Cross-bridge model for predicting muscle short-range stiffness during movement
4 affiliations
- 1Department of Movement Sciences, KU Leuven, Leuven, Belgium
- 2Wallace H. Coulter Department of Biomedical Engineering, Emory University and The Georgia Institute of Technology, Atlanta, Georgia, United States of America
- 3Division of Cardiovascular Medicine, University of Kentucky, Lexington, Kentucky, United States of America
- 4Department of Rehabilitation Medicine, Division of Physical Therapy, Emory University, Atlanta, Georgia, United States of America
Abstract
Musculoskeletal simulations can offer valuable insight into how the properties of our musculoskeletal system influence the biomechanics of our daily movements. One such property is muscle’s initial resistance to stretch, also known as short-range stiffness, which is key to stabilizing movements in response to external perturbations. Short-range stiffness is poorly captured by existing musculoskeletal simulations since they employ phenomenological Hill-type models lacking activation-dependent stiffness properties. Existing simulations also do not capture the history-dependent reduction in short-range stiffness after muscle shortening, known as muscle thixotropy. While cross-bridge models can reproduce muscle short-range stiffness, it remains unclear which model properties are necessary to capture its history dependence. Here, we tested the ability of various cross-bridge models to reproduce empirical short-range stiffness and its history-dependent changes across a broad range of behaviorally relevant length changes and activation levels, using an existing dataset on 11 permeabilized rat soleus muscle fibers. We quantified muscle thixotropy using the ratio between the observed short-range stiffnesses after and before shortening. We computed the root-mean-square deviation (σSRS) between the predicted short-range stiffness ratio of various muscle models and the measured stiffness ratio. We found that cross-bridge models captured short-range stiffness changes across conditions with both small and large history-dependent stiffness reductions (σSRS ≤ 0.1), but only when including cooperative activation of both thin and thick myofilaments. In contrast, Hill-type models and a cross-bridge model without cooperative myofilament activation underestimated short-range stiffness and did not capture its change across conditions with large history-dependent stiffness reductions (σSRS > 0.2). Similar results were obtained when using a Gaussian-approximated solution method to simulate the cross-bridge distribution, but at an approximately eightfold lower computational cost. We therefore propose to implement Gaussian-approximated cross-bridge models with cooperative myofilament activation into musculoskeletal simulations to improve the prediction of short-range stiffness during movements. Author summary: Computer simulations based on musculoskeletal models can help us understand how muscles support and control everyday movements such as walking. One important muscle feature is its initial resistance to stretch, which helps stabilize the body when it is suddenly perturbed. This initial resistance is considerably reduced after muscle shortening, facilitating subsequent movements. Current musculoskeletal simulations rely on ‘Hill-type’ muscle models that do not capture muscle’s initial resistance to stretch nor its reduction after shortening. Here, we evaluated muscle forces during stretch and shortening predicted by both Hill-type models and ‘cross-bridge’ models of muscle’s contractile machinery. We compared model forces to those measured from isolated rat soleus muscle fibers and found that only cross-bridge models with cooperative interactions between muscle’s thin and thick filaments could accurately predict both muscle’s initial resistance to stretch and its reduction after shortening. Simpler models consistently underestimated muscle’s initial resistance to stretch and its reduction after shortening at submaximal activation levels. We also showed that a simplified mathematical solution method could produce similar results ~8 times faster, making these detailed models practical for larger musculoskeletal simulations. Incorporating these cooperative cross-bridge muscle models could lead to more realistic predictions of muscle behavior in everyday movements.
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BibTeX
@article{zee2026crossbridge,
title = {{Cross-bridge model for predicting muscle short-range stiffness during movement}},
author = {Tim J. van der Zee and Surabhi N. Simha and Gregory N. Milburn and Kenneth S. Campbell and Lena H. Ting and Friedl De Groote},
journal = {PLOS Computational Biology},
year = {2026},
month = sep,
doi = {10.1371/journal.pcbi.1014748},
url = {https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1014748},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Cross-bridge model for predicting muscle short-range stiffness during movement
AU - Tim J. van der Zee
AU - Surabhi N. Simha
AU - Gregory N. Milburn
AU - Kenneth S. Campbell
AU - Lena H. Ting
AU - Friedl De Groote
PY - 2026
DA - 2026/09/01
JO - PLOS Computational Biology
DO - 10.1371/journal.pcbi.1014748
UR - https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1014748
AB - Musculoskeletal simulations can offer valuable insight into how the properties of our musculoskeletal system influence the biomechanics of our daily movements. One such property is muscle’s initial resistance to stretch, also known as short-range stiffness, which is key to stabilizing movements in response to external perturbations. Short-range stiffness is poorly captured by existing musculoskeletal simulations since they employ phenomenological Hill-type models lacking activation-dependent stiffness properties. Existing simulations also do not capture the history-dependent reduction in short-range stiffness after muscle shortening, known as muscle thixotropy. While cross-bridge models can reproduce muscle short-range stiffness, it remains unclear which model properties are necessary to capture its history dependence. Here, we tested the ability of various cross-bridge models to reproduce empirical short-range stiffness and its history-dependent changes across a broad range of behaviorally relevant length changes and activation levels, using an existing dataset on 11 permeabilized rat soleus muscle fibers. We quantified muscle thixotropy using the ratio between the observed short-range stiffnesses after and before shortening. We computed the root-mean-square deviation (σSRS) between the predicted short-range stiffness ratio of various muscle models and the measured stiffness ratio. We found that cross-bridge models captured short-range stiffness changes across conditions with both small and large history-dependent stiffness reductions (σSRS ≤ 0.1), but only when including cooperative activation of both thin and thick myofilaments. In contrast, Hill-type models and a cross-bridge model without cooperative myofilament activation underestimated short-range stiffness and did not capture its change across conditions with large history-dependent stiffness reductions (σSRS > 0.2). Similar results were obtained when using a Gaussian-approximated solution method to simulate the cross-bridge distribution, but at an approximately eightfold lower computational cost. We therefore propose to implement Gaussian-approximated cross-bridge models with cooperative myofilament activation into musculoskeletal simulations to improve the prediction of short-range stiffness during movements. Author summary: Computer simulations based on musculoskeletal models can help us understand how muscles support and control everyday movements such as walking. One important muscle feature is its initial resistance to stretch, which helps stabilize the body when it is suddenly perturbed. This initial resistance is considerably reduced after muscle shortening, facilitating subsequent movements. Current musculoskeletal simulations rely on ‘Hill-type’ muscle models that do not capture muscle’s initial resistance to stretch nor its reduction after shortening. Here, we evaluated muscle forces during stretch and shortening predicted by both Hill-type models and ‘cross-bridge’ models of muscle’s contractile machinery. We compared model forces to those measured from isolated rat soleus muscle fibers and found that only cross-bridge models with cooperative interactions between muscle’s thin and thick filaments could accurately predict both muscle’s initial resistance to stretch and its reduction after shortening. Simpler models consistently underestimated muscle’s initial resistance to stretch and its reduction after shortening at submaximal activation levels. We also showed that a simplified mathematical solution method could produce similar results ~8 times faster, making these detailed models practical for larger musculoskeletal simulations. Incorporating these cooperative cross-bridge muscle models could lead to more realistic predictions of muscle behavior in everyday movements.
SN - 1553-7358
ER -