FRAP analysis

Liposomes containing concentrated BSA and IVCE were prepared using the droplet-transfer method [153, 173] with 1 mM eggPC in mineral oil (Nacalai Tesque, Kyoto,

Japan, product code: 23306-84). Initial macromolecule concentrations of BSA and IVCE were 0.14 and 0.11 g/mL in NH buffer (220 mosM), respectively. After for-mation, liposome solutions were mixed with equal volumes of 220, 660, and 1100 mM aqueous sucrose in order to osmotically compress and concentrate the contents.

Macromolecule concentrations inside the liposomes were estimated from the fluores-cence intensity of GFP (0.2 mg/mL) measured by confocal microscopy (Olympus FV1200, Tokyo, Japan). After photobleaching with a circular spot of a radius r ∼ 1 µm for 1 s at full laser power, the fluorescence recovery was recorded. For convenience, the recovery intensity as a function of time was fitted to the following conventionally used equation: f(t) = exp(-2τ/t) × [I0(2τ/t) + I1(2τ/t)], where I0 and I1 indicate modified Bessel functions, and τ is the characteristic time scale of diffusion [174].

This formula is merely valid for 2D diffusion, typically when photobleaching was per-formed with an unfocused laser in bulk solution. The formula r²/4τ then gives the diffusion coefficient of GFP. In this study, however, FRAP was performed with a focused laser in samples with limited dimensions (spheroplasts and liposomes). Be-cause the measured value of r²/4τ merely reflects the scale of the diffusion in this case, it is crucial to normalize it to the value measured for pure solvents with all other conditions unchanged.

11.4 Measurement of ATP concentration

Figure 58. Chemical equation in Firefly-Luciferase Luminescence.

Fluolocence is emitted, after the luciferase reacts with ATP. This figure is from the protocol of ATP measurement kit (Toyo Binet Co., Ltd.)

To measure the mechanics in ATP depleted cells, ATP depleted cells were prepared by adding NaN3 and 2-Deoxy-D-Glucose to the culture media in accordance with the previous study [170]. Here, intracelular ATP concentration both ATP depleted cells and untreated cells are actually measured by using an intracellular ATP measurement kit (manufactured by Toyo Binet Co., Ltd.). In the intracellular ATP measurement

11 MATERIALS AND METHODS

of this kit, ATP are rapidly extracted from cultured HeLa cells and concentration of these ATP are measured by Firefly-Luciferase Luminescence. In the firefly luciferase luminescence reaction, an luciferase which is enzyme reacts with ATP and oxygen molecules (O2) to form excited state oxyluciferin. When this oxyluciferin transits to the ground state, fluolocence whose maximum emission wavelength of 562 nm is emitted (Fig. 58). Since ATP is essential to emit the fluolocence in this reaction, the amount of ATP in the sample is measurable from the amount of luminescence.

Figure 59. Relative emission intensity of ATP in untreated cells and ATP de-pleted cells.

The vertical axis shows the amount proportional to the emission intensity, and the horizontal axis represents the emission wavelength. Red lines show the emission wavelength spectrum of each untreated HeLa cell (n= 4) and the blue dotted line shows the emission wavelength spectrum of ATP depleted cells (n= 4). A yellow vertical line shows the maximum emission wavelength 562 nm of the emission in this reaction.

As a measurement procedure, 5000 cells were dispersed on each of 96 well plates and cultured for 24 hours. After 24 hours, the culture medium was replaced with ATP-depleting medium and cultured more 12 hours to prepare ATP depleted cells.

For measuring ATP concentration. after each culture medium was all taken out, 100 µL/well of ATP extraction reagent was added, and the mixture was kept at room temperature. After 5 minutes, 100µL of the mixture which extracted ATP from each untreated cells and ATP depleted cells were dispensed to the cuvette, and 100 µL of the luminescent reagent was added to the cuvette to be measured. After this reaction,

the cuvette was swiftly shaken and the emission was measured with a luminometer.

