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Sedimentation Equilibrium - Models:

Ideal (Single Ideal Species):

For a single ideal species at sedimentation equilibrium the data obtained from an analytical ultracentrifugation experiment can be described by the equation:

where S(r,l) is the radially-dependent signal at wavelength l. This signal can correspond to absorbance, refractive index increment, or fluorescence intensity.  dl is the baseline offset at wavelength l, el is the molar extinction coefficient at wavelength l.  c0 is the molar concentration of the monomer at the arbitrary reference distance r0, M* is the monomer buoyant molecular weight and f is given by

where w is the angular velocity of the rotor in radians/sec, R is the molar gas constant (83,144,000 erg/mol/K) and T is the absolute temperature.  The buoyant molecular weight, M*, is defined by:

where M is the monomer molar mass, is the partial specific volume, and r is the solvent density.

Nonideal (Single Nonideal Species):

This model incorporates thermodynamic nonideality of a macromolecule. In this case, the observed signal, S(r, l), is given by

where B is the second virial coefficient, w(r) refers to the concentration on a weight/volume scale at distance r and w(r0) refer to the concentration on a weight/volume scale at the reference distance, r0.  Internally, the program uses molar concentrations, and the weight concentrations are derived from molar concentration by w(r) = MC(r).  Thus, for this model it is necessary to know the correct value of the molar mass.

Monomer- Nmer self association:

The model incorporates a single self-association reaction where a monomer is in equilibrium with an N-mer.  Ideal behavior is assumed for both the monomer and the oligomer.  The fitting function is given by:

Where N is the stoichiometry of the association reaction and lnK is the natural logarithm of the association constant, K.  The extinction coefficient should be expressed in typical units of signal/Molar, so that the association constant K is defined is in units of .  For example, in a monomer-dimer system K is obtained in units of  and is defined by 

Monomer- Nmer - Mmer Self Association:  

This model accommodates two independent self-association reactions where the monomer is in equilibrium with two oligomers with association states of N1 and N2. 

Incompetent Monomer / Monomer- Nmer Self -Association:

Preparations of self-associating proteins may be heterogeneous and contain more than one thermodynamic component where the association behavior is altered. Two limiting cases are contamination by monomer that cannot associate or oligomer incapable of dissociation, referred to as incompetent monomer and incompetent oligomer, respectively. In the former case, the weight fraction of incompetent monomer, a, is given by 

Where [M'] refers to the molar concentration of the incompetent monomer and [O] refers to the molar concentration of the oligomer of degree N.  The fitting function used for this model is

where refers to the molar concentration of incompetent monomer at the reference distance. The value of is calculated using a and integration of the concentration gradients of the reversibly associating component.

Incompetent Oligomer  / Monomer- Nmer Self -Association:

Here, a corresponds to the weight fraction of incompetent oligomer and is given by

and the fitting function is

where refers to the molar concentration of incompetent oligomer at the reference distance.

Hetero-Interaction (A+B <=> AB): 

This model accommodates the simple hetero-interaction of two components, A and B, to form a 1:1 complex, designated AB.  The fitting function is

where eA,l is the extinction coefficient of component A at wavelength l and CA,0 is the molar concentration of A at the reference distance. The parameters for component B are defined analogously.  It is assumed that there is no hypo- or hyper-chromism and the extinction coefficient of AB is taken as the sum of the individual extinction coefficients of A and B.  Similarly, it is assumed that there is no volume change upon formation of the AB complex such that the buoyant molecular mass of AB is the sum of the components . 

Hetero-Interaction (A+B <=> AB+B <=> AB2): 

This model corresponds to a bivalent interaction where A binds two molecules of B according the the stepwise association constants K1 and K2.

The fitting function is

Hetero-Interaction (A+B <=> AB+B <=> AB2+B <=> AB3): 

This model corresponds to an interaction where A binds three molecules of B according the the stepwise association constants K1, K2 , and K3.

The fitting function is

Isodesmic Self-Association:

Isodesmic self-association is one where the free energy for the addition of a monomer unit to any oligomer, or another monomer, is constant. 

The total concentration, CT, can be expressed in terms of the monomer concentration, C1, and the isodesmic association constant, k ( expressed in units of [L/g] ) as long as the product kC1 < 1.

The radial distribution of the monomer is described by the following equation,

The fitting function for the observed signal is

Hetero-Interaction (nA <=> An+B <=> AnB): 

This model corresponds to a bivalent interaction where A can self-associate according to the self-association constant Ks, and then the oligomer, An, can interact with B to form a heterocomplex AnB according to the hetero-association constant Kh.

The fitting function is