X-ray Crystallography & Cryo-EM

Definition and Derivation

The scattered amplitude for reflection $(hkl)$ is the Fourier transform of the electron density $\rho(\mathbf{r})$:

$$F_{hkl} = \int_{\text{cell}} \rho(\mathbf{r})\, e^{2\pi i \mathbf{G}_{hkl} \cdot \mathbf{r}}\, dV$$

For a discrete set of $N$ atoms at positions $\mathbf{r}_j = (x_j, y_j, z_j)$(in fractional coordinates), this becomes:

$$F_{hkl} = \sum_{j=1}^{N} f_j\, e^{2\pi i(hx_j + ky_j + lz_j)} \cdot e^{-B_j \sin^2\theta/\lambda^2}$$

where $f_j$ is the atomic scattering factor (proportional to the number of electrons for forward scattering) and $B_j$ is the Debye-Waller (temperature) factor accounting for atomic displacement.

The structure factor is complex: $F_{hkl} = |F_{hkl}|e^{i\phi_{hkl}}$. The measured intensity is proportional to $|F_{hkl}|^2$, giving the amplitude but losing the phase.

Full Derivation: Structure Factor from Unit Cell Scattering

X-rays are scattered by the electrons in each atom. The scattering amplitude from a single atom$j$ at position $\mathbf{r}_j$ relative to the unit cell origin is proportional to its atomic scattering factor$f_j$ (which depends on the element and scattering angle), multiplied by a phase factor encoding the atom's position:

$$A_j = f_j \, e^{2\pi i \, \mathbf{G}_{hkl} \cdot \mathbf{r}_j}$$

The total scattered amplitude from the unit cell is the coherent sum over all $N$atoms. Expressing positions in fractional coordinates $\mathbf{r}_j = x_j \mathbf{a} + y_j \mathbf{b} + z_j \mathbf{c}$and using the orthogonality $\mathbf{a}^*_i \cdot \mathbf{a}_j = \delta_{ij}$:

$$\mathbf{G}_{hkl} \cdot \mathbf{r}_j = (h\mathbf{a}^* + k\mathbf{b}^* + l\mathbf{c}^*) \cdot (x_j\mathbf{a} + y_j\mathbf{b} + z_j\mathbf{c}) = hx_j + ky_j + lz_j$$

Therefore the structure factor is:

$$\boxed{F(hkl) = \sum_{j=1}^{N} f_j \, \exp\!\bigl[2\pi i (h x_j + k y_j + l z_j)\bigr]}$$

This is a complex number: $F(hkl) = |F(hkl)|\, e^{i\phi(hkl)}$. The amplitude$|F|$ determines the intensity of the reflection, and the phase $\phi$encodes positional information about the atoms.

Friedel's Law

An important symmetry property of the structure factor follows directly from the definition. Consider the reflection $(-h, -k, -l)$:

$$F(-h,-k,-l) = \sum_j f_j \, e^{-2\pi i(hx_j + ky_j + lz_j)} = \left[\sum_j f_j \, e^{2\pi i(hx_j + ky_j + lz_j)}\right]^* = F^*(hkl)$$

Therefore $|F(-h,-k,-l)| = |F(hkl)|$ and $\phi(-h,-k,-l) = -\phi(hkl)$. This is Friedel's law: the reflections$(hkl)$ and $(\bar{h}\bar{k}\bar{l})$ form a "Friedel pair" with equal intensities. This holds exactly when$f_j$ is real (no anomalous scattering). Violations of Friedel's law near absorption edges provide phase information in anomalous dispersion methods (MAD/SAD).

Why Phases Matter More Than Amplitudes

A classic demonstration: if you compute the Fourier transform using the amplitudesof one image with the phases of another, the result resembles the image that contributed the phases. This is because phases encode the positions of features (atoms), while amplitudes encode their relative scattering strength.

