Dynamics in Crowded Environments: Is Non-Gaussian Brownian Diffusion Normal?

Anomalous lateral diffusion of lipids during the fluid/gel phase transition of a lipid membrane

A lipid membrane undergoes a phase transition from fluid to gel phase upon changing external thermodynamic conditions, such as decreasing temperature and increasing pressure. Extremophilic organisms face the challenge of preventing this deleterious phase transition. The main focus of their adaptive strategy is to facilitate effective temperature sensing through sensor proteins, relying on the drastic changes in packing density and membrane fluidity during the phase transition. Although the changes in packing density parameters due to the fluid/gel phase transition are studied in detail, the impact on membrane fluidity is less explored in the literature. Understanding the lateral diffusive dynamics of lipids in response to temperature, particularly during the fluid/gel phase transition, is albeit crucial. Here we have simulated the phase transition of a single component lipid membrane composed of dipalmitoylphosphatidylcholine (DPPC) lipids using a coarse-grained (CG) model and studied the changes of the structural and dynamical properties. It is observed that near the phase transition point, both fluid and gel phase domains coexist together. The dynamics remains highly non-Gaussian for a long time even when the mean square displacement reaches the Fickian regime at a much earlier time. This Fickian yet non-Gaussian diffusion (FnGD) is a characteristic of a highly heterogeneous system, previously observed for the lateral diffusion of lipids in raft mimetic membranes having liquid-ordered and liquid-disordered phases co-existing together. We have analyzed the molecular trajectories and calculated the jump-diffusion of the lipids, stemming from sudden jump translations, using a translational jump-diffusion (TJD) approach. An overwhelming contribution of the jump-diffusion of the lipids is observed suggesting anomalous diffusion of lipids during fluid/gel phase transition of the membrane. These results are important in unravelling the intricate nature of lipid diffusion during the phase transition of the membrane and open up a new possibility of investigating the most significant change of membrane properties during phase transition, which can be effectively sensed by proteins.

Nonmonotonic Modulation of the Protein-Ligand Recognition Event by Inert Crowders

The ubiquitous event of a protein recognizing small molecules or ligands at its native binding site is crucial for initiating major biological processes. However, how a crowded environment, as is typically represented by a cellular interior, would modulate the protein-ligand search process is largely debated. Excluded volume-based theory suggests that the presence of an inert crowder would reinforce a steady stabilization and enhancement of the protein-ligand recognition process. Here, we counter this long-held perspective via the molecular dynamics simulation and Markov state model of the protein-ligand recognition event in the presence of inert crowders. Specifically, we demonstrate that, depending on concentration, even purely inert crowders can exert a nonmonotonic effect via either stabilizing or destabilizing the protein-ligand binding event. Analysis of the kinetic network of binding pathways reveals that the crowders would either modulate precedent non-native on-pathway intermediates or would devise additional ones in a multistate recognition event across a wide range of concentrations. As an important insight, crowders gradually shift the relative transitional preference of these intermediates toward a native-bound state, with ligand residence time at the binding pocket dictating the trend of nonmonotonic concentration dependence by simple inert crowders.

Effects of Homogeneous and Heterogeneous Crowding on Translational Diffusion of Rigid Bovine Serum Albumin and Disordered Alfa-Casein

Intracellular environment includes proteins, sugars, and nucleic acids interacting in restricted media. In the cytoplasm, the excluded volume effect takes up to 40% of the volume available for occupation by macromolecules. In this work, we tested several approaches modeling crowded solutions for protein diffusion. We experimentally showed how the protein diffusion deviates from conventional Brownian motion in artificial conditions modeling the alteration of medium viscosity and rigid spatial obstacles. The studied tracer proteins were globular bovine serum albumin and intrinsically disordered α-casein. Using the pulsed field gradient NMR, we investigated the translational diffusion of protein probes of different structures in homogeneous (glycerol) and heterogeneous (PEG 300/PEG 6000/PEG 40,000) solutions as a function of crowder concentration. Our results showed fundamentally different effects of homogeneous and heterogeneous crowded environments on protein self-diffusion. In addition, the applied "tracer on lattice" model showed that smaller crowding obstacles (PEG 300 and PEG 6000) create a dense net of restrictions noticeably hindering diffusing protein probes, whereas the large-sized PEG 40,000 creates a "less restricted" environment for the diffusive motion of protein molecules.

