A unifying hypothesis for the structure of microbial biofilms based on cellular automaton models

Authors

  • Julian W.T. Wimpenny,

    Corresponding author
    1. School of Pure and Applied Biology, University of Wales at Cardiff, Cardiff CF1 3TL, UK
      Corresponding author. Tel.: +44 (1222) 874 974; fax: +44 (1222) 874 305; e-mail: sabjw@cardiff.ac.uk
    Search for more papers by this author
  • Ric Colasanti

    1. Department of Plant Ecology, University of Sheffield, Sheffield, UK
    Search for more papers by this author

Corresponding author. Tel.: +44 (1222) 874 974; fax: +44 (1222) 874 305; e-mail: sabjw@cardiff.ac.uk

Abstract

A review of the basic structure of microbial biofilm reveals that at least three conceptual models exist: (i) heterogeneous mosaic biofilm, (ii) penetrated water-channel biofilm and (iii) dense confluent biofilm. When consideration is given to the effects of growth resource, it may be that all three variants are correct but form at widely different substrate concentrations. Experimental research with bacterial colonies and models of the latter using cellular automata have confirmed this view. Use of cellular automata to model biofilm growth give results which strongly suggest that biofilm structure is largely determined by substrate concentration.

1Introduction

Research into biofilm structure and function has increased steadily and exponentially with a doubling time for new publications of ∼4 years over the last decade or so. Whilst hitherto biofilms have been regarded as smooth, predominantly two-dimensional structures [1, 2], recent use of new and sophisticated methods has revealed a much more complex pattern of behaviour.

Use of the confocal laser scanning microscope (CLSM), together with digital imaging, has been applied very succesfully to biofilm problems [3–5]. CSLM has a number of very significant advantages in biofilm research: (i) the sample may be examined in its natural hydrated state, avoiding problems of fixation and dehydration, (ii) because of its narrow depth of focus, thin optical sections can be recorded across the biofilm profile. These can then be processed digitally to generate three-dimensional maps of the structure or to monitor the distribution of chemical and biological elements within it, in the xy or xz dimensions. The use of a large family of fluorescent markers has helped to determine the location of species when coupled to antibodies or to nucleic acid probes or to monitor chemical changes, including pH, viability and redox potential.

Microsensors, in particular for dissolved oxygen, have helped to confirm the notion that water channels exist between microcolonial structures in some biofilms. For example, a microoxygen electrode was used to demonstrate that conditions within a cell cluster were anaerobic. If the tip of the probe was located a small distance away from the cluster, but in the water-channel surrounding it, oxygen was detectable, though at lower than atmospheric concentrations [6, 7].

Further evidence confirming the existence of water channels came from nuclear resonance magnetic imaging [8] and also by tracking fluorescent particles using a combination of CSLM and normal optical microscopy [9].

2Conceptual models of biofilm structure

2.1The water-channel model

Many of the studies described in Section 1 have led to a new conceptual model of biofilms in general. After colonising the surface, the cells undergo phenotypic changes and produce, among other structural molecules, exopolysaccharides. Some microcolonies form cone-shaped structures. As the film proliferate mushroom forms attached by stalks of EPS and microorganisms are seen, some of which may fuse together leaving water channels that penetrate almost to the base of the film. Much smaller branching channels can also be seen within the microlonies [6, 10–12]. Most of the films observed have been of natural living biofilm generated under a range of environmental conditions. From a considerable amount of experimental work, the consensus structure shown in Fig. 1 was produced. The importance of the water channels has been stressed as representing:’… a primitive circulatory system analagous to that of a higher organism. Because of this remarkable biofilm architecture, bacterial cells within a microcolony have a degree of homeostasis, optimal spatial relationships with cooperative organisms, and an effective means of exchanging nutrients and metabolites with the bulk fluid phase.’ (Costerton et al. [13]).

Figure 1.

Water channel model according to Costerton et al. [6–10].

2.2The heterogeneous mosaic biofilm model

An extreme form of the water-channel model is the heterogeneous mosaic model described by Keevil et al. at Porton (Wiltshire, UK). These workers have examined the structure of natural biofilms forming in water distribution systems. They used a modified differential interference contrast microscope (Nomarski optics) which generates an image with a wide in-focus range, allowing the structure of the biofilm as a whole to be examined. Their view of such biofilms is of microcolonies forming stacks attached to the substratum at the base but generally well separated from their neighbours [14–16]. Often there is, in addition to the stacks, a background of individual cells attached to the surface forming a very thin ‘film’∼5 μm thick (Fig. 2). The main difference between the two models is the sparseness of the observed stacks which form as unconnected towers surrounded by the water phase. In other words, there are fewer fused mushroom structures and the films cannot be said to have channels through them.

Figure 2.

Heterogeneous mosaic biofilm model according to Keevil et al. [16, 17].

2.3Dense biofilm models

The models discussed above may not be representative of all biofilm systems. One obvious exception is dental plaque which forms in different regions around the tooth surface. There are four main types of plaque depending on the precise location of this biofilm. Smooth surface plaque forms on exposed tooth surfaces whilst fissure plaque is found in the deep fissures that form on the biting surface of teeth. Approximal plaque appears at the contact point between teeth. These three are supra-gingival plaques since they form above the gum margin. Sub-gingival plaque forms between the hard tooth surface and the gum epithelium.

Nutrients that lead to the growth of plaque microbes are derived from three sources. (i) Saliva which consists of mucin-like glycoproteins plus other proteins, such as the enzyme amylase and peptides and amino acids. (ii) Gingival crevicular fluid. The latter is serum-like exudate that bathes the gingival crevice and contains many different proteins, including immunoglobulins. (iii) Exogenous nutrients used as food by the individual. Of these fermentable carbohydrates are most important in influencing plaque structure. These can lead to caries in the enamel layer of the teeth due to the accumulation of acidic fermentation products [17].

The structure of dental plaque has been investigated directly using scanning and transmission electron microscopy and indirectly using coaggregation studies. For instance, the two-dimensional structure of 2-week-old dental plaque from three individuals (Fig. 3) was demonstrated by Nyvad and Fejersdorf [18]. The three pictures show significant differences but the one unifying theme was that the structure was a dense biofilm with no evidence of water channels or porous structures within it. There is, however, some structural organisation present. This includes numerous microcolonies of similar shaped bacteria, presumably clones from a single cell. In addition, there are examples of specific associations which are described as ‘corn cobs’ or ‘rosettes’ or ‘bristle brush’ associations as well as pallisades of parallel orientated bacteria [17, 19, 20]

Figure 3.

Dense biofilm system. Dental plaque from three different subjects showing dense structure, microcolonies and some cell associations [18].

The possibility of a complex structure in which different bacteria are associated physically with one another in oral biofilms was raised by the work of Kolenbrander and London [21]. The complex interactions were deduced from simple test tube coaggregation studies where almost every combination of oral bacteria were matched in pairs to see which organisms were capable of binding with which. Although this work raises exciting possibilities there has been little direct evidence that oral bacteria do bind together in vivo in the same manner; it is, however, clear that oral bacteria appear to collaborate physiologically in degrading glycoproteins like mucin [22].

The pictures in Fig. 3 are all transmission electron micrographs and the criticism can be made that they suffer from artifacts of preparation. This has been discussed in detail by Handley et al. [23] who have examined oral biofilm using a range of microscopic methods. For instance, environmental electron microscopy shows a confluent biofilm with no obvious channels from the surface downwards. This technique, like atomic force microscopy, gives little structural detail from above the biofilm which resembles an irregular cloudy surface consisting mostly of cells embedded in EPS.

Another important biofilm is that which forms on the inner surfaces of in-dwelling medical catheters. These have been examined by scanning and transmission electron microscopy and show no evidence of water channels [24–26].