Full Derivation: Loss of Phase Information

A detector measures the intensity of each diffraction spot, which is proportional to the squared modulus of the structure factor:

$$I(hkl) \propto |F(hkl)|^2 = F(hkl) \cdot F^*(hkl)$$

Writing $F(hkl) = |F(hkl)|\, e^{i\phi(hkl)}$, the intensity gives us$|F(hkl)|^2$ but discards the phase $\phi(hkl)$ entirely. To compute the electron density:

$$\rho(\mathbf{r}) = \frac{1}{V} \sum_{hkl} |F(hkl)|\, e^{i\phi(hkl)} \, e^{-2\pi i(hx+ky+lz)}$$

we need both $|F(hkl)|$ (from $\sqrt{I}$) and$\phi(hkl)$ (unknown). For a protein with $\sim 10{,}000$unique reflections, this means $\sim 10{,}000$ unknown phases. The problem is severely underdetermined from diffraction data alone — this is thephase problem of crystallography.

Without correct phases, even perfect amplitudes yield noise. Conversely, correct phases with approximate amplitudes still produce a recognisable electron density map. The phases encode the positions of structural features, while amplitudes encode their scattering strength.

Molecular Replacement: Rotation and Translation Search

When a homologous structure (the "search model") is available, its known coordinates provide trial phases. The challenge is to find the correct orientation$\mathbf{R}$ and position$\mathbf{t}$ of the model in the new unit cell. This is decomposed into two steps:

Rotation function: Intramolecular vectors (short vectors in the Patterson map) depend only on orientation, not position. The rotation function correlates the observed and model Patterson maps:

$$RF(\mathbf{R}) = \int P_{\text{obs}}(\mathbf{u}) \, P_{\text{model}}(\mathbf{R}\,\mathbf{u}) \, d\mathbf{u}$$

The peak in $RF$ as a function of Euler angles $(\alpha, \beta, \gamma)$gives the correct orientation.

Translation function: With rotation fixed, translate the model by $\mathbf{t}$ and compute structure factors. The translation function maximises the agreement between observed and calculated amplitudes:

$$TF(\mathbf{t}) = \frac{\sum_{hkl} |F_{\text{obs}}(hkl)| \cdot |F_{\text{calc}}(hkl; \mathbf{R}, \mathbf{t})|}{\sqrt{\sum |F_{\text{obs}}|^2 \cdot \sum |F_{\text{calc}}|^2}}$$

Success requires the search model to share $\gtrsim 30\%$ sequence identity with the target. The positioned model provides initial phases for electron density map calculation and iterative refinement.

Derivation: The Debye-Waller Factor

Atoms in a crystal are not stationary — they undergo thermal vibrations about their equilibrium positions. If atom $j$ is displaced by$\Delta\mathbf{r}_j$ from its mean position, the time-averaged scattering factor becomes:

$$\langle f_j \, e^{2\pi i \mathbf{G} \cdot (\mathbf{r}_j + \Delta\mathbf{r}_j)} \rangle = f_j \, e^{2\pi i \mathbf{G} \cdot \mathbf{r}_j} \cdot \langle e^{2\pi i \mathbf{G} \cdot \Delta\mathbf{r}_j} \rangle$$

For a harmonic (Gaussian) distribution of displacements with mean-square displacement$\langle u^2 \rangle$ along the scattering vector direction, the thermal average evaluates to:

$$\langle e^{2\pi i \mathbf{G} \cdot \Delta\mathbf{r}} \rangle = e^{-\frac{1}{2}(2\pi)^2 \langle u^2 \rangle |\mathbf{G}|^2} = e^{-2\pi^2 \langle u^2 \rangle / d^2}$$

Using $|\mathbf{G}| = 1/d = 2\sin\theta / \lambda$ from Bragg's law, and defining the B-factor as$B = 8\pi^2 \langle u^2 \rangle$:

$$\boxed{f_{\text{eff}} = f_0 \, \exp\!\left(-B \frac{\sin^2\theta}{\lambda^2}\right)}$$

This is the Debye-Waller factor. Its physical meaning is clear: thermal motion "smears" the electron density, reducing the effective scattering power most severely at high angles (large $\sin\theta / \lambda$), where the scattering is most sensitive to precise atomic positions.

