Growth of walled cells: from shells to vesicles

Cell wall dynamics stabilize tip growth in a filamentous fungus

Hyphal tip growth allows filamentous fungi to colonize space, reproduce, or infect. It features remarkable morphogenetic plasticity including unusually fast elongation rates, tip turning, branching, or bulging. These shape changes are all driven from the expansion of a protective cell wall (CW) secreted from apical pools of exocytic vesicles. How CW secretion, remodeling, and deformation are modulated in concert to support rapid tip growth and morphogenesis while ensuring surface integrity remains poorly understood. We implemented subresolution imaging to map the dynamics of CW thickness and secretory vesicles in Aspergillus nidulans. We found that tip growth is associated with balanced rates of CW secretion and expansion, which limit temporal fluctuations in CW thickness, elongation speed, and vesicle amount, to less than 10% to 20%. Affecting this balance through modulations of growth or trafficking yield to near-immediate changes in CW thickness, mechanics, and shape. We developed a model with mechanical feedback that accounts for steady states of hyphal growth as well as rapid adaptation of CW mechanics and vesicle recruitment to different perturbations. These data provide unprecedented details on how CW dynamics emerges from material secretion and expansion, to stabilize fungal tip growth as well as promote its morphogenetic plasticity.

Lockhart with a twist: Modelling cellulose microfibril deposition and reorientation reveals twisting plant cell growth mechanisms

Plant morphology emerges from cellular growth and structure. The turgor-driven diffuse growth of a cell can be highly anisotropic: significant longitudinally and negligible radially. Such anisotropy is ensured by cellulose microfibrils (CMF) reinforcing the cell wall in the hoop direction. To maintain the cell's integrity during growth, new wall material including CMF must be continually deposited. We develop a mathematical model representing the cell as a cylindrical pressure vessel and the cell wall as a fibre-reinforced viscous sheet, explicitly including the mechano-sensitive angle of CMF deposition. The model incorporates interactions between turgor, external forces, CMF reorientation during wall extension, and matrix stiffening. Using the model, we reinterpret some recent experimental findings, and reexamine the popular hypothesis of CMF/microtubule alignment. We explore how the handedness of twisting cell growth depends on external torque and intrinsic wall properties, and find that cells twist left-handedly 'by default' in some suitable sense. Overall, this study provides a unified mechanical framework for understanding left- and right-handed twist-growth as seen in many plants.

Mechanical force-induced morphology changes in a human fungal pathogen

Background

The initial step of a number of human or plant fungal infections requires active penetration of host tissue. For example, active penetration of intestinal epithelia by Candida albicans is critical for dissemination from the gut into the bloodstream. However, little is known about how this fungal pathogen copes with resistive forces upon host cell invasion.

Results

In the present study, we have used PDMS micro-fabrication to probe the ability of filamentous C. albicans cells to penetrate and grow invasively in substrates of different stiffness. We show that there is a threshold for penetration that corresponds to a stiffness of ~ 200 kPa and that invasive growth within a stiff substrate is characterized by dramatic filament buckling, along with a stiffness-dependent decrease in extension rate. We observed a striking alteration in cell morphology, i.e., reduced cell compartment length and increased diameter during invasive growth, that is not due to depolarization of active Cdc42, but rather occurs at a substantial distance from the site of growth as a result of mechanical compression.

Conclusions

Our data reveal that in response to this compression, active Cdc42 levels are increased at the apex, whereas active Rho1 becomes depolarized, similar to that observed in membrane protrusions. Our results show that cell growth and morphology are altered during invasive growth, suggesting stiffness dictates the host cells that C. albicans can penetrate.

Osmolyte homeostasis controls single-cell growth rate and maximum cell size of Saccharomyces cerevisiae

Cell growth is well described at the population level, but precisely how nutrient and water uptake and cell wall expansion drive the growth of single cells is poorly understood. Supported by measurements of single-cell growth trajectories and cell wall elasticity, we present a single-cell growth model for yeast. The model links the thermodynamic quantities, such as turgor pressure, osmolarity, cell wall elasto-plasticity, and cell size, applying concepts from rheology and thin shell theory. It reproduces cell size dynamics during single-cell growth, budding, and hyper-osmotic or hypo-osmotic stress. We find that single-cell growth rate and final size are primarily governed by osmolyte uptake and consumption, while bud expansion requires additionally different cell wall extensibilities between mother and bud. Based on first principles the model provides a more accurate description of size dynamics than previous attempts and its analytical simplification allows for easy combination with models for other cell processes.

