Mixed and uniform co-operativity of ligand binding to multisite proteins: the co-operativity types allowed by the adair equation and conditions for them

It is proved for the first time that the macroscopic co-operativity of binding to a protein with q binding sites may change signs over a single binding curve any number of times from 0 to (q-2), but no more than (q-2). n changes of sign of macroscopic co-operativity requires as a necessary condition at least n changes of sign of microscopic co-operativity, but this necessary condition is not a sufficient one. The necessary and sufficient condition that decides whether there are two changes of sign in a four-site protein is obtained. There are no changes when K₁K₃(K₂-K₄)+K₁K₂(K₃-K₂)+K₂K₃(K₄-K₃) is positive, and two changes when it is negative, presuming the above mentioned necessary conditions to be satisfied. The K's of this formula are the “intrinsic” per-site Adair constants. As a result, the conditions for all six co-operativity types possible with a four-site protein are now known.     

Protein symmetry and the co-operative ligand-binding behaviour predicted by allosteric Koshland models

It is shown that the nearest-neighbour interaction two-conformation allosteric models of Koshland, Nemethy & Filmer (1966) predict binding curves with a centre of symmetry when the protein is also symmetrical and induced-fit is assumed. When nonexclusive binding to both conformations is assumed, the models predict that the family of homotropic binding curves obtained by varying the heterotropic ligand has a centre of symmetry. It is argued that the symmetry or asymmetry of binding curves is the main experimentally verifiable prediction of allosteric models insofar as they are models of interaction between protein subunits.Symmetry in a binding curve greatly simplifies the analysis of cooperative behaviour. The co-operative features possible with a symmetric binding curve for a four-site protein are analysed. The sign of co-operativity may either be uniform or change twice as saturation increases; the conditions for the various possibilities are given. For example, in terms of the intrinsic binding constants per site A^∥₁, A2, etc. the necessary and sufficient condition for positive macroscopic co-operativity over the whole symmetric binding curve is A^∥₁≤ A2, A ^∥₁ ≤ A3 which should be contrasted with the obvious A^∥₁ ≤ Al, AZ ≤A^∥₃ (positive microscopic co-operativity) which is only a sufficient but not a necessary condition. A symmetric curve may have one or three, but no more, extrema of the “Hill coefficient” h. For three extrema a change of sign of microscopic (but not necessarily macroscopic) co-operativity is necessary but not sufficient. In the case where there are off-centre maxima of h, then h < 2 everywhere on the curve.The Koshland models predict qualitative and quantitative restrictions on the forms of binding curves additional to that of symmetry. In tetrameric induced fit models, negative co-operativity in the mid-region of the curve and positive co-operativity in the outside regions is possible, but not the opposite, and three extrema of h are possible with uniform positive but not with uniform negative co-operativity.Thus by recognising the importance of symmetry it has been possible to describe and categorise all the co-operativity behaviour possible with the most plausible Koshland tetrameric models. Several experimental examples of probable non-exclusive binding to proteins and enzymes are discussed, and it is shown how the symmetry point of view illuminates their interpretation.     

Linkage between carbon dioxide binding and four-step oxygen binding to hemoglobin

In order to elucidate the molecular mechanism of the effect of carbon dioxide on the four-step oxygenation equilibria of hemoglobin, accurate oxygen equilibrium curves of human adult hemoglobin were determined at different concentrations of CO₂ and in the presence and absence of chloride (Cl^?), 2,3-diphosphoglycerate (P₂G), and/or inositol hexaphosphate (IHP) and were analyzed according to Adair's stepwise oxygenation scheme to evaluate the four Adair constants, k_i (i = 1 to 4). The effects of CO₂ on oxygen affinity and co-operativity are influenced by H⁺, Cl^?, P₂G and IHP. The shape of the oxygen equilibrium curve varies with changes of CO₂ concentration; the four Adair constants are affected by CO₂ non-uniformly. Hence, the number of CO₂ molecules released upon oxygenation is not the same in the individual oxygenation steps. In the absence of added Cl^?, CO₂ lowers the overall oxygen affinity expressed by median oxygen pressure (p_m) and increases the co-operativity expressed by Hill's coefficient (n_max) by reducing k₁, k₂ and k₃ without changing k₄. significantly. The effect of CO₂ on oxygen affinity becomes smaller with decrease in pH, disappearing below pH 6.5. The alkaline Bohr effect is reduced by CO₂. The first oxygenation step contributes to the reduction of the Bohr effect more than the fourth step. When log p_m is plotted against log [CO₂] at several constant Cl^? concentrations, the plots converge to a common point that is named “iso-effective point”. When log p_p is plotted against log [Cl^?] at several constant CO₂ concentrations, the plots also converge to an iso-effective point. This phenomenon can be explained in terms of linkage relations in oxygen-linked competitive binding of CO₂ and Cl^?. It was found to be useful to consider in this analysis that the bicarbonate ion introduced by added CO₂ exerts a heterotropic effect equivalent to that of Cl^?. The combined effects of Cl^?, CO₂ and IHP were not explained satisfactorily by the present analysis using linkage relations.     

