Abstract
Cells precisely control their mechanical properties to organize and differentiate into tissues. The architecture and connectivity of cytoskeletal filaments changes in response to mechanical and biochemical cues, allowing the cell to rapidly tune its mechanics from highly-crosslinked, elastic networks to weakly-crosslinked viscous networks. While the role of actin crosslinking in controlling actin network mechanics is clear in purified actin networks, its mechanical role in the cytoplasm of living cells remains unknown. Here, we probe the frequency-dependent viscoelastic properties of the cytoplasm in living cells using multiharmonic excitation and in situ optical trap calibration. At previously unexplored long timescales, we observe that the cytoskeleton fluidizes. The mechanics are well-captured by a model in which actin filaments are dynamically connected by a single dominant crosslinker. A disease-causing point mutation (K255E) of the actin crosslinker α-actinin 4 (ACTN4) causes its binding kinetics to be insensitive to tension. Under normal conditions, the viscoelastic properties of wild type (WT) and K255E+/− cells are similar. However, when tension is reduced through myosin II inhibition, WT cells relax 3x faster to the fluid-like regime while K255E+/− cells are not affected. These results indicate that dynamic actin crosslinking enables the cytoplasm to flow at long timescales.
The actin cytoskeleton is a network of filaments crosslinked by actin binding proteins. The cytoskeleton provides mechanical support and drives cell motility and morphological changes1. Crosslinking proteins undergo continuous cycles of binding and unbinding, enabling the cell to act as an elastic solid at short timescales and as a viscous fluid at long timescales2,3. To characterize the mechanics of the active, viscoelastic cellular environment, we developed methods to probe cytoplasmic mechanics in living cells over timescales ranging from 0.02 to 500 Hz using an optical trap (OT). Active microrheology as applied here is insensitive to non-equilibrium active cellular processes like motor protein-based transport and cytoskeletal dynamics because only the response that is coherent with the applied displacement is analyzed4,5. OTs can apply a local deformation directly in the cytoplasm of living cells, while other commonly used active methods such as atomic force microscopy (AFM) and magnetic twisting cytometry apply a local deformation at the cell surface. The viscoelastic response is described by the frequency-dependent complex shear modulus G*(f) = G’(f) + iG’’(f) where the real part G’(f) and the imaginary part G’’(f) correspond to the elastic (storage) and viscous (loss) modulus, respectively. While the magnitudes of G’(f) and G’’(f) reported in the literature vary greatly between different measurements, their frequency dependence is well conserved across different cell types and experimental conditions4. G’(f) and G’’(f) increase with frequency following a weak power law G*(f) ~ fα with α = 0.05-0.35 from ~1 Hz to ~100 Hz4–18, and a stronger power law G*(f) ~ fβ with β = 0.5-1.0 for frequencies above ~100 Hz4,7,13,19,20. The weak power law has been attributed phenomenologically to the soft glassy material (SGM) theory12,13,21, while the strong power law has been explained by entropic filament bending fluctuations13,22,23. However, at timescales longer than 1s (i.e. at frequencies below ~1 Hz), relevant to cell division, motility and morphological changes, active microrheological data is scarce especially in the cytoplasm of living cells and microrheology from thermally driven particles is prone to errors from the contribution of active processes in the cell4,5. At these long timescales, transient actin crosslinking by various actin crosslinkers govern the viscoelastic properties of purified actin networks24–31. When these networks are deformed at frequencies slower than the crosslinker’s unbinding rate (koff), the crosslinkers have enough time to unbind, allowing the filaments to freely slide past one another resulting in a more fluid-like, viscous network (Fig. 1a). At deformation frequencies faster than koff, the actin filaments remain highly interconnected resulting in filament bending and a more solid-like, elastic network (Fig. 1b). We hypothesize that this role of dynamic actin crosslinking may allow the actin cytoskeleton to fluidize during movement and morphological changes, while also maintaining structural integrity. Active microrheological studies on the cell surface have reported G*(f) at frequencies at least one decade below 1 Hz11–18 but the distinct4 mechanical properties in the intracellular environment at these long timescales remain unexplored. OTs are well-suited to measure these mechanical properties and while several studies have reported intracellular viscoelasticity5,6,8–10,19,20,32 the mechanics at ~0.01-1 Hz have not been quantified in detail. Experimental measurements of slow dynamics are challenging due to the long observations required and, while OT calibration in the cell provides more accurate measurements33–35, it has often been neglected.
