Graphical analysis of nonideal monomer N-mer, isodesmic, and type II indefinite self-associating systems by equilibrium ultracentrifugation.

A graphical procedure is described by which one can obtain in principle the monomer molecular weight, stoichiometry, equilibrium constant, and second virial coefficient of nonideal monomer N-mer, isodesmic, and type II indefinite self-associating systems. In addition, a method is presented for obtaining both the equilibrium constant and the second virial coefficient from the maximum in a plot of apparent molecular weight vs. concentration if the monomer molecular weight and stoichiometry are known. The usefulness and limitations of the methods are discussed, as well as the quality and range of data required for determination of the relevant parameters. The techniques described are applicable to analysis of self-associating systems by osmotic pressure and light scattering, as well as equilibrium ultracentrifugation measurements.     

Assessment of the validity of the Adams and Fujita approximation for the higher oligomers of human spectrin

Analysing the self-association behaviour of human erythrocyte spectrin is complicated by a large degree of nonideality. Adams and Fujita [1] proposed that, as a first order approximation, the logarithm of the activity coefficient of the protomer of a self-associating system can be considered to be linearly dependent on the total concentration of the protein, and that the same second virial coefficient could be considered to apply to all species. As a consequence of the Adams and Fujita approximation, the apparent equilibrium constant is equal to the thermodynamic equilibrium constant. The equilibrium concentrations at 30°C of each oligomer spectrin species up to the 14-mer were determined after electrophoresis at low temperature. An apparent equilibrium constant for forming tetramer (K_2,4) of (1.2 ± 0.1) × 10⁶ l/mol was obtained, a value of (9.4 ± 0.7) × 10⁴ l/mol was obtained for K_4,6 and for all reactions forming oligomers higher than the hexamer an average approximate value of (2.7 ± 0.4) × 10⁵ l/mol was obtained. The apparent equilibrium constants for the formation of all oligomer species of spectrin up to the tetrakaidecamer (14-mer) remain relatively independent of total spectrin concentration, and indicate that within the precision of the measurements a single virial coefficient is sufficient to account for the nonideality of spectrin self-association over the range 2–42 g/l, thus further justifying the use of the Adams and Fujita approximation for this protein over this concentration range.     

Determination of the parameters of self-association by direct fitting of the omega function

Nonlinear regression is used to fit the omega function vs. protein concentration curves (first described by B.K. Milthorpe, P.D. Jeffrey and L.W. Nichol, Biophys. Chem. 3 (1975) 169) obtained from sedimentation equilibrium experiments on self-associating macromolecules. Nonlinear regression allows the direct fit of these curves with discrete or indefinite self-association reaction models in order to obtain values for the equilibrium constants and second virial coefficient. The method is independent of the choice of reference concentration and avoids the original method of extrapolating an omega function curve to zero concentration and then using the extrapolated value to construct a monomer activity curve used for analysis. This extrapolation can become very difficult for mild to strong self-associations where incorrectly extrapolated values lead to systematic error in the monomer activity curves. The method is applied to results from a mild, indefinite self-association, exemplified by the self-association of human spectrin, and to computer-simulated data of weak, mild and strong, indefinite self-associations.     

Analysis of interacting biopolymer systems by analytical ultracentrifugation

Many of the functions of biological macromolecules are based on specific interactions. Extended concentration dependent studies of sedimentation coefficients or molecular masses of biopolymers are highly useful for describing the different kinds of association phenomena. These studies allow one to determine the partial concentrations of monomers and associates or reactants and complexes in self-associating systems or heterologous associations, respectively. Furthermore, in combination with corresponding measurements of biological activity these data allow one to estimate the individual activity parameters of components involved in equilibrium processes. The study of self-association and heterologous association using analytical ultracentrifugation, some recent developments therein, and its application to different examples are outlined here. Accepted: 18 October 1996     

Studies on human serum high density lipoproteins. Self-association of apolipoprotein A-I in aqueous solutions.

Human serum apolipoprotein A-I (apo-A-I), the major protein component of the human serum high density lipoproteins, was studied in aqueous solutions of differing ionic strength and pH by the techniques of sedimentation equilibrium ultracentrifugation and frontal analysis gel chromatography. The ultracentrifugal studies indicate the apo-A-I is a self-associating system that is dependent upon protein concentration, but relatively independent of the nature of the medium. The apparent weight average molecular weights obtained from solutions of initial apo-A-I concentration between 0.2 and 0.9 mg/ml were in the range of 3.0 to 16.7 x 10(4) (monomer molecular weight = 28,014). Of the several models of self-associated examined, that which gave the best theoretical fit was for the monomer-dimertetramer-octamer model. The self-association of apo-A-I in aqueous solutions was further documented by frontal analysis gel chromatography, which not only corroborated the ultracentrifugal results, but also indicated that the multiple species of apo-A-I in solution attain equilibrium rather rapidly. Besides having intrinsic importance, these results indicate that the solution properties of apo-A-I must be established before ligand binding studies are conducted and interpreted.     

