Slime Mold: A True Biological Computer

Audrey Pepin, Ethan Levene, Olivia Cozariuc, Tamir Kegeles

Abstract

Placed in a petri dish with oats replicating Tokyo’s most visited locations, Physarum polycephlum was able to optimize Tokyo’s subway system in only a few days. Despite its lack of a centralized control center, P. polycephalum was able to enhance solutions for a problem engineers have been working on for decades. Slime mold’s apparent intelligent behavior was therefore studied to understand the process behind such clever decision-making. How can a brainless amoeboid organism find the shortest way out of a maze? Not only can it optimize travel routes, but it is able to forage effectively without exploring the same region twice. Intricate relations with its environment allow the plasmodium to assess new problems with real-time solutions. Heuristics guide the slime mold’s decision making. Complex algorithms and mathematical models are inspired by P. polycephalum’s behavior. Slime mold algorithm (SMA), the Reaction-Diffusion theory and the Keller-Segel model are explored in the following paper to understand how P. polycephalum offers mathematical solutions.

Introduction

To solve complex optimizing problems, humans have been relying on computers, algorithms and equations. Only, investigators have found a way to model complex optimization problems with living organisms. Physarum polycephalum has been found to optimize urban designs, transportation and communication networks much faster than engineers can (Kay et al, 2022). Also defined as a biological computer, P. polycephalum owns up to its title despite lacking a centralized neural system or vision. This species of myxomycetes can tackle various problems from maze solving to effective foraging. In fact, Physarum can evaluate its environment to find the shortest path to different food sources without retracing its “steps” (Oettmeier et al., 2022). The slime mold’s intelligent behavior is uncommon for such a low complexity level entity. Physarum polycephalum remains a giant unicellular organism (Oettmeier et al., 2022) resembling mold which might arouse disgust. However, one should never judge a book by its cover or a slime mold by its looks since behind the insignificant appearance lies a complex and powerful design tool.

Studies on slime molds behavior were able to generate mathematical models describing the plasmodium’s lattice structure. The plasmodium is a complex network of protoplasmic tubes which allows it to forage simultaneously in different directions (Oettmeier et al., 2022). Being highly sensitive, the plasmodium reacts to external stimuli. Favorable condition generates a positive response and an accelerated growth while poor conditions lead to decay. Optimal pathways are therefore reinforced which creates visible patterns (Kay et al, 2022).

CELL model and the Slime Mold Algorithm (SMA) were designed to improve network optimization. Transportation, communication and even robotics benefit from this contribution. Physarum polycephalum exhibits signs of memory and cognition despite the lack of a brain. It can strategize its growth and explore new environments effectively. Computational models of competitive slime mold investigate the relationship between different plasmodia. Reaction-diffusion theory and the Keller-Segel model evaluate aggregation models which rely on the plasmodium’s chemotaxis. Dynamic Physarum polycephalum can easily adapt to its environment. This asset has different mathematical uses which will be explored in this paper.

Network Optimization

Through understanding graph theory, and network optimization in slime mold, the organism can guide algorithms to optimize mathematical systems and further apply that to various computational and modeling purposes. There are various applications of these models and algorithms that will be explored.

Why Slime Mold?

Slime mold is highly effective at solving complex network optimization problems. Examples from its natural behavior include intelligent lattice and network topology organization and the ability to effectively cooperate with the Steiner tree theory. In computer science, the Steiner tree graph theory analyzes the minimum ‘weight’ network to connect a set of vertices in a graph form. Slime mold also naturally clusters to form more internal connections in each group than external connections with other parts of the slime mold, an important concept in graph theory referred to as clustering. Clustering helps professionals understand where the product or technology will be most effective in terms of consumers or science. It is important to note that because of the biomimetic nature of slime mold network mathematics and computation, there is not as strong a theoretical background, as the mechanisms simply exist in the organism (Li et al., 2019).

Slime mold exhibits intelligent behavior regarding extension, foraging, and navigating dynamic environments. In comparison to ant inspired network optimization, it is non-static and constantly changing in its physical form. For example, slime mold can effectively navigate, provided the right stimulants, the traveling salesman problem which aims to visit certain nodes exactly once, with only certain connectivity’s, in the most efficient manner possible. In the maze solving example, when given food sources at the entrance and exit of the maze, slime mold will solve it effectively. The simultaneous extension and retraction mechanism is demonstrated, which can be utilized for energy conservation and equilibrium maintenance (Gao et al., 2019).

