Self-propulsion and interactions of catalytic particles in a chemically active medium

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.

Perspective: a stirring role for metabolism in cells

Based on recent findings indicating that metabolism might be governed by a limit on the rate at which cells can dissipate Gibbs energy, in this Perspective, we propose a new mechanism of how metabolic activity could globally regulate biomolecular processes in a cell. Specifically, we postulate that Gibbs energy released in metabolic reactions is used to perform work, allowing enzymes to self-propel or to break free from supramolecular structures. This catalysis-induced enzyme movement will result in increased intracellular motion, which in turn can compromise biomolecular functions. Once the increased intracellular motion has a detrimental effect on regulatory mechanisms, this will establish a feedback mechanism on metabolic activity, and result in the observed thermodynamic limit. While this proposed explanation for the identified upper rate limit on cellular Gibbs energy dissipation rate awaits experimental validation, it offers an intriguing perspective of how metabolic activity can globally affect biomolecular functions and will hopefully spark new research.

Interactions in active colloids

The past two decades have seen a remarkable progress in the development of synthetic colloidal agents which are capable of creating directed motion in an unbiased environment at the microscale. These self-propelling particles are often praised for their enormous potential to self-organize into dynamic nonequilibrium structures such as living clusters, synchronized super-rotor structures or self-propelling molecules featuring a complexity which is rarely found outside of the living world. However, the precise mechanisms underlying the formation and dynamics of many of these structures are still barely understood, which is likely to hinge on the gaps in our understanding of how active colloids interact. In particular, besides showing comparatively short-ranged interactions which are well known from passive colloids (Van der Waals, electrostatic etc), active colloids show novel hydrodynamic interactions as well as phoretic and substrate-mediated 'osmotic' cross-interactions which hinge on the action of the phoretic field gradients which are induced by the colloids on other colloids in the system. The present article discusses the complexity and the intriguing properties of these interactions which in general are long-ranged, non-instantaneous, non-pairwise and non-reciprocal and which may serve as key ingredients for the design of future nonequilibrium colloidal materials. Besides providing a brief overview on the state of the art of our understanding of these interactions a key aim of this review is to emphasize open key questions and corresponding open challenges.

Receptor-Ligand Rebinding Kinetics in Confinement

Rebinding kinetics of molecular ligands plays a key role in the operation of biomachinery, from regulatory networks to protein transcription, and is also a key factor in design of drugs and high-precision biosensors. In this study, we investigate initial release and rebinding of ligands to their binding sites grafted on a planar surface, a situation commonly observed in single-molecule experiments and that occurs in vivo, e.g., during exocytosis. Via scaling arguments and molecular dynamic simulations, we analyze the dependence of nonequilibrium rebinding kinetics on two intrinsic length scales: the average separation distance between the binding sites and the total diffusible volume (i.e., height of the experimental reservoir in which diffusion takes place or average distance between receptor-bearing surfaces). We obtain time-dependent scaling laws for on rates and for the cumulative number of rebinding events. For diffusion-limited binding, the (rebinding) on rate decreases with time via multiple power-law regimes before the terminal steady-state (constant on-rate) regime. At intermediate times, when particle density has not yet become uniform throughout the diffusible volume, the cumulative number of rebindings exhibits a novel, to our knowledge, plateau behavior because of the three-dimensional escape process of ligands from binding sites. The duration of the plateau regime depends on the average separation distance between binding sites. After the three-dimensional diffusive escape process, a one-dimensional diffusive regime describes on rates. In the reaction-limited scenario, ligands with higher affinity to their binding sites (e.g., longer residence times) delay entry to the power-law regimes. Our results will be useful for extracting hidden timescales in experiments such as kinetic rate measurements for ligand-receptor interactions in microchannels, as well as for cell signaling via diffusing molecules.

Rotational motion of a camphor disk in a circular region

In a two-dimensional axisymmetric system, the system symmetry allows rotational or oscillatory motion as stable stationary motion for a symmetric self-propelled particle. In the present paper, we studied the motion of a camphor disk confined in a two-dimensional circular region. By reducing the mathematical model describing the dynamics of the motion of a camphor disk and the concentration field of camphor molecules on a water surface, we analyzed the reduced equations around a bifurcation point where the rest state at the center of the system becomes unstable. As a result, we found that rotational motion is stably realized through the double-Hopf bifurcation from the rest state. The theoretical results were confirmed by numerical calculation and corresponded well to the experimental results.