Whilst the TEM photographs are two-dimensional, results with environmental and atomic force microscopes add a third dimension to this information and suggest that the idea of a dense confluent biofilm with no water channels at these sites is most likely to be correct.

2.4The connection between substrate concentration and biofilm structure.

Clearly at least three conceptual models of biofilm structure exist. At one extreme, is the heterogeneous mosaic model described by Keevil et al. Here, individual microbial stacks well separated from one another are surrounded by water. The second form is the water-channel model constructed by Costerton, Lewandowski et al. in which microcolonies form mushroom-like structures which may coalesce and are penetrated by branching water channels. Last is a dense biofilm model apparent in some medically important biofilms, including the oral environment and in indwelling catheters. Here, there is no evidence of water channels.

These three views have led to some argument both in the literature and at scientific meetings. Proponents of each model have set out their stalls and made assumptions that all biofilms must resemble their own particular paradigm.

How can the three models be reconciled? One possible factor is substrate concentration. Keevil and his group used natural unamended water in their biofilm experiments whilst Lewandowski used river water to which some nutrients were added. At the other extreme, are the oral biofilms with occasionally very high concentrations of nutrients. A number of biofilm systems are presented as a function of estimated substrate concentration in Fig. 4. It is obvious that the range of possible substrate concentrations is enormous and can approach 1 million-fold. Lowest concentrations appear in water purification and fresh water systems, where available nutrients might be <1 mg l−1. Highest concentrations are associated with food systems, internal and external animal surfaces and with oral biofilms. In the human mouth, concentrations can reach 10% (w/v) sucrose as in many soft drinks and could be much higher for short periods whilst the subject is consuming sweets and other confectionary. However, even when plaque is exposed to resting saliva, available nutrients can approach 15–20 g l−1.

Figure 4.

A possible relationship between prevailing substrate concentrations and types of biofilm.

Although the data are not conclusive on their own there is a suggestion that structure and prevailing substrate concentration are associated. Substrate transport in and around biofilm has two components, eddy diffusion due to water movement and molecular diffusion where transport occurs down a diffusion gradient through a matrix by the nearest neighbour exchange of molecules. In the presence of low concentrations of nutrients, demand for the latter may vastly exceed supply and so a zone around a group of organisms can become severely depleted of the substrates they need for growth. Because such zones are devoid of essential molecules they represent a no-go area for further growth.

Is there any relevant information from other areas of biology? Bacterial colonies can often exhibit diffusion limitation. Cooper et al. [27] described the ‘snowflake’ form of colonial growth in colonies of 6-week-old Aerobacter cloacae. This has been extended and confirmed recently [28, 29] using a strain of Bacillus subtilis isolated from food. This organism was cultured over a wide range of agar and substrate concentrations. At high concentrations, colonies were approximately circular but became increasingly branched rhizoid structures as the concentration fell to very low levels. We (J.W.T. Wimpenny, S. Walsh, L. Ryder and W.A. Venables, unpublished observations) have investigated similar phenomena using Bacillus licheniformis. The morphology of this bacterium follows similar rules becoming less circular and more clearly dendritic as the substrate concentration is reduced (Fig. 5a,b).

Figure 5.

Colonial growth of Bacillus licheniformis. (a) Colonies grown on 1.2% w/v agar containing (A) 0.01%, (B) 0.5%, (C) 2.0% and (D) 10.0% peptone. (b) The effects of peptone and agar concentration on B. licheniformis colony morphology (unpublished data from S. Walsh, L. Ryder, W.A. Venables and J.W.T. Wimpenny).

Colonial morphology has been the subject of scrutiny by mathematicians and a number of computer models have been created. Some of these will be described in the next section.