Graphing Slime Mold

Slime mold can be analyzed through graph considerations. The network of slime mold forms a regular classic graph, always with degree 3 nodes (Fig 1.), which demonstrates an elevated level of complexity and replicability in the organism. The complexity comes from varying lengths, widths, and areas of the segments: the distribution of length and area are exponential but logarithmic for width. These specific dependencies are constant for the ever-evolving slime mold network (Baumgarten et al., 2010). This makes mathematical algorithmic work with this organism incredibly efficient, effective, and standardized.

Fig 1. The diamond represents a degree 3 node, as found in the graphed network of slime mold. Many degree 3 nodes connect to form the entire network.

CELL Model (Morphology)

This type of model system closely follows the extension-retraction mechanism of slime mold, where the organism will extend towards a food source, and retract away from its least nutrient-rich area simultaneously. This allows slime mold to remain efficient and optimize its morphology. In the CELL model, cells of the computational system are defined as different segments of the slime mold. Active zones or the bubble are the food source. Internal cells represent the initial area of the slime mold. The external cells are the initially unoccupied environment. The active zones exhibit, like slime molds expansion, movement, generation, and replacement. The random movement of the active zone ends with an optimized path of the internal cells to utilize this site (the food for slime mold). This creates an efficient pathway. It models the graph theory concept of shortest path, which identifies the most efficient holistic path. This model, however, does not account for thickness variations. One major application is enhancing and planning for transportation. In fig. 2 a detailed explanation of the CELL model and the automated system it follows is demonstrated (Gao et al., 2019).

Fig 2. Analysis of the CELL models automation of Physarum‘s path finding capabilities. A) shows an experiment that demonstrates slime mold pathfinding capabilities between 3 sources. B), shows the initial set up of the CELL model. In C), the lifecycle of the active zone or bubble is shown as the path is optimized (Gao et al., 2019).

Current Reinforcement Models

In slime mold during food scavenging or solving a maze, simultaneous retractions of uninvolved tubes correlate with involved tubes thickening and lengthening. There is a feedback relationship between flux and flow inside the tubes and the thickness of the tube. Further, shorter paths of tubes have more flow and are then encouraged to be places of growth in terms of thickness. The aforementioned is measured as increased conductivity for the shorter path (maximum) and eventually zero conductivity for other paths as they disappear. This is what leads to slime mold path optimization and a positive feedback loop. This closely follows current reinforcement models in a more multi-dimensional sense than the CELL model. These models can increase insight into network and route planning on more scales, such as modeling flow (volume), rather than planar, area driven models. Fig. 3 demonstrates this exact concept regarding this modeling system (Gao et al., 2019).

Fig 3. D represents conductivity between 2 nodes and Q, the flux in the same region. In A), tube conductivities begin randomized but become increasingly polarized as the short path converges to a maximum conductivity and all others become 0 (those tubes disappear). In B), the same is shown over time in the maze for slime mold (Gao et al., 2019).

There are a few relevant formulas that can help quantify this model. Foremost, the tubes are the edges connecting a set of nodes in a graph, with $N_i$ and $N_j$ as two nodes, $D_{ij}$ as the conductivity between the two nodes $e_{ij}$ the edge (or tube) and flux $Q_{ij}$ . This provides the following equation (Gao et al., 2019).

(1) $Q_{ij}=\frac{D_{ij}}{L_{ij}}(p_i-p_j)$

$L_{ij}$ represents the length of $e_{ij}$ and $p_i$ and $p_j$ are the pressures at $N_i$ and $N_j$ respectively. This formula can be used to simulate conductivities at different intervals of path optimization (Gao et al., 2019).

Additionally, with u and a as situation specific parameters, the following equation A can be used when searching for the shortest path length only, and equation B for designing entire efficient networks (Gao et al., 2019).

(2.1) $A: f(Q) = Q^u$

(2.2) $B: f(Q) = \frac{(1+a)Q^u}{1+aQ^u}$

This model is a particularly effective slime mold-inspired system for optimization of networks (Gao et al., 2019). This algorithm considers the intelligent branching, splitting, and thickening of slime mold (in response to stimulus) and how they help build useful computational systems.

