Statistical mechanics of the Huxley-Simmons model

Stretching multistate flexible chains and loops

Polymer loop structure commonly appears in biological phenomena, such as DNA looping and DNA denaturation. When a chain forms a loop, its elastic behavior differs from that of an open chain due to the loss of entropy. In the case of reversible loop formation, interesting behavior emerges related to the multistate nature of the conformations. In this study, we model a multistate reversible loop as a looping Gaussian chain, which can bind (close) reversibly at one or several points to form a loop, or a zipping Gaussian loop, which can zip reversibly to form a double-stranded chain. For each model, we calculate the force-extension relations in the fixed-extension (Helmholtz) and the fixed-force (Gibbs) statistical ensembles. Unlike the single Gaussian chain or loop, the multilevel systems demonstrate qualitatively distinct tensile elasticity and ensemble inequivalence. In addition, we investigate a Gaussian necklace consisting of reversible alternating blocks of the zipped chain and loop and obtain the force-temperature phase diagram. The phase diagram implies a force-induced phase transition from a completely looped (denatured) state to a mixed (bound) state.

Coordination, cooperation, competition, crowding and congestion of molecular motors: Theoretical models and computer simulations

Cytoskeletal motor proteins are biological nanomachines that convert chemical energy into mechanical work to carry out various functions such as cell division, cell motility, cargo transport, muscle contraction, beating of cilia and flagella, and ciliogenesis. Most of these processes are driven by the collective operation of several motors in the crowded viscous intracellular environment. Imaging and manipulation of the motors with powerful experimental probes have been complemented by mathematical analysis and computer simulations of the corresponding theoretical models. In this article, we illustrate some of the key theoretical approaches used to understand how coordination, cooperation and competition of multiple motors in the crowded intra-cellular environment drive the processes that are essential for biological function of a cell. In spite of the focus on theory, experimentalists will also find this article as an useful summary of the progress made so far in understanding multiple motor systems.

Elasticity of a Grafted Rod-like Filament with Fluctuating Bending Stiffness

Quite often polymers exhibit different elastic behavior depending on the statistical ensemble (Gibbs vs. Helmholtz). This is an effect of strong fluctuations. In particular, two-state polymers, which locally or globally fluctuate between two classes of microstates, can exhibit strong ensemble inequivalence with negative elastic moduli (extensibility or compressibility) in the Helmholtz ensemble. Two-state polymers consisting of flexible beads and springs have been studied extensively. Recently, similar behavior was predicted in a strongly stretched wormlike chain consisting of a sequence of reversible blocks, fluctuating between two values of the bending stiffness (the so called reversible wormlike chain, rWLC). In this article, we theoretically analyse the elasticity of a grafted rod-like semiflexible filament which fluctuates between two states of bending stiffness. We consider the response to a point force at the fluctuating tip in both the Gibbs and the Helmholtz ensemble. We also calculate the entropic force exerted by the filament on a confining wall. This is done in the Helmholtz ensemble and, under certain conditions, it yields negative compressibility. We consider a two-state homopolymer and a two-block copolymer with two-state blocks. Possible physical realizations of such a system would be grafted DNA or carbon nanorods undergoing hybridization, or grafted F-actin bundles undergoing collective reversible unbinding.

Temperature dependent model for the quasi-static stick-slip process on a soft substrate

The classical Prandtl-Tomlinson model is the most famous and efficient method to describe the stick-slip phenomenon and the resulting friction between a slider and a corrugated substrate. It is widely used in all studies of frictional physics and notably in nanotribology. However, it considers a rigid or undeformable substrate and therefore is hardly applicable for investigating the physics of soft matter and in particular biophysics. For this reason, we introduce here a modified model that is capable of taking into consideration a soft or deformable substrate. It is realized by a sequence of elastically bound quadratic energy wells, which represent the corrugated substrate. We study the quasi-static behavior of the system through the equilibrium statistical mechanics. We thus determine the static friction and the deformation of the substrate as a function of temperature and substrate stiffness. The results are of interest for the study of cell motion in biophysics and for haptic and tactile systems in microtechnology.

