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

International Tables for Crystallography (2006). Vol. F, ch. 5.2, pp. 118-121   | 1 | 2 |

Section 5.2.6. Methods for measuring crystal density

E. M. Westbrooka*

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

5.2.6. Methods for measuring crystal density

| top | pdf |

Density measurements of macromolecular crystals are complicated by their delicate constitution. These crystals tolerate neither dehydration nor thermal or physical shock or stress. Furthermore, since macromolecular crystals contain free solvent, their densities will change as the density of the solvent in which they are suspended is changed. They cannot be picked up with tweezers, nor rinsed with arbitrary solvents, nor placed out to dry on the table.

The other experimental problem with these crystals is that they are very small. Typically, their linear dimensions are 0.1–0.2 mm, volumes are 1–10 nl and weights are 1–10 µg. Molecular structures can now be determined from even smaller crystals (linear dimensions as small as 20 µm) using synchrotron radiation, so density-measurement methods compatible with very small crystals are required. With such small samples, it is far easier and more accurate to measure densities than to measure directly volumes and weights.

The physical properties of macromolecular crystals constrain the methods by which their densities can be measured accurately. In all circumstances, great care must be taken to avoid artifacts such as air bubbles or particulate matter which often adhere to these crystals. All measurements should be made at one tightly controlled temperature, since thermal expansion can change densities and thermal convection can corrupt density gradients. Because crystals contain solvent, it is bad to dry them, since this process usually disrupts them, changing all parameters in unpredictable ways. Yet many density-measurement methods require that all external solvent first be removed from the crystals, since the measured densities will be some average of crystal and any remaining solvent. This can be an almost insurmountable problem for crystals containing cavities and voids. Unfortunately, many crystallization mother liquors are viscous and difficult to remove, for example if they contain polyethylene glycol (PEG).

Richards & Lindley in Chapter 3.2[link] of IT C (2004[link]) list six methods for measuring crystal densities: pycnometry, the method of Archimedes, volumenometry, the immersion microbalance, flotation and the gradient tube. The first three methods require direct weighing of the crystal and are therefore of limited value for crystals as small as those used in macromolecular diffraction, although these methods are used in various applications, such as mineralogy and the sugar industry. The latter three methods measure densities and density differences, and can therefore be used in macromolecular crystallography. A new method specifically for protein crystals has recently been described (Kiefersauer et al., 1996[link]), involving direct tomographic measurement of crystal volumes coupled with quantitative amino-acid analysis. Because the gradient-tube method remains the method of choice for most crystal-density measurements, it will be discussed last and most thoroughly here. Pycnometry

| top | pdf |

Pycnometry measures the density of a liquid by weighing a calibrated volumetric flask before and after it is filled with the liquid. To measure a crystal's density, the pycnometer is first calibrated and weighed, and then the crystal sample, from which all external liquid has been removed, is introduced. The pycnometer is now reweighed, thus determining the crystal's weight. Next, liquid of known density is added, and the pycnometer is reweighed. The crystal volume is derived from the difference in volumes of the pycnometer with and without the crystal present. The method requires direct measurement of the crystal's weight, yet it is difficult to make microbalances with sensitivity and accuracy limits better than about 0.01 mg. Micropycnometry methods have been developed to determine mineral densities with as little as 5 mg of material (Syromyatnikov, 1935[link]), but typical macromolecular crystals are 1000 times smaller than that. Volumenometry

| top | pdf |

This technique measures the increase in gas pressure due to changes in the volume of a calibrated container into which the crystal, having previously been weighed, is introduced (Reilly & Rae, 1954[link]). Almost any gas can be used for this technique if it is compatible with the apparatus and does not interact with the crystal. The empty, calibrated chamber is pressurized by adding a measured gas bolus, and this pressure is measured. After it is weighed, the crystal sample is placed in the container, which is repressurized by adding the same volume of gas. The difference in pressures between the two measurements is due to the change in volume of the container due to the crystal's presence. In principle, this method is compatible with powdered crystal samples, multiple crystals, or irregularly shaped crystals. As with micropycnometry, this technique is not appropriate for macromolecular crystals. The crystal must be free of external solvent, yet not dried, and the microbalance must be able to measure the crystal's weight precisely and accurately. The useful lower limit of crystal size for this technique, reported by Richards & Lindley in Chapter 3.2[link] of IT C (2004[link]), is 0.01 ml. The method of Archimedes

