Solid-body trajectoids shaped to roll along desired pathways.

In everyday life, rolling motion is typically associated with cylindrical (for example, car wheels) or spherical (for example, billiard balls) bodies tracing linear paths. However, mathematicians have, for decades, been interested in more exotically shaped solids such as the famous oloids1, sphericons2, polycons3, platonicons4 and two-circle rollers5 that roll downhill in curvilinear paths (in contrast to cylinders or spheres) yet indefinitely (in contrast to cones, Supplementary Video 1). The trajectories traced by such bodies have been studied in detail6-9, and can be useful in the context of efficient mixing10,11 and robotics, for example, in magnetically actuated, millimetre-sized sphericon-shaped robots12,13, or larger sphericon- and oloid-shaped robots translocating by shifting their centre of mass14,15. However, the rolling paths of these shapes are all sinusoid-like and their diversity ends there. Accordingly, we were intrigued whether a more general problem is solvable: given an infinite periodic trajectory, find the shape that would trace this trajectory when rolling down a slope. Here, we develop an algorithm to design such bodies-which we call 'trajectoids'-and then validate these designs experimentally by three-dimensionally printing the computed shapes and tracking their rolling paths, including those that close onto themselves such that the body's centre of mass moves intermittently uphill (Supplementary Video 2). Our study is motivated largely by fundamental curiosity, but the existence of trajectoids for most paths has unexpected implications for quantum and classical optics, as the dynamics of qubits, spins and light polarization can be exactly mapped to trajectoids and their paths16.

Exceptions to Fourier's Law at the Macroscale.

The usual basis to analyze heat transfer within materials is the equation formulated 200 years ago, Fourier's law, which is identical mathematically to the mass diffusion equation, Fick's law. Revisiting this assumption regarding heat transport within translucent materials, performing the experiments in vacuum to avoid air convection, we compare the model predictions to infrared-based measurements with nearly mK temperature resolution. After heat pulses, we find macroscale non-Gaussian tails in the surface temperature profile. At steady state, we find macroscale anomalous hot spots when the sample is topographically rough, and this is validated by using two additional independent methods to measure surface temperature. These discrepancies from Fourier's law for translucent materials suggest that internal radiation whose mean-free-path is millimeters interacts with defects to produce small heat sources that by secondary emission afford an additional, non-local mode of heat transport. For these polymer and inorganic glass materials, this suggests unique strategies of heat management design.

The ergodicity question when imaging DNA conformation using liquid cell electron microscopy.

Assessing the ergodicity of graphene liquid cell electron microscope measurements, we report that loop states of circular DNA interconvert reversibly and that loop numbers follow the Boltzmann distribution expected for this molecule in bulk solution, provided that the electron dose is low (80-keV electron energy and electron dose rate 1-20 e- Å-2 s-1). This imaging technique appears to act as a "slow motion" camera that reveals equilibrated distributions by imaging the time average of a few molecules without the need to image a spatial ensemble.

Anatomy of cage formation in a two-dimensional glass-forming liquid.

As a glass-forming liquid is cooled, the dynamics of its constituent particles changes from being liquid-like to more solid-like. The solidity of the resulting glassy material is believed to be due to a cage-formation process, whereby the motion of individual particles is increasingly constrained by neighbouring particles. This process begins at the temperature (or particle density) at which the glass-forming liquid first shows signs of glassy dynamics; however, the details of how the cages form remain unclear1-4. Here we study cage formation at the particle level in a two-dimensional colloidal suspension (a glass-forming liquid). We use focused lasers to perturb the suspension at the particle level and monitor the nonlinear dynamic response of the system using video microscopy. All observables that we consider respond non-monotonically as a function of the particle density, peaking at the density at which glassy dynamics is first observed. We identify this maximum response as being due to cage formation, quantified by the appearance of domains in which particles move in a cooperative manner. As the particle density increases further, these local domains become increasingly rigid and dominate the macroscale particle dynamics. This microscale rheological deformation approach demonstrates that cage formation in glass-forming liquids is directly related to the merging of such domains, and reveals the first step in the transformation of liquids to glassy materials1,5.

Oil-on-water droplets faceted and stabilized by vortex halos in the subphase.

