A model for cell motility on soft bio-adhesive substrates

Mechanical stiffness of bio-adhesive substrates has been recognized as a major regulator of cell motility. We present a simple physical model to study the crawling locomotion of a contractile cell on a soft elastic substrate. The mechanism of rigidity sensing is accounted for using Schwarz's two-spring model Schwarz et al. (2006). The predicted dependency between the speed of motility and substrate stiffness is qualitatively consistent with experimental observations. The model demonstrates that the rigidity dependent motility of cells is rooted in the regulation of actomyosin contractile forces by substrate deformation at each anchorage point. On stiffer substrates, the traction forces required for cell translocation acquire larger magnitude but show weaker asymmetry which leads to slower cell motility. On very soft substrates, the model predicts a biphasic relationship between the substrate rigidity and the speed of locomotion, over a narrow stiffness range, which has been observed experimentally for some cell types.     

The consequence of substrates of large-scale rigidity on actin network tension in adherent cells

Abstract

There is compelling evidence that substrate stiffness affects cell adhesion as well as cytoskeleton organization and contractile activity. This work was designed to study the cytoskeletal contractile activity of single cells plated on micropost substrates of different stiffness using a numerical model simulating the intracellular tension of individual cells. We allowed cells to adhere onto micropost substrates of various rigidities and used experimental traction force data to infer cell contractility using a numerical model. The model shows that higher substrate stiffness leads to an increase in intracellular tension. The strength of this model is its ability to calculate the mechanical state of each cell in accordance to its individual cytoskeletal structure. This is achieved by regenerating a numerical cytoskeleton based on microscope images of the actin network of each cell. The resulting numerical structure consequently represents pulling characteristics on its environment similar to those generated by the cell in-vivo. From actin imaging we can calculate and better understand how forces are transmitted throughout the cell.     

Dynamic Mechanisms of Cell Rigidity Sensing: Insights from a Computational Model of Actomyosin Networks

Cells modulate themselves in response to the surrounding environment like substrate elasticity, exhibiting structural reorganization driven by the contractility of cytoskeleton. The cytoskeleton is the scaffolding structure of eukaryotic cells, playing a central role in many mechanical and biological functions. It is composed of a network of actins, actin cross-linking proteins (ACPs), and molecular motors. The motors generate contractile forces by sliding couples of actin filaments in a polar fashion, and the contractile response of the cytoskeleton network is known to be modulated also by external stimuli, such as substrate stiffness. This implies an important role of actomyosin contractility in the cell mechano-sensing. However, how cells sense matrix stiffness via the contractility remains an open question. Here, we present a 3-D Brownian dynamics computational model of a cross-linked actin network including the dynamics of molecular motors and ACPs. The mechano-sensing properties of this active network are investigated by evaluating contraction and stress in response to different substrate stiffness. Results demonstrate two mechanisms that act to limit internal stress: (i) In stiff substrates, motors walk until they exert their maximum force, leading to a plateau stress that is independent of substrate stiffness, whereas (ii) in soft substrates, motors walk until they become blocked by other motors or ACPs, leading to submaximal stress levels. Therefore, this study provides new insights into the role of molecular motors in the contraction and rigidity sensing of cells.     

The Effect of Substrate Stiffness,Thickness, and Cross-Linking Density on Osteogenic Cell Behavior

Osteogenic cells respond to mechanical changes in their environment by altering their spread area, morphology, and gene expression profile. In particular, the bulk modulus of the substrate, as well as its microstructure and thickness, can substantially alter the local stiffness experienced by the cell. Although bone tissue regeneration strategies involve culture of bone cells on various biomaterial scaffolds, which are often cross-linked to enhance their physical integrity, it is difficult to ascertain and compare the local stiffness experienced by cells cultured on different biomaterials. In this study, we seek to characterize the local stiffness at the cellular level for MC3T3-E1 cells plated on biomaterial substrates of varying modulus, thickness, and cross-linking concentration. Cells were cultured on flat and wedge-shaped gels made from polyacrylamide or cross-linked collagen. The cross-linking density of the collagen gels was varied to investigate the effect of fiber cross-linking in conjunction with substrate thickness. Cell spread area was used as a measure of osteogenic differentiation. Finite element simulations were used to examine the effects of fiber cross-linking and substrate thickness on the resistance of the gel to cellular forces, corresponding to the equivalent shear stiffness for the gel structure in the region directly surrounding the cell. The results of this study show that MC3T3 cells cultured on a soft fibrous substrate attain the same spread cell area as those cultured on a much higher modulus, but nonfibrous substrate. Finite element simulations predict that a dramatic increase in the equivalent shear stiffness of fibrous collagen gels occurs as cross-linking density is increased, with equivalent stiffness also increasing as gel thickness is decreased. These results provide an insight into the response of osteogenic cells to individual substrate parameters and have the potential to inform future bone tissue regeneration strategies that can optimize the equivalent stiffness experienced by a cell.     

Rolling adhesion of leukocytes on soft substrates: Does substrate stiffness matter?

Cell rolling on vascular endothelium under hydrodynamic blood flow is critical for realization of many physiological and pathological processes, such as inflammatory response and tumor metastasis. The blood-borne cells are in direct contact with the inner layer of endothelium, formed by a highly compliant layer of endothelial cells. The effect of endothelial stiffness on the adhesion and motion of rolling cells is poorly understood. Inspired by recent in vitro studies, here we implemented a computational method to model the specific adhesion of a rolling cell onto a soft substrate, subjected to a creeping shear flow. The substrate is modeled as an elastic half-space, coated with P- and E-selectin receptors with specific affinity for the complementary ligands located on the moving cell. Of particular importance is to predict the effect of substrate stiffness on cell adhesion and its kinematics and kinetics of motion. Simulation results show that the effect of substrate compliance is minimal when coated with P-selectin. Conversely, the trajectory of rolling cells on E-selectin coated substrates is sensitive to the substrate compliance. This is attributed to the moderation of binding forces applied by the soft substrate which leads to a higher average translational velocity of cells.     

Traction Force Measurements of Human Aortic Smooth Muscle Cells Reveal a Motor-Clutch Behavior

The contractile behavior of smooth muscle cells (SMCs) in the aorta is an important determinant of growth, remodeling, and homeostasis. However, quantitative values of SMC basal tone have never been characterized precisely on individual SMCs. Therefore, to address this lack, we developed an in vitro technique based on Traction Force Microscopy (TFM). Aortic SMCs from a human lineage at low passages (4-7) were cultured 2 days in conditions promoting the development of their contractile apparatus and seeded on hydrogels of varying elastic modulus (1, 4, 12 and 25 kPa) with embedded fluorescent microspheres. After complete adhesion, SMCs were artificially detached from the gel by trypsin treatment. The microbeads movement was tracked and the deformation fields were processed with a mechanical model, assuming linear elasticity, isotropic material, plane strain, to extract the traction forces formerly applied by individual SMCs on the gel. Two major interesting and original observations about SMC traction forces were deduced from the obtained results: 1. they are variable but driven by cell dynamics and show an exponential distribution, with 40% to 80% of traction forces in the range 0-10 μN. 2. They depend on the substrate stiffness: the fraction of adhesion forces below 10 μN tend to decrease when the substrate stiffness increases, whereas the fraction of higher adhesion forces increases. As these two aspects of cell adhesion (variability and stiffness dependence) and the distribution of their traction forces can be predicted by the probabilistic motor-clutch model, we conclude that this model could be applied to SMCs. Further studies will consider stimulated contractility and primary culture of cells extracted from aneurysmal human aortic tissue.     

