Physarum Polycephalum (Slime Mold): the Mazerunner

Key words: Physarum polycephalum; Slime mold; Tube network; Locomotion; External stimuli; Navigation; Structural integrity; Peristaltic waves; Chemotaxis; Lattice structure

Abstract

Numerous nuclei enclosed in one cellular membrane is an uncommon concept. Still, Physarum Polycephalum, more commonly known as slime molds, are roughly summarized by this structure and even demonstrate among the foremost of surprising behaviors. One such example is navigating a maze, which is done quite impressively by the "supercell." Indeed, three different sections will encompass the capabilities of the slime mold to obtain a comprehensive overview of the structural components, the responses to external stimuli, and the locomotion mechanisms. The investigations of the previously mentioned sections of our research allowed for the discovery of the principal elements describing how the slime mold can navigate through a maze. Physarum Polycephalum in ideal environments has a polygonal lattice structure early on, then a hexagonal lattice structure, allowing structural strength and transport of nutrients after oscillatory expansion. Additionally, it has positive phototaxis linked to sporulation and negative phototaxis, with reactions dependent on the wavelength and under considered energy sources. Slime mold is also influenced by temperature, chemical, electronic fields, and magnetic field changes, where external stimuli assure the cell's survival through the guidance of positive and negative taxis. Lastly, slime mold has the aptitude to synchronize its tubular network to the peristaltic waves around, with a wavelength equivalent to its own body length, augmenting material transportation and avoiding flow in a reverse direction.

Introduction

There are several types of slime mold that are differentiated by their structure change and contact with varying nutrient sources. On the one hand, the cellular slime mold will always keep individual cellular separation and on the other hand, the plasmodial or acellular slime mold merges individual cells to behave as a supercell, a cellular membrane uniting multiple nuclei. Furthermore, gene expression differentiates acellular slime mold into two categories: amoebae and plasmodia (Bailey, 2005), allowing the recognition of differences in cellular organization and behavior. The peculiar structural properties of the slime mold provide access to revolutionary research in a vast array of fields to solve optimization problems. This research paper will investigate the behavior of the latter, as it is the only of the two species capable of navigating a maze and doing so with the shortest distance possible to attain the food source without the help of a brain of any kind.

The research on the physics of slime mold through the maze navigation will be divided as follows. Firstly, the structural integrity must be thoroughly understood to assimilate the reasoning behind the different aptitudes of the slime mold. More precisely, the components related to the core form, the polygonal and hexagonal lattice structure, its spores, and how these structural aspects impact the slime mold’s ability to complete complex tasks often related to food sources. The variability of the slime mold’s structure with regards to its dependence on food sources is visually introduced in Figure 1. This is because the species’ capabilities are primarily associated with nutritional sources, as will be demonstrated in this paper. Then, going into detail about the reaction of the slime mold to external stimuli will be essential in discovering the possible influence present during movement. The reasoning behind achieving such intelligent behavior while missing neurological capabilities will be further understood by explaining spatial awareness and response to external stimuli guiding the slime mold towards different objectives. Phototaxis, sporulation, thermotaxis, and other forms of taxis such as chemotaxis, electrotaxis, and magnetotaxis all represent further developments in understanding slime mold’s physical responses. Lastly, the growth and organization of the network necessary to allow locomotion will be related to waves spanning the organism's body. Moreover, slime mold has coordinated tubular contraction that uses amoeboid locomotion in addition to peristaltic waves, allowing it to transport materials across its network. This is how Physarum Polycephalum navigates through a maze, and the next sections will develop these concepts further.

Fig. 1. Illustrations of Physarum Polycephalum networks, showing the structural network’s extension towards food source. (Dussutour & Arson, 2024)

Assembly

Core Form

To analyze the physical properties of slime mold, the basic structure and form are quintessential. Plasmodial slime mold cell size ranges from ten microns to one meter, a vast span. Further, as a coenocytic organism with thousands of nuclei, almost any arbitrary split or breaking of slime mold into pieces will still survive (Ito et al., 2011). Even amid physical trauma to slime mold, the organism can continue to thrive and regenerate which has contributed to its success in most environments.

Fig. 2. Images of the coenocytic nature of slime mold, demonstratively in a petri dish and in nature (Briggs).

