Diffusion trajectory of an asymmetric object: information overlooked by the mean square displacement

Anisotropic diffusion of membrane proteins at experimental timescales

Single-particle tracking (SPT) experiments of lipids and membrane proteins provide a wealth of information about the properties of biomembranes. Careful analysis of SPT trajectories can reveal deviations from ideal Brownian behavior. Among others, this includes confinement effects and anomalous diffusion, which are manifestations of both the nanoscale structure of the underlying membrane and the structure of the diffuser. With the rapid increase in temporal and spatial resolution of experimental methods, a new aspect of the motion of the particle, namely, anisotropic diffusion, might become relevant. This aspect that so far received only little attention is the anisotropy of the diffusive motion and may soon provide an additional proxy to the structure and topology of biomembranes. Unfortunately, the theoretical framework for detecting and interpreting anisotropy effects is currently scattered and incomplete. Here, we provide a computational method to evaluate the degree of anisotropy directly from molecular dynamics simulations and also point out a way to compare the obtained results with those available from SPT experiments. In order to probe the effects of anisotropic diffusion, we performed coarse-grained molecular dynamics simulations of peripheral and integral membrane proteins in flat and curved bilayers. In agreement with the theoretical basis, our computational results indicate that anisotropy can persist up to the rotational relaxation time [τ=(2D_r)^-1], after which isotropic diffusion is observed. Moreover, the underlying topology of the membrane bilayer can couple with the geometry of the particle, thus extending the spatiotemporal domain over which this type of motion can be detected.

Orientation of motion of a flat folding nano-swimmer in soft matter

It is well established that anisotropic molecules do have a preferential direction of motion at short time scales that is washed out at larger times by Brownian noise. Anisotropic molecular motors are able to move at lower temperatures when Brownian noise is smaller suggesting the possibility of oriented motion for larger time scales. We use molecular dynamics simulations to investigate that possibility, calculating the displacements of a simple flat folding molecular nano-swimmer embedded in soft matter. We find actually that the motor displacement is oriented in the direction of its length. We note that the observed orientation of the displacement explains the experimental polarization effect in surface relief gratings formation in agreement with the caterpillar model for azobenzene SRG formation mechanism. That result also suggests a simple route for the creation of molecular motors with oriented displacements.

Direct Single Molecule Imaging of Enhanced Enzyme Diffusion

Recent experimental results have shown that enzymes can diffuse faster when they are in the presence of their reactants (substrate). This faster diffusion has been termed enhanced diffusion. Fluorescence correlation spectroscopy (FCS), which has been employed as the only method to make these measurements, relies on analyzing the fluctuations in fluorescence intensity to measure the diffusion coefficient of particles. Recently, artifacts in FCS measurements due to its sensitivity to environmental conditions have been evaluated, calling prior enhanced diffusion results into question. It behooves us to adopt complementary and direct methods to measure the mobility of enzymes. Herein, we use a technique of direct single molecule imaging to observe the diffusion of individual enzymes in solution. This technique is less sensitive to intensity fluctuations and deduces the diffusion coefficient directly based on the trajectory of the enzyme. Our measurements recapitulate that enzyme diffusion is enhanced in the presence of its substrate and find that the relative increase in diffusion of a single enzyme is even higher than those previously reported using FCS. We also use this complementary method to test if the total enzyme concentration affects the relative increase in diffusion and if the enzyme oligomerization state changes during its catalytic turnover. We find that the diffusion increase is independent of the total concentration of enzymes and the presence of substrate does not change the oligomerization state of enzymes.

Estimation of diffusive states from single-particle trajectory in heterogeneous medium using machine-learning methods

We propose a novel approach to analyze random walks in heterogeneous medium using a hybrid machine-learning method based on a gamma mixture and a hidden Markov model. A gamma mixture and a hidden Markov model respectively provide the number and the most probable sequence of diffusive states from the time series position data of particles/molecules obtained by single-particle/molecule tracking (SPT/SMT) method. We evaluate the performance of our proposed method for numerically generated trajectories. It is shown that our proposed method can correctly extract the number of diffusive states when each trajectory is long enough to be frame averaged. We also indicate that our method can provide an indicator whether the assumption of a medium consisting of discrete diffusive states is appropriate or not based on the available amount of trajectory data. Then, we demonstrate an application of our method to the analysis of experimentally obtained SPT data.

