Simple Cellular Automaton Model Research Articles

Simulation of Somatic Evolution Through the Introduction of Random Mutation to the Rules of Conway’s Game of Life

Abstract Introduction Conway’s Game of Life (GOL), and related cellular automata (CA) models, have served as interesting simulations of complex behaviors resulting from simple rules of interactions between neighboring cells, that sometime resemble the growth and reproduction of living things. Thus, CA has been applied towards understanding the interaction and reproduction of single-cell organisms, and the growth of larger, disorganized tissues such as tumors. Surprisingly, however, there have been few attempts to adapt simple CA models to recreate the evolution of either new species, or subclones within a multicellular, tumor-like tissue. Methods In this article, I present a modified form of the classic Conway’s GOL simulation, in which the three integer thresholds that define GOL (number of neighboring cells, below which a cell will “die of loneliness”; number of neighboring cells, above which a cell will die of overcrowding; and number of neighboring cells that will result in spontaneous birth of a new cell within an empty lattice location) are occasionally altered with a randomized mutation of fractional magnitude during new “cell birth” events. Newly born cells “inherit” the current mutation state of a neighboring parent cell, and over the course of 10,000 generations these mutations tend to accumulate until they impact the behaviors of individual cells, causing them to transition from the sparse, small patterns of live cells characteristic of GOL into a more dense, unregulated growth resembling a connected tumor tissue. Results The mutation rate and mutation magnitude were systematically varied in repeated randomized simulation runs, and it was determined that the most important mutated rule for the transition to unregulated, tumor-like growth was the overcrowding threshold, with the spontaneous birth and loneliness thresholds being of secondary importance. Spatial maps of the different “subclones” of cells that spontaneously develop during a typical simulation trial reveal that cells with greater fitness will overgrow the lattice and proliferate while the less fit, “wildtype” GOL cells die out and are replaced with mutant cells. Conclusions This simple modeling approach can be easily modified to add complexity and more realistic biological details, and may yield new understanding of cancer and somatic evolution.

Cellular automaton model of self-healing

We propose a simple cellular automaton model of a self-healing system and investigate its properties. In the model, the substrate is a two-dimensional checkerboard configuration which can be damaged by changing values of a finite number of sites. The cellular automaton we consider is a checkerboard voting rule, a binary rule with Moore neighborhood which is topologically conjugate to the majority voting rule. For a single-color damage (when only cells in the same state are modified), the rule always fixes the damage. For a general damage, when it is localized inside a 3 × 3 square, the rule also fixes it always. When the damage is inside of a larger n × n square, the efficiency of the rule in fixing the damage becomes smaller than 100 % , but it remains better than 98 % for n ≤ 5 and better than 75 % for n ≤ 7 . We show that in the limit of infinite n the efficiency tends to zero.

Assessment of the effectiveness of restrictive epidemic control measures using original models of cellular automaton

Background. The ongoing COVID-19 pandemic, the human casualties caused by it, and the possibility of new epidemical threats make the search for effective countermeasures actual. One of the most effective tools, as the experience of the COVID-19 pandemic has shown, is restrictive measures of various types, which are especially significant with medical countermeasures being unavailable or insufficient. At the same time, the topic of restrictive measures and their mathematical modeling, especially given its importance, is not sufficiently disclosed in the scientific literature.The aim. To determine the possibility of assessing the effectiveness of restrictive epidemic control measures using original models of cellular automaton with intercellular boundaries.Methods. To determine the impact of restrictive measures on the dynamics of the daily increase in infected people, an original cellular automaton with intercellular boundaries was developed, which makes it possible to simulate epidemic control measures of varying stringency. In the simulations carried out using the Monte Carlo method with subsequent statistical processing, we studied the impact of restrictive measures of varying stringency on the number of infected people, the duration of the epidemic, and the quality of forecasting. The final series of experiments simulated the spread of the COVID-19 virus in Germany in the first half of 2020.The results show that even a simple cellular automaton model with boundaries successfully describes the course of the epidemic and allows us to assess the effectiveness of restrictive measures. The dependence of the daily increase in infected people on the stringency of measures is presented; it is shown what characteristics of the population can influence this dependence. It was found that the measures of medium stringency (40–50 % according to the Stringency Index) have the least predictable effect; they can cause both rapid localization of the focus and the spread of the epidemic to a large part of the population. Weak and strong measures give a more predictable effect.Conclusion. Cellular automaton models with intercellular boundaries have great potential for modeling the impact of restrictive measures on the course of an epidemic, making it possible to predict the dynamics of infected people based on the population data and the restrictive measures being introduced.

