Traction calculation

Before traction calculation, 3D confocal images were deconvoluted (AutoQuant X3), and the separated free nanoparticles from the microgel were discarded using pencil tool in ImageJ (NIH). This step will enhance the image quality and prevent artifacts in the recovered displacements. Cell-induced displacement fields of embedded fluorescent nanoparticles were obtained by FIDVC technique. This technique can capture the deformation using an iterative deformation method and refining interrogation window size in each iteration. Once the displacement field was determined, cell-induced infinitesimal strain tensor was calculated using displacement gradient tensor (∇u), calculated with Optimal-11 tap finite difference kernel.

Based on the experimentally measured alginate mechanical properties and strain range (<25%), isotropic and linear constitutive model was assumed to calculate the Cauchy stress tensor.

where σ and E are the Cauchy stress and strain tensor, respectively; μ and λ are the Lamé parameters. Since some samples in multilayered cells condition showed maximum principal strain of up to 40% in small regions of their surface, the Neo-Hookean model was proposed to estimate the extent of underestimation based on the linear assumption.

Where J and B are the jacobian of the deformation gradient tensor (F = ∇u+I) and the left Cauchy Green’s tensor, respectively.

Microgel surface tractions were computed at inner surface (~1 µm offset from the microgel surface) to remove boundary effect on the recovered tractions (Supplementary Fig. ²⁰). Briefly, the surface of the microgels was computed using the concept of convex hull. In this approach, centers of nanoparticles were located, and the point cloud of the nanoparticles was created using their spatial coordinates. Then, the smallest convex envelope of the set of discrete points was calculated using built in convex hull function in MATLAB. The boundary points were then removed for four layers, and the new exposed boundary points were used to create surface mesh using triangulation. Stress tensor at triangle midpoint was used to calculate surface traction: