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Mechanical cell-matrix feedback explains pairwise and collective endothelial cell behavior in vitro
René F. M. van Oers, Elisabeth G. Rens, Danielle J. LaValley, Cynthia Reinhart-King, Roeland M. H. Merks
TL;DR
The paper addresses the incomplete understanding of how endothelial cells coordinate in vitro angiogenesis across single-cell, pairwise, and collective scales. Using a hybrid cellular Potts–finite element model, it tests whether cell traction, matrix strain, and strain-guided responses suffice to reproduce these behaviors. The model reproduces experimentally observed single-cell responses, pairwise interactions, network formation, and spheroid sprouting on compliant matrices.
Problem
Existing models of in vitro angiogenesis often focus on whole networks or sprouts without testing whether the same mechanisms explain single-cell responses and pairwise interactions.
Method
The study uses a hybrid cellular Potts and finite element model combining endothelial cell traction, ECM strain generation, and cellular responses to strain.
Results
The same mechanical rules reproduce endothelial behavior at single-cell, pairwise, and collective scales, including network formation and spheroid sprouting.
Takeaways & Limitations
Cell traction, strain stiffening, and durotaxis suffice in the model to reproduce several experimentally observed endothelial behaviors on compliant matrices.
Takeaways & Limitations
The model resets the matrix to an undeformed state after each Monte Carlo step and therefore omits cell memory of substrate strains.
Abstract
from arXiv · showhide
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.
Author Summary
The paper shows that endothelial cells communicate biomechanically through the extracellular matrix during morphogenesis. It introduces computational tools for modeling mechanical interactions between cells and the matrix.
- Endothelial cells organize into blood-vessel-like structures in culture through biomechanical signaling.
- The work introduces computational tools for modeling mechanical interactions between cells and the extracellular matrix.
Introduction
The study addresses how mechanical communication through the extracellular matrix coordinates endothelial self-organization. In vitro cultures provide a tractable system, but existing models invoke diverse mechanisms and often do not distinguish among them.
- Mechanical signals through the extracellular matrix can integrate information across tissues and mediate short-range cell-cell communication.
- Endothelial cultures are used to study vascular-like networks from dispersed cells and sprouts from spheroids in defined extracellular matrices.
- Existing network models commonly use density-dependent attractive forces that weaken at higher cellular densities.
- Related models incorporate chemotaxis, ECM-bound growth factors, secreted factors, or contact inhibition to explain vascular-like structures.
- Because diverse models can fit experiments, a single mathematical representation may correspond to several alternative underlying mechanisms.
- Endothelial traction deforms and reorients the matrix, producing local strains that can affect nearby-cell motility.
Results
The model reproduces endothelial responses to matrix strain and stiffness at single-cell, pairwise, and collective scales. Traction-generated strains and durotaxis produce stiffness-dependent morphology, motility, cell contacts, networks, and spheroid sprouts.
- Single-cell responses: The model approximates strain stiffening by making stiffness increase linearly with local strain.
- Single-cell responses: Simulated cells align with applied uniaxial strain, and stronger durotaxis produces greater elongation.Average cell orientation follows the stretch orientation across the tested range.
- Single-cell responses: Traction forces are converted into ECM strains, which bias cell movement through a traction–strain feedback loop.Each simulation step calculates traction, relaxes the matrix with finite elements, and then advances cell motility.
- Single-cell responses: At 0.5 kPa cells round up, at 32 kPa they spread isotropically, and elongation peaks around 12 kPa.The biphasic morphology results from fluctuating-to-mature pseudopod transitions and strain-stiffening feedback.
- Single-cell responses: Motility is biphasic, with the highest dispersion around 12 kPa, where elongated cells generate more persistent movement.
- Pairwise interactions: At intermediate stiffness, cell pairs disperse more slowly than individual cells; this difference is absent on soft and stiff substrates.At 12 kPa the paired-versus-individual difference is significant at 5000 MCS (p < 0.05).
