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

Abstract

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)(Δθ)>-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> ≈ 0.9. The latter corresponds to an ≈14° average deviation of the individual trajectories from the mean trajectory (and mesopore) orientation in each domain.

Single-Molecule Investigations of Morphology and Mass Transport Dynamics in Nanostructured Materials

“…These materials offer 1D trajectories (such as in Figure 4). Recent investigations of MPS (71,(91)(92)(93)(94)(95) provide numerous examples, but similar results have also been obtained from phase-separated BCPs (90,102), MCPs (88), and LCs (82).…”

“…Immobile molecules produced a U-shaped distribution owing to Step angle (°) When the diffusive step angles are measured relative to a fixed direction, these same data can be used both to detect 1D diffusion and to determine the average direction of the 1D motions. Tran Ba et al (93) employed such an analysis for initial assessment of molecular motions in CTABfilled MPS films (Figure 7a,b). Unfortunately, such determinations are of limited utility because the localization precision and single-frame step size are frequently of similar magnitude, yielding relatively broad step-angle distributions.…”

Trajectory-Profile-Guided Single Molecule Tracking for Assignment of One-Dimensional Diffusion Trajectories

“…29,32,48,49,51 Such assessments require the accurate measurement of the inplane orientations of 1D trajectories. Therefore, trajectory angle accuracy also represents an important parameter upon which to judge the efficacy of TPG tracking.…”

“…While some of the aforementioned methods have been applied in studies of diffusion in 1D nanostructured materials, 29,32,48,49 they are not well suited to tracking strongly anisotropic motions. Most 1D SMT studies to date have employed cost-functional methods with varying success.…”

Following Single Molecules to a Better Understanding of Self-Assembled One-Dimensional Nanostructures

“…11,44,47 Figure 3 provides representative trajectory data obtained under common SMT conditions. In these particular data, the angle of a single step can only be determined to within ∼±26°.…”