Toward Understanding of Self-Electrophoretic Propulsion under Realistic Conditions: From Bulk Reactions to Confinement Effects

Active matter concerns itself with the study of particles that convert energy into work, typically motion of the particle itself. This field saw a surge of interest over the past decade, after the first micrometer-sized, man-made chemical motors were created. These particles served as a simple model system for studying in a well-controlled manner complex motion and cooperative behavior as known from biology. In addition, they have stimulated new efforts in understanding out-of-equilibrium statistical physics and started a revolution in microtechnology and robotics. Concentrated effort has gone into realizing these ambitions, and yet much remains unknown about the chemical motors themselves. The original designs for self-propelled particles relied on the conversion of the chemical energy of hydrogen peroxide into motion via catalytic decomposition taking place heterogeneously over the surface of the motor. This sets up gradients of chemical fields around the particle, which allow it to autophorese. That is, the interaction between the motor and the heterogeneously distributed solute species can drive fluid flow and the motor itself. There are two basic designs: the first relies on redox reactions taking place between the two sides of a bimetal, for example, a gold-platinum Janus sphere or nanorod. The second uses a catalytic layer of platinum inhomogeneously vapor-deposited onto a nonreactive particle. For convenience's sake, these can be referred to as redox motors and monometallic half-coated motors, respectively. To date, most researchers continue to rely on variations of these simple, yet elegant designs for their experiments. However, there is ongoing debate on the exact way chemical energy is transduced into motion in these motors. Many of the experimental observations on redox motors were successfully modeled via self-electrophoresis, while for half-coated motors there has been a strong focus on self-diffusiophoresis. Currently, there is mounting evidence that self-electrophoresis provides the dominant contribution to the observed speeds of half-coated motors, even if the vast majority of the reaction products are electroneutral. In this Account, we will summarize the most common electrophoretic propulsion model and discuss its strengths and weaknesses in relation to recent experiments. We will comment on the possible need to go beyond surface reactions and consider the entire medium as an "active fluid" that can create and annihilate charged species. This, together with confinement and collective effects, makes it difficult to gain a detailed understanding of these swimmers. The potentially dominant effect of confinement is highlighted on the basis of a recent study of an electro-osmotic pump that drives fluid along a substrate. Detailed analysis of this system allows for identification of the electro-osmotic driving mechanism, which is powered by micromolar salt concentrations. We will discuss how our latest numerical solver developments, based on the lattice Boltzmann method, should enable us to study collective behavior in systems comprised of these and other electrochemical motors in realistic environments. We conclude with an outlook on the future of modeling chemical motors that may facilitate the community's microtechnological ambitions.

Artificial Chemotaxis of Self-Phoretic Active Colloids: Collective Behavior

Microorganisms use chemotaxis, regulated by internal complex chemical pathways, to swim along chemical gradients to find better living conditions. Artificial microswimmers can mimic such a strategy by a pure physical process called diffusiophoresis, where they drift and orient along the gradient in a chemical density field. Similarly, for other forms of taxis in nature such as photo- or thermotaxis the phoretic counterpart exists. In this Account, we concentrate on the chemotaxis of self-phoretic active colloids. They are driven by self-electro- and diffusiophoresis at the particle surface and thereby acquire a swimming speed. During this process, they also produce nonuniform chemical fields in their surroundings through which they interact with other colloids by translational and rotational diffusiophoresis. In combination with active motion, this gives rise to effective phoretic attraction and repulsion and thereby to diverse emergent collective behavior. A particular appealing example is dynamic clustering in dilute suspensions first reported by a group from Lyon. A subtle balance of attraction and repulsion causes very dynamic clusters, which form and resolve again. This is in stark contrast to the relatively static clusters of motility-induced phase separation at larger densities. To treat chemotaxis in active colloids confined to a plane, we formulate two Langevin equations for position and orientation, which include translational and rotational diffusiophoretic drift velocities. The colloids are chemical sinks and develop their long-range chemical profiles instantaneously. For dense packings, we include screening of the chemical fields. We present a state diagram in the two diffusiophoretic parameters governing translational, as well as rotational, drift and, thereby, explore the full range of phoretic attraction and repulsion. The identified states range from a gaslike phase over dynamic clustering states 1 and 2, which we distinguish through their cluster size distributions, to different types of collapsed states. The latter include a full chemotactic collapse for translational phoretic attraction. Turning it into an effective repulsion, with increasing strength first the collapsed cluster starts to fluctuate at the rim, then oscillates, and ultimately becomes a static collapsed cloud. We also present a state diagram without screening. Finally, we summarize how the famous Keller-Segel model derives from our Langevin equations through a multipole expansion of the full one-particle distribution function in position and orientation. The Keller-Segel model gives a continuum equation for treating chemotaxis of microorganisms on the level of their spatial density. Our theory is extensible to mixtures of active and passive particles and allows to include a dipolar correction to the chemical field resulting from the dipolar symmetry of Janus colloids.

