Profiling Chromosome Topological Features by Super-Resolution 3D Structured Illumination Microscopy.

Three-dimensional structured illumination microscopy (3D-SIM) and fluorescence in situ hybridization on three-dimensional preserved cells (3D-FISH) have proven to be robust and efficient methodologies for analyzing nuclear architecture and profiling the genome's topological features. These methods have allowed the simultaneous visualization and evaluation of several target structures at super-resolution. In this chapter, we focus on the application of 3D-SIM for the visualization of 3D-FISH preparations of chromosomes in interphase, known as Chromosome Territories (CTs). We provide a workflow and detailed guidelines for sample preparation, image acquisition, and image analysis to obtain quantitative measurements for profiling chromosome topological features. In parallel, we address a practical example of these protocols in the profiling of CTs 9 and 22 involved in the translocation t(9;22) in Chronic Myeloid Leukemia (CML). The profiling of chromosome topological features described in this chapter allowed us to characterize a large-scale topological disruption of CTs 9 and 22 that correlates directly with patients' response to treatment and as a possible potential change in the inheritance systems. These findings open new insights into how the genome structure is associated with the response to cancer treatments, highlighting the importance of microscopy in analyzing the topological features of the genome.

Genome folding principles uncovered in condensin-depleted mitotic chromosomes.

During mitosis, condensin activity is thought to interfere with interphase chromatin structures. To investigate genome folding principles in the absence of chromatin loop extrusion, we codepleted condensin I and condensin II, which triggered mitotic chromosome compartmentalization in ways similar to that in interphase. However, two distinct euchromatic compartments, indistinguishable in interphase, emerged upon condensin loss with different interaction preferences and dependencies on H3K27ac. Constitutive heterochromatin gradually self-aggregated and cocompartmentalized with facultative heterochromatin, contrasting with their separation during interphase. Notably, some cis-regulatory element contacts became apparent even in the absence of CTCF/cohesin-mediated structures. Heterochromatin protein 1 (HP1) proteins, which are thought to partition constitutive heterochromatin, were absent from mitotic chromosomes, suggesting, surprisingly, that constitutive heterochromatin can self-aggregate without HP1. Indeed, in cells traversing from M to G1 phase in the combined absence of HP1α, HP1ß and HP1γ, constitutive heterochromatin compartments are normally re-established. In sum, condensin-deficient mitotic chromosomes illuminate forces of genome compartmentalization not identified in interphase cells.

Transcription-induced active forces suppress chromatin motion.

The organization of interphase chromosomes in a number of species is starting to emerge thanks to advances in a variety of experimental techniques. However, much less is known about the dynamics, especially in the functional states of chromatin. Some experiments have shown that the motility of individual loci in human interphase chromosome decreases during transcription and increases upon inhibiting transcription. This is a counterintuitive finding because it is thought that the active mechanical force (F) on the order of ten piconewtons, generated by RNA polymerase II (RNAPII) that is presumably transmitted to the gene-rich region of the chromatin, would render it more open, thus enhancing the mobility. We developed a minimal active copolymer model for interphase chromosomes to investigate how F affects the dynamical properties of chromatin. The movements of the loci in the gene-rich region are suppressed in an intermediate range of F and are enhanced at small F values, which has also been observed in experiments. In the intermediate F, the bond length between consecutive loci increases, becoming commensurate with the distance at the minimum of the attractive interaction between nonbonded loci. This results in a transient disorder-to-order transition, leading to a decreased mobility during transcription. Strikingly, the F-dependent change in the locus dynamics preserves the organization of the chromosome at [Formula: see text]. Transient ordering of the loci, which is not found in the polymers with random epigenetic profiles, in the gene-rich region might be a plausible mechanism for nucleating a dynamic network involving transcription factors, RNAPII, and chromatin.

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.

Fluorescence in situ hybridisation for interphase chromosomal aberration-based biological dosimetry.