To perform simultaneous measurements over a wide frequency range, we optically trap a 500-nm polyethylene glycol (PEG)-coated bead located in the cytoplasm (Fig. 1c,d, Methods) and apply a multiharmonic excitation input to the trap position using an acousto-optic deflector (AOD). Because there are significant elastic contributions from the cytoplasm and non-equilibrium active processes, the trap stiffness, ktrap, and the photodiode constant, βpd, cannot be calibrated using traditional methods relying on thermal fluctuations. Here, a simple viscoelastic model is fit to the measured response to calibrate the optical trap in situ34. From the time domain signal of the photodetector and the trap position (Fig. 2a), the frequency response function (FRF) is obtained (Fig. 2b). The magnitude and phase of the FRF, as well as the power spectrum are simultaneously fit to the model to extract ktrap and βpd. The magnitude response at frequencies above ~1 Hz is well-captured by this simple viscoelastic model, with a first inflection point at ~4 Hz representing the stiffness of the cytoplasm and a second inflection point at ~200 Hz representing the combined stiffnesses of the cytoplasm and the OT (Fig. 2b, Supplementary Information). The resulting ktrap and βpd vary between cells and between experiments due to variable optical properties (Supplementary Fig. 2), reinforcing the necessity for in situ calibration. Despite variability among cells, the product of ktrap and βpd falls within the same constant range, as reported previously36 (Supplementary Fig. 2). The response measured in a 20% PEG water solution agrees well with data obtained from a rheometer (Supplementary Fig. 3). Once calibrated, ktrap and βpd are used, along with the FRF, to obtain the complex shear modulus G*(f)37. Combined with multiharmonic excitation, this calibration method allows ktrap, βpd and G*(f) to be obtained in a single continuous measurement, thereby minimizing sources of error (see Supplementary Information). This approach has advantages over other methods used to calibrate the OT in active, viscoelastic media. Methods based on light momentum changes rely on capturing most of the diffracted and scattered light36. However, organelles and other structures result in substantial refraction. An alternative method uses the active response to sinusoidal inputs combined with the passive response for in situ calibration in viscoelastic medium19,20,33–35,38–40. ktrap can be obtained from the average ktrap values at each input frequency based on the ratio of the real part of the active spectrum to the passive thermal spectrum at the same frequency19,20,33,35,38–40. βpd can be calibrated separately using an active oscillation of the stage or the laser combined with the position readings from the camera or photodiode. However, both ktrap and βpd are sensitive to sources of error as they are derived from a single frequency point.
Using multiharmonic excitation and in situ calibration, we characterize the viscoelastic properties of the cytoplasm of living cells at frequencies ranging from 0.02 to 500 Hz. In WT fibroblasts, the measured moduli can be fit to a double power law (Fig. 3a, magenta), yielding a weak power law (α = 0.116 ± 0.021) at frequencies between ~1 Hz and ~20 Hz and a stronger power law above ~20 Hz (β = 0.717 ± 0.062), in excellent agreement with previous work5,8,13,19. However, at frequencies below ~1 Hz, we observe a pronounced transition to a fluid-like response, apparent in the loss tangent (G’’/G’) which reaches a minimum at ~1 Hz and increases steadily with decreasing frequencies. This behavior is not captured by the power law model (Fig. 3a, inset). This fluidization occurs at timescales relevant to slower processes such as cell migration2,41 and cell division3,42. Fluid-like relaxation has been observed in mixtures of actin filaments and actin crosslinkers in vitro, where the timescales of relaxation are determined by the crosslinker unbinding kinetics27–31. The observation of similar fluidization in cells is surprising as many crosslinking proteins with different binding kinetics likely contribute to the viscoelastic response. The first term of the phenomenological double power law model describes a SGM at longer timescales and the second term describes frequency scaling from entropic filament fluctuations at shorter timescales13 (Fig. 3b, top). The rheology of a SGM has been empirically observed as elastically dominant and weakly scaling with frequency with a constant loss tangent