Off-lattice models are a well-established approach in multicellular modeling, where cells are represented as points that are free to move in space. The representation of cells as point objects is useful in a wide range of settings, particularly when large populations are involved; however, a purely point-based representation is not naturally equipped to deal with objects that have length, such as cell boundaries or external membranes. Here we introduce an off-lattice modeling framework that exploits rigid body mechanics to represent objects using a collection of conjoined one-dimensional edges in a viscosity-dominated system. This framework can be used to represent cells as free moving polygons, to allow epithelial layers to smoothly interact with themselves, to model rod-shaped cells such as bacteria and to robustly represent membranes. We demonstrate that this approach offers solutions to the problems that limit the scope of current off-lattice multicellular models.
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Cross-platform MATLAB code implementing the rigid body framework can be found at https://github.com/luckyphill/EdgeBased (ref. 45). An introductory guide for using the code is provided in the repository. The code for all exemplars can be found in the v.1.0.1 release (the code for the spheroid, epithelial layer and bacterial exemplars can also be found in the in the v.1.0.0 release). The code is released under an open source GNU General Public Licence.
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P.J.B. acknowledges J. Krokiewski for his extremely thorough text on Mechanics of a Rigid Body, which was provided free of charge to students of mechanical engineering at The University of Melbourne, without which this work would not have been possible. This work was supported with supercomputing resources provided by the Phoenix HPC service at the University of Adelaide. B.J.B. and P.J.B. acknowledge funding from the ARC (grant number DP160102644).
The authors declare no competing interests.
Peer review information Nature Computational Science thanks Paul Macklin and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Handling editor: Ananya Rastogi, in collaboration with the Nature Computational Science team.
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Supplementary Figs. 1–6, Tables 1–3, derivations and discussion.
This video shows the polygon cell model producing a tumor spheroid. Four decagon cells are placed in the center of a plane and left to proliferate. As the cell population expands, cells towards the center become too compressed to start growing, leaving a narrow band of proliferating cells around the outer radius. Occasionally, inner cells start growing due to local pressure relief.
This video shows the progression of an unconstrained epithelial ring. The ring starts off being perfectly circular with equal size cells. As it grows, local areas of high pressure cause the ring to buckle. After enough buckling, the ring starts to contact itself in numerous places. The node–edge interaction mechanism along with the viscous rigid body laws of motion, allow contact forces to be transferred across the contact points, causing dynamic restructuring of the layer seen as secondary buckling
This video demonstrates the overlapping rods model replicating the experiment due to Volfson et al. for a single random seed. Initially 20 randomly oriented cells are placed in the center of the channel. The color of a given cell indicates its angle from horizontal, from yellow (horizontal) to blue (vertical). Cells start to grow in localized clusters with roughly the same orientation. As the clusters grow and merge, the cells interact through the node–edge interaction mechanism, causing them to smoothly move and reorient themselves. As the proliferating front travels down the channel, the walls influence the orientation of the new cells, keeping them largely horizontal.
This video shows the progression of tumor development in a duct constrained by a membrane. The membrane starts off as a ring, with overlapping spheres cells covering the inner surface. When the simulation starts, the internal pressure forces the membrane to expand to a point where the tension and pressure are in equilibrium. As the cells proliferate, they start to fill the lumen of the duct. At a certain point, the duct becomes completely filled with cells, and the proliferative pressure causes the membrane to expand. Under greater internal constriction, the cells gradually halt due to contact inhibition, with a few remaining cells starting their growth phase when they have enough space locally.
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Brown, P.J., Green, J.E.F., Binder, B.J. et al. A rigid body framework for multicellular modeling. Nat Comput Sci 1, 754–766 (2021). https://doi.org/10.1038/s43588-021-00154-4