Phasing & Substructure Solution

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1 CCP4 Workshop Chicago 2011 Phasing & Substructure Solution Tim Grüne Dept. of Structural Chemistry, University of Göttingen June CCP4 Chicago 2011: Substructure Solution 1/50

2 Macromolecular Crystallography (in brief) < 1h < 1h days weeks mosflm, xds, HKL2000,... xprep, shelxc, solve,... shelxd, SnB, HySS,... shelxe, resolve, pirate,... arp/warp, coot, refmac5,phenix, YOU! Data Integration (Anomalous) Differences Substructure Solution Phasing & Density Modification Building & Refinement Molecular Replacement CCP4 Chicago 2011: Substructure Solution 2/50

3 Motivation: The Phase Problem An electron density map is required to create a crystallographic model. The electron density is calculated from the complex structure factor F (hkl): ρ(x, y, z) = 1 V cell h,k,l F (h, k, l) e iφ(h,k,l) e 2πi(hx+ky+lz) (1) Experiment measures: I(hkl) i.e. real numbers Model Building requires: F (hkl) i.e. complex numbers: F (hkl) = F (hkl) e iφ(hkl) Connection: F (hkl) = I(hkl) φ(hkl) =? CCP4 Chicago 2011: Substructure Solution 3/50

4 Motivation: The Phase Problem The phase problem is one of the critical steps in macromolecular crystallography: A diffraction experiment only delivers the amplitude F (hkl), but not the phase φ(hkl) of the structure factor. Without phases, a map cannot be calculated and no model can be built. CCP4 Chicago 2011: Substructure Solution 4/50

5 Solutions to the Phase Problem The main methods to overcome the phase problem in macromolecular crystallography (i.e. phases): to obtain initial (multi/ single) wavelength anomalous dispersion (MAD, SAD) (multiple/ single) isomorphous replacement (MIR, SIR) molecular replacement (MR) and combinations thereof (SIRAS, MR-SAD,... ). The first two methods are so-called experimental phasing methods. They require the solution of a substructure. CCP4 Chicago 2011: Substructure Solution 5/50

6 What s a Substructure? a The Substructure of a (crystal-) structure are the coordinates of a subset of atoms within the same unit cell. c b CCP4 Chicago 2011: Substructure Solution 6/50

7 What s a Substructure? a The Substructure of a (crystal-) structure are the coordinates of a subset of atoms within the same unit cell. c b It can be any part of the actual structure. In the usual sense, substructure refers to the marker atoms that are used for phasing. A real crystal with the properties of the substructure (atom distribution vs. unit cell dimensions) cannot exist: the atoms are too far apart for a stable crystal. CCP4 Chicago 2011: Substructure Solution 7/50

8 Small Molecule Crystallography The substructure consists of a small number of atoms in a large unit cell. In small molecule crystallography, the phase problem is usually easily solved: A structure with not too many atoms (< 2000 non-hydrogen atoms) can be solved by means of ab initio methods from a single data set provided the resolution is better than 1.2Å (this is called Sheldrick s rule [2]). CCP4 Chicago 2011: Substructure Solution 8/50

9 The Terms Ab initio and Direct Methods ab initio Methods: phase determination directly from amplitudes, without prior knowledge of any atomic positions. Includes direct methods and the Patterson method. The most widely used of all ab initio methods are: Direct Methods: phase determination using probabilistic phase relations usually the tangent formula (Nobel prize for H. A. Hauptman and J. Karle in Chemistry, 1985). CCP4 Chicago 2011: Substructure Solution 9/50

10 The Substructure Central to phasing based on both anomalous dispersion and isomorphous replacement is the determination of the substructure. The steps involve 1. Collect data set(s) 2. Create an artificial substructure data set 3. Determine the substructure coordinates with direct methods 4. Calculate phases (estimates) for the actual data 5. Improve phases (density modification) CCP4 Chicago 2011: Substructure Solution 10/50

11 What about Sheldrick s Rule? The Sheldrick s Rule says that direct methods only work when the data resolution is 1.2 Å or better. Macromolecules seldomly diffract to this resolution. So why can we still use direct methods to determine the substructure? CCP4 Chicago 2011: Substructure Solution 11/50

