Fundamental Limits in Measuring the Anisotropic Rotational Diffusion of Single Molecules

Abstract

Many biophysical techniques, such as single-molecule fluorescence correlation spectroscopy, Förster resonance energy transfer, and fluorescence anisotropy, measure the translation and rotation of biomolecules to quantify molecular processes at the nanoscale. These methods often simplify data analysis by assuming isotropic rotational diffusion, e.g., that molecules wobble within a circular cone. This simplification ignores the anisotropy present in many biological contexts that may cause molecules to exhibit different degrees of diffusion in different directions. Here, we loosen this assumption and establish a theoretical framework for describing and measuring anisotropic rotational diffusion using fluorescence imaging. We show that anisotropic wobble is directly quantified by the eigenvalues of a 3-by-3 positive-semidefinite Hermitian matrix $M$ consisting of the second-order moments of a molecule’s transition dipole $\mathit{\mu }$. This formalism enables us to model the influence of unavoidable shot noise using a Hermitian perturbation matrix $E$; the eigenvalues of $E$ directly bound errors in measurements of wobble via Weyl’s inequality. Quantifying various perturbations $E$ reveals that anisotropic wobble measurements are generally more sensitive to errors compared to quantifying isotropic wobble. Moreover, severe shot noise can induce negative eigenvalues in estimates of $M$, thereby causing the anisotropic wobble measurement to fail. Our analysis, using Fisher information, shows that techniques with worse orientation measurement sensitivity experience stronger perturbations $E$ and require larger signal to background ratios to measure anisotropic rotational diffusion accurately. Our work provides deep insights for improving the state of the art in imaging the orientations and anisotropic rotational diffusion of single molecules.

Graphical Abstract

Introduction

Since the first observations of the translations and rotations of molecular motors,^1–3 biophysists have been captivated by visualizing biomolecular motions with single-molecule (SM) sensitivity and nanoscale resolution.^4–6 Numerous technologies have been recently demonstrated^7–14 to resolve molecular positions and orientations with resolution beyond the diffraction limit, termed single-molecule orientation-localization microscopy (SMOLM).¹⁵ Moreover, several studies have established frameworks for measuring a molecule’s rotational diffusion, or wobble, during a camera’s integration time.^16–20 Often, the measurement is simplified by assuming that a molecule is rotating within a rotationally symmetric cone or potential well-an isotropic wobble model. Importantly, this simplification may not be appropriate in a variety of biological situations, e.g., when molecules are diffusing within a cell membrane.⁹

In this study, we describe anisotropic wobble using a 3-by-3 positive-semidefinite Hermitian matrix $M$ consisting of the second-order moments of a molecule’s transition dipole.^16,18 This formalism enables us to study how errors in the measurement $\hat{M}$, e.g., caused by Poisson shot noise, affect the precision and accuracy of estimating the molecule’s anisotropic wobble. Importantly, we find that the simpler isotropic wobble model is robust when noise is severe, while noise can cause an anisotropic wobble measurement to fail completely. However, bright molecules enable anisotropic wobble to be measured precisely and accurately, especially if a sensitive SMOLM technique is used. Our work establishes fundamental insights into the limits of measurement performance and provides a framework for future technological developments that will surpass the accuracy and precision of current technologies.

Methods

Modelling anisotropic rotational diffusion

A fluorescent molecule can be modelled as an oscillating electric dipole with orientation $\mathit{\mu }=\left({\mu }_x,{\mu }_y,{\mu }_z\right)=(\text{sin}\theta \text{cos}\varphi ,\text{sin}\theta \text{sin}\varphi ,\text{cos}\theta )$, where ${\mu }_z$ is the component of $\mathit{\mu }$ parallel to the optical axis, $\theta \in \left[0,{90}^{\circ }\right]$ is the polar angle, and $\varphi \in \left[0,{360}^{\circ }\right)$ is the azimuthal angle (Figure 1a). In biological imaging, fluorescent labels are generally neither rotationally fixed nor entirely free to rotate. Thus, we model anisotropic rotational diffusion using a hard-edged cone with a mean orientation $(\bar{\theta },\bar{\varphi })$ and two wobble cone half angles $\alpha$ and $\beta \left(\{\alpha ,\beta \}\in \left[0,{90}^{\circ }\right]\right.$ and $\alpha \le \beta )$. We assume the emission dipole is rotationally constrained within the cone and visits all orientations with uniform probability. When $\alpha \ne \beta$, the cone’s aperture is elliptical, and an angle $\psi \in \left[0,{180}^{\circ }\right)$ describes the rotation of this ellipse about the cone’s mean orientation (Figure 1a). When $\alpha =\beta$, the molecule undergoes isotropic rotational diffusion equivalent to wobble within a circular cone. When the cone is oriented along the $z$ axis with its elliptical minor and major axes aligned along ${\mu }_x$ and ${\mu }_y$, respectively, we term it an aligned cone (e.g., Figure 1b, purple). Otherwise, we term it a general wobble cone (Figure 1b, pink and yellow).