Systematic mapping of cell wall mechanics in the regulation of cell morphogenesis

Walled cells of plants, fungi, and bacteria come with a large range of shapes and sizes, which are ultimately dictated by the mechanics of their cell wall. This stiff and thin polymeric layer encases the plasma membrane and protects the cells mechanically by opposing large turgor pressure derived mechanical stresses. To date, however, we still lack a quantitative understanding for how local and/or global mechanical properties of the wall support cell morphogenesis. Here, we combine subresolution imaging and laser-mediated wall relaxation to quantitate subcellular values of wall thickness (h) and bulk elastic moduli (Y) in large populations of live mutant cells and in conditions affecting cell diameter in the rod-shaped model fission yeast. We find that lateral wall stiffness, defined by the surface modulus, σ = hY, robustly scales with cell diameter. This scaling is valid across tens of mutants spanning various functions-within the population of individual isogenic strains, along single misshaped cells, and even across the fission yeasts clade. Dynamic modulations of cell diameter by chemical and/or mechanical means suggest that the cell wall can rapidly adapt its surface mechanics, rendering stretched wall portions stiffer than unstretched ones. Size-dependent wall stiffening constrains diameter definition and limits size variations; it may also provide an efficient means to keep elastic strains in the wall below failure strains, potentially promoting cell survival. This quantitative set of data impacts our current understanding of the mechanics of cell walls and its contribution to morphogenesis.

Forces that shape fission yeast cells

One of the major challenges of modern cell biology is to understand how cells are assembled from nanoscale components into micrometer-scale entities with a specific size and shape. Here I describe how our quest to understand the morphogenesis of the fission yeast Schizosaccharomyces pombe drove us to investigate cellular mechanics. These studies build on the view that cell shape arises from the physical properties of an elastic cell wall inflated by internal turgor pressure. Consideration of cellular mechanics provides new insights into not only mechanisms responsible for cell-shape determination and growth, but also cellular processes such as cytokinesis and endocytosis. Studies in yeast can help to illuminate approaches and mechanisms to study the mechanobiology of the cell surface in other cell types, including animal cells.

Probing the effect of tip pressure on fungal growth: Application to Aspergillus nidulans

The study of fungal cells is of great interest due to their importance as pathogens and as fermenting fungi and for their appropriateness as model organisms. The differential pressure between the hyphal cytoplasm and the bordering medium is essential for the growth process, because the pressure is correlated with the growth rate. Notably, during the invasion of tissues, the external pressure at the tip of the hypha may be different from the pressure in the surrounding medium. We report the use of a method, based on the micropipette-aspiration technique, to study the influence of this external pressure at the hyphal tip. Moreover, this technique makes it possible to study hyphal growth mechanics in the case of very thin hyphae, not accessible to turgor pressure probes. We found a correlation between the local pressure at the tip and the growth rate for the species Arpergillus nidulans. Importantly, the proposed method allows one to measure the pressure at the tip required to arrest the hyphal growth. Determining that pressure could be useful to develop new medical treatments for fungal infections. Finally, we provide a mechanical model for these experiments, taking into account the cytoplasm flow and the wall deformation.

Dimensionless number is central to stress relaxation and expansive growth of the cell wall

Experiments demonstrate that both plastic and elastic deformation of the cell wall are necessary for wall stress relaxation and expansive growth of walled cells. A biophysical equation (Augmented Growth Equation) was previously shown to accurately model the experimentally observed wall stress relaxation and expansive growth rate. Here, dimensional analysis is used to obtain a dimensionless Augmented Growth Equation with dimensionless coefficients (groups of variables, or Π parameters). It is shown that a single Π parameter controls the wall stress relaxation rate. The Π parameter represents the ratio of plastic and elastic deformation rates, and provides an explicit relationship between expansive growth rate and the wall's mechanical properties. Values for Π are calculated for plant, algal, and fungal cells from previously reported experimental results. It is found that the Π values for each cell species are large and very different from each other. Expansive growth rates are calculated using the calculated Π values and are compared to those measured for plant and fungal cells during different growth conditions, after treatment with IAA, and in different developmental stages. The comparison shows good agreement and supports the claim that the Π parameter is central to expansive growth rate of walled cells.