of ~ 0.143. In contrast, our data indicate the loss tangent of the cytoplasm varies at frequencies below ~1 Hz and reaches 1.0, indicating an equal viscous and elastic contribution at low frequencies (Fig. 3a, inset). Fitting a model that assumes mechanics are governed by the unbinding kinetics of a dominant crosslinker28 captures the data well (Fig. 3a, gray). The semi-phenomenological dynamic crosslinking model has a first term derived from the Fourier transform of the exponentially decaying lifetime of crosslinked points. This first term decreases the plateau elastic modulus G0 and increases the amount of viscous dissipation in isotropically crosslinked actin networks28. The second term describes filament bending fluctuations at shorter timescales as before. The dominant crosslinker’s characteristic timescale, koff, appears as a local maximum in the viscous modulus (Fig. 3a) at the frequency of relaxation (fr = koff/2π). Fitting this model to the measured moduli yields an estimate of koff = 0.24 Hz (Fig. 4b) for WT fibroblasts. This estimate is in agreement with unbinding rates measured for several passive actin crosslinkers44,45. The dynamic crosslinking model was derived from and studied in bulk rheology of actin polymer gels where the unbinding of crosslinkers was related to bulk properties through the assumption that G0 ~ Nx with x = 1 (where N is the number of crosslinked points) in a homogeneous and isotropically crosslinked network28. However, the cytoplasm is likely a composite network with the coexistence of bundled, branched, and isotropically crosslinked actin filaments, despite our efforts to probe the most homogenous region of the cytoplasm (Fig. 1c,d). Additionally, cell orientation and deformation axis also result in anisotropic mechanical properties9. The same model with G0 ~ Nx and x not equal to 1 may account for some of this anisotropy. In addition, in bulk rheology, the shear stresses are assumed to be evenly distributed at the microscale. Thus, the resulting mechanical response is a summation of the contributions of individual crosslinkers across the sample. However, in microrheology, an increased viscous dissipation from the depletion of crosslinked points in the vicinity of the probe is expected, causing a crossover of elastic and viscous moduli at lower frequencies28. Thus, it may be possible to further improve the fit to the model by including a stronger frequency-dependent loss of crosslinking points. All in all, our data along with the fit suggest that dynamic crosslinking of the cytoskeleton results in the fluidization of the cytoplasm. These findings suggest that the cell could tune its viscoelastic properties to suit its mechanical needs by modulating the crosslinking dynamics of key crosslinkers. Correspondingly, a misregulation in crosslinking dynamics may cause the onset of diseases46. To investigate the role of actin crosslinker binding kinetics in governing cytoplasmic mechanics, we probe the viscoelastic properties of cells heterozygous for an ACTN4 mutation. ACTN4 is a ubiquitous crosslinker that dynamically crosslinks actin filaments with koff ~0.4 Hz44 and is involved in the formation of actin bundles47. The K255E mutation has been associated with focal segmental glomerulosclerosis (FSGS)48. In FSGS, podocytes are less able to resist cyclic loading, suggesting a misregulation of the mechanical properties of the cytoskeleton49. Indeed, this mutation alone reduces cell motility while increasing cellular forces in heterozygous cell lines through an interplay with tension-generating myosin II motors46. This effect is believed to be due to the exposure of a cryptic actin binding site in the K255E mutant, resulting in its increased affinity of for actin (i.e. slower koff)47. In WT ACTN4, the cryptic actin binding site is thought to only be exposed when the crosslinker is under tension, resulting in a catch-bond behavior24 (Fig. 4b, top inset). The difference in the unbinding rates of the WT ACTN4 and the K255E mutant results in a shift of fr in purified networks24,25,31. In contrast, our results show no significant difference in the mechanical properties of WT and K255E+/− cells (Fig. 4a), possibly due to lower expression of K255E mutant and other mechanical anisotropies in the region probed. Overall, with koff WT = 0.24 Hz and koff K255E = 0.37 Hz (Fig. 4b) from the average best fit, our results suggest similar crosslinking dynamics in the WT and K255E+/− cells under normal conditions (Fig. 4b, top and bottom insets).