Figure 1:

Model of a dipole exhibiting anisotropic rotational diffusion. (a) Angular parameters $(\bar{\theta },\bar{\varphi },\alpha ,\beta ,\psi )$ quantifying anisotropic rotational diffusion, where $\bar{\theta }$ and $\bar{\varphi }$ are the polar angle and the azimuthal angle of the mean orientation, respectively, $\alpha$ and $\beta$ are two wobble cone half angles, and $\psi$ describes rotation of the cone around the mean orientation vector ${\mathbf{r}}_3$. (b) Three example anisotropic wobble cones. Purple: oriented along the $z$-axis $\left(0^{\circ },0^{\circ },{15}^{\circ },{45}^{\circ },0^{\circ }\right)$. Pink: oriented within the $xz$ plane $\left({45}^{\circ },0^{\circ },{15}^{\circ },{45}^{\circ },0^{\circ }\right)$. Yellow: a skewed orientation $\left({45}^{\circ },{30}^{\circ },{15}^{\circ },{45}^{\circ },{20}^{\circ }\right.$). (c) Relationship between the wobble cone parameters and the second-order moments $m_{ij},\{i,j\}\in \{x,y,z\}$, of a molecule’s transition dipole as a function of $\beta$, holding all other parameters fixed as in (b).

During an experiment, fluorescence emitted by the SM is collected and projected by the microscope onto a camera.²¹ The resulting $N$-pixel intensity image $\mathit{I}\in {\mathbb{R}}^{N\times 1}$ is linearly proportional to the second-order dipole moments $\mathit{m}$, not the first-order transition dipole moments $\mathit{\mu }$, and is given by

$$\begin{gathered} I=sBm+b \\ =s\left[\begin{array}{llllll}{B}_{xx}\hfill & B_{yy}\hfill & B_{zz}\hfill & B_{xy}\hfill & B_{xz}\hfill & B_{yz}\hfill \end{array}\right]m+b, \end{gathered}$$ (1)

where $s$ is the total signal photons detected from an emitter and $\mathit{b}\in {\mathbb{R}}^{N\times 1}$ is the background photons detected at each pixel. The basis images $B_{ij}\in {\mathbb{R}}^{N\times 1},\{i,j\}\in \{x,y,z\}$, are determined by the imaging system.^21,22 The second-moment vector $\mathit{m}=\left(m_{xx},m_{yy},m_{zz},m_{xy},m_{xz},m_{yz}\right)$ is related to the transition dipole $\mathit{\mu }$ via $m_{ij}=⟨{\mu }_i{\mu }_j⟩$, where $⟨\cdot ⟩$ represents a temporal average or an equivalent integration taken over the orientation domain. Thus, $\mathit{m}$ quantifies both a molecule’s mean orientation and its wobble during a camera’s integration time.

We may express the second-order moments $\mathit{m}$ using an equivalent 3-by-3 Hermitian matrix $M$ and its eigendecomposition, i.e.,

$$M=\left[\begin{array}{ccc}{m}_{xx}& m_{xy}& m_{xz}\\ m_{xy}& m_{yy}& m_{yz}\\ m_{xz}& m_{yz}& m_{zz}\end{array}\right]=R{D}_e{R}^{†},$$ (2)

where $R$ is an orthonormal matrix, $D_e$ is a diagonal matrix, and ^† represents a Hermitian transpose operator.^16,18 The rotation matrix $R$ is related to the parameters of the wobble cone via

$$R=R_{{\mu }_z}\left(\overline{\varphi }\right)R_{{\mu }_y}\left(\overline{\varphi }\right)R_{{\mu }_z}\left(\psi \right)=\left[\begin{array}{lll}{r}_1\hfill & r_2\hfill & r_3\hfill \end{array}\right],$$ (3)

where

$$r_1=\left[\begin{array}{c}\text{cos}\overline{\theta }\text{cos}\overline{\varphi }\text{cos}\psi -\text{sin}\overline{\varphi }\text{sin}\psi \\ \text{cos}\overline{\theta }\text{sin}\overline{\varphi }\text{cos}\psi +\text{cos}\overline{\varphi }\text{sin}\psi \\ -\text{sin}\overline{\theta }\text{cos}\psi \end{array}\right],$$ (4)

$$r_2=\left[\begin{array}{c}-\text{cos}\overline{\theta }\text{cos}\overline{\varphi }\text{sin}\psi -\text{sin}\overline{\varphi }\text{cos}\psi \\ -\text{cos}\overline{\theta }\text{sin}\overline{\varphi }\text{sin}\psi +\text{cos}\overline{\varphi }\text{cos}\psi \\ \text{sin}\overline{\theta }\text{sin}\psi \end{array}\right],$$ (5)

$$r_3=\left[\begin{array}{c}\text{sin}\overline{\theta }\text{cos}\overline{\varphi }\\ \text{sin}\overline{\theta }\text{sin}\overline{\varphi }\\ \text{cos}\overline{\theta }\end{array}\right],$$ (6)

${\mathbf{r}}_1,{\mathbf{r}}_2$, and ${\mathbf{r}}_3$ are the eigenvectors of $M$, and $R_{{\mu }_j}$ represents a rotation about the ${\mu }_j$ axis. The mean orientation $(\bar{\theta },\bar{\varphi })$ of the dipole is given by ${\mathbf{r}}_3$, while the remaining two vectors, ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, are functions of $\psi$ and represent the orientation of the ellipse (Figure 1a). Thus, the eigenvector matrix $R$ rotates an aligned wobble cone (oriented along ${\mu }_z$) to a mean orientation $(\bar{\theta },\bar{\varphi })$, with a subsequent counterclockwise rotation $\psi$ about the mean orientation ${\mathbf{r}}_3$.

