Brownian motion and nonequilibrium statistical mechanics

The fluctuation-dissipation relation holds for a macroscopic tracer in an active bath

The fluctuation-dissipation relation (FDR) links thermal fluctuations and dissipation at thermal equilibrium through temperature. Extending it beyond equilibrium conditions in pursuit of broadening thermodynamics is often feasible, albeit with system-dependent specific conditions. We demonstrate experimentally that a generalized FDR holds for a harmonically trapped tracer colliding with self-propelled walkers. The generalized FDR remains valid across a large spectrum of active fluctuation frequencies, extending from underdamped to critically damped dynamics, which we attribute to a single primary channel for energy input and dissipation in our system.

Soluble Model of a Nonequilibrium Steady State: The Van Kampen Objection and Other Lessons

A simple model of charge transport is provided by a classical particle in a random potential and a dissipative coupling to the environment. The corresponding nonequilibrium steady state (NESS) can be determined analytically when both the disorder and dissipation are weak. We use it to illuminate some foundational issues in nonequilibrium statistical mechanics. We show that at nonlinear level dissipative response sensitively depends on the system-environment coupling, and the range of validity of the linear response theory is set by this coupling. This validates the Van Kampen objection. We also show that the principle of minimum entropy production does not determine the NESS beyond linear order in the electric field, while entropy maximization fails to produce the correct NESS already at linear order.

Thermodynamic uncertainty theorem

Thermodynamic uncertainty relations (TURs) express a fundamental lower bound on the precision (inverse scaled variance) of any thermodynamic charge-e.g., work or heat-by functionals of the average entropy production. Relying on purely variational arguments, we significantly extend TUR inequalities by incorporating and analyzing the impact of higher statistical cumulants of the entropy production itself within the general framework of time-symmetrically-controlled computation. We derive an exact expression for the charge that achieves the minimum scaled variance, for which the TUR bound tightens to an equality that we name the thermodynamic uncertainty theorem (TUT). Importantly, both the minimum scaled variance charge and the TUT are functionals of the stochastic entropy production, thus retaining the impact of its higher moments. In particular, our results show that, beyond the average, the entropy production distribution's higher moments have a significant effect on any charge's precision. This is made explicit via a thorough numerical analysis of "swap" and "reset" computations that quantitatively compares the TUT against previous generalized TURs.

Generalized Langevin equation with shear flow and its fluctuation-dissipation theorems derived from a Caldeira-Leggett Hamiltonian

We provide a first-principles derivation of the Langevin equation with shear flow and its corresponding fluctuation-dissipation theorems. Shear flow of simple fluids has been widely investigated by numerical simulations. Most studies postulate a Markovian Langevin equation with a simple shear drag term in the manner of Stokes. However, this choice has never been justified from first principles. We start from a particle-bath system described by a classical Caldeira-Leggett Hamiltonian modified by adding a term proportional to the strain-rate tensor according to Hoover's DOLLS method, and we derive a generalized Langevin equation for the sheared system. We then compute, analytically, the noise time-correlation functions in different regimes. Based on the intensity of the shear rate, we can distinguish between close-to-equilibrium and far-from-equilibrium states. According to the results presented here, the standard, simple, and Markovian form of the Langevin equation with shear flow postulated in the literature is valid only in the limit of extremely weak shear rates compared to the effective vibrational temperature of the bath. For even marginally higher shear rates, the (generalized) Langevin equation is strongly non-Markovian, and nontrivial fluctuation-dissipation theorems are derived.

Anisotropic Diffusion in Driven Convection Arrays

We numerically investigate the transport of a Brownian colloidal particle in a square array of planar counter-rotating convection rolls at high Péclet numbers. We show that an external force produces huge excess peaks of the particle's diffusion constant with a height that depends on the force orientation and intensity. In sharp contrast, the particle's mobility is isotropic and force independent. We relate such a nonlinear response of the system to the advection properties of the laminar flow in the suspension fluid.

On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems

We review generalized fluctuation-dissipation relations, which are valid under general conditions even in "nonstandard systems," e.g., out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperturbed dynamics. In these relations, typically, one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in nonstandard cases, including driven granular media, systems with a multiscale structure, active matter, and systems showing anomalous diffusion.

