Abstract
Single Particle Tracking (SPT) is a powerful class of methods for studying the dynamics of biomolecules inside living cells. The techniques reveal the trajectories of individual particles, with a resolution well below the diffraction limit of light, and from them the parameters defining the motion model, such as diffusion coefficients and confinement lengths. Most existing algorithms assume these parameters are constant throughout an experiment. However, it has been demonstrated that they often vary with time as the tracked particles move through different regions in the cell or as conditions inside the cell change in response to stimuli. In this work, we propose an estimation algorithm to determine time-varying parameters of systems that discretely switch between different linear models of motion with Gaussian noise statistics, covering dynamics such as diffusion, directed motion, and Ornstein–Uhlenbeck dynamics. Our algorithm consists of three stages. In the first stage, we use a sliding window approach, combined with Expectation Maximization (EM) to determine maximum likelihood estimates of the parameters as a function of time. These results are only used to roughly estimate the number of model switches that occur in the data to guide the selection of algorithm parameters in the second stage. In the second stage, we use Change Detection (CD) techniques to identify where the models switch, taking advantage of the off-line nature of the analysis of SPT data to create non-causal algorithms with better precision than a purely causal approach. Finally, we apply EM to each set of data between the change points to determine final parameter estimates. We demonstrate our approach using experimental data generated in the lab under controlled conditions.
Highlights
Single Particle Tracking (SPT) is a class of experimental techniques and mathematical algorithms for following sub diffraction-limit sized particles moving inside living cells, including viruses, proteins, and strands of RNA [1,2,3]
While results have been promising, there is a need to train the underlying neural networks and as a result there are concerns about transfer learning when applying the methods to different model classes than those used for training. As with these prior works, the present paper focuses on systems that discretely jump between different parameter values and builds on our prior efforts using sliding windows and optimal estimation to produce an Maximum Likelihood (ML) estimate of the parameters [25,26]
Where k is the discrete time index, xk, yk, wk, and vk are scalars, qk = 2Dk ∆t is the variance of the process noise defined by the diffusion coefficient Dk and the sampling time ∆t, and rk is the variance of the measurement noise as generated by a variety of processes, including shot noise due to the physics of photon generation in fluorescence and read-out noise in the camera
Summary
Single Particle Tracking (SPT) is a class of experimental techniques and mathematical algorithms for following sub diffraction-limit sized particles moving inside living cells, including viruses, proteins, and strands of RNA [1,2,3]. Given noisy observations of such a model (such as from a trajectory estimated by localizing a fluorescent particle in each frame of an image sequence), the most common technique to estimate the model parameters is to fit the chosen model to the Mean Square Displacement (MSD) curve. This very simple and popular approach has been enormously successful in probing biomolecular dynamics [6,7].
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