Cell polarity: quantitative modeling as a tool in cell biology

Where physics and biology meet

As part of this special issue on physics and biology, we invited several leading experts that bridge these disciplines to provide their views on the reciprocal contributions of each field and the benefits and challenges of working across physics and biology: introduction provided by Wallace Marshall.

Optimized PAR-2 RING dimerization mediates cooperative and selective membrane binding for robust cell polarity

Cell polarity networks are defined by quantitative features of their constituent feedback circuits, which must be tuned to enable robust and stable polarization, while also ensuring that networks remain responsive to dynamically changing cellular states and/or spatial cues during development. Using the PAR polarity network as a model, we demonstrate that these features are enabled by the dimerization of the polarity protein PAR-2 via its N-terminal RING domain. Combining theory and experiment, we show that dimer affinity is optimized to achieve dynamic, selective, and cooperative binding of PAR-2 to the plasma membrane during polarization. Reducing dimerization compromises positive feedback and robustness of polarization. Conversely, enhanced dimerization renders the network less responsive due to kinetic trapping of PAR-2 on internal membranes and reduced sensitivity of PAR-2 to the anterior polarity kinase, aPKC/PKC-3. Thus, our data reveal a key role for a dynamically oligomeric RING domain in optimizing interaction affinities to support a robust and responsive cell polarity network, and highlight how optimization of oligomerization kinetics can serve as a strategy for dynamic and cooperative intracellular targeting.

Myeloid Deletion of Cdc42 Protects Liver From Hepatic Ischemia-Reperfusion Injury via Inhibiting Macrophage-Mediated Inflammation in Mice

BACKGROUND & AIMS

Hepatic ischemia-reperfusion injury (HIRI) often occurs in liver surgery, such as partial hepatectomy and liver transplantation, in which myeloid macrophage-mediated inflammation plays a critical role. Cell division cycle 42 (Cdc42) regulates cell migration, cytoskeleton rearrangement, and cell polarity. In this study, we explore the role of myeloid Cdc42 in HIRI.

METHODS

Mouse HIRI models were established with 1-hour ischemia followed by 12-hour reperfusion in myeloid Cdc42 knockout (Cdc42mye) and Cdc42flox mice. Myeloid-derived macrophages were traced with RosamTmG fluorescent reporter under LyzCre-mediated excision. The experiments for serum or hepatic enzymic activities, histologic and immunologic analysis, gene expressions, flow cytometry analysis, and cytokine antibody array were performed.

RESULTS

Myeloid deletion of Cdc42 significantly alleviated hepatic damages with the reduction of hepatic necrosis and inflammation, and reserved hepatic functions following HIRI in mice. Myeloid Cdc42 deficiency suppressed the infiltration of myeloid macrophages, reduced the secretion of proinflammatory cytokines, restrained M1 polarization, and promoted M2 polarization of myeloid macrophages in livers. In addition, inactivation of Cdc42 promoted M2 polarization via suppressing the phosphorylation of STAT1 and promoting phosphorylation of STAT3 and STAT6 in myeloid macrophages. Furthermore, pretreatment with Cdc42 inhibitor, ML141, also protected mice from hepatic ischemia-reperfusion injury.

CONCLUSIONS

Inhibition or deletion of myeloid Cdc42 protects liver from HIRI via restraining the infiltration of myeloid macrophages, suppressing proinflammatory response, and promoting M2 polarization in macrophages.

Computer models of cell polarity establishment in plants

Plant development is a complex task, and many processes involve changes in the asymmetric subcellular distribution of cell components that strongly depend on cell polarity. Cell polarity regulates anisotropic growth and polar localization of membrane proteins and helps to identify the cell's position relative to its neighbors within an organ. Cell polarity is critical in a variety of plant developmental processes, including embryogenesis, cell division, and response to external stimuli. The most conspicuous downstream effect of cell polarity is the polar transport of the phytohormone auxin, which is the only known hormone transported in a polar fashion in and out of cells by specialized exporters and importers. The biological processes behind the establishment of cell polarity are still unknown, and researchers have proposed several models that have been tested using computer simulations. The evolution of computer models has progressed in tandem with scientific discoveries, which have highlighted the importance of genetic, chemical, and mechanical input in determining cell polarity and regulating polarity-dependent processes such as anisotropic growth, protein subcellular localization, and the development of organ shapes. The purpose of this review is to provide a comprehensive overview of the current understanding of computer models of cell polarity establishment in plants, focusing on the molecular and cellular mechanisms, the proteins involved, and the current state of the field.

Spatiotemporal Coordination of Rac1 and Cdc42 at the Whole Cell Level during Cell Ruffling

Rho-GTPases are central regulators within a complex signaling network that controls cytoskeletal organization and cell movement. The network includes multiple GTPases, such as the most studied Rac1, Cdc42, and RhoA, along with their numerous effectors that provide mutual regulation through feedback loops. Here we investigate the temporal and spatial relationship between Rac1 and Cdc42 during membrane ruffling, using a simulation model that couples GTPase signaling with cell morphodynamics and captures the GTPase behavior observed with FRET-based biosensors. We show that membrane velocity is regulated by the kinetic rate of GTPase activation rather than the concentration of active GTPase. Our model captures both uniform and polarized ruffling. We also show that cell-type specific time delays between Rac1 and Cdc42 activation can be reproduced with a single signaling motif, in which the delay is controlled by feedback from Cdc42 to Rac1. The resolution of our simulation output matches those of time-lapsed recordings of cell dynamics and GTPase activity. Our data-driven modeling approach allows us to validate simulation results with quantitative precision using the same pipeline for the analysis of simulated and experimental data.

