Persistence-Driven Durotaxis: Generic, Directed Motility in Rigidity Gradients

Durotaxis and extracellular matrix degradation promote the clustering of cancer cells

Early stages of metastasis depend on the collective behavior of cancer cells and their interaction with the extracellular matrix (ECM). Cancer cell clusters are known to exhibit higher metastatic potential than single cells. To explore clustering dynamics, we developed a calibrated computational model describing how motile cancer cells biochemically and biomechanically interact with the ECM during the initial invasion phase, including ECM degradation and mechanical remodeling. The model reveals that cluster formation time, size, and shape are influenced by ECM degradation rates and cellular compliance to external stresses (durotaxis). The results align with experimental observations, demonstrating distinct cell trajectories and cluster morphologies shaped by biomechanical parameters. The simulations provide valuable insights into cancer invasion dynamics and may suggest potential therapeutic strategies targeting early-stage invasive cells.

Competing elastic and viscous gradients determine directional cell migration

Cell migration regulates central life processes including embryonic development, tissue regeneration, and tumor invasion. To establish the direction of migration, cells follow exogenous cues. Durotaxis, the directed cell migration towards elastic stiffness gradients, is the classical example of mechanical taxis. However, whether gradients of the relaxation properties in the extracellular matrix may also induce tactic responses (viscotaxis) is not well understood. Moreover, whether and how durotaxis and viscotaxis interact with each other has never been investigated. Here, we integrate clutch models for cell adhesions with an active gel theory of cell migration to reveal the mechanisms that govern viscotaxis. We show that viscotaxis is enabled by an asymmetric expression of cell adhesions that further polarize the intracellular motility forces to establish the cell front, similar to durotaxis. More importantly, when both relaxation and elastic gradients coexist, durotaxis appears more efficient in controlling directed cell migration, which we confirm with experimental results. However, the presence of opposing relaxation gradients to an elastic one can arrest or shift the migration direction. Our model rationalizes for the first time the mechanisms that govern viscotaxis and its competition with durotaxis through a mathematical model.

Elastic interactions compete with persistent cell motility to drive durotaxis

Many animal cells that crawl on extracellular substrates exhibit durotaxis, i.e., directed migration toward stiffer substrate regions. This has implications in several biological processes including tissue development and tumor progression. Here, we introduce a phenomenological model for single-cell durotaxis that incorporates both elastic deformation-mediated cell-substrate interactions and the stochasticity of cell migration. Our model is motivated by a key observation in an early demonstration of durotaxis: a single, contractile cell at a sharp interface between a softer and a stiffer region of an elastic substrate reorients and migrates toward the stiffer region. We model migrating cells as self-propelling, persistently motile agents that exert contractile traction forces on their elastic substrate. The resulting substrate deformations induce elastic interactions with mechanical boundaries, captured by an elastic potential. The dynamics is determined by two crucial parameters: the strength of the cellular traction-induced boundary elastic interaction (A), and the persistence of cell motility (Pe). Elastic forces and torques resulting from the potential orient cells perpendicular (parallel) to the boundary and accumulate (deplete) them at the clamped (free) boundary. Thus, a clamped boundary induces an attractive potential that drives durotaxis, while a free boundary induces a repulsive potential that prevents antidurotaxis. By quantifying the steady-state position and orientation probability densities, we show how the extent of accumulation (depletion) depends on the strength of the elastic potential and motility. We compare and contrast crawling cells with biological microswimmers and other synthetic active particles, where accumulation at confining boundaries is well known. We define metrics quantifying boundary accumulation and durotaxis, and present a phase diagram that identifies three possible regimes: durotaxis, and adurotaxis with and without motility-induced accumulation at the boundary. Overall, our model predicts how durotaxis depends on cell contractility and motility, successfully explains some previous observations, and provides testable predictions to guide future experiments.

Positive and negative durotaxis - mechanisms and emerging concepts

Cell migration is controlled by the coordinated action of cell adhesion, cytoskeletal dynamics, contractility and cell extrinsic cues. Integrins are the main adhesion receptors to ligands of the extracellular matrix (ECM), linking the actin cytoskeleton to the ECM and enabling cells to sense matrix rigidity and mount a directional cell migration response to stiffness gradients. Most models studied show preferred migration of single cells or cell clusters towards increasing rigidity. This is referred to as durotaxis, and since its initial discovery in 2000, technical advances and elegant computational models have provided molecular level details of stiffness sensing in cell migration. However, modeling has long predicted that, depending on cell intrinsic factors, such as the balance of cell adhesion molecules (clutches) and the motor proteins pulling on them, cells might also prefer adhesion to intermediate rigidity. Recently, experimental evidence has supported this notion and demonstrated the ability of cells to migrate towards lower rigidity, in a process called negative durotaxis. In this Review, we discuss the significant conceptual advances that have been made in our appreciation of cell plasticity and context dependency in stiffness-guided directional cell migration.

