Single or Multiple Isomorphous Replacement

The phase problem in the case of proteins was overcome initially by the method of Isomorphous Replacement [166]. This technique depends on the preparation of protein crystals into which additional heavy atoms that scatter X-rays very strongly (e.g. uranium, platinum or mercury) have been introduced at a few specific positions, without otherwise affecting the crystal struc-ture. The modified (derivative) crystals must have the same crystal lattice (i.e. be isomorphous with) the unmodified (native) crystals, so that their diffraction patterns are sampled at the same points. The diffraction pattern then is the crystallographic sum of the protein and the heavy atoms. The heavy atoms must be in only one or a few positions of each asymmetric unit so that their positions in the unit cell can be deduced from the way they alter the protein diffraction pattern.

The preparation of heavy-atom derivatives generally relies on trial-and-error testing. The affinity of heavy metal ions for thiol groups makes Cys residues logical targets but they are useful only if there are just one or two that can react with the heavy metal without disturbing the protein structure or the crystal lattice. Considerable experience over the many years of crys-tallography of biological macromolecules has yielded a collection of compounds of uranium,

2.2 Basics of crystallography

lead, platinum, mercury, gold, silver and the lanthanoids that have been successful in the past and can be ordered asheavy atom kitsby different suppliers.

The phase problem is solved in the case of Isomorphous Replacement by determining the position(s) of the introduced heavy atoms by use ofDifference Patterson maps. As described in section2.2.9, only the intensities of the reflections are needed for the calculation of a Patter-son map and such a map shows the vectors between all the atoms of the unit cell. To determine the positions of the heavy atoms, the patterson map is calculated using the differences in the diffraction intensities between the native crystal and the derivative crystal.

If two isomorphous data sets of structure factors, F_PH for the derivative crystal and F_P for the native crystal, are given, then the difference in the two Patterson functions

P_PH(u)−P_P(u) = 1 V

∑

h

|F_PH(h)|²− |F_P(h)|²

exp(2πih·u) (2.17) will deliver information about the heavy-atom structure. Perutz [167] showed that equation 2.17 gives rise to non-negligible peaks arising from interference between F_H (the structure factor contribution of the heavy atoms) andF_P(the structure factor contribution of the protein) terms. This led to a slightly modified Patterson function that eliminates this “noise” [168].

P_H(u) = 1 V

∑

h

|F_PH(h)−F_P(h)|²

exp(2πih·u) (2.18) Using the Patterson vector superposition technique [169], the heavy atom positions in the unit cell and thus their phase angles can be determined (see figure2.14).

From the heavy atom parameters, the corresponding structure factor F_H(h) can be calcu-lated. To determine the phase angle,ϕ, ofF_P(h)a set of phase circles is constructed as proposed by [170]. From a chosen originO(see figure2.15), the vectorOAis drawn equal to−F_H. Cir-cles of radii|F_P| and|F_PH| are then drawn aroundO andA, respectively. The intersections at BandB⁰define two possible phase angles forF_P. The phase angleϕ is symmetrical aboutF_H. The ambiguity ofϕcan in principle be resolved by two ways:

• By using a second heavy atom derivative (see section2.2.10.1).

• By utilizing the anomalous scattering effects for the first isomorph (see section2.2.11).

2.2.10.1 Multiple Isomorphous Replacement (MIR)

The phase information provided by a second isomorph is illustrated in figure2.16. Theoreti-cally, the three phase circles will intersect exactly at one point and the phase ambiguity will be

2 Introduction to the methods of crystallography

Figure 2.14:The vector superposition method. The Patterson map of a unit cell can be regarded as the superpo-sition of the structure (and its inverse), with each of its atoms placed alternately at the origin. By shifting each peak of the Patterson function to the origin and calculating the correlation of all re-maining peaks with the unshifted map, it is possible to deconvolute the Patterson function. From [162]

Figure 2.15:Harker construction [170] for a single Isomorphous Replacement.ϕ1andϕ2are the most probable phases forF_P. From [162]

2.2 Basics of crystallography

Figure 2.16:Harker construction for a double Isomorphous Replacement.ϕM is the most probable phase for F_P. From [162]

resolved. In reality, there will be errors in the observed amplitudes|F_P|and|F_H|and inF_H due to errors in the heavy atom parameters. Additionally, the isomorphism can be imperfect leading to further errors. To deal with those problems, a model was proposed by [171].

2.2.10.2 Treatment of errors

Owing to errors, the triangle formed by F_P,F_PH andF_H fails to close (see figure2.17). This lack of closure errorε is a function of the calculated phase angleϕ_p

ε(ϕ_P) =|F_PH|_obs− |F_PH|_calc

After an initial set of heavy atoms has been found, their parameters (x,y,z, occupancy, thermal parameters) have to be refined. This is achieved by minimization of

∑

ε² E

whereE is the estimated error (∼=h(|F_PH|_obs− |F_PH|_calc)²i)) [172]. Phase refinement can be monitored by the

phasing power= ∑|F_H|_calc

∑||F_H|_obs− |F_H|_calc|

2 Introduction to the methods of crystallography

Figure 2.17:Phase errors and their treatment. The calculated heavy-atom structure results in a calculated value for both the phase and the magnitude ofF_H (red). According to the value ofϕP, the triangleF_P− F_H−F_PH will fail to close by an amountε. This gives rise to a bimodel phase distribution for a single derivative. The phase probability combined from a series of derivatives has a most probable phase and a best phase (for which the overall phase error is minimal). From [162].

The phasing power should be bigger than 1 (if less than 1, then the phase triangle can not be closed viaF_H). The resulting phase probability is given by

P(ϕ_P) =exp

−ε²(ϕ_P) 2E²

It can be shown, that the error of the phases is minimum, if thebest phaseϕ_best, i.e. the centroid of the phase distribution

ϕ_best = Z

ϕ_PP(ϕ_P)dϕ_P

is used instead of the most probable phase. Thefigure of merit m=

RP(ϕ_P)exp(iϕ_P)dϕ_P RP(ϕ_P)dϕ_P

is used as a quality measurement of the phases. A value of 1 formindicates no phase error, a value ofm=0.5 represents a phase error of about 60^◦, while a value of 0 means that all phases are equally probable.

2.2 Basics of crystallography

Figure 2.18:In case of anomalous scattering the scattering factor becomes complex with a real anomalous con-tribution f⁰and an imaginary contributioni f⁰⁰. From [162].

2.2.11 Single Isomorphous Replacement with Anomalous