@article{CherstvySafdariMetzler2021, author = {Cherstvy, Andrey G. and Safdari, Hadiseh and Metzler, Ralf}, title = {Anomalous diffusion, nonergodicity, and ageing for exponentially and logarithmically time-dependent diffusivity}, series = {Journal of physics. D, Applied physics}, volume = {54}, journal = {Journal of physics. D, Applied physics}, number = {19}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {0022-3727}, doi = {10.1088/1361-6463/abdff0}, pages = {18}, year = {2021}, abstract = {We investigate a diffusion process with a time-dependent diffusion coefficient, both exponentially increasing and decreasing in time, D(t)=D-0(e +/- 2 alpha t). For this (hypothetical) nonstationary diffusion process we compute-both analytically and from extensive stochastic simulations-the behavior of the ensemble- and time-averaged mean-squared displacements (MSDs) of the particles, both in the over- and underdamped limits. Simple asymptotic relations derived for the short- and long-time behaviors are shown to be in excellent agreement with the results of simulations. The diffusive characteristics in the presence of ageing are also considered, with dramatic differences of the over- versus underdamped regime. Our results for D(t)=D-0(e +/- 2 alpha t) extend and generalize the class of diffusive systems obeying scaled Brownian motion featuring a power-law-like variation of the diffusivity with time, D(t) similar to t(alpha-1). We also examine the logarithmically increasing diffusivity, D(t)=D(0)log[t/tau(0)], as another fundamental functional dependence (in addition to the power-law and exponential) and as an example of diffusivity slowly varying in time. One of the main conclusions is that the behavior of the massive particles is predominantly ergodic, while weak ergodicity breaking is repeatedly found for the time-dependent diffusion of the massless particles at short times. The latter manifests itself in the nonequivalence of the (both nonaged and aged) MSD and the mean time-averaged MSD. The current findings are potentially applicable to a class of physical systems out of thermal equilibrium where a rapid increase or decrease of the particles' diffusivity is inherently realized. One biological system potentially featuring all three types of time-dependent diffusion (power-law-like, exponential, and logarithmic) is water diffusion in the brain tissues, as we thoroughly discuss in the end.}, language = {en} } @article{DieterichLindemannMoskoppetal.2022, author = {Dieterich, Peter and Lindemann, Otto and Moskopp, Mats Leif and Tauzin, Sebastien and Huttenlocher, Anna and Klages, Rainer and Chechkin, Aleksei V. and Schwab, Albrecht}, title = {Anomalous diffusion and asymmetric tempering memory in neutrophil chemotaxis}, series = {PLoS Computational Biology : a new community journal}, volume = {18}, journal = {PLoS Computational Biology : a new community journal}, number = {5}, publisher = {PLoS}, address = {San Fransisco}, issn = {1553-734X}, doi = {10.1371/journal.pcbi.1010089}, pages = {26}, year = {2022}, abstract = {Neutrophil granulocytes are essential for the first host defense. After leaving the blood circulation they migrate efficiently towards sites of inflammation. They are guided by chemoattractants released from cells within the inflammatory foci. On a cellular level, directional migration is a consequence of cellular front-rear asymmetry which is induced by the concentration gradient of the chemoattractants. The generation and maintenance of this asymmetry, however, is not yet fully understood. Here we analyzed the paths of chemotacting neutrophils with different stochastic models to gain further insight into the underlying mechanisms. Wildtype chemotacting neutrophils show an anomalous superdiffusive behavior. CXCR2 blockade and TRPC6-knockout cause the tempering of temporal correlations and a reduction of chemotaxis. Importantly, such tempering is found both in vitro and in vivo. These findings indicate that the maintenance of anomalous dynamics is crucial for chemotactic behavior and the search efficiency of neutrophils. The motility of neutrophils and their ability to sense and to react to chemoattractants in their environment are of central importance for the innate immunity. Neutrophils are guided towards sites of inflammation following the activation of G-protein coupled chemoattractant receptors such as CXCR2 whose signaling strongly depends on the activity of Ca2+ permeable TRPC6 channels. It is the aim of this study to analyze data sets obtained in vitro (murine neutrophils) and in vivo (zebrafish neutrophils) with a stochastic mathematical model to gain deeper insight into the underlying mechanisms. The model is based on the analysis of trajectories of individual neutrophils. Bayesian data analysis, including the covariances of positions for fractional Brownian motion as well as for exponentially and power-law tempered model variants, allows the estimation of parameters and model selection. Our model-based analysis reveals that wildtype neutrophils show pure superdiffusive fractional Brownian motion. This so-called anomalous dynamics is characterized by temporal long-range correlations for the movement into the direction of the chemotactic CXCL1 gradient. Pure superdiffusion is absent vertically to this gradient. This points to an asymmetric 'memory' of the migratory machinery, which is found both in vitro and in vivo. CXCR2 blockade and TRPC6-knockout cause tempering of temporal correlations in the chemotactic gradient. This can be interpreted as a progressive loss of memory, which leads to a marked reduction of chemotaxis and search efficiency of neutrophils. In summary, our findings indicate that spatially differential regulation of anomalous dynamics appears to play a central role in guiding efficient chemotactic behavior.