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.2017 Sep 5;114(36):9529-9534.
doi: 10.1073/pnas.1702488114. Epub 2017 Aug 21.

Size- and speed-dependent mechanical behavior in living mammalian cytoplasm

Affiliations

Size- and speed-dependent mechanical behavior in living mammalian cytoplasm

Jiliang Hu et al. Proc Natl Acad Sci U S A..

Abstract

Active transport in the cytoplasm plays critical roles in living cell physiology. However, the mechanical resistance that intracellular compartments experience, which is governed by the cytoplasmic material property, remains elusive, especially its dependence on size and speed. Here we use optical tweezers to drag a bead in the cytoplasm and directly probe the mechanical resistance with varying sizea and speedV We introduce a method, combining the direct measurement and a simple scaling analysis, to reveal different origins of the size- and speed-dependent resistance in living mammalian cytoplasm. We show that the cytoplasm exhibits size-independent viscoelasticity as long as the effective strain rateV/a is maintained in a relatively low range (0.1 s-1 <V/a < 2 s-1) and exhibits size-dependent poroelasticity at a high effective strain rate regime (5 s-1 <V/a < 80 s-1). Moreover, the cytoplasmic modulus is found to be positively correlated with onlyV/a in the viscoelastic regime but also increases with the bead size at a constantV/a in the poroelastic regime. Based on our measurements, we obtain a full-scale state diagram of the living mammalian cytoplasm, which shows that the cytoplasm changes from a viscous fluid to an elastic solid, as well as from compressible material to incompressible material, with increases in the values of two dimensionless parameters, respectively. This state diagram is useful to understand the underlying mechanical nature of the cytoplasm in a variety of cellular processes over a broad range of speed and size scales.

