Growth and metabolism in mycelial fungi

A mathematical model for the growth and morphogenesis in colonies of mycelial fungi is given. The model consists of partial differential equations for accumulation of hyphae by apical growth, uptake of nutrient, and redistribution of a derived metabolite within the mycelium. Mechanisms for nutrient absorption and for metabolite translocation are discussed. An explanation for growth in the form of concentric mycelial rings is offered, based on the hypothesis that repeated metabolite buildup and depletion gives rise to different local branching rates, and thus distinct bands of hyphal densities.     

Growth velocities of branched actin networks

The growth of an actin network against an obstacle that stimulates branching locally is studied using several variants of a kinetic rate model based on the orientation-dependent number density of filaments. The model emphasizes the effects of branching and capping on the density of free filament ends. The variants differ in their treatment of side versus end branching and dimensionality, and assume that new branches are generated by existing branches (autocatalytic behavior) or independently of existing branches (nucleation behavior). In autocatalytic models, the network growth velocity is rigorously independent of the opposing force exerted by the obstacle, and the network density is proportional to the force. The dependence of the growth velocity on the branching and capping rates is evaluated by a numerical solution of the rate equations. In side-branching models, the growth velocity drops gradually to zero with decreasing branching rate, while in end-branching models the drop is abrupt. As the capping rate goes to zero, it is found that the behavior of the velocity is sensitive to the thickness of the branching region. Experiments are proposed for using these results to shed light on the nature of the branching process.     

Tests of a cellular model for constant branch distribution in the filamentous fungus Neurospora crassa

The growth of mycelial fungi is characterized by the highly polarized extension of hyphal tips and the formation of subapical branches, which themselves extend as new tips. In Neurospora crassa, tip growth and branching are crucial elements for this saprophyte in the colonization and utilization of organic substrates. Much research has focused on the mechanism of tip extension, but a cellular model that fully explains the known phenomenology of branching by N. crassa has not been proposed. We described and tested a model in which the formation of a lateral branch in N. crassa was determined by the accumulation of tip-growth vesicles caused by the excess of the rate of supply over the rate of deposition at the apex. If both rates are proportional to metabolic rate, then the model explains the known lack of dependence of branch interval on growth rate. We tested the model by manipulating the tip extension rate, first by shifting temperature in both the wild type and hyperbranching (colonial) mutants and also by observing the behavior of both tipless colonies and colonyless tips. We found that temperature shifts in either direction result in temporary changes in branching. We found that colonyless tips also pass through a temporary transition phase of branching. The tipless colonies produced a cluster of new tips near the point of damage. We also found that branching in colonial mutants is dependent on growth rate. The results of these tests are consistent with a model of branching in which branch initiation is controlled by the dynamics of tip growth while being independent of the actual rate of this growth.     

Tests of a Cellular Model for Constant Branch Distribution in the Filamentous Fungus Neurospora crassa

The growth of mycelial fungi is characterized by the highly polarized extension of hyphal tips and the formation of subapical branches, which themselves extend as new tips. In Neurospora crassa, tip growth and branching are crucial elements for this saprophyte in the colonization and utilization of organic substrates. Much research has focused on the mechanism of tip extension, but a cellular model that fully explains the known phenomenology of branching by N. crassa has not been proposed. We described and tested a model in which the formation of a lateral branch in N. crassa was determined by the accumulation of tip-growth vesicles caused by the excess of the rate of supply over the rate of deposition at the apex. If both rates are proportional to metabolic rate, then the model explains the known lack of dependence of branch interval on growth rate. We tested the model by manipulating the tip extension rate, first by shifting temperature in both the wild type and hyperbranching (colonial) mutants and also by observing the behavior of both tipless colonies and colonyless tips. We found that temperature shifts in either direction result in temporary changes in branching. We found that colonyless tips also pass through a temporary transition phase of branching. The tipless colonies produced a cluster of new tips near the point of damage. We also found that branching in colonial mutants is dependent on growth rate. The results of these tests are consistent with a model of branching in which branch initiation is controlled by the dynamics of tip growth while being independent of the actual rate of this growth.     

