From Particle Currents to Tracer Diffusion: Universal Correlation Profiles in Single-File Dynamics.

Single-file transport refers to the motion of particles in a narrow channel, such that they cannot bypass each other. This constraint leads to strong correlations between the particles, described by correlation profiles, which measure the correlation between a generic observable and the density of particles at a given position and time. They have recently been shown to play a central role in single-file systems. Up to now, these correlations have only been determined for diffusive systems in the hydrodynamic limit. Here, we consider a model of reflecting point particles on the infinite line, with a general individual stochastic dynamics. We show that the correlation profiles take a simple universal form, at arbitrary time. We illustrate our approach by the study of the integrated current of particles through the origin, and apply our results to representative models such as Brownian particles, run-and-tumble particles and Lévy flights. We further emphasise the generality of our results by showing that they also apply beyond the 1D case, and to other observables.

Chemotactic particles as strong electrolytes: Debye-Hückel approximation and effective mobility law.

We consider a binary mixture of chemically active particles that produce or consume solute molecules and that interact with each other through the long-range concentration fields they generate. We analytically calculate the effective phoretic mobility of these particles when the mixture is submitted to a constant, external concentration gradient, at leading order in the overall concentration. Relying on an analogy with the modeling of strong electrolytes, we show that the effective phoretic mobility decays with the square root of the concentration: our result is, therefore, a nonequilibrium counterpart to the celebrated Kohlrausch and Debye-Hückel-Onsager conductivity laws for electrolytes, which are extended here to particles with long-range nonreciprocal interactions. The effective mobility law we derive reveals the existence of a regime of maximal mobility and could find applications in the description of nanoscale transport phenomena in living cells.

Absolute Negative Mobility of an Active Tracer in a Crowded Environment.

Absolute negative mobility (ANM) refers to the situation where the average velocity of a driven tracer is opposite to the direction of the driving force. This effect was evidenced in different models of nonequilibrium transport in complex environments, whose description remains effective. Here, we provide a microscopic theory for this phenomenon. We show that it emerges in the model of an active tracer particle submitted to an external force and which evolves on a discrete lattice populated with mobile passive crowders. Resorting to a decoupling approximation, we compute analytically the velocity of the tracer particle as a function of the different parameters of the system and confront our results to numerical simulations. We determine the range of parameters where ANM can be observed, characterize the response of the environment to the displacement of the tracer, and clarify the mechanism underlying ANM and its relationship with negative differential mobility (another hallmark of driven systems far from the linear response).

Driven Tracer in the Symmetric Exclusion Process: Linear Response and Beyond.

Tracer dynamics in the symmetric exclusion process (SEP), where hard-core particles diffuse on an infinite one-dimensional lattice, is a paradigmatic model of anomalous diffusion. While the equilibrium situation has received a lot of attention, the case where the tracer is driven by an external force, which provides a minimal model of nonequilibrium transport in confined crowded environments, remains largely unexplored. Indeed, the only available analytical results concern the means of both the position of the tracer and the lattice occupation numbers in its frame of reference and higher-order moments but only in the high-density limit. Here, we provide a general hydrodynamic framework that allows us to determine the first cumulants of the bath-tracer correlations and of the tracer's position in function of the driving force, up to quadratic order (beyond linear response). This result constitutes the first determination of the bias dependence of the variance of a driven tracer in the SEP for an arbitrary density. The framework presented here can be applied, beyond the SEP, to more general configurations of a driven tracer in interaction with obstacles in one dimension.

Isotropic active colloids: explicit vs. implicit descriptions of propulsion mechanisms.

Modeling the couplings between active particles often neglects the possible many-body effects that control the propulsion mechanism. Accounting for such effects requires the explicit modeling of the molecular details at the origin of activity. Here, we take advantage of a recent two-dimensional model of isotropic active particles whose propulsion originates from the interactions between solute particles in the bath. The colloid catalyzes a chemical reaction in its vicinity, which results in a local phase separation of solute particles, and the density fluctuations of solute particles cause the enhanced diffusion of the colloid. In this paper, we investigate an assembly of such active particles, using (i) an explicit model, where the microscopic dynamics of the solute particles is accounted for; and (ii) an implicit model, whose parameters are inferred from the explicit model at infinite dilution. In the explicit solute model, the long-time diffusion coefficient of the active colloids strongly decreases with density, an effect which is not captured by the derived implicit model. This suggests that classical models, which usually decouple pair interactions from activity, fail to describe collective dynamics in active colloidal systems driven by solute-solute interactions.

