Lifetime and strength of periodic bond clusters between elastic media under inclined loading

A viscoelastic-stochastic model of cell adhesion considering matrix morphology and medium viscoelasticity

Quantitative investigation of the adhesive behavior between cells and the extracellular matrix (ECM) through molecular bonds is essential for cell culture and bio-medical engineering in vitro. Cell adhesion is a complex multi-scale behavior that includes temporal and spatial scales. However, the influence of the cell and matrix creep effect and the complex spatial morphology characteristics of the matrix on the cell adhesion mechanism is unclear. In the present study, an idealized theoretical model has been considered, where the adhesion of cells and the matrix is simplified into a planar strain problem of homogeneous viscoelastic half-spaces. Furthermore, a new viscoelastic-stochastic model that considers the morphological characteristics of the matrix, the viscoelasticity of the cell and the viscoelasticity of the substrate was developed under the action of a constant external force. The model characterizes the matrix topographical features by fractal dimension (FD), interprets the effects of FD and medium viscoelasticity on the molecular bond force and the receptor-ligand bond re-association rate and reveals a new mechanism for the stable adhesion of molecular bond clusters by Monte Carlo simulation. Based on this model, it was identified that the temporal and spatial distribution of molecular bond force was affected by the matrix FD and the lifetime and stability of the molecular bond cluster could be significantly improved by tuning the FD. At the same time, the viscoelastic creep effect of the cell and matrix increased the re-association rate of open bonds and could expand the window of stable adhesion more flexibly.

Effects of Contact Surface Shape on Dynamic Lifetime and Strength of Molecular Bond Clusters under Displacement- and Force-Controlled Loading Conditions

Many experimental and theoretical studies have shown that the mechanical properties of cells and the extracellular matrix can significantly affect the lifetime and strength of the adhesion clusters of molecular bonds. However, there are few studies on how the shape of the contact surface affects the lifetime and strength of the adhesion clusters of molecular bonds, especially theoretical studies in this area. An idealized model of focal adhesion is adopted, in which two rigid media are bonded together by an array of receptor-ligand bonds modeled as Hookean springs on a complex surface topography, which is described by three parameters: the surface shape factor β, the length of a single identical surface shape L, and the amplitude of surface shapes w. In this study, systematic Monte Carlo simulations of this model are conducted to study the lifetime of the molecular bond cluster under linear incremental force loading and the strength of the molecular bond cluster under linear incremental displacement loading. We find that both small surface shape amplitudes and large surface shape factors will increase the lifetime and strength of the adhesion cluster, whereas the length of a single surface shape causes oscillations in the lifetime and strength of the cluster, and this oscillation amplitude is affected by the surface shape amplitude and the factor. At the same time, we also find that the pretension in the cluster will play a dominant role in the adhesion strength under large amplitudes and small factors of surface shapes. The physical mechanisms behind these phenomena are that the changes of the length of a single surface shape, the amplitude of surface shapes, and the surface shape factor cause the changes of stress concentration in the adhesion region, bond affinity, and the number of similar affinity bonds.

A multi-scale clutch model for adhesion complex mechanics

Cell-matrix adhesion is a central mechanical function to a large number of phenomena in physiology and disease, including morphogenesis, wound healing, and tumor cell invasion. Today, how single cells respond to different extracellular cues has been comprehensively studied. However, how the mechanical behavior of the main individual molecules that form an adhesion complex cooperatively responds to force within the adhesion complex is still poorly understood. This is a key aspect of cell adhesion because how these cell adhesion molecules respond to force determines not only cell adhesion behavior but, ultimately, cell function. To answer this question, we develop a multi-scale computational model for adhesion complexes mechanics. We extend the classical clutch hypothesis to model individual adhesion chains made of a contractile actin network, a talin rod, and an integrin molecule that binds at individual adhesion sites on the extracellular matrix. We explore several scenarios of integrins dynamics and analyze the effects of diverse extracellular matrices on the behavior of the adhesion molecules and on the whole adhesion complex. Our results describe how every single component of the adhesion chain mechanically responds to the contractile actomyosin force and show how they control the traction forces exerted by the cell on the extracellular space. Importantly, our computational results agree with previous experimental data at the molecular and cellular levels. Our multi-scale clutch model presents a step forward not only to further understand adhesion complexes mechanics but also to impact, e.g., the engineering of biomimetic materials, tissue repairment, or strategies to arrest tumor progression.

Receptor-Ligand Binding: Effect of Mechanical Factors

Gaining insight into the in situ receptor-ligand binding is pivotal for revealing the molecular mechanisms underlying the physiological and pathological processes and will contribute to drug discovery and biomedical application. An important issue involved is how the receptor-ligand binding responds to mechanical stimuli. This review aims to provide an overview of the current understanding of the effect of several representative mechanical factors, such as tension, shear stress, stretch, compression, and substrate stiffness on receptor-ligand binding, wherein the biomedical implications are focused. In addition, we highlight the importance of synergistic development of experimental and computational methods for fully understanding the in situ receptor-ligand binding, and further studies should focus on the coupling effects of these mechanical factors.

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.

