Lattice-gas Cellular Automaton Model Research Articles

Modeling the interplay of oscillatory synchronization and aggregation via cell–cell adhesion

We present a model of systems of cells with intracellular oscillators (‘clocks’). This is motivated by examples from developmental biology and from the behavior of organisms on the threshold to multicellularity. Cells undergo random motion and adhere to each other. The adhesion strength between neighbors depends on their clock phases in addition to a constant baseline strength. The oscillators are linked via Kuramoto-type local interactions. The model is an advection-diffusion partial differential equation with nonlocal advection terms. We demonstrate that synchronized states correspond to Dirac-delta measure solutions of a weak version of the equation. To analyze the complex interplay of aggregation and synchronization, we then perform a linear stability analysis of the incoherent, spatially uniform state. This lets us classify possibly emerging patterns depending on model parameters. Combining these results with numerical simulations, we determine a range of possible far-from equilibrium patterns when baseline adhesion strength is zero: There is aggregation into separate synchronized clusters with or without global synchrony; global synchronization without aggregation; or unexpectedly a ‘phase wave’ pattern characterized by spatial gradients of clock phases. A 2D Lattice-Gas Cellular Automaton model confirms and illustrates these results.

Vectorial active matter on the lattice: polar condensates and nematic filaments

We introduce a novel lattice-gas cellular automaton (LGCA) for compressible vectorial active matter with polar and nematic velocity alignment. Interactions are, by construction, zero-range. For polar alignment, we show the system undergoes a phase transition that promotes aggregation with strong resemblance to the classic zero-range process. We find that above a critical point, the states of a macroscopic fraction of the particles in the system coalesce into the same state, sharing the same position and momentum (polar condensate). For nematic alignment, the system also exhibits condensation, but there exist fundamental differences: a macroscopic fraction of the particles in the system collapses into a filament, where particles possess only two possible momenta. Furthermore, we derive hydrodynamic equations for the active LGCA model to understand the phase transitions and condensation that undergoes the system. We also show that generically the discrete lattice symmetries—e.g. of a square or hexagonal lattice—affect drastically the emergent large-scale properties of on-lattice active systems. The study puts in evidence that aligning active matter on the lattice displays new behavior, including phase transitions to states that share similarities to condensation models.

Lattice-gas Cellular Automaton Models for the Analysis of Pattern Formation in Interacting Cell Populations

Lattice-gas Cellular Automaton Models for the Analysis of Pattern Formation in Interacting Cell Populations

Extracting cellular automaton rules from physical Langevin equation models for single and collective cell migration

Cellular automata (CA) are discrete time, space, and state models which are extensively used for modeling biological phenomena. CA are "on-lattice" models with low computational demands. In particular, lattice-gas cellular automata (LGCA) have been introduced as models of single and collective cell migration. The interaction rule dictates the behavior of a cellular automaton model and is critical to the model's biological relevance. The LGCA model's interaction rule has been typically chosen phenomenologically. In this paper, we introduce a method to obtain lattice-gas cellular automaton interaction rules from physically-motivated "off-lattice" Langevin equation models for migrating cells. In particular, we consider Langevin equations related to single cell movement (movement of cells independent of each other) and collective cell migration (movement influenced by cell-cell interactions). As examples of collective cell migration, two different alignment mechanisms are studied: polar and nematic alignment. Both kinds of alignment have been observed in biological systems such as swarms of amoebae and myxobacteria. Polar alignment causes cells to align their velocities parallel to each other, whereas nematic alignment drives cells to align either parallel or antiparallel to each other. Under appropriate assumptions, we have derived the LGCA transition probability rule from the steady-state distribution of the off-lattice Fokker-Planck equation. Comparing alignment order parameters between the original Langevin model and the derived LGCA for both mechanisms, we found different areas of agreement in the parameter space. Finally, we discuss potential reasons for model disagreement and propose extensions to the CA rule derivation methodology.

Analysis of individual cell trajectories in lattice-gas cellular automaton models for migrating cell populations.

