Scattering by two Electrons

Transcription

1 Scattering by two Electrons p = -r k in k in p r e 2 q k in /λ θ θ k out /λ S q = r k out p + q = r (k out - k in ) e 1 Phase difference of wave 2 with respect to wave 1: 2π λ (k out - k in ) r= 2π S r S = (k out - k in ) / λ S = 2sinθ / λ k out 1 (Comparison with Bragg s law gives: d = 1/S) 2

2 Scattering by an Atom Assuming the electron density around an atom is centrosymmetric, the scattering by an atom is always real: f = 2 ρ(r) cos [2π r S] dr r f is the atomic form factor: f is independent of the direction of S f depends on the length of S: S = 2 sin θ / λ At θ = 0: f corresponds to the number of electrons of the respective atom

3 Atomic Scattering Factors

4 Scattering by a Unit Cell The unit cell is composed of n atoms at position r j. The scattering of each atom with respect to the origin of the cell is given by: f j = f j exp [2π r j S] The total scattering from the unit cell is given by the structure factor F(S): N F(S) = Σ f j exp [2π r j S] j=1 f 2 f 3 f 4 f 1 F(S)

5 Scattering by a Crystal (I) S 1 S 2 S a=1 S b=2 a a b Cancelation Amplification = Reflection Im Im Re Re

6 Scattering by a Crystal II Scattering from a crystal will only occur, if the Laue conditions are fulfilled: a S = h b S = k c S = l The amplitude of the total scattered wave is proportional to the structure factor F(S), which describes the scattering by a single unit cell (remember: a typical crystal has on the order of = unit cells). Larger crystals will diffract X-rays more strongly.

7 A Lattice and its Diffraction Image Next 8 slides: taken from K. Cowtan s home page

8 Representation of Phases and Amplitudes Amplitude is represented by color saturation and brightness (small amplitudes: faded colors, large amplitudes: saturated colors) Phase is given by hue. The positive real (horizontal) axis to the right represents a phase of 0. Phase increases anti-clockwise 0 degrees is red 120 degrees is green 240 degrees is blue.

9 From Molecules to Crystals

10 Duck Tales The diffraction pattern of any object can be computed.

11 Animal Magic I A duck and its Fourier Transform A cat and its Fourier Transform

12 Animal Magic II Duck magnitudes and cat phases reveal a: Cat Cat magnitudes and duck phases reveal a: Duck

13 Fourier Transforms I 1-D crystal with three atoms: 2 carbon and 1 oxygen A sine wave with periodicity 2 A sine wave with periodicity 3 A sine wave with periodicity 5

14 Fourier Transforms II The three sine waves and their superposition (red), which approximates the unit cell quite well. The Fourier transform of the unit cell. Additional sine waves (not included in the reconstruction) show up as minor peaks. Only the amplitudes, but not the phases are shown. What happens in the diffraction experiment can be mathematically described by a Fourier transformation. Fourier transformations can also be used to calculate the structure of the molecule from the diffraction pattern, but only if the phases are known.

15 Resolution I A duck and its Fourier transform High resolution details omitted and inverse Fourier transformation: It is still a duck, right? Medium and high resolution details omitted and inverse Fourier transformation: What is it? A cloud? What if we know it should be a duck, can we fit a duck to the picture?

16 Resolution II Electron density maps at (A) 1.16 Å, (B) 2 Å, (C) 3 Å and (D) 4 Å resolution

17 Structure Factor and Electron Density Equations h is the vector (h,k,l) and x is the vector (x,y,z) h x = hx + ky + lz Both operations are Fourier transformations, which are mathematical vehicles between real space (ρ(x)) and reciprocal space (F(h)) ρ(x) has units of electrons per Å 3 and F(h) of electrons. Specifically, F(000) is equal to the number of electrons in the unit cell. F(000) can not be measured since it coincides with the primary beam Fourier transformation of the electron density yields the structure factor and Fourier transformation of the structure factor yields the electron density These operations are sometimes also referred to as Fourier synthesis and Fourier analysis

18 Overview - Macromolecular Crystallography 1. Overexpression and crystallization 2. Crystal characterization and data collection 3. The diffraction experiment 4. Phase problem 1. MIR (Multiple Isomorphous Replacement) - No homologous structure known - Incorporate heavy atoms - Locate bound heavy atoms - Phase refinement 2. MAD (Multiwavelength Anomalous Diffraction) - No homologous structure known - Incorporate anomalous scatterers - Locate anomalous scatterers - Phase refinement 3. MR (Molecular Replacement) - Requires homologous structure - Rotation function (determine orientation of known molecule in unknown crystal) - Translation function (determine position of search molecule in unknown crystal) - Rigid body refinement 5. Phase improvement 6. Model building 7. Refinement

