Fig 1.
Schematic diagram of cell-ECM interaction.
A: Each cell is represented by a mesh structure consisting of multiple nodes which are indicated by the yellow spheres. The ECM is modeled as a network of many fibers connected through a large number of nodes. The i-th membrane node is attached to the j-th ECM node through a focal adhesion connection. B: The forces acting on each membrane node include the cortical tension and membrane elastic energy force (), focal adhesion force (
), lamellipodium force (
), and frictional damping force(
). The forces acting on each node within the ECM fiber network include the elastic energy force (
), focal adhesion force (
), and damping force (
).
Fig 2.
Comparison of nonlinear computational model, latent variable model, trajectory piece-wise linearization model, and taylor series expansion model for predicting a cell interacting with ECM.
A: The cell morphologies over time for the original full nonlinear simulation (green), the latent variable simulation using 100 latent variables (blue), the trajectory piecewise-linear model (TPWL) with 100 linearization points (purple) and the Taylor series expansion model (red). The Taylor series and TPWL model are not capable of representing cell morphologies without significant error (represented by *). B: Comparison of root mean squared error of latent variable model using 100 latent variables, TPWL model using κ = 100 linearization points and first order Taylor expansion model. C: The computation time and root mean squared error (σLV) of the latent variable model as a function of the number of latent variables used to create the model.
Fig 3.
Block diagram of latent variable superposition model and schematic of the relationship between polarity direction, leading edge and direction of maximum stiffness.
A: The ECM changes its latent state with the autoregressive feedback through matrix G as well as with the feedforward path which collects the latent variable states of all the individual cells zc,k(k = 1, …, ncell) through matrices Dk. Each cell changes its latent variable state with autoregressive feedback through A and are exposed to the ECM forces represented by latent vector ze in two separate paths. The path through the cell polarity block and matrix B can be viewed as an “active input”. This feedback path includes a cell’s internal decision as to which direction it extends lamellipodia. In contrast, the other feedback path through a gain matrix C does not have a high-level cell decision, but is reactive, playing a “passive role”. B: The cell polarity direction rotates dynamically in such a way that the polarity vector may align with the direction of the maximum stiffness
. The leading edge of the cell is indicated by a right circular cone with apex angle
having its centerline aligned with the polarity direction. The membrane nodes of the k-th cell within the cone have nonzero lamellipodial forces (
). Membrane nodes outside this cone have zero lamellipodial forces (
).
Fig 4.
Comparison of ECM compaction between nonlinear computational model and linear latent variable models.
A: Two cells placed 30 μm apart are embedded in a 3-D cylindrical ECM that measures 40 μm in diameter and 100 μm in length. B: The ECM is subdivided along the longitudinal axis into 10 μm length cylinders. The volume (V) of each subdivided cylinder may be estimated at multiple time points during the compaction simulation. C: ECM comparison at the subdivided segments at time t = 10 min, 30 min, and 50 min. Compaction predicted by the latent variable model simulations (blue) agrees well with the full nonlinear simulations (green). The compaction volume is normalized with the initial volume of each segment. The compaction is most significant in-between the cells (the region between the dashed lines in the plots). This is further verified by the corresponding cross-sectional images of the 2-cell cylindrical ECM simulations. Polarity directions of both cells (red arrows initially pointing in arbitrary directions) shift to point inward, indicating that larger stresses are detected in the area between the cells. D: Comparison of single cell (cyan) and two cell (blue) compaction predicted by the latent variable superposition model. As can be seen, the single cell model predicts more localized shrinkage of the ECM volume from its original unstressed state whereas the two cell model shows more global shrinkage extended to within the region between cells.
Fig 5.
Effect of cell spacing and density on ECM compaction.
(A) ECM compaction by two cells embedded within a cylindrical ECM spaced at 100μm, 50μm, and 30 μm. As the spacing between cells increases, compaction is less pronounced between them. (B) The average ECM elastic force in-between cells spaced at 100μm is an order of magnitude less than the elastic force in-between cells spaced at 30μm. (C) The latent variable superposition simulation (upper) can be used to reflect the behavior of heterogeneous planar distribution of MC3T3-E1 osteoblasts as shown in supplementary video in [5]. Whereas the group of cells at the left edge contract the gel, the isolated cell at the right edge did not contract the gel, indicating the importance of cell density for compaction. (D) For the latent variable superposition simulation the maximum displacement along the ECM edge noticeably increases over time for the group of cells and is minimal for the isolated cell.