Exploring deep phylogenies using protein structure : a dissertation submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Biochemistry, Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand

Recent times have seen an exponential growth in protein sequence and structure data. The most popular way of characterising newly determined protein sequences is to compare them to well characterised sequences and predict the function of novel sequences based on homology. This practice has been highly successful for a majority of proteins. However, these sequence based methods struggle with certain deeply diverging proteins and hence cannot always recover evolutionary histories. Another feature of proteins, namely their structures, has been shown to retain evolutionary signals over longer time scales compared to the respective sequences that encode them. The structure therefore presents an opportunity to uncover the evolutionary signal that otherwise escapes conventional sequence-based methods. Structural phylogenetics refers to the comparison of protein structures to extract evolutionary relationships. The area of structural phylogenetics has been around for a number of years and multiple approaches exist to delineate evolutionary relationships from protein structures. However, once the relationships have been recovered from protein structural data, no methods exist, at present, to verify the robustness of these relationships. Because of the nature of the structural data, conventional sequence-based methods, e.g. bootstrapping, cannot be applied. This work introduces the first ever use of a molecular dynamics (MD)-based bootstrap method, which can add a measure of significance to the relationships inferred from the structure-based analysis. This work begins in Chapter 2 by thoroughly investigating the use of a protein structural comparison metric Qscore, which has previously been used to generate structural phylogenies, and highlights its strengths and weaknesses. The mechanistic exploration of the structural comparison metric reveals a size difference limit of no more than 5-10% in the sizes of protein structures being compared for accurate phylogenetic inference to be made. Chapter 2 also explores the MD-based bootstrap method to offer an interpretation of the significance values recovered. Two protein structural datasets, one relatively more conserved at the sequence level than the other and with different levels of structural conservation are used as controls to simplify the interpretation of the statistics recovered from the MD-based bootstrap method. Chapter 3 then sees the application of the Qscore metric to the aminoacyl-tRNA synthetases. The aminoacyl-tRNA synthetases are believed to have been present at the dawn of life, making them one of the most ancient protein families. Due to the important functional role they play, these proteins are conserved at both sequence and structural levels and well-characterised using both sequence and structure-based comparative methods. This family therefore offered inferences which could be informed with structural analysis using an automated method. Successful recovery of known relationships raised confidence in the ability of structural phylogenetic analysis based on Qscore to detect evolutionary signals. In Chapter 4, a structural phylogeny was created for a protein structural dataset presenting either the histone fold or its ancestral precursor. This structural dataset comprised of proteins that were significantly diverged at a sequence level, however shared a common structural motif. The structural phylogeny recovered the split between bacterial and non-bacterial proteins. Furthermore, TATA protein associated factors were found to have multiple points of origin. Moreover, some mismatch was found between the classifications of these proteins between SCOP and PFam, which also did not agree with the results from this work. Using the structural phylogeny a model outlining the evolution of these proteins was proposed. The structural phylogeny of the Ferritin-like superfamily has previously been generated using the Qscore metric and supported qualitatively. Chapter 5 recovers the structural phylogeny of the Ferritin-like superfamily and finds quantitative support for the inferred relationships from the first ever implementation of the MD-based bootstrap method. The use of the MD-based bootstrap method simultaneously allows for the resolution of polytomies in structural databases. Some limitations of the MD-based bootstrap method, highlighted in Chapter 2, are revisited in Chapter 5. This work indicates that evolutionary signals can be successfully extracted from protein structures for deeply diverging proteins and that the MD-based bootstrap method can be used to gauge the robustness of relationships inferred

Abelian 2-form gauge theory: superfield formalism

We derive the off-shell nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for {\it all} the fields of a free Abelian 2-form gauge theory by exploiting the geometrical superfield approach to BRST formalism. The above four (3 + 1)-dimensional (4D) theory is considered on a (4, 2)-dimensional supermanifold parameterized by the four even spacetime variables x^\mu (with \mu = 0, 1, 2, 3) and a pair of odd Grassmannian variables \theta and \bar\theta (with \theta^2 = \bar\theta^2 = 0, \theta \bar\theta + \bar\theta \theta = 0). One of the salient features of our present investigation is that the above nilpotent (anti-)BRST symmetry transformations turn out to be absolutely anticommuting due to the presence of a Curci-Ferrari (CF) type of restriction. The latter condition emerges due to the application of our present superfield formalism. The actual CF condition, as is well-known, is the hallmark of a 4D non-Abelian 1-form gauge theory. We demonstrate that our present 4D Abelian 2-form gauge theory imbibes some of the key signatures of the 4D non-Abelian 1-form gauge theory. We briefly comment on the generalization of our supperfield approach to the case of Abelian 3-form gauge theory in four (3 + 1)-dimensions of spacetime.Comment: LaTeX file, 23 pages, journal versio

An Alternative To The Horizontality Condition In Superfield Approach To BRST Symmetries

