In
computational
A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms.
Mechanical or electronic devices (or, historic ...
and
mathematical biology
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development ...
, a biological lattice-gas cellular automaton (BIO-LGCA) is a discrete model for moving and interacting biological agents, a type of
cellular automaton
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
. The BIO-LGCA is based on the
lattice-gas cellular automaton (LGCA) model used in fluid dynamics. A BIO-LGCA model describes cells and other motile biological agents as point particles moving on a discrete lattice, thereby interacting with nearby particles. 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. BIO-LGCA applications include cancer invasion and
cancer progression.
Model definition
As are all cellular automaton models, a BIO-LGCA model is defined by a lattice
, a state space
, a neighborhood
, and a rule
.
* The lattice (
) defines the set of all possible particle positions. Particles are restricted to occupy only certain positions, typically resulting from a regular and periodic
tesselation of space. Mathematically,
is a discrete subset of
-dimensional space.
* The state space (
) describes the possible states of particles within every lattice site
. In BIO-LGCA, multiple particles with different velocities may occupy a single lattice site, as opposed to classic cellular automaton models, where typically only a single cell can reside in every lattice node simultaneously. This makes the state space slightly more complex than that of classic cellular automaton models (see below).
* The neighborhood (
) indicates the subset of lattice sites which determines the dynamics of a given site in the lattice. Particles only interact with other particles within their neighborhood. Boundary conditions must be chosen for neighborhoods of lattice sites at the boundary of finite lattices. Neighborhoods and boundary conditions are identically defined as those for regular cellular automata (see
Cellular automaton
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
).
* The rule (
) dictates how particles move, proliferate, or die with time. As every cellular automaton, BIO-LGCA evolves in discrete time steps. In order to simulate the
system dynamics
System dynamics (SD) is an approach to understanding the nonlinear behaviour of complex systems over time using stocks, flows, internal feedback loops, table functions and time delays.
Overview
System dynamics is a methodology and mathematical ...
, the rule is synchronously applied to every lattice site at every time step. Rule application changes the original state of a lattice site to a new state. The rule depends on the states of lattice sites in the interaction neighborhood of the lattice site to be updated. In BIO-LGCA, the rule is divided into two steps, a probabilistic interaction step followed by a deterministic transport step. The interaction step simulates reorientation, birth, and death processes, and is defined specifically for the modeled process. The transport step translocates particles to neighboring lattice nodes in the direction of their velocities. See below for details.
State space

For modeling particle velocities explicitly, lattice sites are assumed to have a specific substructure. Each lattice site
is connected to its neighboring lattice sites through vectors called "velocity channels",
,
, where the number of velocity channels
is equal to the number of nearest neighbors, and thus depends on the lattice geometry (
for a one-dimensional lattice,
for a two-dimensional hexagonal lattice, and so on). In two dimensions, velocity channels are defined as
. Additionally, an arbitrary number
of so-called "rest channels" may be defined, such that
,
. A channel is said to be occupied if there is a particle in the lattice site with a velocity equal to the velocity channel. The occupation of channel
is indicated by the occupation number
. Typically, particles are assumed to obey an
exclusion principle, such that no more than one particle may occupy a single velocity channel at a lattice site simultaneously. In this case, occupation numbers are Boolean variables, i.e.
, and thus, every site has a maximum
carrying capacity
The carrying capacity of an ecosystem is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as the ...
. Since the collection of all channel occupation numbers defines the number of particles and their velocities in each lattice site, the vector
describes the state of a lattice site, and the state space is given by
.
Rule and model dynamics
The states of every site in the lattice are updated synchronously in discrete time steps to simulate the model dynamics. The rule is divided into two steps. The probabilistic interaction step simulates particle interaction, while the deterministic transport step simulates particle movement.
Interaction step
Depending on the specific application, the interaction step may be composed of reaction and/or reorientation operators.
The reaction operator
replaces the state of a node
with a new state
following a
transition probability
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, ...
, which depends on the state of the neighboring lattice sites
to simulate the influence of neighboring particles on the reactive process. 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 reorientation operator
also replaces a state
with a new state
with probability
. However, this operator conserves particle number and therefore only models changes in particle velocity by redistributing particles among velocity channels. 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
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Lange ...
describing the reorientation dynamics), and typically takes the form
where
is a normalization constant (also known as the
partition function),
is an energy-like function which particles will likely minimize when changing their direction of motion,
is a free parameter inversely proportional to the randomness of particle reorientation (analogous to the
inverse temperature
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).
Thermodynamic beta has units recipr ...
in thermodynamics), and
is a
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
which ensures that particle number before
and after reorientation
is unchanged.
The state resulting form applying the reaction and reorientation operator
is known as the post-interaction configuration and denoted by
.
Transport step
After the interaction step, the deterministic transport step is applied synchronously to all lattice sites. The transport step simulates the movement of agents according to their velocity, due to the
self-propulsion
Self-propulsion is the autonomous displacement of nano-, micro- and macroscopic natural and artificial objects, containing their own means of motion. Self-propulsion is driven mainly by interfacial phenomena. Various mechanisms of self-propelling ...
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.
.
A new time step begins when both interaction and transport steps have occurred. Therefore, the dynamics of the BIO-LGCA can be summarized as the stochastic
finite-difference microdynamical equation
Example interaction dynamics
The transition probability for the reaction and/or reorientation operator must be defined to appropriately simulate the modeled system. Some elementary interactions and the corresponding transition probabilities are listed below.
Random walk
In the absence of any external or internal stimuli, cells may move randomly without any directional preference. In this case, the reorientation operator may be defined through a transition probability
where
. Such transition probability allows any post-reorientation configuration
with the same number of particles as the pre-reorientation configuration
, to be picked
uniformly
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
.
Simple birth and death process
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
with a transition probability given by