To isolate the role of tension on ACTN4 binding kinetics, we treat the cells with 50 μM Blebbistatin (BB). BB reversibly locks myosin II in the low-affinity ADP-Pi state, reducing tension in the actin cytoskeleton50. With 50 μM BB, a sufficient number of mildly affected cells (Supplementary Fig. 6) remain viable for mechanical measurements. Surprisingly, we do not observe significant softening of the cytoplasm (Fig. 4b) as reported previously5. This may be due to differences in BB treatment response levels inducing more local heterogeneity, as depicted by the greater variability in the data (Fig. 4a, bottom) and the wider distribution of koff (Fig. 4b, top), or differences in the in situ calibrated ktrap and βpd values (Supplementary Fig. 2). It is also likely that the local cytoplasmic microenvironment is less sensitive to actomyosin activity, and thus BB treatment, compared to on the cortex where a much more dramatic effect on magnitude is expected14. In terms of frequency-dependence, we observe a ~3-fold faster relaxation in BB-treated WT cells, with koff increasing from 0.24 Hz to 0.89 Hz in the presence of BB (Fig. 4a, bottom). Interestingly, koff in K255E+/− cells remained similar, with koff K255E = 0.37 Hz and koff K255E BB = 0.52 Hz. The results suggest that with reduced tension (through myosin II inhibition), actin binding kinetics differ in a manner that is consistent with the differences between the binding kinetics of WT and K255E ACTN4 (Fig. 4b, top and bottom insets). While tension levels vary between cells and within a single cell51, it is expected that, on average, the tension levels are reduced with BB treatment. The active crosslinker myosin II may also directly play a role in regulating the observed mechanics as shown previously in purified networks52,53 at intermediate and long timescales. The effect of crosslinking dynamics on relaxation is also readily apparent in the differences in loss tangent (Fig. 4c). Latrunculin A (Lat A) depolymerizes actin filaments by sequestering free actin monomers55. Interestingly, treating WT cells with Lat A shifts the relaxation to faster timescales (Supplementary Fig. 5), suggesting actin filaments disruption plays a similar role in relaxation as the inhibition of intracellular tension7. In Lat A-treated cells, the presence of fewer, shorter actin filaments is expected to reduce the ability of ACTN4 and myosin II to interact with and crosslink actin filaments, reducing the tension generated and stored in the network53. This reduced number of crosslinked points does not alter the frequency of relaxation in purified networks25,28,29, suggesting that the shift in relaxation that we observe in Lat A-treated WT cells results from changes in binding kinetics, similar to what is observed with BB treatment. In both BB- and Lat A-treated cells (Fig. 4 and Supplementary Fig. 5), the loss tangent indicates that the network is more fluid-like at lower frequencies, consistent with previous observations using AFM7. However, similar to BB treatment, and in contrast with previous studies7,18, we observe negligible differences in the magnitudes of G* with Lat A treatment. Intermediate filaments, another component of the cytoskeletal network, contribute approximatively half of the mechanical resistance in the intracellular environment10 and thus help maintain mechanical integrity even when the actin cytoskeleton is disrupted. Furthermore, cells that show severe signs of actin cytoskeleton disruption are not viable for mechanical measurements. Consequently, healthier cells (Supplementary Fig. 6) were chosen preferentially to yield more consistent and quantifiable results, which may partly explain the similar magnitudes observed under BB and Lat A treatment. Overall, our results show a shift in relaxation to faster timescales when myosin II is inhibited, particularly in the case of WT cells, suggesting an earlier transition to the fluid-like regime consistent with the tension-dependent differences in binding kinetics between WT and K255E ACTN4.
Mechanical properties play an important role in the regulation of many cellular functions and allow the cell to be solid enough to provide structural support but also fluid enough to reorganize and perform dynamic functions. By developing a novel method to measure intracellular mechanics, our measurements reveal relaxation of the cytoskeleton at long timescales across all conditions, pointing to a fundamental mechanical property of the cytoplasm. In addition, a model that includes the binding kinetics of a dominant actin crosslinker captures these relaxation dynamics well. However, while our data suggest a relationship between mechanical relaxation and the crosslinking dynamics of ACTN4, many other crosslinkers and motor proteins contribute and it is unlikely that ACTN4 alone governs the viscoelasticity at these timescales. Other cytoskeletal components with their associated proteins may also contribute to cytoplasmic relaxation through mechanisms that are yet to be identified. Overall, we performed calibrated mechanical measurements in the cytoplasm of living cells using optical tweezers and related the relaxation dynamics observed to the binding kinetics of ACTN4. Future work focusing on other crosslinkers and other components of the cytoskeleton, as well as on simplified in vitro systems, will add to the results reported here and provide a more complete understanding of the mechanisms governing cytoplasmic fluidization.
