Compressibility

Compressibility is an important measure of the atomic packing and flexibility of protein molecules. Measurement of this quantity is possible, in principle, only for proteins in solution. The partial specific compressibility of a protein in solution b is defined as the change in the partial specific volume v₂ ⁰ with increasing pressure. Since the atomic van der Waals volume, v_c, may be assumed to be incompressible, the experimentally determined b of a protein can be mainly attributed to its cavities v_cav, and to changes in hydration Dv_sol: 

The first term on the right-hand side contributes positively, and the second term negatively, to b. Thus, a positive b value can be ascribed to a large cavity effect overcoming the hydration effect. There exist two types of compressibilities, depending on the experimental conditions: adiabatic and isothermal. Most compressibility studies of proteins have been of adiabatic compressibility because of its accuracy and technical convenience.

 1. Adiabatic Compressibility

The partial specific adiabatic compressibility b_s is calculated with the Laplace equation, b_s = 1/u²d, where u is the sound velocity and d the density of the protein solution. The sound velocity is usually measured with an accuracy of 1 cm sec^–1 by using a resonance or sing-around pulse method at 3 to 6 MHz. All amino acids show large negative b_s, with values between –62.5 × 10^–12 cm2 dyn^–1 (for glycine) and –21 × 10^–12 cm² dyn^–1 (for tryptophan) at 25°C, because they have a negligibly small amount of cavity. Data for b_s have been reported for about 40 different proteins; some values in water (or in dilute buffer at neutral pH) are listed in Table 1. Fibrous proteins show negative b_s, indicating the dominant effects of the surface on hydration relative to internal cavity volume. On the contrary, most globular proteins have positive b_s, due to the large contribution of the internal cavities overcoming the hydration effect. This is consistent with the fact that more than 50% of the total surface area is buried in the interior of a folded protein molecule.

Table 1. Partial specific volume (v₂⁰), compressibility (b_s and b_T), and volume fluctuation (dV_rms) of proteins in water at 25°C (1, 4)

^a The values in parentheses represent the experimental data; the other values were calculated with Equation 2 of the text.

^b The values in parentheses represent the ratio (%) of dV_rms to the total protein volume.

Some assumptions are necessary for estimating separately the contributions of cavities and hydration to b_s. A rough estimation suggests that the intrinsic compressibility of globular proteins free from hydration is of the order of (10 to 20) × 10^–12 cm2 dyn^–1. This value is comparable to the adiabatic compressibility of normal ice, suggesting that an internal protein structure is as rigid as ice.

As can be seen in Table 1, the value of b_s varies over a wide range and is sensitive to the structural characteristics of individual proteins. For example, lysozyme has a considerably smaller b_s than does a-lactalbumin, in spite of their great similarities in primary and tertiary structures. However, the compressibility-structure relationships of proteins have been scarcely discussed on a molecular level, because of the complicated contribution of hydration. The contributions of some structural factors have been deduced by the statistical analyses of b_s data on globular proteins (Ref. 1). From the ratio of accessible surface area to volume, b_s would be expected to increase with increasing molecular weight, but there appears no definite correlation between bs and molecular weight. Instead, a positive correlation is found between b_s and the partial specific volume. Hydrophobic proteins show large positive b_s, probably due to an enhanced imperfect packing of nonpolar residues localized in the interior of the molecule. Typical a-helical proteins, such as myoglobin and bovine serum albumin have very large b_s, even though the a-helix itself is rigid, predominantly nonhelical proteins, such as trypsin and soybean trypsin inhibitor, have small b_s. These results suggest that the a-helix could be a dynamic domain for the thermal fluctuation of proteins. Four amino acids (Leu, Glu, Phe, and His( show statistically a strong ability to increase bs, whereas another four (Asn, Gly, Ser, and Thr( decrease it, although the meaning of this is obscure. A single amino acid substitution can bring about a noticeable change in b_s, probably due to modified atomic packing, even though it is rare to observe any visible changes in the tertiary structures of mutants by X-ray crystallography.

Adiabatic compressibility data for nonnative conformations and for unfolding processes, in combination with partial specific volume, also present useful information on the principles of protein structure that cannot be obtained by spectroscopic techniques. Unfolding of a protein with strong denaturants such as guanidinium chloride decreases the compressibility and specific volume, but it is known that the compressibility increases and the partial specific volume decreases on thermal and pressure denaturation.

2. Isothermal Compressibility

 The volume fluctuations and the pressure-dependent properties of proteins are theoretically related to the partial specific isothermal compressibility b_T, rather than the adiabatic one b_s. To determine b_T, the solution density or partial specific volume must be measured as a function of pressure, under hydrostatic pressure or centrifugal force. Such an experiment is very difficult, however, and few b_T data are available for proteins, because high pressure may cause protein denaturation and modify preferential solvent interactions. Approximate values of b_T may be estimated from b_s by utilizing the relationship 

if the thermal expansion coefficient a and the heat capacity at constant pressure C_p, are known or can be reasonably inferred. Such assumed b_T values are listed in Table 1, along with some experimental data. Despite many assumptions, the calculated values of b_T are not so very different from the experimentally observed ones. The value of b_T is greater than b_s by (3 to 4) × 10^-12 cm2 dyn^–1; these differences are comparable to those found for the amino acids.

 There are some other methods to evaluate the isothermal compressibility of proteins, although the value obtained is not a partial quantity in solution. Kundrot and Richard (2) reported 4.7 × 10^–12 cm² dyn^–1 for b_T of lysozyme by comparing its X-ray crystallographic volume at atmospheric pressure and 1000 atmospheres. Contraction of the molecule was distributed nonuniformly; one domain was essentially incompressible, but another had a large compressibility. Computer simulations, such as molecular dynamics, Monte Carlo, and normal mode analyses, are also available. Normal mode analysis of myoglobin yielded bT = 9.37 × 10^–12 cm² dyn^–1.

According to statistical thermodynamics, the volume fluctuation dV_p of a protein with volume V_p is related to its isothermal compressibility b_T (Ref. 3):

where k is the Boltzmann constant and T the absolute temperature. The root-mean-square fluctuation of the partial molar volume dV_rms, which was estimated by using b_T instead of b_T, is listed in the last column of Table 1. The volume fluctuation is only about 0.3% of the overall dimensions of protein, but the dV_rms values estimated by this method are the lower limit of the probable fluctuations, since the solvation factor is still included in the value of b_T that was used. If the intrinsic isothermal compressibility of a protein itself, which is not easy to determine, is used instead of b_T, a greater volume fluctuation (up to 50% greater) might be expected for a protein without hydration. Although many assumptions are used in estimating b_T and the fluctuation of volume, the values obtained are considered to be reasonable; if concentrated in one area at one particular moment, the fluctuation in volume could produce sufficient cavities or channels to allow the entry of solvent or probe molecules to account for phenomena such as hydrogen exchange and fluorescence quenching observed with folded proteins.

References

1. K. Gekko and Y. Hasegawa (1986) Biochemistry 25, 6563–6571. 

2. E. Kundrot and F. M. Richards (1987) J. Mol. Biol. 193, 157–170. 

3. A. Cooper (1976) Proc. Natl. Acad. Sci. U.S.A. 73, 2740–2741. 

4. K. Gekko (1991) In Water Relationship in Food (H. Levine and L. Slade, eds.), Plenum Press, New York, pp. 753–771.