The Patterson function is a powerful tool in crystallography for solving crystal structures without phase information. It uses intensity data to create a vector map between atoms, revealing interatomic distances and helping locate heavy atoms in complex structures.

Patterson maps and techniques like isomorphous replacement and anomalous dispersion are crucial for solving the phase problem in protein crystallography. These methods enable structure determination by locating heavy atoms and providing initial phase estimates for electron density calculations.

The Patterson Function

Mathematical Derivation and Properties

• Patterson function analyzes diffraction data to determine crystal structures without prior knowledge of atomic positions
• Derived from Fourier transform of intensity data representing vector map between atoms in crystal structure
• Calculated using square of structure factor amplitudes ($|F|^2$) instead of structure factors ($F$) eliminating need for phase information
• Patterson peaks correspond to interatomic vectors with height proportional to product of atomic numbers of atoms involved
• Function remains centrosymmetric regardless of crystal structure symmetry
• Origin peak represents vector between each atom and itself with height proportional to sum of squares of atomic numbers of all atoms in unit cell
• Limitations include peak overlap and difficulty in interpretation for structures with many atoms

Applications in Crystallography

• Enables determination of atomic positions without prior phase knowledge addressing phase problem
• Utilized in heavy atom methods to locate heavy atoms for initial phase estimation
• Employed in isomorphous replacement method to identify heavy atom positions in derivative structures
• Patterson superposition techniques generate trial structures by overlapping shifted Patterson maps
• Facilitates molecular replacement method to determine orientation and position of known molecular fragments
• Incorporated into direct methods to improve phase estimates and structure solution
• Success depends on data quality resolution and presence of distinct atomic features

Interpreting Patterson Maps

Map Characteristics and Symmetry

• Three-dimensional representations of Patterson function displaying peaks corresponding to interatomic vectors
• Always centrosymmetric belonging to one of 24 symmorphic space groups (Patterson symmetry)
• Harker sections and lines contain peaks from symmetry-related atoms in crystal structure
• Size and shape of peaks provide information about relative atomic numbers and spatial relationships
• Identify molecular fragments or structural motifs within crystal structure
• Interpretation becomes challenging with increasing number of atoms due to peak overlap and background
• Advanced techniques (symmetry minimum function vector superposition methods) facilitate complex map interpretation

Analysis Techniques for Complex Structures

• Cross-difference Patterson maps verify consistency of heavy atom positions between datasets
• Combination with density modification techniques improves initial phase estimates and electron density maps
• Automated structure solution pipelines incorporate Patterson-based methods for heavy atom location and initial phasing
• Vector superposition methods reveal true crystal structure by overlapping shifted Patterson maps
• Symmetry minimum function reduces complexity by focusing on symmetry-related peaks
• Patterson correlation methods compare observed and calculated Patterson functions to evaluate trial structures
• Difference Patterson techniques highlight specific structural features by subtracting Patterson functions

Solving the Phase Problem

Isomorphous Replacement Methods

• Single Isomorphous Replacement (SIR) determines heavy atom positions in derivative protein crystal
• Multiple Isomorphous Replacement (MIR) uses multiple heavy atom derivatives improving phase accuracy
• Cross-difference Patterson maps verify consistency between different isomorphous datasets
• Isomorphous difference Patterson maps highlight changes between native and derivative structures
• Phase triangulation techniques combine information from multiple derivatives to resolve phase ambiguity
• Non-crystallographic symmetry can be exploited to improve phase estimates in isomorphous replacement
• Solvent flattening and histogram matching often applied to enhance phases from isomorphous replacement

Anomalous Dispersion Techniques

• Locates anomalous scatterers for phase determination in SAD (Single-wavelength Anomalous Dispersion) and MAD (Multi-wavelength Anomalous Dispersion)
• Anomalous difference Patterson maps reveal positions of anomalous scatterers
• Bijvoet difference Patterson maps highlight differences between Friedel pairs due to anomalous scattering
• MAD technique utilizes data collected at multiple wavelengths to maximize anomalous signal
• SAD combines anomalous signal with density modification for phase determination
• Sulfur-SAD exploits weak anomalous signal from native sulfur atoms in proteins
• Radiation-induced phasing utilizes changes in anomalous signal during data collection

Heavy Atom Determination for Protein Structures

Locating Heavy Atoms

• Patterson function effective for locating heavy atoms due to high scattering power and distinct peaks
• Harker sections in Patterson maps reveal symmetry-related heavy atom positions
• Cross-vectors between heavy atoms and protein atoms aid in confirming heavy atom locations
• Difference Fourier methods complement Patterson analysis for refining heavy atom positions
• Automated heavy atom search algorithms (SHELXD SnB) utilize Patterson function principles
• Dual-space recycling techniques combine direct and Patterson space for robust heavy atom location
• Native Patterson function can reveal presence of unexpected heavy atoms in protein structure

Phasing and Refinement Strategies

• Initial phases from heavy atoms used to calculate first electron density map for protein structure
• Iterative phase improvement combines heavy atom refinement with protein model building
• Density modification techniques (solvent flattening non-crystallographic symmetry averaging) enhance phases
• Maximum-likelihood refinement incorporates heavy atom parameter uncertainty in phase calculation
• Multi-crystal averaging exploits non-isomorphism between crystals to improve phase estimates
• Ab initio phasing methods (charge flipping) can complement Patterson-based approaches for difficult structures
• Phase combination techniques merge experimental phases with those from partial models during refinement