Anomalous diffusion due to hindering by mobile obstacles undergoing Brownian motion or Orstein-Ulhenbeck processes

Simulation Studies of Dynamical Heterogeneity in a Dense Two-Dimensional Dimer-Solvent System with Obstacles

A coarse-grained model of a two-dimensional colloidal suspension was designed. The model was athermal and, in addition, a lattice approximation was introduced. It consisted of solvent (monomer) molecules, dimer molecules, and immobile impenetrable obstacles that introduced additional heterogeneity into the system. Dynamic properties were determined by a Monte Carlo simulation using the dynamic lattice liquid simulation algorithm. It is shown that there is a range of obstacle concentrations in which different diffusion characteristics were observed for dimers and solvents. In the system studied, it is possible to define the ranges of concentrations of individual components (solvent, dimers, and obstacles), in which the nature of the movement of dimers and solvents is different (normal diffusion vs. subdiffusion). The ratio of diffusion coefficients of solvent molecules and dimers for short times does not depend on the concentration of obstacles, while for long times, the ratio increases but remains independent of the concentration of the dimer.

Simulation study on electroporation of cancer cells in multicellular system

Electroporation (EP) of the normal cell and cancer cell both in single-cell and multicellular models was investigated by the meshed transport network method (MTNM) in this paper. The simulation results suggest that the cancer cell undergoes faster and more significant local EP than that of the corresponding normal cell induced by nanosecond pulsed electric fields (nsPEFs) both in single-cell and multicellular models. Furthermore, the results of the multicellular model indicate that there is a unidirectional neighboring effect in the multicellular model, meaning that cells at the center are affected and their pore formation is significantly reduced, but this effect is very weak for cells at the edges of the system. This means that the electric field selectively kills cells in different distribution locations. This work can provide guidance for the selection of parameters for the cancer cell EP process.

Amino acid-dependent phase equilibrium and material properties of tetrapeptide condensates

The rules of how amino acids dictate the physical properties of biomolecular condensates are still incomplete. Here, we study condensates formed by tetrapeptides of the form XXssXX. Eight peptides form four types of condensates at different concentrations and pHs: droplets (X = F, L, M, P, V, and A), amorphous dense liquids (X = L, M, P, V, and A), amorphous aggregates (X = W), and gels (X = I, V, and A). The peptides exhibit differences in phase equilibrium and material properties, including a 368-fold range in the threshold concentration for phase separation and a 3,856-fold range in viscosity. All-atom molecular dynamics simulations provide physical explanations of these results. The present work also reveals widespread critical behaviors-including critical slowing down manifested by amorphous dense liquids and critical scaling obeyed by fusion speed-with broad implications for condensate functions.

Cell-Size Space Regulates the Behavior of Confined Polymers: From Nano- and Micromaterials Science to Biology

Polymer micromaterials in a liquid or gel phase covered with a surfactant membrane are widely used materials in pharmaceuticals, cosmetics, and foods. In particular, cell-sized micromaterials of biopolymer solutions covered with a lipid membrane have been studied as artificial cells to understand cells from a physicochemical perspective. The characteristics and phase transitions of polymers confined to a microscopic space often differ from those in bulk systems. The effect that causes this difference is referred to as the cell-size space effect (CSE), but the specific physicochemical factors remain unclear. This study introduces the analysis of CSE on molecular diffusion, nanostructure transition, and phase separation and presents their main factors, i.e., short- and long-range interactions with the membrane surface and small volume (finite element nature). This serves as a guide for determining the dominant factors of CSE. Furthermore, we also introduce other factors of CSE such as spatial closure and the relationships among space size, the characteristic length of periodicity, the structure size, and many others produced by biomolecular assemblies through the analysis of protein reaction-diffusion systems and biochemical reactions.

