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APPENDIX 14. B THE SIR CASE: THE ONE-STEP PROCEDURE

APPENDIX 14. B THE SIR CASE: THE ONE-STEP PROCEDURE

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332



Isomorphous replacement techniques

the number of heavy atoms in the unit cell is much smaller than the number

of protein atoms, NPeq >> NHeq . The consequence is that for usual proteins,

the Cochran contribution to G is negligible with respect to the second. This is

quite a relevant result; it is shown that supplementary prior knowledge of the

derivative diffraction data reduces the complexity of the phasing problem from

order NPeq to order NHeq .

The expected sign of the triplet cosine mainly depends on the sign of the

product 1 2 3 ; a positive value of this product indicates positive cosine

values. Negative values of 1 2 3 are also frequent and suggest P values

close to π.

According to equation (14.B.3), the native structure phases may be found as

in traditional DM (see Chapter 6); i.e. the reflections with the largest values

of | | are selected, triplet invariants are found among them, random phases

are generated and submitted to the tangent formula, and the most promising

solutions are selected via suitable figures of merit.

The above approach allows us to solve the native protein structure in one

step; preliminary solution of the heavy-atom substructure is no longer necessary. Even if the one-step approach is highly appealing, the two-step approach

seems more effective, because it confronts the phasing difficulties gradually.

An interesting exercise is to establish a relation between the two-step SIR

approach and the above one-step procedure. Let us suppose that 1 > 0,

2 > 0 and 3 > 0; in this case, the distribution (14.B.3) suggests the relation,

P



≈ 0.



(14.B.5)



Classical two-step SIR techniques suggest the following relationships:

φP1 ≈ φH1 , φP2 ≈ φH2 , φP3 ≈ φH3 ,

which, summed, give

P







H,



(14.B.6)



where H = φH1 + φH2 + φH3 .

Expressions (14.B.6), provided by the two-step SIR relations, and (14.B.5),

provided by traditional direct methods, coincide if H ≈ 0. This will occur if

| 1 |, | 2 |,| 3 | are sufficiently large and if the number of heavy atoms in the

unit cell is small (as is usual in SIR cases).

Suppose now that

1 > 0,

2 > 0 and

3 < 0. Then, distribution

(14.B.3) suggests

P



≈ π.



(14.B.7)



Two-step SIR relations provide the following indications:

φP1 ≈ φH1 , φP2 ≈ φH2 , φP3 ≈ φH3 + π

which, summed, give

P



Since



H







H



+ π.



(14.B.8)



≈ 0, again (14.B.7) and (14.B.8) provide similar phase indications.



About methods for estimating the scattering power of the heavy-atom



A P P E N D I X 14 . C A B O U T M E T H O D S F O R

E S T I M AT I N G T H E S C AT T E R I N G

P O W E R O F T H E H E AV Y-AT O M

SUBSTRUCTURE

We will describe two simple procedures for estimating the scattering power

of a heavy-atom substructure. The first has an algebraic basis and is due to

Crick and Magdoff (1956); the second has a probabilistic background and was

suggested by Giacovazzo et al. (2002).

The algebraic relation. Let us suppose that one or more heavy atoms have

been added to the native protein, so giving rise to a perfect isomorph derivative.

Let < IP > be the average intensity for the protein at a given sin θ/λ, < IH >the

corresponding average intensity for the heavy atom structure, and < Id > the

value for the derivative. An estimate of the average relative change in intensity

is given by the ratio,

[< (Id − IP )2 >]1/2

.

(14.C.1)

< IP >

Firstly, let us consider the acentric reflections. In accordance with equation

(14.C.1),

<



>=



Id = |FP + FH |2 = |FP |2 + |FH |2 − 2|FP FH | cos(φP − φH ),

from which

Id − IP = |FH |2 − 2|FP FH | cos(φP − φH ).



(14.C.2)



Introducing (14.C.2) into (14.C.1) gives,

{ < |FH |4 > + < 4|FP FH |2 cos2 (φP − φH ) > − < 4|FP ||FH |3 cos(φP − φH ) > }1/2

< IP >

Since FP and FH may be considered to be uncorrelated, < cos(φP − φH ) > is

expected to be close to zero and < cos2 (φP − φH ) > close to 1/2. Then,

<



>=



<



>≈



(< |FH |4 > + 2 < |FP |2 >< |FH |2 >)1/2

.

