and identify those conformations that are sterically allowed versus those that are not as a function of the backbone dihedral angles ? and ψ. We compare our results with the observed distributions of side-chain dihedral angles in proteins of known structure. With the hard-sphere plus stereochemical constraint model, we obtain agreement between the model predictions and the observed side-chain dihedral angle distributions for Leu and Ile. These results quantify the extent to which local, geometrical constraints determine protein side-chain conformations.