Protein-coated nanoparticles exhibit Lévy flights on a suspended lipid bilayer

Lateral diffusion of nano-objects on lipid membranes is a crucial process in cell biology. Recent studies indicate that nanoparticle lateral diffusion is affected by the presence of membrane proteins and deviates from Brownian motion. Gold nanoparticles (Au NPs) stabilized by short thiol ligands were dispersed near a free-standing bilayer formed in a 3D microfluidic chip. Using dark-field microscopy, the position of single NPs at the bilayer surface was tracked over time. Numerical analysis of the NP trajectories shows that NP diffusion on the bilayer surface corresponds to Brownian motion. The addition of bovine serum albumin (BSA) protein to the solution led to the formation of a protein corona on the NP surface. We found that protein-coated NPs show anomalous superdiffusion and that the distribution of their relative displacement obeys Lévy flight statistics. This superdiffusive motion is attributed to a drastic reduction in adhesive energies between the NPs and the bilayer in the presence of the protein corona. This hypothesis was confirmed by numerical simulations mimicking the random walk of a single particle near a weakly adhesive surface. These results may be generalized to other classes of nano-objects that experience adsorption-desorption behaviour with a weakly adhesive surface.

Normal and anomalous diffusion in the disordered wind-tree model

Ehrenfests' wind-tree model (EWTM) refers to a two-dimensional system where noninteracting point tracer particles move through a random arrangement of overlapping or nonoverlapping square-shaped scatterers. Here, extensive event-driven molecular dynamics simulations of the EWTM at different reduced scatterer densities ρ are presented. For nonoverlapping scatterers, the asymptotic motion of the tracer particles is diffusive. We compare their diffusion coefficient D, as obtained from the simulation, with that predicted by kinetic theory where D^{-1} is expanded up to the second order in the scatterer density. While at low density quantitative agreement between theory and simulation is found, we show that beyond the low-density regime deviations to the theory are associated with the emergence of a maximum in the non-Gaussian parameter at intermediate times. For the case of overlapping scatterers, in agreement with a theoretical prediction, the asymptotic motion of the tracer particles is subdiffusive, i.e., the mean-squared displacement at long times t grows like t^{1-2ρ/3}. We propose a model of the van Hove correlation function that describes the density dependence of the tracer particles' asymptotic subdiffusive transport on a quantitative level.

An overview of diffusion models for intracellular dynamics analysis

We present an overview of diffusion models commonly used for quantifying the dynamics of intracellular particles (e.g. biomolecules) inside eukaryotic living cells. It is established that inference on the modes of mobility of molecules is central in cell biology since it reflects interactions between structures and determines functions of biomolecules in the cell. In that context, Brownian motion is a key component in short distance transportation (e.g. connectivity for signal transduction). Another dynamical process that has been heavily studied in the past decade is the motor-mediated transport (e.g. dynein, kinesin and myosin) of molecules. Primarily supported by actin filament and microtubule network, it ensures spatial organization and temporal synchronization in the intracellular mechanisms and structures. Nevertheless, the complexity of internal structures and molecular processes in the living cell influence the molecular dynamics and prevent the systematic application of pure Brownian or directed motion modeling. On the one hand, cytoskeleton density will hinder the free displacement of the particle, a phenomenon called subdiffusion. On the other hand, the cytoskeleton elasticity combined with thermal bending can contribute a phenomenon called superdiffusion. This paper discusses the basics of diffusion modes observed in eukariotic cells, by introducing the essential properties of these processes. Applications of diffusion models include protein trafficking and transport and membrane diffusion.

Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.

Classification of particle trajectories in living cells: Machine learning versus statistical testing hypothesis for fractional anomalous diffusion

Single-particle tracking (SPT) has become a popular tool to study the intracellular transport of molecules in living cells. Inferring the character of their dynamics is important, because it determines the organization and functions of the cells. For this reason, one of the first steps in the analysis of SPT data is the identification of the diffusion type of the observed particles. The most popular method to identify the class of a trajectory is based on the mean-square displacement (MSD). However, due to its known limitations, several other approaches have been already proposed. With the recent advances in algorithms and the developments of modern hardware, the classification attempts rooted in machine learning (ML) are of particular interest. In this work, we adopt two ML ensemble algorithms, i.e., random forest and gradient boosting, to the problem of trajectory classification. We present a new set of features used to transform the raw trajectories data into input vectors required by the classifiers. The resulting models are then applied to real data for G protein-coupled receptors and G proteins. The classification results are compared to recent statistical methods going beyond MSD.

