Tech Note: (De-)Multiplexing: Multi-Trap Force Measurements and Sample Manipulation with JPK’s NanoTracker 2

Trap multiplexing in the time-sharing mode of Bruker’s NanoTracker 2 relies on ultra-fast high precision beam steering and further expands the range of possible trap configurations for a multitude of applications. Trap stiffness values can be adjusted for each trap individually by the user through the convenient dwell time set function integrated in the NanoTracker 2 control software. All multiplexed traps can be positioned independently or arranged in complex patterns stored in custom configuration files. This freedom in configuring and dynamically rearranging tens to hundreds of parallel traps enables users to perform complex manipulations of microscopic objects.

The new de-multiplexing feature is based on quick and reliable real-time data analysis and allows separating the signals from up to eight parallel traps. The isolated signals can be used for the subsequent calibration of detection sensitivity and trap stiffness which in turn in the prerequisite for high resolution displacement and force measurements. This unique combination of precision and flexibility will enable researchers to gain more detailed insight into the interactions of complex nanoscale systems.

Readers can expect to learn about:

• The principle of force measurements and trapping with Optical Tweezers;
• The challenges of (de-)multiplexing, and calibration methods such as viscous drag measurements showcasing the outstanding flexibility and precision of Bruker's NanoTracker system; and
• The benefits and potential of the new de-multiplexing feature for investigating complex nanoscale systems.

_{KEYWORDS: Optical Tweezers; NanoTracker; Force Measurements; Multiplexing; Viscous Drag Measurements; Nanomanipulation; Life Science}

Optical tweezers utilize the fundamental property of light to carry linear and angular momentum in order to trap objects with a highly focused laser beam. This technique allows the direct manipulation of microscopic objects with sizes from few tens of nanometers up to several micrometers. Furthermore, optical tweezers are a quantitative tool to apply well-defined forces and, at the same time, accurately measure displacements of and forces acting on the trapped object. This combination of non-invasive microscopic manipulation and simultaneous high precision measurements has made optical tweezers a useful and versatile tool in a variety of applications ranging from micro-rheology and colloidal hydrodynamics to molecular biochemistry, biophysics and cell biology. 

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While many measurements (e.g. hydrodynamics, micro­rheology, tracking of protein motion) can be performed with single traps in combination with a piezo-electric high­precision sample scanner, other applications like whole cell rheology and the manipulation of complex multi­molecular systems often require multiple traps that can be positioned individually. Example experimental setups are shown in figure 1. In many cases, the requirements even go beyond mere trap positioning and include the parallel measurement of forces acting on multiple trapped particles. More details on diverse optical trapping applications and techniques can be found in the comprehensive review by Moffit et al. ¹. 

Dual beam traps are typically generated by polarization­-dependent beam splitting of a single laser source ². Here the incoming light is linearly polarized and subsequently split in two beams with perpendicular orientation. This minimizes interference between the two traps in the sample and enables the separate detection of signals from the traps. In JPK's Nano Tracker™ 2, at least one of these traps can be positioned individually via piezo-electric mirrors or acousto-optical deflectors (AODs). This provides the necessary degrees of freedom for a wide range of measurements like single molecule stretching or basic whole cell manipulation. Due to the polarization-based separation of the two beams, the highest possible measurement accuracy is also ensured for dual-trap setups. 

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For many applications, it is favorable or even necessary to manipulate a sample optically at many locations at the same time. Prominent examples would be the handling of multi-molecular constructs with several attached micro­bead handles (see figure 1), the manipulation of suspended cells and particles in order to investigate their interactions, or the deformation of cells and other objects in complex patterns.

For these purposes, multiple (more than two) independent traps can be set up from one single laser source. This method termed multiplexing can be implemented by moving the beam quickly from one desired trap position to the next where it resides for a defined time (dwell time, t_dw) and thus establishes the conditions for optical trapping ³. The time course of this beam position switching is illustrated in figure 2. This approach to multiplexing is called time-sharing (or beam-sharing) and, depending on the laser power, enables the creation of tens to hundreds of parallel traps at independent positions in the sample. The method is technically very demanding and requires precise and fast control electronics as well as high speed beam positioning. For stable trapping, the dwell time has to be longer than the time scale of the particle's Brownian motion under the given conditions. For Polystyrene (PS) beads in the size range of 1 - 2 µm in water, the lower limit for tdw is typically between 20 and 50 µs. 

