I. Singular perturbation problems involving singular points and turning points. II. On the averaged Lagrangian technique for nonlinear dispersive waves

In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.

Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.

Fluorescence optofluidic microscopy and fluorescence microscopy based on the Talbot effect

Light microscopy has been one of the most common tools in biological research, because of its high resolution and non-invasive nature of the light. Due to its high sensitivity and specificity, fluorescence is one of the most important readout modes of light microscopy. This thesis presents two new fluorescence microscopic imaging techniques: fluorescence optofluidic microscopy and fluorescent Talbot microscopy. The designs of the two systems are fundamentally different from conventional microscopy, which makes compact and portable devices possible. The components of the devices are suitable for mass-production, making the microscopic imaging system more affordable for biological research and clinical diagnostics.

Fluorescence optofluidic microscopy (FOFM) is capable of imaging fluorescent samples in fluid media. The FOFM employs an array of Fresnel zone plates (FZP) to generate an array of focused light spots within a microfluidic channel. As a sample flows through the channel and across the array of focused light spots, a filter-coated CMOS sensor collects the fluorescence emissions. The collected data can then be processed to render a fluorescence microscopic image. The resolution, which is determined by the focused light spot size, is experimentally measured to be 0.65 μm.

Fluorescence Talbot microscopy (FTM) is a fluorescence chip-scale microscopy technique that enables large field-of-view (FOV) and high-resolution imaging. The FTM method utilizes the Talbot effect to project a grid of focused excitation light spots onto the sample. The sample is placed on a filter-coated CMOS sensor chip. The fluorescence emissions associated with each focal spot are collected by the sensor chip and are composed into a sparsely sampled fluorescence image. By raster scanning the Talbot focal spot grid across the sample and collecting a sequence of sparse images, a filled-in high-resolution fluorescence image can be reconstructed. In contrast to a conventional microscope, a collection efficiency, resolution, and FOV are not tied to each other for this technique. The FOV of FTM is directly scalable. Our FTM prototype has demonstrated a resolution of 1.2 μm, and the collection efficiency equivalent to a conventional microscope objective with a 0.70 N.A. The FOV is 3.9 mm × 3.5 mm, which is 100 times larger than that of a 20X/0.40 N.A. conventional microscope objective. Due to its large FOV, high collection efficiency, compactness, and its potential for integration with other on-chip devices, FTM is suitable for diverse applications, such as point-of-care diagnostics, large-scale functional screens, and long-term automated imaging.

Gene enrichment using antibodies to DNA/RNA hybrids : mapping the ribosomal DNA of slime mold and rat

A novel method for gene enrichment has been developed and applied to mapping the rRNA genes of two eucaryotic organisms. The method makes use of antibodies to DNA/RNA hybrids prepared by injecting rabbits with the synthetic hybrid poly(rA)•poly(dT). Antibodies which cross-react with non-hybrid nucleic acids were removed from the purified IgG fraction by adsorption on columns of DNA-Sepharose, oligo(dT)-cellulose, and poly(rA)-Sepharose. Subsequent purification of the specific DNA/RNA hybrid antibody was carried out on a column of oligo(dT)-cellulose to which poly(rA) was hybridized. Attachment of these antibodies to CNBr-activated Sepharose produced an affinity resin which specifically binds DNA/RNA hybrids.

In order to map the rDNA of the slime mold Dictyostelium discoideum, R-loops were formed using unsheared nuclear DNA and the 178 and 268 rRNAs of this organism. This mixture was passed through a column containing the affinity resin, and bound molecules containing R- loops were eluted by high salt. This purified rDN A was observed directly in the electron microscope. Evidence was obtained that there is a physical end to Dictyostelium rDN A molecules approximately 10 kilobase pairs (kbp) from the region which codes for the 268 rRNA. This finding is consistent with reports of other investigators that the rRNA genes exist as inverse repeats on extra-chromosomal molecules of DNA unattached to the remainder of the nuclear DNA in this organism.

