An Active Biomechanical Model of Cell Adhesion Actuated by Intracellular Tensioning-Taxis

Abstract

Cell adhesion to the extracellular matrix (ECM) is highly active and plays a crucial role in various physiological functions. The active response of cells to physicochemical cues has been universally discovered in multiple microenvironments. However, the mechanisms to rule these active behaviors of cells are still poorly understood. Here, we establish an active model to probe the biomechanical mechanisms governing cell adhesion. The framework of cells is modeled as a tensional integrity that is maintained by cytoskeletons and extracellular matrices. Active movement of the cell model is self-driven by its intrinsic tendency to intracellular tensioning, defined as tensioning-taxis in this study. Tensioning-taxis is quantified as driving potential to actuate cell adhesion, and the traction forces are solved by our proposed numerical method of local free energy adaptation. The modeling results account for the active adhesion of cells with dynamic protruding of leading edge and power-law development of mechanical properties. Furthermore, the morphogenesis of cells evolves actively depending on actin filaments alignments by a predicted mechanism of scaling and directing traction forces. The proposed model provides a quantitative way to investigate the active mechanisms of cell adhesion and holds the potential to guide studies of more complex adhesion and motion of cells coupled with multiple external cues.

Significance

Adhesion of cells to the extracellular matrix is highly active and influences many physiological functions. However, the driving mechanisms in cell adhesion are still poorly understood. A biomechanical model is presented here to mimic the active behaviors and probe the inherent mechanisms of cell adhesion. Previous study indicates that mechanical stability is a prerequisite for cell survival and functioning. The preference of cells to tensioning (tensioning-taxis) is thus employed as the driving potential in active adhesion of cells. The modeled adhesion of cells exhibits characteristic spread dynamics and power-law dependent mechanics, which is in good agreement with experimental observations. This cell model provides a quantitative tool for further investigation of active adhesion and motion of cells coupled with multiple external cues.

Introduction

Adhesion of cells to the extracellular matrix (ECM) is highly active and influences many physiological phenomena such as wound healing, embryonic morphogenesis, and tumor metastasis (1). Cells continuously sense and actively adhere on ECM mediated by molecular events of globular actin polymerization, myosin II contraction, and integrin activation. Briefly, protruding and branching of polymerized actins initiate cell adhesion by pushing against the cell membrane at leading edge. The new territories of membrane in contact with ECM are anchored by recruiting integrin, focal adhesion (FA) kinase, paxillin, and talin, etc. Then, integrin complexes are reinforced with conformation changes under the contraction of myosin II, and finally matured into FAs (2). This treadmill of molecular movements consequently drives cells to spread, adhere, and migrate on ECM. The active response of cells to external cues using the conservative molecular events have been thoroughly documented; however, it is still largely unknown about the intrinsic mechanisms ruling the active adhesion of living cells.

Growing experiments evidence that cells on ECM maintain mechanical stability by establishing an architecture of tensional integrity (denominated as tensegrity) (3) in which the cellular framework is sustained by prestressed strings (e.g., actin filaments (AFs) and intermediate filaments (IFs)) and rigid struts (e.g., microtubules (MTs)) (4). In adherent cells, considerable tension is transmitted to ECM through FAs and cytoskeletons (CSKs) (5). On the rigid surface, most cells such as endothelial cells (6), muscle cells (7), fibroblasts, and epithelial cells (8) tend to develop a spread and functional architecture by forming large FAs, whereas on the soft surface FAs of cells are relatively small and cells retain a limited-spread morphology. Cellular function of rapid signal transduction was also found to rely on the prestressed CSKs (9). In particular, cells were observed to preferentially migrate from soft to stiff substrates (10) where they could form a spread, tensed, and functional architecture analogous to that in vivo. In the dynamic spread of cells, the fish epithelial keratocytes were identified to locate leading edge at the regions with a larger membrane tension (11). In addition, the protruding of leading edge in a nematode sperm cell was greatly impaired by lowering the local membrane tension (12). Furthermore, molecular recruitment of the nascent adhesions as well as the subsequent FAs formation at the leading edge was promoted by the increased membrane tension (13). All these observations suggest that survival and functioning of cells are built on the mechanical integrity in which cells hold the preference to coordinate the local movement toward increasing cellular tension. Thus, the tendency of cells to intracellular tensioning, defined as tensioning-taxis in this study, is a potential rule to guide the active adhesion of cells.

Theoretical modeling of cells greatly advances our knowledge on biophysical mechanisms of cell adhesion. The spatiotemporal morphogenesis of cell adhesion on the flat surface (14) and in the three-dimensional ECM (15,16) was well presented using modeling methods. Moreover, the function of cell division influenced by adhesion space was successfully predicted by a delicate biochemomechanical model (17). Most of the current models actually discover the adhesive response of cells to external cues, for which cells were routinely dealt with passive material. The active mechanism to govern cell adhesion is absent in most cases; instead, analogous active forces were introduced by setting additional constraints. For example, the traction forces were artificially directed to be along the extending CSKs (14), normal to the constraint surface (17), or parallel with the gradient of chemical (15) or rigidity (18) cues. Although the models obtained reasonable predictions of cell adhesion, additional constrains certainly introduced extra restrictions and underestimated the active performance of cells. Therefore, probing and quantifying the active mechanisms are crucial in deciphering the intrinsic behaviors of cell adhesion.

