A musculo-skeletal model of rat ankle motion and its experimental test on rat

A biomechanical musculo-skeletal model of functional electrical stimulation (FES)-induced rat ankle motion was implemented and tested in rat experiments. The muscle model is a new Hill-based model which includes established physiological relations of force-velocity and force-length-frequency. However, the series-elastic component and the activation component of previous Hill-based models are replaced by a new component which accounts for dynamic time delays and recruitment that occur in real muscle force generation during limb movements. The skeletal model includes gravity and dynamic forces that occur in real rat ankle motions. In computer simulations, various FES patterns were applied to the tibialis anterior (TA) and soleus (SO) model muscles to produce walk-like ankle motions. In lab experiments, the same stimulation patterns were applied by epimysial electrodes implanted in the TA and SO muscles of live rats cordotomized at level T7. The resulting rat motions were recorded by video camera. Video data was converted to ankle angle-vs-time files for comparison with corresponding model angle-vs-time files. Over a physiologically significant range of ankle motions, model parameters were adjustable to yield model motions that agreed with rat motions to within 2 degrees (root mean square differences of rat and model ankle angles). This is shown in plots of model and rat motions presented here for representative cases of FES. The accuracy of our model in reproducing real ankle motions supports the hypothesis that our new muscle model generates correct muscle forces over a useful range of limb motions. It suggests that the model may be useful in the design of FES neural prostheses.     

Side-chain flexibility in protein-ligand binding: the minimal rotation hypothesis

The goal of this work is to learn from nature about the magnitudes of side-chain motions that occur when proteins bind small organic molecules, and model these motions to improve the prediction of protein-ligand complexes. Following analysis of protein side-chain motions upon ligand binding in 63 complexes, we tested the ability of the docking tool SLIDE to model these motions without being restricted to rotameric transitions or deciding which side chains should be considered as flexible. The model tested is that side-chain conformational changes involving more atoms or larger rotations are likely to be more costly and less prevalent than small motions due to energy barriers between rotamers and the potential of large motions to cause new steric clashes. Accordingly, SLIDE adjusts the protein and ligand side groups as little as necessary to achieve steric complementarity. We tested the hypothesis that small motions are sufficient to achieve good dockings using 63 ligands and the apo structures of 20 different proteins and compared SLIDE side-chain rotations to those experimentally observed. None of these proteins undergoes major main-chain conformational change upon ligand binding, ensuring that side-chain flexibility modeling is not required to compensate for main-chain motions. Although more frugal in the number of side-chain rotations performed, this model substantially mimics the experimentally observed motions. Most side chains do not shift to a new rotamer, and small motions are both necessary and sufficient to predict the correct binding orientation and most protein-ligand interactions for the 20 proteins analyzed.     

hdANM: a new comprehensive dynamics model for protein hinges

Hinge motions are essential for many protein functions, and their dynamics are important to understand underlying biological mechanisms. The ways that these motions are represented by various computational methods differ significantly. By focusing on a specific class of motion, we have developed a new hinge-domain anisotropic network model (hdANM) that is based on the prior identification of flexible hinges and rigid domains in the protein structure and the subsequent generation of global hinge motions. This yields a set of motions in which the relative translations and rotations of the rigid domains are modulated and controlled by the deformation of the flexible hinges, leading to a more restricted, specific view of these motions. hdANM is the first model, to our knowledge, that combines information about protein hinges and domains to model the characteristic hinge motions of a protein. The motions predicted with this new elastic network model provide important conceptual advantages for understanding the underlying biological mechanisms. As a matter of fact, the generated hinge movements are found to resemble the expected mechanisms required for the biological functions of diverse proteins. Another advantage of this model is that the domain-level coarse graining makes it significantly more computationally efficient, enabling the generation of hinge motions within even the largest molecular assemblies, such as those from cryo-electron microscopy. hdANM is also comprehensive as it can perform in the same way as the well-known protein dynamics models (anisotropic network model, rotations-translations of blocks, and nonlinear rigid block normal mode analysis), depending on the definition of flexible and rigid parts in the protein structure and on whether the motions are extrapolated in a linear or nonlinear fashion. Furthermore, our results indicate that hdANM produces more realistic motions as compared to the anisotropic network model. hdANM is an open-source software, freely available, and hosted on a user-friendly website.     

