Adherent cells undergo rate-softening mediated by actomyosin kinetics

Abstract

Emerging studies suggest that a wide range of chronic diseases can be linked to prior physical trauma and, in some cases, to the supraphysiological deformation rates experienced by cells during injury. However, the mechanical behavior of cells during these deformations is poorly understood. Here, we studied the strain rate dependent mechanics of vascular smooth muscle cells over rates spanning five orders of magnitude, from physiological to supraphysiological. We find that cells deformed at increasing rates undergo substantial rate-softening in tension but have no rate-dependence when returned to zero strain. This reversible rate-softening is mediated by actin-myosin binding kinetics. Further, we find that at supraphysiological strain rates, cells experience actin-myosin binding mediated disruption of contractile force and alteration of gene expression. Our results suggest a mechanism by which cells shield themselves from excessive forces through cytoskeletal relaxation that loses efficacy at high strain rates like those experienced during mechanical trauma.

Introduction

Major physical trauma affects millions every year. The immediate impact of broken bones and damaged organs results in 25 million emergency department visits annually (1, 2). The longer term effects of trauma can linger for years (3, 4) and are not always immediately apparent. For example, neurodegenerative diseases like chronic traumatic encephalopathy (CTE) are directly linked to mild traumatic brain injuries (TBIs) that do not necessarily involve injury necessitating acute treatment (5). There are many other chronic diseases that are correlated with trauma whose links to injury are less well understood. For example, years after serious orthopedic injuries, patients have an increased likelihood of developing cancer, chronic heart failure, type two diabetes, and stroke, compared to the general population (6). Similarly, patients with mild and moderate TBIs have increased incidence of vascular inflammation, hypertension, coronary artery disease, and diabetes (7, 8). The mechanisms linking physical trauma to these seemingly unrelated diseases are not known. Notably, recent results have shown that exposure to a single high strain rate deformation in neurons is sufficient to initiate tauopathy associated with CTE (9), suggesting the possibility that other trauma-related chronic disorders might also be directly linked to mechanical forces present during injury. To date, however, the mechanobiology of cells exposed to high strain rate deformation, like those experienced during trauma, are not well-defined enough to determine such a link.

Fundamentally, cells are polymeric networks, with the actin-myosin stress fibers acting as the primary, but not only, structural polymers. Cells are typically mechanically characterized as elastic (10), hyperelastic (11), or viscoelastic (12). However, following high strain rate deformations, cells have been observed to fluidize, recovering their initial properties over minutes-to-hours post-stretch (13, 14). Additionally, unlike typical polymers, cells can acutely modify their mechanical state through actin-myosin contraction and relaxation or active remodeling of the cytoskeletal network. These mechano-adaptive processes, which can act to optimize stress distribution throughout the cell and shield important organelles from mechanical damage are influenced by deformation rates, as exemplified by frequency of dependence in cyclically stretched cells (15, 16) and tissues (17–19), observed both experimentally and theoretically. Together, these findings suggest that cell mechanical properties during and after deformation are dependent on the applied strain rate but are not consistent with the typical strain rate dependence of viscoelastic materials. Thus, we sought to investigate how the mechanical behavior of cells is influenced by high strain rate deformations, like those that cells are exposed to during trauma, and asked how high strain-rate dependent mechanics influence cell function following injury.

Materials & Methods

Substrate prep.

A silicone elastomer membrane was clamped between custom metal brackets and held under slight tension. A fluorescent bead-doped polyacrylamide (PA) gel of 13.5 kPa Young’s modulus was bonded to the silicone elastomeric membrane as previously described (11, 20). The PA gel was micropatterned with fibronectin rectangles with aspect ratio 1:4 (AR4) (127 μm × 32 μm) for cell adhesion (Fig. 1b).

Figure 1: Human vascular smooth muscle cells rate soften at both physiological and supraphysiological strain rates.

a, Micropatterned vascular smooth muscle cell on deformable membrane. b, Microscope-mounted high-speed CμBS device for performing traction force microscopy on deformed cells. c, Cells are stretched at strain rates of 0.001 s⁻¹ (slow, black), 0.005 s⁻¹ (medium, purple), 0.05 s⁻¹ (fast, red), or 10 s⁻¹ (trauma, yellow) and held at 0.15 strain. 5 minutes after initiation of stretch, cells are returned to 0 strain at the same rate and held static. This cycle is repeated 3 times. Inset: First 0.5 s of deformation to visualize trauma strain rate. d, Brightfield image (top), underlying fluorescent bead layer image (2^nd from top), and traction stress maps (bottom 5) at 5 timepoints for a representative cell stretched at the slow strain rate. Scale bar, 20 μm. e, Average normalized axial PK1 stress ($P_x$) vs time for cells stretched according to the protocol in c. Stress decreases and cells soften with increasing strain rate, and stress drops below basal stress in the trauma rate. n = 16, 14, 20, 15 cells, respectively. f, Normalized peak (immediately after completion of stretch) and decay (5 minutes after initiation of stretch) stress for first deformation cycle. Solid bars: mean, open circles: individual measurement. *p < 0.05, **p < 0.01, ***p < 0.001. Error bars represent standard error of the mean (s.e.m.).

Cell culture & reagents.

Human umbilical artery smooth muscle cells (HUASMCs) (Lonza, Walkersville, MD) were obtained at passage 3 and cultured in growth media containing Medium 199 (Gibco, Billings, MT) supplemented with 10% fetal bovine serum (Gibco), 10 mM HEPES (Gibco), 3.5 g/L glucose (Avantor, Radnor Township, PA, USA), 2 mg/L vitamin B12 (Sigma-Aldrich, St. Louis, MO), 2 mM penicillin-streptomycin (Sigma-Aldrich), 10 mM MEM nonessential amino acids (Gibco), and 2 mM L-glutamine (Life Technologies, Waltham, MA). Cells between passages 5 and 7 were used for experiments and were seeded onto prepared micropatterned substrates at a density of 25 cells/mm². Cells were allowed to adhere to the substrates for 24 hours in an incubator and were then serum-starved for 24 hours to induce a contractile phenotype prior to performing experiments (21). Experiments were performed at 37 °C and 5% CO₂ to preserve cell viability.

High speed cellular microbiaxial stretching (CμBS).