The result of the ATP measurement is shown in Fig. 59. In the ATP extract from untreated cell specimen, the emission wavelength spectrum of light emitted when oxyluciferin returns to the ground state can be observed. On the other hand, no emission wavelength spectrum was observed from specimens of ATP extract from ATP depleted cells. From this results, it was confirmed that ATP was depleted well in the cells in which we tried to deplete ATP.

Chapter IV

Discussion

We measured mechanical properties of in vitrocytoplasm and in living cells to study the effect of molecular crowding. The viscosity η of in vitro cytoplasm obtained from three different organisms (E. coli, HeLa cells , and Xenopus eggs) was quanti-tatively similar, increased superexponentially in the concentration of macromolecules c mg/ml. Development of the feedback-controlled microrheology enabled us to mea-sure the mechanical properties in living cells. Viscosities η in three different kinds of living cells measured with the novel technique was quantitatively similar; they increased exponentially with c. Our novel technology allowed us to study the effect of nonthermal activation on cell mechanics; results were consistent to the concept of glassy cytoplasm. Energy depleted cells, for instance, showed less FDT violation and became 10-times more elastic than untreated cells, most possibly due to glass transition.

In this chapter, we discuss general physics on the novel concept on living cytoplasm that we propose as actively driven glass-forming material or actively-driven colloidal glass formers . This thesis study made so many of unprecedented novel experimental findings and raises so many of physical questions. Why do we not see enormously heterogeneous mechanics in cells as expected in prior studies? Why totally different living organisms show quantitatively similar glass-forming behavior? How metabolic activity alters cytoplasm from fragile to strong glass formers? And many others. Simply contemplating on our experimental findings in the light of the concept active glass formers , we show that all these questions are explained beautifully and consistently. Before that, we need to start with a more basic question whether the living cytoplasm is truly a glass-forming material. As described above, viscosity η of in vitro cytoplasm and in living cells increase rapidly with the increase of con-centration of macromolecules c. The rapid increase of viscosity is typically observed when microscopic kinetics critically slowed down upon crowding. On the other hand, similar critical increase of viscosity occurs in gelation as well. The difference between glass formation and gelation are subtle especially for colloids and soft matters; thus requires further investigation and discussion.

12 Glass or gel

The drastic increase of viscosity is observed in glass formation and also in gelation.

Kinetics are frozen and cross-linked structures are permanently fixed in a gel. On the other hand, mechanics can relax in glassy state if you were patient enough to wait and observe long enough, possibly longer than your life time. Glasses and gels could be distinguished theoretically by observing the terminal relaxation in the relaxation spectrum. Unfortunately, however, it is not possible to observe very slow relaxation in glass in actual experiments. Note that the terminal relaxation time of a glass is too long to measure, which is another kind of a typical behavior of a glass. On the other hand, both glass-forming liquids and sols flow in measurable time scales; they show various complex relaxation spectra, depending on their constituents, dynamics and interactions. Therefore, there is no general practical way to distinguish gels and glasses from their viscoelastic relaxation spectra. Besides, gels and glassy states share similar features. That is why clarifying the distinction between the two types of nonergodic states (glasses and gels) is one of the main issues debated in the field [175–179]. However, in our case, it is possible to collect various indirect evidences as we show below.

We tested the possibility of gelation by conducting force clamp MR (Fig.35). When forces less than 1 pN were applied, all beads started moving without showing yielding behavior. Besides, probes in BSA and IVCEs show uniform motion at constant veloc-ity, and all samples behaved as Newtonian under a minute-force application (Fig.35B).

These results mean that samples are homogeneous in the smaller length scales than the probe size (2a = 1 µm). Since the size of intracellular bio-polymeric colloids are several hundreds of nm, all bio-polymeric colloids disperse respectively without con-structing the structures in cells [55]. For comparison, we conducted the Force clamp MR in polymer gels. Trajectories of probes that were embedded in actin gels (1.3 mg/mL) crosslinked with H-meromyosin (0.04 mg/mL) and were subjected to 2.5, 3.4 and 4.3 pN forces are shown in Fig.44A as a function of time. In this entangled gel, probes pulled by optical-trap forces either stay at a rested position for a smaller force application (2.5 pN) or move with intermittent jumps and hops for larger forces (3.4, 4.3 pN). These results indicate that gels can flow only after yielding at certain

12 GLASS OR GEL

threshold force. After yields, probes move with intermittent jumps and hops. Probes dispersed in cell extracts did not show any indication of such intermittent yielding.

