Patterns and Proliferation: The Geometry Behind Fungal Systems 

Edison Luke, Felix Gamper, Lea Bleibel, Vasily Ignatov

Abstract

Fungi are masters of optimization in the mathematical world, utilizing intricate geometry phenomena, following complicated patterns, and creating complex networks to maximize their survivability and growth. In this paper, we first investigate fungi’s elegant solution of maximizing fungal growth and spore production through symmetry breaking. By “breaking symmetry,” or introducing slight asymmetries during development, fungi develop gill patterns throughout its maturation process, allowing for greater surface area for its reproduction, overcoming the challenge of efficient and minimal energy-cost spore dispersal. Next, fungi follow phenotypic change pattern cycles, called stochastic switching, to optimize growth in rapidly changing environments or new niches. Similarly, some types of fungi follow population dynamic patterns to ensure a source of nutrition; in particular, pathogenic fungi follow population patterns that can be modelled by the SIR model, and they continually evolve new strategies, such as an ever-changing genome, to ensure plant hosts remain susceptible to infection to ensure a constant nutrition and growth. Finally, through forming symbiotic networks with plants, fungi can maximize nutrient sourcing and selective event resiliency by optimizing how they form their networks. By connecting with nearby hosts and fit hosts, fungi can form networks that highlight resiliency to random selection by ensuring redundancy in connections, while ensuring efficient nutrient distribution. Through optimization of network topologies, fungi maximize growth through constant nutrient sourcing, while its hosts benefit from being able to distribute nutrients over an ecosystem, thus increasing the overall health of a fungal niche. 

Introduction

Studying fungal systems reveals a fascinating world where geometry and biology intersect. Fungi, with their diverse forms and remarkable adaptability, exhibit intricate patterns that are aesthetically pleasing and crucial for their survival and proliferation. Patterns and proliferation intertwine to create a resilient and thriving kingdom of life. This paper delves into the mathematical principles governing fungi’s growth and structure. 

One aspect that explains fungi patterns is the phenomenon of symmetry breaking. In the inanimate world, symmetry signifies stability, but for fungi, breaking symmetry is a vital process that enables growth and adaptation. The geometric arrangement of gills in mushrooms is another key example of this phenomenon. Moreover, the precise patterning of gills, governed by mathematical principles, maximizes the surface area for spore release. This entropic increase allows for reduced energy costs. As mushrooms mature, their gills arrange themselves to maximize spore production, demonstrating a formulaic elegance in how nature orchestrates growth.  

 Similarly to how symmetrical breaking is vital for the fitness and growth of individual fungi, random phenotypic distributions contribute to the survivability and proliferation rates of fungal colonies. Instead of creating complex signaling and response pathways to defy environmental stress, certain fungi simply bet on every trait in hopes that one will be fit for the new environment. This is a simple use of probability that reduces energy costs, while being a robust solution to fitness issues that arise from genome homogeneity usually present within populations with limited locomotion capacity. Moreover, certain opportunistic fungi use this mechanism to great effect to survive and grow within hosts. Most members of a certain fungal colony cannot survive a dog’s internal temperature of 38.3-39.2°C, but random phenotypic distribution ensures that some cells will optimize their growth within the host. 

Furthermore, in contrast to the homeostatic and organized growth of some species, fungal proliferation may also be virulent and opportune. The spread of fungal disease can be characterized by models using differential equations, such as the Susceptible-Infected-Recovered model. Simulations of this model reveal the importance of certain fungal strategies, including varying behavior in primary and secondary hosts, adopting complex reproductive cycles involving the production of sexual and asexual spores, and maintenance of large genomic sequences promoting mutation. These dynamics reveal how lifecycle complexity and genetic diversity, recurring motifs in fungal success, align with mathematical patterns, particularly in phytopathogen spread and adaptation.  

Finally, the hidden networks of fungal hyphae, known as hyphal networks, are analyzed through the lens of graph theory. These networks span vast areas and are crucial in nutrient distribution and communication within ecosystems. The largest known fungal network, created by the Armillaria gallica, spans over 91 acres, highlighting these systems’ extensive reach and complexity. By forming symbiotic relationships with plants, fungi enhance nutrient distribution and communication across vast distances, benefiting entire ecosystems. By using a branch of mathematics called graph theory, information regarding a fungal network’s health and survivability can be made by analyzing the patterns of growth for network formation. For example, a fungal network that forms several symbiotic connections with a single tree reveals that the fungus has can distribute nutrients efficiently, while fungal networks that form several connections with multiple trees exemplify resilience to selective events through network redundancy. Overall, through analyzing the geometry of fungal networks, specifically by analyzing the manner in how fungi form networks, much can be learned about the growth and resilience of fungus and their ecosystems. 

Symmetry Breaking  

Fungi are an impressively diverse kingdom of life, known for their remarkable resilience and ability to thrive in various environments. Effectively, fungal structures and development patterns are not present solely for aesthetic purposes; rather, they are great examples of nature’s ingenious design solutions for efficient and low energy use in the production of spores. 

A fundamental concept to cover in the pursuit of understanding developmental fungal morphology is the phenomenon of symmetry-breaking. Cells, organs, and organisms exhibit some form of symmetry, which establishes their developmental axis and morphological pattern. The symmetry-breaking phenomenon entails the transition from symmetrical to asymmetrical states following energy consideration, often used to explain fungal phenomena like cell polarity and directional growth (Homblé & Léonetti, 2007) [Link to physics paper]. To understand symmetry breaking in simpler terms, Van der Gucht and Sykes (2009) propose a case study of a clown balancing on a ball: When the clown stands on top of the ball (Fig. 1 A), the system has cylindrical symmetry. However, this state is unstable: The slightest perturbation will cause the clown to fall in some direction (Fig. 1 B), breaking the cylindrical symmetry. Now, if the ball is slightly flat on its base, it gives more stability to the clown. This state is metastable (a system that appears stable but can transition to a more stable state if slightly disturbed): The clown can make small excursions safely (Fig 1. C, D), but if he moves too much (i.e., generates too much “noise”), he will eventually also fall (Fig. 1 E, F). 

Fig. 1 Illustration of symmetry breaking with a clown standing on a balloon (van der Gucht & Sykes, 2009). A The clown is in unstable equilibrium and the situation is symmetrical. B In symmetrical state, the clown falls due to slight perturbation, C and E Same situation as A, but the balloon is slightly flat on its base (the system is metastable). D The slight perturbation of the clown does not make him fall due to the metastable state. F In a metastable state, the clown falls due to strong perturbation. 

Similarly, fungi are initially in a symmetrical state that, much like the clown on the ball, can be unstable or metastable, depending on the conditions, but can develop asymmetrically due to internal fluctuations and environmental triggers. This explains the impact of symmetry breaking on fungal growth and adaptation (van der Gucht & Sykes, 2009). In nature, this concept can be observed in gilled mushrooms and, more particularly, in comparing gill configurations in fruit bodies and fully developed gilled mushrooms (Fig. 2). The gills in mushrooms allow the production of a large number of spores due to their specialized structure. Gills are thin, layered structures found on the underside of the mushroom cap. They are covered in tiny structures called basidia, which produce and release the spores. The large surface area of the gills and the presence of numerous basidia increase the number of spores that can be produced. Each basidium contains four spores, so when millions of basidia release their spores, it results in a high number of spores being produced overall. 

Fig. 2 Different arrangements of gills. (A), Single array of gills in Marasmius siccus (Agaricales). (B), Primary, secondary, tertiary, and quaternary gills (lamellae and lamellulae) in Lactarius subplinthogalus (Russulales). (C), Forked gills in Russula variata (Russulales). Photographs courtesy of Michael Kuo (Fischer & Money, 2010). 

In the presence of gills, the surface area of fungi increases, allowing more efficient spore production [Link to physics paper]. A commonly observed pattern is the addition of lamellulae (short gills) every second gill. Interestingly, this pattern is particular to gilled fungi with larger cap sizes, in other words, more mature fungi (Fischer & Money, 2010).  

Gill arrangement model 

As mushrooms mature, their gills precisely arrange themselves, radiating from the central stipe (stem) in a pattern that maximizes their spore-producing potential. Each gill is a thin, blade-like structure that houses the hymenium, the specialized tissue responsible for spore generation. This arrangement is not random; instead, it is carefully governed by the circumference of the stipe and the interplay between gill thickness () and the spacing between each gill(s). 

The maximum number of gills depends on the stipe radius (), revealing a formulaic elegance in how nature orchestrates growth. As the mushroom cap expands, the gills’ surface area increases correspondingly, forming a sprawling network that serves as an efficient platform for spore release. With these concepts in mind, the ratio of these areas can be derived from the formula:  

This ratio can be modeled by the above simplified formula involving the cap radius and the stipe radius  demonstrating the relationship between cap growth and gill area. As the cap grows, the gill-covered surface expands to accommodate more spore-producing tissue without excessive energy costs (Fischer & Money, 2010). 

Fungi adopt the patterns of gill arrangement optimize spore production are worth mentioning. In fact, the gill arrangement in fungi is an evolutionary adaptation that maximizes spore production and dispersal by increasing the hymenium surface area, which is the spore-producing surface of fungi. This increase can be simulated through the doubling model. It involves the addition of free-standing lamellulae (short gills) or the branching (bifurcation) of existing primary gills, effectively doubling the number of gills and the surface area. This increased surface area allows the fungi to reproduce more efficiently and spread their spores over a wider area, enhancing their chances of survival and colonization. This model is commonly adopted by mushroom-forming fungi (agaricomycetes) (Fig. 3).  

Fig. 3 Segment of a transverse section of a fruit body cap of Coprinopsis cinerea at a late stage of development (Trinci, 2019).  

Effectively, the patterning of gills arrangements, seen in nature, is essential to a gilled fungus’ efficient spore production by optimizing the space distribution and therefore the surface area of hymenium. 