| top | pdf |

Known for thousands of years, this method measures the difference in weight of an object in air and in a liquid of known density. The difference divided by the liquid density gives the object's volume. The crystal is suspended by a vertical fibre or wire from a microbalance as it is dipped into the liquid. The surface tension of the liquid acting on the supporting fibre must be accounted for and corrected. The accuracy of this method improves as the density of the liquid used approaches that of the crystal. This method has been used with crystals as small as 25 mg (Berman, 1939[link]; Graubner, 1986[link]), but it does not lend itself well to density measurements of objects as small as macromolecular crystals. Immersion microbalance

| top | pdf |

Barbara Low and Fred Richards developed this ingenious method, which permits the crystal to be weighed in a liquid environment. A microbalance (consisting of a thin horizontal quartz fibre, free at one end) is kept entirely within the liquid bath. Its vertical deflection, observed with a microscope, is initially calibrated as a function of weight (Low & Richards, 1952b[link]; Richards, 1954[link]). The density of the liquid can be determined with high precision by standard techniques. Each crystal's volume (in the 1952–1954 studies) was calculated from two orthogonal photomicrographs (this required that the crystal morphology be regular). The crystal density can then be calculated: [\rho_{c} = \rho_{\rm liquid} + {\hbox{apparent crystal weight} \over \hbox{crystal volume}}. \eqno(]

The easiest and most accurate part of the method is measuring the liquid density. Therefore, experimental error in determining crystal weight and volume can be minimized by using a liquid with a density close to that of the crystal. In the limit where the crystal and liquid densities are the same, this method is equivalent to the flotation method – the fibre deflection is zero and the accuracy of the crystal-density measurement should be high. As originally implemented, the method is useful only for crystals with simple shapes, for which orthogonal photomicrographs can yield good estimates for the volume. Perhaps the method might be generalized if the tomographic volume-measuring method were adopted, as described by Kiefersauer et al. (1996[link]). Richards & Lindley in Chapter 3.2[link] of IT C (2004[link]) state that the method is only suitable for large crystals (volumes of 0.1 mm3 or greater). Flotation

| top | pdf |

The crystal must first be wiped completely free of external liquid and then immersed in a mixture of organic solvents, the density of which is adjusted (by addition of denser or lighter solvents) until the crystal neither rises nor sinks. Note that if the liquid used were aqueous, the crystal density would change as the surrounding liquid density is changed (e.g. by adding salt), since the crystal's free-solvent compartment would exchange with the external liquid. In this case, the equilibrium density, [\rho_{e}], is a function only of the hydration number, w, and the macromolecule's partial specific volume, [\overline{\upsilon}_{m}]: [\rho_{e} = (w + 1)/(w + \overline{\upsilon}_{m}). \eqno(] [\rho_{e}] is about [1.25 \hbox{ g ml}^{-1}] for all protein crystals, regardless of packing arrangements or molecular weights, since [w \simeq 0.25] and [\overline{\upsilon}_{m} \simeq 0.74\hbox{ cm}^{3}\hbox{ g}^{-1}].