Given the change of basis provided by $R$, we may view the diagonal eigenvalue matrix

$$D_e=\left[\begin{array}{ccc}{a}_e& 0& 0\\ 0& b_e& 0\\ 0& 0& c_e\end{array}\right]$$ (7)

as expressing the second-moment vector $\mathit{m}$ as an equivalent $\tilde{\mathit{m}}=\left({\tilde{m}}_{xx},{\tilde{m}}_{yy},{\tilde{m}}_{zz},{\tilde{m}}_{xy},{\tilde{m}}_{xz},{\tilde{m}}_{yz}\right)=\left(a_e,b_e,c_e,\mathrm{0,0},0\right)$ in a rotated coordinate system $\left({\tilde{\mu }}_x,{\tilde{\mu }}_y,{\tilde{\mu }}_z\right)$ with ${\tilde{\mu }}_z$ parallel to ${\mathbf{r}}_3$. The diagonal entries of $D_e$ are related to the spherical “cap” of the wobble cone $\tilde{S}$ via

$$a_e=\frac{1}{\Omega }{\iint }_{\tilde{S}}{\tilde{\mu }}_x^{2}d\tilde{S}=\frac{1}{\Omega }{\iint }_{\tilde{S}}{\text{sin}}^2\tilde{\theta }{\text{cos}}^2\tilde{\varphi }d\tilde{S},$$ (8)

$$b_e=\frac{1}{\Omega }{\iint }_{\tilde{S}}{\tilde{\mu }}_y^{2}d\tilde{S}=\frac{1}{\Omega }{\iint }_{\tilde{S}}{\text{sin}}^2\tilde{\theta }{\text{sin}}^2\tilde{\varphi }d\tilde{S},$$ (9)

$$c_e=\frac{1}{\Omega }{\iint }_{\tilde{S}}{\tilde{\mu }}_z^{2}d\tilde{S}=\frac{1}{\Omega }{\iint }_{\tilde{S}}{\text{cos}}^2\tilde{\theta }d\tilde{S},\text{and}$$ (10)

$$\Omega ={\iint }_{\tilde{S}}d\tilde{S}={\iint }_{\tilde{S}}\text{sin}\tilde{\theta }d\tilde{\theta }d\tilde{\varphi },$$ (11)

where the solid angle subtended by the cone is given by $\Omega$. Thus, the three eigenvalues directly quantify the extent of anisotropic rotational diffusion. Importantly, the purple, pink, and yellow wobble cones in Figure 1b share an identical eigenvalue matrix $D_e$ because these cones share the same shape and only differ in their mean orientation $(\bar{\theta },\bar{\varphi })$ and rotation $\psi$ (eq 3). Since the aligned wobble cone represented by $D_e$ has an average orientation pointing along ${\tilde{\mu }}_z$, it is straightforward to show that $c_e\ge a_e$ and $c_e\ge b_e$. See Supporting Information for additional properties. Equations 7–11 are general and applicable for any arbitrary degree of rotational diffusion.

Relationship between angular parameters $(\mathit{\alpha },\mathit{\beta })$ of an anisotropic wobble cone and the eigenvalues of ${\mathit{D}}_{\mathit{e}}$

We may relate the eigenvalues $\left(a_e,b_e,c_e\right)$ (eq 7) to the parameters of the aligned anisotropic cone $(\alpha ,\beta )$ (Figure 1a) using

$$a_e=\frac{1}{3\Omega }{\int }_0^{2\pi }\left(t^3-3t+2\right){\text{cos}}^2\varphi d\varphi ,$$ (12)

$$b_e=\frac{1}{3\Omega }{\int }_0^{2\pi }\left(t^3-3t+2\right){\text{sin}}^2\varphi d\varphi ,$$ (13)

$$c_e=\frac{1}{3\Omega }{\int }_0^{2\pi }1-t^{3}d\varphi ,$$ (14)

$$\Omega =2\pi -{\int }_0^{2\pi }td\varphi ,\text{and}$$ (15)

$$t=\text{cos}\left(\text{max}\theta \left(\varphi \right)\right)=\sqrt{1-\frac{{\text{sin}}^2\alpha {\text{sin}}^2\beta }{{\text{sin}}^2\beta {\text{cos}}^2\varphi +{\text{sin}}^2\alpha {\text{sin}}^2\varphi }}.$$ (16)

where $\text{max}\theta \left(\varphi \right)$ represents the boundary of the anisotropic wobble cone, given by the maximum value of $\theta$ as a function of $\varphi$.