Metaphase spreads stained with Giemsa or painted with chromosome-specific probes by fluorescence in situ hybridisation (FISH) have been in use since long for retrospective dose assessment (biological dosimetry). However, in cases of accidental exposure to ionising radiation, the culturing of lymphocytes to obtain metaphase chromosomes and analysis of chromosomal aberrations is time-consuming and problematic after high radiation doses. Similarly, analysing chromosomal damage in G0/G1 cells or nondividing cells by premature chromosome condensation is laborious. Following large-scale radiological emergencies, the time required for analysis is more important than precision of dose estimate. Painting of whole chromosomes using chromosome-specific probes in interphase nuclei by the FISH technique will eliminate the time required for cell culture and allow a fast dose estimate, provided that a meaningful dose-response can be obtained by scoring the number of chromosomal domains visible in interphase nuclei. In order to test the applicability of interphase FISH for quick biological dosimetry, whole blood from a healthy donor was irradiated with 8 Gy of gamma radiation. Irradiated whole blood was kept for 2 h at 37°C to allow DNA repair and thereafter processed for FISH with probes specific for Chromosomes-1 and 2. Damaged chromosomal fragments, distinguished by extra color domains, were observed in interphase nuclei of lymphocytes irradiated with 8 Gy. These fragments were efficiently detected and quantified by the FISH technique utilising both confocal and single plane fluorescence microscopy. Furthermore, a clear dose-response curve for interphase fragments was achieved following exposure to 0, 1, 2, 4 and 8 Gy of gamma radiation. These results demonstrate interphase FISH as a promising test for biodosimetry and for studying cytogenetic effects of radiation in nondividing cells.

Analysis of human invasive cytotrophoblasts demonstrates mosaic aneuploidy.

A total of 24 chromosome-specific fluorescence in situ hybridization probes for interphase nucleus analysis were developed to determine the chromosomal content of individual human invasive cytotrophoblasts derived from in vitro cultured assays. At least 75% of invasive cytotrophoblasts were hyperdiploid and the total number of chromosomes ranged from 47 to 61. The results also demonstrated that these hyperdiploid invasive cytotrophoblasts showed significant heterogeneity. The most copy number gains were observed for chromosomes 13, 14, 15, 19, 21, and 22 with average copy number greater than 2.3. A parallel study using primary invasive cytotrophoblasts also showed a similar trend of copy number changes. Conclusively, 24-chromosome analysis of human non-proliferating cytotrophoblasts (interphase nuclei) was achieved. Hyperdiploidy and chromosomal heterogeneity without endoduplication in invasive cytotrophoblasts may suggest a selective advantage for invasion and short lifespan during normal placental development.

Mesoscale, long-time mixing of chromosomes and its connection to polymer dynamics.

Chromosomes are arranged in distinct territories within the nucleus of animal cells. Recent experiments have shown that these territories overlap at their edges, suggesting partial mixing during interphase. Experiments that knock-down of condensin II proteins during interphase indicate increased chromosome mixing, which demonstrates control of the mixing. In this study, we use a generic polymer simulation to quantify the dynamics of chromosome mixing over time. We introduce the chromosome mixing index, which quantifies the mixing of distinct chromosomes in the nucleus. We find that the chromosome mixing index in a small confinement volume (as a model of the nucleus), increases as a power-law of the time, with the scaling exponent varying non-monotonically with self-interaction and volume fraction. By comparing the chromosome mixing index with both monomer subdiffusion due to (non-topological) intermingling of chromosomes as well as even slower reptation, we show that for relatively large volume fractions, the scaling exponent of the chromosome mixing index is related to Rouse dynamics for relatively weak chromosome attractions and to reptation for strong attractions. In addition, we extend our model to more realistically account for the situation of the Drosophila chromosome by including the heterogeneity of the polymers and their lengths to account for microphase separation of euchromatin and heterochromatin and their interactions with the nuclear lamina. We find that the interaction with the lamina further impedes chromosome mixing.

Regulation of loop extrusion on the interphase genome.

In the human cell nucleus, dynamically organized chromatin is the substrate for gene regulation, DNA replication, and repair. A central mechanism of DNA loop formation is an ATPase motor cohesin-mediated loop extrusion. The cohesin complexes load and unload onto the chromosome under the control of other regulators that physically interact and affect motor activity. Regulation of the dynamic loading cycle of cohesin influences not only the chromatin structure but also genome-associated human disorders and aging. This review focuses on the recently spotlighted genome organizing factors and the mechanism by which their dynamic interactions shape the genome architecture in interphase.

