## Main

Observing the native structure and behaviour of biomolecules is challenging owing to their architectural complexity and dynamic nature. Additionally, biomolecules can adopt multiple interchanging conformational states. Protein structure determination is progressing rapidly thanks to recent advances in cryo electron microscopy (cryo-EM) and X-ray crystallography. However, these structures represent static snapshots of averaged ensembles acquired from molecules incorporated into crystals and/or imaged at cryogenic temperature, whereas individual molecules at physiological temperature are highly dynamic. In contrast to cryo-EM, which provides three-dimensional (3D) volume data, AFM is restricted to surface analysis. Nevertheless, AFM images molecules in a native-like environment: (i) at ambient temperature, (ii) at ambient pressure, (iii) in physiological buffer and (iv) in membranes (in the case of membrane proteins). Furthermore, the AFM measurement mechanism and the openness of the fluid cell allow for (v) buffer exchanges, (vi), temperature changes and (vii) force changes during image acquisition2,4.

High-speed AFM (HS-AFM)5 has an additional advantage in that it yields real-time nanometre topographical information of single biomolecules at unprecedented spatiotemporal resolutions6,7,8,9,10,11,12,13, through the integration of short cantilevers14 and the development of faster scanners15 and feedback operation16. Although this is proving powerful in revealing conformational changes of proteins4,17, it is often not possible to resolve sub-molecular structural features on protein surfaces, primarily owing to the finite size of the AFM tip. For probes typically used to image biological samples, the resolution in the z direction (topography) is about 1 Å, whereas the lateral resolution in the x, y directions is about 1 nm, fundamentally limited by the probe geometry and the probe–sample interaction forces. The lateral resolution is further reduced when imaging softer samples, owing to an increased contact area between the tip and flexible protein structures18. Because of these limitations, sub-nanometre lateral resolution of biological samples has only been reported for two-dimensional (2D) crystals19,20, and was evidenced to be an overestimation due to periodic tip convolution effects21. In an attempt to circumvent such limitations, tip deconvolution algorithms were proposed22,23, which produced sharpened images but could introduce artefacts.

Localization microscopy methods, also known as super-resolution fluorescence microscopies, such as stochastic optical reconstruction microscopy (STORM)24 and photoactivated localization microscopy (PALM)3, have provided insights into the architecture and macromolecular assemblies of cells. By isolating and pinpointing the source of excited fluorescence signals with high spatial precision in many images, high-lateral-resolution maps can be reconstructed, taking the ~400 nm resolution limit set by the diffraction limit of light down to about 20 nm (refs. 25,26).

Here, inspired by these fluorescence localization microscopy methods (Extended Data Fig. 1a–d), we develop LAFM, whereby localization algorithms are applied to the spatial fluctuations of topographic features in AFM and HS-AFM images (Extended Data Fig. 1e–h). Comparisons with X-ray structures and molecular dynamics (MD) simulations show that this approach can reveal angstrom-range high-resolution details on protein surfaces.

## Breaking the resolution limit

Under specific conditions; that is, with an atomically sharp tip and rapidly decaying tip–sample interaction forces, atomic resolution is attainable on flat incompressible materials such as mica by conventional AFM imaging18. Achieving and maintaining such conditions on biological samples, which are not only soft and dynamic, but also immersed in liquid at room temperature, is not possible. Typically, the tip geometry from the apex up to the height of the objects being imaged is much larger than the separation distance between the features of interest (Fig. 1a, b, surface). The finite tip radius results in convoluted lateral dimensions. The signal is further obstructed by noise in the z direction and stochastic fluctuations of flexible protein surface features (Supplementary Video 1) in the x, y and z directions (Fig. 1b, AFM traces). Averaging several of these traces removes noise and results in a noise-free topography trace but the tip convolution remains a limiting factor (Fig. 1b, average AFM). By applying localization algorithms that detect the local maxima in the same series of traces (Fig. 1b, AFM traces, crosses), extracting the location-specific heights (Fig. 1b, LAFM height) and merging the individual detections in a peaking-probability map (Fig. 1b, LAFM probability), the surface structures are reconstructed with greater lateral resolution in an LAFM map (Fig. 1b, LAFM). Local peak-search algorithms to identify and accumulate local maxima in AFM data have previously been used to create probability density maps, from which energy landscapes were calculated to sample the conformational space of protein moieties27 and to derive stiffness maps7. Here we build on this concept and extend the approach, leveraging the methodological knowledge generated by the development of super-resolution fluorescence localization microscopies3,24. Localization-based fluorescence microscopy methods taught us that a resolution superior to the physical limitations can be achieved when the localization of isolated signals is determined with high spatial precision in many images, which are later merged in a compiled map3, which has higher lateral resolution than the initial data. Advantage is taken of the fact that the peak position of signals with wide intensity distributions can be determined with astonishing precision. Here, we adapt this transformative rationale to AFM data (Extended Data Fig. 1e–h). First, the pixel- and/or AFM-restricted low-lateral-resolution data are oversampled to allow peak positions to be determined with increased spatial localization resolution. Peak positions are measured and localization data are then merged to give a reconstructed map with higher lateral resolution than the initial pixel sampling and/or technique allowed (Fig. 1b; compare LAFM with average AFM).

