Investigation of blind tip estimation

End-to-end differentiable blind tip reconstruction for noisy atomic force microscopy images

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. Codes are available at https://github.com/matsunagalab/differentiable_BTR .

Rigid-body fitting to atomic force microscopy images for inferring probe shape and biomolecular structure

Atomic force microscopy (AFM) can visualize functional biomolecules near the physiological condition, but the observed data are limited to the surface height of specimens. Since the AFM images highly depend on the probe tip shape, for successful inference of molecular structures from the measurement, the knowledge of the probe shape is required, but is often missing. Here, we developed a method of the rigid-body fitting to AFM images, which simultaneously finds the shape of the probe tip and the placement of the molecular structure via an exhaustive search. First, we examined four similarity scores via twin-experiments for four test proteins, finding that the cosine similarity score generally worked best, whereas the pixel-RMSD and the correlation coefficient were also useful. We then applied the method to two experimental high-speed-AFM images inferring the probe shape and the molecular placement. The results suggest that the appropriate similarity score can differ between target systems. For an actin filament image, the cosine similarity apparently worked best. For an image of the flagellar protein FlhA_C, we found the correlation coefficient gave better results. This difference may partly be attributed to the flexibility in the target molecule, ignored in the rigid-body fitting. The inferred tip shape and placement results can be further refined by other methods, such as the flexible fitting molecular dynamics simulations. The developed software is publicly available.

Observation of functional dynamics of individual biomolecules is important to understand molecular mechanisms of cellular phenomena. High-speed (HS) atomic force microscopy (AFM) is a powerful tool that enables us to visualize the real-time dynamics of working biomolecules under near-physiological conditions. However, the information available by the AFM images is limited to the two-dimensional surface shape detected via the force to the probe. While the surface information is affected by the shape of the probe tip, the probe shape itself cannot be directly measured before each AFM measurement. To overcome this problem, we have developed a computational method to simultaneously infer the probe tip shape and the molecular placement from an AFM image. We show that our method successfully estimates the effective AFM tip shape and visualizes a structure with a more accurate placement. The estimation of a molecular placement with the correct probe tip shape enables us to obtain more insights into functional dynamics of the molecule from HS-AFM images.

Investigation on blind tip reconstruction errors caused by sample features

Precision measurements of a nanoscale sample surface using an atomic force microscope (AFM) require a precise quantitative knowledge of the 3D tip shape. Blind tip reconstruction (BTR), established by Villarrubia, gives an outer bound with larger errors if the tip characterizer is not appropriate. In order to explore the errors of BTR, a series of simulation experiments based on a conical model were carried out. The results show that, to reconstruct the tip precisely, the cone angle of the tip characterizer must be smaller than that of the tip. Furthermore, the errors decrease as a function of the tip cone angle and increase linearly with the sample radius of curvature, irrespective of the tip radius of curvature. In particular, for sharp (20 nm radius) and blunt (80 nm radius) tips, the radius of curvature of the tip characterizer must be smaller than 5 nm. Based on these simulation results, a local error model of BTR was established. The maximum deviation between the errors derived from the model and the simulated experiments is 1.22 nm. Compared with the lateral resolution used in the above simulated experiments (4 nm/pixel), it is valid to ignore the deviations and consider the local error model of BTR is indeed in quantitative agreement with the simulation results. Finally, two simulated ideal structures are proposed here, together with their corresponding real samples. The simulation results show they are suitable for BTR.