Entropic Elasticity Research Articles

Multiple sources of passive stress relaxation in muscle fibres

The forces developed during stretch of nonactivated muscle consist of velocity-sensitive (viscous/viscoelastic) and velocity-insensitive (elastic) components. At the myofibrillar level, the elastic-force component has been described in terms of the entropic-spring properties of the giant protein titin, but entropic elasticity cannot account for viscoelastic properties, such as stress relaxation. Here we examine the contribution of titin to passive stress relaxation of isolated rat-cardiac myofibrils depleted of actin by gelsolin treatment. Monte Carlo simulations show that, up to ∼5 s after a stretch, the time course of stress relaxation can be described assuming unfolding of 1–2 immunoglobulin domains per titin molecule. For extended periods of stress relaxation, the simulations failed to correctly describe the myofibril data, suggesting that in situ, titin-Ig domains may be more stable than predicted in earlier single-molecule atomic-force-microscopy studies. The reasons behind this finding remain unknown; simply assuming a reduced unfolding probability of domains—an effect found here by AFM force spectroscopy on titin-Ig domains in the presence of a chaperone, alpha-B-crystallin—did not help correctly simulate the time course of stress relaxation. We conclude that myofibrillar stress relaxation likely has multiple sources. Evidence is provided that in intact myofibrils, an initial, rapid phase of stress relaxation results from viscous resistance due to the presence of actin filaments.

Elastic properties of single amylose chains in water: a quantum mechanical and AFM study.

Recent single-molecule atomic force microscopy (AFM) experiments have revealed that some polysaccharides display large deviations from force-extension relationships of other polymers which typically behave as simple entropic springs. However, the mechanism of these deviations has not been fully elucidated. Here we report the use of novel quantum mechanical methodologies, the divide-and-conquer linear scaling approach and the self-consistent charge density functional-based tight binding (SCC-DFTB) method, to unravel the mechanism of the extensibility of the polysaccharide amylose, which in water displays particularly large deviations from the simple entropic elasticity. We studied the deformations of maltose, a building block of amylose, both in a vacuum and in solution. To simulate the deformations in solution, the TIP3P molecular mechanical model is used to model the solvent water, and the SCC-DFTB method is used to model the solute. The interactions between the solute and water are treated by the combined quantum mechanical and molecular mechanical approach. We find that water significantly affects the mechanical properties of maltose. Furthermore, we performed two nanosecond-scale steered molecular dynamics simulations for single amylose chains composed of 10 glucopyranose rings in solution. Our SCC-DFTB/MM simulations reproduce the experimentally measured force-extension curve, and we find that the force-induced chair-to-boat transitions of glucopyranose rings are responsible for the characteristic plateau in the force-extension curve of amylose. In addition, we performed single-molecule AFM measurements on carboxymethyl amylose, and we found that, in contrast to the results of an earlier work by others, these side groups do not significantly affect amylose elasticity. By combining our experimental and modeling results, we conclude that the nonentropic elastic behavior of amylose is governed by the mechanics of pyranose rings themselves and their force-induced conformational transitions.

Micromechanics-based thermoviscoelastic constitutive equations for rubber-like matrix composites at finite strains

Global constitutive equations that model the response of multiphase materials undergoing finite deformations in which any phases behaves as a rubber-like thermoviscoelastic material are derived. The monolithic thermoviscoelastic constituent is modeled by a free-energy function which is given by a sum of a long-term contribution, that is based on the entropic elasticity for thermoelastic polymers, plus a non-equilibrium part which characterizes the viscoelastic (dissipative) mechanism. The global constitutive relations that govern the behavior of the composite are derived by using a micromechanical analysis in conjunction with the homogenization technique. Applications are given that illustrate the response of a rubber-like thermoviscoelastic matrix reinforced by continuous elastic nylon fibers. Results exhibit the effect of the viscoelasticity on the response of the composite when it is subjected to thermal and mechanical loadings, as well as its creep and relaxation behavior at room and elevated temperature.

