Refine
Year of publication
Language
- English (297)
Keywords
- anomalous diffusion (51)
- diffusion (42)
- stochastic processes (12)
- living cells (9)
- nonergodicity (7)
- ageing (6)
- first passage time (6)
- fractional Brownian motion (6)
- geometric Brownian motion (6)
- models (6)
- Brownian motion (5)
- Levy flights (5)
- dynamics (5)
- first passage (5)
- first-passage time (5)
- infection pathway (5)
- random-walks (5)
- single-particle tracking (5)
- superstatistics (5)
- first-passage (4)
- physiological consequences (4)
- random diffusivity (4)
- stochastic resetting (4)
- subdiffusion (4)
- transport (4)
- weak ergodicity breaking (4)
- Debye screening (3)
- Fokker-Planck equation (3)
- Langevin equation (3)
- Levy walks (3)
- Mittag-Leffler functions (3)
- aspect ratio (3)
- critical phenomena (3)
- cylindrical geometry (3)
- diffusing diffusivity (3)
- electrostatic interactions (3)
- financial time series (3)
- first-hitting time (3)
- fluctuation-dissipation theorem (3)
- fractional dynamics (3)
- intracellular-transport (3)
- langevin equation (3)
- polyelectrolyte adsorption (3)
- polymers (3)
- power spectral analysis (3)
- probability density function (3)
- protein search (3)
- scaled Brownian motion (3)
- time averaging (3)
- Anomalous diffusion (2)
- Antibiotics (2)
- Bacterial biofilms (2)
- Biofilms (2)
- Biological defense mechanisms (2)
- Boltzmann distribution (2)
- Brownian yet non-Gaussian diffusion (2)
- Chebyshev inequality (2)
- Cystic fibrosis (2)
- Fokker-Planck equations (2)
- Fractional moments (2)
- Lévy flights (2)
- Lévy walks (2)
- Ornstein–Uhlenbeck process (2)
- Pseudomonas aeruginosa (2)
- Sinai diffusion (2)
- Sputum (2)
- active transport (2)
- adenoassociated virus (2)
- approximate methods (2)
- autoregressive models (2)
- behavior (2)
- biological physics (2)
- brownian-motion (2)
- cambridge cb4 0wf (2)
- cambs (2)
- channel (2)
- codifference (2)
- coefficient (2)
- coefficients (2)
- continuous time random walk (2)
- continuous time random walk (CTRW) (2)
- crowded fluids (2)
- cytoplasm (2)
- dna coiling (2)
- dynamics simulation (2)
- endosomal escape (2)
- england (2)
- ensemble and time averaged mean squared displacement (2)
- equation approach (2)
- escherichia-coli (2)
- exact results (2)
- excluded volume (2)
- expanding medium (2)
- extremal values (2)
- fastest first-passage time of N walkers (2)
- first-passage time distribution (2)
- first-reaction time (2)
- flight search patterns (2)
- fluorescence photobleaching recovery (2)
- folding kinetics (2)
- fractional dynamics approach (2)
- gene regulatory networks (2)
- gene-regulation kinetics (2)
- generalised langevin equation (2)
- in-vitro (2)
- intermittent chaotic systems (2)
- large-deviation statistic (2)
- levy flights (2)
- lipid bilayer membrane dynamics (2)
- maximum and range (2)
- mean versus most probable reaction times (2)
- membrane (2)
- membrane channel (2)
- milton rd (2)
- mixed boundary conditions (2)
- mixtures (2)
- monte-carlo (2)
- motion (2)
- nanoparticles (2)
- narrow escape problem (2)
- non-Gaussian diffusion (2)
- non-Gaussianity (2)
- osmotic-pressure (2)
- photon-counting statistics (2)
- posttranslational protein translocation (2)
- power spectral density (2)
- power spectrum (2)
- random-walk (2)
- reaction cascade (2)
- reflecting boundary conditions (2)
- royal soc chemistry (2)
- science park (2)
- shell-like geometries (2)
- single trajectories (2)
- single trajectory analysis (2)
- single-stranded-dna (2)
- single-trajectory analysis (2)
- solid-state nanopores (2)
- space-dependent diffusivity (2)
- spatial-organization (2)
- stationary stochastic process (2)
- stochastic processes (theory) (2)
- stochastic time series (2)
- structured polynucleotides (2)
- thomas graham house (2)
- time random-walks (2)
- time series analysis (2)
- time-averaged mean squared displacement (2)
- trafficking (2)
- truncated power-law correlated noise (2)
- 15 (1)
- 4 (1)
- Absorption (1)
- Adam-Delbruck scenario (1)
- Ageing (1)
- Asymptotic expansions (1)
- Bayesian inference (1)
- Biological Physics (1)
- Black– Scholes model (1)
- Brownian diffusion (1)
- Bulk-mediated diffusion (1)
- Bulk-mediated diffusion; (1)
- Cattaneo equation (1)