Another qualitative criterion to distinguish glasses and gels was proposed based on the rheological behaviors close to sol-gel or liquid-glass transitions. Compared to glass formers, viscosity of gel-forming materials usually increases immediately in the narrow range of concentrations [120, 180] following the form consistent to the percolation theory asη =A(cg−c)^−β with 0.7< β <1.4 [181,182]. The same formula η = A(cg − c)^−β has been also used to describe the glass-forming behavior [183], which was derived based on Mode-coupling theory [184] and is the analog of Doolittle equation (Eq.(70) in section 8) if it is not too close to the jamming concentration.

Although the apparent formula is the same, the power-law exponents in glass-forming liquids are higher than those in gels (Hard spheres: β = 2.5 andα-crystallin: β = 3.2) [152, 185]. As seen in Fig.60, our data fit with this equation only with the exponent (β = 3.6) appropriate for glass-forming liquids.

Figure 60. Power-law fitting for the viscosity of cell extracts.

Green triangles are viscosityη of cell extracts taken fromE. coli. Concentration dependence of η can be fitted by power-law as η/η_w∝(0.26−c)^−3.56.

Thirdly and most importantly, we believe that the universal behavior that we found in mechanics of living andin vitro cytoplasm by itself strongly indicates their glass-forming property. Cell-type dependent elasticity should be expected if the cytoplasm are solidified with attractive interactions (i.e., gelation by cross-linking). That is because most of the attractive interactions (e.g. hydrogen or covalent bonding) need

specific chemical combination of functional groups. In that case, mechanics would largely depend on the slight difference of these chemical constituents. Therefore, that our experiments did not show cell-type dependent mechanics by itself implies that gelation is not the main cause of the super-exponential increase of viscosity. On the other hand, repulsive interactions are mostly non-specific. Therefore, repulsive forces that act between components are more affected by their physical properties such as their shape, size, and stiffness, rather than their chemical properties. It is reasonable to expect that the distribution of these physical properties of cell constituents does not vary much for different cell types, and that can explain the observed universality in the glass-forming behaviors of cytoplasm. As the last minor comment, van der Waals forces are attractive and not specific. Though they are weak in molecular scales, they act additive; they could be effective for aggregations/phase separations that are typically formed at extremely high concentrations. Note that the constituents of cytoplasm were well dispersed at conditions in our experiments. We therefore believe that the effect of van der Waals forces were marginal for the cytoplasmic mechanics measured in this study.

Finally, we have additional evidence thatin vitroandlivingcytoplasm are glass for-mers. Though we discuss in more detail in the next section, we pick up some relevant discussions here. Inlivingcells, we observe high-frequency viscoelasticities show expo-nential increase with concentration c(Fig.52) and power-law increase with frequency (Fig.24A) in the form of G(ω) = 1/6πaα ∝ exp(Ac)×(−iω)^0.5 when cytoskeletons are expressed sparse close to a probe particle observed. The similar frequency de-pendence was also observed for concentrated in vitro cytoplasm (Fig.43D). This is typical response for dense colloidal suspensions with slippery interfaces, as we discuss later. On the other hand, it is well known that G(ω) of gels made of semi-flexible polymers increase linearly to the increase of the concentration of polymerscand with different power-law form with frequency as given G(ω) = 1/6πaα∝c(−iω)^0.75 [186].

Therefore, both concentration dependence and frequency dependence of living cell mechanics contradicts to the theory for the mechanics of gels made of semi-flexible polymer.

Based on these various observations, we believe that living and in vitrocytoplasm show glass-forming property rather than that of gels.