In this study, a biomechanical model is established to probe the active mechanisms of cell adhesion. Framework of the model is set up based on coarse-graining molecular dynamics and constituted with discrete particles, which are constrained by subcellular components, including CSKs, cell membrane, nucleus, and cytoplasm. The contact dynamics between the cell membrane and substrate is defined using the stochastic ligand-receptor bonding kinetics. Active development of cells is modeled by quantifying intracellular tensioning-taxis of cells. Free energy stored in subcellular components is employed to evaluate the degree of tensioning. To promote the tensioning in adhesion, cells coordinate active forces to increase local free energy (LFE) of protrusions at the leading edge. A function of LFE adaptation is formulated and solved numerically to obtain the traction vectors by maximizing LFE in each adhesion step. Under the active positioning of leading particles (LPs) and generation of traction forces, the model efficiently orientates and adapts adherent framework into a tensional integrity. The modeling results capture characteristic behaviors of living cells spreading on homogeneous substrate. Self-driven filopodial and lamlipodial protrusions are observed at the leading edge in adhesion dynamics of the model. The typical two-phase adhesion of cells with initial rapid increase followed by steady plateau is quantified in force generation and spreading dynamics. Furthermore, AFs are predicted to promote cellular morphogenesis through an active mechanism of scaling and directing traction forces.

Methods

An active biomechanical model is established to probe the mechanisms of cell adhesion. The model framework is formulated by extending our previous work (19,20). It is composed by the main mechanical components of eukaryotic cells as shown in Fig. 1 A, including cell membrane, cortical AFs, long AFs, IFs, MTs, nucleus, and cytoplasm. The contact dynamics between cell membrane and substrate is mediated by FAs whose behavior is modeled using stochastic ligand-receptor bonding kinetics (16,21). The probability of ligand-receptor rupturing is determined by bond tension as shown in Fig. 1 B, and bonding probability of the ligand-receptor is calculated according to ligand density as shown in Fig. 1 C. The active force is introduced in this study by quantifying intracellular tensioning-taxis of cells. The magnitude of active force is generated by polymerization of actin monomers at leading edge (22) and calculated using elastic the Brownian ratchet model (23). The unit vector of active force is determined by our proposed method of LFE adaptation by which cells spread to a tensional integrity in the most efficient way as shown in Fig. 1 C. Using the above modeling setup, the cell model is actively evolved from initial suspended and spherical morphology to a spread and tensed architecture.

Figure 1.

Schematic of cell adhesion on substrate. (A) The framework of cells in contact with substrate is shown here. (B) Extending and rupturing dynamics of the ligand-receptor by contraction of stress fibers (AFs bundles) at the rear of cells is shown here. (C) Bonding of the ligand-receptor driven by actin polymerization at leading edge is shown here. It is noticed that the traction force is directed by the proposed formula of LFE adaptation.

Modeling of the active force

Active adhesion of the model is actuated by active force modeling at leading edge, including dynamic positioning of leading edge (composed by LPs) and self-driven generation of traction forces. Previous studies indicate that adhesion of cells is highly active on the interface between cell membrane and ECM surface. FAs, as anchors between cell membrane and ECM, are the mechanical base for retrograde flow and outward protruding of AFs (24). The spatial architectures of FAs are immobilized on ECM with a nanoscale core (∼40 nm) of integrin and functioned with multiple protein layers (50–60 nm) including focal adhesion kinase, paxillin, and talin, etc. (25). Then AFs are polymerized to push membrane protrusion (∼200 nm) out at the thin leading edge by anchoring on FAs (26). In this study, the leading edge is discretized into LPs, which are dynamically positioned in each adhesion step. First, the particles in contact with substrate are employed as FAs. Then LPs are sought out within a spherical space centered with FAs and in a radius of d_on doubling the mean distance of initial modeling particles. This search radius is a coarse-graining value relying on the model complexity. The twofold is a dimensionless scale-up value estimated by dividing membrane protrusion size (∼200 nm) with FAs size (∼100 nm). The mathematical function on positioning of LPs is achieved by introducing an index matrix LP. As shown in Eq. 1, LP is dynamically determined as 1 (LPs) when searched distance (d) of the particle is less than d_on, and 0 (not LPs) when d is larger than d_on.

$$L{P}_i=\{\begin{array}{l}1\left(d_i\le d_{on}\right)\\ 0\left(d_i\text{>}{d}_{on}\right)\end{array}.$$ (1)

Traction force on leading edge is the main drive to overcome physical barrier maintained by cell membrane and cortical AFs network, so as to promote cell adhesion. Traction force is composed by the magnitude and unit vector. Actin polymerization on leading edge was identified as the source of generating traction forces (27). Neglecting the complicate molecular signaling pathways, magnitude of the traction forces (F_p) is derived from elastic Brownian ratchet model (23) and has the form of Eq. 2, as previously modeled (17,23). Here, we introduce a parameter N, the number of AFs connected with LPs, into our discrete system to model the increase of traction magnitude contributed by AFs connections.

$$F_p=-N\frac{k_{B}T}{\delta }\ln \left(\frac{v_m+v_{-}}{v_{+}}\right),$$ (2)

where δ is the size of actin monomer (28). v₊ and v₋ are the rates of free polymerization and depolymerization (29), respectively. v_m is the growth rate of AFs scaled by multiplying a dimensionless coefficient (equal to step time interval Δt) with the instantaneous velocity of LPs.

The molecular mechanism of traction force generation is discovered as actin polymerization at leading edge, and magnitude of the traction force can be estimated in molecular level (elastic Brownian ratchet model is applied in this study). The direction of traction force is determined by systematical processing of intracellular signals and extracellular cues of live cells. The mechanism to direct traction force is still absent in the cellular level, though the assembly mechanisms of actin at leading edge have been continuously developing in molecular level. In this study, a driving potential of cell adhesion is proposed by quantifying intracellular tensioning-taxis of cells by which cells spread to a tensional integrity in the most efficient way. Total free energy stored in subcellular components is employed as driving potential to quantify the tensional integrity. Thus, cells efficiently coordinate traction forces to increase total free energy by maximizing LFE of each protrusion at leading edge. Mathematically, a function of LFE adaptation is formulated as the spatial gradient (G_s) of LFE derivative with respect to time as shown in Eq. 3. LFE (U_loc) is defined by the free energy of the local pyramidal protrusion that employs LPs as the vertex. Then, unit vector (r) is inversely derived from Eq. 3. Thus, unit vector of the traction force is determined from Eq. 4 by equalizing the norm of G_s to its maximum value. The explicit form of r was not obtained analytically, so a numerical solution is proposed to solve Eq. 4. Briefly, discrete particles are generated on the unit spherical surface centered with LPs. The LFE along r, which is directed from LPs to each spherical particle, is calculated and screened to obtain the unit vector (r_p). Finally, traction forces (F_p) with the calculated magnitude and unit vector are obtained by Eq. 5. The detailed parameters setting of traction force is presented in Table 1. In the column of parameter value, the description of “Dynamic” denotes that the instantaneous value is computed dynamically at each iteration step of model development.