Chiral active particles are sensitive reporters to environmental geometry

Chiral active particles (CAPs) are self-propelling particles that break time-reversal symmetry by orbiting or spinning, leading to intriguing behaviors. Here, we examined the dynamics of CAPs moving in 2D lattices of disk obstacles through active Brownian dynamics simulations and granular experiments with grass seeds. We find that the effective diffusivity of the CAPs is sensitive to the structure of the obstacle lattice, a feature absent in achiral active particles. We further studied the transport of CAPs in obstacle arrays under an external field and found a reentrant directional locking effect, which can be used to sort CAPs with different activities. Finally, we demonstrated that parallelogram lattices of obstacles without mirror symmetry can separate clockwise and counter-clockwise CAPs. The mechanisms of the above three novel phenomena are qualitatively explained. As such, our work provides a basis for designing chirality-based tools for single-cell diagnosis and separation, and active particle-based environmental sensors.

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.

Positive, negative and controlled durotaxis

Cell migration is a physical process central to life. Among others, it regulates embryogenesis, tissue regeneration, and tumor growth. Therefore, understanding and controlling cell migration represent fundamental challenges in science. Specifically, the ability of cells to follow stiffness gradients, known as durotaxis, is ubiquitous across most cell types. Even so, certain cells follow positive stiffness gradients while others move along negative gradients. How the physical mechanisms involved in cell migration work to enable a wide range of durotactic responses is still poorly understood. Here, we provide a mechanistic rationale for durotaxis by integrating stochastic clutch models for cell adhesion with an active gel theory of cell migration. We show that positive and negative durotaxis found across cell types are explained by asymmetries in the cell adhesion dynamics. We rationalize durotaxis by asymmetric mechanotransduction in the cell adhesion behavior that further polarizes the intracellular retrograde flow and the protruding velocity at the cell membrane. Our theoretical framework confirms previous experimental observations and explains positive and negative durotaxis. Moreover, we show how durotaxis can be engineered to manipulate cell migration, which has important implications in biology, medicine, and bioengineering.

Adipogenesis or osteogenesis: destiny decision made by mechanical properties of biomaterials

Regenerative medicine affords an effective approach for restoring defect-associated diseases, and biomaterials play a pivotal role as cell niches to support the cell behavior and decide the destiny of cell differentiation. Except for chemical inducers, mechanical properties such as stiffness, pore size and topography of biomaterials play a crucial role in the regulation of cell behaviors and functions. Stiffness may determine the adipogenesis or osteogenesis of mesenchymal stem cells (MSCs) via the translocation of yes-associated protein (YAP) and the transcriptional coactivator with a PDZ-binding motif (TAZ). External forces transmit through cytoskeleton reorientation to assist nuclear deformation and molecule transport, meanwhile, signal pathways including the Hippo, FAK/RhoA/ROCK, and Wnt/β-catenin have been evidenced to participate in the mechanotransduction. Different pore sizes not only tailor the scaffold stiffness but also conform to the requirements of cell migration and vessels in-growth. Topography guides cell geometry along with mobility and determines the cell fate ascribed to micro/nano-scale contact. Herein, we highlight the recent progress in exploring the regulation mechanism by the physical properties of biomaterials, which might lead to more innovative regenerative strategies for adipose or bone tissue repair.

Mechanobiology in the Comorbidities of Ehlers Danlos Syndrome

Ehlers-Danlos Syndromes (EDSs) are a group of connective tissue disorders, characterized by skin stretchability, joint hypermobility and instability. Mechanically, various tissues from EDS patients exhibit lowered elastic modulus and lowered ultimate strength. This change in mechanics has been associated with EDS symptoms. However, recent evidence points toward a possibility that the comorbidities of EDS could be also associated with reduced tissue stiffness. In this review, we focus on mast cell activation syndrome and impaired wound healing, comorbidities associated with the classical type (cEDS) and the hypermobile type (hEDS), respectively, and discuss potential mechanobiological pathways involved in the comorbidities.

