Tracking single fluorescent particles in three dimensions via extremum seeking

Information optimization of laser scanning microscopes for real-time feedback-driven single particle tracking

Real-time feedback-driven single particle tracking (RT-FD-SPT) is a class of microscopy techniques that uses measurements of finite excitation/detection volume in a feedback control loop to actuate that volume and track with high spatio-temporal resolution a single particle moving in three dimensions. A variety of methods have been developed, each defined by a set of user-defined choices. Selection of those values is typically done through ad hoc, off-line tuning for the best perceived performance. Here we present a mathematical framework, based on optimization of the Fisher information, to select those parameters such that the best information is acquired for estimating parameters of interest, such as the location of the particle, specifics of the excitation beam such as its dimensions or peak intensity, or the background noise. For concreteness, we focus on tracking of a fluorescently-labeled particle and apply this framework to determine the optimal parameters for three existing fluorescence-based RT-FD-SPT techniques with respect to particle localization.

Theoretical comparison of real-time feedback-driven single-particle tracking techniques

Real-time feedback-driven single-particle tracking is a technique that uses feedbackcontrol to enable single-molecule spectroscopy of freely diffusing particles in nativeor near-native environments. A number of different RT-FD-SPT approaches exist,and comparisons between methods based on experimental results are of limited usedue to differences in samples and setups. In this study, we used statistical calcu-lations and dynamical simulations to directly compare the performance of differentmethods. The methods considered were the orbital method, the Knight's Tour (gridscan) method and MINFLUX, and we considered both fluorescence-based and inter-ferometric scattering (iSCAT) approaches. There is a fundamental trade-off betweenprecision and speed, with the Knight's Tour method being able to track the fastestdiffusion but with low precision, and MINFLUX being the most precise but onlytracking slow diffusion. To compare iSCAT and fluorescence, different biologicalsamples were considered, including labeled and intrinsically fluorescent samples. Thesuccess of iSCAT as compared to fluorescence is strongly dependent on the particlesize and the density and photophysical properties of the fluorescent particles. Usinga wavelength for iSCAT that is negligibly absorbed by the tracked particle allowsan increased illumination intensity, which results in iSCAT providing better trackingfor most samples. This work highlights the fundamental aspects of performance inRT-FD-SPT and should assist with the selection of an appropriate method for a par-ticular application. The approach used can easily be extended to other RT-FD-SPTmethods.

Real-Time Feedback-Driven Single-Particle Tracking: A Survey and Perspective

Real-time feedback-driven single-particle tracking (RT-FD-SPT) is a class of techniques in the field of single-particle tracking that uses feedback control to keep a particle of interest in a detection volume. These methods provide high spatiotemporal resolution on particle dynamics and allow for concurrent spectroscopic measurements. This review article begins with a survey of existing techniques and of applications where RT-FD-SPT has played an important role. Each of the core components of RT-FD-SPT are systematically discussed in order to develop an understanding of the trade-offs that must be made in algorithm design and to create a clear picture of the important differences, advantages, and drawbacks of existing approaches. These components are feedback tracking and control, ranging from simple proportional-integral-derivative control to advanced nonlinear techniques, estimation to determine particle location from the measured data, including both online and offline algorithms, and techniques for calibrating and characterizing different RT-FD-SPT methods. Then a collection of metrics for RT-FD-SPT is introduced to help guide experimentalists in selecting a method for their particular application and to help reveal where there are gaps in the techniques that represent opportunities for further development. Finally, this review is concluded with a discussion on future perspectives in the field.

Analysis of an Extremum Seeking Controller Under Bounded Disturbance

One of the applications of Extremum Seeking (ES) is to localize the source of a scalar field by using a mobile agent that can measure this field at its current location. While the scientific literature has presented many approaches to this problem, a formal analysis of the behavior of ES controllers for source seeking in the presence of disturbances is still lacking. This paper aims to fill this gap by analyzing a specific version of an ES control algorithm in the presence of source movement and measurement disturbances. We define an approximate version of this controller that captures the main features but allows for a simplified analysis and then formally characterize the convergence properties of this approximation. Through simulations and physical experiments, we compare the theoretically-predicted regions of attraction of the simplified system with the behavior of the full system and show that the simplified version is a good predictor of the behavior of the initial ES controller.

Information Optimal Control for Single Particle Tracking Microscopy

We consider the problem of designing a control policy for a laser scanning microscope (LSM) which will minimize the estimation uncertainty when identifying the state and motion model of a fluorescent biological particle. Using the information optimal design framework we pose an optimization problem which seeks to maximize the Fisher information of the particle's state. We then apply optimal control methods to determine the laser trajectory that maximizes a criterion based on the Fisher information. The resulting optimal control policy is a Bang-Singular control which moves the laser to the set of measurement locations that maximize the rate of information accumulation. Simulations demonstrate the ability of the resulting control system to position the laser to measure the particles location with a minimum uncertainty.