Slime Mold Algorithms for Autonomous Mobile Robot Path Length Optimization

Autonomous robots have the capability to positively influence various industries like manufacturing, agriculture, and healthcare. Effective path planning without being able to ‘see’ in a traditional sense is one of the major obstacles to making these devices effective. The slime mold algorithm (SMA) aims to replicate the shortest path finding capabilities of slime mold. It was found that with just the SMA, there was a 3.92% reduction in the robot’s path length, which is statistically significant for its proposed uses.

The SMA algorithm specifically simulates the path network behavior of slime mold while scavenging. Uniquely, the algorithm aims to have dynamic adjustments of flow rates in different areas under stimuli. Additionally, the SMA balances exploration and exploitation, never remaining stagnant in its approach. It continuously looks for the best path. This is summarized in Fig. 4, with some more stepwise specifics of the SMA (Zheng et al., 2023).

Fig 4. After initialization, the SMA explores the region randomly and then calculates the efficiency of said routes through metrics like fitness, weight coefficient, and local best. Then it updates the global best (current best option). If below the maximum search level, it returns to updating the metrics until the ideal solution is found (Zheng et al., 2023).

Traveling Salesman Problem

A fundamental component of graph theory is choosing the next move correctly and efficiently. By using light as stimuli (to mimic areas that might have food for the slime mold), the experiment was set up to assess the slime molds effectiveness with eight cities in the traveling salesman problem. For the traveling salesman problem each city must be visited exactly once, the final stop must be the same as the first, and the shortest route is desired. A 64-lane stellate chip (radiating pattern) was prepared on an agar plate. Certain lanes are marked as the cities, with light stimulants. Initially slime mold growth is only in the center of the plate. Once it is allowed to grow into the lanes, it solved the salesman problem optimally as seen in fig. 5 (Zhu et al., 2013).

Fig 5. A) shows the slime mold in the center of the plate and empty lanes. In B) the slime mold has grown effectively to fill the correct lanes in the most efficient manner. In B) you can see the areas the slime mold has branched to in the lanes (Zhu et al., 2013).

The slime mold had a 91.3% success rate in these trials. The concept of the traveling salesman problem can be harnessed for real world applications, and this experimental set up for testing slime molds capabilities in that framework can also be utilized (Zhu et al., 2013). By basing algorithms, such as the CELL and Current Reinforcement Models based on this ability of slime mold, useful optimization can occur. For example, communication networks, or designing routes for deliveries. The advantage of doing this with bio simulation is that it is decentralized (less bias) and is quite efficient at solving complex problems, without manually checking every option.

Transportation Applications

Using similar algorithms to the previously defined SMA algorithm, aimed at finding the most effective path optimization, slime mold can be used to optimize or design transportation networks. For example, subway systems, highways, bike paths and more. Stimulants can be used to replicate population densities or help set parameters for the slime mold. In an exemplary model, slime mold designed various highway networks in Mexico, that all in practice would be efficient and effective options (Cai et al., 2019).

Fig 6. An effective, optimized, and theoretical network designed by slime mold for Mexico’s highway network. It hits population centers and is geographically reasonable (Cai et al., 2019).

The networking design solutions of slime mold can be harnessed to compute and create mathematical algorithms, with a wide breadth of functionalities and applications.

Memory and Cognition

Basic cognition can be defined as a process with three parts: sensory perception, information processing, and motor action. Basal cognition is thought to have originated as a metabolism-centric function, as food finding is the most essential and immediate function that cognition is advantageous for (Smith-Ferguson & Beekman, 2020). Many of the experiments highlighting slime mold’s complex problem-solving involved pathfinding to find food. For example, in the study by Tero et al. in 2010 where Physarum polycephalum emulated Tokyo’s rail network, seen in Fig. 7., this simulation was achieved by placing oats where various cities are on a map of Tokyo’s metropolitan area.

Fig 7. Physarum polycephalum emulating Tokyo’s rail network. The slime mold found paths to the different food sources on the map corresponding to urban centers in Tokyo. In A, P. polycephalum was free to roam, while in B, geographic constraints of the Tokyo area were emulated using slime mold’s photophobia to create low-altitude (shaded) and high altitude (lit) regions (Tero et al. 2010).

