Dynamics of cell wall elasticity pattern shapes the cell during yeast mating morphogenesis

Recovery of Sustainable Yeast Proteins: A Mechanistic Study of Autolysis and Enzyme Proteolysis as Pretreatments for Mechanical Cell Rupture

Single-cell protein from yeast can provide a sustainable means to meet increasing protein demands. For yeast protein to be digestible it must be released from within the cells using methods such as autolysis, enzyme proteolysis, and mechanical rupture. This study investigates the combination of autolysis or proteolysis and subsequent mechanical rupture on the protein release. The effects of autolysis and proteolysis using papain on the size, wall thickness, and internal structure of freshly harvested and 7 day stored Saccharomyces cerevisiae cells were investigated using particle sizing and electron microscopy. Samples were then subjected to high-pressure homogenization (single pass, 800 bar) and the extent of cell rupture and protein solubilization determined. During autolysis and proteolysis, intracellular contents were released, shrinking the cells, allowing the elastic walls to contract and thicken. Autolyzed cells were mostly deformed rather than ruptured by high-pressure homogenization, which expelled cell contents. Intact untreated cells with thin, stretched cell walls burst during high-pressure homogenization, with the proteins mostly remaining unsolubilized. Stored cells were smaller with thicker cell walls and generally more difficult to rupture. These findings can assist in developing efficient processes for recovering sustainably produced proteins from yeast and other microorganisms.

Analysis of Cell Wall Mechanics in Fission Yeast

The growth and shape of fungal cells, such as fission yeast, are strongly constrained by the mechanics of their cell wall (CW). The cell wall encases the plasma membrane and defines instantaneous cell shapes by opposing turgor pressure-derived stress on the cell surface. Measuring cell wall mechanical properties may thus bring key insights into the regulation of cell morphogenesis, cell growth, but also cell surface integrity and survival. The fission yeast cell wall has a thickness of a few tens to hundreds of nanometers, and bulk elasticity similar to that of rubber (tens of MPa). These mechanical properties vary locally around single cells, for instance, at the new vs. old growing ends, or birth scars, and may also largely depend on growth conditions and life cycle phases. While cell wall thickness and mechanics have been traditionally measured by complex methodologies including electron microscopy and atomic force microscopy, we here propose a method based on light microscopy to infer with medium-throughput cell wall mechanical properties, as well as turgor pressure in time and space in living cells. This analysis will enhance our appreciation of the mechanical regulation of fission yeast cell morphogenesis and may be directly transferable to the study of other fungal cells.

The "weaken-fill-repair" model for cell budding: Linking cell wall biosynthesis with mechanics

The interplay between cellular mechanics and biochemical processes in the cell cycle is not well understood. We propose a quantitative model of cell budding in Saccharomyces cerevisiae as a "weaken-fill-repair" process, linking Newtonian mechanics of the cell wall with biochemical changes that affect its properties. Our model reveals that (1) oscillations in mother cell size during budding are an inevitable outcome of the process; (2) asymmetric division is necessary for the daughter cell to maintain mechanical stiffness; and (3) although various aspects of the cell are constrained and interconnected, the budding process is governed by a single reduced parameter, ψ, which balances osmolyte accumulation with enzymatic wall-weakening to ensure homeostasis. This model provides insights into the evolution of cell walls and their role in cell division, offering a system-level perspective on cell morphology.