Detection of Anomalous Diffusion with Deep Residual Networks

Identification of the diffusion type of molecules in living cells is crucial to deduct their driving forces and hence to get insight into the characteristics of the cells. In this paper, deep residual networks have been used to classify the trajectories of molecules. We started from the well known ResNet architecture, developed for image classification, and carried out a series of numerical experiments to adapt it to detection of diffusion modes. We managed to find a model that has a better accuracy than the initial network, but contains only a small fraction of its parameters. The reduced size significantly shortened the training time of the model. Moreover, the resulting network has less tendency to overfitting and generalizes better to unseen data.

Enhanced Diffusive Transport in Fluctuating Porous Media

Mass transport within porous structures is a ubiquitous process in biological, geological, and technological systems. Despite the importance of these phenomena, there is no comprehensive theory that describes the complex and diverse transport behavior within porous environments. While the porous matrix itself is generally considered a static and passive participant, many porous environments are in fact dynamic, with fluctuating walls, pores that open and close, and dynamically changing cross-links. While diffusion has been measured in fluctuating structures, notably in model biological systems, it is rarely possible to isolate the effect of fluctuations because of the absence of control experiments involving an identical static counterpart, and it is generally impossible to observe the dynamics of the structure. Here we present a direct comparison of the diffusion of nanoparticles of various sizes within a trackable, fluctuating porous matrix and a geometrically equivalent static matrix, in conditions spanning a range of regimes from obstructed to highly confined. The experimental system comprised a close-packed layer of colloidal spheres that were either immobilized to a planar surface or allowed to fluctuate locally, within the space defined by their nearest neighbors. Interestingly, the effective long-time diffusion coefficient was approximately 35-65% greater in the fluctuating porous matrix than in the static one (depending on the size of the nanoparticle probes), regardless of the geometric regime. This was explained by considering the enhancing effects of matrix fluctuations on the short-time diffusion coefficient and cooperative "gate-opening" motions of matrix particles and nanoparticle probes.

Impact of Feature Choice on Machine Learning Classification of Fractional Anomalous Diffusion

The growing interest in machine learning methods has raised the need for a careful study of their application to the experimental single-particle tracking data. In this paper, we present the differences in the classification of the fractional anomalous diffusion trajectories that arise from the selection of the features used in random forest and gradient boosting algorithms. Comparing two recently used sets of human-engineered attributes with a new one, which was tailor-made for the problem, we show the importance of a thoughtful choice of the features and parameters. We also analyse the influence of alterations of synthetic training data set on the classification results. The trained classifiers are tested on real trajectories of G proteins and their receptors on a plasma membrane.

A sequential algorithm to detect diffusion switching along intracellular particle trajectories

MOTIVATION

Recent advances in molecular biology and fluorescence microscopy imaging have made possible the inference of the dynamics of single molecules in living cells. Changes of dynamics can occur along a trajectory. Then, an issue is to estimate the temporal change-points that is the times at which a change of dynamics occurs. The number of points in the trajectory required to detect such changes will depend on both the magnitude and type of the motion changes. Here, the number of points per trajectory is of the order of 102, even if in practice dramatic motion changes can be detected with less points.

RESULTS

We propose a non-parametric procedure based on test statistics computed on local windows along the trajectory to detect the change-points. This algorithm controls the number of false change-point detections in the case where the trajectory is fully Brownian. We also develop a strategy for aggregating the detections obtained with different window sizes so that the window size is no longer a parameter to optimize. A Monte Carlo study is proposed to demonstrate the performances of the method and also to compare the procedure to two competitive algorithms. At the end, we illustrate the efficacy of the method on real data in 2D and 3D, depicting the motion of mRNA complexes-called mRNA-binding proteins-in neuronal dendrites, Galectin-3 endocytosis and trafficking within the cell.

