[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123939-en":3,"doc-seo-123939-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},123939,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Trajectory Analysis in Single-Particle Tracking - From Mean Squared Displacement to Machine Learning Approaches","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.","Review  \nTrajectory Analysis in Single-Particle Tracking: From Mean Squared Displacement to Machine Learning Approaches  \nChiara Schirripa Spagnolo 1, * and Stefano Luin 1,2, *  \nCitation: 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  \nAcademic Editor: Ben Ovryn  \nReceived: 24 June 2024  \nRevised: 1 August 2024  \nAccepted: 7 August 2024  \nPublished: 8 August 2024  \nCopyright: © 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/)) .  \n1 NEST Laboratory, Scuola Normale Superiore, Piazza San Silvestro 12, I-56127 Pisa, Italy  \n2 NEST Laboratory, Istituto Nanoscienze-CNR, Piazza San Silvestro 12, I-56127 Pisa, Italy  \n* Correspondence: [chiara.schirripaspagnolo@sns.it](chiara.schirripaspagnolo@sns.it) (C.S.S.); [s.luin@sns.it](s.luin@sns.it) (S.L.)  \nAbstract: 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.  \nKeywords: 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  \n1. Introduction  \nSingle-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] .  \nThe 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","cbCaitDcNwkTLdWb","https://ap.wps.com/l/cbCaitDcNwkTLdWb","pdf",3484803,1,27,"English","en",105,"# Introduction\n## Single-particle tracking overview\n## Trajectory reconstruction and analysis workflow\n## Motivation for track-analysis review\n# MSD-based trajectory analysis\n## Precision and accuracy factors\n## Measurement uncertainty and trajectory length\n## Challenges in anomalous motion\n# Alternative parameter distributions and state models\n## Angles, velocities, times, and reach probabilities\n## Hidden Markov Models for state detection\n# Machine learning approaches for trajectory analysis\n## Classification using motion models or feature sets\n## Random forests and deep learning\n## Software resources and best-practice perspective","[{\"question\":\"What is the purpose of trajectory analysis in single-particle tracking?\",\"answer\":\"Trajectory analysis extracts parameters such as motion type, diffusion coefficient, velocity, and interaction events, enabling connections between observed motion and underlying processes.\"},{\"question\":\"What are key limitations of mean squared displacement (MSD) analysis?\",\"answer\":\"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.\"},{\"question\":\"How do machine learning methods improve trajectory motion classification?\",\"answer\":\"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.\"}]","Trajectory Analysis in Single-Particle Tracking - From Mean Squared Displacement to Machine Learning Approaches | PDF",1785819348,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"trajectory-analysis-in-single-particle-tracking-from-mean-squared-displacement-to-machine-learning-approaches","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/trajectory-analysis-in-single-particle-tracking-from-mean-squared-displacement-to-machine-learning-approaches/123939/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the purpose of trajectory analysis in single-particle tracking?","Question",{"text":75,"@type":76},"Trajectory analysis extracts parameters such as motion type, diffusion coefficient, velocity, and interaction events, enabling connections between observed motion and underlying processes.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are key limitations of mean squared displacement (MSD) analysis?",{"text":80,"@type":76},"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.",{"name":82,"@type":73,"acceptedAnswer":83},"How do machine learning methods improve trajectory motion classification?",{"text":84,"@type":76},"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.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]