A model for the stability and creep of organic materials

A model is presented for the thermally assisted breaking of a number of bonds arranged in parallel and stressed by an individual soft spring each. Using a simplified potential for the bond it is shown that in equilibrium there are two definite regions of elastic behavior: one with all bonds intact, the other with a variable fraction of bonds broken, therefore with a tangent modulus steadily decreasing with applied stress. Criteria are given for the existence of these regions. Beyond these regions time-dependent creep to rupture is found, limited, in turn, by the theoretical fracture strength, the stress necessary for fracture without any thermal assistance, beyond which a bound state is impossible. The time-to-fracture for creep rupture is calculated and an example of the time evolution of the accelerating creep given. The results of the calculations are applied to experimental data on Wallaby tendons by Wang and Ker (J. Exp. Biol. 198 (1995) 831) and data estimated for the bond potential depth, the theoretical fracture strength and the number density of bonds involved as well as the elastic modulus of the ensemble. Values are derived under the assumption of one deformation mechanism being dominant--e.g., (sub-)fibril sliding or sliding of collagen molecules along one another--but the model cannot definitely distinguish between mechanisms.     

Graphical method for force analysis: macromolecular mechanics with atomic force microscopy

We present a graphical method for a unifying, quantitative analysis of molecular bonding-force measurements by atomic force microscopy (AFM). The method is applied to interpreting a range of phenomena commonly observed in the experimental AFM measurements of noncovalent, weak bonds between biological macromolecules. The analysis suggests an energy landscape underlying the intermolecular force and demonstrates that many observations, such as "snaps-on," "jumps-off," and hysteresis loops, are different manifestations of a double-well energy landscape. The analysis gives concrete definitions for the operationally defined "attractive" and "adhesive" forces in terms of molecular parameters. It is shown that these operationally defined quantities are usually functions of the experimental setup, such as the stiffness of the force probe and the rate of its movement. The analysis reveals a mechanical instability due to the multistate nature of molecular interactions and provides new insight into macromolecular viscosity. The graphical method can equally be applied to a quantitative analysis of multiple unfolding of subunits of the giant muscle protein titin under AFM.     

Entropic elasticity controls nanomechanics of single tropocollagen molecules

We report molecular modeling of stretching single molecules of tropocollagen, the building block of collagen fibrils and fibers that provide mechanical support in connective tissues. For small deformation, we observe a dominance of entropic elasticity. At larger deformation, we find a transition to energetic elasticity, which is characterized by first stretching and breaking of hydrogen bonds, followed by deformation of covalent bonds in the protein backbone, eventually leading to molecular fracture. Our force-displacement curves at small forces show excellent quantitative agreement with optical tweezer experiments. Our model predicts a persistence length xi(p) approximately 16 nm, confirming experimental results suggesting that tropocollagen molecules are very flexible elastic entities. We demonstrate that assembly of single tropocollagen molecules into fibrils significantly decreases their bending flexibility, leading to decreased contributions of entropic effects during deformation. The molecular simulation results are used to develop a simple continuum model capable of describing an entire deformation range of tropocollagen molecules. Our molecular model is capable of describing different regimes of elastic and permanent deformation, without relying on empirical parameters, including a transition from entropic to energetic elasticity.     

The modelling of interactions between organs and medical tools: a volumetric mass-spring chain algorithm

This paper adds volume deformation capability to the mass-spring chain method using tetrahedral elements in order to obtain more realistic deformations, which occur during the interactions between medical tools and soft tissues. The mass-spring chain method originally does not consider volume information and performs deformation by moving and deforming individual springs of a deformable model. However, most of the applications in computer graphics require volume modelling using tetrahedrons. In the proposed method, the deformation algorithm loops through tetrahedrons and performs deformation based on defined rules similar to those of the original mass-spring chain method. This method can handle not only ordinary deformation applications but also those with topology changes, such as cutting and tearing, as it does not rely on any pre-computed quantities. A method to preserve the volume and the shape of the tetrahedral elements is also developed. In order to speed up the new version of the algorithm, a tetrahedral propagation for deformation is developed. The detailed implementation of the algorithm and the various applications of the organ–surgery tool interactions are presented. The paper also provides the animations of the different models obtained by the proposed method.     

