New methodology for computer-aided modelling of biomolecular structure and dynamics. 2. Local deformations and cycles

A new methodology for the conformational modelling of biomolecular systems (1) is extended to local deformations of chain molecules and to flexible molecular rings. It is shown that these two cases may be reduced to considering an equivalent molecular model with a regular tree-like topology. A simple procedure is developed to analyze any flexible rings (the five- and six-membered sugar rings of carbohydrates and nucleic acids, in particular) and local deformation regions by energy minimization. Dynamic equations are also derived for such molecular systems. As a result, a unified approach is proposed for the efficient energy minimization and simulation of dynamic behavior of multimolecular systems having any set of variable internal coordinates, local deformation regions and cycles. Advantages and domains of applicability of the approach are discussed.     

New Methodology for Computer-Aided Modelling of Biomolecular Structure and Dynamics 2. Local Deformations and Cycles

Abstract

A new methodology for the conformational modelling of biomolecular systems (1) is extended to local deformations of chain molecules and to flexible molecular rings. It is shown that these two cases may be reduced to considering an equivalent molecular model with a regular tree-like topology. A simple procedure is developed to analyze any flexible rings (the five- and six-membered suguar rings of carbohydrates and nucleic acids, in particular) and local deformation regions by energy minimization. Dynamic equations are also derived for such molecular systems. As a result, a unified approach is proposed for the efficient energy minimization and simulation of dynamic behavior of multimolecular systems having any set of variable internal coordinates, local deformation regions and cycles. Advantages and domains of applicability of the approach are discussed.     

Plasma equilibrium in axisymmetric poloidal magnetic field configurations in flux coordinates

A simple derivation is given of equilibrium equations in flux coordinates in the general case of an anisotropic-pressure plasma. The issue of how to formulate the boundary conditions for these equations is discussed for two types of configurations—a straight system and a system with an internal conductor. Examples of numerical solutions to the equilibrium problem for these configurations are presented.     

A simple procedure for simulating samples of tissue using Voronoi diagrams

OBJECTIVE: To describe a simple and quick procedure for modeling samples of tissue with Voronoi diagrams. STUDY DESIGN: Instead of calculating the centers of the so-called Dirichlet domains (i.e., the polygonal areas occupied by individual cells), the centroid of such areas is used to generate Voronoi diagrams. The coordinates of the centroids are calculated by simply averaging the coordinates of the points of the cell contours; that is much simpler and faster than any geometric procedure for locating the Dirichlet centers. Using the centroids as centers, circles are allowed to grow until no space on the surface is available. With this procedure it is easy to control the rate of growth of individual cells or groups of cells according to any rule or rules. It is also possible to simulate the effects of removing > or = 1 cells from the sample. CONCLUSION: The procedure was successfully applied to modeling some of the changes that can occur in a real sample of human corneal endothelium. The procedure is simple, completely automated, efficient and flexible and can be easily implemented on a personal computer. It can be used to test growth or communication strategies among cells.     

Muscle moment-arms: a key element in muscle-force estimation

A clear and rigorous definition of muscle moment-arms in the context of musculoskeletal systems modelling is presented, using classical mechanics and screw theory. The definition provides an alternative to the tendon excursion method, which can lead to incorrect moment-arms if used inappropriately due to its dependency on the choice of joint coordinates. The definition of moment-arms, and the presented construction method, apply to musculoskeletal models in which the bones are modelled as rigid bodies, the joints are modelled as ideal mechanical joints and the muscles are modelled as massless, frictionless cables wrapping over the bony protrusions, approximated using geometric surfaces. In this context, the definition is independent of any coordinate choice. It is then used to solve a muscle-force estimation problem for a simple 2D conceptual model and compared with an incorrect application of the tendon excursion method. The relative errors between the two solutions vary between 0% and 100%.     

Planning and execution of multijoint movements

This paper reviews some recent studies related to the generation of simple multijoint arm movements. Two principal issues are considered. The first concern is how movements are represented internally by the central nervous system. There are many possible sets of coordinates that could be used to represent arm movements. Two of the possibilities are reviewed: representation in terms of joint angular motions versus representation in terms of motions of the hand in external space coordinates. A second concern is the transformation from intention to action: how is an internal representation of motion expressed by the neuromuscular system? The computational complexity of this problem is reviewed. A way in which the mechanics of the neuromuscular system could be exploited to simplify this problem is discussed.     