Contact guidance via heterogeneity of substrate elasticity

Contact guidance, the widely-known phenomenon of cell alignment, is an essential step in the organization of adherent cells. This guidance is known to occur by, amongst other things, anisotropic features in the environment including elastic heterogeneity. To understand the origins of this guidance we employed a novel statistical thermodynamics framework, which recognises the non-thermal fluctuations in the cellular response, for modelling the response of the cells seeded on substrates with alternating soft and stiff stripes. Consistent with observations, the modelling framework predicts the existence of three regimes of cell guidance: (i) in regime I for stripe widths much larger than the cell size guidance is primarily entropic; (ii) for stripe widths on the order of the cell size in regime II guidance is biochemically mediated and accompanied by changes to the cell morphology while (iii) in regime III for stripe widths much less than the cell size there is no guidance as cells cannot sense the substrate heterogeneity. Guidance in regimes I and II is due to "molli-avoidance" with cells primarily residing on the stiff stripes. While the molli-avoidance tendency is not lost with decreasing density of collagen coating the substrate, the reduced focal adhesion formation with decreasing collagen density tends to inhibit contact guidance. Our results provide clear physical insights into the interplay between cell mechano-sensitivity and substrate elastic heterogeneity that ultimately leads to the contact guidance of cells in heterogeneous tissues. STATEMENT OF SIGNIFICANCE: Cellular morphology and organization play a crucial role in the micro-architecture of tissues and dictates their biological and mechanical functioning. Despite the importance of cellular organization in all facets of tissue biology, the fundamental question of how a cell organizes itself in an anisotropic environment is still poorly understood. We employ a novel statistical thermodynamics framework which recognises the non-thermal fluctuations in the cellular response to investigate cell guidance on substrates with alternating soft and stiff stripes. The propensity of cells to primarily reside on stiff stripes results in strong guidance when the period of the stripes is larger than the cell size. For smaller stripe periods, cells sense a homogeneous substrate and guidance is lost.

Durotaxis of Passive Nanoparticles on Elastic Membranes

The transport of macromolecules and nanoscopic particles to a target cellular site is a crucial aspect in many physiological processes. This directional motion is generally controlled via active mechanical and chemical processes. Here we show, by means of molecular dynamics simulations and an analytical theory, that completely passive nanoparticles can exhibit directional motion when embedded in nonuniform mechanical environments. Specifically, we study the motion of a passive nanoparticle adhering to a mechanically nonuniform elastic membrane. We observe a nonmonotonic affinity of the particle to the membrane as a function of the membrane's rigidity, which results in the particle transport. This transport can be both up or down the rigidity gradient, depending on the absolute values of the rigidities that the gradient spans across. We conclude that rigidity gradients can be used to direct average motion of passive macromolecules and nanoparticles on deformable membranes, resulting in the preferential accumulation of the macromolecules in regions of certain mechanical properties.

Durotaxis of Passive Nanoparticles on Elastic Membranes

The transport of macromolecules and nanoscopic particles to a target cellular site is a crucial aspect in many physiological processes. This directional motion is generally controlled via active mechanical and chemical processes. Here we show, by means of molecular dynamics simulations and an analytical theory, that completely passive nanoparticles can exhibit directional motion when embedded in nonuniform mechanical environments. Specifically, we study the motion of a passive nanoparticle adhering to a mechanically nonuniform elastic membrane. We observe a nonmonotonic affinity of the particle to the membrane as a function of the membrane's rigidity, which results in the particle transport. This transport can be both up or down the rigidity gradient, depending on the absolute values of the rigidities that the gradient spans across. We conclude that rigidity gradients can be used to direct average motion of passive macromolecules and nanoparticles on deformable membranes, resulting in the preferential accumulation of the macromolecules in regions of certain mechanical properties.

Submersed micropatterned structures control active nematic flow, topology, and concentration

Coupling between flows and material properties imbues rheological matter with its wide-ranging applicability, hence the excitement for harnessing the rheology of active fluids for which internal structure and continuous energy injection lead to spontaneous flows and complex, out-of-equilibrium dynamics. We propose and demonstrate a convenient, highly tunable method for controlling flow, topology, and composition within active films. Our approach establishes rheological coupling via the indirect presence of fully submersed micropatterned structures within a thin, underlying oil layer. Simulations reveal that micropatterned structures produce effective virtual boundaries within the superjacent active nematic film due to differences in viscous dissipation as a function of depth. This accessible method of applying position-dependent, effective dissipation to the active films presents a nonintrusive pathway for engineering active microfluidic systems.

Effects of substrate stiffness on mast cell migration

Mast cells (MCs) play important roles in multiple pathologies, including fibrosis; however, their behaviors in different extracellular matrix (ECM) environments have not been fully elucidated. Accordingly, in this study, the migration of MCs on substrates with different stiffnesses was investigated using time-lapse video microscopy. Our results showed that MCs could appear in round, spindle, and star-like shapes; spindle-shaped cells accounted for 80-90 % of the total observed cells. The migration speed of round cells was significantly lower than that of cells with other shapes. Interestingly, spindle-shaped MCs migrated in a jiggling and wiggling motion between protrusions. The persistence index of MC migration was slightly higher on stiffer substrates. Moreover, we found that there was an intermediate optimal stiffness at which the migration efficiency was the highest. These findings may help to improve our understanding of MC-induced pathologies and the roles of MC migration in the immune system.