Keywords: cell mechanics; cytoplasmic state diagram; poroelasticity; viscoelasticity.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Experimental setup measuring the mechanical resistance in living mammalian cytoplasm. (A) Bright-field image of an NRK cell with a 1-µm-diameter particle inside. (Scale bar, 5 µm.) (B) Schematic showing the experimental setup. A probe bead is dragged at a constant speed toward the cell boundary to obtain a force-displacement curve. (Inset) The crowed environment around the bead in cells.
Fig. S1.
Fig. S1.
Representative force-displacement curve and relaxation curve obtained in the cytoplasm of NRK cells. The semitransparent bars represent SE (n = 20 forA,n = 15 forB). (A) The force-displacement curve is obtained by stretching a 0.5-μm-diameter bead at a speed of 1 μm/s in the cytoplasm. (B) The relaxation curve is obtained by stretching a 0.5-μm-diameter bead to 0.1 μm displacement at a high speed of 30 μm/s in the cytoplasm, then maintaining the location of the bead and recording the force as a function of time. Because the duration of the initial stretching is much shorter than the relaxation time recorded, the loading process can be regarded as instantaneous. The apparent relaxation timeτ obtained through exponential fitting of the relaxation curves is 0.79 ± 0.13 s (n = 15, mean ± SD). The orange dashed line is the single term exponential fitting of the relaxation curve.
Fig. 2.
Fig. 2.
Validation of the scaling argument through experiments in classical materials and with finite element simulation. (A) Normalized force-displacement curves obtained in a classical poroelastic material, PA gel, by dragging beads with 1 µm (blue lines) and 0.5 µm (orange lines) in diameter at different speeds. Curves with the sameVa (20 µm2/s and 5 µm2/s, respectively) are close to each other. (B) Normalized force-displacement curves obtained in a classical viscoelastic material, alginate gel (5 mg/mL alginate cross-linked with 3 mM calcium sulfate), by dragging beads with 1 µm (blue lines) and 0.5 µm (orange lines) in diameter at different speeds. Curves with the sameV/a (2, 1, and 0.4 s−1, respectively) are close to each other. Each force displacement curve shown inA andB is averaged over at least 20 individual experiments. (C andD) Finite element simulation of dragging a bead in a large cubic poroelastic material (C) and viscoelastic material (D). Two dimensionless bead sizes are used, 5/5,000 of the overall sample length (orange lines), and 2/5,000 of the sample length (purple dashed lines). (C) If the material is defined as poroelastic, curves with the sameVa/D (10 and 0.1, respectively) collapse. (D) If the material is defined as viscoelastic, curves with the sameVτ/a (100, 0.1, and 0.01, respectively) collapse.
Fig. S2.
Fig. S2.
The complex modulusG* of bulk PA gel measured using a rheometer with a parallel plate geometry (Gemini HRnano; Malvern Instruments). We can calculate the value ofE* = 75.4 Pa through the relationE* = 2(1 +ν)G*. This value is consistent withE* = 80 Pa measured with optical tweezers whenVa = 5 µm2/s (Fig. 2A).
Fig. 3.
Fig. 3.
Normalized force-displacement curves measured in the cytoplasm of NRK cells show that the cytoplasm exhibits viscoelasticity under small effective strain rate and exhibits poroelasticity under large effective strain rate. Each curve shown here is averaged over at least 20 individual experiments. (A) Under low effective strain rate, the normalized force-displacement curves obtained with different bead sizes (0.5, 1, and 1.5 µm) are close to each other when the sameV/a is maintained (e.g., 1 s−1 and 0.2 s−1). (Inset) Normalized force-displacement curves with different bead sizes are clearly separated when we compare experiments with the sameVa in the low-speed regime. (B) Under high effective strain rate, the normalized force-displacement curves with different bead sizes (0.5, 1, and 1.5 µm) are close to each other when the sameVa is maintained (e.g., 15 µm2/s and 10 µm2/s). (Inset) Normalized force-displacement curves with different bead sizes are clearly separated when we compare experiments with the sameV/a in the high-speed regime.
Fig. S3.
Fig. S3.
Normalized force-displacement curves measured in the cytoplasm of NRK cells show that the cytoplasm exhibits viscoelasticity under low loading speed and exhibits poroelasticity under high loading speed. Each curve shown here is averaged over at least 20 individual experiments. (A) Under low loading speeds, the normalized force-displacement curves obtained with different bead sizes (0.5 µm, 1 µm, and 1.5 µm) in the cytoplasm are close to each other when the sameV/a is maintained (0.5 s−1). (B) Under high moving speeds, the normalized force-displacement curves with different bead sizes (0.5, 1, and 1.5 µm) in the cytoplasm are close to each other when the sameVa is maintained (20 µm2/s).
Fig. S4.
Fig. S4.
The cytoplasm of NRK cells behaves as a viscous fluid when the strain rate is very low (V/a < 0.1 s−1). When we drag a 1.5-μm-diameter bead at the speed of 0.1 μm/s the resistant force roughly remains constant with increasing displacement. Through the Stokes equation, the effective viscosity is 12.1 ± 6.3 Pa·s (n = 15, mean ± SD).
Fig. S5.
Fig. S5.
Normalized force-displacement curves measured in the cytoplasm of HeLa cells show that the cytoplasm exhibits viscoelasticity under small loading speed and exhibits poroelasticity under large loading speed. Each curve shown here is averaged over at least 20 individual experiments. (A) Under low loading speeds, the normalized force-displacement curves obtained with different bead sizes (0.5 µm and 1 µm) in the cytoplasm are close to each other when the sameV/a is maintained (1 s−1). (B) Normalized force-displacement curves with different bead sizes in the cytoplasm are clearly separated when we compare experiments with the sameVa in the low-speed regime (e.g., 0.5 µm2/s). (C) Under high moving speeds, the normalized force-displacement curves with different bead sizes (0.5 µm and 1 µm) in the cytoplasm are close to each other when the sameVa is maintained (10 µm2/s). (D) Normalized force-displacement curves with different bead sizes in the cytoplasm are clearly separated when we compare experiments with the sameV/a in the high-speed regime (e.g., 20 s−1).
Fig. S6.
Fig. S6.