Further Investigation of Equilibrium Modulus of Diblock Copolymer Micellar Lattice

Polystyrene–polybutadiene (PS–SB) and polystyrene–polyisoprene (PS–PI) diblock copolymers form spherical micelles with PS cores and PB/PI corona in a PB/PI-selective solvent, n-tetradecae (C14). At high concentrations, the micelles are further arranged on bcc lattices. Structural factors governing the equilibrium modulus Ge of these PS–PB and PS–PI micellar lattices are discussed in this study. It turned out that Ge is primarily determined by the number density of the corona blocks νcorona and proportional to νcorona while the number density of the bcc lattice cell, D−3 with D being the lattice spacing (determined from SAXS), has only secondary effects on Ge. The proportionality between Ge and νcorona reflects the thermodynamic origin of the lattice elasticity, the entropic elasticity of the corona PB/PI blocks having osmotically correlated conformations. These results suggest a necessity of re-examination of a previous hypothesis that the cell number density primarily determines Ge of bulk copolymer systems forming cubic phases [Kossuth et al., J. Rheol., 43, 167 (1999)]. Furthermore, the Ge/νcorona ratio was found to be moderately different for the PS–PB/C14 and PS–PI/C14 micellar lattices. This difference can be related to a difference in the magnitude of the osmotic correlation for the PB and PI corona blocks due to a slight difference of their solubilities in C14.

Spontaneous Curvature of Comblike Polymers at a Flat Interface

Spontaneous curvature of molecular brushes adsorbed on a flat substrate was observed by scanning force microscopy for comblike polymers of poly(butyl acrylate). This phenomenon was analyzed theoretically for a 2D model of a brush molecule in the regime of strong adsorption. Although all side chains were confined to the substrate plane, they were allowed to flip over and change their position with respect to the backbone. Instability of the straight backbone was caused solely by the entropic elasticity of the side chains: smaller extension of the side chains was attained upon their uneven distribution relative to the backbone. An equilibrium, i.e., spontaneous, curvature resulted from competition between elasticity of the side chains and stabilizing factors such as inherent stiffness of the backbone, mixing entropy of the side chains and excluded volume interactions of distant parts of the brush. The curvature radius is found to be proportional to the degree of polymerization of the side chains. The effect of polydispersity of the side chains is discussed.

Flaw tolerant bulk and surface nanostructures of biological systems.

Bone-like biological materials have achieved superior mechanical properties through hierarchical composite structures of mineral and protein. Gecko and many insects have evolved hierarchical surface structures to achieve extraordinary adhesion capabilities. We show that the nanometer scale plays a key role in allowing these biological systems to achieve their superior properties. We suggest that the principle of flaw tolerance may have had an overarching influence on the evolution of the bulk nanostructure of bone-like materials and the surface nanostructure of gecko-like animal species. We demonstrate that the nanoscale sizes allow the mineral nanoparticles in bone to achieve optimum fracture strength and the spatula nanoprotrusions in Gecko to achieve optimum adhesion strength. In both systems, strength optimization is achieved by restricting the characteristic dimension of the basic structure components to nanometer scale so that crack-like flaws do not propagate to break the desired structural link. Continuum modeling and atomistic simulations have been conducted to verify the concept of flaw tolerance at nanoscale. A simple tension-shear chain model has been developed to model the stiffness and fracture energy of biocomposites. It is found that, while the problem of low toughness of mineral crystals is alleviated by restricting the crystal size to nanoscale, the problem of low modulus of protein has been solved by adopting a large aspect ratio for the mineral platelets. The fracture energy of biocomposites is found to be proportional to the effective shear strain and the effective shear stress in protein along its path of deformation to fracture. The bioengineered mineral-protein composites are ideally suited for fracture energy dissipation as the winding paths of protein domain unfolding and slipping along protein-mineral interfaces lead to very large effective strain before fracture. The usual entropic elasticity of biopolymers may involve relatively small effective stress and may not be able to ensure simultaneous domain unfolding and interface slipping. Cross-linking mechanisms such as Ca++ induced sacrificial bonds in bone can increase the shear stress in protein and along the protein-mineral interface, effectively converting the behavior of entropic elasticity to one that resembles metal plasticity. The sacrificial bond mechanism not only builds up a large effective stress in protein but also allows protein deformation and interface slipping to occur simultaneously under similar stress levels, making it possible to engineer a very long range of deformation under significant stress in order to maximize energy absorption. Optimization of mineral platelets near theoretical strength is found to be crucial for allowing a large effective stress to be built up in protein via cross-linking mechanisms such as Ca++ induced sacrificial bonds. Similarly, for gecko adhesion, the strength optimization of individual spatulas is found to play a critical role in enhancing adhesion energy at the higher hierarchical level.

