On the spectrum of the normalized graph Laplacian

The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this spectrum under local and global operations like motif doubling, graph joining or splitting. The eigenvalue 1 plays a particular role, and we therefore emphasize those constructions that change its multiplicity in a controlled manner, like the iterated duplication of nodes.Comment: 9 pages, no figure

Probabilistic methods in the analysis of protein interaction networks

Imperial Users onl

An analysis of age-related alterations in functional memory networks

The human brain is the most complex organ of the human body and many aspects of its functioning have not yet been understood. One of the most fascinating abilities of the human brain is the skill to store and retrieve information, which is what we refer to as memory. One attempt to get a deeper insight into the functioning of memory is to analyze the complex activity pattern of the human brain that emerges while a memory task is being processed. The understanding of memory is epistemologically very intriguing since it is this ability that enables us to collect, to store and to recall ideas, emotions and thoughts - hence, it builds our own identity. This thesis analyses age-related changes in functional connectivity networks related to episodic and working memory processing. The data for this study were measured using fMRI technique and the sample set consisted of healthy individuals aging from 20 up to over 80 years. Based on the fMRI data we construct correlation networks by correlating pairwisely the measured voxel activity, the nodes of the network being brain voxels, the edges being correlations. These networks are thresholded, anatomically clustered and analyzed by computing statistical network measures, using spectral methods, computing network entropy and calculating persistent homology. The main findings are: elderly individuals exhibit expanded neural networks with less differentiation between episodic and working memory tasks. However, we observe compensatory mechanisms that accompany this dedifferentiation process. Network synchronizability is higher for elderly individuals. Network entropy increases as well with age, yielding a lower network vulnerability for elderly individuals. Aging processes leave traces in the homology pattern of the networks, whereas all brain networks exhibit different persistent homology features

High-quality, high-throughput measurement of protein-DNA binding using HiTS-FLIP

In order to understand in more depth and on a genome wide scale the behavior of transcription factors (TFs), novel quantitative experiments with high-throughput are needed. Recently, HiTS-FLIP (High-Throughput Sequencing-Fluorescent Ligand Interaction Profiling) was invented by the Burge lab at the MIT (Nutiu et al. (2011)). Based on an Illumina GA-IIx machine for next-generation sequencing, HiTS-FLIP allows to measure the affinity of fluorescent labeled proteins to millions of DNA clusters at equilibrium in an unbiased and untargeted way examining the entire sequence space by Determination of dissociation constants (Kds) for all 12-mer DNA motifs. During my PhD I helped to improve the experimental design of this method to allow measuring the protein-DNA binding events at equilibrium omitting any washing step by utilizing the TIRF (Total Internal Reflection Fluorescence) based optics of the GA-IIx. In addition, I developed the first versions of XML based controlling software that automates the measurement procedure. Meeting the needs for processing the vast amount of data produced by each run, I developed a sophisticated, high performance software pipeline that locates DNA clusters, normalizes and extracts the fluorescent signals. Moreover, cluster contained k-mer motifs are ranked and their DNA binding affinities are quantified with high accuracy. My approach of applying phase-correlation to estimate the relative translative Offset between the observed tile images and the template images omits resequencing and thus allows to reuse the flow cell for several HiTS-FLIP experiments, which greatly reduces cost and time. Instead of using information from the sequencing images like Nutiu et al. (2011) for normalizing the cluster intensities which introduces a nucleotide specific bias, I estimate the cluster related normalization factors directly from the protein Images which captures the non-even illumination bias more accurately and leads to an improved correction for each tile image. My analysis of the ranking algorithm by Nutiu et al. (2011) has revealed that it is unable to rank all measured k-mers. Discarding all the clusters related to previously ranked k-mers has the side effect of eliminating any clusters on which k-mers could be ranked that share submotifs with previously ranked k-mers. This shortcoming affects even strong binding k-mers with only one mutation away from the top ranked k-mer. My findings show that omitting the cluster deletion step in the ranking process overcomes this limitation and allows to rank the full spectrum of all possible k-mers. In addition, the performance of the ranking algorithm is drastically reduced by my insight from a quadratic to a linear run time. The experimental improvements combined with the sophisticated processing of the data has led to a very high accuracy of the HiTS-FLIP dissociation constants (Kds) comparable to the Kds measured by the very sensitive HiP-FA assay (Jung et al. (2015)). However, experimentally HiTS-FLIP is a very challenging assay. In total, eight HiTS-FLIP experiments were performed but only one showed saturation, the others exhibited Protein aggregation occurring at the amplified DNA clusters. This biochemical issue could not be remedied. As example TF for studying the details of HiTS-FLIP, GCN4 was chosen which is a dimeric, basic leucine zipper TF and which acts as the master regulator of the amino acid starvation Response in Saccharomyces cerevisiae (Natarajan et al. (2001)). The fluorescent dye was mOrange. The HiTS-FLIP Kds for the TF GCN4 were validated by the HiP-FA assay and a Pearson correlation coefficient of R=0.99 and a relative error of delta=30.91% was achieved. Thus, a unique and comprehensive data set of utmost quantitative precision was obtained that allowed to study the complex binding behavior of GCN4 in a new way. My Downstream analyses reveal that the known 7-mer consensus motif of GCN4, which is TGACTCA, is modulated by its 2-mer neighboring flanking regions spanning an affinity range over two orders of magnitude from a Kd=1.56 nM to Kd=552.51 nM. These results suggest that the common 9-mer PWM (Position Weight Matrix) for GCN4 is insufficient to describe the binding behavior of GCN4. Rather, an additional left and right flanking nucleotide is required to extend the 9-mer to an 11-mer. My analyses regarding mutations and related delta delta G values suggest long-range interdependencies between nucleotides of the two dimeric half-sites of GCN4. Consequently, models assuming positional independence, such as a PWM, are insufficient to explain these interdependencies. Instead, the full spectrum of affinity values for all k-mers of appropriate size should be measured and applied in further analyses as proposed by Nutiu et al. (2011). Another discovery were new binding motifs of GCN4, which can only be detected with a method like HiTS-FLIP that examines the entire sequence space and allows for unbiased, de-novo motif discovery. All These new motifs contain GTGT as a submotif and the data collected suggests that GCN4 binds as monomer to these new motifs. Therefore, it might be even possible to detect different binding modes with HiTS-FLIP. My results emphasize the binding complexity of GCN4 and demonstrate the advantage of HiTS-FLIP for investigating the complexity of regulative processes