Biomechanics of cranial kinesis in birds: Testing linkage models in the white-throated sparrow (Zonotrichia albicollis)

Cranial kinesis in sparrows refers to the rotation of the upper jaw around its kinetic joint with the braincase. Avian jaw mechanics may involve the coupled motions of upper and lower jaws, in which the postorbital ligament transfers forces from the lower jaw, through the quadrate, pterygoid, and jugal bones, to the upper jaw. Alternatively, jaw motions may be uncoupled, with the upper jaw moving independently of the lower jaw. We tested hypotheses of cranial kinesis through the use of quantitative computer models. We present a biomechanical model of avian jaw kinetics that predicts the motions of the jaws under assumptions of both a coupled and an uncoupled mechanism. In addition, the model predicts jaw motions under conditions of force transfer by either the jugal or the pterygoid bones. Thus four alternative models may be tested using the proposed model (coupled jugal, coupled pterygoid, uncoupled jugal, uncoupled pterygoid). All models are based on the mechanics of four-bar linkages and lever systems and use morphometric data on cranial structure as the basis for predicting cranial movements. Predictions of cranial motions are tested by comparison to kinematics of white-throated sparrows (Zonotrichia albicollis) during singing. The predicted relations between jaw motions for the coupled model are significantly different from video observations. We conclude that the upper and lower jaws are not coupled in white-throated sparrows. The range of jaw motions during song is consistent with a model in which independent contractions of upper and lower jaw muscles control beak motion. © 1996 Wiley-Liss, Inc.     

Development of a computational model for astronaut reorientation

The ability to model astronaut reorientations computationally provides a simple way to develop and study human motion control strategies. Since the cost of experimenting in microgravity is high, and underwater training can lead to motions inappropriate for microgravity, these techniques allow for motions to be developed and well-understood prior to any microgravity exposure. By including a model of the current space suit, we have the ability to study both intravehicular and extravehicular activities. We present several techniques for rotating about the axes of the body and show that motions performed by the legs create a greater net rotation than those performed by the arms. Adding a space suit to the motions was seen to increase the resistance torque and limit the available range of motion. While rotations about the body axes can be performed in the current space suit, the resulting motions generated a reduced rotation when compared to the unsuited configuration.     

Energy landscape of a native protein: Jumping-among-minima model

We have investigated energy landscape of human lysozyme in its native state by using principal component analysis and a model, jumping-among-minima (JAM) model. These analyses are applied to 1 nsec molecular dynamics trajectory of the protein in water. An assumption embodied in the JAM model allows us to divide protein motions into intra-substate and inter-substate motions. By examining intra-substate motions, it is shown that energy surfaces of individual conformational substates are nearly harmonic and mutually similar. As a result of principal component analysis and JAM model analysis, protein motions are shown to consist of three types of collective modes, multiply hierarchical modes, singly hierarchical modes, and harmonic modes. Multiply hierarchical modes, the number of which accounts only for 0.5% of all modes, dominate contributions to total mean-square atomic fluctuation. Inter-substate motions are observed only in a small-dimensional subspace spanned by the axes of multiplyhierarchical and singly hierarchical modes. Inter-substate motions have two notable time components: faster component seen within 200 psec and slower component. The former involves transitions among the conformational substates of the low-level hierarchy, whereas the latter involves transitions of the higher level substates observed along the first four multiply hierarchical modes. We also discuss dependence of the subspace, which contains conformational substates, on time duration of simulation. Proteins 33:496–517, 1998. © 1998 Wiley-Liss, Inc.     

Two-arm trajectory planning in a manipulation task

This paper addresses the trajectory planning problem for a task which requires positioning and orienting an object firmly grasped by two hands at a visually specified goal configuration in the horizontal plane. The motor task involves three degrees of freedom (two translational and one rotational), and the motions of the arms are constrained by the physical coupling through the held object. Experimentally measured trajectories of two arms in the coordinated positioning/orienting task are presented. The hypothesis that the rotational and translational components of motions are decoupled and independently planned is tested. Two explicit mathematical models to account for the kinematic features of the two-arm motions are formulated, and the predictions of the models are compared with the experimental data. Both models extend the minimum-jerk model to the two-arm coordinated motions case. The trajectories predicted by the models were found to be in qualitative agreement with the experimental data. However, neither model could account for the observed configuration dependence of the motions, nor for some of the properties of the measured velocity components of the motions. Our findings support the idea that the rotational and translational components of two-arm motions in the positioning/orienting task are independently planned in extra-personal space, and are further combined in a hierarchical fashion to produce the observed motions. The tested models may serve as a basis for further investigations of issues pertinent to the generation of two-arm trajectories. Received: 27 March 1995 / Accepted in revised form: 17 September 1996     

