Anomalous dynamics of cell migration

Physical Review Letters, 2008

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement δ 2 of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that δ 2 differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable δ 2 . Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed. 05.40.Fb,87.10.Mn An ensemble of non interacting Brownian particles spreads according to Fick's law as a Gaussian packet. The ensemble averaged mean square displacement (MSD) is x 2 (t) = 2D 1 t where D 1 is the diffusion constant. By an Einstein relation D 1 is expressed in terms of statistical properties of the microscopic jumps according to D 1 = δx 2 /2 τ where τ is the average time between jumps and δx 2 is the variance of the jump lengths. Instead one can analyze the time series x(t) of the particle trajectory and determine the time averaged (TA) MSD

Soft Matter, 2014

The motility of the Dictyostelium discoideum (DD) cell is studied by video microscopy when the cells are plated on top of an agar plate at different densities, n. It is found that the fluctuating kinetics of the cells can be divided into two normal directions: the cell's forward-moving direction and its normal direction.

Reports on Progress in Physics, 2008

We review the progress in the field of front propagation in recent years. We survey many physical, biophysical and crossdisciplinary applications, including reduced-variable models of combustion flames, Reid's paradox of rapid forest range expansions, the European colonization of North America during the XIX century, the Neolithic transition in Europe from 13,000 to 5,000 years before present, the description of subsistence boundaries, the formation of cultural boundaries, the spread of genetic mutations, theory and experiments on virus infections, models of cancer tumors, etc. Recent theoretical advances are unified in a single framework, encompassing very diverse systems such as those with biased random walks, distributed delays, sequential reaction and dispersion, cohabitation models, age structure, systems with several interacting species, etc. Directions for future progress are outlined.

The Journal of Cell Biology, 2009

PLoS ONE, 2013

The movement of organisms is subject to a multitude of influences of widely varying character: from the bio-mechanics of the individual, over the interaction with the complex environment many animals live in, to evolutionary pressure and energy constraints. As the number of factors is large, it is very hard to build comprehensive movement models. Even when movement patterns in simple environments are analysed, the organisms can display very complex behaviours. While for largely undirected motion or long observation times the dynamics can sometimes be described by isotropic random walks, usually the directional persistence due to a preference to move forward has to be accounted for, e.g., by a correlated random walk. In this paper we generalise these descriptions to a model in terms of stochastic differential equations of Langevin type, which we use to analyse experimental search flight data of foraging bumblebees. Using parameter estimates we discuss the differences and similarities to correlated random walks. From simulations we generate artificial bumblebee trajectories which we use as a validation by comparing the generated ones to the experimental data.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2014

We extend the kinematic matrix ("kinematrix") formalism [Phys. Rev. E 89, 062304 (2014)], which via simple matrix algebra accesses ensemble properties of self-propellers influenced by uncorrelated noise, to treat Gaussian correlated noises. This extension brings into reach many real-world biological and biomimetic self-propellers for which inertia is significant. Applying the formalism, we analyze in detail ensemble behaviors of a 2D self-propeller with velocity fluctuations and orientation evolution driven by an Ornstein-Uhlenbeck process. On the basis of exact results, a variety of dynamical regimes determined by the inertial, speed-fluctuation, orientational diffusion, and emergent disorientation time scales are delineated and discussed.

PloS one, 2011

Therapeutic application of mesenchymal stem cells (MSC) requires their extensive in vitro expansion. MSC in culture typically grow to confluence within a few weeks. They show spindle-shaped fibroblastoid morphology and align to each other in characteristic spatial patterns at high cell density. We present an individual cell-based model (IBM) that is able to quantitatively describe the spatio-temporal organization of MSC in culture. Our model substantially improves on previous models by explicitly representing cell podia and their dynamics. It employs podia-generated forces for cell movement and adjusts cell behavior in response to cell density. At the same time, it is simple enough to simulate thousands of cells with reasonable computational effort. Experimental sheep MSC cultures were monitored under standard conditions. Automated image analysis was used to determine the location and orientation of individual cells. Our simulations quantitatively reproduced the observed growth dyna...