The LAFM map reconstruction is best illustrated in the simulation, where several features of varying height are contoured next to each other (Fig. 1b, bottom row). Simulations show that the LAFM algorithm detects features that are hidden to theoretical and average topographies (Fig. 1b, Extended Data Fig. 3). However, the detection probability performs best on flat samples, and is nonlinear with the protrusion height if there are closely neighbouring higher features (Extended Data Fig. 2). Each pixel in these maps contains both height and probability information (Fig. 1b, bottom right). Further simulations with varying tip radius and shape on simple (Extended Data Fig. 4, Supplementary Video 2) and more complex (Extended Data Fig. 5) model 3D surfaces showed that the LAFM algorithm outperformed averaging methods within 10–100 images, showing the greatest improvement in resolution (about 1/5) for tip radii greater than the separation of structural features. These analyses corroborate that the quality of the LAFM map increases with increasing number of observations until it plateaus, at around 50 (for a sharp tip) and around 500 (for a blunt tip) particles are analysed.

On real AFM data, detection of local height maxima is performed after image expansion (Fig. 1c). Image expansion using bicubic interpolation (see Methods) does not increase the lateral resolution of the topography but allows the detection of local maxima with far greater spatial precision (Fig. 1c; compare panels (ii) and (v) with (iii) and (vi)). Merging the high-precision local maxima from several particles results in resolving structural features with separation distances shorter than the initial pixel sampling. To retain the topographic structural information, the topography height value from each peak location is carried into the LAFM reconstruction, where height and peaking probability are encoded by a 2D false-colour scale in which the green/red ratio scales linearly with height h and probability P from white at P = 1 to black at P = 0 (Extended Data Fig. 1i, j). Furthermore, each peaking detection, originating from an atomic tip–sample interaction, is assigned a 2D Gaussian density function decaying from 1 to 0 over 1.4 Å to approximate atomic solvent-accessible surface areas. A reconstructed LAFM map thus compiles, from many particles, the average topography height refined by the peaking probability (Fig. 1b, right), where each pixel carries the full information about topography and its likelihood of being detected at this location. In merging many particles, randomly distributed apparatus noise does not merge into consistent height/probability data. Conversely, peaking detections that emerge from protein surface fluctuations will merge into strong localized signals in high-resolution reconstructed LAFM maps.

## Single amino acids on protein surfaces

To illustrate the power of the LAFM approach, we first applied it to a former conventional AFM dataset20. After extraction and alignment of aquaporin-Z (AqpZ) tetrameric channels, the LAFM map revealed details comparable to the surface of the X-ray structure (Fig. 2a, Supplementary Video 3), resolving single amino-acids on surface protruding loops (Fig. 2b). Line profile analysis and image comparison between the average AFM topography, previous peak probability mapping methods27 and LAFM probability maps of independent dataset half-maps show the ability of LAFM to detect previously hidden structural features (separated by 2.6 Å) well beyond the details resolved by previous averaging and peak probability methods (11 Å) and the Nyquist frequency of the raw data (1/(6.6 Å)) (Extended Data Fig. 6a–i). Interestingly, among the AqpZ X-ray structures, E31 in the central a-loop is in different orientations, and the LAFM map indicates that in physiological buffer the E31 rotamer configuration, as found in PDB 2ABM, is preferred (Extended Data Fig. 6j). We also applied the LAFM approach to annexin-V (A5) trimers extracted from HS-AFM videos5,9,28 (Fig. 2d, Supplementary Video 4) and found that the LAFM map resolved fine structural details (whereas the average resolved only the protein envelope) along the backbone of the molecule (Fig. 2b).

To quantitatively assess the resolution of the LAFM maps, we applied the Fourier ring correlation (FRC) method, developed for electron microscopy29 and more recently adapted for super-resolution fluorescence microscopy26. The FRC method splits the datasets into halves and assesses their statistical resemblance as a function of the resolution range. This analysis resulted in a resolution of 4.0 Å for AqpZ, 5.1 Å for A5 and 4.5 Å for A5 P13W-G14W (Fig. 2c, f, Extended Data Fig. 7a, b, h). The FRC curve of AqpZ has, in addition to the signal power up to about 4.0 Å, a second information-containing range in the 2-Å regime. Thus, both the real-space (Extended Data Fig. 6h, i) and the statistical analysis of AqpZ LAFM half-maps report resolution at distances shorter than the Nyquist frequency of the raw data. Accordingly, LAFM maps of both AqpZ and A5 resolve details down to the amino acid size range (around 5 Å to 4 Å), and some signal power on the quasi-atomic scale (around 2 Å) in the case of AqpZ (Fig. 2b, c, Extended Data Fig. 6). We also capitalized on the serendipitous co-existence of two differently oriented A5 trimers in the A5 lattice. LAFM of the two trimer datasets, independent from each other and acquired through different relative AFM scan-directions, agree in great detail (Extended Data Fig. 7c–e). Finally, we cloned, expressed and purified a mutant A5, replacing two amino acids in the N terminus with tryptophans (P13W, G14W), and imaged the A5-mutant by HS-AFM (Extended Data Fig. 7f–h). LAFM maps of the A5 mutant show overall rearrangements of the N terminus with increased height and peaking probability at the mutation site.