Study of the dynamics of human serum albumin by coherent Rayleigh dispersion of Mossbauer radiation]

The measurements of angle dependencies of total and elastic Rayleigh scattering of Mossbauer radiation intensities have been performed for human serum albumin (HSA) with hydration degrees h = 0.13 and h = 0.4. The extended model was developed for calculating the inelastic intensity of Rayleigh scattering. Original data for HSA and published data on met-Mb were fitted within the frame of this model. The best agreement with experiment was obtained when two types of intraglobular motions were taken into account: individual motions of small side-chain groups and cooperative (mechanical) motions of segments (most probable alpha-helices). Long-range correlated motions are essential at low hydration degree. The possibilities of application of the coherent version of RSMS technique are described.     

An analysis of the influence of protein intrinsic dynamical properties on its thermal unfolding behavior

The influence of the protein topology-encoded dynamical properties on its thermal unfolding motions was studied in the present work. The intrinsic dynamics of protein topology was obtained by the anisotropic network model (ANM). The ANM has been largely used to investigate protein collective functional motions, but it is not well elucidated if this model can also reveal the preferred large-scale motions during protein unfolding. A small protein barnase is used as a typical case study to explore the relationship between protein topology-encoded dynamics and its unfolding motions. Three thermal unfolding simulations at 500 K were performed for barnase and the entire unfolding trajectories were sampled and partitioned into several windows. For each window, the preferred unfolding motions were investigated by essential dynamics analysis, and then associated with the intrinsic dynamical properties of the starting conformation in this window, which is detected by ANM. The results show that only a few slow normal modes imposed by protein structure are sufficient to give a significant overlap with the preferred unfolding motions. Especially, the large amplitude unfolding movements, which imply that the protein jumps out of a local energy basin, can be well described by a single or several ANM slow modes. Besides the global motions, it is also found that the local residual fluctuations encoded in protein structure are highly correlated with those in the protein unfolding process. Furthermore, we also investigated the relationship between protein intrinsic flexibility and its unfolding events. The results show that the intrinsic flexible regions tend to unfold early. Several early unfolding events can be predicted by analysis of protein structural flexibility. These results imply that protein structure-encoded dynamical properties have significant influences on protein unfolding motions.     

An alternative derivation of the equations of motion in torsion space for a branched linear chain.

Torsion space molecular dynamics may be more efficiently encoded if the global motions are separated from the internal motions. The equations of motion for single, non-cyclic chains are shown to be first order in the backbone angle parameters when the global frame of reference is ignored and second order otherwise. Adding a simple heuristic substitute for the global motions enables the encoding of dynamics for mixed constrained/unconstrained model systems.     

The dynamical Matryoshka model: 3. Diffusive nature of the atomic motions contained in a new dynamical model for deciphering local lipid dynamics

In accompanying papers [Bicout et al., BioRxiv https://doi.org/10.1101/2021.09.21.461198 (2021); Cissé et al., BioRxiv https://doi.org/10.1101/2022.03.30.486370 (2022)], a new model called Matryoshka model has been proposed to describe the geometry of atomic motions in phospholipid molecules in bilayers and multilamellar vesicles based on their quasielastic neutron scattering (QENS) spectra. Here, in order to characterize the relaxational aspects of this model, the energy widths of the QENS spectra of the samples were analyzed first in a model-free way. The spectra were decomposed into three Lorentzian functions, which are classified as slow, intermediate, and fast motions depending on their widths. The analysis provides the diffusion coefficients, residence times, and geometrical parameters for the three classes of motions. The results corroborate the parameter values such as the amplitudes and the mobile fractions of atomic motions obtained by the application of the Matryoshka model to the same samples. Since the current analysis was carried out independently of the development of the Matryoshka model, the present results enhance the validity of the model. The model will serve as a powerful tool to decipher the dynamics of lipid molecules not only in model systems, but also in more complex systems such as mixtures of different kinds of lipids or natural cell membranes.     