Population Dynamics

While symmetry breaking increases spore production which contributes to the spread of fungal colonies to new niches, fungi also rely on other types of patterns to contribute to population dynamics. For example, patterns pertaining to phenotypic distribution help fungal populations optimize growth in new environments. These patterns are important factors that impact the dynamics (or change over time) of fungal populations.  

Stochastic Switching

Various organisms stochastically (or randomly) transition between phenotypes to ensure that specific cells are always ready for unforeseen environmental challenges. Unlike mutations which arise randomly and irreversibly change the cell’s genome, stochastic switching is reversible and only impacts the phenotype. Organisms with lesser locomotion capability, such as fungi, benefit from this survival strategy since it promotes phenotypic heterogeneity in genetically similar populations, thereby increasing the resilience of the whole colony. This allows fungi to occupy new niches when their surrounding conditions change (Acar et al., 2008). For instance, the yeast Cryptococcus neoformans utilizes stochastic transitions to infiltrate and thrive within animal hosts which allows it to attack the respiratory tracts, central nervous systems, and eyes of cats, dogs, or cattle (Gull, 2023). Interestingly, fungi like the yeast Saccharomyces cerevisiae can even modify the frequency of phenotype switches depending on their surroundings. Switch rates can be categorized as fast when the stochastic transition between phenotypes is faster than the switch rate of the surroundings, and slow when the inverse is true. Both categories of rates provide distinct benefits depending on the nature of environmental stress. Understanding the patterns that arise from the random process of stochastic switching is critical for studying the survival strategies of fungi on the population level (Acar et al., 2008). 

Switching and Growth Rate

In an experiment conducted by Acar and colleagues, phenotypic distributions, and growth rates of fast and slow-switching populations of S. cerevisiae were compared in different media. Firstly, a bistable gene network (or a network with two equally expression states) of galactose utilization was chosen so that the cells could stochastically express a basal (OFF) and 100-fold increased (ON) chemical pathway activity depending on surrounding galactose concentration. The rate of phenotypic transition was engineered such that there were two samples with fast and slow rates. The activity of the galactose utilization pathway was illustrated by enzyme GAL1-mediated yellow fluorescent protein (YFP) synthesis. GAL1 one was also modified to be solely responsible for uracil nucleotide production. This setup coupled YFP production to growth rate via uracil synthesis.  

Fig. 4 Effect of switch rate on YFP amount in non-selective substrate. (A) Fast switchers. (B) Slow switchers (Acar et al., 2008).    

First, Acar et al., (2008) observed the growth of fast/slow switchers in a non-selective substrate. As expected, the growth rate of the two populations was the same. However, the phenotypic distribution of the colonies followed two distinct patterns. Both fast and slow switching populations had a bimodal distribution of the ON and OFF phenotypes, but the behavior of the modes was different. As seen in Fig. 4, the peaks of the slow switchers are equal in height, which indicates that ON and OFF expression are the same. In contrast, the right peak of the fast-switching colony is taller than the left, illustrating that ON expression is more pronounced in this data set. These observations demonstrate a pattern which shows that slow-switching populations are more phenotypically stable than fast-switchers. Although both samples have equally random probabilities of being either ON or OFF, the speed of the transitions impacts the distribution at any given point in time. In non-selective conditions, this conclusion has no implications on growth rate or the fitness of the population. However, in environments where only one of the phenotypes is favorable, stability plays an important role in growth. Fast switchers may have a higher expression of favorable characteristics at any one point but can just as easily switch to an unfavorable distribution. Inversely, the slow population will have an equal distribution throughout, resulting in more steady growth. This notion is explored further in the subsequent experiment by Acar and colleagues, who introduce selectivity to the substrate (Acar et al., 2008).     

Indeed, two distinct environments were created: one lacking uracil (E1) and the other with a mixture of uracil and 5-fluoroorotic acid (E2) that is converted into a toxic intermediate in the presence of uracil synthesis enzymes. Given this, the ON phenotype would thrive in E1 since there is lack of uracil, but OFF would grow in E2 since uracil production is unfavorable. Then, the fast and slow switching samples were incubated in the two substrates and transferred into inverse environments where growth rate was observed via YFP concentration. The data demonstrated that after transition to a new substrate, fast switchers have increased growth rates compared to slow switchers during transition to steady state, but slow switchers excel at growth during the steady state itself. These results, seen in Fig. 5, follow the logic of our prior observations since fast switching yeast cells transition quickly between phenotypes such that the distribution of ON and OFF cells can more quickly favor either of the phenotypes as opposed to slow switchers. In contrast, lower transition frequency allows cells to decrease the fluctuation in phenotypic distribution pattern, thereby keeping a steady growth rate in the long term. However, although different transition frequencies impact the growth of S. cerevisiae, it is not yet clear which, if either, strategy is more viable for the population (Acar et al., 2008).  