A new method for the evaluation of antibody affinity based on serial dilutions of studied antibody-antigen mixture

A new method for the evaluation of the affinity of bivalent antibodies were suggested. This method is based on the previously published by the author the idea of using so-called coordinates of dilutions. It was shown that the suggested method allows to evaluate the affinity of antibodies with high accuracy using this simple approach. It is supposed that at some conditions the suggested method could have substantial advantages in comparison to the traditional methods. This method allows to analyze situations when antibodies are already in a mixture with antigen, for example in the bloodstream in the case of infections or autoimmune diseases. The method provides useful approach for the evaluation not only antibody affinity, but also the concentration of circulated antigen.     

Ligand-Induced Changes of the Apparent Transition-State Position in?Mechanical Protein Unfolding

Force-spectroscopic measurements of ligand-receptor systems and the unfolding/folding of nucleic acids or proteins reveal information on the underlying energy landscape along the pulling coordinate. The slope Δx^‡ of the force-dependent unfolding/unbinding rates is interpreted as the distance from the folded/bound state to the transition state for unfolding/unbinding and, hence, often related to the mechanical compliance of the sample molecule. Here we show that in ligand-binding proteins, the experimentally inferred Δx^‡ can depend on the ligand concentration, unrelated to changes in mechanical compliance. We describe the effect in single-molecule, force-spectroscopy experiments of the calcium-binding protein calmodulin and explain it in a simple model where mechanical unfolding and ligand binding occur on orthogonal reaction coordinates. This model predicts changes in the experimentally inferred Δx^‡, depending on ligand concentration and the associated shift of the dominant barrier between the two reaction coordinates. We demonstrate quantitative agreement between experiments and simulations using a realistic six-state kinetic scheme using literature values for calcium-binding kinetics and affinities. Our results have important consequences for the interpretation of force-spectroscopic data of ligand-binding proteins.     

A kinetic model describing the processivity of myosin-V

The precise details of how myosin-V coordinates the biochemical reactions and mechanical motions of its two head elements to engineer effective processive molecular motion along actin filaments remain unresolved. We compare a quantitative kinetic model of the myosin-V walk, consisting of five basic states augmented by two further states to allow for futile hydrolysis and detachments, with experimental results for run lengths, velocities, and dwell times and their dependence on bulk nucleotide concentrations and external loads in both directions. The model reveals how myosin-V can use the internal strain in the molecule to synchronize the motion of the head elements. Estimates for the rate constants in the reaction cycle and the internal strain energy are obtained by a computational comparison scheme involving an extensive exploration of the large parameter space. This scheme exploits the fact that we have obtained analytic results for our reaction network, e.g., for the velocity but also the run length, diffusion constant, and fraction of backward steps. The agreement with experiment is often reasonable but some open problems are highlighted, in particular the inability of such a general model to reproduce the reported dependence of run length on ADP concentration. The novel way that our approach explores parameter space means that any confirmed discrepancies should give new insights into the reaction network model.     

On the computation of stress in affine versus nonaffine fibril kinematics within planar collagen network models

Some recent analyses modeled the response of collagenous tissues, such as epicardium, using a hypothetical network consisting of interconnected springlike fibers. The fibers in the network were organized such that internal nodes served as the connection point between three such collagen springs. The results for assumed affine and nonaffine deformations are contrasted after a homogeneous deformation at the boundary. Affine deformation provides a stiffer mechanical response than nonaffine deformation. In contrast to nonaffine deformation, affine deformation determines the displacement of internal nodes without imposing detailed force balance, thereby complicating the simplest intuitive notion of stress, one based on free body cuts, at the single node scale. The standard notion of stress may then be recovered via average field theory computations based on large micromesh realizations. An alternative and by all indications complementary viewpoint for the determination of stress in these collagen fiber networks is discussed here, one in which stress is defined using elastic energy storage, a notion which is intuitive at the single node scale. It replaces the average field theory computations by an averaging technique over randomly oriented isolated simple elements. The analytical operations do not require large micromesh realizations, but the tedious nature of the mathematical manipulation is clearly aided by symbolic algebra calculation. For the example case of linear elastic deformation, this results in material stiffnesses that relate the infinitesimal strain and stress. The result that the affine case is stiffer than the nonaffine case is recovered, as would be expected. The energy framework also lends itself to the natural inclusion of changes in mechanical response due to the chemical, electrical, or thermal environment.     