Dynamics of self-propelled tracer particles inside a polymer network

The transport of tracer particles through mesh-like environments such as biological hydrogels and polymer matrices is ubiquitous in nature. These tracers can be passive, such as colloids, or active (self-propelled), for example, synthetic nanomotors or bacteria. Computer simulations in principle could be extremely useful in exploring the mechanism of the active transport of tracer particles through mesh-like environments. Therefore, we construct a polymer network on a diamond lattice and use computer simulations to investigate the dynamics of spherical self-propelled particles inside the network. Our main objective is to elucidate the effect of the self-propulsion on the tracer particle dynamics as a function of the tracer size and the stiffness of the polymer network. We compute the time-averaged mean-squared displacement (MSD) and the van-Hove correlations of the tracer. On the one hand, in the case of a bigger sticky particle, the caging caused by the network particles wins over the escape assisted by the self-propulsion. This results an intermediate-time subdiffusion. On the other hand, smaller tracers or tracers with high self-propulsion velocities can easily escape from the cages and show intermediate-time superdiffusion. The stiffer the network, the slower the dynamics of the tracer, and bigger tracers exhibit longer lived intermediate time superdiffusion, since the persistence time scales as ∼σ³, where σ is the diameter of the tracer. At the intermediate time, non-Gaussianity is more pronounced for active tracers. At the long time, the dynamics of the tracer, if passive or weakly active, becomes Gaussian and diffusive, but remains flat for tracers with high self-propulsion, accounting for their seemingly unrestricted motion inside the network.

Crowding and confinement act in concert to slow DNA diffusion within cell-sized droplets

Dynamics of biological macromolecules, such as DNA, in crowded and confined environments are critical to understanding cellular processes such as transcription, infection, and replication. However, the combined effects of cellular confinement and crowding on macromolecular dynamics remain poorly understood. Here, we use differential dynamic microscopy to investigate the diffusion of large DNA molecules confined in cell-sized droplets and crowded by dextran polymers. We show that confined and crowded DNA molecules exhibit universal anomalous subdiffusion with scaling that is insensitive to the degree of confinement and crowding. However, effective DNA diffusion coefficients Deff decrease up to 2 orders of magnitude as droplet size decreases—an effect that is enhanced by increased crowding. We mathematically model the coupling of crowding and confinement by combining polymer scaling theories with confinement-induced depletion effects. The generality and tunability of our system and models render them applicable to elucidating wide-ranging crowded and confined systems.

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DNA diffusion measured in cell-sized droplets with differential dynamic microscopy

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Combination of crowding and confinement leads to subdiffusion and slowing

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Diffusion coefficients of DNA decrease strongly with decreasing droplet size

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Polymer scaling theories and depletion effects predict observed dynamics

Structural relaxation dynamics of colloidal nanotrimers

By Molecular Dynamics simulation, we investigate the dynamics of isotropic fluids of colloidal nanotrimers whose interactions are described by varying the strength of attractive and repulsive terms of the Mie potential. To provide a consistent comparison between the systems described by different force fields, we determine the phase diagram and critical points of each system, characterize the morphology of high-density liquid phases at the same reduced temperature and density, and finally investigate their long-time relaxation dynamics. In particular, we detect an especially complex dynamics that reveals the existence of slow and fast nanotrimers and the resulting occurrence of non-Gaussianity, which develops at intermediate timescales. Deviations from Gaussianity are temporary and vanish within the timescales of the system's density fluctuations decay, when a Fickian-like diffusion regime is eventually observed.

Motion of an active particle with dynamical disorder

We propose a model for investigating the motion of a single active particle in a heterogeneous environment where the heterogeneity may arise due to crowding, conformational fluctuations and/or slow rearrangement of the surroundings. Describing the active particle in terms of the Ornstein-Uhlenbeck process (OUP) and incorporating heterogeneity in a thermal bath using two separate models, namely "diffusing diffusivity" and "switching diffusion", we explore the essential dynamical properties of the particle for its one-dimensional motion. In addition, we show how the dynamical behavior is controlled by dynamical variables associated with active noise such as strength and persistence time. Our model is relevant in the context of single particle dynamics in a crowded environment, driven by activity.

Transport and phase separation of active Brownian particles in fluctuating environments

In this work, we study the dynamics of a single active Brownian particle, as well as the collective behavior of interacting active Brownian particles, in a fluctuating heterogeneous environment. We employ a variant of the diffusing diffusivity model where the equation of motion of the active particle involves a time-dependent motility and diffusivities. Within our model, those fluctuations are coupled to each other. Using analytical methods, we obtain the probability distribution function of particle displacement and its moments for a single particle. We then investigate the impact of the environmental fluctuations on the collective behavior of the active Brownian particles by means of extensive numerical simulations. Our results show that the fluctuations hinder the motility-induced phase separation, accompanied by a significant change of the density dependence of particle velocities. These effects are interpreted using our analytical results for the dynamics of a single particle.

