Intracellular nonequilibrium fluctuating stresses indicate how nonlinear cellular mechanical properties adapt to microenvironmental rigidity

Living cells are known to be in thermodynamically nonequilibrium, which is largely brought about by intracellular molecular motors. The motors consume chemical energies to generate stresses and reorganize the cytoskeleton for the cell to move and divide. However, since there has been a lack of direct measurements characterizing intracellular stresses, questions remained unanswered on the intricacies of how cells use such stresses to regulate their internal mechanical integrity in different microenvironments. This report describes a new experimental approach by which we reveal an environmental rigidity-dependent intracellular stiffness that increases with intracellular stress - a revelation obtained, surprisingly, from a correlation between the fluctuations in cellular stiffness and that of intracellular stresses. More surprisingly, by varying two distinct parameters, environmental rigidity and motor protein activities, we observe that the stiffness-stress relationship follows the same curve. This finding provides some insight into the intricacies by suggesting that cells can regulate their responses to their mechanical microenvironment by adjusting their intracellular stress.

with inhibitory drugs. This result suggests that cells can regulate their mechanical properties by adjusting their intracellular stress. The data reveal the relationship between molecular dynamics and emergent mesoscale material properties in living cells, thus inspiring further research on how living systems can take advantage of fluctuations in nonlinear systems.

