Directing Multicellular Organization by Varying the Aspect Ratio of Soft Hydrogel Microwells

Abstract Multicellular organization with precise spatial definition is essential to various biological processes, including morphogenesis, development, and healing in vascular and other tissues. Gradients and patterns of chemoattractants are well‐described guides of multicellular organization, but the influences of 3D geometry of soft hydrogels are less well defined. Here, the discovery of a new mode of endothelial cell self‐organization guided by combinatorial effects of stiffness and geometry, independent of protein or chemical patterning, is described. Endothelial cells in 2 kPa microwells are found to be ≈30 times more likely to migrate to the edge to organize in ring‐like patterns than in stiff 35 kPa microwells. This organization is independent of curvature and significantly more pronounced in 2 kPa microwells with aspect ratio (perimeter/depth) < 25. Physical factors of cells and substrates that drive this behavior are systematically investigated and a mathematical model that explains the organization by balancing the dynamic interaction between tangential cytoskeletal tension, cell–cell, and cell–substrate adhesion is presented. These findings demonstrate the importance of combinatorial effects of geometry and stiffness in complex cellular organization that can be leveraged to facilitate the engineering of bionics and integrated model organoid systems with customized nutrient vascular networks.


Supporting Text
Note S1: General mathematical model description To study the mechanism of multicellular organization in microwells, we developed a mathematical model based on force interactions between the cells and substrate. As illustrated in Figure 5A and Figure S21A, the model consists of cells in a microwell projected on a 2D x-z plane. The cell dynamics predicted by the 2D model can be restored to 3D geometries by considering the 2D geometry as a crosssection of a 3D shape along appropriate cell orientation.
For hemispherical microwells, the translation is straightforward due to symmetrical geometric properties and only requires adjusting the steady-state cell length. In a cylindrical microwell, the 3D model would have to consider aspect ratios along two planes. Therefore, the forcebalance mechanism explained by the 2D model provides insights into the observed cellular organization, regardless of the 3D microwell shape (i.e., cylinder vs. hemisphere). We assume the cells to be moving upwards where the cell is modeled as a single-line element with all the forces only acting on each cell's endpoints.
The geometric shape of the microwell, along the cell movement dimension, with crossectional depth H and radius R can be described as: where is the local height which ranges from 0 to H and r is the local radius at z, ranging from 0 to R. ( ( = 0) = 0; ( = ) = ). The region outside the microwell is infinitely flat, or: = for > . The angle between the local tangential line (red dotted line in Figure S21A) and x-axis can be described as: tan = = 2 √ 1− 2 2 for < ; and tan = 0 for ≥ , or = . This simplified 2D geometry can illustrate a more complex 3D microwell curvature along the cell's movement direction, as the local curvature is a function of the first and second derivative of r in respect to z.
By considering cytoskeletal tension (Fk), which regulates the cell shape and size, cell-substrate interaction, which is the frictional force (Fη), and cell-cell interaction (Fw), we can write the force-balance equation at i th node of j th cell, xj,i , as: [ 2 0 ( − 0 ) 2 ] + , + 0 ∑ ( , , , ) , where each cell is described by two nodes (i = 1,2). We describe the intracellular cytoskeletal tension using spring potential with the stiffness of k. The equilibrium length of the cell is 0 , and the current length of the cell j is dj = | xj,1xj,2| The cell constantly generates active protrusion and contraction forces, fj,i. This force, scaled by a constant A, follows the random Gaussian distribution with zero mean and is inversely proportional to the cell length, : , = ( ) , < ( 1 ) ( 2 ) > = 2 ( 1 − 2 ) where σ is the standard deviation of the Gaussian distribution. This force is along the local tangential direction of the substrate. Interaction between the i th node of the j th cell and the k th node of the m th cell is w(xj,i, xm,k), and follows the van der Waals relation. This interaction becomes repulsive when the distance between the two nodes, |xj,ixm,k|, is less than s0 (see Table S3) and attractive if the distance is greater than s0 and fades to zero when |xj,ixm,k|→ ∞. To circumvent the singularity caused by the nature of van der Waals relation when |xj,ixm,k|→ 0, we fitted a continuous stepwise function to the van der Waals equation so that w(xj,i, xm,k) goes from -1 to 1 ( Figure S21B). w(xj,i, xm,k) is scaled by a constant F0, and is along the linear direction between these two nodes (green dotted line in Figure S21A). F0 represents cell-cell interaction strength, a function of VE-cadherin, or other intracellular protein activities. Therefore, our model covers cell-cell attractions at long distances and cell-cell repulsion at short distances. We assume the cell-cell interaction is only significant between the top and bottom nodes of the cell since the distances between any other two nodes are much further than s0. η is the frictional coefficient between the cell and substrate. The righthand side of the force balance (Equation 2) describes the residual force resulting from tension, random protrusions and contractions, and cell-cell interaction, and the left-hand side describes the sliding frictional force between the cell and the substrate. This force balance equation indicates the unbalanced force at each node compensated by the local sliding motion along the microwell.

