Mitotic chromosomes are self-entangled and disentangle through a topoisomerase-II-dependent two-stage exit from mitosis.

The topological state of chromosomes determines their mechanical properties, dynamics, and function. Recent work indicated that interphase chromosomes are largely free of entanglements. Here, we use Hi-C, polymer simulations, and multi-contact 3C and find that, by contrast, mitotic chromosomes are self-entangled. We explore how a mitotic self-entangled state is converted into an unentangled interphase state during mitotic exit. Most mitotic entanglements are removed during anaphase/telophase, with remaining ones removed during early G1, in a topoisomerase-II-dependent process. Polymer models suggest a two-stage disentanglement pathway: first, decondensation of mitotic chromosomes with remaining condensin loops produces entropic forces that bias topoisomerase II activity toward decatenation. At the second stage, the loops are released, and the formation of new entanglements is prevented by lower topoisomerase II activity, allowing the establishment of unentangled and territorial G1 chromosomes. When mitotic entanglements are not removed in experiments and models, a normal interphase state cannot be acquired.

Communities in C. elegans connectome through the prism of non-backtracking walks.

The fundamental relationship between the mesoscopic structure of neuronal circuits and organismic functions they subserve is one of the major challenges in contemporary neuroscience. Formation of structurally connected modules of neurons enacts the conversion from single-cell firing to large-scale behaviour of an organism, highlighting the importance of their accurate profiling in the data. While connectomes are typically characterized by significant sparsity of neuronal connections, recent advances in network theory and machine learning have revealed fundamental limitations of traditionally used community detection approaches in cases where the network is sparse. Here we studied the optimal community structure in the structural connectome of Caenorhabditis elegans, for which we exploited a non-conventional approach that is based on non-backtracking random walks, virtually eliminating the sparsity issue. In full agreement with the previous asymptotic results, we demonstrated that non-backtracking walks resolve the ground truth annotation into clusters on stochastic block models (SBM) with the size and density of the connectome better than the spectral methods related to simple random walks. Based on the cluster detectability threshold, we determined that the optimal number of modules in a recently mapped connectome of C. elegans is 10, which precisely corresponds to the number of isolated eigenvalues in the spectrum of the non-backtracking flow matrix. The discovered communities have a clear interpretation in terms of their functional role, which allows one to discern three structural compartments in the worm: the Worm Brain (WB), the Worm Movement Controller (WMC), and the Worm Information Flow Connector (WIFC). Broadly, our work provides a robust network-based framework to reveal mesoscopic structures in sparse connectomic datasets, paving way to further investigation of connectome mechanisms for different functions.

Topological and nontopological mechanisms of loop formation in chromosomes: Effects on the contact probability.

Chromosomes are crumpled polymer chains further folded into a sequence of stochastic loops via loop extrusion. While extrusion has been verified experimentally, the particular means by which the extruding complexes bind DNA polymer remains controversial. Here we analyze the behavior of the contact probability function for a crumpled polymer with loops for the two possible modes of cohesin binding, topological and nontopological mechanisms. As we show, in the nontopological model the chain with loops resembles a comb-like polymer that can be solved analytically using the quenched disorder approach. In contrast, in the topological binding case the loop constraints are statistically coupled due to long-range correlations present in a nonideal chain, which can be described by the perturbation theory in the limit of small loop densities. As we show, the quantitative effect of loops on a crumpled chain in the case of topological binding should be stronger, which is translated into a larger amplitude of the log-derivative of the contact probability. Our results highlight a physically different organization of a crumpled chain with loops by the two mechanisms of loop formation.

Crumpled polymer with loops recapitulates key features of chromosome organization.

Chromosomes are exceedingly long topologically-constrained polymers compacted in a cell nucleus. We recently suggested that chromosomes are organized into loops by an active process of loop extrusion. Yet loops remain elusive to direct observations in living cells; detection and characterization of myriads of such loops is a major challenge. The lack of a tractable physical model of a polymer folded into loops limits our ability to interpret experimental data and detect loops. Here, we introduce a new physical model - a polymer folded into a sequence of loops, and solve it analytically. Our model and a simple geometrical argument show how loops affect statistics of contacts in a polymer across different scales, explaining universally observed shapes of the contact probability. Moreover, we reveal that folding into loops reduces the density of topological entanglements, a novel phenomenon we refer as "the dilution of entanglements". Supported by simulations this finding suggests that up to ~ 1 - 2Mb chromosomes with loops are not topologically constrained, yet become crumpled at larger scales. Our theoretical framework allows inference of loop characteristics, draws a new picture of chromosome organization, and shows how folding into loops affects topological properties of crumpled polymers.

Stretching of a Fractal Polymer around a Disc Reveals Kardar-Parisi-Zhang Scaling.