Faster Convergence of Diffusion Anisotropy Detection by Three-Step Relation of Single-Particle Trajectory

We focus on the issue of limited number of samples in the single particle tracking (SPT) when trying to extract the diffusion anisotropy that originates from the particle asymmetry. We propose a novel evaluation technique of SPT making use of the relation of the consecutive three steps. More specifically, the trend of the angle comprised of the three positions and the displacements are plotted on a scatter diagram. The particle anisotropy dependence of the shape of the scatter diagram is examined through the data from the standard numerical model of anisotropic two-dimensional Brownian motion. Comparison with the existing method reveals the faster convergence in the evaluation. In particular, our proposed method realizes the detection of diffusion anisotropy under the conditions of not only less number of data but also larger time steps. This is of practical importance not only when the abundant data is hard to achieve but also when the rotational diffusion is fast compared to the frame rate of the camera equipment, which tends to be more common for smaller particles or molecules of interest.

High-precision tracking of brownian boomerang colloidal particles confined in quasi two dimensions

In this article, we present a high-precision image-processing algorithm for tracking the translational and rotational Brownian motion of boomerang-shaped colloidal particles confined in quasi-two-dimensional geometry. By measuring mean square displacements of an immobilized particle, we demonstrate that the positional and angular precision of our imaging and image-processing system can achieve 13 nm and 0.004 rad, respectively. By analyzing computer-simulated images, we demonstrate that the positional and angular accuracies of our image-processing algorithm can achieve 32 nm and 0.006 rad. Because of zero correlations between the displacements in neighboring time intervals, trajectories of different videos of the same particle can be merged into a very long time trajectory, allowing for long-time averaging of different physical variables. We apply this image-processing algorithm to measure the diffusion coefficients of boomerang particles of three different apex angles and discuss the angle dependence of these diffusion coefficients.

Brownian motion of boomerang colloidal particles

We investigate the Brownian motion of boomerang colloidal particles confined between two glass plates. Our experimental observations show that the mean displacements are biased towards the center of hydrodynamic stress (CoH), and that the mean-square displacements exhibit a crossover from short-time faster to long-time slower diffusion with the short-time diffusion coefficients dependent on the points used for tracking. A model based on Langevin theory elucidates that these behaviors are ascribed to the superposition of two diffusive modes: the ellipsoidal motion of the CoH and the rotational motion of the tracking point with respect to the CoH.

Detection of diffusion anisotropy due to particle asymmetry from single-particle tracking of Brownian motion by the large-deviation principle

We show that the diffusion anisotropy due to the asymmetry of the particle can be extracted from the trajectory data without the information of the particle orientation. The subject of analysis is typical in single-particle tracking (SPT) experiments, and the analysis is based on the large-deviation principle in mathematics. We consider the model system of Langevin equations in two dimensions where a particle diffusion shows anisotropy depending on a single parameter defined by the two diffusion coefficients in the perpendicular directions of the frame fixed to the particle. We show how the large-deviation quantities depend on this parameter so that it can be used for the detection of the diffusion anisotropy. We also illustrate how the discreteness of the sampling interval in the SPT and the finiteness of the number of samples influence the results of the analysis.

Brownian motion of a self-propelled particle

Overdamped Brownian motion of a self-propelled particle is studied by solving the Langevin equation analytically. On top of translational and rotational diffusion, in the context of the presented model, the 'active' particle is driven along its internal orientation axis. We calculate the first four moments of the probability distribution function for displacements as a function of time for a spherical particle with isotropic translational diffusion, as well as for an anisotropic ellipsoidal particle. In both cases the translational and rotational motion is either unconfined or confined to one or two dimensions. A significant non-Gaussian behaviour at finite times t is signalled by a non-vanishing kurtosis γ(t). To delimit the super-diffusive regime, which occurs at intermediate times, two timescales are identified. For certain model situations a characteristic t(3) behaviour of the mean-square displacement is observed. Comparing the dynamics of real and artificial microswimmers, like bacteria or catalytically driven Janus particles, to our analytical expressions reveals whether their motion is Brownian or not.

Multiphoton fluorescence recovery after photobleaching in bounded systems

Multiphoton fluorescence recovery after photobleaching (MP-FRAP) is a laser microscopy technique used to measure diffusion coefficients of macromolecules in biological systems. The three-dimensional resolution and superior depth penetration within scattering samples offered by MP-FRAP make it an important tool for investigating both in vitro and in vivo systems. However, biological systems frequently confine diffusion within solid barriers, and to date the effect of such barriers on the measurement of absolute diffusion coefficients via MP-FRAP has not been studied. We have used Monte Carlo simulations of diffusion and MP-FRAP to understand the effect of barriers of varying geometries and positions relative to the two-photon focal volume. Furthermore, we supply ranges of barrier positions within which MP-FRAP can confidently be employed to measure accurate diffusion coefficients. Finally, we produce two new MP-FRAP models that can produce accurate diffusion coefficients in the presence of a single plane boundary or parallel infinite plane boundaries positioned parallel to the optical axis, up to the resolution limit of the multiphoton laser scanning microscope.

