Identifying intermolecular interactions in single-molecule localization microscopy

Abstract

Intermolecular interactions underlie all cellular functions, yet visualizing these interactions at the single-molecule level remains challenging. Single-molecule localization microscopy (SMLM) offers a potential solution. Given a nanoscale map of two putative interaction partners, it should be possible to assign molecules either to the class of coupled pairs or to the class of noncoupled bystanders. Here, we developed a probabilistic algorithm that allows accurate determination of both the absolute number and the proportion of molecules that form coupled pairs. The algorithm calculates interaction probabilities for all possible pairs of localized molecules, selects the most likely interaction set, and corrects for any spurious colocalizations. Benchmarking this approach across a set of simulated molecular localization maps with varying densities (up to ∼55 molecules μm^-2) and localization precisions (1 to 50 nm) showed typical errors in the identification of correct pairs of only a few percent. At molecular densities of ∼5 to 10 molecules μm^-2 and localization precisions of 20 to 30 nm, which are typical parameters for SMLM imaging, the recall was ∼90%. The algorithm was effective at differentiating between noninteracting and coupled molecules both in simulations and experiments. Finally, it correctly inferred the number of coupled pairs over time in a simulated reaction-diffusion system, enabling determination of the underlying rate constants. The proposed approach promises to enable direct visualization and quantification of intermolecular interactions using SMLM.

Keywords: biomolecular interactions; inverse problem; probabilistic model; single-molecule.

Fig. 1.

Defining and estimating the proximity probability $P_{\text{prox}}$. (A) Illustration of three conformational states of a complex between two fluorescently labeled membrane proteins, A and B, with a variable distance $d_{\text{true}}$ between the dyes, where the random variable $d_{\text{true}}$ depends on the conformational state of the proteins and the flexibility of the dye linkers. Created using Biorender. EC and IC indicate extracellular and intracellular regions, respectively. (B) The observed distance $d_{\text{obs}}$ is a random variable. Left: The magenta rings represent the Gaussian distribution centered at the true position ${\overrightarrow{x}}_{A\text{,true}}$ of the fluorophore A with localization precision σ_A. Similarly, the green rings represent the Gaussian distribution at the location of the fluorophore B. The black line denotes the true distance $d_{\text{true}}$. Right: The observed positions ${\overrightarrow{x}}_{A\text{,obs}}$, ${\overrightarrow{x}}_{B\text{,obs}}$ are random variables drawn from these distributions. Magenta and green halos illustrate signals obtained from the two fluorophores. The black line denotes the observed distance $d_{\text{obs}}$. (C) Illustration of how the proximity probability $P_{\text{prox}}$ is estimated by sampling points from a Gaussian distribution and counting the fraction of points that land in an annulus. The Gaussian $\mathcal{N}\left({\overrightarrow{\mathit{x}}}_{A\text{,obs}}-{\overrightarrow{\mathit{x}}}_{B\text{,obs}},{\sigma }_A^2+{\sigma }_B^2\right)$, represented by the yellow rings, depends on the observed distance (black line) and localization precisions. The annulus, represented by the gray region, depends on a structural model of the complex AB and its fluorophores, where the lower bound $d_{\text{lower}}$ and upper bound $d_{\text{upper}}$ are constraints on $d_{\text{obs}}$ consistent with colocalization. (D) $P_{\text{prox}}$ can be estimated for arbitrary $d_{\text{obs}}$, σ_A, σ_B, $d_{\text{lower}}$, and $d_{\text{upper}}$. For $d_{\text{lower}}=0$, $P_{\text{prox}}$ approaches 1 as $d_{\text{obs}}$, σ_A, and ${\sigma }_B\to 0$.

Fig. 2.

Graph Matching Optimization (GMO) selects the most probable configuration of molecular pairing. (A) A simulated SMLM image of proteins A (magenta) and B (green). Size of marker correlates with localization precision. (B) A connected component of the bipartite graph constructed for the dataset in (A). A node represents a localization of either A or B. An edge (dashed lines) connects A_i and B_j if their proximity probability p_ij is positive and their normalized observed distance is less than a data-driven threshold. (C) GMO selects a maximal weight matching (indicated by blue lines), a subgraph that maximizes the sum of p_ij, and where each node is selected at most once. The matching represents a possible configuration of molecular pairing. (D) Matchings selected in four example scenarios. (I): $(A_1,B_1)$ is chosen over A₁ and B₂ if $p_{11}>p_{12}$; (II): $(A_1,B_1)$ and $(A_2,B_2)$ are pairs if $p_{11}+p_{22}>p_{12}+p_{21}$; (III): $(A_2,B_1)$ is a pair if $p_{21}>p_{11}+p_{22}$; and (IV): $(A_1,B_1)$ and $(A_2,B_2)$ are pairs if $p_{11}+p_{22}>p_{21}$.

