Lattice Gas Cellular Automata 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.

Mesoscale numerical simulation of the multiple step reaction in hydrogen reduction of iron oxides

This work designed a mesoscale method of multi-step reduction of Fe2O3 with H2 based on Lattice Gas Cellular Automata (LGCA). Particles in the LGCA were marked to distinguish all the reactants, intermediates, products, and inert. A self-organized evolutionary mechanism is designed. The hydrogen reduction of Fe2O3 in the temperature range of 460–550 °C and 650–800 °C was simulated. Results show that the intermediate product Fe3O4 was found in both conditions, and its mass fraction quickly reach a peak at ∼0.8 and gradually decrease thereafter. An “induction period” can be identified in the initial stage of the reaction at 650–800 °C, where the reduction rate is slow due to the hindered formation of FeO. The porosity inside the Fe2O3 particles increases continuously through the reduction process, resulting in significant changes in the flow field around the particles. Consequently, the wake vortex gradually decreases and disappears eventually.

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.

The dichotomous role of anisotropic sensing in pattern generation and disruption

In this work, we explore the effects of an anisotropic sensing range on the behavior of migrating particles. We use a lattice-gas cellular automaton (LGCA) with liquid-crystal and ferromagnetic velocity alignment, and a sensing range modeled by a von Mises distribution. By performing simulations and mathematical analysis, we study the evolution of the system and the different patterns formed in each case. We observe that, increasing the anisotropy of the sensing range can have different effects on the density patterns. When particles interact ferromagnetically, sensing anisotropy has a deleterious effect which destabilizes patterns observed with an isotropic sensing range; while patterns change from thick bands to small clusters when anisotropy is increased with liquid-crystal interactions. The results suggest that organisms which coordinate through visual information could either benefit from, or be hidered by a restricted vision field depending on the nature of their pairwise interactions.

A mesoscale model for heterogeneous reactions in porous media applied to reducing porous iron oxide

A mesoscale model based on the lattice gas cellular automata (LGCA) is proposed for heterogeneous reaction in porous media. This model features an entirely discretized reaction system, including discrete space, time, gas phases, and solid particle. To describe gas diffusion at solid nodes and gas–solid heterogeneous reaction on the phase boundary, two local probabilities are proposed, which are verified by the analytical solutions for gas diffusion and gas–solid reaction problems. This model was then applied to the hydrogen reduction of porous Fe3O4. The effects of the porosity, reaction rate constant (kl), and initial H2 concentration on the reduction were analyzed. The results indicate that the reaction rate increased with increasing porosity, kl, and H2 concentration. The solid product Fe was randomly distributed on the framework of Fe3O4 at smaller kl and concentrated around the unreacted Fe3O4 at higher kl, which is consistent with our experimental results.

Lattice Fluid Dynamics: Thirty-five Years Down the Road

We discuss the role and impact of lattice fluid dynamics, namely Lattice Gas Cellular Automata and lattice Boltzmann, on the general framework of Computational Fluid Dynamics.

Cellular Automata Machine Modeling to obtain optimal conditions and productivity of microalgae biomass

Microalgae is a promising source of superior biomass for sustainable food sources, high value products, and 3rd generation biofuel. The main factors affecting growth, carbon fixation and biomass production of microalgae are species type, CO2 supply, nutrients, light, temperature, pH and mixing. This paper discusses the precision farming approach in microalgae biomass production using the Cellular Automata Machine modeling, i.e., Probabilistic Cellular Automata. Cellular Automata is a decentralized computing model that provides a useful platform to model the complex dynamic systems such as growth of cells. The application of this method on a large-scale microalgae production system can simulate and model the characteristics of microalgae growth and biomass. In conclusion, the Biological Lattice Gas Cellular Automaton (BIO-LGCA) has been proposed as modeling tools to obtain the optimal conditions and maximum productivity in microalgae biomass production.

BIO-LGCA: A cellular automaton modelling class for analysing collective cell migration.

Collective dynamics in multicellular systems such as biological organs and tissues plays a key role in biological development, regeneration, and pathological conditions. Collective tissue dynamics—understood as population behaviour arising from the interplay of the constituting discrete cells—can be studied with on- and off-lattice agent-based models. However, classical on-lattice agent-based models, also known as cellular automata, fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. To overcome drawbacks of classical on-lattice models, we introduce an on-lattice, agent-based modelling class for collective cell migration, which we call biological lattice-gas cellular automaton (BIO-LGCA). The BIO-LGCA is characterised by synchronous time updates, and the explicit consideration of individual cell velocities. While rules in classical cellular automata are typically chosen ad hoc, rules for cell-cell and cell-environment interactions in the BIO-LGCA can also be derived from experimental cell migration data or biophysical laws for individual cell migration. We introduce elementary BIO-LGCA models of fundamental cell interactions, which may be combined in a modular fashion to model complex multicellular phenomena. We exemplify the mathematical mean-field analysis of specific BIO-LGCA models, which allows to explain collective behaviour. The first example predicts the formation of clusters in adhesively interacting cells. The second example is based on a novel BIO-LGCA combining adhesive interactions and alignment. For this model, our analysis clarifies the nature of the recently discovered invasion plasticity of breast cancer cells in heterogeneous environments.

