Structural Biology from Zero · 7: Seeing Molecules — X-ray, Cryo-EM, and NMR

Part 1 established that visible light cannot resolve atomic features in a protein. The coordinates rendered throughout this series instead come from measurements interpreted through physical and computational models. This article introduces three major sources of those measurements: X-ray crystallography, cryo-electron microscopy, and nuclear magnetic resonance. Understanding their principles and limitations is necessary for evaluating the structures they produce, which is the subject of Part 8.

Let me restate the wall precisely, because the number is the whole motivation.

Why light fails, quantitatively

To resolve two features you need a probe whose wavelength is comparable to, or smaller than, the distance between them. This is the diffraction limit: roughly, the smallest resolvable separation is on the order of $\lambda/2$ (more precisely $\lambda/2\mathrm{NA}$ for a lens of numerical aperture $\mathrm{NA}$ , but the factor of order one doesn't change the argument). You cannot use a meter stick to measure the width of a hair.

The features we care about in a protein are atoms and the bonds between them, separated by about $1.5\ \text{Å} = 0.15\ \text{nm}$ . Visible light has a wavelength of $400$ – $700\ \text{nm}$ . The ratio is brutal:

$\frac{\lambda_{\text{visible}}}{d_{\text{bond}}} \approx \frac{500\ \text{nm}}{0.15\ \text{nm}} \approx 3300.$

Visible light is more than three thousand times too coarse to resolve atoms — it washes over a molecule the way ocean swell washes over a pebble, diffracting around it without ever registering its shape. To see atoms we need a probe with a wavelength of about an ångström. Two kinds of probe can deliver that: X-ray photons and electrons. The difference between how they get to an ångström is the difference between the two dominant imaging methods, so it's worth doing both calculations.

For a photon, wavelength and energy are tied by $E = hc/\lambda$ . Using the convenient constant $hc \approx 1240\ \text{eV·nm}$ , an ångström-wavelength photon needs

$E = \frac{hc}{\lambda} = \frac{1240\ \text{eV·nm}}{0.1\ \text{nm}} = 12{,}400\ \text{eV} \approx 12.4\ \text{keV}.$

That is a hard X-ray — exactly the energy range synchrotrons are built to deliver. For an electron, the wavelength comes instead from de Broglie's relation $\lambda = h/p$ , with the momentum set by an accelerating voltage. We'll do that calculation properly in a moment, because at the voltages electron microscopes use, the electron is relativistic and the naive formula is wrong.

X-ray crystallography: diffraction from a lattice

X-ray crystallography is the oldest of the three methods and remains a major source of high-resolution structures in the PDB. Its central experimental challenge is signal. A single protein molecule scatters X-rays too weakly for conventional structure determination, so crystallography uses a periodic three-dimensional lattice containing many identically oriented copies. Their scattered waves add coherently in specific directions, producing a measurable pattern of sharp spots. Growing a suitable protein crystal is often the most difficult step, and many proteins do not crystallize readily.

Why sharp spots, and where do they point? The condition is Bragg's law, and it is worth deriving because it is a two-line piece of wave physics that entirely explains the geometry of diffraction. Picture the crystal as families of parallel planes of atoms, spaced a distance $d$ apart. An X-ray beam strikes at a grazing angle $\theta$ to the planes; some scatters off the top plane, some off the next plane down. The wave that reflects off the lower plane travels an extra distance — down to that plane and back up — and simple trigonometry gives that extra path length as $2d\sin\theta$ . The two reflected waves reinforce each other (constructive interference, a bright spot) only when that extra path is a whole number of wavelengths:

$n\lambda = 2d\sin\theta, \qquad n = 1, 2, 3, \dots$

That is Bragg's law. Read it as a machine that converts an angle into a spacing: measure the angle $\theta$ at which a spot appears, and you learn the spacing $d$ of a family of planes in the crystal. The full set of spots encodes the geometry of the lattice — their positions give you the unit cell and symmetry, and their intensities encode how the electron density is distributed within it. In principle, a Fourier transform of the diffraction pattern reconstructs the electron density of the molecule, from which you build an atomic model.

