efflux

Binding Models

Binding models describe the adsorption and desorption kinetics governing how solutes interact with the stationary phase in a chromatography column. Efflux implements six binding models of varying complexity to cover ion exchange, hydrophobic interaction, and colloidal chromatography.

Salt vs Non-salt Components

Most binding models distinguish between salt (modulator) components and non-salt (protein/solute) components:

  • Salt components: Modulator species (e.g. NaCl) that influence adsorption/desorption rates but do not bind themselves.
  • Non-salt components: Proteins or solutes that undergo adsorption/desorption kinetics.

The total salt concentration at a given point is:

csalt=∑s∈saltcp,sc_{\text{salt}} = \sum_{s \in \text{salt}} c_{p,s}

Langmuir

Multi-component competitive Langmuir kinetics. The simplest binding model, operating on all components with no salt modulation.

dqidt=ka,i cp,i qmax⁡,i(1−∑j=0nc−1qjqmax⁡,j)−kd,i qi\frac{dq_i}{dt} = k_{a,i} \, c_{p,i} \, q_{\max,i} \left(1 - \sum_{j=0}^{n_c - 1} \frac{q_j}{q_{\max,j}}\right) - k_{d,i} \, q_i
SymbolDescriptionUnitTypical Range
ka,ik_{a,i}Adsorption rate constant1/(M⋅s)\text{1/(M}{\cdot}\text{s)}10−7 – 10710^{-7} \text{ – } 10^{7}
kd,ik_{d,i}Desorption rate constant1/s\text{1/s}10−7 – 10710^{-7} \text{ – } 10^{7}
qmax⁡,iq_{\max,i}Maximum binding capacityM\text{M}0.001 – 1.80.001 \text{ – } 1.8

The saturation term (1−∑qj/qmax⁡,j)\left(1 - \sum q_j / q_{\max,j}\right) creates multi-component competition. Desorption is first-order in qiq_i with no saturation dependence.

Steric Mass Action (SMA)

Salt-modulated binding with steric shielding. Widely used for ion exchange chromatography. Operates on non-salt components only.

dqidt=ka,i cp,i(q^0qref) ⁣νi−kd,i qi(csaltcref) ⁣νi\frac{dq_i}{dt} = k_{a,i} \, c_{p,i} \left(\frac{\hat{q}_0}{q_{\text{ref}}}\right)^{\!\nu_i} - k_{d,i} \, q_i \left(\frac{c_{\text{salt}}}{c_{\text{ref}}}\right)^{\!\nu_i}

where:

q0=Λ−∑j∈non-saltνj qjq_0 = \Lambda - \sum_{j \in \text{non-salt}} \nu_j \, q_j
q^0=max⁡ ⁣(0,  q0−∑j∈non-saltσj qj)\hat{q}_0 = \max\!\left(0, \; q_0 - \sum_{j \in \text{non-salt}} \sigma_j \, q_j\right)
SymbolDescriptionUnitTypical Range
ka,ik_{a,i}Adsorption rate constant1/(M⋅s)\text{1/(M}{\cdot}\text{s)}10−7 – 10710^{-7} \text{ – } 10^{7}
kd,ik_{d,i}Desorption rate constant1/s\text{1/s}10−7 – 10710^{-7} \text{ – } 10^{7}
νi\nu_iCharacteristic charge—0 – 200 \text{ – } 20
σi\sigma_iShielding factor—0 – 500 \text{ – } 50
Λ\LambdaTotal ionic capacity of the resinM\text{M}0.001 – 1.80.001 \text{ – } 1.8

q0q_0 tracks remaining binding capacity after accounting for sites blocked by bound molecules (νj\nu_j sites per molecule). q^0\hat{q}_0 further reduces available capacity via steric shielding (σj\sigma_j). Adsorption scales as a power law in available capacity; desorption scales as a power law in salt concentration — higher salt drives elution.

MPM Langmuir

Mobile phase modulated Langmuir with exponential adsorption modulation and power-law desorption modulation.

dqidt=ka,i eγi csalt cp,i qmax⁡,i(1−∑j∈nsqjqmax⁡,j)−kd,i csaltβi qi\frac{dq_i}{dt} = k_{a,i} \, e^{\gamma_i \, c_{\text{salt}}} \, c_{p,i} \, q_{\max,i} \left(1 - \sum_{j \in \text{ns}} \frac{q_j}{q_{\max,j}}\right) - k_{d,i} \, c_{\text{salt}}^{\beta_i} \, q_i
SymbolDescriptionUnitTypical Range
ka,ik_{a,i}Adsorption rate constant1/(M⋅s)\text{1/(M}{\cdot}\text{s)}10−7 – 10710^{-7} \text{ – } 10^{7}
kd,ik_{d,i}Desorption rate constant1/s\text{1/s}10−7 – 10710^{-7} \text{ – } 10^{7}
qmax⁡,iq_{\max,i}Maximum binding capacityM\text{M}0.001 – 1.80.001 \text{ – } 1.8
γi\gamma_iExponential salt modulation of adsorption—−1 – 0-1 \text{ – } 0
βi\beta_iPower-law salt modulation of desorption—0 – 100 \text{ – } 10

The saturation sum is over non-salt components only. γi\gamma_i can be positive (salt promotes adsorption) or negative (salt inhibits). βi\beta_i controls the strength of salt-driven desorption.

