## 3 Results  
### Quantification of the Computational Efficiency Gain  
To quantify the efficiency gain of the developed atom-filtering methods and GPU calculations, we considered exhaustive sampling, which is a procedure employed during the fitting of structural data into AFM topographic images. This method samples rigid-body orientations of a given structural template in 3D space, each time computing the corresponding simulated AFM image, which is compared with an experimental target image. As structural templates, we used proteins which differ in size and shape:  
* The hepatitis B virus capsid cryo-EM ball-shaped structure with a diameter of \(\sim\)36 nm and a total of 270,960 atoms (constructed from PDB 6 BVF);* A model of the actin filament structure containing 24 subunits and having a rod-shape with \(\sim\)10 nm diameter, \(\sim\)70 nm length, and a total of 70,464 atoms (from Ref. [25]);
* The rotor-less F1-ATPase motor cube-like structure with lengths of \(\sim\)10 and \(\sim\)12.5 nm and a total of 21,867 atoms (PDB 1 SKY);
* The small global-shaped relaxin protein with lengths of \(\sim\)3 nm and \(\sim\)5 nm with just 755 atoms (PDB 6 RLX). The molecular structures are shown in Figure 3A. For the four different protein structures, rigid-body orientations were sampled uniformly in 3D space along a grid with a 5 degree spacing, resulting in \((360/5)^{3}\) = 373,248 conformations that were placed on a fixed plate and scanned in the lateral direction by the scanning tip across the XY scanning grid. For each orientation, we then recorded the computation time required to execute the filtering method and to calculate the corresponding simulated image (without visualizing it). We considered the three filtering methods described before, i.e., primitive XY filtering (XY-F), filtering including the lateral scanning direction Z (XYZ-F), and the latter with GPU calculations (XYZ-F2). The corresponding distributions of computation times for the four proteins are shown in Figure 3A and discussed below. For the virus capsid structure, the calculation of an S-AFM image by the XYZ-F method (238 \(\pm\) 12 ms) is faster than the XY-F method (1438 \(\pm\) 61 ms) by a factor larger than 5 and is further enhanced by the XYZ-F2 method (19 \(\pm\) 0.8 ms) by a factor larger than 10. To put those numbers into perspective, this means that the computation of S-AFM images during the rigid-body fitting of the capsid structure into an actual AFM image (with our chosen parameters and hardware setup) would on average take \(\sim\)149 h (i.e., 6.2 days) using the inefficient XY-F methods, versus only \(\sim\)2 h required with the XYZ-F2 implementation. For the F-actin structure, the computation time of an S-AFM image by the XY-F method shows a very wide distribution (499 \(\pm\) 386 ms), which is due to the special rod-shape of the molecule. For upright orientations with respect to the AFM substrate, the XY contact grid of the tip for each atom is generally large, because most atoms are located far from the substrate, hence resulting in a large number of tip-atom collisions to be calculated. In contrast, for orientations placed flat on the substrate, the XY contact grids of atoms have much smaller sizes, and computation is therefore faster. In such cases, speed improvements by additional Z-filtering are less relevant--distributions of the XY-F and XYZ-F methods even overlap--whereas, for filament positions oriented towards the upright shape, Z-filtering becomes increasingly important and the XYZ-F method significantly enhances the computation speed, as shown by the time distribution (117 \(\pm\) 63 ms). The XYZ-F2 method supersedes others by improving the computation speed further by one order of magnitude (9.5 \(\pm\) 1.5 ms). For the relatively small F1-ATPase protein, the calculation of a simulated AFM image by the XYZ-F method (8.8 \(\pm\) 0.7 ms) is faster than the XY-F method (34 \(\pm\) 6 ms) by a factor of about four, and is further enhanced by the XYZ-F2 method (4 \(\pm\) 2 ms) by a factor of about two.