Defining an olfactory receptor function in airway smooth muscle cells

Pathways that control, or can be exploited to alter, the increase in airway smooth muscle (ASM) mass and cellular remodeling that occur in asthma are not well defined. Here we report the expression of odorant receptors (ORs) belonging to the superfamily of G-protein coupled receptors (GPCRs), as well as the canonical olfaction machinery (G_olf and AC3) in the smooth muscle of human bronchi. In primary cultures of isolated human ASM, we identified mRNA expression for multiple ORs. Strikingly, OR51E2 was the most highly enriched OR transcript mapped to the human olfactome in lung-resident cells. In a heterologous expression system, OR51E2 trafficked readily to the cell surface and showed ligand selectivity and sensitivity to the short chain fatty acids (SCFAs) acetate and propionate. These endogenous metabolic byproducts of the gut microbiota slowed the rate of cytoskeletal remodeling, as well as the proliferation of human ASM cells. These cellular responses in vitro were found in ASM from non-asthmatics and asthmatics, and were absent in OR51E2-deleted primary human ASM. These results demonstrate a novel chemo-mechanical signaling network in the ASM and serve as a proof-of-concept that a specific receptor of the gut-lung axis can be targeted to treat airflow obstruction in asthma.

111 years of Brownian motion

We consider the Brownian motion of a particle and present a tutorial review over the last 111 years since Einstein's paper in 1905. We describe Einstein's model, Langevin's model and the hydrodynamic models, with increasing sophistication on the hydrodynamic interactions between the particle and the fluid. In recent years, the effects of interfaces on the nearby Brownian motion have been the focus of several investigations. We summarize various results and discuss some of the controversies associated with new findings about the changes in Brownian motion induced by the interface.

Effects of translation-rotation coupling on the displacement probability distribution functions of boomerang colloidal particles

Prior studies have shown that low symmetry particles such as micro-boomerangs exhibit behaviour of Brownian motion rather different from that of high symmetry particles because convenient tracking points (TPs) are usually inconsistent with their center of hydrodynamic stress (CoH) where the translational and rotational motions are decoupled. In this paper we study the effects of the translation-rotation coupling on the displacement probability distribution functions (PDFs) of the boomerang colloid particles with symmetric arm length. By tracking the motions of different points on the particle symmetry axis, we show that as the distance between the TP and the CoH is increased, the effects of translation-rotation coupling becomes pronounced, making the short-time 2D PDF for fixed initial orientation to change from elliptical, to bean and then to crescent shape, and the angle averaged PDFs change from ellipsoidal-particle-like PDF to a shape with a Gaussian top and long displacement tails. We also observed that at long times the PDFs revert to Gaussian. These 2D PDF shapes provide a clear physical picture of the non-zero mean displacements observed in boomerangs particles.

Universality and nonuniversality of mobility in heterogeneous single-file systems and Rouse chains

We study analytically the tracer particle mobility in single-file systems with distributed friction constants. Our system serves as a prototype for nonequilibrium, heterogeneous, strongly interacting Brownian systems. The long time dynamics for such a single-file setup belongs to the same universality class as the Rouse model with dissimilar beads. The friction constants are drawn from a density ϱ(ξ), and we derive an asymptotically exact solution for the mobility distribution P[μ0(s)], where μ0(s) is the Laplace-space mobility. If ϱ is light tailed (first moment exists), we find a self-averaging behavior: P[μ0(s)]=δ[μ0(s)-μ(s)], with μ(s)∝s1/2. When ϱ(ξ) is heavy tailed, ϱ(ξ)≃ξ-1-α(0<α<1) for large ξ, we obtain moments 〈[μs(0)]n〉∝sβn, where β=1/(1+α) and there is no self-averaging. The results are corroborated by simulations.

Non-monotonic crossover from single-file to regular diffusion in micro-channels

The diffusion behavior of interacting particles determines the behavior of a large number of systems ranging from pedestrians crossing a road to ions passing through channels in living cells. Here we present a system in which the nature of the diffusion process varies with changes in the external conditions. We find this special behavior in a colloidal model system, consisting of micron sized particles which are confined to narrow channels and interact via induced magnetic dipoles. When the density of these particles is changed, diffusion alternates between normal Fickian behavior and single-file diffusion. This anomalous behavior is induced by the order of the particles in the restricted geometry and does not depend on the exact nature of the inter-particle interactions.