Active interaction switching controls the dynamic heterogeneity of soft colloidal dispersions

We employ Reactive Dynamical Density Functional Theory (R-DDFT) and Reactive Brownian Dynamics (R-BD) simulations to investigate the dynamics of a suspension of active soft Gaussian colloids with binary interaction switching, i.e., a one-component colloidal system in which every particle stochastically switches at predefined rates between two interaction states with different mobility. Using R-DDFT we extend a theory previously developed to access the dynamics of inhomogeneous liquids [Archer et al., Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 75, 040501] to study the influence of the switching activity on the self and distinct part of the Van Hove function in bulk solution, and determine the corresponding mean squared displacement of the switching particles. Our results demonstrate that, even though the average diffusion coefficient is not affected by the switching activity, it significantly modifies the non-equilibrium dynamics and diffusion coefficients of the individual particles, leading to a crossover from short to long times, with a regime for intermediate times showing anomalous diffusion. In addition, the self-part of the van Hove function has a Gaussian form at short and long times, but becomes non-Gaussian at intermediates ones, having a crossover between short and large displacements. The corresponding self-intermediate scattering function shows the two-step relaxation patters typically observed in soft materials with heterogeneous dynamics such as glasses and gels. We also introduce a phenomenological Continuous Time Random Walk (CTRW) theory to understand the heterogeneous diffusion of this system. R-DDFT results are in excellent agreement with R-BD simulations and the analytical predictions of CTRW theory, thus confirming that R-DDFT constitutes a powerful method to investigate not only the structure and phase behavior, but also the dynamical properties of non-equilibrium active switching colloidal suspensions.

Interdomain exchange and the flip-flop of cholesterol in ternary component lipid membranes and their effects on heterogeneous cholesterol diffusion

Cell membranes are heterogeneous with a variety of lipids, cholesterol, and proteins and are composed of domains of different compositions. Such heterogeneous environments make the transport of cholesterol complicated: cholesterol not only diffuses within a particular domain but also travels between domains. Cholesterol also flip-flops between upper and lower leaflets such that cholesterol may reside both within leaflets and in the central region between two leaflets. How the presence of multiple domains and the interdomain exchange of cholesterol would affect the cholesterol transport, however, remains elusive. In this study, therefore, we perform molecular dynamics simulations up to 100μs for ternary component lipid membranes, which consist of saturated lipids (dipalmitoylphosphatidylcholine, DPPC), unsaturated lipids (dilinoleylphosphatidylcholine, DIPC), and cholesterol. The ternary component membranes in our simulations form two domains readily: DPPC and DIPC domains. We find that the diffusion of cholesterol molecules is much more heterogeneous and non-Gaussian than expected for binary component lipid membranes of lipids and cholesterol. The non-Gaussian parameter of the cholesterol molecules is about four times larger in the ternary component lipid membranes than in the binary component lipid membranes. Such non-Gaussian and heterogeneous transport of cholesterol arises from the interplay among the interdomain kinetics, the different diffusivity of cholesterol in different domains, and the flip-flop of cholesterol. This suggests that in cell membranes that consist of various domains and proteins, the cholesterol transport can be very heterogeneous. We also find that the mechanism of the interdomain exchange differs for different domains: cholesterol tends to exit the DIPC domain along the central region of the membrane for the DIPC-to-DPPC transition, while the cholesterol is likely to exit the DPPC domain within the membrane leaflet for the DPPC-to-DIPC transition. Also, the interdomain exchange kinetics of cholesterol for the DPPC-to-DIPC transition is up to 7.9 times slower than the DIPC-to-DPPC transition.

Tracer Diffusion in Tightly-Meshed Homogeneous Polymer Networks: A Brownian Dynamics Simulation Study

We report Brownian dynamics simulations of tracer diffusion in regularly crosslinked polymer networks in order to elucidate the transport of a tracer particle in polymer networks. The average mesh size of homogeneous polymer networks is varied by assuming different degrees of crosslinking or swelling, and the size of a tracer particle is comparable to the average mesh size. Simulation results show subdiffusion of a tracer particle at intermediate time scales and normal diffusion at long times. In particular, the duration of subdiffusion is significantly prolonged as the average mesh size decreases with increasing degree of crosslinking, for which long-time diffusion occurs via the hopping processes of a tracer particle after undergoing rattling motions within a cage of the network mesh for an extended period of time. On the other hand, the cage dynamics and hopping process are less pronounced as the mesh size decreases with increasing polymer volume fractions. The interpretation is provided in terms of fluctuations in network mesh size: at higher polymer volume fractions, the network fluctuations are large enough to allow for collective, structural changes of network meshes, so that a tracer particle can escape from the cage, whereas, at lower volume fractions, the fluctuations are so small that a tracer particle remains trapped within the cage for a significant period of time before making infrequent jumps out of the cage. This work suggests that fluctuation in mesh size, as well as average mesh size itself, plays an important role in determining the dynamics of molecules and nanoparticles that are embedded in tightly meshed polymer networks.

Escape of a passive particle from an activity-induced energy landscape: emergence of slow and fast effective diffusion

Spontaneous persistent motions driven by active processes play a central role in maintaining living cells far from equilibrium. In the majority of research studies, the steady state dynamics of an active system has been described in terms of an effective temperature. By contrast, we have examined a prototype model for diffusion in an activity-induced rugged energy landscape to describe the slow dynamics of a tagged particle in a dense active environment. The expression for the mean escape time from the activity-induced rugged energy landscape holds only in the limit of low activity and the mean escape time from the rugged energy landscape increases with activity. The precise form of the active correlation will determine whether the mean escape time will depend on the persistence time or not. The activity-induced rugged energy landscape approach also allows an estimate of the non-equilibrium effective diffusivity characterizing the slow diffusive motion of the tagged particle due to activity. On the other hand, in a dilute environment, high activity augments the diffusion of the tagged particle. The enhanced diffusion can be attributed to an effective temperature higher than the ambient temperature and this is used to calculate the Kramers' mean escape time, which decreases with activity. Our results have direct relevance to recent experiments on tagged particle diffusion in condensed phases.