Characterization of a small apolar anion binding site of human serum albumin.

Small unbranehed fatty acid anions inhibit the fast reaction between p-nitrophenylacetate and human serum albumin. Plots of reactivity versus fatty acid anion-albumin ratios resemble simple binding isotherms from which corresponding dissociation constants have been calculated. For the homologous fatty acid anions, butyrate through decanoate, dissociation constants decrease from 3.2 × 10^?4 to 1 × 10^?7m, respectively, by uniform increments per methylene group according to the relationship ?ΔG °(kcal) = 0.804_n + 2.30, where n is the number of constituent methylene groups. Small fatty acid anions thus appear to interact primarily with a single, relatively uniform apolar binding site with a capacity sufficient for nine methylene groups. Fatty acid anions larger than decanoate interact significantly with other sites and do not obey the same relationship. The reactivity of diluted human serum with p-nitrophenylacetate was found to be one-third to one-half of that expected for its content of serum albumin, but as in vitro, it could be completely inhibited by small amounts of decanoate.     

Minimal models of multi-site ligand-binding kinetics

Mathematical models based on the current understanding of co-operativity in ligand binding to the (macro) molecule and relating the dose-response (saturation) curve of the (macro) molecule ligation to intrinsic dissociation constants characterizing the affinities of ligand for binding sites of both unliganded and partly liganded (macro) molecule have been developed. The simplified models disregarding the structural properties and considerations concerning conformational changes of the (macro) molecule retain the ability to yield sigmoid curves of ligand binding and reflect the co-operativity. Model 1 contains only three parameters, parameter κ (a multiplier characterising the change in the affinity) reflects also the existence and type of co-operativity of ligand binding: κ<1 corresponds to positive co-operativity, κ>1 to the negative and κ=1 to the absence of any co-operativity. Model 2 contains an extra parameter, ω, equilibrium constant for the T₀↔R₀ transition but fails to produce dose-response, which would suggest negative co-operativity. For any fixed n>1, the deviation of the dose-response (saturation) curve from the Henri hyperbola depends either solely on parameter κ (Model 1) or also on parameter ω (Model 2). The (macro) molecule being a receptor, both models yield a diversity of dose-response curves due to possible variety of efficacies of the (macro) molecule. The models may be considered as extensions of the Henri model: in case the dissociation constants remain unchanged, the proposed models are reduced to the latter.     

Nucleic acid binding properties of Escherichia coli ribosomal protein S1. II. Co-operativity and specificity of binding site II.

The following properties characterize the interaction of nucleic acid binding site II of Escherichia coli ribosomal protein S1 with oligo- and polyribonucleotides; all have been determined with site I complexed with oligo- or polydeoxyribonucleotides. (1) The intrinsic binding constant (K) of site II to single-stranded polyribonucleotides is fairly independent of base composition, though cytidinecontaining polymers bind with approximately threefold higher intrinsic affinities than do the comparable adenine-containing species. (2) Poly(rC) is bound to site II co-operatively; the co-operativity parameter (ω) ? 31. Poly(rA) shows no binding co-operativity. The site size (n) for both polyribonucleotides binding at site II is about ten nucleotide residues. (3) The K value for site II is ? 4 × 10⁵m^?1 for poly(rA), and ? 1 × 10⁶m^?1 for poly(rC), in 0.12 m-Na⁺. Unlike site I, the binding affinity of site II increases somewhat with increasing salt concentration, suggesting that phosphate—basic protein residue contacts are not involved. (4) Varying Mg^{2 +} concentration has no effect on K, and changes in the concentration of either Mg²⁺ or Na⁺ do not affect the magnitude of site II co-operativity. (5) Reaction of the exocyclic amino groups of poly (rC) with formaldehyde drastically reduces the affinity of site II for this polynucleotide, while the affinity of poly (rC) for site I is not altered by this treatment. (6) No major sequence specificity of K for site II is found with either homogeneous polynucleotides or the 3′ terminal dodecanucleotide of 16 S ribosomal RNA; we conclude that selectivity of S1 binding via site II depends largely on the presence or absence of base compositiondependent binding co-operativity.The binding properties of site II probably account for the ability of S1 to inhibit translation at high S1 to ribosome ratios (“factor i” activity). Possible mechanisms for the role of S1 protein as a part of the phage Qβ replicase complex and in protein synthesis are discussed in relation to the binding properties of site I and site II.     