Cellular automaton model with the multi-anticipative effect to reproduce the empirical findings of Kerner’s three-phase traffic theory

As the important spatiotemporal state of traffic flow discovered by Kerner’s three-phase traffic theory, the synchronized traffic flow describes a new traffic phase in congested traffic. However, until now most models within the standard traffic theories cannot reproduce it. The average space gap model (ASGM) is a simple cellular automaton model aimed to reproduce various empirical findings discovered by Kerner’s three-phase traffic theory by incorporating the influence of the multi-anticipative effect on vehicle’s deceleration. However, this paper shows that the simulated synchronized flow by ASGM is not consistent with the reality. To this end, the Multi-Anticipative Model (MAM) based on ASGM is proposed, which describes the influence of the multi-anticipative effect on both the accelerations and the decelerations. Simulations indicate that the empirical consistent synchronized flow and related congested patterns can be well reproduced by MAM. Moreover, MAM can reproduce the speed drop in the car following vehicle platoons reported by the empirical observations. Generally, MAM indicates that the multi-anticipative effect can shed light on the understanding and capturing the complex characteristics of traffic flow especially reported by Kerner’s three-phase traffic theory.

Finite-size effects, demographic noise, and ecosystem dynamics

Strong positive feedback is considered a necessary condition to observe abrupt shifts in ecosystems. A few previous studies have shown that demographic noise—arising from the probabilistic and discrete nature of birth and death processes in finite systems—makes the transitions gradual. In this paper, we investigate the impact of demographic noise on finite ecological systems. We use a simple cellular automaton model with births and deaths influenced by positive feedback processes. We present our methods in a tutorial like format. Using the approach of van Kampen’s system-size expansion, we derive a stochastic differential equation that describes how local probabilistic rules scale to stochastic population dynamics in finite systems. We illustrate that as a consequence of enhanced demographic noise, finite-sized ecological systems can show an ‘effective abrupt transition’ even with weak positive interactions. Numerical simulations of our spatially explicit model confirm this analytical expectation. Thus, we predict that small-sized populations and ecosystems, in response to environmental drivers, are prone to abrupt collapse while larger systems—with the same microscopic interactions—show a smooth response.

Flux-density relation for traffic of army ants in a 3-lane bi-directional trail

Many ant species form extensive trail systems between the nest and food sources. Previously we have proposed a simple cellular automaton model, based on the asymmetric simple exclusion process, that tries to capture the essential traffic properties of such an ant trail. Here we extend this model to the case of trails of army ants. Army ants often form 3-lane trails where ants move on the outer trails in the opposite direction of those on the inner trail. We find, depending on the model parameters, interesting new features in the fundamental diagrams, e.g. a non-monotonic dependence of the flux on the density for large densities.

A simple cellular automaton model with dual cruise-control limit in the framework of Kerner’s three-phase traffic theory

In this study, a new cellular automata traffic flow model with dual cruise-control limit, where the vehicles with their velocities v=1,5 are not affected by noise and the slow-to-start rule is also introduced for standing vehicles with just one free cell, is established. Computer simulations are used to identify three typical phases from the fundamental diagram: free flow, synchronized flow, and wide moving jam. However, in the original cruise-control limit cellular automata traffic model, there are only two kinds of traffic phases, namely, free flow and congested traffic flow. Furthermore, in the ”linear” synchronous flow region, the ratio of flow to density depends on the randomization probability. The synchronous flow with scattered data is firstly found under the dense distribution. Compared to the previous models, the rules of our model are simpler, but it can present many features of Kerner’s three-phase theory, in particular, 2Z-characteristic for phase transitions.