Durotaxis

In this Primer, Sunyer and Trepat introduce durotaxis, the mode of migration by which cells follow gradients of extracellular matrix stiffness.

Deriving time-varying cellular motility parameters via wavelet analysis

Cell migration, which is regulated by intracellular signaling pathways (ICSP) and extracellular matrix (ECM), plays an indispensable role in many physiological and pathological process such as normal tissue development and cancer metastasis. However, there is a lack of rigorous and quantitative tools for analyzing the time-varying characteristics of cell migration in heterogeneous microenvironment, resulted from, e.g. the time-dependent local stiffness due to microstructural remodeling by migrating cells. Here, we develop a wavelet-analysis approach to derive the time-dependent motility parameters from cell migration trajectories, based on the time-varying persistent random walk model. In particular, the wavelet denoising and wavelet transform are employed to analyze migration velocities and obtain the wavelet power spectrum. Subsequently, the time-dependent motility parameters are derived via Lorentzian power spectrum. Our results based on synthetic data indicate the superiority of the method for estimating the intrinsic transient motility parameters, robust against a variety of stochastic noises. We also carry out a systematic parameter study and elaborate the effects of parameter selection on the performance of the method. Moreover, we demonstrate the utility of our approach via analyzing experimental data ofin vitrocell migration in distinct microenvironments, including the migration of MDA-MB-231 cells in confined micro-channel arrays and correlated migration of MCF-10A cells due to ECM-mediated mechanical coupling. Our analysis shows that our approach can be as a powerful tool to accurately derive the time-dependent motility parameters, and further analyze the time-dependent characteristics of cell migration regulated by complex microenvironment.

Shannon entropy for time-varying persistence of cell migration

Cell migration, which can be significantly affected by intracellular signaling pathways and extracellular matrix, plays a crucial role in many physiological and pathological processes. Cell migration is typically modeled as a persistent random walk, which depends on two critical motility parameters, i.e., migration speed and persistence time. It is generally very challenging to efficiently and accurately quantify the migration dynamics from noisy experimental data. Here, we introduce the normalized Shannon entropy (SE) based on the FPS of cellular velocity autocovariance function to quantify migration dynamics. The SE introduced here possesses a similar physical interpretation as the Gibbs entropy for thermal systems in that SE naturally reflects the degree of order or randomness of cellular migration, attaining the maximal value of unity for purely diffusive migration (i.e., SE = 1 for the most "random" dynamics) and the minimal value of 0 for purely ballistic dynamics (i.e., SE = 0 for the most "ordered" dynamics). We also find that SE is strongly correlated with the migration persistence but is less sensitive to the migration speed. Moreover, we introduce the time-varying SE based on the WPS of cellular dynamics and demonstrate its superior utility to characterize the time-dependent persistence of cell migration, which typically results from complex and time-varying intra- or extracellular mechanisms. We employ our approach to analyze experimental data of in vitro cell migration regulated by distinct intracellular and extracellular mechanisms, exhibiting a rich spectrum of dynamic characteristics. Our analysis indicates that the SE and wavelet transform (i.e., SE-based approach) offers a simple and efficient tool to quantify cell migration dynamics in complex microenvironment.

Matrix Stiffness Modulates Mechanical Interactions and Promotes Contact between Motile Cells

The mechanical micro-environment of cells and tissues influences key aspects of cell structure and function, including cell motility. For proper tissue development, cells need to migrate, interact, and form contacts. Cells are known to exert contractile forces on underlying soft substrates and sense deformations in them. Here, we propose and analyze a minimal biophysical model for cell migration and long-range cell–cell interactions through mutual mechanical deformations of the substrate. We compute key metrics of cell motile behavior, such as the number of cell-cell contacts over a given time, the dispersion of cell trajectories, and the probability of permanent cell contact, and analyze how these depend on a cell motility parameter and substrate stiffness. Our results elucidate how cells may sense each other mechanically and generate coordinated movements and provide an extensible framework to further address both mechanical and short-range biophysical interactions.

Gum Tragacanth (GT): A Versatile Biocompatible Material beyond Borders

The use of naturally occurring materials in biomedicine has been increasingly attracting the researchers’ interest and, in this regard, gum tragacanth (GT) is recently showing great promise as a therapeutic substance in tissue engineering and regenerative medicine. As a polysaccharide, GT can be easily extracted from the stems and branches of various species of Astragalus. This anionic polymer is known to be a biodegradable, non-allergenic, non-toxic, and non-carcinogenic material. The stability against microbial, heat and acid degradation has made GT an attractive material not only in industrial settings (e.g., food packaging) but also in biomedical approaches (e.g., drug delivery). Over time, GT has been shown to be a useful reagent in the formation and stabilization of metal nanoparticles in the context of green chemistry. With the advent of tissue engineering, GT has also been utilized for the fabrication of three-dimensional (3D) scaffolds applied for both hard and soft tissue healing strategies. However, more research is needed for defining GT applicability in the future of biomedical engineering. On this object, the present review aims to provide a state-of-the-art overview of GT in biomedicine and tries to open new horizons in the field based on its inherent characteristics.

