## Estimation of Frictional Coefficient

The frictional coefficient (f/fo) relates the hydro-dynamic volume of a protein molecule to its molecular mass. According to Siegel and Monty, the Stokes radius (RS) of a protein is related to its molecular mass (Mr) by the following equation:

where RS (m) is the Stokes radius, f/fo is the frictional coefficient (equivalent to the quotient of the half axes of a rotational ellipsoid), v (m3g_1) is the partial specific volume (the reciprocal of the average density of a protein, (v = 0.75 x 10~6), NA (mol_1) is Avogadro's number (NA = 6.022 x 1023), and Mr (Da =gmol~1) is the molecular mass of a protein. By substituting the actual values one obtains:

The geometric mean radius of a molecular mass equivalent sphere is defined as Rm (m). It is obtained

Figure 11 Plot of transformed migration distances (ln(ln D)) against transformed migration times (r1/2) of (A) marker proteins and (B) five carbonic anhydrase variants. Migration distances and times of electrophoresis as listed in Table 2. Abbreviations as in Figure 10. The common point of intersection of the various straight lines marked I-V on the ln(ln D) axis indicates that the investigated enzymes are size isomers. Reproduced with permission from Rothe (1991).

Figure 11 Plot of transformed migration distances (ln(ln D)) against transformed migration times (r1/2) of (A) marker proteins and (B) five carbonic anhydrase variants. Migration distances and times of electrophoresis as listed in Table 2. Abbreviations as in Figure 10. The common point of intersection of the various straight lines marked I-V on the ln(ln D) axis indicates that the investigated enzymes are size isomers. Reproduced with permission from Rothe (1991).

This means that RS and Rm are interrelated through the frictional coefficient:

The frictional coefficient can be obtained from the experimentally obtained Mr and RS values and eqn [13].

Extremely high frictional ratios are to be expected for molecules with rod-like or fibrous structures, which are characterized by a high axial ratio such as fibrinogen or myosin or by bulky and voluminous globular molecules with normal axial ratios. Examples of the latter are the spider-like immunoglobulin M, the shell-like apoferritin or the branched a-macro-globulin. Usually, native proteins and enzymes do not belong to these groups of proteins.

In eqn [11] the frictional coefficient of native proteins is assumed to be constant. However, when analysing the molecular mass (Mr) and Stokes radius (RS) of more than 60 native proteins it became apparent that the frictional coefficient increases with increasing protein size (see Further Reading). A more precise equation relating RS and Mr is the following:

According to this expression the frictional coefficient of globular proteins equals f/fo = M<0'0225 and increases with molecular masses of 103 to 9 x 106 from f/fo = 1.17 to f/fo = 1.43 while the factor 66 x 10~12 of the expression of Siegel and Monty (RS (nm) = f/fo x 66.1 x 10~12 x M1/3) increases from 61 x 10~12 to 67 x 10~12.

As an average, the frictional ratio of globular proteins sized 45-100 kDa is f/fo = 1.23, for those in the range of 100-500 kDa f/fo = 1.28 and in the range of

Table 2 Time-dependent migration distances of marker proteins and carbonic anhydrase (iso)enzymes from erythrocytes of four mammalian species in a porosity gradient gel from 4 to 30% T

Protein Time t (h) ofelectrophoresis (1 /J tgiven in brackets)

(1. 41421) (1. GGGGG) (G. 7G711) (G. 5GGGG) (G.35355) (G. 2SSßS) (G. 25GGG) (G.22361)

Ovalbumin |
D (mm) |
13.05 |
20.25 |
31.5 |
44.Q |
54.Q |
61.0 |
67.5 | |

Bovine serum albumin D (mm) |
11.7 |
17.S |
26.5 |
36.3 |
43.5 |
47.5 |
5Q.5 |
53.2 | |

l-lactate |
D (mm) |
7.5 |
11.9 |
17.5 |
24.5 |
3Q.2 |
33.3 |
35.5 |
37.5 |

dehydrogenase | |||||||||

Catalase |
D (mm) |
5.5 |
S.S |
13.2 |
^ |
23.5 |
26.6 |
2S.5 |
3Q.Q |

Ferritin |
D (mm) |
3.7 |
6.5 |
9.Q |
12.Q |
14.3 |
16.6 |
17.9 |
1S.9 |

Thyroglobulin |
D (mm) |
1.9 |
3.5 |
4.7 |
6.5 |
S.Q |
1Q.Q |
1Q.S |
11.6 |

Bovine I |
D (mm) |
7.3 |
12.5 |
21.5 |
35.5 |
4S.2 |
56.0 |
62.5 | |

Bovine II |
D (mm) |
6.3 |
11.Q |
19.Q |
32.5 |
45.Q |
52.3 |
5S.S |
6S.0 |

Bovine, rabbit III |
D (mm) |
5.Q |
s.S |
15.5 |
27.5 |
4Q.1 |
47.5 |
52.Q |
5S.Q |

Rabbit IV |
D (mm) |
3.S |
6.7 |
11.S |
21.5 |
33.6 |
41.Q |
45.7 |
5Q.2 |

Canine, Human V |
D (mm) |
3.5 |
6.3 |
11.2 |
2Q.Q |
32.5 |
39.S |
44.5 |
4S.5 |

D(mm), Time-dependent migration distances of marker proteins and carbonic anhydrase (EC 4.2.1.1) variants. Gel length (D (mm)) and gel concentration (T(%)) are interrelated by the equation T= aD# ) where a = 0.3528 + 0.0054 and 0 = 4.1116 + 0.2344; the correlation coefficient is r= 0.9985. Reproduced with permission from Rothe (1991).

500-1000 kDa f/f0 = 1.43. From these data and the Stokes radius of a globular protein its molecular mass can be estimated:

This can be exemplified by mammalian liver alcohol dehydrogenase (EC 1.1.1.1), which has a molecular mass of 80 kDa and a Stokes radius of 3.5 nm; the average frictional coefficient of globular proteins in that range is f/fo = 1.23. By inserting these values into eqn [15] one obtains: Mr (Da) = (1/1.23)3 x 3463 x 3.53 = 79 791.

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