Exact Localisations of Feedback Sets

The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform a given multi digraph $G=(V,E)$ into an acyclic graph by deleting as few arcs (vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one of the classic NP-complete problems. An important contribution of this paper is that the subgraphs $G_{\mathrm{el}}(e)$, $G_{\mathrm{si}}(e)$ of all elementary cycles or simple cycles running through some arc $e \in E$, can be computed in $\mathcal{O}\big(|E|^2\big)$ and $\mathcal{O}(|E|^4)$, respectively. We use this fact and introduce the notion of the essential minor and isolated cycles, which yield a priori problem size reductions and in the special case of so called resolvable graphs an exact solution in $\mathcal{O}(|V||E|^3)$. We show that weighted versions of the FASP and FVSP possess a Bellman decomposition, which yields exact solutions using a dynamic programming technique in times $\mathcal{O}\big(2^{m}|E|^4\log(|V|)\big)$ and $\mathcal{O}\big(2^{n}\Delta(G)^4|V|^4\log(|E|)\big)$, where $m \leq |E|-|V| +1$, $n \leq (\Delta(G)-1)|V|-|E| +1$, respectively. The parameters $m,n$ can be computed in $\mathcal{O}(|E|^3)$, $\mathcal{O}(\Delta(G)^3|V|^3)$, respectively and denote the maximal dimension of the cycle space of all appearing meta graphs, decoding the intersection behavior of the cycles. Consequently, $m,n$ equal zero if all meta graphs are trees. Moreover, we deliver several heuristics and discuss how to control their variation from the optimum. Summarizing, the presented results allow us to suggest a strategy for an implementation of a fast and accurate FASP/FVSP-SOLVER

Quadratic Chabauty for (bi)elliptic curves and Kim's conjecture

We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets $\mathcal{X}(\mathbb{Z}_p)_2$ containing the integral points $\mathcal{X}(\mathbb{Z})$ of an elliptic curve of rank at most $1$. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference $\mathcal{X}(\mathbb{Z}_p)_2\setminus \mathcal{X}(\mathbb{Z})$. We also consider some algorithmic questions arising from Balakrishnan--Dogra's explicit quadratic Chabauty for the rational points of a genus-two bielliptic curve. As an example, we provide a new solution to a problem of Diophantus which was first solved by Wetherell. Computationally, the main difference from the previous approach to quadratic Chabauty is the use of the $p$-adic sigma function in place of a double Coleman integral.Comment: Replaced Conjecture 4.12 with Theorem 1.8; rewrote the introduction and fixed minor issues according to the referee's and PhD examiners' suggestions; 42 page