Lattice Gas Cellular Automata Fluid Dynamics Case Study

The Navier–Stokes equations are the basis for describing the flow of a viscus material and are used to model fluid motion from weather to air flow over a wing. These equations, however, tend to be notoriously difficult to solve, whether analytically or computationally, even for small systems due to their highly nonlinear nature. Thus, less complicated methods that are more computationally tractable are desirable in domains as varied as hydroelectric power to race car construction. At a fundamental level, fluids are composed of interacting molecules. Lattice Gas Cellular Automata (LGCA) represents an efficient way to simulate these interacting fluid particles on a lattice. LGCA captures the microscopic behavior of the fluid by applying simple collision and propagation rules at each lattice site. This leads to realistic macroscopic behavior that can be used to build insight about real fluid flow. Here, we show the Hardy, Pomeau, and de Pazzis model for simulating a lattice gas. The computational framework presented in this case study can also be expanded to more complicated LGCA.

Simulating the Gluing of Wood Particles by Lattice Gas Cellular Automata and Random Walk

In this work a mathematical model and simulation for the gluing of wood particles designated for particleboards is presented. The aim is to obtain a better understanding of the gluing process. Thus, the behaviour of wood particles during gluing is investigated and the resulting adhesive distribution across the surface of the wood particles is analysed. For developing a mathematical model, the modelling methods “lattice gas cellular automata” and “random walk” were used. The model was implemented in MATLAB and different scenarios were simulated for answering the main questions of the behaviour during gluing. The influences of different parameters on the adhesive distribution were investigated and quantitatively determined by several key figures. Based on these key figures, the effects of the mixing arm, realistic size distributions of wood particles and adhesive droplets, the transfer of adhesive, and the total mass of adhesive are discussed. Furthermore the results are compared with experimental measurements. The simulation results show that the model can feasibly be used for studying the gluing of wood particles. For a possible industrial application, additional research for developing a three-dimensional model is needed.

Simulation of Structure Change in Porous Media During Gas–Solid Reactions Using Cellular Automata Model

In some gas–solid reactions, a new solid substance is produced. The product acts as a shield and prevents the collision between gas and solid reactants which further causes an incomplete reaction. If the molar volume of the new product differs from the solid reactant, the inner structure of porous media is changed as well. In this paper, we discuss such gas–solid reactions in porous media using the two-dimensional lattice gas cellular automata FHP-III model. We simulate the fluid flow and chemical reaction in different porous media. We also show the effects of porosity and morphology of the solid, and reaction probability on the reaction process. Results obtained from the simulations agree closely with the theory of gas–solid reactions and diffusion theories. Hence, the proposed model is a good choice to simulate gas–solid chemical reactions in porous media at the mesoscopic level.

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

Modeling and simulation of moving wood chips and resin droplets within a resinating mixer using lattice gas cellular automata

In this paper a model for the movement of wood chips and resin droplets within a resinating mixer is presented. The mixing of wood chips and resin droplets is a sub-process, which is called gluing or resin metering, of the manufacturing process of particleboards. The aim of this sub-process is to equally distribute the resin on the wood chips. As the process of gluing is not completely understood, two arising research questions are “How is the resin distributed across the surface of the wood chips?” and “How much resin penetrated into the wood particles?”. These aspects can hardly be determined using real experiments. A mathematical model in two dimensions is developed. A modified lattice gas cellular automaton (LGCA) is used to model the moving resin droplets and wood chips. Due to the difference in size of the wood and resin particles, some modifications regarding the definition of states, collisions and streaming for the movement of the wood chips are necessary. In the common LGCA a particle is located at a cell of a specific node, whereas in the modified LGCA presented, a wood particle occupies several nodes. Due to the different states (e.g. wood or resin), several collision rules had to be defined. For the streaming, each node of the same wood particle has the same underlying velocity. The simulation is carried out in MATLAB. Finally, first simulation results are presented. Based on the simulation the percentage of glued wood chips is calculated.

