Quantifying protein densities on cell membranes using super-resolution optical fluctuation imaging

TL;DR

This paper introduces a robust, model-free super-resolution imaging method to quantify the distribution of surface molecules on cell membranes, providing insights into their organization under various conditions.

Contribution

It presents a novel SOFI-based approach for measuring molecular densities on cell membranes that is resilient to blinking and high labeling densities, unlike traditional counting methods.

Findings

Validated with simulated data confirming robustness.

Applied to T cells to study CD4 protein distribution.

Demonstrated effectiveness in physiological conditions.

Abstract

Surface molecules, distributed in diverse patterns and clusters on cell membranes, influence vital functions of living cells. It is therefore important to understand their molecular surface organisation under different physiological and pathological conditions. Here, we present a model-free, quantitative method to determine the distribution of cell surface molecules based on TIRF illumination and super-resolution optical fluctuation imaging (SOFI). This SOFI-based approach is robust towards single emitter multiple-blinking events, high labelling densities and high blinking rates. In SOFI, the molecular density is not based on counting events, but results as an intrinsic property due to the correlation of the intensity fluctuations. The effectiveness and robustness of the method was validated using simulated data, as well as experimental data investigating the impact of palmitoylation on CD4 protein nanoscale distribution in the plasma membrane of resting T cells.

Equations

$$I(\mathbf{r},t)=\sum _{k=1}^N{ϵ}_{k}U({\mathbf{r}-\mathbf{r}}_k)s_k(t)+b(\mathbf{r})+n(\mathbf{r},t),$$

$${\kappa }_n\{I(\mathbf{r},t)\}(\tau )={\kappa }_n\left\{\sum _{k=1}^M{ϵ}_{k}U(\mathbf{r}-{\mathbf{r}}_k)s_k(t)+b(\mathbf{r})+n(\mathbf{r},t)\right\}(\tau ).$$

$${\kappa }_n\{I(\mathbf{r},t)\}=\sum _{k=1}^N{ϵ}_k^n{U}^n({\mathbf{r}-\mathbf{r}}_k){\kappa }_n\{s_k(t)\}+{\kappa }_n\{b(\mathbf{r})\}+{\kappa }_n\{n(\mathbf{r},t)\}.$$

$$\rho =\frac{{\tau }_{\mathrm{on}}}{{\tau }_{\mathrm{on}}+{\tau }_{\mathrm{off}}},$$

$$f_n({\rho }_{\mathrm{on}})={\rho }_{\mathrm{on}}(1-{\rho }_{\mathrm{on}})\frac{\partial f_{n-1}}{\partial {\rho }_{\mathrm{on}}}.$$

$${\kappa }_n\{I(\mathbf{r},\mathbf{t})\}\approx {ϵ}^n{f}_n({\rho }_{\mathrm{on}})\sum _{k=1}^N{U}^n({\mathbf{r}-\mathbf{r}}_k).$$

$$g_n(\mathbf{r})\approx {ϵ}^n(\mathbf{r})f_n({\rho }_{\mathrm{on}})N(\mathbf{r}){\mathcal{E}}_V\{U^n(\mathbf{r})\}.$$

$${\mathcal{E}}_V\{U_{\mathrm{TIR}}^n(\mathbf{r})\}=\frac{c({\sigma }_{x,y},{\sigma }_z,d_z)}{n^2},$$

$$K_1=\frac{g_{n-1}}{g_{n-2}}=\frac{f_{n-1}({\rho }_{\mathrm{on}}(\mathbf{r}))}{f_{n-2}({\rho }_{\mathrm{on}}(\mathbf{r}))}ϵ(\mathbf{r})\frac{{\mu }_{n-1}}{{\mu }_{n-2}}$$

$$K_2=\frac{g_n}{g_{n-2}}=\frac{f_n({\rho }_{\mathrm{on}}(\mathbf{r}))}{f_{n-2}({\rho }_{\mathrm{on}}(\mathbf{r}))}{ϵ}^2(\mathbf{r})\frac{{\mu }_n}{{\mu }_{n-2}}$$

$$K_3=\frac{g_n}{g_{n-1}}=\frac{f_n({\rho }_{\mathrm{on}}(\mathbf{r}))}{f_{n-1}({\rho }_{\mathrm{on}}(\mathbf{r}))}ϵ(\mathbf{r})\frac{{\mu }_n}{{\mu }_{n-1}},$$

$$\frac{g_n{g}_{n-2}}{g_{n-1}^2}=\frac{f_n({\rho }_{\mathrm{on}}(\mathbf{r}))f_{n-2}({\rho }_{\mathrm{on}}(\mathbf{r}))}{f_{n-1}^2({\rho }_{\mathrm{on}}(\mathbf{r}))}.$$

$$\left\{ϵ(\mathbf{r})=\frac{g_n}{g_{n-1}}\frac{f_3({\rho }_{\mathrm{on}}(\mathbf{r}))}{f_n({\rho }_{\mathrm{on}}(\mathbf{r}))}\frac{{\mu }_{n-1}}{{\mu }_n},N(\mathbf{r})=\frac{g_n(\mathbf{r})}{{ϵ}^n(\mathbf{r})f_n({\rho }_{\mathrm{on}}){\mu }_n}\right\}.$$