12 Sheldrick s Rule Sheldrick s Rule refers the real structures and the 1.2 Å-limit is about the distance where single atoms can be resolved in the data. The substructure is an artificial crystal and atom distances within the substructure are well above the usual diffraction limits of macromolecules, even for 4 Å data. CCP4 Chicago 2011: Substructure Solution 12/50

13 Solving a Macromolecular Phase Problem with a Substructure The substructure represents only a very small fraction of all atoms in the unit cell (one substructure atom per a few thousand atoms). Yet, knowing the coordinates of the substructure allows to solve the phase problem for the full structure. CCP4 Chicago 2011: Substructure Solution 13/50

14 The Independent Atom Model (IAM) Ordinary crystallography (other than ultra-high resolution charge density studies) is based on the Independent Atom Model : The total structure factor F (hkl) of each reflection (hkl) is the sum the independent contributions from each atom in the unit cell. (x, y, z) Im(F ) φ = 2π(hx + ky + lz) = rel. distance to Miller-plane y Re(F ) x (hkl) = (430) (independent contributions per atom) CCP4 Chicago 2011: Substructure Solution 14/50

15 The Independent Atom Model (IAM) x y (hkl) = (430) The (vectorial) sum from each atom contribution yields the total scattering factor F (hkl). The experiment delivers only its modulus F (hkl) = I(hkl), but not its phase. CCP4 Chicago 2011: Substructure Solution 15/50

16 Isomorphous Replacement (1/2) The Harker Construction represents one way to determine the phase of F (hkl) from an isomorphous replacement experiment. It requires two data sets: I nat (hkl) I der (hkl) Native: lacking one or several atoms (the substructure) Derivative: Same unit cell, same atoms plus substructure atoms CCP4 Chicago 2011: Substructure Solution 16/50

17 Isomorphous Replacement (2/2) The substructure coordinates can derived from the difference data set ( I der (hkl) I nat (hkl)). The substructure phases φ(hkl) can be calculated from the substructure coordinates. The phases of the native (or derivative) data set can be derived from measured native intensities I nat (hkl) measured derivative intensities I der (hkl) calculated substructure structure factor F (hkl) CCP4 Chicago 2011: Substructure Solution 17/50

18 Harker Construction (1/4) Im(F (hkl)) Draw the structure factor F (hkl) of the substructure. Re(F (hkl)) CCP4 Chicago 2011: Substructure Solution 18/50

19 Harker Construction (2/4) I nat (hkl) Draw the structure factor F (hkl) of the substructure. F der (hkl) points somewhere on the circle around the tip of F (hkl) with radius I nat (hkl) because F der = F + F + F + F } {{ } F nat CCP4 Chicago 2011: Substructure Solution 19/50

20 Harker Construction (3/4) Draw the structure factor F (hkl) of the substructure. F der (hkl) points somewhere on the circle around the tip of F (hkl) with radius I nat (hkl) At the same time, F der (hkl) also lies somewhere on the circle around the origin with radius I der (hkl), because F + F + F + F = F + F + F + F CCP4 Chicago 2011: Substructure Solution 20/50

21 Harker Construction (4/4) The two intersections of the two circles are the two possible structure factors F der (hkl). How to select the correct one: MIR a second Harker construction with F der-2 (hkl) has exactly one of the two possibilities in common with the first Harker construction. SIR the mean phase of the two possibilities serve as starting point for density modification, but if the substructure coordinates are centrosymmetric the two-fold ambiguity cannot always be resolved (this problem is specific to SIR). CCP4 Chicago 2011: Substructure Solution 21/50

22 Substructure Contribution to Data The electron density ρ(x, y, z) can be calculated from the (complex) structure factors F (hkl). Inversely the structure factor amplitudes can be calculated once the atom content of the unit cell is known: F (hkl) = atoms j in unit cell f j (θ hkl )e B j sin2 θ hkl λ 2 e 2πi(hx j+ky j +lz j ) (2) f j atomic scattering factor: depends on atom type and scattering angle θ. Values tabulated in [3], Tab B j atom displacement parameter, ADP: reflects vibrational motion of atoms 2π(hx j + ky j + lz j ) phase shift: relative distance from origin (x j, y j, z j ): fractional coordinates of j th atom CCP4 Chicago 2011: Substructure Solution 22/50