Collective dynamics of migrating cell populations drive key processes in tissue formation and maintenance under normal and diseased conditions. Collective cell behavior at the tissue level is typically characterized by considering cell density patterns such as clusters and moving cell fronts. However, there are also important observables of collective dynamics related to individual cell behavior. In particular, individual cell trajectories are footprints of emergent behavior in populations of migrating cells. Lattice-gas cellular automata (LGCA) have proven successful to model and analyze collective behavior arising from interactions of migrating cells. There are well-established methods to analyze cell density patterns in LGCA models. Although LGCA dynamics are defined by cell-based rules, individual cells are not distinguished. Therefore, individual cell trajectories cannot be analyzed in LGCA so far. Here, we extend the classical LGCA framework to allow labeling and tracking of individual cells. We consider cell number conserving LGCA models of migrating cell populations where cell interactions are regulated by local cell density and derive stochastic differential equations approximating individual cell trajectories in LGCA. This result allows the prediction of complex individual cell trajectories emerging in LGCA models and is a basis for model-experiment comparisons at the individual cell level.

Interface dynamics of immiscible two-phase lattice-gas cellular automata: A model with random dynamic scatterers and quenched disorder in two dimensions

We use a lattice gas cellular automata model in the presence of random dynamic scattering sites and quenched disorder in the two-phase immiscible model with the aim of producing an interface dynamics similar to that observed in Hele-Shaw cells. The dynamics of the interface is studied as one fluid displaces the other in a clean lattice and in a lattice with quenched disorder. For the clean system, if the fluid with a lower viscosity displaces the other, we show that the model exhibits the Saffman-Taylor instability phenomenon, whose features are in very good agreement with those observed in real (viscous) fluids. In the system with quenched disorder, we obtain estimates for the growth and roughening exponents of the interface width in two cases: viscosity-matched fluids and the case of unstable interface. The first case is shown to be in the same universality class of the random deposition model with surface relaxation. Moreover, while the early-time dynamics of the interface behaves similarly, viscous fingers develop in the second case with the subsequent production of bubbles in the context of a complex dynamics. We also identify the Hurst exponent of the subdiffusive fractional Brownian motion associated with the interface, from which we derive its fractal dimension and the universality classes related to a percolation process.

The role of adsorption in passivation phenomena modelled by discrete lattice gas automata

Simulation results are presented for a simple modification of recent lattice gas cellular automata model of passivation. In the previous model, the passive layer sites produced at the corroding surface, neither adhere to nor are repelled from the surface. As a result, no net adsorption or desorption, at equilibrium, is possible for this model. Nonetheless, the crucial features of passivation appear due to the kinetic crowding of the passive layer sites. This crowding effect is further enhanced by the aggregation of the passive layer sites introduced via asymmetric exclusion rules. The results of this model challenge a common view that adsorption is crucial for passivation. This analysis calls for the evaluation of the role of adsorption. Therefore, in this paper, the model takes into account the attraction of the passive layer sites with the metallic substrate. The present model is compared with the model examined in the previous paper. It demonstrates that although adsorption does not modify essentially the overall image of passivation, as seen in the polarization curves, there are important quantitative and qualitative new features in the passivation phenomenology introduced by adsorption. These new features result from the appearance of two regimes of passivation: a thin layer regime at the passivation transition followed by a thick layer regime at more anodic potentials.

Analysis of lattice-gas cellular automaton models for tumor growth by means of fractal scaling

Analysis of lattice-gas cellular automaton models for tumor growth by means of fractal scaling

Parameter estimation with a novel gradient-based optimization method for biological lattice-gas cellular automaton models

Lattice-gas cellular automata (LGCAs) can serve as stochastic mathematical models for collective behavior (e.g. pattern formation) emerging in populations of interacting cells. In this paper, a two-phase optimization algorithm for global parameter estimation in LGCA models is presented. In the first phase, local minima are identified through gradient-based optimization. Algorithmic differentiation is adopted to calculate the necessary gradient information. In the second phase, for global optimization of the parameter set, a multi-level single-linkage method is used. As an example, the parameter estimation algorithm is applied to a LGCA model for early in vitro angiogenic pattern formation.

Lattice-Gas Cellular Automaton Models for Biology: From Fluids to Cells

Lattice-gas cellular automaton (LGCA) and lattice Boltzmann (LB) models are promising models for studying emergent behaviour of transport and interaction processes in biological systems. In this chapter, we will emphasise the use of LGCA/LB models and the derivation and analysis of LGCA models ranging from the classical example dynamics of fluid flow to clotting phenomena in cerebral aneurysms and the invasion of tumour cells.