The same general procedure was used to map the rRNA genes of the rat. Molecules of DNA which contained R-loops formed with the 188 and 288 rRNAs were enriched approximately 150- fold from total genomal rat DNA by two cycles of purification on the affinity column. Electron microscopic measurements of these molecules enabled the construction of an R-loop map of rat rDNA. Eleven of the observed molecules contained three or four R-loops or else two R-loops separated by a long spacer. These observations indicated that the rat rRNA genes are arranged as tandem repeats. The mean length of the repeating units was 37.2 kbp with a standard deviation of 1.3 kbp. These eleven molecules may represent repeating units of exactly the same length within the errors of the measurements, although a certain degree of length heterogeneity cannot be ruled out. If significantly shorter or longer repeating units exist, they are probably much less common than the 37.2 kbp unit.

The last section of the thesis describes the production of antibodies to non-histone chromosomal proteins which have been exposed to the ionic detergent sodium dodecyl sulfate (SDS). The presence of low concentrations of SDS did not seem to affect either production of antibodies or their general specificity. Also, a technique is described for the in situ immunofluorescent detection of protein antigens in polyacrylamide gels.

Fluorescence microscopy of nicotinic acetylcholine receptors

Neuronal nicotinic acetylcholine receptors (nAChRs) are pentameric ligand gated ion channels abundantly expressed in the central nervous system. Changes in the assembly and trafficking of nAChRs are pertinent to disease states including nicotine dependence, autosomal dominant nocturnal frontal lobe epilepsy (ADNFLE), and Parkinson’s disease (PD). Here we investigate the application of high resolution fluorescence techniques for the study of nAChR assembly and trafficking. We also describe the construction and validation of a fluorescent α5 subunit and subsequent experiments to elucidate the cellular mechanisms through which α5 subunits are expressed, assembled into mature receptors, and trafficked to the cell surface. The effects of a known single nucleotide polymorphism (D398N) in the intracellular loop of α5 are also examined.

Additionally, this report describes the development of a combined total internal reflection fluorescence (TIRF) and lifetime imaging (FLIM) technique and the first application of this methodology for elucidation of stochiometric composition of nAChRs. Many distinct subunit combinations can form functional receptors. Receptor composition and stoichiometry confers unique biophysical and pharmacological properties to each receptor sub-type. Understanding the nature of assembly and expression of each receptor subtype yields important information about the molecular processes that may underlie the mechanisms through which nAChR contribute to disease and addiction states.

Engineered discoidin domain from factor VIII binds αvβ3 integrin with antibody-like affinity

Alternative scaffolds are non-antibody proteins that can be engineered to bind new targets. They have found useful niches in the therapeutic space due to their smaller size and the ease with which they can be engineered to be bispecific. We sought a new scaffold that could be used for therapeutic ends and chose the C2 discoidin domain of factor VIII, which is well studied and of human origin. Using yeast surface display, we engineered the C2 domain to bind to αvβ3 integrin with a 16 nM affinity while retaining its thermal stability and monomeric nature. We obtained a crystal structure of the engineered domain at 2.1 Å resolution. We have christened this discoidin domain alternative scaffold the “discobody.”

Computationally Guided Monomerization of Red Fluorescent Proteins of the Class Anthozoa