A computational modeling of invadopodia protrusion into an extracellular matrix fiber network

Invadopodia are dynamic actin-rich membrane protrusions that have been implicated in cancer cell invasion and metastasis. In addition, invasiveness of cancer cells is strongly correlated with invadopodia formation, which are observed during extravasation and colonization of metastatic cancer cells at secondary sites. However, quantitative understanding of the interaction of invadopodia with extracellular matrix (ECM) is lacking, and how invadopodia protrusion speed is associated with the frequency of protrusion-retraction cycles remains unknown. Here, we present a computational framework for the characterization of invadopodia protrusions which allows two way interactions between intracellular branched actin network and ECM fibers network. We have applied this approach to predicting the invasiveness of cancer cells by computationally knocking out actin-crosslinking molecules, such as α-actinin, filamin and fascin. The resulting simulations reveal distinct invadopodia dynamics with cycles of protrusion and retraction. Specifically, we found that (1) increasing accumulation of MT1-MMP at tips of invadopodia as the duration of protrusive phase is increased, and (2) the movement of nucleus toward the leading edge of the cell becomes unstable as duration of the retractile phase (or myosin turnover time) is longer than 1 min.

Effect of Cyclic Stretch on Neuron Reorientation and Axon Outgrowth

The directional alignment and outgrowth of neurons is a critical step of nerve regeneration and functional recovery of nerve systems, where neurons are exposed to a complex mechanical environment with subcellular structures such as stress fibers and focal adhesions acting as the key mechanical transducer. In this paper, we investigate the effects of cyclic stretch on neuron reorientation and axon outgrowth with a feasible stretching device that controls stretching amplitude and frequency. Statistical results indicate an evident frequency and amplitude dependence of neuron reorientation, that is, neurons tend to align away from stretch direction when stretching amplitude and frequency are large enough. On the other hand, axon elongation under cyclic stretch is very close to the reference case where neurons are not stretched. A mechanochemical framework is proposed by connecting the evolution of cellular configuration to the microscopic dynamics of subcellular structures, including stress fiber, focal adhesion, and microtubule, yielding theoretical predictions that are consistent with the experimental observations. The theoretical work provides an explanation of the neuron's mechanical response to cyclic stretch, suggesting that the contraction force generated by stress fiber plays an essential role in both neuron reorientation and axon elongation. This combined experimental and theoretical study on stretch-induced neuron reorientation may have potential applications in neurodevelopment and neuron regeneration.

Switching behaviour in vascular smooth muscle cell-matrix adhesion during oscillatory loading

Integrins regulate mechanotransduction between smooth muscle cells (SMCs) and the extracellular matrix (ECM). SMCs resident in the walls of airways or blood vessels are continuously exposed to dynamic mechanical forces due to breathing or pulsatile blood flow. However, the resulting effects of these forces on integrin dynamics and associated cell-matrix adhesion are not well understood. Here we present experimental results from atomic force microscopy (AFM) experiments, designed to study the integrin response to external oscillatory loading of varying amplitudes applied to live aortic SMCs, together with theoretical results from a mathematical model. In the AFM experiments, a fibronectin-coated probe was used cyclically to indent and retract from the surface of the cell. We observed a transition between states of firm adhesion and of complete detachment as the amplitude of oscillatory loading increased, revealed by qualitative changes in the force timecourses. Interestingly, for some of the SMCs in the experiments, switching behaviour between the two adhesion states is observed during single timecourses at intermediate amplitudes. We obtain two qualitatively similar adhesion states in the mathematical model, where we simulate the cell, integrins and ECM as an evolving system of springs, incorporating local integrin binding dynamics. In the mathematical model, we observe a region of bistability where both the firm adhesion and detachment states can occur depending on the initial adhesion state. The differences are seen to be a result of mechanical cooperativity of integrins and cell deformation. Switching behaviour is a phenomenon associated with bistability in a stochastic system, and bistability in our deterministic mathematical model provides a potential physical explanation for the experimental results. Physiologically, bistability provides a means for transient mechanical stimuli to induce long-term changes in adhesion dynamics-and thereby the cells' ability to transmit force-and we propose further experiments for testing this hypothesis.

Mechanical Model for Catch-Bond-Mediated Cell Adhesion in Shear Flow

Catch bond, whose lifetime increases with applied tensile force, can often mediate rolling adhesion of cells in a hydrodynamic environment. However, the mechanical mechanism governing the kinetics of rolling adhesion of cells through catch-bond under shear flow is not yet clear. In this study, a mechanical model is proposed for catch-bond-mediated cell adhesion in shear flow. The stochastic reaction of bond formation and dissociation is described as a Markovian process, whereas the dynamic motion of cells follows classical analytical mechanics. The steady state of cells significantly depends on the shear rate of flow. The upper and lower critical shear rates required for cell detachment and attachment are extracted, respectively. When the shear rate increases from the lower threshold to the upper threshold, cell rolling became slower and more regular, implying the flow-enhanced adhesion phenomenon. Our results suggest that this flow-enhanced stability of rolling adhesion is attributed to the competition between stochastic reactions of bonds and dynamics of cell rolling, instead of force lengthening the lifetime of catch bonds, thereby challenging the current view in understanding the mechanism behind this flow-enhanced adhesion phenomenon. Moreover, the loading history of flow defining bistability of cell adhesion in shear flow is predicted. These theoretical predictions are verified by Monte Carlo simulations and are related to the experimental observations reported in literature.

Adaptive Liquid Interfacially Assembled Protein Nanosheets for Guiding Mesenchymal Stem Cell Fate

There is a growing interest in the development of dynamic adaptive biomaterials for regulation of cellular functions. However, existing materials are limited to two-state switching of the presentation and removal of cell-adhesive bioactive motifs that cannot emulate the native extracellular matrix (ECM) in vivo with continuously adjustable characteristics. Here, tunable adaptive materials composed of a protein monolayer assembled at a liquid-liquid interface are demonstrated, which adapt dynamically to cell traction forces. An ultrastructure transition from protein monolayer to hierarchical fiber occurs through interfacial jamming. Elongated fibronectin fibers promote formation of elongated focal adhesion structures, increase focal adhesion kinase activation, and enhance neuronal differentiation of stem cells. Cell traction force results in spatial rearrangement of ECM proteins, which feeds back to alter stem cell fate. The reported biomimetic adaptive liquid interface enables dynamic control of stem cell behavior and has potential translational applications.