Effects of randomization of characteristic times on spiral wave generation in a simple cellular automaton model of excitable media

Spiral waves are self-repeating waves that can form in excitable media, propagating outward from their center in a spiral pattern. Spiral waves have been observed in different natural phenomena and have been linked to medical conditions such as epilepsy and atrial fibrillation. We used a simple cellular automaton model to study propagation in excitable media, with a particular focus on understanding spiral wave behavior. The main ingredients of this cellular automaton model are an excitation condition and characteristic excitation and refractory periods. The literature shows that fixed excitation and refractory periods together with specific initial conditions generate stationary and stable spiral waves. In the present work, we allowed the activation and refractory periods to fluctuate uniformly over a range of values. Under these conditions, this very simplistic model can recreate the meandering and breakup of spiral waves that were observed in more elaborate models in the literature.

A Model of Vegetation Cover in Conditions of Resource Scarcity: Fairy Rings in Namibia

A mosaic of barren grounds (bars) and areas covered by vegetation often forms in grassy or shrubby communities in conditions of scarcity of environmental resources (e.g., water). Such landscapes have been named “two-phase mosaics”; this term emphasizes the discreteness of vegetation cover units. Fairy rings (FRs) are an example of such structures. There are two sets of competing hypotheses on the reasons behind FR formation: insect activity and self-organization. One of the main arguments against the second set of hypotheses is their inability to explain the FR life cycle: their emergence, ripening, maturation, and disappearance (death). The article proposes a simple cellular-automaton model of the vegetation cover formation. The model imitates the scarcity of resources (water and ash constituents) and generates different variants of the two-phase mosaic. The results depend on two parameters: limitation of resource inflow and plant growth rate. Sixteen percent of combinations of these parameters result in the formation of structures similar to FRs. The model reproduces not only the vegetation distribution pattern observed in the nature but also individual FR life cycles that are consistent with field descriptions. Thus, the model removes the main objection to self-organization as the main mechanism of FR formation.

Calibration and validation of cellular automaton traffic flow model with empirical and experimental data

For traffic flow models, calibration and validation are essential. Cellular automaton (CA) models are a special class of models, describing the movement of vehicles in discretised space and time. However, the previous work on calibration and validation does not discuss CA models systematically. This study calibrates and validates a stochastic CA model. The authors use a simple CA model, which only has two important parameters to be calibrated. The methodology for optimisation is to minimise the relative root mean square error between two properties: the averaged velocity and the variation of velocities in a platoon at a given density. Three different sites are used as cases to show the methodology, for which different types of data (video trajectories or GPS data) are available. The authors find that the best model parameters vary for the different locations. This may result from various driving strategies and potential tendencies. Thus, it is concluded that for CA models, various traffic flow phenomena need to be simulated by various parameters.

Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty

A major conceptual step forward in understanding the logical architecture of living systems was advanced by von Neumann with his universal constructor, a physical device capable of self-reproduction. A necessary condition for a universal constructor to exist is that the laws of physics permit physical universality, such that any transformation (consistent with the laws of physics and availability of resources) can be caused to occur. While physical universality has been demonstrated in simple cellular automata models, so far these have not displayed a requisite feature of life—namely open-ended evolution—the explanation of which was also a prime motivator in von Neumann’s formulation of a universal constructor. Current examples of physical universality rely on reversible dynamical laws, whereas it is well-known that living processes are dissipative. Here we show that physical universality and open-ended dynamics should both be possible in irreversible dynamical systems if one entertains the possibility of state-dependent laws. We demonstrate with simple toy models how the accessibility of state space can yield open-ended trajectories, defined as trajectories that do not repeat within the expected Poincaré recurrence time and are not reproducible by an isolated system. We discuss implications for physical universality, or an approximation to it, as a foundational framework for developing a physics for life.

