Boundary Conditions

The default periodic boundary conditions let cells wrap around the grid edges, which is appropriate for modelling bulk tissue but unsuitable for confined geometries such as a cell monolayer growing in a well or a tissue slice pressed against a rigid substrate. PottsToolkit exposes the boundary condition through the topology keyword of PottsProblem. Swapping a single type changes the behaviour of every attempted copy event at the boundary — no other code needs to change.

Packages

using PottsToolkit
using MakiePotts
using Statistics

Shared model components

Both simulations use an identical energy landscape; only the topology differs.

Cell = CellType(:Cell)
Medium = CellType(:Medium, is_background = true)

sys = PottsSystem(
    cell_types = [Medium, Cell],
    penalties = [
        VolumeComponent(
            Cell => (λ = 5.0f0, target = 300),
            Medium => (λ = 0.0f0, target = 0)
        ),
        SurfaceAreaComponent(
            Cell => (λ = 1.5f0, target = 70),
        ),
        AdhesionComponent(
            (Cell, Medium) => 20.0f0,
            (Cell, Cell) => 4.0f0
        )
    ]
)

alg = CheckerboardMetropolis(T = 2.0f0, sweeps_per_step = 10)

Periodic topology (VonNeumannTopology{2})

The 4-connected periodic neighbourhood is the standard Potts topology. Cells that reach the right edge reappear on the left; top wraps to bottom. This is ideal for bulk tissue models where edge effects should not matter.

prob_periodic = PottsProblem(
    sys,
    HypersphereLayout(Cell, (200, 100), 20),
    (200, 200);
    tspan = (0, 600),
    topology = VonNeumannTopology{2}()
)

sol_periodic = solve(prob_periodic, alg; saveat = 20)

No-flux (rigid wall) topology (NoFluxVonNeumannTopology{2})

The no-flux topology reflects copy attempts at boundaries: a lattice site at the edge cannot copy from a site outside the grid, and copy attempts that would move a cell across the boundary are rejected. This mimics cells growing against a rigid substrate or the walls of a culture dish. Cells therefore pile up, elongate, and crowd near the walls rather than smoothly wrapping.

prob_noflux = PottsProblem(
    sys,
    HypersphereLayout(Cell, (200, 100), 20),
    (200, 200);
    tspan = (0, 600),
    topology = NoFluxVonNeumannTopology{2}()
)

sol_noflux = solve(prob_noflux, alg; saveat = 20)

Extended neighbourhoods

ExtendedVonNeumannTopology{2,2}() uses a radius-2 neighbourhood (up to 12 neighbours in 2-D) instead of the standard 4. Longer-range interactions smooth the lattice artefacts, relevant when modelling signalling molecules that diffuse a few pixels before acting. The extended topology is also available in no-flux form:

ExtendedVonNeumannTopology{2,2}()
ExtendedMooreTopology{2,2}()
NoFluxVonNeumannTopology{2}()
NoFluxMooreTopology{2}()

Side-by-side visualisation

We record both simulations, then assemble them into a single figure for comparison using the Makie layout system.

record_potts("periodic_boundary.mp4", sol_periodic; framerate = 15, resolution = (800, 800))
record_potts("noflux_boundary.mp4", sol_noflux; framerate = 15, resolution = (800, 800))

Build a static side-by-side comparison from the final frames

using CairoMakie
CairoMakie.activate!()

fig = Figure(resolution = (1400, 700))

ax1 = Axis(fig[1, 1]; title = "Periodic (VonNeumann)")
ax2 = Axis(fig[1, 2]; title = "Rigid Walls (NoFlux)")

pottsplot!(ax1, sol_periodic.u[end])
pottsplot!(ax2, sol_noflux.u[end])

save("boundary_conditions_comparison.png", fig)
fig

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