Simulations of passivation phenomena based on discrete lattice gas automata

We present simulation results for a simple lattice gas cellular automata model of passivation. The lattice sites representing the corrosion product are produced at the corroding surface and diffuse executing a random walk. Asymmetric simple exclusion rules of the random walk account for an attractive potential between the corrosion product particles. The particles can aggregate and when sufficiently numerous form a compact phase on the corroding surface. The model predicts a transition from the active to passive state when increasing the reactivity of the surface. The transition is characterized by a sudden increase in the surface coverage of the corrosion product interpreted as a passive layer formation. The layer blocks contact of the metal surface with the environment and reduces the corrosion rate. The model reproduces the known paradox of passivity—the surface must be reactive enough for the layer to form. A further increase in the bare reactivity reduces largely the observed reaction rate. The simulations yield information on the morphological changes of the surface layer before and after the transition. In terms of the corrosion current, the active state is described by the current increase with the polarizing potential according to the Tafel law while in the passive state the current is independent of the anodic potential. Our simple model reproduces principal features of passivation.

A cellular-automata method for studying porous media properties

A cellular-automata (CA) approach for investigating properties of porous media with tortuous channels and different smoothness of pore walls is proposed. This approach is aimed at combining two different CA models: the first one is intended for constructing the morphology of a porous material; the second, for simulating a fluid flow through it. The porous media morphology is obtained as a result of evolution of a cellular automaton, forming a “steady pattern.” The result is then used for simulating a fluid flow through a porous medium by applying the Lattice Gas CA model. The method has been tested on a small fragment of a porous material and implemented for investigating a carbon electrode of a hydrogen fuel cell on a multiprocessor cluster.

Prediction of traveling front behavior in a lattice-gas cellular automaton model for tumor invasion

Cancer invasion is the process of cells detaching from a primary tumor and infiltrating the healthy tissue. Cancer invasion has been recognized as a complex system, since a tumor’s invasive behavior emerges from the combined effect of tumor cell proliferation, tumor cell migration and cell–microenvironment interactions. Cellular automata (CA) provide simple models of self-organizing complex systems in which collective behavior can emerge out of an ensemble of many interacting “simple” components. Here, we introduce a lattice-gas cellular automaton (LGCA) model of tumor cell proliferation, necrosis and tumor cell migration. The impact of the tumor environment on tumor cells has been investigated in a previous study. Our analysis aims at predicting the velocity of the traveling invasion front, which depends upon fluctuations that arise from the motion of the discrete cells at the front. We find an excellent agreement between the velocities measured in simulations of the LGCA and an analytical estimate derived in the cut-off mean-field approximation via the discrete Lattice Boltzmann equation and its linearization. In particular, we predict the front velocity to scale with the square root of the product of probabilities for mitosis and the migration coefficient. Finally, we calculate the width of the traveling front which is found to be proportional to the front velocity.

Setting the length and time scales of a cellular automaton dune model from the analysis of superimposed bed forms

We present a new 3‐D cellular automaton model for bed form dynamics in which individual physical processes such as erosion, deposition, and transport are implemented by nearest neighbor interactions and a time‐dependent stochastic process. Simultaneously, a lattice gas cellular automaton model is used to compute the flow and quantify the bed shear stress on the topography. Local erosion rates are assumed to be proportional to the shear stress in such a way that there is a complete feedback mechanism between flow and bed form dynamics. In the numerical simulations of dune fields, we observe the formation and the evolution of superimposed bed forms on barchan and transverse dunes. Using the same model under different initial conditions, we perform the linear stability analysis of a flat sand bed disturbed by a small sinusoidal perturbation. Comparing the most unstable wavelength in the model with the characteristic size of secondary bed forms in nature, we determine the length and time scales of our cellular automaton model. Thus, we establish a link between discrete and continuous approaches and open new perspectives for modeling and quantification of complex patterns in dune fields.

