Double-Helix based Real-Time Single Particle Tracking

TL;DR

The paper addresses 3D tracking of nanometer-scale particles in native environments by introducing DH-RT-SPT, which uses the DH-PSF to encode axial position into a single-plane PSF. A circular scanning strategy combined with a simple proportional controller enables real-time 3D tracking without multi-plane detection, while PSO-based optimization tunes key DH design, scanning, and control parameters to maximize tracking duration and minimize photon budget. Simulation results show robust tracking of diffusion coefficients up to 10 μm^2/s, with tracking times often exceeding the corresponding first passage times, albeit with increasing errors at higher diffusion. The approach promises faster, simpler RT-FD-SPT and lays groundwork for hardware realization and experimental evaluation in living-cell contexts.

Abstract

In Real-Time, Feedback-Driven Single Particle Tracking methods, measurements of the emission intensity from a labeled, nanometer-scale particle are used in a feedback loop to track the motion of the particle as it moves inside its native environment, including within living cells. In this work, we take advantage of Point Spread Function (PSF) engineering techniques that encode the axial position of the particle into the shape of the PSF in the focal plane to eliminate the need for out-of-focal-plane measurements, reducing the complexity of implementation and decreasing the overall measurement time of the control loop. Specifically, we used the Double Helix PSF (DH-PSF) in which a single fluorescent source gives rise to two lobes in the image plane with the lobes rotating in the plane as the particle moves along the optical axis. We designed simple estimators of the relative error between the particle and the tracker, and a simple proportional feedback controller to regulate that error to zero. We explored the efficacy of the approach through simulation studies, demonstrating the tracking of fast-moving particles (with diffusion coefficients up to 10 {μ\text{m}^2/\text{s}}) over long time periods (multiple seconds).

Figures (5)

• Figure 1: Illustration of the DH-RT-SPT setup.
• Figure 2: DH-PSF scanning mechanism using parameters $R=1000$ nm, $\sigma_{DH}$ = 200 nm, $r_{scan} = 900$ nm, $G$ = 50, and $N_B = 1.$ (a)-(c) Intensity plots representing the orientation of the DH-PSF along the Z-axis locations of 500 nm, 0 nm, and -500 nm, respectively. The white circle is the scanning radius, and the white dashed lines are the best lines along the double helix. (d)-(f) shows the particle locations surrounded by the two lobes of the double helix. The tiny green sphere in the middle represents the particle, and the three green dots are the projections of that particle in the xy, yz, and xz planes. The red and blue curves are the trajectories of the two lobes of the DH-PSF. (g) The segmented scanning ring and its intensity variation when z = 0 nm. (h) Intensity variation as a function of angle, $\theta$, at z = -500 nm.
• Figure 3: Feedback control block diagram of a DH-RT-SPT system.
• Figure 4: DH-FD-SPT based microscopy technique. (a) 3D trajectory of a particle diffusing at D = 1 $\mu\text{m}^2/\text{s}$ tracked for 30 sec. The color coding of the curve represents elapsed time. (b) Movement of the particle and the scanning center of DH-PSF in the time frame of 18.5-18.75 seconds. (c) The total measured intensity around each circular scan and (d) the lateral and axial error over time.
• Figure 5: Performance of DH-PSF-based tracking microscope evaluated from 500 independent simulations. (a) Average tracking duration for different diffusion coefficients (1- 10 $\mu\text{m}^2/\text{s}$). (b) Histogram of tracking times for diffusion coefficients of 5, 7, and 9 $\mu\text{m}^2/\text{s}$. $\tau$ is the first passage time of a diffusing particle out of the scan circle. The 0-$\tau$ column shows the runs that do not result in successful tracking. (c) First passage time ($\tau$) out of sensing area as a function of diffusion coefficient. (d) Average intensity as counts per scan for different diffusion coefficients. (e) Average mean lateral (blue) and combined 3D tracking error (purple). (f) Mean absolute error (MAE) in the individual axes over the change of diffusion coefficients.

Theorems & Definitions (1)

• Remark 1