Fig. 5 Yeast cells growth rates of fast and slow phenotype switchers after environment switch. (A) Transition from E2 to E1. (B) Transition from E1 to E2. (Acar et al., 2008).   

Switching Frequency and Survival

As previously postulated, rates of stochastic transitions influence the growth dynamics of S. cerevisiae by modifying phenotypic patterns of the population. However, this mechanism is also a vital but simple tool that allows cells to optimize growth and fitness depending on environmental conditions without relying on complex signaling and sensing pathways. The stochastic nature of the switches combined with the ability of the cells to modify the transition rates yields a cost-effective method of preparation for changes in the environment (Acar et al., 2008). The mechanism of this design is illustrated in the next set of experiments by Acar and colleagues, where fast and slow switching cells are subjected to fast and slow changes in the environment. As expected, fast switchers were shown to have higher average growth rates and fitness in rapidly changing environments, while slow switchers thrived when the surrounding changes were slower. In nature, changes in the surroundings can occur on vastly different time scales which can complicate adaptation and growth optimization. Random mutations, which are the universal fitness strategy for organisms, are too slow to be effective in rapidly changing environments. Yeast cells combat this challenge by not only randomly switching their phenotypes such that an optimal fraction of the population is always ready for unforeseen stress but also modifying the rate of transition so that it is like the frequency of the environmental changes.  

Mathematics of Fungal Epidemics

Fungi’s chemoheterotrophic lifestyle often involves utilizing resources derived from plants and animals. Sometimes, fungal evolution drives the extraction of resources at the cost of other organisms’ health. Phytopathogenic fungi, which cause diseases in host plants, can spread rapidly through population. While the physical transfer mechanism can be quite complicated, approximating the likelihood of the event allows the modeling of pathogen spread at large scales. [Link to Physics Paper] 

The Susceptible-Infected-Removed (SIR) epidemiological model predicts the spread of pathogens using three qualitative classes: S (susceptible hosts), I (infected individuals), and R (recovered individuals). Through three ordinary differential equations, it computes the rate of change in these populations over time: S'(t), I'(t), and R'(t). The simplified SIR model can represent this spread with three differential equations:  

Where rate constant α is the per capita disease transmission rate and rate constant 𝛳 is the per capita rate of removal, where the removal consists of immunity, recovery, isolation, or death. These equations emphasize that fungi require a basal number of reinfections to survive.  

Analyzing these equations, a quantity known as the basic reproductive number (R₀) can be obtained. R₀ is the ratio of the proportionality constants α/𝛳. For a disease to spread R₀ ≤ 1, that is, for each infected individual at least one new individual must be infected. Yet at some point in a fungal epidemic, the underlying assumption that individuals do not pass on immunity begins to break down. As a result, the true number of infections per infected person differs, known as the effective reproductive number. As time goes on, fungal infection rates decline and fungi risk bottlenecking events. This breaking point of the SIR model is essential to the dynamic nature of pathogenic fungi living equilibrium with their host plants. Chance circumstances could prevent enough infections to ensure survival (Jarroudi et al., 2020). This threat of spatial and temporal variation in fungi infection has driven extreme morphological changes. 

Further mathematical models based on these principles provide greater insight into fungal epidemics. The spread of phytopathogenic species is weather dependent. For example, rusts are a parasitic plant pathogen from the fungal division Basidiomycota. The time between wheat rust infections and quantity of spores produced relies on temperature, travel speed via winds, and the susceptibility of plants due to recent rainfall.  

Fig. 6 From a few inputs such as wind speed, position, time of deposition, it is possible to model maximum density curves of healthy host individuals (left) and spore dispersal (right) for wheat stripe rust caused by Puccinia striiformis (Jarroudi et al., 2020). 

These disease cycles propagate over entire continents, following temperature and crop density conditions. Due to the omnipresence of these phytopathogenic diseases, plants have coevolved resistance to these attacks. When plants experience an infection related to stress response, pattern recognition receptors are trained to recognize certain molecular patterns. Following the adaptation of this pattern triggered immune system, colonizers that fail to change molecular patterns will be subject to antimicrobial lytic enzymes and the release of oxygen radical species (Shu et al., 2023).  

Although fungi initially infect plants with little resistance, progressive herd immunization drives R₀ below 1. Yet, the disease host relationship is self-balancing. Pathogenic fungi have acquired several strategies to overcome plant resistance tactics. For rusts, this entails a complex lifestyle (Fig.7) and huge genome. 

Fig. 7 Life cycles of rust fungi. (A) Macrocyclic rust fungi produce five spore stages with varying quantities of genetic material (B) though some rusts species only follow a subset of these reproductive stages (Lorraine et al., 2019). 