Red fluorescent proteins (RFPs) have attracted significant engineering focus because of the promise of near infrared fluorescent proteins, whose light penetrates biological tissue, and which would allow imaging inside of vertebrate animals. The RFP landscape, which numbers ~200 members, is mostly populated by engineered variants of four native RFPs, leaving the vast majority of native RFP biodiversity untouched. This is largely due to the fact that native RFPs are obligate tetramers, limiting their usefulness as fusion proteins. Monomerization has imposed critical costs on these evolved tetramers, however, as it has invariably led to loss of brightness, and often to many other adverse effects on the fluorescent properties of the derived monomeric variants. Here we have attempted to understand why monomerization has taken such a large toll on Anthozoa class RFPs, and to outline a clear strategy for their monomerization. We begin with a structural study of the far-red fluorescence of AQ143, one of the furthest red emitting RFPs. We then try to separate the problem of stable and bright fluorescence from the design of a soluble monomeric β-barrel surface by engineering a hybrid protein (DsRmCh) with an oligomeric parent that had been previously monomerized, DsRed, and a pre-stabilized monomeric core from mCherry. This allows us to use computational design to successfully design a stable, soluble, fluorescent monomer. Next we took HcRed, which is a previously unmonomerized RFP that has far-red fluorescence (λemission = 633 nm) and attempted to monomerize it making use of lessons learned from DsRmCh. We engineered two monomeric proteins by pre-stabilizing HcRed’s core, then monomerizing in stages, making use of computational design and directed evolution techniques such as error-prone mutagenesis and DNA shuffling. We call these proteins mGinger0.1 (λem = 637 nm / Φ = 0.02) and mGinger0.2 (λem = 631 nm Φ = 0.04). They are the furthest red first generation monomeric RFPs ever developed, are significantly thermostabilized, and add diversity to a small field of far-red monomeric FPs. We anticipate that the techniques we describe will be facilitate future RFP monomerization, and that further core optimization of the mGingers may allow significant improvements in brightness.

Finite differences and a coupled analytic technique with applications to explosions and earthquakes

An analytic technique is developed that couples to finite difference calculations to extend the results to arbitrary distance. Finite differences and the analytic result, a boundary integral called two-dimensional Kirchhoff, are applied to simple models and three seismological problems dealing with data. The simple models include a thorough investigation of the seismologic effects of a deep continental basin. The first problem is explosions at Yucca Flat, in the Nevada test site. By modeling both near-field strong-motion records and teleseismic P-waves simultaneously, it is shown that scattered surface waves are responsible for teleseismic complexity. The second problem deals with explosions at Amchitka Island, Alaska. The near-field seismograms are investigated using a variety of complex structures and sources. The third problem involves regional seismograms of Imperial Valley, California earthquakes recorded at Pasadena, California. The data are shown to contain evidence of deterministic structure, but lack of more direct measurements of the structure and possible three-dimensional effects make two-dimensional modeling of these data difficult.

Structural Characterizations of the Dimeric Anti-HIV Antibody 2G12 and the HIV-2 Envelope Glycoprotein

More than thirty years after the discovery that Human Immunodeficiency Virus (HIV) was the causative agent of Acquired Immunodeficiency Syndrome (AIDS), the disease remains pandemic as long as no effective universal vaccine is found. Over 34 million individuals in the world are infected with the virus, and the vast majority of them have no access to the antiretroviral therapies that have largely reduced HIV to a chronic disease in the developed world. The first chapter of this thesis introduces the history of the virus. The key to the infectious mechanism of the virus lies in its envelope glycoprotein (Env), a trimeric spike on the viral surface that utilizes host T cell receptors for entry. Though HIV-1 Env is immunogenic, most infected patients do not mount an effective neutralizing antibody response against it. Broadly-neutralizing anti-Env antibodies (bNAbs) present in the serum of a minority of infected individuals are usually sufficient to prevent the progression to full blown AIDS. Thus, the molecular details of these bNAbs as well as the antibody-antigen interface are of prime interest for structural studies, as insight gained would contribute to the design of a more effective immunogen and potential vaccine candidate. The second chapter of this thesis describes the low-resolution crystal structure of one such antibody, 2G12 dimer, which targets a high mannose epitope on the surface of Env. Patients infected with HIV-2, a related virus with ~35% sequence identity in the Env region, can generally mount a robust antibody response sufficient for viral control for reasons still unknown. The final two chapters of this thesis focus on the first reported structural studies of HIV-2 Env, the molecular details of which may inform HIV-1 therapy and immunogen design.

