Computation of the Nett Charge

Estimation of the number of unit charges (Z) in a nondenatured protein requires prior knowledge of its Stokes radius (RS) and its apparent free electrophoretic mobility (¿u) or its free electrophoretic mobility (U). In addition to this, the ionic strength (I) and viscosity (|) of the buffer system used to estimate Z and RS must be known. Time-dependent gradient gel electrophoresis can be used to determine the Stokes radius of a protein and its free electrophoretic mobility.

At a first approximation, the free electrophoretic mobility, unhindered by a gel matrix (U (m2 V-1 s-1)), can be described by eqn [30]:

where Z is the number of unit charges (1); s is the unit charge (protonic charge) = 1.602 x 10-19 (C); n = 3.14 ...; | is the dynamic viscosity of the medium (Pa s); RS is the Stokes radius (m) and the following coherences 1 C = 1 A s, 1 Pa = 1 N m-2, 1 V A = 1 W and 1 W s = 1 N m.

Since migration of proteins is studied in buffered solutions, there are also positive and negative buffer ions present, in addition to the protein ions. The small ions of sign opposite to that of the protein, also called counterions, are present in excess and to be found in the vicinity of the protein molecules. The electric field which drives the protein molecules also acts on the counterions, but in the opposite direction and since the migrating counterions drag solvent along with them and the solvent in turn acts on the protein, the nett effect is a secondary force on the protein opposite in direction to the primary force. The migration velocity of the protein molecules towards the electric field may therefore be reduced well below that predicted by eqn [30], an effect known as the electrophoretic effect. This is why eqn [30] must be corrected by a retardation factor (F),

Table 6 Free electrophoretic mobility (U) and net negative charge (valence, Z; charge, Q) of several marker proteins and carbonic anhydrase (iso)enzymes from mammalia at pH 8.4

(molm"3) / = 0.1 x 103b Z Q (C molecule-1) x 10"

(molm"3) / = 0.1 x 103b Z Q (C molecule-1) x 10"

Ovalbumin

3.45

5.99

13.06

20.92

Bovine serum albumin

4.40

7.85

22.42

35.92

Lactate dehydrogenase

3.27

6.00

22.63

35.25

Catalase

2.60

4.94

21.43

34.33

Ferritin

3.28

6.38

43.81

70.18

Thyroglobulin

2.78

5.62

68.46

109.67

CA I

1.58

2.69

4.93

7.90

CA II

1.17

1.99

3.58

5.74

CA III

1.05

1.79

3.32

5.32

CA IV

0.851

1.46

2.75

4.41

CA V

0.734

1.25

2.31

3.70

aIonic strength of electrophoretic buffer system.

bFree electrophoretic mobility at ionic strength 0.1 x 103 (m2 (Vs)-1).

CA, Carbonic anhydrase (iso)enzymesfrom mammalian erythrocytes: (bovine, I, II), bovine, rabbit (III), rabbit (IV) and canine, human (V). Conditions of electrophoresis: linear polyacrylamide gradient from 4 to 27% T(acrylamide-Bis =24 : 1); gel length 73 mm; buffer system 90 mmol L-1 Tris; 80 mmol L-1 boric acid; 1.25 mmol L-1 EDTA-Na2, pH 8.4 (/= 529 (mol m-3); field strength: 41 Vcm-1; 4°C.

aIonic strength of electrophoretic buffer system.

bFree electrophoretic mobility at ionic strength 0.1 x 103 (m2 (Vs)-1).

CA, Carbonic anhydrase (iso)enzymesfrom mammalian erythrocytes: (bovine, I, II), bovine, rabbit (III), rabbit (IV) and canine, human (V). Conditions of electrophoresis: linear polyacrylamide gradient from 4 to 27% T(acrylamide-Bis =24 : 1); gel length 73 mm; buffer system 90 mmol L-1 Tris; 80 mmol L-1 boric acid; 1.25 mmol L-1 EDTA-Na2, pH 8.4 (/= 529 (mol m-3); field strength: 41 Vcm-1; 4°C.

the quantity of which depends on the composition and strength of the small ions of the buffer used. Henry proposed a method for computing this factor using the formula:

where X1 is a function of kx Rs. Introducing this factor into eqn [30] yields eqn [32]:

The function X1(k x Rs) is complicated but always gives values between 1.0 and 1.5, as shown in Figure 14. According to Henry, three different equations must be used to compute the values of the function X1. If k x Rs > 24 then the first of the three equations indicated in Figure 14 must be used. When kxRs ) 5 the last of the three equations in Figure 14 is applied. In the range between the two border values 5 and 24, a linear equation is taken, which is also given in Figure 14. It is somewhat difficult to calculate the X1 values when kxRs)5. Therefore, Table 7 provides a number of values in the range of kxRs = 0.01-5. Kappa (k (m"1)) represents the reciprocal of the radius of the ion cloud, i.e. the radius of the cloud of counterions surrounding the protein. Depending on the ionic composition, ionic strength and temperature of the solution, k acquires values ranging from zero to infinity, and at increasing ionic strengths the value of k increases whilst the radius of the ionic cloud decreases and vice versa. In a salt-free solution, k = 0 so that the electrophoretic mobility U is not influenced at all, whilst conversely it decreases permanently in solutions with increasing salt concentrations. The value of kappa can be obtained from the equation:

where NA = 6.025 x 1023 (mol-1); s is the unit charge (protonic charge) = 1.602 x 10"19 (C); D0 represents