Nonlinear Systems and Complexity, 2013

This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift. Using the intermittent Pomeau-Manneville map the problem of weak chaos and infinite ergodic theory is outlined, defining a very recent mathematical field of research. Considering a spatially extended version of the Pomeau-Manneville map leads us to the phenomenon of anomalous diffusion. This problem will be discussed by applying stochastic continuous time random walk theory and by deriving a fractional diffusion equation. Another important topic within modern nonequilibrium statistical physics are fluctuation relations, which we investigate for anomalous dynamics. The chapter concludes by showing the importance of anomalous dynamics for understanding experimental results on biological cell migration.

Physical Review E, 2011

Proliferation and migration dichotomy of the tumor cell invasion is examined within two non-Markovian models. We consider the tumor spheroid, which consists of the tumor core with a high density of cells and the outer invasive zone. We distinguish two different regions of the outer invasive zone and develop models for both zones. In model I we analyze the near-core-outer region, where biased migration away from the tumor spheroid core takes place. We suggest non-Markovian switching between the migrating and proliferating phenotypes of tumor cells. Nonlinear master equations for mean densities of cancer cells of both phenotypes are derived. In anomalous switching case we estimate the average size of the near-core-outer region that corresponds to sublinear growth r(t) ∼ t μ for 0 < μ < 1. In model II we consider the outer zone, where the density of cancer cells is very low. We suggest an integrodifferential equation for the total density of cancer cells. For proliferation rate we use the classical logistic growth, while the migration of cells is subdiffusive. The exact formulas for the overall spreading rate of cancer cells are obtained by a hyperbolic scaling and Hamilton-Jacobi techniques.

Frontiers in Physics

In the last decade, the cellular Potts model has been extensively used to model interacting cell systems at the tissue-level. However, in early applications of this model, cell movement was taken as a consequence of membrane fluctuations due to cell-cell interactions, or as a response to an external chemotactic gradient. Recent findings have shown that eukaryotic cells can exhibit persistent displacements across scales larger than cell size, even in the absence of external signals. Persistent cell motion has been incorporated to the cellular Potts model by many authors in the context of collective motion, chemotaxis and morphogenesis. In this paper, we use the cellular Potts model in combination with a random field applied over each cell. This field promotes a uniform cell motion in a given direction during a certain time interval, after which the movement direction changes. The dynamics of the direction is coupled to a first order autoregressive process. We investigated statistical properties, such as the mean-squared displacement and spatio-temporal correlations, associated to these self-propelled in silico cells in different conditions. The proposed model emulates many properties observed in different experimental setups. By studying low and high density cultures, we find that cell-cell interactions decrease the effective persistent time.

Physical Review E

Cellular movement is a complex dynamic process, resulting from the interaction of multiple elements at the intra-and extracellular levels. This epiphenomenon presents a variety of behaviors, which can include normal and anomalous diffusion or collective migration. In some cases, cells can get neighborhood information through chemical or mechanical cues. A unified understanding about how such information can influence the dynamics of cell movement is still lacking. In order to improve our comprehension of cell migration we have considered a cellular Potts model where cells move actively in the direction of a driving field. The intensity of this driving field is constant, while its orientation can evolve according to two alternative dynamics based on the Ornstein-Uhlenbeck process. In one case, the next orientation of the driving field depends on the previous direction of the field. In the other case, the direction update considers the mean orientation performed by the cell in previous steps. Thus, the latter update rule mimics the ability of cells to perceive the environment, avoiding obstacles and thus increasing the cellular displacement. Different cell densities are considered to reveal the effect of cell-cell interactions. Our results indicate that both dynamics introduce temporal and spatial correlations in cell velocity in a friction-coefficient and cell-density-dependent manner. Furthermore, we observe alternating regimes in the mean-square displacement, with normal and anomalous diffusion. The crossovers between diffusive and directed motion regimes are strongly affected by both the driving field dynamics and cell-cell interactions. In this sense, when cell polarization update grants information about the previous cellular displacement, the duration of the diffusive regime decreases, particularly in high-density cultures.

