Protein effective rotational correlation times from translational self-diffusion coefficients measured by PFG-NMR

Molecular rotational correlation times are of interest for many studies carried out in solution, including characterization of biomolecular structure and interactions. Here we have evaluated the estimates of protein effective rotational correlation times from their translational self-diffusion coefficients measured by pulsed-field gradient NMR against correlation times determined from both collective and residue-specific (15)N relaxation analyses and those derived from 3D structure-based hydrodynamic calculations. The results show that, provided the protein diffusive behavior is coherent with the Debye-Stokes-Einstein model, translational diffusion coefficients provide rapid estimates with reasonable accuracy of their effective rotational correlation times. Effective rotational correlation times estimated from translational diffusion coefficients may be particularly beneficial in cases where i) isotopically labelled material is not available, ii) collective backbone (15)N relaxation rates are difficult to interpret because of the presence of flexible termini or loops, or iii) a full relaxation analysis is practically difficult because of limited sensitivity owing to low protein concentration, high molecular mass or low temperatures.     

Impact of ankle foot orthosis stiffness on Achilles tendon and gastrocnemius function during unimpaired gait

Ankle foot orthoses (AFOs) are designed to improve gait for individuals with neuromuscular conditions and have also been used to reduce energy costs of walking for unimpaired individuals. AFOs influence joint motion and metabolic cost, but how they impact muscle function remains unclear. This study investigated the impact of different stiffness AFOs on medial gastrocnemius muscle (MG) and Achilles tendon (AT) function during two walking speeds. We performed gait analyses for eight unimpaired individuals. Each individual walked at slow and very slow speeds with a 3D printed AFO with no resistance (free hinge condition) and four levels of ankle dorsiflexion stiffness: 0.25 Nm/°, 1 Nm/°, 2 Nm/°, and 3.7 Nm/°. Motion capture, ultrasound, and musculoskeletal modeling were used to quantify MG and AT lengths with each AFO condition. Increasing AFO stiffness increased peak AFO dorsiflexion moment with decreased peak knee extension and peak ankle dorsiflexion angles. Overall musculotendon length and peak AT length decreased, while peak MG length increased with increasing AFO stiffness. Peak MG activity, length, and velocity significantly decreased with slower walking speed. This study provides experimental evidence of the impact of AFO stiffness and walking speed on joint kinematics and musculotendon function. These methods can provide insight to improve AFO designs and optimize musculotendon function for rehabilitation, performance, or other goals.     

Knee joint kinematics and kinetics during the hop and cut after soft tissue artifact suppression: Time to reconsider ACL injury mechanisms?

The recent development of a soft tissue artifact (STA) suppression method allows us to re-evaluate the tibiofemoral kinematics currently linked to non-contact knee injuries. The purpose of this study was therefore to evaluate knee joint kinematics and kinetics in six degrees of freedom (DoF) during the loading phases of a jump lunge and side cut using this in silico method. Thirty-five healthy adults completed these movements and their surface marker trajectories were then scaled and processed with OpenSim’s inverse kinematics (IK) and inverse dynamics tools. Knee flexion angle-dependent kinematic constraints defined based on previous bone pin (BP) marker trajectories were then applied to the OpenSim model during IK and these constrained results were then processed with the standard inverse dynamics tool. Significant differences for all hip, knee, and ankle DoF were observed after STA suppression for both the jump lunge and side cut. Using clinically relevant effect size estimates, we conclude that STA contamination had led to misclassifications in hip transverse plane angles, knee frontal and transverse plane angles, medial/lateral and distractive/compressive knee translations, and knee frontal plane moments between the NoBP and the BP IK solutions. Our results have substantial clinical implications since past research has used joint kinematics and kinetics contaminated by STA to identify risk factors for musculoskeletal injuries.     

A musculoskeletal model of the human lower extremity: the effect of muscle, tendon, and moment arm on the moment-angle relationship of musculotendon actuators at the hip, knee, and ankle

We have developed a musculoskeletal model of the human lower extremity for computer simulation studies of musculotendon function and muscle coordination during movement. This model incorporates the salient features of muscle and tendon, specifies the musculoskeletal geometry and musculotendon parameters of 18 musculotendon actuators, and defines the active isometric moment of these actuators about the hip, knee, and ankle joints in the sagittal plane. We found that tendon slack length, optimal muscle-fiber length, and moment arm are different for each actuator, thus each actuator develops peak isometric moment at a different joint angle. The joint angle where an actuator produces peak moment does not necessarily coincide with the joint angle where: (1) muscle force peaks, (2) moment arm peaks, or (3) the in vivo moment developed by maximum voluntary contractions peaks. We conclude that when tendon is neglected in analyses of musculotendon force or moment about joints, erroneous predictions of human musculotendon function may be stated, not only in static situations as studied here, but during movement as well.     

