Analytical results for chromatin polymer models with enhancer-promoter interactions

Complex chromosomal organizations can be currently measured by experimental technologies, but spatiotemporal dynamics of the chromatin remain elusive. Here we analyze a chromatin polymer model with long-range interactions that account for the communications between multiple enhancer and promoter (E-P) pairs. We analytically show that the relaxation times of the nucleosomes emerges in hierarchy and the mean square displacement of every nucleosome grows over time in a power law. We find that more E-P pairs change neither the relaxation time hierarchy nor the diffusion mode of the nucleosomes. We also derive the analytical expressions for the joint probability distribution of nucleosome spatial positions and for the distribution of the spatial distance between any two loci, finding that the latter is a Maxwell-Boltzmann distribution rather than the previously assumed Gauss distribution. In addition, we present a method for calculating the encounter probability between any two nucleosomes, and numerically verify that this probability is approximately inversely proportional to the mean spatial distance between the two nucleosomes. These analytical results reveal the essence of chromatin organization and lay a solid foundation for further studying transcriptional dynamics regulated by E-P communications.