Surface Tension Controls the Hydraulic Fracture of Adhesive Interfaces Bridged by Molecular Bonds

Biological function requires cell-cell adhesions to tune their cohesiveness; for instance, during the opening of new fluid-filled cavities under hydraulic pressure. To understand the physical mechanisms supporting this adaptability, we develop a stochastic model for the hydraulic fracture of adhesive interfaces bridged by molecular bonds. We find that surface tension strongly enhances the stability of these interfaces by controlling flaw sensitivity, lifetime, and optimal architecture in terms of bond clustering. We also show that bond mobility embrittles adhesions and changes the mechanism of decohesion. Our study provides a mechanistic background to understand the biological regulation of cell-cell cohesion and fracture.

Effect of Loading History on Airway Smooth Muscle Cell-Matrix Adhesions

Integrin-mediated adhesions between airway smooth muscle (ASM) cells and the extracellular matrix (ECM) regulate how contractile forces generated within the cell are transmitted to its external environment. Environmental cues are known to influence the formation, size, and survival of cell-matrix adhesions, but it is not yet known how they are affected by dynamic fluctuations associated with tidal breathing in the intact airway. Here, we develop two closely related theoretical models to study adhesion dynamics in response to oscillatory loading of the ECM, representing the dynamic environment of ASM cells in vivo. Using a discrete stochastic-elastic model, we simulate individual integrin binding and rupture events and observe two stable regimes in which either bond formation or bond rupture dominate, depending on the amplitude of the oscillatory loading. These regimes have either a high or low fraction of persistent adhesions, which could affect the level of strain transmission between contracted ASM cells and the airway tissue. For intermediate loading, we observe a region of bistability and hysteresis due to shared loading between existing bonds; the level of adhesion depends on the loading history. These findings are replicated in a related continuum model, which we use to investigate the effect of perturbations mimicking deep inspirations (DIs). Because of the bistability, a DI applied to the high adhesion state could either induce a permanent switch to a lower adhesion state or allow a return of the system to the high adhesion state. Transitions between states are further influenced by the frequency of oscillations, cytoskeletal or ECM stiffnesses, and binding affinities, which modify the magnitudes of the stable adhesion states as well as the region of bistability. These findings could explain (in part) the transient bronchodilatory effect of a DI observed in asthmatics compared to a more sustained effect in normal subjects.

Mechanisms of Plastic Deformation in Collagen Networks Induced by Cellular Forces

Contractile cells can reorganize fibrous extracellular matrices and form dense tracts of fibers between neighboring cells. These tracts guide the development of tubular tissue structures and provide paths for the invasion of cancer cells. Here, we studied the mechanisms of the mechanical plasticity of collagen tracts formed by contractile premalignant acinar cells and fibroblasts. Using fluorescence microscopy and second harmonic generation, we quantified the collagen densification, fiber alignment, and strains that remain within the tracts after cellular forces are abolished. We explained these observations using a theoretical fiber network model that accounts for the stretch-dependent formation of weak cross-links between nearby fibers. We tested the predictions of our model using shear rheology experiments. Both our model and rheological experiments demonstrated that increasing collagen concentration leads to substantial increases in plasticity. We also considered the effect of permanent elongation of fibers on network plasticity and derived a phase diagram that classifies the dominant mechanisms of plasticity based on the rate and magnitude of deformation and the mechanical properties of individual fibers. Plasticity is caused by the formation of new cross-links if moderate strains are applied at small rates or due to permanent fiber elongation if large strains are applied over short periods. Finally, we developed a coarse-grained model for plastic deformation of collagen networks that can be employed to simulate multicellular interactions in processes such as morphogenesis, cancer invasion, and fibrosis.

Cellular mechanosensing of the biophysical microenvironment: A review of mathematical models of biophysical regulation of cell responses

Cells in vivo reside within complex microenvironments composed of both biochemical and biophysical cues. The dynamic feedback between cells and their microenvironments hinges upon biophysical cues that regulate critical cellular behaviors. Understanding this regulation from sensing to reaction to feedback is therefore critical, and a large effort is afoot to identify and mathematically model the fundamental mechanobiological mechanisms underlying this regulation. This review provides a critical perspective on recent progress in mathematical models for the responses of cells to the biophysical cues in their microenvironments, including dynamic strain, osmotic shock, fluid shear stress, mechanical force, matrix rigidity, porosity, and matrix shape. The review highlights key successes and failings of existing models, and discusses future opportunities and challenges in the field.

Tuning interfacial patterns of molecular bonds via surface morphology

Many studies have demonstrated that the mechanical properties of the extracellular matrix can significantly influence the morphology, strength and lifetime of focal adhesions. However, how the morphology of the contact surface affects the pattern formation of the molecular bonds still remains largely unknown. Here, by simplifying the cell and extracellular matrix to two opposing elastic bodies and considering the lateral diffusion as well as the bonding/debonding of molecular bonds, we study the clustering behavior of receptor-ligand bonds between curved surfaces and the phase diagrams of cluster patterns. We reveal the important role of surface morphology and bond kinetics in regulating the patterns of bond clusters. We further investigate the segregation dynamics of the interfacial bonds under various loading speeds, and we show that the average interfacial stress is rate-dependent while the rupture stress is rate-independent. Finally, we demonstrate that programmable patterning of bond clusters can be achieved through the designed surface morphology.