Viscoelasticity of rigid macromolecules with irregular shapes in the limit of overwhelming Brownian motion

We consider viscoelastic properties of complex rigid macromolecules in fluids undergoing steady or sinusoidal linear shearing. An arbitrary body is hydrodynamically described by six tensors to allow for irregular shapes that couple rotational and translational motions to each other and to the shear field. The viscosity increment ν is obtained for infinitely dilute suspensions in the limit of overwhelming Brownian motion by balancing drag forces and torques with entropic forces and torques. For sinusoidal shearing, ν is a complex number that exhibits five resonances at frequencies matching the j = 2 eigenvalues of the rotational diffusion equation for a stationary fluid. The resonance amplitudes are conveniently expressed in terms of a second-rank body-fixed tensor χ that characterizes alignment by the shear field, and special cases of symmetry are considered which reduce the number of contributing terms. Steadyshear methods, which assume a body matches the translational and rotational motions of the fluid element it replaces, are shown to slightly overestimate ν when complex shapes are involved. An algebraic criterion is found to locate the center of viscosity needed in these other methods although our treatment is independent of the choice of position in the body used for calculations. Bead-model expressions are derived in order to provide numerical treatments of complicated structures. As an example, we examine a long bent rod.     

Quasielastic light scattering from solutions of filamentous viruses. II. Comparison between theories and experiments

We have compared four theoretical effects of rodlike macromolecules with the fast components, i.e., components other than translational diffusion, of our experimental data, which are presented as amplitude autocorrelation functions of electric field scattered from dilute solutions of monodisperse rodlike viruses with lengths from 3300 Å for tobacco mosaic virus to 20,000 Å for Pf1. The four effects are (1) the optic anisotropy treated by Aragón and Pecora, (2) coupled translational–rotational diffusion due to anisotropy in translational mobility recently reformulated by Gierke, (3) anisotropic rotational diffusion with respect to the direction of translational displacement first discussed by Berne and Pecora, and (4) the bending mode of a rod by Fujime and Maruyama. We show that both the first and second effects are required to explain the enhancement of amplitude of the translational diffusion at the expense of fast components. The experimental decay rates of the fast component exceed that of the rotational diffusions. In order to explain the excessive decay rate in the fast component, we need to include a minute amount (～1%) of bending mode of rodlike viruses, especially in longer viruses such as M13 and Pf1.     

Hydrodynamic properties of macromolecular complexes. IV. Intrinsic viscosity theory,with applications to once-broken rods and multisubunit proteins

We have extended our previous theories of the translational and rotational frictional properties of multisubunit complexes to calculate the intrinsic viscosity of such structures. Our theory is similar to those recently construced by McCammon and Deutch, and by Nakajima and Wada, in that it uses a modified hydrodynamic interaction tensor and solves the system of simultaneous interaction equations by digital computation rather than by successive approximations. However, there are some differences in the formulation and averaging of these equations. Extensive numerical comparison is made between this theory and others that are available—associated with the names of Hearst and Tagami, Abdel-Khalik and Bird, and Tsuda—using as a basis exact results for prolate ellipsoids of revolution. For large axial ratios, only our theory asymptotically approaches the correct limit; but for small axial ratios, only the Tsuda “shell-model” theory is adequate, because the other theories neglect the preponderant influence of the sphere located at the center of rotation. Intrinsic viscosities, translational frictional coefficients, and Scheraga-Mandelkern β parameters, are tabulated for a large number of polygonal and polyhedral subunit structures, with up to eight elements, using both our theory and Tsuda's. Particular application is made to hemerythrin and aspartate transcarbamylase. Finally, the viscosities and friction coefficients o once-broken rods are calculated and compared with an approximate theory by Wilenski.     

In vivo measurement of translational stiffness of rabbit knees

This paper describes the design, evaluation, and preliminary results of a specialized testing device and surgical protocol to determine translational stiffness of a rabbit knee, replicating the clinical anterior drawer test. Coronal-plane transverse pins are inserted through the rabbit leg, two in the tibia and one in the distal femur, to hold and reproducibly position the leg in the device for tests at multiple time points. A linear stepper motor draws the tibia upward then returns to the home position, and a load cell measures the resisting force; force-displacement knee stiffness is then calculated. Initial evaluation of this testing device determined the effects of preconditioning, intra-operator repeatability, rabbit-to-rabbit variability, knee flexion angle (90 degrees vs. 135 degrees ), and anterior cruciate ligament (ACL) sectioning (0%, 25%, 50%, 75%, 100%). Knee stiffness generally decreased as ACL sectioning increased. This testing device and surgical protocol provide an objective and efficient method of determining translational rabbit knee stiffness in vivo, and are being used in an ongoing study to evaluate the effect of knee instability (via partial to complete ACL sectioning) on the development of post-traumatic osteoarthritis.     

