Abstract:
A new method is presented for characterizing the associative properties of a solution of macromolecules at high concentration. Sample aliquots spanning a range of concentrations are injected sequentially into a light scattering photometer. Equilibrium association constants and association stoichiometry are derived from an analysis of the angular and concentration dependence of the scattering signals. Thermodynamic nonideality, which becomes important at high concentrations, is dealt with in the analysis in a simplified manner which is applicable to multiple associated species.

Description:
BACKGROUND 
       [0001]    Macromolecules, e.g. proteins, polysaccharides, synthetic polymers, etc., undergo different types of interactions in solution. Some of these interactions which are non-specific and repulsive are commonly referred to as ‘thermodynamic non-ideality’. Some, whether specific or non-specific, are attractive and lead to reversible association, wherein the ratio of partial concentrations of monomer and associated complexes reaches an equilibrium state that depends on the overall concentration of the molecules in solution, the specific properties of the molecules, and the solvent. The reversible association state in turn impacts the functionality of the solution. For example, in the pharmaceutical industry, the presence of reversible antibody oligomers can increase solution viscosity, which adversely affects manufacture and delivery of therapeutics containing these antibodies. Oligomeric forms of an antibody drug may present an increased immunogenic risk. Determination of the association state, in the form of equilibrium association constants and association stoichiometries, of such macromolecules is an essential step in understanding and controlling the underlying interactions. 
         [0002]    The standard methods for characterizing irreversible aggregates or tightly bound complexes in solution, such as size-exclusion chromatography (SEC), SEC in combination with multi-angle static light scattering (SEC-MALS), or sedimentation velocity (SV), are not applicable to weak reversible associations since these characterization techniques lead to dilution, dissociation of the complexes, and thus deviation from equilibrium. 
         [0003]    Common methods known for characterizing strongly interacting reversible association include sedimentation equilibrium (SE), isothermal titration calorimetry (ITC), and composition gradient multi-angle static light scattering (CG-MALS). In SE, a solution of molecules is placed in an analytical ultracentrifuge and rotated at extremely high speed so as to form an equilibrium distribution which is recorded and analyzed. The specific shape of the distribution contains information on the amounts of self-associating monomer and complexes as well as the association stoichiometries. In ITC, the solution is titrated with respect to the solvent, in the case of self-association, or a solution initially containing only one macromolecule is titrated with respect to the second macromolecule, in the case of hetero-association, and the amount of heat given off or taken up by the solution is measured. This heat measurement and the corresponding shape of the titration curve, i. e. heat vs. titration volume and concentration, contain the requisite information on interaction strength and association stoichiometry, though typically only a single stoichiometry may be characterized. In CG-MALS, a series of dilutions is prepared and delivered to a MALS detector. The dependence of the scattered light intensity on concentration may be analyzed to determine association stoichiometries and equilibrium association constants. CG-MALS is a particularly useful technique since it does not require the very long equilibration times of sedimentation equilibrium. In addition, it is superior to ITC in determining associations resulting in multiple oligomeric states and simultaneous self- and hetero-association. When the associative interactions are strong, the solution is usually characterized at low concentration. However, when the associative interactions are weak, as is typically the case for antibody drug formulations, the solution must be characterized at high concentration, often in the range of 10-100 g/L. 
         [0004]    At molecular concentrations above ˜1 g/L, non-specific, typically repulsive interactions become significant and will affect the reversible association measurements of SE, ITC and CG-MALS. Hence characterization of weak association must often be carried out in a concentration range at which the non-specific interactions must also be accounted for in the analysis. 
         [0005]    The theory describing static scattering of light by multiple species in solution subject to non-specific interactions, known as ‘fluctuation theory’, was described by Kirkwood and Goldberg in J. Chem. Phys. 18, 54-59 (1950). The application of fluctuation theory to light scattering under dilute conditions to a single non-associating species is well known, and involves only three parameters—the molar mass, mean square radius and second virial coefficient. As the solution concentration increases, higher-order virial coefficients must be included, making the analysis successively more difficult. A simplification described by Minton in Biophys. J. 93, 1321-1328 (2007), known as the ‘effective hard sphere approximation’ or EHSA, assumes that the non-specific interactions between molecules are equivalent to those of impenetrable spheres whose effective radii may differ from the actual radii of the molecules. The EHSA framework is useful in interpreting CG-MALS and other data at high concentrations of 10-150 g/L, which may otherwise be intractable. Fernandez and Minton, Biophysical J. 96, 1992-1998 (2009), have shown that fluctuation theory in combination with EHSA may be applied successfully to CG-MALS analysis of the reversible self-association of a protein to one or two oligomeric states at concentrations up to 70 g/L. 
         [0006]    While the combined fluctuation/EHSA theory is rigorous, it becomes very complex mathematically if it must deal with more than two or three species that include monomers and complexes. Yet many important systems exhibit multiple stoichiometries of association, i.e. attain equilibrium between the monomers and several complexes simultaneously. One example is progressive self-association, forming dimer, trimer, tetramer, etc., up to high order oligomers. Another example is simultaneous self- and heteroassociation. Just writing out the full fluctuation theory equations for these cases is prohibitively difficult, let alone analyzing data in terms of the complex equation of Equation 1 below. 
         [0007]    Hence a method for analyzing the reversible association of macromolecules at high concentration, using CG-MALS, which employs a more tractable representation of thermodynamic non-ideality yet accurately determines the sought for association constants and stoichiometries, would be advantageous. 
       SUMMARY OF THE INVENTION 
       [0008]    This invention provides a method of representing CG-MALS data from a reversibly associating solution at high concentration in a highly tractable form amenable to simplified analysis of complex multiple stoichiometries. 
         [0009]    In addition, using the inventive method provides for the determination of the equilibrium association parameters, i.e. the stoichiometry of associating complexes, the equilibrium association constants, and a measure of incompetent fractions present, of a macromolecular solution at high concentration via CG-MALS by means of the aforementioned tractable representation. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]      FIG. 1  is a flow diagram of a method for representing the non-ideality of a solution of a reversibly self-associating species. 
           [0011]      FIG. 2  is a flow diagram of a method for representing the non-ideality of a solution of two constituent species forming reversible complexes. 
           [0012]      FIG. 3  is a schematic of an apparatus used in determining the equilibrium association parameters of one or more species undergoing reversible association. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0013]    Theoretical Description 
         [0014]    The result of a generalized fluctuation theory for multi-component static light scattering, assuming that the incident light is vertically plane polarized and the observation is made in the horizontal plane at an angle θ relative to the direction of the incident light, is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                        
                       
                         ( 
                         
                           
                             c 
                             X 
                           
                           , 
                           θ 
                         
                         ) 
                       
                     
                     K 
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         , 
                         n 
                       
                     
                      
                     
                       
                         { 
                         
                           
                             
                               
                                 
                                   Ψ 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 c 
                                 n 
                               
                                
                               
                                 M 
                                 m 
                               
                                
                               
                                 
                                   Q 
                                   
                                     1 
                                     , 
                                     m 
                                     , 
                                     n 
                                   
                                 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             
                                
                               
                                 
                                   ψ 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                             
                           
                            
                           
                             ( 
                             
                               
                                  
                                 n 
                               
                               
                                  
                                 
                                   c 
                                   m 
                                 
                               
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               
                                  
                                 n 
                               
                               
                                  
                                 
                                   c 
                                   n 
                                 
                               
                             
                             ) 
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0015]    Here R(c X , θ) represents the excess Rayleigh ratio detected at any scattering angle from a solution of macromolecules at a composition c X , where X represents the various monomeric species and c X  represents the totality of weight/volume concentrations [c 1 , c 2 , c 3  . . . ] of each species; the excess Rayleigh ratio is the difference between the Rayleigh ratio of the solution and that of the pure solvent; the Rayleigh ratio of a solution is 
         [0000]    
       
         
           
             
               
                 Ir 
                 s 
                 2 
               
               Iv 
             
             ; 
           
         
       
     
         [0000]    is the intensity of scattered light per unit solid angle observed at a distance r s  from the point of scattering due to an incident intensity I; v is the scattering volume; 
         [0000]    
       
         
           
             
               K 
               = 
               
                 
                   
                     ( 
                     
                       2 
                        
                       π 
                        
                       
                           
                       
                        
                       
                         n 
                         0 
                       
                     
                     ) 
                   
                   2 
                 
                 
                   
                     N 
                     A 
                   
                    
                   
                     λ 
                     0 
                     4 
                   
                 
               
             
             ; 
           
         
       
     
         [0000]    n 0  is the refractive index of the solution; N A  is Avogadro&#39;s number; λ 0  is the wavelength of the incident light in vacuum; m and n represent the different species present, including free monomers and complexes; c n  is the weight concentration, in units of mass per unit volume, of the n th  species; Q l,m,n  is some function of the scattering angle θ which generally depends on the size and mass distributions within the m and n molecules, and approaches a value of 1 as either θ approaches zero or the overall size is much smaller than 
         [0000]    
       
         
           
             
               
                 
                   λ 
                   0 
                 
                 
                   n 
                   0 
                 
               
               ; 
               
                 
                   
                     ψ 
                     
                       m 
                       , 
                       n 
                     
                   
                    
                   
                     ( 
                     θ 
                     ) 
                   
                 
                 = 
                 
                   
                     δ 
                     
                       m 
                       , 
                       n 
                     
                   
                   + 
                   
                     
                       c 
                       m 
                     
                      
                     
                       
                         
                           ∂ 
                           ln 
                         
                          
                         
                             
                         
                          
                         
                           γ 
                           n 
                         
                       
                       
                         ∂ 
                         
                           c 
                           m 
                         
                       
                     
                      
                     
                       
                         Q 
                         
                           2 
                           , 
                           m 
                           , 
                           n 
                         
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where δ m,n  equals 1 if m=n, and equals zero otherwise, γ n  is the thermodynamic activity of component n, and Q 2,m,n  is some function of θ which generally depends on the size and mass distributions within the m and n molecules, and approaches a value of 1 as either θ approaches zero or the size is much smaller than 
         [0000]    
       
         
           
             
               
                 λ 
                 0 
               
               
                 n 
                 0 
               
             
             ; 
           
         
       
     
         [0000]    |ψ m,n (θ)| is the determinant of ψ m,n (θ); Ψ m,n (θ) is the m,n cofactor, or subdeterminant, of ψ m,n (θ); and dn/dc m  is the differential refractive index increment of the m th  species. If the m th  species is a heterocomplex consisting of i X  monomers of type X, i Y  monomers of type Y, etc., then dn/dc m  is the weight average of the contributing refractive index increments of the constitutent molecules. The weight average of the refractive increment is 
         [0000]    
       
         
           
             
               
                 
                   ∑ 
                   X 
                 
                  
                 
                   
                     i 
                     X 
                   
                    
                   
                     M 
                     X 
                   
                    
                   
                     
                        
                       n 
                     
                     
                        
                       
                         c 
                         X 
                       
                     
                   
                 
               
               
                 
                   ∑ 
                   X 
                 
                  
                 
                   
                     i 
                     X 
                   
                    
                   
                     M 
                     X 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the subscript X refers to the different constituent monomers. Equation (1) becomes very complex if more than two or three species are present, owing to the many terms incorporated in the determinant and subdeterminants. 
         [0016]    The expression 
         [0000]    
       
         
           
             
               
                 ∂ 
                 ln 
               
                
               
                   
               
                
               
                 γ 
                 n 
               
             
             
               ∂ 
               
                 c 
                 m 
               
             
           
         
       
     
         [0000]    may be understood to represent the essential specific interaction volume V interaction /(M m +M n ) between macromolecular species m and n that leads to thermodynamic non-ideality. Contributions to 
         [0000]    
       
         
           
             
               
                 ∂ 
                 ln 
               
                
               
                   
               
                
               
                 γ 
                 n 
               
             
             
               ∂ 
               
                 c 
                 m 
               
             
           
         
       
     
         [0000]    include the hard-core repulsion as well as various electrostatic and fluctuating dipole interactions. In a solution of at least intermediate ionic strength, long-range interactions are well-screened, and the non-ideality is dominated by short range interactions. At this condition the specific interaction volume is approximately proportional to the sum of the molecular volumes divided by the sum of masses, which may be written in terms of effective molecular density ρ m : 
         [0000]    
       
         
           
             
               
                 
                   ∂ 
                   ln 
                 
                  
                 
                     
                 
                  
                 
                   γ 
                   n 
                 
               
               
                 ∂ 
                 
                   c 
                   m 
                 
               
             
             ∝ 
             
               
                 
                   
                     
                       M 
                       m 
                     
                     / 
                     
                       ρ 
                       m 
                     
                   
                   + 
                   
                     
                       M 
                       n 
                     
                     / 
                     
                       ρ 
                       n 
                     
                   
                 
                 
                   
                     M 
                     m 
                   
                   + 
                   
                     M 
                     n 
                   
                 
               
               . 
             
