Controlled Deposition of 3D Matrices to Direct Single Cell Functions

Abstract Advances in engineered hydrogels reveal how cells sense and respond to 3D biophysical cues. However, most studies rely on interfacing a population of cells in a tissue‐scale bulk hydrogel, an approach that overlooks the heterogeneity of local matrix deposition around individual cells. A droplet microfluidic technique to deposit a defined amount of 3D hydrogel matrices around single cells independently of material composition, elasticity, and stress relaxation times is developed. Mesenchymal stem cells (MSCs) undergo isotropic volume expansion more rapidly in thinner gels that present an Arg‐Gly‐Asp integrin ligand. Mathematical modeling and experiments show that MSCs experience higher membrane tension as they expand in thinner gels. Furthermore, thinner gels facilitate osteogenic differentiation of MSCs. By modulating ion channels, it is shown that isotropic volume expansion of single cells predicts intracellular tension and stem cell fate. The results suggest the utility of precise microscale gel deposition to control single cell functions.


Supporting Text
Stress of the gel surrounding an expanding rigid cell

Analytical solution in the case of linear elasticity (Figure 3A, i and ii)
Since the problem is spherically symmetric (uθ = uφ = 0), in the spherical coordinate (r,θ,φ), we have: (1) where u is the displacement vector, r is the radius between cell surface and gel surface, and a is a constant to be determined. Eq. 1 was derived from the fundamental equation of conservation of linear momentum (Cauchy equation) in the case of spherical symmetry.
Then, by integrating Eq. 1, we have: where a, b are constants to be determined by boundary conditions.
The strains are given as: Plugging them into Eq. (6), we obtain: At r = r1, the tension stress is given as: Divide the numerator and denominator in Eq. (12) by % L (1 + ν) and rearrange the expression, we have: In terms of gel thickness (dgel), Eq. (13) becomes: Since 0 ≤ ν ≤ 0.5, with fixed values of r1 and u0, the tension stress σ 88 increases with decreasing dgel.
In other words, the tension increases with thinner gels, regardless of n.
When the material is incompressible, i.e. Poisson's ratio ν = 0.5, we have: When r2 = ∞, i.e. a bulk gel, the minimum tension stress is given by: In addition, the volume strain is given as: .
Since 0 ≤ ν ≤ 0.5, b is constant and positive everywhere-i.e. the volume is expanded and the polymer density is decreased with the same amount everywhere. In other words, if ν < 0.5, gel volume is expected to expand as a result of cell volume expansion.

Finite element solution in the case of large deformation with rubberlike elasticity (Figure 3A, iii)
In the case of large deformation with rubber-like elasticity, no analytical solution is available. We applied nonlinear finite element method to solve the boundary value problem using the commercial finite element package Abaqus. Since the problem is axisymmetric, we used the axisymmetric formulation to solve the 3D problem. Approximately 2000 quadrilateral axisymmetric elements were used, and the convergence was reached at such resolution. We applied the displacement boundary condition (u = u0) at the inner boundary, and stress-free boundary condition (σrr(r = r2) = 0) at the outer boundary. We used the incompressible neo-Hookean material to consider the rubber-like elasticity of the gel. The strain energy potential of neo-Hookean material is given as where % = λ % 6 + λ 6 6 + λ L 6 is the first invariant of deformation, λ1, λ2, λ3 are the principal stretches, and G is the shear modulus.
We considered three different cases with varied gel thicknesses (5, 15, 100 µm). We used 100 µm to approximate the bulk gel case. The cell (the inner sphere) has an initial radius of 7.82 µm and expands its volume by 50%.