In terms of cognition, Physarum polycephalum exhibits low-level (on the hierarchy of learning) non-associative learning, such as habituation and transfer of habituated knowledge. The slime mold was able to learn, for example, to cross a bridge containing a deterrent to arrive at a food source on the other side. In other words, it habituated to the negative stimulus and learned to ignore it. A Physarum plasmodium was separated into two, with one half exposed to experimental conditions and habituated to negative stimulus while the other was not. Recombining the two for 3 hours or more led to the unhabituated Physarum obtaining the behavior of the habituated half post-separation (Smith-Ferguson & Beekman, 2020).

In addition to habituated learning, Physarum’s decision-making also displays a behavior common to more cognitive animals such as humans, birds, and bees: comparative valuation. A 2010 experiment by Beekman and Latty shows that Physarum makes irrational economic decisions: when presented with two food sources, it consistently chose the best one, but when a third less nutritional (decoy) choice was introduced, Physarum changed its preferences, despite the same options being present from before (Smith-Ferguson & Beekman, 2020). Another similarity to other more cognitively complex animals is the true slim mold’s foraging strategy. Shirakawa and Sato (2016) observe that Physarum exhibits Lévy flight, a random-walk strategy also presents in both deer and ants.

When it comes to memory, it can be argued that the slime mold’s extracellular slime trail serves as an external spatial memory of where it has been. When presented with two paths (see Fig 8.), one with a blank agar and one with extracellular slime, equal in distance and leading to identical food disks, a foraging slime mold decided most of the time (39 out of 40) to travel down the blank agar (Reid et al. 2012). The utility of this function, as well as slime mold’s dependency on it, is displayed in a U-trap, where a U-shaped barrier blocks the slime mold from directly accessing the food source on the other side of the agar plate. When the slime mold hits the bottom of the U-trap, it starts moving alongside the obstacle, using its secreted slime to prevent itself from returning to the bottom of the U (where the signal from the food source is strongest due to diffusion), as can be seen in Fig 9. In a U-trap set up with slime, the slime mold took significantly longer to reach the goal.

Fig 8. Slime mold prefers to avoid extracellular slime. When arm A contains slime and arm B does not, Physarum shows preference towards arm B. It should be noted that Physarum shows no preference when both arms contain slime; it chooses a path randomly (Reid et al. 2012).

Fig 9. U-trap experimental setup containing extracellular slime. A glucose solution is set at the “goal”, diffusing and creating a gradient (white arrows) leading the slime mold into the trap. Optimal theoretical path is shown in the dotted red line (Reid et al. 2012).

This behavior also has the benefit of maximizing area coverage during foraging, as Tröger et al. (2024) highlight that self-avoidance behavior in large plasmodia leads to larger area coverage. In fact, self-avoidance behavior was not observed in smaller plasmodia.

Game Theory and Evolutionarily Stable Strategy

Game theory is a branch of mathematics that models decision-making of rational agents in game-settings and can be used to model and explain survival strategies of populations in evolutionary settings. The break-through of game-theory into biology was in 1973, when Smith and Price penned the term “evolutionary stable strategy” (Hammerstein & Selten 1994) in a paper explaining why animals engage in ritualistic and safe (as opposed to lethal) combat (Smith & Price 1973). An evolutionary stable strategy is a strategy that, once adopted by most of a population within a species, cannot be displaced by a novel mutant strategy. It is like a Nashville equilibrium, a state in game theory where both agents in a competitive setting cannot change strategies to improve their outcome (Chen, 2021).

There is evidence that Physarum polycephalum has evolved a set social and competitive strategy. For example, Physarum rigidum, a close cousin of polycephalum, is capable of allorecognition, that is the recognition of other organisms of the same species as distinct from itself. A P. rigidum encountering another P. rigidum will fuse or avoid the other (see Fig 10.) depending on whether the two have similar genotypes. Divided selves often fuse after separation, and so do geographically similar plasmodia, but geographically distinct plasmodia avoid each other (Masui et al. 2018)

Fig 10. Allorecognition in slime molds. Two slime molds B1 and B2 are shown fusing together in a, while slime molds Ei and Tk avoid each other in b (Masui et al. 2018).