Multiple cells of slime mold can also fuse together to behave like an individual. As slime mold expands, it’s thickness changes to maintain structural integrity (Ito et al., 2011). The unicellular but multinucleated form of slime mold is physically efficient, allowing for lightness while maintaining incredible mobility and a large amount of genetic information. Slime mold, contrary to widespread belief is not a fungus but rather has properties of several types of organisms including various plants, fungi, and animal species. One demonstrative example is the sclerotium which slime mold forms in the absence of nutrients. This dormant inactive state of slime mold ends with the reintroduction of water or nutrients. The slime mold can survive this way for years due to strong physical elements and robust walls which contain cellulose, a typical plant property. One other essential element of slime mold is the structural contractile fibers that contain f-actin, and contribute to its locomotion which will be discussed (Oettmeier et al., 2017). In its most core form, slime mold is simple enough to study physical processes but also chemical or biological behaviors on a relatively macro scale. Additionally, the predominant growth form of slime mold is a tubular vein network, and the overall slime mold cell can form in many shapes and sizes. Slime mold utilizes membrane bounding which forms a physical barrier for the inner fibers. The cortex is the primary membrane and is found in all plasmodial slime mold (P. Polycephalum) and helps the organism manage mechanical force and maintenance of intracellular pressure relative to the environment. On top of the cortex is a slime layer which provides chemical protections (rather than the physics related protection from the cortex) from harmful surrounding influences. One other physical property of slime mold is its invaginations and pores which are uniform and evenly spaced out in a pattern as seen in figure 3. The pores form as unusually perfect circles at a rating of 0.74 +/- 0.14 with 1 being no flaws (Oettmeier et al., 2018).

Fig. 3. Pores on slime mold are evenly spaced and uniform. This plays a role in its mechanical flexibility (Oettmeier et al., 2018).

Lattice Structure

Slime molds complex lattice structure serves two major roles, transport of nutrients and structural integrity and strength. To achieve these goals, it also vies to accomplish this as effectively as possible with the lowest energetic cost. Further, the lattice structure has an adaptable form that maximizes the aforementioned based on environmental conditions. In one relevant analysis, vertices were defined at the bifurcation or splitting points of plasmodial slime mold pathways. This allowed for analysis of the lattice structure patterns. Uniquely, when the environmental conditions were strong, and nutrients were easily accessible, a polygonal lattice structure formed in early life (usually with over six edges per vertex) and later in growth a consistent hexagonal lattice structure. This is a strong physical form for the slime mold. In less ideal conditions the original polygonal lattice structure developed into a tree-graph structure, in search of the most nutrients and fertile ground. A tree graph is a far more randomized network that is not as physically efficient. These observations can be demonstrated in the images shown in figure 4. Another observation in this study found that in nutrient rich environments slime mold spread in a rounded thin sheet whereas it got thicker in certain locations for the nutrient deficient environment, presumably where it could find food (Ito et al., 2011).

Fig. 4. A-C are photos of slime molds lattice structure in attractive conditions while D-G are in repulsive conditions. The uniformity of the lattice structure in attractive conditions provides structural integrity while the randomness of the lattice structure in less ideal conditions helps it find nutrients (Ito et al., 2011).

There are many physical reasons that slime mold has adapted to form in a lattice structure. Some of the general benefits of a lattice structures include low use of material, efficiency in transportation, low weight, and optimized structural strength. Many engineering lattice structure applications include slime mold like networks due to these many helpful features, and it is therefore remarkable that slime mold has developed this way. The lattice structure is also relevant for many other impressive slime mold capabilities, such as locomotion, which will be discussed later. In terms of nutrient collection, lattice structures also maximize surface area. Many strong biological structures also utilize lattice structures because of its immense structural strength and energetic efficiency, like human bone (Helou & Kara, 2017).

Fig. 5. Bone forms a lattice structure for similar reasons to slime mold, and bone is out of necessity also a remarkably strong structure. This image of slime molds lattice mirrors the typical image of bone structure closely (Oettmeier et al., 2018).