"In principle," because there is a famous catch: the phase problem. To run that Fourier transform you need both the amplitude and the phase of each diffracted wave. But a detector only counts photons — it measures intensity, which is amplitude squared, and throws the phase away entirely. Half the information needed to reconstruct the image is missing from the raw data. Recovering the lost phases is the central intellectual difficulty of crystallography, and it is solved by additional tricks: molecular replacement (borrow phases from a known, similar structure), or anomalous scattering methods like MAD and SAD (introduce heavy atoms that scatter in a phase-shifted way and use them as reference points). Only once the phases are recovered does the electron-density map appear.

The quality of that map is its resolution, reported in ångströms and equal to the smallest plane spacing $d$ that still produces measurable spots — so, confusingly, a smaller number means a better structure. At $1\ \text{Å}$ you can see individual atoms and even some hydrogens; at $2$ – $2.5\ \text{Å}$ (typical for proteins) you resolve the backbone and most side chains confidently; beyond $3.5\ \text{Å}$ you see the fold's overall shape but must infer side-chain details, and the risk of error climbs. That distinction — and the map quality behind it — is exactly what Part 8 is about.

Cryo-EM: imaging single molecules with electrons

For decades, if a protein wouldn't crystallize — and huge, floppy complexes and membrane proteins notoriously won't — you were often stuck. Cryo-electron microscopy removed that requirement, and in doing so it has reshaped structural biology over the last decade. Instead of a crystal, you flash-freeze a thin film of protein solution so fast that the water forms a glass (vitreous ice) rather than crystalline ice that would destroy the sample, trapping the molecules in random orientations. Then you image them directly with a beam of electrons in a transmission electron microscope.

The probe is now electrons, so let's finish that de Broglie calculation, relativistically, because the numbers are striking. An electron accelerated through a voltage $V$ gains kinetic energy $eV$ ; at the $200$ – $300\ \text{kV}$ used in cryo-EM this is a substantial fraction of the electron's rest energy $m_e c^2 = 511\ \text{keV}$ , so we must use the relativistic relation between momentum and energy. The wavelength works out to

$\lambda = \frac{hc}{\sqrt{eV\left(2 m_e c^2 + eV\right)}}.$

Plugging in $V = 300\ \text{kV}$ (so $eV = 300\ \text{keV}$ ) with $hc = 1240\ \text{eV·nm}$ and $m_e c^2 = 511\ \text{keV}$ :

$\lambda = \frac{1240\ \text{eV·nm}}{\sqrt{(3.0\times10^{5})\,(1.022\times10^{6} + 3.0\times10^{5})}\ \text{eV}} \approx \frac{1240}{6.30\times10^{5}}\ \text{nm} \approx 0.00197\ \text{nm} = 0.0197\ \text{Å}.$

Two hundredths of an ångström — the electron's wavelength is about fifty times smaller than the atomic spacing we're trying to resolve. So wavelength is emphatically not the limiting factor in cryo-EM; the resolution is instead limited by radiation damage (electrons destroy the delicate sample as they image it, so each molecule can only be exposed to a tiny, noisy dose), by the detectors, and by the reconstruction. The problem is that each individual image is buried in noise and shows the molecule from an unknown angle.

The solution is statistical, and it is beautiful: single-particle reconstruction. Collect images of hundreds of thousands to millions of individual molecules, each frozen in a random orientation. Computationally determine each molecule's orientation, group images taken from similar viewpoints, and average within each group to beat down the noise; then combine the different views — which are effectively two-dimensional projections through the object from many directions — into a single three-dimensional density map, the same tomographic logic behind a medical CT scan. The "resolution revolution" around 2013, driven by direct electron-detector cameras and better software, pushed cryo-EM from producing blurry blobs to near-atomic maps of $2$ – $4\ \text{Å}$ , now routinely better, and it did so for exactly the enormous, crystallization-resistant machines — ribosomes, spliceosomes, membrane receptors — that crystallography struggled with. The 2017 Nobel Prize in Chemistry recognized its development.