COHESIVE FINITE ELEMENT SIMULATIONS OF CONTRACTION AND SHAPE EFFECTS ON CELL DE-ADHESION

Cohesive-interface-based finite element simulations were conducted to investigate the critical shear stress required for cell de-adhesion from extracellular substrates. The interface ligand–receptor bonds are modeled by a cohesive interface model with initial stiffness, interface strength, and fracture energy as the governing parameters. The ratio of the cell modulus to the interface stiffness defines a length scale. If this length is much less than the contact size, the de-adhesion process can be modeled by the linear elastic fracture mechanics, while the opposite limit leads to the concurrent sliding of the cell or, equivalently, debonding of all the interface ligand–receptor pairs. Since it generates additional shear-stress concentration along the interface, cell contraction generally reduces the critical de-adhesion stress. Cell de-adhesion is more prone to take place for three-dimensional irregular cell shapes because of the much easier failure in the anti-plane Mode III shear, as well as the additional stress concentration in these geometric irregularities.

The Effect of Thermal Fluctuation on the Receptor-Mediated Adhesion of a Cell Membrane to an Elastic Substrate

Mechanics of the bilayer membrane play an important role in many biological and bioengineering problems such as cell-substrate and cell-nanomaterial interactions. In this work, we study the effect of thermal fluctuation and the substrate elasticity on the cell membrane-substrate adhesion. We model the adhesion of a fluctuating membrane on an elastic substrate as a two-step reaction comprised of the out-of-plane membrane fluctuation and the receptor-ligand binding. The equilibrium closed bond ratio as a function of substrate rigidity was computed by developing a coupled Fourier space Brownian dynamics and Monte Carlo method. The simulation results show that there exists a crossover value of the substrate rigidity at which the closed bond ratio is maximal.

A viscoelastic-stochastic model of the effects of cytoskeleton remodelling on cell adhesion

Cells can adapt their mechanical properties through cytoskeleton remodelling in response to external stimuli when the cells adhere to the extracellular matrix (ECM). Many studies have investigated the effects of cell and ECM elasticity on cell adhesion. However, experiments determined that cells are viscoelastic and exhibiting stress relaxation, and the mechanism behind the effect of cellular viscoelasticity on the cell adhesion behaviour remains unclear. Therefore, we propose a theoretical model of a cluster of ligand-receptor bonds between two dissimilar viscoelastic media subjected to an applied tensile load. In this model, the distribution of interfacial traction is assumed to follow classical continuum viscoelastic equations, whereas the rupture and rebinding of individual molecular bonds are governed by stochastic equations. On the basis of this model, we determined that viscosity can significantly increase the lifetime, stability and dynamic strength of the adhesion cluster of molecular bonds, because deformation relaxation attributed to the viscoelastic property can increase the rebinding probability of each open bond and reduce the stress concentration in the adhesion area.

A Chemomechanical Model of Matrix and Nuclear Rigidity Regulation of Focal Adhesion Size

In this work, a chemomechanical model describing the growth dynamics of cell-matrix adhesion structures (i.e., focal adhesions (FAs)) is developed. We show that there are three regimes for FA evolution depending on their size. Specifically, nascent adhesions with initial lengths below a critical value that are yet to engage in actin fibers will dissolve, whereas bigger ones will grow into mature FAs with a steady state size. In adhesions where growth surpasses the steady state size, disassembly will occur until their sizes are reduced to the equilibrium state. This finding arises from the fact that polymerization of adhesion proteins is force-dependent. Under actomyosin contraction, individual integrin bonds within small FAs (i.e., nascent adhesions or focal complexes) must transmit higher loads while the phenomenon of stress concentration occurs at the edge of large adhesion patches. As such, an effective stiffness of the FA-extracellular matrix complex that is either too small or too large will be relatively low, resulting in a limited actomyosin pulling force developed at the edge that is insufficient to prevent disassembly. Furthermore, it is found that a stiffer extracellular matrix and/or nucleus, as well as a stronger chemomechanical feedback, will induce larger adhesions along with a higher level of contraction force. Interestingly, switching the extracellular side from an elastic half-space, corresponding to some widely used in vitro gel substrates, to a one-dimensional fiber (as in the case of cells anchoring to a fibrous scaffold in vivo) does not qualitative change these conclusions. Our model predictions are in good agreement with a variety of experimental observations obtained in this study as well as those reported in the literature. Furthermore, this new model, to our knowledge, provides a framework with which to understand how both intracellular and extracellular perturbations lead to changes in adhesion structure number and size.

The glycocalyx promotes cooperative binding and clustering of adhesion receptors

Cell adhesion plays a pivotal role in various biological processes, e.g., immune responses, cancer metastasis, and stem cell differentiation. The adhesion behaviors depend subtly on the binding kinetics of receptors and ligands restricted at the cell-substrate interfaces. Although much effort has been directed toward investigating the kinetics of adhesion molecules, the role of the glycocalyx, anchored on cell surfaces as an exterior layer, is still unclear. In this paper, we propose a theoretical approach to study the collective binding kinetics of a few and a large number of binders in the presence of the glycocalyx, representing the cases of initial and mature adhesions of cells, respectively. The analytical results are validated by finding good agreement with our Monte Carlo simulations. In the force loading case, the on-rate and affinity increase as more bonds form, whereas this cooperative effect is not observed in the displacement loading case. The increased thickness and stiffness of the glycocalyx tend to decrease the affinity for a few bonds, while they have less influence on the affinity for a large number of bonds. Moreover, for a flexible membrane with thermally-excited shape fluctuations, the glycocalyx is exhibited to promote the formation of bond clusters, mainly due to the cooperative binding of binders. This study helps to understand the cooperative kinetics of adhesion receptors under physiologically relevant loading conditions and sheds light on the novel role of the glycocalyx in cell adhesion.

