Demo calculation of the isoelectric point of a protein

Thus the α–carboxylic acid group has -0.885 charges. The case is just the opposite for the side chain carboxylic acid, in which there are 1/0.13 = 7.67 non-ionised groups for each ionised one.

Therefore:

number of ionised groups 1

degree of ionisation 0.115

total number of groups 7.67 1

(3.44)

Thus the side chain carboxylic acid group has -0.115 charges. Indeed, the two carboxylic acid groups contribute unequally to the total negative charge of one and, as much the share of the α–

carboxylic acid is in excess of 0.5 (0.885-0.5=0.385), this will be the same as the extent to which the share of the side chain carboxylic acid (0.5-0.115=0.385) is short of 0.5.

3.4.3. Demo calculation of the isoelectric point of a protein

Let us calculate the pI of a protein that consists of 152 amino acids and has the following amino acid composition:

H2N-Gly-(150 amino acids)-Gly-COOH, containing the following amino acids with ionisable residues: 2Arg, 3Lys, 2Tyr, 1Cys, 5His, 3Asp.

As there are many ionisable groups in such a large molecule, it is obvious that the calculation of pI will be more a complex procedure. Therefore the procedures employed above are suitable only for an approximate estimation of pI. In some very rare cases, however, the calculation can follow that of the pI of aspartic acid (Demo calculation 3.4.2), and will not even be more difficult than that. In these cases the expected pI must be close to a pKa so that, at the same time, it is at least two pH units away from all the other pKa values. In order to determine whether this criterion is met, an estimation must be performed concerning the expected value of the pI. To this end, we must calculate the net charge of the protein at several different pH values. In order to facilitate these calculations, we can construct a charge calculation table listing the amino acids possessing ionisable residues in the order of their increasing pKa values. In the case of our protein, this table will look as follows (Figure 3.5):

62

Figure 3.5. Calculation of the net charge of a protein. * The pKa values (from Figure 3.2) next to the names of the amino acids refer to the ionisable group in their side chain, while the values concerning the α–carboxylic acid and α–amino groups (C- and N-termini) are those of glycine. **

The number of amino acids (i.e. that of the given type of ionisable group) within the protein. (For example, a protein always has one N- and one C-terminus.)

In order to reach our goal more rapidly, prior to the calculations it worth considering the pH range into which the pI is expected to fall. As a conclusion from the reasoning employed at the pI calculation of aspartic acid (Demo calculation 3.4.2), the pI will fall in the acidic or basic range if the number of negatively or positively ionising groups exceeds that of the other, respectively. The more the ratio is shifted towards one type of group, the more the pI will be shifted towards the pH range of the pKa of these groups. In our protein, the number of groups of the two types is the same. Therefore, it is appropriate to start the calculation of charge at pH 7.0.

Calculation of the charge at pH 7.0

63

According to the above considerations regarding the connection between the ratio of concentrations and the fraction of charges, for every charged group there are 0.0025 uncharged ones. Based on this and according to Equation 3.43, the degree of ionisation, i.e. the amount of charge will be:

the number of ionised groups 1

degree of ionisation 0.998

total number of groups 1 0.0025

(3.48)

This value differs negligibly from one. Therefore, we can take it as a full charge on the α–amino group. During approximate calculations we can always apply such rounding whenever the difference between the pH and the pKa is more than two pH units. This is allowed because in these cases the equilibrium of ionisation is shifted so much towards one of the forms of the group that the difference of the charge from one or zero will only be a few thousandths. Applying this simplification, without detailed calculations we can assign a full charge to the residues of aspartic acids, lysines and arginines and to the α–carboxylic acid, as well as zero charge to tyrosine residues.

Following the calculations above, the ratio of forms of the cysteine residue will be

- 7.0 8.3 1.3 While the above ratio regarding histidine residues will be

7.0 6.0 1.0 groups—we can fill the pH = 7.0 column of Figure 3.5. Summing up the charges, we can see that the charge of our protein at this pH will be -1.593. Thus, as it follows from the relationship between the net charge and the pH (see above), the pI must be below 7.0.

Continuing the estimation of the expected value of pI, we can now calculate the net charge of the protein at pH 6.0. As the minimally expected two-unit difference between the pH and the pKa

values again holds for Asp, Lys, Arg and Tyr residues as well as for the α–carboxylic acid and α–

amino groups, their charge can be taken the same as at pH 7.0. In contrast, the charges of cysteine and histidine residues will change. The former will decrease essentially to zero (because now the distance of pH from the pKa will be more than two units), while the latter will increase to +0.5 because now pH = pKa and, by the definition of pKa, here the degree of ionisation will be 50 %.

Filling the pH = 6.0 column of Figure 3.5 and summing up the charges, we get that the net charge

64

of the protein is +0.5. Thus, this pH is below the pI. Therefore, the pI must be between 6.0 and 7.0.

If we now assume (with some reason) that the pI will be close to 6, we can expect that even this value will differ at least two pH units also from the pKa of cysteine. Thus the criterion set above seems to be met, and its significance can now be understood. At several tenths of pH units above 6, the charge of every group—including cysteine—will only negligibly differ form that at pH 6.0.

Therefore we can try to calculate the exact pI by employing the same logic that was used during the pI calculation of aspartic acid. Thus, at pH = pI, the net charge of the protein will be -2 without the charge of histidines. Therefore, the reasoning applied in writing Equation 3.32 will in this case appear as the following: we will look for the pH at which the combined charges of histidines can neutralise the -2 charge resulting from the other ionisable groups. That is,

[His ]

5 2

[His]

(3.53) thus

[His] 2.5 [His ]

(3.54)

Writing the expression of pI as in Equation 3.18 and performing the appropriate substitutions we get:

a [His]

pI pK lg 6 lg 2.5 6.4

[His ]

(3.55)

Thus the isoelectric point we look for is 6.4. After the calculation and summation of the charge on every ionisable group, we must get zero. Performing this control calculation (see the pH = 6.4 column in Figure 3.5) indeed yields zero net charge.

65

4. Spectrophotometry and protein concentration