$${\mathbf{G}}_s\left(\mathbf{r}\text{, }{U}_{loc}\text{, }t\right)=\nabla \frac{\partial U_{loc}}{\partial t},$$ (3)

$${\mathbf{r}}_p={\mathbf{G}}_s^{-1}\left(U_{loc}\text{, }t\right)\text{,}\Vert {\mathbf{G}}_s\left({\mathbf{r}}_p\text{,}{U}_{loc}\text{,}t\right)\Vert =\text{max}\Vert {\mathbf{G}}_s\left(\mathbf{r}\text{,}{U}_{loc}\text{,}t\right)\Vert ,$$ (4)

$${\mathbf{F}}_p=F_p{\mathbf{r}}_p,$$ (5)

where $\nabla$ is Hamiltonian operator.

Table 1.

Parameter Setting of the Traction Force Modeling

Definition	Value	Sources
Free polymerization rate (v+)	50 nm/s	(29)
Free depolymerization rate (v−)	2 nm/s	(77)
Size of actin monomer (δ)	2.75 nm	(28)
Number of AFs on LPs (N)	Dynamic	Current work
Growth rate of AFs (vm)	Dynamic	Current work
Discrete time step (Δt)	2 × 10−7 s	Current work

Framework setup of the model

Framework of the model is established based on the coarse-graining concept of molecular dynamics. The mass of whole cell is simplified as particles uniformly distributed on cell membrane and nuclear envelope. Interactions of the particles are constrained by tension of AFs and IFs, compression of MTs, bending of membrane, area and volume conservation of cell and nucleus, and repulsive interaction between cell membrane and nuclear envelop. The particles are defined in Cartesian coordinates {x_i}, i = 1…N_v, which are vertices in a triangulated network on the cell (or nucleus) surface. The vertices are connected by N_s edges and form N_t triangles. Although the chains maintained by the particles possess considerable configurational freedom, the approximately sixfold junctions of the chains show much more restriction to the motion of particles. Therefore, the chains behave like a low-temperature triangulated network of springs, and the characteristic energy scales of the model is in a low-temperature state as demonstrated by previous work (30,31). Consequently, it is assumed that free energy changes with the model evolution are dominated by the sum potential energy of each subcellular component, and the contribution of entropic changes is negligible. Helmholtz free energy (U_tot) of the coarse-graining system (30,32) is thus computed by Eq. 6.

$$U_{tot}\left(\left\{{\mathbf{x}}_i\right\}\right)=U_{AFs}+U_{IFs}+U_{POW}+U_{MTs}+U_{AB}+U_{Volum}+U_{Potential}.$$ (6)

Tension of AFs plays a crucial role in mechanical integrity of cells (33). In this model, AFs are categorized into long AFs (major candidates of stress fibers (34)) and short AFs (components of cortical CSKs (35)). The constitutive relationship of AFs is computed using the wormlike chain (WLC) model as described in Eq. 7. Although WLC model was fist derived to describe the passive extension of DNA (36), it is capable of capturing the active stiffening characteristics of AFs and regulatory proteins (e.g., myosin II) as observed in growth of living cells (37,38). IFs, which are radiated from nuclear envelop to cell membrane, present similar strain stiffening property (39) with that of AFs when cells bear large stretching strain (larger than 20%) (40). Thus, the constitutive relationship of IFs is modeled by using Eq. 7.

$$U_{AFs/IFs}=\sum _{i\in \mathrm{1...}AFs/IFsnumber}\frac{k_{B}T{L}_c}{L_p}\frac{3x_i^2-2x_i^3}{1-x_i},$$ (7)

where k_B is Boltzmann’s constant and T is the absolute room temperature. l, L_C, x, and L_P represent instantaneous length, contour length, extending ratio (l/L_C), and persistence length of the chain, respectively.

An in-plane power-law repulsion between linked particles is introduced, as shown in Eq. 8, to represent the elastic energy stored in membrane and compensate the compression exerted by WLC model (41). To simplify the initial equilibrium state, a power term with the same form of Eq. 8 is also introduced into long AFs modeling to compensate the initial attractive forces.

$$U_{POW}=\sum _{i\in 1\text{...}{N}_s}\frac{k_{POW}}{l_i},$$ (8)

where k_POW is power-law coefficient of the chain.

MTs were observed to present relatively straight morphology both in intracellular (42) and extracellular (43) environments. In addition, MTs showed periodic buckling morphology by contractile beating in myocardial cells (42), which suggests a potential role of MTs as strut in tensegrity model. In this study, MTs are dynamically generated between centriole (at one lateral end of the nucleus) and adherent particle with a set probability of 50% when LPs are transformed into FAs. MTs are modeled to carry compressive loads as isotropic and elastic cylinder with the experimentally identified dimension and elasticity according to Eq. 9.

$$U_{MTs}=\sum _{i\in 1\text{...}MTsnumber}\frac{EA}{2l_0}{\left(l_i-l_0\right)}^2,$$ (9)

where E, A, l₀, and l are Young’s modulus, cross-sectional area, initial, and instantaneous length of MTs, respectively.

