QUANTITATIVE METHODS FOR HETEROGENEOUS SAMPLE COMPOSITION DETERMINATION AND BIOCHEMICAL CHARACTERIZATION

Abstract

Methods involving the use of mathematical models of competitive ligand-receptor binding to characterize mixtures of ligands in terms of compositions and properties of the component ligands have been developed. The associated mathematical equations explicitly relate component ligand physical-chemical properties and mole fractions to measurable properties of the mixture including steady state binding activity, 1/K.sub.d,apparent or equivalently 1/EC50, and kinetic rate constants k.sub.on,apparent and k.sub.off,apparent allowing; 1) component ligand physical property determination and 2) mixture property predictions. Additionally, mathematical equations accounting for combinatorial considerations associated with ligand assembly are used to compute ligand mole fractions. The utility of the methods developed is demonstrated using published experimental ligand-receptor binding data obtained from mixtures of afucosylated antibodies that bind FcγRIIIa (CD16a) to: 1) extract component ligand physical property information that has hitherto evaded researchers 2) predict experimental observations and 3) provide explanations for unresolved experimental observations.

Claims

1. A method for characterizing a constituent ligand in a mixture of ligands in terms of a property of said ligand comprising: a. measuring said property of said mixture, b. using a mathematical model of competitive ligand-receptor binding with said model using said property of said mixture to compute said property of said ligand.

2. The method in claim 1 wherein said ligand is an antibody molecule.

3. The method in claim 1 wherein said property is an equilibrium constant and said model is $\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_1}{K_1}+\frac{X_2}{K_2}+\mathrm{.Math.}.Math..Math.\frac{X_{m-1}}{K_{m-1}}+\frac{X_m}{K_m}$ or a mathematical equivalent.

4. The method in claim 1 wherein said property is a kinetic rate constant and said model is
k.sub.on,apparent=X.sub.1k.sub.on,1+X.sub.2k.sub.on,2+ . . . +X.sub.m−1k.sub.on,m−1+X.sub.mk.sub.on,m or
k.sub.off,apparent=f.sub.1*k.sub.off,1+f.sub.2*k.sub.off,2+f.sub.m−1*k.sub.off,m−1+f.sub.m*k.sub.off,m, or a mathematical equivalent.

5. The method in claim 1 wherein said mixture comprises antibody molecules and said ligand is an antibody molecule and said model is $\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_1}{K_1}+\frac{X_2}{K_2}+\mathrm{.Math.}.Math..Math.\frac{X_{m-1}}{K_{m-1}}+\frac{X_m}{K_m}$ or a mathematical equivalent further including use of the binomial distribution to compute the composition of said mixture.

6. A method for characterizing a constituent ligand in a mixture of ligands in terms of a property of said ligand in said mixture comprising: a. measuring said property of said mixture, b. using the binomial distribution to compute the composition of said mixture, c. using a mathematical model of competitive ligand-receptor binding with said model using said property of said mixture and said composition to compute said property of said ligand.

7. The method in claim 6 wherein said ligand is an antibody molecule and said property is an equilibrium constant and said model is $\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_1}{K_1}+\frac{X_2}{K_2}+\mathrm{.Math.}.Math..Math.\frac{X_{m-1}}{K_{m-1}}+\frac{X_m}{K_m}$ or a mathematical equivalent.

8. The method in claim 6 wherein said ligand comprises a plurality of subunits with said subunits having identical amino acid sequences.

9. The method in claim 6 wherein said ligand in an antibody molecule and said property is a kinetic rate constant and said model is
k.sub.on,apparent=X.sub.1k.sub.on,1+X.sub.2k.sub.on,2+ . . . +X.sub.m−1k.sub.on,m−1+X.sub.mk.sub.on,m or
k.sub.off=k.sub.off,apparent=f.sub.1*k.sub.off,1+f.sub.2*k.sub.off,2+f.sub.m−1*k.sub.off,m−1+f.sub.m*k.sub.off,m or a mathematical equivalent.

10. The method in claim 6 wherein said mixture comprises homogeneous fucosylated antibody, hemi-afucosylated antibody and homogeneous afucosylated antibody and said ligand is an antibody molecule and said model is $\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_A}{K_A}+\frac{X_F}{K_F}+\frac{X_{\mathrm{AF}}}{K_{\mathrm{AF}}}$ or
k.sub.on=k.sub.on,apparent=X.sub.Ak.sub.on,A+X.sub.Fk.sub.on,F+X.sub.AFk.sub.on,AF or
k.sub.off=k.sub.off,apparent=f.sub.Ak.sub.off,A+f.sub.Fk.sub.off,F+f.sub.AFk.sub.off,AF or a mathematical equivalent.

11. A method for computing a property of a mixture of ligands comprising: a. determining said property of constituent ligands in said mixture, b. determining the composition of said mixture, c. using a mathematical model of competitive ligand-receptor binding with said model using said property of said ligands and said composition to compute said property of said mixture.

12. The method in claim 11 wherein said property of said mixture is an equilibrium constant and said model is $\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_1}{K_1}+\frac{X_2}{K_2}+\mathrm{.Math.}.Math..Math.\frac{X_{m-1}}{K_{m-1}}+\frac{X_m}{K_m}$ or a mathematical equivalent.

13. The method in claim 11 wherein said property of said mixture is a kinetic rate constant and said model is
k.sub.on,apparent=X.sub.1k.sub.on,1+X.sub.2k.sub.on,2+ . . . +X.sub.m−1k.sub.on,m−1X.sub.mk.sub.on,m or
k.sub.off=k.sub.off,apparent=f.sub.1*k.sub.off,1+f.sub.2*k.sub.off,2+f.sub.m−1*k.sub.off,m−1+f.sub.m*k.sub.off,m or a mathematical equivalent.

14. The method in claim 11 wherein said mixture comprises antibody molecules and said model is $\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_1}{K_1}+\frac{X_2}{K_2}+\mathrm{.Math.}.Math..Math.\frac{X_{m-1.Math.}}{K_{m-1}}+\frac{X_m}{K_m}$ or
k.sub.on,apparent=X.sub.1k.sub.on,1+X.sub.2k.sub.on,2+ . . . +X.sub.m−1k.sub.on,m−1+X.sub.mk.sub.on,m or
k.sub.off=k.sub.off,apparent=f.sub.1*k.sub.off,1+f.sub.2*k.sub.off,2+f.sub.m−1*k.sub.off,m−1+f.sub.m*k.sub.off,m or a mathematical equivalent.

