Generalized Haldane Equation and Fluctuation Theorem in the Steady State Cycle Kinetics of Single Enzymes

Enyzme kinetics are cyclic. We study a Markov renewal process model of single-enzyme turnover in nonequilibrium steady-state (NESS) with sustained concentrations for substrates and products. We show that the forward and backward cycle times have idential non-exponential distributions: \QQ_+(t)=\QQ_-(t). This equation generalizes the Haldane relation in reversible enzyme kinetics. In terms of the probabilities for the forward ($p_+$) and backward ($p_-$) cycles, $k_BT\ln(p_+/p_-)$ is shown to be the chemical driving force of the NESS, $\Delta\mu$. More interestingly, the moment generating function of the stochastic number of substrate cycle $\nu(t)$, follows the fluctuation theorem in the form of Kurchan-Lebowitz-Spohn-type symmetry. When $\lambda$ = $\Delta\mu/k_BT$, we obtain the Jarzynski-Hatano-Sasa-type equality: $\equiv$ 1 for all $t$, where $\nu\Delta\mu$ is the fluctuating chemical work done for sustaining the NESS. This theory suggests possible methods to experimentally determine the nonequilibrium driving force {\it in situ} from turnover data via single-molecule enzymology.Comment: 4 pages, 3 figure

Phase-reference VLBI Observations of the Compact Steep-Spectrum Source 3C 138

We investigate a phase-reference VLBI observation that was conducted at 15.4 GHz by fast switching VLBA antennas between the compact steep-spectrum radio source 3C 138 and the quasar PKS 0528+134 which are about 4$^\circ$ away on the sky. By comparing the phase-reference mapping with the conventional hybrid mapping, we demonstrate the feasibility of high precision astrometric measurements for sources separated by 4$^\circ$. VLBI phase-reference mapping preserves the relative phase information, and thus provides an accurate relative position between 3C 138 and PKS 0528+134 of $\Delta\alpha=-9^m46^s.531000\pm0^s.000003$ and $\Delta\delta=3^\circ6^\prime26^{\prime\prime}.90311\pm0^{\prime\prime}.00007$ (J2000.0) in right ascension and declination, respectively. This gives an improved position of the nucleus (component A) of 3C 138 in J2000.0 to be RA=$05^h 21^m 9^s.885748$ and Dec=$16^\circ 38' 22''.05261$ under the assumption that the position of calibrator PKS 0528+134 is correct. We further made a hybrid map by performing several iterations of CLEAN and self-calibration on the phase-referenced data with the phase-reference map as an input model for the first phase self-calibration. Compared with the hybrid map from the limited visibility data directly obtained from fringe fitting 3C 138 data, this map has a similar dynamic range, but a higher angular resolution. Therefore, phase-reference technique is not only a means of phase connection, but also a means of increasing phase coherence time allowing self-calibration technique to be applied to much weaker sources.Comment: 9 pages plus 2 figures, accepted by PASJ (Vol.58 No.6