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Institute
We introduce and study a Lévy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.
Isoflux tension propagation (IFTP) theory and Langevin dynamics (LD) simulations are employed to study the dynamics of channel-driven polymer translocation in which a polymer translocates into a narrow channel and the monomers in the channel experience a driving force fc. In the high driving force limit, regardless of the channel width, IFTP theory predicts τ ∝ f βc for the translocation time, where β = −1 is the force scaling exponent. Moreover, LD data show that for a very narrow channel fitting only a single file of monomers, the entropic force due to the subchain inside the channel does not play a significant role in the translocation dynamics and the force exponent β = −1 regardless of the force magnitude. As the channel width increases the number of possible spatial configurations of the subchain inside the channel becomes significant and the resulting entropic force causes the force exponent to drop below unity.
Anomalous diffusion or, more generally, anomalous transport, with nonlinear dependence of the mean-squared displacement on the measurement time, is ubiquitous in nature. It has been observed in processes ranging from microscopic movement of molecules to macroscopic, large-scale paths of migrating birds. Using data from multiple empirical systems, spanning 12 orders of magnitude in length and 8 orders of magnitude in time, we employ a method to detect the individual underlying origins of anomalous diffusion and transport in the data. This method decomposes anomalous transport into three primary effects: long-range correlations (“Joseph effect”), fat-tailed probability density of increments (“Noah effect”), and nonstationarity (“Moses effect”). We show that such a decomposition of real-life data allows us to infer nontrivial behavioral predictions and to resolve open questions in the fields of single-particle tracking in living cells and movement ecology.
Sprache
Englisch
Modern single-particle-tracking techniques produce extensive time-series of diffusive motion in a wide variety of systems, from single-molecule motion in living-cells to movement ecology. The quest is to decipher the physical mechanisms encoded in the data and thus to better understand the probed systems. We here augment recently proposed machine-learning techniques for decoding anomalous-diffusion data to include an uncertainty estimate in addition to the predicted output. To avoid the Black-Box-Problem a Bayesian-Deep-Learning technique named Stochastic-Weight-Averaging-Gaussian is used to train models for both the classification of the diffusionmodel and the regression of the anomalous diffusion exponent of single-particle-trajectories. Evaluating their performance, we find that these models can achieve a wellcalibrated error estimate while maintaining high prediction accuracies. In the analysis of the output uncertainty predictions we relate these to properties of the underlying diffusion models, thus providing insights into the learning process of the machine and the relevance of the output.
Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian—or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion.
How do different reset protocols affect ergodicity of a diffusion process in single-particle-tracking experiments? We here address the problem of resetting of an arbitrary stochastic anomalous-diffusion process (ADP) from the general mathematical points of view and assess ergodicity of such reset ADPs for an arbitrary resetting protocol. The process of stochastic resetting describes the events of the instantaneous restart of a particle’s motion via randomly distributed returns to a preset initial position (or a set of those). The waiting times of such resetting events obey the Poissonian, Gamma, or more generic distributions with specified conditions regarding the existence of moments. Within these general approaches, we derive general analytical results and support them by computer simulations for the behavior of the reset mean-squared displacement (MSD), the new reset increment-MSD (iMSD), and the mean reset time-averaged MSD (TAMSD). For parental nonreset ADPs with the MSD(t)∝ tμ we find a generic behavior and a switch of the short-time growth of the reset iMSD and mean reset TAMSDs from ∝ _μ for subdiffusive to ∝ _1 for superdiffusive reset ADPs. The critical condition for a reset ADP that recovers its ergodicity is found to be more general than that for the nonequilibrium stationary state, where obviously the iMSD and the mean TAMSD are equal. The consideration of the new statistical quantifier, the iMSD—as compared to the standard MSD—restores the ergodicity of an arbitrary reset ADP in all situations when the μth moment of the waiting-time distribution of resetting events is finite. Potential applications of these new resetting results are, inter alia, in the area of biophysical and soft-matter systems.
We study the probability density function (PDF) of the first-reaction times between a diffusive ligand and a membrane-bound, immobile imperfect target region in a restricted 'onion-shell' geometry bounded by two nested membranes of arbitrary shapes. For such a setting, encountered in diverse molecular signal transduction pathways or in the narrow escape problem with additional steric constraints, we derive an exact spectral form of the PDF, as well as present its approximate form calculated by help of the so-called self-consistent approximation. For a particular case when the nested domains are concentric spheres, we get a fully explicit form of the approximated PDF, assess the accuracy of this approximation, and discuss various facets of the obtained distributions. Our results can be straightforwardly applied to describe the PDF of the terminal reaction event in multi-stage signal transduction processes.
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.
Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories
(2021)
Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.
We consider the emerging dynamics of a separable continuous time random walk (CTRW) in the case when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding Universe. The CTRW in this study can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case when the velocity field is absent. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: in one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter–controlled reactions in real systems are discussed.