A density-dependent switch drives stochastic clustering and polarization of signaling molecules

Multiscale Modeling of Bistability in the Yeast Polarity Circuit

Cell polarity refers to the asymmetric distribution of proteins and other molecules along a specified axis within a cell. Polarity establishment is the first step in many cellular processes. For example, directed growth or migration requires the formation of a cell front and back. In many cases, polarity occurs in the absence of spatial cues. That is, the cell undergoes symmetry breaking. Understanding the molecular mechanisms that allow cells to break symmetry and polarize requires computational models that span multiple spatial and temporal scales. Here, we apply a multiscale modeling approach to examine the polarity circuit of yeast. In addition to symmetry breaking, experiments revealed two key features of the yeast polarity circuit: bistability and rapid dismantling of the polarity site following a loss of signal. We used modeling based on ordinary differential equations (ODEs) to investigate mechanisms that generate these behaviors. Our analysis revealed that a model involving positive and negative feedback acting on different time scales captured both features. We then extend our ODE model into a coarse-grained reaction-diffusion equation (RDE) model to capture the spatial profiles of polarity factors. After establishing that the coarse-grained RDE model qualitatively captures key features of the polarity circuit, we expand it to more accurately capture the biochemical reactions involved in the system. We convert the expanded model to a particle-based model that resolves individual molecules and captures fluctuations that arise from the stochastic nature of biochemical reactions. Our models assume that negative regulation results from negative feedback. However, experimental observations do not rule out the possibility that negative regulation occurs through an incoherent feedforward loop. Therefore, we conclude by using our RDE model to suggest how negative feedback might be distinguished from incoherent feedforward regulation.

Diffusive lensing as a mechanism of intracellular transport and compartmentalization

While inhomogeneous diffusivity has been identified as a ubiquitous feature of the cellular interior, its implications for particle mobility and concentration at different length scales remain largely unexplored. In this work, we use agent-based simulations of diffusion to investigate how heterogeneous diffusivity affects the movement and concentration of diffusing particles. We propose that a nonequilibrium mode of membrane-less compartmentalization arising from the convergence of diffusive trajectories into low-diffusive sinks, which we call 'diffusive lensing,' is relevant for living systems. Our work highlights the phenomenon of diffusive lensing as a potentially key driver of mesoscale dynamics in the cytoplasm, with possible far-reaching implications for biochemical processes.

Computational Tools for Handling Molecular Clusters: Configurational Sampling, Storage, Analysis, and Machine Learning

Computational modeling of atmospheric molecular clusters requires a comprehensive understanding of their complex configurational spaces, interaction patterns, stabilities against fragmentation, and even dynamic behaviors. To address these needs, we introduce the Jammy Key framework, a collection of automated scripts that facilitate and streamline molecular cluster modeling workflows. Jammy Key handles file manipulations between varieties of integrated third-party programs. The framework is divided into three main functionalities: (1) Jammy Key for configurational sampling (JKCS) to perform systematic configurational sampling of molecular clusters, (2) Jammy Key for quantum chemistry (JKQC) to analyze commonly used quantum chemistry output files and facilitate database construction, handling, and analysis, and (3) Jammy Key for machine learning (JKML) to manage machine learning methods in optimizing molecular cluster modeling. This automation and machine learning utilization significantly reduces manual labor, greatly speeds up the search for molecular cluster configurations, and thus increases the number of systems that can be studied. Following the example of the Atmospheric Cluster Database (ACDB) of Elm (ACS Omega, 4, 10965-10984, 2019), the molecular clusters modeled in our group using the Jammy Key framework have been stored in an improved online GitHub repository named ACDB 2.0. In this work, we present the Jammy Key package alongside its assorted applications, which underline its versatility. Using several illustrative examples, we discuss how to choose appropriate combinations of methodologies for treating particular cluster types, including reactive, multicomponent, charged, or radical clusters, as well as clusters containing flexible or multiconformer monomers or heavy atoms. Finally, we present a detailed example of using the tools for atmospheric acid-base clusters.

