Negative feedback enhances robustness in the yeast polarity establishment circuit

Many cells undergo symmetry-breaking polarization toward a randomly oriented "front" in the absence of spatial cues. In budding yeast, such polarization involves a positive feedback loop that enables amplification of stochastically arising clusters of polarity factors. Previous mathematical modeling suggested that, if more than one cluster were amplified, the clusters would compete for limiting resources and the largest would "win," explaining why yeast cells always make one and only one bud. Here, using imaging with improved spatiotemporal resolution, we show the transient coexistence of multiple clusters during polarity establishment, as predicted by the model. Unexpectedly, we also find that initial polarity factor clustering is oscillatory, revealing the presence of a negative feedback loop that disperses the factors. Mathematical modeling predicts that negative feedback would confer robustness to the polarity circuit and make the kinetics of competition between polarity factor clusters relatively insensitive to polarity factor concentration. These predictions are confirmed experimentally.     

Robust Spatial Sensing of Mating Pheromone Gradients by Yeast Cells

Projecting or moving up a chemical gradient is a universal behavior of living organisms. We tested the ability of S. cerevisiae a-cells to sense and respond to spatial gradients of the mating pheromone α-factor produced in a microfluidics chamber; the focus was on bar1Δ strains, which do not degrade the pheromone input. The yeast cells exhibited good accuracy with the mating projection typically pointing in the correct direction up the gradient (∼80% under certain conditions), excellent sensitivity to shallow gradients, and broad dynamic range so that gradient-sensing was relatively robust over a 1000-fold range of average α-factor concentrations. Optimal directional sensing occurred at lower concentrations (5 nM) close to the K_d of the receptor and with steeper gradient slopes. Pheromone supersensitive mutations (sst2Δ and ste2^300Δ) that disrupt the down-regulation of heterotrimeric G-protein signaling caused defects in both sensing and response. Interestingly, yeast cells employed adaptive mechanisms to increase the robustness of the process including filamentous growth (i.e. directional distal budding) up the gradient at low pheromone concentrations, bending of the projection to be more aligned with the gradient, and forming a more accurate second projection when the first projection was in the wrong direction. Finally, the cells were able to amplify a shallow external gradient signal of α-factor to produce a dramatic polarization of signaling proteins at the front of the cell. Mathematical modeling revealed insights into the mechanism of this amplification and how the supersensitive mutants can disrupt accurate polarization. Together, these data help to specify and elucidate the abilities of yeast cells to sense and respond to spatial gradients of pheromone.     

A bistable model of cell polarity

Ultrasensitivity, as described by Goldbeter and Koshland, has been considered for a long time as a way to realize bistable switches in biological systems. It is not as well recognized that when ultrasensitivity and reinforcing feedback loops are present in a spatially distributed system such as the cell plasmamembrane, they may induce bistability and spatial separation of the system into distinct signaling phases. Here we suggest that bistability of ultrasensitive signaling pathways in a diffusive environment provides a basic mechanism to realize cell membrane polarity. Cell membrane polarization is a fundamental process implicated in several basic biological phenomena, such as differentiation, proliferation, migration and morphogenesis of unicellular and multicellular organisms. We describe a simple, solvable model of cell membrane polarization based on the coupling of membrane diffusion with bistable enzymatic dynamics. The model can reproduce a broad range of symmetry-breaking events, such as those observed in eukaryotic directional sensing, the apico-basal polarization of epithelium cells, the polarization of budding and mating yeast, and the formation of Ras nanoclusters in several cell types.     

Modelling of Yeast Mating Reveals Robustness Strategies for Cell-Cell Interactions

Mating of budding yeast cells is a model system for studying cell-cell interactions. Haploid yeast cells secrete mating pheromones that are sensed by the partner which responds by growing a mating projection toward the source. The two projections meet and fuse to form the diploid. Successful mating relies on precise coordination of dynamic extracellular signals, signaling pathways, and cell shape changes in a noisy background. It remains elusive how cells mate accurately and efficiently in a natural multi-cell environment. Here we present the first stochastic model of multiple mating cells whose morphologies are driven by pheromone gradients and intracellular signals. Our novel computational framework encompassed a moving boundary method for modeling both a-cells and α-cells and their cell shape changes, the extracellular diffusion of mating pheromones dynamically coupled with cell polarization, and both external and internal noise. Quantification of mating efficiency was developed and tested for different model parameters. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the α-factor protease Bar1, and the regulation of sensing sensitivity; all were consistent with data in the literature. In addition, we investigated mating discrimination, the ability of an a-cell to distinguish between α-cells either making or not making α-factor, and mating competition, in which multiple a-cells compete to mate with one α-cell. Our simulations were consistent with previous experimental results. Moreover, we performed a combination of simulations and experiments to estimate the diffusion rate of the pheromone a-factor. In summary, we constructed a framework for simulating yeast mating with multiple cells in a noisy environment, and used this framework to reproduce mating behaviors and to identify strategies for robust cell-cell interactions.     