13 MECHANICS OF LIVING CELLS, CYTOPLASM AND CYTOSKELETONS

13 Mechanics of living cells, cytoplasm and cytoskeletons

The high-frequency shearmodulus of the fibroblast cell ∝ (−iω)^0.74 observed in Fig.23B is consistent with theoretical predictions ∝ (−iω)^3/4 for networks of semiflexible polymers such as the cytoskeleton [186, 187]. Isolated fibroblast cells that are sparsely cultured on glass coverslips acquire flattened shapes, adhere well to the substrate, and grow F-actin-rich cross-linked cytoskeletal structures [14], which should resemble pure actin networks in their response characteristics. At low frequencies, mechanical response is predominantly elastic (G^′ ≫G^′′) but shows slow relaxation with small power-law exponents around 0.2±0.1 [53, 88, 137]. A similar behavior has been observed in various random networks of polymers [48, 91] and has been explained by nonaffine relaxations [91] in glassy worm-like chains [94, 138].

These theoretical models have also successfully explained the mechanical responses of cells to forces that were applied from the outside [53, 87, 88, 137]. Our study reveals that the same general physics appears to apply to the viscoelastic response inside of cytoskeleton-rich living cells. HeLa cells in confluent layers, in contrast to well-spread fibroblasts, are laterally confined and polarized. They are roughly as tall as they are wide, and the actin cytoskeleton is mostly confined to the membrane cortex from which our probe particles were sufficiently separated [20, 188]. The response measured in different HeLa cells showed a surprisingly narrow distribution, especially at high frequencies. We could clearly show that the power-law form of the shear modulus G(ω) ∼ G0(−iω)^0.5 (Fig.24) was distinct from the two regimes G(ω) ∝ G₀(−iω)^0.2±0.1 and G(ω) ∝ G₀(−iω)^0.75 observed in surface-adherent fibroblasts (solid lines in Fig.23B). The power-law dependence [G(ω) ∝ G0(−iω)^0.5] is a well known characteristic of glassy suspensions of colloids with nonsticky surfaces such as emulsions, foams, and swollen microgels [113, 114]. Thus, the mechanics of the interior of living confluent epithelial cells that is sparse in cytoskeletal network structure are likely better modeled as a soft glassy material [53, 94]. Note that a model for soft glassy materials was also used earlier to explain the results of low-frequency cell mechanics experiments [53]. In that case, a G(ω) ∝ G₀(−iω)^0.2±0.1 power law was found for the slow response of the actin cytoskeleton, caused by cross-link constraint release. Here, we report the expected three-fourths scaling

G(ω) ∝ G0(−iω)^0.75 at higher frequencies when probing the actin cytoskeleton in flattened fibroblasts. This is a very different physical situation from the one we describe here for the inside of the epithelial cells where we see, in the same frequency range, a quite different response G(ω) ∝ G0(−iω)^0.5 in the highly crowded glassy but not network-like cytoplasm.

A power-law dependence G(ω) ∝ G₀(−iω)^0.5 has also been observed in networks of flexible polymers for which the Rouse model applies [12, 115, 186]. The concentra-tion dependence of the shear modulus prefactor (G0) provides a way to distinguish between a flexible polymer network and colloidal glass. For a Rouse network, G₀ is proportional to the polymer concentration [115, 186]. In contrast, close to the glass transition of glass-forming materials, viscoelastic moduli grow more rapidly than lin-ear increase [57]. To probe concentration dependence, we osmotically compressed or expanded cells that were initially in isotonic conditions and measured the shear modulus as a function of relative intracellular macromolecular concentration (Fig.25).

The measured G₀ increased exponentially with the macromolecular concentration ϕ in HeLa cells, which is inconsistent with Rouse polymer networks [115,115]. The more plausible explanation for this dependence is thus that the mechanics inside confluent HeLa cells are predominantly determined by the glassy cytosol that is tightly packed with biomolecular colloids and approaches a jamming transition [55].

It has been speculated that crowding in cells containing high concentrations of macromolecules (proteins, RNA, and organelles) affects cell mechanics [54, 139]. The exponential dependence of G₀ on the macromolecule concentration is reminiscent of that of strong glass formers made of soft colloids [66]. It was shown that the viscoelas-ticity of extracted cytosol increases much more rapidly than the cytosol of the living cell, namely, with a super-exponential dependence on ϕ. In addition, the absolute value of G0 of extracted cytosol, which lacks nonequilibrium stress fluctuations, is far larger than that we see in living cells (see the asterisk in Fig.25). This compar-ison leads us to speculate that the out-of-equilibrium activity in the cell might be responsible for both the weaker viscoelastic response and the exponential (as opposed to super-exponential) concentration dependence. Super-exponential dependency is a typical consequence of jamming in tightly packed rigid colloids. Frustration of local arrangements leads to inhomogeneous distributions of residual forces and stresses in the sample, which can only relax through rare cooperative rearrangements.