23 Extraction of Substructure Contribution Experimental determination (i.e. other than by molecular replacement) of the contribution of the substructure to the measured intensities is achieved by Anomalous Dispersion: Deviation from Friedel s Law and comparison of F (hkl) with F ( h k l) Isomorphous replacement: Changing the intensities without changing the unit cell and comparison of F nat with F der The alteration is usually achieved by the introduction of some heavy atom type into the crystal, either by soaking or by co-crystallisation. CCP4 Chicago 2011: Substructure Solution 23/50

24 Anomalous Scattering The atomic scattering factors f j (θ) describe the reaction of an atom s electrons to X-rays. They are different for each atom type (C, N, P,... ), but normally they are real numbers and do not vary significantly when changing the wavelength λ. If, for one atom type, the wavelength λ matches the transition energy of one of the electron shells, anomalous scattering occurs. This behaviour can be described by splitting f j (θ) into three parts: fj anom = f j (θ) + f j (λ) + if j (λ) Near the transition energy, the latter two, f and f vary strongly with the wavelength. CCP4 Chicago 2011: Substructure Solution 24/50

25 Anomalous Scattering Since the equation for F (h, k, l) (Eq. 2) is a simple sum, one can group it into sub-sums. In the case of SAD and MAD the following grouping has turned out to be useful, with its phase diagram on the right (ADP e 8π2 sin U 2 θ hkl j λ 2 omitted for clarity): F (hkl) = + + +i non- f µ e 2πihr µ substructure } {{ } F P substructure f ν e 2πihr ν } normal {{ } F A substructure f νe 2πihr ν } anomalous {{ } F substructure f ν e 2πihr ν } anomalous {{ } if Im(F ) F P F Re(F ) F A F if CCP4 Chicago 2011: Substructure Solution 25/50

26 Anomalous Scattering Since the equation for F (h, k, l) (Eq. 2) is a simple sum, one can group it into sub-sums. In the case of SAD and MAD the following grouping has turned out to be useful, with its phase diagram on the right (ADP e 8π2 sin U 2 θ hkl j λ 2 omitted for clarity): F (hkl) = + + +i non- f µ e 2πihr µ substructure } {{ } F P substructure f ν e 2πihr ν } normal {{ } F A substructure f νe 2πihr ν } anomalous {{ } F substructure f ν e 2πihr ν } anomalous {{ } if Im(F ) Normal presentation, because f usually negative F P F F Re(F ) if F A CCP4 Chicago 2011: Substructure Solution 26/50

27 Breakdown of Friedel s Law Now compare F (hkl) with F ( h k l) in the presence of anomalous scattering: F ( h k l) = non- f µ e 2πi( h)r µ substructure } {{ } FP substructure + f ν e 2πi( h)r ν } normal {{ } FA substructure + f νe 2πi( h)r ν } anomalous {{ } +i substructure F f ν e 2πi( h)r ν } anomalous {{ } if Friedel s Law is valid for the f µ, f ν, and f ν parts: Im(F ) F A + F P + F + Re(F ) FP FA F CCP4 Chicago 2011: Substructure Solution 27/50

28 Breakdown of Friedel s Law Now compare F (hkl) with F ( h k l) in the presence of anomalous scattering: F ( h k l) = non- f µ e 2πi( h)r µ substructure } {{ } FP substructure + f ν e 2πi( h)r ν } normal {{ } FA substructure + f νe 2πi( h)r ν } anomalous {{ } +i substructure F f ν e 2πi( h)r ν } anomalous {{ } if The complex contribution of if ν violates Friedel s Law: Im(F ) F + P F P F + F F + A F A F + Re(F ) if F if + F + = F CCP4 Chicago 2011: Substructure Solution 28/50

29 How all this helps Karle (1980) and Hendrickson, Smith, Sheriff (1985) published the following formula : Im(F ) F + F + 2 = F T 2 + a F A 2 + b F A F T + c F A F T sin α F 2 = F T 2 + a F A 2 + b F A F T c F A F T sin α F + P F + A α F + T Re(F ) with F T = F P + F A the non-anomalous contribution of structure and substructure. NB: α(hkl) = α( h k l), F A (hkl) = F A ( h k l), and F T (hkl) = F T ( h k l) a = f 2 +f 2, b = 2f 2f, c = f 2 f f CCP4 Chicago 2011: Substructure Solution 29/50

