Penalties

Penalty terms are the building blocks of the Potts Hamiltonian. Each term models one biophysical constraint; they are summed to give the total energy $H = \sum_k H_k$. Copy attempts are accepted or rejected based on the change in total energy $\Delta H$.

In PottsToolkit you add penalties to a system via Components (e.g. VolumeComponent, AdhesionComponent). Each component internally constructs the corresponding CorePotts penalty object and registers the required trackers. This page describes the underlying penalty mathematics for reference.

Volume Penalty

Biological meaning: Cells resist compression and over-expansion. The volume penalty drives each cell's area (2D) or volume (3D) towards a target value.

Formula (classical quadratic):

\[H_\text{vol} = \sum_{i} \lambda_{V,\tau(i)} \bigl(v_i - V_i^{\rm target}\bigr)^2\]

where $v_i$ is the current area/volume of cell $i$, $V_i^{\rm target}$ is its target, and $\lambda_{V,\tau(i)}$ is the stiffness for cell type $\tau(i)$.

PottsToolkit component:

VolumeComponent(
    A => (λ = 5.0f0, target = 500),
    B => (λ = 3.0f0, target = 300),
)

When to use: Always include a volume penalty unless you are specifically studying unconstrained lattice gas dynamics. Without it cells will spread across the entire grid.

HST Volume Penalty

Biological meaning: Same as the volume penalty, but with a stochastically evolving auxiliary field that restores detailed balance when the target volume changes over time (e.g. during cell growth).

Formula (augmented Hamiltonian):

\[H_\text{HST-vol} = \sum_{i} \lambda_{V,\tau(i)} \bigl(v_i - \phi_i\bigr)^2\]

where $\phi_i$ follows the Ornstein-Uhlenbeck SDE:

\[d\phi_i = -\eta(\phi_i - V_i^{\rm target})\,dt + \sqrt{2\eta T}\,dW_t\]

The auxiliary field $\phi_i$ acts as a "soft" target that fluctuates around $V_i^{\rm target}$ and relaxes on timescale $1/\eta$.

PottsToolkit component:

HSTVolumeComponent(
    A => (λ = 5.0f0, target = 500);
    eta = 1.0,
)

When to use: Use HSTVolumeComponent instead of VolumeComponent whenever target_volume is not fixed — for example when using LinearGrowthCallback or when training with NeuralPotts. See Concepts → HST Penalties for the theoretical justification.

HST Surface Area Penalty

Biological meaning: Cells have a preferred perimeter (2D) or surface area (3D) set by cytoskeletal tension.

Formula:

\[H_\text{HST-surf} = \sum_{i} \lambda_{S,\tau(i)} \bigl(s_i - \psi_i\bigr)^2\]

where $s_i$ is the current surface area (number of sites on the cell boundary) and $\psi_i$ is the HST auxiliary field for surface area.

PottsToolkit component:

SurfaceAreaComponent(A => (λ = 1.0f0, target = 200))

[!NOTE] SurfaceAreaComponent uses the HST formulation by default. The surface area tracker counts boundary contacts with the medium or other cells, not geometric perimeter, so target should be set accordingly.

Adhesion Penalty

Biological meaning: Cells preferentially adhere to cells of the same type (like likes like) or avoid contact with certain neighbours. The adhesion matrix encodes the energetic cost of each pair of cell-type interfaces.

Formula:

\[H_\text{adh} = \sum_{\langle \mathbf{x}, \mathbf{x}' \rangle} J\bigl(\tau(\sigma(\mathbf{x})),\; \tau(\sigma(\mathbf{x}'))\bigr) \cdot \mathbf{1}\bigl[\sigma(\mathbf{x}) \neq \sigma(\mathbf{x}')\bigr]\]

The sum runs over all neighbouring pairs $\langle \mathbf{x}, \mathbf{x}' \rangle$. $J(\tau_1, \tau_2)$ is the adhesion energy between cell types $\tau_1$ and $\tau_2$; it is symmetric ($J(\tau_1,\tau_2) = J(\tau_2,\tau_1)$).

PottsToolkit component:

AdhesionComponent(
    (A, Medium) => 15.0f0,
    (B, Medium) => 15.0f0,
    (A, A)      => 2.0f0,
    (B, B)      => 2.0f0,
    (A, B)      => 10.0f0,
)

Isotropic flag: Pass isotropic=true to the underlying penalty if you want the adhesion energy to be averaged over all neighbour directions (useful for anisotropic grids or when matching analytical results).