The high-speed cellular microbiaxial stretching (CμBS) apparatus is a custom-built, microscope-mounted device used to apply controlled strain to micropatterned cells while simultaneously imaging them (Fig. 1b). The CμBS device consists of two independently controlled voice coil actuators and two static fixtures for fixing deformation in the direction transverse to stretch. Experimental protocols were performed such that cells were strained along their long axis in the direction of voice coil actuator motion. HUASMCs were deformed from a starting stretch ratio $\lambda$ of 1 by applying a stretch of $\lambda =1.15$ to the elastomer membrane. Applied and measured linear strains $ϵ$ were calculated as $ϵ=\lambda -1$. Strains were applied over 2 min, 30 s, 3 s, or 0.015 s, corresponding to strain rates ($\dot{ϵ}$) of 0.001 s⁻¹, 0.005 s⁻¹, 0.05 s⁻¹, and 10 s⁻¹, respectively.

HUASMCs were stretched to 0.15 strain (stretching phase) at a rate of 0.001 s⁻¹, 0.01 s⁻¹, 0.1 s⁻¹, or 10 s⁻¹, then held static at 0.15 strain (relaxation phase). In all cases, the total stretching phase and relaxation phase always equaled 5 min. For example, for the 0.001 s⁻¹ strain rate, the stretching phase took 2 min, and the relaxation phase took 3 min. Then, HUASMCs were returned to a strain of 0 (return phase) at the same strain rate and again held static at a strain of 0 (recovery phase). Similarly, the return phase and recovery phase always totaled 5 minutes (Fig. 1c). This pattern of stretching constituted one cycle. Three cycles of stretching were performed sequentially during each trial. Two categories of straining protocols were used. In the first set of experiments, all three cycles of stretching were performed at the same rate (i.e. three stretches at 0.001 s⁻¹, slow-slow-slow sequence). In the second set of experiments, the first and third cycles of stretch were performed at 0.001 s⁻¹, and the second at 0.05 s⁻¹ (slow-fast-slow sequence) or 10 s⁻¹ (slow-trauma-slow sequence).

Imaging.

The same imaging protocol was followed for both stretching protocols. Before any stretch was applied, cells were imaged once under bright-field microscopy, and the fluorescent bead layer at the cell-gel interface was imaged once. During each relaxation and recovery phase after deformation, cells were imaged once under brightfield microscopy, and then the fluorescent bead layer at the cell-gel interface was imaged once per second for the duration of the relaxation or recovery phase (Fig. 1d). After the completion of the experiment, cells were then lysed using a 0.1% sodium dodecyl sulfate (Sigma-Aldrich) solution. Cell-free images of the undeformed fluorescent bead layer were then acquired at strains of 0 and 0.15 to determine bead displacement.

Cell stress analysis.

The cell-induced bead displacement was calculated using a particle image velocimetry algorithm on the cell-attached and cell-free fluorescent bead layer images at each time point. An unconstrained Fourier transform traction cytometry algorithm (22) (regularization factor: 1×10⁻⁹, Poisson’s ratio: 0.5) was used to form traction stress vector fields from the calculated bead displacements (Fig. 1d). Substrate traction force vectors were defined by ${\mathit{T}}^n{a}^n$, where ${\mathit{T}}^n=T_x^n{\mathit{e}}_x+T_y^n{\mathit{e}}_y$ is the traction vector acting on area $a^n$, and ${\mathit{e}}_i$ is the unit vector in the $i$-direction (Supplementary Fig. S1ai). At the interface of the cell and the PA gel, these tractions are balanced by forces (${\mathit{f}}^n$) applied by the substrate to the cell, such that ${\mathit{f}}^n=f_x^n{\mathit{e}}_x+f_y^n{\mathit{e}}_y=-T_x^n{a}^n{\mathit{e}}_x-T_y^n{a}^n{\mathit{e}}_y$ (Supplementary Fig. S1aii). Forces were defined as tensile (positive) if they were oriented away from the cell midline and the total tensile forces $f_x$ and $f_y$ are given as $2f_x={\sum }_n{f}_x^n{r}_x^n/\left|r_x^n\right|$ and $2f_y={\sum }_n{f}_y^n{r}_y^n/\left|r_y^n\right|$, where ${\mathit{r}}_n=r_x^n{\mathit{e}}_x+r_y^n{\mathit{e}}_y$ is the vector describing the location of $n$ with respect to the center of the cell. The axial first Piola-Kirchhoff stresses $P_x$ were calculated at the midplane of the cell using the total tensile force and the undeformed cross-sectional area of the cell. The axial first Piola-Kirchhoff stress is given by $P_x=f_x/A_x$ (Supplementary Fig. S1b). The cell cross-sectional area was measured in the same cell line from the same supplier cultured under identical conditions in a previous study (11) to be $A_x=78\mu {\mathrm{m}}^2$. All $P_x$ measurements were normalized to the basal stress of the undeformed cell.

Drug treatment.

In experiments probing myosin binding affinity, cells were treated with stimulators or inhibitors of myosin 30 minutes prior to stretching. For CμBS experiments, cells received either 0.1 nM myosin phosphatase inhibitor calyculin (Cell Signaling Technology, Danvers, MA) or 3 μM rho kinase inhibitor Y-27632 (MedChemExpress, Monmouth Junction, NJ). For RNA sequencing experiments, cells received 20 μM Y-27632 or 50 μM myosin II inhibitor blebbistatin (Selleck Chemicals, Cologne, Germany).

Model of cellular rate-softening.

The cell was modeled as discrete actomyosin fibers embedded in an isotropic matrix. The binding kinetics of each fiber depends on the force at the actin-myosin binding site, as given by Bell (23). Fiber tension follows the Hill equation (24) when actin and myosin are engaged, but if the fiber force is sufficient to unbind all myosin heads in the fiber, actin and myosin slip past each other at the rate of cell deformation until rebinding occurs.

Cell deformation.

The cell was assumed to undergo isochoric planar biaxial stretching (shear free) such that the deformation gradient tensor $\mathit{F}$ is given by $\mathit{F}=diag\left[{\lambda }_x,{\lambda }_y,{\lambda }_z\right]$, where ${\lambda }_i$ is the stretch ratio in the $i$ direction ($x$: parallel to the long axis of the cell, $y$: parallel to the short axis of the cell, $z$: perpendicular to the gel surface). ${\lambda }_x$ and ${\lambda }_y$ were taken from experimental measurements and ${\lambda }_z=\frac{1}{{\lambda }_x{\lambda }_y}$.

