BIO-LGCA
In computational and mathematical biology, a biological lattice-gas cellular automaton (BIO-LGCA) is a discrete model for moving and interacting biological agents, 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 and cancer progression. Model definition. As are all cellular automaton models, a BIO-LGCA model is defined by a lattice formula_1, a state space formula_2, a neighborhood formula_3, and a rule formula_4. State space. For modeling particle velocities explicitly, lattice sites are assumed to have a specific substructure. Each lattice site formula_9 is connected to its neighboring lattice sites through vectors called "velocity channels", formula_13, formula_14, where the number of velocity channels formula_15 is equal to the number of nearest neighbors, and thus depends on the  lattice geometry (formula_16 for a one-dimensional lattice, formula_17 for a two-dimensional hexagonal lattice, and so on). In two dimensions, velocity channels are defined as formula_18. Additionally, an arbitrary number formula_19 of so-called "rest channels" may be defined, such that formula_20, formula_21. 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 formula_13 is indicated by the occupation number formula_23. 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. formula_24, and thus, every site has a maximum carrying capacity formula_25. Since the collection of all channel occupation numbers defines the number of particles and their velocities in each lattice site, the vector formula_26 describes the state of a lattice site, and the state space is given by formula_27. 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 formula_28 replaces the state of a node formula_29 with a new state formula_30 following a transition probability formula_31, which depends on the state of the neighboring lattice sites formula_32to 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 formula_33 also replaces a state formula_34 with a new state formula_35 with probability formula_36. 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), and typically takes the form formula_37where formula_38 is a normalization constant (also known as  the partition function), formula_39 is an energy-like function which particles will likely minimize when changing their direction of motion, formula_40 is a free parameter inversely proportional to the randomness of particle reorientation (analogous to the inverse temperature in thermodynamics), and formula_41 is a Kronecker delta which ensures that particle number before formula_42 and after reorientation formula_43 is unchanged. The state resulting form applying the reaction and reorientation operator formula_44 is known as the post-interaction configuration and denoted by formula_45. 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 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. formula_46. 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 formula_47 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 probabilityformula_48 where formula_49. Such transition probability allows any post-reorientation configuration formula_50 with the same number of particles as the pre-reorientation configuration formula_34, to be picked uniformly. 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 byformula_52where formula_53, formula_54 are constant birth and death probabilities, respectively, formula_55 is the Kronecker delta which ensures only one birth/death event happens every time step, and formula_56 is the Heaviside function, which makes sure particle numbers are positive and bounded by the carrying capacity formula_57. Adhesive interactions. 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 by a reorientation operator with transition probabilities defined asformula_58  where formula_59 is a vector pointing in the direction of maximum cell density, defined as formula_60, where formula_61is the configuration of the lattice site formula_62 within the neighborhood formula_3, and formula_64 is the momentum of the post-reorientation configuration, defined as formula_65. This transition probability favors post-reorientation configurations with cells moving towards the cell density gradient. Mathematical analysis. Since an exact treatment of a stochastic agent-based model quickly becomes unfeasible due to high-order correlations between all agents, 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 formula_66 is obtainedformula_67where formula_68 denotes the expected value, and formula_69 is the expected value of the formula_70-th channel occupation number of the lattice site at formula_71 at time step formula_72. However, the term on the right, formula_73 is highly nonlinear on the occupation numbers of both the lattice site formula_71 and the lattice sites within the interaction neighborhood formula_3, due to the form of  the transition probability formula_76 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 formula_77 are random variables, and formula_78 is a function, thenformula_79 under this approximation. Thus, we can simplify the equation toformula_80where formula_81 is a nonlinear  function of the expected lattice site configuration formula_82 and the expected neighborhood configuration formula_83 dependent on the transition probabilities and in-node particle statistics. From this nonlinear FDE, one may identify several homogeneous steady states, or constants formula_84 independent of formula_71 and formula_72 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 asformula_87where formula_88 denotes the homogeneous steady state formula_89, 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 equationformula_90where formula_91 is the imaginary unit, formula_92 is the size of the lattice along one dimension, formula_93 is the Fourier wave number, and formula_94 denotes the discrete Fourier transform. In matrix notation, this equation is simplified to formula_95, where the matrix formula_96 is called the "Boltzmann propagator" and is defined asformula_97The eigenvalues formula_98 of  the Boltzmann propagator dictate the stability properties of the steady state: Applications. 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. BIO-LGCA models have been used to study several cellular, biophyisical and medical phenomena. Some examples include: