Structurally engineered cytochromes c with novel ligand-binding properties

Semisynthesis of horse heart cytochrome c and site-directed mutagenesis of Saccharomyces cerevisiae (S. c.) iso-1-cytochrome c have been utilized to substitute Ala for the cytochrome c heme axial ligand Met80 to yield ligand-binding proteins (horse heart Ala80cyt c and S.c. Ala80cyt c) with spectroscopic properties remarkably similar to those of myoglobin. Both species of Fe(II)Ala80cyt c form exceptionally stable dioxygen complexes with autoxidation rates 10-30x smaller and O₂ binding constants ~ 3x greater than those of myoglobin. The resistance of O₂-Fe(II)Ala80cyt c to autoxidation is attributed in part to protection of the heme site from solvent as exhibited by the exceptionally slow rate of CO binding to the heme as well as the low quantum yield of CO photodissociation.

UV/vis, EPR, and paramagnetic NMR spectroscopy indicate that at pH 7 the Fe(III)Ala80cyt c heme is low-spin with axial His-OH^- coordination and that below pH ~6.5, Fe(III)Ala80cyt cis high-spin with His-H₂O heme ligation. Significant differences in the pH dependence of the ¹H NMR spectra of S.c. Fe(III)Ala80cyt c compared to wild-type demonstrate that the axial ligands influence the conformational energetics of cytochrome c.

¹H NMR spectroscopy has been utilized to determine the solution structure of the cyanide derivative of S.c. Fe(III)Ala80cyt c. 82% of the resonances in the ¹H NMR spectrum of S.c. CN-Fe(III)Ala80cyt c have been assigned through 1D and 2D experiments. The RMSD values after restrained energy minimization of the family of 17 structures obtained from distance geometry calculations are 0.68 ± 0.11 Å for the backbone and 1.32 ± 0.14 Å for all heavy atoms. The solution structure indicates that a tyrosine in the "distal" pocket of CN-Fe(III)Ala80cyt c forms a hydrogen bond with the Fe(III)-CN unit, suggesting that it may play a role analogous to that of the distal histidine in myoglobin in stabilizing the dioxygen adduct.

The theory of long distance electron transfer reactions

The rate of electron transport between distant sites was studied. The rate depends crucially on the chemical details of the donor, acceptor, and surrounding medium. These reactions involve electron tunneling through the intervening medium and are, therefore, profoundly influenced by the geometry and energetics of the intervening molecules. The dependence of rate on distance was considered for several rigid donor-acceptor "linkers" of experimental importance. Interpretation of existing experiments and predictions for new experiments were made.

The electronic and nuclear motion in molecules is correlated. A Born-Oppenheimer separation is usually employed in quantum chemistry to separate this motion. Long distance electron transfer rate calculations require the total donor wave function when the electron is very far from its binding nuclei. The Born-Oppenheimer wave functions at large electronic distance are shown to be qualitatively wrong. A model which correctly treats the coupling was proposed. The distance and energy dependence of the electron transfer rate was determined for such a model.

On reconstruction theorems in noncommutative Riemannian geometry

We present a novel account of the theory of commutative spectral triples and their two closest noncommutative generalisations, almost-commutative spectral triples and toric noncommutative manifolds, with a focus on reconstruction theorems, viz, abstract, functional-analytic characterisations of global-analytically defined classes of spectral triples. We begin by reinterpreting Connes's reconstruction theorem for commutative spectral triples as a complete noncommutative-geometric characterisation of Dirac-type operators on compact oriented Riemannian manifolds, and in the process clarify folklore concerning stability of properties of spectral triples under suitable perturbation of the Dirac operator. Next, we apply this reinterpretation of the commutative reconstruction theorem to obtain a reconstruction theorem for almost-commutative spectral triples. In particular, we propose a revised, manifestly global-analytic definition of almost-commutative spectral triple, and, as an application of this global-analytic perspective, obtain a general result relating the spectral action on the total space of a finite normal compact oriented Riemannian cover to that on the base space. Throughout, we discuss the relevant refinements of these definitions and results to the case of real commutative and almost-commutative spectral triples. Finally, we outline progess towards a reconstruction theorem for toric noncommutative manifolds.

