Diffusion is the process by which molecules spread from areas of high concentratiion, to areas of low concentration. When the molecules are even throughout a space – it is called EQUILIBRIUM Diffusion Diffusion Equation- For sovling this PDE we need to know initial conditions and the boundary conditions Where, C the concentration and D diffusion Constant.

Today We will solve Partial differential equation (PDE) in Vcell. Start from File  New  BioModel 1.Name the compartment 2.Add a species (c) Now we have to define a Geometry to represent the behavior of the system.

Defining Geometries : What to rememebr before starting Geometric Model 1. Each model requires a detailed description of the (cellular) geometry to represent the behavior of the (cellular) system. 2. Geometric model represents the morphometry of a particular Cell, or a portion of the cell. 3. In order to properly define a simulation domain, the geometry must be 2D or 3D segmented images with appropriate scaling information. 4. Membranes are implicitly represented as the boundary between the dissimilar compartments. 5. For regular structures or symmetric cells, analytic geometry is used instead of experimental images.

Defining Geometry To Create a Analytic Geometry go to File>New>Geometry>Analytic>2D A new document called Geometry will show up. Here we will create a simple 2D geometry. In our case, we have a single compartment.

Click in Geometry editor change domain to change the size. A new dialog Geometry size will appear. Double click Subvolume1 to rename the compartment ( optional)

We can assign values of X and Y in Geometry size window and click ok. Lets take for simplicity, X= 10.0 and Y=10.0 to create a square. The origin refers to the coordinates relative to the analytic Geometry

Click Surface Viewer to see the surface of the geometry you just created.

This surface viewer dialog shows surface of your 2D data set and its size.It allows to rotate in space (interesting for 3D). Save the Geometry with a name. It will be saved in your geometry document. Close the Geometry window. We will need this geometry soon.

Go to Application: Application  New  Deterministic Give a name of this application ( This a spatial application, so i gave the name pde ) Click on View/change Geometry

This window will pop up. Click change Geometry.. and select the geometry you just created.

You will see like this in your screen Close the Geometry box. Note your geometry is in your Application Window.

Use to map your Geometry to your Bio Model Now concentrate here. This is to configure the Boundary Condition of the system For solving PDEs we need to know Initial conditions and Boundary conditions Flux Value

Possible boundary conditions: fixed concentration at the boundary no flux (boundary is impermeable) Consider a Box with a drop of blue ink at the middle and no flux at all boundaries. Case 1. Simple Diffusion Now Click Initial condition in application pannel.

double click Here we have to write the expression for the concentration of c. Concentration of c is confiend in that box. Write the expression as follows and click ok.. It means concentration is at the middle of the box and its value is 1. Expression is given by this inequality

See the description. Assign value of diffusion constant as 1.0 and boundary conditions 0.0 Remember: in the structure mapping sector we consider Flux BC and now we have assigned them all zero, that is no flux at the boundaries Now save the Model with a name to see the Math and to run the simulation. Thats all !!!

This is the Code for the PDE. Click View Math Note this portion

Next step is to Run Simulation. Run the simulation at least for 10 sec to see the result. Increase the time to see the diffusion

Let us see qualitatively how the concentration gradient changes with time.

Selection / Drawing Tools. Selection arrow Point tool Line tool Spline tool Add an additional control point We want to see the quantitative results too.

Select Line tool to draw a line like this. Click spatial plot to see the initial concentration gradient. To delete Line or point, just use Back space or delete key

For t=0For t=1sec For t=7sec For t=10 sec

Change the diffusion rate and see how the results changes. Here Diffusion rate =5.0, as you see equillibrium comes faster !!

Usisng the same Geometry will study a few cases. Case 2: Diffusion with C=0 at all boundaries Steps are same as before. Start from New Biomodel. Only Difference: In Structure Mapping sector we have to assign Value instead of Flux. The rest is the same Check for the curiosty what happens if the circle inside the box becomes very small.

Results:

t=1.6 sec t=0 t=30 sec Result: Concentration is confined at middle of the box with C=0 at the boundaries.

Case 3 : Diffusion with concentration is 1 and 0 at left and right boundaries, and no Flux bounary conditions for top and bottom walls. Start as before: Create BioModel  Application  Map Geometry Click for X- and X+ Value, and Y-and Y+ Flux

Initial Condition – same as before. But the BC_Xm =1 and BC_XP=0 and rests are zero.

Qualitative Results:

Line plot for t=0.4 sec

Line plot for t=10 sec

Let us see other what happens with other geometric structure. Again Start a new BioModel like before. Create a new geometry 2D File  New  Geometry  Analytic  2D

Click change domain. X=30.0 and Y=2.0, Save the Geometry.

Next: 1.Application (deterministic) 2.Structure Mapping (click Flux) 3.Initial condition (provide an expression for initial concentration.) 4.Diffusion constant=1, and All BCs=0

Save the Model and go to simulation Run simulation for t=30 Red= High concentration, Blue= zero concentration at t=0

For t=10.6 sec

Line plot for t=0

Line plot for t=10.6secLine plot for t=3.2sec for t=30sec

1.To see the equillibrim we have to run the simulation for long time. or 2. We have to increase the diffusion constant For t= 200sec, and D=1.0 For t= 11.3 sec D= 50 The system is in equillibrium

Another Geometry. Here we will consider this geometry as a cell in ECM Start File  New  Biomodel Double click to give the compart name ECM click to add feature Click here.

It will look like this Add species c in the compartment cell and your Bio Model is complete.

This is Bio Model.

Create this Geometry (2D analytic). Think how to create geometry. we have two compartments. So click Add to add another subvolume Write the expression for the geometry in any of the subvolumes

See what I did to create this geometry. Save this Geometry. It will be saved in your Geometry document. Or if you can not Use shared Geometry from my account File  Open  Geometry  Shared Geometries  Satarupa  diff_sq_rectang  click

Application: create an application (pde) 1.Structure Mapping  Change Geometry 2.Map the geometry with your model

Initial Condition: The initial concentration is confined in the square.

Save the Model and go to simulation. Run simulation D=1.0 T=0T=1.3 T=5.0T=143.0

Line plot t=10.0 sec Line plot t=0.7 sec Line plot t=143.0 secLine plot t=60.0 sec

Time plot for an arbitrary point. Select a point with and click show time plot