A New Interpretation of the Dynamic Changes of the Potassium Conductance in the Squid Giant Axon

The solutions, n(t), of the differential equation dn/dt = α (1 - n) n (4 - 6n + 4n² - n³) - βn² (4 - 6n + 4n² - n³) in which α and β are instantaneous functions of membrane potential, are shown to fit with good accuracy the time courses of the rise of potassium conductance during depolarizing steps in clamp potential, found experimentally by Hodgkin and Huxley and by Cole and Moore. The equation is derived by analysing the dynamic behaviour of a system consisting of a square array of interacting pores. The possible role of Ca⁺⁺ ions in this system is discussed.     

Interactions of bacteriophage T4-coded gene 32 protein with nucleic acids: III. Binding properties of two specific proteolytic digestion products of the protein (G32P1I and G32P1III)

Brief treatment of gene 32 protein with proteolytic enzymes produces two specific digestion products in good yield (Moise & Hosoda, 1976). One, representing the native protein with ~60 amino acid residues removed from the C-terminus, is G32P¹I. The other, for which ~20 amino acid residues have been removed from the N-terminus in addition to the 60 residues from the C-terminus, is G32P¹III. Both of these specific “core” fragments of gene 32 protein have been isolated and purified, and their binding properties to single-stranded oligo- and polynucleotides have been studied. We find that the binding properties of G32P¹I are relatively little changed from those characteristic of the native gene 32 protein: (1) the apparent binding constants to short (l = 2 to 8) oligonucleotides are independent of lattice length and essentially independent of base and sugar composition, but do show an increased salt dependence of binding relative to that of the native protein; (2) the intrinsic association constants (K) for polynucleotides binding in the co-operative mode show the same binding specificities as seen with the native protein, but with absolute values increased two to fourfold; (3) the polynucleotide binding co-operativity parameter (ω^?2 × 10³) and the binding site size (n ～-7 nucleotide residues) are the same as for the native protein; (4) essentially the entire salt dependence of the net affinity (Kω) remains in K. However, unlike native gene 32 protein, G32P¹I can melt native DNA to equilibrium (Hosoda et al., 1974; Greve et al., 1978); this suggests that the kinetic pathways for DNA melting by these two species must differ, since the changes in equilibrium binding parameters measured here are far too small to account for the differences in melting behavior. In contrast to G32P¹I, for G32P¹III we find that: (1) binding is non-cooperative (ω ～-1); (2) the binding site size (n) for the protein has decreased by one to two nucleotide residues relative to that characteristic of the native protein and G32P¹I; (3) binding to short (l = 2 to 8) oligonucleotides is length and salt concentration dependent; (4) while binding to polynucleotides continues to show approximately the same base composition dependence as the native protein, the absolute values of K are somewhat different and the salt concentration dependencies of K are less. Polynucleotide ultraviolet light and circular dichroism spectra obtained in the presence of G32P¹I and G32P¹III are indistinguishable from those measured with the native protein at similar binding densities, indicating that all three protein species distort the polynucleotide lattice to comparable extents.These results are combined with the equilibrium binding data for native gene 32 protein (Kowalczykowski et al., 1980a: Newport et al., 1980) to obtain further insight into the molecular details of the interactions of this protein with its nucleic acid binding substrates.     

