Chemotaxis and Reactions in Anomalous Diffusion Dynamics

Many systems in biology, physics, and finance exhibit anomalous diffusion dynamics where the mean squared displacement grows with an exponent that deviates from one. When studying time series recording the evolution of these systems, it is crucial to precisely measure the anomalous exponent and confidently identify the mechanisms responsible for anomalous diffusion. These tasks are difficult when only few short trajectories are available, a common scenario in non-equilibrium and living systems. We show that long short-time memory (LSTM) recurrent neural networks excel at characterizing anomalous diffusion from a single short trajectory. The method we developed generalizes to experimental data obtained from subdiffusive colloids trapped in speckle light fields and superdiffusive microswimmers. We discuss the performance of the method in comparison to alternative ones in the context of the Anomalous Diffusion Challenge. In closing, we address the interpretability of the method.

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. We here 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]. 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.

The chapter discusses a model for anomalous diffusion processes. Their one-point probability density functions (p.d.f.) are exact solutions of fractional diffusion equations. The model reflects the asymptotic behavior of a jump (anomalous random walk) process with random jump sizes and random inter-jump time intervals with infinite means (and variances) which do not satisfy the law of large numbers. In the case when these intervals have a fractional exponential p.d.f., the fractional Kolmogorov–Feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions are discussed.

Continuous-time random walks (CTRWs) are generic models of anomalous diffusion and fractional dynamics in statistical physics. They are typically defined in the way that their trajectories are discontinuous step functions. In this paper, we propose alternative definition of CTRWs with continuous trajectories. We also give the scaling limit theorem for sequence of such random walks. In general case this result requires the use of strong Skorohod [Formula: see text] topology instead of Skorohod [Formula: see text] topology, which is usually used in limit theorems for ordinary CTRW processes. We also give additional conditions under which convergence of sequence of considered random walks holds in the [Formula: see text] topology.

A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model

Evolutionary equations containing fractional derivatives have been widely used in financial models and can describe anomalous diffusion and transmission dynamics in some complex systems. We assumes that option pricing obeys the infinite jump CGMY process and treats stock price fluctuations as a fractal transmission system, A time fractional CGMY option pricing (TFCGMY) model is obtained. The L1 approximation of the Caputo fractional derivative and the modified GL approximation are used to discretize the time and space fractional order operators; the first order space derivative is discretized using the central difference quotient. The TFCGMY model and the above numerical techniques are used for the pricing analysis of European call options. By comparing and analyzing the option pricing model with the Black-Scholes model and the CGMY model shows that the introduction of time fractional differentiation can better capture the jump characteristics of stock prices of options.

Three areas where time-independent disorder plays a key role in biological dynamics far from equilibrium are reviewed. We first discuss the anomalous localization dynamics that arises when a single species spreads in space and time via diffusion and fluid advection in the presence of frozen heterogeneities in the growth rate. Next we treat the unzipping of double-stranded DNA as a function of force and temperature, a challenge that must be surmounted every time a cell divides. Heterogeneity in the DNA sequence dominates the physics of single-molecule force-extension curves for a broad range of forces upon approaching a sharp unzipping transition. The dynamics of the unzipping fork exhibits anomalous drift and diffusion in a similar range above this transition, with energy barriers that scale as the square root of the genome size. Finally, we describe how activated peptidoglycan strand extension sites, called dislocations in materials science, can mediate the growth of bacterial cell walls. Enzymatically driven circumferential motions of a few dozen of these defects are sufficient to describe the exponential elongation rates observed in experiments on Escherichia coli in a nutrient-rich environment. However, long-range elastic forces transmitted by the peptidoglycan meshwork cause the moving dislocations to interact not only with each other, but also with a disorderly array of frozen, inactivated strand ends.

In this work, we propose a fractional Poisson-Nernst-Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker-Planck equation. Then, it is generalized to the macroscopic fractional Poisson-Nernst-Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.

Non-extensive statistical analysis of magnetic field during the March 2012 ICME event using a multi-spacecraft approach

Cytoskeletal motor proteins are involved in major intracellular transport processes which are vital for maintaining appropriate cellular function. When attached to cytoskeletal filaments, the motor exhibits distinct states of motility: active motion along the filaments, and pause phase in which it remains stationary for a finite time interval. The transition probabilities between motion and pause phases are asymmetric in general, and considerably affected by changes in environmental conditions which influences the efficiency of cargo delivery to specific targets. By considering the motion of individual non-interacting molecular motors on a single filament as well as a dynamic filamentous network, we present an analytical model for the dynamics of self-propelled particles which undergo frequent pause phases. The interplay between motor processivity, structural properties of filamentous network, and transition probabilities between the two states of motility drastically changes the dynamics: multiple transitions between different types of anomalous diffusive dynamics occur and the crossover time to the asymptotic diffusive or ballistic motion varies by several orders of magnitude. We map out the phase diagrams in the space of transition probabilities, and address the role of initial conditions of motion on the resulting dynamics.