There-14 GLASS-FORMING BEHAVIOR IN V IT RO AND LIV IN GCYTOPLASM fore, the exponential increase of viscosity might be explained by active stirring in cells [12] that can release and homogenize the locally frozen stresses. Active stirring should also make the living HeLa cells respond more homogeneously than expected, explaining the surprisingly narrow distribution of measured viscoelasticities (Fig.24).

14 Glass-forming behavior in vitro and living cytoplasm

We found that multiplein vitro cell extracts (IVCEs) obtained from organisms distant in evolutionary and developmental stages show quantitatively similar glass-forming behavior that is fragile (Fig.39). The chemical details of the intracellular components of somatic vertebrate cells are largely different from those in prokaryotic cells (E.

coli) as well as in germ cells (Xenopus egg) [14, 189]. Constituents of macromolecules in living cells dynamically change during the cell cycle [190] and at different stages of embryonic development. For instance, germ cells change their chemical composi-tion during maturacomposi-tion [191]. Biological materials usually change their mechanical properties sensitively with small additives such as crosslinkers. Quantitative agree-ment of IVCE viscosities was therefore not expected. We also found intracellular viscosity of living cells taken from different tissues (HeLa: malignant cancer, MDCK:

epithelial cells, NIH3T3: mesenchymal) shows quantitatively similar glass-forming behavior that is strong. For colloidal glass transitions, the physical properties of constituent objects such as their shape, size, and stiffness are more important than their chemical details [149]. Indeed, the universal glass-forming behavior of cyto-plasm can be slightly disrupted by artificially changing the physical parameters of constituent colloids. We did this by selectively collecting smaller biomacromolecules when preparingin vitrocytoplasm by the procedures written in the caption of Fig.61 below. When smaller biomacromolecules were selectively collected (big yellow trian-gles and big pink diamonds in Fig.61), e.g., by purification with ultracentrifugation, the viscosity of cell extracts was slightly affected. It is widely known that ribosome, which is made of proteins and ribosomal RNA in a large complex, is major component among intracellular macromolecules from archaea and bacteria (prokaryote) to cells (eukaryote) [14]. Due to its larger molecular weight, one can assume that the separa-tion of ribosomal complex is possible from other smaller proteins and solutes in crude extracts under long-lasting ultracentrifugation. Fig.39 shows relative increase of

vis-cosity is suppressed in the extract exposed to the ultracentrifugation, implying that large macromolecules such as ribosome may be a molecular component responsible for universal glass-forming behavior.

Figure 61. The effects of centrifugation on viscosity in cell extract.

Concentration dependence of relative viscosityη/ηw (ηw: viscosity of water) of cell extracts (big yellow triangles: E. coli extract prepared by using Lysozyme and freeze-thaw, big pink diamonds: HeLa cells S100 extract, small green triangles and small black diamonds: E.

coli, HeLa cells extracted with normal procedure). The broken line is the fit of Eq.(70) to cell extracts data. Small green triangles, small black diamonds and the broken line are the same as those in Fig.39. HeLa cells S100 extract (big pink diamonds) were purchased from CILBIOTECH (CC-01-41-50, Mons, Belgium). According to the manufacturer s instruction, supernatants after another 100,000×gfor 2 hours centrifugation of normal Hela cell extracts (CC-01-40-50, small black diamonds) were collected. Larger macromolecules are therefore less in these extracts. Cell extracts of E. coliprepared by using Lysozyme and freeze-thaw (big yellow triangles): Cell walls and membranes of E. coli were partially disrupted by sequential treatments of Lysozyme and freeze-thaw for the extraction of cytoplasm. Since several fractions of large macromolecules still remaining in cells, this gentle extraction collects smaller molecules compared to the extracts prepared by complete disruption of cells with sonication (Fig.62).