When the crystal just floats, the liquid's density (which now equals the crystal density) can be measured by standard techniques with high accuracy. Flotation measurements can be made with small samples (Bernal & Crowfoot, 1934[link]) and with slurries of microcrystals. Centrifugation should be used to accelerate the crystal settling rate each time the liquid density is altered. The method can be tedious, so its practitioners rarely achieve an accuracy better than 0.2–1.0% (Low & Richards, 1952a[link]). Tomographic crystal-volume measurement

| top | pdf |

Recently, a new method for density measurement which is specific for protein crystals has been reported (Kiefersauer et al., 1996[link]). The crystal volume is calculated tomographically from a set of optical-shadow back projections of the crystal, with the crystal in many ([\gt\! 30]) orientations. This measurement is analogous to methods used in electron microscopy (Russ, 1990[link]). The crystal is mounted on a thin fibre which is in turn mounted on a goniostat capable of positioning it in many angular orientations. The crystal must remain bathed in a humidity-regulated air stream to avoid drying. The uncertainty of the volume measurement improves asymptotically as the number of orientations increases (estimated to be 10–15%). The images are captured by a digital charge-coupled device camera, transferred to a computer and processed with the program package EM (Hegerl & Altbauer, 1982[link]). This same crystal must then be recovered and subjected to quantitative amino-acid analysis (the authors used a Beckman 6300 amino-acid analyser). With a lower limit of 100 pmol for each amino acid, the uncertainty of this measurement was estimated to be 10–20% for typical protein crystals. The method appears to work for crystals with volumes ranging between 4–50 nl. Errors in the determined values of n ranged from 4–30%.

Implementation of the method requires complex equipment and considerable commitment (in terms of hardware and software) by the research laboratory. The accuracy of the method is sufficient to determine n unambiguously in many cases, but it is not as high as can be obtained with gradient-tube or flotation methods if care is taken. The method has the virtue that once established in a research laboratory, it might lend itself to a considerable degree of automation, thereby reducing the activation barrier to measuring crystal densities for members of the research group. Gradient-tube method

| top | pdf |

This is the most commonly used method for measuring densities of macromolecular crystals. It is simple and inexpensive to implement. It can be used to measure densities of very small crystals and crystalline powders. Practised with care, the gradient-tube method is capable of measuring crystal densities with a precision and accuracy of [\pm 0.002 \hbox{ g ml}^{-1}].

Although density gradients were used earlier for other purposes, the application of the gradient-tube method for crystal-density measurement was first described by Low & Richards (1952a[link]). The gradient can be formed in a long glass column (preferably with volume markings, such as a graduated cylinder), in which case the crystal sample will settle by gravity in the tube; or in a transparent centrifuge tube, in which case the crystal's approach to its equilibrium density may be accelerated by centrifugation. The gradient may be made by two organic liquids (Table[link]) with different densities, or it may be made by a salt concentration gradient in water (Table[link]). In either case, formation of the gradient is simplified with a standard double-chamber `gradient maker' – however, a glass gradient maker should be used if the gradient is made of organic solvents! Be aware that all these substances are toxic, particularly to the liver, and some are listed as carcinogens, so avoid prolonged exposure.

Table| top | pdf |
Organic liquids for density determinations

NameDensity (g ml−1)
Carbon tetrachloride (tetrachloromethane), CCl4 1.5940
Bromobenzene, CH5Br 1.4950
Chloroform (trichloromethane), CHCl3 1.4832
Methylene chloride (dichloromethane), CH2Cl2 1.3266
Chlorobenzene, CH5Cl 1.1058
Benzene, CH6 0.8765
m-Xylene, 1,3-(CH3)2C6H4 0.8642
Iso-octane (2-methylheptane), C5H11CH(CH3)2 0.6980

Table| top | pdf |
Inorganic salts for density determinations

The densities are approximate values for aqueous solutions at 20 °C.

SoluteDensity (g ml−1)
Sodium chloride 1.20
Potassium tartrate 1.40
Potassium iodide 1.63
Iron(III) sulfate 1.80
Zinc bromide 2.00
Zinc iodide 2.39

Desired upper and lower density limits for the gradient can be made by mixing two of these liquids in appropriate ratios. The sensitivity and resolution of the measurement can be enhanced by using a shallow gradient covering the expected density. These organic liquids have a nontrivial capacity to dessicate the crystal sample, so it is important that they be water-saturated before use. Also, when an alcohol is the precipitant of a crystal, organic solutions may be inappropriate for density measurements.

