What is a topological domain?

What is a topological domain?

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I found this definition on Wikipedia, but I didn't quit undestand it :

Topologically associating domains (TADs) are genomic regions ("chromosome neighborhoods") used to summarize the three-dimensional nuclear organization of mammalian genomes.

Can anyone explain it more, and give examples if it's possible ?

TADs were initially discovered by computing contact probabilities between regions of the genome using HiC (a chromosome conformation capture method, that try to provide an idea on how the genome is organized inside the nucleus by computing the probability of each contact to be located nearby another locus). People have found that instead of being random, some loci were in contact with some regions of the genome rather than the others, implying some form of functional significance.

In short: a TAD is a genomic region of increased contact probability. They are of functional relevance (for instance, Enhance-Promoter gene expression regulation occurs primarily within one TAD rather than between two (adjacent TADs).

You might be interested in the following papers:

  • A 3D Map of the Human Genome at Kilobase Resolution Reveals Principles of Chromatin Looping (might be behind paywall)
  • Topological Domains in Mammalian Genomes Identified by Analysis of Chromatin Interactions

Finally, the following picture (extracted from the latter publication) provides an example of two TADS:

The bottom panel shows the contact probability: two regions of the genome appear to be isolated whereas they are nearby. They show significant self-interaction. One interpretation for that (top panel) is to imagine that the genome exhibit some reproducible condensation, explaining the increased contacts.

Topological aspects of DNA structure arise primarily from the fact that the two DNA strands are repeatedly intertwined. Untangling these two strands, which occurs in all major genetic processes may prove rather difficult. In the simplest case of a linear DNA in solution, untangling is possible due to the free rotation of the ends of the DNA. However, for all natural DNAs, free end rotation is either restricted or forbidden altogether. Consequently, untangling the two DNA strands becomes topologically impossible (Figure 1 illustrates this for the imaginary case of a circular DNA molecule where the two strands are tangled only once).

A DNA segment constrained so that the free rotation of its ends is impossible is called a topological domain (Figure 2). A canonical example of a topological domain is circular DNA, which is typical of bacteria, mitochondria, chloroplasts, many viruses, etc. In this case, there are obviously no DNA ends at all, since both DNA strands are covalently closed. Although eukaryotic chromosomes are linear overall, they consist of large DNA loops firmly attached to the nuclear matrix. These loops represent topological domains, i.e. they are equivalent to circular DNA topologically. The ends of linear DNA can also be affixed to the membrane, as has been shown for some viruses, making this DNA topologically closed. Finally, a stretch of DNA situated between the two massive protein bodies can also be considered a topological domain.

Source : DNA Topology Fundamentals

Topological domain structure of the Escherichia coli chromosome

The circular chromosome of Escherichia coli is organized into independently supercoiled loops, or topological domains. We investigated the organization and size of these domains in vivo and in vitro. Using the expression of >300 supercoiling-sensitive genes to gauge local chromosomal supercoiling, we quantitatively measured the spread of relaxation from double-strand breaks generated in vivo and thereby calculated the distance to the nearest domain boundary. In a complementary approach, we gently isolated chromosomes and examined the lengths of individual supercoiled loops by electron microscopy. The results from these two very different methods agree remarkably well. By comparing our results to Monte Carlo simulations of domain organization models, we conclude that domain barriers are not placed stably at fixed sites on the chromosome but instead are effectively randomly distributed. We find that domains are much smaller than previously reported, ∼10 kb on average. We discuss the implications of these findings and present models for how domain barriers may be generated and displaced during the cell cycle in a stochastic fashion.


The Escherichia coli chromosome is 4.6 Mb long, circular, and contains a single origin of replication. These characteristics present challenges for replication and segregation, which can nonetheless be completed as quickly as every 20 min. In addition, the chromosome must be compacted ∼10 3 -fold to fit inside the cell (Holmes and Cozzarelli 2000), and its structure must ensure that entanglements between replicated sister chromosomes and subsequent guillotining at the septum are prevented. Thus, the bacterial chromosome is folded in a manner that confers a high level of both organization and compaction (Sherratt 2003).