Tension-compression asymmetry in the binding affinity of membrane-anchored receptors and ligands

Cell adhesion plays a crucial role in many biological processes of cells, e.g., immune responses, tissue morphogenesis, and stem cell differentiation. An essential problem in the molecular mechanism of cell adhesion is to characterize the binding affinity of membrane-anchored receptors and ligands under different physiological conditions. In this paper, a theoretical model is presented to study the binding affinity between a large number of anchored receptors and ligands under both tensile and compressive stresses, and corroborated by demonstrating excellent agreement with Monte Carlo simulations. It is shown that the binding affinity becomes lower as the magnitude of the applied stress increases, and drops to zero at a critical tensile or compressive stress. Interestingly, the critical compressive stress is found to be substantially smaller than the critical tensile stress for relatively long and flexible receptor-ligand complexes. This counterintuitive finding is explained by using the Euler instability theory of slender columns under compression. The tension-compression asymmetry in the binding affinity of anchored receptors and ligands depends subtly on the competition between the breaking and instability of their complexes. This study helps in understanding the role of mechanical forces in cell adhesion mediated by specific binding molecules.

Modeling Active Mechanosensing in Cell-Matrix Interactions

Cells actively sense the mechanical properties of the extracellular matrix, such as its rigidity, morphology, and deformation. The cell-matrix interaction influences a range of cellular processes, including cell adhesion, migration, and differentiation, among others. This article aims to review some of the recent progress that has been made in modeling mechanosensing in cell-matrix interactions at different length scales. The issues discussed include specific interactions between proteins, the structure and mechanosensitivity of focal adhesions, the cluster effects of the specific binding, the structure and behavior of stress fibers, cells' sensing of substrate stiffness, and cell reorientation on cyclically stretched substrates. The review concludes by looking toward future opportunities in the field and at the challenges to understanding active cell-matrix interactions.

ALIGNMENT OF CELLULAR FOCAL CONTACTS AND THEIR SHAPES BY SUBSTRATE ANISOTROPY

Cell adhesion to the extracellular matrix is accomplished by the clustering of receptor–ligand bonds into focal contacts on the cell-substrate interface. The contractile forces applied onto these focal contacts lead to elastic deformation of the surrounding, which results into a cellular mechanosensory capability that plays a key role in cell adhesion, spreading, and migration, among many others. The mechanosensitivity can be manipulated by the substrate anisotropy, by which focal contacts may align into certain directions so to minimize the total mechanical potential energy. Using the elastic anisotropic contact analysis, this work systematically analyzes the dependence of the alignment on the elastic anisotropy, and more importantly, the direction of the inclined contractile forces. The contact displacement fields are a complex function of the elastic constants, so simple analysis based on tensile or shear softest direction cannot properly predict the alignment orientation. It is also proved that if these focal contacts are of elongated shape, the major axis will be parallel to the alignment direction.

Non-uniform breaking of molecular bonds, peripheral morphology and releasable adhesion by elastic anisotropy in bio-adhesive contacts

Biological adhesive contacts are usually of hierarchical structures, such as the clustering of hundreds of sub-micrometre spatulae on keratinous hairs of gecko feet, or the clustering of molecular bonds into focal contacts in cell adhesion. When separating these interfaces, releasable adhesion can be accomplished by asymmetric alignment of the lowest scale discrete bonds (such as the inclined spatula that leads to different peeling force when loading in different directions) or by elastic anisotropy. However, only two-dimensional contact has been analysed for the latter method (Chen & Gao 2007 J. Mech. Phys. Solids 55, 1001-1015 (doi:10.1016/j.jmps.2006.10.008)). Important questions such as the three-dimensional contact morphology, the maximum to minimum pull-off force ratio and the tunability of releasable adhesion cannot be answered. In this work, we developed a three-dimensional cohesive interface model with fictitious viscosity that is capable of simulating the de-adhesion instability and the peripheral morphology before and after the onset of instability. The two-dimensional prediction is found to significantly overestimate the maximum to minimum pull-off force ratio. Based on an interface fracture mechanics analysis, we conclude that (i) the maximum and minimum pull-off forces correspond to the largest and smallest contact stiffness, i.e. 'stiff-adhere and compliant-release', (ii) the fracture toughness is sensitive to the crack morphology and the initial contact shape can be designed to attain a significantly higher maximum-to-minimum pull-off force ratio than a circular contact, and (iii) since the adhesion is accomplished by clustering of discrete bonds or called bridged crack in terms of fracture mechanics terminology, the above conclusions can only be achieved when the bridging zone is significantly smaller than the contact size. This adhesion-fracture analogy study leads to mechanistic predictions that can be readily used to design biomimetics and releasable adhesives.

Aggregation dynamics of molecular bonds between compliant materials

In this paper, we develop a mechanochemical modeling framework in which the spatial-temporal evolution of receptor-ligand bonds takes place at the interface between two compliant media in the presence of an externally applied tensile load. Bond translocation, dissociation and association occur simultaneously, resulting in dynamic aggregation of molecular bonds that is regulated by mechanical factors such as material compliance and applied stress. The results show that bond aggregation is energetically favorable in the out-of-equilibrium process with convoluted time scales from bond diffusion and reaction. Material stiffness is predicted to contribute to adhesion growth and an optimal level of applied stress leads to the maximized size of bond clusters for integrin-based adhesion, consistent with related experimental observations on focal adhesions of cell-matrix interactions. The stress distribution within bond clusters is generally non-uniform and governed by the stress concentration index.