Dynamic Shear Modulus of Isotropic Elastomers

We have investigated the dynamic mechanical behavior of two cross-linked polymer networks with very different topologies: one made of backbones randomly linked along their length; the other with fixed-length strands uniformly cross-linked at their ends. The samples were analyzed using oscillatory shear, at very small strains corresponding to the linear regime. This was carried out at a range of frequencies, and at temperatures ranging from the glass plateau, through the glass transition, and well into the rubbery region. Through the glass transition, the data obeyed the time-temperature superposition principle, and could be analyzed using WLF treatment. At higher temperatures, in the rubbery region, the storage modulus was found to deviate from this, taking a value that is independent of frequency. This value increased linearly with temperature, as expected for the entropic rubber elasticity, but with a substantial negative offset inconsistent with straightforward enthalpic effects. Conversely, the loss modulus continued to follow time-temperature superposition, decreasing with increasing temperature, and showing a power-law dependence on frequency.

Some aspects of thermodynamic effects in finite amplitude wave propagation in rubber-like solids

In this study, we apply the modified entropic elasticity theory to investigate the thermodynamic effects in finite amplitude wave propagation in a stretched hyperelastic string. In formulating the problem, we assume that the string is composed of a non-conducting rubber-like material, and that the deformation is isochoric. Adiabatic stress–stretch relations are obtained for strictly entropic and piezotropic materials, which are the two limiting cases of modified entropic elasticity. The Lagrangian system of governing equations for finite amplitude wave propagation in a stretched hyperelastic string is derived in conservation form. Similarity solutions are presented for two strain energy functions. The errors in adopting an isothermal stress–stretch relation instead of an adiabatic stress–stretch relation, and the validity of the isentropic approximation, are investigated. The treatise developed in this study, and the results obtained, appear to be new.

The elastic theory of a single DNA molecule

We study the elastic responses of double-(ds) and single-stranded (ss) DNA at external force fields. A double-strand-polymer elastic model is constructed and solved by path integral methods and Monte Carlo simulations to understand the entropic elasticity, cooperative extensibility, and supercoiling property of dsDNA. The good agreement with experiments indicates that short-ranged base-pair stacking interaction is crucial for the stability and the high deformability of dsDNA. Hairpin-coil transition in ssDNA is studied with generating function method. A threshold force is needed to pull the ssDNA hairpin patterns, stabilized by base pairing and base-pair stacking, into random coils. This phase transition is predicted to be of first order for stacking potential higher than some critical level, in accordance with experimental observations.

Rate-Dependent Transition From Thermal Softening to Hardening in Elastomers

The thermal-mechanical properties of the materials currently used in packaging are being reexamined as the electronic packaging industry moves towards chip scale packages and wafer scale packages. The rate-dependent transition of elastic modulus and viscosity from thermal softening to thermal hardening with rising temperature, which does not involve any phase change, has been observed in certain elastomers. An explanation about this interesting phenomenon is given based on thermodynamic considerations. A theoretical analysis is performed to show the limitation of existing viscoelastic models in predicting the transition. It appears that macroscopic material properties should be reexamined based on the physics behind the interaction between ordinary elasticity and entropic elasticity.

Nanoscale Intermolecular Interactions between Human Serum Albumin and Alkanethiol Self-Assembled Monolayers

To study the molecular origins of hemocompatibility, the blood plasma protein human serum albumin (HSA) was covalently grafted to a nanosized probe tip at the end of a soft, microfabricated cantilever force transducer. The net force versus separation distance between the HSA-modified probe tip and three different model surfaces, including (1) gold; (2) a hydrophobic, CH3-terminated alkanethiol self-assembling monolayer (SAM); and (3) a hydrophilic, COO--terminated alkanethiol SAM in aqueous sodium phosphate buffer solution (PBS, ionic strength (IS) = 0.01 M, pH = 7.4), was recorded and compared to the values of various theoretical models. The approach interaction of the HSA probe tip on the COO--terminated SAM and Au substrates was found to be purely repulsive for D < 15 nm, nonlinear with decreasing separation distance, and consistent with electrostatic double layer repulsion. The approach interaction of the HSA probe tip on the CH3-terminated SAM substrate was found to be purely attractive, long range (D < 80 nm), nonlinear with decreasing separation distance, and much greater in magnitude and range than that known for van der Waals interactions between hydrocarbon SAMs terminated with hydrophilic chemical groups on Au. Large adhesive energies were observed for the HSA probe tip on both the CH3-terminated SAM and Au surfaces (≤−29 mN/m, −22 to −73kBT/protein), while smaller adhesive energies were observed on the COO--terminated SAM surface (≤ −4.9 mN/m, −5.3 to −12kBT/protein). It was shown that short-range adhesive contacts between the HSA chain segments and these surfaces give rise to energy dissipating mechanisms, such as HSA entropic molecular elasticity and enthalpic unfolding forces (deformation and rupture of noncovalent intramolecular bonds) and noncovalent bond rupture of the HSA chain segments adsorbed to the surface.