Study of impacts of two types of cellular aging on the yeast bud morphogenesis

Understanding the mechanisms of the cellular aging processes is crucial for attempting to extend organismal lifespan and for studying age-related degenerative diseases. Yeast cells divide through budding, providing a classical biological model for studying cellular aging. With their powerful genetics, relatively short cell cycle, and well-established signaling pathways also found in animals, yeast cells offer valuable insights into the aging process. Recent experiments suggested the existence of two aging modes in yeast characterized by nucleolar and mitochondrial declines, respectively. By analyzing experimental data, this study shows that cells evolving into those two aging modes behave differently when they are young. While buds grow linearly in both modes, cells that consistently generate spherical buds throughout their lifespan demonstrate greater efficacy in controlling bud size and growth rate at young ages. A three-dimensional multiscale chemical-mechanical model was developed and used to suggest and test hypothesized impacts of aging on bud morphogenesis. Experimentally calibrated model simulations showed that during the early stage of budding, tubular bud shape in one aging mode could be generated by locally inserting new materials at the bud tip, a process guided by the polarized Cdc42 signal. Furthermore, the aspect ratio of the tubular bud could be stabilized during the late stage as observed in experiments in this work. The model simulation results suggest that the localization of new cell surface material insertion, regulated by chemical signal polarization, could be weakened due to cellular aging in yeast and other cell types, leading to the change and stabilization of the bud aspect ratio.

Study of Impacts of Two Types of Cellular Aging on the Yeast Bud Morphogenesis

Understanding the mechanisms of cellular aging processes is crucial for attempting to extend organismal lifespan and for studying age-related degenerative diseases. Yeast cells divide through budding, providing a classical biological model for studying cellular aging. With their powerful genetics, relatively short lifespan and well-established signaling pathways also found in animals, yeast cells offer valuable insights into the aging process. Recent experiments suggested the existence of two aging modes in yeast characterized by nucleolar and mitochondrial declines, respectively. In this study, by analyzing experimental data it was shown that cells evolving into those two aging modes behave differently when they are young. While buds grow linearly in both modes, cells that consistently generate spherical buds throughout their lifespan demonstrate greater efficacy in controlling bud size and growth rate at young ages. A three-dimensional chemical-mechanical model was developed and used to suggest and test hypothesized mechanisms of bud morphogenesis during aging. Experimentally calibrated simulations showed that tubular bud shape in one aging mode could be generated by locally inserting new materials at the bud tip guided by the polarized Cdc42 signal during the early stage of budding. Furthermore, the aspect ratio of the tubular bud could be stabilized during the late stage, as observed in experiments, through a reduction on the new cell surface material insertion or an expansion of the polarization site. Thus model simulations suggest the maintenance of new cell surface material insertion or chemical signal polarization could be weakened due to cellular aging in yeast and other cell types.

Quantifying turgor pressure in budding and fission yeasts based upon osmotic properties

Walled cells, such as plants, fungi, and bacteria cells, possess a high internal hydrostatic pressure, termed turgor pressure, that drives volume growth and contributes to cell shape determination. Rigorous measurement of turgor pressure, however, remains challenging, and reliable quantitative measurements, even in budding yeast are still lacking. Here, we present a simple and robust experimental approach to access turgor pressure in yeasts based upon the determination of isotonic concentration using protoplasts as osmometers. We propose three methods to identify the isotonic condition - three-dimensional cell volume, cytoplasmic fluorophore intensity, and mobility of a cytGEMs nano-rheology probe - that all yield consistent values. Our results provide turgor pressure estimates of 1.0 ± 0.1 MPa for Schizosaccharomyces pombe, 0.49 ± 0.01 MPa for Schizosaccharomyces japonicus, 0.5 ± 0.1 MPa for Saccharomyces cerevisiae W303a and 0.31 ± 0.03 MPa for Saccharomyces cerevisiae BY4741. Large differences in turgor pressure and nano-rheology measurements between the Saccharomyces cerevisiae strains demonstrate how fundamental biophysical parameters can vary even among wild-type strains of the same species. These side-by-side measurements of turgor pressure in multiple yeast species provide critical values for quantitative studies on cellular mechanics and comparative evolution.