Matrix confinement plays a pivotal role in regulating neutrophil-generated tractions, speed, and integrin utilization

Neutrophils are capable of switching from integrin-dependent motility on two-dimensional substrata to integrin-independent motion following entry into the confined three-dimensional matrix of an afflicted tissue. However, whether integrins still maintain a regulatory role for cell traction generation and cell locomotion under the physical confinement of the three-dimensional matrix is unknown, and this is challenging to deduce from motility studies alone. Using three-dimensional traction force microscopy and a double hydrogel sandwich system, we determined the three-dimensional spatiotemporal traction forces of motile neutrophils at unprecedented resolution and show, for the first time, that entry into a highly confined space (2.5D) is a sufficient trigger to convert to integrin-independent migration. We find that integrins exert a significant regulatory role in determining the magnitude and spatial distribution of tractions and cell speed on confined cells. We also find that 90% of neutrophil tractions are in the out-of-plane axis, and this may be a fundamental element of neutrophil traction force generation.

Effects of cellular viscoelasticity in multiple-bond force spectroscopy

Receptor-ligand bonds are often subjected to forces that regulate their detachment via modulating off-rates. Though the dynamics of detachment is primarily controlled by the physical chemistry of adhesion molecules cellular features such as cell deformability and microvillus viscoelasticity have been shown to have an effect on it as well. In this work, Monte Carlo simulation of the rupture of multiple receptor-ligand bonds between substrate and a polymorphonuclear leukocyte (PMN) cell suspended in a Newtonian fluid is performed. It is demonstrated via various micromechanical models of the PMN cell adhered to the substrate by multiple receptor-ligand bonds that viscous drag caused by relative motion of cell suspended in a Newtonian fluid and cellular viscoelasticity modulate transmission of an applied external load to receptor-ligand bonds. It is demonstrated that due to cellular viscoelasticity the instantaneous intermolecular bond force is lower than the instantaneous applied force. It is also demonstrated that due to cellular viscoelasticity, the mean intermolecular bond rupture forces are lowered while the mean bond lifetime increases.

Active mechanics and dynamics of cell spreading on elastic substrates

The spreading area of cells has been shown to play a central role in the determination of cell fate and tissue morphogenesis; however, a clear understanding of how spread cell area is determined is still lacking. The observation that cell area and force generally increase with substrate rigidity suggests that cell area is dictated mechanically, by means of a force-balance between the cell and the substrate. A simple mechanical model, corroborated by experimental measurements of cell area and force is presented to analyze the temporal force balance between the cell and the substrate during spreading. The cell is modeled as a thin elastic disc that is actively pulled by lamellipodia protrusions at the cell front. The essential molecular mechanisms of the motor activity at the cell front, including, actin polymerization, adhesion kinetics, and the actin retrograde flow, are accounted for and used to predict the dynamics of cell spreading on elastic substrates; simple, closed-form expressions for the evolution of cell size and force are derived. Time-resolved, traction force microscopy, combined with measurements of cell area are performed to investigate the simultaneous variations of cell size and force. We find that cell area and force increase simultaneously during spreading but the force develops with an apparent delay relative to the increase in cell area. We demonstrate that this may reflect the strain-stiffening property of the cytoskeleton. We further demonstrate that the radial cell force is a concave function of spreading speed and that this may reflect the strengthening of cell-substrate adhesions during spreading.

USNCTAM perspectives on mechanics in medicine

Over decades, the theoretical and applied mechanics community has developed sophisticated approaches for analysing the behaviour of complex engineering systems. Most of these approaches have targeted systems in the transportation, materials, defence and energy industries. Applying and further developing engineering approaches for understanding, predicting and modulating the response of complicated biomedical processes not only holds great promise in meeting societal needs, but also poses serious challenges. This report, prepared for the US National Committee on Theoretical and Applied Mechanics, aims to identify the most pressing challenges in biological sciences and medicine that can be tackled within the broad field of mechanics. This echoes and complements a number of national and international initiatives aiming at fostering interdisciplinary biomedical research. This report also comments on cultural/educational challenges. Specifically, this report focuses on three major thrusts in which we believe mechanics has and will continue to have a substantial impact. (i) Rationally engineering injectable nano/microdevices for imaging and therapy of disease. Within this context, we discuss nanoparticle carrier design, vascular transport and adhesion, endocytosis and tumour growth in response to therapy, as well as uncertainty quantification techniques to better connect models and experiments. (ii) Design of biomedical devices, including point-of-care diagnostic systems, model organ and multi-organ microdevices, and pulsatile ventricular assistant devices. (iii) Mechanics of cellular processes, including mechanosensing and mechanotransduction, improved characterization of cellular constitutive behaviour, and microfluidic systems for single-cell studies.

Predicted rupture force of a single molecular bond becomes rate independent at ultralow loading rates

We present for the first time a theoretical model of studying the saturation of the rupture force of a single molecular bond that causes the rupture force to be rate independent under an ultralow loading rate. This saturation will obviously bring challenges to understanding the rupture behavior of the molecular bond using conventional methods. This intriguing feature implies that the molecular bond has a nonzero strength at a vanishing loading rate. We find that the saturation behavior is caused by bond rebinding when the loading rate is lower than a limiting value depending on the loading stiffness.