Silencing Notch4 promotes tumorigenesis and inhibits metastasis of triple-negative breast cancer via Nanog and Cdc42

Elucidation of individual Notch protein biology in specific cancer is crucial to develop safe, effective, and tumor-selective Notch-targeting therapeutic reagents for clinical use [1]. Here, we explored the Notch4 function in triple-negative breast cancer (TNBC). We found that silencing Notch4 enhanced tumorigenic ability in TNBC cells via upregulating Nanog expression, a pluripotency factor of embryonic stem cells. Intriguingly, silencing Notch4 in TNBC cells suppressed metastasis via downregulating Cdc42 expression, a key molecular for cell polarity formation. Notably, downregulation of Cdc42 expression affected Vimentin distribution, but not Vimentin expression to inhibit EMT shift. Collectively, our results show that silencing Notch4 enhances tumorigenesis and inhibits metastasis in TNBC, indicating that targeting Notch4 may not be a potential strategy for drug discovery in TNBC.

Spatiotemporal coordination of Rac1 and Cdc42 at the whole cell level during cell ruffling

Rho-GTPases are central regulators within a complex signaling network that controls the cytoskeletal organization and cell movement. This network includes multiple GTPases, such as the most studied Rac1, Cdc42, and RhoA, and their numerous effectors that provide mutual regulation and feedback loops. Here we investigate the temporal and spatial relationship between Rac1 and Cdc42 during membrane ruffling using a simulation model which couples GTPase signaling with cell morphodynamics to capture the GTPase behavior observed with FRET-based biosensors. We show that membrane velocity is regulated by the kinetic rate of GTPase activation rather than the concentration of active GTPase. Our model captures both uniform and polarized ruffling. We also show that cell-type specific time delays between Rac1 and Cdc42 activation can be reproduced with a single signaling motif, in which the delay is controlled by feedback from Cdc42 to Rac1. The resolution of our simulation output matches those of the time-lapsed recordings of cell dynamics and GTPase activity. This approach allows us to validate simulation results with quantitative precision using the same pipeline for the analysis of simulated and experimental data.

Learning Interactions in Reaction Diffusion Equations by Neural Networks

Partial differential equations are common models in biology for predicting and explaining complex behaviors. Nevertheless, deriving the equations and estimating the corresponding parameters remains challenging from data. In particular, the fine description of the interactions between species requires care for taking into account various regimes such as saturation effects. We apply a method based on neural networks to discover the underlying PDE systems, which involve fractional terms and may also contain integration terms based on observed data. Our proposed framework, called Frac-PDE-Net, adapts the PDE-Net 2.0 by adding layers that are designed to learn fractional and integration terms. The key technical challenge of this task is the identifiability issue. More precisely, one needs to identify the main terms and combine similar terms among a huge number of candidates in fractional form generated by the neural network scheme due to the division operation. In order to overcome this barrier, we set up certain assumptions according to realistic biological behavior. Additionally, we use an L2-norm based term selection criterion and the sparse regression to obtain a parsimonious model. It turns out that the method of Frac-PDE-Net is capable of recovering the main terms with accurate coefficients, allowing for effective long term prediction. We demonstrate the interest of the method on a biological PDE model proposed to study the pollen tube growth problem.

Sensing the shape of a cell with reaction diffusion and energy minimization

Some dividing cells sense their shape by becoming polarized along their long axis. Cell polarity is controlled in part by polarity proteins, like Rho GTPases, cycling between active membrane-bound forms and inactive cytosolic forms, modeled as a "wave-pinning" reaction-diffusion process. Does shape sensing emerge from wave pinning? We show that wave pinning senses the cell's long axis. Simulating wave pinning on a curved surface, we find that high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, a simple rule of minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, when we introduce roughness to our surfaces, shape sensing can be disrupted, and high-activity domains can become localized to locations other than the global peaks and valleys of the surface. On rough surfaces, the domains of the wave-pinning model are more robust in finding the peaks and troughs than the minimization rule, although both can become trapped in steady states away from the peaks and valleys. We can control the robustness of shape sensing by altering the Rho GTPase diffusivity and the domain size. We also find that the shape-sensing properties of cell polarity models can explain how domains localize to curved regions of deformed cells. Our results help to understand the factors that allow cells to sense their shape-and the limits that membrane roughness can place on this process.