Shape dynamics of growing cell walls

We introduce a general theoretical framework to study the shape dynamics of actively growing and remodeling surfaces. Using this framework we develop a physical model for growing bacterial cell walls and study the interplay of cell shape with the dynamics of growth and constriction. The model allows us to derive constraints on cell wall mechanical energy based on the observed dynamics of cell shape. We predict that exponential growth in cell size requires a constant amount of cell wall energy to be dissipated per unit volume. We use the model to understand and contrast growth in bacteria with different shapes such as spherical, ellipsoidal, cylindrical and toroidal morphologies. Coupling growth to cell wall constriction, we predict a discontinuous shape transformation, from partial constriction to cell division, as a function of the chemical potential driving cell wall synthesis. Our model for cell wall energy and shape dynamics relates growth kinetics with cell geometry, and provides a unified framework to describe the interplay between shape, growth and division in bacterial cells.

Mechanics and morphogenesis of fission yeast cells

The integration of biochemical and biomechanical elements is at the heart of morphogenesis. While animal cells are relatively soft objects which shape and mechanics is mostly regulated by cytoskeletal networks, walled cells including those of plants, fungi and bacteria are encased in a rigid cell wall which resist high internal turgor pressure. How these particular mechanical properties may influence basic cellular processes, such as growth, shape and division remains poorly understood. Recent work using the model fungal cell fission yeast, Schizosaccharomyces pombe, highlights important contribution of cell mechanics to various morphogenesis processes. We envision this genetically tractable system to serve as a novel standard for the mechanobiology of walled cell.

Morphogenesis of the Fission Yeast Cell through Cell Wall Expansion

The shape of walled cells such as fungi, bacteria, and plants are determined by the cell wall. Models for cell morphogenesis postulate that the effects of turgor pressure and mechanical properties of the cell wall can explain the shapes of these diverse cell types. However, in general, these models await validation through quantitative experiments. Fission yeast Schizosaccharomyces pombe are rod-shaped cells that grow by tip extension and then divide medially through formation of a cell wall septum. Upon cell separation after cytokinesis, the new cell ends adopt a rounded morphology. Here, we show that this shape is generated by a very simple mechanical-based mechanism in which turgor pressure inflates the elastic cell wall in the absence of cell growth. This process is independent of actin and new cell wall synthesis. To model this morphological change, we first estimate the mechanical properties of the cell wall using several approaches. The lateral cell wall behaves as an isotropic elastic material with a Young's modulus of 50 ± 10 MPa inflated by a turgor pressure estimated to be 1.5 ± 0.2 MPa. Based upon these parameters, we develop a quantitative mechanical-based model for new end formation that reveals that the cell wall at the new end expands into its characteristic rounded shape in part because it is softer than the mature lateral wall. These studies provide a simple example of how turgor pressure expands the elastic cell wall to generate a particular cell shape.

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.

Intergenerational continuity of cell shape dynamics in Caulobacter crescentus

We investigate the intergenerational shape dynamics of single Caulobacter crescentus cells using a novel combination of imaging techniques and theoretical modeling. We determine the dynamics of cell pole-to-pole lengths, cross-sectional widths, and medial curvatures from high accuracy measurements of cell contours. Moreover, these shape parameters are determined for over 250 cells across approximately 10000 total generations, which affords high statistical precision. Our data and model show that constriction is initiated early in the cell cycle and that its dynamics are controlled by the time scale of exponential longitudinal growth. Based on our extensive and detailed growth and contour data, we develop a minimal mechanical model that quantitatively accounts for the cell shape dynamics and suggests that the asymmetric location of the division plane reflects the distinct mechanical properties of the stalked and swarmer poles. Furthermore, we find that the asymmetry in the division plane location is inherited from the previous generation. We interpret these results in terms of the current molecular understanding of shape, growth, and division of C. crescentus.

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.

