Computational model predicts cell orientation in response to a range of mechanical stimuli

To build anisotropic, mechanically functioning tissue, it is essential to understand how cells orient in response to mechanical stimuli. Therefore, a computational model was developed which predicts cell orientation, based on the actin stress fiber distribution inside the cell. In the model, the stress fiber distribution evolves dynamically according to the following: (1) Stress fibers contain polymerized actin. The total amount of depolymerized plus polymerized actin is constant. (2) Stress fibers apply tension to their environment. This active tension is maximal when strain rate and absolute strain are zero and reduces with increasing shortening rate and absolute strain. (3) A high active fiber stress in a direction leads to a large amount of fibers in this direction. (4) The cell is attached to a substrate; all fiber stresses are homogenized into a total cell stress, which is in equilibrium with substrate stress. This model predicts that on a substrate of anisotropic stiffness, fibers align in the stiffest direction. Under cyclic strain when the cellular environment is so stiff that no compaction occurs (1 MPa), the model predicts strain avoidance, which is more pronounced with increasing strain frequency or amplitude. Under cyclic strain when the cellular environment is so soft that cells can compact it (10 kPa), the model predicts a preference for the cyclically strained compared to the compacting direction. These model predictions all agree with experimental evidence. For the first time, a computational model predicts cell orientation in response to this range of mechanical stimuli using a single set of parameters.     

Maturation of Filopodia Shaft Adhesions Is Upregulated by Local Cycles of Lamellipodia Advancements and Retractions

While cell-substrate adhesions that form between the protruding edge of a spreading cell and flat surfaces have been studied extensively, processes that regulate the maturation of filopodia adhesions are far less characterized. Since little is known about how the kinetics of formation or disassembly of filopodia adhesions is regulated upon integration into the lamellum, a kinetic analysis of the formation and disassembly of filopodia adhesions was conducted at the leading edge of β3-integrin-EGFP-expressing rat embryonic fibroblasts spreading on fibronectin-coated glass or on soft polyacrylamide gels. Filopodia β3-integrin adhesions matured only if the lamellipodium in their immediate vicinity showed cyclic protrusions and retractions. Filopodia β3-integrin shaft adhesions elongated rapidly when they were overrun by the advancing lamellipodium. Subsequently and once the lamellipodium stopped its advancement at the distal end of the filopodia β3-integrin adhesion, these β3-integrin shaft adhesions started to grow sidewise and colocalize with the newly assembled circumferential actin stress fibers. In contrast, the suppression of the cyclic protrusions and retractions of the lamellipodium by blocking myosin light chain kinase suppressed the growth of filopodia adhesion and resulted in the premature disassembly of filopodia adhesions. The same failure to stabilize those adhesions was found for the advancing lamellipodium that rapidly overran filopodia shaft adhesions without pausing as seen often during fast cell spreading. In turn, plating cells on soft polyacrylamide gels resulted in a reduction of lamellipodia activity, which was partially restored locally by the presence of filopodia adhesions. Thus filopodia adhesions could also mature and be integrated into the lamellum for fibroblasts on soft polyacrylamide substrates.     

Combining dynamic stretch and tunable stiffness to probe cell mechanobiology in vitro

Cells have the ability to actively sense their mechanical environment and respond to both substrate stiffness and stretch by altering their adhesion, proliferation, locomotion, morphology, and synthetic profile. In order to elucidate the interrelated effects of different mechanical stimuli on cell phenotype in vitro, we have developed a method for culturing mammalian cells in a two-dimensional environment at a wide range of combined levels of substrate stiffness and dynamic stretch. Polyacrylamide gels were covalently bonded to flexible silicone culture plates and coated with monomeric collagen for cell adhesion. Substrate stiffness was adjusted from relatively soft (G′ = 0.3 kPa) to stiff (G′ = 50 kPa) by altering the ratio of acrylamide to bis-acrylamide, and the silicone membranes were stretched over circular loading posts by applying vacuum pressure to impart near-uniform stretch, as confirmed by strain field analysis. As a demonstration of the system, porcine aortic valve interstitial cells (VIC) and human mesenchymal stem cells (hMSC) were plated on soft and stiff substrates either statically cultured or exposed to 10% equibiaxial or pure uniaxial stretch at 1Hz for 6 hours. In all cases, cell attachment and cell viability were high. On soft substrates, VICs cultured statically exhibit a small rounded morphology, significantly smaller than on stiff substrates (p<0.05). Following equibiaxial cyclic stretch, VICs spread to the extent of cells cultured on stiff substrates, but did not reorient in response to uniaxial stretch to the extent of cells stretched on stiff substrates. hMSCs exhibited a less pronounced response than VICs, likely due to a lower stiffness threshold for spreading on static gels. These preliminary data demonstrate that inhibition of spreading due to a lack of matrix stiffness surrounding a cell may be overcome by externally applied stretch suggesting similar mechanotransduction mechanisms for sensing stiffness and stretch.     

