A stochastic description of Dictyostelium chemotaxis

Methods and computational tools to study eukaryotic cell migration in vitro

Cellular movement is essential for many vital biological functions where it plays a pivotal role both at the single cell level, such as during division or differentiation, and at the macroscopic level within tissues, where coordinated migration is crucial for proper morphogenesis. It also has an impact on various pathological processes, one for all, cancer spreading. Cell migration is a complex phenomenon and diverse experimental methods have been developed aimed at dissecting and analysing its distinct facets independently. In parallel, corresponding analytical procedures and tools have been devised to gain deep insight and interpret experimental results. Here we review established experimental techniques designed to investigate specific aspects of cell migration and present a broad collection of historical as well as cutting-edge computational tools used in quantitative analysis of cell motion.

Data-driven classification of individual cells by their non-Markovian motion

We present a method to differentiate organisms solely by their motion based on the generalized Langevin equation (GLE) and use it to distinguish two different swimming modes of strongly confined unicellular microalgae Chlamydomonas reinhardtii. The GLE is a general model for active or passive motion of organisms and particles that can be derived from a time-dependent general many-body Hamiltonian and in particular includes non-Markovian effects (i.e., the trajectory memory of its past). We extract all GLE parameters from individual cell trajectories and perform an unbiased cluster analysis to group them into different classes. For the specific cell population employed in the experiments, the GLE-based assignment into the two different swimming modes works perfectly, as checked by control experiments. The classification and sorting of single cells and organisms is important in different areas; our method, which is based on motion trajectories, offers wide-ranging applications in biology and medicine.

Learning dynamical models of single and collective cell migration: a review

Single and collective cell migration are fundamental processes critical for physiological phenomena ranging from embryonic development and immune response to wound healing and cancer metastasis. To understand cell migration from a physical perspective, a broad variety of models for the underlying physical mechanisms that govern cell motility have been developed. A key challenge in the development of such models is how to connect them to experimental observations, which often exhibit complex stochastic behaviours. In this review, we discuss recent advances in data-driven theoretical approaches that directly connect with experimental data to infer dynamical models of stochastic cell migration. Leveraging advances in nanofabrication, image analysis, and tracking technology, experimental studies now provide unprecedented large datasets on cellular dynamics. In parallel, theoretical efforts have been directed towards integrating such datasets into physical models from the single cell to the tissue scale with the aim of conceptualising the emergent behaviour of cells. We first review how this inference problem has been addressed in both freely migrating and confined cells. Next, we discuss why these dynamics typically take the form of underdamped stochastic equations of motion, and how such equations can be inferred from data. We then review applications of data-driven inference and machine learning approaches to heterogeneity in cell behaviour, subcellular degrees of freedom, and to the collective dynamics of multicellular systems. Across these applications, we emphasise how data-driven methods can be integrated with physical active matter models of migrating cells, and help reveal how underlying molecular mechanisms control cell behaviour. Together, these data-driven approaches are a promising avenue for building physical models of cell migration directly from experimental data, and for providing conceptual links between different length-scales of description.

Nonlinear Growth Dynamics of Neuronal Cells Cultured on Directional Surfaces

During the development of the nervous system, neuronal cells extend axons and dendrites that form complex neuronal networks, which are essential for transmitting and processing information. Understanding the physical processes that underlie the formation of neuronal networks is essential for gaining a deeper insight into higher-order brain functions such as sensory processing, learning, and memory. In the process of creating networks, axons travel towards other recipient neurons, directed by a combination of internal and external cues that include genetic instructions, biochemical signals, as well as external mechanical and geometrical stimuli. Although there have been significant recent advances, the basic principles governing axonal growth, collective dynamics, and the development of neuronal networks remain poorly understood. In this paper, we present a detailed analysis of nonlinear dynamics for axonal growth on surfaces with periodic geometrical patterns. We show that axonal growth on these surfaces is described by nonlinear Langevin equations with speed-dependent deterministic terms and gaussian stochastic noise. This theoretical model yields a comprehensive description of axonal growth at both intermediate and long time scales (tens of hours after cell plating), and predicts key dynamical parameters, such as speed and angular correlation functions, axonal mean squared lengths, and diffusion (cell motility) coefficients. We use this model to perform simulations of axonal trajectories on the growth surfaces, in turn demonstrating very good agreement between simulated growth and the experimental results. These results provide important insights into the current understanding of the dynamical behavior of neurons, the self-wiring of the nervous system, as well as for designing innovative biomimetic neural network models.

Three-component contour dynamics model to simulate and analyze amoeboid cell motility in two dimensions

Amoeboid cell motility is relevant in a wide variety of biomedical processes such as wound healing, cancer metastasis, and embryonic morphogenesis. It is characterized by pronounced changes of the cell shape associated with expansions and retractions of the cell membrane, which result in a crawling kind of locomotion. Despite existing computational models of amoeboid motion, the inference of expansion and retraction components of individual cells, the corresponding classification of cells, and the a priori specification of the parameter regime to achieve a specific motility behavior remain challenging open problems. We propose a novel model of the spatio-temporal evolution of two-dimensional cell contours comprising three biophysiologically motivated components: a stochastic term accounting for membrane protrusions and two deterministic terms accounting for membrane retractions by regularizing the shape and area of the contour. Mathematically, these correspond to the intensity of a self-exciting Poisson point process, the area-preserving curve-shortening flow, and an area adjustment flow. The model is used to generate contour data for a variety of qualitatively different, e.g., polarized and non-polarized, cell tracks that visually resemble experimental data very closely. In application to experimental cell tracks, we inferred the protrusion component and examined its correlation to common biomarkers: the F-actin density close to the membrane and its local motion. Due to the low model complexity, parameter estimation is fast, straightforward, and offers a simple way to classify contour dynamics based on two locomotion types: the amoeboid and a so-called fan-shaped type. For both types, we use cell tracks segmented from fluorescence imaging data of the model organism Dictyostelium discoideum. An implementation of the model is provided within the open-source software package AmoePy, a Python-based toolbox for analyzing and simulating amoeboid cell motility.