15 LIVING CYTOPLASM AS ACTIVITY-DRIVEN GLASS FORMER

Figure 62. Amounts of large macromolecules in cell extract prepared by combi-nation of Lysozyme and freeze-thaw are smaller than those by sonication.

Cells at late log-phase were treated with Lysozyme, immersed in double-distilled water, and then disrupted by freeze-thaw. The obtained cell extracts retained efficient activity of cell-free protein expression similar to IVCE in this study. Protein levels in the cell extracts prepared by the Lysozyme treatment and sonication were estimated from the levels of blue intensities derived from coomassie brilliant blue staining analyzed by ImageJ software. Frac-tions of proteins larger than superfolder GFP dimer in each sample evaluated by native agarose gel electrophoresis [192] were shown (n = 3, standard deviations were indicated by the error bars).

15 Living cytoplasm as activity-driven glass former

In recent years, the concept of the jamming phase diagram has been proposed, high-lighting the significance of mechanical forces for the behavior of glass-forming materi-als [56]. For dense colloidal suspensions, thermal hopping away from the cages formed by surrounding colloids is suppressed upon crowding. The system is fluidized if the energy injected by the mechanical loading instead activates the hopping process. The steady shearing of the crowded suspension, for instance, greatly reduces its viscosity to 1/1000 or less [193] (referred to as shear thinning). The FDT violation and super diffusion seen in Fig.49 clearly show that nonthermal forces are present in living cells.

In cells, forces are likely generated more randomly on a smaller length scale by e.g.

motor proteins. In this sense, the role of metabolic activities in reducing cellular vis-cosity can be qualitatively similar to that of temperature and/or mechanical loading

in dense colloidal suspensions; the structural relaxation is enhanced by nonthermal activity [141–143]. Note that not only the IVCEs but inactive cells are glassy (or less fluidized at least) at biologically-relevant concentrations (Fig.48B). We found that the glass transition points of IVCEs correspond to physiological concentration of living cells. For efficient biochemical reactions, both concentrated situations and efficient transportation of materials are preferred. These requirements would conflict in non-living systems, especially for fragile glasses. It is then reasonable for non-living cells to regulate their concentrations close to the glass transition point for IVCEs and thereby simultaneously satisfy these requirements.

Nonthermal activation is the key to understand the difference of fragility of cell ex-tracts and in living cells (Fig.46). Being subjected to nonthermal activation in excess of thermal one, free volume that macromolecules in living cells can explore should be larger than that inin vitro cytoplasm; the point of jammingc^∗ can shift to higher con-centrations [56]. It is also known that the fragility of colloidal glass formers correlates with the slope of theE≡[elastic energy per colloids (stiffness of colloids)]/[activation energy] as a function of colloidal concentration; a greater slope leads to fragile glass formers [66]. Activation is solely thermal in fragile IVCEs whereas nonthermal activa-tions in living cells make the slope (dE/dc) smaller. Without nonthermal activation, colloidal glasses are strong only when constituent colloids can be compressed more than 10-fold in their volume during glass formation [66, 194]. Such extreme deforma-bility is unlikely for a large part of cell constituents such as ribosomes. Therefore, the softness of cell constituents alone does not explain the non-fragile behavior of living cells; the metabolic activity is essential.

It is recently acknowledged that fragility might be related to the dynamic hetero-geneity; more fragile glasses tend to have larger dynamic heterogeneity [75,76]. When dE/dc is large, colloids in suspension cannot relax the frustration in local arrange-ments independently. This leads to inhomogeneous distributions of residual forces and stresses in the sample which can relax only through rare cooperative rearrange-ments [72]. On the other hand, soft gels, emulsions and foams form strong glasses [66].

