91

Once the Scatchard analysis of binding parameters is completed, a second graphic analysis is accomplished. De Meyts and Roth 4 proposed a new parameter, the average affinity of the receptor sites (I~), calculated as (B/F)/ (Ro - B), where Ro is the concentration of binding sites, B is the concentration of bound hormone and F is the concentration of free hormone. Plotting I~ as a function of fractional occupancy (B/Ro), gives the result that: (1) at very low occupancy a limiting high I~ is obtained ( 'empty sites' conformation); (2) when the fraction of sites filled increases above a certain threshold, I~ begins to fall due to increasing site-site interactions, until (3) a limiting low I~ is obtained ('filled sites' conformation). Fig 3 shows a typical plot for a binding system presenting negative cooperativity. The student is then offered a choice of performing further rounds of the program, in order to cover all the possibilities.

I0,00 9,00 8.06 7,09 6.00 5,00 4,00 3,00 2,00 1,00

0

\

\ I , i

,51 Fractional occupation

Figure 3 Average affinity o f the receptor sites, analyzed versus fractional occupancy for a system presenting negative cooperativity (kl = k2 = 10 mM-1; Pt = 50 nM and cr = 0.1)

Computer-assisted Calculation of Isoelectric Points of Proteins

MICHAEL CAPLICE and JAMES J A HEFFRON

Department o f Biochemistry University College Cork Ireland

Introduction The isoelectric point (pI) of a protein is defined as the pH at which the net charge on the protein is zero. Knowing a protein's pI may be important in such instances as planning the purification of proteins (eg calmodulina), the determination of the most suitable pH gradient for the isoelectric- or chromatofocussing of a particular protein, 2 and the identification of protein spots on two dimensional electrophoretograms of crude extracts. The pI of a protein is usually determined experimentally by isoelectric focuss- ing. 3 In this paper we describe how an estimate of the pI can be obtained by calculation, using a simple iterative computer program. This program also allows students to compare the effects of single or multiple amino acid substitutions on the pls of specific or hypothetical pro- teins.

Theoretical background to the program The pI is the pH at which the net charge on a protein is zero and the program determines that pH by making use of the equations described below. The calculation is re- iterated as the pH estimate is continuously incremented or decremented until the pH value used in the equations is within 0.0001 units of the pH at which the net charge on the protein is zero. This pH is then taken to be the pI.

The calculation of the net charge on the proteins is based on the Henderson-Hasselbalch Equation, which may be written:

[Salt] = [Acid] antiloglo (pH - pK.)

In our experience, students were frustrated that the theory of ligand-receptor interaction was explained with- out 'seeing' what happened after one or more binding parameters were changed. We feel that at least some of the uneasiness can be relieved using computer simulation.

We think this program will be of interest for biochem- istry courses and is readily adaptable to other machines using BASIC. A listing of this program is available from the authors by writing to: Instituto de Biologia y Medicina Experimental, Obligado 2490, (1428) Buenos Aires, Repfiblica Argentina.

Let [Salt] + [Acid] = 1 so that Salt and Acid can be expressed as a proportion (represented by { }) rather than as a concentration, ie

or

and

{Acid} antilogl0 (pH - pKa) + {Acid} = 1

{Acid} = 1/(1 + [antilogl0 (pH - pKa)])

References 1Berson, S A and Yalow, R S (1959) J Clin Invest 38, 1996-2016 eScatchard, G (1949) Ann NY Acad Sci 51,660-672 3Tinoco, I, Sauer, K and Wang, J C (1978) 'Physical Chemistry Principles and Applications in Biological Sciences', Prentice Hall, New Jersey

4De Meyts, P and Roth J (1975) Biochem Biophys Res Commun 66, 1118-1126

{Salt} = 1/(1 - [antilogao (pH - pKa)])

Using these formulae the size of the charge on any ionizable group at any pH can be calculated if the pK, of that group is known. In other words the charge on a single histidine, lysine, arginine or amino terminus is in fact the proportion of that residue or group in the acid (proton-

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ated) form, and the charge on a single aspartate, glutamate, cysteine, tyrosine or carboxyl terminus is the proportion of that residue or group in the salt (deproton- ated) form. The net charge on an individual protein molecule can be calculated from:

Net charge = 5; [(naa ) x (fr chg)]pos - ]~ [(n~a) x (fr chg)]neg

where naa = number of amino acids, fr chg = fractional charge, pos = positively charged residues, neg = nega- tively charged residues.

