Modelling hyphal networks

Exploring discrete space-time models for information transfer: Analogies from mycelial networks to the cosmic web

Fungal mycelium networks are large scale biological networks along which nutrients, metabolites flow. Recently, we discovered a rich spectrum of electrical activity in mycelium networks, including action-potential spikes and trains of spikes. Reasoning by analogy with animals and plants, where travelling patterns of electrical activity perform integrative and communicative mechanisms, we speculated that waves of electrical activity transfer information in mycelium networks. Using a new discrete space-time model with emergent radial spanning-tree topology, hypothetically comparable mycelial morphology and physically comparable information transfer, we provide physical arguments for the use of such a model, and by considering growing mycelium network by analogy with growing network of matter in the cosmic web, we develop mathematical models and theoretical concepts to characterise the parameters of the information transfer.

Network efficiency of spatial systems with fractal morphology: a geometric graphs approach

The functional features of spatial networks depend upon a non-trivial relationship between the topological and physical structure. Here, we explore that relationship for spatial networks with radial symmetry and disordered fractal morphology. Under a geometric graphs approach, we quantify the effectiveness of the exchange of information in the system from center to perimeter and over the entire network structure. We mainly consider two paradigmatic models of disordered fractal formation, the Ballistic Aggregation and Diffusion-Limited Aggregation models, and complementary, the Viscek and Hexaflake fractals, and Kagome and Hexagonal lattices. First, we show that complex tree morphologies provide important advantages over regular configurations, such as an invariant structural cost for different fractal dimensions. Furthermore, although these systems are known to be scale-free in space, they have bounded degree distributions for different values of an euclidean connectivity parameter and, therefore, do not represent ordinary scale-free networks. Finally, compared to regular structures, fractal trees are fragile and overall inefficient as expected, however, we show that this efficiency can become similar to that of a robust hexagonal lattice, at a similar cost, by just considering a very short euclidean connectivity beyond first neighbors.

Multimodal correlative imaging and modelling of phosphorus uptake from soil by hyphae of mycorrhizal fungi

Phosphorus (P) is essential for plant growth. Arbuscular mycorrhizal fungi (AMF) aid its uptake by acquiring P from sources distant from roots in return for carbon. Little is known about how AMF colonise soil pore-space, and models of AMF-enhanced P-uptake are poorly validated. We used synchrotron X-ray computed tomography to visualize mycorrhizas in soil and synchrotron X-ray fluorescence/X-ray absorption near edge structure (XRF/XANES) elemental mapping for P, sulphur (S) and aluminium (Al) in combination with modelling. We found that AMF inoculation had a suppressive effect on colonisation by other soil fungi and identified differences in structure and growth rate between hyphae of AMF and nonmycorrhizal fungi. Our results showed that AMF co-locate with areas of high P and low Al, and preferentially associate with organic-type P species over Al-rich inorganic P. We discovered that AMF avoid Al-rich areas as a source of P. Sulphur-rich regions were found to be correlated with higher hyphal density and an increased organic-associated P-pool, whilst oxidized S-species were found close to AMF hyphae. Increased S oxidation close to AMF suggested the observed changes were microbiome-related. Our experimentally-validated model led to an estimate of P-uptake by AMF hyphae that is an order of magnitude lower than rates previously estimated - a result with significant implications for the modelling of plant-soil-AMF interactions.

Challenges and approaches in assessing the interplay between microorganisms and their physical micro-environments

Spatial structure over scales ranging from nanometres to centimetres (and beyond) varies markedly in diverse habitats and the industry-relevant settings that support microbial activity. Developing an understanding of the interplay between a structured environment and the associated microbial processes and ecology is fundamental, but challenging. Several novel approaches have recently been developed and implemented to help address key questions for the field: from the use of imaging tools such as X-ray Computed Tomography to explore microbial growth in soils, to the fabrication of scratched materials to examine microbial-surface interactions, to the design of microfluidic devices to track microbial biofilm formation and the metabolic processes therein. This review discusses new approaches and challenges for incorporating structured elements into the study of microbial processes across different scales. We highlight how such methods can be pivotal for furthering our understanding of microbial interactions with their environments.