For aqueous gradients, the salts listed in Table[link] may be added to water to create a dense liquid.

A widely used variant of the method has been to form aqueous gradients with Ficoll, a sucrose polymer cross-linked with epichlorhydrin (Westbrook, 1976[link], 1985[link]; Bode & Schirmer, 1985[link]). Manufactured by the Pharmacia Corporation specifically for making density gradients used in the separation of intracellular organelles or intact cells, Ficoll is a large polymer ([M_{r} = 400\ 000]) which is very hydrophilic and soluble, and has chemical properties similar to sucrose. Since it is highly cross-linked, each Ficoll molecule tends to be globular and is so large that it is effectively excluded from the crystal. Ficoll precipitates protein from solution on a per-weight basis as effectively as polyethylene glycol and can prevent protein crystals from dissolving, even in the absence of other solutes. A 60% ([w/w]) solution of Ficoll has a density of about [1.26 \hbox{ g ml}^{-1}], sufficiently dense that almost all protein crystals will float in this solution (nucleic acid crystals are usually too dense for Ficoll). Used with care (see below), Ficoll gradients seem to yield the most reproducible crystal-density measurements. Concentrated Ficoll solutions are quite viscous, so these gradients are usually made by manually overlaying small volumes (0.5 ml each) of decreasing density, rather than with a gradient maker. In a standard cellulose nitrate centrifuge tube of about 5 ml capacity, this procedure makes an almost continuous gradient which works satisfactorily.

The density column must be calibrated once it has been formed. This is performed by introducing small items of known density into the column and noting their vertical positions. The density of the gradient as a function of vertical position can then be defined by interpolating between adjacent calibrated points. Usually, the calibrating points are made from small drops of immiscible liquid. Thus, in an organic solvent gradient, the drops are made of salt water; in an aqueous gradient, the drops are made of mixed organics (previously saturated with water). To make each calibration drop, a solution is made up with approximately the desired density, and its exact density ([\pm 0.002 \hbox{ g ml}^{-1}]) is measured pycnometrically or by refractive index (Midgley, 1951[link]). The drops can be inserted into the gradient with a flame-narrowed Pasteur pipette (this takes practice). Once calibrated, these gradients tend to be extremely stable over many months.

With an organic liquid gradient, two methods have been used to introduce the crystal sample to be measured. It can be extracted from its mother liquor with a pipette and extruded onto filter paper, which wicks away all exterior aqueous liquid. When free of moisture, but before it dessicates, the crystal must be shaken, flipped, or scraped onto the gradient top surface and allowed to sink to its equilibrium position. The second method involves injection of the crystal sample, in an aqueous droplet, into the gradient solution with a Pasteur pipette. A very thin syringe (home-made or commercial) is then used to draw off all extraneous liquid, while the crystal remains submerged in the organic liquid. Either method requires considerable manual dexterity and practice, especially with very small crystals. A significant advantage of Ficoll and aqueous salt gradients is that the crystal does not need to be manipulated at all: any liquid surrounding the crystal, which was introduced into the gradient at the start, rapidly dilutes into the aqueous solution and does not appear to interfere with further measurements.

With very small crystals, the approach to equilibrium is so slow that it is wise to use centrifugation, especially if it is suspected that the density is changing with time (see below). Nitrocellulose centrifuge tubes compatible with swinging-bucket rotors are typically 1 cm diameter, 5 cm long cylinders and are suitably transparent for this work. Centrifugation at 2500–5000 r.p.m. for as little as five minutes is sufficient for most crystals to reach a stable position in the gradient. It can be difficult to find the crystal after centrifugation, so the one or two most likely density values should be calculated in advance, and looked for first. The positions of calibration drops and of crystals in these centrifuge tubes can be measured with a hand-held ruler to a resolution of about 0.5 mm.

