Isospectrality and heat content

We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schr\"{o}dinger operators acting in $L^2[0,1]$ with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content.Comment: 18 page

Functional Consequences of Protein Dynamics

This thesis explores the possible uses of dynamical fluctuations in protein structure for ligand binding and catalysis. Particular emphasis is placed on the dynamic interpretation of allosteric interactions and a statistical mechanical model of dynamic allostery is presented. This model shows that changes in either low frequency collective motions or random uncorrelated atomic fluctuations of the protein induced by the binding of a ligand can alter the binding properties of other remote ligand binding sites giving rise to allostery. Increases in the frequency of vibrational modes and reductions in uncorrelated motions gives rise to positive cooperativity and is equivalent to a stiffening of the protein structure. Small changes in the dynamics can be treated classically with many changes being required to give observed cooperative free energies. Large shifts in low frequency vibrational modes give much larger contributions to the cooperativity and the use of quantum mechanics is required. The dynamic allostery model predicts that cooperativity arising from dynamic changes is predominantly entropic in origin and complements the more conventional models of allostery which invoke changes in the static conformation of the protein involving domain movement, bond rearrangements and electrostatic effects with consequent effects on the enthalpy. The predictions of this model are tested using laser Raman spectroscopy of solid samples to study low frequency modes in proteins and an allosteric model compound. The small organic molecule which displays positive cooperativity between its two binding sites, shows sizeable shifts to higher frequencies in the low frequency spectrum in agreement with the model. The vibrational shifts seen require only the classical version of the model which when combined with changes in the uncorrelated motions of the atoms in the molecule can account for the observed cooperative free energy. The cooperativity is solely entropic in origin in agreement with published results. High frequency spectra of the molecule in various states of ligation are presented and analysed in terms of localised vibrations of atoms and groups of atoms. The low frequency Raman spectra of lysozyme and its complex with the small inhibitor tri-N-acetyl glucosamine, and of trypsin and its complex with pancreatic trypsin inhibitor all displayed a broad band at 20cm. This band is a superposition of a large number of low frequency modes of the protein and the expected shift in frequency of some modes on inhibitor binding is not visible within such a broad band. The allosteric enzyme glyceraldehyde 3-phosphate dehydrogenase and its complex with the cofactor NAD also shows no changes in its low frequency spectrum. These results and their implications are discussed. High frequency Raman spectra of these enzymes are also presented and analysed

Analysis of the subunit assembly of the type IC restriction-modification enzyme EcoR124I

Type I restriction-modification (R-M) enzymes are composed of three different subunits, of which HsdS determines DNA specificity, HsdM is responsible for DNA methylation and HsdR is required for restriction. The HsdM and HsdS subunits can also form an independent DNA methyltransferase with a subunit stoichiometry of M2S1. We found that the purified Eco R124I R-M enzyme was a mixture of two species as detected by the presence of two differently migrating specific DNA-protein complexes in a gel retardation assay. An analysis of protein subunits isolated from the complexes indicated that the larger species had a stoichiometry of R2M2S1and the smaller species had a stoichiometry of R1M2S1. In vitro analysis of subunit assembly revealed that while binding of the first HsdR subunit to the M2S1complex was very tight, the second HsdR subunit was bound weakly and it dissociated from the R1M2S1complex with an apparent K d of approximately 2.4 x 10(-7) M. Functional assays have shown that only the R2M2S1complex is capable of DNA cleavage, however, the R1M2S1complex retains ATPase activity. The relevance of this situation is discussed in terms of the regulation of restriction activity in vivo upon conjugative transfer of a plasmid-born R-M system into an unmodified host cell

Alleviation of restriction by DNA condensation and non-specific DNA binding ligands