A PINK1 input threshold arises from positive feedback in the PINK1/Parkin mitophagy decision circuit

Mechanisms that prevent accidental activation of the PINK1/Parkin mitophagy circuit on healthy mitochondria are poorly understood. On the surface of damaged mitochondria, PINK1 accumulates and acts as the input signal to a positive feedback loop of Parkin recruitment, which in turn promotes mitochondrial degradation via mitophagy. However, PINK1 is also present on healthy mitochondria, where it could errantly recruit Parkin and thereby activate this positive feedback loop. Here, we explore emergent properties of the PINK1/Parkin circuit by quantifying the relationship between mitochondrial PINK1 concentrations and Parkin recruitment dynamics. We find that Parkin is recruited to mitochondria only if PINK1 levels exceed a threshold and then only after a delay that is inversely proportional to PINK1 levels. Furthermore, these two regulatory properties arise from the input-coupled positive feedback topology of the PINK1/Parkin circuit. These results outline an intrinsic mechanism by which the PINK1/Parkin circuit can avoid errant activation on healthy mitochondria.

Particle-based simulations reveal two positive feedback loops allow relocation and stabilization of the polarity site during yeast mating

Many cells adjust the direction of polarized growth or migration in response to external directional cues. The yeast Saccharomyces cerevisiae orient their cell fronts (also called polarity sites) up pheromone gradients in the course of mating. However, the initial polarity site is often not oriented towards the eventual mating partner, and cells relocate the polarity site in an indecisive manner before developing a stable orientation. During this reorientation phase, the polarity site displays erratic assembly-disassembly behavior and moves around the cell cortex. The mechanisms underlying this dynamic behavior remain poorly understood. Particle-based simulations of the core polarity circuit revealed that molecular-level fluctuations are unlikely to overcome the strong positive feedback required for polarization and generate relocating polarity sites. Surprisingly, inclusion of a second pathway that promotes polarity site orientation generated relocating polarity sites with properties similar to those observed experimentally. This pathway forms a second positive feedback loop involving the recruitment of receptors to the cell membrane and couples polarity establishment to gradient sensing. This second positive feedback loop also allows cells to stabilize their polarity site once the site is aligned with the pheromone gradient.

Single-cell directional sensing from just a few receptor binding events

Identifying the directionality of signaling sources from noisy input to membrane receptors is an essential task performed by many cell types. A variety of models have been proposed to explain directional sensing in cells. However, many of these require significant computational and memory capacities for the cell. We propose and analyze a simple mechanism in which a cell adopts the direction associated with the first few membrane binding events. This model yields an accurate angular estimate to the source long before steady state is reached in biologically relevant scenarios. Our proposed mechanism allows for reliable estimates of the directionality of external signals using temporal information and assumes minimal computational capacities of the cell.

Oligomerization of peripheral membrane proteins provides tunable control of cell surface polarity

Asymmetric distributions of peripheral membrane proteins define cell polarity across all kingdoms of life. Non-linear positive feedback on membrane binding is essential to amplify and stabilize these asymmetries, but how specific molecular sources of non-linearity shape polarization dynamics remains poorly understood. Here we show that the ability to oligomerize, which is common to many peripheral membrane proteins, can play a profound role in shaping polarization dynamics in simple feedback circuits. We show that size-dependent binding avidity and mobility of membrane-bound oligomers endow polarity circuits with several key properties. Size-dependent membrane binding avidity confers a form of positive feedback on the accumulation of oligomer subunits. Although insufficient by itself, this sharply reduces the amount of additional feedback required for spontaneous emergence and stable maintenance of polarized states. Size-dependent oligomer mobility makes symmetry breaking and stable polarity more robust with respect to variation in subunit diffusivities and cell sizes, and slows the approach to a final stable spatial distribution, allowing cells to "remember" polarity boundaries imposed by transient external cues. Together, these findings reveal how oligomerization of peripheral membrane proteins can provide powerful and highly tunable sources of non-linear feedback in biochemical circuits that govern cell surface polarity. Given its prevalence and widespread involvement in cell polarity, we speculate that self-oligomerization may have provided an accessible path to evolving simple polarity circuits.