Application of cellular automata and Lattice Boltzmann methods for modelling of additive layer manufacturing

PurposeThe holistic numerical model based on cellular automata (CA) and lattice Boltzmann method (LBM) are being developed as part of an integrated modelling approach applied to study the interaction of different physical mechanisms in laser-assisted additive layer manufacturing (ALM) of orthopaedic implants. Several physical events occurring in sequence or simultaneously are considered in the holistic model. They include a powder bed deposition, laser energy absorption and heating of the powder bed by the moving laser beam, leading to powder melting or sintering, fluid flow in the melted pool and flow through partly or not melted material, and solidification. The purpose of this study is to develop a structure of the holistic numerical model based on CA and LBM applicable for studying the interaction of the different physical mechanisms in ALM of orthopaedic implants. The model supposed to be compatible with the earlier developed CA-based model for the generation of the powder bed.Design/methodology/approachThe mentioned physical events are accompanied by heat transfer in solid and liquid phases including interface heat transfer at the boundaries. The sintering/melting model is being developed using LBM as an independent numerical method for hydrodynamic simulations originated from lattice gas cellular automata. It is going to be coupled with the CA-based model of powder bed generation.FindingsThe entire laser-assisted ALM process has been analysed and divided on several stages considering the relevant physical phenomena. The entire holistic model consisting of four interrelated submodels has currently been developed to a different extent. The submodels include the CA-based model of powder bed generation, the LBM-CA-based model of heat exchange and transfer, the thermal solid-liquid interface model and the mechanical solid-liquid interface model for continuous liquid flow.Practical implicationsThe results obtained can be used to explain the interaction of the different physical mechanisms in ALM, which is an intensively developing field of advanced manufacturing of metal, non-metal and composite structural parts, for instance, in bio-engineering. The proposed holistic model is considered to be a part of the integrated modelling approach being developed as a numerical tool for investigation of the co-operative relationships between multiphysical phenomena occurring in sequence or simultaneously during heating of the powder bed by the moving high energy heat source, leading to selective powder sintering or melting, fluid flow in the melted pool and through partly (or not) melted material, as well as solidification. The model is compatible with the earlier developed CA-based model for the generation of the powder bed, allowing for decrease in the numerical noise.Originality/valueThe present results are original and new for the study of the complex relationships between multiphysical phenomena occurring during ALM process based on selective laser sintering or melting, including fluid flow and heat transfer, identified as crucial for obtaining the desirable properties.

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.

On the iota-delta function: a link between cellular automata and partial differential equations for modeling advection–dispersion from a constant source

Describing complex phenomena by means of cellular automata (CAs) has shown to be a very effective approach in pure and applied sciences. Most of the applications, however, rely on multidimensional CAs. For example, lattice gas CAs and lattice Boltzmann methods are widely used to simulate fluid flow and both share features with two-dimensional CAs. One-dimensional CAs, on the other hand, seem to have been neglected for modeling physical phenomena. In the present paper, we demonstrate that some one-dimensional CAs are equivalent to a stable linear finite difference scheme used to solve advection---diffusion partial differential equations (PDEs) by relying on the so-called iota-delta representation. Consequently, this work shows an important link between continuous and discrete models in general, and PDEs and CAs more in particular.

CellularAutomataLib2: improving the support for cellular automata modelling in Modelica

ABSTRACTCellularAutomataLib is a library developed by the authors to facilitate the description of Cellular Automata (CA) models in Modelica. It supports the description of 1D and 2D CA and their combination with other Modelica models. Modelling versatility and scalability are the main focus in the design. The internal behaviour of CA models is programmed in C, that is consequently hidden to the modelling tool and not considered in the causalization and manipulation of the model. A new version, named CellularAutomataLib2, is presented in this manuscript. The library has been extended to improve the simulation performance, by only evaluating active cells, and to support Lattice Gas Cellular Automata (LGCA) models. The library design and use are discussed. Two models, forest fire spread and the ARGESIM C17 ‘SIR-type Epidemic Spread’, are used to illustrate the functionality of the library. CellularAutomataLib2 is freely available at http://www.euclides.dia.uned.es (under the Modelica License 2).

Computational Complexity of Predicting Periodicity in the Models of Lorentz Lattice Gas Cellular Automata

Computational Complexity of Predicting Periodicity in the Models of Lorentz Lattice Gas Cellular Automata

Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results

In this paper we present an overview of agent-based models that are used to simulate mechanical and physiological phenomena in cells and tissues, and we discuss underlying concepts, limitations, and future perspectives of these models. As the interest in cell and tissue mechanics increase, agent-based models are becoming more common the modeling community. We overview the physical aspects, complexity, shortcomings, and capabilities of the major agent-based model categories: lattice-based models (cellular automata, lattice gas cellular automata, cellular Potts models), off-lattice models (center-based models, deformable cell models, vertex models), and hybrid discrete-continuum models. In this way, we hope to assist future researchers in choosing a model for the phenomenon they want to model and understand. The article also contains some novel results.

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.