Quantifying turgor pressure in budding and fission yeasts based upon osmotic properties

Walled cells, such as plants, fungi, and bacteria cells, possess a high internal hydrostatic pressure, termed turgor pressure, that drives volume growth and contributes to cell shape determination. Rigorous measurement of turgor pressure, however, remains challenging, and reliable quantitative measurements, even in budding yeast are still lacking. Here, we present a simple and robust experimental approach to access turgor pressure in yeasts based upon the determination of isotonic concentration using protoplasts as osmometers. We propose three methods to identify the isotonic condition - 3D cell volume, cytoplasmic fluorophore intensity, and mobility of a cytGEMs nano-rheology probe - that all yield consistent values. Our results provide turgor pressure estimates of 1.0 ± 0.1 MPa for S. pombe, 0.49 ± 0.01 MPa for S. japonicus, 0.5 ± 0.1 MPa for S. cerevisiae W303a and 0.31 ± 0.03 MPa for S. cerevisiae BY4741. Large differences in turgor pressure and nano-rheology measurements between the S. cerevisiae strains demonstrate how fundamental biophysical parameters can vary even among wildtype strains of the same species. These side-by-side measurements of turgor pressure in multiple yeast species provide critical values for quantitative studies on cellular mechanics and comparative evolution.

Analytical Models for Measuring the Mechanical Properties of Yeast

The mechanical properties of yeast play an important role in many biological processes, such as cell division and growth, maintenance of internal pressure, and biofilm formation. In addition, the mechanical properties of cells can indicate the degree of damage caused by antifungal drugs, as the mechanical parameters of healthy and damaged cells are different. Over the past decades, atomic force microscopy (AFM) and micromanipulation have become the most widely used methods for evaluating the mechanical characteristics of microorganisms. In this case, the reliability of such an estimate depends on the choice of mathematical model. This review presents various analytical models developed in recent years for studying the mechanical properties of both cells and their individual structures. The main provisions of the applied approaches are described along with their limitations and advantages. Attention is paid to the innovative method of low-invasive nanomechanical mapping with scanning ion-conductance microscopy (SICM), which is currently starting to be successfully used in the discovery of novel drugs acting on the yeast cell wall and plasma membrane.

On a finite strain modeling of growth in budding yeast

Cell's ability to proliferate constitutes one of the most defining features of life. The proliferation occurs through a succession of events; the cell cycle, whereby the cell grows and divides. In this paper, focus is made on the growth step and we deal specifically with Saccharomyces cerevisiae yeast that reproduces by budding. For this, we develop a theoretical model to predict the growth powered by the turgor pressure. This cell is herein considered as a thin-walled structure with almost axisymmetrical shape. Due to its soft nature, the large deformation range is a priori assumed through a finite growth modeling framework. The used kinematics is based on the multiplicative decomposition of the deformation gradient into an elastically reversible part and a growth part. Constitutive equations are proposed where use is made of hyperelasticity together with a local evolution equation, this latter to describe the way growth takes place. In particular, two essential parameters are involved: a stress-like threshold, and a characteristic time. The developed model is extended to a shell approach as well. In a finite element context, representative numerical simulations examining stress-dependent growth are given and a parametric study is conducted to show the sensitivity with respect to the above mentioned parameters. Finally, a suggestion for natural contractile ring modeling closes this study.

Bending stiffness of Candida albicans hyphae as a proxy of cell wall properties

The cell wall is a key component of fungi. It constitutes a highly regulated viscoelastic shell which counteracts internal cell turgor pressure. Its mechanical properties thus contribute to define cell morphology. Measurements of the elastic moduli of the fungal cell wall have been carried out in many species including Candida albicans, a major human opportunistic pathogen. They mainly relied on atomic force microscopy, and mostly considered the yeast form. We developed a parallelized pressure-actuated microfluidic device to measure the bending stiffness of hyphae. We found that the cell wall stiffness lies in the MPa range. We then used three different ways to disrupt cell wall physiology: inhibition of beta-glucan synthesis, a key component of the inner cell wall; application of a hyperosmotic shock triggering a sudden decrease of the hyphal diameter; deletion of two genes encoding GPI-modified cell wall proteins resulting in reduced cell wall thickness. The bending stiffness values were affected to different extents by these environmental stresses or genetic modifications. Overall, our results support the elastic nature of the cell wall and its ability to remodel at the scale of the entire hypha over minutes.