Adhesion strength and contractility enable metastatic cells to become adurotactic

Significant changes in cell stiffness, contractility, and adhesion, i.e., mechanotype, are observed during a variety of biological processes. Whether cell mechanics merely change as a side effect of or driver for biological processes is still unclear. Here, we sort genotypically similar metastatic cancer cells into strongly adherent (SA) versus weakly adherent (WA) phenotypes to study how contractility and adhesion differences alter the ability of cells to sense and respond to gradients in material stiffness. We observe that SA cells migrate up a stiffness gradient, or durotax, while WA cells largely ignore the gradient, i.e., adurotax. Biophysical modeling and experimental validation suggest that differences in cell migration and durotaxis between weakly and strongly adherent cells are driven by differences in intra-cellular actomyosin activity. These results provide a direct relationship between cell phenotype and durotaxis and suggest how, unlike other senescent cells, metastatic cancer cells navigate against stiffness gradients.

Enhanced persistence and collective migration in cooperatively aligning cell clusters

Most cells possess the capacity to locomote. Alone or collectively, this allows them to adapt, to rearrange, and to explore their surroundings. The biophysical characterization of such motile processes, in health and in disease, has so far focused mostly on two limiting cases: single-cell motility on the one hand and the dynamics of confluent tissues such as the epithelium on the other. The in-between regime of clusters, composed of relatively few cells moving as a coherent unit, has received less attention. Such small clusters are, however, deeply relevant in development but also in cancer metastasis. In this work, we use cellular Potts models and analytical active matter theory to understand how the motility of small cell clusters changes with N, the number of cells in the cluster. Modeling and theory reveal our two main findings: cluster persistence time increases with N, whereas the intrinsic diffusivity decreases with N. We discuss a number of settings in which the motile properties of more complex clusters can be analytically understood, revealing that the focusing effects of small-scale cooperation and cell-cell alignment can overcome the increased bulkiness and internal disorder of multicellular clusters to enhance overall migrational efficacy. We demonstrate this enhancement for small-cluster collective durotaxis, which is shown to proceed more effectively than for single cells. Our results may provide some novel, to our knowledge, insights into the connection between single-cell and large-scale collective motion and may point the way to the biophysical origins of the enhanced metastatic potential of small tumor cell clusters.

Durotaxis: The Hard Path from In Vitro to In Vivo

Durotaxis, the process by which cells follow gradients of extracellular mechanical stiffness, has been proposed as a mechanism driving directed migration. Despite the lack of evidence for its existence in vivo, durotaxis has become an active field of research, focusing on the mechanism by which cells respond to mechanical stimuli from the environment. In this review, we describe the technical and conceptual advances in the study of durotaxis in vitro, discuss to what extent the evidence suggests durotaxis may occur in vivo, and emphasize the urgent need for in vivo demonstration of durotaxis.

Cell Shape and Durotaxis Explained from Cell-Extracellular Matrix Forces and Focal Adhesion Dynamics

Many cells are small and rounded on soft extracellular matrices (ECM), elongated on stiffer ECMs, and flattened on hard ECMs. Cells also migrate up stiffness gradients (durotaxis). Using a hybrid cellular Potts and finite-element model extended with ODE-based models of focal adhesion (FA) turnover, we show that the full range of cell shape and durotaxis can be explained in unison from dynamics of FAs, in contrast to previous mathematical models. In our 2D cell-shape model, FAs grow due to cell traction forces. Forces develop faster on stiff ECMs, causing FAs to stabilize and, consequently, cells to spread on stiff ECMs. If ECM stress further stabilizes FAs, cells elongate on substrates of intermediate stiffness. We show that durotaxis follows from the same set of assumptions. Our model contributes to the understanding of the basic responses of cells to ECM stiffness, paving the way for future modeling of more complex cell-ECM interactions.