Constitutional Contributions to Complex Tasks (Maze Solving)

The aforementioned physical properties of slime mold contribute to its incredible abilities, such as solving mazes. One structure related advantage for the maze example is the oscillatory expansion of the lattice structure which operates by propelling towards positive environments or food sources including through constricted spaces (Ito et al., 2011). In the early 2000’s, Japanese researcher Tosyki Nakagaki first demonstrated plasmodial slime mold solving a maze. Shown in figure 6, he placed oats at the entrance and exit, and the slime mold perfectly rearranged its lattice structure to find the quickest path, oscillating its thickness and branching in the previously discussed hexagonal form (Oettmeier et al., 2017). While there are other contributing factors to how slime mold can do this, its basic form is uniquely built to adapt in these environments.

Fig. 6. In the maze solving experiment with Physarum polycephalum the following analysis was completed (Nakagaki et al., 2000). In a, the slime mold structure is shown before being given time to solve the maze with the blue lines being the shortest available path. In b, four hours later the slime mold has shrunk and is exploring path options. By c, four hours after the b image, the shortest path of the maze has been identified. D demonstrates the frequency of the slime mold in every given coordinate.

Physical Response to Diet and Nutrition

Slime molds survival depends on strong nutrition. The organism also manages its diet without a specialized center to navigate its environment, rather its physical response allows it to make complex nutritional decisions. Particularly, slime mold, as many organisms do, aims to balance carbon and nitrogen-based nutrients. It has been identified that under steady nutrition slime mold grows very steadily and is relatively sedentary whereas it moves a few centimeters per hour when it is looking for better nutrients (Dussutour et al., 2010). When faced with equal nutritional opportunities, the slime mold alters its physical network as discussed (Ito et al., 2011). However, in an even more advanced analysis, it can alter its physical shape to balance its ratio of nutrients. In a test of various nutrient ratios, it was found that slime molds migrate and best live with a ratio of two parts protein to one part carbohydrate. The migration distance, correlating to the growth or energy expenditure of slime mold, showed the least migration distance after feeding on the 2:1 ratio. Further, other properties were observed at different ratios, like the splitting of slime mold with an exceedingly high protein concentration, or mortality of the organism with high ratio of carbohydrates (Dussutour et al., 2010). The minimization of migration distance also lowers the physical exertion of the slime mold.

Fig. 7. When slime mold was tested at various ratios of proteins to carbohydrates, it was found that the 2:1 ratio was heavily favored (Dussutour et al., 2010)

Spores and Mutualism

One other physical property of slime mold is its spores, that bare resemblance to the spores of fungi. It has been discovered that tiny beetle species in Japan, who were known to feed on slime mold, have spores covering their bodies. It can be inferred that the beetles are in a mutualistic, physical relationship with slime mold to spread the spores, which then are able to germinate. Spores were found on 88.8% of adults of these species and it is hypothesized that the slime layer of the slime mold allows them to stick on to the beetles as they feed (Sugiura et al., 2019). This is yet another important property of slime molds assembly, and strongly demonstrates its strong reliance on structural integrity, mechanics, and interactions with its surroundings. The above has and can continue to be applied towards design solutions in physics by mirroring the assembly of aspects of slime mold like its lattice structure.

External Stimuli

In order to accomplish complex tasks such as solving a maze, Physarum polycephalum must effectively navigate and utilize its spatial environment (Oettmeier, Fessel and Döbereiner 2022). In fact, the plasmodium of Physarum polycephalum has the capacity to use space in the most efficient way possible because it could react to its environment (Shirakawa et al., 2012). The plasmodium considers external information to pursue the optimal behavior. There exists an intricate relationship between Physarum plasmodium and external stimuli since they dictate the slime mold’s behavior (Shirakawa et al., 2012). In the absence of that outer data, the plasmodium would grow in all directions without a specific goal (Durham & Ridgway, 1976). This section will present how external stimuli outline P. polycephalum’s behavior.

Phototaxis

Physarum plasmodium is reactive to light, in other words it has phototaxis. More specifically, phototaxis is a physical response to light stimuli. A positive phototaxis causes the cell to move towards the light source, whereas a negative phototaxis causes it to move away (Mead, 2008). The experiment conducted by Hato and co-workers in 1976 still illustrates correctly Physarum polycephalum’s phototaxis since the literature does not contradict the results.

