We have discussed the idea that the structure (ρ(xyz)) and the structure factors (→F(hkl)) are Fourier transforms of each other. That means that an image of the electron density of the structure can be computed by Fourier wave summation once the values of the →F's are known. Recall that each spot or 'reflection' in the diffraction pattern from the crystal represents one structure factor with a particular set of indices, hkl. This means we can measure the structure factors from the observed diffraction pattern. Specifically, the structure factor magnitude |F(hkl)| is obtained as the square root of the recorded intensity of the spot (i.e. the darkness of the spot on the film or the number of x-ray photons detected). But you'll recall that the the structure factors are complex values, which is why they are denoted here as vector quantities. Each structure factor carries a magnitude |F(hkl)| and a phase α(hkl), the latter which you'll recall describes how the corresponding Fourier wave needs to be shifted when doing the Fourier summation to compute ρ(xyz). Critically, the phases α(hkl) of the reflections are not directly measurable in the diffraction experiment. It is important in various contexts to keep in mind that the phases of the structure factors

can

be calculated readily (either as the Fourier transform of ρ(xyz)

or using the structure factor equation to sum over atomic contributions, depending on the details of the particular calculation) once the structure is known, or at least partially modeled. But the phases are not obtainable by direct experimental measurement. The inability to measure the phases for the structure factors presents an obvious, major obstacle, since we cannot compute an image of the structure in order to model it without that knowledge. This is known as the phase problem.

There are essentially two different ways that the phase problem can be overcome in typical macromolecular crystallography applications. In one approach, known as 'molecular replacement', a structure must already be known for a similar molecule; perhaps the structure of a homologous protein from another species, or the structure of a protein in the 'apo' form when the structure of a bound complex has been crystallized. In the molecular replacement approach, the previously determined structure (once its orientation and position in the unit cell of the unknown crystal can be established) provides a powerful workaround to the phase problem, since approximate phases for the unknown structure can be calculated using the previously determined structure as a starting model. Later we will discuss the mechanics of molecular replacement and its usefulness, along with the limitations one faces in overcoming bias from the starting model.

The second approach to overcoming the phase problem encompasses a variety of 'heavy atom' methods. In these applications, a small number of much heavier atoms are present among the thousands of lighter atoms in the crystal of the macromolecule. In the most traditional variation, which the Perutz and Kendrew laboratories pioneered when determining the first protein structures of myoglobin and hemoglobin, a diffraction data set (i.e. a set of structure factor magnitudes) is measured for 'native' crystals, and then separate data sets are measured for crystals that have been 'derivatized' by the binding of one or a few heavy atoms (Hg or Pt for example) to each protein molecule in the crystal by soaking the crystal specimen in various heavy metal reagents. In another variation, pioneered by Wayne Hendrickson's laboratory, the special atoms need not be much heavier that the protein atoms if they scatter x-rays 'anomalously' at certain x-ray wavelengths. In other variations, the heavy atoms or anomalously scattering atoms may already be present in the native protein, e.g. if it contains metal clusters or if the protein can be synthesized with unnatural amino acids containing anomalously scattering atoms.

How do heavy atom methods allow phases to be determined? A general idea that applies accross the various heavy atom phasing approaches is that additional experiments (i.e. measurements from crystals with and without heavy atoms contributing) provide additional equations concerning each structure factor. And this additional informational ultimately makes it possible to recover what was missing, i.e. the phase value α(hkl)

for each reflection. We will discuss how that extra information gives us the missing phases later. Regarding the mechanics of how the phase problem gets circumvented in the different variations of heavy atom methods, the unifying strategic idea is that one must first determine the 'heavy atom substructure'. That is, the positions of the heavy atoms in the crystal unit cell must be determined as a first order of business. How can the heavy atom substructure be determined? Setting aside some mathematical details to be discussed later, the short answer is that measuring structure factor magnitudes from crystals with and without the heavy atoms and then taking their difference gives us estimates for what the structure factor magnitudes would be for a

hypothetical

crystal consisting of only the heavy atoms. To be more concrete, if |FP(hkl)| is the structure factor magnitude from the native crystal and |FPH(hkl)| is the structure factor magnitude from the crystal derivatized with a heavy atom, then the structure factor magnitudes from the hypothetical heavy atom substructure can be approximated as |fH(hkl)|≈||FPH(hkl)|−|FP(hkl)||. With these estimates of structure factor magnitudes from a crystal containing

just a few atoms

, we are faced with what turns out to be a simpler problem: what few-atom structure would give us structure factor magnitudes |fH(hkl)|? As we will discuss shortly, there are a variety of methods for sidestepping the phase problem and solving the structures of small molecules (i.e. structures that contain just a few atoms) more or less directly. We use those methods to obtain the heavy atom substructure as a first step in solving the structure of a macromolecular crystal using heavy atom methods. Because solving the structure of a crystal with just a few atoms plays a critical role in macromolecular crystallography, we consider that topic next.