The study of multiple polymerization equilibria by glass bead exclusion chromatography with allowance for thermodynamic non-ideality effects

Two related aspects are explored of the frontal exclusion chromatography of proteins employing controlled-pore glass beads as the stationary phase. First, it is shown theoretically that, despite the absence of osmotic shrinkage effects previously encountered with Sephadex matrices, the experimentally measurable partition coefficient of a single non-associating solute will be dependent on its concentration due to the differing ratios of activity coefficients in mobile and stationary phases at different total concentrations. The effect is demonstrated with results obtained using ovalbumin in phosphate buffer of pH 7.4, and is Shown to be consistent (up to a solute concentration of 5 $\text{g}\text{litre}$) with theoretical prediction formulated in terms of a single virial coefficient. Secondly, it is shown for self-associating systems that it is possible to determine the monomer concentration as a function of total concentration, provided the stationary phase is selected to ensure exclusion of all oligomeric species except monomer: the relation derived for this purpose accounts for the concentrationdependence of the partition coefficient of monomer, again as a first approximation involving one virial coefficient. Such information on the monomer concentration permits elucidation of the polymerization characteristics of the system in terms of the types of species present and the relevant equilibrium constants. The feasibility of the method, its likely sources of error and the relative contribution of the non-ideality effect are investigated using bovine glutamate dehydrogenase (up to a total concentration of 5.4 $\text{g}\text{litre}$) in phosphate buffer of pH 6.9. This system was selected since comparison was possible with results obtained by other methods, which have established the enzyme polymerization pattern as an isodesmic indefinite self-association. The isodesmic equilibrium constant of 1.5 ± 0.3 $\text{litre}\text{g}$ found in this work is in reasonable agreement with previous findings.     

Partitioning of a fluorescent phospholipid between fluid bilayers: dependence on host lipid acyl chains

The partition coefficient Kp was measured for a headgroup-labeled phospholipid (12:0,12:0)-N-(7-nitrobenz-2-oxa-1,3-diazol-4-yl)-PE (12-NBD-PE), equilibrated between LUV of a series of phosphatidylcholines (PC). Fluorescence resonance energy transfer between the 12-NBD-PE and a headgroup-rhodamine-labeled PE was used to find the equilibrium concentration of the 12-NBD-PE in the different LUV. Reliable equilibrium concentrations were obtained by monitoring the approach to equilibrium starting from a concentration below and from a concentration above the ultimate values. Using (16:0,18:1delta9)-PC as the reference lipid, Kp ranged from a high value of 1.65 favoring (16:0,18:1delta9)-PC over (16:1delta9,16:1delta9)-PC, to a low value of 0.90, favoring (22:1delta13,22:1delta13)-PC over (16:0,18:1delta9)-PC. The Kp values enabled calculation of the acyl chain contribution to the excess free energy of mixing for (12:0,12:0) acyl chains at infinite dilution in the L alpha phase of PC having acyl chains of (16:0,18:1delta9), (16:1delta9,16:1delta9), (18:1delta9,18:1delta9), (18:1delta6,18:1delta6), (20:1delta11,20:1delta11), and (22:1delta13,22:1delta13). (14:1delta9,14:1delta9)-PC was found to transfer so rapidly between LUV as to preclude reliable Kp measurement.     

The self-association of human spectrin at high concentration.

The self-association of purified human spectrin has been studied at sedimentation equilibrium over a wide range of concentration (0-20 g/L) at 30 degrees C and pH 7.5. Coincidence of apparent weight average molecular weight and omega (r) plots as a function of total spectrin concentration indicated that equilibrium was attained and that no significant concentration of solute was incapable of participating in the self-association reaction. Under these conditions, no significant dissociation of the heterodimer to component polypeptide chains could be detected. The behavior of spectrin between 0 and 20 g/L can be described reasonably well by a cooperative isodesmic model, in which the protomer for association is the alpha beta heterodimer. With this model, the equilibrium constant for the heterodimer-tetramer step, K24, is 2 x 10(6) M-1, and K(iso), the equilibrium constant describing all other steps, is approximately 0.2 x 10(6) M-1. The returned value of the second virial coefficient for this model, 1.0 x 10(-7) L mol g-2, is consistent with the lower limit of values calculated for the heterodimer from the charge and Stokes radius of spectrin. On the other hand, the attenuated indefinite association model fails to describe the self-association of spectrin adequately over the range 0-20 g/L. Systematic decreases in the estimates of the second virial coefficient and the equilibrium constants for association beyond the tetramer suggest that the assumption of a single value of the second virial coefficient may not be appropriate for spectrin, and that non-ideality would best be taken into account by consideration of the detailed solution composition.     

