Protein Adsorption

This application simulates an ion-exchange column for protein adsorption. The fluid phase contains four components: two proteins, one solvent, and one salt. The adsorption/desorption kinetics is described by two equilibrium reactions where proteins displace ions adsorbed at the surface and vice versa.

The example highlights how reactions at chemical equilibrium can be studied in a 0D reactor system in Reaction Engineering. In addition, it also shows how the kinetics from the 0D setup is exported to a 3D model where the reacting surface in the column can be studied in detail. The 3D model incorporates mass transport through diffusion and convection, and the reactions at the surface of the ion-exchange mass with Transport of Diluted Species, Laminar Flow, and Surface Reactions interfaces.

Introduction Protein Ion-Exchange

The binding of proteins to ion exchangers can be described within the steric mass action approximation (SMA). This approach assumes that the adsorption of a protein can be considered as an exchange reaction of the protein with a given number of adsorbed ions, the process is schematically depicted in Figure 1.

Figure 1: Proteins A and B displacing salt ions S at an ion-exchanger surface.

The equilibrium describing the adsorption/desorption reactions is

(1)

Here, S denotes the salt ion, P stands for either protein A and B. Once P is adsorbed, P(ads), salt ions are displaced, reducing the concentration of adsorbed salt ions, S(ads).

Model Definition

The system is modeled both in 0D and in 3D. The former model setup is adequate to investigate the kinetics of the equilibrium reactions within the column. The latter makes it possible to study the surface of the ion-exchange beads that make up the porous structure of the ion-exchange mass. In Figure 2, the two model approaches are presented.

Figure 2: Ion-exchange column and model geometries. The 0D model approximates the entire column, while the 3D geometry is a detailed representation of a section at the top of the column.

The 0D model utilizes the reactor type CSTR, constant volume in the Reaction Engineering interface. Ideal conditions are assumed within the reactor, meaning that well-mixed conditions with no concentration gradients apply. The reaction kinetics are described by Equation 1 for proteins A and B are entered directly into the interface as equilibrium surface reactions. The equilibrium constants for reaction 1, adsorption of A, is K1eq = 2 and for reaction 2, adsorption of B, is K2eq = 5. These are entered as well, relating the concentrations as followed

(2)

(3)

In order to compute the concentrations, both protein surface concentrations need to be set as dependent in the Equilibrium Species Vector section.

The proteins enter the reactor with a Feed Inlet with concentrations that vary in accordance to a 10 s Gaussian pulse with a maximum of 0.05 mol/m3. The outlet flow rate regulates so that the volume of the reactor is constant. Initially, no protein is available in the column and the ion-exchange mass is set to be completely adsorbed with salt, that is, the initial site density, γ0, is equal to the initial surface concentration of S, S(ads).

For the 3D model, as shown in Figure 3, only one quarter of the top section of the column is simulated due to symmetry. The proteins enter the top with a constant concentration of 0.05 mol/m3. In the column initially, the pores (the bulk) is filled with solvent and the ion-exchange beads are saturated with adsorbed salt.

Figure 3: 3D model geometry.

The reaction kinetics are taken directly from the 0D model with the Generate Space-Dependent Model feature and are collected in a Chemistry interface that also supplies computed diffusion coefficients and fluid density for the 3D setup.

The mass balances describing the species mass transport in the pores of the column are set up with the Transport of Diluted Species interface with diffusion and convection accounted for

(4)

On the right hand side of Equation 4 no reaction source is present for the bulk. However, the surface of the ion-exchange beads produces a reaction source that needs to be coupled to this equation. This is simply done with a Surface Reaction interface. Such an interface is automatically generated with the use of the Generate Space-Dependent Model feature if surface reactions are present.

The 3D model also computes the convective velocity, u, with a Laminar Flow interface. The velocity is based on the assumption that the velocity, U, within the reactor is 0.1 mm/s and that the inlet is open to the surroundings, that is, exposed to the atmospheric pressure.

Study settings

The 3D problem is solved in a two-step study. First, the Laminar Flow interface is first solved with a Stationary study step. Second, the rest of the interfaces (Chemistry, Transport of Diluted Species, and Surface Reactions) are solved with a Time Dependent study step. A Fully Coupled Direct solver is required in the second step to obtain stable computations.

Study 1 — Space Independent

Figure 4 shows how the concentrations of the reacting species change with the time. Initially, only adsorbed salt species are present in the column. The concentrations of proteins A and B are seen to change with the Gaussian concentration pulse feed inlet. A stronger adsorption affinity of protein B compared to protein A is readily observed. Note also how the concentration of bulk salt species S increases as the proteins adsorb at the surface. Toward the end of the pulse, most proteins have been adsorbed in the column and the bulk salt species have exited the system.

Figure 4: Concentrations of the reacting species as functions of time (s).

Study 2 — Space Dependent

The 3D model is solved for 30 s and a selection of the results are displayed in this section. Figure 5, Figure 6, Figure 7, and Figure 8 show the behavior of protein B after 5 s and 30 s.