## Localization AFM of CLC antiporters

The AFM data of A5 and AqpZ were acquired on 2D lattices, however, a considerable advantage of LAFM is that the biomolecules do not need to be confined in a crystal for analysis, but can be sparsely populating a native-like environment. Furthermore, the buffer conditions inside the fluid cell can be changed to assess structural changes in response to environmental changes. Therefore, we studied CLC-ec1 (a Cl/H+ antiporter from E. coli)30,31, which, to our knowledge, has not been observed by AFM, and whose transport mechanism remains unresolved. Mutations in human CLC family homologues have been associated with diseases32.

HS-AFM of CLC-ec1 in membranes formed through proteo-liposome fusion showed a dispersed population of proteins protruding 1.2 nm from the membrane (Fig. 3a–c, Supplementary Video 5). CLC-ec1 was predominantly dimeric, with small populations of monomers and higher-order oligomers assembled from multiple dimers (Fig. 3b). The topography and lateral dimensions of the dimers (Fig. 3c) were consistent with the 5.5 nm × 9.6 nm dimensions of the extracellular face of CLC-ec1 (Extended Data Fig. 8a–e)33,34. Because the dimers were not confined, they exhibited translational and rotational freedom (Fig. 3c, Supplementary Video 5), which led us to establish a generalized LAFM workflow (Fig. 3d; see Methods): (1) a HS-AFM video is acquired and (2) low-pass-filtered, so that (3) particles can automatically be detected. Particles are thus (4) tracked throughout the HS-AFM observation and (5) selected and extracted in a gallery. (6) Bicubic image expansion allows for (7) precise particle centring and (8) rotational alignment to an arbitrary molecule reference. A second cycle of (9.1) lateral and (9.2) rotational alignment, this time with respect to an ensemble average, prepares particles for (10) application of the LAFM method (Supplementary Video 6). As described in Fig. 1, (10.1) local maxima peaks are detected and (10.2) the height at these locations is extracted with a 1.4-Å-wide probability radius. Finally, all detections are merged in a height–probability LAFM map (Fig. 3e). The particle gallery (step 5) can be assembled from many molecule observations in one or several frames. Alternatively, an LAFM map can be reconstructed from one molecule observed over time, which gives this method unique possibilities to access high-resolution information of individual molecules.

## Conformational changes in CLC-ec1

The exchange pathway in the CLC-ec1 Cl/H+-antiporter has been proposed to have two separate entrances/exits for H+ and Cl on the intracellular face, converging to a central binding region from which both ions follow the same path to the extracellular side. However, there is debate about whether the gating mechanism requires only localized side-chain motions in the Cl pathway based on X-ray structures, or if greater movements occur, as evidenced by nuclear magnetic resonance (NMR)35,36, computational37 and helix-crosslinking studies38. Findings by these non-crystallographic methods35,36,37,38,39 have led to suggestions that confinement of CLC in 3D lattices inhibit large conformational movements (Extended Data Table 1, Extended Data Fig. 8f), similar to other transporters40,41,42,43. Cl transport by CLC-ec1 is maximal at acidic pH and stalled at neutral and basic pH (owing to pH-dependent activation and lack of H+ as substrate)44. A more recent structure of a protonation-mimicking triple mutant also indicates conformational rearrangements45. Therefore, we performed HS-AFM of transporters sparsely packed in lipid membranes and in physiological buffer. Subsequent LAFM of the pH 7.6 (inactive state) and pH 4.5 (active state) observations should inform on whether large-scale conformational changes occur.

On the basis of the X-ray structure surface (Fig. 4a), we assigned the protruding residues expected to give signals in AFM: Asp73 in loop B-C, Glu235, Asp240 and Lys243 in the long loop I–J, Asn327 in loop L–M, and Gln381 and His383 in loop N–O. To refine the interpretation of LAFM reconstructions, we used MD simulations to convert the static X-ray structure into a dynamic molecular system fluctuating at room temperature and at pH 7 (Supplementary Video 1). Similar to the LAFM method, we plotted a population density map of the distribution of the z-coordinate local maxima on the CLC-ec1 extracellular face from MD trajectories, which reflected side-chain motions of membrane-protruding residues (Fig. 4b, Extended Data Fig. 8g, h). The MD trajectories show how structural fluctuations that are probed (in AFM) and merged (in LAFM) allow extraction of high-resolution information of amino acid residues on protein surfaces.