Cell membrane and nuclear envelop are composed by lipid bilayers and resist extending and bending when cells are deformed. The resistance of membrane is modeled using linear constraint method as shown in Eq. 10.

$$U_{AB}=\sum _{i\in 1\text{...}{N}_t}\frac{k_a{\left(A_i-A_0\right)}^2{k}_{B}T}{2A_0{L}_0^2}+\sum _{j\in 1\text{...}{N}_s}{k}_b\left[1-\cos \left({\theta }_j-{\theta }_0\right)\right],$$ (10)

where k_a is the constraint coefficient of area conservation. A₀ and A are the initial and instantaneous area, respectively. L₀ is the average equilibrium length of triangular links. k_b is bending resistance of the membrane. θ₀ and θ are initial and instantaneous angles of the adjacent triangulated membrane surfaces, respectively.

The cytoplasm and nuclear inclusions are simplified as nearly incompressible fluids. The linear constraint method is applied to compute the cellular conservation of volume as described in Eq. 11.

$$U_{Volume}=\sum _{i\in 1\text{...}{N}_t}\frac{k_v{\left(V_i-V_0\right)}^2{k}_{B}T}{2V_0{L}_0^3},$$ (11)

where k_v is the volume constraint coefficient. V₀ and V are initial and instantaneous volume of the cell (or nucleus), respectively.

The interaction between cell membrane and nucleus is physically constrained by CSKs and cytoplasm. Here, a repulsive potential as shown in Eq. 12 is introduced to maintain the repulsive positioning of cell membrane and nucleus (44).

$$U_{Potential}=\sum _{i\in 1\text{...}{N}_v}\sum _{j\in 1\text{...}{N}_v}\{\begin{array}{ll}{k}_p\left[\frac{\pi }{2}{\delta }_{ij}-\tan \left(\frac{\pi }{2}{\delta }_{ij}\right)\right]& \left(-1\le {\delta }_{ij}<0\right)\\ 0& \left({\delta }_{ij}\ge 0\right)\end{array},$$ (12)

where k_p is the repulsive coefficient. δ is the ratio of changed distance between the i^th particle on cell membrane and j^th particle on nuclear envelope. The detailed parameter setting of cellular framework is presented in Table 2.

Table 2.

Parameter Setting of Cellular Framework

Definition	Value	Sources
Initial radius of cells	1 × 10−5 m	Current work
Initial radius of nucleus	3.33 × 10−6 m	Current work
Initial length of cortical AFs (L0)	1.51 × 10−6 m	Current work
Contour length of cortical AFs (Lc)	Triple of L0	(32)
Persistent length of cortical AFs (Lp)	1.8 × 10−10 m	Current work
Initial length of long AFs (L0)	9.46 × 10−6–2 × 10−5 m	Current work
Contour length of long AFs (Lc)	Triple of L0	(32)
Persistent length of long AFs (Lp)	1.8 × 10−10 m	Current work
Power-law coefficient of long AFs (kPOW)	3.07 × 10−21–6.88 × 10−22	Current work
Initial length of nuclear filaments (L0)	5.02 × 10−7 m	Current work
Contour length of nuclear filaments (Lc)	Triple of L0	(32)
Persistent length of nuclear filaments (Lp)	1.8 × 10−10 m	Current work
Initial length of IFs (L0)	6.67 × 10−6 m	Current work
Contour length of IFs (Lc)	Triple of L0	(32)
Persistent length of IFs	1.8 × 10−10 m	Current work
Power-law coefficient of long IFs (kPOW)	1.82 × 10−10	Current work
Initial length of MTs (l0)	Dynamic	Current work
Young’s modulus of MTs (E)	120 kPa	Current work
Cross section area of MTs (A)	1.90 × 10−16 m2	(43)
Initial angle of adjacent membrane surface (θ0)	0.107	Current work
Initial global area of cells (A0)	1.25 × 10−9 m2	Current work
Initial local area of cells (A0)	9.8 × 10−13 m2	Current work
Initial volume of cells (V0)	4.15 × 10−15 m3	Current work
Initial global area of nucleus (A0)	1.38 × 10−10 m2	Current work
Initial local area of nucleus (A0)	1.10 × 10−13 m2	Current work
Initial volume of nucleus (V0)	1.54 × 10−15 m3	Current work
Number of particles on cell membrane	638	Current work
Number of particles on nuclear envelop	638	Current work
Number of cortical AFs	1272	Current work
Number of filaments on nuclear envelop	1272	Current work
Number of long AFs	32–638	Current work
Number of IFs	638	Current work
Number of MTs	Dynamic	Current work
In-plane shear modulus of cell membrane	6.30 × 10−6 N/m	(32,78,79)
In-plane shear modulus of nucleus envelop	1.89 × 105 N/m	(80)
Viscosity of lipid bilayer viscosity (η)	5.1 × 10−2 N · s/m2.	Current work
Global area constraint of cells (ka)	6.17 × 10−5	Current work
Global area constraint of nucleus (ka)	1.85 × 10−4	Current work
Local area constraint of cells (ka)	1.26 × 10−5	Current work
Local area constraint of nucleus (ka)	3.78 × 10−5	Current work
Volume constraint of cells (kv)	836.18	Current work
Volume constraint of nucleus (kv)	7525.70	Current work
Bending rigidity of cell membrane (kb)	2.77 × 10−19 J	(32,81)
Bending rigidity of nuclear envelop (kb)	5.54 × 10−19 J	Current work
Power-law coefficient of cells (kPOW)	1.47 × 10−23	Current work
Power law coefficient of nucleus (kPOW)	1.63 × 10−24	Current work
Power law coefficient of nucleus (kPOW)	1.63 × 10−24	Current work
Repulsive coefficient (kp)	5 × 10−14 J	Current work
Boltzmann’s constant (kB)	1.38065 × 10−23 J/K	Current work
Absolute room temperature (T)	298.15 K	Current work

Contact dynamics of the model

Contact of the cell model with substrate is described as a stochastic process of ligand-receptor bonding kinetics by using Monte Carlo method (45). The probability of ligand-receptor bonding (P_b) in Δt is calculated according to Eqs. 13 and 14.

$$P_b=1-e^{-k_{on}\Delta t},$$ (13)

$$k_{on}=k_f{A}_b{C}_b,$$ (14)

where k_f is the forward reaction rate (16). A_b is the protruding area of the leading edge. C_b is the density of the receptors (e.g., integrin) on cell membrane (17), in which the number of ligands on substrate is considered to be large enough to saturate the receptors on cell membrane.

