Size and Shape of Protein Molecules-3
The Kirkwood/Bloomfield Calculation
The understanding of how protein shape affects hydrodynamics is elegantly extended by an analysis originally developed by Kirkwood (9) and later extended by Bloomfield and Garcia De La Torres (10–12). In its simplest application, it calculates the sedimentation coefficient of a rigid oligomeric protein composed of subunits of known S and known spacing relative to each other. In more complex applications, a protein of any complex shape can be modeled as a set of nonoverlapping spheres or beads. See Byron (13) for a comprehensive review of the principals and applications of hydrodynamic bead modeling of biological macromolecules.
The basis of the Kirkwood/Bloomfield analysis is to account for how each bead shields the others from the effect of solvent flow and thereby determine the hydrodynamics of the ensemble from its component beads. Figure 2 shows a simple example of the bead modeling approach and provides an instructive look at how size and shape affect sedimentation. There are several important conclusions.
| • | A rod of three beads has about a twofold higher S than a single bead. |
| • | S max/S is 1.18 for the single bead (the effect of the assumed shell of water), 1.34 for the three-bead rod, and 1.93 for the straight 11-bead rod. This is consistent with the principals given in Section 4 for globular, somewhat elongated, and very elongated particles. |
| • | Bending the rod at 90° in the middle causes only a small increase in S. Bending it into a U-shape with the arms about one bead diameter apart increases S a bit more. Bending this same 11-bead structure more sharply, so the two arms are in contact, causes a substantial increase in S, from 5.05 to 5.58. The guiding principle is that folding affects S when one part of the molecule is brought close enough to another to shield it from water flow. |
Fig. 2 Each bead models a 10-kDa domain, with an assumed sedimentation coefficient of 1.42 S. The radius of the bead is 1.67 nm, using R min = 1.42 nm, and adding 0.25 nm for a shell of water. The beads are an approximation to FN-III or Ig domains, which are ∼1.7 × 2.8 × 3.5 nm. The sedimentation coefficients of multibead structures were calculated by the formula of Kirkwood/Bloomfield.
Gel Filtration Chromatography and the Stokes Radius
“Gel filtration chromatography is widely used for determining protein molecular weight.” This quote from Sigma-Aldrich bulletin 891A is a widely held misconception. The fallacy is obscurely corrected by a later note in the bulletin that “Once a calibration curve is prepared, the elution volume for a protein of similar shape, but unknown weight, can be used to determine the MW.” The key issue is “of similar shape”. Generally, the calibration proteins are all globular, and if the unknown protein is also globular, the calibrated gel filtration column does give a good approximation of its molecular weight. The problem is that the shape of an unknown protein is generally unknown. If the unknown protein is elongated, it can easily elute at a position twice the molecular weight of a globular protein.
The gel filtration column actually separates proteins not on their molecular weight but on their frictional coefficient. Since the frictional coefficient, f, is not an intuitive parameter, it is usually replaced by the Stokes radius R s . R s is defined as the radius of a smooth sphere that would have the actual f of the protein. This is much more intuitive since it allows one to imagine a real sphere approximately the size of the protein, or somewhat larger if the protein is elongated and has bound water.
As mentioned above for Eq. 4.2 , Stokes calculated theoretically the frictional coefficient of a smooth sphere to be:
| (6.1) |
The Stokes radius R s is larger than R min because it is the radius of a smooth sphere whose f would match the actual f of the protein. It accounts for both the asymmetry of the protein and the shell of bound water. More quantitatively, f/f min = S max/S = R s/R min.
Siegel and Monte (4) argued convincingly that the elution of proteins from a gel filtration column correlates closely with the Stokes radius, R s, presenting experimental data from a wide range of globular and elongated proteins. The Stokes radius is known for large number of proteins, including ones convenient for calibrating gel filtration columns (Table 5 ). Figure 3 shows an example where the R s of the unknown protein SMC protein from B. subtilis was determined by gel filtration.