Results
Intracellular fluctuating forces increase with substrate rigidity. To examine how cells adjust their intracellular mechanical properties to adapt their microenvironment, we use active microrheology (AMR) approach 12,21-24 by using optical tweezers to trap and oscillate harmonically a 1 μm engulfed polystyrene particle inside a living HeLa cell. From the displacements during the particle oscillation, cellular internal responses to the applied forces are determined. Internal responses are captured in the effective spring constant of the following linear relation: F = k * u where F is the force acting on the particle, u is the position of the particle and k * is a complex effective spring constant. The real and imaginary parts of the complex effective spring indicate an stiffness, k′ and dissipative resistance, k″, respectively. Both k′ and k″ are determined from the experimentally measured displacement magnitude and phase of the particle 21,24 . We find that cells attached to a stiffer substrate (20 kPa polyacrylamide gel, coated with collagen) are significantly stiffer than those attached to a more compliant substrate (0.35 kPa polyacrylamide gel, coated with collagen) as shown in Fig. 1B. These results are qualitatively consistent with previous extracellular measurements using atomic force microscopy to determine whole cell stiffness to response substrate rigidity 25 . Additionally, the intracellular stiffnesses follow a power-law dependence on frequency, similar to those of a soft glass 22,24 . In the frequency range 0.1 to 10 Hz, the HeLa cells exhibit a relatively solid-like response (i.e., k′/ k″ > 1, shown in the insert of Fig. 1B), similar to the cellular cortex 26 and the synthesized active cytoskeletal networks 12 measurements. Notably, the solid-like response depends on frequency, which is consistent with the typical release rate of myosin filaments observed in synthesized cytoskeletal networks 12 . However, it is surprising that this solid-like response weakly depends on the microenvironmental rigidity, which indicates both that the viscoelastic properties are adjusted through intracellular force. In active microrheology, the effective complex spring constant k * = k′ + ik″ is measured directly by an oscillatory optical tweezers (OOT) acting on a trapped intracellular probe particle. In passive microrheology, a CCD camera tracks fluctuation in the probe-particle position. (B) Intracellular stiffness k′ for cells cultured on substrates with elastic moduli varying from 0.35 kPa to 20 kPa. k′ follows a weak power-law dependence on frequency. Insert shows a decreasing solid-like behavior k′/k″ (ratio of stiffness to dissipative resistance) as a function of frequency. (C) The total intracellular fluctuating force spectra f total 2 increase with substrate rigidity. Insert shows a total fluctuation spectrum as a function of frequency. (D) C total /C equ. as a function of frequency varies with the substrate rigidity. (E) Nonequilibrium fluctuating force spectra . f nonequ 2 as a function of frequency varies with the substrate rigidity (F) Nonequilibrium fluctuating force at 1 Hz increases with substrate rigidity. Colors indicate elastic moduli of the cell-culture substrates (see Fig. 1D legend). G sub. is the elastic modulus of the substrate. Error bars are standard deviation of the mean from ten independent measurements from individual probed particles in each cell.
To determine how intracellular force responds to substrate rigidity, we combine both active and passive cellular microrheology approaches. The passive method 19,20 is based on a statistical analysis of thermal fluctuations of internalized particle motion. Here, the total fluctuating force spectrum 20 f total 2 can be determined as , where k * is measured by AMR and C total is the total fluctuation of the probe particle position (the insert of Fig. 1C) measured by passive microrheology (PMR). We refer to fluctuations measured by PMR as "total" fluctuations, since they represent the response of the probe particle to an intracellular medium that contains both passive thermal-equilibrium and active forces (see below). The total fluctuating force as a function of frequency follows a power law with exponent about −1.5 at the lower (0.1~10 Hz) frequencies and about −0.5 at the higher frequencies (10~100 Hz) 27 as shown in Fig. 1C. The power-law behavior implies that the microscopic processes responsible for active stress have a broad distribution of activation rates 28 . Compared with previous reports showing ω − f total 2 2 28-30 , our low-frequency exponent of −1.5 indicates that intracellular motors 29 are less abundant and/or less active than motors on the cellular cortex 28,30 , implying that the locally intracellular tension is smaller than tension on the cellular cortex. The behaviors of total fluctuating force spectra indicate that the thermal fluctuation effect would be comparable with the nonequilibrium contribution from intracellular forces inside a living cell.
To address the nonequilibrium contribution, we combine AMR and PMR to distinguish the power spectrum of nonequilibrium forces from that of a thermal force 28,29,31 . Here, the fluctuations of a probe particle measured by PMR show "total" fluctuations including both equilibrium and nonequilibrium fluctuations. The thermal-equilibrium fluctuation spectrum (C equ. ) of a probe particle can be found using AMR measurements of the imaginary part k″ of the effective spring constant k* 24 . In an equilibrium system, only thermal-equilibrium forces act on the probe, and the power spectral density of the displacement fluctuations is related to the mechanical response of the material by the fluctuation-dissipation theorem C equ. = 2k B T/k″ω, where k B is Boltzmann's constant and T is the absolute temperature. Fig. 1D shows the ratio of the total-fluctuation power spectrum, measured by passive microrheology, to that of equilibrium fluctuations estimated by active microrheology. This ratio is defined in previous studies as the ratio of the effective energy (or effective temperature) of the system to the thermal energy 28,29 . Using the assumption by Mizuno et al. 20 that the magnitudes of the equilibrium and nonequilibrium fluctuation energies are additive, we can determine the chemically produced, active fluctuating force spectrum 20 . f nonequ 2 by subtracting the thermal-equilibrium spectrum (C equ. ; estimated by use of AMR) from the total fluctuation spectrum (C total ; measured by PMR). In addition, we also assume that time-averaged local environmental stiffnesses, <k*>, remains constant over time. We examine this additional assumption in the next section. Under both assumptions, we disentangle the equilibrium and non-equilibrium force spectra, even though our measurements of active and passive microrheology are not made time-synchronously.
We find that at frequencies lower than about 1 Hz the fluctuations measured by PMR (C total ) have a magnitude larger than the expected equilibrium fluctuations based on AMR (C equ. ) for cells cultured on the stiffer substrate 27 (Fig. 1D). At frequencies higher than 10 Hz the PMR and AMR results coincide, showing that at high frequencies the system is in equilibrium, with only equilibrium forces acting on the probe particle. Our measurement of the comparison between PMR and AMR is consistent with previous studies 21,32 . Here, we find that stronger nonequilibrium fluctuating forces, . f nonequ 2 , are dependent on cells cultured on stiffer substrates (Fig. 1E) and the nonequilibrium fluctuating forces at 1 Hz increase with substrate rigidity (Fig. 1F). This result indicates that cells can generate stronger nonequilibrium fluctuating forces, which might be regulated by molecular-motor dynamics, to sense the mechanics of their microenvironments.
The correlation between intracellular nonequilibrium fluctuating stresses and local stochastic intracellular stiffness modulus. To understand how living cells adjust their mechanical responses to the active, nonequilibrium fluctuating forces, we use oscillatory optical tweezers and phase lock-in detection method to measure intracellular stiffness (k′). Then, we determine the variance of k′ (i.e., the second moment or the standard deviation of k′ at 1 Hz over a 300-second duration as shown in Fig. 2A) and the probability histogram of k′ over the period, as shown in the insert of Fig. 2A. To examine if <k′> remains constant over time, we calculate the mean of k′ at 1Hz from the different measuring time duration. When a cell was cultured on the stiffer substrate, <k′> at 1Hz , is 633 Pa μm and 641 Pa μm averaged from 0~150 s and 150~300 s, respectively (as shown in Fig. 2A). It shows that the "time-averaged" local environmental stiffnesses have a 1% variation from the different measuring time durations. Our data support our assumption that the "time-averaged" local environmental stiffness spectrum <k*> remains constant.
We find larger fluctuations of intracellular stiffness (Δk′) for cells cultured on substrates with larger elastic moduli (Fig. 2B). We define the fluctuating stiffness modulus (ΔG′) through a generalization of the Stokes relation (ΔG′=Δk′/6π a) 10,20,29 (Fig. 1F), where "a" is the radius of the probe particle. Here, we find that as cell cultured on stiffer substrates, an intracellular nonequilibrium fluctuating stress increase with the increasing fluctuations of intracellular stiffness modulus 27 (Fig. 2C).