Controlling mechanisms in directional growth of aggregated archaeal cells

Members of the family Methanosarcinaceae are important archaeal representatives due to their broad functionality, ubiquitous presence, and functionality in harsh environments. A key characteristic is their multicellular (packet) morphology represented by aggregates of spatially confined cells. This morphology is driven by directed growth of cells in confinement with sequential variation in growth direction. To further understand why spatially confined Methanosarcina cells (and in general, confined prokaryotes) change their direction of growth during consecutive growth-division stages, and how a particular cell senses its wall topology and responds to changes on it a theoretical model for stress dependent growth of aggregated archaeal cells was developed. The model utilizes a confined elastic shell representation of aggregated archaeal cell and is derived based on a work-energy principle. The growth law takes into account the fine structure of archaeal cell wall, polymeric nature of methanochondroitin layer, molecular-biochemical processes and is based on thermodynamic laws. The developed model has been applied to three typical configurations of aggregated cell in 3D. The developed model predicted a geometry response with delayed growth of aggregated archaeal cells explained from mechanistic principles, as well as continuous changes in direction of growth during the consecutive growth-division stages. This means that cell wall topology sensing and growth anisotropy can be predicted using simple cellular mechanisms without the need for dedicated cellular machinery.

Symmetry breaking in spore germination relies on an interplay between polar cap stability and spore wall mechanics

The morphogenesis of single cells depends on their ability to coordinate surface mechanics and polarity. During germination, spores of many species develop a polar tube that hatches out of a rigid outer spore wall (OSW) in a process termed outgrowth. However, how these awakening cells reorganize to stabilize this first growth axis remains unknown. Here, using quantitative experiments and modeling, we reveal the mechanisms underlying outgrowth in fission yeast. We find that, following an isotropic growth phase during which a single polarity cap wanders around the surface, outgrowth occurs when spores have doubled their volume, concomitantly with the stabilization of the cap and a singular rupture in the OSW. This rupture happens when OSW mechanical stress exceeds a threshold, releases the constraints of the OSW on growth, and stabilizes polarity. Thus, outgrowth exemplifies a self-organizing morphogenetic process in which reinforcements between growth and polarity coordinate mechanics and internal organization.

Role of hysteresis in stomatal aperture dynamics

Stomata are pores responsible for gas exchange in leaves. Several experiments indicate that stomata synchronize into clusters or patches. The patches' coordination may produce oscillations in stomatal conductance. Previous studies claim to reproduce some experimental results. However, none was able to explain the variety of behavior observed in the stomatal dynamics. Recently, Ferraz and Prado suggested a realistic geometry of vein distribution. Although it reproduces the patches, no oscillation was observed and the patches remain static. Without exploring significant details, the authors stated that hysteresis in stomatal aperture could explain several experimental features. In this paper, the hysteresis hypothesis is further explored through the concept of hysteretic operators. We have shown that the hysteresis assumption is sufficient to obtain dynamical patches and oscillations in stomatal conductance. The robustness of this hypothesis is tested by using different hysteresis operators. The model analysis reveals a dependence between the period of oscillation in stomatal conductance and the water deficit between the leaf and the environment. This underlying feature of the model might inspire further experiments to test this hypothesis.

The pollen tube paradigm revisited

The polar growth process characterizing pollen tube elongation has attracted numerous modeling attempts over the past years. While initial models focused on recreating the correct cellular geometry, recent models are increasingly based on experimentally assessed cellular parameters such as the dynamics of signaling processes and the mechanical properties of the cell wall. Recent modeling attempts have therefore substantially gained in biological relevance and predictive power. Different modeling methods are explained and the power and limitations of individual models are compared. Focus is on several recent models that use closed feedback loops in order to generate limit cycles representing the oscillatory behavior observed in growing tubes.

Hierarchical helical order in the twisted growth of plant organs

The molecular and cellular basis of left-right asymmetry in plant morphogenesis is a fundamental issue in biology. A rapidly elongating root or hypocotyl of twisting mutants of Arabidopsis thaliana exhibits a helical growth with a handedness opposite to that of the underlying cortical microtubule arrays in epidermal cells. However, how such a hierarchical helical order emerges is currently unknown. We propose a model for investigating macroscopic chiral asymmetry in Arabidopsis mutants. Our elastic model suggests that the helical pattern observed is a direct consequence of the simultaneous presence of anisotropic growth and tilting of cortical microtubule arrays. We predict that the root helical pitch angle is a function of the microtubule helical angle and elastic moduli of the tissues. The proposed model is versatile and is potentially important for other biological systems ranging from protein fibrous structures to tree trunks.