Biased Random Walk Model of Neuronal Dynamics on Substrates with Periodic Geometrical Patterns

Neuronal networks are complex systems of interconnected neurons responsible for transmitting and processing information throughout the nervous system. The building blocks of neuronal networks consist of individual neurons, specialized cells that receive, process, and transmit electrical and chemical signals throughout the body. The formation of neuronal networks in the developing nervous system is a process of fundamental importance for understanding brain activity, including perception, memory, and cognition. To form networks, neuronal cells extend long processes called axons, which navigate toward other target neurons guided by both intrinsic and extrinsic factors, including genetic programming, chemical signaling, intercellular interactions, and mechanical and geometrical cues. Despite important recent advances, the basic mechanisms underlying collective neuron behavior and the formation of functional neuronal networks are not entirely understood. In this paper, we present a combined experimental and theoretical analysis of neuronal growth on surfaces with micropatterned periodic geometrical features. We demonstrate that the extension of axons on these surfaces is described by a biased random walk model, in which the surface geometry imparts a constant drift term to the axon, and the stochastic cues produce a random walk around the average growth direction. We show that the model predicts key parameters that describe axonal dynamics: diffusion (cell motility) coefficient, average growth velocity, and axonal mean squared length, and we compare these parameters with the results of experimental measurements. Our findings indicate that neuronal growth is governed by a contact-guidance mechanism, in which the axons respond to external geometrical cues by aligning their motion along the surface micropatterns. These results have a significant impact on developing novel neural network models, as well as biomimetic substrates, to stimulate nerve regeneration and repair after injury.

The aerotaxis of Dictyostelium discoideum is independent of mitochondria, nitric oxide and oxidative stress

Spatial and temporal variations of oxygen environments affect the behaviors of various cells and are involved in physiological and pathological events. Our previous studies with Dictyostelium discoideum as a model of cell motility have demonstrated that aerotaxis toward an oxygen-rich region occurs below 2% O₂. However, while the aerotaxis of Dictyostelium seems to be an effective strategy to search for what is essential for survival, the mechanism underlying this phenomenon is still largely unclear. One hypothesis is that an oxygen concentration gradient generates a secondary oxidative stress gradient that would direct cell migration towards higher oxygen concentration. Such mechanism was inferred but not fully demonstrated to explain the aerotaxis of human tumor cells. Here, we investigated the role on aerotaxis of flavohemoglobins, proteins that can both act as potential oxygen sensors and modulators of nitric oxide and oxidative stress. The migratory behaviors of Dictyostelium cells were observed under both self-generated and imposed oxygen gradients. Furthermore, their changes by chemicals generating or preventing oxidative stress were tested. The trajectories of the cells were then analyzed through time-lapse phase-contrast microscopic images. The results indicate that both oxidative and nitrosative stresses are not involved in the aerotaxis of Dictyostelium but cause cytotoxic effects that are enhanced upon hypoxia.

Kinetic modeling of the chemotactic process in run-and-tumble bacteria

The chemotactic process of run-and-tumble bacteria results from modulating the tumbling rate in response to changes in chemoattractant gradients felt by the bacteria. The response has a characteristic memory time and is subject to important fluctuations. These ingredients are considered in a kinetic description of chemotaxis, allowing the computation of the stationary mobility and the relaxation times needed to reach the steady state. For large memory times, these relaxation times become large, implying that finite-time measurements give rise to nonmonotonic currents as a function of the imposed chemoattractant gradient, contrary to the stationary regime where the response is monotonic. The case of an inhomogeneous signal is analyzed. Contrary to the usual Keller-Segel model, the response is nonlocal, and the bacterial profile is smoothed with a characteristic length that grows with the memory time. Finally, the case of traveling signals is considered, where appreciable differences appear compared to memoryless chemotactic descriptions.

Understanding the Cell's Response to Chemical Signals: Utilisation of Microfluidic Technology in Studies of Cellular and Dictyostelium discoideum Chemotaxis

Cellular chemotaxis has been the subject of a variety of studies due to its relevance in physiological processes, disease pathogenesis, and systems biology, among others. The migration of cells towards a chemical source remains a closely studied topic, with the Boyden chamber being one of the earlier techniques that has successfully studied cell chemotaxis. Despite its success, diffusion chambers such as these presented a number of problems, such as the quantification of many aspects of cell behaviour, the reproducibility of procedures, and measurement accuracy. The advent of microfluidic technology prompted more advanced studies of cell chemotaxis, usually involving the social amoeba Dictyostelium discoideum (D. discoideum) as a model organism because of its tendency to aggregate towards chemotactic agents and its similarities to higher eukaryotes. Microfluidic technology has made it possible for studies to look at chemotactic properties that would have been difficult to observe using classic diffusion chambers. Its flexibility and its ability to generate consistent concentration gradients remain some of its defining aspects, which will surely lead to an even better understanding of cell migratory behaviour and therefore many of its related biological processes. This paper first dives into a brief introduction of D. discoideum as a social organism and classical chemotaxis studies. It then moves to discuss early microfluidic devices, before diving into more recent and advanced microfluidic devices and their use with D. discoideum. The paper then closes with brief opinions about research progress in the field and where it will possibly lead in the future.

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.