Cellular contractility and substrate elasticity: a numerical investigation of the actin cytoskeleton and cell adhesion

Numerous experimental studies have established that cells can sense the stiffness of underlying substrates and have quantified the effect of substrate stiffness on stress fibre formation, focal adhesion area, cell traction, and cell shape. In order to capture such behaviour, the current study couples a mixed mode thermodynamic and mechanical framework that predicts focal adhesion formation and growth with a material model that predicts stress fibre formation, contractility, and dissociation in a fully 3D implementation. Simulations reveal that SF contractility plays a critical role in the substrate-dependent response of cells. Compliant substrates do not provide sufficient tension for stress fibre persistence, causing dissociation of stress fibres and lower focal adhesion formation. In contrast, cells on stiffer substrates are predicted to contain large amounts of dominant stress fibres. Different levels of cellular contractility representative of different cell phenotypes are found to alter the range of substrate stiffness that cause the most significant changes in stress fibre and focal adhesion formation. Furthermore, stress fibre and focal adhesion formation evolve as a cell spreads on a substrate and leading to the formation of bands of fibres leading from the cell periphery over the nucleus. Inhibiting the formation of FAs during cell spreading is found to limit stress fibre formation. The predictions of this mutually dependent material-interface framework are strongly supported by experimental observations of cells adhered to elastic substrates and offer insight into the inter-dependent biomechanical processes regulating stress fibre and focal adhesion formation.     

Theoretical study of intracellular stress fiber orientation under cyclic deformation

We studied stress fiber orientation under a wide range of uniaxial cyclic deformations. We devised and validated a hypothesis consisting of two parts, as follows: (1) a stress fiber aligns to avoid a mechanical stimulus in the fiber direction under cyclic deformation. This means that, among all allowable directions, a stress fiber aligns in the direction which minimizes the stimulus, i. e., the summation of the changes in length of the stress fiber over one stretch cycle; and (2) there is a limit in the sensitivity of the cellular response to the mechanical stimulus. Due to this sensing limit, the orientation angle in stress fibers is distributed around the angle corresponding to the minimum stimulus. To validate this hypothesis, we approximated an anisotropic deformation of the membrane on which cells were to be cultured. We then obtained the relationships between the stretch range and the fiber angle in the undeformed state which minimize the mechanical stimuli, assuming that the membrane on which stress fibers and cells adhered was homogeneous and incompressible. Numerical simulation results showed that the proposed hypothesis described our previous experimental results well and was consistent with the experimental results in the literature. The simulation results, taking account of the second part of the hypothesis with a small value for the limit in sensitivity to the mechanical stimulus, could explain why cell orientation is distributed so widely with cyclic stretch ranges of <10%. The proposed hypothesis can be applied to various types of deformation because the mechanical stimulus is always sensed and accumulates under cyclic deformation without the necessity of a reference state to measure the stimulus.     

ADAM12/syndecan-4 signaling promotes beta 1 integrin-dependent cell spreading through protein kinase Calpha and RhoA