On analytical theories for conductivity and self-diffusion in concentrated electrolytes.

Describing analytically the transport properties of electrolytes, such as their conductivity or the self-diffusion of the ions, has been a central challenge of chemical physics for almost a century. In recent years, this question has regained some interest in light of Stochastic Density Field Theory (SDFT) - an analytical framework that allows the approximate determination of density correlations in fluctuating systems. In spite of the success of this theory to describe dilute electrolytes, its extension to concentrated solutions raises a number of technical difficulties, and requires simplified descriptions of the short-range repulsion between the ions. In this article, we discuss recent approximations that were proposed to compute the conductivity of electrolytes, in particular truncations of Coulomb interactions at short distances. We extend them to another observable (the self-diffusion coefficient of the ions) and compare them to earlier analytical approaches, such as the mean spherical approximation and mode-coupling theory. We show how the treatment of hydrodynamic effects in SDFT can be improved, that the choice of the modified Coulomb interactions significantly affects the determination of the properties of the electrolytes, and that comparison with other theories provides a guide to extend SDFT approaches in this context.

Cooperatively enhanced reactivity and "stabilitaxis" of dissociating oligomeric proteins.

Many functional units in biology, such as enzymes or molecular motors, are composed of several subunits that can reversibly assemble and disassemble. This includes oligomeric proteins composed of several smaller monomers, as well as protein complexes assembled from a few proteins. By studying the generic spatial transport properties of such proteins, we investigate here whether their ability to reversibly associate and dissociate may confer on them a functional advantage with respect to nondissociating proteins. In uniform environments with position-independent association-dissociation, we find that enhanced diffusion in the monomeric state coupled to reassociation into the functional oligomeric form leads to enhanced reactivity with localized targets. In nonuniform environments with position-dependent association-dissociation, caused by, for example, spatial gradients of an inhibiting chemical, we find that dissociating proteins generically tend to accumulate in regions where they are most stable, a process that we term "stabilitaxis."

Microscopic Theory for the Diffusion of an Active Particle in a Crowded Environment.

We calculate the diffusion coefficient of an active tracer in a schematic crowded environment, represented as a lattice gas of passive particles with hardcore interactions. Starting from the master equation of the problem, we put forward a closure approximation that goes beyond trivial mean field and provides the diffusion coefficient for an arbitrary density of crowders in the system. We show that our approximation is accurate for a very wide range of parameters, and that it correctly captures numerous nonequilibrium effects, which are the signature of the activity in the system. In addition to the determination of the diffusion coefficient of the tracer, our approach allows us to characterize the perturbation of the environment induced by the displacement of the active tracer. Finally, we consider the asymptotic regimes of low and high densities, in which the expression of the diffusion coefficient of the tracer becomes explicit, and which we argue to be exact.

Conditions for the propulsion of a colloid surrounded by a mesoscale phase separation.

We study a two-dimensional model of an active isotropic colloid whose propulsion is linked to the interactions between solute particles of the bath. The colloid catalyzes a chemical reaction in its vicinity, that yields a local phase separation of solute particles. The density fluctuations of solute particles result in the enhanced diffusion of the colloid. Using numerical simulations, we thoroughly investigate the conditions under which activity occurs, and we establish a state diagram for the activity of the colloid as a function of the parameters of the model. We use the generated data to unravel a key observable that controls the existence and the intensity of activity: The filling fraction of the reaction area. Remarkably, we finally show that propulsion also occurs in three-dimensional geometries, which confirms the interest of this mechanism for experimental applications.

Generalized Correlation Profiles in Single-File Systems.

Single-file diffusion refers to the motion in narrow channels of particles which cannot bypass each other, and leads to tracer subdiffusion. Most approaches to this celebrated many-body problem were restricted to the description of the tracer only. Here, we go beyond this standard description by introducing and providing analytical results for generalized correlation profiles (GCPs) in the frame of the tracer. In addition to controlling the statistical properties of the tracer, these quantities fully characterize the correlations between the tracer position and the bath particles density. Considering the hydrodynamic limit of the problem, we determine the scaling form of the GCPs with space and time, and unveil a nonmonotonic dependence with the distance to the tracer despite the absence of any asymmetry. Our analytical approach provides several exact results for the GCPs for paradigmatic models of single-file diffusion, such as Brownian particles with hardcore repulsion, the symmetric exclusion process and the random average process. The range of applicability of our approach is further illustrated by considering (i) extensions to general interactions between particles, (ii) the out-of-equilibrium situation of an initial step of density, and (iii) beyond the hydrodynamic limit, the GCPs at arbitrary time in the dense limit.