< |FP |4 >



(14.C.3)



Since < |FH |4 > is very much smaller than 2 < |FP |2 >< |FH |2 >, we can

approximate (14.C.3) by,

<



>≈







2



< IH >

,

< IP >



(14.C.4)



which is the Crick and Magdoff relation. Let us apply (14.C.4) to two proteins of quite different size: the first with 1000 atoms per molecule, the second

20 000 atoms per molecule. On assuming Zj = 7 for the average protein atom,

one mercury atom per molecule, we have at sin θ/λ = 0:

1. For the first protein,

< IP > = 1000 × 49 = 49000

< IH > = 1 × 6400 = 6400

< > = 0.51.



333



334



Isomorphous replacement techniques

2. For the second protein,

< IP > = 20000 × 49 = 980000

< IH > = 1 × 6400 = 6400

< > = 0.11.

We see that the average change in intensity is quite detectable, even for large

proteins. It is also clear that a partial occupancy for the mercury atom gives

less favourable values of < >.

It is easy to show that for centric reflections,

<



< IH >

< IP >



>= 2



(14.C.5)



If we compare (14.C.4) with (14.C.5), we see that < > is larger for restricted

phases than for general reflections.

Since < IH > / < IP > ≈ H / P, Equation (14.C.4) (and obviously equation (14.C.5)), may also be applied to estimate the scattering power of the

heavy-atom substructure: i.e.

>2 .

H

P

4

The probabilistic relation. In Section 7.2, we illustrated the joint probability

distribution P(E, Ep ), where E was the normalized structure factor of the target structure, and Ep was the normalized structure factor of a model structure.

The same distribution (say equation (7.3)) is valid if E and Ep are replaced

by structure factors EP and Ed of the native protein and of the derivative,

respectively.

From P(EP ,Ed ), the conditional distribution





<



P(| |) = c−1 exp(−| |/c)



(14.C.6)



may be derived, where

2



= R d − R2P ,



c=



H



d



,



Rd = |Fd |



1/2

P



.



Distribution (14.C.6) only depends on the scattering power of the unknown

heavy-atom substructure ( d is a known parameter). Therefore, the best c

value is that for which the theoretical distribution (14.C.6) fits the experimental

| | distribution. Finding c is equivalent to estimating the scattering power of

the heavy-atom substructure.



15



Anomalous dispersion

techniques



15.1 Introduction

The term anomalous scattering originates from the first research on light dispersion in transparent materials. It was found that, in general, the index of

refraction increases when the wavelength decreases (this was considered to be

normal). It was also found that, close to the absorption edges, the refractive

index shows a negative slope, and this effect was called anomalous.

Today, it is clear that anomalous dispersion is a resonance effect. Indeed,

atomic electrons may be considered to be oscillators with natural frequencies; they are bound to the nucleus by forces which depend on the atomic

field strength and on the quantum state of the electron. If the frequency of the

primary beam is near to some of these natural frequencies, resonance will take

place (the concept of dispersion involves a change of property with frequency).

The scattering is then called anomalous, it occurs in correspondence with the

so-called absorption edges of a chemical element, and is expressed analytically

via the complex quantity,

f = f0 +



f + if ,



(15.1)



where f0 is the scattering factor of the atom in the absence of anomalous scattering. f and f (with f > 0) are called the real and imaginary dispersion

corrections.

f is proportional to the absorption coefficient of the atom, μλ , at the given

X-ray energy Eλ :

f λ = mc/4π e2



Eλ μλ ,



where m and e are the electronic mass and charge respectively, λ is the

wavelength, c is the speed of light, and h = 2π , the Planck constant.

f may be obtained from an absorption scan via the Kramers–Kronig

transform relating the real to the imaginary component:

f (λ) =



2

π







E

0



f (E )

E2 − E 2



dE .



An important question is whether f and f vary with diffraction angle. Some

theoretical treatments suggest changes of a few percent with sin θ/λ, but no



336



Anomalous dispersion techniques

rigorous experimental check has been described; therefore, in most of the

applications (and also in this book), f and f are considered to be constant.

As a consequence, the relative modification of the scattering process due to

anomalous dispersion is stronger at high sin θ/λ values, where f0 is smaller,

while f and f remain constant.

