Session V17: Focus Session: Nonequilibrium Thermodynamics of Small Systems

Beyond the second law.

According to the second law, the work needed to bring a system from one state to another one, when in contact with a heat bath, is at least equal to the difference in free energies of these states. By a refinement of the work theorem, we derive an exact microscopic expression for the extra dissipation. It is expressed in terms of the relative entropy of the phase space density. Furthermore, lower bounds for the extra dissipation are obtained when only limited information is available. The result is illustrated on various Hamiltonian and stochastic models. The connection to the thermodynamics of computation is briefly discussed. Ref R. Kawai, JMR. Parrondo and C. Van den Broeck, Phys Rev Lett 98, 080602 (2007)   

Dynamics of Microtubule Growth and Catastrophe

We investigate a simple dynamical model of a microtubule that evolves by attachment of guanosine triphosphate (GTP) tubulin to its end, irreversible conversion of GTP to guanosine diphosphate (GDP) tubulin by hydrolysis, and detachment of GDP at the end of a microtubule. As a function of rates of these processes, the microtubule can grow steadily or its length can fluctuate wildly. In the regime where detachment can be neglected, we find exact expressions for the tubule and GTP cap length distributions, as well as power-law length distributions of GTP and GDP islands. In the opposite limit of instantaneous detachment, we find the time between catastrophes, where the microtubule shrinks to zero length, scales as $e^\lambda$. We also determine the size distribution of avalanches (sequence of consecutive GDP detachment events). We obtain the phase diagram for general rates and verify our predictions by numerical simulations.   

Work fluctuations in modulated systems: what is and what is not universal in the steady state

We study work fluctuations in periodically modulated nonlinear systems. Such systems often have coexisting stable periodic states. We show that the standard relations of the steady-state work fluctuation theorem derived for linear systems do not generally apply to nonlinear systems. Work fluctuations sharply increase near a kinetic phase transition where populations of the coexisting periodic states are close to each other. The work variance in this range is inversely proportional to the rate of fluctuational interstate switching. It exponentially decreases with the increasing distance to the phase transition point. We also show that the work variance in a metastable state displays scaling with the distance to the bifurcation point where this state disappears. The critical exponent in the dependence of the variance on the distance to a saddle-node bifurcation point is -1. The results apply to a broad range of vibrational systems of current interest, from trapped electrons to Josephson junctions and to nano- and micromechanical resonators.   

Recognizing the role of microscopic reversibility in stochastic systems with optical trapping experiments as an example

With optical tweezer experiments it is possible to confine a particle and observe its Brownian motion in a region of known potential. Recognizing that microscopic reversibility can lead to interesting relationships involving the particle's motion, for example, the ratio of the conditional probability of making a transition between two points with its spatial reverse is equal to the difference between two Boltzmann factors ($e^{-\Delta U/k_BT}$), or that the time to make an up-well transition is identical to the time to make a down-well transition. With these relationships in mind, and the Onsager-Machlup action description of a path, we consider a particle in an optical tweezer of varying strength and investigate relationships involving the conditional probabilities, the actions associated with specific paths, and the external work done by varying the strength of the trap.   

What Do We Measure With Single-Molecule Force Spectroscopy?

Single-molecule force spectroscopy is a powerful technique for studying detailed intra- and inter- molecular interactions by manipulating single biomolecules at the nanometer scale. Force is measured while one pulls on the molecules. However, relating the measured information to equilibrium thermodynamic properties is challenging. Jarzynksi's equality allows one to reconstruct the free energy surface as a function of molecular end-to-end distance$^{1,2}$. Using protein folding as an example, we studied the parameters that influence the unfolding process, such as pulling velocity, tempearture, and chemical denaturant concentration in the solution, to demonatrate that valuable equailibrium thermodynamic information can be obtained using this technique. 1. N.\ C.\ Harris, Y.\ Song, and C.-H.\ Kiang, {\em Phys.\ Rev.\ Lett.}, {\bf 99} 068101 (2007). 2.``Pulling Strings: Stretching Proteins Can Reveal How They Fold,'' {\bf Science News}, {\bf 172} 22 (2007).   

Holographic microrheology of biofilms

We present microrheological measurements of polymeric matrices, including the extra-cellular polysaccharide gel synthesized by the dental pathogen S. mutans. As part of this study, we introduce the use of precision three-dimensional particle tracking based on video holographic microscopy. This technique offers nanometer-scale resolution at video rates, thereby providing detailed information on the gels' complex viscoelastic moduli, including insights into their heterogeneity. The particular application to dental biofilms complements previous studies based on macroscopic rheology, and demonstrates the utility of holographic microrheology for soft-matter physics and biomedical research.   

Fluctuation relations in a micromechanical oscillator driven far from thermal equilibrium

We explore fluctuation relations in a micromechanical torsional oscillator. In the linear regime when the modulation is weak, we verify that the ratio of the work variance to the mean work is independent of the driving frequency, consistent with standard fluctuation relations for a steady state system near thermal equilibrium. We then apply a strong drive to force the system into nonlinearity. Here the system displays bistability and the relationship between the work and variance is expected to deviate from the linear regime. For a bistable system the total variance has two distinct contributing components. The first results from small fluctuations about a stable state. The work variance is expected to diverge as the system approaches the bifurcation point where the state disappears. The other part of the variance results from the system switching between coexisting states. This part of the variance is expected to show a sharp peak near the kinetic phase transition when the populations of the two states are comparable. We compare our experimental results to theoretical predictions that distinguish nonlinear oscillators from equilibrium systems.   

