Fig. 2.

Parameterizations of single-molecule orientation and rotational mobility. (a) A rotationally fixed single molecule may be modeled as a fixed dipole with polar orientation $\text{Θ}$ and azimuthal orientation $\text{Φ}$. Alternatively, orientation may be described as a unit vector $\mu$, with x, y and z components ${\mu }_x$, ${\mu }_y$ and ${\mu }_z$ respectively. (b) Experimental schematic: A single molecule is placed a distance d from the focal plane of the objective, and a single widefield image is acquired. (c) Rotation within a cone model: A single molecule undergoes constrained rotation about some mean orientation $\left\{{\text{Φ}}_0,{\text{Θ}}_0\right\}$. The molecule may deviate by an angle $\alpha$ from the mean. (d) A molecule rotating in a cone may be alternatively parameterized by three orthogonal dipoles. One dipole will have amplitude equal to the square root of the largest eigenvalue of the M matrix, as defined in the main text. The other two dipoles will have amplitudes equal to the square root of the second largest eigenvalue. (e) In a more general case, a single molecule’s rotation may be confined to an elliptical region of the unit hemisphere, parameterized by two angles $\alpha$ and $\beta$. (f) For rotation within an elliptic region, the equivalent eigenvectors provide three different dipoles, each with a distinct amplitude determined from the square roots of the eigenvalues of the M matrix.