3Cellular automaton (CA) models

CA models originated with Conway's Game of Life (for a description, see Berlekamp et al. [30]). This was a simple rule based system involving location of ‘cells’1 on an array of compartments each of which is scanned by a computer many times over. If the compartment is empty, the computer moves to the next: if this is occupied, the computer refers to a rule base which indicates the fate of the cell. In the Game of Life, generation of a new cell, death of an old cell or no change, is judged solely on the occupancy of neighbouring compartments. It follows that CA models are defined as spatially and temporally discrete systems where the state of an automaton is determined by a set of rules that act locally but apply globally.

The morphology of a large-scale microbial structure, such as a colony or a biofilm, emerges from the actions of individual microbial cells [31, 32]. These actions are in turn derived from inputs from the microbe's external environment, its own current state and the set of all possible states for that cell. Emergence, where the gross behaviour of a system arises out of the local rules that direct its constituent parts, is the domain of a branch of mathematics which is concerned with CAs [33].

Despite their apparent simplicity, CAs are capable of true complex behaviour [34]. This makes them particularly suited to simulating biological systems which exhibit a high degree of spatial non-linearity and are difficult to represent as a set of differential equations [33, 35, 36].

3.1Applications of CA models to bacterial colonies

CAs have been used in a wide range of biological simulations, including the morphology of bacterial colonies [28, 37–39]. The methods used may be based on physico-chemical principles. For example, the diffusion-limited aggregation (DLA) model aims to reproduce a system where a component diffuses and aggregates to form ‘cells’ in a growing structure. DLA models are part of a subset of CAs known as gas-lattice systems. Gas lattices are based on the same grid system as standard CAs, but in gas lattices the array contains particles that can move between the squares in a prescribed fashion. The microbial colony models of Matsushita and Fujikawa are solidification models where the particles within the grid move under the influence of a random walk algorithm until they come into contact with a box that contains a stationary particle. When this occurs, the colliding particle becomes stationary within the box it occupies. This stationary particle can in turn ‘solidify’ other moving particles. Matsushita and Fijikawa showed that by varying the number and diffusion rates of particles, which they presented as an analogue of nutrients, they could reproduce a range of morphologies similar to those found in colonies of B. subtilis grown on agar plates [28].

Ben-Jacob et al. [39] have produced perhaps the most detailed models of bacterial colony growth using CA systems. Their model incorporated diffusible substrates and mesoscopic groups of cells called ‘walkers’ which could move over a triangular lattice within an envelope. Movement of the cells cause expansion of the envelope in the direction of motion. This feature accounts for ‘cooperativity’ between walkers. Each walker uses resource which is replaced from the environment. Given sufficient resource the cells reproduce. If the resource quotient within the walker falls to zero, it ceases moving and ‘sporulates’. This model generates patterns of growth that broadly reproduce experimental observations of the growth of B. subtilis strain 168 as a function of substrate and agar concentration. This model invoked elements of cell-cell communication and chemotaxis and begins to reflect some of the complexity of a microbial colony.

4Investigating biofilm structure using a CA model

It is clear that a range of different CA models have thrown light on the generation of pattern by growing bacterial colonies. The consensus, both in the experimental work and in modelling colony growth, is that colonies in the presence of excess nutrients generate simple circular structures but that at low concentrations growth becomes diffusion-limited and a branching rhizoid structure develops. Since we have raised the possibility that structures apparent in biofilm formation may also be related to nutrient concentration it seemed appropriate to apply CA modelling techniques to biofilm. It must be emphasised here that this type of modelling as far as the authors are aware, has not so far been applied in biofilm research.

The CA model used here was developed from an earlier version [40] that added biological rules to DLA models. The stationary particle is replaced with a microbial cell. A microbial cell occupies a single square and can produce copies of itself that occupy neighbouring squares. Any action of the microbial cell is mediated by the consumption of resource units. The resource units behave in the same way as the particles in a gas-lattice model, i.e. they can diffuse randomly over a pre-determined range of neighbouring compartments. Specific biological objectives are easy to incorporate into a CA model. Examples include maintenance requirement, growth rate and secondary metabolite production. The uses of object-oriented programming techniques make it straightforward to include a range of microbial types each having different functional characteristics, in the same simulation. Pseudocode for the model is listed in the Appendix.