Multichain aggregates in dilute solutions of associating polyelectrolyte keeping a constant size at the increase in the chain length of individual macromolecules

Multichain aggregates together with individual macromolecules were detected by light scattering in dilute aqueous solutions of chitosan and of its hydrophobic derivatives bearing 4 mol % of n-dodecyl side groups. It was demonstrated that the size of aggregates and their aggregation numbers increase at the introduction of hydrophobic side groups into polymer chains. The key result concerns the effect of the chain length of individual macromolecules on the aggregation behavior. It was shown that for both unmodified and hydrophobically modified (HM) chitosan, the size of aggregates is independent of the length of single chains, which may result from the electrostatic nature of the stabilization of aggregates. At the same time, the number of macromolecules in one aggregate increases significantly with decreasing length of single chains to provide a sufficient number of associating groups to stabilize the aggregate. The analysis of the light scattering data together with TEM results suggests that the aggregates of chitosan and HM chitosan represent spherical hydrogel particles with denser core and looser shell covered with dangling chains.     

The nonlinear mechanical response of the red blood cell

We measure the dynamical mechanical properties of human red blood cells. A single cell response is measured with optical tweezers. We investigate both the stress relaxation following a fast deformation and the effect of varying the strain rate. We find a power-law decay of the stress as a function of time, down to a plateau stress, and a power-law increase of the cell's elasticity as a function of the strain rate. Interestingly, the exponents of these quantities violate the linear superposition principle, indicating a nonlinear response. We propose that this is due to the breaking of a fraction of the crosslinks during the deformation process. The soft glassy rheology model accounts for the relation between the exponents we observe experimentally. This picture is consistent with recent models of bond remodeling in the red blood cell's molecular structure. Our results imply that the blood cell's mechanical behavior depends critically on the deformation process.     

Origin of low-frequency motions in biological macromolecules. A view of recent progress in the quasi-continuity model

The recent progress in the quasi-continuity model and its applications in studying the low-frequency internal motions of biological macromolecules have been surveyed. Emphasis is placed on revealing the origin of this kind of internal collective motion, which involves many atoms and has significant biological functions. In light of such a line, the low-frequency motions in alpha-helix structure, beta-structure (including beta-sheet and beta-barrel), and DNA double-helix structure, the three most fundamental component elements in biological macromolecules, are discussed, and the corresponding physical pictures described. It turns out that the low-frequency motion in biological macromolecules originates from their two common intrinsic characteristics, i.e., they possess a series of weak bonds, such as hydrogen bonds and salt bridges, and a substantial mass distributed over the region containing those weak bonds.     

Filtrability of erythrocytes in "hypersplenic" rats.

The values of haemoglobin, reticulocytes and half-times of the filtrability of non-washed and washed erythrocytes were examined in male albino rats, Wistar strain, after i.p. injections of methylcellulose (MC) and compared with controls. In individual experiments the rats received 2 to 32 injections of MC. In injected animals, the filtrability of non-washed erythrocytes was altered. The filtrability half-times of the washed erythrocytes did not differ from the controls. Thus, the filtrability is altered for extracorpuscular reasons. "Hypersplenism" being completely developed, (after 32 MC injections), the filtrability of non-washed erythrocytes repaired when the application of MC had been discontinued, the reticulocyte values remained however increased. Problems of the mechanism of anaemia in experimental "hypersplenism" after MC injections in rats and relations between the altered filtrability of the erythrocytes and the haemolysis are discussed.     

