A minimal physical model for curvotaxis driven by curved protein complexes at the cell's leading edge.

Cells often migrate on curved surfaces inside the body, such as curved tissues, blood vessels, or highly curved protrusions of other cells. Recent in vitro experiments provide clear evidence that motile cells are affected by the curvature of the substrate on which they migrate, preferring certain curvatures to others, termed "curvotaxis." The origin and underlying mechanism that gives rise to this curvature sensitivity are not well understood. Here, we employ a "minimal cell" model which is composed of a vesicle that contains curved membrane protein complexes, that exert protrusive forces on the membrane (representing the pressure due to actin polymerization). This minimal-cell model gives rise to spontaneous emergence of a motile phenotype, driven by a lamellipodia-like leading edge. By systematically screening the behavior of this model on different types of curved substrates (sinusoidal, cylinder, and tube), we show that minimal ingredients and energy terms capture the experimental data. The model recovers the observed migration on the sinusoidal substrate, where cells move along the grooves (minima), while avoiding motion along the ridges. In addition, the model predicts the tendency of cells to migrate circumferentially on convex substrates and axially on concave ones. Both of these predictions are verified experimentally, on several cell types. Altogether, our results identify the minimization of membrane-substrate adhesion energy and binding energy between the membrane protein complexes as key players of curvotaxis in cell migration.

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.

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.

Cargo Size Limits and Forces of Cell-Driven Microtransport.

The integration of motile cells into biohybrid microrobots offers unique properties such as sensitive responses to external stimuli, resilience, and intrinsic energy supply. Here, biohybrid cell-cargo systems that are driven by amoeboid Dictyostelium discoideum cells are studied and how the cargo speed and the resulting viscous drag force scales with increasing radius of the spherical cargo particle are explored. Using a simplified geometrical model of the cell-cargo interaction, the findings toward larger cargo sizes, which are not accessible with the experimental setup, are extrapolated and a maximal cargo size is predicted beyond which active cell-driven movements will stall. The active forces exerted by the cells to move a cargo show mechanoresponsive adaptation and increase dramatically when challenged by an external pulling force, a mechanism that may become relevant when navigating cargo through complex heterogeneous environments.

From actin waves to mechanism and back: How theory aids biological understanding.

Actin dynamics in cell motility, division, and phagocytosis is regulated by complex factors with multiple feedback loops, often leading to emergent dynamic patterns in the form of propagating waves of actin polymerization activity that are poorly understood. Many in the actin wave community have attempted to discern the underlying mechanisms using experiments and/or mathematical models and theory. Here, we survey methods and hypotheses for actin waves based on signaling networks, mechano-chemical effects, and transport characteristics, with examples drawn from Dictyostelium discoideum, human neutrophils, Caenorhabditis elegans, and Xenopus laevis oocytes. While experimentalists focus on the details of molecular components, theorists pose a central question of universality: Are there generic, model-independent, underlying principles, or just boundless cell-specific details? We argue that mathematical methods are equally important for understanding the emergence, evolution, and persistence of actin waves and conclude with a few challenges for future studies.

Black and gray box learning of amplitude equations: Application to phase field systems.

We present a data-driven approach to learning surrogate models for amplitude equations and illustrate its application to interfacial dynamics of phase field systems. In particular, we demonstrate learning effective partial differential equations describing the evolution of phase field interfaces from full phase field data. We illustrate this on a model phase field system, where analytical approximate equations for the dynamics of the phase field interface (a higher-order eikonal equation and its approximation, the Kardar-Parisi-Zhang equation) are known. For this system, we discuss data-driven approaches for the identification of equations that accurately describe the front interface dynamics. When the analytical approximate models mentioned above become inaccurate, as we move beyond the region of validity of the underlying assumptions, the data-driven equations outperform them. In these regimes, going beyond black box identification, we explore different approaches to learning data-driven corrections to the analytically approximate models, leading to effective gray box partial differential equations.

Role of the Two Flagellar Stators in Swimming Motility of Pseudomonas putida.