Actomyosin fibers were assumed to deform affinely with the matrix. Fibers were considered separate, and cross-linking was not included in the model. Fiber stretch ratio ${\lambda }_f$, which depends on the angle of the fiber with to the $x$-axis (${\theta }_f$), is given by:

$${\lambda }_f^2={\lambda }_x^2{cos}^2\left({\theta }_f\right)+{\lambda }_y^2{sin}^2\left({\theta }_f\right).$$ (Eq. 1)

Cell stress.

Cells were treated as a distribution of discrete contractile actomyosin fibers embedded within an interior isotropic and homogeneous matrix. The total stress in the cell is given by the sum of all fiber stresses (${\mathit{\sigma }}_{\mathit{f}}$) and the stress of the bulk matrix material (${\mathit{\sigma }}_{\mathit{b}}$):

$$\mathit{\sigma }={\mathit{\sigma }}_{\mathit{f}}+{\mathit{\sigma }}_{\mathit{b}}$$ (Eq. 2)

For comparison to the experimental measurements, the First Piola-Kirchhoff stress $\mathit{P}$ was then calculated using

$$\mathit{P}={\mathit{F}}^T\mathit{\sigma },$$ (Eq. 3)

Cell constitutive description.

All non-fiber components of the cell were taken as neo-Hookean, such that the normal stresses ${\sigma }_{ii}$ are given by

$${\sigma }_{ii}=\mu \left({\lambda }_i^2-{\lambda }_z^2\right),$$ (Eq. 4)

where $\mu$ is the shear modulus

Fibers were assumed to be viscoelastic and actively contractile. Stress was assumed to be distributed uniformly along the length of each fiber, though the stress of each fiber can vary depending on its stretch and orientation. The stress of the fibers ${\mathit{\sigma }}_{\mathit{f}}$ is calculated by averaging the microfilament stress ${\sigma }_{mf}$ and scaling appropriately, such that

$${\mathit{\sigma }}_{\mathit{f}}=\frac{1}{n_f}\sum _{k=1}^{n_f}{\mathit{\sigma }}_{\mathit{m}\mathit{f}}^{\mathit{k}},$$ (Eq. 5)

where $n_f$ is the number of fibers and the superscript $k$ represents the $k$ th fiber. The stress ${\sigma }_{mf}$ depends on the actin-myosin binding state and degree of contraction, as described below.

Fiber tension was modeled by altering the fiber zero-stress configuration of the fiber (25). Fiber deformation in the stress-free configuration due to contraction is defined by the active stretch ratio ${\lambda }_a$. The active strain is given by ${ϵ}_a={\lambda }_a-1$ and the total strain of the fiber is the sum of the active strain and the elastic strains $\left({ϵ}_i^{\mathrm{*}}\right)$ of all components of the fiber. Here, we will consider the strain of the actin and myosin fibers together as ${ϵ}_{mf}^{\mathrm{*}}$ and the strain of the myosin head ensemble as ${ϵ}_h^{\mathrm{*}}$, giving a total strain ($ϵ$) and strain rate $\left(\dot{ϵ}\right)$ of a fiber of:

$$ϵ={ϵ}_a+{ϵ}_{mf}^{\mathrm{*}}+{ϵ}_h^{\mathrm{*}},$$ (Eq. 6)

$$\dot{ϵ}=\dot{{ϵ}_a}+\dot{{ϵ}_{mf}^{\mathrm{*}}}+\dot{{ϵ}_h^{\mathrm{*}}},$$ (Eq. 7)

Each myosin and actin filament was considered a linear spring, and each myosin head as a Kelvin solid (see Fig 3b). The strain of an individual myosin head is dependent on both the strain of the ensemble and the amount that head has been strained while bound to actin. The strain of the $j$ th head ${ϵ}_h^j$ is given by

$${ϵ}_h^j={ϵ}_h^{\mathrm{*}}+(\raisebox{1ex}{${\lambda }_f$}\!\left/ \!\raisebox{-1ex}{${\lambda }_{on}^j$}\right.-1),$$ (Eq. 8)

where ${\lambda }_{on}^j$ is the fiber stretch at the time the $j$ th myosin head binds to actin.

Figure 3: Rate-dependent binding kinetics simulations recapitulate strain rate softening.

a, Schematic of 3D cell adhered to a substrate. The cell is decomposed into active contractile filaments and a passive Neo-Hookean matrix representing the nucleus and cytoplasm. b, Spring and dashpot representation of actin and myosin filaments in a simulated cell. c, Model incorporates reversible, force-dependent binding kinetics of myosin and actin. d, Fiber overlap length $L_o$ reduces when myosin heads unbind, which can result in inactive fibers. e, Simulation predictions for three cycles of stress relaxation at slow (black), medium (purple), fast (red), and trauma (yellow) strain rates. Lines represent mean of 100 simulations. Shaded regions represent 95% confidence interval. f, Violin plot of simulated normalized PK1 stress distribution after first simulated cycle of stretch. Colored horizontal lines mark the median of the distribution. Violin plots of peak and decay stress distributions after 1 cycle of simulated stretch for increased affinity (calyculin) simulations (g) and decreased affinity (Y-27632) simulations (h). i, Ratio of mean PK1 stress during fast stretch to slow stretch for untreated (black), Y-27632 (blue), and calyculin (cyan) simulations. Error bars represent 95% confidence interval.

The stress of each myosin head ${\sigma }_h^j$ is given by

$${\sigma }_h^j=E_h{ϵ}_h^j+{\eta }_h\raisebox{1ex}{$d{ϵ}_h^j$}\!\left/ \!\raisebox{-1ex}{$dt$}\right.,$$ (Eq. 9)

where $E_h$ is the effective Young’s modulus of the myosin head, ${\eta }_h$ is the effective coefficient of viscosity of the myosin head. The stress in the microfilaments ${\sigma }_{mf}$ is given by

$${\sigma }_{mf}=E_{eq}{ϵ}_{mf}^{\mathrm{*}},$$ (Eq. 10)

where $E_{\text{eq}}$ is the equivalent Young’s modulus of the actin and myosin microfilaments. The microfilaments and myosin head ensemble are in series within the fiber, so the microfilament stress ${\sigma }_{mf}$ must be equal to the sum of the stress in all the myosin heads ${\sigma }_h$ such that

$${\sigma }_{mf}=\sum _j{\sigma }_h^j.$$ (Eq. 11)

Fiber contraction.