- Characteristic function (1)
- Complete Bernstein function (1)
- Completely monotone function (1)
- Composite fractional derivative (1)
- Distributed order diffusion-wave equations (1)
- Econophysics (1)
- Fokker-Planck-Smoluchowski equation (1)
- Fokker– Planck equation (1)
- Fox H-function (1)
- Fox H-functions (1)
- Fractional calculus (primary) (1)
- Fractional diffusion equation (1)
- Grunwald-Letnikov approximation (1)
- Interdisciplinary Physics (1)
- Levy flight (1)
- Levy foraging hypothesis (1)
- Levy walk (1)
- Lipid bilayer (1)
- Markov additive processes (1)
- Mellin transform (1)
- Mittag-Leffler (1)
- Non-Gaussian (1)
- Pareto analysis (1)
- Pareto law (1)
- Protein crowding (1)
- Riesz-Feller fractional derivative (1)
- Scaling exponents (1)
- Scher-Montroll transport (1)
- Simulations (1)
- Statistical Physics (1)
- Statistical and Nonlinear Physics (1)
- Stochastic modelling (1)
- Stochastic optimization (1)
- Wealth and income distribution (1)
- and surface diffusion (1)
- anomalous (or non-Fickian) diffusion (1)
- anomalous heat conduction (1)
- asymmetric Levy flights (1)
- asymptotic analysis (1)
- autocorrelation function (1)
- barrier escape (1)
- bulk (1)
- cellular signalling (1)
- chemical relaxation (1)
- clustering (1)
- coloured (1)
- comb-like model (1)
- complex (1)
- confinement (1)
- conformational properties (1)
- conservative random walks (1)
- continuous time random (1)
- continuous time random walks (1)
- correlated noise (1)
- coupled initial boundary value problem (1)
- critical adsorption (1)
- crossover anomalous diffusion dynamics (1)
- crossover dynamics (1)
- crowding (1)
- density (1)
- dimensional reduction (1)
- dimerization kinetics (1)
- disordered media (1)
- driven diffusive systems (theory) (1)
- dynamical systems (1)
- ecological (1)
- econophysics (1)
- electrostatics (1)
- escence correlation spectroscopy (1)
- exclusion process (1)
- exclusion processes (1)
- first arrival (1)
- first passage process (1)
- first-arrival density (1)
- first-passage times (1)
- fluctuations (theory) (1)
- fluorescence correlation spectroscopy (1)
- fractional dynamic equations (1)
- fractional generalized Langevin equation (1)
- frictional memory kernel (1)
- function (1)
- gel network (1)
- generalised Langevin equation (1)
- generalized diffusion equation (1)
- heterogeneous diffusion (1)
- heterogeneous diffusion process (1)
- income inequality (1)
- income mobility (1)
- inhomogeneous-media (1)
- kinetic-theory (1)
- large deviation function (1)
- lattice gas (1)
- linear response theory (1)
- local equilibrium (1)
- mean square displacement (1)
- mean squared displacement (1)
- mechanisms (1)
- memory kernel (1)
- mobile-immobile model (1)
- molecular overcrowding (1)
- movement data (1)
- multi-scaling (1)
- multidimensional fractional diffusion equation (1)
- noise (1)
- noise in biochemical signalling (1)
- non-Gaussian (1)
- non-Gaussian distribution (1)
- non-Gaussian probability (1)
- non-ergodicity (1)
- non-exponential relaxation (1)
- non-extensive statistics (1)
- nonequilibrium stationary state (1)
- nonstationary diffusivity (1)
- option pricing (1)
- path integration (1)
- persistence (1)
- phase-transition boundary (1)
- plasma-membrane (1)
- polyelectrolytes (1)
- polymer translocation (1)
- potential landscape (1)
- predator-prey model (1)
- probability distribution function (1)
- quenched energy landscape (1)
- random search process (1)
- random search processes (1)
- random walks (1)
- reaction kinetics theory (1)
- reaction rate constants (1)
- recurrence (1)
- resetting (1)
- rotational diffusion (1)
- search dynamics (1)
- search efficiency (1)
- search optimization (1)
- sensitivity analysis (1)
- single particle tracking (1)
- single-file diffusion (1)
- statistics (1)
- stochastic dynamics (1)
- stochastic simulation algorithm (1)
- superdiffusion and (1)
- susceptibility (1)
- tau proteins (1)
- telegrapher's equation (1)
- time-series analysis (1)
- van Hove correlation (1)
- variances (1)
- walks (1)
- water diffusion in the brain (1)
Institute
We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g. a signal transduction proceeds via a series of consecutive 'messengers': the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.