Approximating anisotropic rotational diffusion using an isotropic cone model

Equations 12–16 describe how the eigenvalues of the second-moment matrix $M$ (eq 2) are related to the first-moment parameters $(\alpha ,\beta )$ of an anisotropic wobble cone. For an isotropic wobble cone, we have an additional constraint $\alpha =\beta$, i.e., the cone aperture is a perfect circle. Therefore, the eigenvalues of $M$ become

$$a_e=b_e=\frac{\pi }{3\Omega }\left({\text{cos}}^3\alpha -3\text{cos}\alpha +2\right),$$ (17)

$$c_e=\frac{2\pi }{3\Omega }\left(1-{\text{cos}}^3\alpha \right),\text{and}$$ (18)

$$\Omega =2\pi \left(1-\text{cos}\alpha \right).$$ (19)

We observe that all three eigenvalues, $a_e,b_e$, and $c_e$, depend on the half angle $\alpha$ of the isotropic cone.

If we use an isotropic cone model (eqs 17–19) to approximate second-moments $M$ that are consistent with anisotropic rotational diffusion, then an inaccurate estimate of $\alpha$ will result due to model mismatch. For all calculations in this work, we assume that the largest eigenvalue $c_e$ is most accurate, and therefore, we may “fit” an isotropic cone to any general second-moment matrix $M$ via eigendecomposition and eqs 2, 18, and 19. The resulting Jaccard index of using an isotropic cone $(\alpha =\beta )$ to approximate anisotropic rotational diffusion is shown in Figure S1.

Simulating measurement performance under noisy conditions

We simulated three imaging systems: the standard $x$- and $y$- polarized (xyPol),²³ polarized vortex,⁹ and radially and azimuthally polarized multi-view reflector (raMVR)¹³ SMOLMs. We assume the emission wavelength is 593 nm, NA = 1.4, and a spatially uniform background. Our simulations model fluorescent molecules located at a refractive index interface between glass and water (RI = 1.334). Typical images of molecules rotating within the purple, pink, and yellow cones in Figure 1b are shown in Figures S2–S4.

To compute an average range of eigenvalues in Figure 4(c), the wobble parameters were sampled extensively: $\alpha \in \left[0,{90}^{\circ }\right]$ sampled in 10° increments, $\beta \in \left[\alpha ,{90}^{\circ }\right]$ sampled in 10° increments, and $\psi \in \left[0,{180}^{\circ }\right)$ sampled in 30° increments. For each $(\alpha ,\beta ,\psi )$, we uniformly sampled the orientation hemisphere using 2513 mean orientations.

Figure 4:

Performance of measuring anisotropic wobble, as quantified by the eigenvalue range of $E_{\mathrm{C}\mathrm{R}\mathrm{B}}$ (eq 20) when using the (i) $x$- and $y$-polarized (xyPol),²³ (ii) polarized vortex,⁹ and (iii) multi-view reflector¹³ microscopes. Eigenvalue ranges $\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\rho }_1,{\rho }_2,{\rho }_3\right\}-\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\rho }_1,{\rho }_2,{\rho }_3\right\}$ (eq 31) are shown as a function of mean orientation (${\mu }_x,{\mu }_y$) for ground-truth wobble (a) $(\alpha ,\beta ,\psi )=\left({15}^{\circ },{45}^{\circ },0^{\circ }\right)$ and (b) $(\alpha ,\beta ,\psi )=\left({15}^{\circ },{45}^{\circ },{20}^{\circ }\right)$ and (c) an average of all possible wobble values $(\alpha ,\beta ,\psi )$ (see Methods for details). The orientations of the purple, pink, and yellow wobble cones in Figure 1b are shown as colored circles. A single colormap is used for each technique (i)-(iii).

Results and Discussion

Impact of second-moment perturbations on wobble measurement accuracy

In practical experiments, the detected images ${\mathit{I}}_{\text{noisy}}$, and consequently our measurements of rotational diffusion, are corrupted by unavoidable photon shot noise. To quantify the impact of noise-induced errors, we investigate estimates $\hat{M}$ that are corrupted by Hermitian perturbations $E$, i.e.,

$$\hat{M}=M+E,$$ (20)

where

$$E=\left[\begin{array}{ccc}\Delta m_{xx}& \Delta m_{xy}& \Delta m_{xz}\\ \Delta m_{xy}& \Delta m_{yy}& \Delta m_{yz}\\ \Delta m_{xz}& \Delta m_{yz}& \Delta m_{zz}\end{array}\right].$$ (21)

We first consider the second-order moments of an aligned wobble cone (Figure 1b, purple), where $R$ is the identity matrix, and $M=D_e$. A perturbation of the diagonal entry $m_{xx}$ gives a corrupted estimate of

$$\hat{M}=\left[\begin{array}{ccc}{a}_e+\Delta m_{xx}& 0& 0\\ 0& b_e& 0\\ 0& 0& c_e\end{array}\right].$$ (22)

In this case, only one eigenvalue is perturbed, and the accuracy of a rotational diffusion measurement gets worse monotonically as $\left|\Delta m_{xx}\right|$ gets larger.

