Guided example: ligand-receptor binding

Let’s revisit our ligand – receptor binding example, but this time using the ODE modelling framework.

We still consider the same three components:

• Ligand (free ligand in solution),
  
• Inactive receptor (unbound receptor),
  
• Active receptor (ligand-bound receptor complex).
  

The binding reaction is:

 Ligand + Receptor ⇌ Active receptor

As before, we apply mass action kinetics:

The forward reaction (binding) occurs at rate:

   r_forward ​= k_bind​ ⋅ ligand(t) ⋅ receptor_in​(t)

The reverse reaction (dissociation) occurs at rate:

 r_back ​= k_dissociate ​⋅ receptor_a​(t)

From this, we can directly write the ODEs:

These three equations describe how the concentrations of ligand, inactive receptor, and active receptor, respectively, change continuously over time, driven by the reversible binding reaction. You will notice that the right hand side of the ODEs is equivalent to the part of the difference equation that describes the change during the time interval \dt. However, conceptually there is a difference: In difference equations, the rate indicates how much the quantity changes over each time step, while in differential equations, it describes how quickly the quantity is changing at any given instant.

As always in ODE models, to simulate the system we also need initial conditions i.e. the starting concentrations of all three components.

Exploring model behaviour

We can use a numeral solver to get the a time course simulation of our ligand receptor binding ODE model. Figure 15 shows the dynamics of the 3 model components over a duration of 10 seconds.