Response of intracellular stiffness modulus to intracellular stress is nonlinear.
To study how the relationship between intracellular stiffness modulus and intracellular stress (σ) varies with microenvironment rigidity, we determine the intracellular stress via variations of the fluctuations in stiffness modulus and nonequilibrium stress in living cells. The ratio of the fluctuating intracellular stiffness modulus to the intracellular nonequilibrium fluctuating stress (ΔG′/Δσ) is not zero and increases with the substrate rigidity (Fig. 3A). The ratio of ΔG′ to Δσ also increases with intracellular average stiffness modulus, indicating stress-depended stiffness nonlinear mechanical behavior (Fig. 3B). To determine the intracellular stress, we integrate the ratio of the nonequilibrium fluctuating stress (Δσ) to the fluctuations of intracellular stiffness (ΔG′) over all values of intracellular stiffness modulus (G′) (∫(Δσ/ΔG′) dG′) 27 . Here, the value for the linear stiffness modulus G 0 ′ in the absence of intracellular stress is determined to be 5 Pa 15 , which is also in the range of unstressed cross-linked actin networks 33 .
We find that the intracellular stress (σ) of cells attached to a stiff substrate is significantly larger than those attached to a soft substrate (Fig. 3C). Our results also show an increasing intracellular stiffness modulus as a function of increasing intracellular stress (Fig. 3D). This shows a non-linearity of the dependence of cell rigidity on cell stress, with a power 1.2 ± 0.02 (black dash line in Fig. 3D) 27 , which is similar to that of active networks with exponent 1.3 12 and cellular cortex with exponent 1.13 14,15 . To compare with previous results, all of the results show a similar nonlinear mechanical behavior (Fig. 3E) with a strong non-linearity of stiffness modulus for flexibly cross-linked actin networks versus extra-stress 12 and for different cell types versus cellular-traction stress 6,7 , or extracellular stress 15,17 . Instead of application of an external stress, our results indicate that cytoskeletal networks can be turned into a contractile material in living cells by motor activity. The data show that cells and cytoskeletal polymers are a stress-stiffening material in which the network stress controls the stiffness. This indicates that the stiffness of adherent cells would increase with intracellular stress and contractile tension.
To investigate how intracellular stress regulates cell mechanical properties, we vary intracellular stress with drugs (i.e., ML-7, Y-27632, and blebbistatin) that alter motor proteins. Cells treated with Y-27632 and blebbistatin exhibit a decrease in time-averaged intracellular stiffness modulus and intracellular stress (the insert of Fig. 3A,C). There is no significant difference in intracellular stress and time-averaged intracellular stiffness modulus for cells treated with ML-7. Treatments with Y-27632 and blebbistatin inhibit the cell's response to substrate rigidity 27 (Fig. 3D), allowing the intracellular stresses to mimic the mechanical properties of the microenvironment. Our measurements of intracellular stress in response to substrate rigidity have a similar trend to those of previous measurements by traction-force microscopy using either fluorescent particles embedded in a substrate 7 or micropillars 34 to observe substrate deformations. With traction-force microscopy, pulling or contraction by cells would be measured as a time-averaged cellular stress that balances the traction stresses exerted on the substrate by the cells. The results show that fibroblasts tend to match their internal stiffness to that of their substrates up to 20 kPa. Also actin remodeling in cells is enhanced with increasing substrate rigidity 35 , suggesting that actin stress fibers may act as force sensors that transmit tension to focal adhesion complexes, possibly via contribution from myosin motors. Our results are qualitatively consistent with previous studies that show cellular traction stress is mainly regulated by ROCK, but not by myosin light chain kinase [36][37][38][39][40] .

