BIO-LGCA

The BIO-LGCA is based on the lattice-gas cellular automaton (LGCA) model used in fluid dynamics.

Contrary to classic cellular automaton models, particles in BIO-LGCA are defined by their position and velocity.

This allows to model and analyze active fluids and collective migration mediated primarily through changes in momentum, rather than density.

[4] For modeling particle velocities explicitly, lattice sites are assumed to have a specific substructure.

is connected to its neighboring lattice sites through vectors called "velocity channels",

is equal to the number of nearest neighbors, and thus depends on the lattice geometry (

The states of every site in the lattice are updated synchronously in discrete time steps to simulate the model dynamics.

Depending on the specific application, the interaction step may be composed of reaction and/or reorientation operators.

The reaction operator does not conserve particle number, thus allowing to simulate birth and death of individuals.

The reaction operator's transition probability is usually defined ad hoc form phenomenological observations.

The transition probability for this operator can be determined from statistical observations (by using the maximum caliber principle) or from known single-particle dynamics (using the discretized, steady-state angular probability distribution given by the Fokker-Planck equation associated to a Langevin equation describing the reorientation dynamics),[5][6] and typically takes the form

is an energy-like function which particles will likely minimize when changing their direction of motion,

The state resulting form applying the reaction and reorientation operator

The transport step simulates the movement of agents according to their velocity, due to the self-propulsion of living organisms.

During this step, the occupation numbers of post-interaction states will be defined as the new occupation states of the same channel of the neighboring lattice site in the direction of the velocity channel, i.e.

Therefore, the dynamics of the BIO-LGCA can be summarized as the stochastic finite-difference microdynamical equation

The transition probability for the reaction and/or reorientation operator must be defined to appropriately simulate the modeled system.

In the absence of any external or internal stimuli, cells may move randomly without any directional preference.

If organisms reproduce and die independently of other individuals (with the exception of the finite carrying capacity), then a simple birth/death process can be simulated[3] with a transition probability given by

is the Heaviside function, which makes sure particle numbers are positive and bounded by the carrying capacity

The formation of cell aggregates via adhesive biomolecules can be modeled[7] by a reorientation operator with transition probabilities defined as

is a vector pointing in the direction of maximum cell density, defined as

Since an exact treatment of a stochastic agent-based model quickly becomes unfeasible due to high-order correlations between all agents,[8] the general method of analyzing a BIO-LGCA model is to cast it into an approximate, deterministic finite difference equation (FDE) describing the mean dynamics of the population, then performing the mathematical analysis of this approximate model, and comparing the results to the original BIO-LGCA model.

This non-linearity would result in high-order correlations and moments among all channel occupations involved.

Instead, a mean-field approximation is usually assumed, wherein all correlations and high order moments are neglected, such that direct particle-particle interactions are substituted by interactions with the respective expected values.

From this nonlinear FDE, one may identify several homogeneous steady states, or constants

After applying the shift theorem and isolating the term with a temporal increment on the left, one obtains the lattice-Boltzmann equation[4]

of the Boltzmann propagator dictate the stability properties of the steady state:[4] Constructing a BIO-LGCA for the study of biological phenomena mainly involves defining appropriate transition probabilities for the interaction operator, though precise definitions of the state space (to consider several cellular phenotypes, for example), boundary conditions (for modeling phenomena in confined conditions), neighborhood (to match experimental interaction ranges quantitatively), and carrying capacity (to simulate crowding effects for given cell sizes) may be important for specific applications.

While the distribution of the reorientation operator can be obtained through the aforementioned statistical and biophysical methods, the distribution of the reaction operators can be estimated from the statistics of in vitro experiments, for example.

[9] BIO-LGCA models have been used to study several cellular, biophysical and medical phenomena.

The substructure of a BIO-LGCA lattice site with six velocity channels (corresponding to a 2D hexagonal lattice) and a single rest channel. In this case $b=6$ , $a=1$ , and the carrying capacity $K=7$ . Channels 2, 3, 6 and 7 are occupied, thus the lattice configuration is $\mathbf {s} =(0,1,1,0,0,1,1)$ , and the number of particles is $n\left(\mathbf {s} \right)=\sum _{i=1}^{K}s_{i}=4$ .

Dynamics of the BIO-LGCA model. Every time step, the occupation numbers are changed stochastically by the reaction and/or reorientation operators in all lattice sites simultaneously during the interaction step. Subsequently, particles are deterministically moved to the same velocity channel on a neighboring node in the direction of their velocity channel, during the transport step. Colors in the sketch are used to track the dynamics of the particles of individual nodes. This sketch assumes a particle-conserving rule (no reaction operator).

A hexagonal BIO-LGCA model of polar swarming. In this model, cells preferentially change their velocities to be parallel to the neighborhood's momentum. Lattice sites are colored according to their orientation, following the color wheel . Empty sites are white. Periodic boundary conditions were used.

A hexagonal BIO-LGCA model of excitable media. In this model, the reaction operator favors the rapid reproduction of particles within velocity channels, and the slow death of particles within rest channels. Particles in rest channels inhibit the reproduction of particles in velocity channels. The reorientation operator is the random walk operator in the text. Lattice sites are brightly colored the more motile particles are present. Resting particles are not shown. Periodic boundary conditions were used.

A square BIO-LGCA model of cells interacting adhesively. Cells move preferentially in the direction of the cell density gradient. Lattice sites are colored with increasingly darker blue colors with increasing cell density. Empty nodes are colored white.Periodic boundary conditions are used.

A square BIO-LGCA model of cells indirectly interacting chemotactically . In this model, cells produce a diffusing chemoattractant with a certain half-life . Cells preferentially move in the direction of the chemoattractant gradient. Lattice sites are additively colored with a darker blue tint with increasing cell density, and with a darker yellow tint with increasing chemoattractant concentration. Empty lattice sites are colored white. Periodic boundary conditions were used.