Conformal geometry processing

This thesis introduces fundamental equations and numerical methods for manipulating surfaces in three dimensions via conformal transformations. Conformal transformations are valuable in applications because they naturally preserve the integrity of geometric data. To date, however, there has been no clearly stated and consistent theory of conformal transformations that can be used to develop general-purpose geometry processing algorithms: previous methods for computing conformal maps have been restricted to the flat two-dimensional plane, or other spaces of constant curvature. In contrast, our formulation can be used to produce---for the first time---general surface deformations that are perfectly conformal in the limit of refinement. It is for this reason that we commandeer the title Conformal Geometry Processing.

The main contribution of this thesis is analysis and discretization of a certain time-independent Dirac equation, which plays a central role in our theory. Given an immersed surface, we wish to construct new immersions that (i) induce a conformally equivalent metric and (ii) exhibit a prescribed change in extrinsic curvature. Curvature determines the potential in the Dirac equation; the solution of this equation determines the geometry of the new surface. We derive the precise conditions under which curvature is allowed to evolve, and develop efficient numerical algorithms for solving the Dirac equation on triangulated surfaces.

From a practical perspective, this theory has a variety of benefits: conformal maps are desirable in geometry processing because they do not exhibit shear, and therefore preserve textures as well as the quality of the mesh itself. Our discretization yields a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications. We also present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces.

Coupled effects of mechanics, geometry, and chemistry on bio-membrane behavior

Lipid bilayer membranes are models for cell membranes--the structure that helps regulate cell function. Cell membranes are heterogeneous, and the coupling between composition and shape gives rise to complex behaviors that are important to regulation. This thesis seeks to systematically build and analyze complete models to understand the behavior of multi-component membranes.

We propose a model and use it to derive the equilibrium and stability conditions for a general class of closed multi-component biological membranes. Our analysis shows that the critical modes of these membranes have high frequencies, unlike single-component vesicles, and their stability depends on system size, unlike in systems undergoing spinodal decomposition in flat space. An important implication is that small perturbations may nucleate localized but very large deformations. We compare these results with experimental observations.

We also study open membranes to gain insight into long tubular membranes that arise for example in nerve cells. We derive a complete system of equations for open membranes by using the principle of virtual work. Our linear stability analysis predicts that the tubular membranes tend to have coiling shapes if the tension is small, cylindrical shapes if the tension is moderate, and beading shapes if the tension is large. This is consistent with experimental observations reported in the literature in nerve fibers. Further, we provide numerical solutions to the fully nonlinear equilibrium equations in some problems, and show that the observed mode shapes are consistent with those suggested by linear stability. Our work also proves that beadings of nerve fibers can appear purely as a mechanical response of the membrane.

Multipole moments in general relativity and dynamical perturbations of black-hole magnetospheres

This thesis consists of two parts. In Part I, we develop a multipole moment formalism in general relativity and use it to analyze the motion and precession of compact bodies. More specifically, the generic, vacuum, dynamical gravitational field of the exterior universe in the vicinity of a freely moving body is expanded in positive powers of the distance r away from the body's spatial origin (i.e., in the distance r from its timelike-geodesic world line). The expansion coefficients, called "external multipole moments,'' are defined covariantly in terms of the Riemann curvature tensor and its spatial derivatives evaluated on the body's central world line. In a carefully chosen class of de Donder coordinates, the expansion of the external field involves only integral powers of r ; no logarithmic terms occur. The expansion is used to derive higher-order corrections to previously known laws of motion and precession for black holes and other bodies. The resulting laws of motion and precession are expressed in terms of couplings of the time derivatives of the body's quadrupole and octopole moments to the external moments, i.e., to the external curvature and its gradient.