From molecules to organs : Microscopy and multi-scale nature of development

Morphogenesis is a phenomenon of intricate balance and dynamic interplay between processes occurring at a wide range of scales (spatial, temporal and energetic). During development, a variety of physical mechanisms are employed by tissues to simultaneously pattern, move, and differentiate based on information exchange between constituent cells, perhaps more than at any other time during an organism's life. To fully understand such events, a combined theoretical and experimental framework is required to assist in deciphering the correlations at both structural and functional levels at scales that include the intracellular and tissue levels as well as organs and organ systems. Microscopy, especially diffraction-limited light microscopy, has emerged as a central tool to capture the spatio-temporal context of life processes. Imaging has the unique advantage of watching biological events as they unfold over time at single-cell resolution in the intact animal. In this work I present a range of problems in morphogenesis, each unique in its requirements for novel quantitative imaging both in terms of the technique and analysis. Understanding the molecular basis for a developmental process involves investigating how genes and their products- mRNA and proteins-function in the context of a cell. Structural information holds the key to insights into mechanisms and imaging fixed specimens paves the first step towards deciphering gene function. The work presented in this thesis starts with the demonstration that the fluorescent signal from the challenging environment of whole-mount imaging, obtained by in situ hybridization chain reaction (HCR), scales linearly with the number of copies of target mRNA to provide quantitative sub-cellular mapping of mRNA expression within intact vertebrate embryos. The work then progresses to address aspects of imaging live embryonic development in a number of species. While processes such as avian cartilage growth require high spatial resolution and lower time resolution, dynamic events during zebrafish somitogenesis require higher time resolution to capture the protein localization as the somites mature. The requirements on imaging are even more stringent in case of the embryonic zebrafish heart that beats with a frequency of ~ 2-2.5 Hz, thereby requiring very fast imaging techniques based on two-photon light sheet microscope to capture its dynamics. In each of the hitherto-mentioned cases, ranging from the level of molecules to organs, an imaging framework is developed, both in terms of technique and analysis to allow quantitative assessment of the process in vivo. Overall the work presented in this thesis combines new quantitative tools with novel microscopy for the precise understanding of processes in embryonic development.

Depth-sensitive analysis of fluorine on lunar sample surfaces and in carbonaceous chondrites by a nuclear resonant reaction technique

The nuclear resonant reaction ¹⁹F(ρ,αγ)¹⁶O has been used to perform depth-sensitive analyses of fluorine in lunar samples and carbonaceous chondrites. The resonance at 0.83 MeV (center-of-mass) in this reaction is utilized to study fluorine surface films, with particular interest paid to the outer micron of Apollo 15 green glass, Apollo 17 orange glass, and lunar vesicular basalts. These results are distinguished from terrestrial contamination, and are discussed in terms of a volcanic origin for the samples of interest. Measurements of fluorine in carbonaceous chondrites are used to better define the solar system fluorine abundance. A technique for measurement of carbon on solid surfaces with applications to direct quantitative analysis of implanted solar wind carbon in lunar samples is described.

Applications of a nuclear technique for depth-sensitive hydrogen analysis : trapped H in lunar samples and the hydration of terrestrial obsidian

The resonant nuclear reaction ¹⁹F(p,αy)¹⁶O has been used to perform depth-sensitive analyses for both fluorine and hydrogen in solid samples. The resonance at 0.83 MeV (center-of-mass) in this reaction has been applied to the measurement of the distribution of trapped solar protons in lunar samples to depths of ~1/2µm. These results are interpreted in terms of a redistribution of the implanted H which has been influenced by heavy radiation damage in the surface region. Fluorine determinations have been performed in a 1-µm surface layer on lunar and meteoritic samples using the same ¹⁹F(p,αy)¹⁶O resonance. The measurement of H depth distributions has also been used to study the hydration of terrestrial obsidian, a phenomenon of considerable archaeological interest as a means of dating obsidian artifacts. Additional applications of this type of technique are also discussed.

A scaling technique for the design of idealized electromagnetic lenses

A technique is developed for the design of lenses for transitioning TEM waves between conical and/or cylindrical transmission lines, ideally with no reflection or distortion of the waves. These lenses utilize isotropic but inhomogeneous media and are based on a solution of Maxwell's equations instead of just geometrical optics. The technique employs the expression of the constitutive parameters, ɛ and μ, plus Maxwell's equations, in a general orthogonal curvilinear coordinate system in tensor form, giving what we term as formal quantities. Solving the problem for certain types of formal constitutive parameters, these are transformed to give ɛ and μ as functions of position. Several examples of such lenses are considered in detail.