AVAILABILITY AND IMPLEMENTATION

A user-friendly Matlab package containing examples and the code of the simulations used in the paper is available at http://serpico.rennes.inria.fr/doku.php? id=software:cpanalysis:index.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Molecular transport in systems containing binding obstacles

We studied the movement of particles in crowded environments by means of extensive Monte Carlo simulations. The dynamic lattice liquid model was employed for this purpose. It is based on the cooperative movement concept and allows the study of systems at high densities. The cooperative model of molecular transport is assumed: the motion of all moving particles is highly correlated. The model is supposed to mimic lateral motion in a membrane and therefore the system is two-dimensional with moving objects and traps placed on a triangular lattice. In our study the interaction (binding) of traps with moving particles was assumed. The conditions in which subdiffusive motion appeared in the system were analysed. The influence of the strength of binding on the dynamic percolation threshold was also shown. The differences in the dynamics compared to systems with impenetrable obstacles and with systems without correlation in motion were presented and discussed. It was shown that in the case of correlated motion the influence of deep traps is similar to that of impenetrable obstacles. If the traps are shallow a recovery to normal diffusion was observed for longer time periods.

A novel computational framework for D(t) from Fluorescence Recovery after Photobleaching data reveals various anomalous diffusion types in live cell membranes

Diffusion of proteins and lipids in lipid membranes plays a pivotal role in almost all aspects of cellular biology, including motility, exo-/endocytosis and signal transduction. For this reason, gaining a detailed understanding of membrane structure and function has long been a major area of cell biology research. To better elucidate this structure-function relationship, various tools have been developed for diffusion measurements, including Fluorescence Recovery After Photobleaching (FRAP). Because of the complexity of cellular microenvironments, biological diffusion is often correlated over time and described by a time-dependent diffusion coefficient, D(t), although the underlying mechanisms are not fully understood. Since D(t) provides important information regarding cellular structures, such as the existence of subresolution barriers to diffusion, many efforts have been made to quantify D(t) by FRAP assuming a single power law, D(t) = Γt ^α - 1 where Γ and α are transport coefficient and anomalous exponent. However, straightforward approaches to quantify a general form of D(t) are lacking. In this study, we develop a novel mathematical and computational framework to compute the mean square displacement of diffusing molecules and diffusion coefficient D(t) from each individual time point of confocal FRAP data without the single power law assumption. Additionally, we developed an auxiliary equation for D(t) which can readily distinguish normal diffusion or single power law anomalous diffusion from other types of anomalous diffusion directly from FRAP data. Importantly, by applying this approach to FRAP data from a variety of membrane markers, we demonstrate the single power law anomalous diffusion assumption is not sufficient to describe various types of D(t) of membrane proteins. Lastly, we discuss how our new approaches can be applied to other fluorescence microscopy tools such as Fluorescence Correlation Spectroscopy (FCS) and Single Particle Tracking (SPT).

Single-Particle Diffusion Characterization by Deep Learning

Diffusion plays a crucial role in many biological processes including signaling, cellular organization, transport mechanisms, and more. Direct observation of molecular movement by single-particle-tracking experiments has contributed to a growing body of evidence that many cellular systems do not exhibit classical Brownian motion but rather anomalous diffusion. Despite this evidence, characterization of the physical process underlying anomalous diffusion remains a challenging problem for several reasons. First, different physical processes can exist simultaneously in a system. Second, commonly used tools for distinguishing between these processes are based on asymptotic behavior, which is experimentally inaccessible in most cases. Finally, an accurate analysis of the diffusion model requires the calculation of many observables because different transport modes can result in the same diffusion power-law α, which is typically obtained from the mean-square displacements (MSDs). The outstanding challenge in the field is to develop a method to extract an accurate assessment of the diffusion process using many short trajectories with a simple scheme that is applicable at the nonexpert level. Here, we use deep learning to infer the underlying process resulting in anomalous diffusion. We implement a neural network to classify single-particle trajectories by diffusion type: Brownian motion, fractional Brownian motion and continuous time random walk. Further, we demonstrate the applicability of our network architecture for estimating the Hurst exponent for fractional Brownian motion and the diffusion coefficient for Brownian motion on both simulated and experimental data. These networks achieve greater accuracy than time-averaged MSD analysis on simulated trajectories while only requiring as few as 25 steps. When tested on experimental data, both net and ensemble MSD analysis converge to similar values; however, the net needs only half the number of trajectories required for ensemble MSD to achieve the same confidence interval. Finally, we extract diffusion parameters from multiple extremely short trajectories (10 steps) using our approach.