Effect of charge distribution on the translocation of an inhomogeneously charged polymer through a nanopore

We investigate the voltage-driven translocation of an inhomogeneously charged polymer through a nanopore by utilizing discrete and continuous stochastic models. As a simplified illustration of the effect of charge distribution on translocation, we consider the translocation of a polymer with a single charged site in the presence and absence of interactions between the charge and the pore. We find that the position of the charge that minimizes the translocation time in the absence of pore-polymer interactions is determined by the entropic cost of translocation, with the optimum charge position being at the midpoint of the chain for a rodlike polymer and close to the leading chain end for an ideal chain. The presence of attractive or repulsive pore–charge interactions yields a shift in the optimum charge position towards the trailing end and the leading end of the chain, respectively. Moreover, our results show that strong attractive or repulsive interactions between the charge and the pore lengthen the translocation time relative to translocation through an inert pore. We generalize our results to accommodate the presence of multiple charged sites on the polymer. Our results provide insight into the effect of charge inhomogeneity on protein translocation through biological membranes.   

Heat Transport in spin chains

The Projection operator technique, usually used to study the relaxation toward thermal equilibrium, is extended to investigate non-equilibrium but stationary processes. Particularly, in this work we apply it to study heat transport in short spin chains with Heisenberg, anisotropic Heisenberg and XY couplings plus a large magnetic field. One long-standing question is under what circumstance the relation between thermal current and local temperature obeys Fourier's Law. Our results suggest that for certain parameters and short chains the heat transport does obey Fourier's Law. The evolution towards a diverging thermal conductivity, expected in the bulk limit, is also elucidated.   

Athermal dynamics of strongly coupled stochastic three-state oscillators

We study the collective behavior of a globally coupled ensemble of $N$ cyclic stochastic three-state systems with rates of transition from state $i-1$ to state $i$ proportional to the number of systems already in state $i$. While the mean field theory predicts only decaying oscillations in this system, direct numerical simulations indicate that the mean field exhibits stochastic oscillations even in the large $N$ limit. We characterize the regularity of oscillations by the coherence parameter which has a well-defined maximum at the coupling constant of order 1. In contrast, the order parameter characterizing the level of synchrony among oscillators, increases monotonously with the coupling strength. We derive the exact solution of the full master equation for the stationary probability distribution and find the analytical expression for the order parameter.   

Entropy-driven hysteresis in a model of DNA overstretching

When pulled along its axis, double-stranded DNA elongates abruptly at a force of about 65 pN. Two physical pictures have been developed to describe this overstretched state. The first proposes that strong forces induce a phase transition to a molten state consisting of unhybridized single strands. The second picture instead introduces an elongated hybridized phase, called S-DNA, structurally and thermodynamically distinct from standard B-DNA. Little thermodynamic evidence exists to discriminate directly between these competing pictures. Here we show that within a microscopic model of DNA we can distinguish between the dynamics associated with each. In experiment, considerable hysteresis in a cycle of stretching and shortening develops as temperature is increased. Since there are few possible causes of hysteresis in a system whose extent is appreciable in only one dimension, such behavior offers a discriminating test of the two pictures of overstretching. Most experiments are performed upon nicked DNA, permitting the detachment (`unpeeling') of strands. We show that the long-wavelength motion accompanying strand separation generates hysteresis, the character of which agrees with experiment only if we assume the existence of S-DNA.   

Replica and extreme-value analysis of the Jarzynski free-energy estimator

We analyze the Jarzynski estimator of free-energy differences from nonequilibrium work measurements. By a simple mapping onto Derrida's Random Energy Model, we obtain a scaling limit for the expectation of the bias of the estimator. We then derive analytical approximations in three different regimes of the scaling parameter x = log(N)/W, where N is the number of measurements and W the mean dissipated work. Our approach is valid for a generic distribution of the dissipated work, and is based on a replica symmetry breaking scheme for x >> 1, the asymptotic theory of extreme value statistics for x << 1, and a direct approach for x near one. The combination of the three analytic approximations describes well Monte Carlo data for the expectation value of the estimator, for a wide range of values of N, from N=1 to large N, and for different work distributions. Based on these results, we introduce improved free-energy estimators and discuss the application to the analysis of experimental data.   

Hand-Over-Hand Binding in a Tethered-Particle Method Study of DNA Hybridization

The tethered-particle method is used to probe the complex process of DNA hybridization. This experiment analyzes tethers formed between PEGylated polystyrene microspheres and PEGylated glass surfaces, to which DNA strands have been grafted. Previous studies have looked at single duplex dissociation in isolation. We seek to understand cooperativity effects of short duplexes in multiple tethered states. This experiment is performed near the duplex melting temperature, resulting in dynamic binding and unbinding. Single molecular tethers are analyzed using high performance particle tracking with a high-speed camera. By analyzing 2D trajectories of the particle's centroid versus time, we can distinguish single binding and unbinding events from multiple bridged states. The trajectories' analysis suggests that multiple bridged states are less stable than expected for a zero cooperativity model. We hypothesize that this instability may be due to extreme force sensitivity of individual DNA duplexes.