The second limiting time factor is the so called revisiting time t_rev , that is the time that passes until the laser beam has cycled through all trap positions and returns to the same spot (i.e. the time a trapped particle is without laser). Uninterrupted manipulation requires that the object cannot escape the trap volume while the laser is scanning the other trap positions. Just like t_dw, t_rev depends on particle size as well as medium temperature and viscosity and needs to be determined for each experimental setup. Typical values are in the range of a few milliseconds. 

Naturally, the achievable revisiting times also depend on the number of traps N to be implemented: 

Large numbers of parallel traps with revisiting times shorter than 10 ms in turn require very high switching rates f_s of the beam positioning devices: 

For t_rev  ≤ 10ms and N = 250 parallel traps, this yields

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In JPK's NanoTracker™ 2, an ultra-fast AOD ensures that the above conditions for stable trapping and manipulation are maintained even for hundreds of simultaneous traps. An example of trap multiplexing with 80 parallel traps is shown in figure 3.

The stiffness of an optical trap denotes the restoring forces acting on a trapped particle per displacement from the trap center and depends on numerous parameters. Among them, the most important are laser power and beam shape, the size of the trapped object, the ratio of refractive indices (n_particte/n_medium), and the steepness of the intensity gradient in the focal spot. In multiplexed traps, the gradient force F_G responsible for centering the trapped particle in the focused laser beam is only acting on each object during the dwell time tdw (i.e. while the laser is present at the respective position) and is absent for trev· Thus, the trap stiffness is also "shared" among multiplexed traps. For scanning frequencies above the corner frequency of the particle's Brownian motion power spectrum (≈ 250 - 1500 Hz for micrometer-sized particles, see figure 7 and JPK's technical note on quantitative force measurements with optical tweezers), the effective trap stiffness decreases linearly with the number of parallel traps ⁴. By varying t_dw for different positions, traps with different stiffness values can be generated from one single laser beam. This function is conveniently integrated in the NanoTracker™ 2 control software where the position and dwell time for each of the multiplexed traps can be set separately or optionally all traps can be synchronized to have the same effective trap stiffness (see figure 4). A widely used application of optical tweezers is the manipulation and deformation of live suspended cells (e.g. red blood cells) for rheological measurements. With the different force levels available for multiplexed traps in the NanoTracker™ 2, cells can be held and positioned with minimum forces in order not to disturb the deformation-force measurements that are conducted simultaneously with two stronger traps. Moreover, multi-molecular arrangements, macro­molecules or nano-fabricated devices can be stably held at constant positions with multiple stronger traps while a weaker trap is used to probe their properties or interactions with other objects. 

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Since multiplexed traps are all derived from the same beam, they do not differ in polarization and cannot be separated easily for individual analysis like the two beams used for dual trap configurations. This means that the signals from all multiplexed traps are recorded with one quadrant photo diode (QPD). This renders the standard noise spectrum based calibration method (and thus the determination of detection sensitivity and trap stiffness) unfeasible without further data processing. The challenge is to separate the signals originating from different trap positions in order to make sure that only Brownian motion data of the same trap and particle are used for calibration of the individual traps. The graph in figure 5 shows the raw signal of three multiplexed traps as it is recorded by the QPD. 

The NanoTracker™ 2 detection system and data processing electronics operate at a frequency of 800 kHz which allows to time­resolve the different phases of trap multiplexing following the approach introduced by Guilford et al. ⁵. It is clearly visible that the traps produce slightly different plateaus of thermal noise signals which are interrupted by higher amplitude signals caused by the position switching process. These plateau phases as well as the characteristic shape of the switching transition help to identify the times when the laser beam is present and stable at the positions of trap 1 to 3. Time delays inherent to the system can be estimated and taken into account. In the de-multiplexing control panel shown in figure 4, values for the delay time tlag and the time required for position switching (invalid time) can be set matching the current trap configuration. 