The CA model was built around a two-dimensional array of up to 100×100 compartments. To set up the model, the array is first populated with ‘substrate’ molecules which are distributed randomly and allowed to diffuse also into random positions around their origin. More than one substrate molecule can occupy the same compartment. Diffusion rates can be varied over a 10-fold range. This is implemented in terms of the distance from its origin that a particular molecule can move. The system is then ‘inoculated’ with ‘cells’. For the biofilm model, they are placed on the lower edge of the array corresponding to the ‘substratum’ on which biofilm grows.

For this simulation, cells remain at fixed positions but in the presence of substrate, they are able to grow. Each microbial cell searches its neighbouring eight squares for the unoccupied square with the most nutrient units. If that square contains the minimum number of nutrient units defined by the yield coefficient, then a new cell ‘grows’ into that compartment. The ‘yield coefficient’ can be varied over a range from 1 to 10, that is each new cell requires 1–10 units of substrate for growth. Cells can accumulate resource units until they have sufficient to generate another cell. Maintenance is not incorporated into this model so that cells do not ‘die’.

Growth continues over a number of iterations until the mature ‘biofilm’ is formed. Individual cell ‘clones’ can be identified by choosing one of seven different colours to mark them.

4.1Biofilm growth

Fifty cells were inoculated at random positions on the bottom edge of the arrays which held 1000, 2000, 8000 and 40 000 U of substrate, respectively. As is clear from Fig. 6, the highest concentration led to a dense uniform film whilst the lowest concentration generated stacked dendritic structures. Results indicate that in an array of the size selected, relatively few stacked structures form. This is because substrate is quickly exhausted at the base of the film where ‘competition’ amongst cells is most intense. Once a stack forms, it can overgrow others or be overgrown and eliminated.

Figure 6.

CA model of biofilm growth at four different ‘substrate’ concentrations. The array was 100×100 and 70 cells were used to inoculate the substratum. (A) 1000; (B) 4000; (C) 8000; (D) 40 000. For further details, see text.

This stochastic element in the simulation is due to the random diffusion of substrate molecules, but it only serves to increase the relevance of the simulation since it must be assumed that chance also plays a significant part in the formation of real biofilms. This seems to be true in models of acutely substrate-limited films or of dense films formed in the presence of excess nutrient. Simulations of either of the above types were established using seven differently coloured organisms as inocula. At each substrate concentration, 12 cells were used as inoculum for three replicate simulations on a 35×35 element array. The small array was chosen so that the ancestry of each stack can easily be traced by its colour. Though the form of the ‘films’ is the same for each type of growth the fate of each of the different original cells is entirely different in each replicate run (Fig. 7a,b). It should be noted that the two sets of simulations were run one after the other during the same session.

Figure 7.

CA models of biofilm growth at low and high substrate concentrations. Each set was inoculated with 12 cells of seven different colours onto the bottom edge of a 35×35 array. The diffusion rate and the growth-yield factors were set at 1. (A) A control showing the position of each of the inoculum cells and the ‘unused’ substrate resource before the program was run. (B–D) Three separate simulations performed one after the other. In a, 245 units of resource were used and the program run for 1100 iterations. In b, 3675 units of resource were used in 20 program iterations.

Changing the diffusion rate and/or the yield factor significantly affects the form of the structure. Where the diffusion rate and the yield factor are both set to 3 the film formed more closely resembles the mushroom like structures reported by Costerton and his group [6–10](Fig. 8). To make the simulation more accurate, the relevance of the CA factors to real diffusion and yield coefficients needs to be established more exactly.

Figure 8.