Feedback-controlled dynamics of neuronal cells on directional surfaces

The formation of neuronal networks is a complex phenomenon of fundamental importance for understanding the development of the nervous system. The basic process underlying the network formation is axonal growth, a process involving the extension of axons from the cell body and axonal navigation toward target neurons. Axonal growth is guided by the interactions between the tip of the axon (growth cone) and its extracellular environmental cues, which include intercellular interactions, the biochemical landscape around the neuron, and the mechanical and geometrical features of the growth substrate. Here, we present a comprehensive experimental and theoretical analysis of axonal growth for neurons cultured on micropatterned polydimethylsiloxane (PDMS) surfaces. We demonstrate that closed-loop feedback is an essential component of axonal dynamics on these surfaces: the growth cone continuously measures environmental cues and adjusts its motion in response to external geometrical features. We show that this model captures all the characteristics of axonal dynamics on PDMS surfaces for both untreated and chemically modified neurons. We combine experimental data with theoretical analysis to measure key parameters that describe axonal dynamics: diffusion (cell motility) coefficients, speed and angular distributions, and cell-substrate interactions. The experiments performed on neurons treated with Taxol (inhibitor of microtubule dynamics) and Y-27632 (disruptor of actin filaments) indicate that the internal dynamics of microtubules and actin filaments plays a critical role for the proper function of the feedback mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which high-curvature geometrical features impart high traction forces to the growth cone. These results have important implications for our fundamental understanding of axonal growth as well as for bioengineering novel substrate to guide neuronal growth and promote nerve repair.

Axonal growth on surfaces with periodic geometrical patterns

The formation of neuron networks is a complex phenomenon of fundamental importance for understanding the development of the nervous system, and for creating novel bioinspired materials for tissue engineering and neuronal repair. The basic process underlying the network formation is axonal growth, a process involving the extension of axons from the cell body towards target neurons. Axonal growth is guided by environmental stimuli that include intercellular interactions, biochemical cues, and the mechanical and geometrical features of the growth substrate. The dynamics of the growing axon and its biomechanical interactions with the growing substrate remains poorly understood. In this paper, we develop a model of axonal motility which incorporates mechanical interactions between the axon and the growth substrate. We combine experimental data with theoretical analysis to measure the parameters that describe axonal growth on surfaces with micropatterned periodic geometrical features: diffusion (cell motility) coefficients, speed and angular distributions, and axon bending rigidities. Experiments performed on neurons treated Taxol (inhibitor of microtubule dynamics) and Blebbistatin (disruptor of actin filaments) show that the dynamics of the cytoskeleton plays a critical role in the axon steering mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which high-curvature geometrical features impart high traction forces to the growth cone. These results have important implications for our fundamental understanding of axonal growth as well as for bioengineering novel substrates that promote neuronal growth and nerve repair.

Temporal fluctuations in chemotaxis gain implement a simulated-tempering strategy for efficient navigation in complex environments

Bacterial chemotaxis is a major testing ground for systems biology, including the role of fluctuations and individual variation. Individual bacteria vary in their tumbling frequency and adaptation time. Recently, large cell-cell variation was also discovered in chemotaxis gain, which determines the sensitivity of the tumbling rate to attractant gradients. Variation in gain is puzzling, because low gain impairs chemotactic velocity. Here, we provide a functional explanation for gain variation by establishing a formal analogy between chemotaxis and algorithms for sampling probability distributions. We show that temporal fluctuations in gain implement simulated tempering, which allows sampling of attractant distributions with many local peaks. Periods of high gain allow bacteria to detect and climb gradients quickly, and periods of low gain allow them to move to new peaks. Gain fluctuations thus allow bacteria to thrive in complex environments, and more generally they may play an important functional role for organism navigation.

Neuronal Growth and Formation of Neuron Networks on Directional Surfaces

The formation of neuron networks is a process of fundamental importance for understanding the development of the nervous system and for creating biomimetic devices for tissue engineering and neural repair. The basic process that controls the network formation is the growth of an axon from the cell body and its extension towards target neurons. Axonal growth is directed by environmental stimuli that include intercellular interactions, biochemical cues, and the mechanical and geometrical properties of the growth substrate. Despite significant recent progress, the steering of the growing axon remains poorly understood. In this paper, we develop a model of axonal motility, which incorporates substrate-geometry sensing. We combine experimental data with theoretical analysis to measure the parameters that describe axonal growth on micropatterned surfaces: diffusion (cell motility) coefficients, speed and angular distributions, and cell-substrate interactions. Experiments performed on neurons treated with inhibitors for microtubules (Taxol) and actin filaments (Y-27632) indicate that cytoskeletal dynamics play a critical role in the steering mechanism. Our results demonstrate that axons follow geometrical patterns through a contact-guidance mechanism, in which geometrical patterns impart high traction forces to the growth cone. These results have important implications for bioengineering novel substrates to guide neuronal growth and promote nerve repair.

Non-Markovian data-driven modeling of single-cell motility

Trajectories of human breast cancer cells moving on one-dimensional circular tracks are modeled by the non-Markovian version of the Langevin equation that includes an arbitrary memory function. When averaged over cells, the velocity distribution exhibits spurious non-Gaussian behavior, while single cells are characterized by Gaussian velocity distributions. Accordingly, the data are described by a linear memory model which includes different random walk models that were previously used to account for various aspects of cell motility such as migratory persistence, non-Markovian effects, colored noise, and anomalous diffusion. The memory function is extracted from the trajectory data without restrictions or assumptions, thus making our approach truly data driven, and is used for unbiased single-cell comparison. The cell memory displays time-delayed single-exponential negative friction, which clearly distinguishes cell motion from the simple persistent random walk model and suggests a regulatory feedback mechanism that controls cell migration. Based on the extracted memory function we formulate a generalized exactly solvable cell migration model which indicates that negative friction generates cell persistence over long timescales. The nonequilibrium character of cell motion is investigated by mapping the non-Markovian Langevin equation with memory onto a Markovian model that involves a hidden degree of freedom and is equivalent to the underdamped active Ornstein-Uhlenbeck process.