Development, growth & differentiation, 2017

Cell movement and intercellular signaling occur simultaneously to organize morphogenesis during embryonic development. Cell movement can cause relative positional changes between neighboring cells. When intercellular signals are local such cell mixing may affect signaling, changing the flow of information in developing tissues. Little is known about the effect of cell mixing on intercellular signaling in collective cellular behaviors and methods to quantify its impact are lacking. Here we discuss how to determine the impact of cell mixing on cell signaling drawing an example from vertebrate embryogenesis: the segmentation clock, a collective rhythm of interacting genetic oscillators. We argue that comparing cell mixing and signaling timescales is key to determining the influence of mixing. A signaling timescale can be estimated by combining theoretical models with cell signaling perturbation experiments. A mixing timescale can be obtained by analysis of cell trajectories from live i...

Physical Review E, 2011

Using Monte Carlo simulations we examine the diffusive properties of the greedy algorithm in the d-dimensional traveling salesman problem. Our results show that for d = 3 and 4 the average squared distance from the origin r 2 is proportional to the number of steps t. In the d = 2 case such a scaling is modified with some logarithmic corrections, which might suggest that d = 2 is the critical dimension of the problem. The distribution of lengths also shows marked differences between d = 2 and d > 2 versions. A simple strategy adopted by the salesman might resemble strategies chosen by some foraging and hunting animals, for which anomalous diffusive behavior has recently been reported and interpreted in terms of Lévy flights. Our results suggest that broad and Lévy-like distributions in such systems might appear due to dimension-dependent properties of a search space.

Soft matter, 2017

Chemotaxis is a ubiquitous biological phenomenon in which cells detect a spatial gradient of chemoattractant, and then move towards the source. Here we present a position-dependent advection-diffusion model that quantitatively describes the statistical features of the chemotactic motion of the social amoeba Dictyostelium discoideum in a linear gradient of cAMP (cyclic adenosine monophosphate). We fit the model to experimental trajectories that are recorded in a microfluidic setup with stationary cAMP gradients and extract the diffusion and drift coefficients in the gradient direction. Our analysis shows that for the majority of gradients, both coefficients decrease over time and become negative as the cells crawl up the gradient. The extracted model parameters also show that besides the expected drift in the direction of the chemoattractant gradient, we observe a nonlinear dependency of the corresponding variance on time, which can be explained by the model. Furthermore, the results...

Cellular movement is a complex dynamic process, resulting from the interaction of multiple elements at the intra and extra-cellular levels. This epiphenomenon presents a variety of behaviors, which can include normal and anomalous diffusion or collective migration. In some cases cells can get neighborhood information through chemical or mechanical cues. A unified understanding about how such information can influence the dynamics of cell movement is still lacking. In order to improve our comprehension of cell migration we consider a cellular Potts model where cells move actively in the direction of a driving field. The intensity of this driving field is constant, while its orientation can evolves according to two alternative dynamics based on the Ornstein-Uhlenbeck process. In the first case, the next orientation of the driving field depends on the previous direction of the field. In the second case, the direction update considers the mean orientation performed by the cell in previo...

Pflügers Archiv - European Journal of Physiology, 2009

The glycocalyx consists of proteoglycans, glycoproteins, glycosaminoglycans, associated plasma proteins, and soluble glycosaminoglycans and covers the surface of all eukaryotic cells. It mediates specific recognition events, modulates biological processes such as ligand-receptor interactions, and has been proposed to affect tumor metastasis. Here, we studied whether the glycocalyx is required for melanoma cell migration. We diminished the glycocalyx of human melanoma cells by inhibiting posttranslational N-glycosylation or by enzymatic digestion of the N-glycosides. This partial destruction of the glycocalyx reduced melanoma cell migration by up to 60%. It was accompanied by the disintegration of a characteristic pH nanoenvironment typically surrounding migrating cells. Restoring this pH profile by stimulating the activity of the Na + /H + exchanger NHE1 rescued cell migration even in the absence of an intact glycocalyx. The effects of partially removing the glycocalyx compared to those of knocking down β 1-integrin expression points to a close functional correlation between glycocalyx, integrins, and cell surface pH nanoenvironment. We conclude that the glycocalyx is required for tumor cell migration. It stabilizes the cell surface pH nanoenvironment allowing a concerted pHdependent interaction of adhesion receptors and extracellular matrix.

Physiology, 2008

Cell motility is a prerequisite for the creation of new life, and it is required for maintaining the integrity of an organism. Under pathological conditions, “too much” motility may cause premature death. Studies over the past few years have revealed that ion channels are essential for cell motility. This review highlights the importance of K+ channels in regulating cell motility.

