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Colloidal Particle Adsorption

The colloidal particle adsorption (CPA) model describes protein adsorption based on colloidal interaction theory. The model captures three key contributions to adsorption:

  • electrostatic protein-adsorber interactions,

  • lateral protein-protein interactions on the surface,

  • steric exclusion effects via scaled-particle theory (hard-disc available surface function).

A defined proton component \(c^p_{\mathrm{H}+}\) (by default the first component, index configurable via CPA_PROTON_IDX) acts as a non-binding pH state. The proton component must be non-binding.

The kinetic formulation reads for each binding component \(i\):

\[\frac{\mathrm{d} c^{s}_{i}}{\mathrm{d} t} = k_{\mathrm{kin},i} \left( K_{v,i} \, c^p_i - c^{s}_{i} \right),\]

where \(c^{s}_{i}\) is the volumetric solid phase concentration, \(c^p_i\) is the pore liquid phase concentration, \(K_{v,i}\) is the volumetric equilibrium constant, and \(k_{\mathrm{kin},i}\) is the kinetic rate constant [13].

In rapid-equilibrium mode, the corresponding bound-state equation is algebraic:

\[0 = c^{s}_{i} - K_{v,i} \, c^p_i.\]

The adsorption mode is selected by IS_KINETIC.

Multiple bound states per component are not supported.

Ionic strength and activity coefficients

The ionic strength \(I_m\) can be supplied in two ways:

  • Fixed parameter (default): \(I_m\) is read from CPA_IONIC_STRENGTH and the pH is computed directly from the proton component concentration. Since the standard definition of pH is based on concentration in mol/L, \(\mathrm{mol}/\mathrm{m}^3\) is converted to mol/L via the factor \(10^{-3}\):

    \[\mathrm{pH} = -\log_{10}\!\left(c^p_{\mathrm{H}+} \cdot 10^{-3}\right).\]

    This corresponds to the setting of [14], where pH and \(I_m\) are prescribed properties of the mobile phase.

  • Computed from concentrations: if component charges \(z_i\) are provided via CPA_IONIC_VALENCE, the ionic strength is computed from the pore-phase concentrations at each time step, and the Davies activity correction is applied to the proton activity,

    \[I_m = \frac{1}{2} \sum_i z_i^2 \, c^p_i.\]

    Components that should not contribute to the ionic strength, should be assigned a charge of zero. This also switches off the activity correction for that component; in particular, assigning \(z = 0\) to the proton component reduces \(\gamma_{\mathrm{H}^+}\) to one. For the Davies model, this value is converted from \(\mathrm{mol}/\mathrm{m}^3\) to \(\mathrm{mol\,/ L}\) as \(I_M = 10^{-3} I_m\). The activity correction is then

    \[\log_{10}\gamma_i = -0.509 \, z_i^2 \left( \frac{\sqrt{I_M}}{1 + \sqrt{I_M}} - 0.3 \, I_M \right),\]

    so that

    \[\mathrm{pH} = -\log_{10}\!\left(\gamma_{\mathrm{H}^+} \, c^p_{\mathrm{H}+} \cdot 10^{-3}\right).\]

    The constant 0.509 is the Debye–Hückel slope for water at 25 °C. The factor \(10^{-3}\) converts the pore-phase concentration from \(\mathrm{mol}/\mathrm{m}^3\) to mol/L, as required by the pH definition.

In both cases, a proton activity less than or equal to \(10^{-14}\mathrm{mol}/\mathrm{m}^3\) is mapped to pH 14.

Inverse Debye length

The inverse Debye length \(\kappa\) characterises the range of electrostatic interactions:

\[\kappa = e \sqrt{\frac{2 \, I_m \, N_A}{k_B \, T \, \varepsilon \, \varepsilon_0}},\]

where \(e\) is the elementary charge, \(I_m\) the ionic strength, \(N_A\) Avogadro’s number, \(k_B\) the Boltzmann constant, \(T\) the absolute temperature, \(\varepsilon\) the relative permittivity, and \(\varepsilon_0\) the vacuum permittivity.