It is interesting to note that Physarum polycephalum is the most studied type of slime mold since it is easy to grow in a laboratory (Oettmeier et al., 2022). Therefore, there has been a good understanding of the basics of P. polycephalum since the 1970s and research papers from that decade are still relevant (with some exceptions) such that papers from 2024 still quote 50-year-old research.

The results can be interpreted as follows; Physarum polycephalum behaves differently according to the wavelength of the light illuminating its environment. In other words, the slime mold’s phototaxis depends on the wavelength (Hato et al., 1976). Physarum polycephalum’s plasmodia were tested with blue light (500nm), red light (650 nm) and far-red light (720 nm) in controlled chambers. The cell’s response in those three specific scenarios is illustrated in Figure 8 where the new motive force during light stimulation is illustrated by a horizontal line. The motive force refers to the sum of all the forces applied on the plasmodium forcing it to move in a certain direction. The motive force of the plasmodium in response to red light shifts from zero to +9 cm H2O which means the plasmodium is moing towards the light source. Initially no light is emitted, the oscillations are then not modified by phototaxis, and the motive force is equal to zero which represents the natural state of Physarum polycephalum (Hato et al., 1976). After 5 minutes, light stimulation begins which is shown by the down pointing arrow in Fig. 8. The plasmodium then either migrates toward the light source resulting in a positive motive force like in Fig. 8b or away from it like in Fig. 8a and c.

Fig. 8. Dynamoplasmograms of plasmodium’s phototaxis in reaction to blue light (500 nm, Fig. 5a), to red light (650 nm, Fig. 5b) and to far-red light (720 nm, Fig. 5c). There is positive phototaxis only in situation b. The down pointing arrows represent the moment light was projected into the chamber, and the up-pointing arrows indicate the moment the light stimulation turns off (Hato et al., 1976).

The initial and the final state depict the light’s influence since there is an obvious change in the motive force when light is projected. Only, that change does not lead to further modifications. After 10 to 15 minutes, when the light stimulation turns off, the plasmodium regains its natural state and the motive force returns to zero (Hato et al., 1976). These results imply that light triggers a physical response but is not a source of energy because once removed everything goes back to its initial state. There is no energy transfer meaning light does not act as an energy source (Hato et al., 1976). Even if the intensity is increased, the results stay the same. At a certain point, the response reaches a plateau, indicating that increasing the light intensity further does not induce a stronger motive force or cause the plasmodium to move faster or farther (Hato et al., 1976). Therefore, beyond a certain intensity, the plasmodium is incapable of reacting accordingly to the severeness of the situation. It would not be able to move specifically faster to avoid deadly stimuli which can lead to its death.

Sporulation

A common fact about Physarum polycephalum is its required time of starvation, which provokes sporulation to assure survival of the species. During sporulation, plasmodium usually seeks for light, and it must be illuminated for a couple of hours to assure an effective process (Schreckenbach et al., 1980). However, a study showed that sporulation can happen in dark environments (Renzel et al., 2000). In that case, calcium ions and malate are involved in the process to make up for the absence of photons. Considering the general case where light does play a role in sporulation, certain wavelengths appear more favorable; however, there is no consensus on which specific wavelength is optimal. Sporulation is activated by specific photoreceptors which detect certain wavelengths (Shirakawa et al., 2012). In a study using a white mutant strain of P. polycephalum which has the same characteristic as the yellow P. polycephalum only it lacks the yellow pigmentation, sporulation was found to be induced by illumination of blue light (450 nm) and red light (over 600 nm) (Schreckenbach et al., 1980). In another study, red light (650-700 nm) and blue light (350-580 nm) seemed to provoke sporulation as well, but a precision was made for the blue spectrum. Blue light (400-470 nm) specifically was found to have the opposite effect and not trigger sporulation (Hato et al., 1976). The precision of the optimal wavelength remains unclear because of some contradictions in the literature. Nevertheless, wavelengths are a piece of the puzzle to understand the behavior of P. polycephalum.