The rupturing probability of ligand-receptor (P_r) in Δt is calculated according to Eqs. 15 and 16.

$$P_r=1-e^{-k_{off}\Delta t},$$ (15)

$$k_{off}=k_0{e}^{F_T{x}_b/k_{B}T},$$ (16)

where k_off is the kinetic dissociation rate of ligand-receptor under tension of F_T. k₀ is the unstressed kinetic dissociation rate (16). x_b is the transition distance of bond (16).

The adhesion between cells and substrate, including bonding adhesion (F_b) and rupturing adhesion (F_r), is determined by Eqs. 17 and 18.

$${\mathbf{F}}_b=P_b{A}_b{C}_b{k}_{LR}\left(l-l_0\right)\mathbf{k},$$ (17)

$${\mathbf{F}}_r=P_r{k}_{LR}\left(l-l_0\right){\mathbf{r}}_{\mathbf{L}\mathbf{R}},$$ (18)

where k_LR is the spring constant of ligand-receptor (16). l is the instantaneous length of bound ligand-receptor. l₀ is the equilibrium length of bound ligand-receptor (25). k is the unit vector denoting that the adhesive force of bonding exerted on LPs by ligand-receptor is in the vertical direction. The detailed parameter setting of contact mechanics is presented in Table 3.

Table 3.

Parameter Setting of the Contact Dynamics

Definition	Value	Sources
Forward reaction rate (kf)	100 molecule−1s−1	Current work
Protruding area of leading edge (Ab)	Dynamic	Current work
Density of bound ligand-receptor (Cb)	1000 μm−2	(17)
Unstressed kinetic dissociation rate (k0)	1 s−1	(16,18)
Transition distance of bond (xb)	0.02 nm	(16,18)
Spring constant of ligand-receptor (kLR)	1 pN/nm	(16,18)
Equilibrium length of ligand-receptor (l0)	25 nm	(16,18)
Unit vector of ligand-receptor extending (rLR)	Dynamic	Current work

Evolution of the model

The cell model is evolved from initial spherical morphology to a spread and tensed architecture by the movement of particles on cell membrane and nuclear envelop. The force exerted on particles is categorized into conservative force (F_C) maintained by subcellular framework, dissipative force by membrane and fluid damping (F_D), traction force by actin polymerization (F_p), and adhesive force (F_b and F_r) provided by substrate.

To calculate the conservative force, total free energy of the model is obtained by the sum of each subcellular free energy as shown in Eq. 6, and F_C on the particle is derived by virtual work theory (44) according to Eq. 19.

$${\mathbf{F}}_C=-\frac{\partial U_{tot}}{\partial \mathbf{r}}.$$ (19)

The membrane of cells is composed by lipid bilayer and exhibits viscous properties (46). The dashpots between linked particles are integrated to model the viscosity of cell membrane and nuclear envelope. The dissipative force resulting from membrane viscosity is calculated by Eq. 20.

$${\mathbf{F}}_D=\eta \Delta \mathbf{v},$$ (20)

where η is the membrane-damping coefficient and set as 0.0508 N · s/m. Δv is the relative velocity between linked particles on cell membrane or nuclear envelope.

Eventually, the resultant force (F_tot) exerted on particles is obtained according to Eq. 21. It should be noticed that F_p and F_b are only exerted on the LPs, and F_r only exists on FAs.

$${\mathbf{F}}_{tot}={\mathbf{F}}_C+{\mathbf{F}}_D+{\mathbf{F}}_p+{\mathbf{F}}_b+{\mathbf{F}}_r.$$ (21)

The movement of particles is governed by momentum theorem and calculated according to Eqs. 22, 23, and 24. The simulation work is performed by using the home-made programs on MATLAB (The MathWorks, Natick, MA).

$${\mathbf{F}}_{tot}=m\frac{d\mathbf{v}}{dt},$$ (22)

$${\mathbf{v}}^{t+\Delta t}={\mathbf{v}}^t+\frac{d\mathbf{v}}{dt}\Delta t$$ (23)

$${\mathbf{r}}^{t+\Delta t}={\mathbf{r}}^t+{\mathbf{v}}^t\Delta t+\frac{1}{2}\frac{d\mathbf{v}}{dt}\Delta t^2$$ (24)

where m, v, and r are the mass, velocity vector, and position vector of the particles, respectively.

Results

Adhesion dynamics of the model

Cell adhesion to ECM is actively mediated by molecular treadmill of polymerized actin protruding and recruited FAs anchoring at leading edge (47). In this model, the active FAs formation and LPs search are introduced to decipher the adhesion dynamics of cells on homogeneous and rigid substrate. As shown in Fig. 2 A, spherical cell is initialized to make contact with substrate and generate initial FAs (highlighted as black dots in the inset of Fig. 2 A). Then, LPs are sought out around FAs using the method presented in Eq. 1, and traction forces (indicated by the blue arrows in Fig. 2) determined by Eqs. 2, 3, 4, and 5 are loaded into LPs to drive the movement of leading edge. Eventually, the model is gradually developed and actively spreads on substrate as shown in Fig. 2, B–F. It is noticed that the spread of the cell model on substrate is a self-driven process in which thed driving field is actively generated according to the proposed intracellular tensioning-taxis. The biological contact of FAs with substrate is modeled using stochastic ligand-receptor-bonding kinetics. The evolutional adhesion of cells is obtained by dynamic turnover of FAs on substrate, as observed in previous work (48). Typically, lamlipodial (shown in the inset of Fig. 2, B–D) and filopodial (shown in the inset of Fig. 2, E and F) protruding dynamics, consistent with previous observations in studies of cell adhesion (47), is identified at leading edge of the model.

Figure 2.