To quantify measurement performance, we compute the Jaccard index, i.e., the intersection area of the ground truth and measured wobble cones after perturbation, normalized by the area of their union.²⁴ A Jaccard index of 1 represents a perfect measurement, while an index of 0 indicates 100% error. We observe that the Jaccard index drops dramatically when the perturbation is strong or if the perturbation causes an eigenvalue of $\hat{M}$ to become negative (Figure 2a). If an eigenvalue is negative, then the corresponding wobble cone is ill-defined (eqs 7–11). Similar conclusions can be drawn when perturbations are added to $m_{yy}$ (Figure S5b) and $m_{zz}$ (Figure 2b). Naturally, the larger the eigenvalue, the more robust it is to identical perturbations (Figure S5a–c).

Figure 2:

Impact of independent perturbations (a) $\Delta m_{xx}$, (b) $\Delta m_{zz}$, and (c) $\Delta m_{xy}$ on the accuracy of measuring anisotropic wobble when using anisotropic (red) and isotropic (grey) wobble models. A single perturbation is added to a diagonal entry ($m_{xx}$ in (a) (eq 22), $m_{zz}$ in (b)) or an off-diagonal entry ($m_{xy}$ in (c) (eq 23)). The ground truth for all cases is the $z$-oriented wobble cone $(\bar{\theta },\bar{\varphi },\alpha ,\beta ,\psi )=\left(0^{\circ },0^{\circ },{15}^{\circ },{45}^{\circ },0^{\circ }\right)$ (Figure 1b, purple). The maximum extent of each perturbation is limited by the bounds on the associated second-order moments $m_{ij}$ (see Supporting Information for details).

Perturbations of off-diagonal entries, which are equal to zero for the $z$-oriented cone in Figure 1b, have more dramatic effects. For example, perturbing $m_{xy}$ yields

$$\hat{M}=\left[\begin{array}{ccc}{a}_e& \Delta m_{xy}& 0\\ \Delta m_{xy}& b_e& 0\\ 0& 0& c_e\end{array}\right]$$ (23)

and eigenvalues $c_e$ and $\frac{1}{2}\left(a_e+b_e\pm {ϵ}_{xy}\right)$, where

$$a_e\ge \frac{1}{2}\left(a_e+b_e-{ϵ}_{xy}\right),$$ (24)

$$b_e\le \frac{1}{2}\left(a_e+b_e+{ϵ}_{xy}\right),\text{and}$$ (25)

$${ϵ}_{xy}=\sqrt{a_e^2-2a_e{b}_e+b_e^2+4\Delta m_{xy}^2.}$$ (26)

Thus, the measurement of anisotropic wobble is sensitive to errors $\Delta m_{xy}$ (Figure 2c). In contrast, since an isotropic cone’s wobble angle $\alpha$ can be estimated solely using the maximum eigenvalue $c_e$ (eq 18), its accuracy is largely unaffected by $\Delta m_{xy}$. Therefore, measurements of isotropic wobble exhibit relatively poor accuracy for small errors $\Delta m_{xy}$ due to model mismatch but become more accurate than the anisotropic model for large errors $\Delta m_{xy}$.

Similarly, perturbing $m_{xz}$ results in a skewed orientation estimate

$$\hat{M}=\left[\begin{array}{ccc}{a}_e& 0& \Delta m_{xz}\\ 0& b_e& 0\\ \Delta m_{xz}& 0& c_e\end{array}\right]$$ (27)

and eigenvalues $b_e$ and $\frac{1}{2}\left(a_e+c_e\pm {ϵ}_{xz}\right)$, where

$$a_e\ge \frac{1}{2}\left(a_e+c_e-{ϵ}_{xz}\right),$$ (28)

$$c_e\le \frac{1}{2}\left(a_e+c_e+{ϵ}_{xz}\right),\text{and}$$ (29)

$${ϵ}_{xz}=\sqrt{a_e^2-2a_e{c}_e+c_e^2+4\Delta m_{xz}^2}.$$ (30)

In this case, $c_e$ is affected instead of $b_e$. Thus, measurements of both anisotropic and isotropic rotational diffusion are impacted by errors $\Delta m_{xz}$ (Figure S5e). Similar conclusions can be drawn when $m_{yz}$ is perturbed (Figure S5f). Importantly, we observe that larger eigenvalues are more robust to errors than smaller ones. Among the diagonal perturbations, errors in $m_{zz}$ have the smallest impact on the Jaccard index (Figure S5a–c). Among the off-diagonal errors, $\Delta m_{xy}$, which affects only the smaller eigenvalues $a_e$ and $b_e$ (eqs 23–26), degrades the Jaccard index of the anisotropic wobble estimates more severely than others (Figure S5d–f).