Figure 14 Graphical representation of Henry's function Xi (kRs). Depending on the value of kRs three different equations must be used to compute the values of X1. If kRs > 24 (case 1), the first of the three equations given is used. The second equation (case 2) comes into use if 5 < kRs < 24 while the third equation (case 3) is applied if kRs ) 5. In the latter case, Table 6 provides a number of values. Reproduced with permission from Rothe (1991).

Figure 14 Graphical representation of Henry's function Xi (kRs). Depending on the value of kRs three different equations must be used to compute the values of X1. If kRs > 24 (case 1), the first of the three equations given is used. The second equation (case 2) comes into use if 5 < kRs < 24 while the third equation (case 3) is applied if kRs ) 5. In the latter case, Table 6 provides a number of values. Reproduced with permission from Rothe (1991).

Table 7

Values of Henry's function X(k x RS)) if k x RS

< 5 (cf. Figure 14)a

K X RS

l°910 (k x Rs)

X1 according to

kxRs

log™ (k x Rs)

X according to

Overbeek's modification

Overbeek's modification

ofHenry's equation

ofHenry's equation

0.01

-2

1.0000062

1.95

0.2900346

1.0632127

0.05

- 1.30103

1.0001452

2.00

0.30103

1.0651048

0.10

- 1

1.0005451

2.05

0.3117539

1.0669887

0.15

- 0.8239087

1.0011577

2.10

0.3222193

1.0688642

0.20

- 0.69897

1.001951

2.15

0.3324385

1.0707308

0.25

- 0.60206

1.0028994

2.20

0.3424227

1.0725882

0.30

- 0.5228787

1.003982

2.25

0.3521825

1.0744361

0.35

- 0.455932

1.005181

2.30

0.3617278

1.0762744

0.40

- 0.39794

1.0064817

2.35

0.3710679

1.0781027

0.45

- 0.3467875

1.0078712

2.40

0.3802112

1.0799208

0.50

-0.30103

1.0093387

2.45

0.3891661

1.0817286

0.55

- 0.2596373

1.0108744

2.50

0.39794

1.0835259

0.60

-0.2218487

1.0124701

2.55

0.4065402

1.0853126

0.65

-0.1870866

1.0141185

2.60

0.4149733

1.0870886

0.70

-0.154902

1.0158129

2.65

0.4232459

1.0888537

0.75

-0.1249387

1.0175476

2.70

0.4313638

1.0906078

0.80

- 0.09691

1.0193175

2.75

0.4393327

1.0923509

0.85

-0.0705811

1.0211181

2.80

0.447158

1.094083

0.90

- 0.0457575

1.0229452

2.85

0.4548449

1.0958039

0.95

- 0.0222764

1.0247952

2.90

0.462398

1.0975136

1.00

0.0

1.0266648

2.95

0.469822

1.0992121

1.05

0.0211893

1.028551

3.00

0.4771213

1.1008994

1.10

0.0413927

1.0304511

3.05

0.4842998

1.1025754

1.15

0.0606978

1.0323626

3.10

0.4913617

1.1042402

1.20

0.0791812

1.0342836

3.15

0.4983106

1.1058938

1.25

0.09691

1.0362118

3.20

0.50515

1.1075361

1.30

0.1139434

1.0381455

3.25

0.5118834

1.1091672

1.35

0.1303338

1.0400832

3.30

0.5185139

1.1107871

1.40

0.146128

1.0420233

3.35

0.5250448

1.1123958

1.45

0.161368

1.0439644

3.40

0.5314789

1.1139934

1.50

0.1760913

1.0459054

3.45

0.5378191

1.11558

1.55

0.1903317

1.0478451

3.50

0.544068

1.1171554

1.60

0.20412

1.0497825

3.55

0.5502284

1.1187199

1.65

0.2174839

1.0517167

3.60

0.5563025

1.1202734

1.70

0.2304489

1.0536469

3.65

0.5622929

1.1218159

1.75

0.243038

1.0555723

3.70

0.5682017

1.1233477

1.80

0.2552725

1.0574921

3.75

0.5740313

1.1248686

1.85

0.2671717

1.0594059

3.80

0.5797836

1.1263788

1.90

0.2787536

1.0613129

3.85

0.5854607

1.1278783

3.90

0.5910646

1.1293672

4.45

0.64836

1.1450647

3.95

0.5965971

1.1308456

4.50

0.6532125

1.1464318

4.00

0.60206

1.1323134

4.55

0.6580114

1.1477892

4.05

0.607455

1.1337709

4.60

0.6627578

1.1491371

4.10

0.6127839

1.135218

4.65

0.667453

1.1504754

4.15

0.6180481

1.1366549

4.70

0.6720979

1.1518043

4.20

0.6232493

1.1380816