For most substances, at most X-ray wavelengths from conventional sources,

dispersion corrections are rather small. Calculated values for CrKα (λ =

2.229 Å), CuKα (λ = 1.542 Å), and MoKα (λ = 0.7107 Å) are listed in the

International Tables for Crytallography. Only in some special cases can

ordinary X-ray sources generate relevant dispersion effects. For example, the

following dispersion corrections are calculated for holmium, which has the L3

absorption edge, (∼1.5368 Å) close to CuKα radiation:

Cu Kα1 (λ = 1.5406 Å);



f = −15.41,



f = 3.70



Cu Kα2 (λ = 1.5444 Å);



f = −14.09,



f = 3.72.



Larger anomalous effects are obtained by using synchrotron radiation; from its

intense continuous spectrum, specific wavelengths may be selected with high

precision, in order to provoke, in several cases, exceptionally large anomalous

scattering.

Why do anomalous dispersion effects help to solve protein structures

(Peederman and Bijvoet, 1956)? We will see in the following that they simulate

isomorphism and more derivatives are simulated when multiwave techniques

are used. A pioneering study by Hoppe and Jakubowski (1975) on erythrocruorin, an iron-containing protein, showed the feasibility of this method.

These authors used NiKα (1.66 Å) and CoKα (1.79 Å) radiation to vary f

and f sizes of Fe around its K edge (1.74 Å) and obtained phase values with

a mean phase error of 50◦ .

What are the limits and the advantages of anomalous dispersion with respect

to isomorphous derivative techniques? We notice two disadvantages, in (a) and

(b) in the following, and a big advantage in (c):

(a) The signal provoked by anomalous scattering is inferior to that usually obtained by isomorphous replacement. The case of praseodynium

(Templeton et al., 1980), for which f = −26, 2f = 55 and of gadolinium (Templeton et al., 1982), for which f = 31.9, 2f = 62.4, are

exceptional. Therefore, the use of anomalous effects requires high measurement accuracy.

(b) Since the modulus of f is proportional to the absorption coefficient,

the corresponding correction for absorption should be carefully calculated. Luckily, heavy atoms usually have absorption edges in the short

wavelength range (say 0.6 Å < λ < 1.1 Å), for which absorption is greatly

reduced.

(c) The isomorphism of the derivatives with respect to the protein is never

ideal. We saw in Chapter 14 how challenging bad derivatives may be for

the success of the phasing process. If anomalous dispersion techniques

are used, the scattering structure coincides with the target structure, and

therefore no lack of isomorphism occurs.



Violation of the Friedel law as basis of the phasing method



337



Furthermore, in case of MAD, the anomalous scattering substructure does

not change with the wavelength (in the MIR approach, the heavy-atom

substructure changes by changing the heavy atom). As a result, the substructure

is overdetermined by MAD data.

The above considerations suggest that we should deal with different cases:

1. The SAD (single wavelength anomalous scattering) case;

2. The SIRAS (single isomorphous replacement combined with anomalous

scattering) case. Typically, protein and heavy-atom derivative data are

simultaneously available, with heavy atoms as anomalous scatterers;

3. The MAD (multiple wavelengths technique) case;

4. The MIRAS (multiple isomorphous replacement combined with anomalous

scattering) case.



15.2 Violation of the Friedel law as basis

of the phasing method

Suppose that an n.cs. crystal contains NP non-H atoms in the unit cell and that

all of them are anomalous scatterers. Usually, the number of efficient anomalous scatterers is a very small fraction of the scatterers in the unit cell; to

include this hypothesis in our mathematical treatment, the reader may set to

zero the anomalous scattering of the atoms the contribution of which they want

to neglect. In the following, for shortness, we will indicate Fh and F−h by

F + = |F + | exp(iφ + ) and F − = |F − | exp(iφ − ), respectively. Then,

NP



F+ =

=



j=1



F0+



+



f0j +



f j + ifj



+



+



exp 2πi h · rj

(15.2)



F



+F



=F



+



+F



+



,



where

F0+ =



NP

j=1 f0j



F+=

F

F



+

+



NP

j=1

NP

j=1



=

=



exp 2πi h · rj = |F0+ | exp iφ0+ ,



F0+



fj exp 2πi h · rj ,

NP

j=1



ifj exp 2πi h · rj =



+



+



=



NP

j=1



F− =



NP



F



f j exp i(2π h · rj + π/2) = |F

+



( f0j +



f j ) exp 2π ih · rj = |F | exp iφ



+



+



| exp iφ



+



,



.