- Pairwise interactions: At 12 kPa, adjacent cells repeatedly touch and separate in response to matrix strains, whereas contact interactions are weaker at lower or higher stiffnesses.Stronger adhesion reduces contact counts and increases contact duration without removing the intermediate-stiffness trend.
- Collective behavior: The same mechanical rules produce dynamically stable networks with remodeling and strain-guided bridging events.A network formed within a few hundred MCS on a 10 kPa substrate.
Discussion
The model links cell traction, strain stiffening, and durotaxis to endothelial behavior from single cells and pairs through collective network formation. Its agreements with experiments support mechanical feedback as a useful explanation, while its simplifying assumptions and missing properties define important limits.
- Model behavior: Cell traction, strain stiffening, and durotaxis reproduced endothelial spreading, contraction, elongation, and pairwise coordination across substrate stiffnesses.On intermediate-stiffness matrices, cells slowed one another and repeatedly touched and retracted.
- Mechanistic interpretation: Elongated cell shapes can create oriented matrix strains that let cells sense one another at a distance, connecting pairwise interactions with collective behavior.The proposed feedback is between stretch-guided extension and cell traction.
- Model behavior: Intermediate substrate stiffness produced the strongest pairwise interactions and enabled network formation, whereas softer or stiffer substrates did not assemble networks.The model also generated dynamic connections through strain bridges while other connections broke.
- Limitations: The model lacks some properties of in vitro angiogenesis, including the experimentally observed faster endothelial movement on stiff rather than soft substrates.The simulated single cells dispersed faster on soft gels, contradicting those observations.
- Limitations: The current formulation assumes uniform ECM density and thickness, although endothelial cells can pull the matrix underneath them and create density or thickness gradients.This restricts the model's representation of culture conditions involving spatially varying matrix structure.
- Limitations: The model resets matrix deformation after each Monte Carlo step and assumes linearly elastic materials with infinitesimal strain, excluding substrate-strain memory and broader material behavior.Future coupling with FEBio is intended to support additional ECM materials, including strain-stiffening materials.
- Interpretation: Alternative mechanical and chemical models can also explain vascular network formation and spheroid sprouting, leaving the underlying mechanism underdetermined.The discussion identifies multiple plausible model families based on mechanical interactions, chemoattraction, and extensions of these mechanisms.
Methods
The model couples stochastic endothelial-cell motility with finite-element mechanics of a compliant extracellular matrix. It alternates cell-shape-dependent traction, ECM deformation, strain-dependent stiffness, and cellular responses to reproduce cell behavior and quantify motility and morphology.
- Model framework: The hybrid model combines a cellular Potts model for stochastic cell motility with a finite-element model for compliant-matrix mechanics.Cells occupy connected lattice sites, while the substrate is represented by finite elements corresponding to CPM pixels.
- Cellular Potts model: The CPM updates cell boundaries through passive forces and active motility, including volume constraints, adhesion, and energetically unfavorable moves accepted with Boltzmann probability.The volume constraint regulates fluctuations around a resting volume, while adhesion is represented by contact costs between neighboring pixels.
- Finite-element substrate: The FEM computes substrate displacements and local strains from cell traction forces by solving Ku = f on an isotropic, linearly elastic substrate.The model uses plane-stress material properties with Poisson’s ratio 0.45 and Young’s moduli from 0.5 kPa to 32 kPa.
- Mechanical cell-substrate coupling: Cell-substrate feedback alternates CPM and FEM steps: cell shapes generate inward traction, FEM calculates ECM strains, and strains influence subsequent cell extensions and retractions.The stiffness matrix is held constant by resetting the substrate to an undeformed reference configuration after each Monte Carlo step.
- Mechanical cell-substrate coupling: Durotaxis favors extensions toward stiffer regions, with local stiffness determined from principal strains and strain orientation through a sigmoid response.Strain stiffening is applied only under extension; compression neither stiffens nor softens the substrate.