Transport properties of particles with segmental flexibility. I. Hydrodynamic resistance and diffusion coefficients of a freely hinged particle

Expressions are derived for the hydrodynamic resistance tensor and the diffusion tensor of a particle consisting of two rigid subunits connected by a free hinge. No restrictions are placed on the shapes of the subunits. The resistance tensor is obtained by using two independent approaches: first, from the Rayleigh dissipation function and, second, from an examination of the generalized forces for the appropriate seven-dimensional coordinate system. For the derivation of the generalized Einstein equation connecting the diffusion and resistance tensors, the Brownian motion is treated as a stochastic process. That derivation is based on the assumption that the restoring force for bending is negligible, and the Einstein relation holds instantaneously only if that assumption is true. The relationship between these tensors and the macroscopically observable parameters is discussed, and it is shown that the separate measurement of resistance and diffusion coefficients can be used to detect macromolecular flexibility. One example is treated, the diffusion of a particle composed of two long rods joined at a free hinge. Those calculations are carried out with the first-order assumption of negligible hydrodynamic interactions between the subunits. For the hinged rod, the bending degree of freedom produces a 34% increase in the translational diffusion coefficient over that of a stiff rod of the same total length, while the rotational diffusion coefficient about the axis perpendicular to the plane of bending is increased by 125%.     

Prediction of apparent trabecular bone stiffness through fourth-order fabric tensors

The apparent stiffness tensor is an important mechanical parameter for characterizing trabecular bone. Previous studies have modeled this parameter as a function of mechanical properties of the tissue, bone density, and a second-order fabric tensor, which encodes both anisotropy and orientation of trabecular bone. Although these models yield strong correlations between observed and predicted stiffness tensors, there is still space for reducing accuracy errors. In this paper, we propose a model that uses fourth-order instead of second-order fabric tensors. First, the totally symmetric part of the stiffness tensor is assumed proportional to the fourth-order fabric tensor in the logarithmic scale. Second, the asymmetric part of the stiffness tensor is derived from relationships among components of the harmonic tensor decomposition of the stiffness tensor. The mean intercept length (MIL), generalized MIL (GMIL), and fourth-order global structure tensor were computed from images acquired through microcomputed tomography of 264 specimens of the femur. The predicted tensors were compared to the stiffness tensors computed by using the micro-finite element method (\(\upmu \)FE), which was considered as the gold standard, yielding strong correlations (\(R^2\) above 0.962). The GMIL tensor yielded the best results among the tested fabric tensors. The Frobenius error, geodesic error, and the error of the norm were reduced by applying the proposed model by 3.75, 0.07, and 3.16 %, respectively, compared to the model by Zysset and Curnier (Mech Mater 21(4):243–250, 1995) with the second-order MIL tensor. From the results, fourth-order fabric tensors are a good alternative to the more expensive \(\upmu \)FE stiffness predictions.     

Diffusion coefficients of segmentally flexible macromolecules with two spherical subunits

The translational and rotational diffusion coefficients have been calculated for a simple, segmentally flexible model: the hinged dumbbell (HD). In the HD, two spherical subunits are attached to an universal joint by means of frictionless connectors. In addition to the case in which hydrodynamic interactions are neglected (NI), we have also considered two more cases, including hydrodynamic interaction by means of the Kirkwood-Riseman approximate treatment (KR) and using accurate procedure based in the series expansions for the two-sphere diffusion tensor (SE). Expressions for the friction coefficients of the HD are given for the three cases, and the diffusion coefficients are evaluted inverting the 9 × 9 resistance matrix, for two HDs with different dimensions. The KR treatment, which includes a contribution from the finite volume of the subunits, is shown to be an excellent approximation to the more rigorous procedure. In the NI case for rotation, the various coefficients present different deviations with respect to the SE results. A rough estimate of the errors of the NI relaxation times indicates that they may be smaller than 15% for a HD with identical beads. However, the influence of hydrodynamic interaction should be more important for the rotational diffusivity of a small sphere attached to a larger one. The error of the NI result for the translational diffusion coefficient is of about 25% for the two HDs.     