Regulation of stem cell dynamics through volume exclusion

The maintenance and regeneration of adult tissues rely on the self-renewal of stem cells. Regeneration without over-proliferation requires precise regulation of the stem cell proliferation and differentiation rates. The nature of such regulatory mechanisms in different tissues, and how to incorporate them in models of stem cell population dynamics, is incompletely understood. The critical birth-death (CBD) process is widely used to model stem cell populations, capturing key phenomena, such as scaling laws in clone size distributions. However, the CBD process neglects regulatory mechanisms. Here, we propose the birth-death process with volume exclusion (vBD), a variation of the birth-death process that considers crowding effects, such as may arise due to limited space in a stem cell niche. While the deterministic rate equations predict a single non-trivial attracting steady state, the master equation predicts extinction and transient distributions of stem cell numbers with three possible behaviours: long-lived quasi-steady state (QSS), and short-lived bimodal or unimodal distributions. In all cases, we approximate solutions to the vBD master equation using a renormalized system-size expansion, QSS approximation and the Wentzel–Kramers–Brillouin method. Our study suggests that the size distribution of a stem cell population bears signatures that are useful to detect negative feedback mediated via volume exclusion.

A positive feedback loop involving the Spa2 SHD domain contributes to focal polarization

Focal polarization is necessary for finely arranged cell-cell interactions. The yeast mating projection, with its punctate polarisome, is a good model system for this process. We explored the critical role of the polarisome scaffold protein Spa2 during yeast mating with a hypothesis motivated by mathematical modeling and tested by in vivo experiments. Our simulations predicted that two positive feedback loops generate focal polarization, including a novel feedback pathway involving the N-terminal domain of Spa2. We characterized the latter using loss-of-function and gain-of-function mutants. The N-terminal region contains a Spa2 Homology Domain (SHD) which is conserved from yeast to humans, and when mutated largely reproduced the spa2Δ phenotype. Our work together with published data show that the SHD domain recruits Msb3/4 that stimulates Sec4-mediated transport of Bud6 to the polarisome. There, Bud6 activates Bni1-catalyzed actin cable formation, recruiting more Spa2 and completing the positive feedback loop. We demonstrate that disrupting this loop at any point results in morphological defects. Gain-of-function perturbations partially restored focal polarization in a spa2 loss-of-function mutant without restoring localization of upstream components, thus supporting the pathway order. Thus, we have collected data consistent with a novel positive feedback loop that contributes to focal polarization during pheromone-induced polarization in yeast.

Orientation of Cell Polarity by Chemical Gradients

Accurate decoding of spatial chemical landscapes is critical for many cell functions. Eukaryotic cells decode local chemical gradients to orient growth or movement in productive directions. Recent work on yeast model systems, whose gradient sensing pathways display much less complexity than those in animal cells, has suggested new paradigms for how these very small cells successfully exploit information in noisy and dynamic pheromone gradients to identify their mates. Pheromone receptors regulate a polarity circuit centered on the conserved Rho-family GTPase, Cdc42. The polarity circuit contains both positive and negative feedback pathways, allowing spontaneous symmetry breaking and also polarity site disassembly and relocation. Cdc42 orients the actin cytoskeleton, leading to focused vesicle traffic that promotes movement of the polarity site and also reshapes the cortical distribution of receptors at the cell surface. In this article, we review the advances from work on yeasts and compare them with the excitable signaling pathways that have been revealed in chemotactic animal cells. Expected final online publication date for the Annual Review of Biophysics, Volume 51 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

A novel stochastic simulation approach enables exploration of mechanisms for regulating polarity site movement

Cells polarize their movement or growth toward external directional cues in many different contexts. For example, budding yeast cells grow toward potential mating partners in response to pheromone gradients. Directed growth is controlled by polarity factors that assemble into clusters at the cell membrane. The clusters assemble, disassemble, and move between different regions of the membrane before eventually forming a stable polarity site directed toward the pheromone source. Pathways that regulate clustering have been identified but the molecular mechanisms that regulate cluster mobility are not well understood. To gain insight into the contribution of chemical noise to cluster behavior we simulated clustering using the reaction-diffusion master equation (RDME) framework to account for molecular-level fluctuations. RDME simulations are a computationally efficient approximation, but their results can diverge from the underlying microscopic dynamics. We implemented novel concentration-dependent rate constants that improved the accuracy of RDME-based simulations, allowing us to efficiently investigate how cluster dynamics might be regulated. Molecular noise was effective in relocating clusters when the clusters contained low numbers of limiting polarity factors, and when Cdc42, the central polarity regulator, exhibited short dwell times at the polarity site. Cluster stabilization occurred when abundances or binding rates were altered to either lengthen dwell times or increase the number of polarity molecules in the cluster. We validated key results using full 3D particle-based simulations. Understanding the mechanisms cells use to regulate the dynamics of polarity clusters should provide insights into how cells dynamically track external directional cues.