Author Contributions
A.G.H., A.J.E., H.K.H. and L.C. designed the research. L.C. and H.K.H. developed experimental protocols and analytical tools. L.C. performed experiments and analyzed data. A.J.E. provided cells and reagents. L.C. and A.G.H. wrote the paper.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information
Methods
Cell culture and preparation
Two dermal fibroblasts cell lines are used from previously immortalized cell lines46. Cells are cultured in DMEM media (Thermo Fisher Scientific), supplemented with 10% FBS (Thermo Fisher Scientific) and 1% Glutamax (Thermo Fisher Scientific). Cells are passaged at 80% confluency and plated 24h before the experiment at low enough confluency to minimize cell-cell contacts. 500 nm diameter fluorescent latex beads (Thermo Fisher Scientific) are PEG-coated following previously protocol56. Prior to the experiment, the beads are washed and resuspended in hypertonic media (~109 beads/mL in 10% PEG 35000, 0.25 M sucrose in supplemented DMEM media). The cells are left in the incubator for one hour to recover from the shock. The coverslips are mounted on microscope slides using double-sided tape to form a flow chamber. Complete media, supplemented with 50 μM BB or 100 nM Lat A in some experiments, is flowed into the chamber. Measurements start 15 minutes after treatment and cells are kept at 37 °C using a heater (World Precision Instruments) in a custom-built environmental chamber during all optical trapping experiments.
Optical trap
The optical trap installed on an inverted microscope (Eclipse Ti-E: Nikon) is used with a 1.49 NA oil-immersion 100x-objective. The near-IR 1064 nm laser beam (10 W, IPG Photonics) is expanded to overfill the back aperture of the objective. The bead position relative to the laser center is measured using a lateral effect photodiode (Thorlabs) positioned conjugate to the objective’s back focal plane. The laser is steered using 2-axis AODs (QuantaTech) controlled through a field-programmable gate array and custom LabVIEW programs (National Instruments). A multiharmonic (~18 frequencies) excitation input wave is applied to a single intracellularly located bead per cell. The frequencies range from 0.02 Hz to 1300 Hz (moduli obtained from 0.02 to 500 Hz), with corresponding amplitudes ranging from ~50 nm to ~1 nm. The frequencies of oscillation are selected to avoid subharmonics generation. The amplitudes are empirically chosen to yield a linear response from the cytoplasm. No significant subharmonic generation was observed from the amplitudes applied, indicating that the measurements were performed in the linear regime57.
Optical trap data analysis
Data is acquired at 200 kHz downsampled to 20 kHz or directly acquired at 20 kHz (Supplementary Fig. 1). ktrap and βpd are obtained for each individual measurement as previously described34. In brief, the FRF is obtained from the cross spectral density of the bead displacement and AOD using Welch’s method on MATLAB. The window length is chosen at each individual frequency to be an integer multiple of the period of the input frequency to avoid leakage33. The integer multiple, is chosen to obtain the best trade-off between frequency resolution and averaging. A simplified viscoelastic model is simultaneously fit to the FRF and the power spectrum of the thermal response to obtain ktrap and βpd34. The elastic and viscous moduli are directly obtained from the FRF knowing ktrap and βpd37. See Fig. 2 and Supplementary Information for more details.
Statistics
1000 bootstrap samples are drawn from the data and each bootstrap sample is fit to either the dynamic crosslinking model or the power law model, allowing distributions of fit parameters to be obtained for each condition. To compare the two model fits, Bayesian Information Criteria (BIC) are used. Fitting bootstrap samples allows statistics to be directly obtained and also allows a more representative comparison of the fit parameters between conditions, as oppose to directly fitting the mean of the data. See Supplementary Information for more details.