For almost 200 y, the dominant approach to understand oil-on-water droplet shape and stability has been the thermodynamic expectation of minimized energy, yet parallel literature shows the prominence of Marangoni flow, an adaptive gradient of interfacial tension that produces convection rolls in the water. Our experiments, scaling arguments, and linear stability analysis show that the resulting Marangoni-driven high-Reynolds-number flow in shallow water overcomes radial symmetry of droplet shape otherwise enforced by the Laplace pressure. As a consequence, oil-on-water droplets are sheared to become polygons with distinct edges and corners. Moreover, subphase flows beneath individual droplets can inhibit the coalescence of adjacent droplets, leading to rich many-body dynamics that makes them look alive. The phenomenon of a "vortex halo" in the liquid subphase emerges as a hidden variable.

Volatile Droplets on Water are Sculpted by Vigorous Marangoni-Driven Subphase Flow.

The shapes of highly volatile oil-on-water droplets become strongly asymmetric when they are out of equilibrium. The unsaturated organic vapor atmosphere causes evaporation and leads to a strong Marangoni flow in the bath, unlike that previously seen in the literature. Inspecting these shapes experimentally on millisecond and submillimeter time and length scales and theoretically by scaling arguments, we confirm that Marangoni-driven convection in the subphase mechanically stresses the droplet edges to an extent that increases for organic droplets of smaller contact angle and accordingly smaller thickness. The viscous stress generated by the subphase overcomes the thermodynamic Laplace pressure. The oil droplets develop copious regularly spaced fingers, and these fingers develop spike-shaped and branched treelike structures. Unlike this behavior for single-component (surfactant-free) oil droplets, droplets composed of two miscible (surfactant-free) organic liquids develop a rim of the less volatile component along the droplet perimeter, from which jets of monodisperse smaller droplets eject periodically due to the Rayleigh-Plateau instability. When evaporation shrinks droplets to µm size, their shapes fluctuate chaotically, and ellipsoidal shapes rupture into smaller daughter droplets when subphase convection flow pulls them in opposite directions. The shape of the evaporating oil droplets is kneaded and sculpted by vigorous flow in the water subphase.

Master curve of boosted diffusion for 10 catalytic enzymes.

Molecular agitation more rapid than thermal Brownian motion is reported for cellular environments, motor proteins, synthetic molecular motors, enzymes, and common chemical reactions, yet that chemical activity coupled to molecular motion contrasts with generations of accumulated knowledge about diffusion at equilibrium. To test the limits of this idea, a critical testbed is the mobility of catalytically active enzymes. Sentiment is divided about the reality of enhanced enzyme diffusion, with evidence for and against. Here a master curve shows that the enzyme diffusion coefficient increases in proportion to the energy release rate-the product of Michaelis-Menten reaction rate and Gibbs free energy change (ΔG)-with a highly satisfactory correlation coefficient of 0.97. For 10 catalytic enzymes (urease, acetylcholinesterase, seven enzymes from the glucose cascade cycle, and one other), our measurements span from a roughly 40% enhanced diffusion coefficient at a high turnover rate and negative ΔG to no enhancement at a slow turnover rate and positive ΔG Moreover, two independent measures of mobility show consistency, provided that one avoids undesirable fluorescence photophysics. The master curve presented here quantifies the limits of both ideas, that enzymes display enhanced diffusion and that they do not within instrumental resolution, and has possible implications for understanding enzyme mobility in cellular environments. The striking linear dependence of ΔG for the exergonic enzymes (ΔG <0), together with the vanishing effect for endergonic enzyme (ΔG >0), are consistent with a physical picture in which the mechanism boosting the diffusion is an active one, utilizing the available work from the chemical reaction.

Intermediate states of molecular self-assembly from liquid-cell electron microscopy.

Traditional single-molecule methods do not report whole-molecule kinetic conformations, and their adaptive shape changes during the process of self-assembly. Here, using graphene liquid-cell electron microscopy with electrons of low energy at low dose, we show that this approach resolves the time dependence of conformational adaptations of macromolecules for times up to minutes, the resolution determined by motion blurring, with DNA as the test case. Single-stranded DNA molecules are observed in real time as they hybridize near the solid surface to form double-stranded helices; we contrast molecules the same length but differing in base-pair microstructure (random, blocky, and palindromic hairpin) whose key difference is that random sequences possess only one stable final state, but the others offer metastable intermediate structures. Hybridization is observed to couple with enhanced translational mobility and torsion-induced rotation of the molecule. Prevalent transient loops are observed in error-correction processes. Transient melting and other failed encounters are observed in the competitive binding of multiple single-stranded molecules. Among the intermediate states reported here, some were predicted but not observed previously, and the high incidence of looping and enhanced mobility come as surprises. The error-producing mechanisms, failed encounters, and transient intermediate states would not be easily resolved by traditional single-molecule methods. The methods generalize to visualize motions and interactions of other organic macromolecules.