           
         
       
     
         [0017]    If the various species in solution are formed as oligomers of just one type of monomer self-associating to form i-mers, then we may reasonably expect that the effective density of all i-mers is approximately a constant ρ, so 
         [0000]    
       
         
           
             
               
                 
                   ∂ 
                   ln 
                 
                  
                 
                     
                 
                  
                 
                   γ 
                   n 
                 
               
               
                 ∂ 
                 
                   c 
                   m 
                 
               
             
             ∝ 
             
               1 
               ρ 
             
           
         
       
     
         [0000]    is independent of m or n. 
         [0000]    
       
         
           
             
               
                 ∂ 
                 ln 
               
                
               
                   
               
                
               
                 γ 
                 n 
               
             
             
               ∂ 
               
                 c 
                 m 
               
             
           
         
       
     
         [0000]    is commonly approximated as a series in powers of the concentration: 
         [0000]    
       
         
           
             
               
                 
                   
                     ∂ 
                     ln 
                   
                    
                   
                       
                   
                    
                   
                     γ 
                     m 
                   
                 
                 
                   ∂ 
                   
                     c 
                     m 
                   
                 
               
               ≅ 
               
                 
                   
                     ∂ 
                     ln 
                   
                    
                   
                       
                   
                    
                   
                     γ 
                     1 
                   
                 
                 
                   ∂ 
                   
                     c 
                     1 
                   
                 
               
             
             = 
             
               
                 2 
                  
                 
                   A 
                   2 
                 
                  
                 M 
               
               + 
               
                 3 
                  
                 
                   A 
                   3 
                 
                  
                 M 
                  
                 
                     
                 
                  
                 
                   c 
                   m 
                 
               
               + 
               … 
             
           
         
       
     
         [0000]    where the coefficients A 2  and A 3  are known as the second and third virial coefficients of the monomer in the particular solvent, respectively. Applying this approximation, Eq. (1) may be reduced to a simplified form heretofore unknown in the scientific literature, wherein all the non-ideal self- and cross-interactions are captured in just the two parameters A 2  and A 3 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                        
                       
                         ( 
                         
                           c 
                           , 
                           θ 
                         
                         ) 
                       
                     
                     K 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                              
                             n 
                           
                           
                              
                             c 
                           
                         
                         ) 
                       
                       2 
                     
                      
                     
                       
                         
                           
                             ∑ 
                             i 
                           
                            
                           
                              
                              
                             
                                 
                             
                              
                             M 
                              
                             
                                 
                             
                              
                             
                               c 
                               i 
                             
                              
                             
                               
                                 P 
                                 i 
                               
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         
                           1 
                           + 
                           
                             2 
                              
                             
                               A 
                               2 
                             
                              
                             M 
                              
                             
                               
                                 ∑ 
                                 i 
                               
                                
                               
                                 
                                   c 
                                   i 
                                 
                                  
                                 
                                   
                                     P 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                           
                           + 
                           
                             3 
                              
                             
                               A 
                               3 
                             
                              
                             M 
                              
                             
                               
                                 ∑ 
                                 i 
                               
                                
                               
                                 c 
                                 i 
                                 2 
                               
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Here R(c, θ) is the excess Rayleigh ratio observed at azimuth angle θ and a total macromolecular concentration c; M is the molar mass of the monomer; dn/dc is the differential refractive index increment of the molecules in the solvent; i is the order of self-association; c i  is the weight concentration at equilibrium of the i-mer;          r g   2            is the angular dependence of the scattered light, within the plane perpendicular to the vertically polarized incident light, for the i-mer; θ is measured relative to the direction of propagation of the beam; and          r g   2            is the mean square radius of the i-mer defined as          r g   2           =∫r 2 dm i /∫dm i  where r is the distance from the center of mass of the molecule to a molecular mass element m i , integrated over all mass elements of the molecule. The validity of Eq. (2) may be illustrated with a relatively simple example as follows:
       For two species—a monomer and one i-mer,       
 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     
                       
                         ψ 
                         
                           m 
                           , 
                           n 
                         
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                      
                   
                   = 
                     
                    
                   
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               c 
                               1 
                             
                              
                             
                               
                                 
                                   ∂ 
                                   ln 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   γ 
                                   1 
                                 
                               
                               
                                 ∂ 
                                 
                                   c 
                                   1 
                                 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               c 
                               i 
                             
                              
                             
                               
                                 
                                   ∂ 
                                   ln 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   γ 
                                   i 
                                 
                               
                               
                                 ∂ 
                                 
                                   c 
                                   i 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         c 
                         1 
                       
                        
                       
                         c 
                         i 
                       
                        
                       
                         
                           
                             ∂ 
                             ln 
                           
                            
                           
                               
                           
                            
                           
                             γ 
                             i 
                           
                         
                         
                           ∂ 
                           
                             c 
                             1 
                           
                         
                       
                        
                       
                         
                           
                             ∂ 
                             ln 
                           
                            
                           
                               
                           
                            
                           
                             γ 
                             1 
                           
                         
                         
                           ∂ 
                           
                             c 
                             i 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     1 
                     + 
                     
                       
                         c 
                         1 
                       
                        
                       
                         
                           
                             ∂ 
                             ln 
                           
                            
                           
                               
                           
                            
                           
                             γ 
                             1 
                           
                         
                         
                           ∂ 
                           
                             c 
                             1 
                           
                         
                       
                     
                     + 
                     
                       
                         c 
                         i 
                       
                        
                       
                         
                           
                             ∂ 
                             ln 
                           
                            
                           
                               
                           
                            
                           
                             γ 
                             i 
                           
                         
                         
                           ∂ 
                           
                             c 
                             i 
                           
                         
                       
                     
                     + 
                     
                       
                         c 
                         1 
                       
                        
                       
                         
                           c 
                           i 
                         
                          
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     ∂ 
                                     ln 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     γ 
                                     1 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     c 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   
                                     ∂ 
                                     ln 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     γ 
                                     i 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     c 
                                     i 
                                   
                                 
                               
                             
                             - 
                             
                               
                                 
                                   
                                     ∂ 
                                     ln 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     γ 
                                     i 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     c 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   
                                     ∂ 
                                     ln 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     γ 
                                     1 
                                   
                                 
                                 
                                   ∂ 
                                   
                                     c 
                                     i 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             Each of the terms 
           
         
       
     
         [0000]    
       
         
           
             
               
                 
                   ∂ 
                   ln 
                 
                  
                 
                     
                 
                  
                 
                   γ 
                   n 
                 
               
               
                 ∂ 
                 
                   c 
                   m 
                 
               
             
             ∝ 
             
               
                 1 
                 ρ 
               
               . 
             