Decoupling Substrate Stiffness, Spread Area, and Micropost Density: A Close Spatial Relationship between Traction Forces and Focal Adhesions

Mechanical cues can influence the manner in which cells generate traction forces and form focal adhesions. The stiffness of a cell's substrate and the available area on which it can spread can influence its generation of traction forces, but to what extent these factors are intertwined is unclear. In this study, we used microcontact printing and micropost arrays to control cell spreading, substrate stiffness, and post density to assess their effect on traction forces and focal adhesions. We find that both the spread area and the substrate stiffness influence traction forces in an independent manner, but these factors have opposite effects: cells on stiffer substrates produce higher average forces, whereas cells with larger spread areas generate lower average forces. We show that post density influences the generation of traction forces in a manner that is more dominant than the effect of spread area. Additionally, we observe that focal adhesions respond to spread area, substrate stiffness, and post density in a manner that closely matches the trends seen for traction forces. This work supports the notion that traction forces and focal adhesions have a close relationship in their response to mechanical cues.     

Unified multiscale theory of cellular mechanical adaptations to substrate stiffness

Rigidity of the extracellular matrix markedly regulates many cellular processes. However, how cells detect and respond to matrix rigidity remains incompletely understood. Here, we propose a unified two-dimensional multiscale framework accounting for the chemomechanical feedback to explore the interrelated cellular mechanosensing, polarization, and migration, which constitute the dynamic cascade in cellular response to matrix stiffness but are often modeled separately in previous theories. By combining integrin dynamics and intracellular force transduction, we show that substrate stiffness can act as a switch to activate or deactivate cell polarization. Our theory quantitatively reproduces rich stiffness-dependent cellular dynamics, including spreading, polarity selection, migration pattern, durotaxis, and even negative durotaxis, reported in a wide spectrum of cell types, and reconciles some inconsistent experimental observations. We find that a specific bipolarized mode can determine the optimal substrate stiffness, which enables the fastest cell migration rather than the largest traction forces that cells apply on the substrate. We identify that such a mechanical adaptation stems from the force balance across the whole cell. These findings could yield universal insights into various stiffness-mediated cellular processes within the context of tissue morphogenesis, wound healing, and cancer invasion.     

Cellular traction stresses increase with increasing metastatic potential

Cancer cells exist in a mechanically and chemically heterogeneous microenvironment which undergoes dynamic changes throughout neoplastic progression. During metastasis, cells from a primary tumor acquire characteristics that enable them to escape from the primary tumor and migrate through the heterogeneous stromal environment to establish secondary tumors. Despite being linked to poor prognosis, there are no direct clinical tests available to diagnose the likelihood of metastasis. Moreover, the physical mechanisms employed by metastatic cancer cells to migrate are poorly understood. Because metastasis of most solid tumors requires cells to exert force to reorganize and navigate through dense stroma, we investigated differences in cellular force generation between metastatic and non-metastatic cells. Using traction force microscopy, we found that in human metastatic breast, prostate and lung cancer cell lines, traction stresses were significantly increased compared to non-metastatic counterparts. This trend was recapitulated in the isogenic MCF10AT series of breast cancer cells. Our data also indicate that increased matrix stiffness and collagen density promote increased traction forces, and that metastatic cells generate higher forces than non-metastatic cells across all matrix properties studied. Additionally, we found that cell spreading for these cell lines has a direct relationship with collagen density, but a biphasic relationship with substrate stiffness, indicating that cell area alone does not dictate the magnitude of traction stress generation. Together, these data suggest that cellular contractile force may play an important role in metastasis, and that the physical properties of the stromal environment may regulate cellular force generation. These findings are critical for understanding the physical mechanisms of metastasis and the role of the extracellular microenvironment in metastatic progression.     

Filamin A Is Essential for Active Cell Stiffening but not Passive Stiffening under External Force

The material properties of a cell determine how mechanical forces are transmitted through and sensed by that cell. Some types of cells stiffen passively under large external forces, but they can also alter their own stiffness in response to the local mechanical environment or biochemical cues. Here we show that the actin-binding protein filamin A is essential for the active stiffening of cells plated on collagen-coated substrates. This appears to be due to a diminished capability to build up large internal contractile stresses in the absence of filamin A. To show this, we compare the material properties and contractility of two human melanoma cell lines that differ in filamin A expression. The filamin A-deficient M2 cells are softer than the filamin A-replete A7 cells, and exert much smaller contractile stresses on the substratum, even though the M2 cells have similar levels of phosphorylated myosin II light chain and only somewhat diminished adhesion strength. In contrast to A7 cells, the stiffness and contractility of M2 cells are insensitive to either myosin-inhibiting drugs or the stiffness of the substratum. Surprisingly, however, filamin A is not required for passive stiffening under large external forces.     