Persistent cell migration emerges from a coupling between protrusion dynamics and polarized trafficking

Migrating cells present a variety of paths, from random to highly directional ones. While random movement can be explained by basal intrinsic activity, persistent movement requires stable polarization. Here, we quantitatively address emergence of persistent migration in (hTERT)–immortalizedRPE1 (retinal pigment epithelial) cells over long timescales. By live cell imaging and dynamic micropatterning, we demonstrate that the Nucleus-Golgi axis aligns with direction of migration leading to efficient cell movement. We show that polarized trafficking is directed toward protrusions with a 20-min delay, and that migration becomes random after disrupting internal cell organization. Eventually, we prove that localized optogenetic Cdc42 activation orients the Nucleus-Golgi axis. Our work suggests that polarized trafficking stabilizes the protrusive activity of the cell, while protrusive activity orients this polarity axis, leading to persistent cell migration. Using a minimal physical model, we show that this feedback is sufficient to recapitulate the quantitative properties of cell migration in the timescale of hours.

Unsupervised logic-based mechanism inference for network-driven biological processes

Modern analytical techniques enable researchers to collect data about cellular states, before and after perturbations. These states can be characterized using analytical techniques, but the inference of regulatory interactions that explain and predict changes in these states remains a challenge. Here we present a generalizable, unsupervised approach to generate parameter-free, logic-based models of cellular processes, described by multiple discrete states. Our algorithm employs a Hamming-distance based approach to formulate, test, and identify optimized logic rules that link two states. Our approach comprises two steps. First, a model with no prior knowledge except for the mapping between initial and attractor states is built. We then employ biological constraints to improve model fidelity. Our algorithm automatically recovers the relevant dynamics for the explored models and recapitulates key aspects of the biochemical species concentration dynamics in the original model. We present the advantages and limitations of our work and discuss how our approach could be used to infer logic-based mechanisms of signaling, gene-regulatory, or other input-output processes describable by the Boolean formalism.

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.

Spatiotemporal dynamics of a reaction-diffusion model of pollen tube tip growth

A reaction-diffusion model is proposed to describe the mechanisms underlying the spatial distributions of ROP1 and calcium on the pollen tube tip. The model assumes that the plasma membrane ROP1 activates itself through positive feedback loop, while the cytosolic calcium ions inhibit ROP1 via a negative feedback loop. Furthermore it is proposed that lateral movement of molecules on the plasma membrane are depicted by diffusion. It is shown that bistable or oscillatory dynamics could exist even in the non-spatial model, and stationary and oscillatory spatiotemporal patterns are found in the full spatial model which resemble the experimental data of pollen tube tip growth.

Reverse and forward engineering of protein pattern formation

Living systems employ protein pattern formation to regulate important life processes in space and time. Although pattern-forming protein networks have been identified in various prokaryotes and eukaryotes, their systematic experimental characterization is challenging owing to the complex environment of living cells. In turn, cell-free systems are ideally suited for this goal, as they offer defined molecular environments that can be precisely controlled and manipulated. Towards revealing the molecular basis of protein pattern formation, we outline two complementary approaches: the biochemical reverse engineering of reconstituted networks and the de novo design, or forward engineering, of artificial self-organizing systems. We first illustrate the reverse engineering approach by the example of the Escherichia coli Min system, a model system for protein self-organization based on the reversible and energy-dependent interaction of the ATPase MinD and its activating protein MinE with a lipid membrane. By reconstituting MinE mutants impaired in ATPase stimulation, we demonstrate how large-scale Min protein patterns are modulated by MinE activity and concentration. We then provide a perspective on the de novo design of self-organizing protein networks. Tightly integrated reverse and forward engineering approaches will be key to understanding and engineering the intriguing phenomenon of protein pattern formation.This article is part of the theme issue 'Self-organization in cell biology'.

"Essentially, all models are wrong, but some are useful"-a cross-disciplinary agenda for building useful models in cell biology and biophysics

Intuition alone often fails to decipher the mechanisms underlying the experimental data in Cell Biology and Biophysics, and mathematical modeling has become a critical tool in these fields. However, mathematical modeling is not as widespread as it could be, because experimentalists and modelers often have difficulties communicating with each other, and are not always on the same page about what a model can or should achieve. Here, we present a framework to develop models that increase the understanding of the mechanisms underlying one's favorite biological system. Development of the most insightful models starts with identifying a good biological question in light of what is known and unknown in the field, and determining the proper level of details that are sufficient to address this question. The model should aim not only to explain already available data, but also to make predictions that can be experimentally tested. We hope that both experimentalists and modelers who are driven by mechanistic questions will find these guidelines useful to develop models with maximum impact in their field.

Dynamic Migration Modes of Collective Cells

Collective cell migration occurs in a diversity of physiological processes such as wound healing, cancer metastasis, and embryonic morphogenesis. In the collective context, cohesive cells may move as a translational solid, swirl as a fluid, or even rotate like a disk, with scales ranging from several to dozens of cells. In this work, an active vertex model is presented to explore the regulatory roles of social interactions of neighboring cells and environmental confinements in collective cell migration in a confluent monolayer. It is found that the competition between two kinds of intercellular social interactions-local alignment and contact inhibition of locomotion-drives the cells to self-organize into various dynamic coherent structures with a spatial correlation scale. The interplay between this intrinsic length scale and the external confinement dictates the migration modes of collective cells confined in a finite space. We also show that the local alignment-contact inhibition of locomotion coordination can induce giant density fluctuations in a confluent cell monolayer without gaps, which triggers the spontaneous breaking of orientational symmetry and leads to phase separation.