Dynamics of proteins in solution

The dynamics of proteins in solution includes a variety of processes, such as backbone and side-chain fluctuations, interdomain motions, as well as global rotational and translational (i.e. center of mass) diffusion. Since protein dynamics is related to protein function and essential transport processes, a detailed mechanistic understanding and monitoring of protein dynamics in solution is highly desirable. The hierarchical character of protein dynamics requires experimental tools addressing a broad range of time- and length scales. We discuss how different techniques contribute to a comprehensive picture of protein dynamics, and focus in particular on results from neutron spectroscopy. We outline the underlying principles and review available instrumentation as well as related analysis frameworks.

Controlling Anomalous Diffusion in Lipid Membranes

Diffusion in cell membranes is not just simple two-dimensional Brownian motion but typically depends on the timescale of the observation. The physical origins of this anomalous subdiffusion are unresolved, and model systems capable of quantitative and reproducible control of membrane diffusion have been recognized as a key experimental bottleneck. Here, we control anomalous diffusion using supported lipid bilayers containing lipids derivatized with polyethylene glycol (PEG) headgroups. Bilayers with specific excluded area fractions are formed by control of PEG lipid mole fraction. These bilayers exhibit a switch in diffusive behavior, becoming anomalous as bilayer continuity is disrupted. Using a combination of single-molecule fluorescence and interferometric imaging, we measure the anomalous behavior in this model over four orders of magnitude in time. Diffusion in these bilayers is well described by a power-law dependence of the mean-square displacement with observation time. Anomaleity in this system can be tailored by simply controlling the mole fraction of PEG lipid, producing bilayers with diffusion parameters similar to those observed for anomalous diffusion in biological membranes.

Statistical analysis of particle trajectories in living cells

Recent advances in molecular biology and fluorescence microscopy imaging have made possible the inference of the dynamics of molecules in living cells. Such inference allows us to understand and determine the organization and function of the cell. The trajectories of particles (e.g., biomolecules) in living cells, computed with the help of object tracking methods, can be modeled with diffusion processes. Three types of diffusion are considered: (i) free diffusion, (ii) subdiffusion, and (iii) superdiffusion. The mean-square displacement (MSD) is generally used to discriminate the three types of particle dynamics. We propose here a nonparametric three-decision test as an alternative to the MSD method. The rejection of the null hypothesis, i.e., free diffusion, is accompanied by claims of the direction of the alternative (subdiffusion or superdiffusion). We study the asymptotic behavior of the test statistic under the null hypothesis and under parametric alternatives which are currently considered in the biophysics literature. In addition, we adapt the multiple-testing procedure of Benjamini and Hochberg to fit with the three-decision-test setting, in order to apply the test procedure to a collection of independent trajectories. The performance of our procedure is much better than the MSD method as confirmed by Monte Carlo experiments. The method is demonstrated on real data sets corresponding to protein dynamics observed in fluorescence microscopy.

