The collective self-organization of cells into three-dimensional structures can give rise to emergent physical properties such as fluid behavior. Here, we demonstrate that tissues growing on curved surfaces develop shapes with outer boundaries of constant mean curvature, similar to the energy minimizing forms of liquids wetting a surface. The amount of tissue formed depends on the shape of the substrate, with more tissue being deposited on highly concave surfaces, indicating a mechano-biological feedback mechanism. Inhibiting cell-contractility further revealed that active cellular forces are essential for generating sufficient surface stresses for the liquid-like behavior and growth of the tissue. This suggests that the mechanical signaling between cells and their physical environment, along with the continuous reorganization of cells and matrix is a key principle for the emergence of tissue shape.

The complex arrangement of the extracellular matrix (ECM) produced by cells during tissue growth, healing and remodelling is fundamental to tissue function. In connective tissues, it is still unclear how both cells and the ECM become and remain organized over length scales much larger than the distance between neighbouring cells. While cytoskeletal forces are essential for assembly and organization of the early ECM, how these processes lead to a highly organized ECM in tissues such as osteoid is not clear. To clarify the role of cellular tension for the development of these ordered fibril architectures, we used an in vitro model system, where pre-osteoblastic cells produced ECM-rich tissue inside channels with millimetre-sized triangular cross sections in ceramic scaffolds. Our results suggest a mechanical handshake between actively contracting cells and ECM fibrils: the build-up of a long-range organization of cells and the ECM enables a gradual conversion of cell-generated tension to pre-straining the ECM fibrils, which reduces the work cells have to generate to keep mature tissue under tension.

Surface curvature both emerges from, and influences the behavior of, living objects at length scales ranging from cell membranes to single cells to tissues and organs. The relevance of surface curvature in biology has been supported by numerous recent experimental and theoretical investigations in recent years. In this review, we first give a brief introduction to the key ideas of surface curvature in the context of biological systems and discuss the challenges that arise when measuring surface curvature. Giving an overview of the emergence of curvature in biological systems, its significance at different length scales becomes apparent. On the other hand, summarizing current findings also shows that both single cells and entire cell sheets, tissues or organisms respond to curvature by modulating their shape and their migration behavior. Finally, we address the interplay between the distribution of morphogens or micro-organisms and the emergence of curvature across length scales with examples demonstrating these key mechanistic principles of morphogenesis. Overall, this review highlights that curved interfaces are not merely a passive by-product of the chemical, biological and mechanical processes but that curvature acts also as a signal that co-determines these processes.

We have studied the collective motion of polar active particles confined to ellipsoidal surfaces. The geometric constraints lead to the formation of vortices that encircle surface points of constant curvature (umbilics). We have found that collective motion patterns are particularly rich on ellipsoids with four umbilics where vortices tend to be located near pairs of umbilical points to minimize their interaction energy. Our results provide a new perspective on the migration of living cells, which most likely use the information provided from the curved substrate geometry to guide their collective motion.Introduction -Active particles are known to spontaneously form complex dynamic patterns at length scales ranging from the molecular [1], to the cellular [2,3] up to macroscopic patterns seen in flocking birds [4], schooling fish [5] or humans in crowded environments [6,7]. The key feature of these active systems is the constant energy input on each individual unit, which renders the system completely out of equilibrium. Collective phenomena in such active systems have been successfully described using so-called self-propelled particle models [8] that are limited to close neighbour interactions only [9]. In unconstrained 2D and 3D systems these models display selforganised pattern formation resembling experimental observations [9]. The behaviour of active particles confined to a surface has been mainly studied on planar surfaces of zero Gaussian curvature. It is known however, that the presence of intrinsic surface curvature frustrates local order giving rise to novel physics [10], as has been shown for 2D fluids confined to curved surfaces [11]. As a consequence of the Poincaré-Hopf theorem, for instance, it is not possible to have continuous fluid flow on the entire surface of a sphere, which requires the presence of two +1 defects (vortices) [12]. The effect of non-zero Gaussian curvature on self-propelled particles remains poorly understood, with only a few recent examples studying the effect of spherical constraints [13,14]. In living systems, cells are influenced by surface curvature as demonstrated by cell movements in the developing corneal epithelium leading to vortex patterns [15] or by the coordinated collective migration of cells during embryonic development [16]. The emergent behaviour of moving cells is not only the result of intercellular interactions, but is crucially influenced by geometrical constraints [2,17,18]. The aim of the current work is to investigate the impact of non-constant Gaussian curvature constraints on the collective behaviour of self-propelled particles. Our restriction of the geometry of the surfaces to ellipsoids allows an analysis of how geometrical cues (represented by the umbilical points of the surface, Fig. 1c) effectively interact with defects in the director-field (e.g., vortices). The strong coupling between vortex position and umbilical points demonstrates the importance of surface geometry on the emergence of patterns in active systems.

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