The CLC-ec1 LAFM reconstructions at pH 7.6 and pH 4.5 display the same set of structural features as the X-ray structure and the MD population map, but in distinctly different configurations (Fig. 4c, d). Peaks 2, 3 and 4, which form a triangle close to the dimer interface, pack more loosely at pH 4.5, and peak 3 moves towards a more lateral position on the dimer, while the most remarkable conformational change is a ~6-Å movement of peak 1 towards the dimer interface at acidic pH. The extracellular Cl/H+ ion pathway lies between Asp73, Asn327 and Glu235 (Fig. 4c, asterisk); thus, under the premise that these displacements are related to movements in the underlying helices, these structural changes might alter accessibility to the extracellular gate. In summary, LAFM reports large pH-dependent conformational changes (Fig. 4e, Supplementary Video 7).

By recording 3D topographic images and videos, AFM and HS-AFM offer rich data, captured through many atomic interactions between tip and sample in liquid and at ambient conditions. By pinpointing peak interaction locations with high spatial precision in oversampled topographies, LAFM produces quasi-atomic resolution maps of protein surfaces from such data. We demonstrate the ability of LAFM to detect amino acid side chains on the surfaces of AqpZ, A5 and CLC-ec1, mutation-related differences in A5, and conformational changes in the angstrom range in CLC-ec1. Our LAFM maps, calculated from CLC-ec1 imaged at physiological and acidic pH, identified substantial differences in the central region, where helices N and O are located, and at the peripheral end of helix B, which moves towards the dimer centre, giving the entire molecule a ~1.2-nm shortened appearance (Fig. 4d, e).

HS-AFM5 operates in amplitude modulation mode using short cantilevers that oscillate at resonance at around 660 kHz (oscillation cycle of around 1.5 μs). The tip touches the surface only during ~10% of an oscillation cycle4, thus about 150 ns. Even though this is a short period in the life of a protein, side-chain fluctuations occur in such time regimes, thus blurring the signal. Hence, LAFM will provide improved data when the next generation of faster HS-AFM systems arrive. Today, amplitude detectors oversample the cantilever5,46, but feedback operation and the z-piezo are limiting (about 100 kHz) factors and need improvement.

The LAFM method can be used in two different ways: LAFM maps can be reconstructed (i) from many molecules recorded in one or several frames or (ii) from a single molecule over time. The first approach allows us to resolve time- or environment-dependent conformational changes. About 50 particles are needed to reconstruct an LAFM map (Extended Data Fig. 5); therefore, the temporal resolution of LAFM is decreased to the time required to accumulate these 50 observations. Faster HS-AFM operation will of course improve time-resolved studies of single molecules. Alternatively, imaging densely packed proteins (with around 50 particles in each frame8,47) would allow LAFM map reconstruction of the conformation of the proteins in each frame, giving high-resolution structural changes as a function of time. The second approach gives the method the unique capability to provide high-resolution information of single molecules or of non-ordered supramolecular assemblies. Altogether, we envisage that LAFM will become the standard method applied to AFM imaging, allowing the extraction of high-resolution information beyond the tip-radius resolution limit in the study of single biomolecules in native-like environments.

## Methods

### HS-AFM

HS-AFM measurements (annexin-V, CLC-ec1) were taken with an amplitude modulation mode HS-AFM (RIBM), as previously described in ref. 28. In brief, we used short cantilevers (USC-F1.2-k0.15, NanoWorld) with a spring constant of 0.15 N m–1, a resonance frequency of ~0.66 MHz and a quality factor of ~1.5 in buffer.

### AFM

AFM data (aquaporin-Z) were taken by contact-mode AFM using a Nanoscope-III AFM (Digital Instruments) equipped with a 120-μm scanner (J-scanner) and oxide-sharpened Si3N4 cantilevers with a length of 120 μm and spring constant of 0.1 N m−1 (Olympus Ltd), as detailed in ref. 20.

### Cloning, expression and purification of annexin-V-P13W-G14W

The P13W-G14W site-directed mutagenesis was performed on an untagged human annexin pET28a expression vector using the Q5 site-directed mutagenesis kit (New England BioLabs) and the following mutagenic primers (mutated nucleotides are in bold): 5′-GACCGATTTTTGGTGGTTTGATGAACGTGCTGATGCC-3′ and 5′-ACGGTACCACGCAGCACTTG-3′.