From Figure 6, in which the bulk concentration is shown, indicates that the beads at the center of the column are less accessible for adsorption or that the protein is more rapidly adsorbed at the center, both phenomena lowering the bulk concentration there.

Figure 5: Bulk concentration of B at 5 s.

Figure 6: Bulk concentration of B at 30 s.

To get a proper understanding of the ion-exchange beads, a comparison of the adsorbed B surface concentration at the two times is also made with the results in Figure 7 and Figure 8. The lower surface concentration of B in the center suggests that less B is adsorbed there and that the porous structure obstructs the species transport. Note that the bead surface cannot adsorb more than the (initial) site density of the ion-exchange material, γ0 (SI unit: mol/m2).

Figure 7: Surface concentration of B at 5 s.

Figure 8: Surface concentration of B at 30 s.

In Figure 9, the velocity field is displayed. It shows that the porous structure causes a quite distorted velocity field. The exception is at the walls where the flow is less obstructed due to the relatively large gap between beads and wall.

Figure 9: Velocity field in the pores of the column section.

Chemical_Reaction_Engineering_Module/Mixing_and_Separation/protein_adsorption

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Import the model parameters from a file from the application’s folder.

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file protein_adsorption_parameters.txt.

Gaussian Pulse 1 (gp1)

Reactants are injected to the system in a pulse. Choose a function to describe the injection.

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 5.

In the text field, type 1.5.

Use the function to set up a time-dependent pulse variable with an amplitude of 1.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
pulse11	3.76*gp1(t/1[s])		Pulse with amplitude 1

Reaction Engineering (re)

Select a constant volume CSTR to model the ion-exchange column in 0D.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Reaction 1

Set up the equilibrium reaction for adsorption/desorption of protein A.

In the toolbar, click

In the window for , locate the section.

In the text field, type S(ads)+A=S+A(ads).

Click .

Locate the section. In the Keq0 text field, type Keq01.

Enter densities and molar masses in the bulk species nodes for .

Surface species: S(ads)

COMSOL will automatically assign the S to be sulfur. However, in this case is S a descriptive variable for the salt ion.

In the window, click .

In the window for , locate the section.

Clear the check box.

In the M text field, type MS.

Species: A

In the window, click .

In the window for , locate the section.

In the M text field, type MA.

In the ρ text field, type rho_p.

Species: S

In the window, click .

In the window for , locate the section.

Clear the check box.

In the M text field, type MS.

In the ρ text field, type rho_S.

Surface species: A(ads)

In the window, click .

In the window for , locate the section.

In the M text field, type MA.

Reaction 2

In the toolbar, click

In the window for , locate the section.

In the text field, type S(ads)+B=S+B(ads).

Click .

Locate the section. In the Keq0 text field, type Keq02.

As before, enter densities and molar masses in the bulk species nodes, but now for .

Species: B

In the window, click .

In the window for , locate the section.

Clear the check box.

In the M text field, type MB.

In the ρ text field, type rho_p.

Surface species: B(ads)

In the window, click .

In the window for , locate the section.

Clear the check box.

In the M text field, type MB.

Replace the default dependent variables and replace these with and . This omits any cyclic dependence of the reaction rates.

In the window, click .

In the window for , locate the section.

In the text field, type A(ads):B(ads).

Locate the section. Find the subsection. From the list, choose .

Set vp=0, to neglect the volumetric production rate.

In the vp text field, type 0.

Select the reactive surface area available in the reactor. This regulates, when multiplied with the site density, how much the ion-exchange column can adsorb.

Find the subsection. In the Ar text field, type Arsurf.

Initial Values 1

In the feature, set all species except adsorbed S () to zero. Also, set the initial density of surface sites for the ion-exchange mass. Note that the latter property sets the upper limit of how many moles can be adsorbed. Assume that each species takes up one reactive site.

In the window, click .

In the window for , click to expand the section.

In the Γs text field, type G0.

In the table, enter the following settings:


Species	Surface concentration (mol/m^2)	Site occupancy number (1)
S(ads)	CS0surf	1

Feed Inlet 1

Last, in the OD model, add the feature containing bulk species A and B. Add the concentration with the Gaussian pulse in the feed stream.

In the toolbar, click

In the window for , locate the section.

In the vf text field, type vfp.

Locate the section. In the table, enter the following settings:


Species	Concentration (mol/m^3)
A	CAmax_inlet*pulse11
B	CBmax_inlet*pulse11

Solve the 0D model for 14 s.

Study 1 - Space Independent

In the window, click .

In the window for , type Study 1 - Space Independent in the text field.

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,0.1,14).

In the toolbar, click

Results

Concentrations 0D model

Create Figure 4 in which all species concentrations are displayed.

In the window for , type Concentrations 0D model in the text field.

Click to expand the section. From the list, choose .

Locate the section. Select the check box.

