Distinct cell shapes determine accurate chemotaxis

Persistent pseudopod splitting is an effective chemotaxis strategy in shallow gradients

Single-cell organisms and various cell types use a range of motility modes when following a chemical gradient, but it is unclear which mode is best suited for different gradients. Here, we model directional decision-making in chemotactic amoeboid cells as a stimulus-dependent actin recruitment contest. Pseudopods extending from the cell body compete for a finite actin pool to push the cell in their direction until one pseudopod wins and determines the direction of movement. Our minimal model provides a quantitative understanding of the strategies cells use to reach the physical limit of accurate chemotaxis, aligning with data without explicit gradient sensing or cellular memory for persistence. To generalize our model, we employ reinforcement learning optimization to study the effect of pseudopod suppression, a simple but effective cellular algorithm by which cells can suppress possible directions of movement. Different pseudopod-based chemotaxis strategies emerge naturally depending on the environment and its dynamics. For instance, in static gradients, cells can react faster at the cost of pseudopod accuracy, which is particularly useful in noisy, shallow gradients where it paradoxically increases chemotactic accuracy. In contrast, in dynamics gradients, cells form de novo pseudopods. Overall, our work demonstrates mechanical intelligence for high chemotaxis performance with minimal cellular regulation.

Local Clustering and Global Spreading of Receptors for Optimal Spatial Gradient Sensing

Spatial information from cell-surface receptors is crucial for processes that require signal processing and sensing of the environment. Here, we investigate the optimal placement of such receptors through a theoretical model that minimizes uncertainty in gradient estimation. Without requiring a priori knowledge of the physical limits of sensing or biochemical processes, we reproduce the emergence of clusters that closely resemble those observed in real cells. On perfect spherical surfaces, optimally placed receptors spread uniformly. When perturbations break their symmetry, receptors cluster in regions of high curvature, massively reducing estimation uncertainty. This agrees in many scenarios with mechanistic models that minimize elastic preference discrepancies between receptors and cell membranes. We further extend our model to motile receptors responding to cell-shape changes and external fluid flow, demonstrating the biological relevance of our model. Our findings provide a simple and utilitarian explanation for receptor clustering at high-curvature regions when high sensing accuracy is paramount.

Gradient sensing limit of an elongated cell with orientational control

Eukaryotic cells perform chemotaxis by determining the direction of chemical gradients based on stochastic sensing of concentrations at the cell surface. To examine the efficiency of this process, previous studies have investigated the limit of estimation accuracy for gradients. However, most studies have treated a circular cell shape, and the few considering elongated shapes assume the elongated direction as fixed. This leaves the question of how adaptive regulation of cell shape affects the estimation limit. Dynamics of cell shape during gradient sensing is biologically ubiquitous and can influence the estimation by altering the way the concentration is measured, and cells may strategically regulate their shape to improve estimation accuracy. To address this gap, we investigate the estimation limits in dynamic situations where elongated cells change their orientation adaptively depending on the sensed signal. We approach this problem by analyzing the stationary solution of the Bayesian nonlinear filtering equation. By applying diffusion approximation to the ligand-receptor binding process and the Laplace method for the posterior expectation under a high signal-to-noise ratio regime, we obtain an analytical expression for the estimation limit. This expression indicates that estimation accuracy can be improved by aligning the elongated direction perpendicular to the estimated direction, which is also confirmed by numerical simulations. Our analysis provides a basis for clarifying the interplay between estimation and control in gradient sensing and sheds light on how cells optimize their shape to enhance chemotactic efficiency.

Cell shape and orientation control galvanotactic accuracy

Eukaryotic cells sense and follow electric fields during wound healing and embryogenesis - this is called galvanotaxis. Galvanotaxis is believed to be driven by the redistribution of "sensors" - potentially transmembrane proteins or other molecules - through electrophoresis and electroosmosis. Here, we update our previous model of the limits of galvanotaxis due to the stochasticity of sensor movements to account for cell shape and orientation. Computing the Fisher information shows that, in principle, cells have more information about the electric field direction when their long axis is parallel to the field. However, for weak fields, maximum-likelihood estimators may have lower variability when the cell's long axis is perpendicular to the field. In an alternate possibility, we find that if cells instead estimate the field direction by taking the average of all the sensor locations as its directional cue ("vector sum"), this introduces a bias towards the short axis, an effect not present for isotropic cells. We also explore the possibility that cell elongation arises downstream of sensor redistribution. We argue that if sensors migrate to the cell's rear, the cell will tend to expand perpendicular the field - as is more commonly observed - but if sensors migrate to the front, the cell will tend to elongate parallel to the field.

The cellular Potts model on disordered lattices

The cellular Potts model, also known as the Glazier-Graner-Hogeweg model, is a lattice-based approach by which biological tissues at the level of individual cells can be numerically studied. Traditionally, a square or hexagonal underlying lattice structure is assumed for two-dimensional systems, and this is known to introduce artifacts in the structure and dynamics of the model tissues. That is, on regular lattices, cells can assume shapes that are dictated by the symmetries of the underlying lattice. Here, we developed a variant of this method that can be applied to a broad class of (ir)regular lattices. We show that on an irregular lattice deriving from a fluid-like configuration, two types of artifacts can be removed. We further report on the transition between a fluid-like disordered and a solid-like hexagonally ordered phase present for monodisperse confluent cells as a function of their surface tension. This transition shows the hallmarks of a first-order phase transition and is different from the glass/jamming transitions commonly reported for the vertex and active Voronoi models. We emphasize this by analyzing the distribution of shape parameters found in our state space. Our analysis provides a useful reference for the future study of epithelia using the (ir)regular cellular Potts model.

