The partial sum process of orthogonal expansion as geometric rough process with Fourier series as an example---an improvement of Menshov-Rademacher theorem

In this paper, we prove that the partial sum process of general orthogonal series is a geometric 2-rough process under the same condition as in Menshov-Rademacher Theorem. For Fourier series, the condition can be improved, and an equivalent condition on the limit function is identified

In Rhodobacter sphaeroides Reaction Centers, Mutation of Proline L209 to Aromatic Residues in the Vicinity of a Water Channel Alters the Dynamic Coupling between Electron and Proton Transfer Processes

The X-ray crystallographic structure of the photosynthetic reaction center from Rhodobactersphaeroides obtained at high resolution has revealed a number of internal water molecules (Ermler, U., Fritzsch, G., Buchanan, S. K., and Michel, H. (1994) Structure2, 925−936; Stowell, M. H. B., McPhillips, T. M., Rees, D. C., Soltis, S. M., Abresch, E., and Feher, G. (1997) Science276, 812−816). Some of them are organized into distinct hydrogen-bonded water chains that connect QB (the terminal quinone electron acceptor of the reaction center) to the aqueous phase. To investigate the role of the water chains in the proton conduction process, proline L209, located immediately adjacent to a water chain, was mutated to the following residues:  F, Y, W, E, and T. We have first analyzed the effects of the mutations on the kinetic and thermodynamic properties of the rate constants of the second electron transfer (kAB(2)) and of the coupled proton uptake (kH+) at the second flash. In all aromatic mutants, kAB(2) and kH+ are notably and concomitantly decreased compared to the wild-type, while no effect is observed in the other mutants. The temperature dependence of these rates shows activation energy values (ΔH⧧) similar for the proton and electron-transfer processes in the wild-type and in most of the mutants, except for the L209PW and L209PF mutants. The analysis of the enthalpy factors related to the electron and proton-transfer processes in the L209PF and the L209PW mutants allows to distinguish the respective effects of the mutations for both transfer reactions. It is noteworthy that in the aromatic mutants a substantial increase of the free energies of activation is observed (ΔG⧧L209PY ⧧L209PF ⧧L209PW) for both proton and electron-transfer reactions, while in the other mutants, ΔG⧧ is not affected. The salt concentration dependence of kAB(2) shows, in the L209PF and L209PW mutants, a higher screening of the protein surface potential experienced by QB. Our data suggest that residues F and W in position L209 increase the polarizability of the internal water molecules and polar residues by altering the organization of the hydrogen-bond network. We have also analyzed the rates of the first electron-transfer reaction (kAB(1)), in the 100 μs time domain. These kinetics have previously been shown to reflect protein relaxation events possibly including proton uptake events (Tiede, D. M., Vazquez, J., Cordova, J., and Marone, P. M. (1996) Biochemistry 35, 10763−10775). Interestingly, in the L209PF and L209PW mutants, kAB(1) is notably decreased in comparison to the wild type and the other mutants, in a similar way as kAB(2) and kH+. Our data imply that the dynamic organization of this web is tightly coupled to the electron transfer process that is kinetically limited by protonation events and/or conformational rearrangements within the protein

Evidence for Delocalized Anticooperative Flash Induced Proton Binding as Revealed by Mutants at the M266His Iron Ligand in Bacterial Reaction Centers

Bacterial reaction centers (RCs) convert light energy into chemical free energy via the double reduction and protonation of the secondary quinone electron acceptor, QB, to the dihydroquinone QBH2. Two RC mutants (M266His → Leu and M266His → Ala) with a modified ligand of the non-heme iron have been studied by flash-induced absorbance change spectroscopy. No important changes were observed for the rate constants of the first and second electron transfers between the first quinone electron acceptor, QA, and QB. However, in the M266HL mutant a destabilization of ∼40 meV of the free energy level of QA- was observed, at variance with the M266HA mutant. The superposition of the three-dimensional X-ray structures of the three proteins in the QA region provides no obvious explanation for the energy modification in the M266HL mutant. The shift of the midpoint redox potential of QA/QA- in M266HL caused accelerated recombination of the charges in the P+QA- state of the RCs where the native QA was replaced by a low potential anthraquinone (AQA). As previously reported for the native RCs, in the M266HL we observed a biphasicity of the P+AQA- → PAQA charge recombination. Interestingly, both phases present a similar acceleration in the M266HL mutant with respect to the wild type. The pH dependencies of the proton uptake upon QA- and QB- formations are superimposable in both mutants but very different from those of native RCs. The data measured in mutants are similar to those that we previously obtained on strains modified at various sites of the cytoplasmic region. The similarity of the response to these different mutations is puzzling, and we propose that it arises from a collective behavior of multiple acidic residues resulting in strongly anticooperative proton binding. The unspecific disappearance of the high pH band of proton uptake observed in all these mutants appears as the natural consequence of removing any member of an interactive proton cluster. This long range interaction also accounts for the similar responses to mutations of the proton uptake pattern induced by either QA- or QB-. We surmise that the presence of an extended protonated water H-bond network providing protons to QB is responsible for these effects