Fluorescence microscopy reveals that the contents of many (membrane-free) nuclear "bodies" exchange rapidly with the soluble pool whilst the underlying structure persists; such observations await a satisfactory biophysical explanation. To shed light on this, we perform large-scale Brownian dynamics simulations of a chromatin fiber interacting with an ensemble of (multivalent) DNA-binding proteins; these proteins switch between two states -active (binding) and inactive (non-binding). This system provides a model for any DNA-binding protein that can be modified post-translationally to change its affinity for DNA (e.g., like the phosphorylation of a transcription factor). Due to this out-of-equilibrium process, proteins spontaneously assemble into clusters of self-limiting size, as individual proteins in a cluster exchange with the soluble pool with kinetics like those seen in photo-bleaching experiments. This behavior contrasts sharply with that exhibited by "equilibrium", or non-switching, proteins that exist only in the binding state; when these bind to DNA non-specifically, they form clusters that grow indefinitely in size. Our results point to post-translational modification of chromatin-bridging proteins as a generic mechanism driving the self-assembly of highly dynamic, non-equilibrium, protein clusters with the properties of nuclear bodies. Such active modification also reshapes intra-chromatin contacts to give networks resembling those seen in topologically-associating domains, as switching markedly favors local (short-range) contacts over distant ones.In all living organisms, from bacteria to man, DNA and chromatin are invariably associated with binding proteins, which organize their structure [1][2][3]. Many of these architectural proteins are molecular bridges that can bind at two or more distinct DNA sites to form loops. For example, bacterial DNA is looped and compacted by the histone-like protein H-NS which has two distinct DNA-binding domains [4]. In eukaryotes, complexes of transcription factors and RNA polymerases stabilize enhancer-promoter loops [5,6,6, 7], while HP1 [9], histone H1 [10], and the polycomb-repressor complex PRC1/2 [5, 11] organize inactive chromatin. Proteins also bind to specific DNA sequences to form larger structures, like nucleoli and the histone-locus, Cajal, and promyeloleukemia bodies [13][14][15][16][17][18]. The selective binding of molecular bridges to active and inactive regions of chromatin has also been highlighted as one possible mechanism underlying the formation of topologically associated domains (TADs) -regions rich in local DNA interactions [6,6,19].From a biophysical perspective, a system made up of DNA and DNA-binding proteins exhibits many kinds of interesting and seemingly counter-intuitive behaviour, such as the clustering of proteins in the absence of any attractive interaction between them. This process is driven by the "bridging-induced attraction" [20]. In conjunction with the specific patterning of binding sites found on a whole human chromoso...