15. The method in claim 11 wherein said mixture comprises homogeneous fucosylated antibody, hemi-afucosylated antibody and homogeneous afucosylated antibody and said model is $\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_A}{K_A}+\frac{X_F}{K_F}+\frac{X_{\mathrm{AF}}}{K_{\mathrm{AF}}}$ or
k.sub.on=k.sub.on,apparent=X.sub.Ak.sub.on,A+X.sub.Fk.sub.on,F+X.sub.AFk.sub.on,AF or
k.sub.off=k.sub.off,apparent=f.sub.Ak.sub.off,A+f.sub.Fk.sub.off,F+f.sub.AFk.sub.off,AF or a mathematical equivalent.

16. The method in claim 11 wherein said property of said mixture is the fraction receptors occupied f wherein said model comprises $f=\underset{i=1}{\overset{m}{.Math.}}.Math.f_i.Math..Math.\mathrm{and}.Math..Math.f_i=\frac{\left[L_i\right]}{\left[L_i\right]+K_i\left(1+\underset{j=1,j\ne i}{\overset{m}{.Math.}}.Math.\frac{\left[L_j\right]}{K_j}\right)}$ or a mathematical equivalent.

17. The method in claim 11 wherein said mixture comprises antibodies and said property of said mixture is the fraction receptors occupied f wherein said model comprises $f=\underset{i=1}{\overset{m}{.Math.}}.Math.f_i.Math..Math.\mathrm{and}.Math..Math.f_i=\frac{\left[L_i\right]}{\left[L_i\right]+K_i\left(1+\underset{j=1,j\ne i}{\overset{m}{.Math.}}.Math.\frac{\left[L_j\right]}{K_j}\right)}$ or a mathematical equivalent further including use of the binomial distribution to compute said composition of said mixture.

Description

DRAWINGS—FIGURES

[0015] FIG. 1: Flow chart for computing ligand properties.

[0016] FIG. 2: Computing the dissociation equilibrium constant K.sub.AF.

[0017] FIG. 3: Computing the dissociation equilibrium constant K.sub.A.

[0018] FIG. 4: Computing kinetic rate constants k.sub.on,AF and k.sub.off,AF.

[0019] FIG. 5: Flow chart for computing mixture properties.

[0020] FIG. 6: Computing 1/K.sub.d,apparent.

[0021] FIG. 7: Computing receptor binding dose-response curves.

DETAILED DESCRIPTION—FIRST EMBODIMENT—FIG. 1

[0022] FIG. 1 shows the general flowchart of how mechanistic mathematical models of competitive ligand-receptor binding are used to characterize mixtures of glycoprotein ligands showing steps that are common with current methods, with solid outlines, and the steps that involved the use of mechanism-based mathematical models, with dashes outlines. The top two “boxes” represent two common orthogonal or independent methods used to characterize mixtures of glycoprotein ligands; biochemical activity and glycoform or carbohydrate analysis. The arrows joining the boxes denote the flow of information or data.

[0023] Measures of biochemical activity may be steady state or kinetic in nature. Steady state measures of biochemical activity include 1/EC50, or equivalently 1/K.sub.d,apparent, obtained from steady state ligand-receptor binding curves where EC50 and K.sub.d,apparent denote the experimental ligand concentration that induces a ½ maximal experimental output in a ligand-receptor binding assay. As discussed later, kinetic rate constants such as k.sub.on,apparent also measure biochemical activity. The subscript “apparent” denotes the fact that in general the value of K.sub.d and k.sub.on obtained from mixtures will depend on mixture composition. As such K.sub.d and k.sub.on for mixtures are denoted K.sub.d,apparent and k.sub.on,apparent respectively. With pure samples, composition is no longer variable and the subscript “apparent” is omitted.

[0024] In addition to biochemical activity, carbohydrate composition information is routinely available for glycoprotein ligands of industrial importance. Glycans or carbohydrates must be physically removed from the amino acid backbone of the glycoprotein before their compositions can be determined so that carbohydrate composition data generally provides sample average information on glycoform structure and composition. For mixtures comprising the three afucosylated antibody glycoforms, glycoform or carbohydrate analysis provides the overall fraction of Fc glycans that are afucosylated, denoted p.

[0025] The dashed boxes in FIG. 1 show how mathematical models are used to extract additional biochemical property information from existing activity and glycoform analysis data. The bottom box depicts the use of mathematical models of competitive ligand-receptor modeling to compute component ligand properties such as the dissociation equilibrium constant K.sub.i from mixture biochemical activity and mole fractions X.sub.i's. K.sub.d,apparent is obtained directly from experimental measurement of receptor binding activity. Mole fractions or mixture composition are obtained by many methods depending on the specific ligand-receptor system including; direct experimental measurements, mass balances or statistical distributions or combinations of such methods.

[0026] The term “competitive” used in the phrase “mathematical model of competitive ligand-receptor binding” refers specifically to the scientifically accepted use of the term “competitive” in the phrases “competition binding” and “competitive inhibition” in the fields of Biochemistry and Biophysics. Competitive binding is mutually exclusive in nature requiring that the binding of ligand i to a specific receptor site is sufficient to prevent the binding of a different ligand j to the same receptor binding site and vice-versa. Thus, the different ligands compete for binding to common receptor binding sites. These concepts are described more formally using mathematics in the following sections.

Steady State Data Analysis

[0027] For a system of m ligands L.sub.1, L.sub.2, . . . , L.sub.m−1, L.sub.m, that compete for binding to a common receptor R, the following m chemical equations apply:

L.sub.1+RRL.sub.1, L.sub.2+RRL.sub.2, L.sub.m+RRL.sub.m.

