A two-state kinetic model for the unfolding of single molecules by mechanical force

We investigate the work dissipated during the irreversible unfolding of single molecules by mechanical force, using the simplest model necessary to represent experimental data. The model consists of two levels (folded and unfolded states) separated by an intermediate barrier. We compute the probability distribution for the dissipated work and give analytical expressions for the average and variance of the distribution. To first order, the amount of dissipated work is directly proportional to the rate of application of force (the loading rate), and to the relaxation time of the molecule. The model yields estimates for parameters that characterize the unfolding kinetics under force in agreement with those obtained in recent experimental results (Liphardt, J., et al. (2002) {\em Science}, {\bf 296} 1832-1835). We obtain a general equation for the minimum number of repeated experiments needed to obtain an equilibrium free energy, to within $k_BT$, from non-equilibrium experiments using the Jarzynski formula. The number of irreversible experiments grows exponentially with the ratio of the average dissipated work, \bar{\Wdis}, to $k_BT$.}Comment: PDF file, 5 page

The Nonequilibrium Thermodynamics of Small Systems

The interactions of tiny objects with their environment are dominated by thermal fluctuations. Guided by theory and assisted by micromanipulation tools, scientists have begun to study such interactions in detail.Comment: PDF file, 13 pages. Long version of the paper published in Physics Toda

Geometric effect and gauge field in nonequilibrium quantum thermostatistics

The concept of work is studied in quantum thermostatistics of a system surrounded by an environment and driven by an external force. It is found that there emerges the gauge theoretical structure in a nonequilibrium process, the field of which is referred to as the work gauge field. The thermodynamic work as the flux of the work gauge field is considered for a cyclic process in the space of the external-force parameters. As an example, the system of a spin-1/2 interacting with an external magnetic field is analyzed. This geometric effect may be observed, for example, in an NMR experiment and can be applied to the problem of cooling/heating of a small system.Comment: 11 page

Mechanical unfolding of RNA hairpins

Mechanical unfolding trajectories, generated by applying constant force in optical tweezer experiments, show that RNA hairpins and the P5abc subdomain of the group I intron unfold reversibly. We use coarse-grained Go-like models for RNA hairpins to explore forced-unfolding over a broad range of temperatures. A number of predictions that are amenable to experimental tests are made. At the critical force the hairpin jumps between folded and unfolded conformations without populating any discernible intermediates. The phase diagram in the force-temperature (f,T) plane shows that the hairpin unfolds by an all-or-none process. The cooperativity of the unfolding transition increases dramatically at low temperatures. Free energy of stability, obtained from time averages of mechanical unfolding trajectories, coincide with ensemble averages which establishes ergodicity. The hopping time between the the native basin of attraction (NBA) and the unfolded basin increases dramatically along the phase boundary. Thermal unfolding is stochastic whereas mechanical unfolding occurs in "quantized steps" with great variations in the step lengths. Refolding times, upon force quench, from stretched states to the NBA is "at least an order of magnitude" greater than folding times by temperature quench. Upon force quench from stretched states the NBA is reached in at least three stages. In the initial stages the mean end-to-end distance decreases nearly continuously and only in the last stage there is a sudden transition to the NBA. Because of the generality of the results we propose that similar behavior should be observed in force quench refolding of proteins.Comment: 23 pages, 6 Figures. in press (Proc. Natl. Acad. Sci.

A charged particle in a magnetic field - Jarzynski Equality

We describe some solvable models which illustrate the Jarzynski theorem and related fluctuation theorems. We consider a charged particle in the presence of magnetic field in a two dimensional harmonic well. In the first case the centre of the harmonic potential is translated with a uniform velocity, while in the other case the particle is subjected to an ac force. We show that Jarzynski identity complements Bohr-van Leeuwen theorem on the absence of diamagnetism in equilibrium classical system.Comment: 5 pages, minor corrections made and journal reference adde

Kinetic barriers in RNA unzipping

We consider a simple model for the unfolding of RNA tertiary structure under dynamic loading. The opening of such a structure is regarded as a two step process, each corresponding to the overcoming of a single energy barrier. The resulting two-barrier energy landscape accounts for the dependence of the unfolding kinetics on the pulling rate. Furthermore at intermediate force, the two barriers cannot be distinguished by the analysis of the opening kinetic, which turns out to be dominated by a single macro-barrier, whose properties depend non-trivially on the two single barriers. Our results suggest that in pulling experiments on RNA molecule containing tertiary structures, the details of the single kinetic barriers can only be obtained using a low pulling rate value, or in the high force regime.Comment: to appear on Eur. Phys. J.

Characterizing dynamical transitions in bistable system using non-equilibrium measurement of work

We show how Jarzynski relation can be exploited to analyze the nature of order-disorder and a bifurcation type dynamical transition in terms of a response function derived on the basis of work distribution over non-equilibrium paths between two thermalized states. The validity of the response function extends over linear as well as nonlinear regime and far from equilibrium situations

Phase diagram for unzipping DNA with long-range interactions

We present a critique and extension of the mean-field approach to the mechanical pulling transition in bound polymer systems. Our model is motivated by the theoretically and experimentally important examples of adsorbed polymers and double-stranded DNA, and we focus on the case in which quenched disorder in the sequence of monomers is unimportant for the statistical mechanics. We show how including excluded volume interactions in the model affects the phase diagram for the critical pulling force, and we predict a re-entrancy phase at low temperatures which has not been previously discussed. We also consider the case of non-equilibrium pulling, in which the external force probes the local, rather than the global structure of the dsDNA or adsorbed polymer. The dynamics of the pulling transition in such experiments could illuminate the polymer's loop structure, which depends on the nature of excluded volume interactions.Comment: 4 pages, 2 figures; this version clarifies Eq. 8, and corrects errors in Fig.

Single-molecule pulling: phenomenology and interpretation

Single-molecule pulling techniques have emerged as versatile tools for probing the noncovalent forces holding together the secondary and tertiary structure of macromolecules. They also constitute a way to study at the single-molecule level processes that are familiar from our macroscopic thermodynamic experience. In this Chapter, we summarize the essential phenomenology that is typically observed during single-molecule pulling, provide a general statistical mechanical framework for the interpretation of the equilibrium force spectroscopy and illustrate how to simulate single-molecule pulling experiments using molecular dynamics.Comment: arXiv admin note: text overlap with arXiv:0908.220