Research ArticleBIOMECHANICS

Feedback-tracking microrheology in living cells

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Science Advances  29 Sep 2017:
Vol. 3, no. 9, e1700318
DOI: 10.1126/sciadv.1700318
  • Fig. 1 Experimental setup for feedback MR.

    (A) Optical trap–based AMR setup with 3D feedback controls. A probe particle in a cell is trapped and oscillated by the drive laser (pink). QPD signals from the probe laser (cyan) are fed into PID controllers, which regulate piezo stage motion in x and y directions such that the trapped probe particle is maintained near the optical trap center. The bright-field image of the probe is simultaneously projected onto a charge-coupled device (CCD) camera [green thick line; light-emitting diode (LED) light source is omitted]. The image is analyzed on a personal computer (PC) to adjust the z position of the objective lens through a z direction piezo scanner. (B) Schematics for bead displacements and laser positions. Positions of fixed probe laser (830 nm, cyan) and acousto-optic deflector (AOD)–steered drive laser (1064 nm, pink) are shown by solid and dashed vertical lines, respectively. The drive laser is drawn displaced to the right of the probe laser. The pale particle indicates the probe starting position.

  • Fig. 2 Diagram of signal flow in AMR under feedback control.

    For AMR, feedback control of the piezo stage (red solid lines) is performed under sinusoidal force application to the probe particle with the drive laser (blue dashed lines). In PMR, the feedback loop is run without force application. V(t) and/or ε(t) is measured with a lock-in amplifier (AMR) or directly digitized and recorded by a PC (PMR).

  • Fig. 3 Feedback AMR in equilibrium sample.

    (A) Real parts of response functions of probes embedded in a cross-linked PAAm gel, measured under feedback control. Open squares and filled circles indicate responses measured with QPD and stage, respectively. Solid and dashed curves show the estimated material response, α′, of the gel, as estimated from Embedded Image and Embedded Image, respectively. (B) Complex shear modulus measured with AMR under feedback control (symbols) and fit of G(ω) = 1.9 + 0.59(−iω)1/2 (solid and dashed curves). Circles and triangles are real and imaginary parts of the modulus, respectively. Filled and open symbols correspond to moduli obtained using stage response and QPD response, respectively.

  • Fig. 4 Feedback PMR and conventional PMR.

    (A) Diagram of signal flow in PMR under feedback control. AFB is the apparent response function in the time domain that relates (thermal) fluctuating force ζ(t′) and total probe displacement u(t) under feedback. uQPD and ustage are correlated via the PID controller. A is the response function of the optically trapped probe in the absence of feedback. ktustage represents the reduction of the optical trapping force due to feedback. (B) Power spectral densities (PSDs) of displacement fluctuations of the probe embedded in entangled actin solution (1 mg/ml) measured with PMR. Open triangles and the dashed curve show PSDs directly measured with and without feedback, respectively. Open circles and the solid curve show 〈|ũ(ω)|2no trap calculated from 〈|ũ(ω)|2FB using Eq. 9 and PSDs calculated from 〈|ũ(ω)|2trap using the conventional PMR method, respectively.

  • Fig. 5 Material properties of fibroblasts and HeLa cells measured with feedback MR.

    (A) 2D trajectory of a probe in a cell measured under feedback. The total period of observation was ~30 min. (B) Shear modulus measured in the peripheral region of an adherent fibroblast, which is rich in actin cytoskeleton. Solid and dashed curves are fits of an empirical model describing low-frequency relaxations and high-frequency power-law behavior of a semiflexible polymer network as G(ω) ~ 140(iω)0.12 + (iω)0.74. (C) Shear modulus measured in confluent HeLa cells (n = 9 cells cultured in different dishes). Bars indicate the unbiased estimate of the SD. Power-law form of viscoelasticities G(ω) ∝ (−iω)0.5 indicates a glassy response of (actin network–poor) cytosol. (D) G0 for osmotically compressed HeLa cells (pink diamonds) plotted against macromolecular concentration relative to isotonic condition (φ = 1). Exponential dependence on macromolecule concentrations φ (solid line, exponential; dashed curve, linear dependence) is typical of strong glass formers rather than polymer networks. The asterisk marks G0 for the cell extract taken from HeLa cells (0.21 g/ml).

  • Fig. 6 Displacements of melamine resin probe bead (2a = 680 nm) in cultured fibroblast cell (NIH-3T3).

    (A) QPD and control signals for piezo stage as output by PID controller were recorded and calibrated to obtain displacements uQPD (green) and ustage (blue), respectively. Total displacement u(t) (red) was obtained by summing these quantities. (B) The displacement PSD measured using QPD with conventional PMR (black curve) agrees with that obtained under feedback by summing uQPD and ustage.

  • Fig. 7 FDT violation and out-of-equilibrium fluctuations in a fibroblast cell.

    (A) Imaginary part of the complex response function measured inside a fibroblast using feedback AMR, compared to the corresponding function ω〈|ũ(ω)|2no trap/2kBT measured with PMR. Curves agree at high frequencies and clearly differ at frequencies less than 10 Hz. The solid curve shows purely nonthermal fluctuations calculated using Eq. 15. Solid lines show power laws for comparison. Inset: Local power-law exponent of nonthermal fluctuations calculated using d logω〈|ũ(ω)|2nonthermal/d logω. Crossover between different scaling regimes can be observed. (B) Autocorrelation function of velocity direction.

  • Fig. 8 3D feedback.

    (A) Typical voltage time series recorded by the QPD with or without feedback, obtained with a probe embedded in a fibroblast cell. Strong fluctuations of the QPD signal (top) are suppressed when feedback is active (bottom). (B) The radial distribution of probe image brightness is used for feedback control of the z position of the objective lens. Top images show probe patterns, and bottom images schematically show typical intensity profiles in radial coordinates. z position is controlled such that the brightness at the fringe and in the center of the pattern remains equal (dotted line).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/9/e1700318/DC1

    note S1. Conventional AMR without feedback control.

    note S2. Response time measurement of the feedback system.

    note S3. Conventional PMR in thermal equilibrium.

    note S4. Inverse proportionality between k and C.

    fig. S1. Experimental test of feedback performance.

    fig. S2. Imaginary part of the response functions of a probe in a PAAm gel.

    fig. S3. Apparent violation of the FDT under feedback.

    fig. S4. Complex shear modulus in HeLa cells measured with feedback AMR.

    fig. S5. PSD obtained under feedback PMR in actin solution.

    fig. S6. Fluctuation and mechanical property in actin/myosin active gel.

    fig. S7. Inversely proportional relationship between trap stiffness and displacement response used for calibration in cells.

  • Supplementary Materials

    This PDF file includes:

    • note S1. Conventional AMR without feedback control.
    • note S2. Response time measurement of the feedback system.
    • note S3. Conventional PMR in thermal equilibrium.
    • note S4. Inverse proportionality between k and C.
    • fig. S1. Experimental test of feedback performance.
    • fig. S2. Imaginary part of the response functions of a probe in a PAAm gel.
    • fig. S3. Apparent violation of the FDT under feedback.
    • fig. S4. Complex shear modulus in HeLa cells measured with feedback AMR.
    • fig. S5. PSD obtained under feedback PMR in actin solution.
    • fig. S6. Fluctuation and mechanical property in actin/myosin active gel.
    • fig. S7. Inversely proportional relationship between trap stiffness and displacement response used for calibration in cells.

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