Single exponential approximation, time averages

Finally, the ensemble-averaged quantity < 1F(Z. .. z r, /)>yy gives the probability of donors surviving to time t. Monte Carlo calculations have shown that in many cases the pair survival probability is approximated well by a single exponential except at very short times [79a, 81]. The rate constant /cd is related to the geometric and dif-fusional characteristics of the system by the equation [79a]... 

Reactions that occur between components in the bulk solution and vesicle-bound components, i.e., reactions occurring across the membrane interface, can be treated mathematically as if they were bimolecular reactions in homogeneous solution. However, kinetic analyses of reactions on the surface of mesoscopic structures are complicated by the finiteness of the reaction space, which may obviate the use of ordinary equations of chemical kinetics that treat the reaction environment as an infinite surface populated with constant average densities of reactant molecules. As was noted above, the kinetics of electron-transfer reactions on the surface of spherical micelles and vesicles is expressed by a sum of exponentials that can be approximated by a single exponential function only at relatively long times [79a, 81], At short times, the kinetics of the oxidative quenching of excited molecules on these surfaces are approximated by the equation [102]... 

This multiexponential approximation (one for each spherical harmonic) is more accurate than a single exponential description, yet still only depends on Dx and equilibrium averages. Equations (35) thereby provide a reasonable way of estimating from simulations D of lipids or other probes in membranes (assuming they can be modeled as rigid cylinders on the time-scale of rotational diffusion). Figure 6 plots these functions for parameters that model the long-axis rotation of a lipid in a bilayer Dx = 10 s, P ) = 0.8836, (P2) = 0.7, and Pi,) = 0.3467 (these Pi) are consistent with a Maier-Saupe potential with a field value of 3.8646 bT). Included (in dashed lines) are Ci(r) and C2(0 for unrestricted rotational diffusion. Note that the two relaxation times are somewhat shorter and closer to each other when diffusion is restricted. 

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