Ecology of Contaminant Biotransformation in the Mycosphere: Role of Transport Processes

Fungi and bacteria often share common microhabitats. Their co-occurrence and coevolution give rise to manifold ecological interactions in the mycosphere, here defined as the microhabitats surrounding and affected by hyphae and mycelia. The extensive structure of mycelia provides ideal "logistic networks" for transport of bacteria and matter in structurally and chemically heterogeneous soil ecosystems. We describe the characteristics of the mycosphere as a unique and highly dynamic bacterial habitat and a hot spot for contaminant biotransformation. In particular, we emphasize the role of the mycosphere for (i) bacterial dispersal and colonization of subsurface interfaces and new habitats, (ii) matter transport processes and contaminant bioaccessibility, and (iii) the functional stability of microbial ecosystems when exposed to environmental fluctuations such as stress or disturbances. Adopting concepts from ecological theory, the chapter disentangles bacterial-fungal impacts on contaminant biotransformation in a systemic approach that interlinks empirical data from microbial ecosystems with simulation data from computational models. This approach provides generic information on key factors, processes, and ecological principles that drive microbial contaminant biotransformation in soil. We highlight that the transport processes create favorable habitat conditions for efficient bacterial contaminant degradation in the mycosphere. In-depth observation, understanding, and prediction of the role of mycosphere transport processes will support the use of bacterial-fungal interactions in nature-based solutions for contaminant biotransformation in natural and man-made ecosystems, respectively.

The Mycelium as a Network

The characteristic growth pattern of fungal mycelia as an interconnected network has a major impact on how cellular events operating on a micron scale affect colony behavior at an ecological scale. Network structure is intimately linked to flows of resources across the network that in turn modify the network architecture itself. This complex interplay shapes the incredibly plastic behavior of fungi and allows them to cope with patchy, ephemeral resources, competition, damage, and predation in a manner completely different from multicellular plants or animals. Here, we try to link network structure with impact on resource movement at different scales of organization to understand the benefits and challenges of organisms that grow as connected networks. This inevitably involves an interdisciplinary approach whereby mathematical modeling helps to provide a bridge between information gleaned by traditional cell and molecular techniques or biophysical approaches at a hyphal level, with observations of colony dynamics and behavior at an ecological level.

Morphological Characterization and Quantification of the Mycelial Growth of the Brown-Rot Fungus Postia placenta for Modeling Purposes

Continuous observation was performed using confocal laser scanning microscopy to visualize the three-dimensional microscopic growth of the brown-rot fungus, Postia placenta, for seventeen days. The morphological characterization of Postia placenta was quantitatively determined, including the tip extension rate, branch angle and branching length, (hyphal length between two adjacent branch sites). A voxel method has been developed to measure the growth of the biomass. Additionally, the tip extension rate distribution, the branch angle distribution and the branching length distribution, which quantified the hyphal growth characteristics, were evaluated. Statistical analysis revealed that the extension rate of tips was randomly distributed in space. The branch angle distribution did not change with the development of the colony, however, the branching length distribution did vary with the development of the colony. The experimental data will be incorporated into a lattice-based model simulating the growth of Postia placenta.

Bayesian model selection framework for identifying growth patterns in filamentous fungi

This paper describes a rigorous methodology for quantification of model errors in fungal growth models. This is essential to choose the model that best describes the data and guide modeling efforts. Mathematical modeling of growth of filamentous fungi is necessary in fungal biology for gaining systems level understanding on hyphal and colony behaviors in different environments. A critical challenge in the development of these mathematical models arises from the indeterminate nature of their colony architecture, which is a result of processing diverse intracellular signals induced in response to a heterogeneous set of physical and nutritional factors. There exists a practical gap in connecting fungal growth models with measurement data. Here, we address this gap by introducing the first unified computational framework based on Bayesian inference that can quantify individual model errors and rank the statistical models based on their descriptive power against data. We show that this Bayesian model comparison is just a natural formalization of Occam׳s razor. The application of this framework is discussed in comparing three models in the context of synthetic data generated from a known true fungal growth model. This framework of model comparison achieves a trade-off between data fitness and model complexity and the quantified model error not only helps in calibrating and comparing the models, but also in making better predictions and guiding model refinements.

L-System model for the growth of arbuscular mycorrhizal fungi, both within and outside of their host roots

Development of arbuscular mycorrhizal fungal colonization of roots and the surrounding soil is the central process of mycorrhizal symbiosis, important for ecosystem functioning and commercial inoculum applications. To improve mechanistic understanding of this highly spatially and temporarily dynamic process, we developed a three-dimensional model taking into account growth of the roots and hyphae. It is for the first time that infection within the root system is simulated dynamically and in a spatially resolved way. Comparison between data measured in a calibration experiment and simulated results showed a good fit. Our simulations showed that the position of the fungal inoculum affects the sensitivity of hyphal growth parameters. Variation in speed of secondary infection and hyphal lifetime had a different effect on root infection and hyphal length, respectively, depending on whether the inoculum was concentrated or dispersed. For other parameters (branching rate, distance between entry points), the relative effect was the same independent of inoculum placement. The model also indicated that maximum root colonization levels well below 100%, often observed experimentally, may be a result of differential spread of roots and hyphae, besides intrinsic plant control, particularly upon localized placement of inoculum and slow secondary infection.

PAM: Particle automata model in simulation of Fusarium graminearum pathogen expansion

The multi-scale nature and inherent complexity of biological systems are a great challenge for computer modeling and classical modeling paradigms. We present a novel particle automata modeling metaphor in the context of developing a 3D model of Fusarium graminearum infection in wheat. The system consisting of the host plant and Fusarium pathogen cells can be represented by an ensemble of discrete particles defined by a set of attributes. The cells-particles can interact with each other mimicking mechanical resistance of the cell walls and cell coalescence. The particles can move, while some of their attributes can be changed according to prescribed rules. The rules can represent cellular scales of a complex system, while the integrated particle automata model (PAM) simulates its overall multi-scale behavior. We show that due to the ability of mimicking mechanical interactions of Fusarium tip cells with the host tissue, the model is able to simulate realistic penetration properties of the colonization process reproducing both vertical and lateral Fusarium invasion scenarios. The comparison of simulation results with micrographs from laboratory experiments shows encouraging qualitative agreement between the two.

A framework for an organelle-based mathematical modeling of hyphae

Background

Although highly desirable, a mechanistic explanation for the outstanding protein secretion capabilities of fungi such as Aspergilli is missing. As a result, a rational and predictive design of strains as cell factories for protein production is still out of reach. The analysis of the secretion apparatus is not only hampered by open issues concerning molecular cell biological processes, but as well by their spatial fragmentation and highly dynamic features. Whereas the former issues are addressed by many groups, an account of the space- and time-dependent processes, which is best done by means of mathematical models, is lacking. Up to now, mathematical models for hyphal organisms mainly focus on one of two extremes. Either macroscopic morphology, such as pellet or mycelium growth, is addressed, or a microscopic picture is drawn predicting, for instance, the form of a hyphal tip. How intra-hyphal transport and organelle distribution works, however, has not been tackled so far mathematically.

Results

The main result of this contribution is a generic modeling framework to describe the space- and time-dependent evolution of intracellular substances and organelles. It takes intrahyphal, passive and active transport of substances into account and explains exponential and then linear length growth by tugor-driven uptake of water. Experimentally observed increasing concentration levels of organelles towards the tip can be well explained within the framework without resorting to complex biological regulations. It is shown that the accumulation can be partly explained by geometrical constraints, besides a necessary deceleration of the active transport velocity. The model is formulated such that more intricate intracellular processes can be included.

Conclusions

Results from steady-state experiments are easy to be interpreted. In a hyphal network, however, new branches are produced at an exponential rate. Moreover, passive and active transport processes give rise to a spatial distribution of organelles and other cytoplasmatic constituents inside hyphae. As a result, most of the data obtained in experiments will be from a non-steady and space dependent state. A quantitative and mechanistic explanation of the processes occurring will only be possible if these dependencies are taking into account while evaluating experimental findings.

