We formulate a model-independent description of time-dependent restricted molecular rotational motion and apply it to the time-resolved fluorescence depolarization signal from fluorescence-labeled molecules following polarized excitation. In this treatment, the time-dependent molecular orientation distribution is equal to the operation of a linear time-development operator on the initial orientation distribution. The formula for the time-development operator is derived from the equation of motion for the orientation distribution. The time-development operator is then expanded in time in terms of a complete set of orthonormal polynomials. When this expression for the orientation distribution is used to calculate the time-resolved fluorescence depolarization signal, properties of the orthonormal polynomials allow the signal to be quantitated uniquely in terms of matrix elements of the differential operators from the equation of motion. We demonstrate how to obtain the experimental value of the matrix elements directly from the fluorescence depolarization signal. In this paper, we also describe the application of the model-independent formalism to fluorescence-labeled myosin cross-bridges in muscle fibers when the fibers are in a variety of physiological states. We show that the analysis of time-resolved fluorescence depolarization data with the new formalism and the rotational diffusion in a potential model results in a slight revision of our previous estimate of the relaxed cross-bridge rotational diffusion constant and the determination of rank 6 order parameters that were ignored in the previous analysis [Burghardt, T. P., & Thompson, N. L. (1985) Biochemistry 24, 3731–3735]. The rank 6 order parameters are shown to make a significant contribution to the proposed steady-state angular distribution of the relaxed cross-bridges. Order parameters of rank 6 do not contribute to the steady-state fluorescence polarization signal and can only be detected with the time-resolved signal [Burghardt, T. P. (1984) Biopolymers 23, 2383–2406]. We have also applied the model-independent formalism to new data from fibers in rigor when the fibers are in a configuration such that the excitation light polarization is perpendicular to the fiber axis and the light propagates along the fiber axis, so that the fluorescence depolarization signal is sensitive to probe rotational motions about the fiber axis. We show that in this configuration the rigor cross-bridge is undergoing rapid rotational motion, probably due to conjugate motion in the actin filament, with a rate that is not significantly affected by the presence of calcium or when the fiber is decorated with extrinsic subfragment 1 of myosin.
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