We now consider the general anisotropic wobble cone, where there is now an intricate relationship between the orientation estimate $\hat{M}$ and its eigenvalues (eq 2) via the rotation matrix $R$ (eq 3). It is almost impossible to give interpretable insights directly from the analytical form of its eigendecomposition.²⁵ However, the effects of additive estimation errors can be quantified using Weyl’s inequality.²⁶ We denote the eigenvalues of the perturbation matrix $E$ as ${\rho }_1,{\rho }_2$, and ${\rho }_3$. The resulting perturbations of the eigenvalues of $\hat{M}$ are bounded by the minimum and maximum eigenvalues of $E$, i.e.,

$${\hat{\lambda }}_i-{\lambda }_i\in \left[\text{min}\left\{{\rho }_1,{\rho }_2,{\rho }_3\right\},\text{max}\left\{{\rho }_1,{\rho }_2,{\rho }_3\right\}\right],$$ (31)

where ${\lambda }_i,i\in \{\mathrm{1,2},3\}$, represents the eigenvalues of $M$, i.e., $a_e,b_e$, and $c_e$, and ${\hat{\lambda }}_i$ represents the corresponding eigenvalues of $\hat{M}$.

Revisiting the case of perturbing a single diagonal element $m_{xx}$ (eqs 2 and 22) for a general anisotropic wobble cone, we find that the eigenvalues of the second-moment estimates $\hat{M}$ are biased in accordance with the sign of $\Delta m_{xx}$. That is, when $\Delta m_{xx}$ is positive, ${\hat{\lambda }}_i-{\lambda }_i\in \left[0,\Delta m_{xx}\right]$, and the accuracy of the wobble measurement decreases. However, when $\Delta m_{xx}<0$, the estimated eigenvalues ${\hat{\lambda }}_i$ tend to be smaller than the ground truth ${\lambda }_i$ and, in the case of a severe perturbation, may become negative. If an eigenvalue is negative, then the wobble cone is ill-defined (eqs 7–11), and one of the estimated wobble cone half angles shrinks to zero. The resulting Jaccard index also drops to zero (e.g., when $\Delta m_{xx}<-0.044$ in Figure S6a, red curve, and $\Delta m_{xx}<-0.1$ in Figure S7a, red curve), and the anisotropic wobble measurement has failed. Similar observations can be found for errors in $m_{yy}$ and $m_{zz}$ (Figures S6b and c, S7b and c).

In contrast, the maximum estimated eigenvalue is relatively less likely to become negative with additive perturbations. Since the half angle $\alpha$ of an isotropic wobble cone can be calculated solely from this value (eq 18), we observe that measurements of isotropic wobble are generally more robust. Failures can still occur if the relative sizes of the eigenvalues swap (e.g., when $\Delta m_{xx}<-0.25$ in Figure S6a), causing the Jaccard indices of anisotropic and isotropic wobble to converge.

An error in the off-diagonal element $m_{xy}$ (eq 23) causes the eigenvalues of $E$ to become $\left\{0,\pm \Delta m_{xy}\right\}$, which then cause the eigenvalues of the measurement $\hat{M}$ to become

$${\hat{\lambda }}_i-{\lambda }_i\in \left[-\left|\Delta m_{xy}\right|,\left|\Delta m_{xy}\right|\right].$$ (32)

Notably, if $\left|\Delta m_{xy}\right|$ is larger than any of the ground-truth eigenvalues ${\lambda }_i$, irregardless of whether the error is positive or negative, then a negative eigenvalue is possible, causing measurement failure (Figures S6d and S7d). Similar effects occur with perturbations to $m_{xz}$ and $m_{yz}$ (Figures S6e and f, S7e and f). Particularly severe perturbations can cause the Jaccard index to drop to zero for both anisotropic and isotropic wobble (e.g., when $\Delta m_{yz}>0.25$ in Figure S7f).

Impact of Poisson shot noise on wobble measurement accuracy

To guide practitioners on how to obtain precise measurements of anisotropic rotational diffusion, we next quantify how shot noise affects measurement accuracy for various SMOLM^9,13,23,27 techniques. The best-possible variance of any unbiased estimator, e.g., of $\hat{M}$, is given by the Cramér-Rao lower bound (CRB), which is the inverse of the associated Fisher information matrix.²⁸ For images $\mathit{I}$ corrupted by Poisson shot noise, the 6-by-6 Fisher information matrix $𝒥$ for measuring the six orientational second-order moments is given by²²

$$𝒥=s^2\sum _{k=1}^N\frac{1}{I_k}{B}_k^{†}{B}_k,\text{where}$$ (33)

$$B_k=\left[\begin{array}{llllll}{B}_{xx,k}\hfill & B_{yy,k}\hfill & B_{zz,k}\hfill & B_{xy,k}\hfill & B_{xz,k}\hfill & B_{yz,k}\hfill \end{array}\right],$$ (34)

$B_{ij,k}$ represents the $k^{\text{th}}$ pixel of each basis image, $I_k$ gives the number of photons collected by pixel $k$ (including signal and background), $N$ is the total number of pixels, and $s$ is total number of photons detected from the SM. As FI decreases and the CRB becomes worse, measurement uncertainties become larger, thus leading to degraded wobble measurement performance.