Dynamic adhesion of the cell model on homogeneous and rigid substrate. The morphology of cells is presented by triangulated cortical AFs (red lines) with subcellular inclusions and FAs (black dots). Long AFs are randomly generated with half of the particle amount on the cell membrane. The model is developed gradually from initial spherical (A) to intermediate spread (B)–(E), and finally, well-extended architecture (F). The inset of figures shows the spread detail of protrusions at leading edge with scalar arrows of traction force. The iteration steps of (A)–(F) are 1, 100, 500, 1000, 2000, and 50,000, respectively. The time interval of each step is 2.5 × 10⁻⁷ s. The axis is in micrometer scale.

To further quantify the dynamics of cell adhesion, evolutional mechanics of the cell model is presented in Fig. 3. The active force modeling is accomplished by dynamic positioning of the LPs and active generation of the traction force on LPs. As shown in Fig. 3 A, the number of positioned LPs (red dash line) shows typical two-phase trend with rapid increase at the beginning and plateau later. It suggests that cells initially expand and anchor on substrate by the rapid recruit of LPs (candidate of FAs) and develop into a relatively tensed and stable architecture with impaired turnover of FAs in late stage. Total traction force (blue solid line in Fig. 3 A) exerted on LPs also presents two-phase increase and couples with LPs positioning to drive the characteristic morphogenesis of cells (shown in Fig. 2). It is noticed that the maximum step of iteration is chosen as 50,000 to obtain an equilibrium state of the modeled system, whereas the system approaches to be equilibrated after ∼5000 steps. Therefore, the following spread dynamics, i.e., spread area and adhesive force shown in Fig. 3 B, is focused within 5000 iteration steps. Growing evidence indicates that mechanical integrity of cells is maintained by the tensioning of CSKs and anchoring of ECM (3,49). Here, the contribution of ECM to the model integrity is evaluated by calculating the adhesive force on substrate. As shown in Fig. 3 C, the adhesive force (blue line with squares) provided by the substrate is capable of balancing considerable tension (∼40 nN compared with ∼16 nN of traction force) and shows the characteristic two-phase increase trend during cell adhesion. The spread area (red line with circles) of cells is finally quantified in Fig. 3 B and follows comparable trend as observed previously in experiments (50,51). Furthermore, the magnitude of spread area is compared with experimental results, and the modeling value is in the reasonable range, as shown in Table 4.

Figure 3.

Quantitative characterization of cell adhesion. (A) Dynamics of LPs positioning (red dash line) and active generation of traction force (blue solid line) are shown here. (B) Dynamic spread area (red line with circles) and adhesive force (blue line with squares) on substrate are shown here.

Table 4.

Comparison of Spread Area between Experimental and Simulated Data

Type of Cells	Magnitude of Spread Area (μm2)
NIH 3T3 cells	500–2200 (50)
Bovine aortic endothelial cells	700–2300 (6)
	1000–6000 (82)
U373-MG human glioma cells	280–1500 (83)
Our model	∼1100

Mechanical changes of cells actuated by AFs density

It has been identified that mechanical properties of cells are largely determined by intracellular bundles of AFs such as stress fibers (52,53). Using the proposed biomechanical model, we first probe the active role of AFs density in cell adhesion. The density of long AFs is characterized by a dimensionless fraction value of the total particle amount on cell membrane. As shown in Fig. 4 A, the model with 5% long AFs presents limited-spread and round morphology. With increase of AFs density, cells exhibit more persistence in extended morphology, as shown in Fig. 4, B and C. Consequently, the cells are fully developed with enhanced mechanical strain of cortical AFs (Fig. 4, D–F) and cell membrane (Fig. 4, G–I). These results indicate that mechanical properties of cells are significantly enhanced by the dense AFs, which is consistent with the observed decrease of cellular mechanical strength by depolymerizing AFs (54,55). The quantitative height (red line with squares) and spread area (blue line with circles) of cells shown in Fig. 5 A further evidence that cell spread is greatly enhanced by dense AFs, which is consistent with the experimental observations (54). Mechanically, the strong positive correlation of morphological development with AFs density is determined by the active force generation. Magnitude of the traction force (red line with squares in Fig. 5 B) is actively increased with the density of AFs, and larger traction force drives cells to a more extended morphology with more mechanical support from the substrate (i.e., adhesive force shown as blue line with circles in Fig. 5 B). The results are consistent with the enhanced promotion of regional protrusion and directional motility of cells at the leading edge rich in AFs (22).

Figure 4.

Mechanical properties of cells determined by AFs density. The spread morphology of cells in (A)–(C), strain of cortical AFs in (D)–(F), and strain of cell membrane in (G)–(I) with different AFs densities are shown. The long AFs are randomly generated between two particles on the cell membrane. AFs density of the model is defined as a fraction of total particle amount on cell membrane and is as follows: 5% in (A), (D), and (G); 50% in (B), (E), and (H); and 100% in (C), (F), and (I). The cell body is encapsulated by cortical AFs (red lines), and the nucleus is encapsulated by nuclear envelop (blue lines). The long filamentous CSKs including AFs (red lines), IFs (blue lines), and MTs (green lines) are distributed in the space between cell membrane and nuclear envelope. The axis is in micrometer scale.

Figure 5.

Quantitative mechanics of the model determined by AFs density. (A) The changes of cell height and spread area with AFs density are shown here. (B) Traction force and adhesive force of the model with different AFs densities are shown here.

