{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T21:46:23Z","timestamp":1769723183156,"version":"3.49.0"},"reference-count":44,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T00:00:00Z","timestamp":1763337600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CMMI-2153744"],"award-info":[{"award-number":["CMMI-2153744"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Robotics"],"abstract":"<jats:p>Numerical simulations of protein folding enable the design of protein-based nanomachines and nanorobots by predicting folded three-dimensional protein structures with high accuracy and revealing the protein conformation transitions during folding and unfolding. In the kinetostatic compliance method (KCM) for folding simulations, protein molecules are represented as ensembles of rigid nano-linkages connected by chemical bonds, and the folding process is driven by the kinetostatic influence of nonlinear interatomic force fields until the system converges to a free-energy minimum of the protein. Despite its strengths, the conventional KCM framework demands an excessive number of iterations to reach folded protein conformations, with each iteration requiring costly computations of interatomic force fields. To address these limitations, this work introduces a family of sign gradient descent (SGD) algorithms for predicting folded protein structures. Unlike the heuristic-based iterations of the conventional KCM framework, the proposed SGD algorithms rely on the sign of the free-energy gradient to guide the kinetostatic folding process. Owing to their faster and more robust convergence, the proposed SGD-based algorithms reduce the computational burden of interatomic force field evaluations required to reach folded conformations. Their effectiveness is demonstrated through numerical simulations of KCM-based folding of protein backbone chains.<\/jats:p>","DOI":"10.3390\/robotics14110167","type":"journal-article","created":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T15:38:09Z","timestamp":1763393889000},"page":"167","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sign Gradient Descent Algorithms for Accelerated Kinetostatic Protein Folding in Nanorobotics Design"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1089-3872","authenticated-orcid":false,"given":"Alireza","family":"Mohammadi","sequence":"first","affiliation":[{"name":"Department Electrical & Computer Engineering, University of Michigan, Dearborn, MI 48128, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0142-1045","authenticated-orcid":false,"given":"Mohammad","family":"Al Janaideh","sequence":"additional","affiliation":[{"name":"Department Mechanical Engineering, University of Guelph, Guelph, ON N1G 2W1, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2415875","DOI":"10.1002\/adfm.202415875","article-title":"Micro\/Nanorobots for Advanced Light-Based Biosensing and Imaging","volume":"35","author":"Neettiyath","year":"2025","journal-title":"Adv. Funct. Mater."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1039\/D3LC00860F","article-title":"Lipid vesicle-based molecular robots","volume":"24","author":"Peng","year":"2024","journal-title":"Lab Chip"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3293","DOI":"10.1038\/s41467-024-46978-2","article-title":"Molecular robotic agents that survey molecular landscapes for information retrieval","volume":"15","author":"Woo","year":"2024","journal-title":"Nat. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1038\/s41563-021-00978-5","article-title":"Integrated computer-aided engineering and design for DNA assemblies","volume":"20","author":"Huang","year":"2021","journal-title":"Nat. Mater."},{"key":"ref_5","unstructured":"Dubey, A., Sharma, G., Mavroidis, C., Tomassone, S.M., Nikitczuk, K., and Yarmush, M.L. (May, January 26). Dynamics and kinematics of viral protein linear nano-actuators for bio-nano robotic systems. Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), New Orleans, LA, USA."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"054411","DOI":"10.1103\/PhysRevE.105.054411","article-title":"Physics of self-rolling viruses","volume":"105","author":"Ruiz","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1521","DOI":"10.1038\/s41565-024-01738-7","article-title":"A modular DNA origami nanocompartment for engineering a cell-free, protein unfolding and degradation pathway","volume":"19","author":"Huang","year":"2024","journal-title":"Nat. Nanotechnol."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Zhang, D., Lin, L., Deng, C., Osman, M.S., Rodriguez, P.E.S., Han, F., Li, M., and Wang, L. (2025). Advanced Imaging Strategies Based on Intelligent Micro\/Nanomotors. Cyborg Bionic Syst., 6.","DOI":"10.34133\/cbsystems.0384"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1177\/0278364908099888","article-title":"Multiscale design and modeling of protein-based nanomechanisms for nanorobotics","volume":"28","author":"Hamdi","year":"2009","journal-title":"Int. J. Robot. Res."