https://doi.org/10.1038/s41598-022-27057-2
“Observing the structural dynamics of biomolecules is vital to deepening our understanding of biomolecular functions. High-speed (HS) atomic force microscopy (AFM) is a powerful method to measure biomolecular behavior at near physiological conditions. In the AFM, measured image profiles on a molecular surface are distorted by the tip shape through the interactions between the tip and molecule. Once the tip shape is known, AFM images can be approximately deconvolved to reconstruct the surface geometry of the sample molecule. Thus, knowing the correct tip shape is an important issue in the AFM image analysis. The blind tip reconstruction (BTR) method developed by Villarrubia (J Res Natl Inst Stand Technol 102:425, 1997) is an algorithm that estimates tip shape only from AFM images using mathematical morphology operators. While the BTR works perfectly for noise-free AFM images, the algorithm is susceptible to noise. To overcome this issue, we here propose an alternative BTR method, called end-to-end differentiable BTR, based on a modern machine learning approach. In the method, we introduce a loss function including a regularization term to prevent overfitting to noise, and the tip shape is optimized with automatic differentiation and backpropagations developed in deep learning frameworks. Using noisy pseudo-AFM images of myosin V motor domain as test cases, we show that our end-to-end differentiable BTR is robust against noise in AFM images. The method can also detect a double-tip shape and deconvolve doubled molecular images. Finally, application to real HS-AFM data of myosin V walking on an actin filament shows that the method can reconstruct the accurate surface geometry of actomyosin consistent with the structural model. Our method serves as a general post-processing for reconstructing hidden molecular surfaces from any AFM images. ”
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Twin experiment: noise-free AFM images
By performing twin experiments, we compared the accuracies of two blind tip reconstruction algorithms, the original and end-to-end differentiable BTRs (Fig. 2). In the experiments, 20 frames of artificial pseudo-AFM images were generated from the structure of myosin V motor domain (PDB ID: 1OE940) using a typical tip shape (a hemisphere combined with a circular frustum of a cone20, which serves as the ground truth tip). The reconstructed tip shape and molecular surface were compared to those of the ground truth (see Methods for details). For each pseudo-AFM image, the structure was randomly oriented. To investigate the relationship between the parameters (thresh and λ) of the algorithms and the identity I∘P=I�∘�=�, we applied wide range of parameter values and monitored the mean square error MSE(p)MSE(�) and reconstructed tip shapes.
Schematic of twin experiments. From the structural model, the images of molecular surfaces are generated. Then, the molecular surfaces are converted to pseudo atomic force microscopy (AFM) images by dilation with a given tip shape. These molecular surfaces and tip shape are used as ground truths in the experiments. The 3D molecular structure is drawn with PyMOL39.
Figure 3 shows the results of BTRs from noise-free AFM images. Note that some of the pixels around the sample molecule in the image are cropped and enlarged for visual clarity. The full-size images are shown in Supplementary Fig. 1. Under the noise-free condition, the original BTR can perfectly reconstruct the ground truth tip with thresh≪1�ℎ���ℎ≪1 (Fig. 3d, left). MSE(p)MSE(�) increases as thresh�ℎ���ℎ increases (Fig. 3b), correlated with the thickness of the reconstructed tip shape (Fig. 3d, right). The rate of the increase is slow for a range of small thresh�ℎ���ℎ, but is gradually accelerated later. The same is true for the differentiable BTR. The differentiable BTR also works perfectly to reconstruct the true tip shape with λ≪1�≪1 (Fig. 3e, left). MSE(p)MSE(�) increases as λ increases (Fig. 3c), reconstructing thicker tips (Fig. 3e, right). Notably, the slope of MSE(p)MSE(�) with respect to λ increases suddenly.
Results of twin experiment in noise-free condition. (a) 1st frame of 20 images used for the twin experiment. (b, c) Mean square errors at various parameter values. (d, e) Cross sections of reconstructed tip shapes along the x-axis with the original and differentiable blind tip reconstructions (indicated by dashed blue lines and dashed green lines, respectively), compared with the ground truth (red line). (f, g) Root mean square deviations (RMSDs) of the reconstructed tips from the ground truth (with the same coloring scheme). (h) Reconstructed molecular surfaces by the deconvolutions with the reconstructed tips. (i) RMSDs of the deconvoluted molecular surfaces of all 20 frames from the ground truth visualized by violin plots.
Figure 3h shows the molecular surfaces reconstructed by deconvolving the 1st frame of the pseudo-AFM images (Fig. 3a) of the data set with the reconstructed tips. Here, the deconvolution was performed by using the erosion. The results show the shape outlines of myosin V motor domain recovered by using both BTRs. Although the reconstructed molecular surfaces S(r)�(�) by the erosion have a property of S(r)⊇S�(�)⊇�, making the reconstructed surface thicker than the ground truth, the reconstructed molecular surfaces look qualitatively similar to that of myosin V motor domain for both BTRs. The accuracies of the reconstructed molecular surfaces are quantitatively evaluated by root mean square deviations (RMSDs) from the ground truth surfaces (Fig. 3i). In the calculation, deconvolutions were performed for all 20 frames of the images and compared with the ground truth. The RMSDs show that both BTRs have good accuracies for reconstructing molecular surfaces from noise-free AFM images.
Twin experiment: noisy AFM images
We then compared the robustness of the two BTRs against noise by adding spatially independent Gaussian noise with a standard deviation of σ=0.3σ=0.3 nm to the 20 frames of pseudo-AFM images used in the previous noise-free condition. Here, σ=0.3σ=0.3 nm is a typical noise size in the current HS-AFM measurements20. 100 sets of noisy pseudo-AFM data (each containing 20 frames) were created using 100 different noise realizations. Again, to investigate the relationship between the parameters (thresh and λ) and the identity I∘P=I�∘�=�, we applied parameter values in a wide range and monitored MSE(p)MSE(�) (Fig. 4b,c). The mean and standard deviation of MSE(p)MSE(�) was calculated only from the single set of 20-frame images by using the cross validation (see “Methods” for details). The accuracies of reconstructed tip shapes and surfaces were evaluated by 100 sets of different noise realizations.
Results of twin experiment in noisy condition. (a) 1st frame of 20 images used for the twin experiment. (b, c) Mean square errors (MSEs) optimized at various parameter values. The mean and standard deviation of MSE was calculated only from the single set of 20-frame images using the cross validation. Shaded area indicates the one standard error bound. (d, e) Cross sections of reconstructed tip shapes along the x-axis with the original and differentiable blind tip reconstructions (indicated by dashed blue lines and dashed green lines, respectively), compared with the ground truth (red line). Shaded area represents the standard deviation. The tip shapes used for deconvolution are indicated by the black lines. (f, g) Root mean square deviations (RMSDs) of the reconstructed tips from the ground truth visualized by violin plots (with the same coloring scheme). (h) Reconstructed molecular surfaces by the deconvolutions with the reconstructed tips. (i) RMSDs of the deconvoluted molecular surfaces of all frames (20 images) from the ground truths visualized by violin plots.
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