Ligand-induced self-association of human luteinizing hormone. Negative cooperativity in the binding of 8-anilino-1-naphthalensulfonate.

The self-association of human luteinizing hormone (hLH) is enhanced in the presence of 8-anilino-1-naphthalenesulfonate (ANS). Sedimentation equilibrium measurements indicate that the hormone exists primarily as a dimer in the presence of excess ANS. It is shown that, for a self-associating protein system in which monomer and dimer have different affinities and/or capacities for ligand, both the shape and the position of the binding curve depend on protein concentration. Gel filtration and fluorescence measurements indicate that the hLH dimer has a single high affintiy site (K greater than 10(6) M-1) for ANS while binding to the monomer is too weak to be observed. This leads to negative cooperativity in the binding and to a shift of the binding curve to lower free ligand concentration with increasing concentration of the hormone.     

Binding of a fluorescent lipid amphiphile to albumin and its transfer to lipid bilayer membranes

Kinetics and thermodynamics of the binding of a fluorescent lipid amphiphile, Rhodamine Green(TM)-tetradecylamide (RG-C(14:0)), to bovine serum albumin were characterized in an equilibrium titration and by stopped-flow fluorimetry. The binding equilibrium of RG-C(14:0) to albumin was then used to reduce its concentration in the aqueous phase to a value below its critical micelle concentration. Under these conditions, the only two species of RG-C(14:0) in the system were the monomer in aqueous solution in equilibrium with the protein-bound species. After previous determination of the kinetic and thermodynamic parameters for association of RG-C(14:0) with albumin, the kinetics of insertion of the amphiphile into and desorption off lipid bilayer membranes in different phases (solid, liquid-ordered, and liquid-disordered phases, presented as large unilamellar vesicles) were studied by stopped-flow fluorimetry at 30 degrees C. Insertion and desorption rate constants for association of the RG-C(14:0) monomer with the lipid bilayers were used to obtain lipid/water equilibrium partition coefficients for this fluorescent amphiphile. The direct measurement of these partition coefficients is shown to provide a new method for the indirect determination of the equilibrium partition coefficient of similar molecules between two defined lipid phases if they coexist in the same membrane.     

Partition of free and monoclonal-antibody-bound horseradish peroxidase in a two-phase aqueous polymer system--novel procedure for the determination of the apparent binding constant of monoclonal antibody to horseradish peroxidase.

The principle that the antigen and the antibody prefer different phases in an aqueous two-phase system is the analytical basis of the work presented here. The antigen horseradish peroxidase, which is bound to a monoclonal antibody (mAb), is separated from free Ag in an aqueous phase system (polyethylene glycol (PEG)/dextran) as a function of the concentration of mAb. The plot of the partition coefficient kappa of horseradish peroxidase versus the concentration of mAb yields a sigmoidal curve similar to the curve obtained by enzyme-linked immunosorbent assay (ELISA). Comparing the plots normally used for ELISA in order to determine the apparent binding constant of mAb and the number of epitopes on the Ag we derived a relationship between the difference in partitioning of the free Ag and the bound Ag (delta kappa) and the concentration of mAb. The new linear plot of reciprocal delta kappa versus reciprocal concentration of mAb gives the apparent binding constant of mAb, which is evaluated from the slope. From the intercept at the ordinate the maximum difference of the partition coefficient of the free and bound antigen is derived and the apparent partition coefficient of the free monoclonal antibody can be calculated.     

Cell surface energy, contact angles and phase partition. I. Lymphocytic cell lines in biphasic aqueous mixtures

The surface energy of cells is the quantity which dominates certain physical interactions of cells such as adhesion to hydrophobic surfaces and phagocytosis. A linear relationship is derived relating the equilibrium constant obtained from phase partition in liquid-liquid systems with the surface energy difference obtained from contact-angle measurements. Using biphasic mixtures of Dextran and poly(ethylene glycol) in a medium of constant salt composition the expression is confirmed for transformed lymphocytic cell lines. The results demonstrate the importance of van der Waals' interactions in the phase-partition process, that phase partition can be used as a direct measure of cell surface hydrophobicity, and that the equilibrium constant of phase partition is directly related to the difference in the surface energy of the partitioned particle between the two phases.     