$$K_1(\mathbf{r})=\frac{{\mu }_2{g}_3}{{\mu }_3{g}_2}(\mathbf{r})=ϵ(\mathbf{r})(1-2{\rho }_{\mathrm{on}}(\mathbf{r}))$$

$$K_2(\mathbf{r})=\frac{{\mu }_2{g}_4}{{\mu }_4{g}_2}(\mathbf{r})={ϵ}^2(\mathbf{r})(1-6{\rho }_{\mathrm{on}}(\mathbf{r})+6{\rho }_{\mathrm{on}}^2(\mathbf{r})),$$

$$\left\{{\rho }_{\mathrm{on}}(\mathbf{r})=\frac{3K_1^2\pm K_1\sqrt{3K_1^2-2K_2}-2K_2}{2(3K_1^2-2K_2)},ϵ(\mathbf{r})=\mp \sqrt{3K_1^2-2K_2}\right\},$$

$$N(\mathbf{r})=\frac{g_2(\mathbf{r})}{{ϵ}^2(\mathbf{r}){\rho }_{\mathrm{on}(\mathbf{r})}(1-{\rho }_{\mathrm{on}}(\mathbf{r}))}.$$

$$K_1(\mathbf{r})=\frac{ϵ(\mathbf{r})(1-6{\rho }_{\mathrm{on}}+6{\rho }_{\mathrm{on}}^2)}{(1-2{\rho }_{\mathrm{on}}(\mathbf{r}))}$$

$$K_2(\mathbf{r})={ϵ}^2(\mathbf{r})(12{\rho }_{\mathrm{on}}^2-12{\rho }_{\mathrm{on}}+1)$$

$${\rho }_{\mathrm{on}}(\mathbf{r}{)}_{1,2}=\frac{12K_1^3\pm \sqrt{3}\sqrt{K_1^2(4K_1^2-3K_2)}\sqrt{4K_1^2\mp 2\sqrt{K_1^2(4K_1^2-3K_2)}-3K_2}-9K_1{K}_2}{6K_1(4K_1^2-3K_2)},$$

$$ϵ(\mathbf{r}{)}_{1,2}=\frac{\sqrt{4K_1^2\mp 2\sqrt{K_1^2(4K_1^2-3K_2)}-3K_2}}{\sqrt{3}}.$$

Full text

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See pages - of LukesSOFI2017ms.pdf

Supplementary Material for the article:

Quantifying protein densities on cell membranes using super-resolution optical fluctuation imaging

Tomáš Lukeš1,2, Daniela Glatzová3,4, Zuzana Kvíčalová3, Florian Levet5,6, Aleš Benda3,7,Tomáš Brdička4, Theo Lasser1 & Marek Cebecauer3

1Laboratoire d’Optique Biomédicale, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

2Department of Radioelectronics, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic

3J. Heyrovsky Institute of Physical Chemistry, Academy of Sciences, Prague, Czech Republic

4Institute of Molecular Genetics, Czech Academy of Sciences, Prague, Czech Republic

5Interdisciplinary Institute for Neuroscience, UMR 5297 CNRS, Université de Bordeaux,

Bordeaux, France

6Bordeaux Imaging Center, UMS 3420 CNRS, Université de Bordeaux, US4 INSERM,

Bordeaux, France

7Imaging methods core facility, BIOCEV, Vestec u Prahy, Czech Republic

Supplementary Note: SOFI density estimation

The technical requirements for SOFI are a classical widefield microscope merged with a fast high sensitivity digital camera. SOFI image processing is based on higher order statistics and exploits the temporal sequence of blinking fluorescent emitters [3, 4]. Calculating spatio-temporal cross-cumulants allows SOFI to obtain a super-resolved, background-free and noise-reduced images. Higher-order cumulants contain information about the photo-physics of the emitters. Combining SOFI images of different cumulant orders, allows one to extract physical parameters like molecular density [5], which we applied to investigate plasma membrane distribution of proteins.

1.1 SOFI principle and theory

As stated by Dertinger et al. [3], the fluctuating emitters should switch between at least two optically distinguishable states (e.g. a dark and a bright state) repeatedly and independently in a stochastic manner. Images of stochastically blinking emitters are recorded such that the point-spread function (PSF) extends over several camera pixels. Acquiring a sequence of images results in a time dependent intensity trace for each pixel. Assuming N independently fluctuating emitters, the detected intensity is given as

[TABLE]

where $\epsilon_{k}$ is the molecular brightness, $U(\mathbf{r-r}_{k})$ is the PSF at the position $\mathbf{r}_{k}$, $s_{k}(t)$ denotes a switching function (normalized fluctuation sequence, $s_{k}(t)\in\{0,1\}$), $b(\mathbf{r})$ is a constant background, and $n(\mathbf{r},t)$ represents an additive noise contribution.