Thermotaxis

Physarum polycephalum also reacts to temperature change. Its reaction time is about 30 minutes implying there is a delay between the moment the temperature is changed and the physical response of the plasmodium. That lag is possibly caused by shock (Tso & Mansour, 1975). When put in an environment with different temperature regions, the plasmodium will not only, stop growing towards the extreme (too cold or too hot) region, but it will also seek and focus its growth towards a more favorable environment. This phenomenon is clearly illustrated in Figure 9 where the slime mold’s leading fronts are much thicker, and its tails are much thinner (Durham & Ridgway, 1976).

Fig. 9. Migration of plasmodia towards warmth. Notice the contrast between bulbous fronts and thin tails (Durham & Ridgway, 1976).

The plasmodia growth is concentrated in the leading fronts since they approach a more favorable temperature region, and the tails are of no interest for the P. polycephalum since they lay in an unfavorable temperature region. Thermotaxis is then an essential key for slime mold’s survival since it gives the plasmodium the information to where to concentrate its growth (Durham & Ridgway, 1976). P. polycephalum’s plasmodia was found to seek hotter environments. The optimal temperature for plasmodia growth is (29 ±1) °C (Tso & Mansour, 1975). The hotter the temperature, the faster plasmodia grow. As temperature rises, the plasmodium movements increase until the temperature reaches 30°C. It can be considered as a peak of stimulation. Once the temperature exceeds 30°C, a decay in the plasmodia motility can be observed. The final turning point is around 35°C where there are no more movements recorded (Tso & Mansour, 1975). In Figure 10E, none of presented temperature zones are optimal. On the left side, the temperature is around 27°C and on the right side, 34°C. Plasmodia converge towards the middle to avoid extreme temperatures that are unfavorable to growth. On that middle line the temperature is around 29°C which is in the ideal temperature rage for P. polycephalum.

Fig. 10. Migration of four individual plasmodia in a temperature gradient, A, B, C, D are photographs at 0,45,60, and 90 minutes after a gradient of 10°C is applied. The left side is 19°C and the right side is 29°C. In E, the migration of the organism between two extreme temperatures at the end of 2 hours (22°C on the left and 37°C on the right; 29°C being the preferred temperature) (Tso & Mansour, 1975).

An interesting fact to note about P. polycephalum is that it never does a U-turn like a car would do. When confronted with a dead-end or to unfavorable conditions it will engage other ends to take the lead into finding another path (Tso & Mansour, 1975). Any area can become the leading front of the plasmodium’s movement. It resembles in a way to a metro where there are two opposite control cabin that alternate in taking the lead. Just like metros, slime mold never turns on itself.

Brief Overview of Other Taxis

Physarum polycephalum presents chemotaxis meaning it changes its morphology in presence of chemicals (Durham & Ridgway, 1976). Chemical substances must have a minimum concentration to be detected. There is also a maximum concentration beyond which no significant changes occur. The same phenomenon has been detected in phototaxis with the intensity of light (Hato et al., 1976). 1% glucose, galactose and mannose are considered attractant chemicals which lead to positive chemotaxis and increase the plasmodium frequency. As for 1% sucrose and ribose, they are repellant chemicals and have the opposite effect (Durham & Ridgway, 1976).

The plasmodium has electrotaxis meaning it reacts to electric fields and magnetotaxis implying it behaves in response to magnetic fields (Shirakawa et al., 2012). There exists a relationship between magnetotaxis, the weight of the plasmodium and the intensity of the magnetic field leading to different orientation (Shirakawa et al., 2012). Plasmodium reacts to magnetic fields of intensity of 1-10 mT which is lower than the intensity on Earth suggesting that geomagnetism affects P. polycephalum. In nature, geomagnetism is useful for directions and in a cell population, magnetotaxis is used to help cells interact with each other which leads to better exploration of the environment since they are guided by the same force (Shirakawa et al., 2012).

External stimuli assure the cells survival since it keeps it away from unfavorable conditions by negative taxis and indicates where it should grow by positive taxis. When the plasmodium senses a stimulus, the inner response results in a modification of the oscillation frequency and amplitude where it happens (Dussutour & Arson, 2024). If the stimulation is considered as positive, the frequency increases locally. The signal then spreads throughout the cell, which leads to physical changes (see Fig. 7), since the plasmodium adapts itself to the environment (Dussutour & Arson, 2024). With the positive taxis the slime mold will in that case be able to direct itself towards a specific goal such as food (Durham & Ridgway, 1976).