The mutated genes were sequenced to confirm that only the desired mutations were inserted into the plasmid. The annexin-V-P13W-G14W plasmid was then transformed into BL21 (DE3) pLysE chemically competent E. coli cells (Invitrogen), and grown overnight at 37 °C for small-scale culture. The overnight culture (50 ml) was inoculated into 2 l fresh Luria–Bertani broth media at 37 °C, and once an optical density (A600) of 0.6–0.8 was achieved, the cells were induced by addition of 0.4 mM isopropyl β-d-1-thiogalactopyranoside. After induction for 4 h, the cells were separated from the culture medium by centrifugation (5,000 g; 20 min) and resuspended in ice-cold calcium buffer (50 mM Tris pH 7.5, 10 mM CaCl2). The suspension was three times tip-sonicated on ice for 5 min (one pulse every 9 s), and centrifuged (23,000 g; 45 min). The supernatant was discarded, and the pellet was resuspended in ice-cold EGTA buffer (50 mM Tris pH 7.5, 60 mM EGTA). After gentle shaking for 30 min, the cell debris were removed by centrifugation (23,000 g; 45 min), and the supernatant containing the soluble Annexin-V-P13W-G14W was dialysed overnight against buffer A (20 mM Tris pH 7.5, 20 mM NaCl). The solution was applied to a HiTrap DEAE FF sepharose column (5 ml) ÄKTA Avant (GE Healthcare Life Sciences), and eluted with a linear gradient of 0–1 M NaCl. Fractions containing annexin-V-P13W-G14W (based on SDS–PAGE analysis) were concentrated to ~1 mg ml−1 using 10 kDa centrifugal filters (Amicon, Millipore), and subjected to a final purification step with a Superdex 200 Increase 10/300 gel filtration column (equilibrated with 20 mM Tris pH 7.5, 100 mM NaCl buffer), reaching a final purity of >95% according to SDS–PAGE analysis.

### CLC-ec1 expression and purification

Expression and purification of CLC-ec1 were carried out as previously described48. BL21-AI E. coli competent cells (Thermo Fisher Scientific) were transformed with the plasmid and then 2 l Terrific Broth supplemented with ampicillin was inoculated and grown at 37 °C. Protein expression was induced with anhydro-tetracycline at OD600 = 1.0. After 3 h of induction, cells were harvested, then lysed by sonication in buffer supplemented with 5 mM reducing agent TCEP (Tris(2-carboxyethyl)phosphine; Soltec Bioscience) and pH adjusted to 7.5. Protein extraction was carried out with 2% n-decyl-β-d-maltopyranoside (DM; Anatrace) for 2 h at room temperature. Cell debris was pelleted down, and the supernatant was run on a 2 ml column volume (CV) TALON cobalt affinity resin (Clontech Laboratories) equilibrated in cobalt column wash buffer (CoWB)/TCEP: 100 mM NaCl, 20 mM Tris, 1 mM TCEP, pH 7.5 with NaOH, 5 mM DM. After binding, the column was washed with 15 CVs of CoWB/TCEP followed by a low-imidazole wash of CoWB/TCEP containing 20 mM imidazole (Sigma-Aldrich). CLC-ec1 was eluted with CoWB/TCEP containing 400 mM imidazole, then concentrated in 30-kDa nominal molecular weight limit (NMWL) centrifugal filters (Amicon, EMD Millipore) to ~500 μl and injected on a Superdex 200 10/30 GL size exclusion column (GE Healthcare) equilibrated in size exclusion buffer (SEB): 150 mM NaCl, 20 mM MOPS pH7.5, 5 mM analytical-grade DM, attached to a medium-pressure chromatography system (NGC, Bio-Rad).

### CLC-ec1 reconstitution and bilayer formation

Lipids were resuspended in 300 mM KCl, 20 mM citrate pH 4.5 with NaOH. CHAPS (35 mM) solubilized lipids were combined with protein at 100 μg CLC-ec1 per 1 mg of lipids, corresponding to a protein/lipid mole fraction of 7.6 × 10−4 (assuming a 50% incorporation yield)48. The protein–lipid–detergent mixture was dialysed in cassettes (NMWL 10 kDa; ThermoFisher Scientific) at 4 °C against 4 l of buffer for 48 h with buffer changes every 8–12 h. After completion of dialysis, the proteo-liposomes were harvested from the cassettes, freeze/thawed and then extruded using an Avanti Polar Lipids Mini Extruder (Alabaster) through a 400-nm membrane. 1.5 μl of the SUV solution with a total lipid concentration of 0.1 mg ml−1 was deposited onto freshly cleaved mica to form supported lipid bilayers (SLBs) through vesicle fusion. The excess lipids, after SLB formation, were rinsed first with deionized water, followed by buffer. For experiments at pH 7.6 the sample was rinsed with 25 mM Tris, 300 mM KCl pH 7.6.