Cancer Cell International

Background Both cell adhesion and matrix metalloproteinase (MMP) activity depend on pH at the cell surface. By regulating extracellular juxtamembrane pH, the Na+/H+ exchanger NHE1 plays a significant part in human melanoma (MV3) cell migration and invasion. Because NHE1, besides its pH-regulatory transport function, also serves as a structural element tying the cortical actin cytoskeleton to the plasma membrane, we investigated whether NHE1 affects cortical stiffness of MV3 cells, and how this makes an impact on their invasiveness. Methods NHE1 overexpressing MV3 cells were compared to the corresponding mock-transfected control cells. NHE1 expression was verified by Western blotting, cariporide (HOE642) was used to inhibit NHE1 activity, cell stiffness was determined by atomic force microscopy, and F-actin was visualized by phalloidin-staining. Migration on, and invasion of, native and glutaraldehyde-fixed collagen I substrates were analyzed using time-lapse video microscopy and Boy...

Physical Review Letters, 2012

We analyze 3D flight paths of bumblebees searching for nectar in a laboratory experiment with and without predation risk from artificial spiders. For the flight velocities we find mixed probability distributions reflecting the access to the food sources while the threat posed by the spiders shows up only in the velocity correlations. The bumblebees thus adjust their flight patterns spatially to the environment and temporally to predation risk. Key information on response to environmental changes is contained in temporal correlation functions, as we explain by a simple emergent model.

Fluctuation Relations and Beyond, 2013

We study Fluctuation Relations (FRs) for dynamics that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of standard Brownian motion. We first briefly review the concept of transient work FRs for stochastic dynamics modeled by the ordinary Langevin equation. We then introduce three generic types of dynamics generating anomalous diffusion: Lévy flights, long-time correlated Gaussian stochastic processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the work probability distributions in the simple nonequilibrium situation of a particle subject to a constant force. This allows us to check the transient FR for anomalous dynamics. We find a new form of FRs, which is intimately related to the validity of fluctuation-dissipation relations. Analogous results are obtained for a particle in a harmonic potential dragged by a constant force. We argue that these findings are important for understanding fluctuations in experimentally accessible systems. As an example, we discuss the anomalous dynamics of biological cell migration both in equilibrium and in nonequilibrium under chemical gradients.

Physical Biology, 2011

We model the motility of Dictyostelium cells in a systematic data-driven manner. We deduce a minimal dynamical model that reproduces the statistical features of experimental trajectories. These are trajectories of the centroid of the cell perimeter, which is more sensitive to pseudopod activity than the usual tracking by centroid or nucleus. Our data account for cell individuality and dictate a

Scientific Reports, 2016

Quantification and discrimination of pharmaceutical and disease-related effects on cell migration requires detailed characterization of single-cell motility. In this context, micropatterned substrates that constrain cells within defined geometries facilitate quantitative readout of locomotion. Here, we study quasi-one-dimensional cell migration in ring-shaped microlanes. We observe bimodal behavior in form of alternating states of directional migration (run state) and reorientation (rest state). Both states show exponential lifetime distributions with characteristic persistence times, which, together with the cell velocity in the run state, provide a set of parameters that succinctly describe cell motion. By introducing PEGylated barriers of different widths into the lane, we extend this description by quantifying the effects of abrupt changes in substrate chemistry on migrating cells. The transit probability decreases exponentially as a function of barrier width, thus specifying a characteristic penetration depth of the leading lamellipodia. Applying this fingerprint-like characterization of cell motion, we compare different cell lines, and demonstrate that the cancer drug candidate salinomycin affects transit probability and resting time, but not run time or run velocity. Hence, the presented assay allows to assess multiple migration-related parameters, permits detailed characterization of cell motility, and has potential applications in cell biology and advanced drug screening.

Physical review. E, 2016

The diffusive behaviors of active colloids with run-and-tumble movement are explored by dissipative particle dynamics simulations for self-propelled particles (force dipole) and external field-driven particles (point force). The self-diffusion of tracers (solvent) is investigated as well. The influences of the active force, run time, and concentration associated with active particles are studied. For the system of self-propelled particles, the normal diffusion is observed for both active particles and tracers. The diffusivity of the former is significantly greater than that of the latter. For the system of field-driven particles, the superdiffusion is seen for both active particles and tracers. In contrast, it is found that the anomalous diffusion exponent of the former is slightly less than that of the latter. The anomalous diffusion is caused by the many-body, long-range hydrodynamic interactions. In spite of the superdiffusion, the sedimentation equilibrium of field-driven partic...

