---
title: "Single-particle Cryo-EM: From Movies to a Map Interactive Lab | Sandbox Physics"
description: "Can more particle images fill in a missing viewing direction? No."
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  zh-CN: "https://sandboxphysics.com/zh/labs/single-particle-cryo-em/"
last_modified: "2026-10-27"
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E43 · Single-particle reconstruction

# Single-particle Cryo-EM: From Movies to a Map

**Can more particle images fill in a missing viewing direction?** No. Averaging more particle images lowers the noise, but the third dimension comes only from seeing the particle from different directions. If no particle is ever seen from a certain direction, that view stays missing however many particles you add.

Find particles in a counted micrograph, estimate the CTF and reconstruct their map. Compare wrong optics, false picks and restricted reference views with genuinely missing viewing directions.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This is a synthetic teaching model in the weak-phase approximation. The particle is a four-lobed phantom on a sixteen-voxel grid, seen from twelve viewing directions at two in-plane angles. Four full-field count frames retain their pixels. Whole-pixel motion, pixel pitch and count normalization are supplied calibration; positions are found from the image. Defocus is fitted from a known amorphous support strip within a positive-underfocus family from 0.5 to 3 micrometres. The assumed white texture spectrum, scalar CTF, finite grid and heuristic fit thresholds limit that inference. Orientations use a supplied broader reference. The isolated inverse uses a padded finite patch and omits weak contributions from distant particles and support texture. Vitrification, ice attenuation and dose damage are only illustrative. The two half-maps share a reference and a calibration, so their agreement does not certify resolution by the gold standard. Atomic interpretation, multiple scattering, detector MTF, continuous orientation search and conformational classification are not included.

Physics tutorial

## Single-particle Cryo-EM: From Movies to a Map

**Background** Single-particle cryo-EM builds a three-dimensional map from many noisy pictures of identical particles frozen in ice. Here you record a full field, find candidates from its pixels, estimate the CTF from a support-film power spectrum, and match the extracted images against a low-resolution reference before reconstructing.

**Why it matters** Recording more particles lowers the noise, so it matters what extra copies can and cannot fix.

Start with the essentials

- **Focus question**: Can more particle images fill in a missing viewing direction?
- **One-sentence intuition**: More independent copies average away noise, but only views from different directions pin down the third dimension.

### Core mathematical model

#### Expected counts in the weak-phase approximation

$$
\begin{aligned}S&=\sum_j T_{r_j}P_{R_j}\rho+\eta\\\mathbb E[C_f]&=b[1+aT_{s_f}(H*S)]\end{aligned}
$$

The full field contains translated particle projections and a support-film texture. The CTF acts on that field before each shifted frame is counted. Motion and count normalization are calibrated; the defocus entering the CTF is estimated later.

#### Electron-optical phase

$$
\chi(q)=\pi\lambda\Delta f\,q^2-\frac{\pi}{2}C_s\lambda^3q^4
$$

Underfocus counts as positive, and spatial frequency is in cycles per nanometre. The damage envelope is an illustrative attenuation set by assumption.

#### CTF from counted texture power

$$
P(q)\approx a_t|H(q)|^2+b_n
$$

The counted support-film spectrum is fitted with nonnegative texture and noise powers. The CTF combines sine phase contrast, amplitude contrast of 0.1 and the assumed dose envelope. The positive-underfocus family is known beforehand; likelihood-gap thresholds are teaching heuristics.

#### Reconstruction from the recorded counts

$$
\begin{aligned}\widehat\rho&=\arg\min_{\rho\geq0}\sum_j\|HP_{\widehat R_j}\rho-y_j\|^2\\&\quad+\alpha\|\rho\|^2\end{aligned}
$$

The solver takes projected gradient steps that keep the density nonnegative, using the same discrete projector and its exact transpose. The reference only supplies orientations; it never adds density to the map.

#### Half-map agreement under a shared reference

$$
F_s=\frac{\operatorname{Re}\sum_{k\in s}V_1(k)\overline{V_2(k)}}{\sqrt{\sum_{k\in s}|V_1(k)|^2\sum_{k\in s}|V_2(k)|^2}}
$$

Candidates alternate between the two halves in picking order before matching. Both halves share the same low-resolution reference and calibration, so their agreement proves neither full direction coverage nor absolute resolution.

### Common difficulties

#### The map is not an atomic structure

**Typical misconception** The reconstructed map shows the protein's atomic structure.

**Better mental model** This is a synthetic teaching model in the weak-phase approximation. The particle is a four-lobed phantom on a sixteen-voxel grid, seen from twelve viewing directions at two in-plane angles. Four full-field count frames retain their pixels. Whole-pixel motion, pixel pitch and count normalization are supplied calibration; positions are found from the image. Defocus is fitted from a known amorphous support strip within a positive-underfocus family from 0.5 to 3 micrometres. The assumed white texture spectrum, scalar CTF, finite grid and heuristic fit thresholds limit that inference. Orientations use a supplied broader reference. The isolated inverse uses a padded finite patch and omits weak contributions from distant particles and support texture. Vitrification, ice attenuation and dose damage are only illustrative. The two half-maps share a reference and a calibration, so their agreement does not certify resolution by the gold standard. Atomic interpretation, multiple scattering, detector MTF, continuous orientation search and conformational classification are not included.

#### A shared reference limits what agreement proves

**Typical misconception** If the shell correlation is high, the map must be right in every direction.

**Better mental model** Both halves share the same assumed low-resolution reference and the same estimated transfer function. The Lab does no joint refinement and does not claim gold-standard independence. It reports the rejected particles and every assigned viewing direction, and it gives no resolution derived from the FSC.

### Run the experiment

1. 01

   #### Collect a movie

   Inspect the default micrograph and its candidate circles, then acquire a new full-field movie. The original first frame remains in the expanded data row.

   **What to observe:** Each particle has four noisy frames, and their counts stay the same when you change the acquisition settings.

2. 02

   #### Correct and classify

   Process the stored movie. Compare the counted power spectrum with the fitted CTF, then click candidate circles to exclude or restore them. Compare the uncorrected crop, corrected crop and reference.

   **What to observe:** Matching picks one of twenty-four discrete orientations for each particle, and rejected windows stay visible.

3. 03

   #### Solve for the volume

   Look at the three orthogonal slices through the reconstruction and watch the data objective fall.

   **What to observe:** Every voxel is solved from the recorded contrast through the projection operator.

4. 04

   #### Test the half-map agreement

   Compare the default with More copies, one direction. Then try the wrong-defocus, false-pick and restricted-reference examples. Each changes a different part of the inference.

   **What to observe:** The two half-maps can agree closely even when every accepted copy is seen from the same direction.

Physical conclusion

Check the sample preparation, count quality and range of viewing directions on their own; half-map agreement does not vouch for them.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)