Adsorber surface potential

The adsorber surface potential \(\psi_{0,A}\) is obtained by solving the electroneutrality condition [14, 15]

\[\sigma_{I,A}(\psi_{0,A}) = \sigma_D(\psi_{0,A})\]

via Newton’s method, where the ionisable surface charge density is

\[\sigma_{I,A} = e \, N_A \, \Gamma_L \left( \zeta_L - \frac{1}{1 + 10^{\mathrm{p}K_L - \mathrm{pH}_0}} \right)\]

with the surface pH

\[\mathrm{pH}_0 = \mathrm{pH} + \frac{e \, \psi_{0,A}}{\ln(10) \, k_B \, T},\]

and the diffuse layer charge density is

\[\sigma_D = 2 \, \varepsilon \, \varepsilon_0 \, \kappa \, \frac{k_B T}{e} \, \sinh\!\left( \frac{e \, \psi_{0,A}}{2 \, k_B \, T} \right).\]

Here, \(\Gamma_L\) is the ligand surface density, \(\zeta_L\) the charge of the fully protonated ligand, and \(\mathrm{p}K_L\) the dissociation constant of the ligand.

The protein surface potential \(\psi_{0,i}\) follows from the same electroneutrality condition, applied to the protein surface charge density \(\sigma_{I,i} = Z_i \, e / (4 \pi a_i^2)\). Since \(\sigma_{I,i}\) does not depend on \(\psi_{0,i}\), the diffuse layer relation can be inverted analytically,

\[\psi_{0,i} = \frac{2 \, k_B T}{e} \operatorname{arcsinh}\!\left( \frac{Z_i \, e^2}{8 \pi \, a_i^2 \, \varepsilon \, \varepsilon_0 \, \kappa \, k_B T} \right),\]

where \(a_i\) is the protein hydrodynamic radius.

Protein net charge and surface potential

The protein net charge \(Z_i\) depends on pH through a polynomial of arbitrary degree \(P\):

\[Z_i(\mathrm{pH}) = z_{i,0} + \sum_{k=1}^{P} z_{i,k} \left(\mathrm{pH} - \mathrm{pH}_{\mathrm{ref}}\right)^k.\]

The coefficients are supplied by CPA_EFFECTIVE_CHARGE_COEF as a polynomial-order-row-major matrix. Its rows correspond to increasing powers, its columns to components, and \(z_{i,0}\) is the charge at \(\mathrm{pH}_{\mathrm{ref}}\). The number of matrix rows determines the polynomial degree.

Distance of closest approach

The distance of closest approach \(\delta_{m,i}\) between protein \(i\) and the adsorber surface is defined as the positive distance at which the protein–adsorber interaction potential \(u_{A,i}(z)\) attains its minimum:

\[\delta_{m,i} := \operatorname*{arg\,min}_{z>0} u_{A,i}(z).\]

Setting \(\partial u_{A,i}(z) / \partial z = 0\) gives the analytical expression

\[\delta_{m,i} = -\frac{1}{\kappa} \ln\!\left( \frac{-2 \, \psi_{0,A} \, \psi_{0,i}}{\psi_{0,A}^2 + \psi_{0,i}^2} \right).\]

The analytical minimum requires opposite signs of \(\psi_{0,A}\) and \(\psi_{0,i}\) and a logarithm argument strictly between zero and one.

