Inverse Problem for Fractional Brownian Motion with Discrete Data

The problem of recovering information from measurement data has already been studied for a long time. In the beginning, the methods were mostly empirical, but already towards the end of the sixties Backus and Gilbert started the development of mathematical methods for the interpretation of geophysical data. The problem of recovering information about a physical phenomenon from measurement data is an inverse problem. Throughout this work, the statistical inversion method is used to obtain a solution. Assuming that the measurement vector is a realization of fractional Brownian motion, the goal is to retrieve the amplitude and the Hurst parameter. We prove that under some conditions, the solution of the discretized problem coincides with the solution of the corresponding continuous problem as the number of observations tends to infinity. The measurement data is usually noisy, and we assume the data to be the sum of two vectors: the trend and the noise. Both vectors are supposed to be realizations of fractional Brownian motions, and the goal is to retrieve their parameters using the statistical inversion method. We prove a partial uniqueness of the solution. Moreover, with the support of numerical simulations, we show that in certain cases the solution is reliable and the reconstruction of the trend vector is quite accurate.

Mechanism of RNA Translocation by a Viral Packaging Motor

Molecular motors are proteins that convert chemical energy into mechanical work. The viral packaging ATPase P4 is a hexameric molecular motor that translocates RNA into preformed viral capsids. P4 belongs to the ubiquitous class of hexameric helicases. Although its structure is known, the mechanism of RNA translocation remains elusive. Here we present a detailed kinetic study of nucleotide binding, hydrolysis, and product release by P4. We propose a stochastic-sequential cooperative model to describe the coordination of ATP hydrolysis within the hexamer. In this model the apparent cooperativity is a result of hydrolysis stimulation by ATP and RNA binding to neighboring subunits rather than cooperative nucleotide binding. Simultaneous interaction of neighboring subunits with RNA makes the otherwise random hydrolysis sequential and processive. Further, we use hydrogen/deuterium exchange detected by high resolution mass spectrometry to visualize P4 conformational dynamics during the catalytic cycle. Concerted changes of exchange kinetics reveal a cooperative unit that dynamically links ATP binding sites and the central RNA binding channel. The cooperative unit is compatible with the structure-based model in which translocation is effected by conformational changes of a limited protein region. Deuterium labeling also discloses the transition state associated with RNA loading which proceeds via opening of the hexameric ring. Hydrogen/deuterium exchange is further used to delineate the interactions of the P4 hexamer with the viral procapsid. P4 associates with the procapsid via its C-terminal face. The interactions stabilize subunit interfaces within the hexamer. The conformation of the virus-bound hexamer is more stable than the hexamer in solution, which is prone to spontaneous ring openings. We propose that the stabilization within the viral capsid increases the packaging processivity and confers selectivity during RNA loading. Finally, we use single molecule techniques to characterize P4 translocation along RNA. While the P4 hexamer encloses RNA topologically within the central channel, it diffuses randomly along the RNA. In the presence of ATP, unidirectional net movement is discernible in addition to the stochastic motion. The diffusion is hindered by activation energy barriers that depend on the nucleotide binding state. The results suggest that P4 employs an electrostatic clutch instead of cycling through stable, discrete, RNA binding states during translocation. Conformational changes coupled to ATP hydrolysis modify the electrostatic potential inside the central channel, which in turn biases RNA motion in one direction. Implications of the P4 model for other hexameric molecular motors are discussed.

Spin and relativistic motion of bound states

The wave functions of moving bound states may be expected to contract in the direction of motion, in analogy to a rigid rod in classical special relativity, when the constituents are at equal (ordinary) time. Indeed, the Lorentz contraction of wave functions is often appealed to in qualitative discussions. However, only few field theory studies exist of equal-time wave functions in motion. In this thesis I use the Bethe-Salpeter formalism to study the wave function of a weakly bound state such as a hydrogen atom or positronium in a general frame. The wave function of the e^-e^+ component of positronium indeed turns out to Lorentz contract both in 1+1 and in 3+1 dimensional quantum electrodynamics, whereas the next-to-leading e^-e^+\gamma Fock component of the 3+1 dimensional theory deviates from classical contraction. The second topic of this thesis concerns single spin asymmetries measured in scattering on polarized bound states. Such spin asymmetries have so far mainly been analyzed using the twist expansion of perturbative QCD. I note that QCD vacuum effects may give rise to a helicity flip in the soft rescattering of the struck quark, and that this would cause a nonvanishing spin asymmetry in \ell p^\uparrow -> \ell' + \pi + X in the Bjorken limit. An analogous asymmetry may arise in p p^\uparrow -> \pi + X from Pomeron-Odderon interference, if the Odderon has a helicity-flip coupling. Finally, I study the possibility that the large single spin asymmetry observed in p p^\uparrow -> \pi(x_F,k_\perp) + X when the pion carries a high momentum fraction x_F of the polarized proton momentum arises from coherent effects involving the entire polarized bound state.