The endocytic protein machinery as an actin-driven membrane-remodeling machine

In clathrin-mediated endocytosis, a principal membrane trafficking route of all eukaryotic cells, forces are applied to invaginate the plasma membrane and form endocytic vesicles. These forces are provided by specific endocytic proteins and the polymerizing actin cytoskeleton. One of the best-studied endocytic systems is endocytosis in yeast, known for its simplicity, experimental amenability, and overall similarity to human endocytosis. Importantly, the yeast endocytic protein machinery generates and transmits tremendous force to bend the plasma membrane, making this system beneficial for mechanistic studies of cellular force-driven membrane reshaping. This review summarizes important protein players, molecular functions, applied forces, and open questions and perspectives of this robust, actin-powered membrane-remodeling protein machine.

Non-linear elastic properties of actin patches to partially rescue yeast endocytosis efficiency in the absence of the cross-linker Sac6

Clathrin mediated endocytosis is an essential and complex cellular process involving more than 60 proteins. In yeast, successful endocytosis requires counteracting a large turgor pressure. To this end, yeasts assemble actin patches, which accumulate elastic energy during their assembly. We investigated the material properties of reconstituted actin patches from a wild-type (WT) strain and a mutant strain lacking the cross-linker Sac6 (sac6Δ), which has reduced endocytosis efficiency in live cells. We hypothesized that a change in the viscous properties of the actin patches, which would dissipate more mechanical energy, could explain this reduced efficiency. There was however no significant difference in the viscosity of both types of patches. However, we discovered a significantly different non-linear elastic response. While WT patches had a constant elastic modulus at different stress values, sac6Δ patches had a lower elastic modulus at low stress, before stiffening at higher ones, up to values similar to those of WT patches. To understand the consequences of this discovery, we performed, in vivo, a precise analysis of actin patch dynamics. Our analysis reveals that a small fraction of actin patches successfully complete endocytosis in sac6Δ cells, provided that those assemble an excess of actin at the membrane compared to WT. This observation indicates that the non-linear elastic properties of actin networks in sac6Δ cells contribute to rescue endocytosis, requiring nevertheless more actin material to build-up the necessary stored elastic energy.

Actin-generated force applied during endocytosis measured by Sla2-based FRET tension sensors

Mechanical forces are integral to many cellular processes, including clathrin-mediated endocytosis, a principal membrane trafficking route into the cell. During endocytosis, forces provided by endocytic proteins and the polymerizing actin cytoskeleton reshape the plasma membrane into a vesicle. Assessing force requirements of endocytic membrane remodeling is essential for understanding endocytosis. Here, we determined actin-generated force applied during endocytosis using FRET-based tension sensors inserted into the major force-transmitting protein Sla2 in yeast. We measured at least 8 pN force transmitted over Sla2 molecule, hence possibly more than 300-880 pN applied during endocytic vesicle formation. Importantly, decreasing cell turgor pressure and plasma membrane tension reduced force transmitted over the Sla2. The measurements in hypotonic conditions and mutants lacking BAR-domain membrane scaffolds then showed the limits of the endocytic force-transmitting machinery. Our study provides force values and force profiles critical for understanding the mechanics of endocytosis and potentially other key cellular membrane-remodeling processes.