Estimation of musculotendon kinematics in large musculoskeletal models using multidimensional B-splines

We present a robust and computationally inexpensive method to estimate the lengths and three-dimensional moment arms for a large number of musculotendon actuators of the human lower limb. Using a musculoskeletal model of the lower extremity, a set of values was established for the length of each musculotendon actuator for different lower limb generalized coordinates (joint angles). A multidimensional spline function was then used to fit these data. Muscle moment arms were obtained by differentiating the musculotendon length spline function with respect to the generalized coordinate of interest. This new method was then compared to a previously used polynomial regression method. Compared to the polynomial regression method, the multidimensional spline method produced lower errors for estimating musculotendon lengths and moment arms throughout the whole generalized coordinate workspace. The fitting accuracy was also less affected by the number of dependent degrees of freedom and by the amount of experimental data available. The spline method only required information on musculotendon lengths to estimate both musculotendon lengths and moment arms, thus relaxing data input requirements, whereas the polynomial regression requires different equations to be used for both musculotendon lengths and moment arms. Finally, we used the spline method in conjunction with an electromyography driven musculoskeletal model to estimate muscle forces under different contractile conditions, which showed that the method is suitable for the integration into large scale neuromusculoskeletal models.     

Platelet Mechanosensing of Collagen Matrices

During vascular injury, platelets adhere to exposed subendothelial proteins, such as collagen, on the blood vessel walls to trigger clot formation. Although the biochemical signalings of platelet-collagen interactions have been well characterized, little is known about the role microenvironmental biomechanical properties, such as vascular wall stiffness, may have on clot formation. To that end, we investigated how substrates of varying stiffness conjugated with the same concentration of Type I collagen affect platelet adhesion, spreading, and activation. Using collagen-conjugated polyacrylamide (PA) gels of different stiffnesses, we observed that platelets do in fact mechanotransduce the stiffness cues of collagen substrates, manifesting in increased platelet spreading on stiffer substrates. In addition, increasing substrate stiffness also increases phosphatidylserine exposure, a key aspect of platelet activation that initiates coagulation on the platelet surface. Mechanistically, these collagen substrate stiffness effects are mediated by extracellular calcium levels and actomyosin pathways driven by myosin light chain kinase but not Rho-associated protein kinase. Overall, our results improve our understanding of how the mechanics of different tissues and stroma affect clot formation, what role the increased vessel wall stiffness in atherosclerosis may directly have on thrombosis leading to heart attacks and strokes, and how age-related increased vessel wall stiffness affects hemostasis and thrombosis.     

REACH Coarse-Grained Normal Mode Analysis of Protein Dimer Interaction Dynamics

The REACH (realistic extension algorithm via covariance Hessian) coarse-grained biomolecular simulation method is a self-consistent multiscale approach directly mapping atomistic molecular dynamics simulation results onto a residue-scale model. Here, REACH is applied to calculate the dynamics of protein-protein interactions. The intra- and intermolecular fluctuations and the intermolecular vibrational densities of states derived from atomistic molecular dynamics are well reproduced by the REACH normal modes. The phonon dispersion relations derived from the REACH lattice dynamics model of crystalline ribonuclease A are also in satisfactory agreement with the corresponding all-atom results. The REACH model demonstrates that increasing dimer interaction strength decreases the translational and rotational intermolecular vibrational amplitudes, while their vibrational frequencies are relatively unaffected. A comparative study of functionally interacting biological dimers with crystal dimers, which are formed artificially via crystallization, reveals a relation between their static structures and the interprotein dynamics: i.e., the consequence of the extensive interfaces of biological dimers is reduction of the intermonomer translational and rotational amplitudes, but not the frequencies.     

Nanoscale protein domain motion and long-range allostery in signaling proteins—a view from neutron spin echo spectroscopy

Many cellular proteins are multi-domain proteins. Coupled domain–domain interactions in these multidomain proteins are important for the allosteric relay of signals in the cellular signaling networks. We have initiated the application of neutron spin echo spectroscopy to the study of nanoscale protein domain motions on submicrosecond time scales and on nanometer length scale. Our NSE experiments reveal the activation of protein domain motions over a long distance of over more than 100 Å in a multidomain scaffolding protein NHERF1 upon binding to another protein, Ezrin. Such activation of nanoscale protein domain motions is correlated with the allosteric assembly of multi-protein complexes by NHERF1 and Ezrin. Here, we summarize the theoretical framework that we have developed, which uses simple concepts from nonequilibrium statistical mechanics to interpret the NSE data, and employs a mobility tensor to describe nanoscale protein domain motion. Extracting nanoscale protein domain motion from the NSE does not require elaborate molecular dynamics simulations, nor complex fits to rotational motion, nor elastic network models. The approach is thus more robust than multiparameter techniques that require untestable assumptions. We also demonstrate that an experimental scheme of selective deuteration of a protein subunit in a complex can highlight and amplify specific domain dynamics from the abundant global translational and rotational motions in a protein. We expect NSE to provide a unique tool to determine nanoscale protein dynamics for the understanding of protein functions, such as how signals are propagated in a protein over a long distance to a distal domain.     