Molecular origin of the internal dipole potential in lipid bilayers: calculation of the electrostatic potential.

The finite difference linearized Poisson-Boltzmann equation was solved for a segment of bilayer for two lipids (phosphatidylcholine dihydrate and phosphatidylethanolamine-acetic acid) in order to obtain the transbilayer electrostatic potential. Atomic coordinates derived from the crystal structures of these lipids were used, and partial changes were assigned to all atoms in the polar parts of the molecules. These calculations confirmed that a dipole potential exists in the uncharged hydrophobic interior of a bilayer. The phosphocholine and phosphoethanolamine groups make negative contributions to the internal potential, and the glycerol acyl esters make positive contributions, but the sum of these terms is negative. The water of hydration in phosphatidylcholine, and the acetic acid which is present in the phosphatidylethanolamine crystal structure, make positive contributions to the internal potential. It is concluded that the water of hydration in fully hydrated lipid bilayers is mainly responsible for the experimentally inferred positive sign of the internal potential.     

Coarse-grained modeling of the actin filament derived from atomistic-scale simulations

A coarse-grained (CG) procedure that incorporates the information obtained from all-atom molecular dynamics (MD) simulations is presented and applied to actin filaments (F-actin). This procedure matches the averaged values and fluctuations of the effective internal coordinates that are used to define a CG model to the values extracted from atomistic MD simulations. The fluctuations of effective internal coordinates in a CG model are computed via normal-mode analysis (NMA), and the computed fluctuations are matched with the atomistic MD results in a self-consistent manner. Each actin monomer (G-actin) is coarse-grained into four sites, and each site corresponds to one of the subdomains of G-actin. The potential energy of a CG G-actin contains three bonds, two angles, and one dihedral angle; effective harmonic bonds are used to describe the intermonomer interactions in a CG F-actin. The persistence length of a CG F-actin was found to be sensitive to the cut-off distance of assigning intermonomer bonds. Effective harmonic bonds for a monomer with its third nearest neighboring monomers are found to be necessary to reproduce the values of persistence length obtained from all-atom MD simulations. Compared to the elastic network model, incorporating the information of internal coordinate fluctuations enhances the accuracy and robustness for a CG model to describe the shapes of low-frequency vibrational modes. Combining the fluctuation-matching CG procedure and NMA, the achievable time- and length scales of modeling actin filaments can be greatly enhanced. In particular, a method is described to compute the force-extension curve using the CG model developed in this work and NMA. It was found that F-actin is easily buckled under compressive deformation, and a writhing mode is developed as a result. In addition to the bending and twisting modes, this novel writhing mode of F-actin could also play important roles in the interactions of F-actin with actin-binding proteins and in the force-generation process via polymerization.     

A simple translation in cortical log-coordinates may account for the pattern of saccadic localization errors

During saccadic eye movements, the visual world shifts rapidly across the retina. Perceptual continuity is thought to be maintained by active neural mechanisms that compensate for this displacement, bringing the presaccadic scene into a postsaccadic reference frame. Because of this active mechanism, objects appearing briefly around the time of the saccade are perceived at erroneous locations, a phenomenon called perisaccadic mislocalization. The position and direction of localization errors can inform us about the different reference frames involved. It has been found, for example, that errors are not simply made in the direction of the saccade but directed toward the saccade target, indicating that the compensatory mechanism involves spatial compression rather than translation. A recent study confirmed that localization errors also occur in the direction orthogonal to saccade direction, but only for eccentricities far from the fovea, beyond the saccade target. This spatially specific pattern of distortion cannot be explained by a simple compression of space around the saccade target. Here I show that a change of reference frames (i.e., translation) in cortical (logarithmic) coordinates, taking into account the cortical magnification factor, can accurately predict these spatial patterns of mislocalization. The flashed object projects onto the cortex in presaccadic (fovea-centered) coordinates but is perceived in postsaccadic (target-centered) coordinates.     