Understanding the Impact of Industrial Stress Conditions on Replicative Aging in Saccharomyces cerevisiae

In yeast, aging is widely understood as the decline of physiological function and the decreasing ability to adapt to environmental changes. Saccharomyces cerevisiae has become an important model organism for the investigation of these processes. Yeast is used in industrial processes (beer and wine production), and several stress conditions can influence its intracellular aging processes. The aim of this review is to summarize the current knowledge on applied stress conditions, such as osmotic pressure, primary metabolites (e.g., ethanol), low pH, oxidative stress, heat on aging indicators, age-related physiological changes, and yeast longevity. There is clear evidence that yeast cells are exposed to many stressors influencing viability and vitality, leading to an age-related shift in age distribution. Currently, there is a lack of rapid, non-invasive methods allowing the investigation of aspects of yeast aging in real time on a single-cell basis using the high-throughput approach. Methods such as micromanipulation, centrifugal elutriator, or biotinylation do not provide real-time information on age distributions in industrial processes. In contrast, innovative approaches, such as non-invasive fluorescence coupled flow cytometry intended for high-throughput measurements, could be promising for determining the replicative age of yeast cells in fermentation and its impact on industrial stress conditions.

Evaluating Young's Modulus of Single Yeast Cells Based on Compression Using an Atomic Force Microscope with a Flat Tip

In this research, atomic force microscopy (AFM) with a flat tip cantilever is utilized to measure Young's modulus of a whole yeast cell (Saccharomyces cerevisiae BY4741). The results acquired from AFM are similar to those obtained using a microfluidic chip compression system. The mechanical properties of single yeast cells are important parameters which can be examined using AFM. Conventional studies apply AFM with a sharp cantilever tip to indent the cell and measure the force-indentation curve, from which Young's modulus can be calculated. However, sharp tips introduce problems because the shape variation can lead to a different result and cannot represent the stiffness of the whole cell. It can lead to a lack of broader meaning when evaluating Young's modulus of yeast cells. In this report, we confirm the differences in results obtained when measuring the compression of a poly(dimethylsiloxane) bead using a commercial sharp tip versus a unique flat tip. The flat tip effectively avoids tip-derived errors, so we use this method to compress whole yeast cells and generate a force–deformation curve. We believe our proposed method is effective for evaluating Young's modulus of whole yeast cells.

Coordinating cell polarization and morphogenesis through mechanical feedback

Many cellular processes require cell polarization to be maintained as the cell changes shape, grows or moves. Without feedback mechanisms relaying information about cell shape to the polarity molecular machinery, the coordination between cell polarization and morphogenesis, movement or growth would not be possible. Here we theoretically and computationally study the role of a genetically-encoded mechanical feedback (in the Cell Wall Integrity pathway) as a potential coordination mechanism between cell morphogenesis and polarity during budding yeast mating projection growth. We developed a coarse-grained continuum description of the coupled dynamics of cell polarization and morphogenesis as well as 3D stochastic simulations of the molecular polarization machinery in the evolving cell shape. Both theoretical approaches show that in the absence of mechanical feedback (or in the presence of weak feedback), cell polarity cannot be maintained at the projection tip during growth, with the polarization cap wandering off the projection tip, arresting morphogenesis. In contrast, for mechanical feedback strengths above a threshold, cells can robustly maintain cell polarization at the tip and simultaneously sustain mating projection growth. These results indicate that the mechanical feedback encoded in the Cell Wall Integrity pathway can provide important positional information to the molecular machinery in the cell, thereby enabling the coordination of cell polarization and morphogenesis.

Role of combined cell membrane and wall mechanical properties regulated by polarity signals in cell budding

Budding yeast, Saccharomyces cerevisiae, serves as a prime biological model to study mechanisms underlying asymmetric growth. Previous studies have shown that prior to bud emergence, polarization of a conserved small GTPase Cdc42 must be established on the cell membrane of a budding yeast. Additionally, such polarization contributes to the delivery of cell wall remodeling enzymes and hydrolase from cytosol through the membrane, to change the mechanical properties of the cell wall. This leads to the hypothesis that Cdc42 and its associated proteins at least indirectly regulate cell surface mechanical properties. However, how the surface mechanical properties in the emerging bud are changed and whether such change is important are not well understood. To test several hypothesised mechanisms, a novel three-dimensional coarse-grained particle-based model has been developed which describes inhomogeneous mechanical properties of the cell surface. Model simulations predict alternation of the levels of stretching and bending stiffness of the cell surface in the bud region by the polarized Cdc42 signals is essential for initiating bud formation. Model simulations also suggest that bud shape depends strongly on the distribution of the polarized signaling molecules while the neck width of the emerging bud is strongly impacted by the mechanical properties of the chitin and septin rings. Moreover, the temporal change of the bud mechanical properties is shown to affect the symmetry of the bud shape. The 3D model of asymmetric cell growth can also be used for studying viral budding and other vegetative reproduction processes performed via budding, as well as detailed studies of cell growth.