A time-dependent phenomenological model for cell mechano-sensing

Adherent cells normally apply forces as a generic means of sensing and responding to the mechanical nature of their surrounding environment. How these forces vary as a function of the extracellular rigidity is critical to understanding the regulatory functions that drive important phenomena such as wound healing or muscle contraction. In recognition of this fact, experiments have been conducted to understand cell rigidity-sensing properties under known conditions of the extracellular environment, opening new possibilities for modeling this active behavior. In this work, we provide a physics-based constitutive model taking into account the main structural components of the cell to reproduce its most significant contractile properties such as the traction forces exerted as a function of time and the extracellular stiffness. This model shows how the interplay between the time-dependent response of the acto-myosin contractile system and the elastic response of the cell components determines the mechano-sensing behavior of single cells.     

Cell-cell mechanical communication through compliant substrates

The role of matrix mechanics on cell behavior is under intense investigation. Cells exert contractile forces on their matrix and the matrix elasticity can alter these forces and cell migratory behavior. However, little is known about the contribution of matrix mechanics and cell-generated forces to stable cell-cell contact and tissue formation. Using matrices of varying stiffness and measurements of endothelial cell migration and traction stresses, we find that cells can detect and respond to substrate strains created by the traction stresses of a neighboring cell, and that this response is dependent on matrix stiffness. Specifically, pairs of endothelial cells display hindered migration on gels with elasticity below 5500 Pa in comparison to individual cells, suggesting these cells sense each other through the matrix. We believe that these results show for the first time that matrix mechanics can foster tissue formation by altering the relative motion between cells, promoting the formation of cell-cell contacts. Moreover, our data indicate that cells have the ability to communicate mechanically through their matrix. These findings are critical for the understanding of cell-cell adhesion during tissue formation and disease progression, and for the design of biomaterials intended to support both cell-matrix and cell-cell adhesion.     

Transmission of Mechanical Stresses within the Cytoskeleton of Adherent Cells: A Theoretical Analysis Based on a Multi-Component Cell Model

How environmental mechanical forces affect cellular functions is a central problem in cell biology. Theoretical models of cellular biomechanics provide relevant tools for understanding how the contributions of deformable intracellular components and specific adhesion conditions at the cell interface are integrated for determining the overall balance of mechanical forces within the cell. We investigate here the spatial distributions of intracellular stresses when adherent cells are probed by magnetic twisting cytometry. The influence of the cell nucleus stiffness on the simulated nonlinear torque-bead rotation response is analyzed by considering a finite element multi-component cell model in which the cell and its nucleus are considered as different hyperelastic materials. We additionally take into account the mechanical properties of the basal cell cortex, which can be affected by the interaction of the basal cell membrane with the extracellular substrate. In agreement with data obtained on epithelial cells, the simulated behaviour of the cell model relates the hyperelastic response observed at the entire cell scale to the distribution of stresses and strains within the nucleus and the cytoskeleton, up to cell adhesion areas. These results, which indicate how mechanical forces are transmitted at distant points through the cytoskeleton, are compared to recent data imaging the highly localized distribution of intracellular stresses.     

Contractile equilibration of single cells to step changes in extracellular stiffness

Extracellular stiffness has been shown to alter long timescale cell behaviors such as growth and differentiation, but the cellular response to changes in stiffness on short timescales is poorly understood. By studying the contractile response of cells to dynamic stiffness conditions using an atomic force microscope, we observe a seconds-timescale response to a step change in extracellular stiffness. Specifically, we observe acceleration in contraction velocity (μm/min) and force rate (nN/min) upon a step decrease in stiffness and deceleration upon a step increase in stiffness. Interestingly, this seconds-timescale response to a change in extracellular stiffness is not altered by inhibiting focal adhesion signaling or stretch-activated ion channels and is independent of cell height and contraction force. Rather, the response timescale is altered only by disrupting cytoskeletal mechanics and is well described by a simple mechanical model of a constant velocity actuator pulling against an internal cellular viscoelastic network. Consistent with the predictions of this model, we find that an osmotically expanding hydrogel responds to step changes in extracellular stiffness in a similar manner to cells. We therefore propose that an initial event in stiffness sensing is establishment of a mechanical equilibrium that balances contraction of the viscoelastic cytoskeleton with deformation of the extracellular matrix.     