The modified Sutherland–Einstein relation for diffusive non-equilibria

There remains a useful relation between diffusion and mobility for a Langevin particle in a periodic medium subject to non-conservative forces. The usual fluctuation–dissipation relation can be easily modified and the mobility matrix is no longer proportional to the diffusion matrix, with a correction term depending explicitly on the (non-equilibrium) forces. We discuss this correction by considering various simple examples and we visualize the various dependencies on the applied forcing and on the time by means of simulations. For example, in all cases the diffusion depends on the external forcing more strongly than does the mobility. We also give an explicit decomposition of the symmetrized mobility matrix as the difference between two positive matrices, one involving the diffusion matrix and the other involving force–force correlations.

Shear rheology of lipid monolayers and insights on membrane fluidity

The concept of membrane fluidity usually refers to a high molecular mobility inside the lipid bilayer which enables lateral diffusion of embedded proteins. Fluids have the ability to flow under an applied shear stress whereas solids resist shear deformations. Biological membranes require both properties for their function: high lateral fluidity and structural rigidity. Consequently, an adequate account must include, in addition to viscosity, the possibility for a nonzero shear modulus. This knowledge is still lacking as measurements of membrane shear properties have remained incomplete so far. In the present contribution we report a surface shear rheology study of different lipid monolayers that model distinct biologically relevant situations. The results evidence a large variety of mechanical behavior under lateral shear flow.

Measurement of the instantaneous velocity of a Brownian particle

Brownian motion of particles affects many branches of science. We report on the Brownian motion of micrometer-sized beads of glass held in air by an optical tweezer, over a wide range of pressures, and we measured the instantaneous velocity of a Brownian particle. Our results provide direct verification of the energy equipartition theorem for a Brownian particle. For short times, the ballistic regime of Brownian motion was observed, in contrast to the usual diffusive regime. We discuss the applications of these methods toward cooling the center-of-mass motion of a bead in vacuum to the quantum ground motional state.

Biophysical basis for airway hyperresponsiveness

Airway hyperresponsiveness is the excessive narrowing of the airway lumen caused by stimuli that would cause little or no narrowing in the normal individual. It is one of the cardinal features of asthma, but its mechanisms remain unexplained. In asthma, the key end-effector of acute airway narrowing is contraction of the airway smooth muscle cell that is driven by myosin motors exerting their mechanical effects within an integrated cytoskeletal scaffolding. In just the past few years, however, our understanding of the rules that govern muscle biophysics has dramatically changed, as has their classical relationship to airway mechanics. It has become well established, for example, that muscle length is equilibrated dynamically rather than statically, and that in a dynamic setting nonclassical features of muscle biophysics come to the forefront, including unanticipated interactions between the muscle and its time-varying load, as well as the ability of the muscle cell to adapt (remodel) its internal microstructure rapidly in response to its ever-changing mechanical environment. Here, we consider some of these emerging concepts and, in particular, focus on structural remodeling of the airway smooth muscle cell as it relates to excessive airway narrowing in asthma.

Open-system nonequilibrium steady state: statistical thermodynamics, fluctuations, and chemical oscillations

Gibbsian equilibrium statistical thermodynamics is the theoretical foundation for isothermal, closed chemical, and biochemical reaction systems. This theory, however, is not applicable to most biochemical reactions in living cells, which exhibit a range of interesting phenomena such as free energy transduction, temporal and spatial complexity, and kinetic proofreading. In this article, a nonequilibrium statistical thermodynamic theory based on stochastic kinetics is introduced, mainly through a series of examples: single-molecule enzyme kinetics, nonlinear chemical oscillation, molecular motor, biochemical switch, and specificity amplification. The case studies illustrate an emerging theory for the isothermal nonequilibrium steady state of open systems.

Cytoskeleton dynamics: fluctuations within the network

Out-of-equilibrium systems, such as the dynamics of a living cytoskeleton (CSK), are inherently noisy with fluctuations arising from the stochastic nature of the underlying biochemical and molecular events. Recently, such fluctuations within the cell were characterized by observing spontaneous nano-scale motions of an RGD-coated microbead bound to the cell surface [Bursac et al., Nat. Mater. 4 (2005) 557-561]. While these reported anomalous bead motions represent a molecular level reorganization (remodeling) of microstructures in contact with the bead, a precise nature of these cytoskeletal constituents and forces that drive their remodeling dynamics are largely unclear. Here, we focused upon spontaneous motions of an RGD-coated bead and, in particular, assessed to what extent these motions are attributable to (i) bulk cell movement (cell crawling), (ii) dynamics of focal adhesions, (iii) dynamics of lipid membrane, and/or (iv) dynamics of the underlying actin CSK driven by myosin motors.