Locomotion

Mechanisms of Locomotion

As previously mentioned, Physarum Polycephalum (referred to as the true slime mold) is infamous for its apparent basal intelligence where it can solve mazes and various logic problems relating to distance. In this section, we discuss the true slime mold’s locomotion which enables this behavior.

Slime mold has a microplasmodial stage at approximately 100micrometers in size, where it can combine with other microplasmodiums to create the macroplasmodial stage, although this transformation can also happen through the growth of a single amoeboid stage over time. The microplasmodium travels like a cell, using amoeboid locomotion made up of constant depolymerization and repolymerization of the actin cortex, which creates filopodia and lameliopodia: extending pseudopods that the cell uses like a treadmill. Combined with an oscillatory flow of the cytoplasm, the amoeboid stage, oriented towards a front, moves (Oettemeir et al., 2022). Accordingly, internal oscillations and a similar locomotion method are used by the macroplasmodial stage (Alim et al., 2013). This stage uses blebbing; rather than expanding the cell boundary by pushing on it with actin polymers, cytosolic streaming creates a pressure-inducing flow, forming blebs (Charras G. and Palush E., 2008), which create the organism’s growth fronts visible in Figure 11.

Blebbing then completes the motion forward. This is done in a similar manner to amoeboid locomotion: by construction of an extension to the actin cortex at the front of the blebs (Charras G. and Palush E., 2008). Although this is like the mechanism responsible for amoeboid locomotion, in which actin at the rear of the cell is depolymerized and then diffused to the front as part of the actin treadmill, there is a key difference. The macroplasmodial stage uses shuttle streaming – a rhythmic, back-and-forth flow of plasm - to transport the actin throughout its network, rather than diffusion. This is because diffusion becomes a progressively less efficient transport mechanism as the cell grows from an amoeboid, microplasmodial stage, to the macroplasmodial stage (Oettemeier et al., 2022).

Fig. 11. Physarum Polycephalum growth fronts. Notice the oozing gel at the end of the veins bordering the growth front [Adapted from Oettemeier et al., 2018].

The slime mold’s network of veins, seen in Figure 11 and Figure 12 is essential to its survival. Along with the shuttle streaming of actin to the front of the organism, the endoplasmic flow through its tubes also transports metabolites and chemical signals (Alim et al., 2013). The contraction of various tubes in the network is the mechanism by which fluid flow is generated within the cell. These contractions are inherently linked to P. Polycephalum’s cellular respiration; as cellular respiration is inhibited, these contractions slow down until they eventually (and irreversibly) stop (Avsievich et al., 2017).

Fig. 12. Another view of Physarum Polycephalum and its tubular network and growth fronts, characteristic of the macroplasmodial stage (Oettmeier et al., 2022).

Tube Structure

In addition to the macro- and microplasmodial stage, there has been a 3rd identified morphology of Physarum Polycephalum; an in-between structure, the mesoplasmodia (Oettmeier et al., 2018). mesoplasmodia contains veins like the macroplasmodia, but has thinner growth fronts and an unorganized tube network. These tubes have the same structure as those in the macroplasmodial stage, with both sharing the same porous outer surface as the outer surface of the amoeboid stage. Thus, the amoeboid stage is used to model the behavior of a single tube within the meso- and macroplasmodia.

Each vein consists of two parts: a harder, more viscous, porous, outer ectoplasm, and the inner, less viscous, endoplasm. The ectoplasm is enmeshed with a helical cytoskeletal actomyosin network (see Figure 13). The actomyosin cortex contracts the organism’s tubes, enabling shuttle streaming. The pattern of tube contractions is thought to be controlled by calcium oscillation in each tube (Whiting et al., 2015). Each plasm contains a different form of actin; G-actin and F-actin for the endoplasm and ectoplasm respectively (Oettmeier et al., 2018). G-actin is monomeric, and F-actin is polymeric, and research shows that the transformation of G-actin to F-actin corresponds to transformation from endoplasm to ectoplasm (Isenberg & Wohlfarth-Bottermann, 1976).

