Size and Shape of Protein Molecules at the Nanometer Level Determined by Sedimentation, Gel Filtrati

An important part of characterizing any protein molecule is to determine its size and shape. Sedimentation and gel filtration are hydrodynamic techniques that can be used for this medium resolution structural analysis. This review collects a number of simple calculations that are useful for thinking about protein structure at the nanometer level. Readers are reminded that the Perrin equation is generally not a valid approach to determine the shape of proteins. Instead, a simple guideline is presented, based on the measured sedimentation coefficient and a calculated maximum S , to estimate if a protein is globular or elongated. It is recalled that a gel filtration column fractionates proteins on the basis of their Stokes radius, not molecular weight. The molecular weight can be determined by combining gradient sedimentation and gel filtration, techniques available in most biochemistry laboratories, as originally proposed by Siegel and Monte. Finally, rotary shadowing and negative stain electron microscopy are powerful techniques for resolving the size and shape of single protein molecules and complexes at the nanometer level. A combination of hydrodynamics and electron microscopy is especially powerful. Key Words: Protein shape - hydrodynamics - gel filtration - sedimentation - electron microscopy Introduction

Most proteins fold into globular domains. Protein folding is driven largely by the hydrophobic effect, which seeks to minimize contact of the polypeptide with solvent. Most proteins fold into globular domains, which have a minimal surface area. Peptides from 10 to 30 kDa typically fold into a single domain. Peptides larger than 50 kDa typically form two or more domains that are independently folded. However, some proteins are highly elongated, either as a string of small globular domains or stabilized by specialized structures such as coiled coils or the collagen triple helix. The ultimate structural understanding of a protein comes from an atomic-level structure obtained by X-ray crystallography or nuclear magnetic resonance. However, structural information at the nanometer level is frequently invaluable. Hydrodynamics, in particular sedimentation and gel filtration, can provide this structural information, and it becomes even more powerful when combined with electron microscopy (EM).

One guiding principle enormously simplifies the analysis of protein structure. The interior of protein subunits and domains consists of closely packed atoms (1 ). There are no substantial holes and almost no water molecules in the protein interior. As a consequence of this, proteins are rigid structures, with a Young’s modulus similar to that of Plexiglas (2 ). Engineers sometimes categorize biology as the science of “soft wet materials”. This is true of some hydrated gels, but proteins are better thought of as hard dry plastic. This is obviously important for all of biology, to have a rigid material with which to construct the machinery of life. A second consequence of the close packed interior of proteins is that all proteins have approximately the same density, about 1.37 g/cm3 . For most of the following, we will use the partial specific volume, v 2 , which is the reciprocal of the density. v 2 varies from 0.70 to 0.76 for different proteins, and there is a literature on calculating or determining the value experimentally. For the present discussion, we will ignore these variations and assume the average v 2  = 0.73 cm3 /g.

How Big Is a Protein Molecule? Assuming this partial specific volume (v 2  = 0.73 cm3 /g), we can calculate the volume occupied by a protein of mass M in Dalton as follows.

The inverse relationship is also frequently useful: M (Da) = 825 V (nm3 ).

What we really want is a physically intuitive parameter for the size of the protein. If we assume the protein has the simplest shape, a sphere, we can calculate its radius. We will refer to this as R min , because it is the minimal radius of a sphere that could contain the given mass of protein

Some useful examples for proteins from 5,000 to 500,000 Da are given in Table   1 . Table 1  R min for proteins of different mass

It is important to emphasize that this is the minimum radius of a smooth sphere that could contain the given mass of protein. Since proteins have an irregular surface, even ones that are approximately spherical will have an average radius larger than the minimum.

How Far Apart Are Molecules in Solution?

It is frequently useful to know the average volume of solution occupied by each molecule, or more directly, the average distance separating molecules in solution. This is a simple calculation based only on the molar concentration.

In a 1-M solution, there are 6 × 1023  molecules/l, = 0.6 molecules/nm3 , or inverting, the volume per molecule is V  = 1.66 nm3 /molecule at 1 M. For a concentration C , the volume per molecule is V =  1.66/C .

