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Molecular Biophysics Solving the phase problem. Der Weg zur Röntgenkristallstruktur eines Proteins. Electron density equation. The electron density equation. Electron density equation. F ( h k l ) =  cell r ( x y z ) exp (2 p i { hx + ky + lz }) d 3 r.

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Molecular Biophysics

Solving the phase problem


Der Weg zur

Röntgenkristallstruktur eines Proteins


Electron density equation
Electron density equation

The electron density equation

Electron density equation

F(hkl) = cellr(xyz)exp (2pi{hx+ ky+lz}) d3r

r(xyz)= ShklF(hkl) exp (-2pi{hx+ ky+lz})

But we can only measure the intensity

I(hkl) = F(hkl) . F*(hkl) = |F(hkl)|2

We have lost the phase information: this is the fundamental problem in X-ray crystallography –

The PHASE PROBLEM




Influence of intensities

Influence of phases

The phases are more important than the amplitudes!!!!




Patterson map
Patterson map

Direct space

Density

and

position

Patterson

map

Fourier

transformation

Fourier

transformation

Amplitudes

and

phases

Intensities

Reciprocal space


Patterson map symmetry
Patterson map symmetry

Patterson map with symmetry

Harker vectors

u, v, w

2x, 1/2, 2z

P21

x, y, z

-x, y+1/2, -z


The crystallographic phase problem can be solved via:

Single isomorphous replacement (SIR)

Multiple isomorphous replacement (MIR)

Single isomorphous replacement with anomalous scattering (SIRAS)

Multiple wavelength anomalous dispersion (MAD)

Molecular replacement (MR)

Difference Fourier methods


Derivative data
Derivative data

Once we have an heavy atom structure rH(r), we can use this to calculate FH(S). In turn, this allows us to calculate phases for FP and FPH for each reflection.


Derivative data1
Derivative data

Once we have an heavy atom structure rH(r), we can use this to calculate FH(S). In turn, this allows us to calculate phases for FP and FPH for each reflection.


Derivative data2
Derivative data

Once we have an heavy atom structure rH(r), we can use this to calculate FH(S). In turn, this allows us to calculate phases for FP and FPH for each reflection.


Harker diagram
Harker diagram

Once we have an heavy atom structure rH(r), we can use this to calculate FH(S). In turn, this allows us to calculate phases for FP and FPH for each reflection.

Harker construction for

single isomorphous replacement (SIR)

The phase probability distribution shows that SIR results in a phase ambiguity


mir

We can use a second derivative to resolve the phase ambiguity

Harker construction for

multiple isomorphous replacement (MIR)


02p

m=




Wave anom
Wave anom

Anomalous scattering involves resonance effects



Anomalous scattering data can also be used to solve the phase ambiguity

Note that the anomalous differences are very small; thus very accurate data are necessary


Phase solution
Phase solution phase ambiguity

The crystallographic phase problem can be solved via:

Single isomorphous replacement (SIR)

Multiple isomorphous replacement (MIR)

Single isomorphous replacement with anomalous scattering (SIRAS)

Multiple wavelength anomalous dispersion (MAD)

Molecular replacement (MR)

Difference Fourier methods


Der Weg zur phase ambiguity

Röntgenkristallstruktur eines Proteins


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