Science & Theory

High-Speed Atomic Force Microscopy

High-Speed Atomic Force Microscopy (HS-AFM) is a scanning probe technique capable of imaging individual protein molecules at near-physiological conditions, in real time. Unlike cryo-EM or X-ray crystallography — which capture static or ensemble-averaged snapshots — HS-AFM movies can directly observe protein conformational dynamics on timescales of milliseconds.

Key Reference

Ando T. et al. (2011). A high-speed atomic force microscope for studying biological macromolecules. PNAS, 98(22):12468–72.

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The Physical Imaging Model

Height Mapping

The fundamental observable in AFM is the topographic height \(H(x, y)\) at each lateral position \((x, y)\) of the sample:

\[H(x, y) = \max_{i} \left( z_i + \sqrt{r_i^2 - \left[(x - x_i)^2 + (y - y_i)^2\right]} \right)\]

where \(i\) indexes over all atoms with position \((x_i, y_i, z_i)\) and van der Waals radius \(r_i\).

This max-over-atoms is a non-differentiable operation. synth-afm approximates it with the Log-Sum-Exp (LSE) soft maximum:

\[\text{logsumexp}_\beta(h_1, \ldots, h_N) = \beta \log \sum_i \exp\!\left(\frac{h_i}{\beta}\right)\]

As the smoothness parameter \(\beta \to 0\), this converges to the exact max. This approximation is smooth and fully differentiable through JAX's autograd system.

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Tip Dilation

A real AFM tip is not an ideal point probe — it has a finite radius \(r_\text{tip}\), typically 10–30 nm for HS-AFM cantilevers. The image recorded is actually the mathematical dilation of the sample surface by the tip shape (Villarrubia, 1997).

For a spherical tip of radius \(r_\text{tip}\), the image height is:

\[H_\text{tip}(x, y) = \max_{i} \left( z_i + \sqrt{(r_i + r_\text{tip})^2 - d_{xy,i}^2} \right) - r_\text{tip}\]

where \(d_{xy,i}^2 = (x - x_i)^2 + (y - y_i)^2\).

This correctly accounts for the broadening of surface features and is the default kernel in synth-afm.

Physical Interpretation

Tip dilation makes narrow features appear wider and rounded. For a protein with features on the 1–5 nm scale, a 10 nm tip radius produces substantial broadening. Always report the tip radius used when publishing synthetic HS-AFM data.

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Scanning Lag

HS-AFM acquires images by raster-scanning a cantilever line-by-line. Each horizontal line is recorded at a slightly different moment in time. If the protein is moving during a scan, the result is a spatially-sheared image — different columns reflect different instantaneous conformations.

The time delay for column \(y\) in a frame starting at time \(t_0\) is:

\[\Delta t(y) = \frac{y}{f_\text{scan} \cdot W}\]

where \(f_\text{scan}\) is the frame rate and \(W\) is the image width in pixels. synth-afm models this by indexing into a molecular dynamics trajectory using the correct time-offset for each column.

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van der Waals Radii

Atomic radii used for the height mapping are from the Bondi (1964) parameter set:

Element	Radius (Å)
H	1.20
C	1.70
N	1.55
O	1.52
S	1.80
P	1.80

Atoms of unknown element fall back to a default radius of 1.70 Å (Carbon).

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Differentiability

Every operation in synth-afm — the LSE height map, tip dilation, scanning lag interpolation, and noise addition — is written in pure JAX and supports jax.grad, jax.jit, and jax.vmap.

This makes synth-afm suitable as the forward model inside a gradient-based structure determination loop:

\[\mathcal{L}(\mathbf{X}) = \|\mathcal{H}(\mathbf{X}) - \mathbf{I}_\text{exp}\|^2\]

\[\mathbf{X}^* = \arg\min_{\mathbf{X}} \mathcal{L}(\mathbf{X})\]

where \(\mathcal{H}\) is the AFMSimulator.scan operator and \(\mathbf{I}_\text{exp}\) is an experimentally acquired HS-AFM frame.