Questions: Crystallographic Symmetry and Space Groups

Question 1

A protein crystal has space group P2(1)2(1)2(1) with one molecule in the asymmetric unit. How many copies of the protein are in the unit cell?

• A. 1 — the asymmetric unit equals the unit cell
• B. 4 — P2(1)2(1)2(1) has four symmetry-equivalent positions (the general multiplicity is 4), so four copies of the asymmetric unit generate the complete unit cell
• C. 21 — the subscript numbers indicate the copies
• D. 8 — orthorhombic cells always contain 8 molecules

Answer: (B)

Every space group has a general position multiplicity — the number of symmetry-equivalent copies of the asymmetric unit in one unit cell. For P2(1)2(1)2(1), the multiplicity is 4 (three mutually perpendicular 2(1) screw axes generate 4 equivalent positions). P2(1)2(1)2(1) is the most common space group for protein crystals (~25% of all protein crystal structures), likely because its screw axes accommodate elongated protein molecules efficiently while maintaining enough contact points for crystal stability. The total molecular content of the unit cell = (number of molecules in asymmetric unit) x (multiplicity of space group).

Question 2

The asymmetric unit of a crystal structure always corresponds to the biologically functional assembly of the protein.

• T. True
• F. False

Answer: False

The asymmetric unit is a mathematical construct defined by the crystal symmetry — it is the minimum set of atoms needed to generate the entire crystal by applying symmetry operations. It may contain less than the biological assembly (a monomer in the asymmetric unit may form a dimer across a crystallographic two-fold axis — the dimer is the biological unit, but only half is in the asymmetric unit), exactly the biological assembly, or more than one biological assembly (two independent monomers that happen to be in the asymmetric unit due to the crystal packing). Determining the biological assembly requires additional information: solution studies (gel filtration, analytical ultracentrifugation), interface analysis (buried surface area, complementarity), and comparison across crystal forms. The PISA server (Protein Interfaces, Surfaces, and Assemblies) provides computational analysis of likely biological assemblies from crystal structures.

Question 3

What is the Matthews coefficient (V_M) and why is it useful during structure determination?

Model answer: The Matthews coefficient (V_M) is the crystal volume per unit of protein molecular weight (in Angstrom^3 per Dalton), calculated as V_M = V_cell / (Z * M_W), where V_cell is the unit cell volume, Z is the number of molecules in the unit cell (space group multiplicity times the number of molecules in the asymmetric unit), and M_W is the molecular weight. Matthews (1968) showed that V_M for protein crystals falls in a characteristic range (1.7-3.5 A^3/Da), corresponding to a solvent content of approximately 30-65%. This allows crystallographers to estimate how many copies of the protein are in the asymmetric unit before solving the structure: given the known unit cell dimensions, space group, and protein molecular weight, the number of molecules per asymmetric unit that gives a V_M in the expected range can be calculated. This information is critical for molecular replacement (knowing how many search models to place) and for data processing (confirming the space group assignment).

The Matthews coefficient reflects the fact that protein crystals are ~50% solvent on average — large solvent channels permeate the lattice, which is why proteins can function in some crystal forms (enzymes can catalyze reactions in the crystal, substrates can diffuse through the channels). Very high V_M (>4.0) suggests the wrong number of molecules per asymmetric unit or an incorrect space group; very low V_M (<1.7) suggests impossibly tight packing.

Question 4

Why are mirror planes and inversion centers forbidden in protein crystal space groups?

Model answer: Mirror planes and inversion centers convert a chiral object into its mirror image (enantiomer). Proteins and nucleic acids are chiral molecules — they are composed of L-amino acids and D-sugars, respectively, and have defined handedness. A mirror plane or inversion center would require that both a protein and its mirror image coexist in the same unit cell, which is physically impossible for a homogeneous sample of a biological macromolecule. This restricts protein crystals to the 65 chiral (Sohncke) space groups out of the 230 total crystallographic space groups. The 165 excluded space groups contain improper symmetry operations (mirrors, glide planes, inversion centers, rotoinversions) that are possible for small achiral molecules but forbidden for biological macromolecules.

In practice, this restriction simplifies space group determination for protein crystals. The most common space groups for proteins are P2(1)2(1)2(1) (~25%), P2(1) (~15%), C2 (~10%), and P1 (~5%). Knowing the allowed space groups helps crystallographers identify the correct symmetry from the diffraction pattern's systematic absences.