},{"key":"ref_10","unstructured":"Chorsi, M.T., Tavousi, P., Mundrane, C., Gorbatyuk, V., Ilie\u015f, H., and Kazerounian, K. (2022, January 26\u201330). One Degree of Freedom 7-R Closed Loop Linkage as a Building Block of Nanorobots. Proceedings of the International Symposium on Advances in Robot Kinematics, Bilbao, Spain."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"021005","DOI":"10.1115\/1.4051472","article-title":"Kinematic design of functional nanoscale mechanisms from molecular primitives","volume":"9","author":"Chorsi","year":"2021","journal-title":"J. Micro Nano-Manuf."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mundrane, C., Chorsi, M., Vinogradova, O., Ilie\u015f, H., and Kazerounian, K. (2022, January 26\u201330). Exploring Electric Field Perturbations as the Actuator for Nanorobots and Nanomachines. Proceedings of the International Symposium on Advances in Robot Kinematics, Bologna, Italy.","DOI":"10.1007\/978-3-031-08140-8_28"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Kazerounian, K., and Ilie\u015f, H. (2024, January 1\u20135). The Evolving Role of Robot Kinematics in Bio-Nanotechnology. Proceedings of the International Symposium on Advances in Robot Kinematics, Ljubljana, Slovenia.","DOI":"10.1007\/978-3-031-64057-5_10"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/j.jmgm.2010.06.003","article-title":"Comparing position and force control for interactive molecular simulators with haptic feedback","volume":"29","author":"Bolopion","year":"2010","journal-title":"J. Mol. Graph. Model."},{"key":"ref_15","unstructured":"Y\u00e1\u00f1ez, M., and Boyd, R.J. (2024). Interactive Molecular Dynamics. Comprehensive Computational Chemistry, Elsevier. [1st ed.]."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"886","DOI":"10.1016\/j.cad.2009.06.010","article-title":"Stable six degrees of freedom haptic feedback for flexible ligand\u2013protein docking","volume":"41","author":"Daunay","year":"2009","journal-title":"Comput.-Aided Des."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4929","DOI":"10.1007\/s11042-023-15385-y","article-title":"Wave space sonification of the folding pathways of protein molecules modeled as hyper-redundant robotic mechanisms","volume":"83","author":"Kacem","year":"2024","journal-title":"Multimed. Tools Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2269","DOI":"10.1109\/TVCG.2024.3372128","article-title":"Molecular docking improved with human spatial perception using virtual reality","volume":"30","author":"Mishra","year":"2024","journal-title":"IEEE Trans. Vis. Comput. Graph."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1038\/d41586-019-03951-0","article-title":"A watershed moment for protein structure prediction","volume":"577","author":"AlQuraishi","year":"2020","journal-title":"Nature"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Fersht, A.R. (2021). AlphaFold\u2014A Personal Perspective on the Impact of Machine Learning. J. Mol. Biol., 433.","DOI":"10.1016\/j.jmb.2021.167088"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.1002\/prot.25759","article-title":"Driven to near-experimental accuracy by refinement via molecular dynamics simulations","volume":"87","author":"Heo","year":"2019","journal-title":"Proteins Struct. Funct. Bioinform."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7547","DOI":"10.1073\/pnas.0502655102","article-title":"Physics-based protein-structure prediction using a hierarchical protocol based on the UNRES force field: Assessment in two blind tests","volume":"102","author":"Czaplewski","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Arkun, Y., and G\u00fcr, M. (2012). Combining optimal control theory and molecular dynamics for protein folding. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0029628"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1089\/cmb.2005.12.1275","article-title":"HOPE: A homotopy optimization method for protein structure prediction","volume":"12","author":"Dunlavy","year":"2005","journal-title":"J. Comput. Biol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.sbi.2012.11.002","article-title":"To milliseconds and beyond: Challenges in the simulation of protein folding","volume":"23","author":"Lane","year":"2013","journal-title":"Curr. Opin. Struct. Biol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"034601","DOI":"10.1115\/1.4032759","article-title":"Protofold II: Enhanced model and implementation for kinetostatic protein folding","volume":"6","author":"Tavousi","year":"2015","journal-title":"ASME J. Nanotechnol. Eng. Med."},{"key":"ref_27","unstructured":"Tavousi, P. (2016). On the Systematic Design and Analysis of Artificial Molecular Machines. [Ph.D Thesis, University of Connecticut]."