Reversible binding kinetics

The objective of this tutorial is to model reversible binding kinetics described by the reaction,

R + L ⇌ C

where R denotes the receptor, L is the ligand and C is the receptor-ligand complex. Consider that the receptor and complex are solid-bound molecules whereas the ligand is a solute in a multiphasic mixture representing a biological tissue. For example, the ligand may represent insulin-like growth factor I (IGF-I) and the receptor may represent the IGF-I binding protein 6 (IGFBP-6). The receptor and complex may be bound to the extracellular matrix of articular cartilage.

This is an intermediate tutorial and assumes some familiarity with creating FEBio models in FEBio Studio. If you are new to FEBio and FEBio Studio, you may want to follow the Intro to FEBio Studio tutorials first.

To start the tutorial, open FEBio Studio and start a new model (menu File\New Model). Select the Multiphasic Analysis option in the New Model dialog box and click OK. Then, follow the steps below.

The completed model can be downloaded here.

Step 1: Defining the model data

Set the units for this model to mm-g-s (mouse right-click, Options…->Units). Multiphasic materials require the specification of global variables. In the Model Viewer, select Globals and enter the temperature and double-check the remaining values:

• Absolute temperature: 293 K
• Gas constant: 8.314 nJ/nmol⋅K
• Faraday’s constant: 96.485 µA·s/nmol

Step 2: Defining the solute and solid-bound molecules

In this step, we’ll define the solute and solid-bound molecules that will interact with each other later in a chemical reaction. The solutes and solid-bound molecules are currently not listed in the model tree. Instead, they are stored in tables that can be accessed via the Physics menu.

First, we’ll add the solute. Select Physics/Solute Table from the menu. In the Solute Table dialog box, click on Add. A new entry will be added to the table with some default values. You can double-click on any value to change it. Change the values to the following:

• Name: ligand
• Molar Mass: 7.649e−6 g/nmol
• Density: 1e-3 g/mm^3

Then close the Solute Table dialog window.

Next, we’ll add the solid-bound molecules. Select Physics/Solid-Bound Molecule Table from the menu. Click Add and enter the following values:

• Name: receptor
• Molar Mass: 23e-6 g/nmol
• Density: 1e−3 g/mm^3

Add one more solid-bound molecule with the following values:

• Name: complex
• Molar Mass: 30.649e−6 g/nmol
• Density: 1e−3 g/mm^3

Then close the dialog window.

Step 2: Creating a multiphasic material

Since we want to solve a multiphasic problem, we need to create a multiphasic material. Select and right-click on the Materials item in the Model Viewer and select Add Material from the popup-menu. Rename the material tissue in the Name entry. Select the multiphasic material and click OK. The material will now be added to the model tree. Modify the material as follows: select an isotropic elastic material for the solid part, perm-const-iso for the permeability and osm-coef-const for the osmotic coefficient.

To add a solute, first set the value of the solute property to 1. (The solute is an array property, and its value is the size of the array.) A new array entry should appear, called solute-1. It is still empty, so add a solute by clicking the dropdown button next to it and click the <select> option. A new dialog box appears. Select the “Solute” option and click OK. This adds the solute to the material. Now, in a similar way, add a constant (diff-const-iso) diffusivity, and a constant (solub-const) solubility properties.

Then add two solid-bound components. First, set the value of the solid_bound property to 2. This adds two empty solid bound entries. For each, now add a “solid_bound” item.

This process only defines the composition of the multiphasic material. We still need to set the properties of all the material components. Enter the following material parameters:

• solid volume fraction (phi0): 0.2

For the solid property:

• Young’s modulus: 1066 Pa

For the permeability property:

• permeability: 1e−9 mm⁴ /µN⋅s

For the solute property:

• solute: ligand
• free diffusivity: 0.005 mm² / s
• diffusivity: 0.004 mm² /s

For the solid bound property (1):

• Solid-bound molecule: receptor
• initial density: 0.92e−9 g/mm³

For the solid bound property (2):

• Solid-bound molecule: complex

All other parameters should be left at their default values.

Step 4: Creating a chemical reaction

In this step, we’ll setup the chemical reaction in the material. Although chemical reactions can be defined directly in the model tree, there is a special tool for defining chemical reactions that simplifies the process a bit. Select Physics/Chemical Reaction Editor from the menu to access the chemical reaction tool. In the dialog box that appears, notice that the top-dropdown is the material for which you wish to define the reaction.

Click the Add button to add a new chemical reaction to the tissue material and set the name to Reversible Binding. Then, pull down the type option and select mass-action-reversible; pull down the Forward Rate menu and select constant reaction rate; pull down the Reverse Rate menu and select constant reaction rate.

Under Reactants you’ll see two panels. The left panel shows all available reactants, and the right panel shows the actual reactants that will be used in this reaction. Now select the ligand and receptor and move them from the left panel to the right panel. The Products panels work similarly. Move the complex product form the left to the right panel. Click Ok to exit this dialog window.