Quantitative effect of scaffold abundance on signal propagation

Protein scaffolds bring together multiple components of a signalling pathway, thereby promoting signal propagation along a common physical ‘backbone’. Scaffolds play a prominent role in natural signalling pathways and provide a promising platform for synthetic circuits. To better understand how scaffolding quantitatively affects signal transmission, we conducted an in vivo sensitivity analysis of the yeast mating pathway to a broad range of perturbations in the abundance of the scaffold Ste5. Our measurements show that signal throughput exhibits a biphasic dependence on scaffold concentration and that altering the amount of scaffold binding partners reshapes this biphasic dependence. Unexpectedly, the wild‐type level of Ste5 is ～10‐fold below the optimum needed to maximize signal throughput. This sub‐optimal configuration may be a tradeoff as increasing Ste5 expression promotes baseline activation of the mating pathway. Furthermore, operating at a sub‐optimal level of Ste5 may provide regulatory flexibility as tuning Ste5 expression up or down directly modulates the downstream phenotypic response. Our quantitative analysis reveals performance tradeoffs in scaffold‐based modules and defines engineering challenges for implementing molecular scaffolds in synthetic pathways.     

A system of counteracting feedback loops regulates Cdc42p activity during spontaneous cell polarization

Cellular polarization is often a response to distinct extracellular or intracellular cues, such as nutrient gradients or cortical landmarks. However, in the absence of such cues, some cells can still select a polarization axis at random. Positive feedback loops promoting localized activation of the GTPase Cdc42p are central to this process in budding yeast. Here, we explore spontaneous polarization during bud site selection in mutant yeast cells that lack functional landmarks. We find that these cells do not select a single random polarization axis, but continuously change this axis during the G1 phase of the cell cycle. This is reflected in traveling waves of activated Cdc42p which randomly explore the cell periphery. Our integrated computational and in vivo analyses of these waves reveal a negative feedback loop that competes with the aforementioned positive feedback loops to regulate Cdc42p activity and confer dynamic responsiveness on the robust initiation of cell polarization.     

Mathematical Analysis of Spontaneous Emergence of Cell Polarity

Cell polarization, in which intracellular substances are asymmetrically distributed, enables cells to carry out specialized functions. While cell polarity is often induced by intracellular or extracellular spatial cues, spontaneous polarization (the so-called symmetry breaking) may also occur in the absence of spatial cues. Many computational models have been used to investigate the mechanisms of symmetry breaking, and it was proved that spontaneous polarization occurs when the lateral diffusion of inactive signaling molecules is much faster than that of active signaling molecules. This conclusion leaves an important question of how, as observed in many biological systems, cell polarity emerges when active and inactive membrane-bound molecules diffuse at similar rates while cycling between cytoplasm and membrane takes place. The recent studies of Rätz and Röger showed that, when the cytosolic and membrane diffusion are very different, spontaneous polarization is possible even if the membrane-bound species diffuse at the same rate. In this paper, we formulate a two-equation non-local reaction-diffusion model with general forms of positive feedback. We apply Turing stability analysis to identify parameter conditions for achieving cell polarization. Our results show that spontaneous polarization can be achieved within some parameter ranges even when active and inactive signaling molecules diffuse at similar rates. In addition, different forms of positive feedback are explored to show that a non-local molecule-mediated feedback is important for sharping the localization as well as giving rise to fast dynamics to achieve robust polarization.     