Some avoidance events happen without direct contact between the two plasmodia. It is hypothesized that the hyaline sheath secreted by the slime mold enables this no-contact decision to happen (Masui et al. 2018). Interestingly, there is another case of Physarum polycephalum altering behavior based on the presence of another individual without coming into direct contact with them. When observing slime mold foraging behavior in competitive settings, Stirrup and Lusseau noted in their 2019 study that the slime mold initiates foraging much more quickly in competitive settings than alone. In their experimental setup, the two slime molds were situated 2 cm apart from each other and from a food source. Time to begin foraging behavior was measured (i.e., time until beginning of movement) for both slime molds. This measured time to initiation was significantly smaller for both slime molds when they were together rather than alone. It should be noted that, since the slime molds were 1 cm² in size, they were not in contact when initiating foraging. The mechanism by which they knew about the presence of the other is unknown (Stirrup & Lusseau, 2019). Additionally, this initiation speed also changed depending on whether the slime mold and its neighbor were hungry/satiated, with hungrier slime molds initiating foraging more quickly.

Slime mold competitive behavior can be modeled computationally to solve complex problems on a faster time scale than in the real world. Awad and co-authors (2019) have proposed a computational model based on principles of reaction diffusion and cellular automata, which can be seen in Fig 11., coupled with the previously mentioned heuristics of Physarum in competitive settings. Heuristics are the loosely defined rules that guide Physarum‘s decision-making. For example, Physarum‘s tendency to avoid extracellular slime can be considered a heuristic. The rules for this model took into consideration the attractive properties of food sources along with the repellant force of competitor slime molds. Additionally, slime molds that are larger in mass would have higher priority in taking food sources (Awad et al. 2019). This is like Physarum‘s observed behavior in inter-species competition, as it has been observed to extend its network into another slime mold (Badhamia utricularis) if its mass is large enough (Awad et al. 2023). The competitive computational model was later used in network optimization problems and discrete multi-objective optimization problems.

Fig 11. Computational model of competitive Physarum polycephalum. Ten slime molds are represented as aggregates of the same number and color on the hexagonal field. Yellow hexagons represent food sources. The cellular automata model is used, wherein at each iteration of the simulation, a cell (hexagon) calculates its value based off the values of its neighbors and the heuristics of the model (Awad et al. 2019).

Convergence of Ideas

We have seen that slime mold cognition and competitive behavior involve decision-making regarding self-avoidance, avoidance and fusing with other Physarum polycephalum, as well as avoidance with competitors. We propose that there is a possible connection between these behaviors: lack of self-avoidant behavior in small plasmodia, as well as fusing with similar plasmodia, could point to an evolutionary strategy where small-scale plasmodia cooperate to become larger plasmodia, achieving a certain threshold size which allows them to maximize foraging efficiency faster than via individual growth. This is like how the plasmodial stage is achieved by amoeboid Physarum spores: by aggregating together.

Mathematical Basis of Pattern Formation

Pattern Formation in Slime Mold

Physarum polycephalum has unique abilities to create complex networks and adapt to environmental stimuli, guiding the studies of decentralized pattern formation. First, the slime mold has an adaptive network that optimizes its connection with food sources, which can allow developments in biological computation (Fernandez-Marquez et al., 2013) and guides decentralized decision-making processes (Horstmann, 2003). Furthermore, dynamic response to light and chemical gradient highlights the slime mold’s local and adaptive behaviors, which are needed to create coherent structures (Yamada et al., 2007).