During conditions of cell stress, the type I restriction and modification enzymes of bacteria show reduced, but not zero, levels of restriction of unmethylated foreign DNA. In such conditions, chemically identical unmethylated recognition sequences also occur on the chromosome of the host but restriction alleviation prevents the enzymes from destroying the host DNA. How is this distinction between chemically identical DNA molecules achieved? For some, but not all, type I restriction enzymes, alleviation is partially due to proteolytic degradation of a subunit of the enzyme. We identify that the additional alleviation factor is attributable to the structural difference between foreign DNA entering the cell as a random coil and host DNA, which exists in a condensed nucleoid structure coated with many non-specific ligands. The type I restriction enzyme is able to destroy the ‘naked’ DNA using a complex reaction linked to DNA translocation, but this essential translocation process is inhibited by DNA condensation and the presence of non-specific ligands bound along the DNA

DNA translocation by type III restriction enzymes: a comparison of current models of their operation derived from ensemble and single-molecule measurements

Much insight into the interactions of DNA and enzymes has been obtained using a number of single-molecule techniques. However, recent results generated using two of these techniques—atomic force microscopy (AFM) and magnetic tweezers (MT)—have produced apparently contradictory results when applied to the action of the ATP-dependent type III restriction endonucleases on DNA. The AFM images show extensive looping of the DNA brought about by the existence of multiple DNA binding sites on each enzyme and enzyme dimerisation. The MT experiments show no evidence for looping being a requirement for DNA cleavage, but instead support a diffusive sliding of the enzyme on the DNA until an enzyme–enzyme collision occurs, leading to cleavage. Not only do these two methods appear to disagree, but also the models derived from them have difficulty explaining some ensemble biochemical results on DNA cleavage. In this ‘Survey and Summary’, we describe several different models put forward for the action of type III restriction enzymes and their inadequacies. We also attempt to reconcile the different models and indicate areas for further experimentation to elucidate the mechanism of these enzymes

Discussion of "Geodesic Monte Carlo on Embedded Manifolds"

Contributed discussion and rejoinder to "Geodesic Monte Carlo on Embedded Manifolds" (arXiv:1301.6064)Comment: Discussion of arXiv:1301.6064. To appear in the Scandinavian Journal of Statistics. 18 page

Verrucous carcinoma of the vulva: patterns of care and treatment outcomes.

Background: Verrucous vulvar carcinoma (VC) is an uncommon and distinct histologic subtype of squamous cell carcinoma (SCC). The available literature on VC is currently limited to case reports and small single institution studies. Aims: The goals of this study were to analyze data from the National Cancer Database (NCDB) to quantitate the incidence of VC and to investigate the effects of patient demographics, tumor characteristics, and treatment regimens on overall survival (OS) in women with verrucous vulvar carcinoma. Methods and results: Patients diagnosed with vulvar SCC or VC between the years of 2004 and 2016 were identified in the NCDB. OS was assessed with Kaplan–Meier curves and the log-rank test. Construction of a Cox model compared survival after controlling for confounding variables. The reported incidence of SCC of the vulva has significantly increased since 2004 (p \u3c .0001). In contrast, the incidence of VC has remained stable (p = .344) since 2004. Compared to SCC, VC was significantly more likely to be diagnosed in older women (p \u3c .0001) and treated with surgery alone (p \u3c .0001). However, on propensity score weighted analysis there was a trend toward improved 5-year OS in women with VC compared to those with SCC (63.4% vs. 57.7%, p = .0794). Multivariable Cox survival analysis showed an improvement in OS in VC patients treated with both primary site and regional lymph node surgery compared to primary site surgery alone (adjusted hazard ratio [aHR] 0.67, 95% confidence interval [CI] 0.46–0.97, p = .0357). Conclusion: Verrucous carcinoma is more likely to present in older women. Regional lymph node surgery in addition to primary site surgery significantly improves OS in VC patients

Spectral geometry of the Steklov problem on orbifolds

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In addition, we give two- imensional examples which show that the Steklov spectrum does \emph{not} detect the presence of interior singularities nor does it determine the orbifold Euler characteristic. In fact, a flat disk is Steklov isospectral to a cone. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components