A fundamental aspect of the folding and compaction of chromosomal DNA is its negative supercoiling (Holmes and Cozzarelli 2000), which is also critical for the numerous processes that require unwinding of DNA (Funnell et al. 1986 Pruss and Drlica 1989). Even a modest reduction of negative supercoiling in bacteria is lethal (Zechiedrich et al. 1997). Supercoiling, however, is fragile. Without additional stabilization, just one DNA break would relax the entire chromosome and kill the cell. Replication itself causes relaxation, as only intact DNA can be supercoiled, and the newly synthesized DNA strands are complete only upon the conclusion of replication.

Fortunately, the effects of DNA interruptions are limited by the binding of proteins, and perhaps RNAs, throughout the chromosome (Drlica 1987). These connections divide the chromosome into topologically isolated loops, that is, domains, which are the subject of this report. The operational definition of a domain is the region relaxed by an interruption in the DNA. We do not explicitly consider loops of chromosomal DNA that are not topologically constrained, but we present evidence below that bacterial chromosome loops are typically isolated topologically.

There is both in vitro and in vivo evidence for topological domains. Chromosomes isolated from E. coli cells were found early on to be folded by unconstrained negative supercoiling (Worcel and Burgi 1972). Unlike plasmids, which are relaxed by a single nick, a number of nicks were required to relax the chromosome completely. The conclusion from this early work was that chromosomes are divided into 12-80 domains. Similar results were found using a different method to measure chromosome relaxation in vivo (Sinden and Pettijohn 1981). Based on these experiments, domains were estimated to be ∼100 kb in length, requiring each genome to be divided into ∼50 domains.

Morphological evidence for the existence of topological domains was provided by electron microscopy (Kavenoff and Bowen 1976 Kavenoff and Ryder 1976). Isolated E. coli chromosomes display many individually supercoiled loops emanating from a central region. These supercoiled loops were hypothesized to be topological domains. This view was supported by the finding that a single loop could be relaxed while the rest of the loops remained supercoiled (Delius and Worcel 1974). Moreover, Kavenoff and coworkers (Kavenoff and Bowen 1976 Kavenoff and Ryder 1976) counted the number of loops visible by electron microscopy and estimated between 65 and 200 loops per nucleoid, in rough agreement with the above estimates for the number of domains.

Although these studies were consistent with the existence of topologically closed domains, there were shortcomings to the methods that have bred some skepticism. For example, the in vitro domain barriers might have been created during chromosome isolation. A giant, highly compacted, twisted DNA could easily have acquired artifactual cross-links through bidentate binding by proteins or RNAs. Conversely, domain barriers that rely on transient attachments could be lost during chromosome isolation. The study of domains in vivo relied on measurements of relatively small differences in trimethylpsoralen binding, an unreliable technique (Sinden et al. 1980 Bliska and Cozzarelli 1987). In addition, the calculation of domain size in the studies cited above was based on the assumption that all domains are equal in size, but there is no evidence to support this view. In spite of these uncertainties, domains are presumed to exist, and the commonly cited figure for their size is 50-100 kb (Drlica 1987 Pettijohn 1996 Trun and Marko 1998).

More recent experiments have suggested that this much-cited domain size is too large. A less invasive approach to studying domains in vivo was taken by Higgins et al. (1996), who used the requirement for negative supercoiling of the γδ resolvase DNA substrate to study domain structure in a region of the Salmonella enterica chromosome. They concluded that topological domains are variable in both size and the placement of barriers, and that domains average 25 kb in length. Although these studies had the advantage of relying neither on the isolation of intact chromosomes nor on the estimation of the efficiency of nicking the genome, there are caveats to this method that could lead to an overestimation of domain size, raising the possibility that domains are even smaller (see Discussion).

Four basic models for how domains might be organized in the bacterial cell are illustrated in Figure 1A. Combinations of these models are also possible. For simplicity, we have shown each chromosome as divided into only six domains. We have depicted three separate chromosomes for each model and have included a marker gene at a specific site, in red, to indicate the sequence specificity of domain barrier placement. These models differ in terms of two unanswered questions regarding domain organization. First, are domain barriers placed at specific sites in the genome? For models in which the placement of the barriers is fixed (models I and III), the distance of the marker gene to its nearest domain barrier is the same in all chromosomes. Conversely, when the site of barriers is variable (models II and IV), the distance of the marker gene to its nearest domain barrier will also vary. Second, are domains of uniform size? If domain length is fixed (models I and II), the distance of the marker gene to its nearest barrier can never exceed half of that fixed length. However, if domain length is variable (models III and IV), the marker gene may be any distance from its nearest barrier.