We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g. a signal transduction proceeds via a series of consecutive 'messengers': the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.
Fixational eye movements show scaling behaviour of the positional mean-squared displacement with a characteristic transition from persistence to antipersistence for increasing time-lag. These statistical patterns were found to be mainly shaped by microsaccades (fast, small-amplitude movements). However, our re-analysis of fixational eye-movement data provides evidence that the slow component (physiological drift) of the eyes exhibits scaling behaviour of the mean-squared displacement that varies across human participants. These results suggest that drift is a correlated movement that interacts with microsaccades. Moreover, on the long time scale, the mean-squared displacement of the drift shows oscillations, which is also present in the displacement auto-correlation function. This finding lends support to the presence of time-delayed feedback in the control of drift movements. Based on an earlier non-linear delayed feedback model of fixational eye movements, we propose and discuss different versions of a new model that combines a self-avoiding walk with time delay. As a result, we identify a model that reproduces oscillatory correlation functions, the transition from persistence to antipersistence, and microsaccades.
Fixational eye movements show scaling behaviour of the positional mean-squared displacement with a characteristic transition from persistence to antipersistence for increasing time-lag. These statistical patterns were found to be mainly shaped by microsaccades (fast, small-amplitude movements). However, our re-analysis of fixational eye-movement data provides evidence that the slow component (physiological drift) of the eyes exhibits scaling behaviour of the mean-squared displacement that varies across human participants. These results suggest that drift is a correlated movement that interacts with microsaccades. Moreover, on the long time scale, the mean-squared displacement of the drift shows oscillations, which is also present in the displacement auto-correlation function. This finding lends support to the presence of time-delayed feedback in the control of drift movements. Based on an earlier non-linear delayed feedback model of fixational eye movements, we propose and discuss different versions of a new model that combines a self-avoiding walk with time delay. As a result, we identify a model that reproduces oscillatory correlation functions, the transition from persistence to antipersistence, and microsaccades.
Fixational eye movements show scaling behaviour of the positional mean-squared displacement with a characteristic transition from persistence to antipersistence for increasing time-lag. These statistical patterns were found to be mainly shaped by microsaccades (fast, small-amplitude movements). However, our re-analysis of fixational eye-movement data provides evidence that the slow component (physiological drift) of the eyes exhibits scaling behaviour of the mean-squared displacement that varies across human participants. These results suggest that drift is a correlated movement that interacts with microsaccades. Moreover, on the long time scale, the mean-squared displacement of the drift shows oscillations, which is also present in the displacement auto-correlation function. This finding lends support to the presence of time-delayed feedback in the control of drift movements. Based on an earlier non-linear delayed feedback model of fixational eye movements, we propose and discuss different versions of a new model that combines a self-avoiding walk with time delay. As a result, we identify a model that reproduces oscillatory correlation functions, the transition from persistence to antipersistence, and microsaccades.
A single predator charging a herd of prey: effects of self volume and predator-prey decision-making
(2016)
We study the degree of success of a single predator hunting a herd of prey on a two-dimensional square lattice landscape. We explicitly consider the self volume of the prey restraining their dynamics on the lattice. The movement of both predator and prey is chosen to include an intelligent, decision making step based on their respective sighting ranges, the radius in which they can detect the other species (prey cannot recognise each other besides the self volume interaction): after spotting each other the motion of prey and predator turns from a nearest neighbour random walk into directed escape or chase, respectively. We consider a large range of prey densities and sighting ranges and compute the mean first passage time for a predator to catch a prey as well as characterise the effective dynamics of the hunted prey. We find that the prey's sighting range dominates their life expectancy and the predator profits more from a bad eyesight of the prey than from his own good eye sight. We characterise the dynamics in terms of the mean distance between the predator and the nearest prey. It turns out that effectively the dynamics of this distance coordinate can be captured in terms of a simple Ornstein–Uhlenbeck picture. Reducing the many-body problem to a simple two-body problem by imagining predator and nearest prey to be connected by an effective Hookean bond, all features of the model such as prey density and sighting ranges merge into the effective binding constant.
