In computational and mathematical biology, a biological lattice-gas cellular automaton (BIO-LGCA) is a discrete model for moving and interacting biological agents, [1] a type of cellular automaton. 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 [2] and cancer progression. [3]
As are all cellular automaton models, a BIO-LGCA model is defined by a lattice , a state space , a neighborhood , and a rule . [4]
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 . 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 .
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.
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 , 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 describing the reorientation dynamics), [5] [6] 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 thermodynamics), and is a Kronecker delta 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 .
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 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
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.
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.
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
where , are constant birth and death probabilities, respectively, is the Kronecker delta which ensures only one birth/death event happens every time step, and is the Heaviside function, which makes sure particle numbers are positive and bounded by the carrying capacity .
Cells may adhere to one another by cadherin molecules on the cell surface. Cadherin interactions allow cells to form aggregates. The formation of cell aggregates via adhesive biomolecules can be modeled [7] by a reorientation operator with transition probabilities defined as
where is a vector pointing in the direction of maximum cell density, defined as , where is the configuration of the lattice site within the neighborhood , and is the momentum of the post-reorientation configuration, defined as . This transition probability favors post-reorientation configurations with cells moving towards the cell density gradient.
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.
First, the expected value of the microdynamical equation is obtained
where denotes the expected value, and is the expected value of the -th channel occupation number of the lattice site at at time step . However, the term on the right, is highly nonlinear on the occupation numbers of both the lattice site and the lattice sites within the interaction neighborhood , due to the form of the transition probability and the statistics of particle placement within velocity channels (for example, arising from an exclusion principle imposed on channel occupations). 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. In other words, if are random variables, and is a function, then under this approximation. Thus, we can simplify the equation to
where is a nonlinear function of the expected lattice site configuration and the expected neighborhood configuration dependent on the transition probabilities and in-node particle statistics. From this nonlinear FDE, one may identify several homogeneous steady states, or constants independent of and which are solutions to the FDE. To study the stability conditions of these steady states and the pattern formation potential of the model, a linear stability analysis can be performed. To do so, the nonlinear FDE is linearized as
where denotes the homogeneous steady state , and a von Neumann neighborhood was assumed. In order to cast it into a more familiar finite difference equation with temporal increments only, a discrete Fourier transform can be applied on both sides of the equation. After applying the shift theorem and isolating the term with a temporal increment on the left, one obtains the lattice-Boltzmann equation [4]
where is the imaginary unit, is the size of the lattice along one dimension, is the Fourier wave number, and denotes the discrete Fourier transform. In matrix notation, this equation is simplified to , where the matrix is called the Boltzmann propagator and is defined as
The eigenvalues 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. Some examples include:
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