Guided example: Ligand receptor binding as a kinetic model

Let’s now revisit our ligand–receptor binding system in the context of a difference equation model with kinetic (mass action) dynamics.

We model a simple biochemical interaction where a ligand binds reversibly to an inactive membrane receptor to form an active, ligand-bound receptor. This interaction can be written as a reversible reaction:

Ligand + Inactive receptor ⇌ Active receptor

Model components

We include the following three dynamic components in our model:

• Ligand (t) : concentration of free ligand at time t
• Receptor_in : concentration of inactive receptor
• Receptor_a : concentration of active (ligand-bound) receptor

Reaction kinetics

We use mass action kinetics, as introduced in the previous section, to describe the reaction rates of the forward (binding) and backward (dissociation) reactions:

Forward rate (binding):

r_bind​ = k_bind ​Ligand (t) Receptor_in​ (t)

Backward rate (dissociation):

r_diss ​= k_diss​⋅ Receptor_a​ (t)

Difference equations

We now write one difference equation for each component, describing how their values change at each time step Δt:

Active receptor:

Receptor_a​ (t + Δt) = Receptor_a​ (t) + r_bind​⋅Δt − r_diss​⋅Δt

Inactive receptor:

Receptor_in​ (t + Δt) = Receptor_in ​(t) − r_bind​⋅Δt + r_diss​⋅Δt

Free ligand:

Ligand (t + Δt) = Ligand (t) − r_bind​⋅Δt + r_diss​⋅Δt

We can interpret these three difference equations as follows:

Active receptor: The number of active receptors increases with binding and decreases with dissociation.

Inactive receptor: The number of inactive receptors decreases with binding and increases with dissociation..

Free ligand: The number of free ligands decreases with binding and increases with dissociation.

Exploring model behaviour

With a model at hand, we can finally explore its dynamic behaviour. Figure 13 shows a simulation of the model dynamic, computed by updating the values of all model components according to their difference equations over 10 time steps. Starting from an initial amount of inactive receptors and ligands, we see how their numbers decrease and the number of active receptors increases.