In the soil bacterium Pseudomonas putida, the motor torque for flagellar rotation is generated by the two stators MotAB and MotCD. Here, we construct mutant strains in which one or both stators are knocked out and investigate their swimming motility in fluids of different viscosity and in heterogeneous structured environments (semisolid agar). Besides phase-contrast imaging of single-cell trajectories and spreading cultures, dual-color fluorescence microscopy allows us to quantify the role of the stators in enabling P. putida's three different swimming modes, where the flagellar bundle pushes, pulls, or wraps around the cell body. The MotAB stator is essential for swimming motility in liquids, while spreading in semisolid agar is not affected. Moreover, if the MotAB stator is knocked out, wrapped mode formation under low-viscosity conditions is strongly impaired and only partly restored for increased viscosity and in semisolid agar. In contrast, when the MotCD stator is missing, cells are indistinguishable from the wild type in fluid experiments but spread much more slowly in semisolid agar. Analysis of the microscopic trajectories reveals that the MotCD knockout strain forms sessile clusters, thereby reducing the number of motile cells, while the swimming speed is unaffected. Together, both stators ensure a robust wild type that swims efficiently under different environmental conditions. IMPORTANCE Because of its heterogeneous habitat, the soil bacterium Pseudomonas putida needs to swim efficiently under very different environmental conditions. In this paper, we knocked out the stators MotAB and MotCD to investigate their impact on the swimming motility of P. putida. While the MotAB stator is crucial for swimming in fluids, in semisolid agar, both stators are sufficient to sustain a fast-swimming phenotype and increased frequencies of the wrapped mode, which is known to be beneficial for escaping mechanical traps. However, in contrast to the MotAB knockout, a culture of MotCD knockout cells spreads much more slowly in the agar, as it forms nonmotile clusters that reduce the number of motile cells.

Spontaneous transitions between amoeboid and keratocyte-like modes of migration.

The motility of adherent eukaryotic cells is driven by the dynamics of the actin cytoskeleton. Despite the common force-generating actin machinery, different cell types often show diverse modes of locomotion that differ in their shape dynamics, speed, and persistence of motion. Recently, experiments in Dictyostelium discoideum have revealed that different motility modes can be induced in this model organism, depending on genetic modifications, developmental conditions, and synthetic changes of intracellular signaling. Here, we report experimental evidence that in a mutated D. discoideum cell line with increased Ras activity, switches between two distinct migratory modes, the amoeboid and fan-shaped type of locomotion, can even spontaneously occur within the same cell. We observed and characterized repeated and reversible switchings between the two modes of locomotion, suggesting that they are distinct behavioral traits that coexist within the same cell. We adapted an established phenomenological motility model that combines a reaction-diffusion system for the intracellular dynamics with a dynamic phase field to account for our experimental findings.

Versatile Patterns in the Actin Cortex of Motile Cells: Self-Organized Pulses Can Coexist with Macropinocytic Ring-Shaped Waves.

Self-organized patterns in the actin cytoskeleton are essential for eukaryotic cellular life. They are the building blocks of many functional structures that often operate simultaneously to facilitate, for example, nutrient uptake and movement of cells. However, identifying how qualitatively distinct actin patterns can coexist remains a challenge. Using bifurcation theory of a mass conserved activator-inhibitor system, we uncover a generic mechanism of how different actin waves-traveling waves and excitable pulses-organize and simultaneously emerge. Live-cell imaging experiments indeed reveal that narrow, planar, and fast-moving excitable pulses may coexist with ring-shaped macropinocytic actin waves in the cortex of motile amoeboid cells.

Wrapped Up: The Motility of Polarly Flagellated Bacteria.

A huge number of bacterial species are motile by flagella, which allow them to actively move toward favorable environments and away from hazardous areas and to conquer new habitats. The general perception of flagellum-mediated movement and chemotaxis is dominated by the Escherichia coli paradigm, with its peritrichous flagellation and its famous run-and-tumble navigation pattern, which has shaped the view on how bacteria swim and navigate in chemical gradients. However, a significant amount-more likely the majority-of bacterial species exhibit a (bi)polar flagellar localization pattern instead of lateral flagella. Accordingly, these species have evolved very different mechanisms for navigation and chemotaxis. Here, we review the earlier and recent findings on the various modes of motility mediated by polar flagella.

Electrophoretic µPAD for Purification and Analysis of DNA Samples.

In this work, the fabrication and characterization of a simple, inexpensive, and effective microfluidic paper analytic device (µPAD) for monitoring DNA samples is reported. The glass microfiber-based chip has been fabricated by a new wax-based transfer-printing technique and an electrode printing process. It is capable of moving DNA effectively in a time-dependent fashion. The nucleic acid sample is not damaged by this process and is accumulated in front of the anode, but not directly on the electrode. Thus, further DNA processing is feasible. The system allows the DNA to be purified by separating it from other components in sample mixtures such as proteins. Furthermore, it is demonstrated that DNA can be moved through several layers of the glass fiber material. This proof of concept will provide the basis for the development of rapid test systems, e.g., for the detection of pathogens in water samples.