Spatiotemporal organisation of protein nanoclusters in adhesion complexes

The main goal of this thesis was to contribute to the understanding of the nanoscale lateral organisation of key proteins in adhesion complexes. For this, we exploited single molecule localisation-based super-resolution microscopy STORM to visualise the lateral organisation of five key proteins of the adhesion complex: the integrins, a5ß1 and avß3, and three of their adaptor proteins: paxillin, talin, and vinculin. We first established that these proteins form nanoclusters of around 50nm size that are preserved across all five proteins. Interestingly, these nanoclusters have similar size and number of localisations regardless of their localisation on the membrane, i.e., in the different adhesion structures studied, namely, FA and fAs as well as outside, and were maintained for different cell seeding times, from 90 min to 24 h. These results suggest that nanoclustering constitutes a general mechanism of adhesion protein organisation, creating nanohubs of functional activity. When studying how protein organisation in nanoclusters changes as a function of adhesion time, we revealed a two- and a four-fold increase in the density of a5ß1 and avß3 clusters, respectively, for cells that spread for 24 h as compared to those that spread for 90 min. Further analysis suggests that the increase in density of integrin nanoclusters is due to selective targeting of new integrin nanoclusters to the basal membrane. Following on from this, we then focus on mapping the distribution of these nanoclusters, first by measuring the nearest neighbour distance; (NND) between clusters of the same protein, and second by considering the shortest distance between clusters of different proteins. We found a clear physical segregation of nanoclusters of the same protein around ~55 nm, which is established at early time points after cell seeding for a5ß1 and the adaptors and maintained after 24 h. Interestingly, avß3 nanoclusters exhibited a more random distribution at earlier seeding times and progressively reached similar lateral segregation at 24 h. Concomitant with this lateral segregation, we observed an enriched of all proteins at distances between 100-200 nm. Our observations are in line with the existence of a critical distance spacing between integrins needed for support adhesion and stabilisation of focal adhesions. Furthermore, we found that the relative distribution of nanoclusters of different proteins is predominantly random, with the exception of a5ß1 and paxillin, which organise with a separation of 50 nm. Such an unexpected random distribution between integrins and their adaptors might reflect the dynamic and short-live active state of integrins. Finally, we evaluated and described the mesoscale organisation of nanoclusters inside adhesions. Specifically, we computed the shortest distance between a nanocluster and the edge of the adhesion and studied how the distance to the edge depends on the NND between clusters of different proteins. Remarkably, we found a preference for a5ß1 nanoclusters to be at the edge of the adhesions and in close proximity to its adaptors in a peripheral belt region of the adhesions. Altogether, the results of this thesis demonstrate a clear lateral and hierarchical organisation of integrins and their adaptors inside focal adhesions. Based on our results (together with extensive literature in the field), we propose that one population of a5ß1 nanoclusters and their adaptors preferentially localise close to the edge of adhesion complexes regulating the process of adhesion. A second population of a5ß1 and most of the avß3 nanoclusters organise more randomly at the centre of the adhesions, with dynamic and brief engagement to their adaptors, likely playing a role in mechanotransduction. As a whole, we postulate that the lateral nano- and meso-scale organisation of adhesion proteins is strictly related to and important for the functions of adhesion, mechanosensing and mechanotransduction.El objetivo de esta tesis ha sido contribuir a la comprensión de la organización lateral a nanoescala de proteínas clave en complejos de adhesión. Para ello, usamos la microscopía de superresolución STORM, para visualizar con resolución espacial nanométrica la organización lateral de cinco proteínas del complejo de adhesión: dos integrinas, a5ß1 y avß3, y las proteínas adaptadoras: paxilin, talin y vinculin. En primer lugar, establecimos que estas proteínas forman nanoagregados de ~50 nm tamaño en las cinco proteínas. Curiosamente, su tamaño y número de localizaciones son similares, independientemente de su localización en la membrana, es decir, tanto en FA y fAs, así como fuera de las adhesiones, manteniéndose constantes durante diferentes tiempos de siembra celular. Estos resultados sugieren que la nanoagregación constituye un mecanismo general de organización de proteínas de adhesión, constituyendo nanocentros de actividad funcional. Además, revelamos un aumento en la densidad de los agregados de a5ß1 y avß3 en células extendidas por 24 h en comparación con 90 min, mientras que la densidad de agregados de las proteínas adaptadoras se mantuvo constante. Esta disparidad en densidades indica que solo una fracción de las integrinas interacciona con sus adaptadores, consistente con estados dinámicos de activación-desactivación de las integrinas. También nos enfocamos en mapear la distribución de estos nanoagregados, midiendo la distancia entre agregados más corta entre grupos de la misma proteína, y luego, considerando la distancia más corta entre grupos de diferentes proteínas. Encontramos una clara segregación física de agregados de la misma proteína alrededor de ~55 nm, que se establece temprano después de la siembra celular para a5ß1 y sus adaptadores, y se mantiene hasta 24 h. Curiosamente, los agregados de avß3 exhibieron una distribución más aleatoria en tiempos tempranos de siembra, alcanzando progresivamente una segregación lateral similar a 24 h. Acompañada a esta segregación lateral, observamos un enriquecimiento de todas las proteínas a distancias entre 100¿200 nm. Nuestras observaciones son consistentes con la existencia de un espaciado de distancia crítico entre las integrinas necesarias para apoyar la adhesión y estabilizar las adhesiones focales. Además, encontramos que la distribución relativa de nanoagregados de diferentes proteínas es aleatoria, lo cual podría reflejar el estado activo dinámico y de corta duración de las integrinas, de modo que con nuestras condiciones de imágenes, actualmente no podemos capturar la participación de aquellas integrinas activas dentro de la población total. Finalmente, evaluamos la organización de mesoescala de nanoagregados en FAs, específicamente, en los bordes y el centro. Sorprendentemente, encontramos una preferencia por nanoagregados de a5ß1 en el borde de las FAs y cerca de sus adaptadores, en una región periférica a los bordes. En conjunto, nuestros resultados demuestran una clara organización lateral y jerárquica de las integrinas y sus adaptadores dentro de las adhesiones focales. Proponemos que una población de nanoagregados de a5ß1 y sus adaptadores se localizan preferentemente cerca del borde de los complejos de adhesión para regular el proceso de adhesión y probablemente interaccionando activamente con la maquinaria de la actomiosina. Una segunda población de a5ß1 y la mayoría de los nano-gregados de avß3 se organizan de forma aleatoria en el centro de las FAs con una interacción dinámica y breve con sus adaptadores, posiblemente comprometidos con el proceso de mecanotransducción. En conjunto, y similar a su organización axial, postulamos que la organización lateral a nano- y meso-escala dentro de las FAs es importante para las funciones de adhesión, mecanosensibilidad y mecanotransducción.Postprint (published version