### Image expansion

AFM topography images were expanded using bicubic interpolation (Catmull–Rom interpolation; implemented in imageJ, scripted using the method of Burger and Burge)49. The method considers values over a 16-pixel surface (4 × 4 pixels) to calculate the new intermediate surface, p(x, y), created by expansion across the central 2 × 2 area. The interpolated values are approximated by 3rd-order polynomials in both the x and y directions:

$$p(x,y)=\mathop{\sum }\limits_{i=0}^{3}\mathop{\sum }\limits_{j=0}^{3}{a}_{ij}{x}^{i}{y}^{j},$$

where i and j are the order of the polynomial for x and y, respectively, and aij are 16 possible corresponding coefficients. The resulting polynomial can be calculated using the values at the four corners of the central 2 × 2 grid (f(x, y)), the gradients at each of those positions in the x and y directions (fx(x, y), fy(x, y)) and the cross-derivatives (fxy(x, y)) requiring the 4 × 4 pixel grid, with the derivatives being calculated numerically. The interpolated surface, p(x, y), between four corner pixels can be described by:

$$p(x,y)=[1\,x\,{x}^{2}\,{x}^{3}]\,\left[\begin{array}{cccc}{a}_{00} & {a}_{01} & {a}_{02} & {a}_{03}\\ {a}_{10} & {a}_{11} & {a}_{12} & {a}_{13}\\ {a}_{20} & {a}_{21} & {a}_{22} & {a}_{23}\\ {a}_{30} & {a}_{31} & {a}_{32} & {a}_{33}\end{array}\right]\,\left[\begin{array}{c}1\\ y\\ {y}^{2}\\ {y}^{3}\end{array}\right],$$

where the 16 coefficients can be calculated using the values and derivatives at the four corners:

$$\begin{array}{c}\left[\begin{array}{cccc}{a}_{00} & {a}_{01} & {a}_{02} & {a}_{03}\\ {a}_{10} & {a}_{11} & {a}_{12} & {a}_{13}\\ {a}_{20} & {a}_{21} & {a}_{22} & {a}_{23}\\ {a}_{30} & {a}_{31} & {a}_{32} & {a}_{33}\end{array}\right]=\left[\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ -3 & 3 & -2 & -1\\ 2 & -2 & 1 & 1\end{array}\right]\\ \,\left[\begin{array}{cccc}f(0,0) & f(0,1) & {f}_{y}(0,0) & {f}_{y}(0,1)\\ f(1,0) & f(1,1) & {f}_{y}(1,0) & {f}_{y}(1,1)\\ {f}_{x}(0,0) & {f}_{x}(0,1) & {f}_{xy}(0,0) & {f}_{xy}(0,1)\\ {f}_{x}(1,0) & {f}_{x}(1,1) & {f}_{xy}(1,0) & {f}_{xy}(1,1)\end{array}\right]\,\left[\begin{array}{cccc}1 & 0 & -3 & 2\\ \\ 0 & 0 & 3 & -2\\ \\ 0 & 1 & -2 & 1\\ \\ 0 & 0 & -1 & 1\end{array}\right].\end{array}$$

Using this method, all our datasets were resampled to 0.5 Å per pixel, as indicated in the figure captions. The reason for expanding to 0.5 Å per pixel is based on approximating the picked maxima features to the solvent-accessible surface of atoms with Gaussian profiling, as detailed in the Methods section ‘Detection probability’. By constructing the interpolant value from continuous piecewise polynomials, the result is always continuous. This works particularly well for interpolation of smooth areas, as in the case of tip-radius-limited imaging, and therefore considerably improves local maxima localization, but does not increase image resolution.

### Detection of local maxima

A local maximum position (Fig. 1c) is defined if a given pixel is higher than all the surrounding eight pixels in a 3 × 3 pixel grid (Figs. 1c, 3d). This 3 × 3 pixel grid is ‘scanned’ pixel by pixel over the image, and thus all pixels (with the exception of those at the image borders) in each particle image are checked for maxima. To reduce the selection of maxima due to noise in certain datasets, a noise tolerance algorithm that selects maxima based on their prominence above surrounding maxima was implemented. The prominence of each maximum, pi, is calculated by the following steps: (i) search for the closest neighbouring maximum hn with higher height than the current maximum hi or closest image boundary; (ii) find the minimum height along the profile between hi and hn or between hi and the image boundary; and (iii) define the peak prominence as:

$${p}_{i}={h}_{i}-{h}_{\min }({h}_{i}\to {h}_{n}).$$

In our method, for a local maximum to be selected, its prominence must be greater than the noise tolerance (typically 1–2 Å). In our plugin, the noise tolerance is defined by the user from 0 to 100%, where the noise tolerance parameter corresponds to the range of height values from lowest to highest in the image. These maxima selection criteria are based on the noise level of the AFM imaging and the typical root-mean-square fluctuations at protein surfaces (Extended Data Fig. 8g, h). An alternative method is to apply a Gaussian filter to the image to reduce noise and use 0% noise tolerance. The repulsive interaction forces between the farthest exposed atoms of the tip and the atoms in protein moieties that protrude most have very steep separation distance dependence. Very strong short-range interactions occur, including Pauli repulsion and van der Waals, hydration, steric and ionic forces, which depend on the surface properties of both the AFM tip and the protein50. As a result, the most exposed atoms dominate local topographic detection and high-resolution information can be obtained through merging many tip–sample atomic interactions at different localizations and time points or on different molecules of the same kind.