This process adds a chemical reaction to the tissue material in your model. Select the tissue material in the Model Viewer and expand it until the reaction components are visible. Then enter the following material parameters:

• forward rate k: 200 mM⁻¹⋅ s⁻¹
• reverse rate k: 0.001 s⁻¹
• ligand vR: 1
• receptor vR: 1
• complex vP: 1

Step 5: Geometry and meshing

The geometry consists of a cube. Open the Create panel, select the box, and accept the following default dimensions in the appropriate editing fields.

• Width = 1 mm
• Height = 1mm
• Depth = 1 mm

Click the Create button to create the geometry. The box, which is named Object01 by default, will now show up in the Model Viewer.

To create a mesh for this box, activate the Mesh panel and enter the following meshing parameters:

• Nx = 1
• Ny = 1
• Nz = 30
• z-Bias = 0.9

Leave the other parameters with their default values and click Apply. This will create a biased hexahedral mesh for the box. Remember to use the ‘m’ shortcut to toggle the mesh in the Graphics View.

Assign the multiphasic material Tissue to this box by first selecting the box then clicking the plus sign in the material’s selection box.

Step 6: Boundary and initial conditions

Use Physics/Add Nodal BC (or Ctrl-B) to prescribe boundary conditions. Select the two faces normal to the x-direction and constrain them to have fixed displacement along x. Similarly, select the two faces normal to the y-direction and constrain them to have fixed displacement along y. Select the z=0 face normal to the z-direction and constrain it to have a fixed displacement along z.

Select the z=1 face normal to the z-direction and add a prescribe concentration boundary condition. Select the ligand as the degree of freedom (dof) paramaeter and set the value to 1. Next, add a prescribed fluid pressure boundary condition and set the value of 1. Note that in both cases, the value parameter has a green, lit icon, which indicates that a load controller was attached by FEBio Studio to these parameters. Next, we’ll need to edit these curves.

Use View/Curve Editor (or F4) to call up the curve editor and select the PrescribedConcentration BC for the ligand; switch the curve type to Step, select the second point (initially at 1,1) and enter the ordinate values 1e−5 mM. Similarly, select the PrescribedFluidPressure BC, switch the curve type to Step, select the second point and enter the ordinate value -2.436e−2 Pa (evaluated from the negated product of the gas constant, the absolute temperature and the prescribed ligand concentration). Close the curve editor.

The initial conditions for the ligand concentration and fluid pressure in this problem are zero, which is the default value for all initial conditions. Therefore it is not necessary to explicitly set the initial conditions here. For completeness however, you may wish to go through the following steps. In the main toolbar click on the Select Part icon and select the box. Select Physics/Add Initial Condition (or Ctrl-I), then select Initial Concentration in the dialog and click Ok. Select the initial condition in the Model Viewer and in the properties list select the ligand solute and set the value to 0.

Step 7: Setting up the analysis

Add a new analysis step to the model using the Physics/Add Step from the menu. Set the Analysis to TRANSIENT. Select the Time Stepping settings as follows:

• Time steps = 10000
• Step size = 1
• Max step size = 1000
• Min step size = 0.1

Set the Nonlinear solver settings to:

• Displacement tolerance (dtol) = 0
• Energy tolerance (etol) =0
• Pressure tolerance  (ptol)= 0
• Concentration tolerance = 0.01
• Max updates (max_ups) = 0

Set the Matrix Storage in the Linear solver settings to Non-symmetric.

Leave other parameters to their default values.

In this tutorial we want to use a load curve for the dtmax parameter. This gives us better control on what the max time step size can be at a particular point in time. Bring up the Curve Editor (F4) and assign a load curve to the dtmax parameter (Model > Steps > Step1 > time_stepper > dtmax). Edit the default load curve by adding two points. The first point is located at (0,0), the second point located at (10000, 2000). This setting will increase the maximum time step linearly from 0 to 2000 s over the span of the analysis time.

Step 8: Running the analysis

This completes the process of setting up the model. You can now proceed to running the model. Make sure you first save the model to a file. Then, use the menu FEBio\Run FEBio to open the Run dialog box. If you wish, you may choose the job’s name and the working directory, or just accept the defaults and click Run. If you are unfamiliar with the Run dialog, please take a look at the tutorial Running your first simulation in FEBio Studio.

The figure below shows the solute 1 (ligand) concentration, the sbm 1 (receptor) concentration, and the sbm 2 (complex) concentration at a select intermediate time step.

(a) ligand concentration (mM)

(b) receptor concentration (mM)

(c) complex concentration (mM)

Was this article helpful?

5 out of 5 stars

2 ratings

5 Stars		100%
4 Stars		0%
3 Stars		0%
2 Stars		0%
1 Stars		0%

5 Submit

How can we improve this article?

Please submit the reason for your vote so that we can improve the article.

Name

Email

Submit