Three-dimensional localization refinement and motion model parameter estimation for confined single particle tracking under low-light conditions

Confined diffusion is an important model for describing the motion of biological macromolecules moving in the crowded, three-dimensional environment of the cell. In this work we build upon the technique known as sequential Monte Carlo - expectation maximization (SMC-EM) to simultaneously localize the particle and estimate the motion model parameters from single particle tracking data. We extend SMC-EM to handle the double-helix point spread function (DH-PSF) for encoding the three-dimensional position of the particle in the two-dimensional image plane of the camera. SMC-EM can handle a wide range of camera models and here we assume the data was acquired using a scientific CMOS (sCMOS) camera. The sensitivity and speed of these cameras make them well suited for SPT, though the pixel-dependent nature of the camera noise presents a challenge for analysis. We focus on the low signal setting and compare our method through simulation to more standard approaches that use the paradigm of localize-then-estimate. To localize the particle under the standard paradigm, we use both a Gaussian fit and a maximum likelihood estimator (MLE) that accounts for both the DH-PSF and the pixel-dependent noise of the camera. Model estimation is then carried out either by fitting the model to the mean squared displacement (MSD) curve, or through an optimal estimation approach. Our results indicate that in the low signal regime, the SMC-EM approach outperforms the other methods while at higher signal-to-background levels, SMC-EM and the MLE-based methods perform equally well and both are significantly better than fitting to the MSD. In addition our results indicate that at smaller confinement lengths where the nonlinearities dominate the motion model, the SMC-EM approach is superior to the alternative approaches.

Computationally efficient application of Sequential Monte Carlo expectation maximization to confined single particle tracking

Single Particle Tracking (SPT) plays a crucial role in biophysics through its ability to reveal dynamic mechanisms and physical properties of biological macromolecules moving inside living cells. Such molecules are often subject to confinement and important information can be revealed by understanding the mobility of the molecules and the size of the domain they are restricted to. In previous work, we introduced a method known as Sequential Monte Carlo-Expectation Maximization (SMC-EM) to simultaneously estimate particle trajectories and model parameters. In this paper, we describe three modifications to SMC-EM aimed at improving its computationally efficiency and demonstrate it through analysis of simulated SPT data of a particle in a three dimensional confined environment. The first two modifications use approximation methods to reduce the complexity of the original motion and measurement models without significant loss of accuracy. The third modification replaces the previous SMC methods with a Gaussian particle filter combined with a backward simulation particle smoother, trading off some level of generality for improved computational performance. In addition, we take advantage of the improved efficiency to investigate the effect of data length on performance in localization and parameter estimation.

Expectation maximization based framework for joint localization and parameter estimation in single particle tracking from segmented images

Single Particle Tracking (SPT) is a well known class of tools for studying the dynamics of biological macromolecules moving inside living cells. In this paper, we focus on the problem of localization and parameter estimation given a sequence of segmented images. In the standard paradigm, the location of the emitter inside each frame of a sequence of camera images is estimated using, for example, Gaussian fitting (GF), and these locations are linked to provide an estimate of the trajectory. Trajectories are then analyzed by using Mean Square Displacement (MSD) or Maximum Likelihood Estimation (MLE) techniques to determine motion parameters such as diffusion coefficients. However, the problems of localization and parameter estimation are clearly coupled. Motivated by this, we have created an Expectation Maximization (EM) based framework for simultaneous localization and parameter estimation. We demonstrate this framework through two representative methods, namely, Sequential Monte Carlo combined with Expectation Maximization (SMC-EM) and Unscented Kalman Filter combined with Expectation Maximization (U-EM). Using diffusion in two-dimensions as a prototypical example, we conduct quantitative investigations on localization and parameter estimation performance across a wide range of signal to background ratios and diffusion coefficients and compare our methods to the standard techniques based on GF-MSD/MLE. To demonstrate the flexibility of the EM based framework, we do comparisons using two different camera models, an ideal camera with Poisson distributed shot noise but no readout noise, and a camera with both shot noise and the pixel-dependent readout noise that is common to scientific complementary metal-oxide semiconductor (sCMOS) camera. Our results indicate our EM based methods outperform the standard techniques, especially at low signal levels. While U-EM and SMC-EM have similar accuracy, U-EM is significantly more computationally efficient, though the use of the Unscented Kalman Filter limits U-EM to lower diffusion rates.