### Detection probability

The peaking probability at a given localization in an LAFM map, is the cumulative probability that a pixel (in the expanded image) is detected within all particles analysed. It is the sum of: picking events (n), multiplied by the power of the 2D Gaussian, g(0 < P < 1) on each pixel, divided by the total number of particles merged (N).

$${P}_{x,y}=\frac{{\sum }_{i=1}^{N}{n}_{xi,yi}{g}_{x,y}}{N}.$$

The 2D Gaussian in all our datasets was set to 1.4 Å width to approximate the solvent-accessible surface of the underlaying atoms (the solvent-accessible surface area is defined as the surface traced out by the centre of a water sphere rolled over the protein atoms)51 while imparting a continuous probability density to each discreetly selected maximum. The application of larger Gaussian radii to approximate the atomic origin of the tip–sample interactions or pre-filtering the data before peaking leads to loss of resolution or loss of peaking detection of lower features, respectively (Extended Data Fig. 9). Because AFM can reproducibly image atoms on solid surfaces, for example, on mica, the piezo-elements that mediate the scanning of the AFM sample stage have sub-atomic xy position precision.

### Height extraction

The real-space topographic height is extracted at each detection to produce a set of N matrices containing height values for each value of n. This matrix is then false-coloured to allow distinction between height and probability information.

### Merging height and detection probability

The false-coloured extracted height values in each image are then multiplied by the greyscale probability values in each image, and then averaged for the whole image set to reconstruct an LAFM map.

### LAFM workflow

The HS-AFM videos were 1st-order flattened to compensate for sample stage tilt, drift-corrected and contrast-adjusted by laboratory-built image analysis software in ImageJ and MATLAB (Mathworks). The workflow used to calculate an LAFM map from molecular HS-AFM raw data is outlined in Fig. 3. The key steps in the preparation for the LAFM method are: extraction of molecular observations from images (Fig. 3d, steps 1–5), image expansion (Fig. 3d, step 6; see Methods section ‘Image expansion’) and creation of a particle gallery with laterally and rotationally well aligned particles (Fig. 3d, steps 7–9). Several image processing packages used for electron microscopy (for example, ref. 52) allow particle extraction and alignment, and could be used for convenience. The particle gallery of pixel-expanded (0.5 Å per pixel) molecular observations is the entry for the LAFM algorithm, which comprises detection of local maxima, height extraction and merging of height and peaking probabilities (Fig. 3d, steps 10–11; Methods sections ‘Detection of local maxima’, ‘Detection probability’, ‘Height extraction’ and ‘Merging height and detection probability’) in the final LAFM map. The LAFM method is available as code in the form of an appendix and as an ImageJ plugin (Supplementary Information).

### LAFM simulations

2D and 3D LAFM simulations were performed using MATLAB. In 2D simulations (x, z), various model surfaces were created with different features depending on the simulation (Fig. 1b; simulation parameters: tip radius, 20 pixels; feature height, 3 pixels; feature width and separation, 2 pixels; scanning noise, 0.05 (standard deviation, σ), feature fluctuation, 0.3 (σ); the Gaussian surface topography (bottom row) has σ = 20; varying parameters are used in Extended Data Figs. 2, 3, given in the figure captions). Each topographic feature was given a height higher than the surrounding baseline surface (set at zero). Normally distributed random numbers with set standard deviation were then generated and added to each x position containing a topographic feature, increasing or decreasing the height. These random fluctuations were added independently of neighbouring x positions. A semicircular tip of defined radius was calculated numerically and then scanned across the simulated 2D surface to create a tip-convoluted topography. To simulate the AFM instrument noise, normally distributed random noise was then added in the z direction to the tip-convoluted topography at all positions. Many randomly generated topographies were then analysed using the LAFM algorithm to produce peaking probability and peaking height traces. 3D simulations were run using a similar methodology, however a hemispherical tip was scanned across 3D model surfaces (Extended Data Figs. 4, 5, Supplementary Video 2).

Simulation data are compared to a theoretical resolution limit (Extended Data Fig. 3) on the basis of geometric considerations, assuming a rigid pair of spikes separated by a distance (d) and a height difference (∆h), contacted by a tip radius (R) without noise or fluctuations. The resolution limit is defined as being resolved if the probe is able to reach a minimum (∆z) below the height of the smallest spike53:

$$d=\sqrt{2R}\,(\sqrt{\varDelta z}+\sqrt{\varDelta z+\varDelta h}).$$

The absolute resolution limit under these considerations occurs when maxima can be detected at both spikes when ∆z = 0.