Synchronization and Enhanced Catalysis of Mechanically Coupled Enzymes.

We examine the stochastic dynamics of two enzymes that are mechanically coupled to each other, e.g., through an elastic substrate or a fluid medium. The enzymes undergo conformational changes during their catalytic cycle, which itself is driven by stochastic steps along a biased chemical free energy landscape. We find conditions under which the enzymes can synchronize their catalytic steps, and discover that the coupling can lead to a significant enhancement in their overall catalytic rate. Both effects can be understood as arising from a global bifurcation in the underlying dynamical system at sufficiently strong coupling. Our findings suggest that, despite their molecular scale, enzymes can be cooperative and improve their performance in metabolic clusters.

Enhanced Diffusion and Chemotaxis at the Nanoscale.

Enzymes have been recently proposed to have mechanical activity associated with their chemical activity. In a number of recent studies, it has been reported that enzymes undergo enhanced diffusion in the presence of their corresponding substrate when this substrate is uniformly distributed in solution. Moreover, if the concentration of the substrate is nonuniform, enzymes and other small molecules have been reported to show chemotaxis (biased stochastic movement in the direction of the substrate gradient), typically toward higher concentrations of this substrate, with a few exceptions. The underlying physical mechanisms responsible for enhanced diffusion and chemotaxis at the nanoscale, however, are still not well understood. Understanding these processes is important both for fundamental biological research, for example, in the context of spatial organization of enzymes in metabolic pathways (metabolon formation), as well as for engineering applications, such as in the design of new vehicles for targeted drug delivery. In this Account, we will review the available experimental observations of both enhanced diffusion and chemotaxis, and we will discuss critically the different theories that have been proposed to explain the two. We first focus on enhanced diffusion, beginning with an overview of the experimental results. We then discuss the two main types of mechanisms that have been proposed, namely, active mechanisms relying on the catalytic step of the enzymatic reaction and equilibrium mechanisms, which consider the reversible binding and unbinding of the substrate to the enzyme. We put particular emphasis on an equilibrium model recently introduced by us, which describes how the diffusion of dumbbell-like modular enzymes can be enhanced in the presence of substrate thanks to a binding-induced reduction of the internal fluctuations of the enzyme. We then turn to chemotaxis, beginning with an overview of the experimental evidence for the chemotaxis of enzymes and small molecules, followed by a description of a number of shortcomings and pitfalls in the thermodynamic and phenomenological models for chemotaxis introduced in those and other works in the literature. We then discuss a microscopic model for chemotaxis including both noncontact interactions and specific binding between enzyme and substrate recently developed by us, which overcomes many of these shortcomings and is consistent with the experimental observations of chemotaxis. Finally, we show that the results of this model may be used to engineer chemically active macromolecules that are directed in space via patterning of the concentrations of their substrates.

Fluctuation-induced hydrodynamic coupling in an asymmetric, anisotropic dumbbell.

We recently introduced a model of an asymmetric dumbbell made of two hydrodynamically coupled subunits as a minimal model for a macromolecular complex, in order to explain the observation of enhanced diffusion of catalytically active enzymes. It was shown that internal fluctuations lead to a negative contribution to the overall diffusion coefficient and that the fluctuation-induced contribution is controlled by the strength of the interactions between the subunits and their asymmetry. We develop the model by studying the effect of anisotropy on the diffusion properties of a modular structure. Using a moment expansion method we derive an analytic form for the long-time diffusion coefficient of an asymmetric, anisotropic dumbbell and show systematically its dependence on internal and external symmetry. The method provides a tractable, analytical route for studying the stochastic dynamics of dumbbell models. The present work opens the way to more detailed descriptions of the effect of hydrodynamic interactions on the diffusion and transport properties of biomolecules with complex structures.

Phoresis and Enhanced Diffusion Compete in Enzyme Chemotaxis.

