Quantifying the degree of persistence in random amoeboid motion based on the Hurst exponent of fractional Brownian motion

Non-Gaussian Displacements in Active Transport on a Carpet of Motile Cells

We study the dynamics of micron-sized particles on a layer of motile cells. This cell carpet acts as an active bath that propels passive tracer particles via direct mechanical contact. The resulting nonequilibrium transport shows a crossover from superdiffusive to normal-diffusive dynamics. The particle displacement distribution is distinctly non-Gaussian even at macroscopic timescales exceeding the measurement time. We obtain the distribution of diffusion coefficients from the experimental data and introduce a model for the displacement distribution that matches the experimentally observed non-Gaussian statistics. We argue why similar transport properties are expected for many composite active matter systems.

Heterogeneous run-and-tumble motion accounts for transient non-Gaussian super-diffusion in haematopoietic multi-potent progenitor cells

Multi-potent progenitor (MPP) cells act as a key intermediary step between haematopoietic stem cells and the entirety of the mature blood cell system. Their eventual fate determination is thought to be achieved through migration in and out of spatially distinct niches. Here we first analyze statistically MPP cell trajectory data obtained from a series of long time-course 3D in vivo imaging experiments on irradiated mouse calvaria, and report that MPPs display transient super-diffusion with apparent non-Gaussian displacement distributions. Second, we explain these experimental findings using a run-and-tumble model of cell motion which incorporates the observed dynamical heterogeneity of the MPPs. Third, we use our model to extrapolate the dynamics to time-periods currently inaccessible experimentally, which enables us to quantitatively estimate the time and length scales at which super-diffusion transitions to Fickian diffusion. Our work sheds light on the potential importance of motility in early haematopoietic progenitor function.

Anomalous diffusion and asymmetric tempering memory in neutrophil chemotaxis

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.

Analysis of protrusion dynamics in amoeboid cell motility by means of regularized contour flows

Amoeboid cell motility is essential for a wide range of biological processes including wound healing, embryonic morphogenesis, and cancer metastasis. It relies on complex dynamical patterns of cell shape changes that pose long-standing challenges to mathematical modeling and raise a need for automated and reproducible approaches to extract quantitative morphological features from image sequences. Here, we introduce a theoretical framework and a computational method for obtaining smooth representations of the spatiotemporal contour dynamics from stacks of segmented microscopy images. Based on a Gaussian process regression we propose a one-parameter family of regularized contour flows that allows us to continuously track reference points (virtual markers) between successive cell contours. We use this approach to define a coordinate system on the moving cell boundary and to represent different local geometric quantities in this frame of reference. In particular, we introduce the local marker dispersion as a measure to identify localized membrane expansions and provide a fully automated way to extract the properties of such expansions, including their area and growth time. The methods are available as an open-source software package called AmoePy, a Python-based toolbox for analyzing amoeboid cell motility (based on time-lapse microscopy data), including a graphical user interface and detailed documentation. Due to the mathematical rigor of our framework, we envision it to be of use for the development of novel cell motility models. We mainly use experimental data of the social amoeba Dictyostelium discoideum to illustrate and validate our approach.

Amoeboid motion is a crawling-like cell migration that plays an important key role in multiple biological processes such as wound healing and cancer metastasis. This type of cell motility results from expanding and simultaneously contracting parts of the cell membrane. From fluorescence images, we obtain a sequence of points, representing the cell membrane, for each time step. By using regression analysis on these sequences, we derive smooth representations, so-called contours, of the membrane. Since the number of measurements is discrete and often limited, the question is raised of how to link consecutive contours with each other. In this work, we present a novel mathematical framework in which these links are described by regularized flows allowing a certain degree of concentration or stretching of neighboring reference points on the same contour. This stretching rate, the so-called local dispersion, is used to identify expansions and contractions of the cell membrane providing a fully automated way of extracting properties of these cell shape changes. We applied our methods to time-lapse microscopy data of the social amoeba Dictyostelium discoideum.