Tetra-, penta- and hexacoordinate copper(II) complexes with N3 donor isoindoline-based ligands: Characterization and SOD-like activity

Reaction of 1,3-bis(2′-Ar-imino)isoindolines (HLⁿ, n = 1-7, Ar = benzimidazolyl, N-methylbenzimidazolyl, thiazolyl, pyridyl, 3-methylpyridyl, 4-methylpyridyl, and benzthiazolyl, respectively) with Cu(OCH₃)₂ yields mononuclear hexacoordinate complexes with Cu(Lⁿ)₂ composition. With cupric perchlorate square-pyramidal [Cu^II(HLⁿ)(NCCH₃)(OClO₃)]ClO₄ complexes (n = 1, 3, 4) were isolated as perchlorate salts, whereas with chloride Cu^II(HLⁿ)Cl₂ (n = 1, 4), or square-planar Cu^IICl₂(HLⁿ) (n = 2, 3, 7) complexes are formed. The X-ray crystal structures of Cu(L³)₂, Cu(L⁵)₂, [Cu^II(HL⁴)(NCCH₃)(OClO₃)]ClO₄, Cu^IICl(L²) and Cu^IICl(L⁷) are presented along with electrochemical and spectral (UV-Vis, FT-IR and X-band EPR) characterization for each compound. When combined with base, the isoindoline ligands in the [Cu^II(HLⁿ)(NCCH₃)(OClO₃)]ClO₄ complexes undergo deprotonation in solution that is reversible and induces UV-Vis spectral changes. Equilibrium constants for the dissociation are calculated. X-band EPR measurements in frozen solution show that the geometry of the complexes is similar to the corresponding X-ray crystallographic structures. The superoxide scavenging activity of the compounds determined from the McCord-Fridovich experiment show dependence on structural features and reduction potentials.     

Synthesis of homodinuclear macrocyclic complexes of lanthanides and phenolic schiff bases

Successful syntheses of the first examples of homodinuclear macrocyclic lanthanide complexes are reported. The complexes were obtained as compounds of the 2:2 Schiff base formed by condensing 2,6-diformyl-p-cresol and triethylenetetramine (L₇) by a template procedure using lanthanide nitrates and perchlorates. When reactant methanolic solutions were concentrated the complexes were deposited as yellow or orange microcrystalline precipitates, Ln₂L₇(NO₃)₄sigma; nH₂O or Ln₂L₇(NO₃)_{4tau; x}(OH)_x, x = 1 or 2, whereas solutions diluted three times deposited complexes as flaky off-white crystalline precipitates of light lanthanides. The orange Ln₂L₇(NO₃)₂(OH)₂ complexes can be converted in quantitative yield to the off-white flaky form of Ln₂L₇(NO₃)₄sigma; nH₂O by refluxing them in methanolic solution containing triethylenetetramine and a three-fold excess of Ln(NO₃)₃. The complexes were characterized by elemental analysis, fast atom bombardment mass spectrometry, UV-Vis and infrared spectroscopy and thermogravimetry. Interesting and mostly new polyatomic oxo clusters, e.g. Ln₂O₃⁺, Ln₃O₄⁺, Ln₄O₆⁺, Ln₅O₇⁺, were dominant in the mass spectra but are treated in detail elsewhere.     

The determination of positive and negative co-operativity with allosteric enzymes and the interpretation of sigmoid curves and non-linear double reciprocal plots for the MWC and KNF models

(i) It is proved that only four independent constants can ever be obtained by extrapolation procedures applied to non-hyperbolic steady-state or binding data, (ii) Analysis of the algebraic graphs $\text{y}\text{x}$, $\text{(1/y)}\text{(1/x)}$, $\text{y}\text{(}\text{y}\text{x}\text{)}$ and ($\text{x}\text{y}\text{)/x}$ is shown to require a knowledge of the sign of six curve shape determinants. In each case, the sign is a necessary and sufficient condition for a specific curve shape feature, (iii) The precise graphical effect of positive and negative co-operativity then requires the definition of two reference curves, the osculating hyperbola at zero substrate concentration, OH(0), and the osculating hyperbola at infinite substrate concentration OH(∞). These are better first order approximations than the Hill equation, (iv) Rules for determining unambiguously the sign of initial, final and overall co-operativity coefficients by inspection of non-hyperbolic binding curves are then possible, (v) These rules require that saturation data for:

$\text{y=}\text{∑}\text{i=1}\text{n}\text{a}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}\text{∑}\text{i=0}\text{n}\text{β}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}$

be fitted by computer for low concentrations to the hyperbola:

$\text{OH}\text{(o)=}\text{(}\text{-a}{}_1{}^2\text{ψ}{}_{11}{}^{20}\text{)x}\text{[(}\text{-a}{}_1\text{β}{}_0\text{ψ}{}_{11}{}^{20}\text{)+x]}$

while regression of high substrate concentration data is to:

