The sedimentation coefficient of a protein is a measure of how fast it moves through the gradient. Increasing the mass of the protein will increase its sedimentation, while increasing its size or asymmetry will decrease its sedimentation. The relationship of S to size and shape of the protein is given by the Svedberg formula:
| (4.1) |
M is the mass of the protein molecule in Dalton; N o is Avogadro’s number, 6.023?×?1023; v 2 is the partial specific volume of the protein; typical value is 0.73 cm3/g; ρ is the density of solvent (1.0 g/cm3 for H2O); η is the viscosity of the solvent (0.01 g/cm?s for H2O).
A critical factor in the equation is the frictional coefficient, f (dimensions gram per second) which depends on both the size and shape of the protein. For a given mass of protein (or given volume), f will increase as the protein becomes elongated or asymmetrical (f can be replaced by an equivalent expression containing Rs, the Stokes radius, to be discussed later). S has the dimensions of time (seconds). For typical protein molecules, S is in the range of 2–20?×?10?13 s, and the value 10?13 s is designated a Svedberg unit, S. Thus, typical proteins have sedimentation coefficients of 2–20 S.
From the above definition of parameters, it is clear that S depends on the solvent and temperature. In classical studies, the solvent-dependent factors were eliminated and the sedimentation coefficient was extrapolated to the value it would have at 20°C in water (for which ρ and η are given above). This is referred to as S 20,w. In the present treatment, we will be referring mostly to standard proteins that have already been characterized, or unknown ones that will be referenced to these in gradient sedimentation, so our use of S will always mean S 20,w.
A useful concept is the minimum value of f, which would obtain if the given mass of protein were packed into a smooth unhydrated sphere. As we have discussed in Section 1 , the radius of this sphere will be Rmin ?=?0.066 M 1/3 (Eq. 2.2 ). In about 1850, G. G. Stokes calculated theoretically the frictional coefficient of a smooth sphere (note that the equation is similar to that for the Stokes radius, to be discussed later, but the parameters here are different):
| (4.2) |
We have now designated f min as the minimal frictional coefficient for a protein of a given mass, which would obtain if the protein were a smooth sphere of radius R min .
The actual f of a protein will always be larger than f min because of two things. First, the shape of the protein normally deviates from spherical, to be ellipsoidal or elongated; closely related to this is the fact that the surface of the protein is not smooth but rather rough on the scale of the water molecules it is traveling through. Second, all proteins are surrounded by a shell of bound water, one–two molecules thick, which is partially immobilized or frozen by contact with the protein. This water of hydration increases the effective size of the protein and thus increases f.
The Perrin Equation Does Not Work for Proteins
If one could determine the amount of water of hydration and factor this out, there would be hope that the remaining excess of f over f min could be interpreted in terms of shape. Algorithms have been devised for estimating the amount of bound water from the amino acid sequence, but these generally do not distinguish between buried residues, which have no bound water and surface residues which bind water. Some attempts have been made to base the estimate of bound water based on polar residues, which are mostly exposed on the surface. A 0.3-g H2O/g protein is a typical estimate, but in fact, this kind of guess is almost useless for analyzing f.
In the older days, when there was some confidence in these estimates of bound water, physical chemists calculated a value called f o, which was the frictional coefficient for a sphere that would contain the given protein, but enlarged by the estimated shell of water (other authors use f o to designate what we term f min(3, 4); we recommend using f min to avoid ambiguity). The measured f for proteins was almost always larger than f o, suggesting that the protein was asymmetrical or elongated. A very popular analysis was to model the protein as an ellipsoid of revolution and calculate the axial ratio from f/f o, using an equation first developed by Perrin. This approach is detailed in most classical texts of physical biochemistry. In fact, the Perrin analysis always overestimates the asymmetry of the proteins, typically by a factor of two to five. It should not be used for proteins.