Thermal effects on fracture and the brittle-to-ductile transition

The fracture behavior of brittle and ductile materials can be strongly influenced by thermal fluctuations, especially in micro- and nanodevices as well as in rubberlike and biological materials. However, temperature effects, in particular on the brittle-to-ductile transition, still require a deeper theoretical investigation. As a step in this direction we propose a theory, based on equilibrium statistical mechanics, able to describe the temperature-dependent brittle fracture and brittle-to-ductile transition in prototypical discrete systems consisting in a lattice with breakable elements. Concerning the brittle behavior, we obtain closed form expressions for the temperature-dependent fracture stress and strain, representing a generalized Griffith criterion, ultimately describing the fracture as a genuine phase transition. With regard to the brittle-to-ductile transition, we obtain a complex critical scenario characterized by a threshold temperature between the two fracture regimes (brittle and ductile), an upper and a lower yield strength, and a critical temperature corresponding to the complete breakdown. To show the effectiveness of the proposed models in describing thermal fracture behaviors at small scales, we successfully compare our theoretical results with molecular dynamics simulations of Si and GaN nanowires.

Passive viscoelastic response of striated muscles

Muscle cells with sarcomeric structure exhibit highly non trivial passive mechanical response. The difficulty of its continuum modeling is due to the presence of long-range interactions transmitted by extended protein skeleton. To build a rheological model for muscle 'material', we use a stochastic micromodel, and derive a linear response theory for a half-sarcomere, which can be extended to the whole fibre. Instead of the first order rheological equation, anticipated by Hill on the phenomenological grounds, we obtain a novel second order equation which shows that tension depends not only on its current length and the velocity of stretching, but also on its acceleration. Expressing the model in terms of elementary rheological elements, we show that one contribution to the visco-elastic properties of the fibre originates in cross-bridges, while the other can be linked to inert elements which move in the sarcoplasm. We apply this model to explain the striking qualitative difference between the relaxation in experiments involving perturbation of length vs. those involving perturbation of force, and we use the values of the microscopic parameters for frog muscles to show that the model is in excellent quantitative agreement with physiological experiments.

Entropy production and Onsager reciprocal relations describing the relaxation to equilibrium in stochastic thermodynamics

We study the relation between stochastic thermodynamics and nonequilibrium thermodynamics by evaluating the entropy production and the relation between fluxes and forces in a harmonic system with N particles in contact with N different reservoirs. We suppose that the system is in a nonequilibrium stationary state in a first phase and we study the relaxation to equilibrium in a second phase. During this relaxation, we can identify the linear relation between fluxes and forces satisfying the Onsager reciprocity and we obtain a nonlinear expression for the entropy production. Only when forces and fluxes are small does the entropic production turn into a quadratic form in the forces, as predicted by the Onsager theory.

Rate-dependent force-extension models for single-molecule force spectroscopy experiments

Single-molecule force spectroscopy techniques allow for the measurement of several static and dynamic features of macromolecules of biological origin. In particular, atomic force microscopy, used with a variable pulling rate, provides valuable information on the folding/unfolding dynamics of proteins. We propose here two different models able to describe the out-of-equilibrium statistical mechanics of a chain composed of bistable units. These latter represent the protein domains, which can be either folded or unfolded. Both models are based on the Langevin approach and their implementation allows for investigating the effect of the pulling rate and of the device intrinsic elasticity on the chain unfolding response. The theoretical results (both analytical and numerical) have been compared with experimental data concerning the unfolding of the titin and filamin proteins, eventually obtaining a good agreement over a large range of the pulling rates.

Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

The statistical mechanics and the thermodynamics of small systems are characterized by the non-equivalence of the statistical ensembles. When concerning a polymer chain or an arbitrary chain of independent units, this concept leads to different force-extension responses for the isotensional (Gibbs) and the isometric (Helmholtz) thermodynamic ensembles for a limited number of units (far from the thermodynamic limit). While the average force-extension response has been largely investigated in both Gibbs and Helmholtz ensembles, the full statistical characterization of this thermo-mechanical behavior has not been approached by evaluating the corresponding probability densities. Therefore, we elaborate in this paper a technique for obtaining the probability density of the extension when force is applied (Gibbs ensemble) and the probability density of the force when the extension is prescribed (Helmholtz ensemble). This methodology, here developed at thermodynamic equilibrium, is applied to a specific chain composed of units characterized by a bistable potential energy, which is able to mimic the folding and unfolding of several macromolecules of biological origin.

