Accurate Reduced Models for the pH Oscillations in the Urea-Urease Reaction Confined to Giant Lipid Vesicles.

This theoretical study concerns a pH oscillator based on the urea-urease reaction confined to giant lipid vesicles. Under suitable conditions, differential transport of urea and hydrogen ion across the unilamellar vesicle membrane periodically resets the pH clock that switches the system from acid to basic, resulting in self-sustained oscillations. We analyze the structure of the phase flow and of the limit cycle, which controls the dynamics for giant vesicles and dominates the pronouncedly stochastic oscillations in small vesicles of submicrometer size. To this end, we derive reduced models, which are amenable to analytic treatments that are complemented by numerical solutions, and obtain the period and amplitude of the oscillations as well as the parameter domain, where oscillatory behavior persists. We show that the accuracy of these predictions is highly sensitive to the employed reduction scheme. In particular, we suggest an accurate two-variable model and show its equivalence to a three-variable model that admits an interpretation in terms of a chemical reaction network. The faithful modeling of a single pH oscillator appears crucial for rationalizing experiments and understanding communication of vesicles and synchronization of rhythms.

Thermally active nanoparticle clusters enslaved by engineered domain wall traps.

The stable assembly of fluctuating nanoparticle clusters on a surface represents a technological challenge of widespread interest for both fundamental and applied research. Here we demonstrate a technique to stably confine in two dimensions clusters of interacting nanoparticles via size-tunable, virtual magnetic traps. We use cylindrical Bloch walls arranged to form a triangular lattice of ferromagnetic domains within an epitaxially grown ferrite garnet film. At each domain, the magnetic stray field generates an effective harmonic potential with a field tunable stiffness. The experiments are combined with theory to show that the magnetic confinement is effectively harmonic and pairwise interactions are of dipolar nature, leading to central, strictly repulsive forces. For clusters of magnetic nanoparticles, the stationary collective states arise from the competition between repulsion, confinement and the tendency to fill the central potential well. Using a numerical simulation model as a quantitative map between the experiments and theory we explore the field-induced crystallization process for larger clusters and unveil the existence of three different dynamical regimes. The present method provides a model platform for investigations of the collective phenomena emerging when strongly confined nanoparticle clusters are forced to move in an idealized, harmonic-like potential.

Stochastic pH Oscillations in a Model of the Urea-Urease Reaction Confined to Lipid Vesicles.

The urea-urease clock reaction is a pH switch from acid to basic that can turn into a pH oscillator if it occurs inside a suitable open reactor. We numerically study the confinement of the reaction to lipid vesicles, which permit the exchange with an external reservoir by differential transport, enabling the recovery of the pH level and yielding a constant supply of urea molecules. For microscopically small vesicles, the discreteness of the number of molecules requires a stochastic treatment of the reaction dynamics. Our analysis shows that intrinsic noise induces a significant statistical variation of the oscillation period, which increases as the vesicles become smaller. The mean period, however, is found to be remarkably robust for vesicle sizes down to approximately 200 nm, but the periodicity of the rhythm is gradually destroyed for smaller vesicles. The observed oscillations are explained as a canard-like limit cycle that differs from the wide class of conventional feedback oscillators.

Collective Directional Locking of Colloidal Monolayers on a Periodic Substrate.

We investigate the directional locking effects that arise when a monolayer of paramagnetic colloidal particles is driven across a triangular lattice of magnetic bubbles. We use an external rotating magnetic field to generate a two-dimensional traveling wave ratchet forcing the transport of particles along a direction that intersects two crystallographic axes of the lattice. We find that, while single particles show no preferred direction, collective effects induce transversal current and directional locking at high density via a spontaneous symmetry breaking. The colloidal current may be polarized via an additional bias field that makes one transport direction energetically preferred.

Enhancing Nanoparticle Diffusion on a Unidirectional Domain Wall Magnetic Ratchet.

The performance of nanoscale magnetic devices is often limited by the presence of thermal fluctuations, whereas in micro- and nanofluidic applications the same fluctuations may be used to spread reactants or drugs. Here, we demonstrate the controlled motion and the enhancement of diffusion of magnetic nanoparticles that are manipulated and driven across a series of Bloch walls within an epitaxially grown ferrite garnet film. We use a rotating magnetic field to generate a traveling wave potential that unidirectionally transports the nanoparticles at a frequency tunable speed. Strikingly, we find an enhancement of diffusion along the propulsion direction and a frequency-dependent diffusion coefficient that can be precisely controlled by varying the system parameters. To explain the reported phenomena, we develop a theoretical approach that shows a fair agreement with the experimental data enabling an exact analytical expression for the enhanced diffusivity above the magnetically modulated periodic landscape. Our technique to control thermal fluctuations of driven magnetic nanoparticles represents a versatile and powerful way to programmably transport magnetic colloidal matter in a fluid, opening the doors to different fluidic applications based on exploiting magnetic domain wall ratchets.