- Morphometry: Single-cell and pair behavior are characterized using mean square displacement, dispersion coefficients, and inertia-tensor-based length, orientation, and eccentricity.The dispersion coefficient is derived from the slope of the mean square displacement and measures random-walker motility.
Endothelial Cell Culture
Bovine aortic endothelial cells were cultured on polyacrylamide hydrogels under controlled temperature, carbon-dioxide, medium, and imaging conditions. Cell behavior was recorded for 24 hours at 30-minute intervals after three days on the gels.
- Cell culture: Bovine aortic endothelial cells were maintained at 37°C and 5% CO2 in supplemented Medium 199 through passage 12.The medium included fetal clone, amino acids, vitamins, and penicillin-streptomycin.
- Cell culture: Approximately 1,375 cells per mm2 were seeded onto polyacrylamide hydrogels and maintained for three days before imaging.The seeding density corresponded to 350,000 cells per gel.
- Imaging: Images were captured every 30 minutes for 24 hours using an inverted spinning-disc microscope with a digital camera.
Figures
The figures depict a hybrid mechanical model linking cellular traction, matrix strain, and cell responses across individual, pairwise, network, and spheroid assays.
- Model mechanics: Traction forces and resulting matrix strains are visualized as black arrows and blue line segments in the hybrid simulation model.The figure presents the mechanical feedback generated by simulated cells and the extracellular matrix.
- Individual cells: Individual-cell simulations measure shape and motility responses across substrate stiffness, including area, length, eccentricity, and dispersion.The measurements use n = 100 for morphology panels and n = 1000 for dispersion coefficients.
- Cell pairs: Pairwise simulations compare cell motion, contact frequency, and head-to-tail alignment on substrates of 4, 12, and 32 kPa.The pairwise analysis includes mean square displacement, contacts over 500 MCS, and alignment over MCS 20-500.
- Network formation: Starting from scattered cells, the model simulates polygonal network formation and reconnection of sprouts on a 10 kPa substrate.The network assay uses 450 cells in a 0.75 × 0.75 mm2 area, while the experimental panel shows endothelial cells forming networks on a 2.5 kPa gel.
- Spheroid assay: Starting from a two-dimensional spheroid of 113 cells, the model simulates sprouting on a 10 kPa substrate.The spheroid assay is conducted in a 0.75 × 0.75 mm2 area and includes close-up views of sprouting.
Supporting Information
The supporting figures vary model parameters and inspect how volume restriction, substrate stiffness, cell contacts, and the response function affect simulated behavior.
- Parameters: Table S1 lists the parameter settings used in the simulation model.It provides the model’s parameter-setting reference for the supporting analyses.
- Parameter sensitivity: Figure S1 tests individual-cell area, length, and eccentricity across volume-restriction values and substrate stiffnesses from 0.5 kPa to 32 kPa.Means and standard deviations are shown for n = 100 after 500 MCS.
- Motility: Figure S2 reports mean square displacement for 1,000 individual cells across substrates ranging from 0.5 kPa to 32 kPa.The figure characterizes simulated individual-cell motility across stiffness conditions.
- Cell-pair motion: Figure S3 compares mean square displacement of individual cells and cell pairs across substrate stiffnesses, with error bars showing standard deviation for n = 100.The comparison evaluates how pairing changes simulated motion across stiffness conditions.
- Contact sensitivity: Figure S4 varies substrate stiffness from 0.5 kPa to 32 kPa and intercellular contact energy from 0.5 for adhesive cells to 4 for repulsive cells.It measures contact number and duration over 500 MCS between cells initially separated by fourteen lattice sites.
- Response-function sensitivity: Figure S5 compares sigmoid, saturated, piecewise-linear, and Gaussian forms of h(E) for simulated cell shapes on substrates of different stiffnesses.The alternatives use different parameter settings for h(E) and λdurotaxis.