When myosin heads are bound to the corresponding myosin filament, fibers behave according to our previously derived Hill-type active fiber model (26) based on the Hill equation for muscular contraction (24). A standard form of the Hill equation is given by

$$V=\frac{b\left(F_0-F\right)}{F+a},$$ (Eq. 12)

where $V$ is the velocity of shortening of the muscle fiber, $F$ is the constant force on the muscle, $F_0$ is the stall force, or force at which a tetanized muscle neither shortens nor lengthens ($V=0$), and $b$ and $a$ are constants. From the Hill equation, we previously derived an expression (26) for the fiber shortening velocity, $\frac{\dot{{\lambda }_a}}{{\lambda }_a}$, that is given by

$$\frac{\dot{{\lambda }_a}}{{\lambda }_a}=\frac{b\left({\sigma }_{mf}-{\sigma }_0\right)}{\left({\sigma }_{mf}+\gamma \right)},$$ (Eq. 13)

where ${\sigma }_0$ is a homeostatic stress wherein the fiber neither contracts nor relaxes and $\gamma$ is a constant parameter.

Fiber slip.

If all myosin heads on a myosin filament are unbound from actin, the fiber detaches in an action we call “slip.” During slip, the fiber undergoes unconstrained one-dimensional deformation (27) such that the differential velocity $dv$ is the sum of the velocity of shortening $d{v}_a$ and the velocity of elastic deformation $d{v}^{\mathrm{*}}$. The change in velocity over the length of the fiber $dx$ can then be expressed as

$$\frac{dv}{dx}=\frac{d{v}_a}{dx}+\frac{d{v}^{\mathrm{*}}}{dx}$$ (Eq. 14)

This can be rewritten as

$$\frac{\dot{{\lambda }_f}}{{\lambda }_f}=\frac{\dot{{\lambda }_a}}{{\lambda }_a}+\frac{{\dot{\lambda }}^{\mathrm{*}}}{{\lambda }^{\mathrm{*}}},$$ (Eq. 15)

If the fibers are sliding freely, then ${\dot{\lambda }}^{\mathrm{*}}=0$. Therefore, during slip, the rate of contraction can be defined as

$$\frac{\dot{{\lambda }_a}}{{\lambda }_a}=\frac{\dot{{\lambda }_f}}{{\lambda }_f},$$ (Eq. 16)

For the purposes of our experiment, ${\dot{\lambda }}_f$ is defined as the rate of deformation applied by the CμBS actuators $\left(\dot{\lambda }\right)$.

Fiber overlap.

We assume that the fibers have a finite length with a finite region of overlap between the actin and myosin filaments. During slip (all myosin heads unbound), the fiber lengthens by length $d_{\text{slip}}$ such that

$$d_{\mathit{slip}}=\dot{\lambda }{L}_0\mathrm{\Delta }t,$$ (Eq. 17)

where $L_0$ is the initial length of the stress fiber. In turn, the length of overlap between the actin and myosin filaments $L_{ov}$ shortens by $d_{\mathit{slip}}$. If $L_{ov}\le 0$, there is no overlap between the actin and myosin filaments, so we set fiber stress ${\sigma }_{mf}=0$ until $L_{ov}$ increases above 0 due to applied compressive deformation of the fiber.

Binding kinetics.

The cell’s active contractile fibers are composed of actin and myosin (Fig. 3b). We simulated stochastic binding kinetics of myosin to actin driven by binding probability. The binding probability $E_{\mathit{bind}}$ of a myosin head is given by

$$E_{\mathit{bind}}=k_{\mathit{on}}\mathrm{\Delta }t,$$ (Eq. 18)

where $k_{\mathit{on}}$ is the association rate of myosin to actin and $\mathrm{\Delta }t$ is the timestep. Unbinding probability $E_{\mathit{unbind}}$ follows the Bell model of force-dependent binding kinetics (23), given by

$$E_{\mathit{unbind}}=k_{\mathit{off}}{e}^{\frac{f_f}{f_b}}\mathrm{\Delta }t,$$ (Eq. 19)

where $k_{off}$ is the dissociation rate of myosin from actin, $f_f$ is the force acting on the fiber defined by $f_f={\sigma }_{mf}{A}_f$ (where $A_f$ is the cross sectional area of the fiber), and $f_b$ is the stress required to break an actomyosin bond.

Solution method.

A random family of 20 fibers matching the measured von Mises distribution of an average cell was generated. Time was discretized into 0.001 s increments over a 32 min runtime. For the first 100 s of simulation, ${\lambda }_x={\lambda }_y={\lambda }_z={\lambda }_f=1$. Initially, all myosin heads on all fibers were assumed to be bound. The number of bound heads was allowed to reach a steady state based on binding kinetics in the absence of force. After 100 s of simulation, deformation was applied to mimic the relevant CμBS experiment.

We used Monte Carlo methods to simulate random binding and unbinding of actin and myosin. We calculated the binding and unbinding probabilities at each timestep. If binding or unbinding probability was greater than a random number generated between 0 and 1, the binding or unbinding step was accepted. If the binding or unbinding probability was less than the random number, the step was rejected, and the system remained the same.