Elastic coupling of nascent apCAM adhesions to flowing actin networks

Adhesions are multi-molecular complexes that transmit forces generated by a cell's acto-myosin networks to external substrates. While the physical properties of some of the individual components of adhesions have been carefully characterized, the mechanics of the coupling between the cytoskeleton and the adhesion site as a whole are just beginning to be revealed. We characterized the mechanics of nascent adhesions mediated by the immunoglobulin-family cell adhesion molecule apCAM, which is known to interact with actin filaments. Using simultaneous visualization of actin flow and quantification of forces transmitted to apCAM-coated beads restrained with an optical trap, we found that adhesions are dynamic structures capable of transmitting a wide range of forces. For forces in the picoNewton scale, the nascent adhesions' mechanical properties are dominated by an elastic structure which can be reversibly deformed by up to 1 µm. Large reversible deformations rule out an interface between substrate and cytoskeleton that is dominated by a number of stiff molecular springs in parallel, and favor a compliant cross-linked network. Such a compliant structure may increase the lifetime of a nascent adhesion, facilitating signaling and reinforcement.

Mathematical modeling of eukaryotic cell migration: insights beyond experiments

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

Nanotopography modulates mechanotransduction of stem cells and induces differentiation through focal adhesion kinase

Regulated biophysical cues, such as nanotopography, have been shown to be integral for tissue regeneration and embryogenesis in the stem cell niche. Tissue homeostasis involves the interaction of multipotent cells with nanoscaled topographical features in their ECM to regulate aspects of cell behavior. Synthetic nanostructures can drive specific cell differentiation, but the sensing mechanisms for nanocues remain poorly understood. Here, we report that nanotopography-induced human mesenchymal stem cell (hMSC) differentiation through cell mechanotransduction is modulated by the integrin-activated focal adhesion kinase (FAK). On nanogratings with 250 nm line width on polydimethylsiloxane, hMSCs developed aligned stress fibers and showed an upregulation of neurogenic and myogenic differentiation markers. The observed cellular focal adhesions within these cells were also significantly smaller and more elongated on the nanogratings compared to microgratings or unpatterned control. In addition, our mechanistic study confirmed that this regulation was dependent upon actomyosin contractility, suggesting a direct force-dependent mechanism. The topography-induced differentiation was observed on different ECM compositions but the response was not indicative of a direct ECM-induced hMSC differentiation pathway. FAK phosphorylation was required for topography-induced hMSC differentiation while FAK overexpression overruled the topographical cues in determining cell lineage bias. The results indicated that FAK activity had a direct impact on topography-induced gene expression, and that this effect of FAK was independent of cell shape. These findings suggest that hMSC sense and transduce nanotopographical signals through focal adhesions and actomyosin cytoskeleton contractility to induce differential gene expression.

A mechanochemical model of cell reorientation on substrates under cyclic stretch

We report a theoretical study on the cyclic stretch-induced reorientation of spindle-shaped cells. Specifically, by taking into account the evolution of sub-cellular structures like the contractile stress fibers and adhesive receptor-ligand clusters, we develop a mechanochemical model to describe the dynamics of cell realignment in response to cyclically stretched substrates. Our main hypothesis is that cells tend to orient in the direction where the formation of stress fibers is energetically most favorable. We show that, when subjected to cyclic stretch, the final alignment of cells reflects the competition between the elevated force within stress fibers that accelerates their disassembly and the disruption of cell-substrate adhesion as well, and an effectively increased substrate rigidity that promotes more stable focal adhesions. Our model predictions are consistent with various observations like the substrate rigidity dependent formation of stable adhesions and the stretching frequency, as well as stretching amplitude, dependence of cell realignment. This theory also provides a simple explanation on the regulation of protein Rho in the formation of stretch-induced stress fibers in cells.

Effect of viscous drag on multiple receptor-ligand bonds rupture force

Monte Carlo simulation of the rupture of multiple receptor-ligand bonds between two PMN cells suspended in a Newtonian fluid is performed. We demonstrate via micro-mechanical model of two cells adhered by multiple receptor-ligand bonds that viscous drag caused by relative motion of cell suspended in a Newtonian fluid modulates transmission of an applied external load to bonds. Specifically, it is demonstrated that at any time the intermolecular bond force is not equivalent to the instantaneous applied force. The difference in the instantaneous applied force and the intermolecular bond force depends on the viscosity of fluid, the size of cell, the applied loading rate, and the number of bonds at any instant of time. Viscous drag acting on cell reduces average bond rupture forces.

United we stand: integrating the actin cytoskeleton and cell-matrix adhesions in cellular mechanotransduction

Many essential cellular functions in health and disease are closely linked to the ability of cells to respond to mechanical forces. In the context of cell adhesion to the extracellular matrix, the forces that are generated within the actin cytoskeleton and transmitted through integrin-based focal adhesions are essential for the cellular response to environmental clues, such as the spatial distribution of adhesive ligands or matrix stiffness. Whereas substantial progress has been made in identifying mechanosensitive molecules that can transduce mechanical force into biochemical signals, much less is known about the nature of cytoskeletal force generation and transmission that regulates the magnitude, duration and spatial distribution of forces imposed on these mechanosensitive complexes. By focusing on cell-matrix adhesion to flat elastic substrates, on which traction forces can be measured with high temporal and spatial resolution, we discuss our current understanding of the physical mechanisms that integrate a large range of molecular mechanotransduction events on cellular scales. Physical limits of stability emerge as one important element of the cellular response that complements the structural changes affected by regulatory systems in response to mechanical processes.