NMR: structures in solution, from a magnet

The third method looks nothing like the other two. Nuclear magnetic resonance spectroscopy determines structures of proteins in solution — no crystal, no freezing, the molecule tumbling in water close to its native state — by exploiting the quantum spin of atomic nuclei. Certain nuclei (notably $^{1}\text{H}$ , and $^{13}\text{C}$ and $^{15}\text{N}$ when the protein is isotopically labeled) behave as tiny magnets. Placed in a strong magnetic field and probed with radio-frequency pulses, each nucleus resonates at a frequency — its chemical shift — exquisitely sensitive to its local electronic environment, so that in effect every atom reports on its own chemical surroundings.

The structural payoff comes from interactions between nuclei. The nuclear Overhauser effect (NOE) produces a signal between two protons whose strength falls off steeply with distance (as $1/r^6$ ), and is therefore most useful for protons within about $5\ \text{Å}$ of each other in space, regardless of their separation in the sequence. Each assigned NOE supplies a short-range distance restraint between specific atoms. Thousands of restraints can define an ensemble of models consistent with the measurements. Variation across those models can reflect conformational heterogeneity, incomplete restraints, or both, so dynamics require additional measurements such as relaxation and exchange experiments. Solution NMR is most straightforward for smaller proteins because spectra become increasingly crowded as molecular size increases, although labeling and advanced experiments extend the practical range.

Choosing a method — and the interactive

The three methods are complementary, not competing, and which one fits depends on the molecule and the question:

X-ray crystallography — often provides high-resolution models for samples that form suitable crystals. Crystal packing and averaging influence the resulting structure.
Cryo-EM — the method for large complexes, membrane proteins, and anything that won't crystallize; near-atomic and improving fast, though historically hard for small (under ~50 kDa) proteins.
NMR — smaller proteins in solution, with unique access to dynamics and disordered or transiently folded regions.

The viewer below makes the physics that opened this article tangible. The first tab lets you dial a probe — a visible photon, an X-ray photon, or an electron at a chosen voltage — and see its wavelength against the $1.5\ \text{Å}$ scale of a chemical bond, so you can watch light fail and X-rays and electrons succeed. The second tab is an interactive Bragg's law: adjust the plane spacing, the wavelength, and the incidence angle, and see the two reflected waves fall in and out of step as the diffraction condition $n\lambda = 2d\sin\theta$ is met.

Interactive. Tab 1: choose a probe (visible photon / X-ray photon / electron) and compare its wavelength — from $E=hc/\lambda$ for photons, the relativistic de Broglie relation for electrons — against the 1.5 Å length of a chemical bond, with a verdict on whether it can resolve atoms. Tab 2: Bragg's law $n\lambda = 2d\sin\theta$ , with the path-length difference between planes drawn and the constructive-interference condition highlighted. Opens full-screen in a new tab.

Where this leaves us

The wall from Part 1 is real but not insurmountable: light is thousands of times too coarse to see an atom, so structural biology reaches for probes with ångström-scale wavelengths and pays the price each method demands. X-ray crystallography amplifies a single molecule's faint scattering into sharp Bragg spots by growing a crystal, then fights the phase problem to turn those spots into an electron-density map. Cryo-EM abandons the crystal entirely, freezing molecules in glassy ice and averaging hundreds of thousands of noisy electron images into a three-dimensional map. NMR reads distance restraints off the spins of nuclei in a magnet, delivering solution-state ensembles that reveal motion. None of them "sees" a protein in the everyday sense; each reconstructs one from a physical signal, through a chain of inference.

And that is precisely why the last article in this arc matters. A structure in the PDB is a model — a human interpretation of experimental data of some finite quality — not a photograph. Some parts of any model are pinned down hard by the data and some are educated guesswork, and the file itself carries the numbers that tell you which is which: the resolution, the R-factors, the per-atom B-factors. In Part 8 we learn to read those numbers and judge, for any structure we pull off the shelf, exactly how much of it to believe.