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.

Random walk and cell morphology dynamics in Naegleria gruberi

Amoeboid cell movement and migration are wide-spread across various cell types and species. Microscopy-based analysis of the model systems Dictyostelium and neutrophils over the years have uncovered generality in their overall cell movement pattern. Under no directional cues, the centroid movement can be quantitatively characterized by their persistence to move in a straight line and the frequency of re-orientation. Mathematically, the cells essentially behave as a persistent random walker with memory of two characteristic time-scale. Such quantitative characterization is important from a cellular-level ethology point of view as it has direct connotation to their exploratory and foraging strategies. Interestingly, outside the amoebozoa and metazoa, there are largely uncharacterized species in the excavate taxon Heterolobosea including amoeboflagellate Naegleria. While classical works have shown that these cells indeed show typical amoeboid locomotion on an attached surface, their quantitative features are so far unexplored. Here, we analyzed the cell movement of Naegleria gruberi by employing long-time phase contrast imaging that automatically tracks individual cells. We show that the cells move as a persistent random walker with two time-scales that are close to those known in Dictyostelium and neutrophils. Similarities were also found in the shape dynamics which are characterized by the appearance, splitting and annihilation of the curvature waves along the cell edge. Our analysis based on the Fourier descriptor and a neural network classifier point to importance of morphology features unique to Naegleria including complex protrusions and the transient bipolar dumbbell morphologies.

Crawling, waving, inch worming, dilating, and pivoting mechanics of migrating cells: Lessons from Ken Jacobson

Research on the locomotion of single cells on hard, flat surfaces brought insight into the mechanisms of leading-edge protrusion, spatially graded adhesion, front-rear coordination, and how intracellular and traction forces are harnessed to execute various maneuvers. Here, we highlight how, by studying a variety of cell types, shapes, and movements, Ken Jacobson and his collaborators made several discoveries that triggered the mechanistic understanding of cell motility. We then review the recent advancements and current perspectives in this field.

Deformability and collision-induced reorientation enhance cell topotaxis in dense microenvironments

In vivo, cells navigate through complex environments filled with obstacles such as other cells and the extracellular matrix. Recently, the term "topotaxis" has been introduced for navigation along topographic cues such as obstacle density gradients. Experimental and mathematical efforts have analyzed topotaxis of single cells in pillared grids with pillar density gradients. A previous model based on active Brownian particles (ABPs) has shown that ABPs perform topotaxis, i.e., drift toward lower pillar densities, due to decreased effective persistence lengths at high pillar densities. The ABP model predicted topotactic drifts of up to 1% of the instantaneous speed, whereas drifts of up to 5% have been observed experimentally. We hypothesized that the discrepancy between the ABP and the experimental observations could be in 1) cell deformability and 2) more complex cell-pillar interactions. Here, we introduce a more detailed model of topotaxis based on the cellular Potts model (CPM). To model persistent cells we use the Act model, which mimics actin-polymerization-driven motility, and a hybrid CPM-ABP model. Model parameters were fitted to simulate the experimentally found motion of Dictyostelium discoideum on a flat surface. For starved D. discoideum, the topotactic drifts predicted by both CPM variants are closer to the experimental results than the previous ABP model due to a larger decrease in persistence length. Furthermore, the Act model outperformed the hybrid model in terms of topotactic efficiency, as it shows a larger reduction in effective persistence time in dense pillar grids. Also pillar adhesion can slow down cells and decrease topotaxis. For slow and less-persistent vegetative D. discoideum cells, both CPMs predicted a similar small topotactic drift. We conclude that deformable cell volume results in higher topotactic drift compared with ABPs, and that feedback of cell-pillar collisions on cell persistence increases drift only in highly persistent cells.

Morphomigrational description as a new approach connecting cell's migration with its morphology

The examination of morphology and migration of cells plays substantial role in understanding the cellular behaviour, being described by plethora of quantitative parameters and models. These descriptions, however, treat cell migration and morphology as independent properties of temporal cell state, while not taking into account their strong interdependence in adherent cells. Here we present the new and simple mathematical parameter called signed morphomigrational angle (sMM angle) that links cell geometry with translocation of cell centroid, considering them as one morphomigrational behaviour. The sMM angle combined with pre-existing quantitative parameters enabled us to build a new tool called morphomigrational description, used to assign the numerical values to several cellular behaviours. Thus, the cellular activities that until now were characterized using verbal description or by complex mathematical models, are described here by a set of numbers. Our tool can be further used in automatic analysis of cell populations as well as in studies focused on cellular response to environmental directional signals.

Traject3d allows label-free identification of distinct co-occurring phenotypes within 3D culture by live imaging

Single cell profiling by genetic, proteomic and imaging methods has expanded the ability to identify programmes regulating distinct cell states. The 3-dimensional (3D) culture of cells or tissue fragments provides a system to study how such states contribute to multicellular morphogenesis. Whether cells plated into 3D cultures give rise to a singular phenotype or whether multiple biologically distinct phenotypes arise in parallel is largely unknown due to a lack of tools to detect such heterogeneity. Here we develop Traject3d (Trajectory identification in 3D), a method for identifying heterogeneous states in 3D culture and how these give rise to distinct phenotypes over time, from label-free multi-day time-lapse imaging. We use this to characterise the temporal landscape of morphological states of cancer cell lines, varying in metastatic potential and drug resistance, and use this information to identify drug combinations that inhibit such heterogeneity. Traject3d is therefore an important companion to other single-cell technologies by facilitating real-time identification via live imaging of how distinct states can lead to alternate phenotypes that occur in parallel in 3D culture.