Non-Gaussian subdiffusion of single-molecule tracers in a hydrated polymer network

Single molecule tracking experiments inside a hydrated polymer network have shown that the tracer motion is subdiffusive due to the viscoelastic environment inside the gel-like network. This property can be related to the negative autocorrelation of the instantaneous displacements at short times. Although the displacements of the individual tracers exhibit Gaussian statistics, the displacement distribution of all the trajectories combined from different spatial locations of the polymer network exhibits a non-Gaussian distribution. Here, we analyze many individual tracer trajectories to show that the central portion of the non-Gaussian distribution can be well approximated by an exponential distribution that spreads sublinearly with time. We explain all these features seen in the experiment by a generalized Langevin model for an overdamped particle with algebraically decaying correlations. We show that the degree of non-Gaussianity can change with the extent of heterogeneity, which is controlled in our model by the experimentally observed distributions of the motion parameters.

Dynamic coupling of a hydration layer to a fluid phospholipid membrane: intermittency and multiple time-scale relaxations

Understanding the coupling of a hydration layer and a lipid membrane is crucial to gaining access to membrane dynamics and understanding its functionality towards various biological processes.

Transport of probe particles in a polymer network: effects of probe size, network rigidity and probe-polymer interaction

Fundamental understanding of the effect of microscopic parameters on the dynamics of probe particles in different complex environments has wide implications. Examples include diffusion of proteins in biological hydrogels, porous media, polymer matrix, etc. Here, we use extensive molecular dynamics simulations to investigate the dynamics of the probe particle in a polymer network on a diamond lattice, which provides substantial crowding to mimic the cellular environment. Our simulations show that the dynamics of the probe increasingly becomes restricted, non-Gaussian and subdiffusive on increasing the network rigidity, binding affinity and probe size. In addition, the velocity autocorrelation functions show negative dips owing to the viscoelasticity and caging due to the surrounding network. These observations go with the general experimental findings. Importantly, for a probe particle of size comparable to the mesh size, unrestricted motion engulfing large length scales has been observed. This happens with a more flexible polymer network, which is easily pushed by the bigger probe. On increasing the rigidity of the network, the bigger probe can not efficiently push the network and as a result the long tail disappears. Our study gives a general qualitative picture of the transport of probes in a gel-like medium, as encountered in different contexts.

Spatial Dependence of Non-Gaussian Diffusion of Nanoparticles in Free-Standing Thin Polymer Films

The addition of nanoparticles (NPs) to a free-standing polymer film affects the properties of the film such as viscosity and glass transition temperature. Recent experiments, for example, showed that the glass transition temperature of thin polymer films was dependent on how NPs were distributed within the polymer films. However, the spatial arrangement of NPs in free-standing polymer films and its effect on the diffusion of NPs and polymers remain elusive at a molecular level. In this study, we employ generic coarse-grained models for polymers and NPs and perform extensive molecular dynamics simulations to investigate the diffusion of polymers and NPs in free-standing thin polymer films. We find that small NPs are likely to stay at the interfacial region of the polymer film, while large NPs tend to stay at the center of the film. On the other hand, as the interaction between a NP and a monomer becomes more attractive, the NP is more likely to be placed at the film center. The diffusion of monomers slows down slightly as more NPs are added to the film. Interestingly, the NP diffusion is dependent strongly on the spatial arrangement of the NPs: NPs at the interfacial region diffuse faster and undergo more non-Gaussian diffusion than NPs at the film center, which implies that the interfacial region would be more mobile and dynamically heterogeneous than the film center. We also find that the mechanism for non-Gaussian diffusion of NPs at the film center differs from that at the interfacial region and that the NP diffusion would reflect the local viscosity of the polymer films.

Fickian yet non-Gaussian diffusion in two-dimensional Yukawa liquids

We investigate Fickian diffusion in two-dimensional (2D) Yukawa liquids using molecular dynamics simulations. We compute the self-van Hove correlation function G_{s}(r,t) and the self-intermediate scattering function F_{s}(k,t), and we compare these functions with those obtained from mean-squared displacement (MSD) using the Gaussian approximation. According to this approximation, a linear MSD with time implies a Gaussian behavior for G_{s}(r,t) and F_{s}(k,t) at all times. Surprisingly, we find that these functions deviate from Gaussian at intermediate timescales, indicating the failure of the Gaussian approximation. Furthermore, we quantify these deviations by the non-Gaussian parameter, and we find that the deviations increase when the temperature of the liquid decreases. The origin of the non-Gaussian behavior may be the heterogeneous dynamics of dust particles observed in 2D Yukawa liquids.