The elastic strain of the microfilaments ${ϵ}_{mf}^{\mathrm{*}}$ and myosin heads ${ϵ}_h^{\mathrm{*}}$ are unknown quantities. The MATLAB command fmincon was used with a Sequential Quadratic Programming (SQP) algorithm (28) to estimate values of ${ϵ}_{mf}^{\mathrm{*}}$ and ${ϵ}_h^{\mathrm{*}}$ at each timepoint. The SQP method uses a quasi-Newton method for constrained optimization (29). At $t=0$, fmincon was given an initial guess of 0.085 strain for both ${ϵ}_{mf}^{\mathrm{*}}$ and ${ϵ}_h^{\mathrm{*}}$. In subsequent timesteps, the estimated values of ${ϵ}_{mf}^{\mathrm{*}}$ and ${ϵ}_h^{\mathrm{*}}$ of the prior timesteps were used as initial guesses. ${ϵ}_{mf}^{\mathrm{*}}$ and ${ϵ}_h^{\mathrm{*}}$ were given a lower bound of 0 strain to prevent guesses that would place the myosin filaments and heads into compression during stretching and an upper bound of $\left({\lambda }_f-1\right)-\left({\lambda }_a-\right.1)$ to prevent strains greater than the total stretch of the fiber ${\lambda }_f$. Optimization was performed subject to the constraints of Eq. 5, Eq. 6, and Eq. 7. Error functions were generated using the square of the difference between ${\sigma }_{mf}$ and $\sum _j{\sigma }_h^j;ϵ(\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{w}\mathrm{n})$ and ${ϵ}_a+{ϵ}_{mf}^{\mathrm{*}}+{ϵ}_h^{\mathrm{*}}$; and $\dot{ϵ}(\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{w}\mathrm{n})$ and $\dot{{ϵ}_a}+{ϵ}_{mf}^{\dot{\mathrm{*}}}+\dot{{ϵ}_h^{\mathrm{*}}}$. The total error of each iteration was the sum of the three error functions. The error functions for Eqs. 6 and 7 were scaled appropriately to give each error function an equal weight in the total error evaluation. The iterative solver terminated once the estimated values of ${ϵ}_{mf}^{\mathrm{*}}$ and ${ϵ}_h^{\mathrm{*}}$ for two subsequent guesses minimized the total error to fall within 0.0001 or once 1 million iterations were reached.

Alternative models considered.

Figures in the body of this paper were generated using the above framework. Model predictions using only the Hill-type model (no simulation of binding kinetics, no slip) or assuming infinitely long fibers with infinite overlap (no change to fiber overlap during slip) are presented in the Supplement (Supplementary Fig. S4).

Trauma-like transient stretch.

For gene expression analysis, HUASMCs were seeded onto constructs at a density of 150 cells/mm² and allowed to expand to confluency over 4 days. Cells were then serum-starved for 24 hours before stretching. Constructs that were pre-treated with myosin inhibitors received either 20 μM Y-27632 or 50 μM blebbistatin. After 30 min of incubation with myosin inhibitors, media was changed in all constructs to remove the inhibitors. Constructs were then stretched to a strain of 0.15 at a rate of either 0.05 s⁻¹ or 10 s⁻¹. All constructs were lysed 24 hours after stretch, and RNA was isolated from the lysates using a Qiagen RNeasy kit (Qiagen, Hilden, Germany).

RNA sequencing.

Sample Quality Assessment:

Total eukaryotic RNA isolates were quantified using a fluorimetric RiboGreen assay. Total RNA integrity was assessed using capillary electrophoresis (Agilent BioAnalyzer 2100), generating an RNA Integrity Number (RIN). For samples to pass the initial QC step, they needed to quantify higher than 500 ng and have a RIN of 8 or greater. Total RNA samples were then converted to Illumina sequencing libraries.

Library Creation:

Total RNA samples were converted to Illumina sequencing libraries using Illumina’s TruSeq Stranded mRNA Sample Preparation Kit. In summary, the mRNA from a normalized input mass of total RNA was isolated using oligo-dT coated magnetic beads, fragmented and then reverse transcribed into cDNA. The cDNA was blunt-ended, A-tailed and indexed by ligating molecularly barcoded adaptors. Libraries were amplified using 15 cycles of PCR. Final library size distribution was validated using capillary electrophoresis and quantified using fluorimetry (PicoGreen) and via Q-PCR. Indexed libraries were then normalized, pooled and size-selected to 320 bp (tight) using the PippinHT instrument.

Cluster generation and sequencing:

Pooled libraries were denatured and diluted to the appropriate clustering concentration. The libraries were loaded onto the NovaSeq paired-end flow cell and clustering occurred on board the instrument. Once clustering was complete, the sequencing reaction immediately began using Illumina’s 2-color SBS chemistry. Upon completion of read 1, 2 separate 8 or 10 base pair index reads were performed. Finally, the clustered library fragments were re-synthesized in the reverse direction thus producing the template for paired-end read 2.

Primary analysis and de-multiplexing:

Base call (.bcl) files for each cycle of sequencing were generated by Illumina Real Time Analysis (RTA) software. The base call files and run folders were streamed to the Minnesota Supercomputing Institute servers. Primary analysis and de-multiplexing were performed using Illumina’s bcl-convert v4.0.3. The result of the bcl-convert workflow was de-multiplexed FASTQ files.

RNA-seq analysis.

Quality control, data alignment, and gene quantification were analyzed using the CHURP pipeline (30) at the University of Minnesota Supercomputing Institute (MSI). 2 × 50bp FASTQ paired-end reads for 21 samples (≥20 million reads per sample) were trimmed using Trimmomatic (v0.33) enabled with the optional “-q” option and 3bp sliding-window trimming from 3’ end requiring minimum Q30. Quality control on raw sequence data for each sample was performed with FastQC. Read mapping was performed via HISAT2 (31) (v2.1.0) using the human genome (GRCh38.p13) as reference. Gene quantification was performed via Feature Counts for raw read counts. Differentially expressed genes were identified using the edgeR (negative binomial, R programming) feature in CLCGWB (Qiagen, Redwood City, CA) using raw read counts. The generated list was filtered based on a minimum 2X absolute fold change and FDR corrected p < 0.05.

Pathway analysis.

Differentially expressed genes (DEGs) identified through RNA-seq analysis above were entered into the NIH Database for Annotation, Visualization, and Integrated Discovery (DAVID) Bioinformatics Functional Information tool. The list of official gene symbols was mapped onto the Homo sapiens genome and compared to the library of KEGG pathways. Enriched pathways were identified using a modified Fisher Exact test. Only pathways with a p-value < 0.05 were considered to be enriched.

Statistics.

All results are expressed as means ± standard error of the mean (s.e.m.). Differences in $P_x$ stress for cells stretched at different strain rates were compared using paired t-tests in SigmaPlot Version 11.0 (Grafiti, Palo Alto, CA). A value of p < 0.05 was considered to indicate statistical significance.

Results

Cells rate-soften at physiological and supraphysiological strain rates.

