Fundamental Limits on Measuring the Rotational Constraint of Single Molecules Using Fluorescence Microscopy

Optical fluorescence imaging is capable of measuring both the translational and rotational dynamics of single molecules. However, unavoidable measurement noise will result in inaccurate estimates of rotational dynamics, causing a molecule to appear to be more rotationally constrained than it actually is. We report a mathematical framework to compute the fundamental limit of accuracy in measuring the rotational mobility of dipolelike emitters. By applying our framework to both in-plane and three-dimensional methods, we provide a means to choose the optimal orientation-measurement technique based on experimental conditions.

Figure 1

Distribution and bias of rotational constraint measurements ${\widehat{\gamma }}_{2\mathrm{D}}$ using in-plane excitation polarization modulation. (a) A noncentral chi PDF describes the distribution of ${\widehat{\gamma }}_{2\mathrm{D}}$ under $s=$ (i) 1000, (ii) 2000, and (iii) 3000 signal photons with ${\mathbf{1}}^{†}\mathit{b}=7290$ background photons per $526.5\times 526.5{\mathrm{nm}}^2$ ($3.5\times \mathrm{FWHM}$ of a diffraction-limited spot) region. Solid line: isotropic emitters, dashed line: emitter whose bias is smaller by $1/e$ compared to an isotropic emitter. Circles and diamonds on the $x$ axis represent the mean of the aforementioned two distributions, respectively. (b) Average bias in ${\widehat{\gamma }}_{2\mathrm{D}}$ versus the ground truth ${\gamma }_{2\mathrm{D}}$. Blue, green, and orange represent 1000, 2000, and 3000 signal photons, respectively. Circles and diamonds correspond to the same data points in (a). (c) Bias in the measured rotational constraint of isotropic emitters scales linearly with the inverse of SNR, $\sqrt{s+{\mathbf{1}}^{†}\mathit{b}}/s$. Solid circles: ${\widehat{\gamma }}_{2\mathrm{D}}$, open circles: ${\widehat{\gamma }}_{3\mathrm{D}}$.

Figure 2

Comparison of in-plane ${\gamma }_{2\mathrm{D}}$ and 3D ${\gamma }_{3\mathrm{D}}$ rotational constraints. (a) The relation between ${\gamma }_{2\mathrm{D}}$ and ${\gamma }_{3\mathrm{D}}$ varies with the average orientation along the $z$ axis, ${\overline{\mu }}_z$ [Eq. (8)]. Black line corresponds to ${\overline{\mu }}_z=\sqrt{1/3}\approx \cos 54.7°$ so that ${\gamma }_{2\mathrm{D}}={\gamma }_{3\mathrm{D}}$. (b) A dipole emitter with an out-of-plane orientation ${\overline{\mu }}_z=0.75$ that has ${\gamma }_{3\mathrm{D}}=0.81$ [a cone half-angle of 30°, open circle in (a)] appears to be more rotationally free in the $xy$ plane (${\gamma }_{2\mathrm{D}}=0.73$, wedge half-angle of 38°). (c) The mean difference $\mathrm{\Delta }\gamma$ across all 3D orientation space ranges from -1 to 0.06. Solid line represents the average across all possible values of ${\gamma }_{3\mathrm{D}}$, and the shaded region represents the range of the difference. (d) 3D enhancement factor ${\mathcal{F}}_{3\mathrm{D}}$ as a function of ${\gamma }_{3\mathrm{D}}$ and ${\overline{\mu }}_z$. Solid line represents the equilibrium point where in-plane and 3D rotational constraints measure changes in rotational dynamics with equal sensitivity. Dashed line represents a $\pm 10\%$ difference in sensitivity.

Figure 3

Bias and precision of rotational constraint measurements using various techniques. (a) Minimum detectable rotational constraint for various numbers of signal photons and 4860 total background photons. Solid line represents the expectation of measured rotational constraint ${\widehat{\gamma }}_{3\mathrm{D}}$ for isotropic emitters; dashed line represents the corresponding cone half-angle $\widehat{\alpha }$ if the molecule is symmetrically diffusing in a cone. (b) Concept of (i) a horizontally and (ii) a vertically oriented fluorescent molecule within a lipid membrane. (c) Distribution of 3D rotational constraint estimates ${\widehat{\gamma }}_{3\mathrm{D}}$. Solid line: true rotational constraint ${\gamma }_{3\mathrm{D}}=0.2$ (cone half-angle $\alpha =72°$); dashed line: ${\gamma }_{3\mathrm{D}}=0.8$ ($\alpha =31°$). (d) Bias of ${\widehat{\gamma }}_{3\mathrm{D}}$ for each technique. Green: in-plane excitation modulation, blue: Tri-spot PSF, red: standard PSF, orange: back focal plane imaging, purple: ideal basis-image matrix.