Comment on "Following Molecular Mobility during Chemical Reactions: No Evidence for Active Propulsion" and "Molecular Diffusivity of Click Reaction Components: The Diffusion Enhancement Question".

We provide arguments why we consider as inaccurate two recent JACS Articles which disagree with our laboratory's report of boosted diffusion during the copper-catalyzed azide-alkyne cycloaddition (CuAAC) "click" reaction. In the first paper (DOI: 10.1021/jacs.1c09455), Fillbrook et al. claim that their diffusion NMR experiments offer no evidence for boosted diffusion, but we argue that their use of Gd3+-chelates to speed up NMR relaxations times is flawed conceptually, the authors interpreting Gd3+-chelates as inert. Actually, the same features that make gadolinium ions useful as contrast agents in magnetic resonance imaging render them unsuitable for diffusion NMR. Nonetheless, by correctly adjusting technical aspects of the measurements, we confirm boosted diffusion even in the presence of this MRI contrast agent. The second paper of which we are skeptical, by Rezaei-Ghaleh et al. (DOI: 10.1021/jacs.1c11754), makes comparisons to a reference state that is not physically meaningful.

Biopolymer Filament Entanglement Softens Then Hardens with Shear.

It is unsatisfactory that regarding the problem of entangled macromolecules driven out of equilibrium, experimentally based understanding is usually inferred from the ensemble average of polydisperse samples. Here, confronting with single-molecule imaging this common but poorly understood situation, over a wide range of shear rate we use single-molecule fluorescence imaging to track alignment and stretching of entangled aqueous filamentous actin filaments in a homebuilt rheo-microscope. With increasing shear rate, tube "softening" is followed by "hardening." Physically, this means that dynamical localization first weakens from molecular alignment, then strengthens from filament stretching, even for semiflexible biopolymers shorter than their persistence length.

Response of vesicle shapes to dense inner active matter.

We consider experimentally the Takatori-Sahu model of vesicle shape fluctuations induced by enclosed active matter, a model till present tested only in the absence of collective motion because few enclosed bacteria were used to generate the desired active motion (S. C. Takatori and A. Sahu, Phys. Rev. Lett., 2020, 124, 158102). Using deformable giant unilamellar vesicles (GUVs) and phase contrast microscopy, we extract the mode-dependence of GUV shape fluctuations when hundreds of E. coli bacteria are contained within each GUV. In the microscope focal plane, patterns of collective bacteria flow include vortex flow, dipolar flow, and chaotic motion, all of which influence the GUV shapes. The Takatori-Sahu model generalizes well to this situation if one considers the moving element to be the experimentally-determined size of the collecively-moving flock.

How to better focus waves by considering symmetry and information loss.

We amend the general belief that waves with extended spherical wavefront focus at their center of curvature. Instead, when the spherical symmetry of waves is broken by propagating them through a finite aperture along an average direction, the forward/backward symmetry is broken and the focal volume shifts its center backward along that direction. The extent of this focal shift increases as smaller apertures are used, up to the point that the nominal focal plane is out of focus. Furthermore, the loss of axial symmetry with noncircular apertures causes distinct focal shifts in distinct axial planes, and the resulting astigmatism possibly degrades the axial focusing resolution. Using experiments and simulations, focal shift with noncircular apertures is described for classical and temporal focusing. The usefulness of these conclusions to improve imaging resolution is demonstrated in a high-resolution optical microscopy application, namely line-temporal focusing microscopy. These conclusions follow from fundamental symmetries of the wave geometry and matter for an increasing number of emerging optical techniques. This work offers a general framework and strategy to understand and improve virtually any wave-based application whose efficacy depends on optimal focusing and may be helpful when information is transmitted by waves in applications from electromagnetic communications, to biological and astronomical imaging, to lithography and even warfare.