Supplementary Information
Data handling
To collect data over several periods of excitation, a single measurement typically lasts 220 seconds or longer and active oscillations across all ~17 frequencies are continuously being applied. The frequency response function (FRF), i.e. the transfer function H(ω), is obtained using Welch’s method, implemented in MATLAB with the “tfestimate” function. The Hamming window function is used and gives a slightly better response in terms of magnitude and coherence compared to the Hann window. The transfer function estimate is obtained at each input frequency separately with separate window lengths (NFFT). The window length is chosen to be an integer multiple of the input frequency to minimize signal leakage into adjacent frequency bins. The per cent overlap between windows is chosen based on empirical observations to yield the maximum magnitude and maximum coherence. At the lowest frequencies, less periods of excitation are contained in the measurement total time and the per cent overlap is thus chosen to be larger to allow more averaging over the ~4 windows. At higher frequencies, the overlap is chosen to be 50% over the 6-8 windows. All windows and overlap amounts are chosen such that at most, ~15 % of the data is truncated at the end. Indeed, as we have a finite sample length, truncation of the data occurs when the windows don’t exactly add up to the total length. Overall, the aforementioned window lengths and overlaps yield the best balance between frequency resolution and averaging. However, a more detailed and quantitative characterization may help for future application of this method. The magnitude and phase obtained are fit to the active part of the viscoelastic model discussed further below.
The power spectrum used for the thermal response is obtained using the same data. The peaks that correspond to the active oscillations are cropped out based on the known excitation frequencies. The remaining power spectrum represents the thermal response. For data that is acquired at 200kHz downsampled to 20kHz using a Field Programmable Gate Array, a pole is added to the viscoelastic model (corner frequency ~ 3000 Hz) to account for this low pass filter coming from the downsampling by averaging. Sampling at 200kHz downsampled to 40kHz shifts this filtering to higher frequencies, confirming that the averaging is acting like a low pass filter (data not shown). Data acquired directly at 20kHz show a reduced signal to noise ratio compared to oversampled and averaged data, where the noise floor is visible on the power spectrum at high frequencies (Supplementary Fig. 1b). In these cases, the power spectrum is fit only up to the point where this upwards shift is observed.
In situ calibration
From the mechanical circuit (Fig. 2b), for the case where the stage is stationary and by making the following substitution: Δx(t) = xbead(t)-xtrap(t) and kcyt(t) = kcyt,0 + kcyt,1t−(α-1), the following equation of motion can be obtained
Substituting the cytoplasmic stiffness (kcyt) by the sum of the constant stiffness (kcyt,0) and the frequency dependent stiffness (kcyt,1t−(α-1)) observed in semiflexible polymer network allows the model to account for some of the viscoelastic properties of the cytoplasm1. Looking at the active part only (i.e. Fthermal = 0), neglecting inertial forces and taking the Laplace transform at steady state (s = jω), the following transfer function H(ω) = ΔX(ω) / Xtrap(ω) is obtained:
Where Vpd(ω)βpd = ΔX(ω) is used and represents the voltage from the photodiode and the photodiode calibration constant respectively. For visualization, if one ignores the complex cytoplasmic stiffness (kcyt,1 = 0), the numerator is a single zero proportional to 1/kcyt,0 and the denominator is a single pole proportional to the inverse of the total stiffness, 1/(ktrap + kcyt,0). The zero and pole are responsible for the two inflection points in the magnitude response, leading to the sigmoidal curve observed in the magnitude response obtained experimentally (Fig. 2b and Supplementary Fig. 1a). The addition of the frequency dependent stiffness shifts the zero and pole positions and changes the sharpness of the transition between the two1. The magnitude and phase of EQ (1) is what we fit to our experimental H(ω) obtained through tfestimate as outlined above.
