التحجيم متعدد الفراكتلات لخرائط Hi-C: من احتمالية التلامس إلى المؤشر الكتلي ✛

الملخص الأصلي

كما كتبه المؤلفون، بالإنجليزية arXiv:2606.31997

The three-dimensional organization of the genome exhibits rich, scale-dependent structure, as revealed by both chromosome contact maps (e.g., Hi-C maps) and chromatin density measured by microscopy. Recent studies have reported multifractal scaling in these data. Yet, the origin of this scaling behavior remains unclear: existing efforts describe it through postulated models. Here, we show that the multifractal structure of Hi-C maps is a direct consequence of the power-law contact probability $$P(s)$$, which is itself an empirical observable measured from Hi-C maps. Starting from $$P(s)$$ with a single exponent $$\gamma$$, we analytically derive the mass exponent $$\tau(q)$$, which characterizes how the $$q$$-th moment of contact density scales with box size $$l$$ used to coarse-grain the genomic coordinate. This multifractal behavior reflects the geometric competition between intra- and inter-segment contacts. We find that the slope of $$\tau(q)$$ at large $$q$$ is given by $$2 -\gamma$$ when $$\gamma <1$$, and by $$1$$ when $$\gamma \geq 1$$. We further show that this behavior is robust to noise and consistent across diverse organisms, indicating that it is a universal feature of chromatin organization. We extend our analysis into double-exponent $$P(s)$$, and show the $$l$$ dependence in multifractal behavior. Taken together, these results provide a physical explanation for multifractal scaling and establish a direct link between the multifractality in Hi-C maps and polymer contact statistics, with the large-$$q$$ slope of $$\tau(q)$$ mapping onto a known polymer contact exponent.