Particularly for crystals with high values of [V_{M}] (i.e., loosely packed) or for crystals of large molecular weight proteins, the apparent crystal density may increase with time: the crystal continues to sink and there is no apparent equilibrium spot. This behaviour is seen in both organic solvent gradients and Ficoll gradients, and the reasons for it are unclear. It may be that, in organic solvent gradients, some of the solvent can dissolve into the crystal; or the crystal may condense from slow dessication. In Ficoll gradients, it may be that sucrose monomers or dimers are present, which diffuse into the crystal over time. A careful study of this behaviour (Bode & Schirmer, 1985[link]) in Ficoll gradients suggested that useful density values can still be obtained for these crystals by fitting the apparent density to an exponential curve: [\rho_{c} (t) = a + b \exp (- \lambda t). \eqno(] In this expression, parameters a, b and λ must be derived from the fitted curve. The crystals were inserted into the gradient with flame-narrowed Pasteur pipettes. Each crystal was initially surrounded by a small amount of mother liquor, which rapidly diffused into the Ficoll solution. Time zero was assigned as the time when centrifugation first began. It was necessary to observe crystal positions within the first minute, and at two- to five-minute intervals thereafter, to obtain a reasonable time curve for the density function. The experimental goal in the Bode & Schirmer experiment was to obtain a good estimate for the density value at time zero, [\rho_{c} (0) = a + b]. This was realized in all six of the crystal forms that manifested time-dependent density drift in the study.


Berman, H. (1939). A torsion microbalance for the determination of specific gravities of minerals. Am. Mineral. 24, 434–440.
Bernal, J. D. & Crowfoot, D. (1934). Use of the centrifuge in determining the density of small crystals. Nature (London), 134, 809–810.
Bode, W. & Schirmer, T. (1985). Determination of the protein content of crystals formed by Mastigocladus laminosus C-phycocyanin, Chroomonas spec. phycocyanin-645, and modified human fibrinogen using an improved Ficoll density gradient method. Biol. Chem. Hoppe–Seyler, 366, 287–295.
Graubner, H. (1986). Densitometer for absolute measurements of the temperature dependence of density, partial volumes, and thermal expansivity of solids and liquids. Rev. Sci. Instrum. 57, 2817–2826.
Hegerl, R. & Altbauer, A. (1982). The EM program system. Ultramicroscopy, 9, 109–116.
Kiefersauer, R., Stetefeld, J., Gomis-Rüth, F. X., Romão, M. J., Lottspeich, F. & Huber, R. (1996). Protein-crystal density by volume measurement and amino-acid analysis. J. Appl. Cryst. 29, 311–317.
Low, B. W. & Richards, F. M. (1952a). The use of the gradient tube for the determination of crystal densities. J. Am. Chem. Soc. 74, 1660–1666.
Low, B. W. & Richards, F. M. (1952b). Determination of protein crystal densities. Nature (London), 170, 412–415.
Midgley, H. G. (1951). A quick method of determining the density of liquid mixtures. Acta Cryst. 4, 565.
Reilly, J. & Rae, W. N. (1954). Determination of the densities of solids by voluminometry. In Physico-chemical methods, Vol. 1, pp. 577–608. New York: van Nostrand.
Richards, F. M. (1954). A microbalance for the determination of protein crystal densities. Rev. Sci. Instrum. 24, 1029–1034.
Richards, F. M. & Lindley, P. F. (2004). Determination of the density of solids. In International tables for crystallography, Vol. C. Mathematical, physical and chemical tables, edited by E. Prince, ch. 3.2. Dordrecht: Kluwer Academic Publishers.
Russ, J. C. (1990). Computer-assisted microscopy: the measurement and analysis of images. New York: Plenum Press.
Syromyatnikov, F. V. (1935). The micropycnometric method for the determination of specific gravities of minerals. Am. Mineral. 20, 364–370.
Westbrook, E. M. (1976). Characterization of a hexagonal crystal form of an enzyme of steroid metabolism, D5–3-ketosteroid isomerase: a new method of crystal density measurement. J. Mol. Biol. 103, 659–664.
Westbrook, E. M. (1985). Crystal density measurements using aqueous Ficoll solutions. Methods Enzymol. 114, 187–196.

to end of page
to top of page