To put the performance of existing techniques in context, we model a hypothetical imaging system with minimum CRB, i.e., one with a diagonal Fisher information (FI) matrix so that measurements of the second-order moments $m_{ij}$ are decoupled from one another.²⁸ Inspired by the ability of spatial mode sorting²⁹ to achieve optimal quantum-limited localization performance, we propose a basis image matrix that maps each second-order moment $m_{ij}$ into its own camera pixel, i.e.,

$$B_{\text{ideal}}=I_6,$$ (35)

where $I_6$ is the 6-by-6 identity matrix. The FI of this idealized instrument for each second-order moment $m_{ij}$ is large, uniform, and diagonal, i.e.,

$$\sum _{k=1}^6{B}_{ij,k}^2=1.$$ (36)

We note that since $m_{ij}\in [-0.5,0.5]$ for $i\ne j$ (see Supporting Information for details), the images ${\mathit{I}}_{\text{ideal}}$ produced by this idealized basis do not satisfy the non-negativity property of photon counting for certain values of $\mathit{m}$, and thus, $B_{\text{ideal}}$ is not physically realizable. However, the performance of this idealized system provides useful context as a bound on the best precision of any practical system.

To quantify the precision of various state-of-art SMOLM techniques, we used Monte Carlo simulations to measure the performance of wobble estimation on noise-corrupted images ${\mathit{I}}_{\text{noisy}}~\mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}\left(\mathit{I}\right)$ (eq 1) for various degrees of rotational wobble (see Methods and Figures S2–S4). Estimates of the second-order moments $\stackrel{\mathit{ˆ}}{\mathit{m}}$ and the equivalent matrix $\hat{M}$ are computed from ${\mathit{I}}_{\text{noisy}}$ using a maximum-likelihood estimator, which is asymptotically unbiased and efficient, i.e., it achieves estimation variances equal to the CRB.²⁸ The eigendecomposition of $\hat{M}$ then yields estimates of rotational diffusion, and the accuracy of the measurement is quantified by the Jaccard index.

When the SBR is low, the isotropic wobble model (Figure 3b) outperforms the anisotropic one (Figure 3a) in estimating the ground-truth wobble (i.e., the aligned wobble cone in Figure 1b, purple from ${\mathit{I}}_{\text{noisy}}$. At these low SBRs, negative eigenvalues are more likely due to Poisson shot noise (Figure 3c), and as these negative eigenvalues occur more frequently, the Jaccard index suffers regardless of imaging technique (Figure S10). The isotropic wobble model is inherently more robust to these negative eigenvalues (Figures S11, S12), given that it only requires a single, precise estimate of the largest eigenvalue of $\hat{M}$ to precisely quantify wobble. These observations are consistent with our explorations of specific perturbations to various second-order moments (Figures 2 and S5).

Figure 3:

Measurement accuracy of the anisotropic and isotropic wobble models using the $x$- and $y$-polarized (xyPol, green solid),²³ polarized vortex (cyan dash-dot),⁹ and multi-view reflector (raMVR, purple dashed)¹³ microscopes at various signal levels. For context, the performance of an idealized microscope (eq 35, red) is included. The ground truth for all cases is the $z$-oriented wobble cone $(\bar{\theta },\bar{\varphi },\alpha ,\beta ,\psi )=\left(0^{\circ },0^{\circ },{15}^{\circ },{45}^{\circ },0^{\circ }\right)$ (Figure 1b, purple). (a) Mean Jaccard index of noisy measurements using the anisotropic wobble model. (b) Same as (a) for the isotropic wobble model. The best Jaccard indices that the anisotropic and isotropic wobble models can achieve are 1 and 0.47, respectively. (c) Probability of the occurrence of negative eigenvalues when using the anisotropic wobble model. Measurements were taken from simulated 1,000 noisy images each at 200 signal levels $s\in \left[500,{10}^5\right]$. Note that the background photon flux originating from the sample is held constant across all methods for fair comparison.

For larger SBRs, measuring rotational dynamics using the anisotropic wobble model is more accurate, as shown by its larger Jaccard indices, than using the isotropic wobble model, whose Jaccard indices saturate at 0.47 (Figures 3a and b, S1, S13, and S14, see Methods for details). As expected, SMOLM designs with superior orientation sensitivities, i.e., smaller CRBs and smaller errors in $\hat{M}$, are better able to resolve anisotropic wobble using fewer photons. For example, when measuring the aligned wobble cone (Figure 1b, purple), ~5430 signal photons are required in order for a standard $x$- and $y$- polarized (xyPol) microscope²³ to quantify anisotropic wobble with greater accuracy than using an isotropic wobble model. More sensitive methods, such as the polarized vortex⁹ and radially and azimuthally polarized multi-view reflector (raMVR) microscopes,¹³ need many fewer photons (~3870 and ~2830, respectively). In contrast, an idealized imaging system using $B_{\text{ideal}}$ (eq 35) only requires ~700 photons to achieve the same goal. Analogous trends are also observed for general anisotropic cones (Figures S6–S9 and S15–S22).