Reviews of Modern Physics, 2015

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The Lévy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to Lévy walks, surveys their existing applications, including latest advances, and outlines further perspectives.

Nature Communications

Metastatic cancer cells differ from their non-metastatic counterparts not only in terms of molecular composition and genetics, but also by the very strategy they employ for locomotion. Here, we analyzed large-scale statistics for cells migrating on linear microtracks to show that metastatic cancer cells follow a qualitatively different movement strategy than their non-invasive counterparts. The trajectories of metastatic cells display clusters of small steps that are interspersed with long "flights". Such movements are characterized by heavytailed, truncated power law distributions of persistence times and are consistent with the Lévy walks that are also often employed by animal predators searching for scarce prey or food sources. In contrast, non-metastatic cancerous cells perform simple diffusive movements. These findings are supported by preliminary experiments with cancer cells migrating away from primary tumors in vivo. The use of chemical inhibitors targeting actin-binding proteins allows for "reprogramming" the Lévy walks into either diffusive or ballistic movements.

Journal of the Mechanics and Physics of Solids

While it is commonly observed that the shape dynamics of mammalian cells can undergo large random fluctuations, theoretical models aiming at capturing cell mechanics often focus on the deterministic part of the motion. In this paper, we present a framework that couples an active gel model of the cell mechanical scaffold with the complex cell metabolic system stochastically delivering the chemical energy needed to sustain an active stress in the scaffold. Our closure assumption setting the magnitude of the fluctuations is that the chemo-mechanical free energy of the cell is controlled at a target homeostatic value. Our model rationalizes the experimental observation that the cell shape fluctuations depend on the mechanical environment that constraints the cell. We apply our framework to the simple case of a cell migrating on a one dimensional track to successfully capture the different regimes of the cell mean square displacement along the track as well as the magnitude of the long time scale effective diffusive motion of the cell.

Advances in Applied Mechanics

Although collective cell migration (CCM) is a highly coordinated and fine-tuned migratory mode, instabilities in the form of cell swirling motion (CSM) often occur. The CSM represents a product of the active turbulence obtained at low Reynolds number which has a feedback impact to various processes such as morphogenesis, wound healing, and cancer invasion. The cause of this phenomenon is related to the viscoelasticity of multicellular systems in the context of cell residual stress accumulation. Particular interest of this work is to: (1) emphasize the roles of cell shear and normal residual stress accumulated during CCM in the appearance of CSM and (2) consider the dynamics of CSM from the standpoint of rheology. Inhomogeneous distribution of the cell residual stress leads to a generation the viscoelastic force which acts to suppress CCM and can induces the system rapid stiffening. This force together with the surface tension force and traction force is responsible for the appearance of the CSM. In this work, a review of existing literature about viscoelasticity caused by CCM is given along with assortment of published experimental findings, in order to invite experimentalists to test given theoretical considerations in multicellular systems.

PLOS Computational Biology, 2021

The dispersal or mixing of cells within cellular tissue is a crucial property for diverse biological processes, ranging from morphogenesis, immune action, to tumor metastasis. With the phenomenon of ‘contact inhibition of locomotion,’ it is puzzling how cells achieve such processes within a densely packed cohesive population. Here we demonstrate that a proper degree of cell-cell adhesiveness can, intriguingly, enhance the super-diffusive nature of individual cells. We systematically characterize the migration trajectories of crawling MDA-MB-231 cell lines, while they are in several different clustering modes, including freely crawling singles, cohesive doublets of two cells, quadruplets, and confluent population on two-dimensional substrate. Following data analysis and computer simulation of a simple cellular Potts model, which faithfully recapitulated all key experimental observations such as enhanced diffusivity as well as periodic rotation of cell-doublets and cell-quadruplets wi...