Post-yield nanomechanics of human cortical bone in compression using synchrotron X-ray scattering techniques

The ultrastructural response to applied loads governs the post-yield deformation and failure behavior of bone, and is correlated with bone fragility fractures. Combining a novel progressive loading protocol and synchrotron X-ray scattering techniques, this study investigated the correlation of the local deformation (i.e., internal strains of the mineral and collagen phases) with the bulk mechanical behavior of bone. The results indicated that the internal strains of the longitudinally oriented collagen fibrils and mineral crystals increased almost linearly with respect to the macroscopic strain prior to yielding, but markedly decreased first and then gradually leveled off after yielding. Similar changes were also observed in the applied stress before and after yielding of bone. However, the collagen to mineral strain ratio remained nearly constant throughout the loading process. In addition, the internal strains of longitudinal mineral and collagen phases did not exhibit a linear relationship with either the modulus loss or the plastic deformation of bulk bone tissue. Finally, the time-dependent response of local deformation in the mineral phase was observed after yielding. Based on the results, we speculate that the mineral crystals and collagen fibrils aligned with the loading axis only partially explain the post-yield deformation, suggesting that shear deformation involving obliquely oriented crystals and fibrils (off axis) is dominant mechanism of yielding for human cortical bone in compression.     

Stability of adhesion clusters and cell reorientation under lateral cyclic tension

This work is motivated by experimental observations that cells on stretched substrate exhibit different responses to static and dynamic loads. A model of focal adhesion that can consider the mechanics of stress fiber, adhesion bonds, and substrate was developed at the molecular level by treating the focal adhesion as an adhesion cluster. The stability of the cluster under dynamic load was studied by applying cyclic external strain on the substrate. We show that a threshold value of external strain amplitude exists beyond which the adhesion cluster disrupts quickly. In addition, our results show that the adhesion cluster is prone to losing stability under high-frequency loading, because the receptors and ligands cannot get enough contact time to form bonds due to the high-speed deformation of the substrate. At the same time, the viscoelastic stress fiber becomes rigid at high frequency, which leads to significant deformation of the bonds. Furthermore, we find that the stiffness and relaxation time of stress fibers play important roles in the stability of the adhesion cluster. The essence of this work is to connect the dynamics of the adhesion bonds (molecular level) with the cell's behavior during reorientation (cell level) through the mechanics of stress fiber. The predictions of the cluster model are consistent with experimental observations.     

Yielding Elastic Tethers Stabilize Robust Cell Adhesion

Many bacteria and eukaryotic cells express adhesive proteins at the end of tethers that elongate reversibly at constant or near constant force, which we refer to as yielding elasticity. Here we address the function of yielding elastic adhesive tethers with Escherichia coli bacteria as a model for cell adhesion, using a combination of experiments and simulations. The adhesive bond kinetics and tether elasticity was modeled in the simulations with realistic biophysical models that were fit to new and previously published single molecule force spectroscopy data. The simulations were validated by comparison to experiments measuring the adhesive behavior of E. coli in flowing fluid. Analysis of the simulations demonstrated that yielding elasticity is required for the bacteria to remain bound in high and variable flow conditions, because it allows the force to be distributed evenly between multiple bonds. In contrast, strain-hardening and linear elastic tethers concentrate force on the most vulnerable bonds, which leads to failure of the entire adhesive contact. Load distribution is especially important to noncovalent receptor-ligand bonds, because they become exponentially shorter lived at higher force above a critical force, even if they form catch bonds. The advantage of yielding is likely to extend to any blood cells or pathogens adhering in flow, or to any situation where bonds are stretched unequally due to surface roughness, unequal native bond lengths, or conditions that act to unzip the bonds.     

Effects of inner solution viscosity and membrane stiffness on erythrocyte deformability on passing through a microchannel: prediction by two-dimensional simulation

We studied erythrocyte deformability in an effort to develop diagnostic methods based on its measurement and thus aid in the development of therapies for circulatory diseases. In the reported work, we performed two-dimensional numerical simulations of blood flow through a microchannel (MC) to evaluate erythrocyte deformability, applying the immersed boundary method to simulate erythrocyte movement and deformation. To evaluate deformability, MC transit capacity and shape recoverability were considered, defined as the time required to pass through the MC and the time constant during the shape-recovery process after exiting the MC, respectively. The simulation results showed that the erythrocyte MC transit time increased when the viscosity of the inner solution or the stiffness of the membrane increased. The time constant for erythrocyte shape recovery increased as the inner solution viscosity increased. In contrast, the time constant decreased as the erythrocyte membrane stiffness increased. These time-constant trends were in agreement with a theoretical equation derived using the Kelvin model and with previous experimental results. This diagnostic method of measuring erythrocyte shape recoverability and MC transit capacity is anticipated to have clinical application.     

