Spatial organisation is a hallmark of all living cells, and recreating it in model systems is a necessary step in the creation of synthetic cells. It is therefore of both fundamental and practical interest to better understand the basic mechanisms underlying spatial organisation in cells. In this work, we use a continuum model of membrane and protein dynamics to study the behaviour of curvature-inducing proteins on membranes of spherical shape, such as living cells or lipid vesicles. We show that the interplay between curvature energy, entropic forces, and the geometric constraints on the membrane can result in the formation of patterns of highly-curved/protein-rich and weakly-curved/proteinpoor domains on the membrane. The spontaneous formation of such patterns can be triggered either by an increase in the average density of curvature-inducing proteins, or by a relaxation of the geometric constraints on the membrane imposed by the membrane tension or by the tethering of the membrane to a rigid cell wall or cortex. These parameters can also be tuned to select the size and number of the protein-rich domains that arise upon pattern formation. The very general mechanism presented here could be related to protein self-organisation in many biological processes, ranging from (proto)cell division to the formation of membrane rafts. protein patterns on the surface of other types of cells, as well as in model systems consisting of lipid vesicles and proteins. In this work, we will explore in full generality and detail the predictions of such a model.The basic idea behind the model is presented in figure 1. A closed, initially spherical membrane contains proteins that impose a spontaneous curvature C p on the membrane (in general, the proteins might be attached to the membrane from the cytoplasmic or the exoplasmic sides, or they might be transmembrane proteins embedded in the membrane) [13,14]. If the proteins did not induce any curvature, a random, homogeneous distribution of proteins would be favoured by thermal fluctuations, that is, entropic forces (in the absence of direct attractive protein-protein interactions). However, if the curvature induced by the proteins is large enough, bending contributions to the free energy of the system can lead to an effective attraction between proteins and to the formation of spatially inhomogeneous patterns in protein distribution and membrane curvature. The details of membrane-mediated protein-protein interactions have been thoroughly studied in the past [15][16][17][18]. Furthermore, we will consider the possibility of geometric constraints on the membrane, such as the tethering of the membrane to a rigid cell wall/cortex or the existence of a membrane area reservoir at nonzero tension. Interestingly, it was recently shown that solid particles such as proteins can sense the local membrane curvature imposed by geometric constraints on the membrane [19].Here, we have found that, in realistic situations, spontaneous pattern formation can be induced either by an increase in the surfac...