Discussion and Conclusions
We report the noise spectra in a nonequilibrium thermodynamic system of the nonlinear mechanical cytoskeleton network in a living cell. We use microrheology to study intracellular stress in response to substrate rigidity (Fig. 3F). We further vary intracellular stress using drugs that inhibit motor activity and produce a single master curve with a power-law dependence. Results of this study, in particular the data shown in Fig. 3D, describe intracellular stiffness modulus as a strongly nonlinear function of intracellular stress. This suggests that the motors induce internal stress and tension that produce a nonequilibrium and nonlinear state. Our finding emphasizes the close analogy of motor-driven internal stress with external shear stress. These aspects of intracellular stress and We show that fluctuations of a nonequilibrium thermodynamic system provide a direct means to characterize the nonlinear mechanical properties of intracellular stiffness modulus as a function of intracellular stress, which have been difficult to obtain by other approaches. Our data provide evidence that cells can modulate their mechanical properties by modulating their inner mechanical stress. It opens the question as to how living systems use these fluctuations as an energy-efficient mechanism to adapt to their microenvironment. Thus, further examination of these fluctuations will advance the understanding of how cells sense and respond to their mechanical environment, leading to new designs in biomaterials and new therapies for diseases linked to cellular mechano-transduction.

Materials and Methods
Preparation of HeLa cells and polyacrylamide thick films. HeLa cells are cultured in Dulbecco's modified eagle medium supplemented with high glucose (Gibco #11965092), 10% fetal bovine serum (Gibco #16140071), 1% penicillin/streptomycin (Invitrogen #15140-122), 7.5% sodium bicarbonate, 200 mM glutamine, and 1% G418 solution (Thermo Fisher #10131027). The cell line is generously provided by Dr. Keiju Kamijo at the National Institutes of Health. Cells are seeded onto polyacrylamide (PA) substrates coated with sulfo-SANPAH cross-linker and collagen type I (0.2 mg/ml) on 22 × 22 mm cover-slips. Different PA substrates, having varying elastic moduli, are prepared 41,42 . Cells are grown under standard culture conditions (37 °C, 5% CO 2 , humidified environment). To explore the internal cell mechanics with respect to the activity of intracellular motors, we treat myosin-inhibitors for 1 hour using (1) ML-7 (20 μM; Sigma-Aldrich #I2764), which is a potent and selective inhibitor of myosin light chain kinase; (2) Y-27632 (10 μM; Sigma-Aldrich #Y0503), which inhibits the Rhoassociated protein kinase (ROCK) and thus inhibits ROCK-mediated myosin light chain phosphorylation; and (3) blebbistatin (20 μM; Sigma-Aldrich #B0560), which binds to the myosin ATPase and slows phosphate release. Passive microrheology. To measure the total fluctuation spectrum using passive microrheology, the fluctuations of a 1 μm polystyrene particle (Thermo Fisher Scientific #4009A) entrapped in HeLa cells are recorded by a fast camera. The fluctuations of the particle position, which is the absence of optical tweezers force, is recorded for 300 seconds with signal acquisition at 500 frames/sec. To combine active and passive microrheology, we compare the results of total fluctuation and the thermal-equilibrium fluctuation from the same probed particle. Then, average the ten independence measurements from individual probed particles in each cell. Note, the measurements of active and passive microrheology are at the same site but are not time-synchronous.

Data analysis.
To determine intracellular stress (σ), we calculate the ratio between Δσ and ΔG′ which is the fluctuation of intracellular stress and the fluctuations of intracellular stiffness, respectively. Both of Δσ and ΔG′ are averaged independent cells cultured on different rigidity substrates. Each averaged value is calculated from ten independent measurements from individual probed particles in each cell. Here, we integrate the Δσ/ΔG′ over all values of intracellular differential stiffness (∫(Δσ/ΔG′)dG′). First, we use a third-order polynomial form to fit the Δσ/Δ G′ as a function of intracellular stiffness (G′), as shown in Fig. 4A. Then, the relative intracellular stress (σ − σ 0 ), as shown in Fig. 4B, is calculated by integrating the polynomial function, Δσ/ΔG′ (G′). σ 0 is the value independent of intracellular stiffness, G′, for the integrating polynomial function. Since G′ is never zero at any σ, we are looking for the value for the linear modulus G 0 ′ in the absence of intracellular stress, σ = 0. Here, σ 0 is calculated from the value for the linear stiffness modulus G 0 ′ in the absence of intracellular stress is determined to be 5 Pa, which is also in the range of unstressed cross-linked actin networks. The stress-dependent stiffness, calculated by integrating the polynomial function, as cells culture on different rigidity substrates shows in Fig. 4C. Using the same protocol, we determine intracellular stress-dependent stiffness for each drug treatment, including ML-7, Y-27632, and blebbistatin, as shown in Fig. 3D.