P. Polycephalum’s tubes also possess conductivity, conserving ~90% of input DC voltage. Experimental results using “Physarum Wires” show that its veins can act as low pass filters, filtering out all DC voltage signals of frequency higher than 19KHz (Whiting et al., 2015). These wires were tested in various uses: connecting circuits, conducting high current waveforms, transmitting digital and analog data in an Arduino circuit, and as voltage dividers (see Figure 14) (Whiting et al., 2016). What is attractive about the notion of Physarum wires is that they self-heal and hold the potential of being electrical wires immune to physical trauma.

Fig. 13. Endoplasm and ectoplasm in P. polycephalum’s veins. Less viscous endoplasm flows through the porous (see (b)) ectoplasmic vein boundary. Notice the helical organization of the actomyosin network within ectoplasm in (c) (Oettmeier et al., 2018).

Fig. 14. LCD screen controlled by microcontrollers and connected via Physarum wires. Physarum wires were grown by letting two colonies sitting on food on metallic plates grow a tube to connect to each other, forming a wire with a high resistance on the order of MΩ (Abbasian and Maddahian 2024).

Tube Network Organization

The structure and mechanics of P. Polycephalum’s tube network poses a design challenge for which the organism, lacking a central nervous system, must find a simple solution: how will it effectively transport the nutrients, polymers, and chemical signals necessary for its survival? To investigate the slime mold’s coordination of its tubular network, studies were first conducted on understanding the mechanism of endoplasm transport within its veins. Cross-sectional analyses of the tubes showed that, over time, P. Polycephalum’s tubes undergo regular contraction and relaxation cycles (Alim et al., 2013). From this regular contraction and relaxation of tube radius, Alim et. al managed to model the flow direction and velocity across space and time throughout an entire slime mould’s network. This was done using the Navier-Stokes equations, with assumptions made about tube outer radius and the cytoplasmic fluid being calculated as water in terms of viscosity and incompressibility. From these calculations, an accurate model of flow direction was produced, verified in Figure 15 by aligning theoretical predictions of fluid-flow reversals and measured values (Alim et al., 2013).

Fig. 15. Comparison of measured endoplasmic flow and direction to model predictions. Model predictions correspond to dashed lines, predicting moments of flow reversal and its direction. Fluid flow to the left of the measured vein (distance = 0) are colored in blue, and flow to the right is colored in red. Alim et al.’s model accurately predicts moments of fluid reversal within the measured P. Polycephalum tube.

With an established regular contraction of its tubes, Physarum Polycephalum is left with four options for organizing its tubular network, which Alim et al. reduce to one characteristic by order of elimination; tube radius distribution is too random throughout a network, with the amplitudes of tube contraction suitably following. The contractile period for each tube throughout the network remains essentially constant, leaving only the pattern of contraction in space as the mode with which P. Polycephalum coordinates its transportation network.

Two key observations follow from this deduction; tubular contractions follow the form of a peristaltic wave, the wavelength of which spans the entire organism. Peristaltic waves are the waves of contraction spanning a tube as something is pushed through the tube; these waves are very common in nature; they’re found in digestive systems (urinary and intestinal tracts) as well as in the locomotion of worms (Abbasian et al., 2024; Yekutieli et al., 2009). Alim et al. apply this concept to the entire network of P. Polycephalum’s tubes. This is supported by observations of the contraction phase, measuring each tube’s contraction relative to its maximum, across the organism over time, as shown in Figure 16. The wavelength of this wave was analyzed at three varied sizes of independent plasmodial networks, and the researchers found that, regardless of size, the wavelength stayed relatively the same, measuring around the same size as the organism’s length (Alim et al., 2013).

Fig. 16. Wave-like spatial pattern of relative tubular contraction over time. Relative contraction is shown via a color spectrum, from 0 to 2π. Notice how the top horizontal layer of the network in (A) shifts from green (π), to yellow (~1.5π) to red (~2π) over time, and how red-black phase (~2π) travels upwards from the middle of the network at t = 0s, to the top of the network at t = 36s. Difference in wave direction was experimentally observed between rounder organisms (isotropic) and more asymmetric (anisotropic) organisms: the wave direction was somewhat random in isotropic networks, while it oriented itself along the longest axis in networks like (B) and (C).