Cellular velocity, electrical persistence and sensing in developed and vegetative cells during electrotaxis

Cells have the ability to detect electric fields and respond to them with directed migratory movement. Investigations identified genes and proteins that play important roles in defining the migration efficiency. Nevertheless, the sensing and transduction mechanisms underlying directed cell migration are still under discussion. We use Dictyostelium discoideum cells as model system for studying eukaryotic cell migration in DC electric fields. We have defined the temporal electric persistence to characterize the memory that cells have in a varying electric field. In addition to imposing a directional bias, we observed that the electric field influences the cellular kinematics by accelerating the movement of cells along their paths. Moreover, the study of vegetative and briefly starved cells provided insight into the electrical sensing of cells. We found evidence that conditioned medium of starved cells was able to trigger the electrical sensing of vegetative cells that would otherwise not orient themselves in the electric field. This observation may be explained by the presence of the conditioned medium factor (CMF), a protein secreted by the cells, when they begin to starve. The results of this study give new insights into understanding the mechanism that triggers the electrical sensing and transduces the external stimulus into directed cell migration. Finally, the observed increased mobility of cells over time in an electric field could offer a novel perspective towards wound healing assays.

Stochastic and Heterogeneous Cancer Cell Migration: Experiment and Theory

Cell migration, an essential process for normal cell development and cancer metastasis, differs from a simple random walk: the mean-square displacement (〈(Δr)²(t)〉) of cells sometimes shows non-Fickian behavior, and the spatiotemporal correlation function (G(r, t)) of cells is often non-Gaussian. We find that this intriguing cell migration should be attributed to heterogeneity in a cell population, even one with a homogeneous genetic background. There are two limiting types of heterogeneity in a cell population: cellular heterogeneity and temporal heterogeneity. Cellular heterogeneity accounts for the cell-to-cell variation in migration capacity, while temporal heterogeneity arises from the temporal noise in the migration capacity of single cells. We illustrate that both cellular and temporal heterogeneity need to be taken into account simultaneously to elucidate cell migration. We investigate the two-dimensional migration of A549 lung cancer cells using time-lapse microscopy and find that the migration of A549 cells is Fickian but has a non-Gaussian spatiotemporal correlation. We find that when a theoretical model considers both cellular and temporal heterogeneity, the model reproduces all of the anomalous behaviors of cancer cell migration.

Trail-mediated self-interaction

A number of microorganisms leave persistent trails while moving along surfaces. For single-cell organisms, the trail-mediated self-interaction will influence the dynamics. It has been discussed recently [Kranz et al., Phys. Rev. Lett. 117, 038101 (2016)] that the self-interaction may localize the organism above a critical coupling χ_c to the trail. Here, we will derive a generalized active particle model capturing the key features of the self-interaction and analyze its behavior for smaller couplings χ < χ_c. We find that fluctuations in propulsion speed shift the localization transition to stronger couplings.

Anomalous diffusion for neuronal growth on surfaces with controlled geometries

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.

Role of geometrical cues in neuronal growth

Geometrical cues play an essential role in neuronal growth. Here, we quantify axonal growth on surfaces with controlled geometries and report a general stochastic approach that quantitatively describes the motion of growth cones. We show that axons display a strong directional alignment on micropatterned surfaces when the periodicity of the patterns matches the dimension of the growth cone. The growth cone dynamics on surfaces with uniform geometry is described by a linear Langevin equation with both deterministic and stochastic contributions. In contrast, axonal growth on surfaces with periodic patterns is characterized by a system of two generalized Langevin equations with both linear and quadratic velocity dependence and stochastic noise. We combine experimental data with theoretical analysis to measure the key parameters of the growth cone motion: angular distributions, correlation functions, diffusion coefficients, characteristics speeds, and damping coefficients. We demonstrate that axonal dynamics displays a crossover from an Ornstein-Uhlenbeck process to a nonlinear stochastic regime when the geometrical periodicity of the pattern approaches the linear dimension of the growth cone. Growth alignment is determined by surface geometry, which is fully quantified by the deterministic part of the Langevin equation. These results provide insight into the role of curvature sensing proteins and their interactions with geometrical cues.

Harnessing Motile Amoeboid Cells as Trucks for Microtransport and -Assembly

Cell-driven microtransport is one of the most prominent applications in the emerging field of biohybrid systems. While bacterial cells have been successfully employed to drive the swimming motion of micrometer-sized cargo particles, the transport capacities of motile adherent cells remain largely unexplored. Here, it is demonstrated that motile amoeboid cells can act as efficient and versatile trucks to transport microcargo. When incubated together with microparticles, cells of the social amoeba Dictyostelium discoideum readily pick up and move the cargo particles. Relying on the unspecific adhesive properties of the amoeba, a wide range of different cargo materials can be used. The cell-driven transport can be directionally guided based on the chemotactic responses of amoeba to chemoattractant gradients. On the one hand, the cargo can be assembled into clusters in a self-organized fashion, relying on the developmentally induced chemotactic aggregation of cells. On the other hand, chemoattractant gradients can be externally imposed to guide the cellular microtrucks to a desired location. Finally, larger cargo particles of different shapes that exceed the size of a single cell by more than an order of magnitude, can also be transported by the collective effort of large numbers of motile cells.