Inhomogeneous assembly of driven nematic colloids.

We present a quantitative analysis of the nonequilibrium assembly of colloidal particles dispersed in a nematic liquid crystal. The driven particles assemble into reconfigurable circular clusters by liquid-crystal-enabled electrokinetic phenomena generated by an AC electric field that provides propulsion along the local director. We identify the coexistence of different aggregation states, including a central, jammed core, where short-range elastic attraction dominates, surrounded by a liquid-like corona where particles retain their mobility but reach a mechanical equilibrium that we rationalize in terms of a balance between centripetal phoretic drive and pairwise repulsion. An analysis of the compressible liquid-like region reveals a linear density profile that can be tuned with the field frequency, and a bond-orientational order that reaches a maximum at intermediate packing densities, where elastic effects are minimized. Since the phoretic propulsion force acts also on assembled particles, we compute the mechanical pressure and show that a hard-disk equation of state can be used to describe the assembly of this driven system.

Bidirectional particle transport and size selective sorting of Brownian particles in a flashing spatially periodic energy landscape.

We demonstrate a size sensitive experimental scheme which enables bidirectional transport and fractionation of paramagnetic colloids in a fluid medium. It is shown that two types of magnetic colloidal particles with different sizes can be simultaneously transported in opposite directions, when deposited above a stripe-patterned ferrite garnet film subjected to a square-wave magnetic modulation. Due to their different sizes, the particles are located at distinct elevations above the surface, and they experience two different energy landscapes, generated by the modulated magnetic substrate. By combining theoretical arguments and numerical simulations, we reveal such energy landscapes, which fully explain the bidirectional transport mechanism. The proposed technique does not require pre-imposed channel geometries such as in conventional microfluidics or lab-on-a-chip systems, and permits remote control over the particle motion, speed and trajectory, by using relatively low intense magnetic fields.

Transport and selective chaining of bidisperse particles in a travelling wave potential.

We combine experiments, theory and numerical simulation to investigate the dynamics of a binary suspension of paramagnetic colloidal particles dispersed in water and transported above a stripe-patterned magnetic garnet film. The substrate generates a one-dimensional periodic energy landscape above its surface. The application of an elliptically polarized rotating magnetic field causes the landscape to translate, inducing direct transport of paramagnetic particles placed above the film. The ellipticity of the applied field can be used to control and tune the interparticle interactions, from net repulsive to net attractive. When considering particles of two distinct sizes, we find that, depending on their elevation above the surface of the magnetic substrate, the particles feel effectively different potentials, resulting in different mobilities. We exploit this feature to induce selective chaining for certain values of the applied field parameters. In particular, when driving two types of particles, we force only one type to condense into travelling parallel chains. These chains confine the movement of the other non-chaining particles within narrow colloidal channels. This phenomenon is explained by considering the balance of pairwise magnetic forces between the particles and their individual coupling with the travelling landscape.

Regulating wave front dynamics from the strongly discrete to the continuum limit in magnetically driven colloidal systems.

The emergence of wave fronts in dissipative driven systems is a fascinating phenomenon which can be found in a broad range of physical and biological disciplines. Here we report the direct experimental observation of discrete fronts propagating along chains of paramagnetic colloidal particles, the latter propelled above a traveling wave potential generated by a structured magnetic substrate. We develop a rigorously reduced theoretical framework and describe the dynamics of the system in terms of a generalized one-dimensional dissipative Frenkel-Kontorova model. The front dynamics is explored in a wide range of field parameters close to and far from depinning, where the discrete and continuum limits apply. We show how symmetry breaking and finite size of chains are used to control the direction of front propagation, a universal feature relevant to different systems and important for real applications.

Colloidal particles driven across periodic optical-potential-energy landscapes.