(A) Models for domain organization. Four models are presented that differ in the variability of barrier placement and domain length. For each model, three different chromosomes are shown, which may depict chromosomes in different cells or in the same cell at different times. Domain barriers are indicated with yellow boxes. A marker gene is shown in red to demonstrate the variability in barrier position. For simplicity, we have divided each chromosome into only six domains. (B) Experimental scheme for microarray experiments. Cleavage of the chromosome will relax domains containing the cut site but leave uncut domains negatively supercoiled. DNA relaxation at numerous sites around the chromosome can be quantified using the expression of supercoiling sensitive genes (SSGs). To measure the expression of SSGs following cleavage, RNA was collected from cells prior to or following the expression of an active restriction endonuclease, labeled with either the dye Cy3 (green, reference) or Cy5 (red, sample), and used to probe microarrays containing PCR products from ORFs in the E. coli genome. The ratio of red to green signal measures the relative abundance of the transcripts from a particular gene. A yellow signal indicates about equal abundance of the reference and sample RNAs. To measure degradation from cut sites, genomic DNA was isolated from cells instead of RNA.

We investigated the organization of topological domains and chromosome structure in E. coli both in vivo and in vitro. First, we showed that unlike in higher organisms, the chromosome in vivo is generally accessible along its entire length to restriction enzymes, and thus there are no extensive stably bound structural components. Next, we used gene microarrays to examine the entire chromosome at once to determine how far DNA relaxation extends from double-strand breaks generated in vivo by a restriction enzyme (Fig. 1B). To measure relaxation, we monitored the transcription of >300 genes that respond to DNA relaxation (B.J. Peter, J. Arsuaga, A.M. Breier, A.B. Khodursky, P.O. Brown, N.R. Cozzarelli, in prep.). We then compared the results with Monte Carlo simulations of the four basic models in Figure 1A using varying mean domain lengths. Finally, we studied directly the sizes of supercoiled loops from isolated chromosomes using electron microscopy. Our results show that domains have a range of sizes, that domain barriers are not located sequence-specifically, and that chromosomes are structurally fluid, as in model IV (variable placement/variable length). In addition, we demonstrate that domains are much smaller than previously believed, with an average length of ∼10 kb. We discuss the implications of these results for chromosome structure and function.


The type II topoisomerase (TOP2) enzymes resolve DNA topology problems in core biological processes such as transcription, replication, recombination, DNA repair, chromatin remodeling, chromosome condensation, and segregation [1–3]. TOP2 enzymes catalyze and rejoin transient DNA double-stranded breaks (DSB) by allowing one of the duplex DNA strands to pass through the other [1–3]. Vertebrates possess two TOP2 genes, TOP2A and TOP2B, that originate from an ancestral gene duplication event [4, 5]. TOP2A and TOP2B are not functionally redundant despite their structural and catalytic similarities [6]. TOP2A is expressed in proliferating cells [7, 8] and knocking out Top2a in mice leads to defects in nuclear division and early embryonic lethality [9–11]. In contrast, TOP2B is ubiquitously expressed and is upregulated during cellular differentiation [7].

The full knockout of Top2b in mice leads to perinatal lethality mediated by defects in neuronal differentiation [12]. Conditional Top2b mouse knockout studies have demonstrated TOP2B’s importance during retinal development [13] and ovulation [14]. Studies using TOP2 poisons have implicated TOP2B in spermatogenesis [15–17] and lymphocyte activation [18]. In contrast to these functional insights, the conditional ablation of TOP2B in the adult heart resulted in few significant gene expression changes [19]. Despite the growing number of tissues and developmental processes that require TOP2B, the mechanisms by which this ubiquitous protein facilitates tissue-specific developmental processes are still not well understood.

It has been proposed that TOP2B’s role in development involves the activation or repression of specific developmental genes [20, 21]. Human TOP2B is required for the activation of hormone sensitive genes through the generation of transient double-stranded DNA breaks at the promoter region [20, 22]. Most recently, TOP2B-generated DSBs have been shown to be essential for the activation of early response genes by neurotransmitters [23]. Moreover, TOP2B has also been implicated in the expression of long genes, presumably through its ability to resolve positive supercoiling that arises during transcription [24].