Impact of growth and uptake patterns of arbuscular mycorrhizal fungi on plant phosphorus uptake—a modelling study

In this paper we present a mathematical model for estimating external mycelium growth of arbuscular mycorrhizal fungi and its effect on root uptake of phosphate (P). The model describes P transport in soil and P uptake by both root and fungi on the single root scale. We investigate differences in soil P depletion and overall P influx into a mycorrhizal root by assuming that different spatial regions of mycelia are active in P uptake. When all external hyphae contribute to P uptake, overall uptake is dominated by the fungus and the most effective growth pattern appears to be the one using a high level of anastomosis. The same is true when only the proportion of external hyphae assumed to be active contributes to uptake. When uptake is restricted to the tips, hyphal contribution to overall P uptake is less dominant; the most effective growth pattern appears to be the one characterised by nonlinear branching where branching stops at a given maximal hyphal tip density. Comparison to measured P depletion in the literature suggests that the scenario where active hyphae are contributing to P uptake is likely to fit the data best. These quantitative predictions promote our understanding of the mycorrhizal symbiosis and its role in plant P nutrition.     

A mutation in the Neurospora crassa actin gene results in multiple defects in tip growth and branching

Actin has a pivotal function in hyphal morphogenesis in filamentous fungi, but it is not certain whether its function is equivalent to that of a morphogen, or if it is simply part of a mechanism that executes orders given by another regulatory entity. To address this question we selected for cytochalasin A resistance and isolated act1, the first actin mutant in Neurospora crassa. This mutant branches apically and shows an altered distribution of actin at the tip. Based on the properties of this mutant, we propose a model of tip growth and branching in which actin effects tip growth by regulating the rate of vesicle flow from proximal to distal regions of a hypha, thereby controlling the tip-high gradient of cytoplasmic calcium. The actin-controlled calcium gradient at the tip is necessary for maintenance of tip growth as well as the dominance of one polarized site at the hyphal tip. The phenotype of act1 indicates that actin controls the balance between lateral and apical branching.     

The 59-kDa polypeptide constituent of 8–10-nm cytoplasmic filaments in Neurospora crassa is a pyruvate decarboxylase

The fungus Neurospora crassa harbors large amounts of cytoplasmic filaments which are homopolymers of a 59-kDa polypeptide (P59Nc). We have used molecular cloning, sequencing and enzyme activity measurement strategies to demonstrate that these filaments are made of pyruvate decarboxylase (PDC, EC 4.1.1.1), which is the key enzyme in the glycolytic-fermentative pathway of ethanol production in fungi, and in certain plants and bacteria. Immunofluorescence analyses of 8–10-nm filaments, as well as quantitative Northern blot studies of P59Nc mRNA and measurements of PDC activity, showed that the presence and abundance of PDC filaments depends on the metabolic growth conditions of the cells. These findings may be of relevance to the biology of ethanol production by fungi, and may shed light on the nature and variable presence of filament bundles described in fungal cells.     

The COT1 homologue CPCOT1 regulates polar growth and branching and is essential for pathogenicity in Claviceps purpurea

Claviceps purpurea, the ergot fungus, is a common grass pathogen attacking exclusively young ovaries. Its pathogenic development involves an early phase of directed growth (with strictly suppressed branching) towards the floral vascular tissue. Since Ser/Thr protein kinases of the NDR family have been shown to be involved in polar growth and branching in fungi, we have analyzed a C. purpurea homologue of the Neurospora crassa cot-1 gene, cpcot1. It encodes a functional homologue of COT1 since it can fully complement the N. crassa cot-1 mutant phenotype. Delta cpcot1 mutants are significantly impaired in vegetative growth properties: they are characterized by hyperbranching, reduced growth rate, and decreased conidiation. Infection studies on rye plants and isolated ovaries show that the delta cpcot1 mutants are apathogenic; microscopical analyses indicate a very early block, probably in penetration. Thus CPCOT1 is not only involved in polarity and branching and hence oriented growth in the host tissue as expected, but it is essential for the initiation of infection.     