Urea-mediated anomalous diffusion in supported lipid bilayers

Diffusion in biological membranes is seldom simply Brownian motion; instead, the rate of diffusion is dependent on the time scale of observation and so is often described as anomalous. In order to help better understand this phenomenon, model systems are needed where the anomalous diffusion of the lipid bilayer can be tuned and quantified. We recently demonstrated one such model by controlling the excluded area fraction in supported lipid bilayers (SLBs) through the incorporation of lipids derivatized with polyethylene glycol. Here, we extend this work, using urea to induce anomalous diffusion in SLBs. By tuning incubation time and urea concentration, we produce bilayers that exhibit anomalous behaviour on the same scale as that observed in biological membranes.

Non-specific interactions govern cytosolic diffusion of nanosized objects in mammalian cells

The diffusivity of macromolecules in the cytoplasm of eukaryotic cells varies over orders of magnitude and dictates the kinetics of cellular processes. However, a general description that associates the Brownian or anomalous nature of intracellular diffusion to the architectural and biochemical properties of the cytoplasm has not been achieved. Here we measure the mobility of individual fluorescent nanoparticles in living mammalian cells to obtain a comprehensive analysis of cytoplasmic diffusion. We identify a correlation between tracer size, its biochemical nature and its mobility. Inert particles with size equal or below 50 nm behave as Brownian particles diffusing in a medium of low viscosity with negligible effects of molecular crowding. Increasing the strength of non-specific interactions of the nanoparticles within the cytoplasm gradually reduces their mobility and leads to subdiffusive behaviour. These experimental observations and the transition from Brownian to subdiffusive motion can be captured in a minimal phenomenological model.

Confined Flow: Consequences and Implications for Bacteria and Biofilms

Bacteria overwhelmingly live in geometrically confined habitats that feature small pores or cavities, narrow channels, or nearby interfaces. Fluid flows through these confined habitats are ubiquitous in both natural and artificial environments colonized by bacteria. Moreover, these flows occur on time and length scales comparable to those associated with motility of bacteria and with the formation and growth of biofilms, which are surface-associated communities that house the vast majority of bacteria to protect them from host and environmental stresses. This review describes the emerging understanding of how flow near surfaces and within channels and pores alters physical processes that control how bacteria disperse, attach to surfaces, and form biofilms. This understanding will inform the development and deployment of technologies for drug delivery, water treatment, and antifouling coatings and guide the structuring of bacterial consortia for production of chemicals and pharmaceuticals.

Effects of soft interactions and bound mobility on diffusion in crowded environments: a model of sticky and slippery obstacles

Crowded environments modify the diffusion of macromolecules, generally slowing their movement and inducing transient anomalous subdiffusion. The presence of obstacles also modifies the kinetics and equilibrium behavior of tracers. While previous theoretical studies of particle diffusion have typically assumed either impenetrable obstacles or binding interactions that immobilize the particle, in many cellular contexts bound particles remain mobile. Examples include membrane proteins or lipids with some entry and diffusion within lipid domains and proteins that can enter into membraneless organelles or compartments such as the nucleolus. Using a lattice model, we studied the diffusive movement of tracer particles which bind to soft obstacles, allowing tracers and obstacles to occupy the same lattice site. For sticky obstacles, bound tracer particles are immobile, while for slippery obstacles, bound tracers can hop without penalty to adjacent obstacles. In both models, binding significantly alters tracer motion. The type and degree of motion while bound is a key determinant of the tracer mobility: slippery obstacles can allow nearly unhindered diffusion, even at high obstacle filling fraction. To mimic compartmentalization in a cell, we examined how obstacle size and a range of bound diffusion coefficients affect tracer dynamics. The behavior of the model is similar in two and three spatial dimensions. Our work has implications for protein movement and interactions within cells.