PLOS ONE, 2019

Geometrical cues are known to play a very important role in neuronal growth and the formation of neuronal networks. Here, we present a detailed analysis of axonal growth and dynamics for neuronal cells cultured on patterned polydimethylsiloxane surfaces. We use fluorescence microscopy to image neurons, quantify their dynamics, and demonstrate that the substrate geometrical patterns cause strong directional alignment of axons. We quantify axonal growth and report a general stochastic approach that quantitatively describes the motion of growth cones. The growth cone dynamics is described by Langevin and Fokker-Planck equations with both deterministic and stochastic contributions. We show that the deterministic terms contain both the angular and speed dependence of axonal growth, and that these two contributions can be separated. Growth alignment is determined by surface geometry, and it is quantified by the deterministic part of the Langevin equation. We combine experimental data with theoretical analysis to measure the key parameters of the growth cone motion: speed and angular distributions, correlation functions, diffusion coefficients, characteristics speeds and damping coefficients. We demonstrate that axonal dynamics displays a cross-over from Brownian motion (Ornstein-Uhlenbeck process) at earlier times to anomalous dynamics (superdiffusion) at later times. The superdiffusive regime is characterized by non-Gaussian speed distributions and power law dependence of the axonal mean square length and the velocity correlation functions. These results demonstrate the importance of geometrical cues in guiding axonal growth, and could lead to new methods for bioengineering novel substrates for controlling neuronal growth and regeneration.

PLOS Computational Biology

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 Brownia...

Physical Review E, 2012

Population dynamics of individuals undergoing birth and death and diffusing by short-or long-range two-dimensional spatial excursions (Gaussian jumps or Lévy flights) is studied. Competitive interactions are considered in a global case, in which birth and death rates are influenced by all individuals in the system, and in a nonlocal but finite-range case in which interaction affects individuals in a neighborhood (we also address the noninteracting case). In the global case one single or few-cluster configurations are achieved with the spatial distribution of the bugs tied to the type of diffusion. In the Lévy case long tails appear for some properties characterizing the shape and dynamics of clusters. Under nonlocal finite-range interactions periodic patterns appear with periodicity set by the interaction range. This length acts as a cutoff limiting the influence of the long Lévy jumps, so that spatial configurations under the two types of diffusion become more similar. By dividing initially everyone into different families and following their descent it is possible to show that mixing of families and their competition is greatly influenced by the spatial dynamics.

Communications physics, 2022

Drosophila melanogaster hemocytes are highly motile cells that are crucial for successful embryogenesis and have important roles in the organism's immunological response. Here we measure the motion of hemocytes using selective plane illumination microscopy. Every hemocyte cell in one half of an embryo is tracked during embryogenesis and analysed using a deep learning neural network. We show that the anomalous transport of the cells is well described by fractional Brownian motion that is heterogeneous in both time and space. LanB1 and SCAR mutants disrupt the collective cellular motion and reduce its persistence due to the modification of laminin and actin-based motility respectively. The anomalous motility of the hemocytes oscillated in time with alternating periods of varying persistent motion. Touching hemocytes appear to experience synchronised contact inhibition of locomotion. A quantitative statistical framework is presented for hemocyte motility which provides biological insights.

Scientific Reports, 2017

Many diffusion processes in nature and society were found to be anomalous, in the sense of being fundamentally different from conventional Brownian motion. An important example is the migration of biological cells, which exhibits non-trivial temporal decay of velocity autocorrelation functions. This means that the corresponding dynamics is characterized by memory effects that slowly decay in time. Motivated by this we construct non-Markovian lattice-gas cellular automata models for moving agents with memory. For this purpose the reorientation probabilities are derived from velocity autocorrelation functions that are given a priori; in that respect our approach is “data-driven”. Particular examples we consider are velocity correlations that decay exponentially or as power laws, where the latter functions generate anomalous diffusion. The computational efficiency of cellular automata combined with our analytical results paves the way to explore the relevance of memory and anomalous di...

Mathematical Modelling of Natural Phenomena, 2013

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We derive the fractional Fokker-Planck equation for the density of cells and apply this equation to the anomalous chemotaxis problem. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.

Physical Review E, 2014

We investigate the dynamical phase diagram of the generalized Langevin equation of the free particle driven by a Mittag-Leffler noise and show critical curves and a critical value of the exponent parameter of the Mittag-Leffler function that mark different dynamical regimes. By considering that the modeling of a Mittag-Leffer memory kernel corresponds to a power-law second-order memory kernel, we show that the generalized Langevin equation of the velocity autocorrelation function (VACF) is transformed in a fractional Langevin equation. In the superdiffusive case our results exhibit oscillations and negative correlations of the VACF that are not provided by the usual power-law noise model.