Analogously:



=



j=1



F0−



+



f0j +



f j + if j exp −2πih · rj

(15.3)



F







+F







=F







+F







,



where

F0− =

F−=

F







=



NP

j=1 f0j

NP

j=1

NP

j=1



F − = F0− +



exp −2π ih · rj = |F0− | exp iφ0− ,

f j exp −2π ih · rj ,



if j exp −2π ih · rj =

F−=



NP

j=1



f0j +



NP

j=1



f j exp i(−2π h · rj + π/2) = |F



f j exp −2π ih · rj = |F − | exp iφ − .







| exp iφ







,



imag. axis



338



Anomalous dispersion techniques



a)



F+



F² +



φ″+

Fo+

ΔF ¢ +



F¢+



ϕ



|F0+ | = |F0− |,



real axis



F–

Fo–

ΔF ¢ –



|F



F+

F¢ + = F¢ –*



Fo+



F ² –*

F –*



φ+−φ–*



φ − = −φ .



The Friedel law does not hold for the pair F

relations



b)



ΔF ¢ +



φ0− = −φ0+ ,



|F + | = |F − |,



F ²–

F¢ –



imag. axis



F + and its component vectors F0+ , F + , F + , F + , as well as F − and its

component vectors F0− , F − , F − and F − , are shown in Fig. 15.1a.

The reader will notice that F0+ and F0− components also contain the normal

scattering contribution of the anomalous scatterers and that they are related

by the Friedel law. The Friedel law also holds for the pairs ( F + , F − ),

( F + , F − ). In detail,



real axis



Fig. 15.1

(a) Relation between F + and F− when

anomalous dispersion is present. (b) Relation between F + and F−∗ when anomalous dispersion is present.



+



| = |F







|,



φ







+







,F



; from definitions, the



+



=π −φ



are easily obtained (see Fig. 15.2). Therefore, while φ + and φ − are symmetrical with respect to the zero angle, φ + and φ − are symmetrical with respect

to π/2.

It may be useful to remember another vectorial relationship. In general conditions, F + and F + are not perpendicular. They are perpendicular only if

the anomalous scattering arises from the same atomic species. Then,

F+=

F



+



f



= if



NP

j=1

NP

j=1



exp(2π ih · rj ),



exp(2π ih · rj ) = f



NP

j=1



exp i 2πh · rj + π/2 ⊥ F + .



The relation between F + and F − is more clearly understood if we compare

F + with F −∗ (the star indicates the complex conjugate), as in Fig. 15.1b. From

definitions, the following phase relations hold:

φ −∗ = φ + ,



φ



−∗



= −φ







+







+ π.



(15.4)



Equivalently, in vectorial form,

F



−∗



= −F



+



and F



+∗







= −F



.



(15.5)



Because of (15.5), from now on we will denote the moduli |F + |, |F − | by |F |

and the moduli |F + |, |F − | by |F |.

From Fig. 15.1a, it is very easy to derive the following relations:

|F + |2 = |F |2 + |F |2 + 2|F F | cos(φ



+



− φ +)



|F − |2 = |F |2 + |F |2 + 2|F F | cos(φ







− φ − ).



(15.6a)



and



In accordance with equation (15.4) and Fig. 15.2b, the above equation may be

modified to,

|F − |2 = |F |2 + |F |2 − 2|F F | cos(φ



+



− φ + ).



(15.6b)



From (15.6a) and (15.6b),

I = |F + |2 − |F − |2 = 4|F F | cos(φ



+



− φ + ).



(15.7)



Violation of the Friedel law as basis of the phasing method

and



339

(a)



|F + |2 + |F − |2

= |F |2 + |F |2

2



(15.8)



are easily obtained.

Relation (15.7) suggests that |Fh | = |F−h | is no longer valid; in other words,



φ′ + ≡ φ′–*

φ′– ≡ φ′+*



in n.cs. space groups the Friedel law is not satisfied in the presence of

anomalous dispersion.

Figure 15.3 suggests that, if F + is perpendicular to F + , the Friedel law

concerning the moduli |Fh | and |F−h | is satisfied, even if the structure is n.cs.;

but this happens only by chance. On the contrary, the Friedel law:

(i) is satisfied in an n.cs. crystal if it is composed entirely of the same

anomalous scatterer;

(ii) is systematically satisfied in cs. structures. Indeed, if the structure is cs.

(see Fig. 15.4), then F + = F − = F −∗ is a real value, while F + ≡ F −

is an imaginary value; indeed, φ + = φ − = ±π/2.