Anomalous transport in the soft-sphere Lorentz model

The sensitivity of anomalous transport in crowded media to the form of the inter-particle interactions is investigated through computer simulations. We extend the highly simplified Lorentz model towards realistic natural systems by modeling the interactions between the tracer and the obstacles with a smooth potential. We find that the anomalous transport at the critical point happens to be governed by the same universal exponent as for hard exclusion interactions, although the mechanism of how narrow channels are probed is rather different. The scaling behavior of simulations close to the critical point confirm this exponent. Our result indicates that the simple Lorentz model may be applicable to describing the fundamental properties of long-range transport in real crowded environments.

Facilitated and Non-Gaussian Diffusion of Cholesterol in Liquid Ordered Phase Bilayers Depends on the Flip-Flop and Spatial Arrangement of Cholesterol

The diffusion of cholesterol in biological membranes is critical to cellular processes such as the formation of cholesterol-enriched domains. The cholesterol diffusion may be complicated especially when cholesterol flip-flops and/or stays at the membrane center. Understanding the diffusion mechanism of cholesterol at a molecular level should be, therefore, a topic of interest. We perform molecular dynamics simulations up to 100 μs for lipid bilayers with various concentrations of cholesterol. We find that cholesterol diffusion in the liquid ordered phase depends on whether it is within leaflets or at the bilayer center, is non-Gaussian for several microseconds, and is enhanced significantly compared to that of lipids. Cholesterol at the bilayer center diffuses fast, while cholesterol in the hydrocarbon region with upright orientation diffuses relatively slowly. Such position-dependent dynamics of cholesterol leads to facilitated and non-Gaussian diffusion.

Anomalous dynamics in tracer-particle motions in an electrohydrodynamically driven oil-in-oil system

We characterize the superdiffusive dynamics of tracer particles in an electrohydrodynamically driven emulsion of oil droplets in an immiscible oil medium, where the amplitude and frequency of an external electric field are the control parameters. In the weakly driven electrohydrodynamic regime, the droplets are trapped dielectrophoretically on a patterned electrode, and the driving is therefore spatially varying. We find excellent agreement with a 〈x^{2}〉∼t^{1.5} power law and find that this superdiffusive dynamics arises from an underlying displacement distribution that is distinctly non-Gaussian and exponential for small displacements and short times. While these results are comparable with a random-velocity field model, the tracer particle speeds are in fact spatially varying in two dimensions, arising from a spatially varying electrohydrodynamic driving force. This suggests that the important ingredient for the superdiffusive t^{1.5} behavior observed is a velocity field that is isotropic in the plane and spatially correlated. Finally, we can extract, from the superdiffusive dynamics, a experimental length scale that corresponds to the lateral range of the hydrodynamic flows. This experimental length scale is non zero only above a threshold ion mobility length.

Fickian yet non-Gaussian behaviour: A dominant role of the intermittent dynamics

We present a study of the dynamics of small solute particles in a solvent medium where the solute is much smaller in size, mimicking the diffusion of small particles in crowded environment. The solute exhibits Fickian diffusion arising from non-Gaussian Van Hove correlation function. Our study shows that there are at least two possible origins of this non-Gaussian behaviour: the decoupling of the solute-solvent dynamics and the intermittency in the solute motion, the latter playing a dominant role. In the former scenario when averaged over time long enough to explore different solvent environments, the dynamics recovers the Gaussian nature. In the case of intermittent dynamics the non-Gaussianity remains even after long averaging and the Gaussian behaviour is obtained at a much longer time. Our study further shows that only for an intermediate attractive solute-solvent interaction the dynamics of the solute is intermittent. The intermittency disappears for weaker or stronger attractions.

CO2 Diffusion in Various Carbonated Beverages: A Molecular Dynamics Study

Carbonated beverages are widely enjoyed in spare time, yet there remain many physical and chemical processes clouded at the molecular level. In this report, we employ molecular dynamics simulations to estimate the diffusion coefficients of CO₂ and the molecular origin of its variations in three model systems with characteristic features of champagnes, sugar-based cola drinks, and club sodas. The computed diffusion coefficients of CO₂ are in good agreement with experimental data. Analyses of hydrogen bonding and the solvent's structural and dynamic properties reveal that the change of CO₂ diffusion coefficient is closely associated with the diffusional behavior of the solvent water itself, as a result of changes in the number and strength of hydrogen bonding interactions among the species and solvent.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Resolving mixed mechanisms of protein subdiffusion at the T cell plasma membrane

The plasma membrane is a complex medium where transmembrane proteins diffuse and interact to facilitate cell function. Membrane protein mobility is affected by multiple mechanisms, including crowding, trapping, medium elasticity and structure, thus limiting our ability to distinguish them in intact cells. Here we characterize the mobility and organization of a short transmembrane protein at the plasma membrane of live T cells, using single particle tracking and photoactivated-localization microscopy. Protein mobility is highly heterogeneous, subdiffusive and ergodic-like. Using mobility characteristics, we segment individual trajectories into subpopulations with distinct Gaussian step-size distributions. Particles of low-to-medium mobility consist of clusters, diffusing in a viscoelastic and fractal-like medium and are enriched at the centre of the cell footprint. Particles of high mobility undergo weak confinement and are more evenly distributed. This study presents a methodological approach to resolve simultaneous mixed subdiffusion mechanisms acting on polydispersed samples and complex media such as cell membranes.

