Helmholtz Ensemble Research Articles

Statistical Mechanics Approaches for Studying Temperature and Rate Effects in Multistable Systems

Systems with a multistable energy landscape are widespread in physics, biophysics, technology, and materials science. They are strongly influenced by thermal fluctuations and external mechanical actions that can be applied at different rates, moving the system from equilibrium to non-equilibrium regimes. In this paper, we focus on a simple system involving a single breaking phenomenon to describe the various theoretical approaches used to study these problems. To begin with, we propose the exact solution at thermodynamic equilibrium based on the calculation of the partition function without approximations. We then introduce the technique of spin variables, which is able to simplify the treatment even for systems with a large number of coordinates. We then analyze the energy balance of the system to better understand its underlying physics. Finally, we introduce a technique based on transition state theory useful for studying the non-equilibrium dynamical regimes of these systems. This method is appropriate for the evaluation of rate effects and hysteresis loops. These approaches are developed for both the Helmholtz ensemble (prescribed extension) and the Gibbs ensemble (applied force) of statistical mechanics. The symmetry and duality of these two ensembles is discussed in depth. While these techniques are used here for a simple system with theoretical purposes, they can be applied to complex systems of interest for several physical, biophysical, and technological applications.

Forced extension of a wormlike chain in the Gibbs and Helmholtz ensembles.

A semiflexible polymer can be stretched by either applying a force to it or by fixing the positions of its endpoints. The two approaches generally yield different results and correspond to experiments performed in either the Gibbs or Helmholtz statistical ensembles. Here, we derive the Helmholtz force-extension relationship for the commonly used wormlike-chain model in the strongly stretched regime. By analyzing it in comparison with the Gibbs ensemble result, we show that equivalence between the two relationships is achieved only in the long-chain thermodynamic limit.

Ensemble inequivalence and negative extensibility in a strongly stretched wormlike chain with fluctuating bending stiffness.

Many semiflexible polymers exhibit fluctuations in the local bending stiffness along their contour. This may be due to intrinsic conformational changes (e.g., denaturation bubble formation in double stranded DNA or helix-coil transition in polypeptides) or the reversible adsorption and desorption of molecules from the polymer's environment (e.g., DNA-protein interactions or hybridization of oligonucleotides). In this article, we analyze the tensile elasticity of a strongly stretched wormlike chain, which consists of N concatenated segments, where each segment can be in one of two states, A or B, which differ in bending stiffness. We call this model the reversible wormlike chain (rWLC) model. In the Gibbs (fixed-force, isotensional) ensemble, we obtain analytic expressions for the force-extension relation and the mean fraction of B segments. We show that, under certain conditions, there is a tension-induced crossover from a mostly A to a mostly B rWLC. In the Helmholtz (fixed-extension, isometric) ensemble, we obtain analytic expressions up to a summation. We show that, for finite N, there is marked ensemble inequivalence. Remarkably, in the Helmholtz ensemble, the rWLC can exhibit negative extensibility and multiple peaks.

Thermal control of nucleation and propagation transition stresses in discrete lattices with non-local interactions and non-convex energy

Non-local and non-convex energies represent fundamental interacting effects regulating the complex behavior of many systems in biophysics and materials science. We study one-dimensional, prototypical schemes able to represent the behavior of several biomacromolecules and the phase transformation phenomena in solid mechanics. To elucidate the effects of thermal fluctuations on the non-convex non-local behavior of such systems, we consider three models of different complexity relying on thermodynamics and statistical mechanics: (i) an Ising-type scheme with an arbitrary temperature-dependent number of interfaces between different domains, (ii) a zipper model with a single interface between two evolving domains, and (iii) an approximation based on the stationary phase method. In all three cases, we study the system under both isometric condition (prescribed extension, matching with the Helmholtz ensemble of the statistical mechanics) and isotensional condition (applied force, matching with the Gibbs ensemble). Interestingly, in the Helmholtz ensemble the analysis shows the possibility of interpreting the experimentally observed thermal effects with the theoretical force–extension relation characterized by a temperature-dependent force plateau (Maxwell stress) and a force peak (nucleation stress). We obtain explicit relations for the configurational properties of the system as well (expected values of the phase fractions and number of interfaces). Moreover, we are able to prove the equivalence of the two thermodynamic ensembles in the thermodynamic limit. We finally discuss the comparison with data from the literature showing the efficiency of the proposed model in describing known experimental effects.

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.