[0028] The general mathematical equation imposed on this system at steady state by the competitive binding mechanism is given by equation (1):

[00001]$\begin{array}{cc}\mathrm{activity}=\frac{1}{\mathrm{EC}.Math..Math.50}=\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_1}{K_1}+\frac{X_2}{K_2}+\mathrm{.Math.}+\frac{X_m}{K_m}& \left(1\right)\end{array}$

with mole fractions X.sub.i:

[00002]$X_i=\frac{\left[L_i\right]}{\underset{j=1}{\overset{m}{.Math.}}.Math.\left[L_j\right]}.$

[L.sub.i] and X.sub.i denote the molar concentration and the mole fraction of unbound component ligand i. Equation 1 is obtained by combining and algebraically manipulating; the definition of the dissociation equilibrium constant for ligand i, K.sub.i, the definition of the apparent dissociation equilibrium constant for mixtures, K.sub.d,apparent, and the molar balances on ligand and ligand-receptor complex:

[00003]$K_{L_1}=\frac{\left[R\right]\left[L_1\right]}{\left[{\mathrm{RL}}_1\right]},K_{L_2}=\frac{\left[R\right]\left[L_2\right]}{\left[{\mathrm{RL}}_2\right]},\mathrm{.Math.}.Math.,K_{L_n}=\frac{\left[R\right]\left[L_m\right]}{\left[{\mathrm{RL}}_m\right]}.Math..Math.\mathrm{and}.Math..Math.$$K_{d,\mathrm{apparent}}=\frac{{\left[R\right]\left[L\right]}_{\mathrm{total}}}{{\left[\mathrm{RL}\right]}_{\mathrm{total}}},$

with molar balances:

[L].sub.total=[L.sub.1]+[L.sub.2]+ . . . +[L.sub.m] (ligand balance)

[RL].sub.total=[RL.sub.1]+[RL.sub.2]+ . . . +[RL.sub.m] (ligand-receptor complex)

[R] denotes the unbound molar concentration of receptor R. [RL.sub.i] denotes the molar concentration of ligand i-receptor complex.

[0029] Equation (1) relates experimental receptor binding activity, defined as 1/EC50, to component ligand mole fractions, X.sub.i's, and dissociation equilibrium constants, s. In terms of the competitive binding model, 1/EC50 is identically the model parameter 1/K.sub.d,apparent. Equation (1) reveals that mixture activity, 1/K.sub.d,apparent, is the sum of the specific activities of the component ligands, 1/K.sub.i's, weighted by their respective mole fractions. Accordingly, equation (1) provides the means to compute the component ligand dissociation equilibrium constants K.sub.i's, or the corresponding binding constants given by 1/K.sub.i's when provided with the appropriate composition and mixture activity data.

[0030] Equation (2) is the specific form of equation (1) that describes the antibody afucosylation system comprising three different antibody glycoform ligands:

[00004]$\begin{array}{cc}\mathrm{activity}=\frac{1}{\mathrm{EC}.Math..Math.50}=\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_A}{K_A}+\frac{X_F}{K_F}+\frac{X_{\mathrm{AF}}}{K_{\mathrm{AF}}}.& \left(2\right)\end{array}$

Equation (2) may be obtained directly from equation (1) using the appropriate subscripts. In terms of this ternary antibody system, K.sub.d,apparent is defined by:

[00005]$K_{d,\mathrm{apparent}}=\frac{{\left[R\right]\left[\mathrm{Ab}\right]}_{\mathrm{total}}}{{\left[\mathrm{RAb}\right]}_{\mathrm{total}}}$

with [Ab].sub.total and [RAb].sub.total denoting the molar concentration of total unbound antibody and the sum of the molar concentrations of all three antibody-receptor complexes respectively. The three afucosylated antibody glycoforms are differentiated by their afucosylated Fc glycan content with the homogeneous fucosylated antibody F containing zero afucosylated Fc glycans, the hemi-afucosylated antibody AF containing one afucosylated Fc glycan, and the homogeneous afucosylated antibody A containing two afucosylated Fc glycans. The Fc region of the three afucosylated antibody ligands compete for binding to the FcγRIIIa (CD16a) receptor with the associated dissociation equilibrium constants K.sub.A, K.sub.AF and K.sub.F, where the subscripts denote the specific glycoforms. X.sub.A, X.sub.F and X.sub.AF denote the three antibody mole fractions with the subscripts denoting the specific glycoform.

[0031] Use of the mathematical models described to analyze experimental receptor binding data requires that the unbound concentrations of antibodies and receptor are known. When such data are not readily available, the general molar excess of ligand over receptor allows the ligand concentrations appearing in the equations to be approximated by the molar ligand concentration added to the experimental samples. Unless otherwise noted, the numerical values used for ligand or antibody concentrations are assumed to be equal to the ligand or antibody concentrations added to the sample.

Kinetic Data Analysis

[0032] The general mathematical constraint imposed by the competitive ligand-receptor binding mechanism on the forward kinetic rate constants of a ligand mixture comprises m components is given by:

k.sub.on=k.sub.on,apparent=X.sub.1k.sub.on,1+X.sub.2k.sub.on,2+ . . . +X.sub.m−1k.sub.on,m−1+X.sub.mk.sub.on,m  (3).

Equation (3) reveals that the forward rate constant for the mixture, k.sub.on,apparent, is the sum of the forward rate constants of the component ligands, k.sub.on,is, weighted by their respective mole fractions. For the afucosylated antibody system involving three glycoform ligands, equation (3) simplifies to:

k.sub.on=k.sub.on,apparent=X.sub.Ak.sub.on,A+X.sub.Fk.sub.on,F+X.sub.AFk.sub.on,AF  (4)

with the component ligand subscripts altered accordingly.

[0033] Equation (3) is obtained by noting that the overall rate of ligand-receptor binding is the sum of the rates of ligand-receptor binding of the component ligands:

[00006]$r_{\mathrm{on}}={k_{\mathrm{on},\mathrm{apparent}}\left[L\right]}_{\mathrm{total}}\left[R\right]=\underset{i=1}{\overset{m}{.Math.}}.Math.r_{\mathrm{on},i}.$

Combining the above equation with component ligand mass action rate laws,

r.sub.on,i=k.sub.on,i[L.sub.i][R] i=1,2, . . . ,m

and solving for k.sub.on,apparent yields equation (3). Equation (4) follows immediately from equation (3) with m=3.