Self-Organization

Self-organization represents the guiding of different biological components to assemble without any external forces involved. Slime mold experiences significant self-organization due to its amoeboid characteristics, where cells interact, leading to the formation of different networks. These critical interactions are possible because environmental stimuli like nutrient availability allow a collective behavior to optimize pathfinding and nutrient distribution throughout the organism (Fernandez-Marquez et al., 2013); (Horstmann, 2003). Since it has been shown that slime mold can sense and respond to chemical signals, the comprehension of the mechanism behind self-organization is easier. This simple process commonly used in slime mold is crucial in computational systems using decentralized algorithms where the related units can act on local information to produce more general results. (Yamada et al., 2007)

Chemotaxis, or movement in response to chemical gradients, is critical to slime mold’s behavior. The cells of Physarum exhibit positive chemotaxis by moving toward areas of higher attractant concentration, such as regions with more nutrients. This chemotactic behavior is vital to understanding how the organism navigates its environment and illustrates how simple local rules can lead to the self-organized construction of efficient pathways (Horstmann, 2003). For example, the slime mold’s exploration for food sources, followed by its removal from unfavorable areas, proves adaptive behavior through chemical signaling and collective response. This decentralized decision-making pattern leads to emergent properties where no single cell coordinates the action, but the group achieves a coherent, optimized structure (Fernandez-Marquez et al., 2013).

Reaction-Diffusion (RD) Theory

The interaction between two substances forming spontaneous pattern formation by diffusion across space can be described by the mathematical framework called the RD (reaction-diffusion) theory (Turing, 1990). This theory consists of an activator and an inhibitor substance, whose behavior is shown in Fig 12. and under certain conditions, they can create some stable spatial patterns as seen in nature. More specifically, this concept is a model for chemical reactions that can produce wave-like behavior, such as animal coat patterns (Fernandez-Marquez et al., 2013).

Fig 12. (a) Schematic representation of the activator-inhibitor system. (b) Spatial representation of local activation and long-range inhibition. (Murray, 2003)

In the case of the slime mold, the RD theory further explains the chemical gradients and the diffusion contributing to the path decision process or how Physarum polycephalum arranges its growth to connect efficiently to nutrient sources (Yamada et al., 2007). In other words, the RD theory can also be applied to demonstrate how the consideration of both chemical reactions and diffusion processes induce self-organized movement as well as resource optimization. Furthermore, attractant molecules interact with slime mold cells, dictating the movement and aggregation of the organism, a relationship that is also simulated by the RD models (Murray, 2003). Overall, this produces visually complex yet functionally intelligent network structures that mimic natural processes like vascular and neural patterning (Zhanga et al., 2022). The RD equations, therefore, could describe every behavior mentioned, showing the significance of adaptive networks in the studies of biological studies and technological applications (Fernandez-Marquez et al., 2013).

General RD Model Structure

To describe how local chemical interactions produce emergent behavior, the RD equation gives a simplified insight into the changes in chemical concentrations over time and space (Yamada et al., 2007). For slime mold, the chemical signals or attractants detected by the organism influence the distribution and movement of cells, which are all described by the RD model and by its ability to further predict the path-like structure connecting areas of higher attractant concentration (Murray, 2003). The equations shown below help us understand how some complex interaction systems can achieve simplified coherence.

(3.1) $\frac{\partial u}{\partial t}=D_u\nabla^2u+f(u,v)$

(3.2) $\frac{\partial U}{\partial t}=D_v\nabla^2v+g(u,v)$

Where u and v are the activator and inhibitor concentrations, respectively therefore, $D_u$ and $D_v$ are the diffusion coefficients, $\nabla^2$ is the Laplacian operator, and 𝑓(𝑢,𝑣) and 𝑔(𝑢,𝑣) are reaction terms describing the interactions between the activator and inhibitor (Murray, 2003), (Fernandez-Marquez et al., 2013).

Moreover, RD equations can also model the migration of slime mold cells by interpreting how local cues can guide a collective movement. One of the main variables of these equations driving cells towards the highly concentrated chemical areas is chemotaxis, which is also the principal explanation behind self-organization (Horstmann, 2003). These models can be used for simple, decentralized systems, consequently solving problems related to network optimization, pathfinding, and environmental adaptation (Fernandez-Marquez et al., 2013). Therefore, the RD model study allows researchers and scientists to predict the behaviors allowing slime mold to adapt and thrive in diverse conditions (Yamada et al., 2007).

Keller-Segel Model

The Keller-Segel model (Horstmann, 2003) can be applied to explain reaction-diffusion models driven by chemotaxis. This model expresses a mathematical approach to the collective movement of cells after their contact with chemical gradients, resulting in aggregation and pattern formation (Horstmann, 2003). Examples of those equations are shown for cell density in A below and chemical attractants in B. The model can more precisely describe the slime mold’s ability to sense and react to its environment by describing how decentralized interactions among individual units can collectively form adaptative networks. This concept is imitated frequently in many natural systems (Fernandez-Marquez et al., 2013).

