Macromolecular X-Ray Crystallography Study Notes

The phase problem is a significant issue in macromolecular X-ray crystallography, primarily due to the difficulty in obtaining phase information compared to intensity data.

Phases:
- Difficult to obtain but provide vital information necessary for structure determination.
Intensities:
- Easier to obtain.
Data Collection:
- Collect intensities through X-ray diffraction, which involves Fourier transforms of the data.

Amplitude:
- Represented as amplitude (A) and amplitude squared (A^2).
- Proportional relationship: Intensity of reflection is proportional to amplitude.
Wavelength:
- Key in determining diffraction properties of the waves.
Phase Differences:
- Waves can differ in phase by various angles, such as 90°.

Methodology:
- The phases of protein reflections can be determined by adding one (or a few) heavy atoms (e.g., Mercury (Hg), Platinum (Pt), Gold (Au)) to each unit cell without altering crystal packing or protein conformation.
Process:
- Use the interference behavior caused by heavy atoms to identify their positions directly from intensity differences in diffraction patterns.
- Only feasible with one or two heavy atoms per asymmetric unit.
Example:
- Two diffraction patterns can be observed: one from native bovine $eta$-lactoglobulin and another from the crystal soaked in a mercury-salt solution.

Overview:
- Certain heavy atoms (e.g., Iron (Fe), Selenium (Se), Mercury (Hg), Platinum (Pt), Iodine (I)) absorb radiation at specific resonance frequencies, leading to differences in diffraction intensities.
Incorporation of Selenomethionine:
- Often included during recombinant expression, which aids in phasing.
Data Collection:
- Three datasets are measured at different wavelengths, requiring a synchrotron radiation source.
- Use intensity differences to calculate positions of anomalous atoms in the unit cell, helping derive phases similar to isomorphous replacement.
- The small size of intensity differences demands high-energy synchrotron radiation for sensitivity.

Requirements:
- Requires the known structure of a similar protein (phasing model) with the same fold.
Procedure:
- Superimpose known structure in the unit cell based on diffraction pattern characteristics.
- Calculate diffraction pattern for the known model and use the phases as phase estimates for the unknown structure.
Applications:
- Particularly effective for homologous proteins, site-directed mutants, and small ligand complexes.
Structural Similarity:
- If protein sequence identity is greater than 30%, it indicates a high probability of exhibiting the same fold.

Objective:
- Calculate electron density maps using phases from the new model combined with native intensities.
Building Models:
- Models are built based on discernible elements, considering known peptide geometries (bond lengths, bond angles).
Assessment of Agreement:
- Degree of agreement between model and native dataset is expressed by the R-factor (residual index).

Definition:
- Structure factor $F(hkl)$ represents both amplitude and phase of any X-ray reflection hkl.
Quality Assessment:
- Comparison between previous and new models’ structure factors is essential to assess refinement quality.

Formulation:
- $p(xyz) = rac{1}{V} imes ext{Σ} (F{ ext{obs}} - F{ ext{calc}}) imes ext{exp}[-2 ext{i} (h x + k y + l z)]$.
Significance:
- These maps reveal where model discrepancies occur, highlighting maxima for missing atoms and minima for wrongly positioned atoms (negative electron density).

Definition:
- Resolution refers to the distance between peaks in electron density maps which can be interpreted as separate atoms.
Mathematical Expressions:
- Maximum achievable resolution is determined by the diffraction equation: $n imes ext{λ} = 2d imes ext{sin}( heta_{ ext{max}})$.
Diffraction Limits:
- Common diffraction limits reached in practice include from 5.5 Å down to 0.8 Å.