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Chromatin stability generated by stochastic binding and unbinding of cross-linkers at looping sites revealed by Markov models

Chromatin loops inside the nucleus can be stable for a very long time, which remains poorly understood. Such a time is crucial for chromatin organization maintenance and stability. We explore here several physical scenarios, where loop maintenance is due to diffusing cross-linkers (cohesin stabilized by two CTCF molecules) that can bind and unbind at the base of chromatin loops. Using a Markov chain approach to coarse-grain the binding and unbinding, we consider that a stable loop disappears when the last cross-linker is unbound. We derive expressions for this last passage time that we use to quantify the loop stability for various parameters, such as the chemical rate constant or the number of cross-linkers. The present analysis suggests that the balance between binding and unbinding events regulates the number of cross-linkers in place, based on a positive feed-back mechanism that stabilizes the loop over long-time. To conclude, we found that short- and long-lasting stable loops can vary from minutes to the entire cell cycle lifetime, when the number of cross-linkers increases from 1 to 10. This result suggests that a large spectrum of loop time scales is expected with such a few numbers of cross-linkers per local binding sites.