Model for the alignment of actin filaments in endothelial cells subjected to fluid shear stress

Cultured vascular endothelial cells undergo significant morphological changes when subjected to sustained fluid shear stress. The cells elongate and align in the direction of applied flow. Accompanying this shape change is a reorganization at the intracellular level. The cytoskeletal actin filaments reorient in the direction of the cells' long axis. How this external stimulus is transmitted to the endothelial cytoskeleton still remains unclear. In this article. we present a theoretical model accounting for the cytoskeletal reorganization under the influence of fluid shear stress. We develop a system of integro-partial-differential equations describing the dynamics of actin filaments, the actin-binding proteins, and the drift of transmembrane proteins due to the fluid shear forces applied on the plasma membrane. Numerical simulations of the equations show that under certain conditions, initially randomly oriented cytoskeletal actin filaments reorient in structures parallel to the externally applied fluid shear forces. Thus, the model suggests a mechanism by which shear forces acting on the cell membrane can be transmitted to the entire cytoskeleton via molecular interactions alone.     

Mathematical model for apical growth, septation, and branching of mycelial microorganisms

A mathematical model for apical growth, septation, and branching of mycelial microorganisms is presented. The model consists of two parts: the determinstic part of the model is based on fundamental cellular and physical mechanisms; it represents the kinetics for growth of hyphal tips and septation of apical as well as intercalary compartments. In regard to random occurrences of hyphal growth and branching, the stochastic part deals with branching processes, tip growth directions, and outgrowth orientations of branches. The model can explain the morphological development of mycelia up to the formation of pellets. The results, as predicted by the model, correspond very closely to those observed in experiments. In addition, some unmeasured states can be ascertained, such as the distribution functions of hyphal length (biomass) and tips along pellet radii.     

Modeling of Branching Patterns in Plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients—we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.     

Mechanistic basis of branch-site selection in filamentous bacteria

Many filamentous organisms, such as fungi, grow by tip-extension and by forming new branches behind the tips. A similar growth mode occurs in filamentous bacteria, including the genus Streptomyces, although here our mechanistic understanding has been very limited. The Streptomyces protein DivIVA is a critical determinant of hyphal growth and localizes in foci at hyphal tips and sites of future branch development. However, how such foci form was previously unknown. Here, we show experimentally that DivIVA focus-formation involves a novel mechanism in which new DivIVA foci break off from existing tip-foci, bypassing the need for initial nucleation or de novo branch-site selection. We develop a mathematical model for DivIVA-dependent growth and branching, involving DivIVA focus-formation by tip-focus splitting, focus growth, and the initiation of new branches at a critical focus size. We quantitatively fit our model to the experimentally-measured tip-to-branch and branch-to-branch length distributions. The model predicts a particular bimodal tip-to-branch distribution results from tip-focus splitting, a prediction we confirm experimentally. Our work provides mechanistic understanding of a novel mode of hyphal growth regulation that may be widely employed.     

Mathematical model for positioning the FtsZ contractile ring in Escherichia coli

During bacterial cytokinesis, a proteinaceous contractile ring assembles in the cell middle. The Z ring tethers to the membrane and contracts, when triggered, to form two identical daughter cells. One mechanism for positioning the ring involves the MinC, MinD and MinE proteins, which oscillate between cell poles to inhibit ring assembly. Averaged over time, the concentration of the inhibitor MinC is lowest at midcell, restricting ring assembly to this region. A second positioning mechanism, called Nucleoid Occlusion, acts through protein SlmA to inhibit ring polymerization in the location of the nucleoid. Here, a mathematical model was developed to explore the interactions between Min oscillations, nucleoid occlusion, Z ring assembly and positioning. One-dimensional advection-reaction-diffusion equations were built to simulate the spatio-temporal concentrations of Min proteins and their effect on various forms of FtsZ. The resulting partial differential equations were numerically solved using a finite volume method. The reduced chemical model assumed that the ring is composed of overlapping FtsZ filaments and that MinC disrupts lateral interactions between filaments. SlmA was presumed to break long FtsZ filaments into shorter units. A term was developed to account for the movement of FtsZ subunits in membrane-bound filaments as they touch and align with other filaments. This alignment was critical in forming sharp stable rings. Simulations qualitatively reproduced experimental results showing the incorrect positioning of rings when Min proteins were not expressed, and the formation of multiple rings when FtsZ was overexpressed.     