A maximum-entropy model to predict 3D structural ensembles of chromatin from pairwise distances with applications to interphase chromosomes and structural variants.

The principles that govern the organization of genomes, which are needed for an understanding of how chromosomes are packaged and function in eukaryotic cells, could be deciphered if the three-dimensional (3D) structures are known. Recently, single-cell imaging techniques have been developed to determine the 3D coordinates of genomic loci in vivo. Here, we introduce a computational method (Distance Matrix to Ensemble of Structures, DIMES), based on the maximum entropy principle, with experimental pairwise distances between loci as constraints, to generate a unique ensemble of 3D chromatin structures. Using the ensemble of structures, we quantitatively account for the distribution of pairwise distances, three-body co-localization, and higher-order interactions. The DIMES method can be applied to both small and chromosome-scale imaging data to quantify the extent of heterogeneity and fluctuations in the shapes across various length scales. We develop a perturbation method in conjunction with DIMES to predict the changes in 3D structures from structural variations. Our method also reveals quantitative differences between the 3D structures inferred from Hi-C and those measured in imaging experiments. Finally, the physical interpretation of the parameters extracted from DIMES provides insights into the origin of phase separation between euchromatin and heterochromatin domains.

Mitotic chromosome condensation resets chromatin to safeguard transcriptional homeostasis during interphase.

Mitotic entry correlates with the condensation of the chromosomes, changes in histone modifications, exclusion of transcription factors from DNA, and the broad downregulation of transcription. However, whether mitotic condensation influences transcription in the subsequent interphase is unknown. Here, we show that preventing one chromosome to condense during mitosis causes it to fail resetting of transcription. Rather, in the following interphase, the affected chromosome contains unusually high levels of the transcription machinery, resulting in abnormally high expression levels of genes in cis, including various transcription factors. This subsequently causes the activation of inducible transcriptional programs in trans, such as the GAL genes, even in the absence of the relevant stimuli. Thus, mitotic chromosome condensation exerts stringent control on interphase gene expression to ensure the maintenance of basic cellular functions and cell identity across cell divisions. Together, our study identifies the maintenance of transcriptional homeostasis during interphase as an unexpected function of mitosis and mitotic chromosome condensation.

Fluorescence In Situ Hybridization of DNA Probes on Mitotic Chromosomes of the Mexican Axolotl.

Fluorescence in situ hybridization (FISH) is used extensively for visual localization of specific DNA fragments (and RNA fragments) in broad applications on chromosomes or nuclei at any stage of the cell cycle: metaphase, anaphase, or interphase. The cytogenetic slides that serve as a target for the labeled DNA probe might be prepared using any approach suitable for obtaining cells with appropriate morphology for imaging and analysis. In this chapter, we focus on the application of molecular cytogenetic methods such as DNA labeling, slide preparation, and in situ hybridization related to cells from Mexican axolotl.

Visualizing Active Replication Regions in S-Phase Chromosomes.

A basic question of cell biology is how DNA folds to chromosome. A number of recently accumulated evidences have suggested that folding of chromosome proceeds tightly coupled with DNA replication progresses. Drug-induced PCC is a useful tool for visualization of the interphase nuclei, in particular, S-phase, as S-phase prematurely condensed chromosomes (S-phase PCC). Active replicating DNA is labeled directly with Cy3-dUTP by bead loading method, and then S-phase nuclei is immediately condensed prematurely by calyculin A to obtain S-phase PCC. Active replicating regions on S-PCC are observed under a scanning confocal microscope. Cy3-dUTP-labeled S-phase PCCs clearly reveal the drastic transitional change of chromosome formation through S-phase, starting from a "cloudy nebula" to numerous numbers of "beads on a string" and finally to "striped arrays of banding structured chromosome" known as G- or R-banding pattern. The number, distribution, and shape of replication foci were also measured in individual subphase of S-phase; maximally ~1400 foci of 0.35 µm average radius size were scored at the beginning of S-phase, and the number is reduced to ~100 at the end of S-phase. Drug-induced PCC clearly provided the new insight that eukaryote DNA replication is tightly coupled with the chromosome condensation/compaction for construction of eukaryote higher-ordered chromosome structure.

