A folded (compact) protein has a much smaller effective radius than its unfolded (coil) form. Inserting a sphere of radius a into a bath of hard-sphere crowders of radius R at volume fraction φ costs a free energy given by scaled-particle theory (Minton):
βμ(a,R,φ) = −ln(1−φ) + 3q·t + [3q²t + 4.5q²t²] + q³t
where q = a/R, t = φ/(1−φ)
Because the unfolded coil's radius is bigger, inserting it costs more free energy than inserting the compact folded state — crowding entropically favours the folded state. The stabilisation is:
ΔΔG_crowd = RT · [μ(a_folded) − μ(a_unfolded)] (kcal/mol, always ≤ 0)
ΔG_total = ΔG_intrinsic + ΔΔG_crowd
K_fold = exp(−ΔG_total / RT), fraction folded = K_fold / (1 + K_fold)
Radii are estimated from chain length N (residues) with standard scaling laws: folded Rg ≈ 0.224·N0.38 nm (compact globule), unfolded Rg ≈ 0.193·N0.598 nm (Flory random coil). Temperature is fixed at 310 K (body temperature).
- Box panel — a 2D schematic slice of the crowded cytoplasm: grey crowder disks fill the box, the teal bead chain morphs between a spiral globule and a random-walk coil. Drag to pan, scroll/pinch to zoom.
- Energy panel — live bars for the insertion free energy of each conformation (βμfolded, βμunfolded), the crowding stabilisation ΔΔG and the total ΔG that sets the fold/unfold odds.
- History strip — a scrolling trace of the chain's radius of gyration over time, shaded by folded (teal) vs. unfolded (grey) state, so you can see the equilibrium being sampled attempt after attempt.
- φ slider — how much of the cytoplasm area is occupied by other macromolecules (crowders). Real cells run ~20–40%.
- rc slider — crowder size (e.g. small ions vs. BSA-sized proteins). Smaller crowders pack more excluded volume per φ, so they stabilise folding more strongly.
- N slider — protein chain length; sets both the on-screen bead-chain length and the two physical radii above.
- ΔG0 slider — the protein's own intrinsic folding stability in dilute buffer, before crowding is added (many real proteins are only marginally stable, ±3 kcal/mol).
- Every few seconds the chain attempts a folding/unfolding transition, sampled from the live equilibrium probability above — this is the same excluded-volume argument used to explain why proteins that barely fold in a test tube fold reliably inside a real, crowded cell.