Interaction layer thickness

The dimensionless interaction layer thickness parameter \(\delta_i\) corresponds to \(\Delta_i = A_{s,i} \left( d_i^* - \delta_{m,i} \right)\) of [13, 14], i.e. the thickness of the interaction boundary layer scaled by the specific adsorber surface area. It is parameterised in terms of the protein surface charge density \(\sigma_{I,i} = Z_i \, e / (4\pi a_i^2)\):

\[\log_{10}(\delta_i) = \log_{10}(\delta_{i,\mathrm{ref}}) + \delta_{i,\mathrm{lin}} \left( |\sigma_{I,i}| - |\sigma_{I,i}^{\mathrm{ref}}| \right),\]

where \(\delta_{i,\mathrm{ref}} > 0\) and \(\sigma_{I,i}^{\mathrm{ref}} = z_{i,0} \, e / (4\pi a_i^2)\). The outer edge of the interaction layer is then \(d_i^* = \delta_{m,i} + \delta_i / A_{s,i}\), so that \(A_{s,i} \left( d_i^* - \delta_{m,i} \right) = \delta_i\).

Protein–adsorber interaction energy

The electrostatic interaction energy between protein \(i\) and the adsorber at distance \(\delta_{m,i}\) follows from the Derjaguin approximation applied to the constant-charge plate–plate interaction free energy of [16], as described in [14, 15]:

\[u_{A,i}(\delta_{m,i}) = \pi \, a_i \, \varepsilon \, \varepsilon_0 \left[ 2 \, \psi_{0,A} \, \psi_{0,i} \ln\!\left(\frac{1 + e^{-\kappa \delta_{m,i}}}{1 - e^{-\kappa \delta_{m,i}}}\right) - \left(\psi_{0,A}^2 + \psi_{0,i}^2\right) \ln\!\left(1 - e^{-2\kappa \delta_{m,i}}\right) \right].\]

Note the sign of the second term: it is the constant-charge counterpart of the constant-potential (Hogg–Healy–Fuerstenau) expression, and it is what makes \(u_{A,i}\) attain a finite minimum at \(\delta_{m,i} > 0\) for oppositely charged surfaces.

Henry coefficient

The Henry adsorption coefficient \(K_{H,i}\) is obtained by linearising \(u_{A,i}(z)\) on \(z \in [\delta_{m,i}, d_i^*]\) and averaging the Boltzmann factor over the interaction layer [15]:

\[K_{H,i} = \frac{k_B T}{u_{A,i}} \left( 1 - \exp\!\left(-\frac{u_{A,i}}{k_B T}\right) \right).\]

In the limit \(u_{A,i} \to 0\), \(K_{H,i} \to 1\).

Available surface function (steric blocking)

The available surface function \(B_i(\Theta)\) follows the two-dimensional scaled-particle theory for hard discs of [17], as used in [14]:

\[B_i(\Theta) = (1 - \Theta) \exp\!\left( -\frac{\pi a_i^2 \, N_A \sum_j \tilde{c}^{s}_{j} + 2\pi a_i \, N_A \sum_j a_j \tilde{c}^{s}_{j}}{1 - \Theta} - \frac{\pi^2 a_i^2 \left(N_A \sum_j a_j \tilde{c}^{s}_{j}\right)^2}{(1 - \Theta)^2} \right),\]

where \(\tilde{c}^{s}_{j} = c^{s}_{j} / A_{s,j}\) denotes the surface concentration, and the total surface coverage is

\[\Theta = \pi \, N_A \sum_j a_j^2 \, \tilde{c}^{s}_{j}.\]

Lateral protein–protein interaction

Adsorbed proteins are assumed to be arranged on a hexagonal lattice whose centre-to-centre distance \(D_{\mathrm{hex}}\) follows from the maximum packing density [14],

\[D_{\mathrm{hex}} = \sqrt{\frac{2\sqrt{3}}{3 \, N_A \sum_j \tilde{c}^{s}_{j}}}.\]

