It might be worth looking into the HIC model used by J.M. Mollerup, T.B. Hansen, S. Kidal, A. Staby: Quality by design-Thermodynamic modelling of chromatographic separation of proteins. In this paper, the isotherm is only written in its equilibrium form, so I have expressed this in the kinetic form shown below

\frac{\partial q_i}{\partial t}=k_{a, i} \left(1-\sum_{j=1}^N \frac{q_j}{q_{\max , j}}\right)^{n_i} \tilde{\gamma}_i c_{p, i}-k_{d, i}q_i

This has been employed in several publications and includes the asymmetric activity coefficient

\tilde{\gamma}_i=\frac{\gamma_i}{\gamma^{\infty, \omega}} \approx \exp \left(K_{p, i} c_{p, i}+K_{s, i} c_s\right)

which is a very powerful term—see our recent paper in JChromA on isotherm models in multimodal chromatography, not HIC but I think the takeaways on modeling hydrophobic interactions are relevant here—and does not include the \beta_i, which is redundant with K_{s,i} in its dependence on salt concentration c_s and comparatively a less useful term.

The one thing I don’t like about this formulation is that it does not include ligand density, but I think for now it is sufficient—especially because ligand density for HIC resins is very rarely measured or provided by the manufacturer. If this is included, it should come along with the steric factor for hydrophobic interactions s_i, which could be pH-dependent. Lastly, pH could be included here by adding the dependence for k_{a,i} and/or k_{d,i}

k_{a,i} = k_{a,i,0} \exp \left(k_{a,i,1}\mathrm{pH} + k_{a,i,2}\mathrm{pH}^2 \right)

to the adsorption term in the \frac{\partial q_i}{\partial t} formulation. The asymmetric activity coefficient has been shown by Mollerup to be pH-dependent, but this need not be included as to avoid redundancy with the aforementioned pH-dependent affinity term.

The pH-dependent parameters and activity coefficient terms can be included in the k_{d,i} term for generality, but I think actually using these would most certainly lead to overparameterization of the isotherm. Actually, I always set k_{d,i} to unity such that k_{a,i} is equivalent to the equilibrium constant.

The stoichiometric term for hydrophobic interactions n_i can also be pH dependent and can be expressed analogously to \nu_i in the GIEX model.

n_i = n_{i,0} + n_{i,1}\mathrm{pH} + n_{i,2}\mathrm{pH}^2

If I could make a feature request, I think this would be a useful one. This model can be used for HIC or for multimodal in the pure HIC regime—when binding affinity and capacity are strictly increasing (electrostatic interactions are negligible) with respect to salt concentration, such as multimodal anion exchange for mAbs with pH << pI. Otherwise, the GIEX model can be employed for multimodal and, of course, for IEC. This way, we have two isotherms in CADET that are powerful and particularly general in their scope. We could call this one the generalized hydrophobic interaction (GHI) isotherm or something like that.

Thoughts?