Phasing - Isomorphous Derivative Method

Add just one or a few heavy atoms to the crystal without disturbing the rest of the structure.

In general, anything that changes the effect scattering power in a place without significant change to the rest of the molecular structure.

This can be done either by:

Substitution -- Ba++ for Ca++ , I for CH3

or Addition -- PtCl4 in space between molecules

or changing wavelength so a particular “heavy” atom scatters more or less -- a small effect but remarkably useful in special circumstances. (Anomalous scattering, MAD, etc. considered in another section.)

**F _{PH}**

|F

(Traditionally the native protein and the derivatives are usually separate crystals.*)

*Same crystal: changing wavelength, or adding heavy atoms to a crystal in a flow cell.

|F

so an approximate |F

We do a Patterson map using ≈ |F_{H}|^{2}, find the heavy atoms, and calculate φ_{H}. This works even though this approximation is only really good if **F _{PH}** and

Fortunately,

1) Sometimes the phases are constrained to be either 0° or 180°.

2) There is a trick we can use: anomalous scattering by the heavy atoms reveals the difference of φ_{H} from φ_{P}.

3) Simple subtraction actually works fairly well because

a) Error can only reduce |FH|, so that term merely contributes less to the Patterson than it should have.

b) We are using hundreds or thousands of measurements to determine a few heavy atom positions, so the Patterson peaks build-up in the right places.

Phasing -- Determination of φ_{P}

|F_{P}| can be measured; we must find φ_{P} (for each *h, k, l* reflection).

|F_{PH}| can be measured; |F_{H}| and |φ_{H}| can be calculated for each *h, k, l*.

Lay out circles of radii |F_{P}| and |F_{PH}| at the ends of the calculated vector F_{H} (arrow in diagrams). The intersections of the circles give possible values for φ_{P}; but there is a twofold ambiguity!

So, get a second heavy-atom derivative: measure |F_{PH}'|, calculate |F_{H}'| and φ_{P}'.

Layout the circles for this second case also: *

This case will give a different twofold ambiguity, and the alternative that agrees with one of the intersections from the first case should give the correct φ_{P}.

Do this for all reflections.

Using |F_{P}(*h, k, l*)| and φ_{P}(*h, k, l*) , plug into the Fourier transform formula to calculate an electron density map.

* For each derivative, calculate a |F_{PH}| - |F_{P}| Patterson map and solve for those heavy atoms. Thus for each derivative |F_{H}| and φ_{H} can be calculated for each *h, k, l*.

The heavy atom models are independent, and before they can be combined, they must be related to the same origin. There are various tricks that take advantage of symmetry and patterns in the diffraction from the native protein molecule to do this. However, the best way to relate the derivatives is to make a crystal with BOTH derivatives at once. The Patterson map from this this combined derivative will give the relative postions of the heavy atoms, even if the crystal is of poorer quality.

**H**eavy**A**tom**I**somorphous**R**eplacement #1

Reference wave **F**_{H1}

Amplitude calculated

Phase calculated from

known heavy atom

position

2 choices for wave **F**_{P}

from all protein atoms,

Amplitude measured

What 2 phases for **F**_{P}

will give the measured

|**F**_{H1}|?

**F**_{H1} + **F**_{P} = **F**_{PH1}

Reference wave +

protein wave =

Amplitude measured

for **F**_{PH1}

**H**eavy**A**tom**I**somorphous**R**eplacement #2

Reference wave **F**_{H2}

Amplitude calculated

Phase calculated from

known heavy atom

position

2 choices for wave **F**_{P}

from all protein atoms,

Amplitude measured

What 2 phases for **F**_{P}

will give the measured

|**F**_{H2}|?

**F**_{H2} + **F**_{P} = **F**_{PH2}

Reference wave +

protein wave =

Amplitude measured

for **F**_{PH2}

**M****I****R** **M**ultiple **I**somorphous **R**eplacement

Reference waves 1 & 2

Amplitude calculated

Phases calculated from

known heavy atom

positions

wave **F**_{P}

from all protein atoms,

Amplitude measured

What phases for **F**_{P}

will give the measured

|**F**_{H}|'s ?

**F**_{H1} + **F**_{P} = **F**_{PH1}

**F**_{H2} + **F**_{P} = **F**_{PH2}

Amplitudes measured

for **F**_{PH1} and **F**_{PH2}

Download: PS-PhaseTriangles.pdf(80KB) Problem for this section

Phase Probability Distributions

Because the data are by no means perfect, one does not try to find a unique φ_{P} where for both derivatives the circles intersect. Instead, a probability of intersection is defined that is related to the distance between these circles (as measured along a F_{PH} radius.)

P = e^{(-dist/ε)2} where ε is an appropriate estimate of the error in the data.

Each probability distribution will be bimodal. They can be plotted on circular graphs, with radial distance out beyond a reference circle representing the probability of a given phase angle for φ_{P}. The probability distributions then can be multiplied together to give a new probability distribution that hopefully will have only one large peak at the true φ_{P}, as shown below for the first example of derivatives discussed above.

Often more than two derivatives are used in solving a protein structure. These probability distributions provide a convenient method of combining several sets of imperfect data.

prob. derivative H prob. derivative H' prob. combined

Triangles and Phase Probability Distributions

Probability distributions correlated with triangles. The F_{P} circle is centered on the arrow head. The probability distributions are drawn around the circumference of the F_{P} circle.

GOOD CASE

This is a good determination of the Phase! If the distribution as shown really measures the density on the circumference of the F_{P} circle, then the center of mass of the resultant probability distribution is very close to the tip of the F_{P} arrowhead. This reflection would have a figure of merit very close to 1.0.

Triangles and Phase Probability Distributions

Probability distributions correlated with triangles. The F_{P} circle is centered on the arrow head. The probability distributions are drawn around the circumference of the F_{P} circle.

NOT SO GOOD CASE

This reflections has a less sure determination of its Phase!
If the distribution as shown really measures the density on the circumference of the F_{P} circle, then the fractional radius of the center of mass of the resultant probability distribution is the figure of merit. This reflection would have a figure of merit about 0.7.

[Note: There is now a more elaborate method called Maximum Likelihood, which explicitly takes into account estimates of possible uncertainty in both the amplitude measurement and in knowledge of the phase of all components.]

bottom 7, Phasing: Isomorphous