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Extrachromosomal DNA (ecDNA) are frequently observed in human cancers and are responsible for high levels of oncogene expression. In glioblastoma (GBM), ecDNA copy number correlates with poor prognosis. It is hypothesized that their copy number, size, and chromatin accessibility facilitate clustering of ecDNA and colocalization with transcriptional hubs, and that this underpins their elevated transcriptional activity. Here, we use super-resolution imaging and quantitative image analysis to evaluate GBM stem cells harbouring distinct ecDNA species ( EGFR, CDK4, PDGFRA ). We find no evidence that ecDNA routinely cluster with one another or closely interact with transcriptional hubs. Cells with EGFR -containing ecDNA have increased EGFR transcriptional output, but transcription per gene copy is similar in ecDNA compared to the endogenous chromosomal locus. These data suggest that it is the increased copy number of oncogene-harbouring ecDNA that primarily drives high levels of oncogene transcription, rather than specific interactions of ecDNA with each other or with high concentrations of the transcriptional machinery.

Cancers arise through the acquisition of oncogenic mutations and grow by clonal expansion 1 , 2 . Here we reveal that most mutagenic DNA lesions are not resolved into a mutated DNA base pair within a single cell cycle. Instead, DNA lesions segregate, unrepaired, into daughter cells for multiple cell generations, resulting in the chromosome-scale phasing of subsequent mutations. We characterize this process in mutagen-induced mouse liver tumours and show that DNA replication across persisting lesions can produce multiple alternative alleles in successive cell divisions, thereby generating both multiallelic and combinatorial genetic diversity. The phasing of lesions enables accurate measurement of strand-biased repair processes, quantification of oncogenic selection and fine mapping of sister-chromatid-exchange events. Finally, we demonstrate that lesion segregation is a unifying property of exogenous mutagens, including UV light and chemotherapy agents in human cells and tumours, which has profound implications for the evolution and adaptation of cancer genomes. Mutagenic lesions such as those that give rise to cancer frequently segregate—unrepaired—during cell division, resulting in phasing of multiple alleles across generations of daughter cells and consequent tumour heterogeneity.

We prove the crepant resolution conjecture for Donaldson–Thomas invariants of hard Lefschetz 3-Calabi–Yau (CY3) orbifolds, formulated by Bryan–Cadman–Young, interpreting the statement as an equality of rational functions. In order to do so, we show that the generating series of stable pair invariants on any CY3 orbifold is the expansion of a rational function. As a corollary, we deduce a symmetry of this function induced by the derived dualising functor. Our methods also yield a proof of the orbifold DT/PT correspondence for multi-regular curve classes on hard Lefschetz CY3 orbifolds.

Single cells are typically typed by clustering in reduced dimensional transcriptome space. Here we introduce Stator, a novel method, workflow and app that reveals cell types, subtypes and states without relying on local proximity of cells in gene expression space. Rather, Stator derives higher-order gene expression dependencies from a sparse gene-by-cell expression matrix. From these dependencies the method multiply labels the same single cell according to type, sub-type and state (activation, differentiation or cell cycle sub-phase). By applying the method to data from mouse embryonic brain, and human healthy or diseased liver, we show how Stator first recapitulates other methods' cell type labels, and then reveals combinatorial gene expression markers of cell type, state, and disease at higher resolution. By allowing multiple state labels for single cells we reveal cell type fates of embryonic progenitor cells and liver cancer states associated with patient survival.Competing Interest StatementThe authors have declared no competing interest.

Let Y be a smooth complex projective Calabi{Yau threefold. Donaldson-Thomas invariants [Tho00] are integer invariants that virtually enumerate curves on Y. They are organised in a generating series DT(Y) that is interesting from a variety of perspectives. For example, well-known series in mathematics and physics appear in explicit computations. Furthermore, closer to the topic of this thesis, the generating series of birational Calabi-Yau threefolds determine one another [Cal16a]. The crepant resolution conjecture for Donaldson-Thomas invariants [BCY12] conjectures another such comparison result. It relates the Donaldson{Thomas generating series of a certain type of three-dimensional Calabi-Yau orbifold to that of a particular resolution of singularities of its coarse moduli space. The conjectured relation is an equality of generating series. In this thesis, I first provide a counterexample showing that this conjecture cannot hold as an equality of generating series. I then verify that both generating series are the Laurent expansion about different points of the same rational function. This suggests a reinterpretation of the crepant resolution conjecture as an equality of rational functions. Second, following a strategy of Bridgeland [Bri11] and Toda [Tod10a, Tod13, Tod16a], I prove a wall-crossing formula in a motivic Hall algebra relating the Hilbert scheme of curves on the orbifold to that on the resolution. I introduce the notion of pair object associated to a torsion pair, putting ideal sheaves and stable pairs on the same footing, and generalise the wall-crossing formula to this setting, essentially breaking the former in many pieces. Pairs, and their wall-crossing formula, are fundamentally objects of the bounded derived category of the Calabi-Yau orbifold. Finally, I present joint work with J. Calabrese and J. Rennemo [BCR] in which we use the wall-crossing formula and Joyce's integration map to prove the crepant resolution conjecture for Donaldson-Thomas invariants as an equality of rational functions. A crucial ingredient is a result of J. Rennemo that detects when two generating functions related by a wall-crossing are expansions of the same rational function.

