This is the 2D companion to the 3D TCR-engineered T-cell simulator, computed independently rather than a flattened render of the same tissue scene. It keeps the exact same Hill-type avidity model that scores every T cell–tumor contact, but instead of animating individual cells patrolling a 3D field, it renders that model directly as two genuinely 2D-native views: an activation-probability landscape over the (antigen density, Kd) plane, and a mean-field population-kinetics chart that integrates the same equation forward in time for two competing sub-populations.
effective density C = pMHC_copies × (1 − MHC_downreg)
P(activation) = C^n / (K_d^n + C^n), n = 2
mean-field kinetics (visible pool v, escaped pool e):
attempt rate = min(T_cells, N_v+N_e) / τ_cycle, split ∝ N_v : N_e
dN_v/dt = −attempt_v · P(antigen, K_d)
dN_e/dt = −attempt_e · P(antigen × 0.06, K_d)
- Heatmap — every pixel is an independent (antigen, Kd) pair fed through the Hill equation above; the two markers show where your current sliders and the escaped sub-population's much lower effective antigen level land on that surface.
- Population chart — solves the kinetic equations above by explicit time-stepping (not per-cell movement), so the visible and escaped tumor sub-populations decline at rates set purely by where their marker sits on the heatmap.
- Kd — lower = a tighter-binding receptor; slides both markers left on the surface (higher kill probability).
- MHC-I downregulation — the escaped marker's effective antigen is only 6% of the visible marker's, the same near-total recognition loss used in the 3D model; watch how far down the surface that marker sits even when the visible marker is deep in the red.
Because both sub-populations are driven by the same Hill equation on the same surface, the chart makes an important point invisible in the 3D per-cell view: the escaped population isn't literally immune, it's just parked in a low-probability region of the landscape — given enough dwell time, even a small activation probability eventually depletes it too.