Impact of Symmetries in Graph Clustering

Diese Dissertation beschäftigt sich mit der durch die Automorphismusgruppe definierten Symmetrie von Graphen und wie sich diese auf eine Knotenpartition, als Ergebnis von Graphenclustering, auswirkt. Durch eine Analyse von nahezu 1700 Graphen aus verschiedenen Anwendungsbereichen kann gezeigt werden, dass mehr als 70 % dieser Graphen Symmetrien enthalten. Dies bildet einen Gegensatz zum kombinatorischen Beweis, der besagt, dass die Wahrscheinlichkeit eines zufälligen Graphen symmetrisch zu sein bei zunehmender Größe gegen Null geht. Das Ergebnis rechtfertigt damit die Wichtigkeit weiterer Untersuchungen, die auf mögliche Auswirkungen der Symmetrie eingehen. Bei der Analyse werden sowohl sehr kleine Graphen (10 000 000 Knoten/>25 000 000 Kanten) berücksichtigt. Weiterhin wird ein theoretisches Rahmenwerk geschaffen, das zum einen die detaillierte Quantifizierung von Graphensymmetrie erlaubt und zum anderen Stabilität von Knotenpartitionen hinsichtlich dieser Symmetrie formalisiert. Eine Partition der Knotenmenge, die durch die Aufteilung in disjunkte Teilmengen definiert ist, wird dann als stabil angesehen, wenn keine Knoten symmetriebedingt von der einen in die andere Teilmenge abgebildet werden und dadurch die Partition verändert wird. Zudem wird definiert, wie eine mögliche Zerlegbarkeit der Automorphismusgruppe in unabhängige Untergruppen als lokale Symmetrie interpretiert werden kann, die dann nur Auswirkungen auf einen bestimmten Bereich des Graphen hat. Um die Auswirkungen der Symmetrie auf den gesamten Graphen und auf Partitionen zu quantifizieren, wird außerdem eine Entropiedefinition präsentiert, die sich an der Analyse dynamischer Systeme orientiert. Alle Definitionen sind allgemein und können daher für beliebige Graphen angewandt werden. Teilweise ist sogar eine Anwendbarkeit für beliebige Clusteranalysen gegeben, solange deren Ergebnis in einer Partition resultiert und sich eine Symmetrierelation auf den Datenpunkten als Permutationsgruppe angeben lässt. Um nun die tatsächliche Auswirkung von Symmetrie auf Graphenclustering zu untersuchen wird eine zweite Analyse durchgeführt. Diese kommt zum Ergebnis, dass von 629 untersuchten symmetrischen Graphen 72 eine instabile Partition haben. Für die Analyse werden die Definitionen des theoretischen Rahmenwerks verwendet. Es wird außerdem festgestellt, dass die Lokalität der Symmetrie eines Graphen maßgeblich beeinflusst, ob dessen Partition stabil ist oder nicht. Eine hohe Lokalität resultiert meist in einer stabilen Partition und eine stabile Partition impliziert meist eine hohe Lokalität. Bevor die obigen Ergebnisse beschrieben und definiert werden, wird eine umfassende Einführung in die verschiedenen benötigten Grundlagen gegeben. Diese umfasst die formalen Definitionen von Graphen und statistischen Graphmodellen, Partitionen, endlichen Permutationsgruppen, Graphenclustering und Algorithmen dafür, sowie von Entropie. Ein separates Kapitel widmet sich ausführlich der Graphensymmetrie, die durch eine endliche Permutationsgruppe, der Automorphismusgruppe, beschrieben wird. Außerdem werden Algorithmen vorgestellt, die die Symmetrie von Graphen ermitteln können und, teilweise, auch das damit eng verwandte Graphisomorphie Problem lösen. Am Beispiel von Graphenclustering gibt die Dissertation damit Einblicke in mögliche Auswirkungen von Symmetrie in der Datenanalyse, die so in der Literatur bisher wenig bis keine Beachtung fanden