Section 5.2.6.7. Gradient-tube method

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 .

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). 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 5.2.6.1) with different densities, or it may be made by a salt concentration gradient in water (Table 5.2.6.2). 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.

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 5.2.6.2 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, 1985; Bode & Schirmer, 1985). 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 () 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% () solution of Ficoll has a density of about , 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 () is measured pycnometrically or by refractive index (Midgley, 1951). 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 (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) in Ficoll gradients suggested that useful density values can still be obtained for these crystals by fitting the apparent density to an exponential curve: 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, . This was realized in all six of the crystal forms that manifested time-dependent density drift in the study.