Cells localize polarity molecules in a small region of the plasma membrane forming a polarity cluster that directs functions such as migration, reproduction, and growth. Guided by external signals, these clusters move across the membrane allowing cells to reorient growth or motion. The polarity molecules continuously and randomly shuttle between the cluster and the cell cytosol and, as a result, the number and distribution of molecules at the cluster constantly changes. Here we present an improved stochastic simulation algorithm to investigate how such molecular-scale fluctuations induce cluster movement across the cell membrane. Unexpectedly, cluster mobility does not correlate with variations in total molecule abundance within the cluster, but rather with changes in the spatial distribution of molecules that form the cluster. Cluster motion is faster when polarity molecules are scarce and when they shuttle rapidly between the cluster and the cytosol. Our results suggest that cells control cluster mobility by regulating the abundance of polarity molecules and biochemical reactions that affect the time molecules spend at the cluster. We provide insights into how cells harness random molecular behavior to perform functions important for survival, such as detecting the direction of external signals.

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.

Mechanistic insights into actin-driven polarity site movement in yeast

Directed cell growth or migration are critical for the development and function of many eukaryotic cells. These cells develop a dynamic "front" (also called "polarity site") that can change direction. Polarity establishment involves autocatalytic accumulation of polarity regulators, including the conserved Rho-family GTPase Cdc42, but the mechanisms underlying polarity reorientation remain poorly understood. The tractable model yeast, Saccharomyces cerevisiae, relocates its polarity site when searching for mating partners. Relocation requires polymerized actin, and is thought to involve actin-mediated vesicle traffic to the polarity site. In this study, we provide a quantitative characterization of spontaneous polarity site movement as a search process and use a mechanistic computational model that combines polarity protein biochemical interactions with vesicle trafficking to probe how various processes might affect polarity site movement. Our findings identify two previously documented features of yeast vesicle traffic as being particularly relevant to such movement: tight spatial focusing of exocytosis enhances the directional persistence of movement, and association of Cdc42-directed GTPase-Activating Proteins with secretory vesicles increases the distance moved. Furthermore, we suggest that variation in the rate of exocytosis beyond simple Poisson dynamics may be needed to fully account for the characteristics of polarity site movement in vivo.

Stochastic activation and bistability in a Rab GTPase regulatory network

The eukaryotic endomembrane system is controlled by small GTPases of the Rab family, which are activated at defined times and locations in a switch-like manner. While this switch is well understood for an individual protein, how regulatory networks produce intracellular activity patterns is currently not known. Here, we combine in vitro reconstitution experiments with computational modeling to study a minimal Rab5 activation network. We find that the molecular interactions in this system give rise to a positive feedback and bistable collective switching of Rab5. Furthermore, we find that switching near the critical point is intrinsically stochastic and provide evidence that controlling the inactive population of Rab5 on the membrane can shape the network response. Notably, we demonstrate that collective switching can spread on the membrane surface as a traveling wave of Rab5 activation. Together, our findings reveal how biochemical signaling networks control vesicle trafficking pathways and how their nonequilibrium properties define the spatiotemporal organization of the cell.

Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations

Reaction-diffusion systems have been widely used to study spatio-temporal phenomena in cell biology, such as cell polarization. Coupled bulk-surface models naturally include compartmentalization of cytosolic and membrane-bound polarity molecules. Here we study the distribution of the polarity protein Cdc42 in a mass-conserved membrane-bulk model, and explore the effects of diffusion and spatial dimensionality on spatio-temporal pattern formation. We first analyze a one-dimensional (1-D) model for Cdc42 oscillations in fission yeast, consisting of two diffusion equations in the bulk domain coupled to nonlinear ODEs for binding kinetics at each end of the cell. In 1-D, our analysis reveals the existence of symmetric and asymmetric steady states, as well as anti-phase relaxation oscillations typical of slow-fast systems. We then extend our analysis to a two-dimensional (2-D) model with circular bulk geometry, for which species can either diffuse inside the cell or become bound to the membrane and undergo a nonlinear reaction-diffusion process. We also consider a nonlocal system of PDEs approximating the dynamics of the 2-D membrane-bulk model in the limit of fast bulk diffusion. In all three model variants we find that mass conservation selects perturbations of spatial modes that simply redistribute mass. In 1-D, only anti-phase oscillations between the two ends of the cell can occur, and in-phase oscillations are excluded. In higher dimensions, no radially symmetric oscillations are observed. Instead, the only instabilities are symmetry-breaking, either corresponding to stationary Turing instabilities, leading to the formation of stationary patterns, or to oscillatory Turing instabilities, leading to traveling and standing waves. Codimension-two Bogdanov-Takens bifurcations occur when the two distinct instabilities coincide, causing traveling waves to slow down and to eventually become stationary patterns. Our work clarifies the effect of geometry and dimensionality on behaviors observed in mass-conserved cell polarity models.

Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast

Oscillations occur in a wide variety of essential cellular processes, such as cell cycle progression, circadian clocks and calcium signaling in response to stimuli. It remains unclear how intrinsic stochasticity can influence these oscillatory systems. Here, we focus on oscillations of Cdc42 GTPase in fission yeast. We extend our previous deterministic model by Xu and Jilkine to construct a stochastic model, focusing on the fast diffusion case. We use SSA (Gillespie's algorithm) to numerically explore the low copy number regime in this model, and use analytical techniques to study the long-time behavior of the stochastic model and compare it to the equilibria of its deterministic counterpart. Numerical solutions suggest noisy limit cycles exist in the parameter regime in which the deterministic system converges to a stable limit cycle, and quasi-cycles exist in the parameter regime where the deterministic model has a damped oscillation. Near an infinite period bifurcation point, the deterministic model has a sustained oscillation, while stochastic trajectories start with an oscillatory mode and tend to approach deterministic steady states. In the low copy number regime, metastable transitions from oscillatory to steady behavior occur in the stochastic model. Our work contributes to the understanding of how stochastic chemical kinetics can affect a finite-dimensional dynamical system, and destabilize a deterministic steady state leading to oscillations.

The Interplay of Structural and Cellular Biophysics Controls Clustering of Multivalent Molecules

Dynamic molecular clusters are assembled through weak multivalent interactions and are platforms for cellular functions, especially receptor-mediated signaling. Clustering is also a prerequisite for liquid-liquid phase separation. It is not well understood, however, how molecular structure and cellular organization control clustering. Using coarse-grained kinetic Langevin dynamics, we performed computational experiments on a prototypical ternary system modeled after membrane-bound nephrin, the adaptor Nck1, and the actin nucleation promoting factor NWASP. Steady-state cluster size distributions favored stoichiometries that optimized binding (stoichiometry matching) but still were quite broad. At high concentrations, the system can be driven beyond the saturation boundary such that cluster size is limited only by the number of available molecules. This behavior would be predictive of phase separation. Domains close to binding sites sterically inhibited clustering much less than terminal domains because the latter effectively restrict access to the cluster interior. Increased flexibility of interacting molecules diminished clustering by shielding binding sites within compact conformations. Membrane association of nephrin increased the cluster size distribution in a density-dependent manner. These properties provide insights into how molecular ensembles function to localize and amplify cell signaling.

Modeling the Self-Assembly of Protein Complexes through a Rigid-Body Rotational Reaction-Diffusion Algorithm

The reaction-diffusion equations provide a powerful framework for modeling nonequilibrium, cell-scale dynamics over the long time scales that are inaccessible by traditional molecular modeling approaches. Single-particle reaction-diffusion offers the highest resolution technique for tracking such dynamics, but it has not been applied to the study of protein self-assembly due to its treatment of reactive species as single-point particles. Here, we develop a relatively simple but accurate approach for building rigid structure and rotation into single-particle reaction-diffusion methods, providing a rate-based method for studying protein self-assembly. Our simplifying assumption is that reactive collisions can be evaluated purely on the basis of the separations between the sites, and not their orientations. The challenge of evaluating reaction probabilities can then be performed using well-known equations based on translational diffusion in both 3D and 2D, by employing an effective diffusion constant we derive here. We show how our approach reproduces both the kinetics of association, which is altered by rotational diffusion, and the equilibrium of reversible association, which is not. Importantly, the macroscopic kinetics of association can be predicted on the basis of the microscopic parameters of our structurally resolved model, allowing for critical comparisons with theory and other rate-based simulations. We demonstrate this method for efficient, rate-based simulations of self-assembly of clathrin trimers, highlighting how formation of regular lattices impacts the kinetics of association.

