Mechanical forces are critical for the emergence of diverse three-dimensional morphologies of multicellular systems. However, it remains unclear what kind of mechanical parameters at cellular level substantially contribute to tissue morphologies. This is largely due to technical limitations of live measurements of cellular forces. Here we developed a framework for inferring and modeling mechanical forces of cell-cell interactions. First, by analogy to coarse-grained models in molecular and colloidal sciences, we approximated cells as particles, where mean forces (i.e. effective forces) of pairwise cell-cell interactions are considered. Then, the forces were statistically inferred by fitting the mathematical model to cell tracking data. This method was validated by using synthetic cell tracking data resembling various in vivo situations. Application of our method to the cells in the early embryos of mice and the nematode Caenorhabditis elegans revealed that cell-cell interaction forces can be written as a pairwise potential energy in a manner dependent on cell-cell distances. Importantly, the profiles of the pairwise potentials were quantitatively different among species and embryonic stages, and the quantitative differences correctly described the differences of their morphological features such as spherical vs. distorted cell aggregates, and tightly vs. non-tightly assembled aggregates. We conclude that the effective pairwise potential of cell-cell interactions is a live measurable parameter whose quantitative differences can be a parameter describing three-dimensional tissue morphologies.

We developed MDGRAPE-4A, a special-purpose computer system for molecular dynamics simulations, consisting of 512 nodes of custom system-on-a-chip LSIs with dedicated processor cores and interconnects designed to achieve strong scalability for biomolecular simulations. To reduce the global communications required for the evaluation of Coulomb interactions, we conducted a co-design of the MDGRAPE-4A and the novel algorithm, tensor-structured multilevel Ewald summation method (TME), which produced hardware modules on the custom LSI circuit for particle-grid operations and for grid-grid separable convolutions on a 3D torus network. We implemented the convolution for the top-level grid potentials by using 3D FFTs on an FPGA, along with an FPGA-based octree network to gather grid charges. The elapsed time for the long-range part of Coulomb is 50 μs, which can mostly overlap with those for the short-range part, and the additional cost is approximately 10 μs/step, which is only a 5% performance loss.