Fig. 3 Determination of R s of BsSMC by gel filtration. The column was calibrated by running standard proteins BSA, catalase, and thyroglobulin over the column, then BsSMC. BsSMC eluted in fraction 14.2, which corresponds to an R s of 10 nm on the extrapolated curve. In repeated experiments, the average R swas 10.3 nm (19).
Table 5 Standards for hydrodynamic analysis
Protein | M r aa seq | S 20,w | S max/S | R s (nm) | Source | M r S-M |
|---|---|---|---|---|---|---|
Ribonuclease A beef pancreas | 14,044 | 2.0a | 1.05a | 1.64 | HBC | 13,791 |
Chymotrypsinogen A beef pancreas | 25,665 | 2.6 | 1.21 | 2.09 | HBC | 22,849 |
Ovalbumin hen egg | 42,910s | 3.5 | 1.27 | 3.05 | HBC | 44,888 |
Albumin beef serum | 69,322 | 4.6a | 1.33 | 3.55 | S-M, HBC | 68,667 |
Aldolase rabbit muscle | 157,368 | 7.3 | 1.45 | 4.81 | HBC | 147,650 |
Catalase beef liver | 239,656 | 11.3 | 1.21 | 5.2 | S-M | 247,085 |
Apo-ferritin horse spleen | 489,324 | 17.6 | 1.28 | 6.1 | HBC | 451,449 |
Thyroglobulin bovine | 606,444 | 19 | 1.37 | 8.5 | HBC | 679,107 |
Fibrinogen, human | 387,344 | 7.9 | 2.44 | 10.7 | S-M | 355,449 |
Gel filtration calibration kits, containing globular proteins of known molecular weight and R s, are commercially available (GE Healthcare, Sigma-Aldrich). These same proteins can be used for sedimentation standards. The proteins in these kits are included in the table along with some others that we have found useful. The values for M r given in the first column are from amino acid sequence data. Values for S 20,wand R s are from the Siegel–Monte paper (indicated S-M under source), or the CRC Handbook of Biochemistry (3) (indicated HBC). They agree with the values listed for R s in the GE Healthcare gel filtration calibration kit, with the exception of ferritin. The “M r calc” in the last column was obtained by our simplification of the Siegel–Monte calculation (M = 4,205 S R s). Note that the worst disagreement with “M raa seq” is about 10%
a S for ribonuclease A is questionable because of the low S max/S (1.05). S values for bovine serum albumin vary in the literature from 4.3 to 4.9. Many sources use 4.3, but we find that 4.6 gives a better fit with other standards (note that the standard curve in Fig. 5 used 4.3, but 4.6 would have placed it closer to the line)
The standard proteins should span R s values above and below that of the protein of interest (but in the case of SMC protein from B. subtilis, a short extrapolation to a larger value was used). The literature generally recommends determining the void and included volumes of the column and plotting a partition coefficient K AV (4). However, we have found it generally satisfactory to simply plot elution position vs R sfor the standard proteins. This generally gives an approximately linear plot, but otherwise, it is satisfactory to draw lines between the points and read the R s of the protein of interest from its elution position on this standard curve.
A gel filtration column can determine R s relative to the R s of the standard calibration proteins. The R s of these standards was generally determined from experimentally measured diffusion coefficients. Some tabulations of hydrodynamic data list the diffusion coefficient, D, rather than R s, so it is worth knowing the relationship:
| (6.2) |
where k = 1.38 × 10−16 g cm2 s−2 K−1 is Boltzman’s constant and T is the absolute temperature. k is given here in centimeter–gram–second units because D is typically expressed in centimeter–gram–second; Rs will be expressed in centimeter in this equation. Typical proteins have D in the range of 10−6 to 10−7 cm2 s−1. Converting to nanometer and for T = 300 K and η = 0.01:
| (6.3) |
where R s is in nanometer and D is in centimeter squared per second.
Simply knowing, R s is not very valuable in itself, except for estimating the degree of asymmetry, but this would be the same analysis developed above for S max/S. However, if one determines both R s and S, this permits a direct determination of molecular weight, which cannot be deduced from either one alone. This is described in the next section.
