Single-particle tracking: Brownian dynamics of viscoelastic materials

Viscoelasticity, Like Forces, Plays a Role in Mechanotransduction

Viscoelasticity and its alteration in time and space has turned out to act as a key element in fundamental biological processes in living systems, such as morphogenesis and motility. Based on experimental and theoretical findings it can be proposed that viscoelasticity of cells, spheroids and tissues seems to be a collective characteristic that demands macromolecular, intracellular component and intercellular interactions. A major challenge is to couple the alterations in the macroscopic structural or material characteristics of cells, spheroids and tissues, such as cell and tissue phase transitions, to the microscopic interferences of their elements. Therefore, the biophysical technologies need to be improved, advanced and connected to classical biological assays. In this review, the viscoelastic nature of cytoskeletal, extracellular and cellular networks is presented and discussed. Viscoelasticity is conceptualized as a major contributor to cell migration and invasion and it is discussed whether it can serve as a biomarker for the cells’ migratory capacity in several biological contexts. It can be hypothesized that the statistical mechanics of intra- and extracellular networks may be applied in the future as a powerful tool to explore quantitatively the biomechanical foundation of viscoelasticity over a broad range of time and length scales. Finally, the importance of the cellular viscoelasticity is illustrated in identifying and characterizing multiple disorders, such as cancer, tissue injuries, acute or chronic inflammations or fibrotic diseases.

Imaging three-dimensional single-atom arrays all at once

Simultaneous imaging of a three-dimensional distribution of point sources is presented. In a two-lens microscope, the point-spreads on the quasi-image plane, which is located between the Fourier and image planes, are spatially distinct, so a set of Fresnel lenslets can perform individual wave-front shaping for axial and lateral rearrangements of the images. In experiments performed with single atoms and holographically programmed lenslets, various three-dimensional arrangements of point sources, including axially aligned atoms, are successfully refocused on the screen, demonstrating the simultaneous and time-efficient detection of the three-dimensional holographic imaging. We expect that non-sequential real-time measurements of three-dimensional point sources shall be in particular useful for quantum correlation measurements and in situ tracking of dynamic particles.

Cell viscoelasticity is linked to fluctuations in cell biomass distributions

The viscoelastic properties of mammalian cells can vary with biological state, such as during the epithelial-to-mesenchymal (EMT) transition in cancer, and therefore may serve as a useful physical biomarker. To characterize stiffness, conventional techniques use cell contact or invasive probes and as a result are low throughput, labor intensive, and limited by probe placement. Here, we show that measurements of biomass fluctuations in cells using quantitative phase imaging (QPI) provides a probe-free, contact-free method for quantifying changes in cell viscoelasticity. In particular, QPI measurements reveal a characteristic underdamped response of changes in cell biomass distributions versus time. The effective stiffness and viscosity values extracted from these oscillations in cell biomass distributions correlate with effective cell stiffness and viscosity measured by atomic force microscopy (AFM). This result is consistent for multiple cell lines with varying degrees of cytoskeleton disruption and during the EMT. Overall, our study demonstrates that QPI can reproducibly quantify cell viscoelasticity.

Optical tracking of nanoscale particles in microscale environments

The trajectories of nanoscale particles through microscale environments record useful information about both the particles and the environments. Optical microscopes provide efficient access to this information through measurements of light in the far field from nanoparticles. Such measurements necessarily involve trade-offs in tracking capabilities. This article presents a measurement framework, based on information theory, that facilitates a more systematic understanding of such trade-offs to rationally design tracking systems for diverse applications. This framework includes the degrees of freedom of optical microscopes, which determine the limitations of tracking measurements in theory. In the laboratory, tracking systems are assemblies of sources and sensors, optics and stages, and nanoparticle emitters. The combined characteristics of such systems determine the limitations of tracking measurements in practice. This article reviews this tracking hardware with a focus on the essential functions of nanoparticles as optical emitters and microenvironmental probes. Within these theoretical and practical limitations, experimentalists have implemented a variety of tracking systems with different capabilities. This article reviews a selection of apparatuses and techniques for tracking multiple and single particles by tuning illumination and detection, and by using feedback and confinement to improve the measurements. Prior information is also useful in many tracking systems and measurements, which apply across a broad spectrum of science and technology. In the context of the framework and review of apparatuses and techniques, this article reviews a selection of applications, with particle diffusion serving as a prelude to tracking measurements in biological, fluid, and material systems, fabrication and assembly processes, and engineered devices. In so doing, this review identifies trends and gaps in particle tracking that might influence future research.

Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion

The discovery of anomalous diffusion of larger biopolymers and submicron tracers such as endogenous granules, organelles, or virus capsids in living cells, attributed to the viscoelastic nature of the cytoplasm, provokes the question whether this complex environment equally impacts the active intracellular transport of submicron cargos by molecular motors such as kinesins: does the passive anomalous diffusion of free cargo always imply its anomalously slow active transport by motors, the mean transport distance along microtubule growing sublinearly rather than linearly in time? Here we analyze this question within the widely used two-state Brownian ratchet model of kinesin motors based on the continuous-state diffusion along microtubules driven by a flashing binding potential, where the cargo particle is elastically attached to the motor. Depending on the cargo size, the loading force, the amplitude of the binding potential, the turnover frequency of the molecular motor enzyme, and the linker stiffness we demonstrate that the motor transport may turn out either normal or anomalous, as indeed measured experimentally. We show how a highly efficient normal active transport mediated by motors may emerge despite the passive anomalous diffusion of the cargo, and study the intricate effects of the elastic linker. Under different, well specified conditions the microtubule-based motor transport becomes anomalously slow and thus significantly less efficient.

Anomalous transport of subdiffusing cargos by single kinesin motors: the role of mechano-chemical coupling and anharmonicity of tether

Here we generalize our previous model of molecular motors trafficking subdiffusing cargos in viscoelastic cytosol by (i) including mechano-chemical coupling between cyclic conformational fluctuations of the motor protein driven by the reaction of ATP hydrolysis and its translational motion within the simplest two-state model of hand-over-hand motion of kinesin, and also (ii) by taking into account the anharmonicity of the tether between the motor and the cargo (its maximally possible extension length). It is shown that the major earlier results such as occurrence of normal versus anomalous transport depending on the amplitude of binding potential, cargo size and the motor turnover frequency not only survive in this more realistic model, but the results also look very similar for the correspondingly adjusted parameters. However, this more realistic model displays a substantially larger thermodynamic efficiency due to a bidirectional mechano-chemical coupling. For realistic parameters, the maximal thermodynamic efficiency can transiently be about 50% as observed for kinesins, and even larger, surprisingly also in a novel strongly anomalous (sub)transport regime, where the motor enzymatic turnovers become also anomalously slow and cannot be characterized by a turnover rate. Here anomalously slow dynamics of the cargo enforces anomalously slow cyclic kinetics of the motor protein.

How molecular motors work in the crowded environment of living cells: coexistence and efficiency of normal and anomalous transport

Recent experiments reveal both passive subdiffusion of various nanoparticles and anomalous active transport of such particles by molecular motors in the molecularly crowded environment of living biological cells. Passive and active microrheology reveals that the origin of this anomalous dynamics is due to the viscoelasticity of the intracellular fluid. How do molecular motors perform in such a highly viscous, dissipative environment? Can we explain the observed co-existence of the anomalous transport of relatively large particles of 100 to 500 nm in size by kinesin motors with the normal transport of smaller particles by the same molecular motors? What is the efficiency of molecular motors in the anomalous transport regime? Here we answer these seemingly conflicting questions and consistently explain experimental findings in a generalization of the well-known continuous diffusion model for molecular motors with two conformational states in which viscoelastic effects are included.

Universal algorithm for identification of fractional Brownian motion. A case of telomere subdiffusion