After identifying t_lag as well as the switching and effective dwelling phases, the signal from one dwelling phase is averaged for further data processing. These average values (one per trap and cycle) are then used to generate the separated signals from the multiplexed traps. A typical example of such an extracted signal for one trap is shown in figure 6. On longer time scales (left graph), the signals appear like regular constant noise as it is typically recorded from spherical particles in an optical trap. Only zooming in the time axis reveals that the signal has been reconstituted from the time separated original signal. After this averaging step, QPD signals for each of the N traps can be generated with a maximum effective sample rate of 

For the example shown in figure 4 with N = 3 and t_dw = 50µs, this means that after Fourier transforming the signal, each trap can be calibrated using a power spectrum recorded up to approximately 6.7 kHz (corresponding to a time resolution of 150 µs). This in turn allows using JPK's well-established and reliable power spectrum fitting routine and to achieve the same high standard for the determination of detection sensitivity and trap stiffness in the multiplexing mode that is known from single or dual trap calibrations (figure 7). Considering the nature of this de-multiplexed signal, it is clear that frequencies higher than the effective sample rate cannot be resolved in the noise signals and are not contained in the respective power spectra. 

As a standard application and trap stiffness calibration method, viscous drag measurements are routinely used in various experiments. Here, the fluid-filled sample chamber is moved periodically with respect to the trapped beads, thus generating well-defined viscous drag forces acting on the particles. For low Reynolds number systems (laminar flow), the drag forces F_d  on spherical objects can be calculated easily from bead radius, r, flow velocity, v, and medium viscosity, η, using Stokes' drag equation: 

In optical traps, this force results in a displacement of the particle from the trap center that can be measured in order to determine the trap stiffness. Vice versa, identical particles in traps with different stiffness values will show different displacements from the trap center when exposed to the same viscous drag as schematically shown in figure 8A. As mentioned above, variations in dwell time tdw of multiplexed traps result in different relative trap parameters: traps with larger tdw are stiffer, following approximately a linear relation. Figure 8B displays the displacement data from two beads trapped in multiplexed traps with t_dw = 50 (blue curve) and 100 µs (green curve). For visualization purposes, the curves have been separated by shifting them ± 100 nm. It is clearly visible that the particle with t_dw = 100µs shows a smaller displacement (corresponding to higher trap stiffness k) while the particle trapped with tdw = 50 µs is displaced approximately twice as far. Using the calibration parameters obtained from individual power spectrum fits for each trap (like the one shown in figure 6) delivers the force-time curves displayed in figure 8C. As expected after successful calibration, the drag forces measured in the three traps agree very well. Due to the linearity of Stokes' drag equation, reducing the flow speed from 200µm/s to 100 µm/s leads to a drop in viscous drag of 50%. This is well reflected by the measured forces displayed in figure 8D. This demonstrates that force measurements with piconewton resolution are also possible with de-multiplexed traps. 

Trap multiplexing in the time-sharing mode of JPK's NanoTracker™ 2 relies on ultra-fast high precision beam steering and further expands the range of possible trap configurations for a multitude of applications. Trap stiffness values can be adjusted for each trap individually by the user through the convenient dwell time set function integrated in the NanoTracker™ 2 control software. All multiplexed traps can be positioned independently or arranged in complex patterns stored in custom configuration files. This freedom in configuring and dynamically rearranging tens to hundreds of parallel traps enables Nano Tracker™ 2 users to perform complex manipulations of microscopic objects. 

The new de-multiplexing feature is based on quick and reliable real-time data analysis and allows separating the signals from up to eight parallel traps. The isolated signals can be used for the subsequent calibration of detection sensitivity and trap stiffness which in turn is the prerequisite for high resolution displacement and force measurements. This unique combination of precision and flexibility will enable researchers to gain more detailed insight into the interactions of complex nano-scaled systems. 

References

1. J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, "Recent Advances in Optical Tweezers," Annu. Rev. Biochem., vol. 77, no. 1, pp.205 - 228, Jun. 2008.
2. H. Misawa, K. Sasaki, M. Koshioka, N. Kitamura, and H. Masuhara, "Multibeam laser manipulation and fixation of microparticles," Appl. Phys. Lett., vol. 60,no.3,p.310, 1992.
3. K. Visscher, G. J. Brakenhoff, and J. J. Krol, "Micromanipulation by 'multiple' optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope," Cytometry, vol. 14, no. 2, pp. 105-114, 1993.
4. K. Visscher, S. P. Gross, and S. M. Block, "Construction of multiple-beam optical traps with nanometer-resolution position sensing," /eee J. Se/. Top. Quantum Electron., vol. 2, no. 4, pp. 1066- 1076, Dec. 1996.
5. W. H. Guilford, J. A. Tournas, D. Dascalu, and D. S. Watson, "Creating multiple time-shared laser traps with simultaneous displacement detection using digital signal processing hardware," Anal. Biochem., vol. 326, no. 2, pp. 153-166, Mar. 2004.