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.

Chemotaxis strategies of bacteria with multiple run modes.

Bacterial chemotaxis-a fundamental example of directional navigation in the living world-is key to many biological processes, including the spreading of bacterial infections. Many bacterial species were recently reported to exhibit several distinct swimming modes-the flagella may, for example, push the cell body or wrap around it. How do the different run modes shape the chemotaxis strategy of a multimode swimmer? Here, we investigate chemotactic motion of the soil bacterium Pseudomonas putida as a model organism. By simultaneously tracking the position of the cell body and the configuration of its flagella, we demonstrate that individual run modes show different chemotactic responses in nutrition gradients and, thus, constitute distinct behavioral states. On the basis of an active particle model, we demonstrate that switching between multiple run states that differ in their speed and responsiveness provides the basis for robust and efficient chemotaxis in complex natural habitats.

Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves.

During the last decade, intracellular actin waves have attracted much attention due to their essential role in various cellular functions, ranging from motility to cytokinesis. Experimental methods have advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complexity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum model of activator-inhibitor type and highlight the qualitative role of mass conservation, which is typically overlooked. Specifically, our interest is to connect between the mathematical mechanisms of pattern formation in the presence of a large-scale mode, due to mass conservation, and distinct behaviors of actin waves.

How cortical waves drive fission of motile cells.

Cytokinesis-the division of a cell into two daughter cells-is a key step in cell growth and proliferation. It typically occurs in synchrony with the cell cycle to ensure that a complete copy of the genetic information is passed on to the next generation of daughter cells. In animal cells, cytokinesis commonly relies on an actomyosin contractile ring that drives equatorial furrowing and separation into the two daughter cells. However, also contractile ring-independent forms of cell division are known that depend on substrate-mediated traction forces. Here, we report evidence of an as yet unknown type of contractile ring-independent cytokinesis that we termed wave-mediated cytofission. It is driven by self-organized cortical actin waves that travel across the ventral membrane of oversized, multinucleated Dictyostelium discoideum cells. Upon collision with the cell border, waves may initiate the formation of protrusions that elongate and eventually pinch off to form separate daughter cells. They are composed of a stable elongated wave segment that is enclosed by a cell membrane and moves in a highly persistent fashion. We rationalize our observations based on a noisy excitable reaction-diffusion model in combination with a dynamic phase field to account for the cell shape and demonstrate that daughter cells emerging from wave-mediated cytofission exhibit a well-controlled size.

Excitable solitons: Annihilation, crossover, and nucleation of pulses in mass-conserving activator-inhibitor media.

Excitable pulses are among the most widespread dynamical patterns that occur in many different systems, ranging from biological cells to chemical reactions and ecological populations. Traditionally, the mutual annihilation of two colliding pulses is regarded as their prototypical signature. Here we show that colliding excitable pulses may exhibit solitonlike crossover and pulse nucleation if the system obeys a mass conservation constraint. In contrast to previous observations in systems without mass conservation, these alternative collision scenarios are robustly observed over a wide range of parameters. We demonstrate our findings using a model of intracellular actin waves since, on time scales of wave propagations over the cell scale, cells obey conservation of actin monomers. The results provide a key concept to understand the ubiquitous occurrence of actin waves in cells, suggesting why they are so common, and why their dynamics is robust and long-lived.

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.

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? Based on the statistical analysis of experimental single-cell tracking data of the two-dimensional motion of the Dictyostelium discoideum amoeboid cells, we quantify their diffusive behaviour based on a number of standard and complementary statistical indicators. We compute the ensemble- and time-averaged mean-squared displacements (MSDs) of the diffusing amoebae cells and observe a pronounced spread of short-time diffusion coefficients and anomalous MSD-scaling exponents for individual cells. The distribution functions of the cell displacements, the long-tailed distribution of instantaneous speeds, and the velocity autocorrelations are also computed. In particular, we observe a systematic superdiffusive short-time behaviour for the ensemble- and time-averaged MSDs of the amoeboid cells. Also, a clear anti-correlation of scaling exponents and generalised diffusivity values for different cells is detected. Most significantly, we demonstrate that the distribution function of the cell displacements has a strongly non-Gaussian shape and-using a rescaled spatio-temporal variable-the cell-displacement data collapse onto a universal master curve. The current analysis of single-cell motions can be implemented for quantifying diffusive behaviours in other living-matter systems, in particular, when effects of active transport, non-Gaussian displacements, and heterogeneity of the population are involved in the dynamics.

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.