TOP2B is also actively studied in the context of cancer. For example, TOP2B-mediated cleavage occurs at known chromosomal breakpoints in prostate cancer [25] and has been observed near translocation breakpoints in leukemia [26]. TOP2 proteins are prominent targets of many widely used chemotherapy agents including doxorubicin, etoposide, and mitoxantrone [27]. However, these chemotherapeutic agents can cause secondary malignancies in non-neoplastic tissues (reviewed in [28]). Whereas TOP2A is the intended target of these widely used chemotherapeutic agents, mechanistic studies in cell lines and animal models show that TOP2B-mediated DNA cleavage is an important player in treatment-related malignancies [19, 25, 29]. Intriguingly, heart-specific ablation of TOP2B significantly reduced the cardiotoxicity that normally occurs from doxorubicin treatment [19].

Identifying the protein–protein and protein–DNA interactions of TOP2B is essential for understanding its roles in development, transcription, and cancer. Here we report a comprehensive proximal protein interaction network for TOP2B that includes several members of the cohesin complex. Using ChIP-seq and ChIP-exo in combination with high-throughput chromosome conformation capture (Hi-C) data, we find that TOP2B interacts with CTCF and the cohesin complex with a distinct spatial organization at the borders of long-range chromosomal domain structures.

Understanding what a topological germ is

I have trouble understanding what is a germ in a topological space.
I have to mention that i don't have knowledge about this concept in any other area (complex analysis or others), so please don't answer by this means.
The definitions that have been given to me are this:

Let $(X, au_1)$ and $(Y, au_2)$ be topological spaces, and $xin X$ .
Let $xi(x)$ denote the set of all neighbourhoods of $x$ .

  • Let $U,Vinxi(x)$ , $f:U ightarrow Y$ and $g:V ightarrow Y$ two maps.
    We say that $f$ and $g$ are germ-equivalent at $x$ if $exists Winxi(x):Wsubseteq Ucap V,forall yin W,f(y)=g(y)$ .
  • Let $A,Bsubseteq X$ .
    We say $A$ and $B$ are germ-equivalent at $x$ if $exists Uinxi(x):Acap U=Bcap U$ .

This two relations define two equivalence relations.
And what i wanted to prove is that $[A]_x=[B]_x$ $Leftrightarrow$ $[chi_A]_x=[chi_B]_x$ .
If this is not true i will be grateful if someone can provide a counterexample.
I have to mention that i'm newbie in this so maybe i have made some mistakes.


Unrestrained DNA supercoiling was measured using a Me3-psoralen photobinding assay within a transcriptionally active hygromycin B phosphotransferase (hph) gene integrated into different chromosomal locations in five transformed human fibrosarcoma cell lines. The level of unrestrained supercoiling in the hph gene varied, from high to low levels, in different chromosomal locations in living human cells. In one cell line, the hph gene contained no unrestrained supercoiling. Consequently, supercoiling was not dictated by the DNA sequence of the active hph gene. The addition of α-amanitin, which can inhibit transcription, reduced unrestrained supercoiling by 75% at one chromosomal location, by 50% at two other locations, and had little, if any, effect at two other chromosomal locations. Different levels of supercoiling in separate regions of the chromosome require that the chromosome be organized into independent topological domains in vivo. Evidence for independent topological domains in living cells is presented. From analysis of the relaxation of supercoiling as a function of the number of breaks introduced into the chromosome, the in vivo topological domain size for the human ribosomal RNA genes was estimated between 30 000 and 45 000 kb.

This work was supported by Public Health Service Grant ES056520 from the National Institute of Environmental Health Sciences.

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Local quasiparticle excitations and topological quasiparticle excitations

To understand and classify anyonic quasiparticles in topologically ordered states, such as FQH states, it is important to understand the notions of local quasiparticle excitations and topological quasiparticle excitations. First let us define the notion of ``particle-like'' excitations.