Select the check box. In the associated text field, type Concentration (mol/m<sup>3</sup>) | Surface concentration (mol/m<sup>2</sup>).

Locate the section. From the list, choose .

Global 1

In the window, expand the node, then click .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Click in the upper-right corner of the section. From the menu, choose .

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
re.csurf_A_surf*1e4	mol/m^2	Surface concentration per square dm
re.csurf_B_surf*1e4	mol/m^2	Surface concentration per square dm
re.csurf_S_surf*1e4	mol/m^2	Surface concentration per square dm

Click to expand the section. From the list, choose .

In the table, enter the following settings:


Legends
A
B
S
Surface A
Surface B
Surface S

In the toolbar, click

Continue setting up a space-dependent model to investigate the top section of the ion-exchange column. Start by adding a solvent to the system. The solvent in these types of systems is often water.

Reaction Engineering (re)

Species 1

In the toolbar, click

In the window for , locate the section.

In the text field, type H2O.

Locate the section. From the list, choose .

Locate the section. In the M text field, type MH2O.

In the ρ text field, type rho_H2O.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Species	Concentration (mol/m^3)
H2O	CH2O

Specify the transport properties of the solvent. These will transferred to the interface in the space-dependent model below.

In the window, click .

In the window for , click to expand the section.

Select the check box.

From the list, choose .

In the μ text field, type muH2O.

Use the feature. Select to create a 3D geometry together with a and a interface and a study type.

Generate Space-Dependent Model 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. Find the subsection. From the list, choose .

Locate the section. Click .

Geometry 1(3D)

Insert the 3D geometry sequence file protein_adsorption_geom_sequence.mph from the application’s folder.

In the window, expand the node, then click .

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file protein_adsorption_geom_sequence.mph.

In the toolbar, click

Click the

button in the toolbar.

Transport of Diluted Species (tds)

Start with the interface.

Transport Properties 1

The interface is Coupled to the velocity field.

A and B enter the column at the top. Select for a more stable computation.

Despite the fact that only solvent makes up the fluid within the column initially, adding initial trace concentrations of all dependent species gives a more stable solution to the equilibrium reactions.

Initial Values 1

In the window, expand the node, then click .

In the window for , locate the section.

In the cA text field, type CAmax_inlet*0.01.

In the cB text field, type CBmax_inlet*0.01.

In the cS text field, type 1e-7.

Inflow 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. In the c0,cA text field, type CAmax_inlet.

In the c0,cB text field, type CBmax_inlet.

Locate the section. From the list, choose .

Outflow 1

Set an condition at the bottom of the geometry. This means that the transport along the z direction (the height) of the reactor is dominated by convection.

In the window, click .

In the window for , locate the section.

From the list, choose .

Surface Equilibrium Reaction 1

The surface of the beads is the boundary where reactions take place.

In the window, click .

In the window for , locate the section.

From the list, choose .

Surface Equilibrium Reaction 2

In the window, click .

In the window for , locate the section.

From the list, choose .

Surface Reactions 1 (sr)

In the surface reactions interface make sure that the reactions take place on the surface of the beads.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Reactions 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

For the same reason as in the interface, add initial trace concentrations of species A and B.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the csA text field, type CS0surf*0.01.

In the csB text field, type CS0surf*0.01.

Laminar Flow 1 (spf)

At the inlet, or the top of the column, atmospheric pressure is applied since the column is open to the surroundings.

Inlet 1

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

The average velocity through the xy-plane of the geometry is assumed to be constant everywhere along the column height. Set a constant velocity within the column, in this case, at the bottom of the 3D model geometry.

Outlet 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the U0 text field, type U_column.

Symmetry 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Multiphysics

Reacting Flow, Diluted Species 1 (rfd1)

In the toolbar, click

and choose .

Mesh 1

Select a mesh that resolves the surface of the beads well.

Free Triangular 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section.

Select the check box. In the associated text field, type 9E-2.

Free Tetrahedral 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section.

Select the check box. In the associated text field, type 5E-1.

This model requires that a two-step study node is used. The first step solves the stationary solution of the interface and the second the time-dependent solution for the rest of the interfaces.

Study 2 - Space Dependent

In the window, click .

In the window for , type Study 2 - Space Dependent in the text field.

Stationary - Laminar Flow

In the toolbar, click

and choose .

In the window for , type Stationary - Laminar Flow in the text field.

Locate the section. In the table, clear the check boxes for , , , and .

Time Dependent - the Rest of the Interfaces

In the window, click .

In the window for , type Time Dependent - the Rest of the Interfaces in the text field.

Locate the section. In the text field, type range(0,1,30).

From the list, choose .

In the text field, type 2e-4.

Locate the section. In the table, clear the check box for .

Step 2: Stationary - Laminar Flow

In the window, under right-click and choose .

Solution 2 (sol2)

In the toolbar, click

Replace the solver with a one since this fits the quick shifts in concentration originating from the equilibrium reactions in the system. Also apply a solver to further improve the stability of the computations.