The ADAMs (a disintegrin and metalloprotease) comprise a large family of multidomain proteins with cell-binding and metalloprotease activities. The ADAM12 cysteine-rich domain (rADAM12-cys) supports cell attachment using syndecan-4 as a primary cell surface receptor that subsequently triggers beta(1) integrin-dependent cell spreading, stress fiber assembly, and focal adhesion formation. This process contrasts with cell adhesion on fibronectin, which is integrin-initiated but syndecan-4-dependent. In the present study, we investigated ADAM12/syndecan-4 signaling leading to cell spreading and stress fiber formation. We demonstrate that syndecan-4, when present in significant amounts, promotes beta(1) integrin-dependent cell spreading and stress fiber formation in response to rADAM12-cys. A mutant form of syndecan-4 deficient in protein kinase C (PKC)alpha activation or a different member of the syndecan family, syndecan-2, was unable to promote cell spreading. GF109203X and G?6976, inhibitors of PKC, completely inhibited ADAM12/syndecan-4-induced cell spreading. Expression of syndecan-4, but not syn4DeltaI, resulted in the accumulation of activated beta(1) integrins at the cell periphery in Chinese hamster ovary beta1 cells as revealed by 12G10 staining. Further, expression of myristoylated, constitutively active PKCalpha resulted in beta(1) integrin-dependent cell spreading, but additional activation of RhoA was required to induce stress fiber formation. In summary, these data provide novel insights into syndecan-4 signaling. Syndecan-4 can promote cell spreading in a beta(1) integrin-dependent fashion through PKCalpha and RhoA, and PKCalpha and RhoA likely function in separate pathways.     

A mathematical model of the coupled mechanisms of cell adhesion,contraction and spreading

Recent research has shown that cell spreading is highly dependent on the contractility of its cytoskeleton and the mechanical properties of the environment it is located in. The dynamics of such process is critical for the development of tissue engineering strategy but is also a key player in wound contraction, tissue maintenance and angiogenesis. To better understand the underlying physics of such phenomena, the paper describes a mathematical formulation of cell spreading and contraction that couples the processes of stress fiber formation, protrusion growth through actin polymerization at the cell edge and dynamics of cross-membrane protein (integrins) enabling cell-substrate attachment. The evolving cell’s cytoskeleton is modeled as a mixture of fluid, proteins and filaments that can exchange mass and generate contraction. In particular, besides self-assembling into stress fibers, actin monomers able to polymerize into an actin meshwork at the cell’s boundary in order to push the membrane forward and generate protrusion. These processes are possible via the development of cell-substrate attachment complexes that arise from the mechano-sensitive equilibrium of membrane proteins, known as integrins. After deriving the governing equation driving the dynamics of cell evolution and spreading, we introduce a numerical solution based on the extended finite element method, combined with a level set formulation. Numerical simulations show that the proposed model is able to capture the dependency of cell spreading and contraction on substrate stiffness and chemistry. The very good agreement between model predictions and experimental observations suggests that mechanics plays a strong role into the coupled mechanisms of contraction, adhesion and spreading of adherent cells.     

Regulation of PD-L1 expression by matrix stiffness in lung cancer cells

Expression of programmed death-ligand 1 (PD-L1) in tumor cells such as lung cancer cells plays an important role in mechanisms underlying evasion of an immune check point system. Lung cancer tissue with increased deposition of extracellular matrix is much stiffer than normal lung tissue. There is emerging evidence that the matrix stiffness of cancer tissue affects the phenotypes and properties of cancer cells. Nevertheless, the effects of substrate rigidity on expression of PD-L1 in lung cancer cells remain elusive. We evaluated the effects of substrate stiffness on PD-L1 expression in HCC827 lung adenocarcinoma cells by using polyacrylamide hydrogels with stiffnesses of 2 and 25?kPa. Expression of PD-L1 protein was higher on the stiffer substrates (25?kPa gel and plastic dish) than on the soft 2?kPa gel. PD-L1 expression was reduced by detachment of cells adhering to the substrate. Interferon-γ enhanced expression of PD-L1 protein cultured on stiff (25?kPa gel and plastic dishes) and soft (2?kPa gel) substrates and in the cell adhesion-free condition. As the stiffness of substrates increased, formation of actin stress fiber and cell growth were enhanced. Transfection of the cells with short interfering RNA for PD-L1 inhibited cell growth without affecting stress fiber formation. Treatment of the cells with cytochalasin D, an inhibitor of actin polymerization, significantly reduced PD-L1 protein levels. Taken together, a stiff substrate enhanced PD-L1 expression via actin-dependent mechanisms in lung cancer cells. It is suggested that stiffness as a tumor environment regulates PD-L1 expression, which leads to evasion of the immune system and tumor growth.     