A CA model of biofilm growth. In this simulation, both the yield and the diffusion rates were increased from 1 to 3. The array is 70×70 and the base was inoculated with a repeating sequence of 70 cells of seven different colours. The starting resource was set at 9800 units.

As a simple control on the biofilm model, exactly the same simulations were carried out to generate colonies by inoculating at the centre rather than at the edge of the array. A single cell was placed near the centre of a 100×100 array and allowed to develop with 1000, 2000, 8000 and 40 000 U of substrate. At the highest concentration, the colony is substantially circular in profile. This becomes more irregular as the substrate concentration decreases whilst very sparse dendritic or rhizoid structures appear at the lowest concentration (Fig. 9).

Figure 9.

CA model of colonial growth at four different ‘substrate’ concentrations. In this simulation, conditions were the same as in Fig. 6: (A) 1000; (B) 4000; (C) 8000; (D) 40 000. The array was 100×100 and a single cell was inoculated approximately at the its centre. For further details, see text.

Although the relevant experiments have yet to be done it seems that the simplest explanation of the observed differences between the three types of biofilm discussed in Section 1 is that they are progressively more substrate-limited as one goes from high to low prevailing substrate concentrations. This conclusion gains added weight from results with colony models which have been confirmed by direct experimentation in at least three different laboratories.

It may be that this simple result will clarify an area which has been dogged by controversy. There have been comments that the porous biofilms having water channels within them are a form of ‘morphogenesis’ designed to generate a simple circulation between different microcolonies. There is no doubt that the end result might be useful to biofilm community members; however, the reason for these channels is probably accidental and based solely on diffusion limitation as a mechanism. The same argument applies to individual stacks. These systems bear an uncanny resemblance to the CA model. For instance, in the real systems, there is usually a layer of individual cells on the substratum surface whilst stacks of microcolonies develop amongst them [16]. From the simulation, we conclude that there is very little nutrient at the surface, it having been scavenged by the original cell population and the growing stacks which at the extremely low prevailing substrate concentration will sweep a large area around them clear of substrate.

Meanwhile, the variable but often very high substrate concentration found in the mouth leads to a confluent dense biofilm. Holes have been seen in oral biofilms and this should not cause any serious alarm since CA models with slightly lower substrate levels also show such holes.

Perhaps, it is germaine at this point to highlight the importance of reaction diffusion phenomena in pattern formation. The subject has been reviewed before by Wimpenny [31].

5Outlook

Our CA models are intrinsically very simple in design and operation yet we believe that they can provide quite a realistic representation of events that take place in real systems at least in terms of patterns of biological structures. The use of a gas-lattice model is mechanistically a close approximation to molecular diffusion although the CA model is quite coarse in scale. The coarseness emphasises the stochastic element in the simulation and does not contradict the relevance of the model to the natural situation.

Our CA model provides a platform for further sophistication. Versions under development include the following. (i) Incorporation of more than one species, for example, differentiation of a film between aerobic and anaerobic organism where oxygen as well as another nutrient is present. (ii) The outcome of competition between cells with different substrate relationships. (iii) The effects of interactions between different bacteria where one acts as a donor of an advantageous substrate for another, or where cross feeding is involved. (iv) Incorporation of a model for polysaccharide production. (v) Incorporating flow into the model. In many natural ecosystems, biofilm forms in the presence of water currents which in the water-channel and the stack models allow nutrient solutions to penetrate between the colonial aggregates. It will also be necessary to develop the model to make it quantitative. Here, the rule base will need to include Monod relationships and growth yield factors as well as relationships defining the movement and consumption of nutrient. Each rule will be implemented by the solution of the relevant differential equations.

We plan in the future to consider three-dimensional systems as well as taking a more quantitative view of CA solutions (e.g. [41]).

Appendix

Appendix A

The morphology program was developed using an object oriented language. The main feature of the latter is that data and functions are intimately linked. The main data type of the program is the living cell (referred to simply as the ‘cell’). Making this object oriented means that data about the cell and functions that alter the data are kept together.