Cell Migration Driven by Self-Generated Integrin Ligand Gradient on Ligand-Labile Surfaces

Integrin-ligand interaction mediates the adhesion and migration of many metazoan cells. Here, we report a unique mode of cell migration elicited by the lability of integrin ligands. We found that stationary cells spontaneously turn migratory on substrates where integrin ligands are subject to depletion by cellular force. Using TGT, a rupturable molecular linker, we quantitatively tuned the rate of ligand rupture by cellular force and tested platelets (anucleate cells), CHO-K1 cells (nucleated cells), and other cell types on TGT surfaces. These originally stationary cells readily turn motile on the uniform TGT surface, and their motility is correlated with the ligand depletion rate caused by cells. We named this new migration mode ligand-depleting (LD) migration. Through both experiments and simulations, we revealed the biophysical mechanism of LD migration. We found that the cells create and maintain a gradient of ligand surface density underneath the cell body by constantly rupturing local ligands, and the gradient in turn drives and guides cell migration. This is reminiscent of the phenomenon that some liquid droplets or solid beads can spontaneously move on homogeneous surfaces by chemically forming and maintaining a local gradient of surface energy. Here, we showed that cells, as living systems, can harness a similar mechanism to migrate. LD migration is beneficial for cells to maintain adhesion on ligand-labile surfaces, and might also play a role in the migration of cancer cells, immune cells, and platelets that deplete adhesive ligands of the matrix.

All Roads Lead to Directional Cell Migration

Directional cell migration normally relies on a variety of external signals, such as chemical, mechanical, or electrical, which instruct cells in which direction to move. Many of the major molecular and physical effects derived from these cues are now understood, leading to questions about whether directional cell migration is alike or distinct under these different signals, and how cells might be directed by multiple simultaneous cues, which would be expected in complex in vivo environments. In this review, we compare how different stimuli are spatially distributed, often as gradients, to direct cell movement and the mechanisms by which they steer cells. A comparison of the downstream effectors of directional cues suggests that different external signals regulate a common set of components: small GTPases and the actin cytoskeleton, which implies that the mechanisms downstream of different signals are likely to be closely related and underlies the idea that cell migration operates by a common set of physical principles, irrespective of the input.

Effects of Microstripe Geometry on Guided Cell Migration

Cell migration on material surfaces is a fundament issue in the fields of biomaterials, cell biology, tissue engineering, regenerative medicine, etc. Herein, we aim to guide cell migration by flat microstripes with significant contrast of cell adhesion and varied geometric features of the adhesive stripes. To this end, we designed and fabricated cell-adhesive arginine-glycine-aspartate (RGD) microstripes on the nonfouling poly(ethylene glycol) (PEG) background and examined the microstripe-guided adhesion and migration of a few cell types. The migration of cell clusters adhering on the RGD regions was found to be significantly affected by the widths and arc radiuses of the guided microstripes. The cells migrated fastest on the straight microstripes with width of about 20 μm, which we defined as single file confined migration (SFCM). We also checked the possible left-right asymmetric bias of cell migration guided by combinatory microstripes with alternative wavy and quasi-straight stripes under a given width, and found that the velocity of CCW (counter clockwise) migration was higher than that of CW (clockwise) migration for primary rat mesenchymal stem cells (rMSCs), whereas no left-right asymmetric bias was observed for NIH3T3 (mouse embryonic fibroblast cell line) and Hela (human cervix epithelial carcinoma cell line) cells. Comparison of migration of cells on the nanotopological stripe and smooth surfaces further confirmed the importance of cell orientation coherence for guided cell migration and strengthened the superiority of SFCM.

The Impact of Elastic Deformations of the Extracellular Matrix on Cell Migration

The mechanical properties of the extracellular matrix, in particular its stiffness, are known to impact cell migration. In this paper, we develop a mathematical model of a single cell migrating on an elastic matrix, which accounts for the deformation of the matrix induced by forces exerted by the cell, and investigate how the stiffness impacts the direction and speed of migration. We model a cell in 1D as a nucleus connected to a number of adhesion sites through elastic springs. The cell migrates by randomly updating the position of its adhesion sites. We start by investigating the case where the cell springs are constant, and then go on to assuming that they depend on the matrix stiffness, on matrices of both uniform stiffness as well as those with a stiffness gradient. We find that the assumption that cell springs depend on the substrate stiffness is necessary and sufficient for an efficient durotactic response. We compare simulations to recent experimental observations of human cancer cells exhibiting durotaxis, which show good qualitative agreement.

Topotaxis of active Brownian particles

Recent experimental studies have demonstrated that cellular motion can be directed by topographical gradients, such as those resulting from spatial variations in the features of a micropatterned substrate. This phenomenon, known as topotaxis, has been extensively studied for topographical gradients at the subcellular scale, but can also occur in the presence of a spatially varying density of cell-sized features. Such a large-scale topotaxis has recently been observed in highly motile cells that persistently crawl within an array of obstacles with smoothly varying lattice spacing. We introduce a toy model of large-scale topotaxis, based on active Brownian particles. Using numerical simulations and analytical arguments, we demonstrate that topographical gradients introduce a spatial modulation of the particles' persistence, leading to directed motion toward regions of higher persistence. Our results demonstrate that persistent motion alone is sufficient to drive large-scale topotaxis and could serve as a starting point for more detailed studies on self-propelled particles and cells.