MOLMOL: A program for display and analysis of macromolecular structures

MOLMOL is a molecular graphics program for display, analysis, and manipulation of three-dimensional structures of biological macromolecules, with special emphasis on nuclear magnetic resonance (NMR) solution structures of proteins and nucleic acids. MOLMOL has a graphical user interface with menus, dialog boxes, and on-line help. The display possibilities include conventional presentation, as well as novel schematic drawings, with the option of combining different presentations in one view of a molecule. Covalent molecular structures can be modified by addition or removal of individual atoms and bonds, and three-dimensional structures can be manipulated by interactive rotation about individual bonds. Special efforts were made to allow for appropriate display and analysis of the sets of typically 20–40 conformers that are conventionally used to represent the result of an NMR structure determination, using functions for superimposing sets of conformers, calculation of root mean square distance (RMSD) values, identification of hydrogen bonds, checking and displaying violations of NMR constraints, and identification and listing of short distances between pairs of hydrogen atoms.     

The key event in force-induced unfolding of Titin's immunoglobulin domains

Steered molecular dynamics simulation of force-induced titin immunoglobulin domain I27 unfolding led to the discovery of a significant potential energy barrier at an extension of approximately 14 A on the unfolding pathway that protects the domain against stretching. Previous simulations showed that this barrier is due to the concurrent breaking of six interstrand hydrogen bonds (H-bonds) between beta-strands A' and G that is preceded by the breaking of two to three hydrogen bonds between strands A and B, the latter leading to an unfolding intermediate. The simulation results are supported by Angstrom-resolution atomic force microscopy data. Here we perform a structural and energetic analysis of the H-bonds breaking. It is confirmed that H-bonds between strands A and B break rapidly. However, the breaking of the H-bond between strands A' and G needs to be assisted by fluctuations of water molecules. In nanosecond simulations, water molecules are found to repeatedly interact with the protein backbone atoms, weakening individual interstrand H-bonds until all six A'-G H-bonds break simultaneously under the influence of external stretching forces. Only when those bonds are broken can the generic unfolding take place, which involves hydrophobic interactions of the protein core and exerts weaker resistance against stretching than the key event.     

Sacrificial bonds and hidden length: unraveling molecular mesostructures in tough materials

Sacrificial bonds and hidden length in structural molecules and composites have been found to greatly increase the fracture toughness of biomaterials by providing a reversible, molecular-scale energy-dissipation mechanism. This mechanism relies on the energy, of order 100 eV, needed to reduce entropy and increase enthalpy as molecular segments are stretched after being released by the breaking of weak bonds, called sacrificial bonds. This energy is relatively large compared to the energy needed to break the polymer backbone, of order a few eV. In many biological cases, the breaking of sacrificial bonds has been found to be reversible, thereby additionally providing a "self-healing" property to the material. Due to the nanoscopic nature of this mechanism, single molecule force spectroscopy using an atomic force microscope has been a useful tool to investigate this mechanism. Especially when investigating natural molecular constructs, force versus distance curves quickly become very complicated. In this work we propose various types of sacrificial bonds, their combination, and how they appear in single molecule force spectroscopy measurements. We find that by close analysis of the force spectroscopy curves, additional information can be obtained about the molecules and their bonds to the native constructs.     