We study the motion of colloidal particles driven by a constant force over a periodic optical potential energy landscape. First, the average particle velocity is found as a function of the driving velocity and the wavelength of the optical potential energy landscape. The relationship between average particle velocity and driving velocity is found to be well described by a theoretical model treating the landscape as sinusoidal, but only at small trap spacings. At larger trap spacings, a nonsinusoidal model for the landscape must be used. Subsequently, the critical velocity required for a particle to move across the landscape is determined as a function of the wavelength of the landscape. Finally, the velocity of a particle driven at a velocity far exceeding the critical driving velocity is examined. Both of these results are again well described by the two theoretical routes for small and large trap spacings, respectively. Brownian motion is found to have a significant effect on the critical driving velocity but a negligible effect when the driving velocity is high.

Microscopic dynamics of synchronization in driven colloids.

Synchronization of coupled oscillators has been scrutinized for over three centuries, from Huygens' pendulum clocks to physiological rhythms. One such synchronization phenomenon, dynamic mode locking, occurs when naturally oscillating processes are driven by an externally imposed modulation. Typically only averaged or integrated properties are accessible, leaving underlying mechanisms unseen. Here, we visualize the microscopic dynamics underlying mode locking in a colloidal model system, by using particle trajectories to produce phase portraits. Furthermore, we use this approach to examine the enhancement of mode locking in a flexible chain of magnetically coupled particles, which we ascribe to breathing modes caused by mode-locked density waves. Finally, we demonstrate that an emergent density wave in a static colloidal chain mode locks as a quasi-particle, with microscopic dynamics analogous to those seen for a single particle. Our results indicate that understanding the intricate link between emergent behaviour and microscopic dynamics is key to controlling synchronization.

Functional colloidal micro-sieves assembled and guided above a channel-free magnetic striped film.

Colloidal inclusions in lab-on-a-chip devices can be used to perform analytic operations in a non-invasive fashion. We demonstrate here a novel approach to realize fast and reversible micro-sieving operations by manipulating and transporting colloidal chains via mobile domain walls in a magnetic structured substrate. We show that this technique allows one to precisely move and sieve non-magnetic particles, to tweeze microscopic cargos or to mechanically compress highly dense colloidal monolayers.

Tunable interactions between paramagnetic colloidal particles driven in a modulated ratchet potential.

We study experimentally and theoretically the interactions between paramagnetic particles dispersed in water and driven above the surface of a stripe patterned magnetic garnet film. An external rotating magnetic field modulates the stray field of the garnet film and generates a translating potential landscape which induces directed particle motion. By varying the ellipticity of the rotating field, we tune the inter-particle interactions from net repulsive to net attractive. For attractive interactions, we show that pairs of particles can approach each other and form stable doublets which afterwards travel along the modulated landscape at a constant mean speed. We measure the strength of the attractive force between the moving particles and propose an analytically tractable model that explains the observations and is in quantitative agreement with experiment.

Zigzag transitions and nonequilibrium pattern formation in colloidal chains.

Paramagnetic colloidal particles that are optically trapped in a linear array can form a zigzag pattern when an external magnetic field induces repulsive interparticle interactions. When the traps are abruptly turned off, the particles form a nonequilibrium expanding pattern with a zigzag symmetry, even when the strength of the magnetic interaction is weaker than that required to break the linear symmetry of the equilibrium state. We show that the transition to the equilibrium zigzag state is always potentially possible for purely harmonic traps. For anharmonic traps that have a finite height, the equilibrium zigzag state becomes unstable above a critical anharmonicity. A normal mode analysis of the equilibrium line configuration demonstrates that increasing the magnetic field leads to a hardening and softening of the spring constants in the longitudinal and transverse directions, respectively. The mode that first becomes unstable is the mode with the zigzag symmetry, which explains the symmetry of nonequilibrium patterns. Our analytically tractable models help to give further insight into the way that the interplay of factors such as the length of the chain, hydrodynamic interactions, thermal fluctuations affects the formation and evolution of the experimentally observed nonequilibrium patterns.

Signatures of nonlinearity in single cell noise-induced oscillations.