Micromachines, 2014

We report a novel approach to study cell migration under physical stresses by utilizing established growth factor chemotaxis. This was achieved by studying cell migration in response to epidermal growth factor (EGF) chemoattraction in a gradually tapered space, imposing mechanical stresses. The device consisted of two 5-mm-diameter chambers connected by ten 600 µm-long and 10 µm-high tapered microchannels. The taper region gradually changes the width of the channel. The channels tapered from 20 µm to 5 µm over a transition length of 50 µm at a distance of 250 µm from one of the chambers. The chemoattractant drove cell migration into the narrow confines of the tapered channels, while the mechanical gradient clearly altered the migration of cells. Cells traversing the channels from the wider to narrow-end and vice versa were observed using time-lapsed imaging. Our results indicated that the impact of physical stress on cell migration patterns may be cell type specific.

Advances in Experimental Medicine and Biology, 2019

We discuss recent advances in interpreting the collective dynamics of cellular assemblies using ideas and tools coming from the statistical physics of materials. Experimental observations suggest analogies between the collective motion of cell monolayers and the jamming of soft materials. Granular media, emulsions and other soft materials display transitions between fluid-like and solid-like behavior as control parameters, such as temperature, density and stress, are changed. A similar jamming transition has been observed in the relaxation of epithelial cell monolayers. In this case, the associated unjamming transition, in which cells migrate collectively, is linked to a variety of biochemical and biophysical factors. In this framework, recent works show that wound healing induce monolayer fluidization with collective migration fronts moving in an avalanche-like behavior reminiscent of intermittent front propagation in materials such as domain walls in magnets, cracks in disordered media or flux lines in superconductors. Finally, we review the ability of discrete models of cell migration, from interacting active particles to vertex and Voronoi models, to simulate the statistical properties observed experimentally.

Journal of Cell Biology, 2021

Rab40b is a SOCS box–containing protein that regulates the secretion of MMPs to facilitate extracellular matrix remodeling during cell migration. Here, we show that Rab40b interacts with Cullin5 via the Rab40b SOCS domain. We demonstrate that loss of Rab40b–Cullin5 binding decreases cell motility and invasive potential and show that defective cell migration and invasion stem from alteration to the actin cytoskeleton, leading to decreased invadopodia formation, decreased actin dynamics at the leading edge, and an increase in stress fibers. We also show that these stress fibers anchor at less dynamic, more stable focal adhesions. Mechanistically, changes in the cytoskeleton and focal adhesion dynamics are mediated in part by EPLIN, which we demonstrate to be a binding partner of Rab40b and a target for Rab40b–Cullin5-dependent localized ubiquitylation and degradation. Thus, we propose a model where Rab40b–Cullin5-dependent ubiquitylation regulates EPLIN localization to promote cell mi...

Frontiers in Immunology

Dendritic cell (DC) migration is crucial for mounting immune responses. Immature DCs (imDCs) reportedly sense infections, while mature DCs (mDCs) move quickly to lymph nodes to deliver antigens to T cells. However, their highly heterogeneous and complex innate motility remains elusive. Here, we used an unsupervised machine learning (ML) approach to analyze long-term, two-dimensional migration trajectories of Granulocyte-macrophage colony-stimulating factor (GMCSF)-derived bone marrow-derived DCs (BMDCs). We discovered three migratory modes independent of the cell state: slow-diffusive (SD), slow-persistent (SP), and fast-persistent (FP). Remarkably, imDCs more frequently changed their modes, predominantly following a unicyclic SD→FP→SP→SD transition, whereas mDCs showed no transition directionality. We report that DC migration exhibits a history-dependent mode transition and maturation-dependent motility changes are emergent properties of the dynamic switching of the three migratory...