Cell mechanics studied by a reconstituted model tissue

Tissue models reconstituted from cells and extracellular matrix (ECM) simulate natural tissues. Cytoskeletal and matrix proteins govern the force exerted by a tissue and its stiffness. Cells regulate cytoskeletal structure and remodel ECM to produce mechanical changes during tissue development and wound healing. Characterization and control of mechanical properties of reconstituted tissues are essential for tissue engineering applications. We have quantitatively characterized mechanical properties of connective tissue models, fibroblast-populated matrices (FPMs), via uniaxial stretch measurements. FPMs resemble natural tissues in their exponential dependence of stress on strain and linear dependence of stiffness on force at a given strain. Activating cellular contractile forces by calf serum and disrupting F-actin by cytochalasin D yield "active" and "passive" components, which respectively emphasize cellular and matrix mechanical contributions. The strain-dependent stress and elastic modulus of the active component were independent of cell density above a threshold density. The same quantities for the passive component increased with cell number due to compression and reorganization of the matrix by the cells.     

Probing Cellular Mechanoadaptation Using Cell-Substrate De-Adhesion Dynamics: Experiments and Model

Physical properties of the extracellular matrix (ECM) are known to regulate cellular processes ranging from spreading to differentiation, with alterations in cell phenotype closely associated with changes in physical properties of cells themselves. When plated on substrates of varying stiffness, fibroblasts have been shown to exhibit stiffness matching property, wherein cell cortical stiffness increases in proportion to substrate stiffness up to 5 kPa, and subsequently saturates. Similar mechanoadaptation responses have also been observed in other cell types. Trypsin de-adhesion represents a simple experimental framework for probing the contractile mechanics of adherent cells, with de-adhesion timescales shown to scale inversely with cortical stiffness values. In this study, we combine experiments and computation in deciphering the influence of substrate properties in regulating de-adhesion dynamics of adherent cells. We first show that NIH 3T3 fibroblasts cultured on collagen-coated polyacrylamide hydrogels de-adhere faster on stiffer substrates. Using a simple computational model, we qualitatively show how substrate stiffness and cell-substrate bond breakage rate collectively influence de-adhesion timescales, and also obtain analytical expressions of de-adhesion timescales in certain regimes of the parameter space. Finally, by comparing stiffness-dependent experimental and computational de-adhesion responses, we show that faster de-adhesion on stiffer substrates arises due to force-dependent breakage of cell-matrix adhesions. In addition to illustrating the utility of employing trypsin de-adhesion as a biophysical tool for probing mechanoadaptation, our computational results highlight the collective interplay of substrate properties and bond breakage rate in setting de-adhesion timescales.     

A predictive model of cell traction forces based on cell geometry

Recent work has indicated that the shape and size of a cell can influence how a cell spreads, develops focal adhesions, and exerts forces on the substrate. However, it is unclear how cell shape regulates these events. Here we present a computational model that uses cell shape to predict the magnitude and direction of forces generated by cells. The predicted results are compared to experimentally measured traction forces, and show that the model can predict traction force direction, relative magnitude, and force distribution within the cell using only cell shape as an input. Analysis of the model shows that the magnitude and direction of the traction force at a given point is proportional to the first moment of area about that point in the cell, suggesting that contractile forces within the cell act on the entire cytoskeletal network as a single cohesive unit. Through this model, we demonstrate that intrinsic properties of cell shape can facilitate changes in traction force patterns, independently of heterogeneous mechanical properties or signaling events within the cell.     