The AFs density-associated adhesion of cells is further quantified by the cortical strain and strain energy. As shown in Fig. 6 A, the strain energy shows an obviously increased trend with the increase of AF’s density with a reasonable magnitude consistent with previous studies (30,56,57). In addition, the results indicate that more elastic energy is stored in the cortical AFs compared with the cell membrane during the active adhesion of cells. Interestingly, the change of cortical strain versus AFs density in Fig. 6 B presents a characteristic two-phase trend with rapid increase at the beginning and plateau at the late state. The data depicts a biologically saturable response of cells to the input signals and defines an adaptive behavior of the cells to the input. Growing studies describe the adaptive response of cells in morphological processes by using the power-law functions (7,58,59). More importantly, theoretical studies based on continuum mechanics derived the analytical solution of time-dependent cell spread as the form of power-law (60,61). The cellular response to the environmental signals during spread was thus concluded to follow the power law universally (61). This function indicates an input-dependent response of cell spread, and the exponent value perhaps defines the affinitive level of cells to the input signals, where a large value may correspond to high affinity and rapid response. The fitted exponential value of 0.5 seems to be typical in characterizing the time-dependent spread of cells at the early stage (60,61). And 0.29–0.37 was obtained as the exponential value in the power-law fit of spread area versus substrate elasticity (7). Therefore, the relationship between strain and AFs density in Fig. 6 B is fitted using a power-law function. The obtained exponential value of ∼0.4 in our model may indicate a relatively large dependence of active adhesion of cells on the AFs density.

Figure 6.

Strain energy (A) and strain (B) of the cortical AFs and membrane area versus AFs density. The strain value of cortical AFs is presented as the instantaneous length of links (L) divided by initial length (L₀). The strain value of cell membrane is presented as the instantaneous area of triangulated membrane (S) divided by initial area (S₀). The AFs density is presented as concentration of AFs (C) divided by constant particle amount (C₀). The strains versus AFs density are fitted with power-law functions.

Morphogenesis of cells driven by AFs alignment

Morphogenesis is a crucial issue in the dynamic adhesion of cells and greatly influenced by AFs. Cells are able to perceive the extracellular environments and adapt their morphology via AFs reorganization, either in single-cell confined space (62) or cell-cell contact condition (63). In particular, the dense AFs bundles (i.e., stress fibers) were clearly observed in the extending direction of adherent cells (62,63). These stress fibers were routinely considered as passive building blocks to balance cellular tension and resist external disturbance (49). Using the developed model, we evaluate the active role of AFs in morphogenesis by varying alignment of the AFs. Three typical alignments of AFs, including random (Fig. 7 A), polar-concentrated (Fig. 7 B), and half-distributed (Fig. 7 C), are preset in the initial spherical framework. The characteristic morphogenesis is developed to be well-spread (osteocyte-like) morphology shown in Fig. 7 D, polarized (neuron-like) morphology shown in Fig. 7 E, and half-polarized (keratocyte-like) morphology (22) shown in Fig. 7 F. The distinct morphogenesis suggests that the preexistence (or considered as polymerizing potential along the alignment) of AFs may play a significant role in cell adhesion. Directional protruding at leading edge is greatly biased by the traction vector of AFs polymerization and the spread of cells is orientated to the direction along AFs. Consequently, the model is fully spread in the direction of AFs alignment and the extended AFs are spontaneously developed as stress fibers to support cellular tension. Particularly, the strain maps of AFs as shown in Fig. 7, G–I evidence that the AFs are extremely extended in AF’s concentrated direction and bundled as stress fibers in cell adhesion.

Figure 7.

Morphogenesis of the model determined by AFs alignment. Schematic alignment of the preset AFs including random (A), polar-concentrated (B), and half-distributed (C) alignments. The spread morphology in (D)–(F) and AFs the strain in (G)–(I) with varied alignments of AFs. AFs density is set as half of the particle amount on cell membrane in the models. The axis is in micrometer scale.

It is noted that the initial shape of cells in the above study is modeled as sphere according to the commonly observed spherical morphology of cells in a suspended state. However, this isotropic shape may underestimate the inherent potential of cellular polarization maintained by the initial anisotropy of membrane morphology. Here, we discuss this anisotropic effect on the adhesion of cells by initializing the cells with an elliptical shape. As shown in Fig. 8, A and E, the morphogenesis of cells without AFs is greatly biased by the initial morphology to spread along the long axis of ellipse, in which the traction forces are more polarized in the direction of long axis. When randomly distributed AFs are introduced, the polarized effect of anisotropic morphology is significantly impaired, and the cell spreads without distinct polarization as shown in Fig. 8, B and F. Moreover, the morphogenesis polarization of cells is restored when introducing aligned AFs and the fully spread of cells is observed along the orientation of AFs alignments shown in Fig. 8, C, D, G, and H. Thus, we conclude that the initial anisotropic morphology of cells may contribute to the polarized spread of cells; however, the effect is relatively low and negligible when dense AFs exist.

Figure 8.

Morphogenesis of the cell models biased by the anisotropy of initial morphology. The initial state of elliptical cells without AFs (A), with the randomly distributed AFs (B), with the long-axis aligned AFs (C), and with the short-axis aligned AFs (D). The corresponding morphologies of cells in (E)–(H) spread under the initial state of (A)–(D), respectively. Schematic representation of the initial cell state is shown in the inset. The AFs density is set as half of the particle amount on cell membrane in the models. The axis is in micrometer scale.

Adhesion of cells influenced by ECM stiffness

The stiffness of ECM is one of the most significant environmental signals to influence the active behaviors of cells such as adhesion and migration (64). The rigidity sensing of cells often actively involves the periodic formation and breakdown of standard sensory modules such as the binding of receptor-ligands, recruitment of adaptor proteins, and reinforcement of FAs (65). By neglecting the complicated details of molecular modules, it is widely accepted that cells on stiffer ECM tend to generate larger traction forces and develop more spread morphology. To explore the active adhesion of cells on elastic ECM, we introduce an engineering coefficient of traction scale (T_scale), which is derived from the modeling of cellular traction forces in steady state (66). T_scale defines the stiffness-dependent changes of traction forces by multiplying into the magnitude of traction forces (F_p). Here, T_scale = k_scale/(1 + k_scale) and k_scale = k_sub/k_FA. And k_sub is a variable to represent the stiffness of ECM. k_FA is a characteristic constant of the modeled system to depict the stiffness of FAs. The value of k_FA in this model is ∼40 nN/μm, which is estimated by the product of area of FAs, density of bound ligand-receptor, and spring constant of single ligand-receptor.