We present a systematic statistical analysis of the recently measured individual trajectories of fluorescently labeled telomeres in the nucleus of living human cells. The experiments were performed in the U2OS cancer cell line. We propose an algorithm for identification of the telomere motion. By expanding the previously published data set, we are able to explore the dynamics in six time orders, a task not possible earlier. As a result, we establish a rigorous mathematical characterization of the stochastic process and identify the basic mathematical mechanisms behind the telomere motion. We find that the increments of the motion are stationary, Gaussian, ergodic, and even more chaotic--mixing. Moreover, the obtained memory parameter estimates, as well as the ensemble average mean square displacement reveal subdiffusive behavior at all time spans. All these findings statistically prove a fractional Brownian motion for the telomere trajectories, which is confirmed by a generalized p-variation test. Taking into account the biophysical nature of telomeres as monomers in the chromatin chain, we suggest polymer dynamics as a sufficient framework for their motion with no influence of other models. In addition, these results shed light on other studies of telomere motion and the alternative telomere lengthening mechanism. We hope that identification of these mechanisms will allow the development of a proper physical and biological model for telomere subdynamics. This array of tests can be easily implemented to other data sets to enable quick and accurate analysis of their statistical characteristics.

Superdiffusion induced by a long-correlated external random force

We consider a particle immersed in a thermal reservoir and simultaneously subjected to an external random force that drives the system to a nonequilibrium situation. Starting from a Langevin equation description, we derive exact expressions for the mean-square displacement and the velocity autocorrelation function of the diffusing particle. An effective temperature is introduced to characterize the deviation from the internal equilibrium situation. Using a power-law force autocorrelation function, the mean-square displacement and the velocity autocorrelation function are analytically obtained in terms of Mittag-Leffler functions. In this case, we show that the present model exhibits a superdiffusive regime as a consequence of the competition between passive and active processes.

Understanding nanoparticle diffusion and exploring interfacial nanorheology using molecular dynamics simulations

We have studied the dynamics of nanoparticles at polydimethylsiloxane (PDMS) oil-water interfaces using molecular dynamics (MD) simulations. The diffusion of nanoparticles in pure water and low-viscosity PDMS oil is found to be reasonably consistent with the prediction by the Stokes-Einstein equation. In addition, we have calculated the shear moduli and viscosities of bulk oil and water, as well as oil-water interfaces from single nanoparticle tracking and demonstrated the potential of probing nanorheology from an MD simulation approach. Surprisingly, we found that the lateral diffusion of nanoparticles as well as apparent interfacial nanorheology at the PDMS oil (low viscosity)-water interface are independent of the position of the nanoparticle at the interface.

Nanosecond motions in proteins impose bounds on the timescale distributions of local dynamics

We elucidate the physics of protein dynamical transition via 10-100-ns molecular dynamics simulations at temperatures spanning 160-300 K. By tracking the energy fluctuations, we show that the protein dynamical transition is marked by a crossover from nonstationary to stationary processes that underlie the dynamics of protein motions. A two-timescale function captures the nonexponential character of backbone structural relaxations. One timescale is attributed to the collective segmental motions and the other to local relaxations. The former is well defined by a single-exponential, nanosecond decay, operative at all temperatures. The latter is described by a set of processes that display a distribution of timescales. Although their average remains on the picosecond timescale, the distribution is markedly contracted at the onset of the transition. It is shown that the collective motions impose bounds on timescales spanned by local dynamical processes. The nonstationary character below the transition implicates the presence of a collection of substates whose interactions are restricted. At these temperatures, a wide distribution of local-motion timescales, extending beyond that of nanoseconds, is observed. At physiological temperatures, local motions are confined to timescales faster than nanoseconds. This relatively narrow window makes possible the appearance of multiple channels for the backbone dynamics to operate.

Viscoelastic subdiffusion: from anomalous to normal

We study viscoelastic subdiffusion in bistable and periodic potentials within the generalized Langevin equation approach. Our results justify the (ultra)slow fluctuating rate view of the corresponding bistable non-Markovian dynamics which displays bursting and anticorrelation of the residence times in two potential wells. The transition kinetics is asymptotically stretched exponential when the potential barrier V0 several times exceeds thermal energy k(B)T [V(0) approximately (2-10)k(B)T] and it cannot be described by the non-Markovian rate theory (NMRT). The well-known NMRT result approximates, however, ever better with the increasing barrier height, the most probable logarithm of the residence times. Moreover, the rate description is gradually restored when the barrier height exceeds a fuzzy borderline which depends on the power-law exponent of free subdiffusion alpha . Such a potential-free subdiffusion is ergodic. Surprisingly, in periodic potentials it is not sensitive to the barrier height in the long time asymptotic limit. However, the transient to this asymptotic regime is extremally slow and it does profoundly depend on the barrier height. The time scale of such subdiffusion can exceed the mean residence time in a potential well or in a finite spatial domain by many orders of magnitude. All these features are in sharp contrast with an alternative subdiffusion mechanism involving jumps among traps with the divergent mean residence time in these traps.

