International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 5.2, pp. 152-153   | 1 | 2 |

## Section 5.2.4. Algebraic concepts

E. M. Westbrooka*

aMolecular Biology Consortium, Argonne, Illinois 60439, USA
Correspondence e-mail: westbrook@anl.gov

### 5.2.4. Algebraic concepts

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Let V be the volume of one unit cell of the crystal. Let be the total mass within one unit cell, and , and be the masses, within one unit cell, of the macromolecule, bound solvent and free solvent, respectively. Let , , and , respectively, be the densities of a complete macromolecular crystal, its unsolvated macromolecule, its bound-solvent compartment and its free-solvent compartment. Let , and , respectively, be the fractions of the crystal volume occupied by the unsolvated macromolecule, the bound solvent and the free solvent. By conservation of mass, The volume fractions must all add to unity: The density of the crystal is the total mass divided by the unit-cell volume: The mass in each solvent compartment is the product of its density and the volume it occupies: The mass of the macromolecule in the cell can be defined either from its partial specific volume, , the unit-cell volume, V, and the molecules' volume fraction, , or from the molar weight, M, the number of molecular copies in the unit cell, n, and Avogadro's number, : Now (5.2.4.3 ) may be rewritten as Define a mean solvent density, : This allows (5.2.4.6 ) to be rewritten as Upon rearrangement, this gives expressions for the volume fraction of a macromolecule and for the molecular-packing number: In (5.2.4.9 ), all terms can be measured directly, except . The treatment of will be discussed in Section 5.2.7 . (5.2.4.9 ) defines the total macromolecular mass in the unit cell, , from a measurement of the crystal density . If M were known from the primary sequence of the molecule, this measurement determines the molecular-packing number, n, with considerable certainty. If the molar weight were not accurately known, it could be determined by measuring the crystal density.