Search in unknown random environments

N searchers are sent out by a source in order to locate a fixed object which is at a finite distance D, but the search space is infinite and D would be in general unknown. Each of the searchers has a finite random lifetime, and may be subject to destruction or failures, and it moves independently of other searchers, and at intermediate locations some partial random information may be available about which way to go. When a searcher is destroyed or disabled, or when it "dies naturally," after some time the source becomes aware of this and it sends out another searcher, which proceeds similarly to the one that it replaces. The search ends when one of the searchers finds the object being sought. We use N coupled brownian motions to derive a closed form expression for the average search time as a function of D which will depend on the parameters of the problem: the number of searchers, the average lifetime of searchers, the routing uncertainty, and the failure or destruction rate of searchers. We also examine the cost in terms of the total energy that is expended in the search.

Synaptic stability and plasticity in a floating world

A fundamental feature of membranes is the lateral diffusion of lipids and proteins. Control of lateral diffusion provides a mechanism for regulating the structure and function of synapses. Single-particle tracking (SPT) has emerged as a powerful way to directly visualize these movements. SPT can reveal complex diffusive behaviors, which can be regulated by neuronal activity over time and space. Such is the case for neurotransmitter receptors, which are transiently stabilized at synapses by scaffolding molecules. This regulation provides new insight into mechanisms by which the dynamic equilibrium of receptor-scaffold assembly can be regulated. We will briefly review here recent data on this mechanism, which ultimately tunes the number of receptors at synapses and therefore synaptic strength.

Spatial structure and diffusive dynamics from single-particle trajectories using spline analysis

Single-particle tracking of biomolecular probes has provided a wealth of information about intracellular trafficking and the dynamics of proteins and lipids in the cell membrane. Conventional mean-square displacement (MSD) analysis of single-particle trajectories often assumes that probes are moving in a uniform environment. However, the observed two-dimensional motion of probe particles is influenced by the local three-dimensional geometry of the cell membrane and intracellular structures, which are rarely flat at the submicron scale. This complex geometry can lead to spatially confined trajectories that are difficult to analyze and interpret using conventional two-dimensional MSD analysis. Here we present two methods to analyze spatially confined trajectories: spline-curve dynamics analysis, which extends conventional MSD analysis to measure diffusive motion in confined trajectories; and spline-curve spatial analysis, which measures spatial structures smaller than the limits of optical resolution. We show, using simulated random walks and experimental trajectories of quantum dot probes, that differences in measured two-dimensional diffusion coefficients do not always reflect differences in underlying diffusive dynamics, but can instead be due to differences in confinement geometries of cellular structures.

Self-assembled autonomous runners and tumblers

A class of artificial microswimmers with combined translational and rotational self-propulsion is studied experimentally. The chemically fueled microswimmers are made of doublets of Janus colloidal beads with catalytic patches that are positioned at a fixed angle relative to one another. The mean-square displacement and the mean-square angular displacement of the active doublets are analyzed in the context of a simple Langevin description, using which the physical characteristics of the microswimmers such as the spontaneous translational and rotational velocities are extracted. Our work suggests strategies for designing microswimmers that could follow prescribed cycloidal trajectories.

Single-particle tracking methods for the study of membrane receptors dynamics

Single-particle tracking (SPT) applications have been growing rapidly in the field of cell biology, and in particular in neurobiology, as a means of unravelling the involvement of diffusion dynamics of neurotransmitter receptors and other synaptic proteins in the regulation of neuronal activity. Suitable probes and technological improvements make SPT more accessible than it used to be and open up broad applications in cellular biology. In this technical highlight, we give an overview of the experimental approach in SPT. The concepts and results in neurobiology have already been the object of detailed reviews. Here, we focus on a qualitative description of the implementation of SPT, from molecule labelling to acquisition, data treatment and analysis of protein diffusion properties. Constraints, limitations and future developments are discussed.

Brownian motion of an asymmetrical particle in a potential field

It is well known that a free ellipsoidal Brownian particle exhibits anisotropic diffusion for short times which changes to isotropic at long times, and, that the long-time diffusion coefficient is an average of the translational diffusion coefficients along the different semiaxes of the particle. We show analytically that in the presence of external forces, the long-time diffusion coefficient is different from that of a free particle. The magnitude of the difference in the two diffusion coefficients is found to increase proportionately with the particle's asymmetry, being zero only for a perfectly spherical Brownian particle. It is also found that, for asymmetrical particles, the application of external forces can amplify the non-Gaussian character of the spatial probability distributions which consequently delays the transition to the classical behavior. We illustrate these phenomena by considering the quasi-two-dimensional Brownian motion of an ellipsoidal rigid particle in linear and harmonic potential fields. These two examples provide insight into the role played by particle asymmetry in electrophoresis and microconfinement due to a laser trap or due to intracellular macromolecular crowding.