We provide an alternative to the gauge covariant horizontality condition which is responsible for the derivation of the nilpotent (anti-)BRST symmetry transformations for the gauge and (anti-)ghost fields of a (3 + 1)-dimensional (4D) interacting 1-form non-Abelian gauge theory in the framework of the usual superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism. The above covariant horizontality condition is replaced by a gauge invariant restriction on the (4, 2)-dimensional supermanifold, parameterized by a set of four spacetime coordinates x^\mu (\mu = 0, 1, 2, 3) and a pair of Grassmannian variables \theta and \bar\theta. The latter condition enables us to derive the nilpotent (anti-)BRST symmetry transformations for all the fields of an interacting 4D 1-form non-Abelian gauge theory where there is an explicit coupling between the gauge field and the Dirac fields. The key differences and striking similarities between the above two conditions are pointed out clearly.Comment: LaTeX file, 20 pages, journal versio

Writing an Escalation Contract Using the Consumer Price Index

[Excerpt] Each year thousands of people write contracts with escalation clauses that are tied to the Consumer Price Index (CPI). Escalation contracts call for an increase in some type of payment in the event of an increase in prices. These contracts are used in a wide variety of ways, from adjusting rent prices to adding cost-of-living adjustments to alimony payments and wage contracts. Unfortunately, many escalation contracts tied to the CPI are vague. For example, a contract may stipulate that “the Consumer Price Index (CPI) be used to escalate an apartment rent, but the Bureau of Labor Statistics (BLS) publishes thousands of CPIs each month, so a more carefully worded contract could minimize ambiguity and the likelihood of future disputes. This issue of BEYOND THE NUMBERS can help those who use the CPI to write escalation clauses to create a more comprehensive contract

Geometrical Aspects Of BRST Cohomology In Augmented Superfield Formalism

In the framework of augmented superfield approach, we provide the geometrical origin and interpretation for the nilpotent (anti-)BRST charges, (anti-)co-BRST charges and a non-nilpotent bosonic charge. Together, these local and conserved charges turn out to be responsible for a clear and cogent definition of the Hodge decomposition theorem in the quantum Hilbert space of states. The above charges owe their origin to the de Rham cohomological operators of differential geometry which are found to be at the heart of some of the key concepts associated with the interacting gauge theories. For our present review, we choose the two $(1 + 1)$-dimensional (2D) quantum electrodynamics (QED) as a prototype field theoretical model to derive all the nilpotent symmetries for all the fields present in this interacting gauge theory in the framework of augmented superfield formulation and show that this theory is a {\it unique} example of an interacting gauge theory which provides a tractable field theoretical model for the Hodge theory.Comment: LaTeX file, 25 pages, Ref. [49] updated, correct page numbers of the Journal are give

Double power series method for approximating cosmological perturbations

We introduce a double power series method for finding approximate analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a non-cosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on sub-horizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding approximate analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new approximate oscillating solutions in linear order cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well known growing and decaying M\'esz\'aros solutions, these oscillating modes provide a complete set of sub-horizon approximations for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our approximations can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.Comment: 22 pages, 10 figures, 2 tables. Mathematica notebook available from Github at https://github.com/AndrewWren/Double-power-series.gi

Gauge Transformations, BRST Cohomology and Wigner's Little Group

We discuss the (dual-)gauge transformations and BRST cohomology for the two (1 + 1)-dimensional (2D) free Abelian one-form and four (3 + 1)-dimensional (4D) free Abelian 2-form gauge theories by exploiting the (co-)BRST symmetries (and their corresponding generators) for the Lagrangian densities of these theories. For the 4D free 2-form gauge theory, we show that the changes on the antisymmetric polarization tensor e^{\mu\nu} (k) due to (i) the (dual-)gauge transformations corresponding to the internal symmetry group, and (ii) the translation subgroup T(2) of the Wigner's little group, are connected with each-other for the specific relationships among the parameters of these transformation groups. In the language of BRST cohomology defined w.r.t. the conserved and nilpotent (co-)BRST charges, the (dual-)gauge transformed states turn out to be the sum of the original state and the (co-)BRST exact states. We comment on (i) the quasi-topological nature of the 4D free 2-form gauge theory from the degrees of freedom count on e^{\mu\nu} (k), and (ii) the Wigner's little group and the BRST cohomology for the 2D one-form gauge theory {\it vis-{\`a}-vis} our analysis for the 4D 2-form gauge theory.Comment: LaTeX file, 29 pages, misprints in (3.7), (3.8), (3.9), (3.13) and (4.14)corrected and communicated to IJMPA as ``Erratum'

Superfield approach to symmetry invariance in QED with complex scalar fields

We show that the Grassmannian independence of the super Lagrangian density, expressed in terms of the superfields defined on a (4, 2)-dimensional supermanifold, is a clear-cut proof for the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST invariance of the corresoponding four (3 + 1)-dimensional (4D) Lagrangian density that describes the interaction between the U(1) gauge field and the charged complex scalar fields. The above 4D field theoretical model is considered on a (4, 2)-dimensional supermanifold parametrized by the ordinary four spacetime variables x^\mu (with \mu = 0, 1, 2, 3) and a pair of Grassmannian variables \theta and \bar\theta (with \theta^2 = \bar\theta^2 = 0, \theta \bar\theta + \bar\theta \theta = 0). Geometrically, the (anti-)BRST invariance is encoded in the translation of the super Lagrangian density along the Grassmannian directions of the above supermanifold such that the outcome of this shift operation is zero.Comment: LaTeX file, 14 pages, minor changes in the title and text, version to appear in ``Pramana - Journal of Physics'