Contractile Torque as a Steering Mechanism for Orientation of Adherent Cells

It is well established that adherent cells change their orientation in response to non-uniform substrate stretching. Most observations indicate that cells orient away from the direction of the maximal substrate strain, whereas in some cases cells also align with the direction of the maximal strain. Previous studies suggest that orientation and steering of the cell may be closely tied to cytoskeletal contractile stress but they could not explain the mechanisms that direct cell reorientation. This led us to develop a simple, mechanistic theoretical model that could predict a direction of cell orientation in response to mechanical nonuniformities of the substrate. The model leads to a simple physical mechanism -- namely the contractile torque -- that directs the cell toward a new orientation in response to anisotropic substrate stretching or substrate material anisotropy. A direction of the torque is determined by a dependence of the contractile stress on substrate strain. Model predictions are tested in the case of simple elongation of the substrate and found to be consistent with experimental data from the literature.     

Nanomechanics controls neuronal precursors adhesion and differentiation

The ability to control the differentiation of stem cells into specific neuronal types has a tremendous potential for the treatment of neurodegenerative diseases. In vitro neuronal differentiation can be guided by the interplay of biochemical and biophysical cues. Different strategies to increase the differentiation yield have been proposed, focusing everything on substrate topography, or, alternatively on substrate stiffness. Both strategies demonstrated an improvement of the cellular response. However it was often impossible to separate the topographical and the mechanical contributions. Here we investigate the role of the mechanical properties of nanostructured substrates, aiming at understanding the ultimate parameters which govern the stem cell differentiation. To this purpose a set of different substrates with controlled stiffness and with or without nanopatterning are used for stem cell differentiation. Our results show that the neuronal differentiation yield depends mainly on the substrate mechanical properties while the geometry plays a minor role. In particular nanostructured and flat polydimethylsiloxane (PDMS) substrates with comparable stiffness show the same neuronal yield. The improvement in the differentiation yield obtained through surface nanopatterning in the submicrometer scale could be explained as a consequence of a substrate softening effect. Finally we investigate by single cell force spectroscopy the neuronal precursor adhesion on the substrate immediately after seeding, as a possible critical step governing the neuronal differentiation efficiency. We observed that neuronal precursor adhesion depends on substrate stiffness but not on surface structure, and in particular it is higher on softer substrates. Our results suggest that cell–substrate adhesion forces and mechanical response are the key parameters to be considered for substrate design in neuronal regenerative medicine. Biotechnol. Bioeng. 2013; 110: 2301–2310. © 2013 Wiley Periodicals, Inc.     

Human Neutrophil Cytoskeletal Dynamics and Contractility Actively Contribute to Trans-Endothelial Migration

Transmigration through the endothelium is a key step in the immune response. In our recent work, the mechanical properties of the subendothelial matrix and biophysical state of the endothelium have been identified as key modulators of leukocyte trans-endothelial migration. Here, we demonstrated that neutrophil contractile forces and cytoskeletal dynamics also play an active biophysical role during transmigration through endothelial cell-cell junctions. Using our previously-established model for leukocyte transmigration, we first discovered that >93% of human neutrophils preferentially exploit the paracellular mode of transmigration in our in vitro model, and that is independent of subendothelial matrix stiffness. We demonstrated that inhibition of actin polymerization or depolymerization completely blocks transmigration, thus establishing a critical role for neutrophil actin dynamics in transmigration. Next, inhibition of neutrophil myosin II-mediated contractile forces renders 44% of neutrophils incapable of retracting their trailing edge under the endothelium for several minutes after the majority of the neutrophil transmigrates. Meanwhile, inhibition of neutrophil contractile forces or stabilization of microtubules doubles the time to complete transmigration for the first neutrophils to cross the endothelium. Notably, the time to complete transmigration is significantly reduced for subsequent neutrophils that cross through the same path as a previous neutrophil and is less dependent on neutrophil contractile forces and microtubule dynamics. These results suggest that the first neutrophil induces a gap in endothelial cell-cell adhesions, which “opens the door” in the endothelium and facilitates transmigration of subsequent neutrophils through the same hole. Collectively, this work demonstrates that neutrophils play an active biophysical role during the transmigration step of the immune response.