Combined Effect of Matrix Topography and Stiffness on Neutrophil Shape and Motility

The crawling behavior of leukocytes is driven by the cell morphology transition, which is a direct manifestation of molecular motor machinery. The topographical anisotropy and mechanical stiffness of the substrates are the main physical cues that affect leukocytes' shape generation and migratory responses. However, their combined effects on the cell morphology and motility have been poorly understood, particularly for neutrophils, which are the fastest reacting leukocytes against infections and wounds. Here, spatiotemporally correlated physical parameters are shown, which determine the neutrophil shape change during migratory processes, in response to surface topography and elasticity. Guided crawling and shape generation of individual neutrophils, activated by a uniform concentration of a chemoattractant, are analyzed by adopting elasticity-tunable micropatterning and live cell imaging techniques. Whole cell-level image analysis is performed based on a planar geometric quantification of cell shape and motility. The findings show that the pattern anisotropy and elastic modulus of the substrate induce synergic effects on the shape anisotropy, deformability, and polarization/alignment of crawling neutrophils. How the morphology-motility relationship is affected by different surface microstructures and stiffness is demonstrated. These results imply that the neutrophil shape-motility correlations can be utilized for controlling the immune cell functions with predefined physical microenvironments.

T cell morphodynamics reveal periodic shape oscillations in three-dimensional migration

T cells use sophisticated shape dynamics (morphodynamics) to migrate towards and neutralize infected and cancerous cells. However, there is limited quantitative understanding of the migration process in three-dimensional extracellular matrices (ECMs) and across timescales. Here, we leveraged recent advances in lattice light-sheet microscopy to quantitatively explore the three-dimensional morphodynamics of migrating T cells at high spatio-temporal resolution. We first developed a new shape descriptor based on spherical harmonics, incorporating key polarization information of the uropod. We found that the shape space of T cells is low-dimensional. At the behavioural level, run-and-stop migration modes emerge at approximately 150 s, and we mapped the morphodynamic composition of each mode using multiscale wavelet analysis, finding 'stereotyped' motifs. Focusing on the run mode, we found morphodynamics oscillating periodically (every approx. 100 s) that can be broken down into a biphasic process: front-widening with retraction of the uropod, followed by a rearward surface motion and forward extension, where intercalation with the ECM in both of these steps likely facilitates forward motion. Further application of these methods may enable the comparison of T cell migration across different conditions (e.g. differentiation, activation, tissues and drug treatments) and improve the precision of immunotherapeutic development.

Physics-informed deep learning characterizes morphodynamics of Asian soybean rust disease

Medicines and agricultural biocides are often discovered using large phenotypic screens across hundreds of compounds, where visible effects of whole organisms are compared to gauge efficacy and possible modes of action. However, such analysis is often limited to human-defined and static features. Here, we introduce a novel framework that can characterize shape changes (morphodynamics) for cell-drug interactions directly from images, and use it to interpret perturbed development of Phakopsora pachyrhizi, the Asian soybean rust crop pathogen. We describe population development over a 2D space of shapes (morphospace) using two models with condition-dependent parameters: a top-down Fokker-Planck model of diffusive development over Waddington-type landscapes, and a bottom-up model of tip growth. We discover a variety of landscapes, describing phenotype transitions during growth, and identify possible perturbations in the tip growth machinery that cause this variation. This demonstrates a widely-applicable integration of unsupervised learning and biophysical modeling.

Comparative mapping of crawling-cell morphodynamics in deep learning-based feature space

Navigation of fast migrating cells such as amoeba Dictyostelium and immune cells are tightly associated with their morphologies that range from steady polarized forms that support high directionality to those more complex and variable when making frequent turns. Model simulations are essential for quantitative understanding of these features and their origins, however systematic comparisons with real data are underdeveloped. Here, by employing deep-learning-based feature extraction combined with phase-field modeling framework, we show that a low dimensional feature space for 2D migrating cell morphologies obtained from the shape stereotype of keratocytes, Dictyostelium and neutrophils can be fully mapped by an interlinked signaling network of cell-polarization and protrusion dynamics. Our analysis links the data-driven shape analysis to the underlying causalities by identifying key parameters critical for migratory morphologies both normal and aberrant under genetic and pharmacological perturbations. The results underscore the importance of deciphering self-organizing states and their interplay when characterizing morphological phenotypes.

Migratory cells that move by crawling do so by extending and retracting their plasma membrane. When and where these events take place determine the cell shape, and this is directly linked to the movement patterns. Understanding how the highly plastic and interconvertible morphologies appear from their underlying dynamics remains a challenge partly because their inherent complexity makes quantitatively comparison against the outputs of mathematical models difficult. To this end, we employed machine-learning based classification to extract features that characterize the basic migrating morphologies. The obtained features were then used to compare real cell data with outputs of a conceptual model that we introduced which describes coupling via feedback between local protrusive dynamics and polarity. The feature mapping showed that the model successfully recapitulates the shape dynamics that were not covered by previous related models and also hints at the critical parameters underlying state transitions. The ability of the present approach to compare model outputs with real cell data systematically and objectively is important as it allows outputs of future mathematical models to be quantitatively tested in an accessible and common reference frame.