Artificial Chemotaxis of Self-Phoretic Active Colloids: Collective Behavior

Microorganisms use chemotaxis, regulated by internal complex chemical pathways, to swim along chemical gradients to find better living conditions. Artificial microswimmers can mimic such a strategy by a pure physical process called diffusiophoresis, where they drift and orient along the gradient in a chemical density field. Similarly, for other forms of taxis in nature such as photo- or thermotaxis the phoretic counterpart exists. In this Account, we concentrate on the chemotaxis of self-phoretic active colloids. They are driven by self-electro- and diffusiophoresis at the particle surface and thereby acquire a swimming speed. During this process, they also produce nonuniform chemical fields in their surroundings through which they interact with other colloids by translational and rotational diffusiophoresis. In combination with active motion, this gives rise to effective phoretic attraction and repulsion and thereby to diverse emergent collective behavior. A particular appealing example is dynamic clustering in dilute suspensions first reported by a group from Lyon. A subtle balance of attraction and repulsion causes very dynamic clusters, which form and resolve again. This is in stark contrast to the relatively static clusters of motility-induced phase separation at larger densities. To treat chemotaxis in active colloids confined to a plane, we formulate two Langevin equations for position and orientation, which include translational and rotational diffusiophoretic drift velocities. The colloids are chemical sinks and develop their long-range chemical profiles instantaneously. For dense packings, we include screening of the chemical fields. We present a state diagram in the two diffusiophoretic parameters governing translational, as well as rotational, drift and, thereby, explore the full range of phoretic attraction and repulsion. The identified states range from a gaslike phase over dynamic clustering states 1 and 2, which we distinguish through their cluster size distributions, to different types of collapsed states. The latter include a full chemotactic collapse for translational phoretic attraction. Turning it into an effective repulsion, with increasing strength first the collapsed cluster starts to fluctuate at the rim, then oscillates, and ultimately becomes a static collapsed cloud. We also present a state diagram without screening. Finally, we summarize how the famous Keller-Segel model derives from our Langevin equations through a multipole expansion of the full one-particle distribution function in position and orientation. The Keller-Segel model gives a continuum equation for treating chemotaxis of microorganisms on the level of their spatial density. Our theory is extensible to mixtures of active and passive particles and allows to include a dipolar correction to the chemical field resulting from the dipolar symmetry of Janus colloids.

Mutual inhibition between PTEN and PIP3 generates bistability for polarity in motile cells

Phosphatidylinositol 3,4,5-trisphosphate (PIP3) and PIP3 phosphatase (PTEN) are enriched mutually exclusively on the anterior and posterior membranes of eukaryotic motile cells. However, the mechanism that causes this spatial separation between the two molecules is unknown. Here we develop a method to manipulate PIP3 levels in living cells and used it to show PIP3 suppresses the membrane localization of PTEN. Single-molecule measurements of membrane-association and -dissociation kinetics and of lateral diffusion reveal that PIP3 suppresses the PTEN binding site required for stable PTEN membrane binding. Mutual inhibition between PIP3 and PTEN provides a mechanistic basis for bistability that creates a PIP3-enriched/PTEN-excluded state and a PTEN-enriched/PIP3-excluded state underlying the strict spatial separation between PIP3 and PTEN. The PTEN binding site also mediates the suppression of PTEN membrane localization in chemotactic signaling. These results illustrate that the PIP3-PTEN bistable system underlies a cell's decision-making for directional movement irrespective of the environment.

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.

Langevin equations for the run-and-tumble of swimming bacteria

The run and tumble motions of a swimming bacterium are well characterized by two stochastic variables: the speed v(t) and the change of direction or deflection x(t) = cos φ(t), where φ(t) is the turning angle at time t. Recently, we have introduced [G. Fier, D. Hansmann and R. C. Buceta, A stochastic model for directional changes of swimming bacteria, Soft Matter, 2017, 13, 3385-3394.] a single stochastic model for the deflection x(t) of an E. coli bacterium performing both types of movement in isotropic media without taxis, based on available experimental data. In this work we introduce Langevin equations for the variables (v, x), which for particular values of a control parameter β correspond to run and tumble motions, respectively. These Langevin equations have analytical solutions, which make it possible to calculate the statistical properties of both movements in detail. Assuming that the stochastic processes x and v are not independent during the tumble, we show that there are small displacements of the center of mass along the normal direction to the axis of the bacterial body, a consequence of the flagellar unbundling during the run-to-tumble transition. Regarding the tumble we show, by means of the directional correlation, that the process is not stationary for tumble-times of the order of experimentally measured characteristic tumble-time. The mean square displacement is studied in detail for both movements even in the non-stationary regime. We determine the diffusion and ballistic coefficients for tumble- and run-times, establishing their properties and relationships.

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?

Extracellular Processing of Molecular Gradients by Eukaryotic Cells Can Improve Gradient Detection Accuracy

Eukaryotic cells sense molecular gradients by measuring spatial concentration variation through the difference in the number of occupied receptors to which molecules can bind. They also secrete enzymes that degrade these molecules, and it is presently not well understood how this affects the local gradient perceived by cells. Numerical and analytical results show that these enzymes can substantially increase the signal-to-noise ratio of the receptor difference and allow cells to respond to a much broader range of molecular concentrations and gradients than they would without these enzymes.

Modelling the collective response of heterogeneous cell populations to stationary gradients and chemical signal relay

The directed motion of cell aggregates toward a chemical source occurs in many relevant biological processes. Understanding the mechanisms that control this complex behavior is of great relevance for our understanding of developmental biological processes and many diseases. In this paper, we consider a self-propelled particle model for the movement of heterogeneous subpopulations of chemically interacting cells towards an imposed stable chemical gradient. Our simulations show explicitly how self-organisation of cell populations (which could lead to engulfment or complete cell segregation) can arise from the heterogeneity of chemotactic responses alone. This new result complements current theoretical and experimental studies that emphasise the role of differential cell-cell adhesion on self-organisation and spatial structure of cellular aggregates. We also investigate how the speed of individual cell aggregations increases with the chemotactic sensitivity of the cells, and decreases with the number of cells inside the aggregates.

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.

A stochastic model for directional changes of swimming bacteria

In this work we introduce a stochastic model to describe directional changes in the movement of swimming bacteria. We use the probability density function (PDF) of turn angles, measured on tumbling wild-type E. coli, to build a Langevin equation for the deflection of the bacterial body swimming in isotropic media. We have solved this equation analytically by means of the Green function method and shown that three parameters are sufficient to describe the movement: the characteristic time, the steady-state solution and the control parameter. We conclude that the tumble motion, which is manifested as abrupt turns, is primarily caused by the rotational boost generated by the flagellar motor and complementarily by the rotational diffusion introduced by noise. We show that in the tumble motion the deflection is a non-stationary stochastic process during times at which the tumbling occurs. By tuning the control parameter our model is able to explain small turns of the bacteria around their centres of mass along the run. We show that the deflection during the run is an Ornstein-Uhlenbeck process, which for typical run times is stationary. We conclude that, along the run, the rotational boosts do not exist and that only the rotational diffusion remains. Thus we have a single model to explain the turns of the bacterium during the run or tumble movements, through a control parameter that can be tuned through a critical value that can explain the transition between the two turn behaviours. This model is also able to explain in a very satisfactory way all available statistical experimental data, such as PDFs and average values of turning angles times, of both run and tumble motions.

