Determination of active forces in actomyosin systems as inverse source problems for the Stokes equation

Emily Klass; Tram Thi Ngoc Nguyen; Nilay Cicek; Yoav G. Pollack; Sarah Köster; Andreas Janshoff; Anne Wald

doi:10.3934/ammc.2026001

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March 2026, Volume 7, 34-58. Doi: 10.3934/ammc.2026001

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Determination of active forces in actomyosin systems as inverse source problems for the Stokes equation

1.
Institute for Numerical and Applied Mathematics, University of Göttingen, Germany
2.
Max Planck Institute for Solar System Research, Göttingen, Germany
3.
Institute for Physical Chemistry, University of Göttingen, Germany
4.
Institute for Dynamics of Complex Systems, University of Göttingen, Germany
5.
Institute for X-Ray Physics, University of Göttingen, Germany

^*Corresponding author: Anne Wald
^*Corresponding author: Anne Wald

Received: January 16, 2026

Published online: March 17, 2026

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Abstract

The identification of forces and stresses is a central task in biophysics research: Knowledge on forces is key to understanding dynamic processes in active biological systems that are able to self-organize and display emergent properties by converting energy into mechanical work. The aim of this paper is to identify forces generated by a filament-motor network of F-actin and myosin – actomyosin – and exerted on the surrounding fluid, therefore causing a fluid flow. In particular, we evaluate optical microscopy data stemming from two different physical settings, confined and non-confined active gels. As a theoretical model, we use the Stokes equation together with an incompressibility condition and suitable boundary conditions reflecting the physical settings. The problem of determining the forces from knowledge on the fluid flow is formulated as an inverse source problem. Due to experimental limitations, only incomplete data are available. We provide a rigorous analysis of the forward problems and the impact of missing data, derive the adjoints of the forward operators needed for regularization, and demonstrate our methods on both synthetic and experimentally measured data.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 35R25, 92C05, 65J22, 65N21; Secondary: 65J10, 65J15, 65J05.

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Figure 1. Illustration of an actomyosin droplet with domain $ \Omega $

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Figure 2. Illustration of an actomyosin network with domain $ \Omega $

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Figure 3. Experiment-to-reconstruction pipeline for actomyosin droplets

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Figure 4. Fluorescence image of actin-myosin networks encapsulated in water-in-oil droplets. Actin is fluorescently labelled

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Figure 5. Velocity field $ {\mathbf v}^{\operatorname{meas}} $, ground truth $ {\mathbf f}^\dagger $, initial guess $ {\mathbf f}_0 $ and reconstruction of force $ {\mathbf f}^* $ for the operator $ A_R $ with Robin boundary conditions

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Figure 6. Velocity field $ {\mathbf v}^{\operatorname{meas}} $, ground truth $ {\mathbf f}^\dagger $, initial guess $ {\mathbf f}_0 $ and reconstructed force $ {\mathbf f}^* $ for the operator $ A_D $ with Dirichlet boundary conditions

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Figure 7. Reconstructions of forces (right) from velocity fields with uniform random noise and differing noise levels (left)

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Figure 8. Reconstruction of a force causing the observed flow of an actin-myosin network encapsulated in a droplet, corresponding to the inverse problem for the operator $ A_R $

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Figure 9. Experimentally obtained velocity field of the flow of a bulk actin-myosin network and reconstructed force $ {\mathbf f}^* $ using $ A_D $

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Figure 10. Experimentally obtained velocity field $ {\mathbf v}^{\operatorname{meas}} $ of the flow of a bulk actin-myosin network and reconstructed force $ {\mathbf f}^* $ using $ A_D $

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