Neurite Branch Retraction Is Caused by a Threshold-Dependent Mechanical Impact

Recent results indicate that, in addition to chemical cues, mechanical stimuli may also impact neuronal growth. For instance, unlike most other cell types, neurons prefer soft substrates. However, the mechanisms responsible for the neuronal affinity for soft substrates have not yet been identified. In this study, we show that, in vitro, neurons continuously probe their mechanical environment. Growth cones visibly deform substrates with a compliance commensurate with their own. To understand the sensing of stiff substrates by growth cones, we investigated their precise temporal response to well-defined mechanical stress. When the applied stress exceeded a threshold of 274 ± 41 pN/μm², neurons retracted and re-extended their processes, thereby enabling exploration of alternative directions. A calcium influx through stretch-activated ion channels and the detachment of adhesion sites were prerequisites for this retraction. Our data illustrate how growing neurons may detect and avoid stiff substrates—as a mechanism involved in axonal branch pruning—and provide what we believe is novel support of the idea that mechanics may act as guidance cue for neuronal growth.     

ARHGAP18, a GTPase-activating protein for RhoA, controls cell shape, spreading, and motility

Rho GTPases are molecular switches that transmit biochemical signals in response to extracellular stimuli to elicit changes in the actin cytoskeleton. Rho GTPases cycle between an active, GTP-bound state and an inactive, GDP-bound state. These states are regulated by two distinct families of proteins-guanine nucleotide exchange factors and GTPase-activating proteins (GAPs). We studied the role of a previously uncharacterized GAP, ARHGAP18 (MacGAP). Overexpression of ARHGAP18 suppressed the activity of RhoA and disrupted stress fiber formation. Conversely, silencing of ARHGAP18 by small interfering RNA transfection-enhanced stress fiber formation and induced rounding of cells. We examined the role of ARHGAP18 in cell spreading and migration. Immunofluorescence analysis revealed that ARHGAP18 was localized to the leading edge during cell spreading and migration. ARHGAP18-knockdown cells showed impaired spreading, premature formation of stress fibers, and sustained activation of RhoA upon cell attachment. In addition, knockdown and overexpression of ARHGAP18 resulted in the inhibition and promotion of cell migration, respectively. Furthermore, ARHGAP18 was required for the polarization of cells for migration. Our results define ARHGAP18 as one of the crucial factors for the regulation of RhoA for the control of cell shape, spreading, and migration.     

An RGD spacing of 440 nm is sufficient for integrin alpha V beta 3- mediated fibroblast spreading and 140 nm for focal contact and stress fiber formation

The synthetic peptide Gly-Arg-Gly-Asp-Tyr (GRGDY), which contains the RGD sequence of several adhesion molecules, was covalently grafted to the surface of otherwise poorly adhesive glass substrates and was used to determine the minimal number of ligand-receptor interactions required for complete spreading of human foreskin fibroblasts. Well-defined adhesion substrates were prepared with GRGDY between 10(-3) fmol/cm2 and 10(4) fmol/cm2. As the adhesion ligand surface concentration was varied, several distinct morphologies of adherent cells were observed and categorized. The population of fully spread cells at 4 h reached a maximum at 1 fmol/cm2, with no further increases up to 10(4) fmol/cm2. Although maximal cell spreading was obtained at 1 fmol/cm2, focal contacts and stress fibers failed to form at RGD surface concentrations below 10 fmol/cm2. The minimal peptide spacings obtained in this work correspond to 440 nm for spreading and 140 nm for focal contact formation, and are much larger than those reported in previous studies with adsorbed adhesion proteins, adsorbed RGD-albumin conjugates, or peptide-grafted polyacrylamide gels. Vitronectin receptor antiserum specific for integrin alpha V beta 3 blocked cell adhesion and spreading on substrates containing 100 fmol/cm2 of surface-bound GRGDY, while fibronectin receptor antiserum specific for alpha 5 beta 1 did not. Furthermore, alpha V beta 3 was observed to cluster into focal contacts in spread cells, but alpha 5 beta 1 did not. It was thus concluded that a peptide-to-peptide spacing of 440 nm was required for alpha V beta 3-mediated cellular spreading, while 140 nm was required for alpha V beta 3-mediated focal contact formation and normal stress fiber organization in human foreskin fibroblasts; these spacings represent much fewer ligands than were previously thought to be required.     