The dynamics of phase partition. A study of parameters affecting rat liver organelle partitioning in aqueous two-polymer phase systems.

Separation of subcellular organelles by two-phase partition is thought to reflect differential partition of the organelles between the two phases or between one of the phases and the interface. Studies by Fisher and colleagues [Fisher & Walter (1984) Biochim. Biophys. Acta 801, 106-110] suggest that cell separation by phase partition is a dynamic process in which the partition changes with time. This is mainly due to association of the cells with sedimenting droplets of one phase in the bulk of the other. Rat liver organelle partition was studied to determine whether the same dynamic behaviour is observed. Partition was clearly time-dependent during 24 h at unit gravity, and was also affected by altering the volume ratio of the two phases and the duration of phase mixing. These results indicate that, as with cells, the partition of organelles between phases is a dynamic process, and is consistent with the demonstration that organelles adhere to the phase droplet surfaces. Optimization of the volume ratio between phases may lead to significant processing economies. Organelle sedimentation in the upper phase was significantly faster than in the isoosmotic sucrose. Theoretical modelling of apparent organelle sizes indicates that aggregation occurs in the poly(ethylene glycol)-rich upper phase. This phenomenon is likely to limit the use of this technique in organelle separations unless means can be found to decrease aggregation.     

Intramolecular electron transfer in cytochrome c oxidase: a cascade of equilibria.

Intramolecular electron redistribution in cytochrome c oxidase after photolysis of the partially reduced CO-bound enzyme was followed at a number of different wavelengths by absorption spectroscopy. Spectra were constructed for the first two phases of this process. The first phase (tau = 3 microseconds) has a spectrum essentially identical to the difference between the Fea and Fea3 reduced-minus-oxidized spectra, indicating a 1:1 stoichiometry between the amount of Fea3 oxidized and Fea reduced. It is not necessary to invoke reduction or oxidation of other redox carriers in this phase. The second phase (tau = 35 microseconds) spectrum appears to be a linear combination of the Fea3 and Fea reduced-minus-oxidized difference spectra, reflecting the oxidation of four parts of Fea3 for every part of Fea oxidized. This process can be described in terms of transfer to CuA of electrons from the Fea3<==>Fea equilibrium system established in the first phase. The relative contributions of Fea3 and Fea in the second phase allow us to calculate the equilibrium constant for Fea3<==>Fea electron exchange, which yields a delta Em of 36 mV for the two centers (Fea3 more positive). Together with the apparent rate constant for the fast phase, this equilibrium constant yields, in turn, the forward (kf) and reverse (kr) rates for electron transfer from Fea to Fea3 as follows: kf = 2.4 x 10(5) s-1 and kr = 6 x 10(4) s-1. kf is much faster than any observed step in the reaction of the reduced enzyme with O2. Thus, the catalytic mechanism of O2 reduction to water is not rate-limited by electron transfer from Fea to the binuclear Fea3/Cu(B) site.     

Self-association mode of a flavoenzyme D-amino acid oxidase from hog kidney. I. Analysis of apparent weight-average molecular weight data for the apoenzyme in terms of models

The self-association of D-amino acid oxidase apoenzyme in 0.1 M sodium pyrophosphate, pH 8.3, at 25 degrees C was studied by low-angle laser light scattering. The concentration (c) dependence of the apparent weight-average molecular weight (Mwapp) was determined over a wide concentration range of 0.04 to 6.1 mg/ml. The extrapolated Mwapp value, to zero enzyme concentration, corresponded to the Mr value of the monomer. The self-association mode of the apoenzyme was systematically explored with nonlinear least-squares analysis of the Mwapp versus c data. The simplest model that fitted the data well was a model of isodesmic indefinite self-association of the monomer with the isodesmic association constant of 0.467 +/- 0.034 liter/g. The monomer-dimer model proposed previously, but only in a low enzyme concentration range of less than 0.9 mg/ml at 5-20 degrees C (Henn, S. W., and Ackers, G. K. (1969) Biochemistry 8, 3829-3838), did not fit the Mwapp versus c data either in the limited low concentration range or in the whole concentration range examined at 25 degrees C. To test the validity of the chosen model, the observed sedimentation boundary profiles were compared with the idealized boundary profiles calculated for the better-fit models. The profile calculated with the model of the isodesmic indefinite self-association mechanism was qualitatively consistent with the observed ones. The utility of the nonlinear least-squares procedure for analyzing self-associating systems was demonstrated.     

