iCasp9 safety switch - 2D culture dish

What is modelled

Engineered T cells can carry an emergency off switch. In the iCasp9 design a human caspase-9 is fused to a modified FKBP12 domain. Caspase-9 only becomes active when two copies are brought together; a small drug with two identical binding ends (a chemical dimerizer) does exactly that. The simulation follows a culture of cells, some carrying the switch at different expression levels and some not, through a dose of the dimerizer: the drug is injected, spreads through two body compartments, binds the switch inside the cells, and kills the switch-positive cells while sparing the rest.

Equations

• Binding: the drug D has two identical ends, each binding an iCasp9-FKBP molecule R with dissociation constant K_d = 1 nM. With free drug D fixed by the plasma, the single-ended complex is RD = 2 R D/Kd, the active dimer is RDR = R²D/Kd² and the total receptor is Rt = R + RD + 2 RDR. This gives the quadratic (2D/Kd²) R² + (1 + 2D/Kd) R − Rt = 0, and the active fraction is φ = 2 RDR/Rt.
• Hook effect: φ first rises with D, then falls again at high D, because every receptor then binds its own drug molecule and no bridges form. A very large dose therefore works worse than a moderate one. The graph “dimer fraction against dose” shows the bell shape for five expression levels.
• Killing: a cell class with dimer fraction φ dies at rate k = kmax φ²/(θ² + φ²) with k_max = 8 per hour and θ = 0.1, standing for the amplification of the caspase cascade.
• Pharmacokinetics: after an injection the plasma level follows two compartments, C1' = −(k10 + k12) C1 + k21 C2 and C2' = k12 C1 − k21 C2, giving a fast distribution phase and a slow elimination phase.
• Populations: five classes of switch-positive cells with iCasp9 levels of 0.02, 0.1, 1, 3 and 10 times the median (weights 10, 20, 40, 20, 10 %), plus untransduced cells. Each grows logistically toward a shared capacity, Ni' = r Ni (1 − N/K) − ki Ni. Low expressers receive a small φ and escape, which is how a residual pool of switch-positive cells can survive a dose.

Checks run in the page

• The closed-form dimer fraction is compared with the steady state of the full mass-action equations (R + D ⇌ RD, RD + R ⇌ RDR) integrated in time for six dose and expression pairs.
• For a vanishing receptor level the peak of the bell curve is known exactly, D = K_d/2; a golden-section search on the full solution is compared with it, and the high-dose collapse is verified.
• The area under the plasma curve from the integrator is compared with dose/k₁₀, the eigenvalues are tied to k₁₀, k₁₂ and k₂₁, and logistic growth without drug is compared with its closed form.

Honest simplifications

• K_d, the expression levels, k_max, θ and the pharmacokinetic rates are illustrative values chosen to be plausible; they are not fitted to clinical data and the dose slider is not a patient dose. In reports on this switch a single dose of the dimerizer removed the large majority of switch-positive T cells within hours and a small residual population remained, which is the qualitative behaviour you can reproduce here.
• The free drug inside a cell is assumed equal to the plasma level; protein binding, membrane transport, FKBP12 competition and tissue distribution are not modelled. Cells are treated as independent and well mixed, and the logistic term stands in for all growth limits.
• The dish is a schematic top view: each dot is one cell unit of the capacity of 220, drawn far larger than a real T cell, and positions are random. The 3D page shows the same model in a culture flask with a dosing syringe.

What to try

• Press Administer at the standard 5 nM dose and watch the high-expression classes disappear within hours while the lowest class, which carries almost no switch, keeps growing.
• Choose Overdose: 1000 nM and compare with 5 nM: the larger dose kills fewer cells because of the hook effect.
• Choose Low expression: clones escape, and the dose-response graph shows why the dimer fraction never reaches the threshold.