The problem is illustrated by an early collaborative study of phosphofructokinase, in which the laboratory of James Lee did hydrodynamics and our laboratory did EM (5). We found by EM that the tetrameric particles were approximately cylinders, 9 nm in diameter and 14 nm long. The shape was therefore like a rugby ball, with an axial ratio of 1.5 for a prolate ellipsoid of revolution. The Lee group measured the molecular weight and sedimentation coefficient, determined f and estimated water of hydration and f o. They then used the Perrin equation to calculate the axial ratio. The ratio was five, which would suggest that the protein had the shape of a hot dog. The EM structure (which was later confirmed by X-ray crystallography) shows that the Perrin equation overestimated the axial ratio by a factor of 3.
Teller et al. (6) summarized the situation: “Frequently the axial ratios resulting from such treatment are absurd in light of the present knowledge of protein structure.” They explained that the major problem with the Perrin equation is that it treats the protein as a smooth ellipsoid, when in fact the surface of the protein is quite rough. Teller et al. went on to show how the frictional coefficient can actually be derived from the known atomic structure of the protein, by modeling the surface of the protein as a shell of small beads of radius 1.4 ?. The shell coated the surface of the protein, modeling its rugosity, and increasing the size of the protein by the equivalent of a single layer of bound water. This analysis has been extended by Garcia De La Torre and colleagues (7).
Interpreting Shape from f/f min?=?S max/S
If the Perrin equation is useless, is there some other way that shape can be interpreted from f? The answer is yes, at a semiquantitative level. We have discovered simple guidelines where the ratio f/f min can provide a good indication of whether a protein is globular, somewhat elongated, or very elongated.
Instead of proceeding with the classical ratio f/f min, where f is in nonintuitive units, we will reformulate the analysis directly in terms of the sedimentation coefficient, which is the parameter actually measured. We will define a value S max as the maximum possible sedimentation coefficient, corresponding to f min . S max is the S value that would be obtained if the protein were a smooth sphere with no bound water. These two ratios are equal: f/f min?=?S max /S. Combining Eqs. 2.2 , 4.1 , and 4.2 , we have
| (4.3a) |
| (4.3b) |
The leading factor of 1013 in Eq. 4.3a converts S max to Svedberg units. The numbers in brackets in Eq. 4.3a are calculated using v 2?=?0.73 cm3/g, ρ?=?1.0 g/cm3, η ?=?0.01 g cm?1 s?1?=?10?9 g nm?1 s?1. The final expression, Eq. 4.3b expresses S max in Svedbergs for a protein of mass M in Daltons. Some typical numerical values of S max for proteins from 10,000 to 1,000,000 Da are given in Table 3 .
Table 3 S max calculated for proteins of different mass
Protein M r (kDa) | 10 | 25 | 50 | 100 | 200 | 500 | 1,000 |
|---|---|---|---|---|---|---|---|
S max Svedbergs | 1.68 | 3.1 | 4.9 | 7.8 | 12.3 | 22.7 | 36.1 |
We have surveyed values of S max/S for a variety of proteins of known structure. Table 4 presents Smax/S for a number of approximately globular proteins and for a range of elongated proteins, all of known dimensions. It turns out that S max/S is an excellent predictor of the degree of asymmetry of a protein. From this survey of known proteins, we can propose the following general principals.
| ? | No protein has S max/S?=?f/f min smaller than ～1.2. |
| ? | For approximately globular proteins: S max/S is typically between 1.2 and 1.3. |
| ? | For moderately elongated proteins: S max/S is in the range of 1.5 to 1.9. |
| ? | For highly elongated proteins (tropomyosin, fibrinogen, extended fibronectin): S max/S is in the range of 2.0 to 3.0. |
| ? | For very long thread-like molecules like collagen, or huge extended molecules like the tenascin hexabrachion (not shown): S max/S can range from 3–4 or more. |
Table 4 S max /S values for representative globular and elongated proteins
![$$S_{{\max }} = 10^{{13}} M{\left( {1 - v_{2} \rho } \right)}/N_{{\text{o}}} {\left( {6\pi \eta R_{{\min }} } \right)} = M{\left[ {2.378\; \times \;10^{{ - 4}} } \right]}/R_{{\min }} $$](http://www.springerprotocols.com/Full/doi/10.1007/12575_2009_9008_Article_Equ6.gif)