To overcome plant resistance, Rusts may have five phases in their life cycle: pycniospores (0), aeciospores (I), urediniospores (II), teliospores (III), and basidiospores (IV). Rusts may infect one their life cycle making them autoecious or heteroecious, respectively. As an example, Fig. 7 illustrates how spore stage and nuclear cycle differ by host. The aecial host – the secondary host – supports the 0 and I phases. Pycniospores, also known as spermatia, are a haploid phase devoted to mating. Aecia structures develop from fertilized spermagona – structures made by pycniospores, forming fruiting bodies that release aeciospores, a dikaryotic spore type (Newcombe, 2004). For example, wheat rusts like P. graminis rely on alternate hosts like the barberry (Berberis vulgaris) when producing pycniospores and aeciospores that are involved in sexual reproduction. During this phase transition, parent cells fuse without fusion of the nuclei, preparing for meiosis (Wang et al., 2015).  

In contrast to the aecia where tissue damage is localized, the telial host is subject to more severe parasitism to fuel high spore production. The telial host is the primary host of the fungi and experiences a much more prolific resource extraction than the secondary host to produce spores. Stages III, IV and V are the primary way these fungi boost their numbers; to feed this growth the fungi must invade the plant, penetrating deep inside its tissues. The process of uredinal vegetative cycles, where fungal growth occurs inside the plant, is particularly harmful and drives plant immune adaptation.  

Urediniospores and teliospores are both dikaryotic. Urediniospores continue to infect a host cell, in a process known as vegetative growth. Chemical and physical stimuli, such as temperature changes from fall to summer drive the transition to teliospores. Teliospores for chainlike structures are known as telia. Teliospores typically have thicker cell walls than other phases and allow rust to winter on the host plant. As it warms, teliospores undergo meiosis producing basidiospores. Dispersed on the surrounding environment, basidiospores continue the cycle of infection (Lorrain et al., 2019). Certain spore stages are absent in species with lifecycles: demi-cyclic and micro-cyclic (fig. 6). Although the life cycle is simplified, the meiotic pathway is preserved.  

By adopting these distinct stages, fungi have a chance to overcome transgenerational immune memory and falling effective reproductive numbers in their primary host. Through strategies such as meiosis and host switching, fungi have adapted to the resistances of recovered hosts. Instead of explosive epidemics and eradication (due to processes such as herd immunity), a pattern of oscillating host and fungal adaptation emerges. Pathogenic fungi optimize their reproductive strategies to address the shifting host environment, ensuring that the cycle of infection continues. 

Genetic Warfare

The evolution of fungal pathogens is dictated by patterns of warfare with their host plants. Fungal phytopathogens rely on the secretion of effector proteins, fungi’s toolkit to suppress plant immunity and colonize their hosts. As plant immune systems eventually acquire resistance to these proteins though, evolving specificity to effectors, pathogens lose and regain genetic sequences, driving evolutionary diversification. The development of AlphaFold2: a program designed to predict the structure and function of proteins based on their genetic sequences, allowed researchers to understand effector variation. Evolutionary analyses by Seong and Krasileva (2023) revealed that though undergoing rapid genetic divergence and duplications, evolution rewrites novel combinations of phytopathogenic protein sequences with differing forms and functions. Seong & Krasileva, 2023. 

Fig. 8 The rapid evolutionary diversification of effectors. Evolution of a pathogenic fungi from non-pathogenic ancestor, effector proteins develop, undergoing divergent evolution and developing novel domains. Effector proteins continue to evolve within the same niche, generating novel sequences and functions by acquiring new domains. Adapted from Seong & Krasileva, 2023. 

This versatility allows fungi to continue to evade the immune system of plants. This versatility also allows fungi to transition between pathogenic and mutualistic species over time, as plant recognition markers lag behind fungal effector evolution. Pathogenic fungi like rusts may have extremely large genomes: P. graminis (wheat stem rust) and Puccinia striiformis (wheat stripe rust) have around a thousand candidate effector protein encoding genes (Duplessis et al., 2011). Increasing the chance of genetic recombination ensures that these fungi continue to have susceptible hosts.  

Genetic variety is extremely important to fungi, yet asexual spore production remains the primary means for fungi like rust. While the SIR model is useful to characterize the spreading of a fungal pathogen over fields or forests, genetic recombination begins to play a much larger role for fungi on larger temporal and spatial scales. The importance of sexual reproduction is emphasized by the rate limiting effect. On a large scale, a reaction diffusion model describes disease propagation. Infectious entities spread through space over time, progressing as a traveling wave.  

Fig. 9 Theoretical models, resembling that of a novel infection: approximating reaction-diffusion reactions, show that traveling wave velocity c as a function of the ratio of asexual to sexual spore production b increase logarithmically (Hamelin et al., 2016). 

This is to be expected, as the requirement of finding a mate makes it less likely that a spore will infect a new host. Yet, most pathogenic fungi maintain sexual reproduction, indicating its value. This is a trait widely observed in the fungal kingdom, particularly in the fungal subkingdom Dikarya, to which many pathogenic fungi such as rusts belong (Wallen & Perlin, 2018). Fungi’s capacity to rapidly adapt to novel environments allows them to capitalize on very specific niches, despite their complexity, they can maintain environmental control by outpacing the evolution of plants, bacteria, and other organisms. 