To study how cell mechanical behavior differs when deformed at different strain rates, we focused on human umbilical artery smooth muscle cells (HUASMCs), which we have previously extensively characterized under quasi-static loading (11, 26, 32). We used fibronectin (FN) to micropattern individual HUASMCs atop a fluorescent bead-doped polyacrylamide (PA) gel (Fig. 1a) bonded to a silicone elastomer membrane mounted between two linear voice coil actuators (Fig. 1b). We performed a uniaxial stretch-and-hold experiment with a maximum strain ($ϵ$) of 0.15. Physiological strain rates of $\dot{ϵ}=0.001{\mathrm{s}}^{-1},0.005{\mathrm{s}}^{-1},\text{or}0.005{\mathrm{s}}^{-1}$ (hereafter referred to as “slow,” “medium,” and “fast” rates) or a supraphysiological rate of $\dot{ϵ}=10{\mathrm{s}}^{-1}$ (hereafter referred to as “trauma”) were used (Fig. 1c). Following stretch, we held cells static at $ϵ=0.15$ (referred to hereafter as the relaxation phase) and used traction force microscopy (TFM) (22) to calculate the axial first Piola-Kirchhoff stress $P_x$ of the cell (11) (Fig. 1d). Cells were then returned to 0 strain at the same strain rate, and TFM was performed to measure stress at 0 strain. This cycle was repeated three times total for each cell. Stress was normalized by the undeformed basal stress of the cell.

Regardless of strain rate, cell stress reached a peak immediately after deformation and decayed with time in a stress-relaxation-like manner (33). This behavior has been observed before using micro tensile testers (34) and atomic force microscopy (35) in multiple cell types (12, 36), leading many to conclude that cells behave like viscoelastic solids (34, 36, 37). Surprisingly, we found that as cells were stretched with increasing strain rates, the cells produced lower stress (Figs. 1e,f), which is opposite what would be expected in a viscoelastic solid (38). This observation held true both for the peak stress produced by the cells and the stress produced at the end of 5 minutes of relaxation (hereafter referred to as “decay stress”) (Fig. 1f). Our results suggest that cells rate-soften as they are deformed at increasing rates, contrary to the behavior of a viscoelastic material.

Interestingly, the effects of rate-softening only appear during the loading phase of the experiments. Regardless of strain rate, upon return to $ϵ=0$, the stresses are nearly identical (Fig. 1e). The stress recovery curve after returning to $ϵ=0$ is strikingly similar to the stress drop and recovery observed in prior studies that measured cell stiffness and traction following an acute stretch (13, 39), a phenomenon that has been attributed to “cytoskeletal fluidization.” These results suggest that the mechanical effects of rate softening are “reset”, perhaps via cytoskeletal fluidization, when the cell is returned to zero strain. This possibility is supported by the cycle-to-cycle repeatability of stresses, which suggest that, within the physiological strain rates at least, there is little to no accumulated damage.

Strain rate-softening depends on actomyosin binding kinetics.

Prior studies of airway smooth muscle tissue have shown that the storage modulus of smooth muscle tissue changes with both strain and strain rate, attributed to a dynamic binding equilibrium of actin and myosin (17–19). These studies suggest that actomyosin interactions could mediate the observed rate-softening, but prior work only considers cross-bridge cycling equilibria and does not explore changes to binding kinetics. To probe this phenomenon, we treated HUASMCs with myosin agonist calyculin or rho kinase inhibitor Y-27632 prior to stretch-and-hold experiments. Following calyculin treatment, which increases actin-myosin binding affinity, the relative rate softening in cells exposed to fast and slow stretch was non-significant (Fig. 2a,b). Treating with Y-27632, which decreases actin-myosin binding affinity, however, exaggerated rate softening (Fig. 2e,f), compared to untreated cells (Fig. 2c,d). To assess the degree of rate softening we compared the ratio of both the peak and decay stresses between the fast and slow stretch results and found that this ratio decreases monotonically with actin-myosin binding affinity (Fig. 2g). Thus, strain rate softening appears to be impacted by actomyosin binding kinetics and may be driven by the disengagement of stress fibers. Notably, while prior observations of cytoskeletal fluidization attribute changes in mechanics to passive severing or depolymerization of f-actin (40, 41), our findings suggest that reduced active stress exerted by actomyosin fibers is also a major contributor.

Figure 2: Strain rate softening depends on actomyosin binding kinetics.

Average axial PK1 stress during first cycle of stress relaxation for cells pretreated with 0.1 nM calyculin (a) to promote myosin activity, untreated cells (c), or 3 μM Y-27632 (e) to inhibit myosin activity and then subjected stress relaxation at slow (black) or fast (red) strain rates. Calyculin: n = 13 cells slow, 12 cells fast. Untreated: reproduced from Fig. 1f. Y-27632: n = 13 cells slow, 9 cells fast. Normalized peak and decay stress during first cycle of stress relaxation for cells pre-treated with calyculin (b), untreated (d), or pre-treated with Y-27632 (f). Stresses between cells stretched at the slow vs the fast rate are not significantly different for cells pre-treated with calyculin. Solid bars represent mean normalized axial PK1 stress, open circles represent individual measurements. Error bars represent s.e.m. *p < 0.05. g, Ratio of stress in cells stretched at fast rate vs slow rate for untreated (black), Y-27632-treated (blue), and calyuclin-treated (cyan) cells. Stimulating myosin activity with calyculin decreases strain rate softening, while inhibiting myosin with Y-27632 increases strain rate softening.

Modeling rate-dependent actomyosin binding kinetics in viscoelastic stress fibers recapitulates cell rate-softening.

To explore how deformation rate influences cell stress after stretch, we developed a computational model of rate-dependent actin-myosin binding kinetics based on the force-dependent binding models proposed by Bell (23) and employed in motor clutch models of cellular adhesion (42). This model considers actin and myosin filaments as simple springs, while the myosin heads behave as viscoelastic Kelvin solids (43) acting in parallel (Fig. 3b). When actin and myosin are bound, they follow the Hill model for force-dependent contraction (26) and when they are unbound, the myosin head force is zero. The binding off-rate $k_{off}$ of each bound myosin head depends on the ratio between the applied force and the rupture force for an actomyosin bond (Fig. 3c). In this model, myosin heads experience more force during high strain rate loading, increasing the binding off-rate. When all myosin heads on a fiber unbind, the fiber experiences a “slip” event, and the actin and myosin filaments slide past each other at the rate of cell deformation. Should the filaments slide past each other such that there is no overlap, the fiber can no longer produce force (Fig. 3d).