Run-and-tumble motion in one dimension with space-dependent speed

We consider a particle performing run-and-tumble dynamics with space-dependent speed. The model has biological relevance as it describes motile bacteria or cells in heterogeneous environments. We give exact expression for the probability density function in the case of free motion in unbounded space. We then analyze the case of a particle moving in a confined interval in the presence of partially absorbing boundaries, reporting the probability density in the Laplace (time) domain and the mean time to absorption. We also discuss the relaxation to the steady state in the case of confinement with reflecting boundaries and drift effects due to direction-dependent tumbling rates, modeling taxis phenomena of cells. The case of diffusive particles with spatially variable diffusivity is obtained as a limiting case.

Substrate-rigidity dependent migration of an idealized twitching bacterium

Mechanical properties of the extracellular matrix are important determinants of cellular migration in diverse processes, such as immune response, wound healing, and cancer metastasis. Moreover, recent studies indicate that even bacterial surface colonization can depend on the mechanics of the substrate. Here, we focus on physical mechanisms that can give rise to substrate-rigidity dependent migration. We study a "twitcher", a cell driven by extension-retraction cycles, to idealize bacteria and perhaps eukaryotic cells that employ a slip-stick mode of motion. The twitcher is asymmetric and always pulls itself forward at its front. Analytical calculations show that the migration speed of a twitcher depends non-linearly on substrate rigidity. For soft substrates, deformations do not lead to build-up of significant force and the migration speed is therefore determined by stochastic adhesion unbinding. For rigid substrates, forced adhesion rupture determines the migration speed. Depending on the force-sensitivity of front and rear adhesions, forced bond rupture implies an increase or a decrease of the migration speed. A requirement for the occurrence of rigidity-dependent stick-slip migration is a "sticky" substrate, with binding rates being an order of magnitude larger than unbinding rates in absence of force. Computer simulations show that small stall forces of the driving machinery lead to a reduced movement on high rigidities, regardless of force-sensitivities of bonds. The simulations also confirm the occurrence of rigidity-dependent migration speed in a generic model for slip-stick migration of cells on a sticky substrate.

Mechanical Model for Durotactic Cell Migration

Cell migration is a fundamental process in biological systems, playing an important role for diverse physiological processes. Cells often exhibit directed migration in a specific direction in response to various types of cues. In particular, cells are able to sense the rigidity of surrounding environments and then migrate toward stiffer regions. To understand this mechanosensitive behavior called durotaxis, several computational models have been developed. However, most of the models employed cell decision making to recapitulate durotactic behaviors, significantly limiting insights provided from these studies. In this study, we developed a computational biomechanical model without any cell decision making to illuminate intrinsic mechanisms of durotactic behaviors of cells migrating on a two-dimensional substrate. The model consists of a simplified cell generating contractile forces and a deformable substrate coarse-grained into an irregular triangulated mesh. Using the model, we demonstrated that durotactic behaviors emerge from purely mechanical interactions between the cell and the underlying substrate. We investigated how durotactic migration is regulated by biophysical properties of the substrate, including elasticity, viscosity, and stiffness profile.

Engineering the cellular mechanical microenvironment - from bulk mechanics to the nanoscale

The field of mechanobiology studies how mechanical properties of the extracellular matrix (ECM), such as stiffness, and other mechanical stimuli regulate cell behaviour. Recent advancements in the field and the development of novel biomaterials and nanofabrication techniques have enabled researchers to recapitulate the mechanical properties of the microenvironment with an increasing degree of complexity on more biologically relevant dimensions and time scales. In this Review, we discuss different strategies to engineer substrates that mimic the mechanical properties of the ECM and outline how these substrates have been applied to gain further insight into the biomechanical interaction between the cell and its microenvironment.

An investigation of the distribution and location of mast cells affected by the stiffness of substrates as a mechanical niche

The distribution and location of mast cells are closely related to their physiological and pathological functions, such as allergic responses, immunity, and fibrosis, and are used in acupuncture. In this study, the distribution of mast cells in vivo was observed, and mechanical clues for understanding their distribution based on mechanical niches were explored. By toluidine blue staining and immunohistochemical staining, we examined the distribution and location of mast cells in rat skin and found that mast cells are distributed in a spatially nonuniform manner, preferring to locate at regions in the tissue and extracellular matrix with stiffness changes. In vitro experiments for studying the distribution of rat basophilic leukemia (RBL-2H3) mast cell line on poly-di-methyl-siloxane (PDMS) substrates with stiffness variations were performed. It was found that RBL-2H3 cells migrate and tend to remain in the areas with stiffness variations. The present research suggests that changing the stiffness of local tissues may stimulate mast cell recruitment, which may be the method by which some traditional Chinese medicine treatments, such as acupuncture. On the basis of the origin of mast cells and our experimental results, we predict that mast cells exist in tissues that contain permeable capillaries and prefer regions with stiffness changes. We discussed this prediction using examples of specific tissues from some cases.