Early Cretaceous record of microboring organisms in skeletons of growing corals

Ko?odziej, B., Golubic, S., Bucur, I.I., Radtke, G. & Tribollet, A. 2011: Early Cretaceous record of microboring organisms in skeletons of growing corals. Lethaia, Vol. 45, pp. 34–45. A spectacularly preserved assemblage of microbial euendoliths, penetrating into skeletons of growing scleractinian corals, has been recognized in Early Aptian (Early Cretaceous) reef limestone of the Rar?u Mountains (East Carpathians, NE Romania). Microboring euendolithic filaments were found in five coral colonies of the suborder Microsolenina. They remained in part well‐preserved, often impregnated with iron oxides, which made them visible even in strongly recrystallized parts of coral skeletons. Filaments of a wide range of sizes (2–40 μm in diameter) were concentrated within medium parts of coral septa, oriented along the septa in the direction of the coral growth. The larger filaments were tubular, occurring in bundles and branched into finer, often tapering branches. Their behaviour and organization were quite similar to the modern euendolithic siphonalean chlorophyte Ostreobium. Filament diameters exceeded those reported for the modern species, but covered a similarly wide size range. Narrower frequently branching filaments, 4–8 μm in diameter, resemble distal branching patterns of modern Ostreobium quekettii. Some very thin filaments (ca. 1–2 μm) observed within skeleton or inside the large tubular filaments, sometimes associated with globular swellings, may represent euendolithic fungi. The recrystallization of coral skeleton had limited effect on preservation of euendoliths due to their impregnation with iron oxides; microbial euendoliths were subjected to different taphonomic changes. □Chlorophytes, Early Cretaceous, fungi, microbial euendoliths, Romania, scleractinian corals.     

Regulation of actin dynamics in rapidly moving cells: a quantitative analysis

We develop a mathematical model that describes key details of actin dynamics in protrusion associated with cell motility. The model is based on the dendritic-nucleation hypothesis for lamellipodial protrusion in nonmuscle cells such as keratocytes. We consider a set of partial differential equations for diffusion and reactions of sequestered actin complexes, nucleation, and growth by polymerization of barbed ends of actin filaments, as well as capping and depolymerization of the filaments. The mechanical aspect of protrusion is based on an elastic polymerization ratchet mechanism. An output of the model is a relationship between the protrusion velocity and the number of filament barbed ends pushing the membrane. Significantly, this relationship has a local maximum: too many barbed ends deplete the available monomer pool, too few are insufficient to generate protrusive force, so motility is stalled at either extreme. Our results suggest that to achieve rapid motility, some tuning of parameters affecting actin dynamics must be operating in the cell.     

Stochastic modelling of tumour-induced angiogenesis

A major source of complexity in the mathematical modelling of an angiogenic process derives from the strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network with a family of interacting underlying fields. The aim of this paper is to propose a novel mathematical approach for reducing complexity by (locally) averaging the stochastic cell, or vessel densities in the evolution equations of the underlying fields, at the mesoscale, while keeping stochasticity at lower scales, possibly at the level of individual cells or vessels. This method leads to models which are known as hybrid models. In this paper, as a working example, we apply our method to a simplified stochastic geometric model, inspired by the relevant literature, for a spatially distributed angiogenic process. The branching mechanism of blood vessels is modelled as a stochastic marked counting process describing the branching of new tips, while the network of vessels is modelled as the union of the trajectories developed by tips, according to a system of stochastic differential equations à la Langevin.     

细胞增长的数学模型

本文研究一个关于细胞增长的数学模型,用微分方程组来确定。通过系统的稳定性的分析指出参数对细胞组织容量的平衡状态的影响。     

Development of mathematical models (Logistic, Gompertz and Richards models) describing the growth pattern of Pseudomonas putida (NICM 2174)

Bacterial growth curve, which is asymptotic after a certain period, is described using three different mathematical models, namely, Logistic model, Gompertz model and Richards model. The equations for these three models are fitted by evaluating the mathematical parameters involved in these models. This is done by applying the method of partial sums to the data in Table 1 containing the optical density values for different cell mass at different time intervals. The sum of square of residuals between the expected optical density values and the experimental values is calculated for each of these models. In the cases tested, the Logistic model was found to be the best fit for the growth curve of Pseudomonas putida (NICM 2174) and was found to be easy to use. These results fit the data very well at 5% level for more than 70% of the readings.