Early endocytosis as a key to understanding mechanisms of plasma membrane tension regulation in filamentous fungi

BACKGROUND INFORMATION

Two main systems regulate plasma membrane tension (PMT) and provide a close connection between the protoplast and the cell wall in fungi: turgor pressure and the actin cytoskeleton. These systems work together with the plasma membrane focal adhesion to the cell wall and their contribution to fungal cell organization and physiology has been partially studied. However, it remains controversial in model filamentous ascomycetes and oomycetes and even less investigated in filamentous basidiomycetes. Early endocytosis can be used to research the mechanisms regulating PMT since the dynamics of early endocytosis is largely dependent on this tension.

RESULTS

This study examined the effects of actin polymerization inhibitors and hyperosmotic shock on early endocytosis and cell morphology in two filamentous basidiomycetes. The main obtained results are: (i) the depolymerisation of F-actin leads to the fast formation of endocytic pits while inhibiting of their scission from the plasma membrane and (ii) the moderate hyperosmotic shock does not affect the dynamics of early endocytosis. These and our other results have allowed suggesting a curtain model for the regulation of PMT in basidiomycetes.

CONCLUSIONS AND SIGNIFICANCE

According to the proposed curtain model, the PMT in many non-apical cells of hyphae is more often regulated not by turgor pressure but by a system of actin driver cables that are associated with the proteins of the focal adhesion sites. The change in PMT occurs similar to the movement of a curtain along the curtain rod using the curtain drivers. This model addresses the fundamental properties of the fungal structure and physiology. It requires confirmation including the currently technically unavailable high-quality labelling of the actin cytoskeleton of the basidiomycetes.

Pulling-force generation by ensembles of polymerizing actin filaments

The process by which actin polymerization generates pulling forces in cellular processes such as endocytosis is less well understood than pushing-force generation. To clarify the basic mechanisms of pulling-force generation, we perform stochastic polymerization simulations for a square array of polymerizing semiflexible actin filaments, having different interactions with the membrane. The filaments near the array center have a strong attractive component. Filament bending and actin-network elasticity are treated explicitly. We find that the outer filaments push on the membrane and the inner filaments pull, with a net balance of forces. The total calculated pulling force is maximized when the central filaments have a very deep potential well, and the outer filaments have no well. The steady-state force is unaffected by the gel rigidity, but equilibration takes longer for softer gels. The force distributions are flat over the pulling and pushing regions. Actin polymerization is enhanced by softening the gel or reducing the filament binding to the membrane. Filament-membrane detachment can occur for softer gels, even if the total binding energy of the filaments to the membrane is 100 [Formula: see text] or more. It propagates via a stress-concentration mechanism similar to that of a brittle crack in a solid, and the breaking stress is determined by a criterion similar to that of the 'Griffith' theory of crack propagation.

Using Atomic Force Microscopy To Illuminate the Biophysical Properties of Microbes

Since its invention in 1986, atomic force microscopy (AFM) has grown from a system designed for imaging inorganic surfaces to a tool used to probe the biophysical properties of living cells and tissues. AFM is a scanning probe technique and uses a pyramidal tip attached to a flexible cantilever to scan across a surface, producing a highly detailed image. While many research articles include AFM images, fewer include force-distance curves, from which several biophysical properties can be determined. In a single force-distance curve, the cantilever is lowered and raised from the surface, while the forces between the tip and the surface are monitored. Modern AFM has a wide variety of applications, but this review will focus on exploring the mechanobiology of microbes, which we believe is of particular interest to those studying biomaterials. We briefly discuss experimental design as well as different ways of extracting meaningful values related to cell surface elasticity, cell stiffness, and cell adhesion from force-distance curves. We also highlight both classic and recent experiments using AFM to illuminate microbial biophysical properties.