Chemotaxis and autochemotaxis of self-propelling droplet swimmers

Chemotaxis and autochemotaxis play an important role in many essential biological processes. We present a self-propelling artificial swimmer system that exhibits chemotaxis as well as negative autochemotaxis. Oil droplets in an aqueous surfactant solution are driven by interfacial Marangoni flows induced by micellar solubilization of the oil phase. We demonstrate that chemotaxis along micellar surfactant gradients can guide these swimmers through a microfluidic maze. Similarly, a depletion of empty micelles in the wake of a droplet swimmer causes negative autochemotaxis and thereby trail avoidance. We studied autochemotaxis quantitatively in a microfluidic device of bifurcating channels: Branch choices of consecutive swimmers are anticorrelated, an effect decaying over time due to trail dispersion. We modeled this process by a simple one-dimensional diffusion process and stochastic Langevin dynamics. Our results are consistent with a linear surfactant gradient force and diffusion constants appropriate for micellar diffusion and provide a measure of autochemotactic feedback strength vs. stochastic forces. This assay is readily adaptable for quantitative studies of both artificial and biological autochemotactic systems.

Physical models of collective cell motility: from cell to tissue

In this article, we review physics-based models of collective cell motility. We discuss a range of techniques at different scales, ranging from models that represent cells as simple self-propelled particles to phase field models that can represent a cell's shape and dynamics in great detail. We also extensively review the ways in which cells within a tissue choose their direction, the statistics of cell motion, and some simple examples of how cell-cell signaling can interact with collective cell motility. This review also covers in more detail selected recent works on collective cell motion of small numbers of cells on micropatterns, in wound healing, and the chemotaxis of clusters of cells.

Inferring the Chemotactic Strategy of P. putida and E. coli Using Modified Kramers-Moyal Coefficients

Many bacteria perform a run-and-tumble random walk to explore their surrounding and to perform chemotaxis. In this article we present a novel method to infer the relevant parameters of bacterial motion from experimental trajectories including the tumbling events. We introduce a stochastic model for the orientation angle, where a shot-noise process initiates tumbles, and analytically calculate conditional moments, reminiscent of Kramers-Moyal coefficients. Matching them with the moments calculated from experimental trajectories of the bacteria E. coli and Pseudomonas putida, we are able to infer their respective tumble rates, the rotational diffusion constants, and the distributions of tumble angles in good agreement with results from conventional tumble recognizers. We also define a novel tumble recognizer, which explicitly quantifies the error in recognizing tumbles. In the presence of a chemical gradient we condition the moments on the bacterial direction of motion and thereby explore the chemotaxis strategy. For both bacteria we recover and quantify the classical chemotactic strategy, where the tumble rate is smallest along the chemical gradient. In addition, for E. coli we detect some cells, which bias their mean tumble angle towards smaller values. Our findings are supported by a scaling analysis of appropriate ratios of conditional moments, which are directly calculated from experimental data.

The movement strategies of bacteria have received increasing attention over the past decade, in particular with respect to the tracking of individual cells and the mathematical description of the resulting trajectories. Bacteria typically move in almost straight runs interrupted by sharp turning events (run-and-tumble). In order to characterize their motion on a single cell level, the tumble events in individual trajectories have to be identified. Traditionally, tumble recognition relies on threshold values that are applied to the swimming speed and the reorientation angle. They are chosen in an ad hoc fashion and introduce a certain degree of arbitrariness to the results of statistical motion analyses. Here, we propose a new stochastic model for the orientation angle of a bacterium and formulate conditonal moments, which we determine both in theory and from experimental trajectories. This provides an alternative way of quantifying the bacterial run-and-tumble strategy and of recognizing tumble events. Our approach no longer relies on arbitrarily chosen segmentation thresholds and rigorously quantifies the uncertainty in tumble recognition. We successfully apply our method not only to the paradigmatic case of E. coli but also to trajectories of the soil bacterium Pseudomonas putida, demonstrating that our approach provides a novel way to reliably characterize the tumbling statistics and chemotaxis strategies of bacterial swimmers across different species.

Large-Scale Patterns in a Minimal Cognitive Flocking Model: Incidental Leaders, Nematic Patterns, and Aggregates

We study a minimal cognitive flocking model, which assumes that the moving entities navigate using the available instantaneous visual information exclusively. The model consists of active particles, with no memory, that interact by a short-ranged, position-based, attractive force, which acts inside a vision cone (VC), and lack velocity-velocity alignment. We show that this active system can exhibit-due to the VC that breaks Newton's third law-various complex, large-scale, self-organized patterns. Depending on parameter values, we observe the emergence of aggregates or millinglike patterns, the formation of moving-locally polar-files with particles at the front of these structures acting as effective leaders, and the self-organization of particles into macroscopic nematic structures leading to long-ranged nematic order. Combining simulations and nonlinear field equations, we show that position-based active models, as the one analyzed here, represent a new class of active systems fundamentally different from other active systems, including velocity-alignment-based flocking systems. The reported results are of prime importance in the study, interpretation, and modeling of collective motion patterns in living and nonliving active systems.

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.