While stretching of a polymer along a flat surface is hardly different from the classical Pincus problem of pulling chain ends in free space, the role of curved geometry in conformational statistics of the stretched chain is an exciting open question. We use scaling analysis and computer simulations to examine stretching of a fractal polymer chain around a disc in 2D (or a cylinder in 3D) of radius R. We reveal that the typical excursions of the polymer away from the surface and curvature-induced correlation length scale as Δ∼R^{ß} and S^{*}∼R^{1/z}, respectively, with the Kardar-Parisi-Zhang (KPZ) growth ß=1/3 and dynamic exponents z=3/2. Although probability distribution of excursions does not belong to KPZ universality class, the KPZ scaling is independent of the fractal dimension of the polymer and, thus, is universal across classical polymer models, e.g., SAW, randomly branching polymers, crumpled unknotted rings. Additionally, our Letter establishes a mapping between stretched polymers in curved geometry and the Balagurov-Vaks model of random walks among traps.

Order and stochasticity in the folding of individual Drosophila genomes.

Mammalian and Drosophila genomes are partitioned into topologically associating domains (TADs). Although this partitioning has been reported to be functionally relevant, it is unclear whether TADs represent true physical units located at the same genomic positions in each cell nucleus or emerge as an average of numerous alternative chromatin folding patterns in a cell population. Here, we use a single-nucleus Hi-C technique to construct high-resolution Hi-C maps in individual Drosophila genomes. These maps demonstrate chromatin compartmentalization at the megabase scale and partitioning of the genome into non-hierarchical TADs at the scale of 100 kb, which closely resembles the TAD profile in the bulk in situ Hi-C data. Over 40% of TAD boundaries are conserved between individual nuclei and possess a high level of active epigenetic marks. Polymer simulations demonstrate that chromatin folding is best described by the random walk model within TADs and is most suitably approximated by a crumpled globule build of Gaussian blobs at longer distances. We observe prominent cell-to-cell variability in the long-range contacts between either active genome loci or between Polycomb-bound regions, suggesting an important contribution of stochastic processes to the formation of the Drosophila 3D genome.

Anomalous one-dimensional fluctuations of a simple two-dimensional random walk in a large-deviation regime.

The following question is the subject of our work: could a two-dimensional (2D) random path pushed by some constraints to an improbable "large-deviation regime" possess extreme statistics with one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though nonuniversal, since the fluctuations depend on the underlying geometry. We consider in detail two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span 〈d(t)〉∼t^{γ} of the trajectory of t steps above the top of the semicircle or the triangle. We show that γ=1/3, i.e., 〈d(t)〉 shares the KPZ statistics for the semicircle, while γ=0 for the triangle. We propose heuristic derivations of scaling exponents γ for different geometries, justify them by explicit analytic computations, and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.

Effect of Architecture on Micelle Formation and Liquid-Crystalline Ordering in Solutions of Block Copolymers Comprising Flexible and Rigid Blocks: Rod-Coil vs Y-Shaped vs Comblike Copolymers.

Micelle formation of amphiphilic block copolymers of various architectures comprising both flexible and rodlike blocks were studied in a selective solvent via dissipative particle dynamics (DPD) simulations. Peculiarities of self-assembly of Y-shaped (insoluble rigid block and two flexible soluble arms) and comblike (soluble flexible backbone with insoluble rigid side chains) copolymers are compared with those of equivalent rod-coil diblock copolymers. We have shown that aggregation of the rigid blocks into the dense core of the micelles is accompanied by their nematic ordering. However, the orientation order parameter and aggregation number of the micelles are strongly dependent on macromolecular architecture. Relatively small micelles of pretty high nematic order parameter, S2 ≈ 0.5-0.8, are the features of the Y-shaped and rod-coil copolymer micelles. They are characterized by different responses to the solvent quality worsening. The aggregation number of the rod-coil diblock copolymer micelles increases and that of the Y-shaped copolymer micelles decreases at the solvent quality worsening. However, the order parameter grows in both cases, achieving a maximum value for the Y-shaped copolymer micelles. Herewith, the core elongates. On the contrary, comblike copolymers self-assemble into bigger spherical micelles whose core possesses a lower nematic order of the rods, S2 ≈ 0.3-0.4. The aggregation number is shown to depend on the length of the combs (on the number of repeating elements in the architecture). Possible physical reasons for such behavior of the systems are discussed.

From geometric optics to plants: the eikonal equation for buckling.

Optimal buckling of a tissue, e.g. a plant leaf, growing by means of exponential division of its peripheral cells, is considered in the framework of a conformal approach. It is shown that the boundary profile of a tissue is described by the 2D eikonal equation, which provides the geometric optic approximation for the wavefront propagating in a medium with an inhomogeneous refraction coefficient. A variety of optimal surfaces embedded in 3D is controlled by spatial dependence of the refraction coefficient which, in turn, is dictated by the local growth protocol.