Catalytic enzymes are active matter.

Using a microscopic theory to analyze experiments, we demonstrate that enzymes are active matter. Superresolution fluorescence measurements-performed across four orders of magnitude of substrate concentration, with emphasis on the biologically relevant regime around or below the Michaelis-Menten constant-show that catalysis boosts the motion of enzymes to be superdiffusive for a few microseconds, enhancing their effective diffusivity over longer timescales. Occurring at the catalytic turnover rate, these fast ballistic leaps maintain direction over a duration limited by rotational diffusion, driving enzymes to execute wormlike trajectories by piconewton forces performing work of a few kBT against viscosity. The boosts are more frequent at high substrate concentrations, biasing the trajectories toward substrate-poor regions, thus exhibiting antichemotaxis, demonstrated here experimentally over a wide range of aqueous concentrations. Alternative noncatalytic, passive mechanisms that predict chemotaxis, cross-diffusion, and phoresis, are critically analyzed. We examine the physical interpretation of our findings, speculate on the underlying mechanism, and discuss the avenues they open with biological and technological implications. These findings violate the classical paradigm that chemical reaction and motility are distinct processes, and suggest reaction-motion coupling as a general principle of catalysis.

Enzyme leaps fuel antichemotaxis.

There is mounting evidence that enzyme diffusivity is enhanced when the enzyme is catalytically active. Here, using superresolution microscopy [stimulated emission-depletion fluorescence correlation spectroscopy (STED-FCS)], we show that active enzymes migrate spontaneously in the direction of lower substrate concentration ("antichemotaxis") by a process analogous to the run-and-tumble foraging strategy of swimming microorganisms and our theory quantifies the mechanism. The two enzymes studied, urease and acetylcholinesterase, display two families of transit times through subdiffraction-sized focus spots, a diffusive mode and a ballistic mode, and the latter transit time is close to the inverse rate of catalytic turnover. This biochemical information-processing algorithm may be useful to design synthetic self-propelled swimmers and nanoparticles relevant to active materials. Executed by molecules lacking the decision-making circuitry of microorganisms, antichemotaxis by this run-and-tumble process offers the biological function to homogenize product concentration, which could be significant in situations when the reactant concentration varies from spot to spot.

Micromotor That Carries Its Own Fuel Internally.

Micromotors enjoy burgeoning interest but a limitation of their design is to require continuous supply of new fuel. The preponderance of extant micromotors depend, for their motion, on irradiation by light or exposure to acid in their environment. Here we demonstrate a motor that carries its own fuel internally, in this sense representing an analogue, in micron-sized objects, of the internal combustion engine. The fuel is DPCP (diphenylcyclopropenone) microcrystal, a solid-state chemical that after ignition by UV light requires no further irradiation to sustain a chemical reaction that emits carbon monoxide gas that can be used to propel the particle on which this chemical resides. It is loaded asymmetrically onto inexpensive microparticles to produce internally fueled propulsion with speed up to ∼20 µm/s over distances up to 15 times the capsule length in water. Once ignited, the motors maintain their direction of motion and move without need for light to follow their path. Possible strategies to extend the idea beyond the current proof of concept are discussed.

Rapid-prototyping a Brownian particle in an active bath.

Particles kicked by external forces to produce mobility distinct from thermal diffusion are an iconic feature of the active matter problem. Here, we map this onto a minimal model for experiment and theory covering the wide time and length scales of usual active matter systems. A particle diffusing in a harmonic potential generated by an optical trap is kicked by programmed forces with time correlation at random intervals following the Poisson process. The model's generic simplicity allows us to find conditions for which displacements are Gaussian (or not), how diffusion is perturbed (or not) by kicks, and quantifying heat dissipation to maintain the non-equilibrium steady state in an active bath. The model reproduces experimental results of tracer mobility in an active bath of swimming algal cells. It can be used as a stochastic dynamic simulator for Brownian objects in various active baths without mechanistic understanding, owing to the generic framework of the protocol.

Apparatus to Measure Subnanometer Fluctuation of Giant Unilamellar Vesicle Membranes.