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.

A Note on GRegNetSim: A Tool for the Discrete Simulation and Analysis of Genetic Regulatory Networks

Discrete simulations of genetic regulatory networks were used to study subsystems of yeast successfully. However, implementations of existing models underlying these simulations do not support a graphic interface, and require computations necessary to analyze their results to be done manually. Furthermore, differences between existing models suggest that an enriched model, encompassing both existing models, is needed. We developed a software tool, GRegNetSim, that allows the end-user to describe genetic regulatory networks graphically. The user can specify various transition functions at different nodes of the network, supporting, for example, threshold and gradient effects, and then apply the network to a variety of inputs. GRegNetSim displays the relationship between the inputs and the mode of behavior of the network in a graphic form that is easy to interpret. Furthermore, it can automatically extract statistical data necessary to analyze the simulations. The discrete simulations performed by GRegNetSim can be used to elucidate and predict the behavior, structure and properties of genetic regulatory networks in a unified manner. GRegNetSim is implemented as a Cytoscape App. Installation files, examples and source code, along with a detailed user guide, are freely available at https://sites.google.com/site/gregnetsim/.

Modeling random crawling, membrane deformation and intracellular polarity of motile amoeboid cells

Amoeboid movement is one of the most widespread forms of cell motility that plays a key role in numerous biological contexts. While many aspects of this process are well investigated, the large cell-to-cell variability in the motile characteristics of an otherwise uniform population remains an open question that was largely ignored by previous models. In this article, we present a mathematical model of amoeboid motility that combines noisy bistable kinetics with a dynamic phase field for the cell shape. To capture cell-to-cell variability, we introduce a single parameter for tuning the balance between polarity formation and intracellular noise. We compare numerical simulations of our model to experiments with the social amoeba Dictyostelium discoideum. Despite the simple structure of our model, we found close agreement with the experimental results for the center-of-mass motion as well as for the evolution of the cell shape and the overall intracellular patterns. We thus conjecture that the building blocks of our model capture essential features of amoeboid motility and may serve as a starting point for more detailed descriptions of cell motion in chemical gradients and confined environments.

Lily Pollen Tubes Pulse According to a Simple Spatial Oscillator

Polar growth is a fundamental mode of cell morphogenesis observed in nearly all major groups of organisms. Among polarly growing cells, the angiosperm pollen tubes have emerged as powerful experimental systems in large part because of their oscillatory growth, which provides a window into the network of interactions regulating morphogenesis. Empirical studies of oscillatory pollen tubes have sought to uncover the temporal sequence of cellular and molecular events that constitutes an oscillatory cycle. Here we show that in lily pollen tubes the distance or wavelength (λ = 6.3 ± 1.7 μm) over which an oscillatory cycle unfolds is more robust than the period of oscillation (τ = 39.1 ± 17.6 s) (n = 159 cells). Moreover, the oscillatory cycle is divided into slow and fast phases, with each phase unfolding over precisely one half of the wavelength. Using these observations, we show that a simple spatial bi-oscillator predicts the most common modes of oscillation observed in pollen tubes. These results call into question the traditional view of pollen tube morphogenesis as a temporal succession of cellular events. Space, not time, may be the most natural metric to inteprete the morphogenetic dynamics of these cells.

Spherical Caps in Cell Polarization

Intracellular symmetry breaking plays a key role in wide range of biological processes, both in single cells and in multicellular organisms. An important class of symmetry-breaking mechanisms relies on the cytoplasm/membrane redistribution of proteins that can autocatalytically promote their own recruitment to the plasma membrane. We present an analytical construction and a comprehensive parametric analysis of stable localized patterns in a reaction-diffusion model of such a mechanism in a spherical cell. The constructed patterns take the form of high-concentration patches localized into spherical caps, similar to the patterns observed in the studies of symmetry breaking in single cells and early embryos.

The effect of cell geometry on polarization in budding yeast

The localization (or polarization) of proteins on the membrane during the mating of budding yeast (Saccharomyces cerevisiae) is an important model system for understanding simple pattern formation within cells. While there are many existing mathematical models of polarization, for both budding and mating, there are still many aspects of this process that are not well understood. In this paper we set out to elucidate the effect that the geometry of the cell can have on the dynamics of certain models of polarization. Specifically, we look at several spatial stochastic models of Cdc42 polarization that have been adapted from published models, on a variety of tip-shaped geometries, to replicate the shape change that occurs during the growth of the mating projection. We show here that there is a complex interplay between the dynamics of polarization and the shape of the cell. Our results show that while models of polarization can generate a stable polarization cap, its localization at the tip of mating projections is unstable, with the polarization cap drifting away from the tip of the projection in a geometry dependent manner. We also compare predictions from our computational results to experiments that observe cells with projections of varying lengths, and track the stability of the polarization cap. Lastly, we examine one model of actin polarization and show that it is unlikely, at least for the models studied here, that actin dynamics and vesicle traffic are able to overcome this effect of geometry.