Noisy Oscillations in the Actin Cytoskeleton of Chemotactic Amoeba

Biological systems with their complex biochemical networks are known to be intrinsically noisy. Here we investigate the dynamics of actin polymerization of amoeboid cells, which are close to the onset of oscillations. We show that the large phenotypic variability in the polymerization dynamics can be accurately captured by a generic nonlinear oscillator model in the presence of noise. We determine the relative role of the noise with a single dimensionless, experimentally accessible parameter, thus providing a quantitative description of the variability in a population of cells. Our approach, which rests on a generic description of a system close to a Hopf bifurcation and includes the effect of noise, can characterize the dynamics of a large class of noisy systems close to an oscillatory instability.

Collective Signal Processing in Cluster Chemotaxis: Roles of Adaptation, Amplification, and Co-attraction in Collective Guidance

Single eukaryotic cells commonly sense and follow chemical gradients, performing chemotaxis. Recent experiments and theories, however, show that even when single cells do not chemotax, clusters of cells may, if their interactions are regulated by the chemoattractant. We study this general mechanism of "collective guidance" computationally with models that integrate stochastic dynamics for individual cells with biochemical reactions within the cells, and diffusion of chemical signals between the cells. We show that if clusters of cells use the well-known local excitation, global inhibition (LEGI) mechanism to sense chemoattractant gradients, the speed of the cell cluster becomes non-monotonic in the cluster's size-clusters either larger or smaller than an optimal size will have lower speed. We argue that the cell cluster speed is a crucial readout of how the cluster processes chemotactic signals; both amplification and adaptation will alter the behavior of cluster speed as a function of size. We also show that, contrary to the assumptions of earlier theories, collective guidance does not require persistent cell-cell contacts and strong short range adhesion. If cell-cell adhesion is absent, and the cluster cohesion is instead provided by a co-attraction mechanism, e.g. chemotaxis toward a secreted molecule, collective guidance may still function. However, new behaviors, such as cluster rotation, may also appear in this case. Co-attraction and adaptation allow for collective guidance that is robust to varying chemoattractant concentrations while not requiring strong cell-cell adhesion.

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.

Emergent Collective Chemotaxis without Single-Cell Gradient Sensing

Many eukaryotic cells chemotax, sensing and following chemical gradients. However, experiments show that even under conditions when single cells cannot chemotax, small clusters may still follow a gradient. This behavior is observed in neural crest cells, in lymphocytes, and during border cell migration in Drosophila, but its origin remains puzzling. Here, we propose a new mechanism underlying this "collective guidance," and study a model based on this mechanism both analytically and computationally. Our approach posits that contact inhibition of locomotion, where cells polarize away from cell-cell contact, is regulated by the chemoattractant. Individual cells must measure the mean attractant value, but need not measure its gradient, to give rise to directional motility for a cell cluster. We present analytic formulas for how the cluster velocity and chemotactic index depend on the number and organization of cells in the cluster. The presence of strong orientation effects provides a simple test for our theory of collective guidance.

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.

How grow-and-switch gravitropism generates root coiling and root waving growth responses in Medicago truncatula

Experimental studies show that plant root morphologies can vary widely from straight gravity-aligned primary roots to fractal-like root architectures. However, the opaqueness of soil makes it difficult to observe how environmental factors modulate these patterns. Here, we combine a transparent hydrogel growth medium with a custom built 3D laser scanner to directly image the morphology of Medicago truncatula primary roots. In our experiments, root growth is obstructed by an inclined plane in the growth medium. As the tilt of this rigid barrier is varied, we find Medicago transitions between randomly directed root coiling, sinusoidal root waving, and normal gravity-aligned morphologies. Although these root phenotypes appear morphologically distinct, our analysis demonstrates the divisions are less well defined, and instead, can be viewed as a 2D biased random walk that seeks the path of steepest decent along the inclined plane. Features of this growth response are remarkably similar to the widely known run-and-tumble chemotactic behavior of Escherichia coli bacteria, where biased random walks are used as optimal strategies for nutrient uptake.

Polarization of cells and soft objects driven by mechanical interactions: consequences for migration and chemotaxis

We study a generic model for the polarization and motility of self-propelled soft objects, biological cells, or biomimetic systems, interacting with a viscous substrate. The active forces generated by the cell on the substrate are modeled by means of oscillating force multipoles at the cell-substrate interface. Symmetry breaking and cell polarization for a range of cell sizes naturally "emerge" from long range mechanical interactions between oscillating units, mediated both by the intracellular medium and the substrate. However, the harnessing of cell polarization for motility requires substrate-mediated interactions. Motility can be optimized by adapting the oscillation frequency to the relaxation time of the system or when the substrate and cell viscosities match. Cellular noise can destroy mechanical coordination between force-generating elements within the cell, resulting in sudden changes of polarization. The persistence of the cell's motion is found to depend on the cell size and the substrate viscosity. Within such a model, chemotactic guidance of cell motion is obtained by directionally modulating the persistence of motion, rather than by modulating the instantaneous cell velocity, in a way that resembles the run and tumble chemotaxis of bacteria.

Geometry-Driven Polarity in Motile Amoeboid Cells

Motile eukaryotic cells, such as leukocytes, cancer cells, and amoeba, typically move inside the narrow interstitial spacings of tissue or soil. While most of our knowledge of actin-driven eukaryotic motility was obtained from cells that move on planar open surfaces, recent work has demonstrated that confinement can lead to strongly altered motile behavior. Here, we report experimental evidence that motile amoeboid cells undergo a spontaneous symmetry breaking in confined interstitial spaces. Inside narrow channels, the cells switch to a highly persistent, unidirectional mode of motion, moving at a constant speed along the channel. They remain in contact with the two opposing channel side walls and alternate protrusions of their leading edge near each wall. Their actin cytoskeleton exhibits a characteristic arrangement that is dominated by dense, stationary actin foci at the side walls, in conjunction with less dense dynamic regions at the leading edge. Our experimental findings can be explained based on an excitable network model that accounts for the confinement-induced symmetry breaking and correctly recovers the spatio-temporal pattern of protrusions at the leading edge. Since motile cells typically live in the narrow interstitial spacings of tissue or soil, we expect that the geometry-driven polarity we report here plays an important role for movement of cells in their natural environment.