It seems that a peristaltic wave spanning the entire body length of Physarum polycephalum is the organism’s design solution for organizing its network of tubes. The elegance of this solution is found in how it optimizes the travel of particles in the network at the same time as organizing the network. This can be done by comparing travel distance and phase difference (in terms of relative tube contraction) between tube contractions following random patterns and those following a peristaltic wave. Figure 17 shows how the theoretically optimized network organization closely matches that of P. polycephalum:

Fig. 17. Optimized vs. random particle displacement and network organization (measured in phase-difference between neighboring tubes). The phase of contraction of each tube is shown via color and means particle displacement is normalized by L the entire length of the network. Network (B) and the dashed red line correspond to the random network, while network (A) and the solid red line correspond to the theoretically optimized network. Notice the similarity between the theoretically optimal network organization in this figure and the slime mold’s network organization in figure 16A (Alim et al. 2013).

In the case of a peristaltic wave applied to a network of tubes, having multiple periods throughout the organism creates barriers for transport. Periods in a peristaltic wave are separated by areas of no flow followed by a reversal of flow, which would hinder the travel of slime mold’s flowing endoplasm. Thus, the slime mold’s tube-organizing peristaltic wave’s wavelength is also a physical design solution for optimizing the transport of plasm, nutrients, chemical signals, and actin polymers.

Conclusion

In conclusion, the navigation of a maze by plasmodial slime mold is possible because of external stimuli, which initiate a blebbing locomotion mechanism similar to amoeboid movement, all thanks to its structural integrity. The research concerning Physarum Polycephalum revealed how the physical properties open the possibility to easily navigate through a maze.

Firstly, the coenocytic structure allows slime mold to survive destruction and allows multiple cells to combine and become one. Furthermore, the tubular vein network forms a lattice structure that has the capability to transport nutrients efficiently while maintaining physical integrity. The format of the lattice structure is a hexagonal shape in optimal environmental conditions and a polygonal shape (randomized) in poorer conditions. This structure is crucial in the formation of oscillatory movements to drive the slime mold through mazes. More importantly, it should be noted that diet has the most significant role in navigation since optimal growth may be attained from a 2:1 protein-to-carbohydrate ratio. Also, slime mold's biggest motivation to get into a state of movement is principally due to the external stimuli projected from food sources.

External stimuli also play a significant role in the explanation of the displacement of the slime mold through a maze. In fact, the influence of light and heat, as well as electric and magnetic fields, on slime mold behavior is of utmost importance for understanding the physics of the species. Phototaxis varies with wavelength, influencing sporulation and movement, while thermotaxis guides growth, with an optimal temperature of 29°C. Slime mold also uses the cues of magnetic and electric field for orientation and exploration towards nutritious sources. Overall, P. polycephalum adapts to its environment continuously in order to survive. It will interact with external stimuli, employing it to find guidance towards favorable conditions for survival.

There were also key findings on locomotion and material transport. Indeed, the slime mold moves based on amoeboid movement combined with peristaltic waves. This basis for movement is located in a tubular network that transports cytoplasm, metabolites, and signals. A certain coordination between the contraction of the network and its material transport assures the organism's survival since the pattern, comparable to peristalsis, guarantees a smooth flow of nutrients and cytoplasm. The system also contains veins that are self-healing and act as low-pass filters. These tubes use shuttle streaming to solve complex problems by the transportation of actin and other materials.

Overall, slime mold solves mazes thanks to its physical properties. More specifically, slime mold needs to find nutrients and maintain structural strength, triggering a structural growth that shows a hexagonal lattice. Moreover, it uses external stimuli to redirect itself and optimize its growth towards a more favorable environment, precisely one that is without light and with a temperature of 29°C. Lastly, the motion is thanks to its blebbing locomotion, which, similar to amoeboid locomotion, uses actin depolymerization and repolymerization. The recycling of actin from one end of the slime mold to the other, along with the transport of nutrients, metabolites, and chemical signals, is done using its tubular network, which displays design solutions for particle transport by applying a peristaltic wave of a specific wavelength to the network.

This is how slime mold, a brainless organism, can navigate through a maze across the most efficient path.