### MD simulations of CLC

#### Construct for MD simulations

The molecular model of the CLC-ec1 dimer used in all MD simulations described in this work was based on the X-ray structure PDB 1OTS31. The protonation states of the titratable residues at pH 7 were determined from constant-pH calculations with the neMD/MC (non-equilibrium MD/Monte Carlo) approach54. The spatial arrangement of the CLC-ec1 dimer in the bilayer was optimized using the Orientations of Proteins in Membranes (OPM) database55 and input to the Membrane Builder module on the CHARMM-GUI web server56 to assemble the protein–membrane system. The CLC-ec1 dimer was embedded in a 629-lipid membrane bilayer containing a ~70:30 mixture of POPE (1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine) and POPG (1-palmitoyl-2-oleoyl-sn-glycero-3-phospho-(1'-rac-glycerol)), solvated in 150 mM KCl explicit water to achieve electroneutrality.

#### MD simulation procedures

The assembled molecular system was subjected to an initial equilibration phase using NAMD57 (version 2.13) following two protocols. The first used the standard six-step equilibration protocol provided by CHARMM-GUI. The other used a laboratory-built multi-step equilibration, in which the backbone of the protein was first fixed58. Backbone constraints were gradually released in three 300-ps steps of force constant change (1 to 0.5 and to 0.1 kcal per mol−1 Å−2). The final structures from the equilibration phases were subjected to short (46 ns and 48 ns) unbiased MD with NAMD (2-fs time steps, vdwForceSwitching option, and PME for electrostatic interactions)59. The runs were in the NPT ensemble under semi-isotropic pressure coupling at 24 °C. The Nose–Hoover–Langevin piston algorithm60 was used to control the target P = 1 atm pressure with LangevinPistonPeriod = 50 fs and LangevinPistonDecay = 25 fs. Van der Waals interactions had a cutoff distance of 12 Å. The first phase of production runs (Production 1) was initiated by all-atom velocity resetting and continued with simulations of the system in 50 independent replicates of ~150 ns each (that is, 100 replicates overall for a cumulative 15 μs) using ACEMD61. At the conclusion of Production 1, the trajectories were analysed to assess the stability of the bound Cl ions, and replicates with the most stably bound Cl ions were identified. The final snapshots from 48 replicates were selected as starting points for the next phase, Production 2, in which the systems were simulated using NAMD with the parameters described above for ~120 ns (cumulative 5.76 μs). Run parameters: timestep 4 fs, vdwforceswitching on, switching on, switchdist 7.5, cutoff 9, fullelectfrequency 2, langevindamping 0.1, pme on, and pmegridspacing 1.0. All the simulations used the latest CHARMM36 force-field parameters for proteins, lipids and ions.

#### Population density maps from the MD trajectories

To analyse the height of protein atoms with respect to the membrane plane during the MD simulations, the symmetry axis of the CLC-ec1 dimer was set perpendicular to the XY plane. In analogy to the LAFM method, the highest z-coordinate values on the CLC extracellular surface were selected for each frame to plot the position distribution map. Maps were constructed by taking the 8, 10 and 16 highest points in each frame, leading to the conclusion that detection of more than 8 points resulted in sampling the neighbouring atoms of residues already included in the 8-point set. Thus, the distribution maps were obtained by pooling the 8 highest-z-coordinate peaks from each frame. Analysis performed separately on Production 1 and Production 2 trajectories did not show notable differences, and in the main text we show the results from the analysis of 5.6 μs with 20-ps time strides of Production 2 trajectories. Because both protomers of CLC-ec1 were considered identical, we symmetrized the data by aligning trajectories of each protomer onto another one.

### MD simulation of annexin-V P13W-G14W

MD simulations of the mutant annexin-V-P13W-G14W were conducted with Gromacs2019.162, using the Amber03 force field63. The initial molecular model of annexin-V-P13W-G14W was generated using the X-ray structure PDB 1HVD, and the double mutation introduced using the program Coot64. This model was then solvated with ~40,000 water molecules in accordance with the Tip3P water model65, and neutralized with Na+ and Cl ions to a concentration of 150 mM. The system was placed in a dodecahedron box, with a minimal distance of 1.0 nm between protein and box wall. Van der Waals interactions were implemented with a cutoff at 1.0 nm, and long-range electrostatic effects were treated with the particle mesh Ewald method. The protein–solvent model was then put through four rounds of geometry optimization and energy minimization, followed by a 50-ps protein position-restrained equilibration and an additional 50 ps of unrestrained equilibration. The system was then heated to 300 K using a velocity-rescaling thermostat66 (50 ps), and equilibrated to a constant pressure of 1 bar using a Parrinello–Rahman barostat (50 ps). Following these equilibration procedures, a time trajectory of 100 ns was simulated at constant temperature and pressure, using time steps of 2 fs and the same thermostat and barostat. The data were then symmetrized along the threefold axis by aligning trajectories of each protomer one onto the other. To build an annexin-V-P13W-G14W mutant structural model that represents the rotamer conformations of the mutated Trp residues, clustering analysis of the simulation trajectories was performed with Gromacs (g_cluster, gromos algorithm)62, with a root-mean-square deviation cut-off of 0.2 with respect to the mutated Trp residues in positions 13–14. Out of the 10 resulting clusters, the most representative structure was extracted from the centre of the most populated cluster (containing ~50% of total protein structures).