Elongation and shape changes in organisms with cell walls: A dialogue between experiments and models

The generation of anisotropic shapes occurs during morphogenesis of almost all organisms. With the recent renewal of the interest in mechanical aspects of morphogenesis, it has become clear that mechanics contributes to anisotropic forms in a subtle interaction with various molecular actors. Here, we consider plants, fungi, oomycetes, and bacteria, and we review the mechanisms by which elongated shapes are generated and maintained. We focus on theoretical models of the interplay between growth and mechanics, in relation with experimental data, and discuss how models may help us improve our understanding of the underlying biological mechanisms.

A computational framework for cortical microtubule dynamics in realistically shaped plant cells

Plant morphogenesis is strongly dependent on the directional growth and the subsequent oriented division of individual cells. It has been shown that the plant cortical microtubule array plays a key role in controlling both these processes. This ordered structure emerges as the collective result of stochastic interactions between large numbers of dynamic microtubules. To elucidate this complex self-organization process a number of analytical and computational approaches to study the dynamics of cortical microtubules have been proposed. To date, however, these models have been restricted to two dimensional planes or geometrically simple surfaces in three dimensions, which strongly limits their applicability as plant cells display a wide variety of shapes. This limitation is even more acute, as both local as well as global geometrical features of cells are expected to influence the overall organization of the array. Here we describe a framework for efficiently simulating microtubule dynamics on triangulated approximations of arbitrary three dimensional surfaces. This allows the study of microtubule array organization on realistic cell surfaces obtained by segmentation of microscopic images. We validate the framework against expected or known results for the spherical and cubical geometry. We then use it to systematically study the individual contributions of global geometry, cell-edge induced catastrophes and cell-face induced stability to array organization in a cuboidal geometry. Finally, we apply our framework to analyze the highly non-trivial geometry of leaf pavement cells of Arabidopsis thaliana, Nicotiana benthamiana and Hedera helix. We show that our simulations can predict multiple features of the microtubule array structure in these cells, revealing, among others, strong constraints on the orientation of division planes.

A cellular automaton based on plasma membrane turnover accurately recapitulates cell mechanics during epithelial scattering

Epithelial cells can be triggered to actively detach from epithelial tissues and become solitary, migratory and invasive. This process occurs repeatedly in development, where it is termed epithelial-mesenchymal transition (EMT), and can be recapitulated as epithelial scattering in cell culture models. Detachment of cell-cell junctions involves changes in contractile forces, actin cytoskeletal organization, changes in cell-substrate adhesion properties, surface presentation of cell-cell adhesion molecules, and gene expression. That these cellular processes affect each other and share molecular components creates difficulties in generating hypotheses and designing experiments to understand the mechanics of epithelial scattering. Computational modeling is proving a powerful too in such instances. Here we develop a cellular automaton to reveal insights into how cells rupture epithelial cell-cell junctions during scattering. The model is optimized for realistic and stable recapitulation of behavior of single cells, then for realistic simulation of multiple cells forming epithelial colonies. With a workable model of epithelial cell behavior, we then alter model parameters and assess whether we can realistically mimic epithelial scattering. Adjusting model parameters to recapitulate epithelial scattering reveals that induction of cell migration is the major driver of epithelial scattering.

Many roads to symmetry breaking: molecular mechanisms and theoretical models of yeast cell polarity

Mathematical modeling has been instrumental in identifying common principles of cell polarity across diverse systems. These principles include positive feedback loops that are required to destabilize a spatially uniform state of the cell. The conserved small G-protein Cdc42 is a master regulator of eukaryotic cellular polarization. Here we discuss recent developments in studies of Cdc42 polarization in budding and fission yeasts and demonstrate that models describing symmetry-breaking polarization can be classified into six minimal classes based on the structure of positive feedback loops that activate and localize Cdc42. Owing to their generic system-independent nature, these model classes are also likely to be relevant for the G-protein–based symmetry-breaking systems of higher eukaryotes. We review experimental evidence pro et contra different theoretically plausible models and conclude that several parallel and non–mutually exclusive mechanisms are likely involved in cellular polarization of yeasts. This potential redundancy needs to be taken into consideration when interpreting the results of recent cell-rewiring studies.

A microtubule-based minimal model for spontaneous and persistent spherical cell polarity

We propose a minimal model for the spontaneous and persistent generation of polarity in a spherical cell based on dynamic microtubules and a single mobile molecular component. This component, dubbed the polarity factor, binds to microtubules nucleated from a centrosome located in the center of the cell, is subsequently delivered to the cell membrane, where it diffuses until it unbinds. The only feedback mechanism we impose is that the residence time of the microtubules at the membrane increases with the local density of the polarity factor. We show analytically that this system supports a stable unipolar symmetry-broken state for a wide range of parameters. We validate the predictions of the model by 2D particle-based simulations. Our model provides a route towards the creation of polarity in a minimal cell-like environment using a biochemical reconstitution approach.

Exploring the inhibitory effect of membrane tension on cell polarization

Cell polarization toward an attractant is influenced by both physical and chemical factors. Most existing mathematical models are based on reaction-diffusion systems and only focus on the chemical process occurring during cell polarization. However, membrane tension has been shown to act as a long-range inhibitor of cell polarization. Here, we present a cell polarization model incorporating the interplay between Rac GTPase, filamentous actin (F-actin), and cell membrane tension. We further test the predictions of this model by performing single cell measurements of the spontaneous polarization of cancer stem cells (CSCs) and non-stem cancer cells (NSCCs), as the former have lower cell membrane tension. Based on both our model and the experimental results, cell polarization is more sensitive to stimuli under low membrane tension, and high membrane tension improves the robustness and stability of cell polarization such that polarization persists under random perturbations. Furthermore, our simulations are the first to recapitulate the experimental results described by Houk et al., revealing that aspiration (elevation of tension) and release (reduction of tension) result in a decrease in and recovery of the activity of Rac-GTP, respectively, and that the relaxation of tension induces new polarity of the cell body when a cell with the pseudopod-neck-body morphology is severed.

Expanding the scale of molecular biophysics

Here, I argue that some of the secrets of complex biological function rely on assemblies of many heterogeneous proteins that together enable sophisticated sensing and actuating processes. Evolution seems to delight in making these structures and in continually elaborating upon their capabilities. Developing tools that can go beyond the few protein limit, both on the experimental frontier and from a theoretical, conceptual framework, should be an extremely high priority for the next generation of molecular biophysicists.

Spatial pattern formation in microtubule post-translational modifications and the tight localization of motor-driven cargo

Microtubule (MT) "age" can be interpreted as nucleotide state, lattice defects, or post-translational modification (PTM) such as acetylation and detyrosination. In all three cases, these have been recently shown to have functionally-important effects on the dynamics of MT arrays, and can present spatial and temporal heterogeneity. While mathematical models for MT array densities are well-established, here we present equations describing MT age, defined as the mean time since the MT's building blocks (tubulin) were polymerized from their soluble dimer state. We derive the age equations using a mean first-passage time calculation and two complementary approaches: The continuum limit of spatial discretization model, and an adjoint operator approach. These equations can recapitulate the observation that the oldest (most de-tyrosinated) tubulin in axons is near the middle of axons during neuronal development in chick embryos. Furthermore, PTMs influence motor kinetics up to approximately twofold for off-rates and velocities. Our simulations demonstrate that this relatively weak dependence of motor kinetics is sufficient to target motor cargo to a specific location along the array. This localization is tightly peaked in a way that magnifies the relatively small signal of PTM spatial heterogeneity. Thus, MT age can produce long-range spatial patterning without feedbacks or diffusing signals.

An Experimental Model for Simultaneous Study of Migration of Cell Fragments, Single Cells, and Cell Sheets

Recent studies have demonstrated distinctive motility and responses to extracellular cues of cells in isolation, cells collectively in groups, and cell fragments. Here we provide a protocol for generating cell sheets, isolated cells, and cell fragments of keratocytes from zebrafish scales. The protocol starts with a comprehensive fish preparation, followed by critical steps for scale processing and subsequent cell sheet generation, single cell isolation, and cell fragment induction, which can be accomplished in just 3 days including a 36-48 h incubation time. Compared to other approaches that usually produce single cells only or together with either fragments or cell groups, this facile and reliable methodology allows generation of all three motile forms simultaneously. With the powerful genetics in zebrafish our model system offers a useful tool for comparison of the mechanisms by which cell sheets, single cells, and cell fragments respond to extracellular stimuli.

Specification of Dendritogenesis Site in Drosophila aCC Motoneuron by Membrane Enrichment of Pak1 through Dscam1

Precise positioning of dendritic branches is a critical step in the establishment of neuronal circuitry. However, there is limited knowledge on how environmental cues translate into dendrite initiation or branching at a specific position. Here, through a combination of mutation, RNAi, and imaging experiments, we found that a Dscam-Dock-Pak1 hierarchical interaction defines the stereotypical dendrite growth site in the Drosophila aCC motoneuron. This interaction localizes the Cdc42 effector Pak1 to the plasma membrane at the dendrite initiation site before the activation of Cdc42. Ectopic expression of membrane-anchored Pak1 overrides this spatial specification of dendritogenesis, confirming its function in guiding Cdc42 signaling. We further discovered that Dscam1 localization in aCC occurs through an inter-neuronal contact that involves Dscam1 in the partner MP1 neuron. These findings elucidate a mechanism by which Dscam1 controls neuronal morphogenesis through spatial regulation of Cdc42 signaling and, subsequently, cytoskeletal remodeling.