4.75

0.6766936

1.1531238

4.25

0.6283889

1.1394981

4.80

0.6812412

1.1544341

4.30

0.6334685

1.1409047

4.85

0.6857417

1.1557352

4.35

0.6384893

1.1423012

4.90

0.6901961

1.1570272

4.40

0.6434527

1.1436879

4.95

0.6946052

1.1583101

5.00

0.69897

1.159584

a Values were calculated using eqn [3] of Figure 14 (cf. Overbeek JTG (1950) Advances in Colloid Science, 3: 97-135). Tolerance of values: 10"6, calculation of integral: 7 digits. Data from Rothe (1991).

the dielectric constant of vacuum = 8.8542 x 10"12 (C V"1 m"1 = C2 N"1 m"2); D is the temperature-dependent dielectric constant of water (without dimension, cf. Table 8), k is Boltzmann's con stant = 1.3805 x 10"23 (J K"1 = N m K"1); T is absolute temperature (K) and I is the ionic strength (mol m"3) of the buffer that was used for electrophor-esis.

Table 8 Dielectric constant (D) of water depending on the temperature t(°C)

t(0C)

D

t(0C)

D

0

87.90

18

80.93

5

85.90

20

80.18

10

83.95

25

78.36

15

82.04

30

76.58

Reproduced with permission from West (1976-1977).

Reproduced with permission from West (1976-1977).

By substituting these values into the equation one obtains:

k = [(2 x 6.025 x 1023 x (1.602 x 10-19)2) x (8.8542 x 10-12 x 1.3805 x 10-23)-1]1/2 (K m (mol)-1)1/2 x(I (D T)-1)1/2(molm-3K-1)1/2 [34]

thus:

At a temperature of 5°C (278 K), the dielectric constant of water is 85.90 (cf. Table 8). Inserting both values into eqn [35] yields eqn [36]:

The ionic strength I (mol m-3) is calculated using the formula:

where (mol m-3) represents the concentrations of the ionic species of the buffer times their squared charges (Zi).

Taking, for example, a 90 mmol L-1 Tris, 80 mmol L-1 boric acid, 1.25 mmol L-1 EDTA-Na2 buffer of pH 8.0, the ionic strength of this buffer is:

thus:

Substituting this value into eqn [36] gives:

which rearranges to:

Taking ferritin as an example, with a Stokes radius of 6.20 x 10-9 (m), then k x RS = 14.67. The log of kxRs equals 1.167 and using this value one obtains from the equation in case 2 shown in Figure 14, a value of 1.293 for the function X1 (kxRs). Thus, inserting these values into eqn [31], it follows that:

F = (X1(kxRS))x(1 + (kxRS))-1 = 1.293 x(1 # 14.67)-1 = 0.08251 [44]

To calculate the number of nett charges in ferritin, eqn [32] must be solved for Z:

Z = ((Ux6xnx|xRS)xs-1)x((1 # (kxRs)) x(X1(kxRS))-1) [45]

From gradient gel electrophoresis results, the free electrophoretic mobility of ferritin was calculated as U = 3.28 x 10-9 (m2 V-1 s-1). Substituting this value, that of factor F and the value for the temperature-dependent dynamic viscosity (i (N s m-2)) of water as taken from Table 9 into eqn [45], the number of unit charges that ferritin acquires under the electrophoretic conditions indicated above can be computed as:

Z = (3.28 x 10-9x 6 xnx 1.519 x 10-3 x 6.20 x 10-9) x (1.602 x 10-19)-1 x ((1 x 0.08251)-1) [46]

which works out to:

The actual charge on the molecule is given by Zx s [C molecule-1] = 44.05 x 1.602 x 10-19 = 7.057x 10-18 (Table 6).

Table 9 Dynamic viscosity (N s m 2)) of water depending on the temperature (t (°C))

t(°C)

(Nsm-2) 10-

t(°C)

(Nsm-2) 10-

0

1.787

16

1.109

1

1.728

17

1.081

2

1.671

18

1.053

3

1.618

19

1.027

4

1.567

20

1.002

5

1.519

21

0.9779

6

1.472

22

0.9548

7

1.428

23

0.9325

8

1.386

24

0.9111

9

1.346

25

0.8904

10

1.307

26

0.8705

11

1.271

27

0.8513

12

1.235

28

0.8327

13

1.202

29

0.8148

14

1.169

30

0.7975

15

1.139

Reproduced with permission from West (1976-1977).

Reproduced with permission from West (1976-1977).

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