Local actin dynamics couple speed and persistence in a cellular Potts model of cell migration

Cell migration is astoundingly diverse. Molecular signatures, cell-cell interactions, and environmental structures each play their part in shaping cell motion, yielding numerous morphologies and migration modes. Nevertheless, in recent years, a simple unifying law was found to describe cell migration across many different cell types and contexts: faster cells turn less frequently. This universal coupling between speed and persistence (UCSP) was explained by retrograde actin flow from front to back, but it remains unclear how this mechanism generalizes to cells with complex shapes and cells migrating in structured environments, which may not have a well-defined front-to-back orientation. Here, we present an in-depth characterization of an existing cellular Potts model, in which cells polarize dynamically from a combination of local actin dynamics (stimulating protrusions) and global membrane tension along the perimeter (inhibiting protrusions). We first show that the UCSP emerges spontaneously in this model through a cross talk of intracellular mechanisms, cell shape, and environmental constraints, resembling the dynamic nature of cell migration in vivo. Importantly, we find that local protrusion dynamics suffice to reproduce the UCSP-even in cases in which no clear global, front-to-back polarity exists. We then harness the spatial nature of the cellular Potts model to show how cell shape dynamics limit both the speed and persistence a cell can reach and how a rigid environment such as the skin can restrict cell motility even further. Our results broaden the range of potential mechanisms underlying the speed-persistence coupling that has emerged as a fundamental property of migrating cells.

Of Cell Shapes and Motion: The Physical Basis of Animal Cell Migration

Motile cells have developed a variety of migration modes relying on diverse traction-force-generation mechanisms. Before the behavior of intracellular components could be easily imaged, cell movements were mostly classified by different types of cellular shape dynamics. Indeed, even though some types of cells move without any significant change in shape, most cell propulsion mechanisms rely on global or local deformations of the cell surface. In this review, focusing mostly on metazoan cells, we discuss how different types of local and global shape changes underlie distinct migration modes. We then discuss mechanical differences between force-generation mechanisms and finish by speculating on how they may have evolved.

An image-based data-driven analysis of cellular architecture in a developing tissue

Quantitative microscopy is becoming increasingly crucial in efforts to disentangle the complexity of organogenesis, yet adoption of the potent new toolbox provided by modern data science has been slow, primarily because it is often not directly applicable to developmental imaging data. We tackle this issue with a newly developed algorithm that uses point cloud-based morphometry to unpack the rich information encoded in 3D image data into a straightforward numerical representation. This enabled us to employ data science tools, including machine learning, to analyze and integrate cell morphology, intracellular organization, gene expression and annotated contextual knowledge. We apply these techniques to construct and explore a quantitative atlas of cellular architecture for the zebrafish posterior lateral line primordium, an experimentally tractable model of complex self-organized organogenesis. In doing so, we are able to retrieve both previously established and novel biologically relevant patterns, demonstrating the potential of our data-driven approach.

Equilibrium structure and deformation response of 2D kinetoplast sheets

The considerable interest in two-dimensional (2D) materials and complex molecular topologies calls for a robust experimental system for single-molecule studies. In this work, we study the equilibrium properties and deformation response of a complex DNA structure called a kinetoplast, a 2D network of thousands of linked rings akin to molecular chainmail. Examined in good solvent conditions, kinetoplasts appear as a wrinkled hemispherical sheet. The conformation of each kinetoplast is dictated by its network topology, giving it a unique shape, which undergoes small-amplitude thermal fluctuations at subsecond timescales, with a wide separation between fluctuation and diffusion timescales. They deform elastically when weakly confined and swell to their equilibrium dimensions when the confinement is released. We hope that, in the same way that linear DNA became a canonical model system on the first investigations of its polymer-like behavior, kinetoplasts can serve that role for 2D and catenated polymer systems.

Persistent and polarized global actin flow is essential for directionality during cell migration

Cell migration is hypothesized to involve a cycle of behaviours beginning with leading edge extension. However, recent evidence suggests that the leading edge may be dispensable for migration, raising the question of what actually controls cell directionality. Here, we exploit the embryonic migration of Drosophila macrophages to bridge the different temporal scales of the behaviours controlling motility. This approach reveals that edge fluctuations during random motility are not persistent and are weakly correlated with motion. In contrast, flow of the actin network behind the leading edge is highly persistent. Quantification of actin flow structure during migration reveals a stable organization and asymmetry in the cell-wide flowfield that strongly correlates with cell directionality. This organization is regulated by a gradient of actin network compression and destruction, which is controlled by myosin contraction and cofilin-mediated disassembly. It is this stable actin-flow polarity, which integrates rapid fluctuations of the leading edge, that controls inherent cellular persistence.