Simple stochastic cellular automaton model for starved beds and implications about formation of sand topographic features in terms of sand flux

A simple stochastic cellular automaton model is proposed for simulating bedload transport, especially for cases with a low transport rate and where available sediments are very sparse on substrates in a subaqueous system. Numerical simulations show that the bed type changes from sheet flow through sand patches to ripples as the amount of sand increases; this is consistent with observations in flume experiments and in the field. Without changes in external conditions, the sand flux calculated for a given amount of sand decreases over time as bedforms develop from a flat bed. This appears to be inconsistent with the general understanding that sand flux remains unchanged under the constant-fluid condition, but it is consistent with the previous experimental data. For areas of low sand abundance, the sand flux versus sand amount (flux–density relation) in the simulation shows a single peak with an abrupt decrease, followed by a long tail; this is very similar to the flux–density relation seen in automobile traffic flow. This pattern (the relation between segments of the curve and the corresponding bed states) suggests that sand sheets, sand patches, and sand ripples correspond respectively to the free-flow phase, congested phase, and jam phase of traffic flows. This implies that sand topographic features on starved beds are determined by the degree of interference between sand particles. Although the present study deals with simple cases only, this can provide a simplified but effective modeling of the more complicated sediment transport processes controlled by interference due to contact between grains, such as the pulsatory migration of grain-size bimodal mixtures with repetition of clustering and scattering.

A rapid method for constructing precaution maps based on a simple virtual ecology model: a case study on the range expansion of the invasive aquatic species Limnoperna fortunei

Before invasion, or in its early stages, information on the invader in target areas is generally extremely limited. In such situations, managers must select focal areas in which to concentrate control and mitigation efforts. Here, we discuss a rapid method for selecting areas in which to control invasive aquatic species based on limited information. We used a simple cellular automata model that does not require species-specific information, but simulates the process of invasive species expansion and includes observed expansion progress to detect keystone areas. As a case study, we simulated the expansion of an invasive aquatic mussel, Limnoperna fortunei, in Ibaraki Prefecture, Japan, and detected the areas in which control efforts should be concentrated. To some extent, our model was able to predict the expansion of L. fortunei from the initial detected invasion to the current distribution. We predicted areas with a high potential of spreading and areas that would suffer from high propagule pressure. Results revealed a mismatch between areas with high spread potential and those with high propagule pressure. Managers should concentrate their invasion prevention efforts in the former because these are likely to have a greater long-term influence. Additionally, we predicted future expansion from the current distribution and showed that current scattered populations could merge naturally. Our approach is useful for establishing a management plan before or in the early stages of invasion.

Cancer Stem Cells: A Minor Cancer Subpopulation that Redefines Global Cancer Features

In recent years cancer stem cells (CSCs) have been hypothesized to comprise only a minor subpopulation in solid tumors that drives tumor initiation, progression, and metastasis; the so-called “cancer stem cell hypothesis.” While a seemingly trivial statement about numbers, much is put at stake. If true, the conclusions of many studies of cancer cell populations could be challenged, as the bulk assay methods upon which they depend have, by, and large, taken for granted the notion that a “typical” cell of the population possesses the attributes of a cell capable of perpetuating the cancer, i.e., a CSC. In support of the CSC hypothesis, populations enriched for so-called “tumor-initiating” cells have demonstrated a corresponding increase in tumorigenicity as measured by dilution assay, although estimates have varied widely as to what the fractional contribution of tumor-initiating cells is in any given population. Some have taken this variability to suggest the CSC fraction may be nearly 100% after all, countering the CSC hypothesis, and that there are simply assay-dependent error rates in our ability to “reconfirm” CSC status at the cell level. To explore this controversy more quantitatively, we developed a simple cellular automaton model of CSC-driven tumor growth dynamics. Assuming CSC and non-stem cancer cells (CC) subpopulations coexist to some degree, we evaluated the impact of an environmentally dependent CSC symmetric division probability and a CC proliferation capacity on tumor progression and morphology. Our model predicts, as expected, that the frequency of CSC divisions that are symmetric highly influences the frequency of CSCs in the population, but goes on to predict the two frequencies can be widely divergent, and that spatial constraints will tend to increase the CSC fraction over time. Further, tumor progression times show a marked dependence on both the frequency of CSC divisions that are symmetric and on the proliferation capacities of CC. Together, these findings can explain, within the CSC hypothesis, the widely varying measures of stem cell fractions observed. In particular, although the CSC fraction is influenced by the (environmentally modifiable) CSC symmetric division probability, with the former converging to unity as the latter nears 100%, the CSC fraction becomes quite small even for symmetric division probabilities modestly lower than 100%. In the latter case, the tumor exhibits a clustered morphology and the CSC fraction steadily increases with time; more so on both counts when the death rate of CCs is higher. Such variations in CSC fraction and morphology are not only consistent with the CSC hypothesis, but lend support to it as one expected byproduct of the dynamical interactions that are predicted to take place among a relatively small CSC population, its CC counterpart, and the host compartment over time.