GDI-Mediated Cell Polarization in Yeast Provides Precise Spatial and Temporal Control of Cdc42 Signaling

Cell polarization is a prerequisite for essential processes such as cell migration, proliferation or differentiation. The yeast Saccharomyces cerevisiae under control of the GTPase Cdc42 is able to polarize without the help of cytoskeletal structures and spatial cues through a pathway depending on its guanine nucleotide dissociation inhibitor (GDI) Rdi1. To develop a fundamental understanding of yeast polarization we establish a detailed mechanistic model of GDI-mediated polarization. We show that GDI-mediated polarization provides precise spatial and temporal control of Cdc42 signaling and give experimental evidence for our findings. Cell cycle induced changes of Cdc42 regulation enhance positive feedback loops of active Cdc42 production, and thereby allow simultaneous switch-like regulation of focused polarity and Cdc42 activation. This regulation drives the direct formation of a unique polarity cluster with characteristic narrowing dynamics, as opposed to the previously proposed competition between transient clusters. As the key components of the studied system are conserved among eukaryotes, we expect our findings also to apply to cell polarization in other organisms.     

Tradeoffs limit the evolution of male traits that are attractive to females

Tradeoffs occur between a variety of traits in a diversity of organisms, and these tradeoffs can have major effects on ecological and evolutionary processes. Far less is known, however, about tradeoffs between male traits that affect mate attraction than about tradeoffs between other types of traits. Previous results indicate that females of the variable field cricket, Gryllus lineaticeps, prefer male songs with higher chirp rates and longer chirp durations. In the current study, we tested the hypothesis that a tradeoff between these traits affects the evolution of male song. The two traits were negatively correlated among full-sibling families, consistent with a genetically based tradeoff, and the tradeoff was stronger when nutrients were limiting. In addition, for males from 12 populations reared in a common environment, the traits were negatively correlated within populations, the strength of the tradeoff was largely invariant across populations, and the within-population tradeoff predicted how the traits have evolved among populations. A widespread tradeoff thus affects male trait evolution. Finally, for males from four populations assayed in the field, the traits were negatively correlated within and among populations. The tradeoff is thus robust to the presence of environmental factors that might mask its effects. Together, our results indicate there is a fundamental tradeoff between male traits that: (i) limits the ability of males to produce multiple attractive traits; (ii) limits how male traits evolve; and (iii) might favour plasticity in female mating preferences.     

Multiple nutrient transporters enable cells to mitigate a rate-affinity tradeoff

Eukaryotic genomes often encode multiple transporters for the same nutrient. For example, budding yeast has 17 hexose transporters (HXTs), all of which potentially transport glucose. Using mathematical modelling, we show that transporters that use either facilitated diffusion or symport can have a rate-affinity tradeoff, where an increase in the maximal rate of transport decreases the transporter’s apparent affinity. These changes affect the import flux non-monotonically, and for a given concentration of extracellular nutrient there is one transporter, characterised by its affinity, that has a higher import flux than any other. Through encoding multiple transporters, cells can therefore mitigate the tradeoff by expressing those transporters with higher affinities in lower concentrations of nutrients. We verify our predictions using fluorescent tagging of seven HXT genes in budding yeast and follow their expression over time in batch culture. Using the known affinities of the corresponding transporters, we show that their regulation in glucose is broadly consistent with a rate-affinity tradeoff: as glucose falls, the levels of the different transporters peak in an order that mostly follows their affinity for glucose. More generally, evolution is constrained by tradeoffs. Our findings indicate that one such tradeoff often occurs in the cellular transport of nutrients.     

Endocytosis optimizes the dynamic localization of membrane proteins that regulate cortical polarity

Diverse cell types require the ability to maintain dynamically polarized membrane-protein distributions through balancing transport and diffusion. However, design principles underlying dynamically maintained cortical polarity are not well understood. Here we constructed a mathematical model for characterizing the morphology of dynamically polarized protein distributions. We developed analytical approaches for measuring all model parameters from single-cell experiments. We applied our methods to a well-characterized system for studying polarized membrane proteins: budding yeast cells expressing activated Cdc42. We found that a balance of diffusion, directed transport, and endocytosis was sufficient for accurately describing polarization morphologies. Surprisingly, the model predicts that polarized regions are defined with a precision that is nearly optimal for measured endocytosis rates and that polarity can be dynamically stabilized through positive feedback with directed transport. Our approach provides a step toward understanding how biological systems shape spatially precise, unambiguous cortical polarity domains using dynamic processes.     