Diffusion in tight confinement: A lattice-gas cellular automaton approach. I. Structural equilibrium properties

The thermodynamic and transport properties of diffusing species in microporous materials are strongly influenced by their interactions with the confining framework, which provide the energy landscape for the transport process. The simple topology and the cellular nature of the alpha cages of a ZK4 zeolite suggest that it is appropriate to apply to the study of the problem of diffusion in tight confinement a time-space discrete model such as a lattice-gas cellular automaton (LGCA). In this paper we investigate the properties of an equilibrium LGCA constituted by a constant number of noninteracting identical particles, distributed among a fixed number of identical cells arranged in a three-dimensional cubic network and performing a synchronous random walk at constant temperature. Each cell of this network is characterized by a finite number of two types of adsorption sites: the exit sites available to particle transfer and the inner sites not available to such transfers. We represent the particle-framework interactions by assuming a differentiation in binding energy of the two types of sites. This leads to a strong dependence of equilibrium and transport properties on loading and temperature. The evolution rule of our LGCA model is constituted by two operations (randomization, in which the number of particles which will be able to try a jump to neighboring cells is determined, and propagation, in which the allowed jumps are performed), each one applied synchronously to all of the cells. The authors study the equilibrium distribution of states and the adsorption isotherm of the model under various conditions of loading and temperature. In connection with the differentiation in energy between exit and inner sites, the adsorption isotherm is described by a conventional Langmuir isotherm at high temperature and by a dual-site Langmuir isotherm at low temperature, while a first order diffuse phase transition takes place at very low temperature.

WITHDRAWN: Characterization of the travelling front behaviour in a lattice gas cellular automaton model of glioma invasion

WITHDRAWN: Characterization of the travelling front behaviour in a lattice gas cellular automaton model of glioma invasion

Sorbing and Non-Sorbing Solute Migration in Rough Fractures with a Multi-Species LGA Model: Dispersion Dependence on Retardation and Roughness

A multi-species Lattice-Gas cellular automaton model was applied to the study of the migration of sorbing and non-sorbing tracers in 2D smooth fractures and in a series of increasing roughness fractures. A tenfold increase of the dispersion of the non-sorbing tracer was calculated in the highest roughness case compared to the smooth fracture. Up to a threefold increase of the dispersion of the sorbing tracers was calculated compared to the non-sorbing tracer. These enhanced dispersions were found to be of a Fickian form and were interpreted in terms of the classical Taylor–Aris dispersion. The effects of roughness and retardation over the increase of dispersion were identified and quantified through a semi-empirical relation. These effects were found additive and independent.

Cancer Cell Invasion of Brain Tissue: Guided by a Prepattern?

The malignant brain tumour Glioblastoma multiforme (GBM) displays a highly invasive behaviour. Spreading of the malignant cells appears to be guided by the white matter fibre tracts within the brain. In order to understand the global growth process we introduce a lattice‐gas cellular automaton model which describes the local interaction between individual malignant cells and their neighbourhood. We consider interactions between cells (brain cells and tumour cells) and between malignant cells and the fibre tracts in the brain, which are considered as a prepattern. The prepattern implies persistent individual cell motion along the fibre structure. Simulations with the model show that only the inclusion of the prepattern results in invading tumour and growing tumour islets in front of the expanding tumour bulk (i.e. the growth pattern observed in clinical practice). Our results imply that the infiltrative growth of GBMs is, in part, determined by the physical structure of the surrounding brain rather than by intrinsic properties of the tumour cells.

A new mechanism of aggregation in a lattice-gas cellular automaton model

A cellular automaton with moving and resting “particles” is introduced as a model of swarming and aggregation. No external (indirect) sources of information (e.g., chemoattractants) are needed. Instead, “particles” are capable of (directly) evaluating orientations as well as the rest particle density in their local vicinity. Simulations exhibit transitions from random movement to collective motion and, furthermore, to aggregation. Mean-field analysis of the “aggregation instability” provides an explanation of spatial pattern formation.

Spectra of spatial fluctuations in lattice gas cellular automata fluids

Spectra or decay rates of spatial fluctuations for different Lattice Gas Cellular Automata (LGCA) Fluids are discussed using the Boltzmann equation. We demonstrate how the spectra provide information on the time and length scales of hydrodynamic behavior, isotropy and existence of spurious conservation laws in different LGCA models. The range of validity of generalized hydrodynamics with wave-number-dependent sound speed and transport coefficients is indicated. These properties give a quantitative explanation of several existing, but unexplained simulation results on the so called “negative bulk viscosities”, and on the strong dispersion in the simulation results on sound damping. The use of the spectra to analyse the spatial instabilities occurring in certain models of LGCA is also indicated.