(b)

φ′′–

φ′′+

φ′′ –*



φ′′+*



Accordingly, F + ≡ F − , and the following statement arises:

in cs. space groups the Friedel law is satisfied even in the presence of

anomalous scattering.

The Friedel law is also satisfied for phase restricted reflections in n.cs. space

groups.

Let us now consider how anomalous dispersion modifies the reciprocal

space symmetry. In the absence of anomalous dispersion, the number of reflections with equal (because of the symmetry) amplitude is fixed by the Laue

group. If an anomalous signal is present in an n.cs. space group of order m, the

following relations hold:



Fig. 15.2

Phase relationships between: (a) φ + and

φ − ; (b) φ + and φ − . φ + and φ − are

the phases of the reflections F + and F − ;

φ +∗ and φ −∗ are the phases of the reflections F +∗ and F −∗ , complex conjugates

of F + and F − , respectively. Since F +

and F − obey the Friedel relationship,

then φ + = φ −∗ and φ − = φ +∗ . F +

and F − do not obey the Friedel relationship: we have φ + = φ −∗ + π and

φ − = φ +∗ + π .



and

|F−h | = |F−hR2 | = . . . . . = |F−hRm | with



|F−h | = |F−h |.



For example, for C2,



imag. axis



|Fh | = |FhR2 | = . . . . . = |FhRm |

F ″+

F+



F ″F –*

F ′+ = F ′–*



|Fhkl | = |Fhk

¯ ¯l | = |Fhkl

¯ | = |Fh¯ k¯ ¯l |.



F –*



It may be concluded that:



imag. axis



for normal scattering, the equivalence of reflections is fixed by the Laue

group symmetry, while, when anomalous scattering occurs, the equivalence

agrees with the point group symmetry (Ramaseshan, 1963).







+ F

F ≡



F ′+ = F ′ –



F ′′ + ≡ F′′ –

real axis



real axis

Fig. 15.3

I = 0 if F + is perpendicular to F



+



.



Fig. 15.4

The cs. case: F + and F − when anomalous

dispersion is present.



340



Anomalous dispersion techniques

The reader will certainly have understood that a violation of the Friedel law

is just the source of information necessary to solve a structure. Indeed, the

anomalous difference,

ano



= |F + | − |F − |,



depends on the nature of the anomalous scatterers and on their positions; conversely, the anomalous substructure may, in principle, be derived from the

anomalous differences.

If the anomalous scattering from light atoms like C, O, N, H, (these are

nearly all the atoms in proteins) may be neglected (see Section 15.3), the

anomalous substructure mainly consists of some (relatively) few anomalous

scatterers. Thus, as for the isomorphous derivative technique, the phasing

approach may be subdivided into two steps: defining the anomalous substructure first, and then phasing the protein. If we compare this just outlined

approach with the isomorphous derivative method, we see that the role of the

isomorphous difference, iso , is now played by ano . This is the reason why

it is usual to state the following (Pepinsky and Okaya, 1956; Ramaseshan and

Venkatesan, 1957; Mitchell, 1957):

anomalous dispersion effects simulate isomorphism.



15.3 Selection of dispersive atoms

and wavelengths

Anomalous dispersion may be used in small- as well as in macro-molecule

crystallography. Since the phase problem is practically solved for small-sized

structures, in this area anomalous effects are mostly used for other purposes

(see Helliwell, 2000). For example:

(i) by tuning the wavelength close to the absorption edge of specific elements, it is possible to distinguish atoms which have close atomic

numbers, even when they occupy the same site.

(ii) in microporous materials, for determining the concentrations of transition

metals incorporated into the frameworks.

(iii) in powder crystallography, as an additional tool for phasing (see

Section 15.9).

In this chapter, we are mainly interested in macromolecular crystallography.

For such structures we will give some practical recipes for designing good

SAD or MAD experiments.

MAD experiments are very demanding in terms of beamline properties. The

question is: which strategy should be chosen in order to collect the minimum

amount of data allowing a straightforward crystal structure solution (Gonzalez,

2003a,b)? To decide which strategy, the characteristics of the crystal sample

(e.g. resistance to radiation damage, diffracting power, chemical composition) and the properties of the X-ray source (e.g. intensity of the beamline

at the wavelengths of interest, ease of tunability, stability, and reproducibility)



Selection of dispersive atoms and wavelengths



341



should be considered. If the beamline does not fulfil one or more of the above

requirements, a SAD experiment is advisable.