The lateral interaction energy for component \(i\) is given by

\[u_{\mathrm{lat},i} = \frac{3\sqrt{3} \, D_{\mathrm{hex}} \, N_A \, e^{-\kappa D_{\mathrm{hex}}}}{1 - \exp\!\left(-\frac{3\sqrt{3}}{2\pi} \kappa D_{\mathrm{hex}}\right)} \sum_j \tilde{c}^{s}_{j} \, \beta_{ij},\]

with

\[\beta_{ij} = \frac{Z_{\mathrm{lat},i} \, Z_{\mathrm{lat},j} \, e^2}{4\pi \, \varepsilon \, \varepsilon_0} \cdot \frac{\exp\bigl(\kappa(a_i + a_j)\bigr)}{(1 + \kappa a_i)(1 + \kappa a_j)},\]

where \(Z_{\mathrm{lat},i}\) is the lateral charge of component \(i\).

Volumetric equilibrium constant

Combining all contributions, the volumetric equilibrium constant is [13]

\[K_{v,i} = A_{s,i} \left(d_i^* - \delta_{m,i}\right) K_{H,i} \, B_i(\Theta) \, \exp\!\left(-\frac{u_{\mathrm{lat},i}}{k_B T}\right) = \delta_i \, K_{H,i} \, B_i(\Theta) \, \exp\!\left(-\frac{u_{\mathrm{lat},i}}{k_B T}\right).\]

Kinetic rate constant

The kinetic rate constant \(k_{\mathrm{kin},i}\) is calculated from the kinetic prefactor \(k^*_{\mathrm{kin},i}\) supplied by CPA_KKIN:

\[k_{\mathrm{kin},i} = \frac{k^*_{\mathrm{kin},i}}{2} \cdot \frac{\left(u_{A,i} / (k_B T)\right)^2}{\cosh\!\left(u_{A,i} / (k_B T)\right) - 1}.\]

This expression is taken from [13]; it approaches \(k^*_{\mathrm{kin},i}\) in the limit \(u_{A,i} \to 0\). The kinetic rate and CPA_KKIN are not evaluated for bound states in rapid-equilibrium mode.

Model assumptions and limitations

  • One component must serve as a non-binding proton/pH state (index configurable, default 0).

  • If CPA_IONIC_VALENCE is provided, the ionic strength is computed from the pore-phase concentrations and the Davies activity correction is applied to the proton activity. Otherwise, CPA_IONIC_STRENGTH is used as a fixed parameter.

  • The degree of the protein charge polynomial is inferred from the number of rows in CPA_EFFECTIVE_CHARGE_COEF and is the same for all components.

  • Kinetic and rapid-equilibrium adsorption can be selected globally or per bound state through IS_KINETIC.

  • Multiple bound states per component are not supported.

  • Physical constants (\(e\), \(N_A\), \(k_B\), \(\varepsilon_0\)) are hard-coded to CODATA 2018 values.

  • The Davies activity correction uses the Debye–Hückel slope for water at 25 °C. Setting CPA_TEMPERATURE to a different value while CPA_IONIC_VALENCE is given triggers a warning.

  • Treating the pH as a transported state derived from a proton component is a CADET-Core extension; in [13, 14, 15] the pH is a prescribed property of the mobile phase.

  • The model is only defined for \(\Theta < 1\) and for \(\psi_{0,A}\) and \(\psi_{0,i}\) of opposite sign and different magnitude. States violating this are reported as a recoverable error and logged, so that the time integrator can retry with a smaller step rather than aborting the simulation. Repeated warnings of this kind indicate an unsuitable parameter set.

For more information on model parameters required to configure in CADET-Core, see Colloidal Particle Adsorption.

Literature

  • [15] — charge regulation, protein–adsorber interaction energy, and the Henry coefficient (linear range).

  • [14] — extension to the nonlinear range: available surface function, lateral interactions, and the pH dependence of \(Z_i\) and \(\delta_i\).

  • [13] — kinetic formulation, \(k_{\mathrm{kin},i}\) and \(K_{v,i}\).

  • [18] — application to protein mixtures.

  • [17] — scaled-particle theory for the hard-disc available surface function.

  • [16] — constant-charge double-layer interaction free energy.