Matrix feedback enables diverse higher-order patterning of the extracellular matrix

The higher-order patterning of extra-cellular matrix in normal and pathological tissues has profound consequences on tissue function. Whilst studies have documented both how fibroblasts create and maintain individual matrix fibers and how cell migration is altered by the fibers they interact with, a model unifying these two aspects of tissue organization is lacking. Here we use computational modelling to understand the effect of this interconnectivity between fibroblasts and matrix at the mesoscale level. We created a unique adaptation to the Vicsek flocking model to include feedback from a second layer representing the matrix, and use experimentation to parameterize our model and validate model-driven hypotheses. Our two-layer model demonstrates that feedback between fibroblasts and matrix increases matrix diversity creating higher-order patterns. The model can quantitatively recapitulate matrix patterns of tissues in vivo. Cells follow matrix fibers irrespective of when the matrix fibers were deposited, resulting in feedback with the matrix acting as temporal 'memory' to collective behaviour, which creates diversity in topology. We also establish conditions under which matrix can be remodelled from one pattern to another. Our model elucidates how simple rules defining fibroblast-matrix interactions are sufficient to generate complex tissue patterns.

Screening by changes in stereotypical behavior during cell motility

Stereotyped behaviors are series of postures that show very little variability between repeats. They have been used to classify the dynamics of individuals, groups and species without reference to the lower-level mechanisms that drive them. Stereotypes are easily identified in animals due to strong constraints on the number, shape, and relative positions of anatomical features, such as limbs, that may be used as landmarks for posture identification. In contrast, the identification of stereotypes in single cells poses a significant challenge as the cell lacks these landmark features, and finding constraints on cell shape is a non-trivial task. Here, we use the maximum caliber variational method to build a minimal model of cell behavior during migration. Without reference to biochemical details, we are able to make behavioral predictions over timescales of minutes using only changes in cell shape over timescales of seconds. We use drug treatment and genetics to demonstrate that maximum caliber descriptors can discriminate between healthy and aberrant migration, thereby showing potential applications for maximum caliber methods in automated disease screening, for example in the identification of behaviors associated with cancer metastasis.

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.

A DISCRETE MATHEMATICAL MODEL FOR SINGLE AND COLLECTIVE MOVEMENT IN AMOEBOID CELLS

In this paper, we develop a new discrete mathematical model for individual and collective cell motility. We introduce a mechanical model for the movement of a cell on a two-dimensional rigid surface to describe and investigate the cell–cell and cell–substrate interactions. The cell cytoskeleton is modeled as a series of springs and dashpots connected in parallel. The cell–substrate attachments and the cell protrusions are also included. In particular, this model is used to describe the directed movement of endothelial cells on a Matrigel plate. We compare the results from our model with experimental data. We show that cell density and substrate rigidity play an important role in network formation.

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.

Collective cell migration: a physics perspective

Cells have traditionally been viewed either as independently moving entities or as somewhat static parts of tissues. However, it is now clear that in many cases, multiple cells coordinate their motions and move as collective entities. Well-studied examples comprise development events, as well as physiological and pathological situations. Different ex vivo model systems have also been investigated. Several recent advances have taken place at the interface between biology and physics, and have benefitted from progress in imaging and microscopy, from the use of microfabrication techniques, as well as from the introduction of quantitative tools and models. We review these interesting developments in quantitative cell biology that also provide rich examples of collective out-of-equilibrium motion.

Local sampling paints a global picture: Local concentration measurements sense direction in complex chemical gradients

Detecting and interpreting extracellular spatial signals is essential for cellular orientation within complex environments, such as during directed cell migration or growth in multicellular development. Although the molecular understanding of how cells read spatial signals like chemical gradients is still lacking, recent work has revealed that stochastic processes at different temporal and spatial scales are at the core of this gradient sensing process in a wide range of eukaryotes. Fast biochemical reactions like those underlying GTPase activity dynamics form a functional module together with slower cell morphological changes driven by membrane remodelling. This biochemical-morphological module explores the environment by stochastic local concentration sampling to determine the source of the gradient signal, enabling efficient signal detection and interpretation before polarised growth or migration towards the gradient source is initiated. Here we review recent data describing local sampling and propose a model of local fast and slow feedback counteracted by gradient-dependent substrate limitation to be at the core of gradient sensing by local sampling.

A critical-like collective state leads to long-range cell communication in Dictyostelium discoideum aggregation

The transition from single-cell to multicellular behavior is important in early development but rarely studied. The starvation-induced aggregation of the social amoeba Dictyostelium discoideum into a multicellular slug is known to result from single-cell chemotaxis towards emitted pulses of cyclic adenosine monophosphate (cAMP). However, how exactly do transient, short-range chemical gradients lead to coherent collective movement at a macroscopic scale? Here, we developed a multiscale model verified by quantitative microscopy to describe behaviors ranging widely from chemotaxis and excitability of individual cells to aggregation of thousands of cells. To better understand the mechanism of long-range cell—cell communication and hence aggregation, we analyzed cell—cell correlations, showing evidence of self-organization at the onset of aggregation (as opposed to following a leader cell). Surprisingly, cell collectives, despite their finite size, show features of criticality known from phase transitions in physical systems. By comparing wild-type and mutant cells with impaired aggregation, we found the longest cell—cell communication distance in wild-type cells, suggesting that criticality provides an adaptive advantage and optimally sized aggregates for the dispersal of spores.

A multiscale model and imaging data show that cells of the slime mold Dictyostelium discoideum maximize their cell—cell communication range during aggregation by a critical-like state known from phase transitions in physical systems.

