Tables for
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, p. 152   | 1 | 2 |

Section 5.2.3. Matthews number

E. M. Westbrooka*

aMolecular Biology Consortium, Argonne, Illinois 60439, USA
Correspondence e-mail:

5.2.3. Matthews number

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In an initial survey of 116 crystals of globular proteins (Matthews, 1968[link]) and in a subsequent survey of 226 protein crystals (Matthews, 1977[link]), Matthews observed that proteins typically occupy between 22 and 70% of their crystal volumes, with a mean value of 51%, although extreme cases exist, such as tropomyosin, whose crystals are 95% solvent (Phillips et al., 1979[link]). The volume fraction occupied by a macromolecule, [\varphi_{m}], is reciprocally related to [V_{M}], the Matthews number, according to [V_{M} = \overline{\upsilon}_{m}/N_{o} \varphi_{m} = V/nM, \eqno(]where [\overline{\upsilon}_{m}] is the partial specific volume of the macromolecule (Tanford, 1961[link]), [N_{o}] is Avogadro's number, V is the volume of the crystal's unit cell, n is the number of copies of the molecule within the unit cell and M is the molar weight of the macromolecule (grams per mole). [V_{M}] is the ratio between the unit-cell volume and the molecular weight of protein contained in that cell. The distribution of VM (2.4 ± 0.5 Å3 Da−1) is asymmetric, being sharply bounded at 1.7 Å3 Da−1, a density limit consistent with spherical close packing. The upper limits to [V_{M}] are much less distinct, particularly for larger proteins. Matthews observed a slight tendency for [V_{M}] to increase as the molecular weight of proteins increases. [V_{M}] values below 1.9, or above 2.9, can occur but are relatively rare (beyond a 1σ cutoff).

The unit-cell volume, V, is determined from crystal diffraction. The partial specific volume of a macromolecule, [\overline{\upsilon}_{m}], is the rate of change in the volume of a solution as the (unhydrated) macromolecule is added. It can be measured in several ways, including by ultracentrifugation (Edelstein & Schachman, 1973[link]) and by measuring the vibrational frequency of a capillary containing a solution of the macromolecule (Kratky et al., 1973[link]). [\overline{\upsilon}_{m}] typically has a value around [0.74 \hbox{ cm}^{3} \hbox{ g}^{-1}] for proteins and around [0.50\ \hbox{cm}^{3}\ \hbox{g}^{-1}] for nucleic acids (Cantor & Schimmel, 1980[link]). Values of [\overline{\upsilon}_{m}] are tabulated for all amino acids and nucleotides, and [\overline{\upsilon}_{m}] of a macromolecule can be estimated with reasonable accuracy as the mean value of its monomers. Commercial density-measuring instruments are available to determine [\overline{\upsilon}_{m}] by the Kratky method. Because M is usually well known from sequence studies, n – the number of copies of the macromolecule in the unit cell – can be calculated thus: [n = V/V_{M}M = VN_{o} \varphi_{m}/\overline{\upsilon}_{m}M. \eqno(]

For proteins, evaluating this expression with [V_{M} = 2.4] usually provides an unambiguous integer value for n – which must be a multiple of the number of general positions in the crystal's space group! Setting n to its integer value then provides the actual value for [V_{M}]. If the calculated [V_{M}] value lies beyond the usual distribution limits, if n has an unexpected value or a large value, or if the crystal contains unusual components or several different kinds of molecular subunits, the crystal density may need to be measured accurately.


Cantor, C. R. & Schimmel, P. R. (1980). Biophysical Chemistry. San Francisco: W. H. Freeman & Co.
Edelstein, S. J. & Schachman, H. (1973). Measurement of partial specific volumes by sedimentation equilibrium in H2O − D2O solutions. Methods Enzymol. 27, 83–98.
Kratky, O., Leopold, H. & Stabinger, H. (1973). The determination of the partial specific volume of proteins by the mechanical oscillator technique. Methods Enzymol. 27, 98–110.
Matthews, B. W. (1968). Solvent content of protein crystals. J. Mol. Biol. 33, 491–497.
Matthews, B. W. (1977). Protein crystallography. In The Proteins, edited by H. Neurath & R. L. Hill, pp. 403–590. New York: Academic Press.
Phillips, G. N., Lattman, E. E., Cummins, P., Lee, K. Y. & Cohen, C. (1979). Crystal structure and molecular interactions of tropomyosin. Nature (London), 278, 413–417.
Tanford, C. (1961). Physical Chemistry of Macromolecules. New York: John Wiley & Sons.

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