}, language = {en} } @article{BaerGrossmannHeidenreichetal.2019, author = {B{\"a}r, Markus and Großmann, Robert and Heidenreich, Sebastian and Peruani, Fernando}, title = {Self-propelled rods}, series = {Annual review of condensed matter physics}, volume = {11}, journal = {Annual review of condensed matter physics}, publisher = {Annual Reviews}, address = {Palo Alto}, issn = {1947-5454}, doi = {10.1146/annurev-conmatphys-031119-050611}, pages = {441 -- 466}, year = {2019}, abstract = {A wide range of experimental systems including gliding, swarming and swimming bacteria, in vitro motility assays, and shaken granular media are commonly described as self-propelled rods. Large ensembles of those entities display a large variety of self-organized, collective phenomena, including the formation of moving polar clusters, polar and nematic dynamic bands, mobility-induced phase separation, topological defects, and mesoscale turbulence, among others. Here, we give a brief survey of experimental observations and review the theoretical description of self-propelled rods. Our focus is on the emergent pattern formation of ensembles of dry self-propelled rods governed by short-ranged, contact mediated interactions and their wet counterparts that are also subject to long-ranged hydrodynamic flows. Altogether, self-propelled rods provide an overarching theme covering many aspects of active matter containing well-explored limiting cases. Their collective behavior not only bridges the well-studied regimes of polar selfpropelled particles and active nematics, and includes active phase separation, but also reveals a rich variety of new patterns.}, language = {en} } @article{WangCherstvyChechkinetal.2020, author = {Wang, Wei and Cherstvy, Andrey G. and Chechkin, Aleksei V. and Thapa, Samudrajit and Seno, Flavio and Liu, Xianbin and Metzler, Ralf}, title = {Fractional Brownian motion with random diffusivity}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {53}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {47}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aba467}, pages = {34}, year = {2020}, abstract = {Numerous examples for a priori unexpected non-Gaussian behaviour for normal and anomalous diffusion have recently been reported in single-particle tracking experiments. Here, we address the case of non-Gaussian anomalous diffusion in terms of a random-diffusivity mechanism in the presence of power-law correlated fractional Gaussian noise. We study the ergodic properties of this model via examining the ensemble- and time-averaged mean-squared displacements as well as the ergodicity breaking parameter EB quantifying the trajectory-to-trajectory fluctuations of the latter. For long measurement times, interesting crossover behaviour is found as function of the correlation time tau characterising the diffusivity dynamics. We unveil that at short lag times the EB parameter reaches a universal plateau. The corresponding residual value of EB is shown to depend only on tau and the trajectory length. The EB parameter at long lag times, however, follows the same power-law scaling as for fractional Brownian motion. We also determine a corresponding plateau at short lag times for the discrete representation of fractional Brownian motion, absent in the continuous-time formulation. These analytical predictions are in excellent agreement with results of computer simulations of the underlying stochastic processes. Our findings can help distinguishing and categorising certain nonergodic and non-Gaussian features of particle displacements, as observed in recent single-particle tracking experiments.}, language = {en} } @article{SeyrichAlirezaeizanjaniBetaetal.2018, author = {Seyrich, Maximilian and Alirezaeizanjani, Zahra and Beta, Carsten and Stark, Holger}, title = {Statistical parameter inference of bacterial swimming strategies}, series = {New journal of physics : the open-access journal for physics}, volume = {20}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/aae72c}, pages = {20}, year = {2018}, abstract = {We provide a detailed stochastic description of the swimming motion of an E. coli bacterium in two dimension, where we resolve tumble events in time. For this purpose, we set up two Langevin equations for the orientation angle and speed dynamics. Calculating moments, distribution and autocorrelation functions from both Langevin equations and matching them to the same quantities determined from data recorded in experiments, we infer the swimming parameters of E. coli. They are the tumble rate lambda, the tumble time r(-1), the swimming speed v(0), the strength of speed fluctuations sigma, the relative height of speed jumps eta, the thermal value for the rotational diffusion coefficient D-0, and the enhanced rotational diffusivity during tumbling D-T. Conditioning the observables on the swimming direction relative to the gradient of a chemoattractant, we infer the chemotaxis strategies of E. coli. We confirm the classical strategy of a lower tumble rate for swimming up the gradient but also a smaller mean tumble angle (angle bias). The latter is realized by shorter tumbles as well as a slower diffusive reorientation. We also find that speed fluctuations are increased by about 30\% when swimming up the gradient compared to the reversed direction.