Characterization of protein dynamics from residual dipolar couplings using the three dimensional Gaussian axial fluctuation model

Residual dipolar couplings are potentially very powerful probes of slower protein motions, providing access to dynamic events occurring on functionally important timescales up to the millisecond. One recent approach uses the three dimensional Gaussian Axial Fluctuation model (3D GAF) to determine the major directional modes and associated amplitudes of motions along the peptide chain. In this study we have used standard and accelerated molecular dynamics simulations to determine the accuracy of 3D GAF-based approaches in characterizing the nature and extent of local molecular motions. We compare modes determined directly from the trajectories with motional parameterization derived from RDCs simulated from the same trajectories. Three approaches are tested, that either suppose a known three-dimensional structure, simultaneously determine backbone structure and dynamics, or determine dynamic modes in the absence of a structural model. The results demonstrate the robustness of the 3D GAF analysis even in the presence of large-scale motions, and illustrate the remarkably quantitative nature of the extracted amplitudes. These observations suggest that the approach can be generally used for the study of functionally interesting biomolecular motions.     

Molecular dynamics simulations of peptides and proteins with amplified collective motions

We present a novel method that uses the collective modes obtained with a coarse-grained model/anisotropic network model to guide the atomic-level simulations. Based on this model, local collective modes can be calculated according to a single configuration in the conformational space of the protein. In the molecular dynamics simulations, the motions along the slowest few modes are coupled to a higher temperature by the weak coupling method to amplify the collective motions. This amplified-collective-motion (ACM) method is applied to two test systems. One is an S-peptide analog. We realized the refolding of the denatured peptide in eight simulations out of 10 using the method. The other system is bacteriophage T4 lysozyme. Much more extensive domain motions between the N-terminal and C-terminal domain of T4 lysozyme are observed in the ACM simulation compared to a conventional simulation. The ACM method allows for extensive sampling in conformational space while still restricting the sampled configurations within low energy areas. The method can be applied in both explicit and implicit solvent simulations, and may be further applied to important biological problems, such as long timescale functional motions, protein folding/unfolding, and structure prediction.     

Anisotropic overall and internal motions of short DNA fragments.

Anisotropic motions of DNA fragments in the size range 6-118 base pairs are studied by the steady-state fluorescence polarization of different excitation transitions in the intercalated ethidium cation. Calculated effective tumbling and twisting times are found to be shorter than predicted for overall motions of rigid DNA, indicating that internal motions and/or dye wobbling contribute to the depolarization. The data are consistent with a model where the DNA fragments are considered to be rigid against bending but torsionally flexible, and where the dye can wobble within the intercalated site. We also discuss the possibility of correlated out-of-plane motions of the dye and the DNA bases.     

Elastic network models capture the motions apparent within ensembles of RNA structures

The role of structure and dynamics in mechanisms for RNA becomes increasingly important. Computational approaches using simple dynamics models have been successful at predicting the motions of proteins and are often applied to ribonucleo-protein complexes but have not been thoroughly tested for well-packed nucleic acid structures. In order to characterize a true set of motions, we investigate the apparent motions from 16 ensembles of experimentally determined RNA structures. These indicate a relatively limited set of motions that are captured by a small set of principal components (PCs). These limited motions closely resemble the motions computed from low frequency normal modes from elastic network models (ENMs), either at atomic or coarse-grained resolution. Various ENM model types, parameters, and structure representations are tested here against the experimental RNA structural ensembles, exposing differences between models for proteins and for folded RNAs. Differences in performance are seen, depending on the structure alignment algorithm used to generate PCs, modulating the apparent utility of ENMs but not significantly impacting their ability to generate functional motions. The loss of dynamical information upon coarse-graining is somewhat larger for RNAs than for globular proteins, indicating, perhaps, the lower cooperativity of the less densely packed RNA. However, the RNA structures show less sensitivity to the elastic network model parameters than do proteins. These findings further demonstrate the utility of ENMs and the appropriateness of their application to well-packed RNA-only structures, justifying their use for studying the dynamics of ribonucleo-proteins, such as the ribosome and regulatory RNAs.     