The role of mechanical forces in plant morphogenesis

The shape of an organism relies on a complex network of genetic regulations and on the homeostasis and distribution of growth factors. In parallel to the molecular control of growth, shape changes also involve major changes in structure, which by definition depend on the laws of mechanics. Thus, to understand morphogenesis, scientists have turned to interdisciplinary approaches associating biology and physics to investigate the contribution of mechanical forces in morphogenesis, sometimes re-examining theoretical concepts that were laid out by early physiologists. Major advances in the field have notably been possible thanks to the development of computer simulations and live quantitative imaging protocols in recent years. Here, we present the mechanical basis of shape changes in plants, focusing our discussion on undifferentiated tissues. How can growth be translated into a quantified geometrical output? What is the mechanical basis of cell and tissue growth? What is the contribution of mechanical forces in patterning?

Morphology, growth, and size limit of bacterial cells

Bacterial cells utilize a living peptidoglycan network (PG) to separate the cell interior from the surroundings. The shape of the cell is controlled by PG synthesis and cytoskeletal proteins that form bundles and filaments underneath the cell wall. The PG layer also resists turgor pressure and protects the cell from osmotic shock. We argue that mechanical influences alter the chemical equilibrium of the reversible PG assembly and determine the cell shape and cell size. Using a mechanochemical approach, we show that the cell shape can be regarded as a steady state of a growing network under the influence of turgor pressure and mechanical stress. Using simple elastic models, we predict the size of common spherical and rodlike bacteria. The influence of cytoskeletal bundles such as crescentin and MreB are discussed within the context of our model.

Back to the future: evolution of computational models in plant morphogenesis

There has been a recent surge of studies in plant biology that combine experimental data with computational modeling. Here, we categorize a diversity of theoretical models and emphasize the need to tailor modeling approaches to the questions at hand. Models can start from biophysical or purely heuristic basic principles, and can focus at several levels of biological organization. Recent examples illustrate that this entire spectrum can be useful to understand plant development, and point to a future direction where more approaches are combined in fruitful ways--either by proving the same result with different basic principles or by exploring interactions across levels, in the so-called multilevel models.

Explaining lengths and shapes of yeast by scaling arguments

Lengths and shapes are approached in different ways in different fields: they serve as a read-out for classifying genes or proteins in cell biology whereas they result from scaling arguments in condensed matter physics. Here, we propose a combined approach with examples illustrated for the fission yeast Schizosaccharomyces pombe.

Mechanical forces of fission yeast growth

Mechanical properties contribute to the control of cell size, morphogenesis, development, and lifestyle of fungal cells. Tip growth can be understood by a viscoplastic model, in which growth is derived by high internal turgor pressure and cell-wall elasticity. To understand how these properties regulate growth in the rod-shaped fission yeast Schizosaccaromyces pombe, we devised femtoliter cylindrical polydimethylsiloxane (PDMS) microchambers with varying elasticity as force sensors for single cells. By buckling cells in these chambers, we determine the elastic surface modulus of the cell wall to be 20.2 +/- 6.1 N.m(-1). By analyzing the growth of the cells as they push against the walls of the chamber, we derive force-velocity relationships and values for internal effective turgor pressure of 0.85 +/- 0.15 MPa and a growth-stalling force of 11 +/- 3 muN. The behavior of cells buckling under the force of their own growth provides an independent test of this model and parameters. Force generation is dependent on turgor pressure and a glycerol synthesis gene, gpd1(+) (glycerol-3-phosphate dehydrogenase), and is independent of actin cables. This study develops a quantitative framework for tip cell growth and characterizes mechanisms of force generation that contribute to fungal invasion into host tissues.

The flagellar cytoskeleton of the spirochetes

The recent discoveries of prokaryotic homologs of all three major eukaryotic cytoskeletal proteins (actin, tubulin, intermediate filaments) have spurred a resurgence of activity in the field of bacterial morphology. In spirochetes, however, it has long been known that the flagellar filaments act as a cytoskeletal protein structure, contributing to their shape and conferring motility on this unique phylum of bacteria. Therefore, revisiting the spirochete cytoskeleton may lead to new paradigms for exploring general features of prokaryotic morphology. This review discusses the role that the periplasmic flagella in spirochetes play in maintaining shape and producing motility. We focus on four species of spirochetes: Borrelia burgdorferi, Treponema denticola, Treponema phagedenis and Leptonema (formerly Leptospira) illini. In spirochetes, the flagella reside in the periplasmic space. Rotation of the flagella in the above species by a flagellar motor induces changes in the cell morphology that drives motility. Mutants that do not produce flagella have a markedly different shape than wild-type cells.