[0034] The reverse kinetic rate constants, k.sub.off,i, for the component ligands can be computed using the dissociation equilibrium constant K.sub.i and the forward rate constant k.sub.on,i using the well-known equation:

[00007]$\begin{array}{cc}{K}_i=\frac{k_{\mathrm{off},i}}{k_{\mathrm{on},i}}.& \left(5\right)\end{array}$

[0035] Alternatively, performing an analysis similar to that used to derive equation (3), one can arrive at:

[00008]$\begin{array}{cc}{k}_{\mathrm{off}}=k_{\mathrm{off},\mathrm{apparent}}=f_1^{*}.Math.k_{\mathrm{off},1}+f_2^{*}.Math.k_{\mathrm{off},2}+f_{m-1}^{*}.Math.k_{\mathrm{off},m-1}+f_m^{*}.Math.k_{\mathrm{off},m}.Math..Math.f_i^{*}=\frac{\left[{\mathrm{RL}}_i\right]}{{\left[\mathrm{RL}\right]}_{\mathrm{total}}}.& \left(6\right)\end{array}$

f.sub.i* denotes the fraction of all the bound receptors [RL].sub.total that include ligand i in complex with receptor R. The ternary component analog of equation (6) applicable to the antibody afucosylation system is given by:

k.sub.off=k.sub.off,apparent=f.sub.A*k.sub.off,A+f.sub.F*k.sub.off,F+f*.sub.AFk.sub.off,AF  (7).

Statistical and Combinatorial Considerations

[0036] The primary sequence of many glycoform ligands can often be complex with ligands comprising multiple subunits. Therefore in general, glycoform molar compositions cannot be deduced from carbohydrate or glycan composition data. However when a ligand comprises more than one subunit with the same primary sequence, with the primary sequence containing a glycosylation site, combinatorial considerations can be used to compute the mole fractions of the glycoform variants of the glycoprotein.

[0037] The compositions of the afucosylated antibody glycoforms may be computed using the binomial distribution with n=2. Antibody assembly involves the dimerization of two antibody heavy chains with identical amino acid sequence. Each antibody heavy chain possesses an Fc bound glycan with one potential fucosylation site. Fucosylated antibody heavy chains have a core fucose molecule attached to the base of the glycan bound to the conserved Asn.sup.297 glycosylation site. Afucosylated antibody heavy chains are devoid of said fucose molecule. The probability that a heavy chain will be afucosylated is given by the fraction of Fc glycan that are afucosylated or p. Therefore the binomial distribution with n=2 can be used to compute the mole fractions of the three afucosylated antibody glycoforms, X.sub.F, X.sub.A and X.sub.AF, in accordance with equation (8):

X.sub.A=p.sup.2

X.sub.F=(1−p).sup.2

X.sub.AF=2p(1−p)  (8).

Combining equation (8) with equation (2) yields:

[00009]$\begin{array}{cc}\mathrm{activity}=\frac{1}{\mathrm{EC}.Math..Math.50}=\frac{1}{K_{d,\mathrm{apparent}}}=\frac{p^2}{K_A}+\frac{2.Math.\left(p-p^2\right)}{K_{\mathrm{AF}}}+\frac{{\left(1-p\right)}^2}{K_F}.& \left(9\right)\end{array}$

Similarly, combining equation (8) and equation (4) yields:

k.sub.on=k.sub.on,apparent=p.sup.2k.sub.on,A+(1−p).sup.2k.sub.on,F+2(p−p.sup.2)k.sub.on,AF  (10)

[0038] Equation (8) reveals that pure samples of homogeneous fucosylated and afucosylated antibodies are obtained in the limiting cases when the fraction of Fc glycans that are afucosylated p approaches either zero or unity respectively. When the afucosylated Fc glycan fraction p.fwdarw.0, the mixture for all practical purposes is considered to be a “pure” homogeneous fucosylated antibody sample and X.sub.F.fwdarw.1. Similarly when p.fwdarw.1, the mixture for all practical purposed is considered to be a “pure” homogeneous afucosylated antibody sample and X.sub.A.fwdarw.1.

OPERATION—FIRST EMBODIMENT—FIGS. 2-4

[0039] The computational flowchart in FIG. 1 shows the general flow of information involved in using mathematical models to computing component ligand properties from experimental data obtained from mixtures. The general approach is summarized in the following steps: [0040] 1) acquire mixture property information such as K.sub.d,apparent, by direct experimental measurements, [0041] 2) acquire mixture compositions or mole fractions using: direct measurement, statistical distributions, mass balances or combinations of these elements, [0042] 3) obtain the specific form the general mathematical model of competitive ligand-receptor binding based on experimental considerations [0043] 4) use mathematical model to compute component ligand property.
These steps allow component ligand properties to be computed from data obtained from mixtures of ligands. When a component ligand cannot be isolated in pure form, the steps outlined provide a means to determine its' properties. Such a means does not currently exist. When a component ligand can be isolate in pure form, determining its' properties is straightforward. The methods comprising the steps listed assume that the properties of glycoforms that are available in pure form are available. The term “pure” refers to a sample of an antibody typically in excess of 95% on a molar basis since 100% purity is rarely achieved in practice.

Example 1: Computing K.SUB.AF

[0044] FIG. 2 shows the flowchart used to compute K.sub.AF for the hemi-afucosylated antibody. Afucosylated Fc glycan content measured by p are used to compute glycoform mole fractions using equation (8). Composition and mixture activities are then used with mathematical models of competitive ligand-receptor binding to compute component ligand properties. The appropriate mathematical models used to compute K.sub.AF from the data published by Chung and coworkers (2012) for low p mixtures are given by:

[00010]$\begin{array}{cc}\mathrm{activity}=\frac{1}{\mathrm{EC}.Math..Math.50}=\frac{1}{K_{d,\mathrm{apparent}}}=2.Math.p\left[\frac{1}{K_{\mathrm{AF}}}-\frac{1}{K_F}\right]+\frac{1}{K_F}.Math..Math..Math.\left(p\le 0.1\right)& \left(11\right)\\ \mathrm{RA}=\frac{\mathrm{EC}.Math..Math.{50}_F}{\mathrm{EC}.Math..Math.50}=\frac{K_F}{K_{d,\mathrm{apparent}}}=2.Math.p\left[\frac{K_F}{K_{\mathrm{AF}}}-1\right]+1.Math..Math.\left(p\le 0.1\right).& \left(12\right)\end{array}$

Equation (12) is obtain from equation (11) by dividing equation (11) by 1/K.sub.F. Equation (11) is obtained directly from equation (9) by noting that when p≦0.1, p.sup.2 terms may be neglected and equation (9) simplifies to yield equation (11).