In part II, we study the interaction of magnetohydrodynamic (MHD) waves in a black-hole magnetosphere with the "dragging of inertial frames" effect of the hole's rotation - i.e., with the hole's "gravitomagnetic field." More specifically: we first rewrite the laws of perfect general relativistic magnetohydrodynamics (GRMHD) in 3+1 language in a general spacetime, in terms of quantities (magnetic field, flow velocity, ...) that would be measured by the ''fiducial observers” whose world lines are orthogonal to (arbitrarily chosen) hypersurfaces of constant time. We then specialize to a stationary spacetime and MHD flow with one arbitrary spatial symmetry (e.g., the stationary magnetosphere of a Kerr black hole); and for this spacetime we reduce the GRMHD equations to a set of algebraic equations. The general features of the resulting stationary, symmetric GRMHD magnetospheric solutions are discussed, including the Blandford-Znajek effect in which the gravitomagnetic field interacts with the magnetosphere to produce an outflowing jet. Then in a specific model spacetime with two spatial symmetries, which captures the key features of the Kerr geometry, we derive the GRMHD equations which govern weak, linealized perturbations of a stationary magnetosphere with outflowing jet. These perturbation equations are then Fourier analyzed in time t and in the symmetry coordinate x, and subsequently solved numerically. The numerical solutions describe the interaction of MHD waves with the gravitomagnetic field. It is found that, among other features, when an oscillatory external force is applied to the region of the magnetosphere where plasma (e⁺e^-) is being created, the magnetosphere responds especially strongly at a particular, resonant, driving frequency. The resonant frequency is that for which the perturbations appear to be stationary (time independent) in the common rest frame of the freshly created plasma and the rotating magnetic field lines. The magnetosphere of a rotating black hole, when buffeted by nonaxisymmetric magnetic fields anchored in a surrounding accretion disk, might exhibit an analogous resonance. If so then the hole's outflowing jet might be modulated at resonant frequencies ω=(m/2) Ω_H where m is an integer and Ω_H is the hole's angular velocity.

Fast numerical methods for mixed, singular Helmholtz boundary value problems and Laplace eigenvalue problems - with applications to antenna design, sloshing, electromagnetic scattering and spectral geometry

This thesis presents a novel class of algorithms for the solution of scattering and eigenvalue problems on general two-dimensional domains under a variety of boundary conditions, including non-smooth domains and certain "Zaremba" boundary conditions - for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the methods for the Zaremba problems on smooth domains concern detailed information, which is put forth for the first time in this thesis, about the singularity structure of solutions of the Laplace operator under boundary conditions of Zaremba type. The new methods, which are based on use of Green functions and integral equations, incorporate a number of algorithmic innovations, including a fast and robust eigenvalue-search algorithm, use of the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities, and newly derived quadrature rules which give rise to high-order convergence even around singular points for the Zaremba problem. The resulting algorithms enjoy high-order convergence, and they can tackle a variety of elliptic problems under general boundary conditions, including, for example, eigenvalue problems, scattering problems, and, in particular, eigenfunction expansion for time-domain problems in non-separable physical domains with mixed boundary conditions.

Photochemical electron transfer at fixed distance : a synthetic model of the photosynthetic primary process

A series of meso-phenyloctamethylporphyrins covalently bonded at the 4'phenyl position to quinones via rigid bicyclo[2.2.2]octane spacers were synthesized for the study of the dependence of electron transfer reaction rate on solvent, distance, temperature, and energy gap. A general and convergent synthesis was developed based on the condensation of ac-biladienes with masked quinonespacer-benzaldehydes. From picosecond fluorescence spectroscopy emission lifetimes were measured in seven solvents of varying polarity. Rate constants were determined to vary from 5.0x10⁹sec^-1 in N,N-dimethylformamide to 1.15x10¹⁰ Sec^-1 in benzene, and were observed to rise at most by about a factor of three with decreasing solvent polarity. Experiments at low temperature in 2-MTHF glass (77K) revealed fast, nearly temperature-independent electron transfer characterized by non-exponential fluorescence decays, in contrast to monophasic behavior in fluid solution at 298K. This example evidently represents the first photosynthetic model system not based on proteins to display nearly temperature-independent electron transfer at high temperatures (nuclear tunneling). Low temperatures appear to freeze out the rotational motion of the chromophores, and the observed nonexponential fluorescence decays may be explained as a result of electron transfer from an ensemble of rotational conformations. The nonexponentiality demonstrates the sensitivity of the electron transfer rate to the precise magnitude of the electronic matrix element, which supports the expectation that electron transfer is nonadiabatic in this system. The addition of a second bicyclooctane moiety (15 Å vs. 18 Å edge-to-edge between porphyrin and quinone) reduces the transfer rate by at least a factor of 500-1500. Porphyrinquinones with variously substituted quinones allowed an examination of the dependence of the electron transfer rate constant κ_ET on reaction driving force. The classical trend of increasing rate versus increasing exothermicity occurs from 0.7 eV≤ |ΔG^0'(R)| ≤ 1.0 eV until a maximum is reached (κ_ET = 3 x 10⁸ sec^-1 rising to 1.15 x 10¹⁰ sec^-1 in acetonitrile). The rate remains insensitive to ΔG⁰ for ~ 300 mV from 1.0 eV≤ |ΔG^0’(R)| ≤ 1.3 eV, and then slightly decreases in the most exothermic case studied (cyanoquinone, κ_ET = 5 x 10⁹ sec^-1).