Normalized force-displacement curves measured in the cytoplasm of NRK cells with F-actin depolymerized show a clear decrease of the characteristic viscoelastic timescale,τ, and a marked increase of the effective diffusivity,D, in the cytoplasm. (A) The cytoplasm of NRK cells with F-actin depolymerized behaves as a viscous fluid when the strain rateV/a is ∼0.2 s−1, and the cytoplasm of normal NRK cells behaves as a fluid whenV/a < 0.1 s−1. This increase in the transition strain rate from a seemingly viscous fluid behavior to a viscoelastic material is attributed to the decrease of the characteristic viscoelastic timescale,τ. (B) The normalized force-displacement curves obtained with different bead sizes (0.5 µm and 1 µm) in the cytoplasm are close to each other when the sameV/a is maintained as 2 s−1; this proves that F-actin polymerized NRK cytoplasm behaves as a viscoelastic material in such a condition. (C andD) Under high loading speeds, the normalized force-displacement curves obtained with different bead sizes (0.5 µm and 1 µm) in the F-actin depolymerized cytoplasm are separated when the sameVa is maintained (e.g., 5 µm2/s inC and 10 µm2/s inD), while force-displacement curves with the same speeds and sizes obtained in the cytoplasm of control NRK cells collapse well (Fig. 3B and Fig. S7A). This proves that the critical strain rate,V/a, at which we observe poroelasticity to start dominating the mechanical resistance increases because of an increase of the cytoplasmic diffusivity,D. (E andF) Bright-field and fluorescent imaging of NRK cells stained with phalloidine in the control cells (E) and cytochalasin D-treated cells (F) shows that F-actin structure is fully inhibited.
Fig. 4.
Fig. 4.
Cytoplasmic modulus under different effective strain rates and probe sizes. (A) Dependence of the apparent cytoplasmic modulus on normalized loading speed and probe size. Each modulus value (black circles) is averaged over at least 20 experiments; the SE bar has a length smaller than the size of the symbols in the plot. The 3D surface is obtained by piecewise linear interpolation of the measured apparent cytoplasmic modulus (EA). The color of the surface represents the value ofEA. (B)EA at different loading conditions does not change with the area of the probe bead at low effective strain rate regime, since viscoelasticity is dominating in this regime. However,EA increases with the area of the bead at the high effective strain rate regime, since poroelasticity is dominating in this regime. (C)EA is solely determined byV/a at low effective strain rate regime (0.1 s−1 <V/a < 2 s−1); this reinforces thatV/a is the control parameter of the mechanical resistance at the low effective strain rate regime where viscoelasticity dominates. At the high effective strain rate regime (V/a > 5 s−1),EA obtained with different probe size starts to deviate even at the sameV/a. Instead,EA has the same value when the sameVa is maintained (indicated by the horizontal dashed lines); this reinforces thatVa is the control parameter of the resistance at the high effective strain rate regime where poroelasticity dominates. (Inset) The logarithmic plot of the dependence ofEA onV/a, which exhibits a seeming power-law behavior with a power of ∼0.25 at the low-speed regime.
Fig. S7.
Fig. S7.
Normalized force-displacement curves measured in the cytoplasm of NRK cells show that the cytoplasm exhibits poro-viscoelasticity when loading speedV = 3.5 μm/s and loading sizea = 1.5 μm. Each curve shown here is averaged over at least 20 individual experiments. (A andB) Under intermediate loading speeds, the normalized force-displacement curves obtained with 0.5-µm beads and 1-µm beads in the cytoplasm are close to each other when the sameVa (5 µm2/s) or the sameV/a (2.2 s−1) is maintained; however, the curves obtained with 0.5-µm beads and 1-µm beads are clearly separated with the curves obtained with 1.5-µm beads at a speed of 3.3 μm/s, when the sameVa (5 µm2/s) orV/a (2.2 s−1) is maintained. This proves that the cytoplasm can exhibit both poroelasticity and viscoelasticity under large loading size and intermediate loading speed.
Fig. 5.
Fig. 5.
The state diagram of living mammalian cytoplasm as a function of two dimensionless numbers,Vτ/a andVa/D. The state diagram shows that as dimensionless numberVτ/a increases from 10−2 to 102 the cytoplasm changes from a viscous fluid to a solid; as dimensionless numberVa/D increases from 10−3 to 101 the cytoplasm changes from a compressible material to an incompressible material. Each data point represents an experimental test with that particular speed and size combination. Different symbol shapes of the data points represent different origins of the mechanical resistance, and the color of the symbol represents the value of the measured apparent modulus. Based on our experiments at different loading speeds and probe sizes, combined with our scaling analysis, the origins of mechanical resistance in the cytoplasm can be divided into eight regions of the sate diagram: I, Viscous fluid; the resistance remains a constant as displacement increases (Fig. S4). II, Incompressible viscoelasticity. III, Poroviscoelasticity, where both viscoelasticity and poroelasticity are important (Fig. S7). IV, Compressible viscoelasticity. V, Incompressible pure elasticity; both viscoelasticity and poroelasticity do not relax in this region, and thus the cytoplasmic modulus always remains constant. VI, Poroelasticity. VII, Compressible pure elasticity (Fig. S8); poroelasticity relaxes completely while viscoelasticity does not relax at all in this region. VIII, Continuous mechanics fails when the object is equal or smaller than cytoskeletal mesh size (∼50 nm). Moreover, we mark a variety of physiological intracellular transport and cellular movement phenomena on the state diagram, to illustrate different origins of the mechanical resistance each of these processes experiences. The characteristic size and speed of these phenomena are taken from literature (details in Table S2). The dashed lines in the state diagram are transitional boundaries of different regions; the location of these dashed lines is determined by experimental results and theoretical analysis.
Fig. S8.
Fig. S8.
Normalized force-displacement curves measured in the cytoplasm of NRK cells show that the cytoplasm exhibits pure elasticity under high strain rate (Vτ/a >> 1) and small loading size (Va/D << 1). Each curve shown here is averaged over at least 20 individual experiments. Under high strain ratesV/a (e.g., 4.4 s−1 and 6.6 s−1) and small loading sizesa (e.g., 0.5 μm), the normalized force-displacement curves obtained in the cytoplasm are close to each other with variedVa andV/a. In this poro-viscoelastic region, cytoplasmic mechanical behavior is independent with both speed and size.
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