Definition and Properties

The Patterson function uses only the measured intensities (no phases required):

$$P(\mathbf{u}) = \frac{1}{V}\sum_{hkl} |F_{hkl}|^2 e^{-2\pi i \mathbf{G}_{hkl} \cdot \mathbf{u}}$$

Equivalently, $P(\mathbf{u}) = \int \rho(\mathbf{r})\, \rho(\mathbf{r} + \mathbf{u})\, dV$— the autocorrelation of the electron density. Peaks in the Patterson map occur at positions $\mathbf{u} = \mathbf{r}_j - \mathbf{r}_k$ (interatomic vectors), with heights proportional to $Z_j \cdot Z_k$.

For $N$ atoms, there are $N(N-1)$ non-origin Patterson peaks, plus a large origin peak. The Patterson function is critical for:

• Locating heavy atoms (isomorphous replacement)
• Finding anomalous scatterers (MAD/SAD phasing)
• Molecular replacement (rotation/translation search)

Derivation: Resolution in Crystallography

The resolution of a crystal structure is defined by the minimum d-spacing of the observed reflections. From Bragg's law, the smallest resolvable spacing is:

$$\boxed{d_{\min} = \frac{\lambda}{2\sin\theta_{\max}}}$$

At the theoretical limit $\theta_{\max} = 90°$, we get$d_{\min} = \lambda/2$. In practice, crystal disorder, radiation damage, and thermal motion limit $\theta_{\max}$ to much smaller values.

The number of independent reflections within a given resolution shell is determined by the volume of the reciprocal-space sphere of radius $1/d_{\min}$, divided by the volume per reciprocal lattice point ($1/V$ where $V$ is the unit cell volume):

$$N_{\text{refl}} \approx \frac{4}{3}\pi \left(\frac{1}{d_{\min}}\right)^3 \cdot V = \frac{4\pi V}{3\, d_{\min}^3}$$

For a typical protein with unit cell volume $V = 100{,}000$ Å$^3$: at 3.0 Å resolution there are $\sim 15{,}500$ reflections; at 2.0 Å there are $\sim 52{,}400$; at 1.0 Å there are $\sim 419{,}000$. Higher resolution provides exponentially more data and correspondingly more detail in the electron density map.

Principle

If a homologous structure (the "search model") is available, molecular replacement finds its orientation and position in the new crystal by:

Step 1 — Rotation search: Find the rotation$R$ that best overlaps the Patterson functions. The rotation function is:

$$RF(R) = \int P_{\text{obs}}(\mathbf{u})\, P_{\text{model}}(R\mathbf{u})\, dV$$

Step 2 — Translation search: For the best rotation, find the translation $\mathbf{t}$ that maximises the correlation between observed and calculated structure factors:

$$TF(\mathbf{t}) = \frac{\sum_{hkl} |F_{\text{obs}}| |F_{\text{calc}}(R, \mathbf{t})| \cos(\phi_{\text{calc}} - \phi_{\text{obs}})}{\sqrt{\sum |F_{\text{obs}}|^2 \sum |F_{\text{calc}}|^2}}$$

The phases from the positioned model provide initial estimates for the unknown phases, allowing computation of the electron density map and iterative model building/refinement.

Image Formation

Each micrograph is a 2D projection of the 3D Coulomb potential of the molecule, convolved with the contrast transfer function (CTF):

$$I(\mathbf{r}) = \int_{-\infty}^{\infty} V(\mathbf{r}, z)\, dz \otimes \text{CTF}(\mathbf{r}) + \text{noise}$$

The CTF oscillates as a function of spatial frequency:

$$\text{CTF}(s) = -\sin\left[\pi\left(\Delta f \lambda s^2 - \frac{C_s \lambda^3 s^4}{2}\right)\right]$$

where $\Delta f$ is the defocus, $\lambda$ the electron wavelength (0.025 Å at 200 kV), $C_s$ the spherical aberration, and$s = |\mathbf{s}|$ the spatial frequency. CTF correction is essential for high-resolution reconstruction.

Key Milestones in Crystallography