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Molecular Biophysics Solving the phase problem

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Molecular Biophysics

Solving the phase problem

Der Weg zur

Röntgenkristallstruktur eines Proteins

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

The phase problem

The phase problem

Influence of intensities

Influence of phases

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

Direct space

Density

and

position

Patterson

map

Fourier

transformation

Fourier

transformation

Amplitudes

and

phases

Intensities

Reciprocal space

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

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.

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.

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

We can use a second derivative to resolve the phase ambiguity

Harker construction for

multiple isomorphous replacement (MIR)

02p

m=

Anomalous scattering involves resonance effects

Anomalous scattering leads to a breakdown of Friedel‘s law

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

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

Röntgenkristallstruktur eines Proteins