Imaging the earth with ambient noise and earthquakes

In this thesis, I develop the velocity and structure models for the Los Angeles Basin and Southern Peru. The ultimate goal is to better understand the geological processes involved in the basin and subduction zone dynamics. The results are obtained from seismic interferometry using ambient noise and receiver functions using earthquake- generated waves. Some unusual signals specific to the local structures are also studied. The main findings are summarized as follows:

(1) Los Angeles Basin

The shear wave velocities range from 0.5 to 3.0 km/s in the sediments, with lateral gradients at the Newport-Inglewood, Compton-Los Alamitos, and Whittier Faults. The basin is a maximum of 8 km deep along the profile, and the Moho rises to a depth of 17 km under the basin. The basin has a stretch factor of 2.6 in the center decreasing to 1.3 at the edges, and is in approximate isostatic equilibrium. This "high-density" (~1 km spacing) "short-duration" (~1.5 month) experiment may serve as a prototype experiment that will allow basins to be covered by this type of low-cost survey.

(2) Peruvian subduction zone

Two prominent mid-crust structures are revealed in the 70 km thick crust under the Central Andes: a low-velocity zone interpreted as partially molten rocks beneath the Western Cordillera – Altiplano Plateau, and the underthrusting Brazilian Shield beneath the Eastern Cordillera. The low-velocity zone is oblique to the present trench, and possibly indicates the location of the volcanic arcs formed during the steepening of the Oligocene flat slab beneath the Altiplano Plateau.

The Nazca slab changes from normal dipping (~25 degrees) subduction in the southeast to flat subduction in the northwest of the study area. In the flat subduction regime, the slab subducts to ~100 km depth and then remains flat for ~300 km distance before it resumes a normal dipping geometry. The flat part closely follows the topography of the continental Moho above, indicating a strong suction force between the slab and the overriding plate. A high-velocity mantle wedge exists above the western half of the flat slab, which indicates the lack of melting and thus explains the cessation of the volcanism above. The velocity turns to normal values before the slab steepens again, indicating possible resumption of dehydration and ecologitization.

(3) Some unusual signals

Strong higher-mode Rayleigh waves due to the basin structure are observed in the periods less than 5 s. The particle motions provide a good test for distinguishing between the fundamental and higher mode. The precursor and coda waves relative to the interstation Rayleigh waves are observed, and modeled with a strong scatterer located in the active volcanic area in Southern Peru. In contrast with the usual receiver function analysis, multiples are extensively involved in this thesis. In the LA Basin, a good image is only from PpPs multiples, while in Peru, PpPp multiples contribute significantly to the final results.

Experimental studies of weak superconductivity

Experimental investigations were made of the nature of weak superconductivity in a structure having well-defined, controllable characteristics and geometry. Controlled experiments were made possible by using a thin-film structure which was entirely metallic and consisted of a superconducting film with a localized section that was weak in the sense that its transition temperature was depressed relative to the rest of the film. The depression of transition temperature was brought about by underlaying the superconductor with a normal metal.

The DC and AC electrical characteristics of this structure were studied. It was found that this structure exhibited a non-zero, time-average supercurrent at finite voltage to at least .2 mV, and generated an oscillating electric potential at a frequency given by the Josephson relation. The DC V-I characteristic and the amplitude of the AC oscillation were found to be consistent with a two- fluid (normal current-supercurrent) model of weak super-conductivity based on e thermodynamically irreversible process of repetitive phase-slip, and featuring a periodic time dependence in the amplitude of the superconducting order parameter.

The observed linewidth of the AC oscillation could be accounted for by incorporating Johnson noise in the two-fluid model.

Experimentally it was found that the behavior of a short (length on the order of the coherence distance) weak superconductor could be characterized by its critical current and normal-state resistance, and an empirical expression was obtained for the time dependence of the super-current and voltage.

It was found that the results could not be explained on the basis of the theory of the Josephson junction.