The domain-within-domain packing of euchromatin can be described as multiplicative cascades

ABSTRACTThe genome is packed into the cell nucleus in the form of chromatin. Biochemical approaches have revealed that chromatin is packed within domains, which group into larger domains, and so forth. Such domain-within-domain packing, also called hierarchical packing, is equally visible in super-resolution microscopy images of large-scale chromatin organization. While previous work has suggested that chromatin is partitioned into distinct domains via microphase separation, it is unclear how these domains organize into a hierarchical packing. A particular challenge is to find an image analysis approach that fully incorporates such hierarchical packing, so that hypothetical governing mechanisms of euchromatin packing can be compared against the results of such an analysis. Here, we obtain 3D STED super-resolution images from pluripotent zebrafish embryos labeled with improved DNA fluorescence stains, and demonstrate how the hierarchical packing of euchromatin in these images can be described as multiplicative cascades. Multiplicative cascades are an established theoretical concept to describe the placement of ever-smaller structures within bigger structures. Importantly, these cascades can generate artificial image data by applying a single rule again and again, and can be fully specified using only four parameters. Here, we show how the typical patterns of euchromatin organization are reflected in the values of these four parameters. In particular, we can pinpoint the values required to mimic a microphase-separated configuration of euchromatin. We suspect that the concept of multiplicative cascades can also be applied to images of other types of chromatin. In particular, cascade parameters could serve as test quantities to assess whether microphase separation or other theoretical models accurately reproduce the hierarchical packing of chromatin.SIGNIFICANCEDNA is stored inside the cell nucleus in the form of chromatin. Chromatin exhibits a striking three-dimensional organization, where small domains group into larger domains, which again group into larger domains, and so forth. While this hierarchical domain-within-domain organization is obvious from microscopy images, it is still not entirely clear how it is established, or how it should be properly characterized. Here, we demonstrate that multiplicative cascades – a concept from theoretical physics used to characterize for example cloud patterns, galaxy locations, or soil patterns – are also ideally suited to describe the domain-within-domain organization of chromatin. This description is rather simple, using only four numbers, and can thus facilitate testing of competing theories for the domain-within-domain organization of chromatin.

In-depth analysis of a discrete <i>p</i> model

Towards the end of the last century, B. Mandelbrot saw the importance, revealed the beauty, and robustly promoted (multi)fractals. Multiplicative cascades are closely related and provide simple models for the study of turbulence and chaos. For pedagogical reasons, but also due to technical difficulties, continuous stochastic models have been favoured over discrete cascades. Particularly important are the α, the β and the p model (Lovejoy and Scherzter 2013, Chapter 3; de Wijs (1951, 1953). It is the aim of this contribution to introduce original concepts that shed new light on the latter paradigmatic cascade and allow key features to be derived in a rather elementary fashion. To this end, we introduce and study a discrete version of the p model which is based on a new kind of sampling. Technical machinery can be kept simple, therefore formulas are explicit, proofs extend standard arguments, and potential extensions are numerous. Thus the proposed line of investigation may enrich and simplify received multifractal analyses.