3.1.

Hardware Configuration

The laser tweezer apparatus shown in Fig. 1 is largely the same as discussed previously⁶¹ and operates as a single-beam 3-D trap. The source was a 0.5-W diode-pumped Nd:YAG continuous-wave single-mode laser. The optical tweezer was aligned to the optical axis of the microscope (Leica DM6000B, Wetzlar, Germany) using a 5-degree-of-freedom ($x$-axis, $y$-axis, $z$-axis, pitch, and yaw) mount. The laser beam was expanded to fill the back aperture of the objective lens. The objective lens used was a Leica 63X NA 0.9 U-V-I HCX long working distance plan apochromat dipping objective. The tweezer couples into the microscope through a lateral port, providing direct optical access to the fluorescence turret. A side-looking 1064-nm dichroic mirror (Chroma Technology, Bellows Falls, Vermont) mounted within the fluorescence turret provides the ability to perform the normal (visible) transillumination microscope viewing while the tweezers are operating. A KG-1 IR cutoff filter (Newport, Irvine, California) inserted above the fluorescence turret prevents the observation of the tweezer spot by a camera during operation. The trapping beam applies a force to objects by moving the microscope sample stage (Prior Proscan II Motorized H101/2 Stage); the trapping beam does not move.

Fig. 1

Hardware configuration of the laser tweezer, microscope, and quadrant photodiode (QPD) data acquisition system. Tweezer beam path is indicated by a solid line. Components labeled as follows: (a) Tweezer module. (b) Side-looking dichroic mirror located in fluorescence turret. (c) Objective lens. (d) Sample plane. (e) Condenser lens. (f) Dichroic mirror. (g) QPD. (h) Tube lens. (i) Camera. (j) Five-degree-of-freedom tweezer mount. (k) Transillumination stage. (l) Incubator. (m) QPD data acquisition module. (n) Computer. Inset: photograph of experimental apparatus.

To record the movement of a trapped object, a QPD (first sensor model QP50-6-SD2) records the location of the trapping beam centroid. The QPD is located downstream from the condenser lens. Use of a dichroic mirror (Qoptic Microbench, Fairport, New York) and laser line filter (1064 nm, Edmund Optics, Barrington, New Jersey) ensures that only the Nd:YAG light is incident on the QPD. Intensities were converted to voltage readouts within the QPD housing and then transferred to a National Instruments data acquisition device. This device recorded voltages at a user-specified frequency and exported a digital record of particle positions to the host computer to be saved and analyzed later.

3.2.

Software Architecture

3.2.1.

Tracking protocol

We created a virtual instrument (VI) environment in LabView 2010 with a front panel that allows for customization of sampling frequency and duration of data acquisition. Our tracking VI recorded voltage outputs from the QPD, normalized the data via a reference voltage to remove the circuit noise, and displayed the particle information in real time on the front panel. Figure 2 shows the flow of data through the VI, and Fig. 3 presents an example of the VI’s voltage/position readout.

Fig. 2

Flowchart detailing data progression within the data acquisition virtual instrument (VI). The incoming signal for the QPD is constantly pinged while the VI is running, allowing the user the ability to see a real-time feed of the laser spot on the surfaces of the QPD.

Fig. 3

A sample of raw QPD output (‘X’ and ‘Y’ represent orthogonal directions). The dip is characteristic of a microsphere entering the trap.

The recorded datasets are then converted and stored as .TDMS binary files, which are used for their low data-loss rate at high sampling frequency and low memory footprint. This file can then be directly imported into our MATLAB analysis algorithm.

Stock data analysis methods are not well suited for the large data files produced by high sampling rates. Therefore, an algorithm was developed to manage the large sets of acquired experimental data. All the analysis procedures are called using the function QPDanalysisBulk(TDMSFolder,TDMSfileID,f). This function has user input variables for the root directory of the .TDMS file to be analyzed and the filename of said .TDMS.

3.2.4.

Remove linear drift of MSD

The MSD curve can be split into roughly two sections. The initial (short-time) section of the curve represents an inertial timescale, whereas the latter section represents a long time limit corresponding to the restoring force of the optical trap. However, the long time limit is often not reached and as a result, the calculated MSD contains a linear drift. This experimental artifact is removed through the use of the function RemoveLinarDrift(MSDArray,maxlagtime,freq). The function first applies a linear regression to the long-time calculated MSD data. The linear regression is then subtracted from the experimentally acquired MSD data. The corrected MSD data allow the fitting of the idealized curve equation to the experimental data. An example of this curve can be seen in Fig. 4, which shows the removal of linear drift and resultant, corrected MSD.