Synthetic Nanomotors: Working Together through Chemistry

Active matter, some of whose constituent elements are active agents that can move autonomously, behaves very differently from matter without such agents. The active agents can self-assemble into structures with a variety of forms and dynamical properties. Swarming, where groups of living agents move cooperatively, is commonly observed in the biological realm, but it is also seen in the physical realm in systems containing small synthetic motors. The existence of diverse forms of self-assembled structures has stimulated the search for new applications that involve active matter. We consider active systems where the agents are synthetic chemically powered motors with various shapes and sizes that operate by phoretic mechanisms, especially self-diffusiophoresis. These motors are able to move autonomously in solution by consuming fuel from their environment. Chemical reactions take place on catalytic portions of the motor surface and give rise to concentration gradients that lead to directed motion. They can operate in this way only if the chemical composition of the system is maintained in a nonequilibrium state since no net fluxes are possible in a system at equilibrium. In contrast to many other active systems, chemistry plays an essential part in determining the properties of the collective dynamics and self-assembly of these chemically powered motor systems. The inhomogeneous concentration fields that result from asymmetric motor reactions are felt by other motors in the system and strongly influence how they move. This chemical coupling effect often dominates other interactions due to fluid flow fields and direct interactions among motors and determines the form that the collective dynamics takes. Since we consider small motors with micrometer and nanometer sizes, thermal fluctuations are strong and cannot be neglected. The media in which the motors operate may not be simple and may contain crowding agents or molecular filaments that influence how the motors assemble and move. The collective motion is also influenced by the chemical gradients that arise from reactions in the surrounding medium. By adopting a microscopic perspective, where the motors, fluid environment, and crowding elements are treated at the coarse-grained molecular level, all of the many-body interactions that give rise to the collective behavior naturally emerge from the molecular dynamics. Through simulations and theory, this Account describes how active matter made from chemically powered nanomotors moving in simple and more complicated media can form different dynamical structures that are strongly influenced by interactions arising from cooperative chemical reactions on the motor surfaces.

Regulation of Pom cluster dynamics in Myxococcus xanthus

Precise positioning of the cell division site is essential for the correct segregation of the genetic material into the two daughter cells. In the bacterium Myxococcus xanthus, the proteins PomX and PomY form a cluster on the chromosome that performs a biased random walk to midcell and positively regulates cell division there. PomZ, an ATPase, is necessary for tethering of the cluster to the nucleoid and regulates its movement towards midcell. It has remained unclear how the cluster dynamics change when the biochemical parameters, such as the attachment rates of PomZ dimers to the nucleoid and the cluster, the ATP hydrolysis rate of PomZ or the mobility of PomZ interacting with the nucleoid and cluster, are varied. To answer these questions, we investigate a one-dimensional model that includes the nucleoid, the Pom cluster and PomZ proteins. We find that a mechanism based on the diffusive PomZ fluxes on the nucleoid into the cluster can explain the latter's midnucleoid localization for a broad parameter range. Furthermore, there is an ATP hydrolysis rate that minimizes the time the cluster needs to reach midnucleoid. If the dynamics of PomZ on the nucleoid is slow relative to the cluster's velocity, we observe oscillatory cluster movements around midnucleoid. To understand midnucleoid localization, we developed a semi-analytical approach that dissects the net movement of the cluster into its components: the difference in PomZ fluxes into the cluster from either side, the force exerted by a single PomZ dimer on the cluster and the effective friction coefficient of the cluster. Importantly, we predict that the Pom cluster oscillates around midnucleoid if the diffusivity of PomZ on the nucleoid is reduced. A similar approach to that applied here may also prove useful for cargo localization in ParABS systems.