We next quantify how the precision of measuring various second-order moments, as quantified by the CRB, are correlated with the performance of measuring anisotropic wobble. Assuming that an efficient unbiased estimator is used, we may model the effects of shot noise on the measurement $\hat{M}$ by letting

$$E_{\text{CRB}}=\left[\begin{array}{ccc}{\sigma }_{xx}& {\sigma }_{xy}& {\sigma }_{xz}\\ {\sigma }_{xy}& {\sigma }_{yy}& {\sigma }_{yz}\\ {\sigma }_{xz}& {\sigma }_{yz}& {\sigma }_{zz}\end{array}\right],$$ (37)

i.e., noise-induced perturbations of each orientational second-order moment are equal to their best-possible precisions, where the diagonal entries of the CRB matrix ${𝒥}^{-1}$ are given by $\left({\sigma }_{xx}^2,{\sigma }_{yy}^2,{\sigma }_{zz}^2,{\sigma }_{xy}^2,{\sigma }_{xz}^2,{\sigma }_{yz}^2\right).$²⁸ We may then compute the eigenvalue range $\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\rho }_1,{\rho }_2,{\rho }_3\right\}-\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\rho }_1,{\rho }_2,{\rho }_3\right\}$ of $E_{\text{CRB}}$. By Weyl’s inequality (eq 31), a smaller eigenvalue range leads to smaller perturbations in the measured eigenvalues under noisy conditions, a reduced occurrence of negative eigenvalues, and superior performance for measuring anisotropic wobble (eqs 12–16).

When the signal is 10,000 photons, we find that the eivenvalue ranges of $E_{\text{CRB}}$ of xyPol, vortex, raMVR, and the idealized microscope are 0.1248, 0.0494, 0.0458, and 0.0095, respectively (Figure 4), for an aligned wobble cone (Figure 1b, purple). Thus, smaller eigenvalue ranges of $E_{\mathrm{C}\mathrm{R}\mathrm{B}}$ lead to more accurate eigenvalue estimates of $\hat{M}$ and larger Jaccard indices for measuring anisotropic wobble (Figure 3); similar trends can be observed for a general wobble cone with $\psi ={20}^{\circ }$ (Figure 4b). In general, larger eigenvalue ranges for xyPol lead to lower Jaccard indices and relatively frequent negative eigenvalues. Additionally, raMVR exhibits the best measurement precisions and smallest eigenvalue ranges of $E_{\text{CRB}}$ over all possible wobble values (Figure 4c) and shows nearly uniform performance across all azimuthal orientations $\bar{\varphi }$ because of its direct detection of azimuthally and radially polarized light.^12,27,30,31

Conclusion

In summary, we propose a model for a rotationally diffusing dipole that expresses a molecule’s average orientation and wobble as an orthonormal rotation matrix $R$ and a diagonal matrix $D_e$, respectively (eq 2). Importantly, this eigendecomposition formalism^16,18 enables us to model anisotropic wobble⁹ and to quantify the impact of shot noise on wobble measurements by considering Hermitian perturbations (eq 20). Under low SBR conditions, we observe that the isotropic wobble model is relatively robust, but its accuracy is limited (Figure 3b). This tradeoff is inherent to choosing a simplified but mismatched wobble model.

In contrast, the anisotropic cone model, with its two additional parameters $\beta$ and $\psi$, is hampered by measurements $\hat{M}$ with negative eigenvalues (Figure 3c) at low SBRs, which cause the Jaccard index to drop to zero. At higher SBRs, the negative eigenvalues disappear, and the anisotropic wobble model becomes more accurate than the isotropic model (Figures 3a and S1). Fortunately, the two models may be evaluated side-by-side using identical data (Figs. S11–S22) to determine which is most suitable for a specific application. Fundamentally, when using sufficiently bright fluorophores, robustly measuring anisotropic rotational dynamics is entirely achievable via a properly calibrated microscope and image analysis algorithm.

We found that, since the uncertainty of any unbiased estimator of $\hat{M}$ is bounded by the CRB, SMOLM techniques with larger CRBs experience stronger perturbations $E$ and require larger SBRs to measure anisotropic rotational diffusion accurately (Figures 3 and 4). While other bounds exist,²⁶ we show that the range of eigenvalue perturbations bounded by Weyl’s inequality (eq 31) is effective for describing the relative performance of various SMOLM techniques for measuring anisotropic wobble; the technique that exhibits minimum perturbation will have the best performance. Of the techniques we studied, the raMVR microscope has the smallest eigenvalue ranges (Figure 4) and thus the best performance overall (Figures 3, S8, and S9).

The theoretical framework and practical simulations explored here open a variety of avenues for accurate and precise imaging of molecular dynamics at the nanoscale. For example, it should be possible to numerically optimize a phase mask¹¹ to measure anisotropic wobble by, for example, minimizing the eigenvalues of a CRB-based perturbation matrix. One should also be able to use both pumping polarization modulation and dipole-spread function engineering^32,33 to improve the accuracy of measuring anisotropic wobble. Perhaps most importantly, the development of brighter fluorophores^34–36 will yield higher SBRs that are necessary for high-precision wobble measurements. We eagerly anticipate continued development of single-molecule imaging technologies to enable cutting-edge biophysical experiments.^37–40

Data Availability Statement

Simulation data and code are available via OSF (https://osf.io/zkvmd/) and from the authors upon reasonable request.