Hyphal Networks

While the growth patterns and morphology of individual fungi play a significant role in the success of fungal survival, the importance of the growth patterns of collective fungi cannot be understated. Around 5000 to 6000 species of mycorrhizal fungi can form networks with host plants (Yih, 2017), networks that can cover several acres. In fact, the largest known organism to man is an Armillaria gallica fungus, whose network spans over 91 acres (Daley, 2018). These networks can be modeled using Graph Theory to better understand the implications of their topologies, or connection patterns, to better understand the competitive advantages and disadvantages fungal networks come with. Overall, how fungi use networks can reveal much about the survival and proliferation of whole ecosystems. 

Brief Introduction to Hyphal Networks 

Oftentimes, mushrooms are what come to mind when fungi are mentioned. However, beneath the fruiting mushroom body lies a thick, interweaving matrix of thin fungal filaments, connecting the flora of whole ecosystems. Hyphal networks [Link to physics article], also called mycorrhizal networks, are the complex, interconnected structures created by the hyphal filaments of mycorrhizal fungi. Mycorrhizal fungi form symbiotic relationships with plants, utilizing their abilities to form networks to act as nutrient conduits and communication channels from one host plant to another. Through extensive connecting of hosts, mycorrhizal fungi form what is called the “Common Mycorrhizal Network” (CMN), giving plants the benefits of long-distance nutrient transport and communication, while also benefiting itself through a steady source of nutrients allocated by the host plants (The Common Mycelial Network (CMN) of Forests, 2022). These symbiotic connections pose an interesting challenge for fungi – In an ecosystem, there are several plants that fungi can form connections with. Therefore, how should fungi arrange and connect themselves in a network to best optimize their growth and survivability from host-plant nutrient sourcing? Since fungi act as connections between host plants, we can analyze hyphal networks using graph theory, which can provide insights on the patterns behind fungal network design to overcome this optimization challenge. 

Graph Theory 

But what is graph theory? Consider Euler’s “Seven Bridges of Königsberg” question. Euler wondered if it was possible to walk through the town of Königsberg and cross each of its seven bridges exactly once, given the following layout of the town, as seen in Fig. 10. 

Fig. 10 The layout of the “Seven Bridges of Königsberg” problem (The Konigsberg Bridge Problem | NRICH, https://nrich.maths.org/articles/konigsberg-bridge-problem) 

Euler determined that the town could be represented as a “graph,” a collection of points called “nodes,” and a collection of connectors called “edges.” Since the riverbanks and islands were locations that could be repeatedly visited, they were labeled as nodes, and the bridges connecting the landmasses became edges. Euler’s graph diagram (Fig. 11) was represented as: 

Fig. 11 Graph representation of the Seven Bridges problem (The Konigsberg Bridge Problem | NRICH, https://nrich.maths.org/articles/konigsberg-bridge-problem) 

To prove that the walk across town using each bridge once was possible, Euler had to show that every edge could be traveled through exactly once. He reasoned that the starting and ending nodes had to have an odd degree – degree, meaning the number of edge connections for a given node – since to enter and leave a node, one edge is required to enter and one more to leave. Thus, exactly zero (if the start and end points are the same) or two (for unique start and end points) nodes could have an odd number of edges. However, since four nodes had an odd degree, Euler proved, using graphs, that this problem was unsolvable. From here, the basis of graph theory was formed. The analysis of systems that could be represented by a collection of nodes and edges, such as those seen in Fig. 12, became the category of mathematics known as graph theory. 

Fig. 12 An example of a simple graph, describing what nodes and edges are (Högberg, 2020) 

Hyphal networks are systems that can be analyzed using graph theory! Groups of genetically similar fungi, called operational taxonomic units (OTU) (Toju et al., 2014) can be modeled as nodes, and host plants can also be modeled as nodes. Any interaction between an OTU and a host can be represented with edges – that is, if a particular fungus or group of fungi connects two plants together, plant nodes will form an edge with the said OTU node. Thus, any mention of nodes will represent an OTU or a host plant, and any edge will represent a connection between an OTU and a host, as seen in Fig. 13 and Fig. 14. 

Fig. 13 A simple graphical representation of an OTU connecting two plants together (Luke, 2024) 

Fig. 14 A graph representing the connections of various fungi OTUs to plants, where plants are red nodes, and fungi are blue, yellow, and pink nodes. 

Topology of Hyphal Networks 

When trying to understand how hyphal networks provide benefits to a fungus as a whole, it is critical to consider the network’s topology – topology meaning the patterns behind node interactions, including what conditions allow OTU nodes to connect with host nodes, how often nodes interact, and how accessible nodes are to each other (Högberg, 2020). Network topologies are important because they provide a basis to describe and compare complex systems, which can help reveal the optimal strategies for fungi to best connect to hosts to increase nutrient gain from symbiosis, as well as convey information about a fungus’ resilience to node-destroying events, such as natural disasters or animal predation (Heaton et al., 2012). Thus, analysis of fungal network topology can predict fungal profitability and survivability through better understanding of a fungi’s robustness – the ability for a fungal network to remain fully connected even after the elimination of nodes (Högberg, 2020). The greater a network’s robustness, the more a fungus can grow through nutrient gain, and how resistant a fungus is nutrient loss via nutrient pathway destruction. 