           
         
       
       
         
           
             Hence the term 
           
         
       
     
         [0000]    
       
         
           
             
               c 
               1 
             
              
             
               
                 c 
                 i 
               
                
               
                 ( 
                 
                   
                     
                       
                         
                           ∂ 
                           ln 
                         
                          
                         
                             
                         
                          
                         
                           γ 
                           1 
                         
                       
                       
                         ∂ 
                         
                           c 
                           1 
                         
                       
                     
                      
                     
                       
                         
                           ∂ 
                           ln 
                         
                          
                         
                             
                         
                          
                         
                           γ 
                           i 
                         
                       
                       
                         ∂ 
                         
                           c 
                           i 
                         
                       
                     
                   
                   - 
                   
                     
                       
                         
                           ∂ 
                           ln 
                         
                          
                         
                             
                         
                          
                         
                           γ 
                           i 
                         
                       
                       
                         ∂ 
                         
                           c 
                           1 
                         
                       
                     
                      
                     
                       
                         
                           ∂ 
                           ln 
                         
                          
                         
                             
                         
                          
                         
                           γ 
                           1 
                         
                       
                       
                         ∂ 
                         
                           c 
                           i 
                         
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             is not only of higher order than the other terms, it is the difference of two quantities that are of comparable magnitude, and hence should be small compared to even one second-order term. It will also be small as one of the concentrations tends to zero. The final expression for the denominator will be 
           
         
       
     
         [0000]    
       
         
           
             
                
               
                 
                   ψ 
                   
                     m 
                     , 
                     n 
                   
                 
                  
                 
                   ( 
                   θ 
                   ) 
                 
               
                
             
             ≈ 
             
               1 
               + 
               
                 
                   c 
                   1 
                 
                  
                 
                   
                     
                       ∂ 
                       ln 
                     
                      
                     
                         
                     
                      
                     
                       γ 
                       1 
                     
                   
                   
                     ∂ 
                     
                       c 
                       1 
                     
                   
                 
               
               + 
               
                 
                   c 
                   i 
                 
                  
                 
                   
                     
                       
                         ∂ 
                         ln 
                       
                        
                       
                           
                       
                        
                       
                         γ 
                         i 
                       
                     
                     
                       ∂ 
                       
                         c 
                         i 
                       
                     
                   
                   . 
                 
               
             
           
         
       
       
         
           
             Likewise, for any number of species, the higher order terms may be ignored to yield 
           
         
       
     
         [0000]    
       
         
           
             
                
               
                 
                   ψ 
                   
                     m 
                     , 
                     n 
                   
                 
                  
                 
                   ( 
                   θ 
                   ) 
                 
               
                
             
             ≈ 
             
               1 
               + 
               
                 
                   ∑ 
                   m 
                 
                  
                 
                   
                     c 
                     m 
                   
                    
                   
                     
                       
                         ∂ 
                         ln 
                       
                        
                       
                           
                       
                        
                       
                         γ 
                         m 
                       
                     
                     
                       ∂ 
                       
                         c 
                         m 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             The 
           
         
       
     
         [0000]    
       
         
           
             
               
                 ∂ 
                 ln 
               
                
               
                   
               
                
               
                 γ 
                 m 
               
             
             
               ∂ 
               
                 c 
                 m 
               
             
           
         
       
       
         
           
             may be expressed in terms of virial coefficients as described above, with the approximation that A 2 M and A 3 M, which are closely related to the inverse density, are approximately constants for the monomer and all oligomers: 
           
         
       
     
         [0000]    
       
         
           
             
               
                  
                 
                   
                     ψ 
                     
                       m 
                       , 
                       n 
                     
                   
                    
                   
                     ( 
                     θ 
                     ) 
                   
                 
                  
               
               ≈ 
               
                 1 
                 + 
                 
                   
                     ( 
                     
                       
                         2 
                          
                         
                           A 
                           2 
                         
                          
                         
                           
                             MP 
                             1 
                           
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       + 
                       
                         3 
                          
                         
                           A 
                           3 
                         
                          
                         M 
                          
                         
                             
                         
                          
                         
                           c 
                           1 
                         
                       
                       + 
                       … 
                     
                      
                     
                         
                     
                     ) 
                   
                    
                   
                     c 
                     1 
                   
                 
                 + 
                 
                   
                     ( 
                     
                       
                         2 
                          
                         
                           A 
                           2 
                         
                          
                         
                           
                             MP 
                             i 
                           
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       + 
                       
                         3 
                          
                         
                           A 
                           3 
                         
                          
                         M 
                          
                         
                             
                         
                          
                         
                           c 
                           i 
                         
                       
                       + 
                       … 
                     
                      
                     
                         
                     
                     ) 
                   
                    
                   
                     c 
                     i 
                   
                 
                 + 
                 … 
               
             
             = 
             
               1 
               + 
               
                 2 
                  
                 
                   A 
                   2 
                 
                  
                 M 
                  
                 
                   
                     ∑ 
                     i 
                   
                    
                   
                     
                       c 
                       i 
                     
                      
                     
                       
                         P 
                         i 
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
               + 
               
                 3 
                  
                 
                   A 
                   3 
                 
                  
                 M 
                  
                 
                   
                     ∑ 
                     i 
                   
                    
                   
                     c 
                     i 
                     2 
                   
                 
               
               + 
               … 
             
           
         
       
       
         
           
             Also, for two species, monomer and one i-mer, 
           
         
       
     
         [0000]    
       
         
           
             
               
                  
                 n 
               
               
                  
                 
                   c 
                   1 
                 
               
             
             = 
             
               
                  
                 n 
               
               
                  
                 
                   c 
                   i 
                 
               
             
           
         
       
       
         
           
             so that the numerator of Eq. (1) becomes: 
           
         
       
     
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         
                           M 
                            
                           
                               
                           
                            
                           
                             c 
                             1 
                           
                            
                           
                             
                               P 
                               1 
                             
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   c 
                                   i 
                                 
                                  
                                 
                                   
                                     
                                       ∂ 
                                       ln 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       γ 
                                       i 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       c 
                                       i 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                            
                            
                           
                               
                           
                            
                           M 
                            
                           
                               
                           
                            
                           
                             c 
                             i 
                           
                            
                           
                             
                               P 
                               i 
                             
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   c 
                                   1 
                                 
                                  
                                 
                                   
                                     
                                       ∂ 
                                       ln 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       γ 
                                       1 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       c 
                                       1 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         - 
                       
                     
                   
                   
                     
                       
                         
                           c 
                           1 
                         
                          
                         
                           
                             c 
                             i 
                           
                            
                           
                             ( 
                             
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     MP 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     
                                       ∂ 
                                       ln 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       γ 
                                       1 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       c 
                                       i 
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     MP 
                                     1 
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     
                                       ∂ 
                                       ln 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       γ 
                                       i 
                                     
                                   
                                   
                                     ∂ 
                                     
                                       c 
                                       1 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ] 
               
                
               
                   
               
                
               
                 
                   ( 
                   
                     
                        
                       n 
                     
                     
                        
                       
                         c 
                         1 
                       
                     
                   
                   ) 
                 
                 2 
               
             
             = 
             
                 
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               M 
                                
                               
                                   
                               
                                
                               
                                 c 
                                 1 
                               
                                
                               
                                 
                                   P 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             + 
                             
                                
                                
                               
                                   
                               
                                
                               M 
                                
                               
                                   
                               
                                
                               
                                 c 
                                 i 
                               
                                
                               