HIC 1

Non-competitive cooperative adsorption model with salt-dependent anomalous desorption kinetics, developed by Wang et al. (2016). Operates on non-salt components only.

dqidt=ka,i(1−qiqmax⁡,i) ⁣nicp,i  −  kd,i qi 1+ni β\frac{dq_i}{dt} = k_{a,i} \left(1 - \frac{q_i}{q_{\max,i}}\right)^{\!n_i} c_{p,i} \;-\; k_{d,i} \, q_i^{\,1 + n_i \, \beta}

where:

β=β0 eβ1 csalt\beta = \beta_0 \, e^{\beta_1 \, c_{\text{salt}}}
SymbolScopeDescriptionUnitTypical Range
ka,ik_{a,i}Per componentAdsorption rate constant1/s\text{1/s}1 – 10,0001 \text{ – } 10{,}000
kd,ik_{d,i}Per componentDesorption rate constant1/s\text{1/s}1 – 1001 \text{ – } 100
nin_iPer componentCooperativity coefficient—5 – 155 \text{ – } 15
qmax⁡,iq_{\max,i}Per componentMaximum binding capacityM\text{M}0.005 – 0.50.005 \text{ – } 0.5
β0\beta_0GlobalDesorption salt modulation constant—0.02 – 0.050.02 \text{ – } 0.05
β1\beta_1GlobalDesorption salt modulation exponent1/M\text{1/M}0.5 – 20.5 \text{ – } 2

Adsorption has per-component saturation (not competitive across components). ni=1n_i = 1 gives simple Langmuir-like adsorption; ni>1n_i > 1 gives cooperative behavior. The desorption exponent depends on salt through β\beta, giving anomalous kinetics. Decreasing salt concentration (typical HIC gradient) reduces β\beta, promoting elution.

HIC 2

The most complex binding model, developed by Jäpel et al. (2025). Multi-factor salt and concentration modulation with competitive saturation and nonlinear desorption. Operates on non-salt components only.

dqidt=ka,i(1−∑j∈nsqjqmax⁡,j) ⁣nicp,i ekp,i cp,i eks csalt  −  kd,i (1+ϵi qi) qi e−ρ csalt ni β0 eβ1 csalt\frac{dq_i}{dt} = k_{a,i} \left(1 - \sum_{j \in \text{ns}} \frac{q_j}{q_{\max,j}}\right)^{\!n_i} c_{p,i} \, e^{k_{p,i} \, c_{p,i}} \, e^{k_s \, c_{\text{salt}}} \;-\; k_{d,i} \, (1 + \epsilon_i \, q_i) \, q_i \, e^{-\rho \, c_{\text{salt}} \, n_i \, \beta_0 \, e^{\beta_1 \, c_{\text{salt}}}}
SymbolScopeDescriptionUnitTypical Range
ka,ik_{a,i}Per componentAdsorption rate constant1/s\text{1/s}1 – 10,0001 \text{ – } 10{,}000
kd,ik_{d,i}Per componentDesorption rate constant1/s\text{1/s}1 – 1001 \text{ – } 100
nin_iPer componentCooperativity coefficient—5 – 155 \text{ – } 15
qmax⁡,iq_{\max,i}Per componentMaximum binding capacityM\text{M}0.005 – 0.50.005 \text{ – } 0.5
kp,ik_{p,i}Per componentSelf-interaction modifier for adsorption1/M\text{1/M}−5 – 10-5 \text{ – } 10
ϵi\epsilon_iPer componentNonlinear desorption parameter1/M\text{1/M}0 – 500 \text{ – } 50
ksk_sGlobalSalt promotion of adsorption1/M\text{1/M}−1 – 1-1 \text{ – } 1
β0\beta_0GlobalDesorption salt modulation constant—0.02 – 0.10.02 \text{ – } 0.1
β1\beta_1GlobalDesorption salt modulation exponent1/M\text{1/M}0.5 – 20.5 \text{ – } 2
ρ\rhoGlobalDesorption strength modifier1/M\text{1/M}0.01 – 10.01 \text{ – } 1