Cytomolecular Organisation of the Nuclear Genome.

Modern molecular cytogenetics allows many aspects of the nuclear genome structure, function, and evolution to be analysed within the topographic context of mitotic and meiotic chromosomes and interphase nuclei [...].

Simulation of Different Three-Dimensional Models of Whole Interphase Nuclei Compared to Experiments - A Consistent Scale-Bridging Simulation Framework for Genome Organization.

The three-dimensional architecture of chromosomes, their arrangement, and dynamics within cell nuclei are still subject of debate. Obviously, the function of genomes-the storage, replication, and transcription of genetic information-has closely coevolved with this architecture and its dynamics, and hence are closely connected. In this work a scale-bridging framework investigates how of the 30 nm chromatin fibre organizes into chromosomes including their arrangement and morphology in the simulation of whole nuclei. Therefore, mainly two different topologies were simulated with corresponding parameter variations and comparing them to experiments: The Multi-Loop-Subcompartment (MLS) model, in which (stable) small loops form (stable) rosettes, connected by chromatin linkers, and the Random-Walk/Giant-Loop (RW/GL) model, in which large loops are attached to a flexible non-protein backbone, were simulated for various loop and linker sizes. The 30 nm chromatin fibre was modelled as a polymer chain with stretching, bending and excluded volume interactions. A spherical boundary potential simulated the confinement to nuclei with different radii. Simulated annealing and Brownian Dynamics methods were applied in a four-step decondensation procedure to generate from metaphase decondensated interphase configurations at thermodynamical equilibrium. Both the MLS and the RW/GL models form chromosome territories, with different morphologies: The MLS rosettes result in distinct subchromosomal domains visible in electron and confocal laser scanning microscopic images. In contrast, the big RW/GL loops lead to a mostly homogeneous chromatin distribution. Even small changes of the model parameters induced significant rearrangements of the chromatin morphology. The low overlap of chromosomes, arms, and subchromosomal domains observed in experiments agrees only with the MLS model. The chromatin density distribution in CLSM image stacks reveals a bimodal behaviour in agreement with recent experiments. Combination of these results with a variety of (spatial distance) measurements favour an MLS like model with loops and linkers of 63 to 126 kbp. The predicted large spaces between the chromatin fibres allow typically sized biological molecules to reach nearly every location in the nucleus by moderately obstructed diffusion and is in disagreement with the much simplified assumption that defined channels between territories for molecular transport as in the Interchromosomal Domain (ICD) hypothesis exist and are necessary for transport. All this is also in agreement with recent selective high-resolution chromosome interaction capture (T2C) experiments, the scaling behaviour of the DNA sequence, the dynamics of the chromatin fibre, the diffusion of molecules, and other measurements. Also all other chromosome topologies can in principle be excluded. In summary, polymer simulations of whole nuclei compared to experimental data not only clearly favour only a stable loop aggregate/rosette like genome architecture whose local topology is tightly connected to the global morphology and dynamics of the cell nucleus and hence can be used for understanding genome organization also in respect to diagnosis and treatment. This is in agreement with and also leads to a general novel framework of genome emergence, function, and evolution.

SPTBN2 Promotes the Progression of Thyroid Cancer by Accelerating G1/S Transition and Inhibiting Apoptosis.