Membrane protein diffusion is affected by distinct mechanisms such as molecular crowding and medium elasticity. Here the authors present an analytical approach to analyse single particle trajectories and distinguish mixed subdiffusive processes affecting membrane protein mobility in living cells.

Diffusing diffusivity: Rotational diffusion in two and three dimensions

We consider the problem of calculating the probability distribution function (pdf) of angular displacement for rotational diffusion in a crowded, rearranging medium. We use the diffusing diffusivity model and following our previous work on translational diffusion [R. Jain and K. L. Sebastian, J. Phys. Chem. B 120, 3988 (2016)], we show that the problem can be reduced to that of calculating the survival probability of a particle undergoing Brownian motion, in the presence of a sink. We use the approach to calculate the pdf for the rotational motion in two and three dimensions. We also propose new dimensionless, time dependent parameters, αrot,2D and αrot,3D, which can be used to analyze the experimental/simulation data to find the extent of deviation from the normal behavior, i.e., constant diffusivity, and obtain explicit analytical expressions for them, within our model.

Single Particle Tracking: From Theory to Biophysical Applications

After three decades of developments, single particle tracking (SPT) has become a powerful tool to interrogate dynamics in a range of materials including live cells and novel catalytic supports because of its ability to reveal dynamics in the structure-function relationships underlying the heterogeneous nature of such systems. In this review, we summarize the algorithms behind, and practical applications of, SPT. We first cover the theoretical background including particle identification, localization, and trajectory reconstruction. General instrumentation and recent developments to achieve two- and three-dimensional subdiffraction localization and SPT are discussed. We then highlight some applications of SPT to study various biological and synthetic materials systems. Finally, we provide our perspective regarding several directions for future advancements in the theory and application of SPT.

Lévy flight with absorption: A model for diffusing diffusivity with long tails

We consider diffusion of a particle in rearranging environment, so that the diffusivity of the particle is a stochastic function of time. In our previous model of "diffusing diffusivity" [Jain and Sebastian, J. Phys. Chem. B 120, 3988 (2016)JPCBFK1520-610610.1021/acs.jpcb.6b01527], it was shown that the mean square displacement of particle remains Fickian, i.e., 〈x^{2}(T)〉∝T at all times, but the probability distribution of particle displacement is not Gaussian at all times. It is exponential at short times and crosses over to become Gaussian only in a large time limit in the case where the distribution of D in that model has a steady state limit which is exponential, i.e., π_{e}(D)∼e^{-D/D_{0}}. In the present study, we model the diffusivity of a particle as a Lévy flight process so that D has a power-law tailed distribution, viz., π_{e}(D)∼D^{-1-α} with 0<α<1. We find that in the short time limit, the width of displacement distribution is proportional to sqrt[T], implying that the diffusion is Fickian. But for long times, the width is proportional to T^{1/2α} which is a characteristic of anomalous diffusion. The distribution function for the displacement of the particle is found to be a symmetric stable distribution with a stability index 2α which preserves its shape at all times.

Population dynamics, information transfer, and spatial organization in a chemical reaction network under spatial confinement and crowding conditions

We investigate, via Brownian dynamics simulations, the reaction dynamics of a generic, nonlinear chemical network under spatial confinement and crowding conditions. In detail, the Willamowski-Rossler chemical reaction system has been "extended" and considered as a prototype reaction-diffusion system. Our results are potentially relevant to a number of open problems in biophysics and biochemistry, such as the synthesis of primitive cellular units (protocells) and the definition of their role in the chemical origin of life and the characterization of vesicle-mediated drug delivery processes. More generally, the computational approach presented in this work makes the case for the use of spatial stochastic simulation methods for the study of biochemical networks in vivo where the "well-mixed" approximation is invalid and both thermal and intrinsic fluctuations linked to the possible presence of molecular species in low number copies cannot be averaged out.