(4.1) $A: \frac{\partial n}{\partial t}=\nabla\cdot\left(D_n\nabla_n-\chi^\eta\nabla_c\right)$

(4.2) $B: \frac{\partial\mathcal{C}}{\partial t}=D_c\nabla^2c-\lambda c+h\left(n\right)$

Where 𝑛(𝑥,𝑡) represents the cell density at location x and time t, 𝑐(𝑥,𝑡) represents the concentration of the chemical attractant, $D_n$ represents the diffusion coefficient for the cell movement, and $D_c$ for the chemical gradients, $\chi^{n}$ represents the chemotactic sensitivity, 𝜆 is the degradation rate of the chemical present, and finally, ℎ(𝑛) represents the production term for the chemical. (Horstmann, 2003; Yamada et al., 2007)

The Keller-Segel model describes how slime mold uses chemotaxis to coordinate movement to achieve a general structural formation (Yamada et al., 2007). This coordination offers a meaningful analogy that inspires innovations in network theory and distributed computing for engineering systems that need robust and adaptable networks (Horstmann, 2003).

Potential for Biomaterial Design

The previously discussed mathematical models used to describe the behaviors of slime mold provide multiple applications for designing biomimetic and adaptive materials. The reaction-diffusion model delivers strategies to obtain self-healing and efficient network structures in biomaterials by inspiration from the slime mold’s natural ability to create paths and respond to environmental stimuli (Zhanga et al., 2022). By designing these materials following the slime mold’s adaptability features, they can obtain autonomous reorganization character when faced with damaging threats, making them valuable to engineering infrastructures that are resilient and flexible.

For example, engineered cement composites (ECCs) are designed to have similarities to the RD principles explained. Indeed, these composites hold self-healing properties due to the reliance on interactions of chemical processes like those of the slime mold. A common example of ECCs is the ancient Roman concrete. This resilient concrete has shown impressive durability thanks to hot mixing with a mixture of aggregate-scale lime casts (Linda et al., 2023). The slime mold naturally adapts and adjusts its path formation, which is reflected by the ability of ECCs to recover from damage without any external intervention. This opens a close relationship between calculated behaviors in biology and material science (Zhanga et al., 2022). Moreover, these biomaterials show that the mathematical models of pattern formation can provide practical solutions for the need for durability and adaptability in engineering (Yamada et al., 2007).

These self-healing materials use reaction-diffusion models, which use the slime mold’s adaptability to external stresses as a standard. These materials can autonomously respond to external stresses by originating chemical reactions to fix their structure, an aptitude that is shown in Fig 13. Thus, the adaptive nature of slime mold described by RD models helps support advancements in innovative sustainable biomaterials (Zhanga et al., 2022).

Fig 13. Healing process in self-healing materials, such as ECCs using chemical reactions (Zhanga et al., 2022).

Conclusion

This paper presented the possible real-life applications of Physarum polycephalum. Its dynamic and decentralized problem-solving mechanism has inspired different models and algorithms which are design solutions. The CELL model is a computational system which can be used to enhance transportation. The SMA mimics slime molds foraging and can improve the travel route of autonomous robots. By solving the traveling salesman problem, P. polycephalum proves it can optimize networks. The slime mold exhibits specific aptitudes such as habituated learning, comparative evaluation and foraging strategies which are usually found in more complex organisms. The plasmodium displays signs of memory detection in the context of the U-trap. Game theory applies to slime mold since the presence of other plasmodia affects its behavior. Strategic aggregation and competitive behavior are ingenious survival skills Physarum polycephalum has developed since it can adapt to its environment. Chemical gradients and external conditions influence the plasmodium distribution and growth. Design solutions such as the reaction-diffusion theory and the Keller-Segel model can predict plasmodium’s chemotaxis and display the structural formation of the organism. Physarum polycephalum is a highly coordinated organism which inspired self-healing biomaterials by means of chemical reactions which is another design solution. The slime mold presents a robust and adaptable network which can sustain changes, a characteristic even more crucial in the context of climate changes.

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