30 Combining Theory and Experiment The diffraction experiment measures the Bijvoet pairs F + and F for many reflections. In order to simulate a small-molecule experiment for the substructure, we must know F A. With the approximation 1 2 ( F + + F ) F T the difference between the two equations above yields F + F c F A sin α CCP4 Chicago 2011: Substructure Solution 30/50

31 Our experiment measures F + = Status Quo A Summary I(hkl) and F = I( h k l). We are looking for (an estimate) of F A for all measured reflections. These F A mimic a small-molecule data set from the substructure alone. The small-molecule data set can be solved with direct methods, even at moderate resolution. With the help of Karle and Hendrickson, Smith, Sheriff, we already derived F + F c F A sin α We are nearly there! CCP4 Chicago 2011: Substructure Solution 31/50

32 The Factor c Before we know F A, we must get rid of c and sin α. c = 2f (λ) f(θ(hkl)) can be calculated for every reflection provided f (λ). During a MAD experiment, f (λ) is usually measured by a fluorescence scan. To avoid the dependency on f (λ), the program shelxd [1] uses normalised structure factor amplitudes E(hkl) instead of F (hkl), which are very common for small-molecule programs. CCP4 Chicago 2011: Substructure Solution 32/50

33 The Angle α In the case of MAD, multi-wavelength anomalous dispersion, we can calculate sin α because there is one equation for each wavelength. In the case of SAD, the program shelxd approximates F A sin(α) F A. Why is this justified? Bijvoet pairs with a strong anomalous difference ( F + F ) have greater impact in direct methods. The difference is large, however, when α is close to 90 or 270, i.e. when sin(α) ±1. This coarse approximation has proven good enough to solve hundreds or thousands of structures with shelxd. CCP4 Chicago 2011: Substructure Solution 33/50

34 The Angle α In the case of MAD, multi-wavelength anomalous dispersion, we can calculate sin α because there is one equation for each wavelength. In the case of SAD, the program shelxd approximates F A sin(α) F A. Why is this justified? Bijvoet pairs with a strong anomalous difference ( F + F ) have greater impact in direct methods. The difference is large, however, when α is close to 90 or 270, i.e. when sin(α) ±1. This coarse approximation has proven good enough to solve hundreds or thousands of structures with shelxd. CCP4 Chicago 2011: Substructure Solution 34/50

35 Fluorescence Scan In order to find the wavelength with the strongest response of the anomalous scatterer, the values of f and f are determined from a fluorescence scan. 4 2 f and f for Br Br f" Br f Signal [e] λ = 1Å * ev/e In order to get the strongest contrast in the different data sets, MAD experiments collect data at 1. maximum for f (peak wavelength) 2. minimum for f (inflection point) Energy [ev] CCP4 Chicago 2011: Substructure Solution 35/50

36 SIR/MIR Single isomorphous replacement requires two data sets: 1. Macromolecule in absence of heavy metal (Au, Pt, Hg, U,... ), called the native data set. Measures F P. 2. Macromolecule in presence of heavy metal, called the derivative data set. Measures F. F A can then be estimated from the difference F F P Im(F (hkl)) F P F A F Re(F (hkl)) CCP4 Chicago 2011: Substructure Solution 36/50

37 SAD and MAD vs. SIR and MIR SAD/ MAD F A from F + vs. F (one data set) signal improved by wavelength selection requires tunable wavelength (synchrotron) multi : one crystal, different wavelengths SIR/ MIR F A from native vs. derivative (two data sets) signal improved by heavy atom (large atomic number Z) independent from wavelength (inhouse data) multi : several derivatives = several crystals CCP4 Chicago 2011: Substructure Solution 37/50

38 SIRAS (1/2) Isomorphous replacement is hardly used for phasing anymore: Incorporation of heavy atom into crystal often alters the unit cell non-isomorphism between native and derivative ruins usability of data sets Heavy atom in structure go to synchrotron and try MAD instead of SIR/MIR CCP4 Chicago 2011: Substructure Solution 38/50

39 SIRAS (2/2) However, one often has a high-resolution data set from a native crystal and one (lower resolution) data set with anomalous signal. They can be combined into a SIRAS experiment: Anomalous data set: SAD Anomalous data set as derivative vs. native: SIR These are two independent sources of phase information and usually work better than either alone. CCP4 Chicago 2011: Substructure Solution 39/50