Unified control of amoeboid pseudopod extension in multiple organisms by branched F-actin in the front and parallel F-actin/myosin in the cortex

The trajectory of moving eukaryotic cells depends on the kinetics and direction of extending pseudopods. The direction of pseudopods has been well studied to unravel mechanisms for chemotaxis, wound healing and inflammation. However, the kinetics of pseudopod extension-when and why do pseudopods start and stop- is equally important, but is largely unknown. Here the START and STOP of about 4000 pseudopods was determined in four different species, at four conditions and in nine mutants (fast amoeboids Dictyostelium and neutrophils, slow mesenchymal stem cells, and fungus B.d. chytrid with pseudopod and a flagellum). The START of a first pseudopod is a random event with a probability that is species-specific (23%/s for neutrophils). In all species and conditions, the START of a second pseudopod is strongly inhibited by the extending first pseudopod, which depends on parallel filamentous actin/myosin in the cell cortex. Pseudopods extend at a constant rate by polymerization of branched F-actin at the pseudopod tip, which requires the Scar complex. The STOP of pseudopod extension is induced by multiple inhibitory processes that evolve during pseudopod extension and mainly depend on the increasing size of the pseudopod. Surprisingly, no differences in pseudopod kinetics are detectable between polarized, unpolarized or chemotactic cells, and also not between different species except for small differences in numerical values. This suggests that the analysis has uncovered the fundament of cell movement with distinct roles for stimulatory branched F-actin in the protrusion and inhibitory parallel F-actin in the contractile cortex.

Stochastic Interpolation of Sparsely Sampled Time Series via Multipoint Fractional Brownian Bridges

We propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractional Brownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of intermediate and nonequidistant points. Determining the optimal value of the Hurst exponent H_{opt}, appropriate to interpolate the sparse signal, is a very important step of our method. We demonstrate the validity of our method on a signal from fluid turbulence in a high Reynolds number flow and discuss the implications of the non-self-similar character of the signal. The method introduced here could be instrumental in several physical problems, including astrophysics, particle tracking, and specific tailoring of surrogate data, as well as in domains of natural and social sciences.

Symmetry Breaking during Cell Movement in the Context of Excitability, Kinetic Fine-Tuning and Memory of Pseudopod Formation

The path of moving eukaryotic cells depends on the kinetics and direction of extending pseudopods. Amoeboid cells constantly change their shape with pseudopods extending in different directions. Detailed analysis has revealed that time, place and direction of pseudopod extension are not random, but highly ordered with strong prevalence for only one extending pseudopod, with defined life-times, and with reoccurring events in time and space indicative of memory. Important components are Ras activation and the formation of branched F-actin in the extending pseudopod and inhibition of pseudopod formation in the contractile cortex of parallel F-actin/myosin. In biology, order very often comes with symmetry. In this essay, I discuss cell movement and the dynamics of pseudopod extension from the perspective of symmetry and symmetry changes of Ras activation and the formation of branched F-actin in the extending pseudopod. Combining symmetry of Ras activation with kinetics and memory of pseudopod extension results in a refined model of amoeboid movement that appears to be largely conserved in the fast moving Dictyostelium and neutrophils, the slow moving mesenchymal stem cells and the fungus B.d. chytrid.

Analytical representation of Gaussian processes in the A-T plane

Closed-form expressions, parametrized by the Hurst exponent H and the length n of a time series, are derived for paths of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) in the A-T plane, composed of the fraction of turning points T and the Abbe value A. The exact formula for A_{fBm} is expressed via Riemann ζ and Hurwitz ζ functions. A very accurate approximation, yielding a simple exponential form, is obtained. Finite-size effects, introduced by the deviation of fGn's variance from unity, and asymptotic cases are discussed. Expressions for T for fBm, fGn, and differentiated fGn are also presented. The same methodology, valid for any Gaussian process, is applied to autoregressive moving average processes, for which regions of availability of the A-T plane are derived and given in analytic form. Locations in the A-T plane of some real-world examples as well as generated data are discussed for illustration.

Statistical and Mathematical Modeling of Spatiotemporal Dynamics of Stem Cells

Statistical and mathematical modeling are crucial to describe, interpret, compare, and predict the behavior of complex biological systems including the organization of hematopoietic stem and progenitor cells in the bone marrow environment. The current prominence of high-resolution and live-cell imaging data provides an unprecedented opportunity to study the spatiotemporal dynamics of these cells within their stem cell niche and learn more about aberrant, but also unperturbed, normal hematopoiesis. However, this requires careful quantitative statistical analysis of the spatial and temporal behavior of cells and the interaction with their microenvironment. Moreover, such quantification is a prerequisite for the construction of hypothesis-driven mathematical models that can provide mechanistic explanations by generating spatiotemporal dynamics that can be directly compared to experimental observations. Here, we provide a brief overview of statistical methods in analyzing spatial distribution of cells, cell motility, cell shapes, and cellular genealogies. We also describe cell-based modeling formalisms that allow researchers to simulate emergent behavior in a multicellular system based on a set of hypothesized mechanisms. Together, these methods provide a quantitative workflow for the analytic and synthetic study of the spatiotemporal behavior of hematopoietic stem and progenitor cells.

