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Fourier transform in X-ray crystallography .ppt

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Fourier transform in X-ray crystallography .ppt

  1. 1. Application of Fourier transform in X-ray crystallography
  2. 2. The Process: An Overview why?? for structure function relationship
  3. 3. A basic understanding of diffraction physics is required if crystal structure solution and refinement is to be understood.
  4. 4. Diffraction • Diffraction is a phenomenon by which wave fronts of propagating waves bend in the neighborhood of obstacles. •The amount of bending depends on the relative size of the wavelength of light to the size of the opening. •If the opening is much larger than the light's wavelength, the bending will be almost unnoticeable
  5. 5. Diffraction at a single slit
  6. 6. Destructive interference: Constructive interference
  7. 7. When X-ray photons collide with matter, the oscillating electric field of the radiation causes the charged particles of the object to oscillate with the same frequency as the incident radiation. Each oscillating dipole returns to a less energetic state by emitting an electromagnetic photon that can, in general, travel in any outward direction. Thus, the emitted photons have the same energy as the incident photons. This process is known as coherent scattering.
  8. 8. When do waves scatter in phase? • the two waves will be in phase if the pathlengths differ by any multiple of the wavelength
  9. 9. • Constructive interference (diffraction) will only occur if CB + BD = 2CB = n d sin = CB CB = n/2 Bragg’s Law 2dsin = n sin  1/d (reciprocal space) sin = (n/2)/d
  10. 10. • sin(θ)/l = 1/(2 λ) • λ = l / (2 sin(θ)) • the bigger the angle of diffraction, the smaller the spacing to which the diffraction pattern is sensitive
  11. 11. (1 1 0) planes (1 -1 0) planes (2 1 0) planes
  12. 12. 15
  13. 13. • For every family of plane (h k l), a vector can be drawn from a common origin having the direction of the plane normal and a length 1/d (d is inter planner distance).This new coordinate space is reciprocal space.Any reciprocal lattice vector is defined by h k l. • Reciprocal lattice is Fourier transform of the real lattice
  14. 14. Lattice plane
  15. 15. Electron source Electron detector Ewald sphere Reciprocal lattice 3D diffraction data collection (high resolution)
  16. 16. • The mathematical description of the diffracted X-ray can be described by the structure factor equation • FT as lens
  17. 17. Fourier transform Mathematical lens Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern X-ray diffraction Diffraction pattern Crystal Fourier transform (Lens) Protein structure Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecule in a crystal
  18. 18. 24
  19. 19. Two Electron system • Phase difference = 2π r.S • Wave can be regarded as being reflected against a plane with as the reflection angle. • S is perpendicular to the imaginary “reflecting plane” • Resultant vector (T sum of 1+2) = 1+1 exp[2π i r.S] • Phase diff = path diff / x 2 p • T = 1 +2 = 1+1 exp[2π i r.S] s0 – incident wave vector {|s0| = 1/ λ } s – diffracted wave vector { |s| = 1/ λ } Path difference = p + q p = λ.r. s0 q = -λ.r. s PD = λ.r (s0 - s) or λ.r.S S = s-s0 S=2sinθ/λ
  20. 20. Shift of Origin by -R • Shift of origin causes an increase of all phase angle by 2π R.S. • Amplitude and intensity of resultant vector (T) do not change. 2π (r+R).S 2πR.S
  21. 21. Consequences of sin  1/d (Reciprocal Space) Oblique Lattice Crystal Translation
  22. 22. Scattering by an atom • Electron cloud scatters X-ray. • Number and position of electrons affect the scattering. • Assume origin at nucleus. atomic scattering factor (f) Electron cloud is assumed spherically symmetric. So the atomic scattering factor is independent of the direction of S, but does depend on the length.
  23. 23. Scattering by a unit cell • n atoms at position rj. • Diffraction origin as unit cell origin. • So • Total scattering from unit cell • F(S) is called structure factor as it depends on the arrangement (structure) of the atoms in unit cell.
  24. 24. Scattering by a crystal • Transition vectors a b c n1  a, n2  b, n3  c • Add scattering by all unit cells with respect to single origin. • If unit cell has origin at (t.a+u.b+v.c), then its scattering is • Total scattering by crystal i.e. K(S)
  25. 25. Electron density from structure factor • Electron density of a crystal is a complicated periodic function and can be described as Fourier series. • Do structure factor equation have any connection with the Fourier series? Our goal is to find electron density m x y z Is there any way to solve these equations for the function ρ(x,y,z) ?