In the case of a weak anomalous signal (e.g. on the S anomalous dispersion), some additional parameters like exposure time and data redundancy may

become more critical. According to Cianci et al. (2008) <| ano| /σ ( ano )>

greater than 1.5 for all resolution shells is a necessary requirement for a

successful phasing attempt.

The above considerations suggest that the phasing process would become

more straightforward if the anomalous signal is maximized; that may be done

by introducing (if necessary) stronger anomalous scatterers into the crystal,

and/or by proper selection of the wavelengths. Let us examine the various

aspects.

The usable energy range for most synchrotrons is in the range 5–15 keV;

to have good anomalous differences, the absorption edges (K, or L, or M) to

exploit should lie in this range. Luckily, most of the elements of the periodic

table show edges in this interval (see Fig. 15.5). In particular, many metalloenzymes, oxidase, reductase, etc. naturally contain transition metals like Fe,

Zn, Cu, etc. with absorption edges in the range 5–15 keV, and also very heavy

atoms (like Hg, Pt, Au, Br) show a strong L-edge in that range. Only elements belonging to the 5th period between Rb and Te lie outside the range; we

will see below, however, that anomalous data collection may also be made at

wavelengths above the absorption edge, even if this choice does not maximize

the anomalous difference (Leonard et al., 2005).

The S absorption edge falls outside the K-range, which is the reason why

special selenoproteins are grown, i.e. proteins where the S atom in methionines

is replaced by Se. There are two main reasons for preferring seleno-methionine

to methionine. The first is that Se is a more efficient anomalous scatterer;

the second is that, at a given wavelength λ, the best achievable resolution is

RES = λ/2. Since the K-edge for sulphur falls at 5.018 Å, the best resolution

experimentally attainable is about 2 Å; the K-edge for Se falls at 0.98 Å, so

allowing more complete experimental datasets.



20



X-ray energy (keV)



15

L1



K



L3



10



5

K



L

GAP



10



20



30



40

50

60

atomic number



70



80



90



Fig. 15.5

Absorption edges as a function of the

atomic number. The interval 5–15 keV

approximately corresponds to the interval

2.48–0.83 Å.



342



Anomalous dispersion techniques

15

f ″ maximized

10

ƒ″



Electrons



5



0



–5



high f ″ remote



minΔf ′

low f ″ remote



Δf ′



–10



Fig. 15.6

Se dispersion pattern close to the K

absorption edge.



–15

12600

(0.9840Å)



12650

(0.9801Å)



12700

(0.9763Å)

Energy (eV)



12750

(0.9724Å)



12800

(0.9686Å)



Let us now consider how the wavelengths may be chosen in a typical MAD

experiment. We will see in the following paragraphs that, besides ano , also the

dispersive differences, disp (say |Fλ+i | − |Fλ+j | or |Fλ−i | − |Fλ−j |) and the mixed

differences, |Fλ+i | − |Fλ−j | may contribute to the phase problem solution.

The first data are usually measured by using a wavelength which maximizes

the f (peak dataset; see Fig. 15.6).

Accordingly, the second wavelength is usually chosen at the negative peak

of the f curve, to maximize the dispersive differences between wavelengths.

This wavelength coincides with the inflection point of the f scan, and the

corresponding data are called the inflection point dataset. It may be useful to

notice that at such an inflection point, the f value is about half the value at

the peak, while the dispersive difference with any other wavelength is very

high. If the radiation damage is low, a third wavelength may be chosen; the

corresponding data are denoted as a high energy remote dataset. This shows a

non-negligible f signal and a good dispersive contrast against the inflection

dataset.

If radiation decay is slowly progressing, a fourth low energy wavelength

may be chosen (then, low energy remote data are collected). Even if there is a

high f contrast with the peak and a still good f contrast with the inflection

point, this wavelength is the last choice because of the large absorption of light

atoms, which may rapidly destroy the crystal.

X-ray absorption edges are very sharp in many cases; thus the energies of

the peak and of the inflection points are seperated by a few eV. In addition,

the exact position of the edge depends on the chemical environment of the

anomalous scatterers. It is therefore mandatory to record the absorption edge

at the time of a MAD experiment.

Sometimes, the anisotropy of anomalous scattering should be taken into

account. Templeton and Templeton (1988) showed that a remarkable anisotropy can be found for the Se K-edge in selenomethionine proteins, caused by



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APPENDIX 14. B THE SIR CASE: THE ONE-STEP PROCEDURE

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