Simple Theory for Affinity Partitioning

A basic theory for affinity partitioning was elaborated by Flanagan and Barondes in 1975. They analysed the combined binding and partition equilibria taking place in and between the two phases, respectively (Figure 3).

In this scheme the ligand-PEG(L), the free protein (P) and the two complexes (PL and PL2) have each their own partition coefficient (KL, KP, KPL and KPL2). Furthermore, in both phases association between protein and ligand-PEG takes place which can be described by the association constants:

one set for each phase. A total association constant for the equilibrium:

can also be used: Ktot = K1K2.

Table 1 Partition coefficients of PEG (KPEG) and dextran (Kdextran) and their logarithmic values (log) at various tie-line lengths of the system in Figure 1

Tie-line length (polymer Kpeg Kiextran log Kpeg log Kiextran concentration scale)

Figure 3 Scheme for affinity partitioning of a protein (P) with two binding sites for a ligand attached to PEG (L). The complexes between protein and ligand-PEG are PL and PL2, respectively.

Figure 4 Increase in the logarithmic partition coefficient of phosphofructokinase (PFK) from bakers' yeast as function of the concentration of Cibacron blue F3G-A PEG (Cb-PEG). System composition: 7%w/w dextran 500, 5%w/w PEG 8000 including Cb-PEG, 50 mM sodium phosphate buffer pH 7.0, 0.5 mM EDTA, 5 mM 2-mercaptoethanol and 4nkatg~1 enzyme. Temperature, 0°C. The inverse plot inserted is used to determine the A log K^.

The maximum partition coefficient of protein,

Kprotein

( = KpL2), is related to Kp, Kh and the K values via the following equations:

protein

Kprotein - KpKL "

protein

Figure 3 Scheme for affinity partitioning of a protein (P) with two binding sites for a ligand attached to PEG (L). The complexes between protein and ligand-PEG are PL and PL2, respectively.

The association constants, Ktot, K1 and K2 may differ between the two phases. According to Flanagan and Barondes, the measured log K value of a protein, log Kprotein, will, theoretically, give rise to a saturation curve when plotted versus the concentration of polymer-bound ligand in the system (compare Figure 4).

The log Kprotein value reaches a plateau when the concentration of L-PEG is so high that practically all the protein is present as the fully saturated complex PL2. The protein molecule is then surrounded by two PEG chains and outwardly shows a PEG atmosphere.

The maximum increase in the logarithmic partition coefficient, A log Kmax, is consequently given by:

Figure 4 Increase in the logarithmic partition coefficient of phosphofructokinase (PFK) from bakers' yeast as function of the concentration of Cibacron blue F3G-A PEG (Cb-PEG). System composition: 7%w/w dextran 500, 5%w/w PEG 8000 including Cb-PEG, 50 mM sodium phosphate buffer pH 7.0, 0.5 mM EDTA, 5 mM 2-mercaptoethanol and 4nkatg~1 enzyme. Temperature, 0°C. The inverse plot inserted is used to determine the A log K^.

= 2 log Kl # log Ktot,T - log Ktot,B If Ktot,T = Ktot,B then A log Kmax = 2 log Kl.

From the values in Table 1 it may therefore be assumed that for proteins with two binding sites A log Kmax can be as high as 3.57 (an increase of 3700 times in K) when PEG is used as ligand carrier with KL = 61. If dextran is used as carrier, in the same system, the A log Kmax should theoretically be around — 8 corresponding to a one hundred million times increase in the affinity of the protein for the lower phase if KL is 0.0001. A higher number of binding sites (n) should then give strongly increasing A log Kmax values with A log Kmax = n log KL. However, the affinity extraction effect may be reduced by a reduction of individual binding strengths.

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