From Kepler’s conjecture and fcc lattice to modelling of crowding in living matter

Up to now, sphere packing has been investigated without any reference to living matter. This study focuses on the void space (VS) of sphere packing to mimic the extracellular spaces of living tissues. It was inspired by the importance of the extracellular matrix, the vehicle of micro and macromolecules involved in cell metabolism, intercellular communication and drug delivery. The analysis of sphere packing evidenced that in uniform random packing VS is about 1.9 times greater than in the face centered cubic (fcc) lattice (thus being very close to the 1.9 volume ratio of the cube to the sphere). This datum is a good reference for cell packing in vivo. The disproportionate increase of VS per sphere in loose packing in vitro is analyzed having in mind the variability in volume and composition of the interstitial spaces in vivo and cell trafficking. Arrangements of lymphocytes mimicking a two-dimensional hexagonal pattern and dense packing of disks generated by numerical procedures, are described in 7 μm-thick haematoxylin and eosin-stained histological slices from a human lymph node. In narrow tubes simulating roundish cells arranged in limited compartments of the interstice, sphere packing is characterized by noticeable increases of VS. The VS of this packing in vitro is compatible with variability in volume and composition of the interstitial spaces and with cell trafficking in vivo. This paper stresses that in mammalian tissues and organs cells can be packed quite more densely than spheres in the fcc lattice. As to pathology, attention is focused: (i) on overcrowding of cell organelles in some diseases, (ii) on shrinking or swelling of high amplitude, whose opposite effects are to concentrate or dilute intracellular structures and crowding of macromolecules, and (iii) on neoplastic tissues.