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

TL;DR

This work addresses identifying forces generated by actomyosin networks from fluid-flow data by formulating it as an inverse source problem for the Stokes equation with spatially varying viscosity. It develops two forward models reflecting confined (Robin boundary) and bulk (Dirichlet boundary) settings, proves forward-problem well-posedness, and derives differentiability and adjoint operators to enable regularized inversion from incomplete velocity data. The authors implement a Landweber-type algorithm and demonstrate reconstructions from synthetic data (including noisy cases) and from experimental PIV measurements, showing that the recoverable part of the active force is the divergence-free component under the assumed viscosity. The approach provides a rigorous, data-driven route to quantify active stresses in actomyosin systems and offers a framework extendable to 3D, time dependence, and pressure-informed curl-free reconstructions.

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.

Figures (10)

• Figure 1: Illustration of an actomyosin droplet with domain $\Omega$
• Figure 2: Illustration of an actomyosin network with domain $\Omega$
• Figure 3: Experiment-to-reconstruction pipeline for actomyosin droplets
• Figure 4: Fluorescence image of actin-myosin networks encapsulated in water-in-oil droplets. Actin is fluorescently labelled.
• 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.

Theorems & Definitions (22)

• Theorem 3.1: Well-posedness of $\mathcal{S}_R$
• proof
• Theorem 3.2: Well-posedness of $\mathcal{S}_D$
• proof
• Remark 4.1: Source inversion with incomplete data
• Lemma 5.1: Well posedness of $A$
• proof
• Lemma 5.2: Fréchet derivative of $S_R$
• proof
• Lemma 5.3: Fréchet derivative of $S_D$