Exploration and stabilization of Ras1 mating zone: A mechanism with positive and negative feedbacks

In mating fission yeast cells, sensing and response to extracellular pheromone concentrations occurs through an exploratory Cdc42 patch that stochastically samples the cell cortex before stabilizing towards a mating partner. Active Ras1 (Ras1-GTP), an upstream regulator of Cdc42, and Gap1, the GTPase-activating protein for Ras1, localize at the patch. We developed a reaction-diffusion model of Ras1 patch appearance and disappearance with a positive feedback by a Guanine nucleotide Exchange Factor (GEF) and Gap1 inhibition. The model is based on new estimates of Ras1-GDP, Ras1-GTP and Gap1 diffusion coefficients and rates of cytoplasmic exchange studied by FRAP. The model reproduces exploratory patch behavior and lack of Ras1 patch in cells lacking Gap1. Transition to a stable patch can occur by change of Gap1 rates constants or local increase of the positive feedback rate constants. The model predicts that the patch size and number of patches depend on the strength of positive and negative feedbacks. Measurements of Ras1 patch size and number in cells overexpressing the Ras1 GEF or Gap1 are consistent with the model.

Unicellular fission yeasts mate by fusing with partners of the opposite mating type. Each pair member grows towards its selected partner that signals its presence through secreted pheromone. The process of partner selection occurs through an exploratory patch (containing activated signaling protein Cdc42 and upstream regulator Ras1) that assembles and disassembles on the cell cortex, stabilizing in regions of higher opposite pheromone concentration. We present a computational model of the molecular mechanisms driving the dynamical pattern of patch exploration and stabilization. The model is based on reaction and diffusion along the curved cell membrane, with diffusion coefficients measured experimentally. In the model, a positive Ras1 activation feedback loop generates a patch containing most of the activating protein (Ras1 GEF). The fast diffusing inhibitor Gap1 that is recruited locally from the cytoplasm spreads on the cell membrane, limiting patch size and causing its decay. Spontaneous reinitiation of Ras1 activation elsewhere on the cortex provides a mechanism for exploration. Transition of the system’s behavior to that of a single stable patch is possible upon simulated pheromone sensing. The computational model provides predictions for the number of patches and patch size dependence on parameters that we tested experimentally.

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.

Particle-based simulations of polarity establishment reveal stochastic promotion of Turing pattern formation

Polarity establishment, the spontaneous generation of asymmetric molecular distributions, is a crucial component of many cellular functions. Saccharomyces cerevisiae (yeast) undergoes directed growth during budding and mating, and is an ideal model organism for studying polarization. In yeast and many other cell types, the Rho GTPase Cdc42 is the key molecular player in polarity establishment. During yeast polarization, multiple patches of Cdc42 initially form, then resolve into a single front. Because polarization relies on strong positive feedback, it is likely that the amplification of molecular-level fluctuations underlies the generation of multiple nascent patches. In the absence of spatial cues, these fluctuations may be key to driving polarization. Here we used particle-based simulations to investigate the role of stochastic effects in a Turing-type model of yeast polarity establishment. In the model, reactions take place either between two molecules on the membrane, or between a cytosolic and a membrane-bound molecule. Thus, we developed a computational platform that explicitly simulates molecules at and near the cell membrane, and implicitly handles molecules away from the membrane. To evaluate stochastic effects, we compared particle simulations to deterministic reaction-diffusion equation simulations. Defining macroscopic rate constants that are consistent with the microscopic parameters for this system is challenging, because diffusion occurs in two dimensions and particles exchange between the membrane and cytoplasm. We address this problem by empirically estimating macroscopic rate constants from appropriately designed particle-based simulations. Ultimately, we find that stochastic fluctuations speed polarity establishment and permit polarization in parameter regions predicted to be Turing stable. These effects can operate at Cdc42 abundances expected of yeast cells, and promote polarization on timescales consistent with experimental results. To our knowledge, our work represents the first particle-based simulations of a model for yeast polarization that is based on a Turing mechanism.