Cells are often coupled to each other in cell collectives, such as aggregates during early development, tissues in the developed organism, and tumors in disease. How do cells communicate over macroscopic distances much larger than the typical cell—cell distance to decide how they should behave? Here, we developed a multiscale model of social amoeba, spanning behavior from individuals to thousands of cells. We show that local cell—cell coupling via secreted chemicals may be tuned to a critical value, resulting in emergent long-range communication and heightened sensitivity. Hence, these aggregates are remarkably similar to bacterial biofilms and neuronal networks, all communicating in a pulselike fashion. Similar organizing principles may also aid our understanding of the remarkable robustness in cancer development.

Coupled excitable Ras and F-actin activation mediates spontaneous pseudopod formation and directed cell movement

Chemotaxis is the mechanism by which cells move in the direction of chemical gradients. The central circuit connecting basal movement and gradient sensing is unknown. Ras activation and F-actin form one coupled excitable system, which is the beating heart of cell movement in both the absence and presence of external cues.

Many eukaryotic cells regulate their mobility by external cues. Genetic studies have identified >100 components that participate in chemotaxis, which hinders the identification of the conceptual framework of how cells sense and respond to shallow chemical gradients. The activation of Ras occurs during basal locomotion and is an essential connector between receptor and cytoskeleton during chemotaxis. Using a sensitive assay for activated Ras, we show here that activation of Ras and F-actin forms two excitable systems that are coupled through mutual positive feedback and memory. This coupled excitable system leads to short-lived patches of activated Ras and associated F-actin that precede the extension of protrusions. In buffer, excitability starts frequently with Ras activation in the back/side of the cell or with F-actin in the front of the cell. In a shallow gradient of chemoattractant, local Ras activation triggers full excitation of Ras and subsequently F-actin at the side of the cell facing the chemoattractant, leading to directed pseudopod extension and chemotaxis. A computational model shows that the coupled excitable Ras/F-actin system forms the driving heart for the ordered-stochastic extension of pseudopods in buffer and for efficient directional extension of pseudopods in chemotactic gradients.

Inference of the drivers of collective movement in two cell types: Dictyostelium and melanoma

Collective cell movement is a key component of many important biological processes, including wound healing, the immune response and the spread of cancers. To understand and influence these movements, we need to be able to identify and quantify the contribution of their different underlying mechanisms. Here, we define a set of six candidate models-formulated as advection-diffusion-reaction partial differential equations-that incorporate a range of cell movement drivers. We fitted these models to movement assay data from two different cell types: Dictyostelium discoideum and human melanoma. Model comparison using widely applicable information criterion suggested that movement in both of our study systems was driven primarily by a self-generated gradient in the concentration of a depletable chemical in the cells' environment. For melanoma, there was also evidence that overcrowding influenced movement. These applications of model inference to determine the most likely drivers of cell movement indicate that such statistical techniques have potential to support targeted experimental work in increasing our understanding of collective cell movement in a range of systems.

Challenges in tissue engineering - towards cell control inside artificial scaffolds

Control of living cells is vital for the survival of organisms. Each cell inside an organism is exposed to diverse external mechano-chemical cues, all coordinated in a spatio-temporal pattern triggering individual cell functions. This complex interplay between external chemical cues and mechanical 3D environments is translated into intracellular signaling loops. Here, we describe how external mechano-chemical cues control cell functions, especially cell migration, and influence intracellular information transport. In particular, this work focuses on the quantitative analysis of (1) intracellular vesicle transport to understand intracellular state changes in response to external cues, (2) cellular sensing of external chemotactic cues, and (3) the cells' ability to migrate in 3D structured environments, artificially fabricated to mimic the 3D environment of tissue in the human body.

Self-Generated Chemoattractant Gradients: Attractant Depletion Extends the Range and Robustness of Chemotaxis

Chemotaxis is fundamentally important, but the sources of gradients in vivo are rarely well understood. Here, we analyse self-generated chemotaxis, in which cells respond to gradients they have made themselves by breaking down globally available attractants, using both computational simulations and experiments. We show that chemoattractant degradation creates steep local gradients. This leads to surprising results, in particular the existence of a leading population of cells that moves highly directionally, while cells behind this group are undirected. This leading cell population is denser than those following, especially at high attractant concentrations. The local gradient moves with the leading cells as they interact with their surroundings, giving directed movement that is unusually robust and can operate over long distances. Even when gradients are applied from external sources, attractant breakdown greatly changes cells' responses and increases robustness. We also consider alternative mechanisms for directional decision-making and show that they do not predict the features of population migration we observe experimentally. Our findings provide useful diagnostics to allow identification of self-generated gradients and suggest that self-generated chemotaxis is unexpectedly universal in biology and medicine.

Know the Single-Receptor Sensing Limit? Think Again

How cells reliably infer information about their environment is a fundamentally important question. While sensing and signaling generally start with cell-surface receptors, the degree of accuracy with which a cell can measure external ligand concentration with even the simplest device-a single receptor-is surprisingly hard to pin down. Recent studies provide conflicting results for the fundamental physical limits. Comparison is made difficult as different studies either suggest different readout mechanisms of the ligand-receptor occupancy, or differ on how ligand diffusion is implemented. Here we critically analyse these studies and present a unifying perspective on the limits of sensing, with wide-ranging biological implications.