Theory of high-force DNA stretching and overstretching.

Single-molecule experiments on single- and double-stranded DNA have sparked a renewed interest in the force versus extension of polymers. The extensible freely jointed chain (FJC) model is frequently invoked to explain the observed behavior of single-stranded DNA, but this model does not satisfactorily describe recent high-force stretching data. We instead propose a model (the discrete persistent chain) that borrows features from both the FJC and the wormlike chain, and show that it resembles the data more closely. We find that most of the high-force behavior previously attributed to stretch elasticity is really a feature of the corrected entropic elasticity; the true stretch compliance of single-stranded DNA is several times smaller than that found by previous authors. Next we elaborate our model to allow coexistence of two conformational states of DNA, each with its own stretch and bend elastic constants. Our model is computationally simple and gives an excellent fit through the entire overstretching transition of nicked, double-stranded DNA. The fit gives the first value for the bend stiffness of the overstretched state. In particular, we find the effective bend stiffness for DNA in this state to be about 12 nm k(B)T, a value quite different from either the B-form or single-stranded DNA.

Transverse fluctuation analysis of single extended DNA molecules

We observed fluctuations of elongated DNA molecules by fluorescence microscopy. The molecules are fixed at both ends and undulate. Mode analysis of the thermally excited undulations of the labeled DNA molecules gives access to the spectral density of the amplitude fluctuations. From these measurements we estimate the tension acting on the DNA molecules. We found the forces to be within the entropic elasticity range of a typical DNA molecule (measured on the single-molecule level).

Unfolding transitions in myosin give rise to the double-hyperbolic force–velocityrelation in muscle

This work presents an extension to a recent model of muscle contractionthat was based on entropic elasticity (Nielsen 2002 J. Theor. Biol. 21999–119). By using entropic elasticity as the origin of muscle force, variouspossibilities emerge that can account for the presence of the double-hyperbolicforce–velocity relation in muscle that was observed by Edman (1988 J. Physiol.404 301–21). In the present work, it will be argued that a slight change(elongation) of the contour length of the entropic springs involved in theirhigh-force regions is sufficient to produce such a double-hyperbolic profile. Asudden elongation would correspond to an unfolding event of a smallregion of the myosin molecule, which causes a sudden reduction of thetension that may be produced by the individual molecule. To obtain thedouble-hyperbolic profile, it is assumed that a gradual transition occurs in theentropic spring array from being mainly composed of non-unfolded myosinsprings that have a short (i.e. normal) contour length to consisting of amixture of myosin springs with short and long (unfolded) contour lengths.

Elastic modulus and relaxation times in telechelic associating polymers

Dynamic mechanical measurements have been performed on solutions of HEUR-associating polymers with different molar mass, end-cap length, and extent of modification. These systems appear as Maxwellian fluids although at high frequencies a short-time relaxation process is evidenced for the systems with a strong hydrophobicity. This relaxation process could be attributed to the Rouse dynamics of active links in the temporary network. The high-frequency elastic modulus G∞ decreases with increasing temperature according to an Arrhenius law. The associated activation energy decreases with increasing concentration. This is interpreted in terms of two contributions to the elastic modulus (osmostic and entropic elasticity).

THE ELASTIC THEORY OF SINGLE-MOLECULE DNA

Our recent work on the elastic responses of double- (ds) and single-stranded (ss) DNA at external force fields is reviewed. By constructing as elastic model of dsDNA in which the base-pair stacking interaction is included, we demonstrate that dsDNA entropic elasticity, cooperative extensibility, and supercoiling property can all be understood from a unified viewpoint. The base-pair stacking interaction is also found to determine the cooperativity of the stretch-induced hairpin-coil transition is ssDNA.