[0045] Equations (11) predicts that activity or 1/K.sub.d,apparent will scale linearly with p when p≦0.1 and that the slope and the y-intercept associated with this linear relationship are given by 2[1/K.sub.AF−1/K.sub.F] and 1/K.sub.F respectively. Similarly, the slope and the y-intercept associated with equation (12) are given by the terms 2[K.sub.F/K.sub.AF−1] and 1 respectively. Therefore the existence of a linear relationship between mixture activity and p when p≦0.1 can be used to compute K.sub.AF.

[0046] Chung and coworkers (2012) reported on the existence of an empirical linear correlation between relative activity (RA), obtained from ELISA's measuring antibody Fc-FcγRIIIa F158 binding, and afucosylated Fc glycan fraction p for IgG1 mixtures characterized by p≦0.1. Experimental RA data is obtained by dividing mixture activity, 1/K.sub.d,apparent, by the activity of pure homogeneous fucosylated antibodies or 1/K.sub.F. Experimental RA data as defined is given by K.sub.d,apparent/K.sub.F in terms of the model parameters. Note that equations (11) and (12) theoretically predict the existence of a linear relationship between activity and afucosylation content. Therefore equation (12) is the appropriate mathematical model for analyzing the experimental relative activity data of Chung and coworkers.

[0047] Using the slope value of the empirical linear correlation reported by Chung and coworkers, 80, K.sub.AF can be computed:

[00011]$\mathrm{slope}=80=2\left[\frac{K_F}{K_{\mathrm{AF}}}-1\right]=2\left[\frac{12.Math.\mathrm{nM}}{K_{\mathrm{AF}}}-1\right]$

yielding K.sub.AF=0.30 nM where K.sub.F=12 nM has been used. The value of K.sub.F used to compute K.sub.AF was independently obtained from activity data gathered for the pure homogeneous fucosylated antibody (Chung et al. 2012). The value of K.sub.AF so computed is the first to appear in the public domain.

Example 2: Computing K.SUB.A

[0048] FIG. 3 shows the flowchart used to compute K.sub.A, the dissociation equilibrium constant for the homogeneous afucosylated antibody, using activity obtained from mixtures. Chung and coworkers (2012) constructed mixtures comprised of defined proportions of the homogeneous fucosylated F and the homogeneous afucosylated A antibodies so as to arbitrarily set p in accordance with the relationship between p and mole fractions given by:

[00012]$\begin{array}{cc}p=X_A+\frac{X_{\mathrm{AF}}}{2}.& \left(13\right)\end{array}$

For example, a sample with p=0.5 was created by preparing an equimolar mixture of pure homogeneous afucosylated, X.sub.A=1, and pure homogeneous fucosylated, X.sub.F=1, antibodies. Since these artificial mixtures are characterized by the absence of the hemi-afucosylated form, or X.sub.AF≈0, the binomial distribution cannot be used to compute mole fractions as in EXAMPLE 1. However, mole fractions may be obtained from afucosylated Fc glycan fractions p immediately from equation (13) by noting that p=X.sub.A when X.sub.AF≈0.

[0049] The appropriate mathematical model for analyzing the data of Chung and coworkers is given by:

[00013]$\begin{array}{cc}\mathrm{Activity}=\frac{1}{\mathrm{EC}.Math..Math.50}=\frac{1}{K_{d,\mathrm{apparent}}}=p\left[\frac{1}{K_A}-\frac{1}{K_F}\right]+\frac{1}{K_F}.Math..Math..Math.\left(X_{\mathrm{AF}}\approx 0\right)& \left(14\right)\\ \mathrm{RA}=\frac{\mathrm{EC}.Math..Math.{50}_F}{\mathrm{EC}.Math..Math.50}=\frac{K_F}{K_{d,\mathrm{apparent}}}=X_A\left[\frac{K_F}{K_A}-1\right]+1.Math..Math.\left(X_{\mathrm{AF}}\approx 0\right).& \left(15\right)\end{array}$

Equation (15) is obtain from equation (14) by dividing equation (14) by 1/K.sub.F. Equation (14) is obtained directly from equation (2) by noting that X.sub.AF≈0 and using the mole fraction constraint

[00014]$\begin{array}{cc}\underset{i=1}{\overset{m}{.Math.}}.Math.X_i=1.& \left(16\right)\end{array}$

[0050] Equations (14) predicts that activity or 1/K.sub.d,apparent will scale linearly with p and that the slope and the y-intercept associated with this linear relationship are given by [1/K.sub.A−1/K.sub.F] and 1/K.sub.F respectively. Similarly, the slope and the y-intercept associated with equation (15) are given by the terms [K.sub.F/K.sub.A−1] and 1 respectively. Therefore the existence of a linear relationship between mixture activity and p for artificial mixtures can be used to compute K.sub.A.

[0051] Chung and coworkers (2012) reported on the existence of an empirical linear correlation between relative activity (RA), obtained from ELISA's measuring antibody Fc-FcγRIIIa F158 binding, and afucosylated Fc glycan fraction p for binary mixtures of homogeneous fucosylated and homogeneous afucosylated IgG1. Experimental RA data is obtained by dividing mixture activity, 1/K.sub.d,apparent, by the activity of pure homogeneous fucosylated antibodies or 1/K.sub.F. Experimental RA as defined is given by K.sub.d,apparent/K.sub.F in terms of the model parameters. Note that equations (14) and (15) theoretically predict the existence of a linear relationship between activity and afucosylation content. Therefore equation (15) is the appropriate mathematical model for analyzing the experimental relative activity data. Using the slope value of the empirical linear correlation reported by Chung and coworkers, 25, K.sub.A can be computed using the relationship:

[00015]$\mathrm{slope}=25=\left[\frac{K_F}{K_A}-1\right]=\left[\frac{12.Math.\mathrm{nM}}{K_A}-1\right]$

to yield K.sub.A=0.46 nM where K.sub.F=12 nM has been used. The value of K.sub.F used to compute K.sub.A was independently obtained from activity data gathered for the pure homogeneous fucosylated antibody (Chung et al. 2012).