A model for growth of a single fungal hypha based on well-mixed tanks in series: simulation of nutrient and vesicle transport in aerial reproductive hyphae

Current models that describe the extension of fungal hyphae and development of a mycelium either do not describe the role of vesicles in hyphal extension or do not correctly describe the experimentally observed profile for distribution of vesicles along the hypha. The present work uses the n-tanks-in-series approach to develop a model for hyphal extension that describes the intracellular transport of nutrient to a sub-apical zone where vesicles are formed and then transported to the tip, where tip extension occurs. The model was calibrated using experimental data from the literature for the extension of reproductive aerial hyphae of three different fungi, and was able to describe different profiles involving acceleration and deceleration of the extension rate. A sensitivity analysis showed that the supply of nutrient to the sub-apical vesicle-producing zone is a key factor influencing the rate of extension of the hypha. Although this model was used to describe the extension of a single reproductive aerial hypha, the use of the n-tanks-in-series approach to representing the hypha means that the model has the flexibility to be extended to describe the growth of other types of hyphae and the branching of hyphae to form a complete mycelium.

Modeling the Growth of Filamentous Fungi at the Particle Scale in Solid-State Fermentation Systems

Solid-state fermentation (SSF) with filamentous fungi is a promising technique for the production of a range of biotechnological products and has the potential to play an important role in future biorefineries. The performance of such processes is intimately linked with the mycelial mode of growth of these fungi: Not only is the production of extracellular enzymes related to morphological characteristics, but also the mycelium can affect bed properties and, consequently, the efficiency of heat and mass transfer within the bed. A mathematical model that describes the development of the fungal mycelium in SSF systems at the particle scale would be a useful tool for investigating these phenomena, but, as yet, a sufficiently complete model has not been proposed. This review presents the biological and mass transfer phenomena that should be included in such a model and then evaluates how these phenomena have been modeled previously in the SSF and related literature. We conclude that a discrete lattice-based model that uses differential equations to describe the mass balances of the components within the system would be most appropriate and that mathematical expressions for describing the individual phenomena are available in the literature. It remains for these phenomena to be integrated into a complete model describing the development of fungal mycelia in SSF systems.

The Cell Factory Aspergillus Enters the Big Data Era: Opportunities and Challenges for Optimising Product Formation

Living with limits. Getting more from less. Producing commodities and high-value products from renewable resources including waste. What is the driving force and quintessence of bioeconomy outlines the lifestyle and product portfolio of Aspergillus, a saprophytic genus, to which some of the top-performing microbial cell factories belong: Aspergillus niger, Aspergillus oryzae and Aspergillus terreus. What makes them so interesting for exploitation in biotechnology and how can they help us to address key challenges of the twenty-first century? How can these strains become trimmed for better growth on second-generation feedstocks and how can we enlarge their product portfolio by genetic and metabolic engineering to get more from less? On the other hand, what makes it so challenging to deduce biological meaning from the wealth of Aspergillus -omics data? And which hurdles hinder us to model and engineer industrial strains for higher productivity and better rheological performance under industrial cultivation conditions? In this review, we will address these issues by highlighting most recent findings from the Aspergillus research with a focus on fungal growth, physiology, morphology and product formation. Indeed, the last years brought us many surprising insights into model and industrial strains. They clearly told us that similar is not the same: there are different ways to make a hypha, there are more protein secretion routes than anticipated and there are different molecular and physical mechanisms which control polar growth and the development of hyphal networks. We will discuss new conceptual frameworks derived from these insights and the future scientific advances necessary to create value from Aspergillus Big Data.

A flexible mathematical model platform for studying branching networks: experimentally validated using the model actinomycete, Streptomyces coelicolor

Branching networks are ubiquitous in nature and their growth often responds to environmental cues dynamically. Using the antibiotic-producing soil bacterium Streptomyces as a model we have developed a flexible mathematical model platform for the study of branched biological networks. Streptomyces form large aggregates in liquid culture that can impair industrial antibiotic fermentations. Understanding the features of these could aid improvement of such processes. The model requires relatively few experimental values for parameterisation, yet delivers realistic simulations of Streptomyces pellet and is able to predict features, such as the density of hyphae, the number of growing tips and the location of antibiotic production within a pellet in response to pellet size and external nutrient supply. The model is scalable and will find utility in a range of branched biological networks such as angiogenesis, plant root growth and fungal hyphal networks.