Empirical data on mycorrhizal networks is unfortunately rare due to the difficulty in accurately measuring underground mycorrhizal networks (Högberg, 2020). However, various topologies of mycorrhizal networks can be simulated by generating probability distributions that describe the degree of any given fungal or plant node based on spatial and fitness parameters. Using these degree distributions, and the information from spatial and fitness parameters, the topology and robustness of a network can be inferred. 

Topology 1: Uniform Spatial Distribution

The first simulation introduced is the uniform spatial distribution, where all fungal and plant nodes are uniformly spaced apart, and their fitness values are also uniformly distributed. This serves as a base case to compare the effects of more varied spatial and fitness values on networks. (Högberg, 2020) The distribution of degrees in this topology is represented by Fig. 15. 

 Fig. 15 Graph representing the probability g(k) of a node being randomly selected over the degree (k) of the selected node, where ɑ represents severity of cost regulation for forming longer connections. (Högberg, 2020) 

In this case, the nodes of this network, as inferred by the distribution, all have low degrees. This occurs because a particular OTU would have no reason to preferentially connect to specific hosts, other than spatial closeness. Due to the uniform spatial distribution, plants only form connections with fungi due to closeness, resulting in a uniform distribution of low degree nodes across the network. This type of network, known as a random network, is characterized by a lack of high-degree nodes, called “hubs.” The lack of hubs can leave fungal networks vulnerable though, since random removal of critical connecting links due to selective events can result in network fragmentation (Heaton et al, 2012).  

Topology 2: Concentrated Spatial Distribution 

For the second simulation, consider a highly concentrated spatial distribution of fungi, meaning all fungal nodes are located extremely close to each other. While this proximity is an unrealistic occurrence in nature, it can serve as a model of highly concentrated point habitats. From this topology, the degree distribution is given by:

where k_Τ represents the average degree of a tree node, r_Τ represents the ability of a tree to form a connection, ɑ represents severity of cost regulation for forming a connection, and θ represents the angular position of a tree. From this, the following degree distribution plot is acquired (Fig. 16) (Högberg, 2020) 

Fig. 16 Logarithmic degree distribution plot for a spatially concentrated fungal network, where ɑ represents severity of cost regulation for forming longer connections (Högberg, 2020). 

This topology results in trees located closer to the concentrated fungus nodes forming more links, allowing for nodes with a larger number of mutualistic connections, or “hubs.” The greater the cost becomes to form networks, represented in Fig. 16 by ɑ, the more prevalent this effect becomes; only trees close to the concentrated fungi habitat will be able to form connections with the fungi. This means trees located further away from a concentrated fungi habitat will have decreased survivability due to a lack of symbiotic connections, and the fungi will have fewer available hosts to interact with. Furthermore, fungi can become vulnerable to random attacks since concentrated networks are highly reliant on their hubs for connectivity. If an attacker randomly happens to eliminate these hubs, the entire network can quickly fragment or collapse (Högberg, 2020).  

Topology 3: Exponential Fitness Distribution 

For the third simulation, reconsider the spatially uniform distribution. However, this time, both fungi and tree nodes have associated fitness values, where a greater fitness η indicates an increased affinity for forming a connection. These fitness values are distributed amongst nodes by the exponential distribution, visualized by Fig. 17. (Högberg, 2020) 

Fig. 17 Logarithmic degree distribution plot for a fitness-weighted fungal network, where fitness is distributed by the exponential distribution, and ɑ represents severity of cost regulation for forming longer connections (Högberg, 2020) 

The effect of adding fitness values means that the graph will have several hubs due to preferential connections with high fitness nodes, while the majority nodes will have a few connections. The key difference between this topology and topology two is that there are lower degreed-hubs rather than a few high degreed-hubs, while still maintaining several smaller degreed node-connections. This provides a greater level of redundancy in the network, increasing robustness in the case of random selection events, as random elimination of multiple hubs is unlikely (Heaton, 2012).  

Topology 4: Heterogeneous Spatial and Fitness Distribution 

For the last simulation, consider variation in both spatial and fitness values. By combining the properties from the second and third simulations, Högberg (2020) obtained the following distribution (Fig. 18): 

Fig. 18 Logarithmic degree distribution plot for a spatially heterogeneous and fitness-weighted fungal network, where ɑ represents severity of cost regulation for forming longer connections (Högberg, 2020) 

Högberg found that when considering both spatial and fitness values, for nodes with small degrees, the fitness simulation distribution dominated, and for nodes with larger degrees, the spatial simulation distribution dominated. This means that these networks benefit from being able to form several mutualistic connections with “hubs,” as seen with spatially concentrated networks, while also benefiting from the robustness provided by forming connections with several fit hosts, as seen with fitness distributed networks. 