A class of theoretical models seeks to explain rhythmic single cell data by postulating that they are generated by intrinsic noise in biochemical systems whose deterministic models exhibit only damped oscillations. The main features of such noise-induced oscillations are quantified by the power spectrum which measures the dependence of the oscillatory signal's power with frequency. In this paper we derive an approximate closed-form expression for the power spectrum of any monostable biochemical system close to a Hopf bifurcation, where noise-induced oscillations are most pronounced. Unlike the commonly used linear noise approximation which is valid in the macroscopic limit of large volumes, our theory is valid over a wide range of volumes and hence affords a more suitable description of single cell noise-induced oscillations. Our theory predicts that the spectra have three universal features: (i) a dominant peak at some frequency, (ii) a smaller peak at twice the frequency of the dominant peak and (iii) a peak at zero frequency. Of these, the linear noise approximation predicts only the first feature while the remaining two stem from the combination of intrinsic noise and nonlinearity in the law of mass action. The theoretical expressions are shown to accurately match the power spectra determined from stochastic simulations of mitotic and circadian oscillators. Furthermore it is shown how recently acquired single cell rhythmic fibroblast data displays all the features predicted by our theory and that the experimental spectrum is well described by our theory but not by the conventional linear noise approximation.

Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators.

The linear noise approximation (LNA) offers a simple means by which one can study intrinsic noise in monostable biochemical networks. Using simple physical arguments, we have recently introduced the slow-scale LNA (ssLNA), which is a reduced version of the LNA under conditions of timescale separation. In this paper we present the first rigorous derivation of the ssLNA using the projection operator technique and show that the ssLNA follows uniquely from the standard LNA under the same conditions of timescale separation as those required for the deterministic quasi-steady-state approximation. We also show that the large molecule number limit of several common stochastic model reduction techniques under timescale separation conditions constitutes a special case of the ssLNA.

The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions.

BACKGROUND: It is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effective reactions. Each of the latter represents a group of elementary reactions in the large network and has associated with it an effective macroscopic rate law. A popular method to achieve model reduction in the presence of intrinsic noise consists of using the effective macroscopic rate laws to heuristically deduce effective probabilities for the effective reactions which then enables simulation via the stochastic simulation algorithm (SSA). The validity of this heuristic SSA method is a priori doubtful because the reaction probabilities for the SSA have only been rigorously derived from microscopic physics arguments for elementary reactions. RESULTS: We here obtain, by rigorous means and in closed-form, a reduced linear Langevin equation description of the stochastic dynamics of monostable biochemical networks in conditions characterized by small intrinsic noise and timescale separation. The slow-scale linear noise approximation (ssLNA), as the new method is called, is used to calculate the intrinsic noise statistics of enzyme and gene networks. The results agree very well with SSA simulations of the non-reduced network of elementary reactions. In contrast the conventional heuristic SSA is shown to overestimate the size of noise for Michaelis-Menten kinetics, considerably under-estimate the size of noise for Hill-type kinetics and in some cases even miss the prediction of noise-induced oscillations. CONCLUSIONS: A new general method, the ssLNA, is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions. The ssLNA provides a simple and accurate means of performing stochastic model reduction and hence it is expected to be of widespread utility in studying the dynamics of large noisy reaction networks, as is common in computational and systems biology.

Communication: limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks.

It is commonly believed that, whenever timescale separation holds, the predictions of reduced chemical master equations obtained using the stochastic quasi-steady-state approximation are in very good agreement with the predictions of the full master equations. We use the linear noise approximation to obtain a simple formula for the relative error between the predictions of the two master equations for the Michaelis-Menten reaction with substrate input. The reduced approach is predicted to overestimate the variance of the substrate concentration fluctuations by as much as 30%. The theoretical results are validated by stochastic simulations using experimental parameter values for enzymes involved in proteolysis, gluconeogenesis, and fermentation.

How accurate are the nonlinear chemical Fokker-Planck and chemical Langevin equations?

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order Ω(-3∕2) for reaction systems which do not obey detailed balance and at least accurate to order Ω(-2) for systems obeying detailed balance, where Ω is the characteristic size of the system. Hence, the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order Ω(-1∕2) and variance estimates accurate to order Ω(-3∕2). This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.

Small-scale particle advection, manipulation and mixing: beyond the hydrodynamic scale.

In this paper we discuss the problems of particle advection, manipulation and mixing at small scales. We start by considering reaction-advection-diffusion systems with the focus on mixing. We show how mixing advection affects the processes of reaction-diffusion and discuss mixing-induced instabilities. Further, we consider the problem of particle manipulation and discuss collective effects in systems comprising solid and compressible particles. We particularly discuss mechanisms of particle entrapment, the role of compressibility in the dynamics of bubbly liquids and nonequilibrium colloidal explosion. Finally, we address two issues related to the problem of wetting. First, we study the role of contact line motion for a sessile droplet (or a bubble) on an oscillating substrate. Second, we discuss an instability of a thin film leading to the formation of a fractal structure of droplets.