Numerical Results
Since the cells can only move along the tangential direction, we can simplify the system by considering only the tangential direction of Equation 2 for each cell node. The dynamic equations for cells 1 (top) to N (bottom) can then be written as: as j ranging from 1 to N. φj denotes the angle between j th cell ( xj,2xj,1) and the x-axis. φjθ(xj,i) denotes the orientation of the cells relative to the microwell's tangential direction at the i th node of the j th cell: xj,i. βj-1,j is the angle between the line connecting the two cell nodes ( xj,1 xj-1,2) and the x-axis. We used a MATLAB package to generate random E1,1, E1,2, E2,1, E2,2, The cells are more likely to stay at the microwell center in the shallower microwells but relocate to the edges of deeper microwells. With the correct set of parameters, our simulation results can predict cellular organization as observed experimentally ( Figure 5C). We could also predict the sensitivity of multicellular organization to geometry in response to cytoskeletal contractility (or tension) influenced by substrate stiffness and cell-cell interactions using this force balance model ( Figure 5C and D), and these predictions are consistent with experimental findings ( Figure S14, S15, and S17). Solving Equation 3 for a 3 or 4-cell system yielded similar trends as the 2-cell system ( Figure S21E-H). The geometric setting of the experiment (d0 ~ 2 ) allows a maximum of 3 or 4 cells in one microwell ( Figure S21I-J). However, the real experimental setting is equivalent to applying periodic boundary conditions to Equation 3, as cells that are pushed out would enter a neighboring microwell. Therefore, we can extend our findings to a monolayer seeded on a periodically microwell-patterned substrate.
The observed spatial localization can be explained by the force-balance feature of the system, caused due to the geometry of the microwell. Cells rely on cytoskeletal activities and forces to move and balance extracellular forces, including cell-cell interactions and cell-substrate friction.
Since the cytoskeletal tension is not precisely along the microwell's local tangential direction, the magnitude of tangential components of cytoskeletal tension depends on the cell's orientation relative to the microwell's tangential direction. The unbalanced part of the force should be balanced by friction, and drives the cell movement along the microwell. Therefore, the cell is less likely to move when the cytoskeletal tension is more likely to balance the extracellular forces. This balance is most likely to occurs at microwell positions where |φj -θj,i| ~ 0.
According to the geometric calculation, |φj -θj,i| is close to zero around the center of shallower microwells or along the edge of deeper microwells ( Figure  In general, there are two popular ways to crosslink gelatin: Photo or chemical crosslinking. Our laboratory has extensive expertise and published papers using photocrosslinked GelMA. [1,2] While our previous work included fabricating flat sheets crosslinked in a chamber constructed using glass, the hydrogels prepared in the current work are micropatterned and rely on crosslinking the hydrogel while molding against a PDMS mold. We observed that the GelMA did not crosslink uniformly when exposed to UV light on the micropatterned PDMS mold with either the Igracure 2959 UV light photoinitiator or a Ruthenium visible light photoinitiator. Due to incomplete crosslinking, the gel surfaces tore and were damaged during peeling after photocrosslinking. This insufficient crosslinking on PDMS and other oxygen-permeable substrates, when photocrosslinked in air, is a well-known problem and has been observed with many hydrogels that are crosslinked via free-radical polymerization in the case of Irgacure 2959. [3] In free-radical polymerization, the initiator molecules form radicals, initiating the reaction with monomers, which then propagates, to form polymer chains. In the presence of oxygen, the active radicals are scavenged, leading to dead chain ends and inhibiting radical polymerization yielding a partially crosslinked hydrogel. Microbial Transglutaminase (mTG) is a well-known chemical crosslinker used to crosslink gelatin for various applications. [4][5][6][7][8] mTG-based crosslinking is not significantly affected by the presence of oxygen, and the crosslinked hydrogels have uniform mechanical properties on the surface and the bulk as long as mTG is mixed correctly with the pregel solution. The reaction does not lead to any harmful byproducts, and the byproducts produced are removed from the gel when the enzyme is deactivated and washed at 60°C. Also, transglutaminase crosslinks the glutamine and lysine amino acids of gelatin, thus keeping the cell adhesive RGD motif intact. [9] Additional References Figure S1: Process flow used for creating hydrogel microwells using reflow lithography and replica molding.