Spontaneous cell polarization: Feedback control of Cdc42 GTPase breaks cellular symmetry

Spontaneous polarization without spatial cues, or symmetry breaking, is a fundamental problem of spatial organization in biological systems. This question has been extensively studied using yeast models, which revealed the central role of the small GTPase switch Cdc42. Active Cdc42-GTP forms a coherent patch at the cell cortex, thought to result from amplification of a small initial stochastic inhomogeneity through positive feedback mechanisms, which induces cell polarization. Here, I review and discuss the mechanisms of Cdc42 activity self-amplification and dynamic turnover. A robust Cdc42 patch is formed through the combined effects of Cdc42 activity promoting its own activation and active Cdc42-GTP displaying reduced membrane detachment and lateral diffusion compared to inactive Cdc42-GDP. I argue the role of the actin cytoskeleton in symmetry breaking is not primarily to transport Cdc42 to the active site. Finally, negative feedback and competition mechanisms serve to control the number of polarization sites.

Modeling large-scale dynamic processes in the cell: polarization, waves, and division

The past decade has witnessed significant developments in molecular biology techniques, fluorescent labeling, and super-resolution microscopy, and together these advances have vastly increased our quantitative understanding of the cell. This detailed knowledge has concomitantly opened the door for biophysical modeling on a cellular scale. There have been comprehensive models produced describing many processes such as motility, transport, gene regulation, and chemotaxis. However, in this review we focus on a specific set of phenomena, namely cell polarization, F-actin waves, and cytokinesis. In each case, we compare and contrast various published models, highlight the relevant aspects of the biology, and provide a sense of the direction in which the field is moving.

Calcium is an organizer of cell polarity in plants

Cell polarity is a fundamental property of pro- and eukaryotic cells. It is necessary for coordination of cell division, cell morphogenesis and signaling processes. How polarity is generated and maintained is a complex issue governed by interconnected feed-back regulations between small GTPase signaling and membrane tension-based signaling that controls membrane trafficking, and cytoskeleton organization and dynamics. Here, we will review the potential role for calcium as a crucial signal that connects and coordinates the respective processes during polarization processes in plants. This article is part of a Special Issue entitled: 13th European Symposium on Calcium.

Every laboratory with a fluorescence microscope should consider counting molecules

Protein numbers in cells determine rates of biological processes, influence the architecture of cellular structures, reveal the stoichiometries of protein complexes, guide in vitro biochemical reconstitutions, and provide parameter values for mathematical modeling. The purpose of this essay is to increase awareness of methods for counting protein molecules using fluorescence microscopy and encourage more cell biologists to report these numbers. We address the state of the field in terms of utility and accuracy of the numbers reported and point readers to references for details of specific techniques and applications.

Building a patchwork - The yeast plasma membrane as model to study lateral domain formation

The plasma membrane (PM) has to fulfill a wide range of biological functions including selective uptake of substances, signal transduction and modulation of cell polarity and cell shape. To allow efficient regulation of these processes many resident proteins and lipids of the PM are laterally segregated into different functional domains. A particularly striking example of lateral segregation has been described for the budding yeast PM, where integral membrane proteins as well as lipids exhibit very slow translational mobility and form a patchwork of many overlapping micron-sized domains. Here we discuss the molecular and physical mechanisms contributing to the formation of a multi-domain membrane and review our current understanding of yeast PM organization. Many of the fundamental principles underlying membrane self-assembly and organization identified in yeast are expected to equally hold true in other organisms, even for the more transient and elusive organization of the PM in mammalian cells. This article is part of a Special Issue entitled: Nanoscale membrane organisation and signalling.

Balance between exocytosis and endocytosis determines the efficacy of sterol-targeting antibiotics

Antifungals targeting membrane ergosterol are longstanding, yet indispensable drugs in clinical use. However, the mechanisms by which the cellular membrane domains recognized by these antibiotics are generated remain largely unknown. Here, we demonstrate that the balance between endocytosis and exocytosis in membrane trafficking is a critical factor in the action of sterol-targeting antibiotics. When fission yeast cells were treated with manumycin A, cellular binding and the action of the antifungals filipin, amphotericin B, and theonellamides, all of which are ergosterol-binders, were abolished. Additionally, manumycin A treatment attenuated Cdc42 activity and inhibited exocytosis, while endocytosis was only moderately suppressed. Similar defects in membrane trafficking could be reproduced by heat shock and genetic perturbation, which also abolished the action of the antibiotics. We propose that exocytosis and endocytosis respectively supply and internalize the specific plasma membrane domains recognized by sterol-targeting antibiotics.

Systems cell biology

Systems cell biology melds high-throughput experimentation with quantitative analysis and modeling to understand many critical processes that contribute to cellular organization and dynamics. Recently, there have been several advances in technology and in the application of modeling approaches that enable the exploration of the dynamic properties of cells. Merging technology and computation offers an opportunity to objectively address unsolved cellular mechanisms, and has revealed emergent properties and helped to gain a more comprehensive and fundamental understanding of cell biology.

Imaging methodologies for systems biology

Systems biology has recently achieved significant success in the understanding of complex interconnected phenomena such as cell polarity and migration. In this context, the definition of systems biology has come to encompass the integration of quantitative measurements with sophisticated modeling approaches. This article will review recent progress in live cell imaging technologies that have expanded the possibilities of quantitative in vivo measurements, particularly in regards to molecule counting and quantitative measurements of protein concentration and dynamics. These methods have gained and continue to gain popularity with the biological community. In general, we will discuss three broad categories: protein interactions, protein quantitation, and protein dynamics.