19 3. Solving the Phase Problem 3a. Multiple Isomorphous Replacement 3b. Multiwavelength Anomalous Diffraction 3c. Molecular Replacement

20 Solving the Phase Problem by MIR or MAD Obtain small differences in diffraction that are due to a simple substructure Multiple Isomorphous Replacement (MIR) Soak heavy atom compounds into the crystal Favorable case: heavy atoms specifically bind at few sites without altering the protein and thus the crystal Bad case 1: no binding Bad case 2: crystals no longer diffract, or are non-isomorphous (too different from native crystals) Multiwavelength Anomalous Diffraction (MAD) Incorporate SeMet instead of Met during protein preparation, or have intrinsic anomalous scatterer (Fe, Zn, etc). Measure one crystal at multiple wavelengths near the absorption edge of the anomalous scatterer. Small differences in diffraction are observed that are due to anomalous scatterers.

21 Multiple Isomorphous Replacement (MIR) A heavy (electron-rich) atom is incorporated at specific sites in the crystal. F PH = F P + F H After heavy atom incorporation the crystals ideally have to remain isomorphous, which means all protein atoms stay at exactly the same position.

22 Harker Construction I Native dataset: blue circle F P Derivative dataset: pink circle F PH F H F P F PH F H is known (both F H and α H) Two vector triangles satisfy: F PH = F H + F P Two possible phase angles for F P This construction has to be repeated for every single reflection

23 Harker Construction II F PH2 F H2 F H1 F P F PH1 A second derivative (green circle) provides an unambiguous solution to the phase problem. More derivatives are even better, because of errors in: F P, F PH1 and F PH2 and F H1 and F H2

24 Patterson Function I P(u v w) = 1/V Σ hkl F(hkl) 2 cos[2π(h u + k v + l w)] The Patterson function is a Fourier summation with intensities as coefficients and all phase angles set to zero. It can be calculated directly from the data. It represents all interatomic vectors weighted by the number of electrons. For a structure with n atoms it contains n(n-1)=(n 2 -n) peaks. It is centrosymmetric. It is highly complex even with just a few atoms. It can be used to solve crystal structures containing heavy atoms: Vitamin B12

25 Patterson Function II u u u u u u u u u A two-dimensional unit cell with two atoms The corresponding Patterson cell with two (2 2-2) peaks

26 Patterson Function III 1- >2 1- >2 3- >2 1- >3 3- >2 1- > >1 2- >1 2- >3 3- >1 2- >1 2- > >2 1- >2 1- >3 3- >2 1- >2 1- >3 3- >1 2- >1 2- >3 3- >1 2- >1 2- >3 A two-dimensional unit cell with three atoms The corresponding Patterson cell with six (3 2-3) peaks

27 Harker Section u DMSO reductase Xe-derivative difference Patterson (w=1/2 section): v u = 3.4/16.4 = v = 5.3/20.1 = Space group P (1) : x, y, z (2) : -x+1/2, -y, z+1/2 (3) : -x, y+1/2, -z+1/2 (4) : x+1/2, -y+1/2, -z (1) - (2): 2x - 1/2, 2y, 1/2 2x - 1/2 = x = y = y = 0.132

28 3. Solving the Phase Problem 3a. Multiple Isomorphous Replacement 3b. Multiwavelength Anomalous Diffraction 3c. Molecular Replacement

29 Anomalous Scattering The photon is either scattered or not, but is not absorbed as it has insufficient energy to excite any of the available electronic transitions. The scattering cross-section of the atom (or if you like, the probability that the photon is scattered) may be adequately described in using the normal atomic scattering coefficient f o only. The photon scatters with no phase delay (imaginary, or f", component is 0). Some photons are scattered normally. Some photons are absorbed and re-emitted at lower energy (fluorescence). Some photons are absorbed and immediately re-emitted at the same energy (strong coupling to absorption edge energy). The scattered photon gains an imaginary component to its phase (f" scattering coefficient becomes non-zero); i.e. it is retarded compared to a normally scattered photon.

30 Atomic Form Factor including Anomalous Scattering f anomalous if f Δf The atomic scattering factor of a completely free electron has no anomalous component. The anomalous contribution consists of two parts: a real part Δf and an imaginary part if : f ano = f - Δf + if