Hypoxia alters biophysical properties of endothelial cells via p38 MAPK- and Rho kinase-dependent pathways

Hypoxia alters the barrier function of the endothelial cells that line the pulmonary vasculature, but underlying biophysical mechanisms remain unclear. Using rat pulmonary microvascular endothelial cells (RPMEC) in culture, we report herein changes in biophysical properties, both in space and in time, that occur in response to hypoxia. We address also the molecular basis of these changes. At the level of the single cell, we measured cell stiffness, the distribution of traction forces exerted by the cell on its substrate, and spontaneous nanoscale motions of microbeads tightly bound to the cytoskeleton (CSK). Hypoxia increased cell stiffness and traction forces by a mechanism that was dependent on the activation of Rho kinase. These changes were followed by p38-mediated decreases in spontaneous bead motions, indicating stabilization of local cellular-extracellular matrix (ECM) tethering interactions. Cells overexpressing phospho-mimicking small heat shock protein (HSP27-PM), a downstream effector of p38, exhibited decreases in spontaneous bead motions that correlated with increases in actin polymerization in these cells. Together, these findings suggest that hypoxia differentially regulates endothelial cell contraction and cellular-ECM adhesion.

Brownian motion and diffusion: from stochastic processes to chaos and beyond

One century after Einstein's work, Brownian motion still remains both a fundamental open issue and a continuous source of inspiration for many areas of natural sciences. We first present a discussion about stochastic and deterministic approaches proposed in the literature to model the Brownian motion and more general diffusive behaviors. Then, we focus on the problems concerning the determination of the microscopic nature of diffusion by means of data analysis. Finally, we discuss the general conditions required for the onset of large scale diffusive motion.

Role of heat shock protein 27 in cytoskeletal remodeling of the airway smooth muscle cell

Remodeling of the airway smooth muscle (ASM) cell has been proposed to play an important role in airway hyperresponsiveness. Using a functional assay, we have assessed remodeling of the cultured rat ASM cell and the role of heat shock protein (HSP) 27 in that process. To probe remodeling dynamics, we measured spontaneous motions of an individual Arg-Gly-Asp-coated microbead that was anchored to the cytoskeleton. We reasoned that the bead could not move unless the microstructure to which it is attached rearranged; if so, then its mean square displacement (MSD) would report ongoing internal reorganizations over time. Each bead displayed a random, superdiffusive motion; MSD increased with time as approximately t(1.7), whereas an exponent of unity would be expected for a simple passive diffusion. Increasing concentrations of cytochalasin-D or latrunculin-A caused marked increases in the MSD, whereas colchicine did not. Treatments with PDGF or IL-1beta, but not transforming growth factor-beta, caused decreases in the MSD, the extent of which rank-ordered with the relative potency of these agents in eliciting the phosphorylation of HSP27. The chemical stressors anisomycin and arsenite each increased the levels of HSP27 phosphorylation and, at the same time, decreased bead motions. In particular, arsenite prevented and even reversed the effects of cytochalasin-D on bead motions. Finally, ASM cells overexpressing phospho-mimicking human HSP27, but not wild-type or phosphorylation-deficient HSP27, exhibited decreases in bead motions that were comparable to the arsenite response. Taken together, these results show that phosphorylated HSP27 favors reduced bead motions that are probably due to stabilization of the actin cytoskeleton.

Relaxation of finite perturbations: beyond the fluctuation-response relation

We study the response of dynamical systems to finite amplitude perturbation. A generalized fluctuation-response relation is derived, which links the average relaxation toward equilibrium to the invariant measure of the system and points out the relevance of the amplitude of the initial perturbation. Numerical computations on systems with many characteristic times show the relevance of the above-mentioned relation in realistic cases.

Single-particle tracking: Brownian dynamics of viscoelastic materials

A unifying theoretical framework for analyzing stochastic data from single-particle tracking (SPT) in viscoelastic materials is presented. A generalization of the bead-spring model for linear polymers is developed from a molecular point of view and from the standpoint of phenomenological linear viscoelasticity. The hydrodynamic interaction in the former is identified as the dashpots in the latter. In elementary terms, the intimate correspondence between time-correlation of the fluctuation measurements and transient relaxation kinetics after perturbation is discussed, and the central role of the fluctuation-dissipation relation is emphasized. The work presented here provides a bridge between the microscopic and the macroscopic views of linear viscoelastic biological materials, and is applicable to membrane protein diffusion, linear DNA chain dynamics, and mechanics of intracellular cytoskeletal networks.