Gradient-driven diffusion and pattern formation in crowded mixtures

Gradient-driven diffusion in crowded, multicomponent mixtures is a topic of high interest because of its role in biological processes such as transport in cell membranes. In partially phase-separated solutions, gradient-driven diffusion affects microstructure, which in turn affects diffusivity; a key question is how this complex coupling controls both transport and pattern formation. To examine these mechanisms, we study a two-dimensional multicomponent lattice gas model, where "tracer" molecules diffuse between a source and a sink separated by a solution of sticky "crowder" molecules that cluster to form dynamically evolving obstacles. In the high-temperature limit, crowders and tracers are miscible, and transport may be predicted analytically. At intermediate temperatures, crowders phase separate into clusters that drift toward the tracer sink. As a result, steady-state tracer diffusivity depends nonmonotonically on both temperature and crowder density, and we observe a variety of complex microstructures. In the low-temperature limit, crowders rapidly aggregate to form obstacles that are kinetically arrested; if crowder density is near the percolation threshold, resulting tracer diffusivity shows scaling behavior with the same scaling exponent as the random resistor network model. Though highly idealized, this simple model reveals fundamental mechanisms governing coupled gradient-driven diffusion, phase separation, and microstructural evolution in crowded mixtures.

Comparison of different models of motion in a crowded environment: a Monte Carlo study

In this paper we investigate the motion of molecules in crowded environments for two dramatically different types of molecular transport. The first type is realized by the dynamic lattice liquid model, which is based on a cooperative movement concept and thus, the motion of molecules is highly correlated. The second one corresponds to a so-called motion of a single agent where the motion of molecules is considered as a random walk without any correlation with other moving elements. The crowded environments are modeled as a two-dimensional triangular lattice with fixed impenetrable obstacles. Our simulation results indicate that the type of transport has an impact on the dynamics of the system, the percolation threshold, critical exponents, and on molecules' trajectories.

Spatial heterogeneity can facilitate the target search of self-propelled particles

A numerical investigation of the target search dynamics of self-propelled particles (SPPs) in heterogeneous environments is presented in this work. We show that the spatial heterogeneity has a dramatic effect on the target search dynamics of SPPs. The relative magnitude of the self-propulsion length l_p and the radius of the circular domain R_c determines how the mean search time of SPPs τ depends on the area fraction of fixed obstacles ϕ_ob. For l_p < R_c, the target search process is diffusion-dominated so that a monotonic increase in τ with increasing ϕ_ob is observed. For l_p > R_c, τ is shown to be a non-monotonic convex function as a function of ϕ_ob due to the interplay of the distribution-dominated and diffusion-dominated dynamic regimes. Furthermore, at fixed ϕ_ob, τ shows a minimum upon increasing the self-propulsion velocity v₀ of a SPP of a slow rotational diffusion when it searches for a target at low ϕ_ob, while it decreases monotonically at high ϕ_ob. The present work highlights that the introduction of spatial heterogeneity causes rich dynamic behaviors of a SPP searching for a target, and deepens our understanding of the transport of active matter in heterogeneous media.

Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

Simulation of Molecular Transport in Systems Containing Mobile Obstacles

In this paper, we investigate the movement of molecules in crowded environments with obstacles undergoing Brownian motion by means of extensive Monte Carlo simulations. Our investigations were performed using the dynamic lattice liquid model, which was based on the cooperative movement concept and allowed to mimic systems at high densities where the motion of all elements (obstacles as well as moving particles) were highly correlated. The crowded environments are modeled on a two-dimensional triangular lattice containing obstacles (particles whose mobility was significantly reduced) moving by a Brownian motion. The subdiffusive motion of both elements in the system was analyzed. It was shown that the percolation transition does not exist in such systems in spite of the cooperative character of the particles' motion. The reduction of the obstacle mobility leads to the longer caging of liquid particles by mobile obstacles.

Modeling of band-3 protein diffusion in the normal and defective red blood cell membrane