Response to “Comment on ‘Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles”’ [J. Chem. Phys. 138, 157101 (2013)

Two different statistical ensembles can be considered for extending a single polymer chain: the Gibbs (or isotensional) ensemble characterized by a deterministic force applied to the free end of the chain (the other being fixed in a given reference frame), and the Helmholtz (or isometric) ensemble obtained with both the ends of the polymers tethered at two different points of the space. When the thermodynamic limit is satisfied (the number of monomers approaches infinity) these ensembles are equivalent from the thermodynamic point of view: it means that the constitutive equations (vector force-extension relations) assume the same mathematical form in both isotensional and isometric conditions. Equivalently, the Helmholtz and Gibbs free energies are linked by a Legendre transform. We explain that this general result is coherent with some “forms of inequivalence” observed by defining different average values of force and position vectors. However, this fact does not indicate thermodynamic inequivalence as largely discussed in the present Response.

Comment on “Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles” [J. Chem. Phys. 136, 154906 (2012)

In this Comment, we point out that (i) Manca et al. err in their finding that the force law obtained for a single polymer chain in the thermodynamic limit using the Helmholtz ensemble is identical to that obtained using the Gibbs ensemble in all force regimes, and (ii) the perpetuation of this misunderstanding can have serious consequences when interpreting the results of polymer-stretching experiments in the weak-force region.

Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles

Stretching experiments on single molecules of arbitrary length opened the way for studying the statistical mechanics of small systems. In many cases in which the thermodynamic limit is not satisfied, different macroscopic boundary conditions, corresponding to different statistical mechanics ensembles, yield different force-displacement curves. We formulate analytical expressions and develop Monte Carlo simulations to quantitatively evaluate the difference between the Helmholtz and the Gibbs ensembles for a wide range of polymer models of biological relevance. We consider generalizations of the freely jointed chain and of the worm-like chain models with extensible bonds. In all cases we show that the convergence to the thermodynamic limit upon increasing contour length is described by a suitable power law and a specific scaling exponent, characteristic of each model.

Semiflexible polymers: Dependence on ensemble and boundary orientations

We show that the mechanical properties of a worm-like-chain (WLC) polymer, of contour length L and persistence length lambda such that t=L/lambda approximately O(1), depend both on the ensemble and the constraint on end orientations. In the Helmholtz ensemble, multiple minima in the free energy near t=4 persists for all kinds of orientational boundary conditions. The qualitative features of projected probability distribution of end-to-end vector depend crucially on the embedding dimensions. A mapping of the WLC model, to a quantum particle moving on the surface of a unit sphere, is used to obtain the statistical and mechanical properties of the polymer under various boundary conditions and ensembles. The results show excellent agreement with Monte Carlo simulations.

Conformational transitions in single polymer molecules modeled with a complete energy landscape: continuous two-state model

An extension of the two-state Freely Jointed Chain model is presented in which the discrete energies of the two conformers are replaced by continuous functions of the conformer length. The statistical mechanics is initially developed in the Gibbs ensemble and leads to a conformational multi-state model. This is used to fit the equilibrium force-extension curve for Dextran. The continuous model also allows the use of Transfer Matrix methods to calculate all statistical properties in the Helmholtz ensemble, including thermal fluctuations. The latter are obtained with near perfect agreement to experiment.

Inequivalence of statistical ensembles in single molecule measurements

We study the role of fluctuations in single molecule experimental measurements of force-extension (f-zeta) curves. We use the worm-like chain (WLC) model to bring out the connection between the Helmholtz ensemble characterized by the free energy [F(zeta)] and the Gibbs ensemble characterized by the free energy [G(f)] . We consider the rigid rod limit of the WLC model as an instructive special case to bring out the issue of ensemble inequivalence. We point out the need for taking into account the free energy of transition when one goes from one ensemble to another. We also comment on the "phase transition" noticed in an isometric setup for semiflexible polymers and propose a realization of its thermodynamic limit. We present general arguments which rule out nonmonotonic force-extension curves in some ensembles and note that these do not apply to the isometric ensemble.

Stretching a Macromolecule in an Atomic Force Microscope: Statistical Mechanical Analysis

We formulate the proper statistical mechanics to describe the stretching of a macromolecule under a force provided by the cantilever of an Atomic Force Microscope. In the limit of a soft cantilever, the generalized ensemble of the coupled molecule–cantilever system reduces to the Gibbs ensemble for an isolated molecule subject to a constant force in which the extension is fluctuating. For a stiff cantilever, one obtains the Helmholtz ensemble for an isolated molecule held at a fixed extension with the force fluctuating. Numerical examples and predictions for experiments with cantilevers of differing stiffness are given for short and long chains of poly (ethylene glycol), based on parameter-free ab initio calculations.

Stretching a macromolecule in an atomic force microscope: statistical mechanical analysis.

We formulate the proper statistical mechanics to describe the stretching of a macromolecule under a force provided by the cantilever of an atomic force microscope. In the limit of a soft cantilever the generalized ensemble of the coupled molecule/cantilever system reduces to the Gibbs ensemble for an isolated molecule subject to a constant force in which the extension is fluctuating. For a stiff cantilever we obtain the Helmholtz ensemble for an isolated molecule held at a fixed extension with the force fluctuating. Numerical examples are given for poly (ethylene glycol) chains.