Looking at the passive part only (Xtrap = 0) and taking the Laplace transform at steady state as before, the following expression is obtained:
This complex response function χ(ω) can be related to the complex shear modulus through Stokes relation2
Here, Gth(ω) represents the theoretical complex shear modulus based on the simple viscoelastic model, and R is the radius of the spherical probe. From the fluctuation-dissipation theorem, the power spectral density of the bead can be related to this response function χ(ω) or equivalently, to Gth(ω) as follows3:
Where the following substitution has been done , Vpd is the voltage measured by the photodiode, kB the Boltzmann constant and T the temperature. EQ (4) is the equation fit to the power spectrum, with Gth(ω) defined by EQ (2) through EQ (3). The corner frequency in the power spectrum is the result from the sum of the total stiffness of the system and corresponds to the pole in the magnitude response described above. α dictates the slope after the corner frequency, with α=1 corresponding to purely diffusive behavior and a slope of 2 in the power spectrum, and α<1 corresponding to constrained diffusion with a slope < 2 in the power spectrum. For more details on the different parameters of the model, see reference4.
Once the calibration is done, the complex shear modulus G(ω) of the real cytoplasm can be directly obtained using the experimental transfer function H(ω) of the measurement following reference5:
Where A(ω) is the apparent complex response function (See equations (4) and (5) in reference5) which can be directly related to the experimental transfer function H(ω)
With the previously calibrated ktrap and βpd calibrated from EQ (1) and EQ (4) and the experimental H(ω), A(ω) can be solved using EQ (6) and the complex shear modulus G(ω) of the actual cytoplasm can be obtained using EQ (5).
Simple viscoelastic model fitting
A global fitting algorithm is used to fit simultaneously EQ (1) and EQ (4) to the FRF and power spectrum of the data respectively. As mentioned previously, only frequencies above ~1 Hz in the FRF and frequencies above ~200 Hz in the power spectrum are included in the fit (Supplementary Figure 1). Occasionally, despite the addition of an extra pole compensating for the filtering effect observed when data acquisition was done with downsampling (Supplementary Figure 1b, right), or direct sampling where filtering effect was not observed (Supplementary Figure 1b, left), the FRF at higher frequencies showed the presence of an extra pole, causing a shift in magnitude and phase. As the simple viscoelastic model did not include a term to correct for this shift, fitting of the FRF was stopped before reaching these frequencies. The last point in the magnitude plot in Supplementary Figure 1a left shows such a dip in magnitude, indicative of filtering at higher frequencies. Further experiments and quantification of this phenomenon may help to isolate the source and correct the data above ~500 Hz allowing the higher frequencies to be reliably obtained (frequencies past ~500 Hz were not included in this manuscript). A weight function was applied to the FRF and power spectrum to account for the large difference in number of points (~11 for the FRF, and thousands for the power spectrum). A second weight function was applied to the FRF to weight the magnitude response more than the phase as the latter appeared to be more sensitive to sources of error. Each fit begins with an initial guess that is manually found to be close to the solution and with tighter bounds to restrict the algorithm from finding a local minimum in a completely separated region in the parameter space. A second fit is done based on the output of the first fit with looser bounds.
Bootstrapping and cell mechanics model fitting
Bootstrapping with replacement is done by generating k numbers randomly drawn from 1 to N. N is the total number of data sets per condition, and k is the minimum amount of data available at any given frequency. For example, in the case of untreated WT, the number of data points available per frequency varied from 14 to 17, due to slightly different excitation inputs. In that case, k = 14, and N = 17. This bootstrapping is repeated 1000 times to obtain 1000 bootstrap samples. The mean of these samples is first manually fit to either the power law or dynamic crosslinking model to obtain an initial guess of the parameters. Then, each bootstrap sample is automatically fit to either models using the corresponding initial guess. Each bootstrap sample is fit 3x, starting at the initial guess with stricter bounds, and passing the best parameters found for the next 2 subsequent fits with increasingly looser bounds.
Acknowledgments
The authors thank Gary Tom and Giancarlo Szymborski for their work on the control software for the optical trap, Chris Sitaras for helpful discussions and hands-on assistance in PEG-coating of the beads, and both Chris Sitaras and Ajinkya Ghagre for bulk rheology data. The authors also thank Malina K. Iwanski, Abdullah R. Chaudhary and Linda Balabanian for helpful discussions and valuable inputs on the manuscript. AGH acknowledges support from NSERC (RGPIN-2014-06380) and the Canadian Foundation for Innovation. AJE acknowledges support from NSERC (RGPIN/05843-2014, EQPEQ/472339-2015, RTI/00348-2018), CIHR (Grant # 143327), and the Canadian Foundation for Innovation (Project #32749).