  26. 26. From Structure Factor to Electron density I  Fhkl 2 The contribution of scattering from a very short length of scattering matter dx , so short that ρ(x) can be considered constant within it , will be proportional to ρ(x)dx. Each structure factor equation can be written as a sum in which each term describes the diffraction by the electron in one volume element Fhkl = f( ρ1) + f( ρ2) + … + f( ρm) + … + f( ρn) + f( ρ1)
  27. 27. Total scattering from unit cell This summation is over j atoms in unit cell, instead of summing over all separate atoms we can integrate over all electrons in unit cell F(S)=
  28. 28. Diffraction data to structure •A simple wave like that of a visible light or x-ray can be described as a periodic function •F(x)= F cos 2π ( hx+α) •F(x)= F sin2π ( hx+α)
  29. 29. Fourier synthesis Figure 1: Fourier series approximation to sq(t) . The number of terms in the Fourier sum is indicated in each plot, and the square wave is shown as a dashed line over two periods. Jean Baptiste Joseph Fourier (1768 – 1830 Most intricate periodic function can be described as the sum of simple sine and cosine function Fourier synthesis is used to comput the sine and cosine terms that describe a complex wave
  30. 30. Structure factor- wave description of x-ray reflection • Each diffracted ray arrives at the film to produce a recorded reflection can be described as sum of contribution of all scatterer. • Sum that describe a diffracted ray is called a structure factor • Fhkl = fA + fB +…….. fA’ + fB’ + fF’ • Every atom in the unit cell contributes to every reflection in the diffraction pattern
  31. 31. Electron-density maps • Diffraction reveals the distribution of the electron density of the molecule • Electron density reflects the molecule shape • Electron density of proteins in crystal can be described mathematicaly by a periodic function • ρ(x,y,z) • Graph of the function is an image of the electron cloud that sourrounds the molecule in unit cell • The goal of crystallography is to obtain the mathematical function whose graph is the desired electron density map!
  32. 32. Fourier series Simple wave form: f (x) = F0 cos 2π ( h +α 0) f (x) = F0 cos 2π ( 0x +α 0) + F1 cos 2π ( 1x+α 1) + F2 cos 2π ( 2x+α 2 )……..+ Fn cos 2π ( nx+α n) Or equivalently f (x) = F0 cos 2π ( 0 x+α 0) Waveform of cosine and sine are combined to make a complex number cos 2π (hx) + i sin 2π ( hx) f (x)= Fh [cos 2π (hx) + i sin 2π ( hx)]
  33. 33. Three dimensional waves Each term is a simple wave with its own amplitude Fh , its own frequency h, and its own phase α Complex number in square brackets can be express exponentially, Cos θ + i sin θ = so the fourier series becomes f (x)= Fh A three dimensional wave has three frequencies one along each of the x, y, and z-axes. So, a general Fourier series for the wave f(x,y,z) is written as f (x,y,z)= Fhkl Fourier series representation of three dimensional wave f(x,y,z) Each term is a simple wave with its own amplitude Fh , its own frequency h, and its own phase α Complex number in square brackets can be express exponentially, Cos θ + i sin θ = so the fourier series becomes f (x)= Fh A three dimensional wave has three frequencies one along each of the x, y, and z-axes. So, a general Fourier series for the wave f(x,y,z) is written as f (x,y,z)= Fhkl
  34. 34. The fourier transform Fourier demonstrated that for any function f(x,y,z) there exist the function F(h,k,l) such that F(h,k,l)= f(x,y,z ) dx dy dz. F(h,k,l) is called Fourier transform of f(x,y,z) and in turn, f(x,y,z) is Fourier transform of F(h,k,l) as follows f(x,y,z)= F(h,k,l) dh dk dl. Information about real space f(x,y,z) can be obtain from information about reciprocal space, F(h,k,l).