We will take the cube root of the volume per molecule as an indication of the average separation.

where C is in molar and d is in nanometer. Table   2 gives some typical values. Table 2  Distance between molecules as function of concentration

Two interesting examples are hemoglobin and fibrinogen. Hemoglobin is 330 mg/ml in erythrocytes, making its concentration 0.005 M. The average separation of molecules (center to center) is 6.9 nm. The diameter of a single hemoglobin molecule is about 5 nm. These molecules are very concentrated, near the highest physiological concentration of any protein (the crystallins in lens cells can be at >50% protein by weight).

Fibrinogen is a large rod-shaped molecule that forms a fibrin blood clot when activated. It circulates in plasma at a concentration of around 2.5 g/l, about 9 μM. The fibrinogen molecules are therefore about 60 nm apart, comparable to the 46-nm length of the rod-shaped molecule.

The Sedimentation Coefficient and Frictional Ratio. Is the Protein Globular or Elongated?

Biochemists have long attempted to deduce the shape of a protein molecule from hydrodynamic parameters. There are two major hydrodynamic methods that are used to study protein molecules―sedimentation and diffusion (or gel filtration, which is the equivalent of measuring the diffusion coefficient).

The sedimentation coefficient, S , can be determined in an analytical ultracentrifuge. This was a standard part of the characterization of proteins in the 1940s and 1950s, and values of S 20,w (sedimentation coefficient standardized to 20°C in water) are collected in references such as the Chemical Rubber Co. (CRC) Handbook of Biochemistry (3 ). Today, S is more frequently determined by zone sedimentation in a sucrose or glycerol gradient, by comparison to standard proteins of known S . Five to twenty percent sucrose gradients have been most frequently used, but we prefer 15�40% glycerol gradients in 0.2 M ammonium bicarbonate, because this is the buffer used for rotary shadowing EM (Section   6 ). The protein of interest is sedimented in one bucket of the swinging bucket rotor, and protein standards of known S (Table   5 ) are sedimented in a separate (or sometimes the same) gradient. Following sedimentation, the gradient is eluted into fractions and each fraction is analyzed by sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE) to locate the standards and the test protein. Figure 1 shows an example determining the sedimentation coefficient of the structural maintenance of chromosome (SMC) protein from Bacillus subtilis . Fig. 1  Glycerol gradient sedimentation analysis of SMC protein from B. subtilis (BsSMC ; upper panel ) and sedimentation standards catalase and bovine serum albumin (lower panel ). A 200-μl sample was layered on a 5.0-ml gradient of 15�40% glycerol in 0.2 M ammonium bicarbonate and centrifuged in a Beckman SW55.1 swinging bucket rotor, 16 h, 38,000 rpm, 20°C. Twelve fractions of 400 μl each were collected from a hole in the bottom of the tube and each fraction was run on SDS-PAGE. Lane SM shows the starting material, and fraction 1 is the bottom of the gradient. The bottom panel shows that the 11.3-S catalase eluted precisely in fraction 4, while the 4.6-S BSA eluted mostly in fraction 8, with some in fraction 9. We estimated the BSA to be centered on fraction 8.2. Experiments with additional standard proteins have demonstrated that the 15�40% glycerol gradients are linear over the range 3�20 S, so a linear interpolation is used to determine S of the unknown protein. BsSMC is in fractions 7 and 8, estimated more precisely at fraction 7.3. Extrapolating from the standards, we determine a sedimentation coefficient of 6.0 S for BsSMC. Other experiments gave an average value of 6.3 S for BsSMC (19 ). 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:

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 H2 O); η is the viscosity of the solvent (0.01 g/cm−s for H2 O).

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 R s , 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 R min  =  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):

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 H2 O/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 Å. 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

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

We have surveyed values of S max /S for a variety of proteins of known structure. Table   4 presents S max /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.

Table 4  S max /S values for representative globular and elongated proteins