Example 3: Deconvolute Activity into Composition & Specific Activity

[0052] The competitive ligand-receptor binding mechanism provides the means to decompose mixture activity into component ligand contributions and to dissect component ligand contributions into composition and specific activity differences. For the afucosylated antibody system, steady state receptor binding activity is given by equation (2):

[00016]$\begin{array}{cc}\mathrm{activity}=\frac{1}{\mathrm{EC}.Math..Math.50}=\frac{1}{K_{d,\mathrm{apparent}}}=\frac{X_A}{K_A}+\frac{X_F}{K_F}+\frac{X_{\mathrm{AF}}}{K_{\mathrm{AF}}}.& \left(2\right)\end{array}$

Equation (2) reveals that the contribution to activity of each constituent antibody is the multiplicative product of the specific activity 1/K, and the mole fraction X.sub.i of the antibody. Therefore knowledge of the three dissociation equilibrium constants K.sub.A, K.sub.F and K.sub.AF and the mole fractions X.sub.A, X.sub.F and X.sub.AF is sufficient to completely and uniquely decomposed mixture activity. Due to the availability of pure homogeneous afucosylated and fucosylated antibodies, K.sub.A and K.sub.F are obtained using standard experimental methods. However the hemi-afucosylated antibody cannot be isolated in pure form so that K.sub.AF can only be obtained using mathematical methods such as described in detail in EXAMPLE 1. Experimental limitations also preclude direct determination of the mole fractions of the different afucosylated antibody glycoforms necessitating use of the binomial distribution to compute compositions based on statistical considerations governing ligand assembly.

[0053] TABLE 1 shows the decomposition of mixture activity into the contributions to activity of the different afucosylated antibody glycoforms for different samples or mixtures. Column one shows the mixture activity for five different samples with different afucosylated Fc glycan fraction p (column two). Since activity is defined as 1/EC50 or 1/K.sub.d,apparent it has units of reciprocal concentration. Columns 3-5 show how mixture activity is decomposed into the contributions of the different ligands. The last three columns show the mole fractions of the different glycoforms in the mixture computed using equation (8). The values of K.sub.A, K.sub.F and K.sub.AF used to compute component glycoform activities are 0.46 nM, 12 nM and 0.30 nM respectively for the FcγRIIIa F158 allotype. K.sub.AF and K.sub.A were computed from experimental data as described in EXAMPLE 1 and EXAMPLE 2. K.sub.F was obtained directly from experimental activity data for pure homogeneous fucosylated antibody. The data reveal that the both the hemi-afucosylated and the homogeneous afucosylated antibodies contribute significantly to activity.

TABLE-US-00001 TABLE 1 Component Antibody Activity for Different Ligand Mixtures Activity X.sub.A/K.sub.A X.sub.F/K.sub.F X.sub.AF/K.sub.AF [nM.sup.−1] p [nm.sup.−1] [nM.sup.−1] [nm.sup.−1] X.sub.A X.sub.F X.sub.AF 0.08 0 0.0 0.08 0.0 0.0 1 0 0.41 0.05 0.0 0.075 0.33 0.0 0.90 0.10 0.74 0.1 0.0 0.07 0.67 0.0 0.80 0.20 2.36 0.9 1.76 0.0 0.6 0.81 0.01 0.18 2.17 1 2.17 0.0 0.0 1 0.0 0.0

Example 4: Computing k.SUB.on,AF .and k.SUB.on,AF

[0054] FIG. 4 shows the flowchart used to compute k.sub.on,AF and k.sub.off,AF, the forward and the reverse kinetic rate constants associated with the binding of the hemi-afucosylated antibody glycoform to receptor R. k.sub.on,apparent and k.sub.off,apparent obtained from mixtures are combined with mole fractions to compute component glycoform rate constants k.sub.on,i and k.sub.on,i. Rates constants for the ternary system comprising the three afucosylated antibody glycoforms are constrained by equation (4) as the direct consequence of the competitive binding mechanism:

k.sub.on=k.sub.on,apparent=X.sub.Ak.sub.on,A+X.sub.Fk.sub.on,F+X.sub.AFk.sub.on,AF  (4)

Using the binomial distribution with n=2, equation (8), to compute component antibody mole fractions yields:

k.sub.on=k.sub.on,apparent=p.sup.2k.sub.on,A(1−p).sup.2k.sub.on,F+2(p−p.sup.2)k.sub.on,AF  (17)

For low p mixtures, p≦0.1, equation (17) simplifies yielding:

k.sub.on,apparent=k.sub.on,F+2pk.sub.on,AF  (18).

Equation (18) predicts that k.sub.on,apparent will vary linearly with afucosylated Fc glycan fraction p and that the slope and the y-intercept of this linear relationship are 2k.sub.on,AF and k.sub.on,F respectively. Therefore a plot of k.sub.on,apparent versus p can be used to compute k.sub.on,AF. k.sub.off,AF may be straightforwardly computed from K.sub.AF and k.sub.on,AF using equation (5).

DETAILED DESCRIPTION—SECOND EMBODIMENT—FIG. 5

[0055] FIG. 5 shows how mathematical models of competitive ligand-receptor binding are used to predict or compute mixture properties, such as K.sub.d,apparent, using component ligand mole fractions, X.sub.i's, and properties. The main difference between this and the previously described embodiment is the flow of information described by the arrows in FIG. 5. The same mathematical models that are used to compute component properties can be used to compute mixture properties. Only the inputs, solid arrows, and the outputs, open arrows, into the mathematical equations have changed. Mole fractions and component ligand properties such as K.sub.1 and k.sub.on,i, are used to compute mixture properties such as the equilibrium constant K.sub.d,apparent and the kinetic rate constant k.sub.on,apparent. Component ligands compositions may be obtained from a number of sources including if possible direct experimental measurements or indirectly using statistical distributions and mass balances.