Background: Thyroid carcinoma (TC) is an increasingly common malignancy of endocrine organs, and its most frequently encountered histotype is papillary thyroid cancer (PTC). Identifying new potential gene alterations is important for completely elucidating the mechanism of PTC initiation and progression. Thus, we performed whole transcriptome sequence analysis (RNA-seq) on 79 PTC tissue samples and paired adjacent nontumor tissue samples to study the molecular mechanism of TC tumorigenesis and progression further. The results of RNA-seq analysis showed that spectrin beta, nonerythrocytic 2 (SPTBN2), was markedly overexpressed in PTC tissues relative to that in the paired nontumor tissues. Additionally, the analysis results for 502 PTC samples and 58 nontumor thyroid samples from The Cancer Genome Atlas dataset were consistent with our RNA-seq results. However, the molecular mechanisms and function of SPTBN2 in TC progression remain unknown. Methods: We examined SPTBN2 gene expression in 48 papillary thyroid tumor tissues and paired adjacent normal thyroid tissues by using qRT-PCR. SPTBN2 expression in the TC cell lines was silenced by small interfering RNA. Then, the transfected TC cells were used to investigate the in vitro function of SPTBN2. Result: The expression of SPTBN2 was significantly upregulated in our RNA-seq cohort, our local validated cohort, and TCGA RNA-seq cohort. The results of the in vitro experiment revealed that in TC cell lines, SPTBN2 downregulation considerably suppressed tumor cell proliferation, the cell cycle, migration, colony formation, and invasion and induced cell apoptosis. Furthermore, the protein levels of CCNE2, CDK2, CDK4, and Bcl-2 were downregulated, and those of P21, Bax, cleaved caspase-8, and cleaved caspase-3 had increased in transfected TC cells relative to in control TC cells. Conclusion: The downregulation of SPTBN2 caused apoptosis and retarded G1/S cell cycle transition in TC cells. Thus, SPTBN2 may be a good candidate gene for TC diagnosis and therapy.

A proposed unified interphase nucleus chromosome structure: Preliminary preponderance of evidence.

Cryoelectron tomography of the cell nucleus using scanning transmission electron microscopy and deconvolution processing technology has highlighted a large-scale, 100- to 300-nm interphase chromosome structure, which is present throughout the nucleus. This study further documents and analyzes these chromosome structures. The paper is divided into four parts: 1) evidence (preliminary) for a unified interphase chromosome structure; 2) a proposed unified interphase chromosome architecture; 3) organization as chromosome territories (e.g., fitting the 46 human chromosomes into a 10-µm-diameter nucleus); and 4) structure unification into a polytene chromosome architecture and lampbrush chromosomes. Finally, the paper concludes with a living light microscopy cell study showing that the G1 nucleus contains very similar structures throughout. The main finding is that this chromosome structure appears to coil the 11-nm nucleosome fiber into a defined hollow structure, analogous to a Slinky helical spring [https://en.wikipedia.org/wiki/Slinky; motif used in Bowerman et al., eLife 10, e65587 (2021)]. This Slinky architecture can be used to build chromosome territories, extended to the polytene chromosome structure, as well as to the structure of lampbrush chromosomes.

Application of Interphase Fluorescent in Situ Hybridization: a Screening Tool for the Diagnosis of Microdeletion Syndrome.

BACKGROUND: Most laboratories adopt the results of metaphase fluorescent in situ hybridization (FISH) for the diagnosis of microdeletion syndromes. To investigate the discrepancy between the results of interphase and metaphase, we compared the quantitative results of FISH for 5 kinds of microdeletion syndrome and gender determination disorders (SDD). METHODS: A total of 282 (135 for DiGeorge syndrome, 20 for Kalmann syndrome, 7 for Miller-Dieker syndrome, 38 for Prader Willi/Angelman syndrome, 62 for Williams syndrome, and 20 for SDD (SRY FISH)) were enrolled. For SRY FISH, we artificially mixed fresh blood of male and female with various ratios and then compared the results of metaphase and interphase SRY FISH. Using a bio-cell chip, we performed interphase FISH in 168 patients with microdeletion syndromes and compared the results with manual interphase. RESULTS: The concordance rate between the results of metaphase and interphase was 100% in microdeletion syndrome. In the disorders of gender development, SRY FISH showed 100% concordance between interphase and metaphase when we counted 50 metaphase cells and 100 interphase cells. Comparison with mixtures of male and female blood at various ratios also showed 100% concordance. The results of bio-cell chip showed 100% concordance between previous interphase FISH results. CONCLUSIONS: Considering the complete concordance between interphase and metaphase in microdeletion syndrome, the application of interphase FISH without performing metaphase FISH can be a screening test for microdeletion syndrome. Confirmation by metaphase FISH can be performed only in cases with abnormal results by interphase FISH.