Fig. 3.

Iterative Monte Carlo Estimation of Molecular Couplings and Background Pairings (iMEC) estimates the number of background pairs among a configuration of molecular pairing. (A) Process overview of iMEC: Starting with the most probable configuration with $N_{\text{pairs}}^{\ast }$ pairings from GMO, distribute the localizations in an experimentally imaged region uniformly random to estimate the number of background pairs $N_{\text{bg}}^i$. For the next iteration, repeat the process with $2\times N_{\text{bg}}^i$ fewer localizations, which is equivalent to deleting $N_{\text{bg}}^i$ pairs from the dataset. The number of couplings $N_{\text{coupled}}^{\ast }$ is estimated successively by $N_{\text{coupled}}^i=N_{\text{pairs}}^{\ast }-N_{\text{bg}}^i$. (B) Two example progressions shown on a plot of $N_{\text{coupled}}^i$ versus $N_{\text{bg}}^i$. Of the 2,000 localizations from A and 2,000 localizations from B, the number of true molecular couplings were $N_{\text{coupled}}^{\ast }=1,545$ (blue) and 900 (pink). The lines of slope $-1$ denote the conserved sum $N_{\text{pairs}}^{\ast }=N_{\text{coupled}}^i+N_{\text{bg}}^i$. Arrows show iterative progression to convergence. After 15 iterations, the estimates were $N_{\text{coupled}}^{15}=1,545$ (blue) and 903 (pink). (C) $N_{\text{bg}}^i$ and $N_{\text{coupled}}^i$ for the examples shown in (B), averaged over 10 Monte Carlo trials.

Fig. 4.

Algorithm performance on simulated datasets. Recall rate (GMO), precision (GMO), and error rate (GMO+iMEC) as a function of (A) localization precision and (B) molecular density. At fixed density, precise localizations lead to better performance. At fixed localization precision, recall and precision are better at lower densities, while error remains mostly unchanged. Error bars indicate ±1 SD across 20 trials.

Fig. 5.

Validation of the algorithm on simulated and experimental SMLM data. (A) Schematics of Halo-TM-SNAP (positive control) and Halo-β2AR:SNAP-CaaX (negative control)—two test systems of membrane proteins. Halo-TM-SNAP was expected to show significantly more colocalization events than Halo-β2AR:SNAP-CaaX. (B) Confocal images of Halo-TM-SNAP (Left) and Halo-β2AR:SNAP-CaaX (Right) expressed in HEK 293FT cells demonstrating localization of proteins to the plasma membrane. Confocal imaging does not provide information about individual molecular localization, and confocal data were not used in colocalization analysis. (C) Sample regions of experimental SMLM images for Halo-TM-SNAP (Left) and Halo-β2AR:SNAP-CaaX (Right). Paired molecules are highlighted with yellow rectangles. Localizations are shown with PA-JF549-Halo in green and PA-JF646-SNAP in magenta. (D) Percentage of identified colocalizations for Halo-TM-SNAP and Halo-β2AR:SNAP-CaaX. Each dot represents data from a cell; bars represent averages over all cells. Error bars indicate $\pm 1$ SD. Dashed horizontal lines denote the means in each scenario. (E) Schematic of imaging planes in confocal imaging and SMLM. As expected, proteins were observed at the plasma membrane. Created using Biorender.

Fig. 6.

Validation of the algorithm on simulated nonequilibrium binding dynamics. (A) Schematic of a hypothetical biological system: Following ligand stimulation at t = 0, membrane protein A (blue) becomes active and can bind to membrane protein B (red). EC and IC indicate extracellular and intracellular regions, respectively. Created using Biorender. (B) Schematic of a hypothetical experimental protocol: Ensembles of cells are stimulated with ligand at t = 0 and are fixed at times $t_1,t_2,$ and t₃ for SMLM imaging. (C) Snapshots from a simulation modeling protein binding, $A+B\rightleftharpoons AB$. AB shown indicate true binding. Snapshots were taken at t = 0, $0.25$ s, and 3 s after ligand addition. (D) Top: Density of AB identified by the algorithm at various time points (each time point consists of 10 simulated datasets). The variations in the shaded regions originate from stochastic simulations, while the variations in the error bars are attributed to the inference algorithm. Bottom: Error rate at various time points. Error bars indicate ±1 SD.