Repeatability and precision of a weighted centroid method for estimating dynamic in vivo tibiofemoral surface interactions in sheep

Persistent changes in joint biomechanics resulting from knee injury are thought to contribute to progressive cartilage damage and post-traumatic osteoarthritis (PTOA). The identification and quantification of in vivo tibiofemoral surface interactions are critical to understanding them, particularly abnormal interactions that are damaging to articular cartilage and other structures of the knee. In this study, we describe an approach for understanding such potential interactions by using a weighted centroid derived from in vivo stifle kinematics in sheep. Collectively, repeatability and sensitivity analyses indicate that the magnitude of the changes in tibiofemoral centroid location resulting from combined ligament transection is greater than the repeatability and precision of the current weighted centroid approach, making this method useful for describing the changes in dynamic surface interactions that may be relevant in the pathogenesis of PTOA in this stifle injury model.     

A direct comparison of spine rotational stiffness and dynamic spine stability during repetitive lifting tasks

Stability of the spinal column is critical to bear loads, allow movement, and at the same time avoid injury and pain. However, there has been a debate in recent years as to how best to define and quantify spine stability, with the outcome being that different methods are used without a clear understanding of how they relate to one another. Therefore, the goal of the present study was to directly compare lumbar spine rotational stiffness, calculated with an EMG-driven biomechanical model, to local dynamic spine stability calculated using Lyapunov analyses of kinematic data, during a series of continuous dynamic lifting challenges. Twelve healthy male subjects performed 30 repetitive lifts under three varying load and three varying rate conditions. With an increase in the load lifted (constant rate) there was a significant increase in mean, maximum, and minimum spine rotational stiffness (p<0.001) and a significant increase in local dynamic stability (p<0.05); both stability measures were moderately to strongly related to one another (r=-0.55 to -0.71). With an increase in lifting rate (constant load), there was also a significant increase in mean and maximum spine rotational stiffness (p<0.01); however, there was a non-significant decrease in the minimum rotational stiffness and a non-significant decrease in local dynamic stability (p>0.05). Weak linear relationships were found for the varying rate conditions (r=-0.02 to -0.27). The results suggest that spine rotational stiffness and local dynamic stability are closely related to one another, as they provided similar information when movement rate was controlled. However, based on the results from the changing lifting rate conditions, it is evident that both models provide unique information and that future research is required to completely understand the relationship between the two models. Using both techniques concurrently may provide the best information regarding the true effects of (in) stability under different loading and movement scenarios, and in comparing healthy and clinical populations.     

Increased Global and Local Efficiency of Human Brain Anatomical Networks Detected with FLAIR-DTI Compared to Non-FLAIR-DTI

Diffusion-weighted MRI (DW-MRI), the only non-invasive technique for probing human brain white matter structures in vivo, has been widely used in both fundamental studies and clinical applications. Many studies have utilized diffusion tensor imaging (DTI) and tractography approaches to explore the topological properties of human brain anatomical networks by using the single tensor model, the basic model to quantify DTI indices and tractography. However, the conventional DTI technique does not take into account contamination by the cerebrospinal fluid (CSF), which has been known to affect the estimated DTI measures and tractography in the single tensor model. Previous studies have shown that the Fluid-Attenuated Inversion Recovery (FLAIR) technique can suppress the contribution of the CSF to the DW-MRI signal. We acquired DTI datasets from twenty-two subjects using both FLAIR-DTI and conventional DTI (non-FLAIR-DTI) techniques, constructed brain anatomical networks using deterministic tractography, and compared the topological properties of the anatomical networks derived from the two types of DTI techniques. Although the brain anatomical networks derived from both types of DTI datasets showed small-world properties, we found that the brain anatomical networks derived from the FLAIR-DTI showed significantly increased global and local network efficiency compared with those derived from the conventional DTI. The increases in the network regional topological properties derived from the FLAIR-DTI technique were observed in CSF-filled regions, including the postcentral gyrus, periventricular regions, inferior frontal and temporal gyri, and regions in the visual cortex. Because brain anatomical networks derived from conventional DTI datasets with tractography have been widely used in many studies, our findings may have important implications for studying human brain anatomical networks derived from DW-MRI data and tractography.