Effect of loading conditions on the dissociation behaviour of catch bond clusters

Under increasing tensile load, the lifetime of a single catch bond counterintuitively increases up to a maximum and then decreases exponentially like a slip bond. So far, the characteristics of single catch bond dissociation have been extensively studied. However, it remains unclear how a cluster of catch bonds behaves under tensile load. We perform computational analysis on the following models to examine the characteristics of clustered catch bonds: (i) clusters of catch bonds with equal load sharing, (ii) clusters of catch bonds with linear load sharing, and (iii) clusters of catch bonds in micropipette-manipulated cell detachment. We focus on the differences between the slip and catch bond clusters, identifying the critical factors for exhibiting the characteristics of catch bond mechanism for the multiple-bond system. Our computation reveals that for a multiple-bond cluster, the catch bond behaviour could only manifest itself under relatively uniform loading conditions and at certain stages of decohesion, explaining the difficulties in observing the catch bond mechanism under real biological conditions.

Motor force homeostasis in skeletal muscle contraction

In active biological contractile processes such as skeletal muscle contraction, cellular mitosis, and neuronal growth, an interesting common observation is that multiple motors can perform coordinated and synchronous actions, whereas individual myosin motors appear to randomly attach to and detach from actin filaments. Recent experiment has demonstrated that, during skeletal muscle shortening at a wide range of velocities, individual myosin motors maintain a force of ~6 pN during a working stroke. To understand how such force-homeostasis can be so precisely regulated in an apparently chaotic system, here we develop a molecular model within a coupled stochastic-elastic theoretical framework. The model reveals that the unique force-stretch relation of myosin motor and the stochastic behavior of actin-myosin binding cause the average number of working motors to increase in linear proportion to the filament load, so that the force on each working motor is regulated at ~6 pN, in excellent agreement with experiment. This study suggests that it might be a general principle to use catch bonds together with a force-stretch relation similar to that of myosin motors to regulate force homeostasis in many biological processes.

Probing mechanical principles of focal contacts in cell-matrix adhesion with a coupled stochastic-elastic modelling framework

Cell-matrix adhesion depends on the collective behaviours of clusters of receptor-ligand bonds called focal contacts between cell and extracellular matrix. While the behaviour of a single molecular bond is governed by statistical mechanics at the molecular scale, continuum mechanics should be valid at a larger scale. This paper presents an overview of a series of recent theoretical studies aimed at probing the basic mechanical principles of focal contacts in cell-matrix adhesion via stochastic-elastic models in which stochastic descriptions of molecular bonds and elastic descriptions of interfacial traction-separation are unified in a single modelling framework. The intention here is to illustrate these principles using simple analytical and numerical models. The aim of the discussions is to provide possible clues to the following questions: why does the size of focal adhesions (FAs) fall into a narrow range around the micrometre scale? How can cells sense and respond to substrates of varied stiffness via FAs? How do the magnitude and orientation of mechanical forces affect the binding dynamics of FAs? The effects of cluster size, cell-matrix elastic modulus, loading direction and cytoskeletal pretension on the lifetime of FA clusters have been investigated by theoretical arguments as well as Monte Carlo numerical simulations, with results showing that intermediate adhesion size, stiff substrate, cytoskeleton stiffening, low-angle pulling and moderate cytoskeletal pretension are factors that contribute to stable FAs. From a mechanistic point of view, these results provide possible explanations for a wide range of experimental observations and suggest multiple mechanisms by which cells can actively control adhesion and de-adhesion via cytoskeletal contractile machinery in response to mechanical properties of their surroundings.

Soft matrices suppress cooperative behaviors among receptor-ligand bonds in cell adhesion

The fact that biological tissues are stable over prolonged periods of time while individual receptor-ligand bonds only have limited lifetime underscores the critical importance of cooperative behaviors of multiple molecular bonds, in particular the competition between the rate of rupture of closed bonds (death rate) and the rate of rebinding of open bonds (birth rate) in a bond cluster. We have recently shown that soft matrices can greatly increase the death rate in a bond cluster by inducing severe stress concentration near the adhesion edges. In the present paper, we report a more striking effect that, irrespective of stress concentration, soft matrices also suppress the birth rate in a bond cluster by increasing the local separation distance between open bonds. This is shown by theoretical analysis as well as Monte Carlo simulations based on a stochastic-elasticity model in which stochastic descriptions of molecular bonds and elastic descriptions of interfacial force/separation are unified in a single modeling framework. Our findings not only are important for understanding the role of elastic matrices in cell adhesion, but also have general implications on adhesion between soft materials.

Mechanical principle of enhancing cell-substrate adhesion via pre-tension in the cytoskeleton

Motivated by our earlier study on the effect of pre-tension in gecko adhesion, here we investigate whether and how pre-tension in cytoskeleton influences cell adhesion by developing a stochastic-elasticity model of a stress fiber attached on a rigid substrate via molecular bonds. By comparing the variations in adhesion lifetime and observing the sequences of bond breaking with and without pre-tension in the stress fiber under the same applied force, we demonstrate that the effect of pre-tension is to shift the interfacial failure mode from cracklike propagation toward uniform bond failure within the contact region, thereby greatly increasing the adhesion lifetime. Since stress fibers are the primary load-bearing components of cells, as well as the basic functional units of cytoskeleton that facilitate cell adhesion, this study suggests a feasible mechanism by which cell adhesion could be actively controlled via cytoskeletal contractility and proposes that pre-tension may be a general principle in biological adhesion.