Effect of finite trajectory length on the correlation function analysis of single molecule data.

The effect of finite trajectory length on single molecule rotational correlation functions has been studied by utilizing time series analysis and numerical simulations. Correlation functions obtained from the trajectories of length less than 100 times the correlation time constant (tau([script-l])) exhibit significant deviations from the true correlation function. The distributions of sample time constants (tau(F)) and stretching exponents (Beta(F)) are mapped by fitting a large number of rotational trajectories to stretched exponentials. As the trajectory length gets smaller, the distributions become broader and asymmetric and their mean values deviate from the true value predicted by pure rotational diffusion. Analysis based on higher order spherical harmonics is suggested as a method for minimizing the effect of the trajectory length. The distributions of time constants for different higher order spherical harmonics are also compared. While the focus of the paper is on rotational correlation functions, the general conclusions apply to any dynamical process that yields an exponentially decaying correlation function.