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

Amoebae explore their environment in a random way, unless external cues like, e.g., nutrients, bias their motion. Even in the absence of cues, however, experimental cell tracks show some degree of persistence. In this paper, we analyzed individual cell tracks in the framework of a linear mixed effects model, where each track is modeled by a fractional Brownian motion, i.e., a Gaussian process exhibiting a long-term correlation structure superposed on a linear trend. The degree of persistence was quantified by the Hurst exponent of fractional Brownian motion. Our analysis of experimental cell tracks of the amoeba Dictyostelium discoideum showed a persistent movement for the majority of tracks. Employing a sliding window approach, we estimated the variations of the Hurst exponent over time, which allowed us to identify points in time, where the correlation structure was distorted ("outliers"). Coarse graining of track data via down-sampling allowed us to identify the dependence of persistence on the spatial scale. While one would expect the (mode of the) Hurst exponent to be constant on different temporal scales due to the self-similarity property of fractional Brownian motion, we observed a trend towards stronger persistence for the down-sampled cell tracks indicating stronger persistence on larger time scales.

Active Brownian agents with concentration-dependent chemotactic sensitivity

We study a biologically motivated model of overdamped, autochemotactic Brownian agents with concentration-dependent chemotactic sensitivity. The agents in our model move stochastically and produce a chemical ligand at their current position. The ligand concentration obeys a reaction-diffusion equation and acts as a chemoattractant for the agents, which bias their motion towards higher concentrations of the dynamically altered chemical field. We explore the impact of concentration-dependent response to chemoattractant gradients on large-scale pattern formation, by deriving a coarse-grained macroscopic description of the individual-based model, and compare the conditions for emergence of inhomogeneous solutions for different variants of the chemotactic sensitivity. We focus primarily on the so-called receptor-law sensitivity, which models a nonlinear decrease of chemotactic sensitivity with increasing ligand concentration. Our results reveal qualitative differences between the receptor law, the constant chemotactic response, and the so-called log law, with respect to stability of the homogeneous solution, as well as the emergence of different patterns (labyrinthine structures, clusters, and bubbles) via spinodal decomposition or nucleation. We discuss two limiting cases, where the model can be reduced to the dynamics of single species: (I) the agent density governed by a density-dependent effective diffusion coefficient and (II) the ligand field with an effective bistable, time-dependent reaction rate. In the end, we turn to single clusters of agents, studying domain growth and determining mean characteristics of the stationary inhomogeneous state. Analytical results are confirmed and extended by large-scale GPU simulations of the individual based model.

Recent developments in microfluidics-based chemotaxis studies

Microfluidic devices can better control cellular microenvironments compared to conventional cell migration assays. Over the past few years, microfluidics-based chemotaxis studies showed a rapid growth. New strategies were developed to explore cell migration in manipulated chemical gradients. In addition to expanding the use of microfluidic devices for a broader range of cell types, microfluidic devices were used to study cell migration and chemotaxis in complex environments. Furthermore, high-throughput microfluidic chemotaxis devices and integrated microfluidic chemotaxis systems were developed for medical and commercial applications. In this article, we review recent developments in microfluidics-based chemotaxis studies and discuss the new trends in this field observed over the past few years.

Excitable behavior in amoeboid chemotaxis

Chemotaxis, the directed motion of cells in response to chemical gradients, is a fundamental process. Eukaryotic cells detect spatial differences in chemoattractant receptor occupancy with high precision and use these differences to bias the location of actin-rich protrusions to guide their movement. Research into chemotaxis has benefitted greatly from a systems biology approach that combines novel experimental and computational tools to pose and test hypotheses. Recently, one such hypothesis has been postulated proposing that chemotaxis in eukaryotic cells is mediated by locally biasing the activity of an underlying excitable system. The excitable system hypothesis can account for a number of cellular behaviors related to chemotaxis, including the stochastic nature of the movement of unstimulated cells, the directional bias imposed by chemoattractant gradients, and the observed spatial and temporal distribution of signaling and cytoskeleton proteins.

Modeling and measuring signal relay in noisy directed migration of cell groups

We develop a coarse-grained stochastic model for the influence of signal relay on the collective behavior of migrating Dictyostelium discoideum cells. In the experiment, cells display a range of collective migration patterns, including uncorrelated motion, formation of partially localized streams, and clumping, depending on the type of cell and the strength of the external, linear concentration gradient of the signaling molecule cyclic adenosine monophosphate (cAMP). From our model, we find that the pattern of migration can be quantitatively described by the competition of two processes, the secretion rate of cAMP by the cells and the degradation rate of cAMP in the gradient chamber. Model simulations are compared to experiments for a wide range of strengths of an external linear-gradient signal. With degradation, the model secreting cells form streams and efficiently transverse the gradient, but without degradation, we find that model secreting cells form clumps without streaming. This indicates that the observed effective collective migration in streams requires not only signal relay but also degradation of the signal. In addition, our model allows us to detect and quantify precursors of correlated motion, even when cells do not exhibit obvious streaming.

Collective cell migration is observed in various biological processes including angiogenesis, gastrulation, fruiting body formation, and wound healing. Dictyostelium discoideum, for example, exhibits highly dynamic patterns such as streams and clumps during its early phases of collective motion and has served as a model organism for the study of collective migration. In this study, facilitated by experiments, we develop a conceptual, minimalistic, computational model to analyze the dynamical processes leading to the emergence of collective patterns and the associated dependence on the external injection of a cAMP signal, the intercellular cAMP secretion rate, and the cAMP degradation rate. We demonstrate that degradation is necessary to reproduce the experimentally observed collective migration patterns, and show how our model can be utilized to uncover basic dependences of migration modes on cell characteristics. Our numerical observations elucidate the different possible types of motion and quantify the onset of collective motion. Thus, the model allows us to distinguish noisy motion guided by the external signal from weakly correlated motion.