Trajectory angle determination in one dimensional single molecule tracking data by orthogonal regression analysis

The quantitative assessment of single molecule diffusion trajectories by orthogonal regression analysis is reported. This analysis is broadly applicable to any single particle tracking experiments in which diffusion along one dimension (1D) is expected. It affords quantitative data on the (in plane) orientation of 1D trajectories, allowing for their absolute orientations to be determined. Histograms depicting the distribution of trajectory angles provide new physical insights into the degree of trajectory alignment, and by inference, materials order. Estimates of the errors in the trajectory angle and particle positioning along each trajectory are also obtained. The angle results are compared to those from single-step angle determinations. The advantages of the regression method include its simplicity and computational efficiency, and the ability to objectively differentiate between 1D and 2D/immobile trajectories. Its utility is demonstrated through analysis of single molecule diffusion trajectories in surfactant-templated mesoporous silica films as probed by wide-field fluorescence microscopy. The trajectory angle histograms obtained provide quantitative data on mean trajectory orientation and the degree of trajectory alignment in distinct populations and sample regions. Mesopore order was quantitatively assessed by implementation of an order parameter, < P > = 2 < cos(2)(Delta theta)> - 1, calculated from the individual trajectory angles in each of four representative sample regions. The results depict the presence of well-ordered domains (from microns to tens of microns in size), all having < P > approximate to 0.9. The latter corresponds to an approximate to 14 degrees average deviation of the individual trajectories from the mean trajectory (and mesopore) orientation in each domain.