Implications of Simulations and Networks 

Based on topology simulations, it could be concluded that fungal networks should exhibit some level of spatial heterogeneity and an inclination to connect with fitter hosts to maximize survival and system robustness. This is true; mycorrhizal networks found within Douglas-fir forests support the idea that fungal networks are scale-free – that is, their distribution follows a power law, meaning there are a few high-degree nodes and many low-degree nodes. Alternatively, this means their degree distribution curve follows the equation, as modelled in Fig. 19: 

where k is the degree of a node, and is a constant that is usually between 2 to 3.

Fig. 19 Power law distribution in a) normal scale and b) log scale (Sözlük, 2015) 

Similarly, topologies 2, 3, and 4 resemble the power-law distribution, making them scale-free networks. Thus, topology analysis reveals how and why fungi form the networks in the way they do; fungi tend to form connections with closer hosts or fitter hosts to ensure their network robustness. (Högberg, 2020) In other words, fungi solve their optimization challenge by forming topologies that balance the need for central resource distribution using hubs by connecting to close hosts, while maintaining a high network resiliency by keeping a high number of simple decentralized connections by connecting to further, but fitter hosts. However, nature is far more complicated than a simulation; many more variables must be considered than just position and fitness, such as targeted selection resiliency or a fungus’ willingness to help weaker hosts. For example, even scale-free networks can disintegrate with enough selective pressure, such as selective removal of high-degree nodes. From an ecological perspective, if there was some event that favored attacking hub trees, the associated fungi may be in trouble. Furthermore, fungi do not only form associations with solely fittest hosts; mycorrhiza enables nutrient transfer between healthy and weaker hosts to increase net ecosystem survivability, so this model is limited in how connections with weaker hosts occur (Högberg, 2020).  

To further analyze fungal networks’ robustness, alternative metrics and models can be used. For example, Heaton et al. (2012) measures robustness by determining the amount of alternate pathways from one node to another rather than measuring node degrees. Furthermore, Heaton et al.’s (2012) model uses the weighting of edges to further increase accuracy, where the edges are “weighted” with a corresponding connection affinity value. With edge weights, greater values indicate a stronger connection, such as a thicker collection of hyphae connecting two hosts. Having a larger number of redundant pathways, such as the network in Fig. 20, as well as differential strengthening of edges, increases a network’s robustness to damage (Heaton, 2012). Thus, multiple models can and should be considered when trying to determine robustness from network topology. 

Fig. 20  A fungal network with multiple paths, displaying a network of increased robustness. These paths are weighted, represented by the numbers on edges. Adapted from Heaton et al., 2012. 

Hyphal networks are incredibly complex, with several graphical models that can be used to understand them. These fascinating networks can grow and connect with several parameters in mind, designing network topologies that consider their proximity to hosts as well as a host’s fitness (or lack of) to create solutions to challenge of optimally connecting to hosts to maximize nutrient gain from hosts and minimize loss of nutrients due to host elimination. Moreover, even with certain connection strategies to increase survival and growth, sometimes preferred network topologies have their own shortcomings, which can make them susceptible to targeted selection events. Overall, though, fungal networks can increase a fungal species’ robustness and general survival, simply due to their network geometry and from the patterns in how they form connections. 

Conclusion

The exploration of fungal systems through mathematical principles can be unveiled by the sophisticated strategies fungi use for growth and survival. By breaking symmetry, fungi transition from stable states to dynamic forms, allowing them to adapt and thrive. The geometric arrangement of gills in mushrooms optimizes spore production, demonstrating nature’s efficiency. Furthermore, fungi like the yeast Saccharomyces cerevisiae utilize stochastic transitions between phenotypes to optimize their growth rates in changing environments. This design allows the population to “blindly” prepare for the unpredictable reality of nature. These cells are also capable of adjusting the rate of their transitions to match the frequency of their surroundings, resulting in a cheap alternative mechanism to the sensing and signaling pathways present in other organisms. Lifecycle and genetic diversity are a repeated motif in fungal successes. From fruiting bodies to meiotic recombination, this adaptive capacity allows fungal diseases to traverse and reinfect vast regions of host plants. Finally, hyphal networks, when analyzed using graph theory, reveal the growth patterns behind the extensive structures that support nutrient distribution and ecosystem communication. Since fungi have so many hosts they can connect to within an ecosystem, fungi must establish a way to connect to plants in a manner that maximizes both the resiliency of the fungi and the amount of nutrients it can source from its host plants. Thus, to maximize fungal network growth and survival, fungi design their topologies such that they connect to hosts that are close by or have exceptional fitness values, because these types of symbiotic connections promote the nutrient gain a fungus can obtain from hosts, as well as insulating fungi from nutrient loss in the case of host plant selection events through the creation of redundant pathways. Overall, fungi cleverly use the mathematical principles that govern growth in nature to increase their survival and proliferation, allowing for the incredible diversity of this superorganism seen today. 

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