[0056] Also of interest is the variable f, the fraction receptors occupied. Experimentally, f represents the normalized dose response output obtained from classical ligand-receptor binding assays such as ELISAs and is defined as the ratio of the molar concentrations of total bound receptors [RL].sub.total to total receptors [R].sub.total or:

[00017]$\begin{array}{cc}f=\frac{{\left[\mathrm{RL}\right]}_{\mathrm{total}}}{{\left[R\right]}_{\mathrm{total}}}.& \left(19\right)\end{array}$

The appropriate equation for computing f from total ligand concentration [L].sub.total is:

[00018]$\begin{array}{cc}f=\frac{{\left[L\right]}_{\mathrm{total}}}{{\left[L\right]}_{\mathrm{total}}+K_{d,\mathrm{apparent}}}.& \left(20\right)\end{array}$

Equation (20) is obtained by combining the definition off the definition of K.sub.d,apparent, the definitions of K.sub.i and the overall receptor molar balance:

[R].sub.total=[R]+[RL].sub.total.

[0057] For an m ligand system, the competitive ligand-receptor binding model allows f to be expressed as the sum of the component ligand contributions:

[00019]$\begin{array}{cc}f=\underset{i=1}{\overset{m}{.Math.}}.Math.f_i.Math..Math.\mathrm{with}& \left(21\right)\\ f_i=\frac{\left[L_i\right]}{\left[L_i\right]+K_i\left(1+\stackrel{m}{\underset{j=1,j\ne i}{.Math.}}.Math.\frac{\left[L_j\right]}{K_j}\right)}.& \left(22\right)\end{array}$

f.sub.i denotes the fraction of overall available receptors occupied by ligand i. Equations (20) and (21) predict identical responses f. However equation (21) shows how f comprises the component ligand contributions. Since each f.sub.i cannot in general be obtained experimentally, equations (21) and (22) are able to extract information from existing data that would otherwise remain unrevealed.

OPERATIONS—SECOND EMBODIMENT—FIGS. 6 AND 7

[0058] Use of mathematical models of competitive ligand-receptor binding to compute mixture properties is outlined in the general following steps: [0059] 1) acquire component property information such as the dissociation equilibrium constants K.sub.i's, [0060] 2) acquire the mole fractions of the mixture components using: direct measurements, statistical distributions, mass balances or combinations of these elements, [0061] 3) obtain the specific form the general mathematical model of competitive ligand-receptor binding using experimental considerations, [0062] 4) use mathematical model to compute mixture property.

Example 1: Mixtures of Homogeneous Antibodies

[0063] For binary mixture of homogeneous fucosylated and homogeneous afucosylated antibodies that compete for the common receptor FcγRIIIa (CD16a), equation (14) is the specific form of equation (2) that applies. Since X.sub.AF≈0 for this system, p is given by X.sub.A. FIG. 6 illustrates how equation (14) is used to compute mixture activity from component antibody equilibrium constants, K.sub.A and K.sub.F, with mole fraction X.sub.A, or equivalently p, variable.

[0064] TABLE 2 shows computed and experimental values of 1/K.sub.d,apparent for the binary homogenenous system with activity determined using ELISA for IgG1 Fc-FcγRIIIa F158 and IgG1 Fc-FcγRIIIa V158 receptor binding. The mixtures comprise define proportions of homogeneous fucosylated and afucosylated IgG1. For binary mixtures comprising the homogeneous fucosylated and afucosylated antibodies, X.sub.A is identically p. X.sub.F is computed from the mole fraction constraint for a binary mixture. K.sub.A=0.46 nM and K.sub.F=12 nM for the FcγRIIIa F158 allotype and K.sub.A=0.167 nM and K.sub.F=167 nM for the FcγRIIIa V158 allotype.

TABLE-US-00002 TABLE 2 1/K.sub.d,apparent Computed for Binary Mixtures of A and F (Fc   RIIIa F158 & V158) 1/K.sub.d,apparent F 1/K.sub.d,apparent F 1/K.sub.d,apparent V 1/K.sub.d,apparent V p X.sub.A X.sub.F (computed) (observed) (computed) (observed) 0 0 1 0.083 0.083 0.6 0.6 0.02 0.02 0.98 0.132 0.12 0.71 0.63 0.05 0.05 0.95 0.20 0.14 0.87 0.75 0.075 0.075 0.925 0.26 0.21 1.0 1.0 0.1 0.1 0.9 0.33 0.19 1.1 1.2 0.2 0.2 0.8 0.57 0.38 1.7 1.4 0.5 0.5 0.5 1.3 1.3 3.3 3 1 1 0 2.5 2.5 6.0 6

Example 2: Fraction Receptors Occupied

[0065] FIG. 7 shows how the fraction receptors occupied f is computed for the ternary afucosylated antibody system. The appropriate model equations are given by equation (20) and equations (21) and (22) with m=3. Adopting the appropriate subscripts to denote the three afucosylated antibody glycoforms, A, F and AF yields:

[00020]$\begin{array}{cc}f=\frac{\left[\mathrm{Ab}\right]}{\left[\mathrm{Ab}\right]+K_{d,\mathrm{apparent}}}=f_A+f_F+f_{\mathrm{AF}},.Math.\mathrm{with}& \left(23\right)\\ f_A=\frac{\left[A\right]}{\left[A\right]+K_A\left(1+\frac{\left[F\right]}{K_F}+\frac{\left[\mathrm{AF}\right]}{K_{\mathrm{AF}}}\right)}.Math..Math.f_F=\frac{\left[F\right]}{\left[F\right]+K_F\left(1+\frac{\left[A\right]}{K_A}+\frac{\left[\mathrm{AF}\right]}{K_{\mathrm{AF}}}\right)}.Math..Math.f_{\mathrm{AF}}=\frac{\left[\mathrm{AF}\right]}{\left[\mathrm{AF}\right]+K_{\mathrm{AF}}\left(1+\frac{\left[A\right]}{K_A}+\frac{\left[F\right]}{K_F}\right)}.& \left(24\right)\end{array}$

When supplied with component ligand equilibrium constants K.sub.A, K.sub.F and K.sub.AF and compositions, equations (23) and (24) can be used to compute both f and the component ligand contributions f.sub.i as of function of overall and individual ligand concentrations.

[0066] The concentrations appearing in equation (23) and equation (24) denote the concentrations of unbound antibody. The general excess of antibody or ligand over receptor allows the antibody concentrations appearing in the equations to be approximated by the antibody concentration added to the experimental samples. Unless otherwise noted, the numerical values used for antibody concentrations are assumed to be equal to the antibody concentrations added to the sample.