Let us consider a system with translation symmetry. The ground state has a uniform energy density. If we have a state with an excitation, we can observe the energy distribution of the state over the space. If for some local area the energy density is higher than ground state, while for the rest area the energy density is the same as ground state, one may say there is a ``particle-like'' excitation, or a quasiparticle, in this area. Quasiparticles defined like this can be further divided into two types. The first type can be created or annihilated by local operators, such as a spin flip. Hence they are not robust under perturbations. The second type are robust states. The higher local energy density cannot be created or removed by any local operators in that area. We will refer the first type of quasiparticles as local quasiparticles, and the second type of quasiparticles as topological quasiparticles.

As an simple example, consider the 1D Ising model with open boundary condition. There are two ground states, spins all up or all down. Simply flipping one spin of the ground state leads to the second excited state, and creates a local quasiparticle. On the other hand, the first excited state looks like a domain wall. For example the spins on the left are all up while those on the right all down, and the domain wall between the up domain and the down domain is a topological quasiparticle. Flipping the spins next to the domain wall moves the quasiparticle but cannot remove it. Such quasiparticles is protected by the boundary condition. As long as as the two edge spins are opposite, there will be at least one domain wall, or one topological quasiparticle in the bulk. Moreover a spin flip can be viewed as two domain walls.

From the notions of local quasiparticles and topological quasiparticles, we can also introduce a notion topological quasiparticle types (ie topological charges), or simply, quasiparticle types. We say that local quasiparticles are of the trivial type, while topological quasiparticles are of non-trivial types. Also two topological quasiparticles are of the same type if and only if they differ by local quasiparticles. In other words, we can turn one topological quasiparticle into the other one by applying some local operators. The total number of the topological quasiparticle types (including the trivial type) is also a topological property. It turns out that this topological property is directly related to another topological property for 2+1D topological states: The number of the topological quasiparticle types equal to the ground state degeneracy on torus. This is one of many amazing and deep relations in topological order.

The distinction between "ordinary" and topological charges comes from the fact that the conservation of the ordinary charges is a consequence of the Noether's theorem, i.e., when the system under consideration possesses a symmetry, then according to the Noether's theorem, the corresponding charge is conserved.

Topological charges, on the other hand, do not correspond to a symmetry of the given system model, and they stem from a procedure that can be called topological quantization. Please see the seminal work by Orlando Alvarez explaining some aspects of this subject. These topological charges correspond to topological invariants of manifolds related to the physical problem.

One of the most basic examples is the Dirac's quantization condition, which implies the quantization of the magnetic charge in units of the reciprocal of the electric charge. This condition is related to the quantization of the first Chern class of the quantum line bundle. It is also possible to obtain the quantization condition from single-valuedness requirement of the path integral. The existence of the topological invariants is related to a nontrivial topology of the manifold under consideration, for example nonvanishing homotopy groups, please, see the following review by V.P. Nair.

Of course, topological charges can also be non-Abelian a basic example of this phenomenon, is the 't Hooft-Polyakov monopole, where these solutions have non-Abelian charges corresponding to weight vectors of the dual of the unbroken gauge group. Please see the following review by Goddard and Olive.

It should be emphasized that the distinction between ordinary charges and topological charges is model dependent, and "ordinary" charges in some model of a system emerge as topological charges in another model of the same system. For example the electric charge of a particle can be obtained as a topological charge in a Kaluza-Klein description. Please see section 7.6 here in Marsden and Ratiu.

Topological charges correspond sometimes to integer parameters of the model, for example, Witten was able to obtain the quantization of the number of colors from the (semiclassical) topological quantization of the coefficient of the Wess-Zumino term of the Skyrme model.

A simple example, where quantum numbers can be obtained as topological charges is the isotropic harmonic oscillator. If we consider an isotropic Harmonic oscillator in two dimensions then its energy hypersurfaces are $3$-spheres, which can be viewed as circle bundles over a $2$-sphere by the Hopf fibration. The $2$-spheres are the reduced phase spaces of the (energy hypersurfaces) of the two dimensional oscillator. In the quantum theory, the areas of these spheres need to be quantized, in order to admit a quantum line bundle. This quantization condition is equivalent to the quantization of the energy of the harmonic oscillator.

Actually, these alternative representations of physical systems, such that ordinary charges emerge as topological charges offer possible explanations for the quantization of these charges in nature (the Kaluza-Klein model for the electric charge, for example).