The design, calibration, and performance of an apparatus are described to study the nanometer-scale thermal or driven fluctuations of free-standing vesicle membranes using a design resembling the position detection system of optical tweezers except for the laser power lower by orders of magnitude to avoid trapping. Over four decades of frequency, 1-10,000 Hz, it reports membrane fluctuation amplitudes 0.01-100 nm by measuring scattering of a laser beam as it passes membranes (∼1 µm cross-section) suspended in the aqueous medium. The low-power laser beam, <100 µW, is sharply focused on the edge of a giant unilamellar vesicle, and fluctuations of position are measured using a position-sensitive photodetector. The central result of this approach is the capability to reach small fluctuations otherwise inaccessible using other techniques. The typical obtained data are fit to the standard Helfrich mechanical model. The applications and limitations of the device are discussed, as well as other potential uses to which the apparatus may be applied by rational extension of the approach presented.

Effective temperature concept evaluated in an active colloid mixture.

Thermal energy agitates all matter, and its competition with ordering tendencies is a fundamental organizing principle in the physical world; this observation suggests that an effective temperature might emerge when external energy input enhances agitation. However, despite the repeated proposal of this concept based on kinetics for various nonequilibrium systems, the value of an effective temperature as a thermodynamic control parameter has been unclear. Here, we introduce a two-component system of driven Janus colloids, such that collisions induced by external energy sources agitate the system, and we demonstrate quantitative agreement with hallmarks of statistical thermodynamics for binary phase behavior: the archetypal phase diagram with equilibrium critical exponents, Gaussian displacement distributions, and even capillarity. The significance is to demonstrate a class of dynamical conditions under which thermodynamic analysis extends quantitatively to systems that are decidedly nonequilibrium except that the effective temperature differs from the physical temperature.

Dynamic cross-correlations between entangled biofilaments as they diffuse.

Entanglement in polymer and biological physics involves a state in which linear interthreaded macromolecules in isotropic liquids diffuse in a spatially anisotropic manner beyond a characteristic mesoscopic time and length scale (tube diameter). The physical reason is that linear macromolecules become transiently localized in directions transverse to their backbone but diffuse with relative ease parallel to it. Within the resulting broad spectrum of relaxation times there is an extended period before the longest relaxation time when filaments occupy a time-averaged cylindrical space of near-constant density. Here we show its implication with experiments based on fluorescence tracking of dilutely labeled macromolecules. The entangled pairs of aqueous F-actin biofilaments diffuse with separation-dependent dynamic cross-correlations that exceed those expected from continuum hydrodynamics up to strikingly large spatial distances of ≈15 µm, which is more than 104 times the size of the solvent water molecules in which they are dissolved, and is more than 50 times the dynamic tube diameter, but is almost equal to the filament length. Modeling this entangled system as a collection of rigid rods, we present a statistical mechanical theory that predicts these long-range dynamic correlations as an emergent consequence of an effective long-range interpolymer repulsion due to the de Gennes correlation hole, which is a combined consequence of chain connectivity and uncrossability. The key physical assumption needed to make theory and experiment agree is that solutions of entangled biofilaments localized in tubes that are effectively dynamically incompressible over the relevant intermediate time and length scales.

Enhanced Diffusion and Oligomeric Enzyme Dissociation.

The concept that catalytic enzymes can act as molecular machines transducing chemical activity into motion has conceptual and experimental support, but experimental support has involved oligomeric enzymes, often studied under conditions where the substrate concentration is higher than biologically relevant and accordingly exceeds kM, the Michaelis constant. Urease, a hexamer of subunits, has been considered to be the gold standard demonstrating enhanced diffusion. Here we show that urease and certain other oligomeric enzymes dissociate above kM into their subunits that diffuse more rapidly, thus providing a simple physical mechanism that contributes to enhanced diffusion in this regime of concentrations. Mindful that this conclusion may be controversial, our findings are supported by four independent analytical techniques: static light scattering, dynamic light scattering (DLS), size-exclusion chromatography (SEC), and fluorescence correlation spectroscopy (FCS). Data for urease are emphasized and the conclusion is validated for hexokinase, acetylcholinesterase, and aldolase. For hexokinase and aldolase no enhanced diffusion is observed except under conditions when these oligomeric enzymes dissociate. At substrate concentration regimes below kM at which acetylcholinesterase and urease do not dissociate, our finding showing up to 10% enhancement of the diffusion coefficient is consistent with various theoretical scenarios in the literature.