  35. 35. Structure factor as Fourier series Fhkl = f( ρ1) + f( ρ2) + … + f( ρm) + … + f( ρn) + f( ρ1) Or equivalently, F(hkl) = ρ(x,y,z) dV Thus, Fhkl is the transform of ρ(x,y,z) on the set of real lattice plane (hkl) Applying reverse fourier transform to the above function ρ(x,y,z)= 1/V F(h,k,l) V is the volume of unit cell This equation tell us how to obtain the ρ(x,y,z)
  36. 36. • The mathematical process of connecting the diffraction of x-rays with the crystal structure is based in Fourier analysis. • crystal lattice function, which • is composed of one point ( Dirac delta function) at a fixed location in each repeating periodic unitof the crystal structure. Its Fourier transform is called the reciprocal lattice function, in which • each point (again a Dirac delta function) represents the wave-vector of a wave of electron density in • the crystal, of a given wavelength and orientation. The second element is the Fourier transform of • the contents (electron density) of one unit cell of the crystal structure, called the basis function • of the structure. This fourier transform is called the structure factor function. As you will • learn, because we can describe a periodic crystal structure as the convolution of the basis function • with the crystal lattice function, the Fourier transform of the crystal structure is the product of • the structure factor function times the reciprocal lattice function. That is, the Fourier transform • of the basis function is “sampled” at each reciprocal lattice point.
  37. 37. • validity of the proposed structure must be tested by comparison of the calculated values of the amplitudes of the structure factor Fc with the observed amplitudes |F0|. This is done by calculating a reliability index or R factor defined by •
  38. 38. ATOMIC SCATTERING FACTOR ATOMIC SCATTERING FACTOR f is - ratio of the amplitude of the wave scattered by the atom to that of an electron under identical conditions.  I  Ione4 2p 2 m2 c4 (1 cos2 ) Thomson’s theory of X-ray scattering Where I = scattered intensity Io = incident intensity n = number of electrons/unit volume m =mass c = speed of light
  39. 39. Another version of atomic scattering factor What happens near the absorption edge -you get resonance scattering - a phase shift of 90o What happens due to thermal motion
  40. 40. STRUCTURE FACTOR - scattering from planes
  41. 41. FOURIER TRANSFORM A complex wave Simple wave with Frequence of 2 Simple wave with Frequence of 3 Simple wave with Frequency of 5 ADD them all up
  42. 42. ELECTRON DENSITY - the result of a crystallographic experiment.
  43. 43. Let us recall what we actually measure. The intensity I of a reflection is proportional to the square of the complex structure factor F: I ~ F2 , and F2 = |F|2 NOW CAN WE DETERMINE THE STRUCTURE? THE PHASE PROBLEM
  44. 44. Structure Factors to Electron Density • The structure factor F(h,k,l) reflects distribution of electrons in crystal; positions of electrons described by x,y,z • If amplitude and phase of structure factors are known, can determine distribution of electrons in crystal,r (x,y,z) where V is volume of unit cell
  45. 45. Reflecting (or Scattering) Planes in Protein Crystals • Lower resolution - whole molecules; units of secondary structure • Higher resolution - residues; atoms
  46. 46. Solving the Phase Problem • Molecular replacement Existing molecular model for initial guess on electronic distribution in crystal • Isomorphous replacement (Perutz) multiple heavy-atom derivatives must not perturb protein conformation • Anomalous dispersion (MAD phasing) also relies on heavy atoms (typically Se in Se-Met) only one derivative needed relies on absorption of x-rays by heavy atoms leads to changes in hkl intensities vs. x-ray wavelength Intensity variations related to positions of heavy atoms in crystal need variable wavelength x-rays (synchrotron sources)
  47. 47. Convolution theory f(X) X X g(X) c(X) = f(X) * g(X) X c(X)
  48. 48. Electron Density Calculation • I(hkl) is proportional to square of F(hkl) • The summation is over all atoms in unit cell. Instead of summing over all separate atoms we can integrate over all electrons in unit cell…. and r.S = (a.x + b.y + c.z). S = a.S.x + b.S.y + c.S.z = hx + ky + lz • So F(S) can be written as F(hkl)….
  49. 49. • F(hkl) is the Fourier transform of ρ(xyz)…. So • Since |F(hkl)| can be derived from I(hkl) but can not be derived straight forward. The Phase Problem
  50. 50. Why Phase Problem • Very high frequency of X-rays. – Changes have to monitored at the time scale 10-18 sec. • Wave length of X-rays. – Measurement have to be at 10-11 m scale. • Actual nature of scattering. – The incident X-ray is incoherent so there is no well defined phase relation between incident and reflected wave. F(hkl) is a complex number and can be represented as:
  51. 51. Technique for solving the phase problem 1. The Isomorphous Replacement Method. • Requires the attachment of heavy atoms to the protein molecules in the crystal. 2. The Multiple Wavelength Anomalous Diffraction Method. • Depends on strong anomalously scattering atoms present in protein. • Anomalous scattering occurs if electron can not be regarded as free electron. 3. The Molecular Replacement Method. • Based on the known homologous structure. 4. Direct Method. • Method of future 