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Multi-State Steric Mass Action

The multi-state steric mass action model adds $M_i-1$ additional bound states $q_{i,j}$ ($j=0,\dots ,M_i-1$) for each component $i$ to the steric mass action model (see Section Steric Mass Action) and allows the exchange between the different bound states $q_{i,0},\dots ,q_{i,M-1}$ of each component. In the multi-state SMA model a variable number of states of the bound molecule (e.g., different orientations on the surface, binding strength of tentacle adsorbers) is added which are more and more strongly bound, i.e.,

$$\begin{array}{r}{\nu }_{i,j}\le {\nu }_{i,j+1}i=1,\dots ,N_{\text{comp}}-1,j=0,\dots ,M_i-1.\end{array}$$

The exchange between the different states of each component is allowed and, since the molecules can potentially bind in all states at the same binding site, competitive effects are present.

$$\begin{array}{r}\begin{array}{rl}\frac{\mathrm{d}{q}_{i,j}}{\mathrm{d}t}=& \phantom{+}{k}_{a,i}^{(j)}{c}_{p,i}{\left(\frac{{\overline{q}}_0}{q_{\text{ref}}}\right)}^{{\nu }_{i,j}}-k_{d,i}^{(j)}{q}_{i,j}{\left(\frac{c_{p,0}}{c_{\text{ref}}}\right)}^{{\nu }_{i,j}}\\ & -\sum _{\ell =0}^{j-1}\underset{\text{to weak state}}{\underbrace{k_{j\ell }^{(i)}{q}_{i,j}{\left(\frac{c_{p,0}}{c_{\text{ref}}}\right)}^{({\nu }_{i,j}-{\nu }_{i,\ell })}}}-\sum _{\ell =j+1}^{M_i-1}\underset{\text{to strong state}}{\underbrace{k_{j\ell }^{(i)}{q}_{i,j}{\left(\frac{{\overline{q}}_0}{q_{\text{ref}}}\right)}^{({\nu }_{i,\ell }-{\nu }_{i,j})}}}\\ & +\sum _{\ell =0}^{j-1}\underset{\text{from weak state}}{\underbrace{k_{\ell j}^{(i)}{q}_{i,\ell }{\left(\frac{{\overline{q}}_0}{q_{\text{ref}}}\right)}^{({\nu }_{i,j}-{\nu }_{i,\ell })}}}+\sum _{\ell =j+1}^{M_i-1}\underset{\text{from strong state}}{\underbrace{k_{\ell j}^{(i)}{q}_{i,\ell }{\left(\frac{c_{p,0}}{c_{\text{ref}}}\right)}^{({\nu }_{i,\ell }-{\nu }_{i,j})}}}& \begin{array}{rl}i& =1,\dots ,N_{\text{comp}}-1,\\ j& =0,\dots ,M_i-1,\end{array}\end{array}\end{array}$$

where $c_{p,0}$ and $q_0$ denote the salt concentrations in the liquid and solid phase of the beads respectively. The number of available salt ions ${\overline{q}}_0$ is given by

$$\begin{array}{r}{\overline{q}}_0=\mathrm{\Lambda }-\sum _{j=1}^{N_{\text{comp}}-1}\sum _{\ell =0}^{M_j-1}({\nu }_{j,\ell }+{\sigma }_{j,\ell })q_{j,\ell }.\end{array}$$

A neutrality condition compensating for the missing equation for $\frac{\mathrm{d}{q}_0}{\mathrm{d}t}$ is required:

$$\begin{array}{r}{q}_0=\mathrm{\Lambda }-\sum _{j=1}^{N_{\text{comp}}-1}\sum _{\ell =0}^{M_j-1}{\nu }_{j,\ell }{q}_{j,\ell }.\end{array}$$

The concept of reference concentrations ($c_{\text{ref}}$ and $q_{\text{ref}}$) is explained in the respective paragraph in Section Reference concentrations.

For more information on model parameters required to define in CADET file format, see Multi-State Steric Mass Action.