X-ray Crystallography Flashcards
What are x-rays
a traveling electromagnetic wave, which has oscillating electric (E) & magnetic fields (B) at right angles to each other
(wavelength=0.2nm)
X-ray crystallography process
- grow crystal
- put in front of x-ray beam
- calculate e density map
- build protein molecule/structure exactly as how it looks in the crystal
What gives diffraction pattern?
interaction of x-rays with e in crystals
Crystals
-shape of molecule defined by shape of e clouds association w/constituent nuclei
-crystal = ordered 3D array of distinctively shaped distributions of e
Electric field (E)
-describes electrostatic force felt by charged particle due to presence/motions of other charged particles
-can be represented as a vector
-tells you which way a +ve charge will move
Diffraction
-objects scattering x-rays at similar wavelengths as that of light
-get diffraction when wavelength = incident ray
Waves
-Phase angle θ = indicates position in the wave cycle
-Travelling waves phase varies with position and time
-One cycle corresponds to phase angle 2π (360º) ie. one full rotation of the line
-Y = maximum displacement from the x-axis
-A = amplitude
2πx/λ = wavelength
Y = A.cos(2πx/λ + Φ)
() = phase angle varies with x
Adding waves
Constructive interference
-Two waves of amplitude A perfectly in phase = results in 2A in phase with components
-Phase shift = 2nπ
Destructive interference
-Two waves of amplitude A perfectly out of phase = amplitude 0 (they cancel out)
-Phase shift = (2n-1)π
Intermediate example
-Two waves (A) partially out of phase = resultant has different phase to components
-Phase shift = a fraction of 2π
X-ray scattering 1e
- Incident x-ray (oscillating E-field)
- E responds by oscillating
- Oscillating electron emits x-rays over a wider angle
X-ray scattering 2e
- 2ē start oscillating and emitting x-rays in all directions
- Waves from 2ē in close proximity interfere/interact with one another so we have to add the waves
-The detector measures the added waves
-Scattering pattern is the result of adding the two scattered waves
-Diffraction pattern depends on structure -contains information about the structure of the 2ē, the spacing and relative position to one another
-X-rays come in, interact w/2ē and x-rays scatter w/angle of 2θ
-Incoming x-rays are in phase; top ray travels further before hitting ē = introduces path difference between scattered rays = results in phase shift
-Phase shift is due to structure of ē
-Structure affects A and phase of resultant wave in direction 2θ
-K0 = incident wave vector; k = scattered wave vector
magnitude = l/λ for both; assume that interaction of ē is not altering the wavelength
r = relative position of ē (structure)
path difference between waves scattered by ē at origin and one at position r = AB-OC
phase shift= -2π ((path difference)/λ) * - sign is just a convention
=2πr ( (cos(α)-cos(β) ) /λ)
Scattering vectors (S)
-Magnitude and direction of S contain information about λ and scattering angle θ, but it does not point in direction of scattering
-At fixed λ, S varies only with θ
2k because k = k0
length of S |S| = 2k.sin = (2/λ).sinθ
Calculating phase shift
-Between ray scattered form O and ray scattered from B: ϕ=2πrS
-Phase shift depends on r and S
-Contains information about the structure (position of 2ē), wavelength, and scattering angle
Wave equation
General wave equation:
Y = A.cos(2πx/λ + Φ)
Wave equation using complex numbers:
Y = Aei(2πx/λ + Φ)
If we are only concerned with phase shifts:
Y = Aei(Φ)
Structures with 2e
Resultant wave:
Y = Aei(0) + Y = A.ei(2πrS)
Wave scattered from origin (r=0) + wave scattered from r
Structures with +2e
-Each ē has a different position and vector (r)
-Add all the waves scattered from all electrons
* assume all ē have same amplitude
J = increment of system
N = denoting amount of electrons present
f(S) = Σ N j=1 e i2πrj.S
-9 scattered waves of same amplitude go in same direction; their phase depends on the position of the electron it scattered from
-Amplitude and phase of resultant wave depends on that of components
Structure factor f(S)
-f(S) describes diffraction pattern = how the diffracted waves in each direction are related to the structure (r)
-f(S) = a complex number
-f(S) is a wave; has amplitude, phase & direction
f(S) = |f(S)| eiΦ(S)
e density function
-E density gives 3D shape of the protein
-E density function = density you can measure at a specific position, r, within the molecule
ρ(r)dxdydz
-By defining ē density at specific position, can map overall ē density on protein molecule
-Protein molecule emits ē in all directions
-Protein consists of different atoms which have different ē clouds
-Proteins have higher ē density at the core
-Amplitude depends on ē density/probability of x-ray interacting with ē/ē cloud at particular positions
Each scattered wave has the form: [ρ(r)dxdydz] e i2πrS
Total scattering by a molecule in one direction
-add up all contributions
-Have different f(S) in different directions
Diffraction pattern of molecules
-Contains information about ē density and phase-shift due to different positions of atoms within molecule
Structure factor of a molecule f(S)
-f(S) = sum of phases at different positions of r
-Describes total scattering by the molecule as a function of direction
-Amplitude has to be taken into account = different parts of molecule have different A
-To calculate for every position of the molecule:
-Need to consider all possible dimensions molecule can scatter x-rays
-d(r) = dxdydz
Fourier transform equation:
f(S) = ∫ ∞ ∞ ρ(r)e i(2πrS) dr
Fourier transform
-Fourier transform = the total scattering in the direction associated with S, from an object described by electron density function is the sum of the waves scattered from every point in the object
-Adding all waves coming from the molecule gives the structure factors
Inverse fourier transform
-If you can measure all possible f(S) values (ie. The diffraction pattern in detector), and we know the phase shift, we can calculate ρ(r) ie. structure
-Can apply this theory as long as you can measure every single possible diffraction spot on detector
p(r) = ∫ ∞ ∞ ρ(r)e -i(2πrS) dS
-Fourier transform gives information about the total scattered waves coming from a protein molecule that contains electron density information = inverse measures all these waves to work out what the structure looks like
Diffraction from crystalline samples
-Diffraction pattern from protein crystal contains same information as scattering from a single molecule = just an enhancement in the signal because you have billion copies of protein in the crystal
-Measure how dark the spots are
Why can you not use single molecule for x-ray diffraction?
-Too small; cannot rotate it in 3D for every position
-Diffraction would be too weak
-X-rays cause ionisation radiation damage so molecule will blow apart when electrons get excited and move to different spin
Advantages of crystals
-Contain billions of molecules
-Amplify the scattering making it detectable
-FACT: some properties of crystals restrict scattering in certain directions (not a problem)
Reflection from semi-transparent crystal planes
-Only ~2% of x-rays are scattered from a protein crystal; most pass through the sample
-Incident angle θ = angle of reflection/scatter θ
-Due to constructive interference, if you have 20,000 waves in phase, you can increase diffraction signal significantly
Braggs Law
-Path difference = 2dsinθ
-Get strong diffraction only if path difference between waves from adjacent layers = nλ
-No scattering when path difference = odd number of a half wavelength = destructive interference = 2dsinθ = (2n-1)/λ-2
-The positions that you measure on the diffraction spots tells us about the position of the electrons within the protein; can measure the spacing that tells us from which plane the crystal diffraction is coming from
Crystalline diffraction
-Structure function of crystal = n x f(S)
-n = amount of molecules in the crystal
-S is only allowed to be discrete positions within the planes, use hkl for distinct space in the crystal
-Hkl = allowable directions that give constructive interference