Macromolecular crowding in living biological cells effects subdiffusion of larger biomolecules such as proteins and enzymes. Mimicking this subdiffusion in terms of random walks on a critical percolation cluster, we here present a case study of EcoRV restriction enzymes involved in vital cellular defence. We show that due to its so far elusive propensity to an inactive state the enzyme avoids non-specific binding and remains well-distributed in the bulk cytoplasm of the cell. Despite the reduced volume exploration capability of subdiffusion processes, this mechanism guarantees a high efficiency of the enzyme. By variation of the non-specific binding constant and the bond occupation probability on the percolation network, we demonstrate that reduced nonspecific binding are beneficial for efficient subdiffusive enzyme activity even in relatively small bacteria cells. Our results corroborate a more local picture of cellular regulation.
We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.
Proteins are capable of locating specific targets on DNA by employing a facilitated diffusion process with intermittent 1D and 3D search steps. Gene colocalisation and coregulation-i.e. the spatial proximity of two communicating genes-is one factor capable of accelerating the target search process along the DNA. We perform Monte Carlo computer simulations and demonstrate the benefits of gene colocalisation for minimising the search time in a model DNA-protein system. We use a simple diffusion model to mimic the search for targets by proteins, produced initially in bursts of multiple proteins and performing the first-passage search on the DNA chain. The behaviour of the mean first-passage times to the target is studied as a function of distance between the initial position of proteins and the DNA target position, as well as versus the concentration of proteins. We also examine the properties of bursty target search kinetics for varying physical-chemical protein-DNA binding affinity. Our findings underline the relevance of colocalisation of production and binding sites for protein search inside biological cells.
Molecular signalling in living cells occurs at low copy numbers and is thereby inherently limited by the noise imposed by thermal diffusion. The precision at which biochemical receptors can count signalling molecules is intimately related to the noise correlation time. In addition to passive thermal diffusion, messenger RNA and vesicle-engulfed signalling molecules can transiently bind to molecular motors and are actively transported across biological cells. Active transport is most beneficial when trafficking occurs over large distances, for instance up to the order of 1 metre in neurons. Here we explain how intermittent active transport allows for faster equilibration upon a change in concentration triggered by biochemical stimuli. Moreover, we show how intermittent active excursions induce qualitative changes in the noise in effectively one-dimensional systems such as dendrites. Thereby they allow for significantly improved signalling precision in the sense of a smaller relative deviation in the concentration read-out by the receptor. On the basis of linear response theory we derive the exact mean field precision limit for counting actively transported molecules. We explain how intermittent active excursions disrupt the recurrence in the molecular motion, thereby facilitating improved signalling accuracy. Our results provide a deeper understanding of how recurrence affects molecular signalling precision in biological cells and novel medical-diagnostic devices.
We study the effects of ageing-the time delay between initiation of the physical process at t = 0 and start of observation at some time t(a) > 0-and spatial confinement on the properties of heterogeneous diffusion processes (HDPs) with deterministic power-law space-dependent diffusivities, D(x) = D-0 vertical bar x vertical bar(alpha). From analysis of the ensemble and time averaged mean squared displacements and the ergodicity breaking parameter quantifying the inherent degree of irreproducibility of individual realizations of the HDP we obtain striking similarities to ageing subdiffusive continuous time random walks with scale-free waiting time distributions. We also explore how both processes can be distinguished. For confined HDPs we study the long-time saturation of the ensemble and time averaged particle displacements as well as the magnitude of the inherent scatter of time averaged displacements and contrast the outcomes to the results known for other anomalous diffusion processes under confinement.
In ageing systems physical observables explicitly depend on the time span elapsing between the original initiation of the system and the actual start of the recording of the particle motion. We here study the signatures of ageing in the framework of ultraslow continuous time random walk processes with super-heavy tailed waiting time densities. We derive the density for the forward or recurrent waiting time of the motion as function of the ageing time, generalise the Montroll-Weiss equation for this process, and analyse the ageing behaviour of the ensemble and time averaged mean squared displacements.
