In eukaryotes, higher-order chromatin organization is spatiotemporally regulated as domains, for various cellular functions. However, their physical nature in living cells remains unclear (e.g., condensed domains or extended fiber loops; liquid-like or solid-like). Using novel approaches combining genomics, single-nucleosome imaging, and computational modeling, we investigated the physical organization and behavior of early DNA replicated regions in human cells, which correspond to Hi-C contact domains with active chromatin marks. Motion correlation analysis of two neighbor nucleosomes shows that nucleosomes form physically condensed domains with ~150-nm diameters, even in active chromatin regions. The mean-square displacement analysis between two neighbor nucleosomes demonstrates that nucleosomes behave like a liquid in the condensed domain on the ~150 nm/~0.5 s spatiotemporal scale, which facilitates chromatin accessibility. Beyond the micrometers/minutes scale, chromatin seems solid-like, which may contribute to maintaining genome integrity. Our study reveals the viscoelastic principle of the chromatin polymer; chromatin is locally dynamic and reactive but globally stable.

Genomes are spatiotemporally organized within the cell nucleus. Genome-wide chromosome conformation capture (Hi-C) technologies have uncovered the 3D genome organization. Furthermore, live-cell imaging experiments have revealed that genomes are functional in 4D. Although computational modeling methods can convert 2D Hi-C data into population-averaged static 3D genome models, exploring 4D genome nature based on 2D Hi-C data remains lacking. Here, we describe a 4D simulation method, PHi-C (polymer dynamics deciphered from Hi-C data), that depicts 4D genome features from 2D Hi-C data by polymer modeling. PHi-C allows users to interpret 2D Hi-C data as physical interaction parameters within single chromosomes. The physical interaction parameters can then be used in the simulations and analyses to demonstrate dynamic characteristics of genomic loci and chromosomes as observed in live-cell imaging experiments. PHi-C is available at https://github.com/soyashinkai/PHi-C.

In networks of spatially distributed fluorescent molecules, Förster resonance energy transfer (FRET) can simultaneously occur over multiple locations and times. Such “FRET networks” have great potential for information-processing and computing applications. To design these applications, the spatiotemporal behavior of FRET networks should be understood. However, studies on their spatiotemporal behavior are scarce. Here, we develop a spatiotemporal model for FRET networks and uncover its temporal characteristic behavior. We theoretically show that our model can generate a distinctive temporal behavior, i.e., the network-induced multicomponent exponential decay of the fluorescence intensity, even for FRET networks of fluorophores with an identical single exponential decay. This theoretical result is supported experimentally using quantum dots.

The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent (exponential dynamical instability) is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov exponent, as a unified characterization of nonexponential and exponential dynamical instabilities in one-dimensional maps. Chaos is classified into three different types, i.e., superexponential, exponential, and subexponential chaos. Using one-dimensional maps, we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the Lyapunov pair. In subexponential chaos, we show superweak chaos, which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth. The scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process, which is related to infinite ergodic theory.

The well-known Hölder’s inequality has been recently utilized as an essential tool for solving several optimization problems. However, such an essential role of Hölder’s inequality does not seem to have been reported in the context of generalized entropy, including Rényi–Tsallis entropy. Here, we identify a direct link between Rényi–Tsallis entropy and Hölder’s inequality. Specifically, we demonstrate yet another elegant proof of the Rényi–Tsallis entropy maximization problem. Especially for the Tsallis entropy maximization problem, only with the equality condition of Hölder’s inequality is the q-Gaussian distribution uniquely specified and also proved to be optimal.