For each pixel, an $n^{th}$ order cumulant is calculated for disentangling emitters inside the PSF. By applying the $n^{th}$ order cumulant to Eq. (1), we obtain

[TABLE]

Using additivity and semi-invariance properties of cumulants [6], the $n^{th}$ order cumulant with zero time lag can be written as

[TABLE]

For ($n\geq 2$), the Gaussian noise ($\kappa\{n(\mathbf{r},t)\}$)) and stationary background ($\kappa\{b(\mathbf{r})\}$ ) terms are eliminated by the cumulant analysis as an intrinsic property of cumulants. For an $n^{th}$ order cumulant, the PSF is raised to the $n^{th}$ power (see Eq. 3). As a consequence, the PSF is narrowed and the spatial resolution is improved by a factor of $\sqrt{n}$ [3]. Therefore, increasing the cumulant order yields an image with an enhanced spatial resolution. Since a multiplication in the spatial domain corresponds to a convolution in the frequency domain, the cut-off frequency of the spectrum $\tilde{U}^{n}\left(\mathbf{k}\right)$ is n-times higher than that of $\tilde{U}\left(\mathbf{k}\right)$. By applying a deconvolution and a subsequent rescaling, the $n^{th}$ order cumulant image exhibits an up to n-fold resolution improvement [4]. As shown in [4], virtual pixels can be calculated in between the physical pixels using cross-cumulants and followed by a flattening operation i.e. assigning proper weights to these virtual pixels [4, 7, 8].

SOFI assumes a blinking model where the fluorophores reversibly switch between a bright and a dark state. In Deschout et al. [9], SOFI was applied to the PALM photo-physical model. In the PALM photo-physical model, the emitter activation is assumed as non-reversible, however, once the emitter is activated, it exhibits several fast blinking events prior to the final bleaching event [10]. The emitter fluctuates between two different states (an on-state $S_{\mathrm{on}}$ and a dark state $S_{\mathrm{off}}$), which is expressed by the on-time ratio as

[TABLE]

where $\tau_{\mathrm{on}}$ and $\tau_{\mathrm{off}}$ are the characteristic lifetimes of the $S_{\mathrm{on}}$ and $S_{\mathrm{off}}$ states. The $n^{th}$ order cumulant $\kappa_{n}\{s_{k}(t)\}$ is in this model described by a Bernoulli distribution with probability $\rho_{\mathrm{on}}$ [5] and approximated by an $n^{th}$ order polynomial function for the on-time ratio as

[TABLE]

Under these conditions, the $n^{th}$ order cumulant can be approximated as [5]

[TABLE]

1.2 Estimation of density maps

Geissbuehler et al. [5] used three cumulant images ($2^{\text{nd}}$, $3^{\text{rd}}$, and $4^{\text{th}}$ order) to estimate molecular parameters: on-time ratio, brightness and molecular density. Here, we generalize this concept to any three cumulant images of distinct orders. If we assume spatially varying but locally constant on-time ratios and molecular brightness, the cumulants (for the cumulant order $n>1$) can be approximated by [5]

[TABLE]

where $\mathcal{E}_{V}\{U^{n}(\mathbf{r})\}$ is the expectation value of $U^{n}(\mathbf{r})$, $N(\mathbf{r})$ is the number of emitters inside a detection volume V. Approximating the PSF near the interface in a total internal reflection (TIR) configuration by a lateral 2D Gaussian profile combined with an axial exponential profile, we obtain

[TABLE]

where $d_{z}$ represents the exponential decay of the TIR illumination [11].

Using 3 consecutive cumulant images of orders $n,(n-1),(n-2)$, we obtain for the ratios

[TABLE]

where $\mu_{n}=\mathcal{E}_{V}\{U^{n}(\mathbf{r})\}$. Substitution of Eq. (11) into Eq. (9) leads to

[TABLE]

[TABLE]

Building up the ratios $K_{1}$ and $K_{2}$ from the Eq. (14) for cumulants of $2^{nd},3^{rd},$ and $4^{th}$ order, we obtain

[TABLE]

Solving for molecular brightness $\epsilon$, on-time ratio $\rho_{\mathrm{on}}$, we obtain two solutions for the on-time ratio $\rho_{\mathrm{on}}$, and molecular brightness $\epsilon$

[TABLE]

where the first solution corresponds to a negative brightness and will be discarded. The molecular density (number of molecules $N$ per pixel area) is

[TABLE]

For cumulants of $3^{rd},4^{th},$ and $5^{th}$ order, the ratios $K_{1}$ and $K_{2}$ become

[TABLE]

which ends in four solutions. Two correspond to positive molecular brightness

[TABLE]

Using a combination of higher order cumulants for molecular parameters can theoretically provide higher spatial resolution of the molecular parameter maps assuming high enough SNR of the cumulant images used. For the combination of $4^{th},5^{th},6^{th}$ order cumulant, it is also possible to find a solution in a closed form, but due to its complexity, a numerical approach might be preferred.

Therefore SOFI extracts density without counting individual events in the image. Density simply results from a correlation/cumulant analysis of intensity time traces.

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