An instance of an object is created and values assigned to its variables in the same way that standard data types such as integers are created and values assigned to them. Each instance of an object will hold separate data values. Calling a function of that instance will only affect the data values associated with it. Object orientation is particularly useful in a cellular automaton model such as this where each cell represents an individual having separate state values but common functionality.

In this program the two main objects are cells and compartments in the array.

The cell contains two unique data values:

  • 1.1.its nutritional state,
  • 1.2.the position of the cell in the array,

and one function:

  • 2.1.to grow into neighbouring compartments.

The compartment contains three unique data values:

  • 1.1.a pointer to a cell,
  • 1.2.the number of resource units in it,
  • 1.3.a temporary store of resources,

and one function:

  • 2.1.to redistribute resources.

The complete program consists of the following:

  • 1Initialise the array of compartments.
  • 2Fill the array with the predetermined amount of resource.
  • 3Inoculate the array with cell(s).
  • 4At each iteration grow those cells that can.
  • 5At each iteration diffuse resource units.
  • 6Repeat steps 4 and 5 until the required number of cycles has been completed.

Each step is described in more detail:

  • 1Initialise the array of compartments. A loop is set up to scan each member of the array in turn creating an instance for each compartment.When a new instance of an object is created it is assigned to a location in the computer's memory. A pointer to that location is then held at the relevant variable. The array is therefore represented by pointers to object memory areas.
  • 2Fill the array with the predetermined amount of resource.A compartment is chosen at random from the array and the resource variable of the location to which it points is incremented by one. This process is repeated until the required amount of resource has been introduced to the grid.
  • 3Inoculate the array with cell(s).A compartment is chosen. An instance of a living cell is created. The position of the chosen compartment is then stored by the cell and a pointer to that cell is in turn stored at the instance of the compartment.A pointer to each new instance of a cell is also added to a separate list. The latter is used by the growth algorithm.
  • 4At each iteration grow those cells that can.The reason for having two sets of pointers to cells, the array and the list, is that they perform different functions. The former represents the landscape that the cells occupy. Generally the majority of compartments in the array will remain unoccupied for most of the simulation. It is inefficient to use this structure to locate cells as this would require that every compartment be visited in turn. It is better to work from a separate list of cells. This is possible because both the compartments and the list only hold pointers to instances of the cells not to the cells themselves. Pointers in the list and in the compartment are linked to the same object, so action by either pointer will affect it. The list can therefore update the cells referenced in the array. A member of the list is chosen at random and the grow function of the cell to which it points is then called.The grow function uses the position data of the instance of the cell to calculate if there is sufficient resource held in the compartment to allow new growth. If there is, then the required amount of resource is removed from the compartment and an empty compartment is located from its neighbours in the array.Growth will not occur if there is no available free space or if the resource level is insufficient.If growth is possible a new instance of a cell is created and a pointer to it is placed into both the vacant compartment and into the holding list. The pointer to the original cell is also placed in here and removed from the main list. This prevents any cell from being called twice.Once the main list is empty, one iteration of the program is complete. Now the temporary holding list is made the main list ready for the next iteration of the growth function.
  • 5At each iteration diffuse resource units.Diffusion is simulated by randomly redistributing the resources from each compartment to the temporary resource store in each of its neighbours. This is performed for each compartment in turn. Finally the number of resource units in the compartment takes on the value of its temporary resources store and the latter is reset to zero.
  • 6Repeat steps 4 and 5 until the required number of cycles has been completed.

A full program listing in the Delphi programming language can be obtained by contacting RC at: ricardoc@cogs.susx.ac.uk

Footnotes

  1. 1In most cellular automaton (CA) research, the term ‘cell’ is used to define a location in an array. To avoid confusion, we will use the term ‘compartment’ to indicate an individual location in the matrix and retain the word ‘cell’ to define an ‘organism’ that can ‘grow’ in it.

Ancillary