Adsorption term:

  • (1−∑qj/qmax⁡,j)ni\left(1 - \sum q_j / q_{\max,j}\right)^{n_i} — competitive saturation with cooperativity
  • ekp,i cp,ie^{k_{p,i} \, c_{p,i}} — concentration-dependent self-interaction
  • eks csalte^{k_s \, c_{\text{salt}}} — global salt promotion of adsorption

Desorption term:

  • (1+ϵi qi) qi(1 + \epsilon_i \, q_i) \, q_i — nonlinear loading dependence
  • e−ρ csalt ni β0 eβ1 csalte^{-\rho \, c_{\text{salt}} \, n_i \, \beta_0 \, e^{\beta_1 \, c_{\text{salt}}}} — complex salt-dependent desorption rate

Colloidal 1

Models lateral interactions between adsorbed molecules on the surface, based on the colloidal energetics framework of Oberholzer & Lenhoff (1999). As surface coverage increases, the energy barrier for further adsorption rises. Operates on non-salt components only.

dqidt=ka,i cp,i e−ϕi−kd,i qi\frac{dq_i}{dt} = k_{a,i} \, c_{p,i} \, e^{-\phi_i} - k_{d,i} \, q_i

where the lateral interaction energy is:

ϕi=exp⁡ ⁣(m1,i θm2,i)−1\phi_i = \exp\!\left(m_{1,i} \, \theta^{m_{2,i}}\right) - 1
θ=∑j∈non-saltqjqmax⁡,j\theta = \sum_{j \in \text{non-salt}} \frac{q_j}{q_{\max,j}}

The coefficients m1m_1 and m2m_2 are precomputed from physical parameters using the Oberholzer correlations:

κa=κ⋅ri\kappa_a = \kappa \cdot r_i
m1,i=α1⋅(ln⁡bpp,i)α2+α3m_{1,i} = \alpha_1 \cdot (\ln b_{pp,i})^{\alpha_2} + \alpha_3
m2,i=(0.075 (ln⁡κa)0.74−0.189)(2.59+ln⁡bpp,i)+0.021 (ln⁡bpp,i)2+1.50m_{2,i} = (0.075 \, (\ln \kappa_a)^{0.74} - 0.189)(2.59 + \ln b_{pp,i}) + 0.021 \, (\ln b_{pp,i})^2 + 1.50

where:

α1=0.010 κa2−0.137 κa+0.825\alpha_1 = 0.010 \, \kappa_a^2 - 0.137 \, \kappa_a + 0.825
α2=0.394 (ln⁡κa)0.304+1.250\alpha_2 = 0.394 \, (\ln \kappa_a)^{0.304} + 1.250
α3=−0.04 κa2+0.233 κa+3.12\alpha_3 = -0.04 \, \kappa_a^2 + 0.233 \, \kappa_a + 3.12
SymbolScopeDescriptionUnitTypical Range
ka,ik_{a,i}Per componentAdsorption rate constant1/(M⋅s)\text{1/(M}{\cdot}\text{s)}10−7 – 10710^{-7} \text{ – } 10^{7}
kd,ik_{d,i}Per componentDesorption rate constant1/s\text{1/s}10−7 – 10710^{-7} \text{ – } 10^{7}
qmax⁡,iq_{\max,i}Per componentMaximum binding capacityM\text{M}0.01 – 0.10.01 \text{ – } 0.1
bpp,ib_{pp,i}Per componentDimensionless pair interaction energy—1 – 301 \text{ – } 30
rir_iPer componentProtein radiusm\text{m}10−9 – 5×10−910^{-9} \text{ – } 5 \times 10^{-9}
κ\kappaGlobalDebye screening parameter1/m\text{1/m}108 – 5×10810^{8} \text{ – } 5 \times 10^{8}

θ\theta is the total fractional surface coverage. ϕi\phi_i is the lateral interaction energy, which increases with coverage. Higher coverage raises the energy barrier for further adsorption (e−ϕie^{-\phi_i} decreases). Desorption follows simple first-order kinetics with no lateral interaction effect.

References

  • Wang, G., et al. (2016). Hydrophobic interaction chromatography model. J. Chromatogr. A, 1465, 71-78.
  • Jäpel, R., et al. (2025). Unified HIC isotherm. J. Chromatogr. A, 1756, 466095.
  • Oberholzer, M. R., & Lenhoff, A. M. (1999). Protein adsorption isotherms through colloidal energetics. Langmuir, 15, 3905-3914.
  • Xu, X., & Lenhoff, A. M. (2009). Lattice Boltzmann simulations. J. Chromatogr. A, 1216, 6177-6195.
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