Even if they are packed, they can still locally relax that the heterogeneous dynamics might be less likely to be observed. In our experiments, heterogeneity of viscosity in cell extracts were not large (Fig.63). Likely, the characteristic length of the hetero-geneity is smaller than probe sizes (Fig.64). Also, we speculate that heterohetero-geneity in

15 LIVING CYTOPLASM AS ACTIVITY-DRIVEN GLASS FORMER

living cells, which were remarkably smaller than expected [55], is reduced since cell interior is actively stirred and homogenized by nonthermal activation (Fig.49). Here, active stirring in cells can relax the residual stresses that would convert the fragile IVCEs to strong glass formers.

Figure 63. Distribution of mechanical properties of cell extracts and in living cells.

η in living HeLa cells (n = 9) and that of concentrated 0.21 g/mL cell extracts (n = 21) were measured with PMR and their averages and standard deviations are shown.

Figure 64. Probe size dependency in highly concentrated Cell extracts (0.21 g/mL).

η of concentrated cell extracts (0.21 g/mL) were measured with PMR. PEG-coated probes with three different sizes (2a= 0.62µm, 1µm and 2µm) were embedded in the same sample.

Averages ofη and standard deviations are obtained from 16 particles for each size of probes.

16 Activity-driven dynamics and nonequilibrium mechanics in cells

We have discussed that activity in cells critically change their glass-forming behaviors.

The mechanics of living cells should be thus understood in the context of nonequi-librium physics. Our feedback MR that can perform active and passive modes si-multaneously in cells have unique ability to characterize the activity (dynamics) and nonequilibrium mechanics (as its consequence).

In cells, molecular motors such as myosins generate forces that induce nonther-mal fluctuations that are tracked by our probe particles. Nonmuscle cytoplasmic myosins bind to actin and progressively build up stress in the network with cor-relation times on the order of second. Myosins then stochastically unbind from the actin filaments and thereby instantaneously release stress stored in the elastic network [12, 90, 195]. These motor-induced fluctuations have been studied in re-constituted actin/myosin gels where the abrupt stress release causes directed mo-tion of probes (Fig.65A) [12, 195]. Considering the standard relamo-tion between step response and the frequency response of a material, the correlation time τ1 of the directed motion can be estimated from the angular frequency ω1(∼ 1/τ1), where G^′(ω₁) = G^′′(ω₁) (Fig.65B). The abrupt release of stress corresponds to a step in f(t) in Eq.(23) and, therefore, on average

⟨f˜(ω)²

⟩

∝ ω⁻² in Eq.(40) if such events occur in an uncorrelated manner [33]. The nonthermal fluctuations derived from measuring both

⟨|u(ω)˜ |²⟩nonthermal

and α(ω) [48, 104] were consistently ex-plained by inserting this power-law dependency into Eq.(40):

⟨|u(ω)˜ |²⟩nonthermal

= |α(ω)|²⟨ f(ω)˜ ²

⟩

. The same discussion was speculated to hold for nonthermal fluctuations in living cells [107]. A viscoelastic response function in the cell of the form |α(ω)|² ∝ 1/|G(ω)|² ∝ ω^−0.36 at frequencies less than 10 Hz (Fig.23B) would lead to ω

⟨|u(ω)˜ |²⟩nonthermal

∝ω|α(ω)|²⟨ f˜(ω)²

⟩

∝ω^−1.36, which was close to the observed relationship ω

⟨|u(ω)˜ |²⟩nonthermal

∝ ω^−1.4. However, note that the actual correlation time of the directed motions observed in cells (>1 s) is much longer than the estimate τ₁ ≤ 1 ms based on the abrupt stress-release model (see the arrow in

16 ACTIVITY-DRIVEN DYNAMICS AND NONEQUILIBRIUM MECHANICS IN CELLS

Fig.23B for the estimate of ω₁). The assumption of uncorrelated motor activity is thus likely to neglect important aspects of intracellular activity. A more realistic in-terpretation of the directed fluctuations (super-diffusion) observed in cells is that the probes are driven by collective and correlated force generation. The collective action of force generators likely persistently stirs the cytoplasm on relatively long time scales, so that the memory of force is not lost immediately after a single motor detaches from the cytoskeleton. At longer time scales (ω/2π <0.1 Hz), the displacement PSDs show power-law behavior (Fig.28A, inset) similar to that of simple diffusion in a viscous medium, ω

⟨|u(ω)˜ |²⟩no trap

/2k_BT= α^′′(ω)∝ ω⁻¹ [11, 49, 111]. However, the slope of -1 here does not imply thermal diffusion because the probe is driven by nonthermal forces, as proven by the breaking of the FDT. Probes moving in cells lose their velocity memory on the same time scale (≥ 1 s).The slope of -1 is thus due to the Markovian nature of probe displacements driven by cells.