Ageing first passage time density in continuous time random walks and quenched energy landscapes
(2015)
We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function Psi (t) similar or equal to t(-1-alpha) (0 <= alpha <= 2). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time t(a), the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with 0 < alpha < 1 describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range 1 < alpha < 2, for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.
We study the properties of ageing Scher-Montroll transport in terms of a biased subdiffusive continuous time random walk in which the waiting times between consecutive jumps of the charge carriers are distributed according to the power law probability with . As we show, the dynamical properties of the Scher-Montroll transport depend on the ageing time span between the initial preparation of the system and the start of the observation. The Scher-Montroll transport theory was originally shown to describe the photocurrent in amorphous solids in the presence of an external electric field, but it has since been used in many other fields of physical sciences, in particular also in the geophysical context for the description of the transport of tracer particles in subsurface aquifers. In the absence of ageing () the photocurrent of the classical Scher-Montroll model or the breakthrough curves in the groundwater context exhibit a crossover between two power law regimes in time with the scaling exponents and . In the presence of ageing a new power law regime and an initial plateau regime of the current emerge. We derive the different power law regimes and crossover times of the ageing Scher-Montroll transport and show excellent agreement with simulations of the process. Experimental data of ageing Scher-Montroll transport in polymeric semiconductors are shown to agree well with the predictions of our theory.
Ageing single file motion
(2014)
We study time averages of single particle trajectories in scale-free anomalous diffusion processes, in which the measurement starts at some time t(a) > 0 after initiation of the process at t = 0. Using aging renewal theory, we show that for such nonstationary processes a large class of observables are affected by a unique aging function, which is independent of boundary conditions or the external forces. Moreover, we discuss the implications of aging induced population splitting: with growing age ta of the process, an increasing fraction of particles remains motionless in a measurement of fixed duration. Consequences for single biomolecule tracking in live cells are discussed.
We discuss a renewal process in which successive events are separated by scale-free waiting time periods. Among other ubiquitous long-time properties, this process exhibits aging: events counted initially in a time interval [0, t] statistically strongly differ from those observed at later times [t(a,) t(a) + t]. The versatility of renewal theory is owed to its abstract formulation. Renewals can be interpreted as steps of a random walk, switching events in two-state models, domain crossings of a random motion, etc. In complex, disordered media, processes with scale-free waiting times play a particularly prominent role. We set up a unified analytical foundation for such anomalous dynamics by discussing in detail the distribution of the aging renewal process. We analyze its half-discrete, half-continuous nature and study its aging time evolution. These results are readily used to discuss a scale-free anomalous diffusion process, the continuous-time random walk. By this, we not only shed light on the profound origins of its characteristic features, such as weak ergodicity breaking, along the way, we also add an extended discussion on aging effects. In particular, we find that the aging behavior of time and ensemble averages is conceptually very distinct, but their time scaling is identical at high ages. Finally, we show how more complex motion models are readily constructed on the basis of aging renewal dynamics.
Aging scaled Brownian motion
(2015)
Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.
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.
Lateral diffusion plays a crucial role in numerous processes that take place in cell membranes, yet it is quite poorly understood in native membranes characterized by, e.g., domain formation and large concentration of proteins. In this article, we use atomistic and coarse-grained simulations to consider how packing of membranes and crowding with proteins affect the lateral dynamics of lipids and membrane proteins. We find that both packing and protein crowding have a profound effect on lateral diffusion, slowing it down. Anomalous diffusion is observed to be an inherent property in both protein-free and protein-rich membranes, and the time scales of anomalous diffusion and the exponent associated with anomalous diffusion are found to strongly depend on packing and crowding. Crowding with proteins also has a striking effect on the decay rate of dynamical correlations associated with lateral single-particle motion, as the transition from anomalous to normal diffusion is found to take place at macroscopic time scales: while in protein-poor conditions normal diffusion is typically observed in hundreds of nanoseconds, in protein-rich conditions the onset of normal diffusion is tens of microseconds, and in the most crowded systems as large as milliseconds. The computational challenge which results from these time scales is not easy to deal with, not even in coarse-grained simulations. We also briefly discuss the physical limits of protein motion. Our results suggest that protein concentration is anything but constant in the plane of cell membranes. Instead, it is strongly dependent on proteins' preference for aggregation.