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.

Molecular mechanisms of force production in clathrin-mediated endocytosis

During clathrin-mediated endocytosis (CME), a flat patch of membrane is invaginated and pinched off to release a vesicle into the cytoplasm. In yeast CME, over 60 proteins-including a dynamic actin meshwork-self-assemble to deform the plasma membrane. Several models have been proposed for how actin and other molecules produce the forces necessary to overcome the mechanical barriers of membrane tension and turgor pressure, but the precise mechanisms and a full picture of their interplay are still not clear. In this review, we discuss the evidence for these force production models from a quantitative perspective and propose future directions for experimental and theoretical work that could clarify their various contributions.

Structural organization and energy storage in crosslinked actin assemblies

During clathrin-mediated endocytosis in yeast cells, short actin filaments (< 200nm) and crosslinking protein fimbrin assemble to drive the internalization of the plasma membrane. However, the organization of the actin meshwork during endocytosis remains largely unknown. In addition, only a small fraction of the force necessary to elongate and pinch off vesicles can be accounted for by actin polymerization alone. In this paper, we used mathematical modeling to study the self-organization of rigid actin filaments in the presence of elastic crosslinkers in conditions relevant to endocytosis. We found that actin filaments condense into either a disordered meshwork or an ordered bundle depending on filament length and the mechanical and kinetic properties of the crosslinkers. Our simulations also demonstrated that these nanometer-scale actin structures can store a large amount of elastic energy within the crosslinkers (up to 10kBT per crosslinker). This conversion of binding energy into elastic energy is the consequence of geometric constraints created by the helical pitch of the actin filaments, which results in frustrated configurations of crosslinkers attached to filaments. We propose that this stored elastic energy can be used at a later time in the endocytic process. As a proof of principle, we presented a simple mechanism for sustained torque production by ordered detachment of crosslinkers from a pair of parallel filaments.

Membrane bending by actin polymerization

Actin polymerization provides driving force to aid several types of processes that involve pulling the plasma membrane into the cell, including phagocytosis, cellular entry of large viruses, and endocytosis. In endocytosis, actin polymerization is especially important under conditions of high membrane tension or high turgor pressure. Recent modeling efforts have shown how actin polymerization can give rise to a distribution of forces around the endocytic site, and explored how these forces affect the shape dynamics; experiments have revealed the structure of the endocytic machinery in increasing detail, and demonstrated key feedback interactions between actin assembly and membrane curvature. Here we provide a perspective on these findings and suggest avenues for future research.

Mechanical feedback coordinates cell wall expansion and assembly in yeast mating morphogenesis

The shaping of individual cells requires a tight coordination of cell mechanics and growth. However, it is unclear how information about the mechanical state of the wall is relayed to the molecular processes building it, thereby enabling the coordination of cell wall expansion and assembly during morphogenesis. Combining theoretical and experimental approaches, we show that a mechanical feedback coordinating cell wall assembly and expansion is essential to sustain mating projection growth in budding yeast (Saccharomyces cerevisiae). Our theoretical results indicate that the mechanical feedback provided by the Cell Wall Integrity pathway, with cell wall stress sensors Wsc1 and Mid2 increasingly activating membrane-localized cell wall synthases Fks1/2 upon faster cell wall expansion, stabilizes mating projection growth without affecting cell shape. Experimental perturbation of the osmotic pressure and cell wall mechanics, as well as compromising the mechanical feedback through genetic deletion of the stress sensors, leads to cellular phenotypes that support the theoretical predictions. Our results indicate that while the existence of mechanical feedback is essential to stabilize mating projection growth, the shape and size of the cell are insensitive to the feedback.