                                 P 
                                 i 
                               
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               c 
                               1 
                             
                              
                             
                               
                                 c 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               MP 
                                               1 
                                             
                                              
                                             
                                               ( 
                                               θ 
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                               
                                                 ∂ 
                                                 ln 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 γ 
                                                 i 
                                               
                                             
                                             
                                               ∂ 
                                               
                                                 c 
                                                 i 
                                               
                                             
                                           
                                         
                                         + 
                                         
                                            
                                            
                                           
                                               
                                           
                                            
                                           
                                             M 
                                             i 
                                           
                                            
                                           
                                             
                                               P 
                                               i 
                                             
                                              
                                             
                                               ( 
                                               θ 
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                               
                                                 ∂ 
                                                 ln 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 γ 
                                                 1 
                                               
                                             
                                             
                                               ∂ 
                                               
                                                 c 
                                                 1 
                                               
                                             
                                           
                                         
                                         - 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                            
                                            
                                           
                                               
                                           
                                            
                                           
                                             
                                               MP 
                                               i 
                                             
                                              
                                             
                                               ( 
                                               θ 
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                               
                                                 ∂ 
                                                 ln 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 γ 
                                                 1 
                                               
                                             
                                             
                                               ∂ 
                                               
                                                 c 
                                                 i 
                                               
                                             
                                           
                                         
                                         - 
                                         
                                           
                                             
                                               MP 
                                               1 
                                             
                                              
                                             
                                               ( 
                                               θ 
                                               ) 
                                             
                                           
                                            
                                           
                                             
                                               
                                                 ∂ 
                                                 ln 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 γ 
                                                 i 
                                               
                                             
                                             
                                               ∂ 
                                               
                                                 c 
                                                 1 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                    
                   
                       
                   
                    
                   
                     
                       ( 
                       
                         
                            
                           n 
                         
                         
                            
                           
                             c 
                             1 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
                 ≈ 
                 
                     
                   
                     
                       [ 
                       
                         
                           M 
                            
                           
                               
                           
                            
                           
                             c 
                             1 
                           
                            
                           
                             
                               P 
                               1 
                             
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         + 
                         
                            
                            
                           
                               
                           
                            
                           M 
                            
                           
                               
                           
                            
                           
                             c 
                             i 
                           
                            
                           
                             
                               P 
                               i 
                             
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       
                         ( 
                         
                           
                              
                             n 
                           
                           
                              
                             
                               c 
                               1 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
               
             
           
         
       
       
         
           
             as again there is a term which includes the difference of high-order terms of very comparable magnitude and so may be ignored. Likewise, for any number of oligomeric species, the final expression would be 
           
         
       
     
         [0000]    
       
         
           
             
               ∑ 
               m 
             
              
             
               mM 
                
               
                   
               
                
               
                 c 
                 m 
               
                
               
                 
                   P 
                   m 
                 
                  
                 
                   ( 
                   θ 
                   ) 
                 
               
                
               
                 
                   ( 
                   
                     
                        
                       n 
                     
                     
                        
                       
                         c 
                         1 
                       
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
       
         
           
             Combining these approximations for the numerator and denominator of Eq. (1) yields Eq. (2) . Numerical studies show that the terms that have been dropped only account for a small fraction of the total non-ideality correction, up to concentrations of tens of g/L, and thus the relationships that are the subject of this invention are applicable at such high concentrations. 
           
         
       
     
         [0029]    Under certain common assumptions, a fixed relationship may be assumed between A 2  and A 3 , so that a single parameter captures all the non-ideal behavior. For example, if the molecules are assumed to behave like hard spheres then 
         [0000]    
       
         
           
             
               A 
               3 
             
             = 
             
               
                 5 
                 8 
               
                
               
                 A 
                 2 
                 2 
               
                
               
                 M 
                 . 
               
             
           
         
       
     
         [0030]    Depending on the relative magnitudes of A 2 , A 3  and the sin 2 (θ/2) terms in the P i , some of the terms in Eq. (2) may be ignored, as will be obvious to those familiar with numerical analysis. For example, for molecules in a solution of only moderately high concentration A 3  may be ignored, and if the complexes are all smaller than about λ/70, the angular dependence may be ignored as well, yielding a very simple form: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                        
                       
                         ( 
                         c 
                         ) 
                       
                     
                     K 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                              
                             n 
                           
                           
                              
                             c 
                           
                         
                         ) 
                       
                       2 
                     
                      
                     
                       
                         
                           ∑ 
                           i 
                         
                          
                         
                            
                            
                           
                               
                           
                            
                           M 
                            
                           
                               
                           
                            
                           
                             c 
                             i 
                           
                         
                       
                       
                         1 
                         + 
                         
                           2 
                            
                           
                             A 
                             2 
                           
                            
                           M 
                            
                           
                               
                           
                            
                           
                             c 
                             tot 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               c 
               tot 
             
             = 
             
               
                 ∑ 
                 i 
               
                
               
                 c 
                 i 
               
             
           
         
       
     
         [0000]    is the total weight/volume concentration of the material in solution. 
         [0031]    If the various species in solution are formed as complexes of two different monomers X and Y, then under similar assumptions to those stated above, Eq. (1) may be reduced to a highly simplified form heretofore unknown in the scientific literature, wherein all the non-ideal self- and cross-interactions are captured in just two parameters A 2   X  and A 2   Y : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                        
                       
                         ( 
                         
                           c 
                           , 
                           θ 
                         
                         ) 
                       
                     
                     K 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           , 
                           j 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     M 
                                     X 
                                   
                                    
                                   
                                     
                                        
                                       n 
                                     
                                     
                                        
                                       
                                         c 
                                         X 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   
                                     M 
                                     
                                       Y 
                                       . 
                                     
                                   
                                    
                                   
                                     
                                        
                                       n 
                                     
                                     
                                        
                                       
                                         c 
                                         Y 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                            
                           
                             
                               
                                 c 
                                 ij 
                               
                                
                               
                                 
                                   P 
                                   ij 
                                 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             
                               M 
                               ij 
                             
                           
                         
                         ] 
                       
                     
                     
                       1 
                       + 
                       
                         2 
                          
                         
                           A 
                           2 
                           X 
                         
                          
                         
                           M 
                           X 
                         
                          
                         
                           
                             ∑ 
                             ij 
                           
                            
                           
                             
                                
                                
                               
                                   
                               
                                
                               
                                 M 
                                 X 
                               
                                
                               
                                 c 
                                 ij 
                               
                                
                               
                                 
                                   P 
                                   ij 
                                 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             
                               M 
                               ij 
                             
                           
                         
                       
                       + 
                       
                         2 
                          
                         
                           A 
                           2 
                           Y 
                         
                          
                         
                           M 
                           Y 
                         
                          
                         
                           
                             ∑ 
                             ij 
                           
                            
                           