Fig. 4

Mean-squared displacement (MSD) data generated from a sample QPD output for a trapped microsphere. Note the linear drift in the Test Data series and its removal in the Corrected Data series due to our algorithm.

3.2.5.

Fit primary dataset to idealized relationship and store fitting parameters

Once the experimental data have been corrected with the linear drift of the long-term limit removed, the algorithm fits the data to the idealized MSD function. Since our data have an arbitrary length and time units, we must introduce two conversion constants, $\alpha$ (normalized volts per meter) and $\beta$ (seconds per desired time unit). The new model can be expressed as

(4)

$$[\mathrm{\Delta }{r}^2(\tau )]=\frac{2k_{B}T{\alpha }^2}{\kappa }(1-e^{-\kappa \beta \tau /\xi })\equiv a(1-e^{-b\tau }).$$

Thus, our model can be expressed in terms of $a$ and $b$

(5)

$$a=\frac{2k_{B}T{\alpha }^2}{\kappa };b=\frac{\kappa \beta }{\xi }.$$

To enhance our fitting routine’s accuracy, we first fit the long-time portion of our data with a linear model to obtain the value of $a$. Next, we fit our data using a robust nonlinear least squares fit, supported by the MATLAB Curve Fitting toolbox. During this process, the stored value of $a$ is substituted into the model so that $b$ is the only free parameter. A graph of this fit is then exported as a MATLAB .FIG file and a high-resolution .PNG to the root folder. Figure 5 details an example of the quality of the fit achieved by the algorithm. Utilizing the relationships of the constants $a$ and $b$, the software then solves for the spring and diffusion constants for the dataset. Finally, these values are stored in their own array to be appended with more values down the line.

Fig. 5

Example of the results from the curve-fitting routine for the MSD data shown in Fig. 4. Note the fitting of the long time-scale portion of the graph lies nearly along the average, allowing for an accurate reading of the fitting constants.

4.1.

Algorithm Calibration

Simulation of a trapped particle executing confined Brownian motion was used as an internal consistency check of our data analysis algorithm. The Langevin equation for a Brownian particle moving within a 3-D harmonic potential (neglecting the inertial term) is given as

Following Refs. 64 and 65, we independently generated a trajectory for a spherical particle in an optical trap using our system specifications and analyzed the simulated data with our toolkit. Our toolkit calculated the MSD fitting parameters for several sets of simulation data with varying sampling frequencies, time spans, trap strengths, and initial positions. The returned constants fit the provided values within a range of 5% from the expected values.

4.2.

Instrument Calibration

We calibrated the optical trap fitting parameters using homogeneous spherical particles, and we will show that once calibrated, the fitting algorithm correctly returns the spring constant for (1) spherical particles trapped near a glass surface, (2) spherical particles immersed in glycerol, and (3) cylindrical objects (E. coli bacteria). Each test was chosen to verify the fitting algorithm using uncertainties in viscous drag (tests 1 and 2), changes in trap beam geometry (test 2), and changes in object geometry and optical properties (test 3).

The optical trap was calibrated using 0.5-, 1-, 1.98-, 2.5-, and 5.46-μm diameter polystyrene microspheres (Bangs Laboratories, Fishers, Indiana), each stock solution diluted (by volume) $1:{10}^9$ in phosphate-buffered saline (PBS, Cellgro, Pittsburgh, Pennsylvania) to prevent the aggregation. Viscosity of PBS was measured with a Cannon-Fenske Routine viscometer (Induchem Lab Glass, Roselle, New Jersey) and pycnometer (Cole Parmer, Vernon Hills, Illinois), resulting in a value of $0.8816\pm 0.0002\mathrm{cP}$ at 25°C. The dilute suspension of (monodisperse) microspheres was placed within a hanging drop-type microscope slide, the laser trap was turned on, and the microscope stage was manipulated to bring a microsphere into the trap far from any solid surface using a camera for visual confirmation that a single microsphere was trapped. The height of the trapped particle relative to the microscope slide was recorded using the microscope’s internal $z$-axis encoder readout. During lateral movements of the stage, the distance between trapped particle and microscope slide remained constant.