When to use: The adhesion penalty is what drives cell sorting and tissue organisation. Low $J$ for same-type pairs relative to cross-type pairs causes cells to cluster; high $J$ for all pairs relative to medium causes cells to round up.

Focal-Point Spring Penalty

Biological meaning: Models a spring-like tether between two cells or between a cell and a fixed anchor point — relevant for cell–substrate adhesion, cell–cell junctions, and tissue attachment.

Formula:

\[H_\text{fps} = \lambda_\text{fps}\; \bigl|\mathbf{c}_i - \mathbf{r}_0\bigr|^2\]

where $\mathbf{c}_i$ is the centroid of cell $i$ and $\mathbf{r}_0$ is the anchor (which may be the centroid of another cell).

When to use: Tissue confinement, modelling substrate adhesion sites, or anchoring cells to an epithelium.

HST Focal-Point Spring Penalty

The HST variant of the focal-point spring uses an auxiliary field for the anchor position, allowing the anchor to diffuse or be driven by an external field without violating detailed balance.

HST Length Penalty

Biological meaning: Cells have a preferred elongation (major-axis length). This drives cells to adopt rod-like or spindle-shaped morphologies, relevant for muscle cells, neurons, and migrating mesenchymal cells.

Formula:

\[H_\text{len} = \sum_{i} \lambda_{L,\tau(i)} \bigl(\ell_i - \phi_{L,i}\bigr)^2\]

where $\ell_i$ is the major-axis length of cell $i$ (computed from the inertia tensor) and $\phi_{L,i}$ is the HST auxiliary field for length.

PottsToolkit component:

LengthComponent(A => (λ = 3.0f0, target = 20.0f0); eta = 1.0)

[!TIP] Combine LengthComponent with AdhesionComponent to produce spontaneous alignment of elongated cells — a simple model of nematic ordering in epithelial monolayers.

Chemotaxis Penalty

Biological meaning: Cells sense a chemical gradient in an external scalar field and bias their motion up or down the gradient (chemotaxis / chemorepulsion).

Formula:

\[\Delta H_\text{chem}(\mathbf{x} \to \mathbf{x}') = -\chi_{\tau(\sigma(\mathbf{x}'))}\; \bigl[c(\mathbf{x}') - c(\mathbf{x})\bigr]\]

where $c(\mathbf{x})$ is the chemical concentration at site $\mathbf{x}$ and $\chi_\tau$ is the chemotactic sensitivity of cell type $\tau$ (negative for chemorepulsion).

PottsToolkit component:

ChemotaxisComponent(A => 0.5f0, chemical_field = my_field)

chemical_field should be an array of the same size as the simulation grid containing the chemical concentrations at each lattice site. It can be updated between MCS steps to model a dynamic field.

When to use: Directed cell migration, wound healing, developmental patterning, tumour invasion assays.

Quick-Reference Table

Custom Penalties and Auxiliary Hooks

If you need to implement a biophysical mechanic that isn't covered by the standard components, you can define your own AbstractPenalty. The CorePotts engine uses a highly optimized, flat loop structure for evaluating these penalties.

Auxiliary Integration Hooks

Some penalties (like the Hydrostatic formulations) require continuous integration of state variables alongside the discrete Monte Carlo sweeps. The engine provides two distinct hooks for updating these auxiliary variables. It is critical to use the correct hook to preserve both algorithmic stability and thermodynamic balance:

• update_step_auxiliary!(penalty, u, p, cache): Evaluates once per SciML macro-step (defined by alg.dt). Use this hook for slowly varying fields, macroscopic updates, or external PDE boundary conditions that do not need to respond instantaneously to lattice changes.
• update_sweep_auxiliary!(penalty, u, p, cache): Evaluates at the end of every Monte Carlo Sweep (MCS) (which occurs sweeps_per_step times per macro-step). You must use this hook for continuous internal variables (like Hydrostatic pressure fields) that need high-frequency relaxation in response to immediate structural cell shape changes.

Failing to hook continuous thermodynamic variables into the high-frequency sweep loop can cause catastrophic variance regressions and numerical instability.