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Affinity Complex Titration

The affinity complex titration (ACT) isotherm is a modified Langmuir isotherm where both the binding capacity and equilibrium constant are dependent on pH via a Hill-type relationship [11]. pH, rather than the proton concentration, is treated as a mobile phase modulator with a concentration. Multiple bound state is not supported. The current implementation requires the first component to be pH and non-binding. Although the original derivation and the equation shown below is based on pH, the mobile phase modulator can also be any type of salt. The ACT isotherm reads:

\[\begin{aligned} \frac{\mathrm{d}q_i}{\mathrm{d}t} = k_{a,i} q_{\text{max},i} \left( f_{A, i}-\sum_{j=1}^{N_{\text{comp}}} \frac{q_j}{q_{\text{max},j}} \right) f_{G,i} c_i - k_{d,i}q_i, \end{aligned}\]

where \(f_{A, i}\) is the modification factor for the binding capacity \(q_{\text{max}, i}\), and \(f_{G,i}\) is the modification factor for the equilibrium constant \(K_{eq, i} = k_{a,i} / k_{d,i}\). The modification factors are defined by:

\[\begin{aligned} f_{A, i} =\frac{1}{1+10^{\eta_{A, i} (\mathrm{p}Ka_{A, i}-\mathrm{pH})}} , \quad f_{G, i} =\frac{1}{1+10^{\eta_{G, i} (\mathrm{p}Ka_{G, i}-\mathrm{pH})}}, \end{aligned}\]

where \(\eta_{A, i}\) and \(\eta_{G, i}\) denote the Hill-type coefficients that control the slope of the \(q_{max, i}\) and \(K_{eq, i}\) responses as a function of the pH, respectively, while \(\mathrm{p}Ka_{A, i}\) and \(\mathrm{p}Ka_{G, i}\) denote the center of their responses. respectively. For more details and interpretations on these parameters, please refer to [11].

For more information on model parameters required to define in CADET file format, see Affinity Complex Titration.