The cytoplasmic domain of the alpha1 integrin subunit influences stress fiber formation via the conserved GFFKR motif

Integrins are heterodimeric transmembrane proteins that mediate substrate adhesion and migration but also the bidirectional transfer of information across the plasma membrane via their cytoplasmic domains. We addressed the question of whether the very short cytoplasmic tail of the alpha1 integrin subunit of alpha1beta1 integrin is required for alpha1beta1-specific adhesion, spreading, and migration. For this purpose we transfected the alpha1 integrin subunit and two cytoplasmically truncated alpha1 subunits into Chinese hamster ovary (CHO) cells. Elimination of the entire cytoplasmic domain of the alpha1 subunit does not affect adhesion but leads to inhibition of spreading and stress fiber formation. The defect in spreading could not be rescued by lysophosphatidic acid, which has been reported to stimulate actin stress fiber formation via Rho. Additionally, deletion of the entire cytoplasmic domain of the alpha1 subunit abolishes migration toward alpha1beta1-specific substrates. Migration and stress fiber formation are similar in CHO-alpha1 cells and CHO cells carrying an alpha1 subunit still containing the conserved GFFKR motif. So, the GFFKR motif of the alpha1 subunit is essential and sufficient for these processes.     

Sialylation of vitronectin regulates stress fiber formation and cell spreading of dermal fibroblasts via a heparin-binding site

Vitronectin (VN) plays an important role in tissue regeneration. We previously reported that VN from partial hepatectomized (PH) rats results in a decrease of sialylation of VN and de-sialylation of VN decreases the cell spreading of hepatic stellate cells. In this study, we analyzed the mechanism how sialylation of VN regulates the properties of mouse primary cultured dermal fibroblasts (MDF) and a dermal fibroblast cell line, Swiss 3T3 cells. At first, we confirmed that VN from PH rats or de-sialylated VN also decreased cell spreading in MDF and Swiss 3T3 cells. The de-sialylation suppressed stress fiber formation in Swiss 3T3 cells. Next, we analyzed the effect of the de-sialylation of VN on stress fiber formation in Swiss 3T3 cells. RGD peptide, an inhibitor for a cell binding site of VN, did not affect the cell attachment of Swiss 3T3 cells on untreated VN but significantly decreased it on de-sialylated VN, suggesting that the de-sialylation attenuates the binding activity of an RGD-independent binding site in VN. To analyze a candidate RGD-independent binding site, an inhibition experiment of stress fiber formation for a heparin binding site was performed. The addition of heparin and treatment of cells with heparinase decreased stress fiber formation in Swiss 3T3 cells. Furthermore, de-sialylation increased the binding activity of VN to heparin, as detected by surface plasmon resonance (SPR). These results demonstrate that sialylation of VN glycans regulates stress fiber formation and cell spreading of dermal fibroblast cells via a heparin binding site.     

Collagen type V modulates fibroblast behavior dependent on substrate stiffness

Collagen type V is highly expressed during tissue development and wound repair, but its exact function remains unclear. Cell binding to collagen V affects various basic cell functions and increased collagen V levels alter the structural organization and the stiffness of the ECM. We studied the combined effects of collagen V and substrate stiffness on the morphology, focal adhesion formation, and actin organization of fibroblasts. We found that a hybrid collagen I/V coating impairs fibroblast spreading on soft substrates (<10 kPa), but not on stiffer substrates (68 kPa or glass). In sharp contrast, a pure collagen I coating does not impair cell spreading on soft substrates. The impairment of cell spreading by collagen V is accompanied by diffuse actin staining patterns and small focal adhesions. These observations suggest that collagen V plays an essential role in modifying cell behavior during development and remodeling, when very soft tissues are present.     

Heparin II domain of fibronectin uses alpha4beta1 integrin to control focal adhesion and stress fiber formation, independent of syndecan-4

Co-signaling events between integrins and cell surface proteoglycans play a critical role in the organization of the cytoskeleton and adhesion forces of cells. These processes, which appear to be responsible for maintaining intraocular pressure in the human eye, involve a novel cooperative co-signaling pathway between alpha5beta1 and alpha4beta1 integrins and are independent of heparan sulfate proteoglycans. Human trabecular meshwork cells isolated from the eye were plated on type III 7-10 repeats of fibronectin (alpha5beta1 ligand) in the absence or presence of the heparin (Hep) II domain of fibronectin. In the absence of the Hep II domain, cells had a bipolar morphology with few focal adhesions and stress fibers. The addition of the Hep II domain increased cell spreading and the numbers of focal adhesions and stress fibers. Cell spreading and stress fiber formation were not mediated by heparan sulfate proteoglycans because treatment with chlorate, heparinase, or soluble heparin did not prevent Hep II domain-mediated cell spreading. Cell spreading and stress fiber formation were mediated by alpha4beta1 integrin because soluble anti-alpha4 integrin antibodies inhibited Hep II domain-mediated cell spreading and soluble vascular cell adhesion molecule-1 (alpha4beta1 ligand)-induced cell spreading. This is the first demonstration of the Hep II domain mediating cell spreading and stress fiber formation through alpha4beta1 integrin. This novel pathway demonstrates a cooperative, rather than antagonistic, role between alpha5beta1 and alpha4beta1 integrins and suggests that interactions between the Hep II domain and alpha4beta1 integrin could modulate the strength of cytoskeleton-mediated processes in the trabecular meshwork of the human eye.     