Counterbalance of ligand‐ and self‐coupled motions characterizes multispecificity of ubiquitin

Date hub proteins are a type of proteins that show multispecificity in a time‐dependent manner. To understand dynamic aspects of such multispecificity we studied Ubiquitin as a typical example of a date hub protein. Here we analyzed 9 biologically relevant Ubiquitin‐protein (ligand) heterodimer structures by using normal mode analysis based on an elastic network model. Our result showed that the self‐coupled motion of Ubiquitin in the complex, rather than its ligand‐coupled motion, is similar to the motion of Ubiquitin in the unbound condition. The ligand‐coupled motions are correlated to the conformational change between the unbound and bound conditions of Ubiquitin. Moreover, ligand‐coupled motions favor the formation of the bound states, due to its in‐phase movements of the contacting atoms at the interface. The self‐coupled motions at the interface indicated loss of conformational entropy due to binding. Therefore, such motions disfavor the formation of the bound state. We observed that the ligand‐coupled motions are embedded in the motions of unbound Ubiquitin. In conclusion, multispecificity of Ubiquitin can be characterized by an intricate balance of the ligand‐ and self‐coupled motions, both of which are embedded in the motions of the unbound form.     

Virtual power based algorithm for decoupling large motions from infinitesimal strains: application to shoulder joint biomechanics

New trends of numerical models of human joints require more and more computation of both large amplitude joint motions and fine bone stress distribution. Together, these problems are difficult to solve and very CPU time consuming. The goal of this study is to develop a new method to diminish the calculation time for this kind of problems which include calculation of large amplitude motions and infinitesimal strains. Based on the Principle of Virtual Power, the present method decouples the problem into two parts. First, rigid body motion is calculated. The bone micro-deformations are then calculated in a second part by using the results of rigid body motions as boundary conditions. A finite element model of the shoulder was used to test this decoupling technique. The model was designed to determine the influence of humeral head shape on stress distribution in the scapula for different physiological motions of the joint. Two versions of the model were developed: a first version completely deformable and a second version based on the developed decoupling method. It was shown that biomechanical variables, as mean pressure and von Mises stress, calculated with the two versions were sensibly the same. On the other hand, CPU time needed for calculating with the new decoupled technique was more than 6 times less than with the completely deformable model.     

A Finite Element Model of the L5-S1 Functional Spinal Unit: Development and Comparison with Biomechanical Tests In Vitro

The main objective of this work is to develop a three-dimensional finite element model of the L5-S1 segment that is able to simulate its passive mobility measured in vitro . Due to their limited role in segment mobility, an isotropic linear elastic constitutive law was used for cartilage, cancellous and cortical bone. The intervertebral disk ground substance was modeled with a non-linear hyperelastic polynomial law. Fibers of the disk, as well as ligaments, were modeled with piecewise linear springs. Flexion-extension, axial rotation, and lateral bending torques were applied to the model. A comparison with the experimental results obtained on the same segment for these three major motions was conducted. The compliance of the segment subjected to pure torques was found to be similar between numerical and experimental results for all major motions. Coupled motions and translations were also similar, even in their amplitude. For lateral bending, the normal coupled motions originate from the geometry of the disk and not from the facet geometry.     

A finite element model of the L5-S1 functional spinal unit: development and comparison with biomechanical tests in vitro

The main objective of this work is to develop a three-dimensional finite element model of the L5-S1 segment that is able to simulate its passive mobility measured in vitro. Due to their limited role in segment mobility, an isotropic linear elastic constitutive law was used for cartilage, cancellous and cortical bone. The intervertebral disk ground substance was modeled with a non-linear hyperelastic polynomial law. Fibers of the disk, as well as ligaments, were modeled with piecewise linear springs. Flexion-extension, axial rotation, and lateral bending torques were applied to the model. A comparison with the experimental results obtained on the same segment for these three major motions was conducted. The compliance of the segment subjected to pure torques was found to be similar between numerical and experimental results for all major motions. Coupled motions and translations were also similar, even in their amplitude. For lateral bending, the normal coupled motions originate from the geometry of the disk and not from the facet geometry.     

A Model Coupling Vibrational and Rotational Motion for the DNA Molecule

We investigate a mechanical model for the DNA molecule using an extension of the Peyrard and Bishop model. In the present model, there are two chains of oscillators linked by a Morse potential, which represent the hydrogen bonds. The rotational and vibrational motions of each base pair are considered and the coupling for these motions are introduced by a nonlinear combination of them in the Morse potential. In this context, thermodynamics and structural properties are discussed.