Geometric computation of finite, disordered mineral lattices through topologically complex active confinements

Mineralization involves surface growth which successively records contour geometries of the crystal in a history dependent manner. It leads to the evolution of a crystal shape towards an energetically favorable equilibrium that is determined by the species concentration and gradients which are dependent on the boundary conditions. Living systems can however continuously alter these boundary conditions and guide mineral growth along desired pathways to create functional structures. Sea cucumbers grow ~100 µm length scale calcite-based skeletal structures called ossicles in diverse forms. Each ossicle can be described as a finite-disordered polygonal lattice with mineral deposits along the edges, making it equivalent to a multi-genus torus. We demonstrate that ossicles grow from an initial seed crystal that transforms into a multi-holed lattice through 4 steps - symmetry breaking in the seed, tip elongation to extend branches, tip splitting or budding to create new branches, and the fusion of two tips create closed loops. We find that the computation of final geometry of such a topologically complex rigid object is achieved through the unique cellular physiology of the cells continuously wrapping the structure as it evolves. This wrapping restricts transport of mineral precursors through microtubule networks along the surface of the existing geometry, thus coupling shape and material transport. The continuous extension of this wrapping around growing tips makes growth of actin filament based membranous projections possible. This endows otherwise passive mineral tips with activity, enabling them to execute a local search algorithm to find other tips nearby and perform fusion based loop closure. We also identify distinct modes of participating cell-cluster dispersion, which acts as an additional layer of control over the underlying growth process. Through reduced-order models of transport on self-closing active branching networks, we demonstrate that globally observed broken symmetries can be replicated through a simple parameter tuning and thus highlight the hidden universality in the diversity of shapes. The system thus serves as a unique playground merging aspects of cellular physiology, non-equilibrium growth processes, and classical branching morphogenesis in living systems.