Philosophical Transactions of the Royal Society B: Biological Sciences, 2020

Collective migration has become a paradigm for emergent behaviour in systems of moving and interacting individual units resulting in coherent motion. In biology, these units are cells or organisms. Collective cell migration is important in embryonic development, where it underlies tissue and organ formation, as well as pathological processes, such as cancer invasion and metastasis. In animal groups, collective movements may enhance individuals&#39; decisions and facilitate navigation through complex environments and access to food resources. Mathematical models can extract unifying principles behind the diverse manifestations of collective migration. In biology, with a few exceptions, collective migration typically occurs at a ‘mesoscopic scale’ where the number of units ranges from only a few dozen to a few thousands, in contrast to the large systems treated by statistical mechanics. Recent developments in multi-scale analysis have allowed linkage of mesoscopic to micro- and macros...

Physical Review E, 2014

We consider the rates of relaxation of a particle in a harmonic well, subject to Lévy noise characterized by its Lévy index µ. Using the propagator for this Lévy Ornstein-Uhlenbeck process (LOUP), we show that the eigenvalue spectrum of the associated Fokker-Planck operator has the form (n+mµ)ν where ν is the force constant characterizing the well, and n, m ∈ N. If µ is irrational, the eigenvalues are all non-degenerate, but rational µ can lead to degeneracy. The maximum degeneracy is shown to be two. The left eigenfunctions of the fractional Fokker-Planck operator are very simple while the right eigenfunctions may be obtained from the lowest eigenfunction by a combination of two different step-up operators. Further, we find that the acceptable eigenfunctions should have the asymptotic behavior |x| −n 1 +n 2 µ as |x| → ∞, with n 1 and n 2 being positive integers, though this condition alone is not enough to identify them uniquely. We also assert that the rates of relaxation of LOUP are determined by the eigenvalues of the associated fractional Fokker-Planck operator and do not depend on the initial state if the moments of the initial distribution are all finite. If the initial distribution has fat tails, for which the higher moments diverge, one would have non-spectral relaxation, as pointed out by Toenjes et. al (Physical Review Letters, 110, 150602 (2013)).

bioRxiv (Cold Spring Harbor Laboratory), 2022

Dendritic cells (DCs) patrol the body as immunological sentinels and search for pathogens. Upon stimulation, immature DCs (imDCs) become mature DCs (mDCs), which migrate to the lymph nodes and present antigens to T cells. The migratory behavior is crucial for initiating and controlling immune responses; however, the properties of the highly heterogeneous and dynamic motility phenotype are not fully understood. Here, we established an unsupervised machine learning (ML) strategy to investigate spatiotemporal motility patterns in long-term, two-dimensional cell migration trajectories, and determined the number of motility patterns and how these are related to the maturation status. We identified three distinct migratory modes independent of the cell state: slowdiffusive (SD), slow-persistent (SP), and fast-persistent (FP). We found that maturation-dependent motility changes are emergent properties of the distribution and dynamic transitions of these three modes. Remarkably, imDCs changed their migration modes more frequently, and predominantly followed the SD→FP→SP→SD unicyclic transition, indicating that imDCs rapidly increase their speed during the shift from diffusive to persistent motility; however, persistence progressively declines when switching back to diffusive motility. In contrast, mDCs show no transition directionality. Our ML-promoted motility pattern analysis and history-dependent mode transition investigation may provide new insights into the complex process of biological motility.

Physics, 2019

In this work, we investigate a series of mathematical aspects for the fractional diffusion equation with stochastic resetting. The stochastic resetting process in Evans–Majumdar sense has several applications in science, with a particular emphasis on non-equilibrium physics and biological systems. We propose a version of the stochastic resetting theory for systems in which the reset point is in motion, so the walker does not return to the initial position as in the standard model, but returns to a point that moves in space. In addition, we investigate the proposed stochastic resetting model for diffusion with the fractional operator of Prabhakar. The derivative of Prabhakar consists of an integro-differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes a series of fractional operators such as Riemann–Liouville–Caputo. We present how the generalized model of stochastic resetting for fractional diffusion implies a rich...

SN Applied Sciences, 2020

Human pluripotent stem cells hold great promise for developments in regenerative medicine and drug design. The mathematical modelling of stem cells and their properties is necessary to understand and quantify key behaviours and develop non-invasive prognostic modelling tools to assist in the optimisation of laboratory experiments. Here, the recent advances in the mathematical modelling of hPSCs are discussed, including cell kinematics, cell proliferation and colony formation, and pluripotency and differentiation.