Comparing QPD output to Stokes drag measurements, shown in Fig. 7, allowed us to obtain a calibrated relation between QPD data and applied force for spherical objects, because the only variable is the particle diameter. The QPD recorded the positions of spheres before the stage was moved. After the QPD data was acquired, the stage was moved laterally at a known speed from its home position to second position approximately 2 mm away and then back to its home position. The test was performed at progressively higher stage speeds until the trapped particle fell out of the laser trap due to viscous drag. The stage speed at which the particle fell out of the laser trap was used to determine the drag force from the PBS solution on the microsphere using Stokes’ law.

4.3.

Validation of Trap Strength Algorithm

As internal consistency checks, QPD data were collected for 60 s at sampling rates of 10, 50, and 100 kHz to determine potential effects from sampling rate. Other datasets were acquired for spot centroids located at various positions on the QPD to determine effects from misalignment. The aforementioned analysis was performed on each dataset, and the calculated spring constant was unchanged, indicating our system is quite robust. Next, a single large dataset was divided into smaller sets of 20,000 to 30,000 data points. The same analysis was performed on these subsets of data. The resulting spring constant of the entire set was compared with the spring constant results of the subsets of data, and the algorithm returned consistent spring constants. Taken together, our internal consistency checks demonstrate internal validation of our trap strength algorithm.

4.4.

Validation of Trap Strength Calculation

Three experimental checks of the fitting algorithm were performed. First, we trapped microspheres near the glass surface as an experimental test of Faxen’s law.⁶⁷ Next, we trapped microspheres suspended in glycerol rather than PBS. Finally, we trapped an E. coli bacterium, a rod-shaped object for which a closed-form solution to Stokes drag can be obtained, and for all tests, we compared the calibrated fitting algorithm output to the (maximum) viscous drag force.

The first validation step was performed by comparing the QPD data to Stokes drag data for particles trapped at various heights above the slide (Fig. 6). This check was performed because while the fluid drag varies with height, approximately given by Faxen’s law,⁶⁸ the trapping force does not (the small amount of light back-reflected off the fluid–glass interface can be neglected). The calculated spring constant did not show a variation in height, as expected.

Fig. 6

Comparison of Stokes drag (diamonds) and fitted spring constant (squares) for 1.98-μm-diameter spheres trapped at fixed distances from a glass slide. The drag force varies with height; the fitted spring constant does not.

Next, we trapped a 1.98-μm-diameter microsphere in glycerol (Fisher Scientific, Pittsburgh, Pennsylvania). Glycerol is approximately 1000 times as viscous as PBS ($\mu =1.416\mathrm{P}$), but because the MSD fit curve allows separate fitting of trap stiffness $\kappa$ and viscous drag $\xi$, the trap force can be distinguished from viscous drag. As seen in Fig. 7, the calibrated algorithm correctly measured the spring constant. Thus, we demonstrated that our calibration curve is not sensitive to variations in solvent viscosity.

Fig. 7

This graph shows the experimental results for the QPD algorithm calibration using microspheres ($N=5$ for each data point) overlaid on results of our algorithm output for $N=5$ microsphere suspended in glycerol and our algorithm output for a trapped ($N=5$) Escherichia coli bacterium. The error bars correspond to measured uncertainty in (vertical) Stokes drag and (horizontal) results of our algorithm.

Additionally, because the objective lens is uncorrected for glycerol (at $\lambda =1064\mathrm{nm}$, $n_{\mathrm{gly}}=1.46$, while $n_{\mathrm{PBS}}=1.33$), the objective lens introduced wavefront aberrations (primarily spherical aberration) to the trapping beam. Thus, this measurement confirmed the results of test 1 and also demonstrated that our fitting algorithm does not require detailed information about the trapping beam wavefront.

We next optically trapped a cylindrical object, a single E. coli bacterium. E. coli is a rod-shaped bacterium 1 μm in diameter and 3 μm in length. E. coli normally swim using a flagellum, so we first killed the bacteria with a brief exposure to intense ultraviolet light to avoid experimental artifacts. Once trapped, the bacterium oriented along the optical axis, and both QPD and Stokes drag measurements were performed. Critically important, the bacterium does not deform during the Stokes drag measurement. Modeling the bacterium as a cylinder of length $L$ and radius $R$ capped with hemispherical ends, the viscous drag at velocity $U$ (fluid density $\rho$ and viscosity $\mu$) is given by

The first term results from the two hemispherical caps, and the second term is the drag force per unit length for a cylindrical object⁵¹ and contains Euler’s constant $\gamma =0.577$. Again, as seen in Fig. 7, the calibrated QPD algorithm correctly computed the spring constant, demonstrating that our algorithm results are insensitive to object shape.