40 F A : Substructure Solution with Direct Methods CCP4 Chicago 2011: Substructure Solution 40/50

41 Direct Methods Having figured out the values F A from our measured data we are actually pretending having collected a data set from a crystal with exactly the same (large) unit cell as our actual macromolecule but with only very few atoms inside. We artificially created a small molecule data set. CCP4 Chicago 2011: Substructure Solution 41/50

42 Normalised Structure Factors Experience shows that direct methods produce better results if, instead of the normal structure factor F (hkl), the normalised structure factor is used. The normalised structure factor is calculated as E(hkl) 2 = F (hkl)2 /ε F (hkl) 2 /ε ε is a statistical constant used for the proper treatment of centric and acentric reflections; the denominator F (hkl) 2 /ε as is averaged per resolution shell. It is calculated per resolution shell ( 20 shells over the whole resolution range). This eliminates the strong fall-off with scattering angle θ. CCP4 Chicago 2011: Substructure Solution 42/50

43 Starting Point for Direct Methods: The Sayre Equation In 1952, Sayre published what now has become known as the Sayre-Equation F (h) = q(sin(θ)/λ) h F h F h h This equation is exact for an equal-atom-structure (like the substructure generally is). It requires, however, complete data including F (000), which is hidden by the beam stop, so per se the Sayreequation is not very useful. CCP4 Chicago 2011: Substructure Solution 43/50

44 Tangent Formula (Karle & Hauptman, 1956) The Sayre-equation serves to derive the tangent formula h tan(φ h ) E h E h h sin(φ h + φ h h ) h E h E h h cos(φ h + φ h h ) which relates three reflections h, h, and h h. H. A. Karle & J. Hauptman were awarded the Nobel prize in Chemistry in 1985 for their work on the tangent formula (and many other contributions). CCP4 Chicago 2011: Substructure Solution 44/50

45 Solving the Substructure Direct methods (and in particular shelxd) do the following: 1. Assign a random phase to each reflection. They will not fulfil the tangent formula. 2. Refine the phases using F A (or rather E A ) to improve their fit to the tangent formula 3. Calculate a map, pick the strong peaks (as putative substructure) 4. Calculate phases from the peak coordinates and return to step 2. CCP4 Chicago 2011: Substructure Solution 45/50

46 Solving the Substructure Steps 1-4 are repeated many times, each time with a new set of random phases. Every such attempt is evaluated and the best attempt is kept as solution for the substructure. The substructure is thus found by chance. It works even if the data quality is only medium but may require more trials. The cycling between steps 2-3, i.e. phase refinement in reciprocal space vs. peak picking in direct space, is called dual space recycling [4, section 16.1] CCP4 Chicago 2011: Substructure Solution 46/50

47 Dual-Space Recycling random atoms or (better) atoms consistent with Patterson selection criterion: Correlation Coefficient Real space: select atoms best CC? phases(map) from atoms dual space FFT and peak search recycling No reciprocal space refine phases repeat a few times between E obs and E calc Yes keep solution CCP4 Chicago 2011: Substructure Solution 47/50

48 Phasing the Rest Once the coordinates of the substructure are found, they serve to calculate the phases for the whole structure, e.g. with the Harker Construction. With those phases, an initial electron density map for the new structure can be calculated: I(hkl) φ(hkl) A φ(hkl) F A (x, y, z) A fµ ρ(x, y, z) CCP4 Chicago 2011: Substructure Solution 48/50

49 Phasing the Rest... this is the field of density modification and not part of this presentation... CCP4 Chicago 2011: Substructure Solution 49/50

50 References 1. G. M. Sheldrick, A short history of SHELX, Acta Crystallogr. (2008), A64, pp R. J. Morris & G. Bricogne, Sheldrick s 1.2Å rule and beyond, Acta Crystallogr. (2003), D59, pp E. Prince (ed.), International Tables for Crystallography, Vol. C, Union of Crystallography 4. M. G. Rossmann & E. Arnold (eds.), International Tables for Crystallography, Vol. F, Union of Crystallography CCP4 Chicago 2011: Substructure Solution 50/50

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