Modeling the role of water inBacillus subtiliscolonies

We propose a simple cellular automaton model for the description of the evolution of a colony of Bacillus subtilis. The originality of our model lies in the fact that the bacteria can move in a pool of liquid. We assume that each migrating bacterium is surrounded by an individual pool, and the overlap of the latter gives rise to a collective pool with a higher water level. The bacteria migrate collectively when the level of water is high enough. When the bacteria are far enough from each other, the level of water becomes locally too low to allow migration, and the bacteria switch to a proliferating state. The proliferation-to-migration switch is triggered by high levels of a substance produced by proliferating bacteria. We show that it is possible to reproduce in a fairly satisfactory way the various forms that make up the experimentally observed morphological diagram of B. subtilis. We propose a phenomenological relation between the size of the water pool used in our model and the agar concentration of the substrate on which the bacteria migrate. We also compare experimental results from cutting the central part of the colony with the results of our simulations.

Strong and weak competitors can coexist in the same niche

AbstractThe competitive exclusion principle postulates that two trophically identical but fitness different species can not stably coexist in the same niche. However, this principle contradicts the observed nature's species richness. This fact is known as the biodiversity paradox. Here, using a simple cellular automaton model, we mechanistically show how two trophically identical, but fitness different species may stably coexist in the same niche. As environment is stable and any trade-offs are absent in this model, it strongly violates the competitive exclusion principle.

Strong and weak competitors can coexist in the same niche

AbstractThe competitive exclusion principle postulates that two trophically identical but fitness different species can not stably coexist in the same niche. However, this principle contradicts the observed nature's species richness. This fact is known as the biodiversity paradox. Here, using a simple cellular automaton model, we mechanistically show how two trophically identical, but fitness different species may stably coexist in the same niche. As environment is stable and any trade-offs are absent in this model, it strongly violates the competitive exclusion principle.

Self-organized spatial pattern determines biodiversity in spatial competition

In a simple cellular automata model it is shown that self-organization of spatial pattern in a community of strong competitors may generate a previously unrecognized mechanism of species richness determination. Employing some well-known general properties of interspecific competition, we elaborate a theoretical framework that generates both spatial mosaics and spiral waves within the same conceptual framework, dependent on the covariance of competition. We demonstrate that the qualitative nature of the spatial pattern depends on the “balance” of competition and that the number of species retained in the community depends on this spatial patterning.

Educational software for the learning of electrical activation in the heart: A simple cellular automaton model

This paper aims to present a bio-computational model that simulates the electrical activation in the heart, mainly designed for people who study or work in the fields of biology, health sciences or medicine with no previous background in programming. The model has been created on a MATLAB environment using the principles of cellular automaton discrete models and Greenberg-Hastings proposal. It visualises the energy propagation on a normal and on a malfunctioning heart tissue during its disturbances. As a result, a computationally inexpensive model has been developed which incorporates various types of cell neighbourhood patterns. In conclusion, the integration of this multi-disciplinary model at the universities could be a very beneficial addition to medical, biological and health education.

The Water Transferred of Meandering River in Cellular Model

We have known that many of the main features of river can be captured in a relatively simple cellular computer model. Here we examine some of the detailed characteristics of this model. We show a new tangential angle and special treatment method in curvature channel in a simple cellular model. The water distribute will be determined by the tangential angle which represent the flow direction. And the same tangential angle cellular in curvature reach will be transport in one group in filially. The results show it can make the flow routing conformed to the real river, and solve the question of curve bending coefficient is too large. The results of the routing scheme are compared with field measurements of cross-section, with the predictions of a more close to the real. It’s indicate that the curvature flow routing scheme outlined here is able to overcome some of the limitations of previous simple cellular automata models and may be suitable for use in curvature reach of river modeling water and sediment transport and channel change in complex fluvial environments. As such this research represents a small and ongoing contribution to the field of numerical simulation of curvature channel processes.