A current direction of research along to these lines is to find topological "explanations" to fractional charges. One of the known examples of is the expanation of the fractional hypercharge of the quarks (in units of $frac<1><3>$), which can be explained from the requirement of the anomaly cancelation (which is topological) of the standard model, where the contribution of the quarks must be multiplied by $3$ (due to the three colors). In addition to anomalies, it is known that the existence of fields of different irreducible representations in the same model and separately knotted configurations may give rise to fractional charges.

Topological domain structure of the Escherichia coli chromosome

The circular chromosome of Escherichia coli is organized into independently supercoiled loops, or topological domains. We investigated the organization and size of these domains in vivo and in vitro. Using the expression of >300 supercoiling-sensitive genes to gauge local chromosomal supercoiling, we quantitatively measured the spread of relaxation from double-strand breaks generated in vivo and thereby calculated the distance to the nearest domain boundary. In a complementary approach, we gently isolated chromosomes and examined the lengths of individual supercoiled loops by electron microscopy. The results from these two very different methods agree remarkably well. By comparing our results to Monte Carlo simulations of domain organization models, we conclude that domain barriers are not placed stably at fixed sites on the chromosome but instead are effectively randomly distributed. We find that domains are much smaller than previously reported, ∼10 kb on average. We discuss the implications of these findings and present models for how domain barriers may be generated and displaced during the cell cycle in a stochastic fashion.

Seeking answers in ferroelectric patterning

Thin-film ferroelectric materials display characteristic evolving patterns. Credit: FLEET

Why do some ferroelectric materials display bubble-shaped patterning, while others display complex, labyrinthine patterns?

A FLEET study finds the answer to the changing patterns in ferroelectric films lies in non-equilibrium dynamics, with topological defects driving subsequent evolution.

Ferroelectric materials can be considered an electrical analogy to ferromagnetic materials, with their permanent electric polarization resembling the north and south poles of a magnet.

Understanding the physics behind their domain-pattern changes is crucial for designing advanced low-energy ferroelectric electronics, or brain-inspired neuromorphic computing.

Labyrinthine vs bubbles: what patterns reveal

The characteristic domain patterns of thin-film ferroelectric materials are strongly influenced by the type of materials, and by the film configuration (substrate, electrode, thickness, structure, etc).

"We wanted to understand what drives the emergence of one pattern rather than another," explains Dr. Qi (Peggy) Zhang (UNSW), who led experiments at UNSW.

"For example: what drives formation of mosaic-shaped domain patterning, instead of labyrinth-shaped patterning. And why would drive a subsequent change to bubble-shaped patterning."

The research team were seeking a common framework or roadmap driving such domain arrangements.

Credit: FLEET

"Is there a topological signature in these states? Is their topology evolutive? And if yes, how so? These are the types of answers we were seeking," says lead author Dr. Yousra Nahas (University of Arkansas).

"We found that self-patterning of ferroelectric polar domains can be understood by examining the non-equilibrium dynamics, and that a common framework is that of phase separation kinetics.

"We also performed topological characterization, and studied pattern evolution under an external, applied electric field, which revealed the crucial role of topological defects in mediating the pattern transformation."

"The results of this study build a roadmap (a phase diagram of polar domain patterns) for researchers to use when wanting to 'navigate' through the plurality of modulated phases in low-dimensional ferroelectrics, says co-lead author Dr. Sergei Prokhorenko (University of Arkansas).

This study is thus interesting in its own field (condensed matter physics, ferroelectrics) but might also be relevant for an interdisciplinary audience in regards to the universality of concepts and results.

Co-author Ph.D. student Vivasha Govinden in the material-science labs at UNSW. Credit: UNSW

Researchers investigated domain features and domain evolutions of thin-film Pb(Zr0.4Ti0.6)O3 (or "PZT') through extensive modeling and experimental study (piezoresponse force microscope).

The researchers found that:

  • Electric-field control of skyrmion density elicits hysteretic conductance, which could be harnessed for solid-state neuromorphic computing
  • Engineering topological order in ferroic systems can enhance functional topological-based properties.

"Topology and control of self-assembled domain patterns in low-dimensional ferroelectrics" was published in Nature Communications in November 2020.

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