Hydrodynamics in Cell Studies

Hydrodynamic phenomena are ubiquitous in living organisms and can be used to manipulate cells or emulate physiological microenvironments experienced in vivo. Hydrodynamic effects influence multiple cellular properties and processes, including cell morphology, intracellular processes, cell–cell signaling cascades and reaction kinetics, and play an important role at the single-cell, multicellular, and organ level. Selected hydrodynamic effects can also be leveraged to control mechanical stresses, analyte transport, as well as local temperature within cellular microenvironments. With a better understanding of fluid mechanics at the micrometer-length scale and the advent of microfluidic technologies, a new generation of experimental tools that provide control over cellular microenvironments and emulate physiological conditions with exquisite accuracy is now emerging. Accordingly, we believe that it is timely to assess the concepts underlying hydrodynamic control of cellular microenvironments and their applications and provide some perspective on the future of such tools in in vitro cell-culture models. Generally, we describe the interplay between living cells, hydrodynamic stressors, and fluid flow-induced effects imposed on the cells. This interplay results in a broad range of chemical, biological, and physical phenomena in and around cells. More specifically, we describe and formulate the underlying physics of hydrodynamic phenomena affecting both adhered and suspended cells. Moreover, we provide an overview of representative studies that leverage hydrodynamic effects in the context of single-cell studies within microfluidic systems.

Mechanoreciprocity in cell migration

Cell migration is an adaptive process that depends on and responds to physical and molecular triggers. Moving cells sense and respond to tissue mechanics and induce transient or permanent tissue modifications, including extracellular matrix stiffening, compression and deformation, protein unfolding, proteolytic remodelling and jamming transitions. Here we discuss how the bi-directional relationship of cell-tissue interactions (mechanoreciprocity) allows cells to change position and contributes to single-cell and collective movement, structural and molecular tissue organization, and cell fate decisions.

Phenomenological modeling of durotaxis

Cells exhibit qualitatively different behaviors on substrates with different rigidities. The fact that cells are more polarized on the stiffer substrate motivates us to construct a two-dimensional cell with the distribution of focal adhesions dependent on substrate rigidities. This distribution affects the forces exerted by the cell and thereby determines its motion. Our model reproduces the experimental observation that the persistence time is higher on the stiffer substrate. This stiffness-dependent persistence will lead to durotaxis, the preference in moving towards stiffer substrates. This propensity is characterized by the durotaxis index first defined in experiments. We derive and validate a two-dimensional corresponding Fokker-Planck equation associated with our model. Our approach highlights the possible role of the focal adhesion arrangement in durotaxis.

Mechanosensing of matrix by stem cells: From matrix heterogeneity, contractility, and the nucleus in pore-migration to cardiogenesis and muscle stem cells in vivo

Stem cells are particularly 'plastic' cell types that are induced by various cues to become specialized, tissue-functional lineages by switching on the expression of specific gene programs. Matrix stiffness is among the cues that multiple stem cell types can sense and respond to. This seminar-style review focuses on mechanosensing of matrix elasticity in the differentiation or early maturation of a few illustrative stem cell types, with an intended audience of biologists and physical scientists. Contractile forces applied by a cell's acto-myosin cytoskeleton are often resisted by the extracellular matrix and transduced through adhesions and the cytoskeleton ultimately into the nucleus to modulate gene expression. Complexity is added by matrix heterogeneity, and careful scrutiny of the evident stiffness heterogeneity in some model systems resolves some controversies concerning matrix mechanosensing. Importantly, local stiffness tends to dominate, and 'durotaxis' of stem cells toward stiff matrix reveals a dependence of persistent migration on myosin-II force generation and also rigid microtubules that confer directionality. Stem and progenitor cell migration in 3D can be further affected by matrix porosity as well as stiffness, with nuclear size and rigidity influencing niche retention and fate choices. Cell squeezing through rigid pores can even cause DNA damage and genomic changes that contribute to de-differentiation toward stem cell-like states. Contraction of acto-myosin is the essential function of striated muscle, which also exhibit mechanosensitive differentiation and maturation as illustrated in vivo by beating heart cells and by the regenerative mobilization of skeletal muscle stem cells.