Crawling and Gliding: A Computational Model for Shape-Driven Cell Migration

Cell migration is a complex process involving many intracellular and extracellular factors, with different cell types adopting sometimes strikingly different morphologies. Modeling realistically behaving cells in tissues is computationally challenging because it implies dealing with multiple levels of complexity. We extend the Cellular Potts Model with an actin-inspired feedback mechanism that allows small stochastic cell rufflings to expand to cell protrusions. This simple phenomenological model produces realistically crawling and deforming amoeboid cells, and gliding half-moon shaped keratocyte-like cells. Both cell types can migrate randomly or follow directional cues. They can squeeze in between other cells in densely populated environments or migrate collectively. The model is computationally light, which allows the study of large, dense and heterogeneous tissues containing cells with realistic shapes and migratory properties.

Cell migration is involved in vital processes like morphogenesis, regeneration and immune system responses, but can also play a central role in pathological processes like metastasization. Computational models have been successfully employed to explain how single cells migrate, and to study how diverse cell-cell interactions contribute to tissue level behavior. However, there are few models that implement realistic cell shapes in multicellular simulations. The method we present here is able to reproduce two different types of motile cells—amoeboid and keratocyte-like cells. Amoeboid cells are highly motile and deform frequently; many cells can act amoeboid in certain circumstances e.g., immune system cells, epithelial cells, individually migrating cancer cells. Keratocytes are (fish) epithelial cells which are famous for their ability to preserve their shape and direction when migrating individually; during wound healing, keratocytes migrate collectively, in sheets, to the site needing reepithelialization. Our method is computationally simple, improves the realism of multicellular simulations and can help assess the tissue level impact of specific cell shapes. For example, it can be employed to study the tissue scanning strategies of leukocytes, the circumstances in which cancer cells adopt amoeboid migration strategies, or the collective migration of keratocytes.

Drink or drive: competition between macropinocytosis and cell migration

The cytoskeleton is utilized for a variety of cellular processes, including migration, endocytosis and adhesion. The required molecular components are often shared between different processes, but it is not well understood how the cells balance their use. We find that macropinocytosis and cell migration are negatively correlated. Heavy drinkers move only slowly and vice versa, fast cells do not take big gulps. Both processes are balanced by the lipid phosphatidylinositol 3,4,5-trisphosphate (PIP3). Elevated PIP3 signalling causes a shift towards macropinocytosis and inhibits motility by redirecting the SCAR/WAVE complex, a major nucleator of actin filaments. High resolution microscopy shows that patches with high levels of PIP3 recruit SCAR/WAVE on their periphery, resulting in circular ruffle formation and engulfment. Results shed new light on the role of PIP3, which is commonly thought to promote cell motility.

Image based validation of dynamical models for cell reorientation

A key feature of directed cell movement is the ability of cells to reorient quickly in response to changes in the direction of an extracellular stimulus. Mathematical models have suggested quite different regulatory mechanisms to explain reorientation, raising the question of how we can validate these models in a rigorous way. In this study, we fit three reaction-diffusion models to experimental data of Dictyostelium amoebae reorienting in response to alternating gradients of mechanical shear flow. The experimental readouts we use to fit are spatio-temporal distributions of a fluorescent reporter for cortical F-actin labeling the cell front. Experiments performed under different conditions are fitted simultaneously to challenge the models with different types of cellular dynamics. Although the model proposed by Otsuji is unable to provide a satisfactory fit, those suggested by Meinhardt and Levchenko fit equally well. Further, we show that reduction of the three-variable Meinhardt model to a two-variable model also provides an excellent fit, but has the advantage of all parameters being uniquely identifiable. Our work demonstrates that model selection and identifiability analysis, commonly applied to temporal dynamics problems in systems biology, can be a powerful tool when extended to spatio-temporal imaging data.

The dynamics of growth cone morphology

BACKGROUND

Normal brain function depends on the development of appropriate patterns of neural connections. A critical role in guiding axons to their targets during neural development is played by neuronal growth cones. These have a complex and rapidly changing morphology; however, a quantitative understanding of this morphology, its dynamics and how these are related to growth cone movement, is lacking.

RESULTS

Here we use eigenshape analysis (principal components analysis in shape space) to uncover the set of five to six basic shape modes that capture the most variance in growth cone form. By analysing how the projections of growth cones onto these principal modes evolve in time, we found that growth cone shape oscillates with a mean period of 30 min. The variability of oscillation periods and strengths between different growth cones was correlated with their forward movement, such that growth cones with strong, fast shape oscillations tended to extend faster. A simple computational model of growth cone shape dynamics based on dynamic microtubule instability was able to reproduce quantitatively both the mean and variance of oscillation periods seen experimentally, suggesting that the principal driver of growth cone shape oscillations may be intrinsic periodicity in cytoskeletal rearrangements.

CONCLUSIONS

Intrinsically driven shape oscillations are an important component of growth cone shape dynamics. More generally, eigenshape analysis has the potential to provide new quantitative information about differences in growth cone behaviour in different conditions.