DESCRIPTION—ADDITIONAL EMBODIMENT

[0067] Ligand-receptor binding is a specific class of protein-protein interactions involving one binding partner that has been termed a ligand and the other binding partner that has been termed a receptor. However equation (1) does not depend on any particular linguistic classification of proteins or entities and describes any system involving m proteins or entities that bind competitively to a common partner. Accordingly, equation (1) may be used to characterize any mixture of proteins or entities that competitively bind to a common entity such as the binding of mixtures of enzymes to a common protein. When the components of the mixture are multimers with each multimer comprised of k monomers with each monomer differentiated by the presence or absence of a defined molecular entity at a specific site on the monomer, then k+1 different multimers exist and the binomial distribution with n=k can be used to compute the mole fractions of the k+1 different forms.

[0068] For example, consider a dimeric enzyme comprised of two monomers with each monomer containing one site that may or may not contain a bound mannose-6-phosphate molecule. Statistical considerations give rise to three different glycoforms with mole fractions given by equation (8). The applicability of equation (8) follows immediately from the fact that an antibody molecule is a dimer. Simply replace the term “antibody” with “dimer.” In an assay where the different glycoforms competitively bind to a common protein, such as a receptor R, the applicability of equation (1) follows immediately.

Mathematical Nomenclature

[0069] EC50 Experimental ligand concentration that induces a 50% maximum response
EC50.sub.F Homo. fucosylated antibody concentration that induces a 50% max. response
f Fraction of total receptor bound to all ligands
f.sub.A Fraction of total receptors bound to homogeneous afucosylated antibody
f.sub.AF Fraction of total receptors bound to hemi-afucosylated antibody
f.sub.F Fraction of total receptors bound to homogenous fucosylated antibody
f.sub.i Fraction of total receptors bound to ligand i
f.sub.A* Fraction of total bound receptors bound to homogeneous afucosylated antibody
f.sub.AF* Fraction of total bound receptors bound to hemi-afucosylated antibody
f.sub.F* Fraction of total bound receptors bound to homogeneous fucosylated antibody
f.sub.i* Fraction of total bound receptors bound to ligand i
k.sub.off dissociation reaction kinetic rate constant
k.sub.off,apparent apparent dissociation reaction kinetic rate constant
k.sub.off,AF dissociation reaction kinetic rate constant for hemi-afucosylated antibody
k.sub.off,F dissociation reaction kinetic rate constant for homogeneous fucosylated antibody
k.sub.off,i dissociation reaction kinetic rate constant for ligand i
k.sub.on binding reaction kinetic rate constant
k.sub.on,apparent apparent binding reaction kinetic rate constant
k.sub.on,A binding reaction kinetic rate constant for homogeneous afucosylated antibody
k.sub.on,AF binding reaction kinetic rate constant for hemi-afucosylated antibody
k.sub.on,F binding reaction kinetic rate constant for homogeneous fucosylated antibody
K.sub.d,apparent apparent dissociation equilibrium constant
K.sub.A dissociation equilibrium constant for homogeneous afucosylated antibody
K.sub.AF dissociation equilibrium constant for hemi-afucosylated antibody
K.sub.F dissociation equilibrium constant for homogeneous fucosylated antibody
K.sub.i dissociation equilibrium constant for ligand i
p fraction of Ig Fc heavy chains that are afucosylated
RA relative activity or activity relative to 1/K.sub.F
r.sub.on rate of binding of all ligands to receptor
r.sub.on,i rate of binding of ligand i to receptor
X.sub.A mole fraction of unbound homogeneous afucosylated antibody
X.sub.AF mole fraction of unbound hemi-afucosylated antibody
X.sub.F mole fraction of unbound homogeneous fucosylated antibody
X.sub.i mole fraction of unbound ligand i
[A] molar concentration of unbound homogeneous afucosylated antibody
[AF] molar concentration of unbound hemi-afucosylated antibody
[F] molar concentration of unbound homogeneous fucosylated antibody
[Ab].sub.total molar concentration of total unbound antibody
[L].sub.total molar concentration of total unbound ligand
[L.sub.i] molar concentration of unbound ligand i
[R] molar concentration of unbound receptor
[R].sub.total molar concentration of total bound and unbound receptor
[RAb].sub.total molar concentration of total bound receptor
[RL.sub.i] molar concentration of receptor bound to ligand i
[RL].sub.total molar concentration of total bound receptor

CONCLUSIONS, RAMIFICATIONS & SCOPE

[0070] The reader will see that use of a mechanism based mathematical model of competitive ligand-receptor binding enhances the ability to characterize mixtures of glycoform ligands providing access to fundamental information on the activities and compositions of important constituent glycoproteins that cannot be obtained by existing methods. Specifically, the methods described provide the means to extract biochemical property information for important biologically activity constituents in mixtures that currently cannot be isolated in pure form. Since mixtures of ligands are routinely employed therapeutically, the methods developed directly address an unmet need associated with biologics characterization. The utility of the methods developed was demonstrated with mixtures of afucosylated antibodies comprised of the three antibody glycoforms that are differentiated by afucosylated Fc glycan content. In addition to serving as a model system for methods development, the antibody afucosylation system is directly relevant to many marketed therapeutics that rely of Ig Fc mediated effector function. The methods developed provide the means to deduce the equilibrium constant K.sub.AF which has evaded researchers hitherto.

[0071] The methods developed have the potential to raise the standard of quality for manufactured biologics worldwide by providing fundamental knowledge on the compositions and the specific activities of the important biochemically active glycoforms in therapeutic mixtures of ligands. Emphasis can now be placed on determining the distribution of glycoproteins or glycoforms in a mixture rather than their bound carbohydrates structures. Since the glycoform is the physical entity that mediates in vivo efficacy, the importance of this information cannot be overstated. Current methods cannot distinguish between glycan and glycoform compositions so that different manufacturers of the same glycoprotein must argue for glycoprotein similarity without fundamental information on the molar concentrations of the important constituent glycoforms. The methods developed in this invention provide manufacturers of biologics with the means to overcome this critical barrier to proper product characterization thus promoting higher standards of product quality.

Although the above descriptions contain many specifications, these should not be construed as limitations on the scope, but rather as examples of several embodiments thereof. Accordingly the scope of this invention should not be determined by the embodiment(s) illustrated, but by the appended claims and their legal equivalents.