Ligand-functionalized nanoparticles are suspended in an extracellular fluid volume (the translucent outer shell) and execute a 3D random walk driven by thermal (Brownian) fluctuations, exactly as real nanocarriers do before they reach a target cell. Each frame every free particle's position is perturbed by a Gaussian step whose size is set by the diffusion coefficient D via the Einstein relation.
Brownian step: Δx = sqrt(2·D·Δt) · N(0,1) (per axis)
Surface receptor binding (Langmuir adsorption):
dθ/dt = k_on·C_local·(1-θ) − k_off·θ
Per-step binding probability (particle within capture radius r_c
of a free receptor): P_bind = 1 − exp(−k_on·Δt)
Per-step unbinding probability (if k_off enabled):
P_unbind = 1 − exp(−k_off·Δt)
When a diffusing particle drifts within the capture radius of an unoccupied receptor on the cell membrane, it binds with probability P_bind each timestep — a discretised, stochastic version of Langmuir kinetics. Bound particles (shown in accent colour) lock onto that receptor site; if dissociation is enabled they can detach again with probability P_unbind and resume diffusing.
- Diffusion coefficient D — sets the thermal step size (Stokes–Einstein regime); larger D means faster, larger nanoparticles diffuse more slowly in reality, but here it directly scales the random-walk amplitude.
- Binding affinity kon — the association rate constant; higher values mean a collision with a free receptor is more likely to result in a bond this frame.
- Nanoparticle count — total carriers released into the extracellular volume.
- koff toggle — switches on reversible binding (dissociation), letting bound-fraction reach a dynamic Langmuir equilibrium instead of saturating irreversibly.
This mirrors real biological-nanotechnology systems such as antibody-functionalized nanoparticles for targeted chemotherapy delivery, where diffusion and receptor-mediated binding kinetics together determine how efficiently a drug payload reaches diseased cells.