Tracer diffusion in a sea of polymers with binding zones: mobile vs. frozen traps

We use molecular dynamics simulations to investigate the tracer diffusion in a sea of polymers with specific binding zones for the tracer. These binding zones act as traps. Our simulations show that the tracer can undergo normal yet non-Gaussian diffusion under certain circumstances, e.g., when the polymers with traps are frozen in space and the volume fraction and the binding strength of the traps are moderate. In this case, as the tracer moves, it experiences a heterogeneous environment and exhibits confined continuous time random walk (CTRW) like motion resulting in a non-Gaussian behavior. Also the long time dynamics becomes subdiffusive as the number or the binding strength of the traps increases. However, if the polymers are mobile then the tracer dynamics is Gaussian but could be normal or subdiffusive depending on the number and the binding strength of the traps. In addition, with increasing binding strength and number of polymer traps, the probability of the tracer being trapped increases. On the other hand, removing the binding zones does not result in trapping, even at comparatively high crowding. Our simulations also show that the trapping probability increases with the increasing size of the tracer and for a bigger tracer with the frozen polymer background the dynamics is only weakly non-Gaussian but highly subdiffusive. Our observations are in the same spirit as found in many recent experiments on tracer diffusion in polymeric materials and question the validity of using Gaussian theory to describe diffusion in a crowded environment in general.

Abnormal polymer transport in crowded attractive micropost arrays

We investigate polymer diffusion in a quasi-two-dimensional environment decorated with attractive cylindrical posts using Langevin dynamics simulation. We find that the polymer diffusivity has non-monotonic dependence on the post array density. This diffusive behavior strongly depends on the adsorption-desorption transition and the critical adsorption strength ε_c. For ε < ε_c, the polymer undergoes normal diffusion and the diffusivity decreases as the post density increases due to the reduction of the void volume. For ε > ε_c, polymer dynamics is strongly mediated by post adsorption, and we observe a regime where the polymer diffusivity increases as the post density increases. The polymer diffusivity reaches a maximum, which can be attributed to cross-post translation enabled by large polymer conformation fluctuations. We find both cross-post transport and polymer conformation fluctuations strongly depend on the post absorption strength and the chain length.

Dynamics of highly polydisperse colloidal suspensions as a model system for bacterial cytoplasm

There are various kinds of macromolecules in bacterial cell cytoplasm. The size polydispersity of the macromolecules is so significant that the crystallization and the phase separation could be suppressed, thus stabilizing the liquid state of bacterial cytoplasm. On the other hand, recent experiments suggested that the macromolecules in bacterial cytoplasm should exhibit glassy dynamics, which should be also affected significantly by the size polydispersity of the macromolecules. In this work, we investigate the anomalous and slow dynamics of highly polydisperse colloidal suspensions, of which size distribution is chosen to mimic Escherichia coli cytoplasm. We find from our Langevin dynamics simulations that the diffusion coefficient (D_{tot}) and the displacement distribution functions (P(r,t)) averaged over all colloids of different sizes do not show anomalous and glassy dynamic behaviors until the system volume fraction ϕ is increased up to 0.82. This indicates that the intrinsic polydispersity of bacterial cytoplasm should suppress the glass transition and help maintain the liquid state of the cytoplasm. On the other hand, colloids of each kind show totally different dynamic behaviors depending on their size. The dynamics of colloids of different size becomes non-Gaussian at a different range of ϕ, which suggests that a multistep glass transition should occur. The largest colloids undergo the glass transition at ϕ=0.65, while the glass transition does not occur for smaller colloids in our simulations even at the highest value of ϕ. We also investigate the distribution (P(θ,t)) of the relative angles of displacement for macromolecules and find that macromolecules undergo directionally correlated motions in a sufficiently dense system.

Characterizing anomalous diffusion in crowded polymer solutions and gels over five decades in time with variable-lengthscale fluorescence correlation spectroscopy

The diffusion of macromolecules in cells and in complex fluids is often found to deviate from simple Fickian diffusion. One explanation offered for this behavior is that molecular crowding renders diffusion anomalous, where the mean-squared displacement of the particles scales as 〈r(2)〉∝t(α) with α < 1. Unfortunately, methods such as fluorescence correlation spectroscopy (FCS) or fluorescence recovery after photobleaching (FRAP) probe diffusion only over a narrow range of lengthscales and cannot directly test the dependence of the mean-squared displacement (MSD) on time. Here we show that variable-lengthscale FCS (VLS-FCS), where the volume of observation is varied over several orders of magnitude, combined with a numerical inversion procedure of the correlation data, allows retrieving the MSD for up to five decades in time, bridging the gap between diffusion experiments performed at different lengthscales. In addition, we show that VLS-FCS provides a way to assess whether the propagator associated with the diffusion is Gaussian or non-Gaussian. We used VLS-FCS to investigate two systems where anomalous diffusion had been previously reported. In the case of dense cross-linked agarose gels, the measured MSD confirmed that the diffusion of small beads was anomalous at short lengthscales, with a cross-over to simple diffusion around ≈1 μm, consistent with a caged diffusion process. On the other hand, for solutions crowded with marginally entangled dextran molecules, we uncovered an apparent discrepancy between the MSD, found to be linear, and the propagators at short lengthscales, found to be non-Gaussian. These contradicting features call to mind the "anomalous, yet Brownian" diffusion observed in several biological systems, and the recently proposed "diffusing diffusivity" model.