Cell cycle dynamics and developmental dynamics of the 3D genome: toward linking the two timescales.

In the mammalian cell nucleus, chromosomes are folded differently in interphase and mitosis. Interphase chromosomes are relatively decondensed and display at least two unique layers of higher-order organization: topologically associating domains (TADs) and cell-type-specific A/B compartments, which correlate well with early/late DNA replication timing (RT). In mitosis, these structures rapidly disappear but are gradually reconstructed during G1 phase, coincident with the establishment of the RT program. However, these structures also change dynamically during cell differentiation and reprogramming, and yet we are surprisingly ignorant about the relationship between their cell cycle dynamics and developmental dynamics. In this review, we summarize the recent findings on this topic, discuss how these two processes might be coordinated with each other and its potential significance.

The success rate of interphase fluorescence in situ hybridization in plasma cell disorders can be improved using unconventional sources of plasma cells.

BACKGROUND: Immunomagnetic cell sorting (IMCS) is a preferred technique for the enrichment of plasma cells (PC) before fluorescence in situ hybridization (FISH). Here, we share our real-world experience regarding the success rate of IMCS, its limitations, and the utility of alternate sources to obtain a successful FISH in various PC disorders. MATERIALS AND METHODS: A retrospective analysis was performed in patients with a PC neoplasm, who underwent bone marrow (BM) examination, and FISH testing over 30 months. In all cases with an unsuccessful IMCS, an attempt was made to identify the cause of failure. RESULTS: Immunomagnetic cell sorting of PCs was successful in 395/450 cases (87.8%; 77/98 cases (78.6%) with <10% PCs and 318/352 (90.3%) with ≥10% PCs in BM aspirate; P = .003). Among cases with unsuccessful IMCS (<10% PCs; n = 21 and ≥10% PCs; n = 34), an alternate source could be used successfully in 34 (62%) patients and includes air-dried trephine biopsy imprint smears (n = 28) with aggregates or sheets of PCs, fine-needle aspiration smears/biopsy from plasmacytoma (n = 5), and ascitic fluid (n = 1). 284/395 (71.9%) patients with successful IMCS and all 34 cases with an alternate source of PCs showed at least one cytogenetic abnormality on four-probe FISH. CONCLUSION: Variations in the sample quality together with significant variation in the number of PCs between BM aspirate and the trephine biopsy imprint smears/biopsy reduce the success rate of IMCS in a real-world scenario and necessitate utilization of patient-specific alternate sources of PCs like a trephine biopsy imprint or cytology smears from extramedullary sources for successful FISH testing in PC neoplasms.

MCPH1 inhibits Condensin II during interphase by regulating its SMC2-Kleisin interface.

Dramatic change in chromosomal DNA morphology between interphase and mitosis is a defining features of the eukaryotic cell cycle. Two types of enzymes, namely cohesin and condensin confer the topology of chromosomal DNA by extruding DNA loops. While condensin normally configures chromosomes exclusively during mitosis, cohesin does so during interphase. The processivity of cohesin's loop extrusion during interphase is limited by a regulatory factor called WAPL, which induces cohesin to dissociate from chromosomes via a mechanism that requires dissociation of its kleisin from the neck of SMC3. We show here that a related mechanism may be responsible for blocking condensin II from acting during interphase. Cells derived from patients affected by microcephaly caused by mutations in the MCPH1 gene undergo premature chromosome condensation. We show that deletion of Mcph1 in mouse embryonic stem cells unleashes an activity of condensin II that triggers formation of compact chromosomes in G1 and G2 phases, accompanied by enhanced mixing of A and B chromatin compartments, and this occurs even in the absence of CDK1 activity. Crucially, inhibition of condensin II by MCPH1 depends on the binding of a short linear motif within MCPH1 to condensin II's NCAPG2 subunit. MCPH1's ability to block condensin II's association with chromatin is abrogated by the fusion of SMC2 with NCAPH2, hence may work by a mechanism similar to cohesin. Remarkably, in the absence of both WAPL and MCPH1, cohesin and condensin II transform chromosomal DNAs of G2 cells into chromosomes with a solenoidal axis.