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kacem, A., Zbiss, K., and Mohammadi, A. (2024). A numerical integrator for kinetostatic folding of protein molecules modeled as robots with hyper degrees of freedom. Robotics, 13.","DOI":"10.3390\/robotics13100150"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"051011","DOI":"10.1115\/1.4068504","article-title":"Predicting Protein Folding Pathways With Quadratic Constraints on Rates of Entropy Change: A Nonlinear Optimization-Based Control Approach","volume":"147","author":"Mohammadi","year":"2025","journal-title":"J. Dyn. Syst. Meas. Control"},{"key":"ref_30","unstructured":"Pascanu, R., Mikolov, T., and Bengio, Y. (2013, January 16\u201321). On the difficulty of training recurrent neural networks. Proceedings of the International Conference on Machine Learning, Atlanta, GA, USA."},{"key":"ref_31","unstructured":"Riedmiller, M., and Braun, H. (April, January 28). A direct adaptive method for faster backpropagation learning: The RPROP algorithm. Proceedings of the IEEE International Conference on Neural Networks, San Francisco, CA, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.ins.2019.04.012","article-title":"Properties of the sign gradient descent algorithms","volume":"492","author":"Moulay","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1732","DOI":"10.1109\/TII.2023.3280938","article-title":"On Faster Convergence of Scaled Sign Gradient Descent","volume":"20","author":"Li","year":"2023","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ciemny, M.P., Badaczewska-Dawid, A.E., Pikuzinska, M., Kolinski, A., and Kmiecik, S. (2019). Modeling of disordered protein structures using monte carlo simulations and knowledge-based statistical force fields. Int. J. Mol. Sci., 20.","DOI":"10.3390\/ijms20030606"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"8035","DOI":"10.1109\/LRA.2024.3438039","article-title":"Gradient Descent-Based Task-Orientation Robot Control Enhanced with Gaussian Process Predictions","volume":"9","author":"Roveda","year":"2024","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Knuth, J., and Barooah, P. (2012, January 14\u201318). Collaborative 3D localization of robots from relative pose measurements using gradient descent on manifolds. Proceedings of the 2012 IEEE International Conference on Robotics and Automation (ICRA), Saint Paul, MN, USA.","DOI":"10.1109\/ICRA.2012.6225066"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Heiden, E., Palmieri, L., Koenig, S., Arras, K.O., and Sukhatme, G.S. (2018, January 21\u201325). Gradient-Informed Path Smoothing for Wheeled Mobile Robots. Proceedings of the 2018 IEEE International Conference on Robotics and Automation (ICRA), Brisbane, Australia.","DOI":"10.1109\/ICRA.2018.8460818"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Mohammadi, A., and Al Janaideh, M. (2023, January 2\u20136). Sign Gradient Descent Algorithms for Kinetostatic Protein Folding. Proceedings of the 2023 International Conference on Manipulation, Automation and Robotics at Small Scales (MARSS), Abu Dhabi, United Arab Emirates.","DOI":"10.1109\/MARSS58567.2023.10294128"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Finkelstein, A.V., and Ptitsyn, O. (2016). Protein Physics: A Course of Lectures, Academic Press. [2nd ed.].","DOI":"10.1016\/B978-0-12-809676-5.00018-1"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3985","DOI":"10.1021\/ct300148f","article-title":"FlexE: Using elastic network models to compare models of protein structure","volume":"8","author":"Perez","year":"2012","journal-title":"J. Chem. Theory Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.sbi.2015.11.013","article-title":"New generation of elastic network models","volume":"37","year":"2016","journal-title":"Curr. Opin. Struct. Biol."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Kountouris, P., and Hirst, J.D. (2010). Predicting \u03b2-turns and their types using predicted backbone dihedral angles and secondary structures. BMC Bioinform., 11.","DOI":"10.1186\/1471-2105-11-407"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3801","DOI":"10.1021\/acs.jpcb.8b00288","article-title":"Analysis of structural stability of chignolin","volume":"122","author":"Maruyama","year":"2018","journal-title":"J. Phys. Chem. B"},{"key":"ref_44","unstructured":"Safaryan, M., and Richt\u00e1rik, P. (2021, January 18\u201324). Stochastic sign descent methods: New algorithms and better theory. Proceedings of the 38th International Conference on Machine Learning, Virtual."}],"container-title":["Robotics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2218-6581\/14\/11\/167\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T10:29:48Z","timestamp":1769682588000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2218-6581\/14\/11\/167"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,17]]},"references-count":44,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["robotics14110167"],"URL":"https:\/\/doi.org\/10.3390\/robotics14110167","relation":{},"ISSN":["2218-6581"],"issn-type":[{"value":"2218-6581","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,17]]}}}