A master equation approach to actin polymerization applied to endocytosis in yeast

We present a Master Equation approach to calculating polymerization dynamics and force generation by branched actin networks at membranes. The method treats the time evolution of the F-actin distribution in three dimensions, with branching included as a directional spreading term. It is validated by comparison with stochastic simulations of force generation by actin polymerization at obstacles coated with actin “nucleation promoting factors” (NPFs). The method is then used to treat the dynamics of actin polymerization and force generation during endocytosis in yeast, using a model in which NPFs form a ring around the endocytic site, centered by a spot of molecules attaching the actin network strongly to the membrane. We find that a spontaneous actin filament nucleation mechanism is required for adequate forces to drive the process, that partial inhibition of branching and polymerization lead to different characteristic responses, and that a limited range of polymerization-rate values provide effective invagination and obtain correct predictions for the effects of mutations in the active regions of the NPFs.

Endocytosis is a dynamic process by which cells internalize substances from outside the cell. Especially in yeast, endocytosis is mechanically demanding due to the high pressure difference across the cell membrane, or turgor pressure. Polymerization of a branched actin network is the major process providing the mechanical force to overcome the turgor pressure. Understanding the kinetics of the actin network, and the mechanical interaction between the actin network and the cell membrane, is thus crucial for the study of endocytosis. We develop an efficient mathematical framework for actin dynamics that can realistically incorporate these two features, thus providing a practical method for quantitatively modeling actin dynamics during endocytosis. The resulting model mechanistically reveals that spontaneous nucleation at the center of the endocytic site is required for successful endocytosis, distinguishes the roles of branching and polymerization, and predicts several other experimentally testable outcomes. The accuracy and efficiency of the method, in describing both mechanics and chemistry, render it applicable to a broad field of membrane-bending processes.

Local Turgor Pressure Reduction via Channel Clustering

The primary drivers of yeast endocytosis are actin polymerization and curvature-generating proteins, such as clathrin and BAR domain proteins. Previous work has indicated that these factors may not be capable of generating the forces necessary to overcome turgor pressure. Thus local reduction of the turgor pressure, via localized accumulation or activation of solute channels, might facilitate endocytosis. The possible reduction in turgor pressure was calculated numerically, by solving the diffusion equation through a Legendre polynomial expansion. It was found that for a region of increased permeability having radius 45 nm, as few as 60 channels with a spacing of 10 nm could locally decrease the turgor pressure by 50%. We identified a key dimensionless parameter, p = P₁a/D, where P₁ is the increased permeability, a is the radius of the permeable region, and D is the solute diffusion coefficient. When p > 0.44, the turgor pressure is locally reduced by >50%. An approximate analytic theory was used to generate explicit formulas for the turgor pressure reduction in terms of key parameters. These findings may also be relevant to plants, where the mechanisms that allow endocytosis to proceed despite high turgor pressure are largely unknown.

Actin growth profile in clathrin-mediated endocytosis

Clathrin-mediated endocytosis in yeast is driven by a protein patch containing close to 100 different types of proteins. Among the proteins are 5000-10000 copies of polymerized actin, and successful endocytosis requires growth of the actin network. Since it is not known exactly how actin network growth drives endocytosis, we calculate the spatial distribution of actin growth required to generate the force that drives the process. First, we establish the force distribution that must be supplied by actin growth, by combining membrane-bending profiles obtained via electron microscopy with established theories of membrane mechanics. Next, we determine the profile of actin growth, using a continuum mechanics approach and an iterative procedure starting with an actin growth profile obtained from a linear analysis. The profile has fairly constant growth outside a central hole of radius 45-50 nm, but very little growth in this hole. This growth profile can reproduce the required forces if the actin shear modulus exceeds 80 kPa, and the growing filaments can exert very large polymerization forces. The growth profile prediction could be tested via electron-microscopy or super-resolution experiments in which the turgor pressure is suddenly turned off.