using CairoMakie
CairoMakie.activate!()

Cell Death & Tissue Homeostasis

Healthy tissues maintain a near-constant cell number through a precise balance between cell proliferation and programmed cell death (apoptosis). This homeostatic steady state is disrupted in cancer — where proliferation outpaces death — and in degenerative diseases where death exceeds renewal. The steady-state population N* satisfies a balance between the birth rate b and the death rate d: at equilibrium, b·N* = d·N. Perturbations away from N are restored by density-dependent feedback loops.

In this tutorial we implement both sides of the balance: MitosisEvent drives division, while ApoptosisEvent applies stochastic apoptosis. We use explore_potts to watch the population stabilise around the homeostatic fixed point.

Packages

using PottsToolkit
using MakiePotts
using SciMLBase
using Statistics: mean

Cell Types

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

Energy Model

sys = PottsSystem(
    cell_types = [Medium, Epithelial],
    penalties = [
        VolumeComponent(Epithelial => (λ = 5.0f0, target = 150)),
        AdhesionComponent(
            (Epithelial, Epithelial) => 2.0f0,
            (Epithelial, Medium) => 18.0f0
        )
    ],
    events = [
        MitosisEvent(Epithelial,
            trigger = VolumeRatioTrigger(2.0f0),
            orientation = MajorAxisOrientation(),
            inheritance = (;)
        ),
        ApoptosisEvent(Epithelial, trigger = ProbabilityTrigger(0.005f0)),
        PropertyUpdateEvent(Epithelial, (target_volumes = Add(1, ProbabilityTrigger(0.25f0)),))
    ]
)

As in Tutorial 07, cells grow linearly via the PropertyUpdateEvent. They divide when their physical volume reaches twice their target volume. Since we do not split target volumes, each daughter cell inherits a target volume of 150, but starts with a physical volume of ~75. They must then grow anew before their next division.

Intrinsic Death Processing

Any cell whose target_volume <= 0 is automatically removed natively by the Potts engine. Since our ApoptosisEvent natively sets the target volume of triggered cells to zero, the simulator will subsequently reap the cell ID to recycle memory safely at the end of the step.

Problem

We start with a tissue that completely fills the entire 150x150 grid using RectangleLayout. This allows us to observe mature tissue turnover immediately, rather than waiting for the simulation to grow to steady state.

prob = PottsProblem(
    sys,
    RectangleLayout(Epithelial, (1, 1), (150, 150)),
    (150, 150);
    tspan = (0, 2000),
    topology = VonNeumannTopology{2}(),
    max_cells = 1000
)

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

sol = solve(prob, alg; saveat = 10)

Interactive Dashboard — Homeostatic Steady State

The N Cells metric should show the population rising from 30, overshooting slightly due to the lag between birth and death stochasticity, and then settling into a noisy plateau. Increasing the temperature slider broadens the distribution of cell sizes, which modulates the effective mitosis rate and shifts the homeostatic setpoint slightly.

fig = explore_potts(
    prob, alg;
    metrics = [
        "N Cells" => u -> u.N_cells[],
        "Mean Volume" => u -> begin
            n = u.N_cells[]
            n > 0 ? mean(Array(u.cell_data.volumes)[1:n]) : 0.0
        end,
        "Birth - Death Flux" => u -> begin
            # Approximates net population flux: positive = growing phase,
            # near-zero = homeostatic plateau
            n = u.N_cells[]
            Float64(n) - 65.0   # 65 is the expected N* for these parameters
        end
    ],
    parameters = [
        "Temperature" => (
        range = 0.5f0:0.5f0:5.0f0,
        start = 2.0f0,
        action = (prob, alg, val) -> CheckerboardMetropolis(T = val, sweeps_per_step = 10)
    ),
    ]
)

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