The change of traction scale (i.e., the traction force) versus ECM stiffness presents the characteristic two-phase increase trend with the fist rapid increase and followed plateau (the red dash line in Fig. 9). The spread area of the cell model is then acquired to quantify the influence of ECM stiffness on cell adhesion. As shown in the blue squares of Fig. 9, the increase of spread area versus ECM stiffness qualitatively follows the similar hyperbolic two-phase trend as the traction scale. However, the power-law fit (the blue solid line in Fig. 9) describes a more precisely qualitative characteristic of spread area especially for the adhesion of cells on soft ECM, and this result is consistent with the previous experimental work of cellular spread on elastic substrate (7).

Figure 9.

Phase diagram of the effect of ECM stiffness on cell adhesion. The relationship between T_scale and k_scale is presented as the red dash line. The spread area of cell model under different ECM stiffness is acquired and plotted as the blue squares. The inset figures show the spreading morphology of cells under specific ECM stiffness. The value of spread area is fitted with power-law function and presented as the blue solid line.

Discussion

Cells in contact with substrate actively perceive surrounding environments and adjust the adhesion and motion in adaptive manners (67). Adhesion of cells to ECM is actuated by multiple cues such as chemical gradient (68) and substrate compliance (7). Considerable models were thus developed by focusing on the identified external cues and provided valuable predictions on chemotaxis (34), haptotaxis (15), and durotaxis (18). Most models routinely employed one or multiple external cues to determine the active force (e.g., traction force) in cell adhesion. However, the intrinsic motivation of living cells, such as adaptive reorganization of intracellular components, was usually underestimated in the modeling of cell adhesion. In this study, we expanded the concept of cellular tensegrity and further quantified it as a driving potential (i.e., tensioning-taxis) of cell adhesion. An active biomechanical model actuated by the intracellular tensioning-taxis is established to probe the active mechanisms in cell adhesion. Our model provides insight into self-driven adhesion of cells on 2D surface, and may be used as a starting point for more complex adhesion and motion of cells coupled with multiple external cues.

The activity of our model arises from modeling the intracellular tensioning-taxis as driving potential in cell adhesion. As observed in most adherent cells, the tensioning structuration is prerequisite for cells to resist external disturbance and enable biological functions (3). Here, we use free energy to quantify tensioning of the modeled system. Tensioning-tax is modeled as functions of the LFE adaptation (Eqs. 3, 4, and 5) which enable active generation of the traction field. The cell model is thus fully tensed in each cycle of protruding, and the whole cell system is actively developed into an architecture of tensegrity. This method is particularly powerful because it allows us to incorporate the self-driven mechanism into a purely mechanical model. The biologically derived models of WLC, elastic Brownian ratchet, and stochastic ligand-receptor bonding kinetics are integrated into this model. A mechanical framework with the constitutive relation of CSKs is thus defined to adhere on substrate with biological contact dynamics. Taken together, the developed model presents the first, to the best of our knowledge, active adhesion of cells in which living cells are modeled to actively adapt on substrate under the driving potential of intracellular tensioning-taxis.

By incorporating the essential biological processes as mentioned above, this model is actively developed with characteristic behaviors of cell adhesion. The sequential process of integrin bonding, FAs maturing and actin protruding has long been characterized as adaptive activities of cells on substrate (69). Using the proposed method of LPs positioning (shown in Eq. 1), this molecular treadmilling behaviors are successfully captured in the modeling of adhesion dynamics. And the quantified adhesion dynamics of traction force and spread area shows typical two-phase increase and is well consistent with experimental results on cell adhesion. AF-dependent mechanics are universally identified in studies of adherent cells (70,71); however, the active mechanism behind this passive mechanical changes is not fully elucidated. Using this model, we propose that AFs may actively drive cell adhesion by modulating the magnitude and direction of traction force. In particular, the predicted stiffening of cells depending on AFs is observed to follow the universal power-law behaviors in cell adhesion and morphogenesis (61). The model developed here is capable of predicting both qualitative and quantitative behaviors of cell adhesion.

Thermal fluctuation is crucial to influence the mechanical and functional properties of the plasma membrane, especially in the flexible red blood cells. The thermal-induced membrane deformation of red blood cells was observed as tens of nanometers both in experiments (72) and simulations (73). In adherent cells, the membrane undulations actuated by actin polymerization was experimentally measured in the micrometer scale (74), which is dominant in the membrane deformation compared with the relatively smaller thermal-induced nanoscale amplitudes (75). In this model, the plasma membrane is physically constrained by the dense cortical CSKs, intracellular CSKs, and tight contact with ECM. The thermal-associated fluctuation of membrane is severely suppressed, and thus neglected to simplify the modeling process as commonly performed in previous coarse-graining models (30,32). Molecularly, the membrane fluctuation in nanoscale may significantly influence the binding and clustering dynamics of the integrin-ligand bonds (76). Our future attention will be focused on improving the theoretical models to incorporate the membrane fluctuation and promote the understanding of active cell adhesion under thermal fluctuation.

Conclusions

In this study, we establish a biomechanical model actuated by intracellular tensioning-taxis to probe the active mechanisms of cell adhesion. The methods of LPs positioning and traction force generating are proposed to model the self-driven mechanism in cell adhesion. By integrating models of WLC, elastic Brownian ratchet, and stochastic ligand-receptor bonding kinetics, an active framework with biological characteristics of living cells is established. The modeling adhesion of cells on substrate exhibits active spread dynamics and power-law-dependent mechanics, which is in good agreement with experimental observations. Furthermore, the model predicts that AFs may actuate morphogenesis of cells by scaling and directing the traction force at leading edge. This study provides a quantitative biomechanical model to investigate the active adhesion of cells and holds the potential for studies on more complex adhesion and motion of cells coupled with multiple external cues.

Author Contributions

Y.F. performed the simulation and analyzed the data. Y.F., H.G., R.Y., K.W.C.L., and M.Q. contributed the ideas and designed the simulation. All authors discussed the results and reviewed the manuscript.

Editor: Guy Genin.