The structure of dynamic GPCR signaling networks

G-protein-coupled receptors (GPCRs) stimulate signaling networks that control a variety of critical physiological processes. Static information on the map of interacting signaling molecules at the basis of many cellular processes exists, but little is known about the dynamic operation of these networks. Here we focus on two questions. First, Is the network architecture underlying GPCR-activated cellular processes unique in comparison with others such as transcriptional networks? We discuss how spatially localized GPCR signaling requires uniquely organized networks to execute polarized cell responses. Second, What approaches overcome challenges in deciphering spatiotemporally dynamic networks that govern cell behavior? We focus on recently developed microfluidic and optical approaches that allow GPCR signaling pathways to be triggered and perturbed with spatially and temporally variant input while simultaneously visualizing molecular and cellular responses. When integrated with mathematical modeling, these approaches can help identify design principles that govern cell responses to extracellular signals. We outline why optical approaches that allow the behavior of a selected cell to be orchestrated continually are particularly well suited for probing network organization in single cells.

The systemic immune response to trauma: an overview of pathophysiology and treatment

Improvements in the control of haemorrhage after trauma have resulted in the survival of many people who would otherwise have died from the initial loss of blood. However, the danger is not over once bleeding has been arrested and blood pressure restored. Two-thirds of patients who die following major trauma now do so as a result of causes other than exsanguination. Trauma evokes a systemic reaction that includes an acute, non-specific, immune response associated, paradoxically, with reduced resistance to infection. The result is damage to multiple organs caused by the initial cascade of inflammation aggravated by subsequent sepsis to which the body has become susceptible. This Series examines the biological mechanisms and clinical implications of the cascade of events caused by large-scale trauma that leads to multiorgan failure and death, despite the stemming of blood loss. Furthermore, the stark and robust epidemiological finding--namely, that age has a profound influence on the chances of surviving trauma irrespective of the nature and severity of the injury--will be explored. Advances in our understanding of the inflammatory response to trauma, the impact of ageing on this response, and how this information has led to new and emerging treatments aimed at combating immune dysregulation and reduced immunity after injury will also be discussed.

Feedback mechanisms in a mechanical model of cell polarization

Directed cell migration requires a spatially polarized distribution of polymerized actin. We develop and treat a mechanical model of cell polarization based on polymerization and depolymerization of actin filaments at the two ends of a cell, modulated by forces at either end that are coupled by the cell membrane. We solve this model using both a simulation approach that treats filament nucleation, polymerization, and depolymerization stochastically, and a rate-equation approach based on key properties such as the number of filaments N and the number of polymerized subunits F at either end of the cell. The rate-equation approach agrees closely with the stochastic approach at steady state and, when appropriately generalized, also predicts the dynamic behavior accurately. The calculated transitions from symmetric to polarized states show that polarization is enhanced by a high free-actin concentration, a large pointed-end off-rate, a small barbed-end off-rate, and a small spontaneous nucleation rate. The rate-equation approach allows us to perform a linear-stability analysis to pin down the key interactions that drive the polarization. The polarization is driven by a positive-feedback loop having two interactions. First, an increase in F at one side of the cell lengthens the filaments and thus reduces the decay rate of N (increasing N); second, increasing N enhances F because the force per growing filament tip is reduced. We find that the transitions induced by changing system properties result from supercritical pitchfork bifurcations. The filament lifetime depends strongly on the average filament length, and this effect is crucial for obtaining polarization correctly.

Multi-state modeling of biomolecules

Multi-state modeling of biomolecules refers to a series of techniques used to represent and compute the behavior of biological molecules or complexes that can adopt a large number of possible functional states. Biological signaling systems often rely on complexes of biological macromolecules that can undergo several functionally significant modifications that are mutually compatible. Thus, they can exist in a very large number of functionally different states. Modeling such multi-state systems poses two problems: the problem of how to describe and specify a multi-state system (the “specification problem”) and the problem of how to use a computer to simulate the progress of the system over time (the “computation problem”). To address the specification problem, modelers have in recent years moved away from explicit specification of all possible states and towards rule-based formalisms that allow for implicit model specification, including the κ-calculus [1], BioNetGen [2]–[5], the Allosteric Network Compiler [6], and others [7], [8]. To tackle the computation problem, they have turned to particle-based methods that have in many cases proved more computationally efficient than population-based methods based on ordinary differential equations, partial differential equations, or the Gillespie stochastic simulation algorithm[9], [10]. Given current computing technology, particle-based methods are sometimes the only possible option. Particle-based simulators fall into two further categories: nonspatial simulators, such as StochSim [11], DYNSTOC [12], RuleMonkey [9], [13], and the Network-Free Stochastic Simulator (NFSim) [14], and spatial simulators, including Meredys [15], SRSim [16], [17], and MCell [18]–[20]. Modelers can thus choose from a variety of tools, the best choice depending on the particular problem. Development of faster and more powerful methods is ongoing, promising the ability to simulate ever more complex signaling processes in the future.