THE ELASTIC THEORY OF A SINGLE DNA MOLECULE

We review our recent efforts in understanding the elastic properties of double-stranded (ds) and single-stranded (ss) DNA macromolecules. A simple geometric model of dsDNA was constructed and solved by path integral methods. The good agreement with experiments on dsDNA's entropic elasticity, cooperative extensibility, and supercoiling property suggested that the short-ranged base-pair stacking interaction is crucial for the stability and the high deformability of dsDNA. For ssDNA at high ionic conditions, base-pairing and base-pair stacking interactions cause the polymer to fold into compact hairpin configurations. The force-induced hairpin-coil transition was studied with the generating function method. In accordance with experiment, this transition was found to be highly cooperative when the average stacking potential is higher than some critical level. At low ionic conditions, the electrostatic repulsive interaction along the ssDNA becomes dominant, and ssDNA can be regarded as model polyelectrolytes. Our MC simulation results suggested an exponential relation between external force and the total extension. This prediction was confirmed experimentally.

Kinesin: a molecular motor with a spring in its step.

A key step in the processive motion of two-headed kinesin along a microtubule is the 'docking' of the neck linker that joins each kinesin head to the motor's dimerized coiled-coil neck. This process is similar to the folding of a protein beta-hairpin, which starts in a highly mobile unfolded state that has significant entropic elasticity and finishes in a more rigid folded state. We therefore suggest that neck-linker docking is mechanically equivalent to the thermally activated shortening of a spring that has been stretched by an applied load. This critical tension-dependent step utilizes Brownian motion and it immediately follows the binding of ATP, the hydrolysis of which provides the free energy that drives the kinesin cycle. A simple three-state model incorporating neck-linker docking can account quantitatively for both the kinesin force-velocity relation and the unusual tension-dependence of its Michaelis constant. However, we find that the observed randomness of the kinesin motor requires a more detailed four-state model. Monte Carlo simulations of single-molecule stepping with this model illustrate the possibility of sub-8 nm steps, the size of which is predicted to vary linearly with the applied load.

Entropic Elasticity in the Generation of Muscle Force – A Theoretical Model

A novel simplified structural model of sarcomeric force production in striate muscle is presented. Using some simple assumptions regarding the distribution of myosin spring lengths at different sliding velocities it is possible to derive a very simple expression showing the main components of the experimentally observed force–velocity relationship of muscle: nonlinearity during contraction (Hill, 1938), maximal force production during stretching equal to two times the isometric force (Katz, 1939), yielding at high stretching velocity, slightly concave force–extension relationship during sudden length changes (Ford et al., 1977; Lombardi & Piazzesi, 1990), accurate reproduction of the rate of ATP consumption (Shirakawa et al., 2000; He et al., 2000) and of the extra energy liberation rate (Hill, 1964a). Different assumptions regarding the force–length relationship of individual cross-bridges are explored [linear, power function and worm-like chain (WLC) model based], and it is shown that the best results are obtained if the individual myosin-spring forces are modelled using a WLC model, thus hinting that entropic elasticity could be the main source of force in myosin undergoing the conformational changes associated with the power stroke.

Entropic Elasticity in the Generation of Muscle Force – A Theoretical Model

A novel simplified structural model of sarcomeric force production in striate muscle is presented. Using some simple assumptions regarding the distribution of myosin spring lengths at different sliding velocities it is possible to derive a very simple expression showing the main components of the experimentally observed force–velocity relationship of muscle: nonlinearity during contraction (Hill, 1938), maximal force production during stretching equal to two times the isometric force (Katz, 1939), yielding at high stretching velocity, slightly concave force–extension relationship during sudden length changes (Ford et al., 1977; Lombardi & Piazzesi, 1990), accurate reproduction of the rate of ATP consumption (Shirakawa et al., 2000; He et al., 2000) and of the extra energy liberation rate (Hill, 1964a). Different assumptions regarding the force–length relationship of individual cross-bridges are explored [linear, power function and worm-like chain (WLC) model based], and it is shown that the best results are obtained if the individual myosin-spring forces are modelled using a WLC model, thus hinting that entropic elasticity could be the main source of force in myosin undergoing the conformational changes associated with the power stroke.