Surfing on Protein Waves: Proteophoresis as a Mechanism for Bacterial Genome Partitioning

Efficient bacterial chromosome segregation typically requires the coordinated action of a three-component machinery, fueled by adenosine triphosphate, called the partition complex. We present a phenomenological model accounting for the dynamic activity of this system that is also relevant for the physics of catalytic particles in active environments. The model is obtained by coupling simple linear reaction-diffusion equations with a proteophoresis, or "volumetric" chemophoresis, force field that arises from protein-protein interactions and provides a physically viable mechanism for complex translocation. This minimal description captures most known experimental observations: dynamic oscillations of complex components, complex separation, and subsequent symmetrical positioning. The predictions of our model are in phenomenological agreement with and provide substantial insight into recent experiments. From a nonlinear physics view point, this system explores the active separation of matter at micrometric scales with a dynamical instability between static positioning and traveling wave regimes triggered by the dynamical spontaneous breaking of rotational symmetry.

Brownian Ratchet Mechanism for Faithful Segregation of Low-Copy-Number Plasmids

Bacterial plasmids are extrachromosomal DNA that provides selective advantages for bacterial survival. Plasmid partitioning can be remarkably robust. For high-copy-number plasmids, diffusion ensures that both daughter cells inherit plasmids after cell division. In contrast, most low-copy-number plasmids need to be actively partitioned by a conserved tripartite ParA-type system. ParA is an ATPase that binds to chromosomal DNA; ParB is the stimulator of the ParA ATPase and specifically binds to the plasmid at a centromere-like site, parS. ParB stimulation of the ParA ATPase releases ParA from the bacterial chromosome, after which it takes a long time to reset its DNA-binding affinity. We previously demonstrated in vitro that the ParA system can exploit this biochemical asymmetry for directed cargo transport. Multiple ParA-ParB bonds can bridge a parS-coated cargo to a DNA carpet, and they can work collectively as a Brownian ratchet that directs persistent cargo movement with a ParA-depletion zone trailing behind. By extending this model, we suggest that a similar Brownian ratchet mechanism recapitulates the full range of actively segregated plasmid motilities observed in vivo. We demonstrate that plasmid motility is tuned as the replenishment rate of the ParA-depletion zone progressively increases relative to the cargo speed, evolving from diffusion to pole-to-pole oscillation, local excursions, and, finally, immobility. When the plasmid replicates, the daughters largely display motilities similar to that of their mother, except that when the single-focus progenitor is locally excursive, the daughter foci undergo directed segregation. We show that directed segregation maximizes the fidelity of plasmid partition. Given that local excursion and directed segregation are the most commonly observed modes of plasmid motility in vivo, we suggest that the operation of the ParA-type partition system has been shaped by evolution for high fidelity of plasmid segregation.

Ionic screening and dissociation are crucial for understanding chemical self-propulsion in polar solvents

Polar solvents like water support the bulk dissociation of themselves and their solutes into ions, and the re-association of these ions into neutral molecules in a dynamic equilibrium, e.g., H₂O₂ ⇌ H⁺ + HO₂^-. Using continuum theory, we study the influence of these association-dissociation reactions on the self-propulsion of colloids driven by surface chemical reactions (chemical swimmers). We find that association-dissociation reactions should have a strong influence on swimmers' behaviour, and therefore should be included in future modelling. In particular, such bulk reactions should permit charged swimmers to propel electrophoretically even if all species involved in the surface reactions are neutral. The bulk reactions also significantly modify the predicted speed of chemical swimmers propelled by ionic currents, by up to an order of magnitude. For swimmers whose surface reactions produce both anions and cations (ionic self-diffusiophoresis), the bulk reactions produce an additional reactive screening length, analogous to the Debye length in electrostatics. This in turn leads to an inverse relationship between swimmer radius and swimming speed, which could provide an alternative explanation for recent experimental observations on Pt-polystyrene Janus swimmers [S. Ebbens et al., Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 85, 020401]. We also use our continuum theory to investigate the effect of the Debye screening length itself, going beyond the infinitely-thin-screening-length approximation used by previous analytical theories. We identify significant departures from this limiting behaviour for micron-sized swimmers under typical experimental conditions and find that the approximation fails entirely for nanoscale swimmers.

‘Fuelled’ motion: phoretic motility and collective behaviour of active colloids

Phoretic motility is a propulsion mechanism used to design active particles which display complex collective behaviours, characterised experimentally and theoretically.