When cell loading is simulated to match the experimentally applied stretch and hold experiments, this model predicts that, as the cell is deformed at increasing rates, unbinding dominates the kinetics of actomyosin and stress fibers disengage. When stretched at the trauma rate, the overlap between actin and myosin filaments reduced to zero in multiple fibers (Supplementary Fig. S2). As a result, rate softening is observed for stretched cells, with the cells stretched at the highest rates having the lowest stress (Fig. 3e,f), similar to findings from the experiment (Fig 1e,f). Additionally, the model correctly predicts that stress in cells stretched at the trauma rate drops below the basal stress, replicates the recovery phase following the return to zero strain, and the illustrates the cycle-to-cycle repeatability seen experimentally.

To simulate calyculin and Y-27632 treatment, we altered the $k_{off}$ for actin-myosin binding in the model, decreasing it for calcyculin and increasing it for Y-27632. For calyculin treatment, much like the observed data (Figs. 2b,d), the model predicts only a slight difference in stress between the cells stretched at the slow strain rate vs. the fast strain rate (Fig. 3g), while for Y-27632 treatment the difference in stress between cells stretched at fast and slow strain rates is significantly increased (Fig. 3h). The consistency between the measured stresses and model predictions (Fig. 3i) supports the idea that actin-myosin binding kinetics mediate rate softening.

Supraphysiological strain rates induce plastic deformation.

In our stretch-and-hold experiments, while cycle-to-cycle stresses were consistent for most conditions, we observed that stress decreased slightly with each cycle in the cells stretched at the trauma rate (Fig. 4a). To explore this effect further, we asked if a single stretch at a trauma-like strain rate was sufficient to alter HUASMC mechanical function. We performed one cycle of stress relaxation at the slow strain rate, followed by one cycle of stress relaxation at either the fast or trauma strain rate, and then returned to the slow strain rate for the final stress relaxation cycle (Fig. 4b,d). Cells stretched with the slow-fast-slow pattern produced lower stress after a fast stretch, as expected, but the stresses during the third cycle were nearly identical to the first, indicating the stretch at the fast rate had minimal impact on cell mechanics in subsequent stretches (Fig. 4c). Conversely, when the second cycle was instead performed at the trauma strain rate (Fig. 4d), stress in the third slow cycle was lower than in the first slow cycle, suggesting the singular stretch at the trauma strain rate is sufficient to disrupt HUASMC mechanics (Fig. 4e). This result is best illustrated by the cycle-3-to-cycle-1 ratio of the stresses (Fig. 4f), where the slow-slow-slow and slow-fast-slow sequences exhibit no changes, while the slow-trauma-slow exhibits a significant drop during the cycle 3 stress. These results suggest that the effect of rate-softening is reversible at physiological strain rates, but as the strain rate is increased to supraphysiological levels the cells begin to undergo plastic deformation.

Figure 4: High strain rates induce plastic deformation.

a, Ratio of stress during third cycle to first cycle for cells deformed in three cycles of stress relaxation at the same rates as in Fig. 1. Stress decreases after trauma-like stretch when all three stretches are at the same rate. b-e, Cells were stretched in alternating patterns of strain rate: slow-fast-slow (black-red-black, b) and slow-trauma-slow (black-yellow-black, d). Average normalized axial PK1 stress for cells subjected to the slow-fast-slow (c) and slow-trauma-slow (e) sequences. Rate softening occurs in the slow-fast-slow stretching sequence but cells recover their stress in subsequent cycles of stretch. A single stretch at the trauma-like rate reduces stress in subsequent slow stretching sequences. n = 17, 14 cells, respectively. f, Ratio of stress during third cycle to first cycle for slow-slow-slow (black), slow-fast-slow (red), and slow-trauma-slow (yellow) strain sequences. Solid bars represent mean cycle 3 to cycle 1 ratio. Open circles are individual measurements. *p < 0.05, **p < 0.01. Error bars represent s.e.m.

A single high strain rate stretch alters gene expression.

Actomyosin fibers play a role in contraction and are also implicated in mechanotransduction (44). The results of our experiments and simulations indicate that stress fibers may disengage during high strain rate deformation, contributing to strain rate softening. We asked whether rate softening plays a role in protecting the cell from injury. We subjected a confluent layer of HUASMCs to a single fast strain rate stretch or a single trauma strain rate stretch and isolated RNA 24 hours later. We sequenced the RNA and found that 365 genes were differentially expressed in the cells plastically deformed at the trauma strain rate (Fig. 5a,b). Many of the upregulated genes were implicated in inflammatory pathways, while many of the downregulated genes were involved in smooth muscle cell contractile phenotype (Supplementary Fig. S3). Only 190 genes were differentially expressed in cells deformed at the fast strain rate, where reversible rate softening is possible. Changes to expression of the 365 differentially expressed genes identified in cells stretched at the trauma strain rate were significantly lower in cells that were only stretched at the fast strain rate (Fig. 5c), indicating that gene expression depends not only on how much a cell is stretched but also how quickly. These results are consistent with our mechanical studies, wherein cells exposed to trauma stretch are plastically deformed, or damaged, while those exposed to fast stretch have little to no damage.

Figure 5: A single high strain rate stretch alters gene expression.

a, The log₂ fold change of the 365 genes differentially expressed in cells experiencing a single trauma-like stretch (blue) compared to unstretched controls. Expression of these same genes in cells stretched at a fast rate (orange) or pre-treated with blebbistatin (green) or Y-27632 (pink) before a trauma-like stretch is superimposed over the expression of untreated cells stretched at a trauma-like rate. b, Differentially expressed genes (DEG) for cells stretched at trauma-like rate only. c, Expression levels of the genes in b for cells stretched at a fast rate. Expression levels of the genes in b for cells pre-treated with 50 μM blebbistatin (d) or 20 μM Y-27632 (e) before a trauma-like stretch. Violin plots of the distribution of upregulated (f) or downregulated (g) genes from b for cells experiencing a trauma-like stretch (blue), fast stretch (orange), or pretreated with blebbistatin (green) or Y-27632 (pink) before a trauma-like stretch. Reversible rate softening that occurs at fast strain rates reduces changes in gene expression. Pre-treating cells with myosin inhibitors to promote rate softening similarly reduces changes to gene expression. Colored horizontal lines mark the median of the distribution. Gray shaded regions indicate gene expression fold changes of less than 2.