On the Gaussian approximation in colloidal hard sphere fluids

We study the behaviour of the self-intermediate scattering function and self-van Hove correlation function for quasi-two-dimensional colloidal hard sphere fluids at a range of area fractions. We compute these functions first directly from the particle coordinates and secondly from the mean squared displacement via the Gaussian approximation. This allows us to test the validity of this approximation over a range of length and time scales, where we find that the Gaussian approximation holds if the hydrodynamic limits are appropriately probed. Surprisingly, only small deviations from Gaussian behaviour are seen at intermediate times, even for dense fluids. We next consider these deviations from Gaussian behaviour firstly via the non-Gaussian parameter and secondly by considering the relaxation times of the intermediate scattering function. From these measurements we develop a scaling relation in order to directly determine the combinations of wavevectors and times at which the non-Gaussian behavior is seen. This allows for the clear identification of the hydrodynamic regimes and thus provides new insight into the crossover between long- and short-time self-diffusion.

Diffusion in a Crowded, Rearranging Environment

It has been found in many experiments that the mean square displacement of a Brownian particle x(T) diffusing in a rearranging environment is strictly Fickian, obeying ⟨(x(T))(2)⟩ ∝ T, but the probability distribution function for the displacement is not Gaussian. An explanation of this is that the diffusivity of the particle itself is changing as a function of time. Models for this diffusing diffusivity have been solved analytically in the limit of small time, but simulations were necessary for intermediate and large times. We show that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and introduce a class of models for which it is possible to find analytical solutions. Our solution gives ⟨(x(T))(2)⟩ ∝ T for all times and at short times the probability distribution function of the displacement is exponential which crosses over to a Gaussian in the limit of long times and large displacements.

Probing Non-Gaussianity in Confined Diffusion of Nanoparticles

Confined diffusion is ubiquitous in nature. Ever since the "anomalous yet Brownian" motion was observed, the non-Gaussianity in confined diffusion has been unveiled as an important issue. In this Letter, we experimentally investigate the characteristics and source of non-Gaussian behavior in confined diffusion of nanoparticles suspended in polymer solutions. A time-varied and size-dependent non-Gaussianity is reported based on the non-Gaussian parameter and displacement probability distribution, especially when the nanoparticle's size is smaller than the typical polymer mesh size. This non-Gaussianity does not vanish even at the long-time Brownian stage. By inspecting the displacement autocorrelation, we observe that the nanoparticle-structure interaction, indicated by the anticorrelation, is limited in the short-time stage and makes little contribution to the non-Gaussianity in the long-time stage. The main source of the non-Gaussianity can therefore be attributed to hopping diffusion that results in an exponential probability distribution with the large displacements, which may also explain certain processes dominated by rare events in the biological environment.

Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes

We investigate the diffusive and ergodic properties of massive and confined particles in a model disordered medium, in which the local diffusivity fluctuates in time while its mean has a power law dependence on the diffusion time.

Diffusive dynamics of nanoparticles in ultra-confined media

Differential dynamic microscopy (DDM) was used to investigate the diffusive dynamics of nanoparticles of diameter 200-400 nm that were strongly confined in a periodic square array of cylindrical nanoposts. The minimum distance between posts was 1.3-5 times the diameter of the nanoparticles. The image structure functions obtained from the DDM analysis were isotropic and could be fit by a stretched exponential function. The relaxation time scaled diffusively across the range of wave vectors studied, and the corresponding scalar diffusivities decreased monotonically with increased confinement. The decrease in diffusivity could be described by models for hindered diffusion that accounted for steric restrictions and hydrodynamic interactions. The stretching exponent decreased linearly as the nanoparticles were increasingly confined by the posts. Together, these results are consistent with a picture in which strongly confined nanoparticles experience a heterogeneous spatial environment arising from hydrodynamics and volume exclusion on time scales comparable to cage escape, leading to multiple relaxation processes and Fickian but non-Gaussian diffusive dynamics.

Differential dynamic microscopy of weakly scattering and polydisperse protein-rich clusters

Nanoparticle dynamics impact a wide range of biological transport processes and applications in nanomedicine and natural resource engineering. Differential dynamic microscopy (DDM) was recently developed to quantify the dynamics of submicron particles in solutions from fluctuations of intensity in optical micrographs. Differential dynamic microscopy is well established for monodisperse particle populations, but has not been applied to solutions containing weakly scattering polydisperse biological nanoparticles. Here we use bright-field DDM (BDDM) to measure the dynamics of protein-rich liquid clusters, whose size ranges from tens to hundreds of nanometers and whose total volume fraction is less than 10(-5). With solutions of two proteins, hemoglobin A and lysozyme, we evaluate the cluster diffusion coefficients from the dependence of the diffusive relaxation time on the scattering wave vector. We establish that for weakly scattering populations, an optimal thickness of the sample chamber exists at which the BDDM signal is maximized at the smallest sample volume. The average cluster diffusion coefficient measured using BDDM is consistently lower than that obtained from dynamic light scattering at a scattering angle of 90°. This apparent discrepancy is due to Mie scattering from the polydisperse cluster population, in which larger clusters preferentially scatter more light in the forward direction.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.