This is a genuine 2D counterpart to the 3D bulk-solution simulator: instead of monomers diffusing through a 3D volume, it models peptide amphiphiles confined to a planar interface (the geometry of a Langmuir monolayer or a substrate-adsorbed PA film — a real experimental self-assembly setting in its own right, not a flattened slice of the 3D scene). Each disc is a monomer with a 2D position and a single in-plane orientation angle θ standing in for its β-strand backbone direction. The same three competing forces act, but the alignment term is re-derived directly in 2D rather than reused from the 3D vector code:
Hydrophobic core burial (Lennard-Jones, same as 3D):
U_LJ(r) = 4ε[(σ/r)^12 − (σ/r)^6]
Screened electrostatic repulsion (Debye–Hückel / Yukawa):
F_elec(r) ∝ exp(−r/λ_D)·(1/r² + 1/(rλ_D)), λ_D ≈ 0.304/√I nm
(headgroups sit at the interface but are screened by the bulk
3D electrolyte above it, so the same exponential form applies)
2D nematic alignment (exact analytic reduction, not a 3D projection):
U_ij = −κ·cos(2(θ_i−θ_j)) (θ and θ+π are the same rod)
dθ_i/dt ∝ β·Σ_j w(r)·sin(2(θ_j−θ_i))
2D nematic order tensor (eigenvalues closed-form, no iteration needed):
Q_xx = ⟨cos 2θ⟩, Q_xy = ⟨sin 2θ⟩ → S = √(Q_xx² + Q_xy²)
Motion is overdamped Langevin dynamics: translational noise scales as √(2D·Δt) per axis, and — because a 2D rod's orientation is a single scalar angle rather than a unit vector on a sphere — the rotational noise √(2D_r·Δt) needs no renormalization step, unlike the 3D version's per-frame unit-vector projection.
- Concentration — below the effective critical aggregation concentration monomers stay dispersed on the interface; above it, fibrils nucleate and grow.
- Ionic strength — added salt shrinks the Debye length λ_D, screening headgroup repulsion so fibrils assemble faster and pack thicker.
- Temperature — higher T adds thermal agitation that can outrun the hydrophobic/H-bond driving forces and disassemble fibrils.
- β-sheet propensity — strength of the backbone hydrogen-bond registry torque; low values give disordered clumps, high values give long aligned fibrils (teal = assembled, grey = free monomer).
- Local bond order — the average of cos(2Δθ) between monomers that are actually touching (within the neighbor radius), a purely local diagnostic distinct from the global nematic order S, which can stay low even while individual fibrils are internally well-ordered but point in different directions.