Figure 65. Fluctuation and mechanical property in actin/myosin active gel.

(A) Position of a probe particle with a diameter of 1µm in an active actin/myosin gel. The arrow indicates a large abrupt jump of the probe that is occasionally observed. The relax-ation is faster than the frame rate (1/30 s) of the video. (B) Shear viscoelastic modulus in actin/myosin active gel. Closed and open circles show real and imaginary part, respectively.

The arrow indicates the frequency f1 = ω1/2π where G^′(ω1) = G^′′(ω1). This frequency provides an estimate of the viscoelastic relaxation timeτ₁≡1/ω₁= 1/2πf₁∼10 ms.

The mechanical response of many biomaterials is not well understood, even in thermodynamic equilibrium. In cells, mechanical responses are often further mod-ulated by cell activity. Motor-generated forces, for instance, stiffen cytoskeletal

networks in vitro and in vivo until the network structure is broken by excessive stress [12,48,90,119]. Cytoskeletal networks in cells are also thought to undergo sol-gel transitions when self-organized contractile stresses drive cytoplasmic flows [196, 197].

Our feedback MR experiments in epithelial-like HeLa cells revealed that the living cy-toplasm exhibits glass-like behavior. This provides a critical role for active stress gen-eration in the cell. The cytoplasm can be fluidized by activity just as colloidal glasses and granular materials are undermechanical loads or external perturbations [55, 56].

It will be crucial to understand in detail how the nonthermal fluctuations in cells determine the mechanical properties of living cells [11, 104]. A precise quantitative analysis of the out-of-equilibrium mechanics in cells, as can be performed with the ap-proach presented here, will be essential to further investigate the fascinating interplay between cellular mechanics and nonequilibrium force generation.

Chapter V

Concluding remarks

In this theses we studied the mechanics of driven and crowded cytoplasm and devel-oped the feedback microrheology technique to measure the mechanics in living cells.

The displacement of an embedded probe particle in living cells are measured by laser interferometry in this developed feedback microrheology. This developed mi-crorheology under three-dimentional feedback of a piezo-mechanical sample stage en-ables a probe particle to be stably tracked in intracellular vigorous flow. With using this feedback technique, the fluctuation of the probe particle was measured (PMR), and the rheology of the cell was obtained from the response after applying a force to the probe particle by using a laser (AMR). By simultaneously measuring in PMR and AMR, we can study the nonthermal drive forces in the living cell.

Direct evidence was obtained for the glass-forming behavior of in vitro cytoplasm and living cytoplasm, by measuring their mechanical properties using developed mi-crorheology techniques. It was crucial to explore the mechanics of simplifiedin vitro model cytoplasm from which effects of cytoskeletal networks and cell metabolism ex-cept crowding were artificially removed, and to compare them with those in living cells. We found that the mechanical properties of the model cytoplasm were identical to those known for fragile colloidal glass formers that steeply increase their viscosity close to glass transition. They showed well-defined Newtonian viscosity below the physiological concentration of living cells (∼ 0.3 g/mL). The critical concentration c^∗ for jamming was found to be nearly identical to the physiological concentration of metabolically active cells. In contrast, microrheology performed in living cells revealed finite fluidity though it is reduced in spontaneously deactivated cells. Viscous mod-ulus in living cells was also increased by the increased macromolecular concentration via osmotic compression, however, more gently in a manner of complete Arrhenius typical of strong glass formers. All these observations support that the non-living cytoplasm at the physiological concentration is on the verge of glass transition. The living cytoplasm gains some fluidity via metabolic activity by changing its glass-forming characters from that of fragile to that of strong glass formers. We also found that ATP-depleted cells which decreased nonthermal fluctuations became more elastic