Experimental Study of the Bending Properties and Deformation Analysis of Web-Reinforced Composite Sandwich Floor Slabs with Four Simply Supported Edges

Web-reinforced composite sandwich panels exhibit good mechanical properties in one-way bending, but few studies have investigated their flexural behavior and deformation calculation methods under conditions of four simply supported edges. This paper studies the bending performance of and deformation calculation methods for two-way web-reinforced composite sandwich panels with different web spacing and heights. Polyurethane foam, two-way orthogonal glass-fiber woven cloth and unsaturated resin were used as raw materials in this study. Vacuum infusion molding was used to prepare an ordinary composite sandwich panel and 5 web-reinforced composite sandwich panels with different spacing and web heights. The panels were subjected to two-way panel bending tests with simple support for all four edges. The mechanical properties of these sandwich panels during the elastic stage were determined by applying uniformly distributed loads. The non-linear mechanical characteristics and failure modes were obtained under centrally concentrated loading. Finally, simulations of the sandwich panels, which used the mechanical model established herein, were used to deduce the formulae for the deflection deformation for this type of sandwich panel. The experimental results show that webs can significantly improve the limit bearing capacity and flexural rigidity of sandwich panels, with smaller web spacing producing a stronger effect. When the web spacing is 75 mm, the limit bearing capacity is 4.63 times that of an ordinary sandwich panel. The deduced deflection calculation formulae provide values that agree well with the measurements (maximum error <15%). The results that are obtained herein can provide a foundation for the structural design of this type of panel.     

Acetaldehyde-ethanol exchange in vivo during ethanol oxidation.

Bovine thrombin was immobilized on sepharose 4B through an oxidized sialic acid group on its B chain. The immobilized thrombin was reduced with β-mercaptoethanol in 8 m urea under conditions that were shown to be sufficient to reduce the disulfide bond connecting the A and B chains. The immobilized B chain that remained after the A chain was washed away was allowed to refold, and the disulfide bonds were reoxidized with a mixture of oxidized and reduced glutathione under anaerobic conditions. The refolded immobilized B chain exhibited 15–25% of the tosyl-l-arginine methyl ester esterase activity of the immobilized thrombin and a significant amount of fibrinogen clotting activity. The immobilized B chain behaved qualitatively like immobilized thrombin towards two oligopeptide fibrinogen-like substrates and showed no activity towards lysine or arginine peptide bonds in a fragment of ribonuclease.Since the recovered activity was greater than that computed for a random refolding of seven -SH groups to form three SS bonds, it is concluded that the B chain retains a sufficient number of interacting groups to refold correctly, and it is suggested that prothrombin might fold in localized domains with only weak interactions between domains. The behavior of the B chain towards the substrates tested suggests that the A chain does not play a significant role in determining the catalytic specificity of thrombin or in distinguishing its specificity from that of trypsin.     

Mechanical Effects of Dynamic Binding between Tau Proteins on Microtubules during Axonal Injury

The viscoelastic nature of axons plays a key role in their selective vulnerability to damage in traumatic brain injury (TBI). Experimental studies have shown that although axons can tolerate 100% strain under slow loading rates, even strain as small as 5% can rupture microtubules (MTs) during the fast loading velocities relevant to TBI. Here, we developed a computational model to examine rate-dependent behavior related to dynamic interactions between MTs and the MT-associated protein tau under varying strains and strain rates. In the model, inverted pairs of tau proteins can dynamically cross-link parallel MTs via the respective MT-binding domain of each tau. The model also incorporates realistic thermodynamic breaking and reformation of the bonds between the connected tau proteins as they respond to mechanical stretch. With simulated stretch of the axon, the model shows that despite the highly dynamic nature of binding and unbinding events, under fast loading rates relevant to TBI, large tensile forces can be transmitted to the MTs that can lead to mechanical rupture of the MT cylinder, in agreement with experimental observations and as inferred in human TBI. In contrast, at slow loading rates, the progressive breaking and reformation of the bonds between the tau proteins facilitate the extension of axons up to ∼100% strain without any microstructural damage. The model also predicts that under fast loading rates, individual MTs detach from MT bundles via sequential breaking of the tau-tau bonds. Finally, the model demonstrates that longer MTs are more susceptible to mechanical rupture, whereas short MTs are more prone to detachment from the MT bundle, leading to disintegration of the axonal MT ultrastructure. Notably, the predictions from the model are in excellent agreement with the findings of the recent in vitro mechanical testing of micropatterned neuronal cultures.