                             
                               j 
                                
                               
                                   
                               
                                
                               
                                 M 
                                 Y 
                               
                                
                               
                                 c 
                                 ij 
                               
                                
                               
                                 
                                   P 
                                   ij 
                                 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                             
                               M 
                               ij 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where M X  and M Y  correspond to the molar masses of the X and Y monomers; dn/dc X  and dn/dc Y  correspond to the differential refractive index increments of the X and Y monomers in the particular solvent; i and j are the number of X and Y monomers in the complex, M ij =iM X +jM Y  is the molar mass and c ij  is the weight concentration, at equilibrium, of the XiYj complex; A 2   X  and A 2   Y  refer to the second virial coefficients of the X and Y monomers in the particular solvent; 
         [0000]    
       
         
           
             
               
                 P 
                 ij 
               
                
               
                 ( 
                 θ 
                 ) 
               
             
             = 
             
               1 
               - 
               
                 
                   
                     16 
                      
                     
                       π 
                       2 
                     
                      
                     
                       n 
                       0 
                       2 
                     
                   
                   
                     3 
                      
                     
                       λ 
                       0 
                       2 
                     
                   
                 
                  
                 
                   
                     〈 
                     
                       r 
                       g 
                       2 
                     
                     〉 
                   
                   ij 
                 
                  
                 
                   
                     sin 
                     2 
                   
                    
                   
                     ( 
                     
                       θ 
                       / 
                       2 
                     
                     ) 
                   
                 
               
               + 
               … 
             
           
         
       
     
         [0000]    and          r g   2            is the mean square radius of the ij complex. The derivation is similar to that described for oligomers of the same monomer. 
         [0032]    If the complexes are smaller than about λ/70 then the angular dependence may be ignored and Eq. (4) may be reduced to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                        
                       
                         ( 
                         c 
                         ) 
                       
                     
                     K 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           , 
                           j 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 
                                    
                                    
                                   
                                       
                                   
                                    
                                   
                                     M 
                                     X 
                                   
                                    
                                   
                                     
                                        
                                       n 
                                     
                                     
                                        
                                       
                                         c 
                                         X 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   
                                     M 
                                     Y 
                                   
                                    
                                   
                                     
                                        
                                       n 
                                     
                                     
                                        
                                       
                                         c 
                                         Y 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                            
                           
                             
                               c 
                               ij 
                             
                             
                               M 
                               ij 
                             
                           
                         
                         ] 
                       
                     
                     
                       1 
                       + 
                       
                         2 
                          
                         
                           A 
                           2 
                           X 
                         
                          
                         
                           M 
                           X 
                         
                          
                         
                           c 
                           X 
                           tot 
                         
                       
                       + 
                       
                         2 
                          
                         
                           A 
                           2 
                           Y 
                         
                          
                         
                           M 
                           Y 
                         
                          
                         
                           c 
                           Y 
                           tot 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               c 
               X 
               tot 
             
             = 
             
               
                 ∑ 
                 ij 
               
                
               
                 
                    
                    
                   
                       
                   
                    
                   
                     M 
                     X 
                   
                    
                   
                     c 
                     ij 
                   
                 
                 
                   M 
                   ij 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               c 
               Y 
               tot 
             
             = 
             
               
                 ∑ 
                 ij 
               
                
               
                 
                   j 
                    
                   
                       
                   
                    
                   
                     M 
                     Y 
                   
                    
                   
                     c 
                     ij 
                   
                 
                 
                   M 
                   ij 
                 
               
             
           
         
       
     
         [0000]    are the total weight/volume concentrations of X and Y in solution. Equations (4) and (5) may be readily generalized to more than two distinct monomeric species. 
         [0033]    Determining A 2    
         [0034]    The non-ideality parameter A 2  may be estimated a priori, or it may be a parameter of the fit of the data to the non-ideality-corrected light scattering equation and the association model equations described below. 
         [0035]    In order to estimate A 2  from a priori information, a known molecular radius may be substituted into the formula for computing A 2  of a hard sphere: 
         [0000]    
       
         
           
             
               A 
               2 
             
             = 
             
               
                 
                   16 
                    
                   π 
                    
                   
                       
                   
                    
                   
                     N 
                     A 
                   
                 
                 3 
               
                
               
                 
                   
                     r 
                     3 
                   
                   
                     M 
                     2 
                   
                 
                 . 
               
             
           
         
       
     
         [0000]    The molecular radius of the monomer may be derived from structural information, e.g. as may be determined by x-ray crystallography, or estimated from a measurement of the hydrodynamic radius r h . The hydrodynamic radius may be calculated from measurements of dynamic light scattering or differential viscometry under dilute conditions, as is known to those familiar with macromolecular characterization. 
         [0036]    Methods for Simplified Representations of Non-Ideal Solutions 
         [0037]    Hence one method for representing CG-MALS data from a reversibly self-associating solution at high concentration, in a highly simplified form amenable to further analysis, consists of the following steps as illustrated in  FIG. 1 :
       1. Determine a suitable estimate of the effective molecular radius r either from the known structure of the molecule, or from a quasi-elastic light scattering or differential viscometry measurements of the hydrodynamic radius r h  taken under non-associating conditions such as suitably low concentration or an appropriate association-restricting solvent;   2. Based on the known molar mass of the monomer M and the estimated effective molecular radius r, estimate the monomer excluded volume value       
 
         [0000]    
       
         
           
             
               A 
               2 
             
             = 
             
               
                 
                   16 
                    
                   π 
                    
                   
                       
                   
                    
                   
                     N 
                     A 
                   
                    
                   
                     r 
                     3 
                   
                 
                 
                   3 
                    
                   
                     M 
                     2 
                   
                 
               
               . 
             
           
         
       
       
         
           
             In some cases, the association is relatively weak and it is possible to estimate A 2  from a series of measurements at low concentrations. 
             3. Given the maximal concentration of interest c max , compute an estimate of the maximum non-ideality contribution ξ=2A 2 Mc max ; if ξ is greater than a predetermined cutoff value, e.g. 0.3, retain the A 3  term in Eq. (2); otherwise, drop the A 3  term; 
             4. Estimate the mean square radius r max  of the largest oligomer expected to form; if r max &gt;λ/70, retain the angular terms in Eq. (2); otherwise, drop them. 
             5. Use the final form of Eq. (2) to represent the CG-MALS data. If some of the macromolecules are expected to be incompetent to reversible association, treat it in the equation as a distinct species that does not associate but has the same virial coefficient as the competent macromolecules. 
           