Next, we repeated the experiment on cells that were pre-treated with 20 μM Y-27632 or 50 μM blebbistatin 30 minutes before stretching to weaken actomyosin binding and encourage rate softening through stress fiber disengagement. The expression of the genes identified in the untreated cells stretched at the trauma strain rate were substantially lower in the myosin binding-inhibited cells (Fig. 5d,e), similar to the cells stretched at the fast strain rate. This pattern held true for both upregulated and downregulated genes from the trauma strain rate case (Fig. 5f,g). Our findings thus demonstrate that reducing actomyosin binding affinity may allow the cell to achieve sufficient rate softening to reduce mechanotransduction during injury at supraphysiological rates and protect the cell from extreme deformations.

Discussion

Our results clarify an unexpected cellular material behavior, where cells mechanically soften in response to deformation at increasing rates. The identified mechanism could help interpret several previous results in the literature, both in vitro and in vivo. Cytoskeletal fluidization after transient stretch has captured the attention of cellular biomechanicians for the better part of three decades (17, 41, 45). Early investigations into cytoskeletal fluidization using transient stretching revealed that stimulation or inhibition of actomyosin can affect the degree of fluidization (13), with these results being further reinforced with other fluidization techniques like low intensity ultrasound (46). Our observations from experiments and simulations of rate-softening in adherent cells follow a similar pattern: cells soften more as they are deformed at increasing rates, but the softening is reduced when actomyosin activity is stimulated. Many solid polymers, such as polystyrene, show rate- and temperature-dependent strain rate softening (47). It has previously been observed that smooth muscle tissue demonstrates strain rate-dependent reductions in storage modulus when cyclically stretched (17), but this phenomenon has not been measured in individual cells. Here, we show, for the first time, that isolated adherent cells under tension produce less stress when deformed at increasing rates.

Though the observed decreases in stress at higher strain rates within the physiological range might suggest material failure, the cycle-to-cycle repeatability tells a different story. Within the physiological range, rate softening appears not to be a one-time aberration leading to sustained mechanical alterations, but instead a built-in characteristic of the cell. It is only when strain rate is pushed beyond the threshold of physiological rates into rates characteristic of trauma that the observed softening becomes irreversible. Other biological materials, such as cortical bone, possess strain rate-dependent yield stresses, meaning that when loaded at a supraphysiological strain rate, bone does not deform plastically until stresses almost double the yield stress of physiological strain rates (48). Adherent cells in our experiment appeared to possess the opposite characteristic: yielding and plastic deformation only occurred at supraphysiological strain rates.

The fact that cells comprising soft tissue plastically deform at trauma-like strain rates, while load bearing structures like bone have evolved to strengthen at high strain rates (48), could suggest that strain rate softening plays a protective role for cells, but is not sufficient to fully protect the cell from trauma. In our RNA sequencing assay, multiple malignant pathways were activated by stretching at trauma-like rates. When cells were stretched the same amount but at a lower rate within the reversible range, activation of these same malignant pathways was substantially reduced, which we attribute to reversible rate softening. When rate softening was promoted by reducing actomyosin affinity, cells could better tolerate trauma-like stretch, further supporting the protective nature of strain rate softening. Further investigation into mechanotransduction at high strain rates is warranted.

There are several limitations in our measurement and modeling of reversible strain rate softening in adherent cells. Our understanding of the mechanisms of strain rate softening would be strengthened with the ability to visualize the motion myosin filaments within the cell in real time. Other investigators have found success visualizing myosin through labeling myosin heads with tagged fluorescent dots (49), but it is unclear whether such a technique could be applied to our CμBS assay. A further shortcoming is our inability to image the cells during stretching, as opposed to immediately post-stretch while static. As a result, though our model suggests that myosin slip occurs during stretching, we cannot observe this phenomenon directly with our current approach. In the model, we assume a two-state representation (i.e. latch bridge) of myosin rather than a four-state representation, including cross bridges, such as those employed in the Huxley, Hai, and Murphy model of myosin cycling (50, 51), which predicts strain rate-dependent stiffness in smooth muscle tissue (17, 18). While simulating myosin cross bridge cycling in addition to the reversible binding of myosin from actin would likely lead to stronger agreement between the model predictions and experimental findings, particularly at the slow and medium rates, the two state model is sufficient to capture the behavior of cells at the fast and trauma strain rates, where the most rate softening and plastic deformation occurs.

While both the role of rate softening and mechanism of injury at high strain rates are not yet clear, one possibility is that cytoskeletal softening at high rates limits the deformation of the nucleus during trauma. Increasing evidence implicates nuclear deformation in cellular sensation and response to force (44, 52). Stress fibers terminate at focal adhesions but are physically connected to the nucleus through linker of nucleoskeleton and cytoskeleton (LINC) protein complexes (53). Strain of the extracellular microenvironment is transmitted to the nucleus through stress fibers (54, 55). In turn, gene activation and transcription depend on the relative location of chromatin gene loci (interior of nucleus vs. periphery) and the degree of chromatin condensation, both of which can be modified through nuclear deformation (56, 57). Thus, rate softening may play a key role in limiting aberrant mechanotransduction.

Conclusion

This study provides direct evidence that adherent cells, while displaying stress relaxation-like characteristics of viscoelastic materials, are fundamentally strain rate softening materials, not unlike prior observations on the tissue scale (17). While the observed strain rate softening is similar to passive cytoskeletal fluidization demonstrated by others (14, 39), we find here that strain rate softening is also mediated by active actomyosin interactions and binding affinity. Further, we found that the rate of deformation impacts gene expression of adherent cells, potentially implicating strain rate softening in mechanotransduction. Understanding cellular rate-softening in the context of trauma may be key to identifying treatments and preventing adverse outcomes in trauma patients.

Statement of significance.

Chronic health conditions like cancer, heart failure, and diabetes, are overrepresented in populations exposed to acute traumatic injury, despite no clear association between these conditions and the injury itself, which suggests a lasting impact of trauma that is maintained in cells and tissues over time. This study explores the mechanisms by which cells tolerate high-speed deformations and establishes a threshold for cellular injury that is above the deformations that are expected physiologically. This cross disciplinary study uses single-cell mechanical measurements, computational modeling, and transcriptomics to unravel a subcellular mechanism for adapting to high-speed deformations, highlighting the essential role of actomyosin interactions in stress fibers in transmitting force throughout the cell.