We used the following pK~ values in our calculations (values were for amino acids in proteins and were taken from Table 1-1 of Creighton4):

Amino Acid pK~ of Ionizing Side Chain

Histidine 6.2* Lysine 10.4" Arginine - 12.0" Aspartate 4.5* Glutamate 4.6* Cysteine 9.1-9.5 (9.3)* Tyrosine 9.7 a-Amino 6.8-7.9 (7.35)* a-Carboxyl 3.5-4.3 (3.9)*

* Values used in program

General Structure of the Program The first part of the program asks the student for the number of each type of ionizable amino acid present in the protein. (The program assumes that both the amino and carboxyl termini are free.) It then calculates the pI of the protein by the following iterative procedure. Initially the pH is set to zero. The net charge on the protein is calculated in a subroutine, which calculates the charge contribution of each of the seven types of charged amino acid and the carboxyl and amino termini. The net charge on the protein is then taken as the sum of all the positive charges minus the sum of all the negative charges. (At pH 0 this will be positive.)

If the charge is positive the value ascribed to the pH is incremented by a factor (1.0 initially) and the net charge on the protein is recalculated. If, after this, the charge is still positive, then the value of the pH is incremented by the same factor again and the incrementation and recal- culation continued until the net charge becomes negative. When this occurs the true pI must be somewhere between the currently set pH and the last value ascribed to the pH. At this point the value of the increment (1.0 initially) is subtracted from the pH, and the size of the increment is then decreased by a factor of 10 (i.e. 0.1, 0.01 . . . . ) and steps 2-4 are repeated. In this way, the value ascribed to the pH gets progressively closer to the pI with each loop. The program exits the loop after five turns.

The first isoelectric point calculated is referred to as the 'plo' and is printed on the screen before the student is

asked if amino acid alterations of the original protein are required. If alterations are required, then the original amino acid content appears on the screen and the student can make changes. The pI is recalculated and referred to as 'pI". Then the pIo, pI' and (pI' - plo) are printed on the screen.

If the number of charged amino acids on the surface of a protein and the pKa values of their side groups are known accurately, then it should be possible to calculate the exact isoelectric point of any protein as described above. This information, however, is not generally available for proteins in their native configuration: an unknown number of charged residues may be buried in the interior of the protein and thus contribute nothing to the surface charge. Other residues may be on the surface in relatively hydrophobic areas: the pKas of these groups may be different from the values operative when they can freely interact with water. Once again the precise value of these pK~s will not normally be known.

Proteins can be completely denatured by the addition of thiol compounds to break disulphide bridges and high concentrations of urea to dissociate the hydrogen bonds. Under these conditions all the charged groups should be exposed and all of the same type of amino acid side group should have the same pKa. Consequently we should be able to calculate the isoelectric points of fully denatured proteins with a greater degree of confidence than we can for native proteins. However, under these conditions, the ionization of groups is reduced (eg causing the pK~s of carboxylate groups to be increased by approx. 1 pH unit2). The pK~s used in this program should therefore be used only as a guide until more precise data on the pK,s of all amino acid side groups in the presence of high concentrations of urea are available.

Future Prospects When this information becomes available, it will be possible accurately to calculate the pls of denatured proteins. This could be a useful tool to molecular biologists as a means of confirming a proposed reading frame in a sequenced gene. For example the amino acid composition of the gene product could be predicted from the nucleotide sequence of the gene read in a certain reading frame. The validity of choosing that reading frame could then be assessed by comparing the predicted pI with the pI measured using isoelectric focusing under denatur- ing conditions. This approach is already being used with respect to calculated and predicted molecular weights.

Copies of the program described may be obtained by writing to J J A H.

References Klee, C B, Crouch, T H and Richman. P G (1980) Ann Rev Biochern 49. 489-515

20'Farrell. P H (1975) J Biol Chem 250. 4007-4021

3'Isoelectric Focussing: Principles and Methods'. Pharmacia, 1986

4Creighton, T C (1984) 'Proteins - - Structures and Molecular Proper- ties', W H Freeman and Co, New York

BIOCHEMICAL EDUCATION 16(2) 1988