         
       
     
         [0044]    A method for representing CG-MALS data from a reversibly hetero-associating solution at high concentration, in a highly simplified form amenable to further analysis, consists of the following steps as illustrated in  FIG. 2 :
       1. Determine a suitable estimate of the effective molecular radii r X  and r Y  either from the known structure of the molecule, or from quasi-elastic light scattering or differential viscometry measurements of the hydrodynamic radii r h,X  and r h,Y ;   2. Based on the known molar masses of the monomers M X , M Y  and the estimated effective molecular radii r X  and r Y , estimate the monomer excluded volume values       
 
         [0000]    
       
         
           
             
               
                 A 
                 
                   2 
                   , 
                   X 
                 
               
               = 
               
                 
                   16 
                    
                   π 
                    
                   
                       
                   
                    
                   
                     N 
                     A 
                   
                    
                   
                     r 
                     X 
                     3 
                   
                 
                 
                   3 
                    
                   
                     M 
                     X 
                     2 
                   
                 
               
             
             , 
             
               
                 A 
                 
                   2 
                   , 
                   Y 
                 
               
               = 
               
                 
                   
                     16 
                      
                     π 
                      
                     
                         
                     
                      
                     
                       N 
                       A 
                     
                      
                     
                       r 
                       Y 
                       3 
                     
                   
                   
                     3 
                      
                     
                       M 
                       Y 
                       2 
                     
                   
                 
                 . 
               
             
           
         
       
       
         
           
             In some cases, the association is relatively weak and it is possible to estimate A 2  from a series of measurements at low concentrations. 
             3. Estimate the mean square radius r max  of the largest complex expected to form; if r max &gt;λ/70, retain the angular terms in Eq. (4); otherwise, drop them. 
             4. Use the final form of Eq. (4) to represent the CG-MALS data. If some of the macromolecules are expected to be incompetent to reversible association, treat them in the equation as a distinct species that does not associate but has the same virial coefficient as the competent macromolecules.
 
Variants on these methods for determining suitable forms of the above equations will be apparent to those familiar with numerical analysis.
 
           
         
       
     
         [0050]    Methods for Characterizing Reversibly Associating Non-Ideal Solutions 
         [0051]    With thermodynamic non-ideality accounted for in a simplified equation according to one of the forms shown above, characterization of the interaction in terms of stoichiometry and equilibrium association constants is straightforward and similar to the methods described by Attri and Minton in Anal. Biochem. 346,132-138 (2005) for ideal solutions and by Fernandez and Minton in Biophys. J. 96, 1992-1998 (2009) for concentrated solutions, but employing Eq. (1) rather than one of the simplified forms described herein. 
         [0052]    The characterization method comprises the steps of: preparing a series of solutions comprising one or more macromolecular species; allowing each solution to reach equilibrium between the free monomers and any reversibly-associating complexes; measuring the light scattering intensity of each solution; reducing the light scattering data to a series of excess Rayleigh ratios; and fitting the data simultaneously to the appropriate simplified representation of non-ideal light scattering and the equations for the specific association model described below. 
         [0053]    The model equations for self-association are:
       1. the equations for mass action, relating each equilibrium oligomer concentration c i  to the corresponding equilibrium association constant K i  for the specific stoichiometry, and the concentration of free monomer c 1 :       
 
         [0000]    
       
         
           
             
               
                 
                   c 
                   i 
                 
                 
                    
                    
                   
                       
                   
                    
                   M 
                 
               
               = 
               
                 
                   
                     K 
                     i 
                   
                    
                   
                     ( 
                     
                       
                         c 
                         1 
                       
                       M 
                     
                     ) 
                   
                 
                 i 
               
             
             ; 
           
         
       
       
         
           
             2. the equation for conservation of mass 
           
         
       
     
         [0000]    
       
         
           
             
               c 
               tot 
             
             = 
             
               
                 
                   ∑ 
                   i 
                 
                  
                 
                   c 
                   i 
                 
               
               + 
               
                 c 
                 inc 
               
             
           
         
       
       
         
           
             where c tot  is known for each solution as determined by the preparation procedure or measured by a concentration detector, and c inc  is the concentration of macromolecules incompetent to associate and is considered a distinct species. 
           
         
       
     
         [0057]    The model equations for heteroassociations of two different monomer species X and Y are:
       1. the equations for mass action, relating each equilibrium complex concentration c ij  to the corresponding equilibrium association constant K ij  for the specific stoichiometry, and the concentration of free monomer c X , c Y :       
 
         [0000]    
       
         
           
             
               
                 
                   c 
                   ij 
                 
                 
                   
                      
                      
                     
                         
                     
                      
                     
                       M 
                       X 
                     
                   
                   + 
                   
                     j 
                      
                     
                         
                     
                      
                     
                       M 
                       Y 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       K 
                       ij 
                     
                      
                     
                       ( 
                       
                         
                           c 
                           X 
                         
                         
                           M 
                           X 
                         
                       
                       ) 
                     
                   
                   i 
                 
                  
                 
                   
                     ( 
                     
                       
                         c 
                         Y 
                       
                       
                         M 
                         Y 
                       
                     
                     ) 
                   
                   j 
                 
               
             
             ; 
           
         
       
       
         
           
             2. the equations for conservation of mass 
           
         
       
     
         [0000]    
       
         
           
             
               
                 c 
                 X 
                 tot 
               
               = 
               
                 
                   
                     ∑ 
                     
                       i 
                       , 
                       j 
                     
                   
                    
                   
                     
                       ( 
                       
                         
                            
                            
                           
                               
                           
                            
                           
                             M 
                             X 
                           
                         
                         
                           
                              
                              
                             
                                 
                             
                              
                             
                               M 
                               X 
                             
                           
                           + 
                           
                             j 
                              
                             
                                 
                             
                              
                             
                               M 
                               Y 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       c 
                       
                         i 
                         , 
                         j 
                       
                     
                   
                 
                 + 
                 
                   c 
                   X 
                   inc 
                 
               
             
             , 
             
               
                 c 
                 Y 
                 tot 
               
               = 
               
                 
                   
                     ∑ 
                     
                       i 
                       , 
                       j 
                     
                   
                    
                   
                     
                       ( 
                       
                         
                           j 
                            
                           
                               
                           
                            
                           
                             M 
                             Y 
                           
                         
                         
                           
                              
                              
                             
                                 
                             
                              
                             
                               M 
                               X 
                             
                           
                           + 
                           
                             j 
                              
                             
                                 
                             
                              
                             
                               M 
                               Y 
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       c 
                       
                         i 
                         , 
                         j 
                       
                     
                   
                 
                 + 
                 
                   c 
                   Y 
                   inc 
                 
               
             
           
         
       
       
         
           
             where c X   tot  and c Y   tot  are known for each solution as determined by the preparation procedure or measured with concentration detectors; and c X   inc  and c Y   inc  are the concentrations of X and Y macromolecules, respectively, incompetent to associate, and considered additional distinct species. 
           