Functionality of Disorder in Muscle Mechanics

A salient feature of skeletal muscles is their ability to take up an applied slack in a microsecond timescale. Behind this fast adaptation is a collective folding in a bundle of elastically interacting bistable elements. Since this interaction has a long-range character, the behavior of the system in force and length controlled ensembles is different; in particular, it can have two distinct order-disorder-type critical points. We show that the account of the disregistry between myosin and actin filaments places the elementary force-producing units of skeletal muscles close to both such critical points. The ensuing "double criticality" contributes to the system's ability to perform robustly and suggests that the disregistry is functional.

SNARE machinery is optimized for ultrafast fusion

SNARE proteins zipper to form complexes (SNAREpins) that power vesicle fusion with target membranes in a variety of biological processes. A single SNAREpin takes about 1 s to fuse two bilayers, yet a handful can ensure release of neurotransmitters from synaptic vesicles much faster: in a 10th of a millisecond. We propose that, similar to the case of muscle myosins, the ultrafast fusion results from cooperative action of many SNAREpins. The coupling originates from mechanical interactions induced by confining scaffolds. Each SNAREpin is known to have enough energy to overcome the fusion barrier of 25-[Formula: see text]; however, the fusion barrier only becomes relevant when the SNAREpins are nearly completely zippered, and from this state, each SNAREpin can deliver only a small fraction of this energy as mechanical work. Therefore, they have to act cooperatively, and we show that at least three of them are needed to ensure fusion in less than a millisecond. However, to reach the prefusion state collectively, starting from the experimentally observed half-zippered metastable state, the SNAREpins have to mechanically synchronize, which takes more time as the number of SNAREpins increases. Incorporating this somewhat counterintuitive idea in a simple coarse-grained model results in the prediction that there should be an optimum number of SNAREpins for submillisecond fusion: three to six over a wide range of parameters. Interestingly, in situ cryoelectron microscope tomography has very recently shown that exactly six SNAREpins participate in the fusion of each synaptic vesicle. This number is in the range predicted by our theory.

Spin variable approach for the statistical mechanics of folding and unfolding chains

The force-extension response of chains composed of bistable (or multistable) units strongly depends on the applied boundary conditions. As a matter of fact, isotensional conditions (soft devices) lead to a plateau-like response, whereas isometric conditions (hard devices) lead to a sawtooth-like pattern. We develop an equilibrium statistical mechanics methodology, based on the introduction of a set of discrete or spin variables, which is able to describe the thermal and mechanical properties of a folding and unfolding chain under arbitrary external conditions. In particular, we will work within the Gibbs and Helmholtz ensembles, which correspond to soft and hard devices, respectively. We introduce a one-dimensional system composed of multistable units and a bistable freely jointed chain. For both systems we obtain explicit expressions for the force-extension relation and we study the spinoidal behavior induced by the isometric conditions.

Dynamics of a mechanical system with multiple degrees of freedom out of thermal equilibrium

Out of thermal equilibrium, an environment imposes effective mechanical forces on nanofabricated devices as well as on microscopic chemical or biological systems. Here we address the question of how to calculate these forces together with the response of the system from first principles. We show that an ideal gaslike environment, even near thermal equilibrium, can enforce a specific steady state on the system by creating effective potentials in otherwise homogeneous space. An example of stable and unstable rectifications of thermal fluctuations is presented using a modified Feynman-Smoluchowski ratchet with two degrees of freedom. Moreover, the stability of a steady configuration depends on its chiral symmetry. The transition rate probabilities and the corresponding kinetic equations are derived for a complex mechanical system with arbitrary degrees of freedom. This work, therefore, extends the applicability of mechanical systems as a toy model playground of statistical physics for active and living matter with multiple degrees of freedom.