We employ a two-component red blood cell (RBC) membrane model to simulate lateral diffusion of band-3 proteins in the normal RBC and in the RBC with defective membrane proteins. The defects reduce the connectivity between the lipid bilayer and the membrane skeleton (vertical connectivity), or the connectivity of the membrane skeleton itself (horizontal connectivity), and are associated with the blood disorders of hereditary spherocytosis (HS) and hereditary elliptocytosis (HE) respectively. Initially, we demonstrate that the cytoskeleton limits band-3 lateral mobility by measuring the band-3 macroscopic diffusion coefficients in the normal RBC membrane and in a lipid bilayer without the cytoskeleton. Then, we study band-3 diffusion in the defective RBC membrane and quantify the relation between band-3 diffusion coefficients and percentage of protein defects in HE RBCs. In addition, we illustrate that at low spectrin network connectivity (horizontal connectivity) band-3 subdiffusion can be approximated as anomalous diffusion, while at high horizontal connectivity band-3 diffusion is characterized as confined diffusion. Our simulations show that the band-3 anomalous diffusion exponent depends on the percentage of protein defects in the membrane cytoskeleton. We also confirm that the introduction of attraction between the lipid bilayer and the spectrin network reduces band-3 diffusion, but we show that this reduction is lower than predicted by the percolation theory. Furthermore, we predict that the attractive force between the spectrin filament and the lipid bilayer is at least 20 times smaller than the binding forces at band-3 and glycophorin C, the two major membrane binding sites. Finally, we explore diffusion of band-3 particles in the RBC membrane with defects related to vertical connectivity. We demonstrate that in this case band-3 diffusion can be approximated as confined diffusion for all attraction levels between the spectrin network and the lipid bilayer. By comparing the diffusion coefficients measured in horizontal vs. vertical defects, we conclude that band-3 mobility is primarily controlled by the horizontal connectivity.

Ergodicity breaking and particle spreading in noisy heterogeneous diffusion processes

We study noisy heterogeneous diffusion processes with a position dependent diffusivity of the form D(x) ∼ D0|x|(α0) in the presence of annealed and quenched disorder of the environment, corresponding to an effective variation of the exponent α in time and space. In the case of annealed disorder, for which effectively α0 = α0(t), we show how the long time scaling of the ensemble mean squared displacement (MSD) and the amplitude variation of individual realizations of the time averaged MSD are affected by the disorder strength. For the case of quenched disorder, the long time behavior becomes effectively Brownian after a number of jumps between the domains of a stratified medium. In the latter situation, the averages are taken over both an ensemble of particles and different realizations of the disorder. As physical observables, we analyze in detail the ensemble and time averaged MSDs, the ergodicity breaking parameter, and higher order moments of the time averages.

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.

Explicit spatiotemporal simulation of receptor-G protein coupling in rod cell disk membranes

Dim-light vision is mediated by retinal rod cells. Rhodopsin (R), a G-protein-coupled receptor, switches to its active form (R(∗)) in response to absorbing a single photon and activates multiple copies of the G-protein transducin (G) that trigger further downstream reactions of the phototransduction cascade. The classical assumption is that R and G are uniformly distributed and freely diffusing on disk membranes. Recent experimental findings have challenged this view by showing specific R architectures, including RG precomplexes, nonuniform R density, specific R arrangements, and immobile fractions of R. Here, we derive a physical model that describes the first steps of the photoactivation cascade in spatiotemporal detail and single-molecule resolution. The model was implemented in the ReaDDy software for particle-based reaction-diffusion simulations. Detailed kinetic in vitro experiments are used to parametrize the reaction rates and diffusion constants of R and G. Particle diffusion and G activation are then studied under different conditions of R-R interaction. It is found that the classical free-diffusion model is consistent with the available kinetic data. The existence of precomplexes between inactive R and G is only consistent with the data if these precomplexes are weak, with much larger dissociation rates than suggested elsewhere. Microarchitectures of R, such as dimer racks, would effectively immobilize R but have little impact on the diffusivity of G and on the overall amplification of the cascade at the level of the G protein.

Non-universal tracer diffusion in crowded media of non-inert obstacles

For tracer motion in an array of attractive obstacles we observe transient, non-ergodic anomalous diffusion depending on the obstacle density.

Enzyme kinetics and transport in a system crowded by mobile macromolecules

The dynamics of an elastic network model for the enzyme 4-oxalocrotonate tautomerase is studied in a system crowded by mobile macromolecules, also modeled by elastic networks.