Spatial Stochastic Dynamics Enable Robust Cell Polarization

Although cell polarity is an essential feature of living cells, it is far from being well-understood. Using a combination of computational modeling and biological experiments we closely examine an important prototype of cell polarity: the pheromone-induced formation of the yeast polarisome. Focusing on the role of noise and spatial heterogeneity, we develop and investigate two mechanistic spatial models of polarisome formation, one deterministic and the other stochastic, and compare the contrasting predictions of these two models against experimental phenotypes of wild-type and mutant cells. We find that the stochastic model can more robustly reproduce two fundamental characteristics observed in wild-type cells: a highly polarized phenotype via a mechanism that we refer to as spatial stochastic amplification, and the ability of the polarisome to track a moving pheromone input. Moreover, we find that only the stochastic model can simultaneously reproduce these characteristics of the wild-type phenotype and the multi-polarisome phenotype of a deletion mutant of the scaffolding protein Spa2. Significantly, our analysis also demonstrates that higher levels of stochastic noise results in increased robustness of polarization to parameter variation. Furthermore, our work suggests a novel role for a polarisome protein in the stabilization of actin cables. These findings elucidate the intricate role of spatial stochastic effects in cell polarity, giving support to a cellular model where noise and spatial heterogeneity combine to achieve robust biological function.     

A diffusion-translocation model for gradient sensing by chemotactic cells

Small chemotactic cells like Dictyostelium and neutrophils transduce shallow spatial chemoattractant gradients into strongly localized intracellular responses. We show that the capacity of a second messenger to establish and maintain localized signals, is mainly determined by its dispersion range, lambda = the square root of D(m)/k(-1), which must be small compared to the cell's length. Therefore, short-living second messengers (high k(-1)) with diffusion coefficients D(m) in the range of 0-5 microm(2) s(-1) are most suitable. Additional to short dispersion ranges, gradient sensing may include positive feedback mechanisms that lead to local activation and global inhibition of second-messenger production. To introduce the essential nonlinear amplification, we have investigated models in which one or more components of the signal transduction cascade translocate from the cytosol to the second messenger in the plasma membrane. A one-component model is able to amplify a 1.5-fold difference of receptor activity over the cell length into a 15-fold difference of second-messenger concentration. Amplification can be improved considerably by introducing an additional activating component that translocates to the membrane. In both models, communication between the front and the back of the cell is mediated by partial depletion of cytosolic components, which leads to both local activation and global inhibition. The results suggest that a biochemically simple and general mechanism may explain various signal localization phenomena not only in chemotactic cells but also those occurring in morphogenesis and cell differentiation.     

Robustness and aging—A systems-level perspective

The theory of robustness describes a system level property of evolutionary systems, which predicts tradeoffs of great interest for the systems biology of aging, such as accumulation of non-heritable damage, occurrence of fragilities and limitations in performance, optimized allocation of restricted resources and confined redundancies. According to the robustness paradigm cells and organisms evolved into a state of highly optimized tolerance (HOT), which provides robustness to common perturbations, but causes tradeoffs generally characterized as “robust yet fragile”. This raises the question whether the ultimate cause of aging is more than a lack of adaptation, but an inherent fragility of complex evolutionary systems. Since robustness connects to evolutionary designs, consideration of this theory provides a deeper connection between evolutionary aspects of aging, mathematical models and experimental data. In this review several mechanisms influential for aging are re-evaluated in support of robustness tradeoffs. This includes asymmetric cell division improving performance and specialization with limited capacities to prevent and repair age-related damage, as well as feedback control mechanisms optimized to respond to acute stressors, but unable to halt nor revert aging. Improvement in robustness by increasing efficiencies through cellular redundancies in larger organisms alleviates some of the damaging effects of cellular specialization, which can be expressed in allometric relationships. The introduction of the robustness paradigm offers unique insights for aging research and provides novel opportunities for systems biology endeavors.     

