Optical tweezer-based microrheology using large trapped beads exhibit multiple nonlinearities not found in measurements made with smaller beads

Megan Keech (1), Ashok Pabbathi (2), Maureen Buckley (1) , Joshua Alper (2) (3) (4)

(1) Department of Bioengineering, Clemson University
(2) Department of Physics and Astronomy, Clemson University
(3) Department of Biological Sciences, Clemson University
(4) Eukaryotic Pathogens Innovation Center (EPIC), Clemson University

Abstract

Materials scientists and engineers use optical tweezer-based microrheology measurements to probe the viscoelastic properties of complex microstructured materials such as gels and filament networks with various structural feature sizes at multiple length scales by trapping and tracking beads of various sizes suspended within the materials. Current procedures for analyzing these data assume that the detection of the bead within the trap and the force of the trap on the bead are linear. These assumptions inherently rely on an approximation that the bead diameter is similar to the beam width at the diffraction-limited focal point of the trapping laser, but less dense materials have larger structural features and require larger diameter beads. We discovered significant deviations from the linear assumption in the detection of beads that are 5-10-fold larger than the laser focal point. Additionally, we found nonlinearities in the force-displacement curves of the trap when we applied known constant forces to 5 µm trapped beads. We customized MATLAB toolboxes to theoretically model and validate our experimental data. Using both the calculation and experimental results, we found that that the magnitude and direction of the slope and linear range of the detector-response graphs are strong functions of bead size for large beads that can be modeled as third-order polynomials. Further, the force-displacement modeling and data that we collected suggest that the trap has a non-linear effective spring constant for large beads. By quantifying these observations with the analysis of large bead optical tweezer experiments, we better understand their interaction with the trap and how to account for these nonlinearities properly. Ultimately, our results will allow researchers to analyze sparse material matrices, use larger beads, and accurately perform calculations with optical tweezer-based microrheology.

Introduction

Optical tweezers use focused light (Figure 1) to exert forces on micron-scale physical objects and hold them within the trap due to the balance of forces (Figure 2). The instrument allows the user to exert and measure forces in the pN range and distances ranging from a few nm to multiple mm [1]. These unique qualities have resulted in the optical tweezer being used in experiments such as DNA study and microrheological measurements.

Mircorheology is particularly useful when studying viscoelastic materials and complex fluids, which exhibit structural heterogeneity causing length scale dependency of rheological properties [2,3]. The use of optical tweezers allows for active microrheology, where the user can apply forces on a tracer bead. The use of optical tweezer based microrheology is dependent on accurate calculations and calibrations of the trap.

Figure 1. A representative schematic of an optical tweezer instrument [8]

Figure 2. Example of force vectors on bead within an optical trap, demonstrating how the bead is pushed toward the center of the trap and reaches an equilibrium position when trap forces are balanced. [7]

Materials and Methods

The following will be used to create physical samples for optical tweezer analysis

  • 5 µm polystyrene microspheres
  • 0-30 mM NaCl solution
  • Glass slide, cover slip, parafilm

Stuck bead experiment:

Constant velocity experiments will be conducted for multiple frequencies. Stokes’ law will be used to calculate the force of fluid drag, which is equal to the force of the trap needed to keep the bead at a constant velocity.

MATLAB and Fiji will be used to analyze data and calculate theoretical models.

Results

Figure 3. Examples of MATLAB program output graphs of best fitting stuck bead calibration data collected from a single bead in the y direction left to right in the left graph and right to left in the right graph.  The best fit line shown has the equation form a(x-x1)^3+b(x-x1)^2)+c(x-x1)+d. Part b shows a representation of the QPD signal during a stuck bead experiment conducted using a 1 µm bead published by Hajizadeh et al [5,6]. The last two photos depict other shapes observed during calculation, likely due to incorrect positioning of the trap, demonstrating the sensitivity of large bead experiments.

Figure 4. Examples of MATLAB program output graphs. Bead position during calibration time of a bead stuck to the slide with average is shown in A. B shows bead position during calibration of a trapped bead with average position indicating trap location. Graph C shows the position of the trapped bead displaced due to stage oscillation and the likely line of travel of the bead. D demonstrates the relationship between the average distance from the trap during motion and the force the trap exerted on the bead. The slope represents the spring constant.

Figure 5. The graph on the left was produced using the most current version of the Optical Tweezer Toolbox (OTT) and is a close replication to the figure on the right published by Jahnel et al. [4]

Conclusions

Our data suggests

  • large beads create stuck bead calibration profiles with an oppositely sloping linear region (Figure 2)
  • the relationship of the trap force and distance from center is nonlinear, indicating the spring behavior is not a constant when using large beads (Figure 3d)

As a result, calibrations and measurements conducted using beads with a diameter significantly larger than that of the focal point of the optical trap are likely inaccurate when using small bead analysis procedure.

Continuing Research & Future Directions

  • Test larger bead sizes
    • Determine bead size correlation with trap behavior
    • Verify graph shapes
  • Determine a numeric equation to predict stuck bead region of interest shape
  • Determine a numeric equation appropriate for nonlinear spring behavior of trap
  • Investigate QPD readings vs tracking
  • Apply information to cytoskeletal networks and other active/passive materials