A dynamical study of antibody-antigen encounter reactions

The effects of internal dynamics in diffusion-driven encounters between macro-molecules represent a problem of broad relevance in molecular biology. In this view, we investigate a typical antigen-antibody reaction chain, based on a coarse-grained mechanical model parameterized directly upon results from single-molecule experiments. We demonstrate that the internal dynamics is a crucial factor in the encounter process. To describe our numerical results, we formulate a simple, intuitive theoretical framework, and we develop it analytically. This enables us to show that the inner dynamics of antibody molecules results in a cooperative behavior of their individual sub-units. Along the same lines, we also investigate the case of double binding to multi-valent antigens. Our results quantify the enhancement of avidity afforded by the double binding in excellent agreement with the available experimental data.     

Focal adhesions as mechanosensors: the two-spring model

Adhesion-dependent cells actively sense the mechanical properties of their environment through mechanotransductory processes at focal adhesions, which are integrin-based contacts connecting the extracellular matrix to the cytoskeleton. Here we present first steps towards a quantitative understanding of focal adhesions as mechanosensors. It has been shown experimentally that high levels of force are related to growth of and signaling at focal adhesions. In particular, activation of the small GTPase Rho through focal adhesions leads to the formation of stress fibers. Here we discuss one way in which force might regulate the internal state of focal adhesions, namely by modulating the internal rupture dynamics of focal adhesions. A simple two-spring model shows that the stiffer the environment, the more efficient cellular force is built up at focal adhesions by molecular motors interacting with the actin filaments.     

The Phan-Thien and Tanner model applied to thin film spherical coordinates: applications for lubrication of hip joint replacement

The Phan-Thien and Tanner (PTT) model is one of the most widely used rheological models. It can properly describe the common characteristics of viscoelastic non-Newtonian fluids. There is evidence that synovial fluid in human joints, which also lubricates artificial joints, is viscoelastic. Modeling the geometry of the total hip replacement, the PTT model is applied in spherical coordinates for a thin confined fluid film. A modified Reynolds equation is developed for this geometry. Several simplified illustrative problems are solved. The effect of the edge boundary condition on load-carrying normal stress is discussed. Solutions are also obtained for a simple squeezing flow. The effect of both the relaxation time and the PTT shear parameter is to reduce the load relative to a Newtonian fluid with the same viscosity. This implies that the Newtonian model is not conservative and may overpredict the load capacity. The PTT theory is a good candidate model to use for joint replacement lubrication. It is well regarded and derivable from molecular considerations. The most important non-Newtonian characteristics can be described with only three primary material parameters.     

Analytical and multibody modeling for the power analysis of standing jumps

Two methods for the power analysis of standing jumps are proposed and compared in this article. The first method is based on a simple analytical formulation which requires as input the coordinates of the center of gravity in three specified instants of the jump. The second method is based on a multibody model that simulates the jumps processing the data obtained by a three-dimensional (3D) motion capture system and the dynamometric measurements obtained by the force platforms. The multibody model is developed with OpenSim, an open-source software which provides tools for the kinematic and dynamic analyses of 3D human body models. The study is focused on two of the typical tests used to evaluate the muscular activity of lower limbs, which are the counter movement jump and the standing long jump. The comparison between the results obtained by the two methods confirms that the proposed analytical formulation is correct and represents a simple tool suitable for a preliminary analysis of total mechanical work and the mean power exerted in standing jumps.     

SURFNET: A program for visualizing molecular surfaces,cavities, and intermolecular interactions

The SURFNET program generates molecular surfaces and gaps between surfaces from 3D coordinates supplied in a PDB-format file. The gap regions can correspond to the voids between two or more molecules, or to the internal cavities and surface grooves within a single molecule. The program is particularly useful in clearly delineating the regions of the active site of a protein. It can also generate 3D contour surfaces of the density distributions of any set of 3D data points. All output surfaces can be viewed interactively, along with the molecules or data points in question, using some of the best-known molecular modeling packages. In addition, PostScript output is available, and the generated surfaces can be rendered using various other graphics packages.