Cyclic Tensile Strain Controls Cell Shape and Directs Actin Stress Fiber Formation and Focal Adhesion Alignment in Spreading Cells

The actin cytoskeleton plays a crucial role for the spreading of cells, but is also a key element for the structural integrity and internal tension in cells. In fact, adhesive cells and their actin stress fiber–adhesion system show a remarkable reorganization and adaptation when subjected to external mechanical forces. Less is known about how mechanical forces alter the spreading of cells and the development of the actin–cell-matrix adhesion apparatus. We investigated these processes in fibroblasts, exposed to uniaxial cyclic tensile strain (CTS) and demonstrate that initial cell spreading is stretch-independent while it is directed by the mechanical signals in a later phase. The total temporal spreading characteristic was not changed and cell protrusions are initially formed uniformly around the cells. Analyzing the actin network, we observed that during the first phase the cells developed a circumferential arc-like actin network, not affected by the CTS. In the following orientation phase the cells elongated perpendicular to the stretch direction. This occurred simultaneously with the de novo formation of perpendicular mainly ventral actin stress fibers and concurrent realignment of cell-matrix adhesions during their maturation. The stretch-induced perpendicular cell elongation is microtubule-independent but myosin II-dependent. In summary, a CTS-induced cell orientation of spreading cells correlates temporary with the development of the acto-myosin system as well as contact to the underlying substrate by cell-matrix adhesions.     

Mechanical stress regulates the mechanotransduction and metabolism of cardiac fibroblasts in fibrotic cardiac diseases

Fibrotic cardiac diseases are characterized by myocardial fibrosis that results in maladaptive cardiac remodeling. Cardiac fibroblasts (CFs) are the main cell type responsible for fibrosis. In response to stress or injury, intrinsic CFs develop into myofibroblasts and produce excess extracellular matrix (ECM) proteins. Myofibroblasts are mechanosensitive cells that can detect changes in tissue stiffness and respond accordingly. Previous studies have revealed that some mechanical stimuli control fibroblast behaviors, including ECM formation, cell migration, and other phenotypic traits. Further, metabolic alteration is reported to regulate fibrotic signaling cascades, such as the transforming growth factor-β pathway and ECM deposition. However, the relationship between metabolic changes and mechanical stress during fibroblast-to-myofibroblast transition remains unclear. This review aims to elaborate on the crosstalk between mechanical stress and metabolic changes during the pathological transition of cardiac fibroblasts.     

Universal Kinetics of the Onset of Cell Spreading on Substrates of Different Stiffness

When plated onto substrates, cell morphology and even stem-cell differentiation are influenced by the stiffness of their environment. Stiffer substrates give strongly spread (eventually polarized) cells with strong focal adhesions and stress fibers; very soft substrates give a less developed cytoskeleton and much lower cell spreading. The kinetics of this process of cell spreading is studied extensively, and important universal relationships are established on how the cell area grows with time. Here, we study the population dynamics of spreading cells, investigating the characteristic processes involved in the cell response to the substrate. We show that unlike the individual cell morphology, this population dynamics does not depend on the substrate stiffness. Instead, a strong activation temperature dependence is observed. Different cell lines on different substrates all have long-time statistics controlled by the thermal activation over a single energy barrier ΔG ≈ 18 kcal/mol, whereas the early-time kinetics follows a power law ～t⁵. This implies that the rate of spreading depends on an internal process of adhesion complex assembly and activation; the operational complex must have five component proteins, and the last process in the sequence (which we believe is the activation of focal adhesion kinase) is controlled by the binding energy ΔG.