Correcting for incomplete saturation and off-resonance effects in multiple-site saturation-transfer kinetic measurements

The effects of incomplete saturation and off-resonance irradiation on nuclear magnetic resonance saturation-transfer measurements of three-site chemical-exchange rates are discussed. A new method that uses double-saturation measurements is compared with two published methods, one that uses single-saturation measurements and one that uses a single-saturation measurement and a double-saturation measurement. Several formulas are compared for measuring the exchange rate constant k(DE) for exchange from a detected spin D to an exchanging spin E in the presence of exchange from spin D to a competing spin C. For each method, formulas are derived with corrections for incomplete saturation or off-resonance effects, with both corrections, and with neither correction. Exact formulas are available for three exchanging sites with incomplete saturation if there are no off-resonance effects. Off-resonance corrections are imperfect even with complete saturation.     

Effects of off-resonance irradiation, cross-relaxation, and chemical exchange on steady-state magnetization and effective spin-lattice relaxation times

In the presence of an off-resonance radiofrequency field, recovery of longitudinal magnetization to a steady state is not purely monoexponential. Under reasonable conditions with zero initial magnetization, recovery is nearly exponential and an effective relaxation rate constant R_1eff = 1/T_1eff can be obtained. Exact and approximate formulas for R_1eff and steady-state magnetization are derived from the Bloch equations for spins undergoing cross-relaxation and chemical exchange between two sites in the presence of an off-resonance radiofrequency field. The relaxation formulas require that the magnetization of one spin is constant, but not necessarily zero, while the other spin relaxes. Extension to three sites with one radiofrequency field is explained. The special cases of off-resonance effects alone and with cross-relaxation or chemical exchange, cross-relaxation alone, and chemical exchange alone are compared. The inaccuracy in saturation transfer measurements of exchange rate constants by published formulas is discussed for the creatine kinase reaction.     

Photophysical properties of a series of electron-donating and -withdrawing platinum acetylide two-photon chromophores

To explore spectroscopic structure-property relationships in platinum acetylides, we synthesized a series of complexes having the molecular formula trans-bis(tributylphosphine)-bis(4-((9,9-diethyl-7-ethynyl-9H-fluoren-2-yl)ethynyl)-R)-platinum. The substituent, R = NH(2), OCH(3), N(phenyl)(2), t-butyl, CH(3), H, F, benzothiazole, CF(3), CN, and NO(2), was chosen for a systematic variation in electron-donating and -withdrawing properties as described by the Hammett parameter σ(p). UV/vis, fluorescence, and phosphorescence spectra, transient absorption spectra on the fs-ps time scale, and longer time scale flash photolysis on the ns time scale were collected. DFT and TDDFT calculations of the T(1) and S(1) energies were performed. The E(S) and E(T) values measured from linear spectra correlate well with the calculated results, giving evidence for the delocalized MLCT character of the S(1) state and confinement of the T(1) exciton on one ligand. The calculated T(1) state dipole moment ranges from 0.5 to 14 D, showing the polar, charge-transfer character of the T(1) state. The ultrafast absorption spectra have broad absorption bands from 575 to 675 nm and long wavelength contribution, which is shown from flash photolysis measurements to be from the T(1) state. The T(1) energy obtained from phosphorescence, the T(1)-T(n) transition energy obtained from flash photolysis measurements, and the triplet-state radiative rate constant are functions of the calculated spin density distribution on the ligand. The calculations show that the triplet exciton of chromophores with electron-withdrawing substituents is localized away from the central platinum atom, red-shifting the spectra and increasing the triplet-state lifetime. Electron-donating substituents have the opposite effect on the location of the triplet exciton, the spectra, and the triplet-state lifetime. The relation between the intersystem crossing rate constant and the S(1)-T(1) energy gap shows a Marcus relationship with a reorganization energy of 0.83 eV. The calculations show that intersystem crossing occurs by conversion from a nonpolar, delocalized S(1) state to a polar, charge-transfer T(1) state confined to one ligand, accompanied by conformation changes and charge transfer, supporting the experimental evidence for Marcus behavior.     

Monotone second-best optimal contracts

We review the two stage nonlinear programming approach to the principal-agent problem proposed by Grossman and Hart (G-H) in 1983. The objective of the principal is to determine an optimal payment schedule to an agent that depends only on observable outcomes when (random) outcomes are influenced by unobservable actions of the agent. We establish that these ‘second-best’ optimal contracts are monotonically relted to observable outputs under fairly mild conditions. Two methods of proof are employed to establish that the monotone likelihood ratio condition (MLRC) on the probabilities of outcomes is sufficient for monotonicity of second-best optimal contracts in a three-state model. The first uses ideas related to the variation-diminishing property of totally positive matrices. The second approach uses a theorem of the alternative from linear programming theory to establish a condition that must hold under MLRC. The paper concludes with a counter-example that is a perturbed version of an example in G-H that shows that MLRC is not sufficient for monotonicity of second-best optimal contracts when there are four or more states.