Protein Crystallography

While the phase problem stood, two kinds of result were obtained around it, and both mattered.

The first was to work on molecules small enough to be solved by trial. For a structure with a few dozen atoms, one can guess an arrangement, compute the diffraction pattern it would give, compare with the measured intensities, and adjust — and if a heavy atom is present its position can often be found from the pattern of intensities alone. Dorothy Hodgkin made this a method. Her structure of penicillin, completed in 1945, settled a chemical argument by showing a strained four-membered ring that senior chemists had declared impossible; her structure of vitamin B12, in 1956, located 181 atoms around a central cobalt in a molecule for which no chemical route to the answer existed at all. The B12 work used one of the first digital computers applied to crystallography, and it demonstrated that X-ray analysis could now outrun chemical inference.

The second route did not use diffraction patterns to find a structure at all; it used them to check one. Linus Pauling had established that the peptide bond is planar, with partial double-bond character that prevents rotation about it, and he knew the hydrogen bond's preferred length and geometry. With Robert Corey and Herman Branson he asked what regular conformations a polypeptide chain can adopt if every backbone amide is to make a hydrogen bond and no bond angle is to be strained. The question has only a few answers. One is a helix with 3.6 residues per turn and a rise of 1.5 Å per residue — the $\alpha$-helix — and another is a pleated sheet in which neighbouring strands hydrogen-bond to each other.

Both were published in 1951, derived from bond lengths and angles on paper, with no protein structure in existence to test them against. Astbury's stretched and unstretched keratin patterns from the 1930s turned out to correspond to the two forms. Seven years later the helices appeared in the first protein structure, at the predicted dimensions. It is the clearest case in structural biology of a prediction from chemical principles arriving before the measurement, and it is worth noting what it did not predict: how the helices are arranged relative to each other, which is the part that required the phases.

The reasoning behind the solution is worth following, because it is a case of extracting missing information by deliberately perturbing the sample.

First, the scale of the problem. Each diffraction spot is one Fourier component, and the number of them grows as the inverse cube of the resolution. Myoglobin's 2 Å map required about ten thousand independent measurements, each needing a phase angle somewhere between 0 and 360 degrees. Guessing is not available: a Fourier synthesis with wrong phases gives noise that looks like nothing.

Max Perutz's idea was to add a heavy atom. Haemoglobin has two reactive sulphydryl groups that will bind mercury, and if the crystal's packing is undisturbed — an isomorphous derivative — then every spot's amplitude becomes the sum of two contributions: the protein's, unknown in phase, and the mercury's, whose phase is calculable once its position is known. The two interfere. Where they are in step the spot gets brighter; where they oppose, dimmer. The change in each amplitude therefore carries information about the protein's phase at that reflection.

Whether the effect would be big enough to measure was not obvious, and Francis Crick and Beatrice Magdoff computed it in 1956. For $N_H$ heavy atoms of scattering factor $f_H$ in a protein of $N_P$ light atoms of average factor $f_P$, the typical fractional change in amplitude is

$$\frac{\Delta F}{F} \approx \sqrt{\frac{2N_H}{N_P}} \cdot \frac{f_H}{f_P}.$$

Haemoglobin has about 4,500 non-hydrogen atoms, mostly carbon, nitrogen and oxygen, so $f_P \approx 7$ electrons. Mercury has $f_H = 80$. With two mercury sites:

$$\frac{\Delta F}{F} \approx \sqrt{\frac{4}{4500}} \times \frac{80}{7} = 0.030 \times 11.4 = 0.34.$$

A third. Intensity measurements of the day were good to a few per cent, so the signal is comfortably above the noise — which is why the method works for a molecule of 65,000 daltons, and why it begins to fail for much larger ones, since the effect falls as $1/\sqrt{N_P}$. For a protein of a million daltons, two mercuries would change the amplitudes by about 7%, and that is the practical ceiling of the technique.

One derivative is not quite enough: the interference fixes the phase to one of two values, so a second derivative with a different heavy atom at a different site is needed to break the tie. Perutz's haemoglobin work used several. With phases in hand, the Fourier synthesis can be computed — ten thousand terms summed at every point of a grid, which on the machines of 1959 took weeks.

The resolution the map achieves follows from Bragg's law, $n\lambda = 2d\sin\theta$. With copper X-rays at $\lambda = 1.54$ Å, resolving $d = 2$ Å needs spots out to

$$\sin\theta = \frac{1.54}{2 \times 2} = 0.385, \qquad \theta = 22.6^{\circ}.$$

Spots at larger angles are weaker and are the first casualties of radiation damage, which is why resolution is a measure of how good the crystal was rather than how good the instrument is.

John Kendrew chose myoglobin because it is a quarter the size of haemoglobin and crystallises well from the muscle of sperm whales, which were then still being hunted. His 6 Å map of 1958 showed a sausage of electron density; the 2 Å map of 1960 showed the chain.

It was a mess. Eight helical segments of varying length, joined by irregular turns, packed into a compact lump with the haem group wedged in a pocket, and no symmetry of any kind. Kendrew wrote that the most striking feature was the almost total absence of the regularity that had been expected. The helical segments were exactly the ones Pauling and Corey had derived on paper seven years earlier, at the predicted pitch. The way those segments were arranged relative to one another was not derivable from anything.

This is the result the field was founded on, and it set the agenda for everything after. A protein's shape is specific, reproducible and irregular, which means it must be determined rather than deduced — and, as the chemistry turned out, it is determined by the amino acid sequence alone, a fact established at almost the same time and discussed under protein structure prediction. The immediate payoff was medical: Perutz's haemoglobin maps located the glutamate at position 6 of the β chain whose replacement by valine causes sickle-cell disease, and showed that the substitution creates a sticky hydrophobic patch on the deoxygenated molecule — a surface defect that makes the molecules polymerise into fibres.

What could then be done with structures of enzymes, membrane proteins and assemblies of hundreds of components is structural biology. The crystals themselves remain the bottleneck: growing one is still done by dispensing thousands of conditions and hoping.