Estimates of biomechanical forces in Magnaporthe grisea

The mechanical actions of the fungus Magnaporthe grisea raise many intriguing questions concerning the forces involved. These include: (1) the material properties of the appressorial wall; (2) the strength of the adhesive that keeps the appressorium anchored to the rice leaf surface; and (3) the forces involved in the penetration process whereby a peg is driven through the host cell wall. In this paper we give order of magnitude estimates for all three of these quantities. A simple Young-Laplace law type argument is used to show that the appressorial wall elastic modulus is of order 10-100 MPa; and an adaptation of standard adhesion theory indicates a lower bound on the strength of the appressorial adhesive to be of the order 500 J/m(2). Drawing on ideas from plasticity theory and ballistics, estimates of the penetration force raise interesting questions about experiments performed on the penetration of inert substrates by the fungus.

Buckling of swelling gels

The patterns arising from the differential swelling of gels are investigated experimentally and theoretically as a model for the differential growth of living tissues. Two geometries are considered: a thin strip of soft gel clamped to a stiff gel, and a thin corona of soft gel clamped to a disk of stiff gel. When the structure is immersed in water, the soft gel swells and bends out of plane leading to a wavy periodic pattern whose wavelength is measured. The linear stability of the flat state is studied in the framework of linear elasticity using the equations for thin plates. The flat state is shown to become unstable to oscillations above a critical swelling rate and the computed wavelengths are in quantitative agreement with the experiment.

Biomechanical model for appressorial design in Magnaporthe grisea

The fungus Magnaporthe grisea, commonly referred to as the rice blast fungus, is responsible for destroying from 10% to 30% of the world's rice crop each year. The fungus attaches to the rice leaf and forms a dome-shaped structure, the appressorium, in which enormous pressures are generated that are used to blast a penetration peg through the rice cell walls and infect the plant. We develop a model of the appressorial design in terms of a bioelastic shell that can explain the shape of the appressorium, and its ability to maintain that shape under the enormous increases in turgor pressure that can occur during the penetration phase.

The diffusive vesicle supply center model for tip growth in fungal hyphae

We propose the diffusive vesicle supply center model for tip growth in fungal hyphae. The model is based on the three-dimensional vesicle supply center (VSC) model [Gierz, G., Bartnicki-García, S., 2001. A three-dimensional model of fungal morphogenesis based on the vesicle supply center concept: J. Theor. Biol. 208, 151-164], but incorporates two aspects of a more realistic vesicle delivery mechanism: vesicle diffusion from the VSC and a finite rate constant for vesicle fusion with the cell membrane. We develop a framework to describe tip growth for a general class of models based on the vesicle supply center concept. Combining this with a method for calculating the steady state distribution of diffusive vesicles we iteratively solve for stationary cell shapes. These show a blunter tip than predicted by the original VSC model, which we attribute to increased forward-directed vesicle delivery via diffusion. The predicted distance between the VSC and the utmost tip of the cell is set by the ratio between the diffusion constant and the rate constant for vesicle exocytosis. Combined with the cell radius, these define the only dimensionless parameter for our model.

Mass flow and pressure-driven hyphal extension in Neurospora crassa

Mass flow of cytoplasm inNeurospora crassatrunk hyphae was directly confirmed by injecting oil droplets into the hyphae. The droplets move in a manner similar to cytoplasmic particles and vacuoles within the hyphae. The direction of mass flow is towards the growing hyphal tips at the colony edge. Based on flow velocities (about 5 μm s−1), hyphal radius and estimates of cytoplasm viscosity, the Reynolds number is about 10−4, indicating that mass flow is laminar. Therefore, the Poiseulle equation can be used to calculate the pressure gradient required for mass flow: 0·0005–0·1 bar cm−1(depending on the values used for septal pore radius and cytoplasmic viscosity). These values are very small compared to the normal hydrostatic pressure of the hyphae (4–5 bar). Mass flow stops after respiratory inhibition with cyanide, or creation of an extracellular osmotic gradient. The flow is probably caused by internal osmotic gradients created by differential ion transport along the hyphae. Apical cytoplasm migrates at the same rate as tip extension, as do oil droplets injected near the tip. Thus, in addition to organelle positioning mediated by molecular motors, pressure-driven mass flow may be an integral part of hyphal extension.