Performance of basic strategies for following gradients in two dimensions

Computer models for following stimulus gradients in two-dimensional space were evaluated to determine the relative advantages of different strategies and to identify the issues that must be addressed in making such a comparison. The simulations were implemented with emphasis on making them as general and free of specific assumptions as possible. Performance was defined as progress along the gradient divided by the cost of the movements and time taken. Plausible values of costs were taken from data on animal energetics. The models also included various kinds of noise that limit performance. These included unintended variations and biases in motor outputs as well as sensory inputs. An initial guess at appropriate noise levels led to performance worse than that observed in experiments with leukocytes. Reduced noise levels gave good agreement. Under these, more appropriate, conditions, peak performance for the various models varied from 24 to 99% of the maximum possible. The threshold gradient required to provide performance equal to 1% of the maximum possible varied from 800 to 5000 searcher diameters per gradient decay length. Some models performed well only over a narrow range of gradients. There was no indication of a tradeoff between sensitivity to shallow gradients and high performance in steep gradients. The model of tropotaxis (simultaneous, spatial comparison) with movement in any direction was superior in having the lowest threshold, the highest maximum performance, requiring the fewest parameters to fit, and performed well over the widest range of gradients. This result suggests that amoeboid cells and echinoderms might be particularly well suited to following gradients. The modeling demonstrates the need to obtain quantitative estimates for a number of parameters (relating costs and noise levels) for a more rigorous understanding of the relative advantages of these different strategies.     

Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed tradeoffs

Theory predicts that interspecific tradeoffs are primary determinants of coexistence and community composition. Using information from empirically observed tradeoffs to augment the parametrisation of mechanism‐based models should therefore improve model predictions, provided that tradeoffs and mechanisms are chosen correctly. We developed and tested such a model for 35 grassland plant species using monoculture measurements of three species characteristics related to nitrogen uptake and retention, which previous experiments indicate as important at our site. Matching classical theoretical expectations, these characteristics defined a distinct tradeoff surface, and models parameterised with these characteristics closely matched observations from experimental multi‐species mixtures. Importantly, predictions improved significantly when we incorporated information from tradeoffs by ‘snapping’ characteristics to the nearest location on the tradeoff surface, suggesting that the tradeoffs and mechanisms we identify are important determinants of local community structure. This ‘snapping’ method could therefore constitute a broadly applicable test for identifying influential tradeoffs and mechanisms.     

A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues

Polarization, a primary step in the response of an individual eukaryotic cell to a spatial stimulus, has attracted numerous theoretical treatments complementing experimental studies in a variety of cell types. While the phenomenon itself is universal, details differ across cell types, and across classes of models that have been proposed. Most models address how symmetry breaking leads to polarization, some in abstract settings, others based on specific biochemistry. Here, we compare polarization in response to a stimulus (e.g., a chemoattractant) in cells typically used in experiments (yeast, amoebae, leukocytes, keratocytes, fibroblasts, and neurons), and, in parallel, responses of several prototypical models to typical stimulation protocols. We find that the diversity of cell behaviors is reflected by a diversity of models, and that some, but not all models, can account for amplification of stimulus, maintenance of polarity, adaptation, sensitivity to new signals, and robustness.     

Ecological stoichiometry and the shape of resource-based tradeoffs

Fitness tradeoffs are expected when organisms must allocate a limited resource between two or more traits. Despite the fact that most resources are made up of multiple elements or components, tradeoff models have traditionally been implicitly or explicitly based on a single currency. Here I propose a model for a resource-based tradeoff between two traits that share two resource elements, and suggest ecological stoichiometry as a critical mechanism controlling the shape of tradeoffs. The tradeoff curve in the model has different shapes depending on the elemental stoichiometry of the traits and the available resource. I further describe a conceptual framework that combines a tradeoff curve with a fitness function to predict an optimal combination of traits in stoichiometrically different environments. The model suggests that even though the fitness function is fixed, an optimal combination of traits can vary depending on the stoichiometry of both the traits and the supplied resource.     

西部山区流域生态系统服务权衡与协同关系——以甘肃白龙江流域为例

甘肃白龙江流域是长江上游重要水源涵养和生态屏障区,生态系统服务研究对该流域生态保护与经济发展的双赢具有重要意义。本研究基于InVEST模型进行4种典型生态系统服务(如土壤保持、水源涵养、食物供给、生境质量)评估,并通过相关性和均方根偏差(RMSD)开展服务间多尺度权衡及其成因分析。结果表明: 土壤保持、水源涵养和生境质量两两之间为显著协同关系,食物供给分别与生境质量、土壤保持和水源涵养间为显著权衡关系。3对协同服务和食物供给与生境质量的权衡强度高值主要集中于文县、迭部和舟曲的中高山阴陡坡林区;食物供给与土壤保持、食物供给与水源涵养间的权衡强度高值主要集中于宕昌和武都的中低山阳缓坡农牧区。人类活动驱使下的土地利用/覆被格局空间差异是影响服务权衡程度及其尺度效应的重要因素。