}, language = {en} } @article{GhoshCherstvyGrebenkovetal.2016, author = {Ghosh, Surya K. and Cherstvy, Andrey G. and Grebenkov, Denis S. and Metzler, Ralf}, title = {Anomalous, non-Gaussian tracer diffusion in crowded two-dimensional environments}, series = {NEW JOURNAL OF PHYSICS}, volume = {18}, journal = {NEW JOURNAL OF PHYSICS}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/18/1/013027}, pages = {16}, year = {2016}, abstract = {A topic of intense current investigation pursues the question of how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. Motivated by recent experiments we report results of extensive simulations of the motion of a finite sized tracer particle in a heterogeneously crowded environment made up of quenched distributions of monodisperse crowders of varying sizes in finite circular two-dimensional domains. For given spatial distributions of monodisperse crowders we demonstrate how anomalous diffusion with strongly non-Gaussian features arises in this model system. We investigate both biologically relevant situations of particles released either at the surface of an inner domain or at the outer boundary, exhibiting distinctly different features of the observed anomalous diffusion for heterogeneous distributions of crowders. Specifically we reveal an asymmetric spreading of tracers even at moderate crowding. In addition to the mean squared displacement (MSD) and local diffusion exponent we investigate the magnitude and the amplitude scatter of the time averaged MSD of individual tracer trajectories, the non-Gaussianity parameter, and the van Hove correlation function. We also quantify how the average tracer diffusivity varies with the position in the domain with a heterogeneous radial distribution of crowders and examine the behaviour of the survival probability and the dynamics of the tracer survival probability. Inter alia, the systems we investigate are related to the passive transport of lipid molecules and proteins in two-dimensional crowded membranes or the motion in colloidal solutions or emulsions in effectively two-dimensional geometries, as well as inside supercrowded, surface adhered cells.}, language = {en} } @article{BodrovaChechkinCherstvyetal.2015, author = {Bodrova, Anna S. and Chechkin, Aleksei V. and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Ultraslow scaled Brownian motion}, series = {New journal of physics : the open-access journal for physics}, volume = {17}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/17/6/063038}, pages = {16}, year = {2015}, abstract = {We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form D(t) similar or equal to 1/t. For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations.}, language = {en} } @article{DieterichKlagesChechkin2015, author = {Dieterich, Peter and Klages, Rainer and Chechkin, Aleksei V.}, title = {Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations}, series = {New journal of physics : the open-access journal for physics}, volume = {17}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/17/7/075004}, pages = {14}, year = {2015}, abstract = {Anomalous dynamics characterized by non-Gaussian probability distributions (PDFs) and/or temporal long-range correlations can cause subtle modifications of conventional fluctuation relations (FRs). As prototypes we study three variants of a generic time-fractional Fokker-Planck equation with constant force. Type A generates superdiffusion, type B subdiffusion and type C both super-and subdiffusion depending on parameter variation. Furthermore type C obeys a fluctuation-dissipation relation whereas A and B do not. We calculate analytically the position PDFs for all three cases and explore numerically their strongly non-Gaussian shapes. While for type C we obtain the conventional transient work FR, type A and type B both yield deviations by featuring a coefficient that depends on time and by a nonlinear dependence on the work. We discuss possible applications of these types of dynamics and FRs to experiments.}, language = {en} } @article{ShinCherstvyMetzler2015, author = {Shin, Jaeoh and Cherstvy, Andrey G. and Metzler, Ralf}, title = {Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions}, series = {New journal of physics : the open-access journal for physics}, volume = {17}, journal = {New journal of physics : the open-access journal for physics}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1367-2630}, doi = {10.1088/1367-2630/17/11/113028}, pages = {12}, year = {2015}, abstract = {We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction phi of the crowders and the crowder-crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational-but not rotational-diffusivity of the particles Dis a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law D(phi) similar to (1 - phi/phi*)(2 ... 2.4) with the area fraction phi occupied by the crowders and the critical value phi*. Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.}, language = {en} } @article{PalyulinChechkinMetzler2014, author = {Palyulin, Vladimir V. and Chechkin, Aleksei V. and Metzler, Ralf}, title = {Levy flights do not always optimize random blind search for sparse targets}, series = {Proceedings of the National Academy of Sciences of the United States of America}, volume = {111}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, number = {8}, publisher = {National Acad. of Sciences}, address = {Washington}, issn = {0027-8424}, doi = {10.1073/pnas.1320424111}, pages = {2931 -- 2936}, year = {2014}, abstract = {It is generally believed that random search processes based on scale-free, Levy stable jump length distributions (Levy flights) optimize the search for sparse targets. Here we show that this popular search advantage is less universal than commonly assumed. We study the efficiency of a minimalist search model based on Levy flights in the absence and presence of an external drift (underwater current, atmospheric wind, a preference of the walker owing to prior experience, or a general bias in an abstract search space) based on two different optimization criteria with respect to minimal search time and search reliability (cumulative arrival probability). Although Levy flights turn out to be efficient search processes when the target is far from the starting point, or when relative to the starting point the target is upstream, we show that for close targets and for downstream target positioning regular Brownian motion turns out to be the advantageous search strategy. Contrary to claims that Levy flights with a critical exponent alpha = 1 are optimal for the search of sparse targets in different settings, based on our optimization parameters the optimal a may range in the entire interval (1, 2) and especially include Brownian motion as the overall most efficient search strategy.}, language = {en} }