Modeling random crawling, membrane deformation and intracellular polarity of motile amoeboid cells

Amoeboid movement is one of the most widespread forms of cell motility that plays a key role in numerous biological contexts. While many aspects of this process are well investigated, the large cell-to-cell variability in the motile characteristics of an otherwise uniform population remains an open question that was largely ignored by previous models. In this article, we present a mathematical model of amoeboid motility that combines noisy bistable kinetics with a dynamic phase field for the cell shape. To capture cell-to-cell variability, we introduce a single parameter for tuning the balance between polarity formation and intracellular noise. We compare numerical simulations of our model to experiments with the social amoeba Dictyostelium discoideum. Despite the simple structure of our model, we found close agreement with the experimental results for the center-of-mass motion as well as for the evolution of the cell shape and the overall intracellular patterns. We thus conjecture that the building blocks of our model capture essential features of amoeboid motility and may serve as a starting point for more detailed descriptions of cell motion in chemical gradients and confined environments.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

Modelling of Dictyostelium discoideum movement in a linear gradient of chemoattractant

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 of the model show that the non-linear term in the mean squared displacement of the cell trajectories can dominate the linear term on large time scales.

Discrete Modeling of Amoeboid Locomotion and Chemotaxis in Dictyostelium discoideum by Tracking Pseudopodium Growth Direction

Dictyostelium discoideum amoeba is a well-established model organism for studying the crawling locomotion of eukaryotic cells. These amoebae extend pseudopodium - a temporary actin-based protrusion of their body membrane to probe the medium and crawl through it. Experiments show highly-ordered patterns in the growth direction of these pseudopodia, which results in persistence cell motility. Here, we propose a discrete model for studying and investigating the cell locomotion based on the experimental evidences. According to our model, Dictyostelium selects its pseudopodium growth direction based on a second-order Markov chain process, in the absence of external cues. Consequently, compared to a random walk process, our model indicates stronger growth in the mean-square displacement of cells, which is consistent with empirical findings. In the presence of external chemical stimulants, cells tend to align with the gradient of chemoattractant molecules. To quantify this tendency, we define a coupling coefficient between the pseudopodium extension direction and the gradient of an external stimulant, which depends on the local stimulant concentration and its gradient. Additionally, we generalize the model to weak-coupling regime by utilizing perturbation methods.

How to connect time-lapse recorded trajectories of motile microorganisms with dynamical models in continuous time

We provide a tool for data-driven modeling of motility, data being time-lapse recorded trajectories. Several mathematical properties of a model to be found can be gleaned from appropriate model-independent experimental statistics, if one understands how such statistics are distorted by the finite sampling frequency of time-lapse recording, by experimental errors on recorded positions, and by conditional averaging. We give exact analytical expressions for these effects in the simplest possible model for persistent random motion, the Ornstein-Uhlenbeck process. Then we describe those aspects of these effects that are valid for any reasonable model for persistent random motion. Our findings are illustrated with experimental data and Monte Carlo simulations.

Transient Superdiffusion and Long-Range Correlations in the Motility Patterns of Trypanosomatid Flagellate Protozoa

We report on a diffusive analysis of the motion of flagellate protozoa species. These parasites are the etiological agents of neglected tropical diseases: leishmaniasis caused by Leishmania amazonensis and Leishmania braziliensis, African sleeping sickness caused by Trypanosoma brucei, and Chagas disease caused by Trypanosoma cruzi. By tracking the positions of these parasites and evaluating the variance related to the radial positions, we find that their motions are characterized by a short-time transient superdiffusive behavior. Also, the probability distributions of the radial positions are self-similar and can be approximated by a stretched Gaussian distribution. We further investigate the probability distributions of the radial velocities of individual trajectories. Among several candidates, we find that the generalized gamma distribution shows a good agreement with these distributions. The velocity time series have long-range correlations, displaying a strong persistent behavior (Hurst exponents close to one). The prevalence of “universal” patterns across all analyzed species indicates that similar mechanisms may be ruling the motion of these parasites, despite their differences in morphological traits. In addition, further analysis of these patterns could become a useful tool for investigating the activity of new candidate drugs against these and others neglected tropical diseases.

Cell-substrate impedance fluctuations of single amoeboid cells encode cell-shape and adhesion dynamics

We show systematic electrical impedance measurements of single motile cells on microelectrodes. Wild-type cells and mutant strains were studied that differ in their cell-substrate adhesion strength. We recorded the projected cell area by time-lapse microscopy and observed irregular oscillations of the cell shape. These oscillations were correlated with long-term variations in the impedance signal. Superposed to these long-term trends, we observed fluctuations in the impedance signal. Their magnitude clearly correlated with the adhesion strength, suggesting that strongly adherent cells display more dynamic cell-substrate interactions.