$\text{OH}\text{(∞)=}\text{(}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)x}\text{[(}\text{φ}{}_{\mathrm{n,n-1}}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)+x]}$

. Comparisons of the best fit pseudo-kinetic constants then gives the type of co-operativity present in an unambiguous way with no assumptions as to molecular mechanism, (vi) These rules are then applied to the MWC and KNF allosteric models of ligand binding and the constraints necessary for specific curve shape effects are given, (vii) The graphical expression of positive or negative final co-operativity depends only on events at high substrate concentration but overall and initial co-operativities produce specific geometric effects depending upon the difference between behaviour of saturation data at both extremes of concentration, (viii) This apparent anomaly is explained by a discussion of the relationships between the osculating hyperbolae, the theoretical parent hyperbola and the Hill plot asymptotes.     

3,5-Bis(diphenylphosphinoethyl)pyrazolate ligand (PNNP) and its dirhodium complexes: Comparison with related quadridentate dinucleating diphenylphosphinomethyl (PNNP) and phthalazine derivatives (PNNP)

Dirhodium carbonyl complex with the 3,5-bis(diphenylphosphinoethyl)pyrazolato ligand (PNNP^C2), [(μ-κ²:κ²-PNNP^C2)Rh₂(CO)₃]BF₄, is prepared and its reactivity is studied as compared with the previously reported 3,5-bis(diphenylphosphinomethyl)pyrazolate (PNNP), [(μ-κ²:κ²-PNNP){Rh(CO)₂}₂]BF₄, and 1,4-bis(diphenylphosphinomethyl)phthalazine (PNNP^Ph) derivatives, [(μ-κ²:κ²-PNNP^Ph){Rh(CO)₂}₂](BF₄)₂. The three quadridentate ligands are different in the size of the central ring and the charge; six-membered ring/neutral (PNNP^C2) vs. five-membered ring/mono-negative (PNNP) vs. six-membered ring/neutral (PNNP^Ph). The number of the carbonyl ligands (n) in the dirhodium carbonyl complexes, [(μ-PNNP)Rh₂(CO)_n](BF₄)_x, is dependent on the dinucleating ligand: n = 2 (PNNP^Ph), 3 (PNNP^C2) and 4 (PNNP^Py). The three dirhodium carbonyl complexes serve as 4e-acceptors, and their reactivities turn out to be very similar as can be seen from formation of the analogous, unique tetranuclear μ₄-acetylide ([(μ-PNNP)₂{Rh(CO)}₄(μ₄-CC-R)](BF₄)_x) and μ₄-dicarbide complexes ([(μ-PNNP)₂{Rh(CO)}₄(μ₄-C₂)](BF₄)_x).     

Letter: Kinetic negative co-operativity in the allosteric model of Monod, Wyman and Changeux

The co-operativity of homotropic interactions between substrate molecules in oligomeric enzymes is analyzed in the frame of the concerted transition theory of Monod et al. (1965). A discussion of the Hill coefficient n_H allows determination of the conditions for negative co-operativity (n_H < 1). This phenonomenon, usually taken as indicative of a sequential mechanism (Koshland et al., 1966), can be accounted for by the concerted model when the enzyme represents a K-V or V system, i.e. when the two protomer conformational states postulated in the theory differ in their catalytic activity. However, only negative co-operativity for catalysis can be explained by the concerted model, not negative co-operativity of binding.     

Effect of urea on the kinetics of ligand combination reactions of hemoglobins A and H, alphaSH chains and myoglobin.

In O₂ equilibrium studies of hemoglobin it was observed that the presence of urea increases f₅₀2 and decreases co-operativity. The results were interpreted on the basis of unfolding and swelling of the peptide chains. The results of the present kinetic investigation indicate that while the CO combination rate constants for α^SH chains, hemoglobin H and myoglobin do not change with increasing concentrations of urea, the corresponding rate constants for deoxy hemoglobin A increase continuously with urea concentration. At urea concentrations of 4 m or more, the reaction time course becomes biphasic. The fast component of the reaction time course yields CO combination rate constants which are in close agreement with the rate constants of dimeric and monomeric hemoglobins. These results indicate that up to 4 m-urea the kinetic and equilibria parameters increase due to weakened constraints imposed by intersubunit contacts and bonds. At higher urea concentrations Hb₄ is significantly dissociated into dimers and monomers, and hence the high ligand affinity and decreased co-operativity of the system. The implications of higher ligand combination rate constants of the deoxyhemoglobin tetramer in the presence of urea on the reaction mechanism are discussed.     

