The three-dimensional structure of eukaryotic genomes is non-random, dynamic, highly regulated, and can be observed to change according to external signals and differentiation state. Disruptions can lead to disease, in which incorrect genomic contacts are responsible for mis-regulation of gene expression. However, the underlying mechanisms that organise the genome are still largely unknown. While most studies focus on the activity of specific proteins which connect two chromatin loci, we seek to understand how the cells makes use of its entire complexity and of physical mechanisms. Using stochastic polymer simulations, which are informed by the analysis of large datasets and verified by quantitative experiments in budding yeast, we discovered a fundamental mechanism: The mobility of the chromatin fibre is not uniform, but heterogeneous, along its length, as a result of the unequal distribution of proteins binding along the genome. This leads to thermodynamically driven self-organisation, which we observe experimentally. It achieves spatial clustering of poised genes and mechanistically contributes to the directed relocalisation of active genes to the nuclear periphery upon heat shock (bioRxiv 106344). I will discuss the implications and present follow-up studies that make use of polymer and particle-based simulations to discover an additional layer of genome organisation.