Trajectory Analysis in Single-Particle Tracking - From Mean Squared Displacement to Machine Learning Approaches

Aug 4, 2026

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Single-particle tracking serves as a high-resolution approach for studying molecular or particle motion, and its core value depends on how reconstructed trajectories are analyzed. This review systematizes methods for extracting motion parameters from tracks, starting with classical mean squared displacement (MSD) analysis and its limitations arising from localization uncertainty, temporal resolution, and short trajectories. It then expands to richer parameter distributions—angles, velocities, times, and target-reach probabilities—along with hidden Markov models for state identification, populations, and switching kinetics. Finally, it surveys machine learning trajectory analysis, including random forests and deep learning, and highlights software resources and the benefit of combining statistical and learning-based strategies for improved accuracy and informativeness.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages27

Reading Time1 h 8 min

AccessFree to download

Introduction

Single-particle tracking overview

Trajectory reconstruction and analysis workflow

Motivation for track-analysis review

MSD-based trajectory analysis

Precision and accuracy factors

Measurement uncertainty and trajectory length

Challenges in anomalous motion

Alternative parameter distributions and state models

Angles, velocities, times, and reach probabilities

Hidden Markov Models for state detection

Machine learning approaches for trajectory analysis

Classification using motion models or feature sets

Random forests and deep learning

Software resources and best-practice perspective

Review Trajectory Analysis in Single-Particle Tracking: From Mean Squared Displacement to Machine Learning Approaches Chiara Schirripa Spagnolo 1, * and Stefano Luin 1,2, * Citation: Schirripa Spagnolo, C.; Luin, S. Trajectory Analysis in SingleParticle Tracking: From Mean Squared Displacement to Machine Learning Approaches. Int. J. Mol. Sci. 2024, 25, 8660. [https://doi.org/10.3390/](https://doi.org/10.3390/)[ ](https://doi.org/10.3390/)ijms25168660 Academic Editor: Ben Ovryn Received: 24 June 2024 Revised: 1 August 2024 Accepted: 7 August 2024 Published: 8 August 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([https://](https://)[ ](https://)[creativecommons.org/licenses/by/](creativecommons.org/licenses/by/)[ ](creativecommons.org/licenses/by/)[4.0/](4.0/)) . 1 NEST Laboratory, Scuola Normale Superiore, Piazza San Silvestro 12, I-56127 Pisa, Italy 2 NEST Laboratory, Istituto Nanoscienze-CNR, Piazza San Silvestro 12, I-56127 Pisa, Italy * Correspondence: [chiara.schirripaspagnolo@sns.it](chiara.schirripaspagnolo@sns.it) (C.S.S.); [s.luin@sns.it](s.luin@sns.it) (S.L.) Abstract: Single-particle tracking is a powerful technique to investigate the motion of molecules or particles. Here, we review the methods for analyzing the reconstructed trajectories, a fundamental step for deciphering the underlying mechanisms driving the motion. First, we review the traditional analysis based on the mean squared displacement (MSD), highlighting the sometimes-neglected factors potentially affecting the accuracy of the results. We then report methods that exploit the distribution of parameters other than displacements, e.g., angles, velocities, and times and probabilities of reaching a target, discussing how they are more sensitive in characterizing heterogeneities and transient behaviors masked in the MSD analysis. Hidden Markov Models are also used for this purpose, and these allow for the identification of different states, their populations and the switching kinetics. Finally, we discuss a rapidly expanding field—trajectory analysis based on machine learning. Various approaches, from random forest to deep learning, are used to classify trajectory motions, which can be identified by motion models or by model-free sets of trajectory features, either previously defined or automatically identified by the algorithms. We also review free software available for some of the analysis methods. We emphasize that approaches based on a combination of the different methods, including classical statistics and machine learning, may be the way to obtain the most informative and accurate results. Keywords: particle dynamics; molecular diffusion; molecular trajectory statistics; single-molecule analysis; single molecule tracking; machine learning in biology; quantitative microscopy; quantitative biology; hidden Markov models; moment scaling spectrum 1. Introduction Single-particle tracking (SPT) is an established technique for observing the behavior of single entities at high spatial and temporal resolution (nanometers and milliseconds) in various scientific fields such as Biology, Chemistry, and Physics. SPT is based on thereconstruction of the trajectories of single particles visualized in real time in the system of interest. In life science, for example, it has been applied to resolve the working mechanisms of molecules, organelles, and viruses [1–4] . The technique requires synergistic efforts in several aspects, such as instrumentation, particle labeling, and data analysis [5–9] . Experimental time-lapse images are processed with two main analysis steps—trajectory reconstruction and trajectory analysis [1,10] . The latter allows for the extraction of various parameters characterizing the behavior of the tracked particles, such as the type of motion, diffusion c

FAQs

What is the purpose of trajectory analysis in single-particle tracking?

Trajectory analysis extracts parameters such as motion type, diffusion coefficient, velocity, and interaction events, enabling connections between observed motion and underlying processes.

What are key limitations of mean squared displacement (MSD) analysis?

MSD-based results can be affected by localization uncertainty, temporal resolution choices, the number of points used in the MSD curve, and issues like too-short trajectories and heterogeneities, especially for anomalous motion.

How do machine learning methods improve trajectory motion classification?

Machine learning approaches, including random forests and deep learning, classify trajectory motions using motion models or model-free feature sets that may be automatically identified, improving sensitivity to heterogeneous and transient behaviors.

Aug 4, 2026

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