Chemotaxis of enzymes in response to gradients in the concentration of their substrate has been widely reported in recent experiments, but a basic understanding of the process is still lacking. Here, we develop a microscopic theory for chemotaxis that is valid for enzymes and other small molecules. Our theory includes both nonspecific interactions between enzyme and substrate as well as complex formation through specific binding between the enzyme and the substrate. We find that two distinct mechanisms contribute to enzyme chemotaxis: a diffusiophoretic mechanism due to the nonspecific interactions and a new type of mechanism due to binding-induced changes in the diffusion coefficient of the enzyme. The latter chemotactic mechanism points toward lower substrate concentration if the substrate enhances enzyme diffusion and toward higher substrate concentration if the substrate inhibits enzyme diffusion. For a typical enzyme, attractive phoresis and binding-induced enhanced diffusion will compete against each other. We find that phoresis dominates above a critical substrate concentration, whereas binding-induced enhanced diffusion dominates for low substrate concentration. Our results resolve an apparent contradiction regarding the direction of urease chemotaxis observed in experiments and, in general, clarify the relation between the enhanced diffusion and the chemotaxis of enzymes. Finally, we show that the competition between the two distinct chemotactic mechanisms may be used to engineer nanomachines that move toward or away from regions with a specific substrate concentration.

Nonequilibrium Fluctuations and Enhanced Diffusion of a Driven Particle in a Dense Environment.

We study the diffusion of a tracer particle driven out of equilibrium by an external force and traveling in a dense environment of arbitrary density. The system evolves on a discrete lattice and its stochastic dynamics is described by a master equation. Relying on a decoupling approximation that goes beyond the naive mean-field treatment of the problem, we calculate the fluctuations of the position of the tracer around its mean value on a lattice of arbitrary dimension, and with different boundary conditions. We reveal intrinsically nonequilibrium effects, such as enhanced diffusivity of the tracer induced by both the crowding interactions and the external driving. We finally consider the high-density and low-density limits of the model and show that our approximation scheme becomes exact in these limits.

'Fuelled' motion: phoretic motility and collective behaviour of active colloids.

Designing microscopic and nanoscopic self-propelled particles and characterising their motion have become a major scientific challenge over the past few decades. To this purpose, phoretic effects, namely propulsion mechanisms relying on local field gradients, have been the focus of many theoretical and experimental studies. In this review, we adopt a tutorial approach to present the basic physical mechanisms at stake in phoretic motion, and describe the different experimental works that led to the fabrication of active particles based on this principle. We also present the collective effects observed in assemblies of interacting active colloids, and the theoretical tools that have been used to describe phoretic and hydrodynamic interactions.

Exothermicity Is Not a Necessary Condition for Enhanced Diffusion of Enzymes.

Recent experiments have revealed that the diffusivity of exothermic and fast enzymes is enhanced when they are catalytically active, and different physical mechanisms have been explored and quantified to account for this observation. We perform measurements on the endothermic and relatively slow enzyme aldolase, which also shows substrate-induced enhanced diffusion. We propose a new physical paradigm, which reveals that the diffusion coefficient of a model enzyme hydrodynamically coupled to its environment increases significantly when undergoing changes in conformational fluctuations in a substrate concentration dependent manner, and is independent of the overall turnover rate of the underlying enzymatic reaction. Our results show that substrate-induced enhanced diffusion of enzyme molecules can be explained within an equilibrium picture and that the exothermicity of the catalyzed reaction is not a necessary condition for the observation of this phenomenon.

Velocity anomaly of a driven tracer in a confined crowded environment.

We consider a discrete model in which a tracer performs a random walk biased by an external force, in a dense bath of particles performing symmetric random walks constrained by hard-core interactions. We reveal the emergence of a striking velocity anomaly in confined geometries: in quasi-1D systems such as stripes or capillaries, the velocity of the tracer displays a long-lived plateau before ultimately dropping to a lower value. We develop an analytical solution that quantitatively accounts for this intriguing behavior. Our analysis suggests that such a velocity anomaly could be a generic feature of driven dynamics in quasi-1D crowded systems.

Geometry-induced superdiffusion in driven crowded systems.

Recent molecular dynamics simulations of glass-forming liquids revealed superdiffusive fluctuations associated with the position of a tracer particle (TP) driven by an external force. Such an anomalous response, whose mechanism remains elusive, has been observed up to now only in systems close to their glass transition, suggesting that this could be one of its hallmarks. Here, we show that the presence of superdiffusion is in actual fact much more general, provided that the system is crowded and geometrically confined. We present and solve analytically a minimal model consisting of a driven TP in a dense, crowded medium in which the motion of particles is mediated by the diffusion of packing defects, called vacancies. For such nonglass-forming systems, our analysis predicts a long-lived superdiffusion which ultimately crosses over to giant diffusive behavior. We find that this trait is present in confined geometries, for example long capillaries and stripes, and emerges as a universal response of crowded environments to an external force. These findings are confirmed by numerical simulations of systems as varied as lattice gases, dense liquids, and granular fluids.