S6K is a morphogenic protein with a mechanism involving Filamin-A phosphorylation and phosphatidic acid binding

Change of cell shape in vivo plays many roles that are central to life itself, such as embryonic development, inflammation, wound healing, and pathologic processes such as cancer metastasis. Nonetheless, the spatiotemporal mechanisms that control the concerted regulation of cell shape remain understudied. Here, we show that ribosomal S6K, which is normally considered a protein involved in protein translation, is a morphogenic protein. Its presence in cells alters the overall organization of the cell surface and cell circularity [(4π × area)/(perimeter)(2)] from 0.47 ± 0.06 units in mock-treated cells to 0.09 ± 0.03 units in S6K-overexpressing macrophages causing stellation and arborization of cell shape. This effect was partially reversed in cells expressing a kinase-inactive S6K mutant and was fully reversed in cells silenced with small interference RNA. Equally important is that S6K is itself regulated by phospholipids, specifically phosphatidic acid, whereby 300 nM 1,2-dioleoyl-sn-glycero-3-phosphate (DOPA), but not the control 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), binds directly to S6K and causes an ∼ 2.9-fold increase in S6K catalytic activity. This was followed by an increase in Filamin A (FLNA) functionality as measured by phospho-FLNA (S(2152)) expression and by a subsequent elevation of actin nucleation. This reliance of S6K on phosphatidic acid (PA), a curvature-inducing phospholipid, explained the extra-large perimeter of cells that overexpressed S6K. Furthermore, the diversity of the response to S6K in several unrelated cell types (fibroblasts, leukocytes, and invasive cancer cells) that we report here indicates the existence of an underlying common mechanism in mammalian cells. This new signaling set, PA-S6K-FLNA-actin, sheds light for the first time into the morphogenic pathway of cytoskeletal structures that are crucial for adhesion and cell locomotion during inflammation and metastasis.

Predicting chemical environments of bacteria from receptor signaling

Sensory systems have evolved to respond to input stimuli of certain statistical properties, and to reliably transmit this information through biochemical pathways. Hence, for an experimentally well-characterized sensory system, one ought to be able to extract valuable information about the statistics of the stimuli. Based on dose-response curves from in vivo fluorescence resonance energy transfer (FRET) experiments of the bacterial chemotaxis sensory system, we predict the chemical gradients chemotactic Escherichia coli cells typically encounter in their natural environment. To predict average gradients cells experience, we revaluate the phenomenological Weber's law and its generalizations to the Weber-Fechner law and fold-change detection. To obtain full distributions of gradients we use information theory and simulations, considering limitations of information transmission from both cell-external and internal noise. We identify broad distributions of exponential gradients, which lead to log-normal stimuli and maximal drift velocity. Our results thus provide a first step towards deciphering the chemical nature of complex, experimentally inaccessible cellular microenvironments, such as the human intestine.

Outside the laboratory, bacteria live in complex microenvironments characterized by competition for space and available nutrients. Although often inaccessible by experiments, understanding the spatio-temporal dynamics of bacterial microenvironments is biomedically important. For instance, the chemical environment that symbiotic Escherichia coli encounter in the human gut relates to health of the gastrointestinal tract, gut metabolism, immune response, and tissue homeostasis. Other complex microenvironments include soil and biofilms. Assuming that bacterial sensory systems have evolved to optimally sense typical gradients, we treat signaling data, the signaling pathway with its architecture and reaction rates, and computer simulations of swimming bacteria in different gradients as “prior knowledge” to “reverse engineer” E. coli's habitat. Our identified gradients are exponentially shaped with wide-ranging rate values. These microenvironments most likely stem from local fluctuating nutrient sources and degradation by competing species, in which bacteria have evolved to swim with optimal performance.

How cells explore shape space: a quantitative statistical perspective of cellular morphogenesis

Through statistical analysis of datasets describing single cell shape following systematic gene depletion, we have found that the morphological landscapes explored by cells are composed of a small number of attractor states. We propose that the topology of these landscapes is in large part determined by cell-intrinsic factors, such as biophysical constraints on cytoskeletal organization, and reflects different stable signaling and/or transcriptional states. Cell-extrinsic factors act to determine how cells explore these landscapes, and the topology of the landscapes themselves. Informational stimuli primarily drive transitions between stable states by engaging signaling networks, while mechanical stimuli tune, or even radically alter, the topology of these landscapes. As environments fluctuate, the topology of morphological landscapes explored by cells dynamically adapts to these fluctuations. Finally we hypothesize how complex cellular and tissue morphologies can be generated from a limited number of simple cell shapes.

Gradient generation platforms: new directions for an established microfluidic technology

Microscale platforms are enabling for cell-based studies as they allow the recapitulation of physiological conditions such as extracellular matrix (ECM) configurations and soluble factors interactions. Gradient generation platforms have been one of the few applications of microfluidics that have begun to be translated to biological laboratories and may become a new "gold standard". Though gradient generation platforms are now established, their full potential has not yet been realized. Here, we will provide our perspective on milestones achieved in the development of gradient generation and cell migration platforms, as well as emerging directions such as using cell migration as a diagnostic readout and attaining mechanistic information from cell migration models.

Memory improves precision of cell sensing in fluctuating environments

Biological cells are often found to sense their chemical environment near the single-molecule detection limit. Surprisingly, this precision is higher than simple estimates of the fundamental physical limit, hinting towards active sensing strategies. In this work, we analyse the effect of cell memory, e.g. from slow biochemical processes, on the precision of sensing by cell-surface receptors. We derive analytical formulas, which show that memory significantly improves sensing in weakly fluctuating environments. However, surprisingly when memory is adjusted dynamically, the precision is always improved, even in strongly fluctuating environments. In support of this prediction we quantify the directional biases in chemotactic Dictyostelium discoideum cells in a flow chamber with alternating chemical gradients. The strong similarities between cell sensing and control engineering suggest universal problem-solving strategies of living matter.