Mechanical cell-matrix feedback explains pairwise and collective endothelial cell behavior in vitro

In vitro cultures of endothelial cells are a widely used model system of the collective behavior of endothelial cells during vasculogenesis and angiogenesis. When seeded in an extracellular matrix, endothelial cells can form blood vessel-like structures, including vascular networks and sprouts. Endothelial morphogenesis depends on a large number of chemical and mechanical factors, including the compliancy of the extracellular matrix, the available growth factors, the adhesion of cells to the extracellular matrix, cell-cell signaling, etc. Although various computational models have been proposed to explain the role of each of these biochemical and biomechanical effects, the understanding of the mechanisms underlying in vitro angiogenesis is still incomplete. Most explanations focus on predicting the whole vascular network or sprout from the underlying cell behavior, and do not check if the same model also correctly captures the intermediate scale: the pairwise cell-cell interactions or single cell responses to ECM mechanics. Here we show, using a hybrid cellular Potts and finite element computational model, that a single set of biologically plausible rules describing (a) the contractile forces that endothelial cells exert on the ECM, (b) the resulting strains in the extracellular matrix, and (c) the cellular response to the strains, suffices for reproducing the behavior of individual endothelial cells and the interactions of endothelial cell pairs in compliant matrices. With the same set of rules, the model also reproduces network formation from scattered cells, and sprouting from endothelial spheroids. Combining the present mechanical model with aspects of previously proposed mechanical and chemical models may lead to a more complete understanding of in vitro angiogenesis.Comment: 25 pages, 6 figures, accepted for publication in PLoS Computational Biolog

From energy to cellular forces in the Cellular Potts Model: An algorithmic approach.

Single and collective cell dynamics, cell shape changes, and cell migration can be conveniently represented by the Cellular Potts Model, a computational platform based on minimization of a Hamiltonian. Using the fact that a force field is easily derived from a scalar energy (F = -∇H), we develop a simple algorithm to associate effective forces with cell shapes in the CPM. We predict the traction forces exerted by single cells of various shapes and sizes on a 2D substrate. While CPM forces are specified directly from the Hamiltonian on the cell perimeter, we approximate the force field inside the cell domain using interpolation, and refine the results with smoothing. Predicted forces compare favorably with experimentally measured cellular traction forces. We show that a CPM model with internal signaling (such as Rho-GTPase-related contractility) can be associated with retraction-protrusion forces that accompany cell shape changes and migration. We adapt the computations to multicellular systems, showing, for example, the forces that a pair of swirling cells exert on one another, demonstrating that our algorithm works equally well for interacting cells. Finally, we show forces exerted by cells on one another in classic cell-sorting experiments

Nodal Signaling Range Is Regulated by Proprotein Convertase-Mediated Maturation

Tissue patterning is established by extracellular growth factors or morphogens. Although different theoretical models explaining specific patterns have been proposed, our understanding of tissue pattern establishment invivo remains limited. In many animal species, left-right patterning is governed by a reaction-diffusion system relying on the different diffusivity of an activator, Nodal, and an inhibitor, Lefty. In a genetic screen, we identified a zebrafish loss-of-function mutant for the proprotein convertase FurinA. Embryological and biochemical experiments demonstrate that cleavage of the Nodal-related Spaw proprotein into a mature form by FurinA is required for Spaw gradient formation and activation of Nodal signaling. We demonstrate that FurinA is required cell-autonomously for the long-range signaling activity of Spaw and no other Nodal-related factors. Combined insilico and invivo approaches support a model in which FurinA controls the signaling range of Spaw by cleaving its proprotein into a mature, extracellular form, consequently regulating left-right patterning

Simulated individual cell responses to mechanical cell-ECM feedback.

<p>(<i>A</i>) Single cells on substrates of varying stiffness after 100 MCS. Line pieces indicate strain magnitude and orientation. (<i>B</i>) cell area () of cells; (<i>C</i>) cell length (length of major axis if the cell is seen as an ellipse) as a function of substrate stiffness (<i>D</i>) cell eccentricity (, with and the lengths of the cell's major and minor semi-axes) as a function of stiffness. Mean and standard deviation shown for in panels B-D. (<i>E</i>) Dispersion coefficients of individual, simulated cells, derived from a linear fit on the mean square displacements (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003774#pcbi.1003774.s002" target="_blank">Figure S2</a>); . Error bars indicate 95% confidence intervals of linear fits.</p

Visualization of simulated traction forces (<i>black arrows</i>) and resulting matrix strains (<i>blue line segments</i>) generated in the proposed hybrid cellular Potts and finite element simulation model.

<p>Visualization of simulated traction forces (<i>black arrows</i>) and resulting matrix strains (<i>blue line segments</i>) generated in the proposed hybrid cellular Potts and finite element simulation model.</p

Simulated network formation assay.

<p>(<i>A</i>) Simulated collective cell behavior on substrates of varying stiffness, with a uniformly distributed initiated configuration of cells. (<i>B</i>) Time lapse showing the development of a polygonal network on a 10kPa substrate (time in MCS). Panels <i>A</i> and <i>B</i> represent a 0.75×0.75 area ( pixels) initiated with 450 cells. (<i>C</i>) Close-up of simulated network formation on a 10 kPa substrate, showing the reconnection of two sprouts. Time in MCS. (<i>D</i>) Time lapse imaging of bovine aortic endothelial cells seeded onto a 2.5 kPa polyacrylamide gel functionalized with RGD-peptide. Arrows indicate cells that join together and elongate into a network. Time scale is in hours. Scale bar is 50<i> µ</i>m.</p

Simulated cellular responses to static strains.

<p>Cells do not generate traction forces in this figure. (<i>A</i>) Cell length as a function of the durotaxis parameter, , on a substrate stretched along the vertical axis. (<i>B</i>) Cell orientation as a function of the stretch orientation (simulated with ). Error bars show standard deviation for . Insets show five simulations per value tested.</p