Chaotic Dynamics of Neuromuscular System Parameters and the Problems of the Evolution of Complexity

The evolution rate v(t) varies among diverse biosystems, but a general theory can be formulated when the dynamics of the biosystem stater x = x(t) = (x₁, x₂, x _m ) ^T is considered in the m-dimensional space of states. A mathematical approach is proposed for evaluating such processes and describes the processes in terms of particular chaos of the statistical distribution functions f(x). In the case of complex multicomponent systems with a high dimension number m (m ?1) of the phase space of states, we propose using pairwise comparison matrices of samples x(t) when homeostasis is constant and calculating the parameters of quasiattractors. The Glensdorff–Prigogine thermodynamic approach to estimating evolution is inefficient in assessing the third-type systems, while it is applicable and the Prigogine theorem works at the level of molecular systems. Alterations in the state of the human neuromuscular system were found to lead to chaotic changes in the statistical functions f(x) in tremor recording samples, while quasiattractor parameters demonstrate a certain regularity.     

Construction of analytically solvable models for interacting species

By observing that the n-tuple of rate functionsQ(c) is orthogonal to the c-space gradients of each of the (n - 1) constants of the motion Φ _v (c), a generic canonical expression for the rate functions is given in terms of the exterior product of the gradients of the (n - 1) Φ _v 's. For models withQ so prescribed from the outset, an analytical general solution is obtainable directly for the system of autonomous ordinary differential equations dc/dt =Q(c). Thus, the generic canonical expression for the rate functions can be utilized to construct analytically solvable models for interacting biological species, as ilIus~rated by examples here.     

5f-Element complexes with a p-tert-butylcalix[4]arene bearing phosphinoyl pendant arms: Separation from rare earths and structural studies

Phosphinoylated calixarenes feature high coordination ability toward f elements and a great potentiality toward actinide/rare earth separation. Here, we report three characteristic properties of a tetra-phosphinoylated p-tert-butylcalix[4]arene, B₄bL⁴ functionalized with phosphinoyl pendant arms: (i) its coordination ability toward Th(IV) complexation in organic medium, (ii) its ability to separate thorium from yttrium, lanthanum, and europium in three different organic media, and (iii) the X-ray crystal structure of the La complex. Thorium(IV) forms 1:1 and 1:2 (M:L) complexes with B₄bL⁴: Th(NO₃)₄(B₄bL⁴)_n·xH₂O (n = 1, x = 1, 1; n = 2, x = 4, 2). Spectroscopic data point to the inner coordination sphere of 1 and 2 containing nitrate ions and water molecules. Molecular modeling of 1 yielded an 8-coordinate species and its coordination polyhedron can be described as a distorted square antiprism while that for 2, a 9-coordinate species, as a distorted tricapped trigonal prism. The extraction study of tetravalent thorium and trivalent rare-earth (Y, La, Eu) ions from acidic nitrate media by B₄bL⁴ in chloroform shows thorium being much more extracted than the rare earths, with selectivity close to 100%. The extraction behavior can be easily modulated by changing the initial conditions (pH, nitrate concentration). The X-ray structure of [LaB₄bL⁴(H₂O)₅] CH₃CN·(ClO₄)₃ points to the La^III ion lying on a C₄ axis and being 9-coordinated by the four O(P) atoms and five O atoms from water molecules. It is located in the middle of the void formed by the four O-CH₂-PO(Me)₂ pendant arms.     

Analysis of ligand binding curves in terms of species fractions

The ligand binding curve for a macromolecular system presents the average number or ligand molecules bound per macromolecule as a function of the chemical potential or the logarithm of the ligand concentration. We show that various observable properties of this curve, for example its asymptotes and derivatives, are expressible in terms of linear combinations of the mole fractions α_i of macromolecules binding i molecules of ligand. Whenever enough such properties of the binding curve are known, the linear equations in α_i can be solved to give the mole fractions of each of the various macromolecular species. An application of these results is that a Hill plot for hemoglobin-ligand equilibrium where the asymptotes approach unit slope can be made to yield the four Adair constants by a simple algebraic method. A second use is that a knowledge of the first and second derivatives of the binding curve at points along the curve can yield the species fractions as functions of the degree of saturation without direct knowledge of the ligand binding constants. These methods are illustrated by some numerical examples.