Almost all processes in life are governed by proteins which are biomolecules consisting of a chain of amino acids. They perform a wide range of functions in organisms, including catalysis of metabolic reactions, transport of molecules, transcription of DNA to RNA, and ensuring the structural stability of cells.This function is intimately linked to a proteins’ three-dimensional structure and its dynamics. Protein dynamics is a complex process comprised of motions on multiple orders of magnitude both in characteristic time-scale and amplitude. Frauenfelder et al. suggested that this complex process can be modeled as a diffusion process in a hierarchically structured free energy landscape. With the rapid improvement in performance of computer hardware protein dynamics can be simulated on multiple time scales. Indeed, it was found that protein dynamics shows anomalous diffusion which is a characteristic feature of diffusion processes in hierarchical free-energy landscapes. The main aim of this work is to investigate the hierarchical structure of protein free-energy landscapes. To that end, we use a d-dimensional generalization of the simple hierarchical lattice model as a reference to translate anomalous diffusion exponents obtained from MD simulations into ruggedness and dimensionality estimates. We were able to estimate the ruggedness and dimensionality estimates of 500 small globular proteins from one-microsecond molecular dynamics simulations based on this d-dimensional hierarchical lattice model. Furthermore, we asked what the cause of anomalous diffusion in hierarchical free energy landscapes is. Based on this, we developed an enhanced sampling algorithm for simulating diffusion in hierarchical free energy landscapes.

Anomalous diffusion in nonlinear dynamical systemsJurgen ParisiPhysical Institute, University of BayreuthBox 10 12 51, 0-95440 Bayreuth, GermanyRuedi StoopPhysics Institute, University of ZurichSchanberggasse 9, CH-8001 Zurich, SwitzerlandIn the present contribution, we investigate anomalous diffusion behaviorfrom the point of view of dynamical systems by using explicitly the con-nection between nonhyperbolicity of the dynamical map and anomalous dif-fusion. Exemplarily, two closely related models are looked at, capable toshow dispersed and enhanced diffusion, respectively.Recently, the phenomenon of anomalous diffusion has attracted enormousinterest within the physical community.1'2 One of the traditional methodsto study the problem is based on random walks. There, as a characteristicquantity, we have the mean-squared displacement r as a function of time t(or the number n of iterations, in the discrete case) that typically fol-lows the scaling t with an exponent p. Regular diffusion, called Browni-an motion, gives p = 1, sublinear, so-called dispersive diffusion, p < 1,and superlinear, so-called enhanced diffusion,

We perform extensive simulations and systematic statistical analyses on the structural dynamics of a model of amorphous silicon. The simulations follow the dynamics introduced by Wooten, Winer and Weaire: the energy is obtained with the Keating potential, and the dynamics consists of bond transpositions proposed at random locations and accepted with the Metropolis acceptance ratio. The structural quantities we track are the variations in time of the lateral lengths (Lx, Ly, Lz) of the cuboid simulation cell. We transform these quantities into the volume V and two aspect ratios B1 and B2. Our analysis reveals that at short times, the mean squared displacement (MSD) for all of them exhibits normal diffusion. At longer times, they cross over to anomalous diffusion (AD), with a temperature-dependent anomalous exponent α<1. We analyze our findings in the light of two standard models in statistical physics that feature anomalous dynamics, viz., continuous time random walker (CTRW) and fractional Brownian motion (fBm). We obtain the distribution of waiting times, and find that the data are consistent with a stretched-exponential decay. We also show that the three quantities, V, B1 and B2 exhibit negative velocity autocorrelation functions. These observations together suggest that the dynamics of the material belong to the fBm class.

A review of recent developments in the field of front dynamics in anomalous diffusion-reaction systems is presented. Both fronts between stable phases and those propagating into an unstable phase are considered. A number of models of anomalous diffusion with reaction are discussed, including models with Lévy flights, truncated Lévy flights, subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators.

Dynamics of confined molecules within porous materials is equally important as local structural order, and it is necessary to quantify it and to reveal the microscopic mechanisms ruling it for better control of adsorption applications. In this study, molecular dynamics simulations were carried out to investigate the translational and the rotational dynamics of methanol trapped into the flexible NH2-MIL-53(Al) metal-organics framework (MOF). Indeed, atomistic simulation is nowadays a relevant tool to explore matter at the nanoscale. Very recently it has been shown that the NH2-MIL-53(Al) MOF material was capable to undergo a reversible structural transition (breathing phenomenon) by combining adsorption and thermal stimuli. This flexibility can drastically affect the dynamics of confined molecules and therefore the successful conduct of adsorption applications such as gas storage and separation. Rotational and translational dynamics of confined methanol through nanoporous flexible NH2-MIL-53(Al) MOF were then deeply investigated by exploring a broad range of dynamical properties to extract the molecular mechanisms ruling them. This study allowed us to shed light on the interplay of dynamics of confined fluids and flexibility of porous material and to highlight the physical insights in diffusion mechanisms of confined molecules. Anomalous translational diffusion was evidenced due to a dynamical heterogeneity caused by a combination of a localized dynamics at the subnanometric scale and translational jumps between nanodomains in a zigzag scheme between the hydroxide group of the NH2-MIL-53(Al). Actually, the non-Fickian dynamics of methanol is the result of the specific host-guest interactions and the MOF flexibility involving the pore opening. Eventually, decoupling between both rotational and translational dynamics related to breaking in the Stokes-Einstein relation was highlighted.