         
       
     
         [0061]    Various fitting algorithms, such as Levenberg-Marquardt nonlinear least squares algorithms and others, are well known from numerical analysis theory. These algorithms may be employed for fitting the data to the system of equations which includes the non-ideal light scattering equation, the mass conservation equations and the mass action equations, thereby obtaining fitted values of the interaction parameters K i  or K ij , A 2 , etc. 
         [0062]    Measurement Apparatus and Procedure—Self-Association 
         [0063]    Referring to  FIG. 3 , a set of aliquots of the macromolecule of interest, at concentrations c 1 , c 2 , . . . c k , are introduced into MALS detector  1 , providing photodetectors at a plurality of scattering angles θ v . One example of a MALS detector is the DAWN-HELEOS®, from Wyatt Technology Corporation, Santa Barbara, Calif. 
         [0064]    In a typical procedure, the concentration series corresponds to c s =αΔc, where s=1, 2, . . . k; Δc is a fixed concentration step, and k is the number of concentrations, usually at least five. Aliquots of each of the k concentrations may be prepared and introduced to the detectors by means of various methods. In one method, these aliquots are prepared manually and placed in the MALS detector by means of scintillation vials or cuvettes. In a second method, the aliquots are prepared manually and injected into the light scattering detector flow cell by means of a pump  2 . In a third method, the aliquots are prepared automatically by means of a dual pump  2  under computer control, which dilutes a stock solution  6  at a maximal concentration c max  with a solvent  7 , and subsequently delivers sequentially each aliquot as produced to the detector. One example of an extant system capable of carrying out the dilution and delivery are the Calypso™ SP3 accessory using the Calypso™ software, from Wyatt Technology Corporation, Santa Barbara, Calif. 
         [0065]    The actual concentrations of the aliquots in the flow cell may differ from the original, as-prepared values c s , as the sample dilutes in the course of flowing through the system and interacting with filters  3 , surfaces, etc. A sufficient injection volume will fully equilibrate the detector flow cell at each injected concentration, so that knowledge of the as-prepared concentrations c s  suffices to determine the actual concentration in the MALS flow cell. Alternatively, the optional in-line concentration detector  4  may be used to measure the actual sample concentrations. 
         [0066]    Various methods are known for determining the concentration of a sample in solution. In one method applicable to manual preparation of the aliquots, appropriate masses of concentrated or lyophilized sample are weighed out and dissolved in a known volume of the solvent. In a second method, the concentration is determined by measuring absorbance with a spectrophotometer. In a third method, the concentrations are determined by means of a suitable in-line concentration detector  4  connected in series with the MALS detector. An example of an in-line concentration detector is the Optilab® rEX™, also from Wyatt Technology Corporation; other in-line concentration detectors are known, including UV/V is absorption and fluorescence detectors. The in-line concentration detector may be connected in series or in parallel with the MALS detector. If the MALS and concentration detectors are connected in series, a sufficient volume of sample must be delivered so as to saturate both flow cells at the desired concentration. If the MALS and concentration detectors are connected in parallel, then the sample flow must be split between them in a controlled fashion so as to ensure that at the completion of each sample injection, the concentrations in the two detectors are the same. Sample flow splitting is typically controlled by a needle valve and monitored by means of suitable flow meters so as to maintain the required ratio. 
         [0067]    Data is acquired from the detectors while the sample is flowing and while it is stopped between injections, then stored and analyzed by a computer  5  performing the fitting procedure described previously. Optimally the data to be used for the analysis is that acquired after flow has stopped and the sample has equilibrated. 
         [0068]    Each successive sample passes through the MALS detector  1 , whereby the values of the excess Rayleigh ratio, R s (c s , θ v ), at each detector angle θ v , are measured at successive sample concentrations c s . The resultant light scattering and concentration signals are then stored and processed by a computer means  5  to calculate, for each injected aliquot s, the values c s , R s (c s , θ v ). Computer  5  also computes the molecular characteristics including M and &lt;r g   2 &gt;, and the molecular interaction characteristics A 2  and K i , by fitting the calculated results to Eq. (2) or a simplified form thereof, together with the association model equations. Various fitting procedures may be implemented to extract the molecular interaction characteristics. In a preferred embodiment, the fitting procedure consists of the Levenberg-Marquardt algorithm as applied to two variables (c and sin 2 (θ/2)), with M and A 2  fixed. 
         [0069]    From numerical analysis theory, the fitting of the measured data to the form of the light scattering equation and association model equations, whether by the Levenberg-Marquardt method, or other algorithms, may include statistical weighting whereby the data used to perform these fits is weighted by their reciprocal measured standard deviations. 
         [0070]    Measurement Apparatus and Procedure—Heteroassociation 
         [0071]    The measurement proceeds as for a single-species measurement, except that each aliquot contains different concentrations of two macromolecular species X and Y in various association states to be determined. 
         [0072]    In one procedure, known as a crossover composition gradient, k aliquots are prepared wherein the composition of the s th  aliquot is [sΔc X , (k−s)Δc Y ], and Δc X  and Δc Y  are fixed concentration step sizes. 
         [0073]    In another procedure, known as a constant-ratio composition gradient, k aliquots are prepared wherein the composition of the s th  aliquot is [sΔc X ,sΔc Y ], and Δc X  and Δc Y  are fixed concentration step sizes. 
         [0074]    The apparatus is similar to those of the single-species measurement, except that a computer-controlled triple-pump system is employed instead of a dual pump system, each pump controllable by means of computer to produce, mix and deliver an aliquot comprising species X, species Y, and solvent at the desired compositions. Such a triple-pump system and suitable controlling software are the Calypso system, of Wyatt Technology Corporation, Santa Barbara, Calif. 
         [0075]    The total concentrations of constituents X and Y, c X   tot  and c Y   tot  respectively, may be determined from the predetermined stock solution concentrations and the mixing ratio as set in the preparation method, or by means of a method for measuring concentrations of two species in solution. 
         [0076]    In one method for measuring the concentrations of two distinct molecules in solution, the total concentration signal is measured by means of an on-line concentration detector  4 , and the constituent concentrations calculated from the known ratio between the two constituent species and the relative contributions of each to the total concentration signal. Such a method has been described by Attri and Minton in Anal. Biochem. 346 (2005) 132-138. 
         [0077]    In a second method for measuring the concentrations of two distinct molecules in solution, at least two different on-line concentration detection means are used, and the constituent concentrations determined from the concentration signals and the known responses of each constituent species to each concentration detection means. For example, the signals of a differential refractometer and UV absorption detector may be analyzed to yield the concentrations of each of two species present in the same solution, if the responses of the molecules to the respective concentration detectors differs for at least one measurement. 
         [0078]    As will be evident to those skilled in the arts of light scattering, macromolecular characterization, and numerical analysis, there are many obvious variations of the methods we have invented and described that do not depart from the fundamental elements that we have listed for their practice; all such variations are but obvious implementations of the invention described hereinbefore and are included by reference to our claims, which follow.