The homological spectrum via definable subcategories

We develop an alternative approach to the homological spectrum through the lens of definable subcategories. This culminates in a proof that the homological spectrum is homeomorphic to a quotient of the Ziegler spectrum. Along the way, we characterise injective objects in homological residue fields in terms of the definable subcategory corresponding to a given homological prime. We use these results to give a purity perspective on the relationship between the homological and Balmer spectrum.Comment: 20 pages; comments welcom

The Value of Seizure Semiology in Epilepsy Surgery: Epileptogenic-Zone Localisation in Presurgical Patients using Machine Learning and Semiology Visualisation Tool

Background Eight million individuals have focal drug resistant epilepsy worldwide. If their epileptogenic focus is identified and resected, they may become seizure-free and experience significant improvements in quality of life. However, seizure-freedom occurs in less than half of surgical resections. Seizure semiology - the signs and symptoms during a seizure - along with brain imaging and electroencephalography (EEG) are amongst the mainstays of seizure localisation. Although there have been advances in algorithmic identification of abnormalities on EEG and imaging, semiological analysis has remained more subjective. The primary objective of this research was to investigate the localising value of clinician-identified semiology, and secondarily to improve personalised prognostication for epilepsy surgery. Methods I data mined retrospective hospital records to link semiology to outcomes. I trained machine learning models to predict temporal lobe epilepsy (TLE) and determine the value of semiology compared to a benchmark of hippocampal sclerosis (HS). Due to the hospital dataset being relatively small, we also collected data from a systematic review of the literature to curate an open-access Semio2Brain database. We built the Semiology-to-Brain Visualisation Tool (SVT) on this database and retrospectively validated SVT in two separate groups of randomly selected patients and individuals with frontal lobe epilepsy. Separately, a systematic review of multimodal prognostic features of epilepsy surgery was undertaken. The concept of a semiological connectome was devised and compared to structural connectivity to investigate probabilistic propagation and semiology generation. Results Although a (non-chronological) list of patients’ semiologies did not improve localisation beyond the initial semiology, the list of semiology added value when combined with an imaging feature. The absolute added value of semiology in a support vector classifier in diagnosing TLE, compared to HS, was 25%. Semiology was however unable to predict postsurgical outcomes. To help future prognostic models, a list of essential multimodal prognostic features for epilepsy surgery were extracted from meta-analyses and a structural causal model proposed. Semio2Brain consists of over 13000 semiological datapoints from 4643 patients across 309 studies and uniquely enabled a Bayesian approach to localisation to mitigate TLE publication bias. SVT performed well in a retrospective validation, matching the best expert clinician’s localisation scores and exceeding them for lateralisation, and showed modest value in localisation in individuals with frontal lobe epilepsy (FLE). There was a significant correlation between the number of connecting fibres between brain regions and the seizure semiologies that can arise from these regions. Conclusions Semiology is valuable in localisation, but multimodal concordance is more valuable and highly prognostic. SVT could be suitable for use in multimodal models to predict the seizure focus