Extrachromosomal DNA (ecDNA) are frequently observed in human cancers and are responsible for high levels of oncogene expression. In glioblastoma (GBM), ecDNA copy number correlates with poor prognosis. It is hypothesized that their copy number, size and chromatin accessibility facilitate clustering of ecDNA and colocalization with transcriptional condensates, and that this underpins their elevated transcriptional activity. Here, we use super-resolution imaging and quantitative image analysis to evaluate GBM stem cells harboring distinct ecDNA species (EGFR, MYC, PDGFR). We found no evidence that ecDNA cluster with one another or closely interact with transcriptional condensates. Cells with EGFR-containing ecDNA have increased EGFR transcriptional output, but transcription per gene copy was similar in ecDNA compared to the endogenous chromosomal locus. These data suggest that is the increased copy number of oncogene-harbouring ecDNA that primarily drives high levels of oncogene transcription, rather than specific interactions of ecDNA with the cellular transcriptional machinery. Competing Interest Statement The authors have declared no competing interest.

The problem of inferring pair-wise and higher-order interactions in complex systems involving large numbers of interacting variables, from observational data, is fundamental to many fields. Known to the statistical physics community as the inverse problem, it has become accessible in recent years due to real and simulated 'big' data being generated. Current approaches to the inverse problem rely on parametric assumptions, physical approximations, e.g. mean-field theory, and ignoring higher-order interactions which may lead to biased or incorrect estimates. We bypass these shortcomings using a cross-disciplinary approach and demonstrate that none of these assumptions and approximations are necessary: We introduce a universal, model-independent, and fundamentally unbiased estimator of all-order symmetric interactions, via the non-parametric framework of Targeted Learning, a subfield of mathematical statistics. Due to its universality, our definition is readily applicable to any system at equilibrium with binary and categorical variables, be it magnetic spins, nodes in a neural network, or protein networks in biology. Our approach is targeted, not requiring fitting unnecessary parameters. Instead, it expends all data on estimating interactions, hence substantially increasing accuracy. We demonstrate the generality of our technique both analytically and numerically on (i) the 2-dimensional Ising model, (ii) an Ising-like model with 4-point interactions, (iii) the Restricted Boltzmann Machine, and (iv) simulated individual-level human DNA variants and representative traits. The latter demonstrates the applicability of this approach to discover epistatic interactions causal of disease in population biomedicine.

We show that the Quot scheme \\(\\text{Quot}_{\\mathbf{A}^3}(\\mathcal{O}^r,n)\\) admits a symmetric obstruction theory, and we compute its virtual Euler characteristic. We extend the calculation to locally free sheaves on smooth \\(3\\)-folds, thus refining a special case of a recent Euler characteristic calculation of Gholampour-Kool. We then extend Toda's higher rank DT/PT correspondence on Calabi-Yau \\(3\\)-folds to a local version centered at a fixed slope stable sheaf. This generalises (and refines) the local DT/PT correspondence around the cycle of a Cohen-Macaulay curve. Our approach clarifies the relation between Gholampour-Kool's functional equation for Quot schemes, and Toda's higher rank DT/PT correspondence.

We show that the moduli space \\(\\overline{M}_X(v)\\) of Gieseker stable sheaves on a smooth cubic threefold \\(X\\) with Chern character \\(v = (3,-H,-H^2/2,H^3/6)\\) is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of \\(X\\) maps it birationally onto the theta divisor \\(\\Theta\\), contracting only a copy of \\(X \\subset \\overline{M}_X(v)\\) to the singular point \\(0 \\in \\Theta\\). We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that \\(X\\) can be recovered from its Kuznetsov component \\(\\operatorname{Ku}(X) \\subset \\mathrm{D}^{\\mathrm{b}}(X)\\). Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that \\(X\\) can be recovered from its intermediate Jacobian.

MicroRNAs (miRNAs) are post-transcriptional regulators of gene expression whose contributions to cell biology remain underexplored. Here, we used Cell Painting to quantify the morphological effect of 2,565 human and 1,900 mouse miRNA mimics on 24 million cells across five cell lines. To do so, we developed a novel single-cell morphological profiling analysis framework, involving stringent batch correction, feature selection, and hit calling. With this, we discovered that 9% of human and 15% of mouse miRNA mimics significantly alter cell morphology in at least one cell line. Eighteen miRNAs caused significant changes in multiple cell lines, including eight orthologous miRNAs that altered morphology in both human and mouse cells. Among the replicating miRNAs were human and mouse miR-155-5p, which affected morphology in at least one replicate of four cell lines. As expected, miRNAs with identical seed sequences induced more similar morphological changes than miRNAs with different seeds. Also, morphological changes were associated with cytotoxicity and annotation confidence. This comprehensive single-cell morphological resource will help elucidate human and mouse miRNA cellular function.