Markov chains are central to the understanding of random processes. This is not only because they pervade the applications of random processes, but also because one can calculate explicitly many quantities of interest. This textbook, aimed at advanced undergraduate or MSc students with some background in basic probability theory, focuses on Markov chains and quickly develops a coherent and rigorous theory whilst showing also how actually to apply it. Both discrete-time and continuous-time chains are studied. A distinguishing feature is an introduction to more advanced topics such as martingales and potentials in the established context of Markov chains. There are applications to simulation, economics, optimal control, genetics, queues and many other topics, and exercises and examples drawn both from theory and practice. It will therefore be an ideal text either for elementary courses on random processes or those that are more oriented towards applications.

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We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs.Comment: Published in at http://dx.doi.org/10.1214/07-PS121 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Photosynthetic reaction centers (RCs) from the photosynthetic bacteria Rhodobacter sphaeroides and Rhodopseudomonas viridis are protein complexes closely related in both structure and function. The structure of the Rps. viridis RC was used to determine the structure of the RC from Rb. sphaeroides. Small but meaningful differences between the positions of the helices and the cofactors in the two complexes were identified. The distances between helices AL and AM, between BL and BM, and between bacteriopheophytins BPL and BPM are significantly shorter in Rps. viridis than they are in Rb. sphaeroides RCs. There are a number of differences in the amino acid residues that surround the cofactors; some of these residues form hydrogen bonds with the cofactors. Differences in chemical properties and location of these residues account in some manner for the different spectral properties of the two RCs. In several instances, the hydrogen bonds, as well as the apparent distances between the histidine ligands and the Mg atoms of the bacteriochlorophylls, were found to significantly differ from the Rb. sphaeroides RC structure previously described by Yeates et al. [(1988) Proc. Natl. Acad. Sci. U.S.A. 85, 7993-7997] and Allen et al. [(1988) Proc. Natl. Acad. Sci. U.S.A. 85, 8487-8491].

essential that in vitro esr observations be interpreted in terms of the chlorophyll species actually present. Chlorophyll is able to act both as electron donor and electron acceptor in chargetransfer complexes. The central Mg atom of chlorophyll is coordinatively unsaturated when it has the coordination number 4, and at least one of the Mg axial positions must always be occupied by an electron donor group. In the absence of other nucleophiles, the ketone C-O function in Ring V of one chlorophyll molecule serves as donor to the Mg atom of another, forming chlorophyll dimers, (Chl2), or oligomers, (ChlW)3. Extraneous nucleophiles (bases) can compete for the coordination site at Mg, with disruption of the chlorophyllchlorophyll interactions, to form chlorophyll-ligand adducts, (Chl-L). The nature of the nucleophile determines whether the chlorophyll-nucleophile adduct is monomeric or polymeric. Bifunctional ligands, such as dioxane, pyrazine, 1, diazobicyclo(2.2.2.)octane, and, in particular, water can cross-link chlorophyll molecules or chlorophyll dimers by coordination to Mg to form large (chlorophyll-nucleophile) micelles (to be published). The chlorophyll species present in a particular experiment are very sensitive to temperature, adventitious nucleophiles such as water, and solvent, factors not always taken into account in previous work. EXPERIMENTAL METHODSChlorophyll samples were dried, prior to solution preparation, by codistillation with CCL (36). The solvents used for in vitro measurements were first dried over Linde 3A molecular sieve and degassed under reduced pressure. Solutions were prepared and oxidant was added in a nitrogen-filled dry box or on the vacuum line.Irradiations were performed with a 150-W Varian Eimac lamp. Infrared components were removed by 2 cm of water and 2 dichroic infrared-rejecting filters. All irradiations were by red light with a Corning 2404 sharp cut-off filter. All in vitro measurements were made in 4-mm quartz esr tubes at -170'C; in vivo measurements were made at room temperature on concentrated slurries of cells held in a Varian water cell (V4548).625