Studies on human serum high-density lipoproteins. Self-association of human serum apolipoprotein A-II in aqueous solutions.

Some of the solution properties of pure preparations of human serum high-density apolipoprotein A-II were studied by sedimentation equilibrium ultracentrifugation, conducted at different apoprotein concentrations and at several speeds. The concentration dependence of the apparent weight average molecular weight indicated that apolipoprotein A-II, when dissolved in 0.02 MEDTA (pH 8.6), undergoes self-association. Over a protein concentration range between 0.8 and 1.5 mg/ml, the self-association could best be described by a monomer-dimer-trimer step association, although indefinite self-association could not be ruled out. The equilibrium constants obtained were sufficient to describe the system over the concentration range investigated.     

Entry into and release of solvents by Escherichia coli in an organic-aqueous two-liquid-phase system and substrate specificity of the AcrAB-TolC solvent-extruding pump

Growth of Escherichia coli is inhibited upon exposure to a large volume of a harmful solvent, and there is an inverse correlation between the degree of inhibition and the log P(OW) of the solvent, where P(OW) is the partition coefficient measured for the partition equilibrium established between the n-octanol and water phases. The AcrAB-TolC efflux pump system is involved in maintaining intrinsic solvent resistance. We inspected the solvent resistance of delta acrAB and/or delta tolC mutants in the presence of a large volume of solvent. Both mutants were hypersensitive to weakly harmful solvents, such as nonane (log P(OW) = 5.5). The delta tolC mutant was more sensitive to nonane than the delta acrAB mutant. The solvent entered the E. coli cells rapidly. Entry of solvents with a log P(OW) higher than 4.4 was retarded in the parent cells, and the intracellular levels of these solvents were maintained at low levels. The delta tolC mutant accumulated n-nonane or decane (log P(OW) = 6. 0) more abundantly than the parent or the delta acrAB mutant. The AcrAB-TolC complex likely extrudes solvents with a log P(OW) in the range of 3.4 to 6.0 through a first-order reaction. The most favorable substrates for the efflux system were considered to be octane, heptane, and n-hexane.     

Heregulin reverses the oligomerization of HER3

We analyzed the propensity of the HER3 receptor and its extracellular domain (ECD) to undergo ligand-independent self-association. The HER3-ECD, purified from Drosophila S2 cells, binds the EGF-like domain of heregulin (hrg) with a K(d) of 1.9 nM as measured by surface plasmon resonance (SPR) studies. In a gel shift assay, the HER3-ECD self-associates into a uniform, slowly migrating species in a concentration-dependent manner, starting at concentrations of <10 nM. In contrast to the HER3-ECD, the ECD from the related HER2 receptor does not oligomerize under the same conditions. The direct interaction of HER3-ECDs was also demonstrated by pull-down assays and SPR measurements under physiological salt conditions. This self-association of the HER3-ECD was reversed by the addition of hrg but not by EGF. The apparent equilibrium dissociation constant for the HER3-ECD self-association is 15 nM, based on SPR measurements. In this analysis, hrg blocks HER3-ECD self-association, and the addition of hrg during the dissociation phase resulted in an accelerated off rate. This finding suggests that hrg can bind to and disrupt preexisting HER3-ECD oligomers. Full-length HER3 likewise exhibited self-association. Under conditions where co-immunoprecipitation and cross-linking of HER2 and HER3 were stimulated by hrg, HER3 self-association and cross-linking were disrupted by hrg. The implication is that the self-association of HER3-ECD favors the formation of catalytically inactive complexes of the HER3 receptor. Binding of hrg releases HER3 which may then form signaling-competent HER3-HER2 heterodimers.     

Computer stimulation of sedimentation in the ultracentrifuge. VII. Solutes undergoing indefinite self-association

The velocity sedimentation of solutes involved in self-association equilibria of the indefinite type was simulated using a computer model. The changes in boundary shape that resulted from varying the association constant and the molecular weight of the self-associating monomer were examined. Both ideally and nonideally sedimenting solutes were considered, and several alternative treatments of the variation of the frictional ratio with molecular size were used. All of the calculated boundaries were skewed, with the leading limb of the gradient profile steeper than the trailing limb. For relatively tightly self-associating solutes, the boundaries were very broad and strongly skewed. No shoulders or subsidiary peaks were observed for any of the model solutes used.