Subdiffusive behavior in a trapping potential: mean square displacement and velocity autocorrelation function

A theoretical framework for analyzing stochastic data from single-particle tracking in viscoelastic materials and under the influence of a trapping potential is presented. Starting from a generalized Langevin equation, we found analytical expressions for the two-time dynamics of a particle subjected to a harmonic potential. The mean-square displacement and the velocity autocorrelation function of the diffusing particle are given in terms of the time lag. In particular, we investigate the subdiffusive case. Using a power-law memory kernel, exact expressions for the mean-square displacement and the velocity autocorrelation function are obtained in terms of Mittag-Leffler functions and their derivatives. The behaviors for short-, intermediate-, and long-time lags are investigated in terms of the involved parameters. Finally, the validity of usual approximations is examined.

How a vicinal layer of solvent modulates the dynamics of proteins

The dynamics of a folded protein is studied in water and glycerol at a series of temperatures below and above their respective dynamical transition. The system is modeled in two distinct states whereby the protein is decoupled from the bulk solvent at low temperatures, and communicates with it through a vicinal layer at physiological temperatures. A linear viscoelastic model elucidates the less-than-expected increase in the relaxation times observed in the backbone dynamics of the protein. The model further explains the increase in the flexibility of the protein once the transition takes place and the differences in the flexibility under the different solvent environments. Coupling between the vicinal layer and the protein fluctuations is necessary to interpret these observations. The vicinal layer is postulated to form once a threshold for the volumetric fluctuations in the protein to accommodate solvents of different sizes is reached. Compensation of entropic-energetic contributions from the protein-coupled vicinal layer quantifies the scaling of the dynamical transition temperatures in various solvents. The protein adapts different conformational routes for organizing the required coupling to a specific solvent, which is achieved by adjusting the amount of conformational jumps in the surface-group dihedrals.

Sampling the cell with anomalous diffusion - the discovery of slowness

Diffusion-mediated searching for interaction partners is an ubiquitous process in cell biology. Transcription factors, for example, search specific DNA sequences, signaling proteins aim at interacting with specific cofactors, and peripheral membrane proteins try to dock to membrane domains. Brownian motion, however, is affected by molecular crowding that induces anomalous diffusion (so-called subdiffusion) of proteins and larger structures, thereby compromising diffusive transport and the associated sampling processes. Contrary to the naive expectation that subdiffusion obstructs cellular processes, we show here by computer simulations that subdiffusion rather increases the probability of finding a nearby target. Consequently, important events like protein complex formation and signal propagation are enhanced as compared to normal diffusion. Hence, cells indeed benefit from their crowded internal state and the associated anomalous diffusion.

Apparent microrheology of oil-water interfaces by single-particle tracking

We investigate the dynamics of charged microparticles at polydimethylsiloxane (oil)-water interfaces using Pickering emulsions as an experimental template. The mobility of the charged particles depends largely on the viscoelastic properties of the oil phase and the wettability of the solid particles. In addition, we have explored the potential of developing microrheology at liquid-liquid interfaces from the single-particle tracking technique. The apparent loss modulus, storage modulus, and relaxation time of the oil-water interfaces obtained from singe-particle microrheology depend strongly on the surface nature of the tracer particles, especially when the oil phase is viscoelastic.

Relative entropy: free energy associated with equilibrium fluctuations and nonequilibrium deviations

Using a one-dimensional macromolecule in aqueous solution as an illustration, we demonstrate that the relative entropy from information theory, Sigma(k)p(k) ln(p(k)/p(*)(k)), has a natural role in the energetics of equilibrium and nonequilibrium conformational fluctuations of the single molecule. It is identified as the free energy difference associated with a fluctuating density in equilibrium, and is associated with the distribution deviate from the equilibrium in nonequilibrium relaxation. This result can be generalized to any other isothermal macromolecular system using the mathematical theories of large deviations and Markov processes, and at the same time provides the well-known mathematical results with interesting physical interpretations.