Tracking Multiple Diffusing Particles Using Information Optimal Control

We study the problem of tracking multiple diffusing particles using a laser scanning fluorescence microscope. The goal is to design trajectories for the laser to maximize the information contained in the measured intensity signal about the particles' trajectories. Our approach consists of a two level scheme: in the lower level we use an extremum seeking controller to track a single particle by first seeking it then orbiting around it. In the higher level controller, we decide which particle should be observed at each instant, with the goal of efficiently estimating each particle position while not losing track of any of them. Using simulations, we show that this technique is able to collect photons efficiently and to track multiple particles with low position estimation error.

Feedforward Control for Single Particle Tracking Synthetic Motion

Single particle tracking (SPT) is a method to study the transport of biomolecules with nanometer resolution. Unfortunately, recent reports show that systematic errors in position localization and uncertainty in model parameter estimates limits the utility of these techniques in studying biological processes. There is a need for an experimental method with a known ground-truth that tests the total SPT system (sample, microscope, algorithm) on both localization and estimation of model parameters. Synthetic motion is a known ground-truth method that moves a particle along a trajectory. This trajectory is a realization of a Markovian stochastic process that represents models of biomolecular transport. Here we describe a platform for creating synthetic motion using common equipment and well-known, simple methods that can be easily adopted by the biophysics community. In this paper we describe the synthetic motion system and calibration to achieve nanometer accuracy and precision. Steady state input-output characteristics are analyzed with both line scans and grid scans. The resulting relationship is described by an affine transformation, which is inverted and used as a prefilter. Model inverse feed forward control is used to increase the system bandwidth. The system model was identified from frequency response function measurements using an integrated stepped-sine with coherent demodulation built into the FPGA controller. Zero magnitude error tracking controller method was used to invert non-minimum phase zeros to achieve a stable discrete time feed forward filter.

Monte Carlo Simulation of Brownian Motion using a Piezo-Actuated Microscope Stage

Single particle tracking is a powerful tool for studying and understanding the motions of biological macromolecules integral to cellular processes. In the past three decades there has been continuous and rapid development of these techniques in both optical microscope design and in algorithms to estimate the statistics and positions of the molecule's trajectory. Although there has been great progress, comparison between different microscope configurations and estimation algorithms has been difficult beyond simulated data. In this paper we explore using a piezo actuated microscope stage to reproduce Brownian motion. Our goal is to use this as a tool to test performance of single particle tracking optical microscopes and estimation algorithms. In this study, Monte Carlo simulations were used to assess the ability of piezo actuated microscope stages for reproducing Brownian motion. Surprisingly, the dynamics of the stage together with configuration of the system allow for preservation of the Brownian motion statistics. Further, feed forward model inverse control allows for low error tracking of Brownian motion trajectories over a wide range of diffusion constants, varying stage response times, and trajectory discrete time steps. These results show great promise in using a piezo actuated microscope stage for testing single particle tracking experimental setups.

High-Speed Single-Molecule Tracking of CXCL13 in the B-Follicle

Soluble factors are an essential means of communication between cells and their environment. However, many molecules readily interact with extracellular matrix components, giving rise to multiple modes of diffusion. The molecular quantification of diffusion in situ is thus a challenging imaging frontier, requiring very high spatial and temporal resolution. Overcoming this methodological barrier is key to understanding the precise spatial patterning of the extracellular factors that regulate immune function. To address this, we have developed a high-speed light microscopy system capable of millisecond sampling in ex vivo tissue samples and submillisecond sampling in controlled in vitro samples to characterize molecular diffusion in a range of complex microenvironments. We demonstrate that this method outperforms competing tools for determining molecular mobility of fluorescence correlation spectroscopy (FCS) and fluorescence recovery after photobleaching (FRAP) for evaluation of diffusion. We then apply this approach to study the chemokine CXCL13, a key determinant of lymphoid tissue architecture, and B-cell-mediated immunity. Super-resolution single-molecule tracking of fluorescently labeled CCL19 and CXCL13 in collagen matrix was used to assess the heterogeneity of chemokine mobility behaviors, with results indicating an immobile fraction and a mobile fraction for both molecules, with distinct diffusion rates of 8.4 ± 0.2 and 6.2 ± 0.3 µm²s⁻¹, respectively. To better understand mobility behaviors in situ, we analyzed CXCL13-AF647 diffusion in murine lymph node tissue sections and observed both an immobile fraction and a mobile fraction with an example diffusion coefficient of 6.6 ± 0.4 µm²s⁻¹, suggesting that mobility within the follicle is also multimodal. In quantitatively studying mobility behaviors at the molecular level, we have obtained an increased understanding of CXCL13 bioavailability within the follicle. Our high-speed single-molecule tracking approach affords a novel perspective from which to understand the mobility of soluble factors relevant to the immune system.