Most spherical virus capsids are built from many copies of one (or a few) coat-protein subunits arranged with icosahedral symmetry. Caspar & Klug (1962) showed that a shell can only close smoothly if its capsomers sit on a geodesic triangulation of the sphere obeying:
T = h² + hk + k² (h,k = 0,1,2,…)
capsomers = 10T + 2 → 12 pentamers (always, 5-fold vertices)
10(T−1) hexamers (6-fold vertices)
This simulator builds the exact geodesic lattice for the selected Class I (k=0) T-number by subdividing each face of a regular icosahedron into a triangular grid of frequency ν=√T, welding shared lattice points, and classifying each resulting vertex by its graph degree — Euler's formula guarantees exactly 12 vertices of degree 5 (pentamers) and the rest of degree 6 (hexamers), for any closed triangulated icosahedral shell.
Growth follows a simplified nucleation-and-growth model consistent with in vitro capsid assembly kinetics (Zlotnick, 1994): an empty site's attachment probability per unit time scales with the number of already-occupied neighbouring sites, so a lone nucleation event is rare but growth accelerates sharply once a nucleus forms —
P(attach in dt) = 1 − exp[−k_on·[subunit]·(0.08 + n_occupied_neighbors)·dt]
P(detach in dt) = 1 − exp[−k_off·θ·exp(−n_occupied_neighbors)·dt]
- T-number — selects the lattice size (12 → 42 → 92 → 162 capsomers), matching real capsids from small satellite viruses (T=1) up toward large icosahedral phages (T≥16).
- kon / concentration — jointly set the effective association rate; higher subunit concentration also enlarges the visible free-protein pool.
- Thermal agitation — raises the off-rate for weakly-bonded (few-neighbour) subunits, so a poorly-nucleated shell can visibly fall apart, matching the reversible, error-correcting character of real capsid self-assembly.
- Binding energy — each capsomer-capsomer contact formed is assigned ≈4 kT, a typical order of magnitude for the weak, reversible interfaces (H-bonds + hydrophobic contacts) that make capsid assembly cooperative rather than a single irreversible reaction.