Many cells need to generate and maintain biochemical signals in specific subcellular regions. This phenomenon is broadly called polarity establishment, and is important in fundamental processes such as cell migration and differentiation. A key polarity factor found in diverse organisms, including yeast and humans, is the protein Cdc42. In yeast, Cdc42-dependent polarization occurs through a self-reinforcing biochemical signaling loop. Directional cues can guide polarity establishment, but interestingly, yeast can polarize in the absence of such a cue. The mechanism thought to underlie this symmetry breaking involves the amplification of inhomogeneities in molecular distributions that arise from molecular-level fluctuations. We investigated the effects of random fluctuations on polarization by performing particle-based simulations of the Cdc42 signaling network. We found that fluctuations can facilitate polarization, allowing faster polarization, and polarization over a broader range of concentrations. Our observations may help understand how polarity works in other systems.

Smoldyn: particle-based simulation with rule-based modeling, improved molecular interaction and a library interface

Motivation

Smoldyn is a spatial and stochastic biochemical simulator. It treats each molecule of interest as an individual particle in continuous space, simulating molecular diffusion, molecule-membrane interactions and chemical reactions, all with good accuracy. This article presents several new features.

Results

Smoldyn now supports two types of rule-based modeling. These are a wildcard method, which is very convenient, and the BioNetGen package with extensions for spatial simulation, which is better for complicated models. Smoldyn also includes new algorithms for simulating the diffusion of surface-bound molecules and molecules with excluded volume. Both are exact in the limit of short time steps and reasonably good with longer steps. In addition, Smoldyn supports single-molecule tracking simulations. Finally, the Smoldyn source code can be accessed through a C/C ++ language library interface.

Availability and Implementation

Smoldyn software, documentation, code, and examples are at http://www.smoldyn.org .

Contact

steven.s.andrews@gmail.com.

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.

Local sampling paints a global picture: Local concentration measurements sense direction in complex chemical gradients

Detecting and interpreting extracellular spatial signals is essential for cellular orientation within complex environments, such as during directed cell migration or growth in multicellular development. Although the molecular understanding of how cells read spatial signals like chemical gradients is still lacking, recent work has revealed that stochastic processes at different temporal and spatial scales are at the core of this gradient sensing process in a wide range of eukaryotes. Fast biochemical reactions like those underlying GTPase activity dynamics form a functional module together with slower cell morphological changes driven by membrane remodelling. This biochemical-morphological module explores the environment by stochastic local concentration sampling to determine the source of the gradient signal, enabling efficient signal detection and interpretation before polarised growth or migration towards the gradient source is initiated. Here we review recent data describing local sampling and propose a model of local fast and slow feedback counteracted by gradient-dependent substrate limitation to be at the core of gradient sensing by local sampling.

A model of cell surface receptor aggregation

In this paper we construct and analyze a model of cell receptor aggregation. Experiments have shown that receptors in an aggregated state have greatly reduced mobility. We model the effects of this reduced mobility with a density dependent diffusion and study the impact of density dependent diffusion on aggregate formation in a one-dimensional domain. Critical values of receptor diffusivity and receptor activation are found and compared with numerical simulations. We find that the role of density dependant diffusion is quite limited in the formation of aggregate structures. In the case of receptor activation, the analytical results agree very well with the numerical calculations. Finally, we consider our model in higher dimensional domains. In this case our analysis is primarily numerical.

Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology

Hybrid deterministic-stochastic methods provide an efficient alternative to a fully stochastic treatment of models which include components with disparate levels of stochasticity. However, general-purpose hybrid solvers for spatially resolved simulations of reaction-diffusion systems are not widely available. Here we describe fundamentals of a general-purpose spatial hybrid method. The method generates realizations of a spatially inhomogeneous hybrid system by appropriately integrating capabilities of a deterministic partial differential equation solver with a popular particle-based stochastic simulator, Smoldyn. Rigorous validation of the algorithm is detailed, using a simple model of calcium 'sparks' as a testbed. The solver is then applied to a deterministic-stochastic model of spontaneous emergence of cell polarity. The approach is general enough to be implemented within biologist-friendly software frameworks such as Virtual Cell.

A framework for discrete stochastic simulation on 3D moving boundary domains

We have developed a method for modeling spatial stochastic biochemical reactions in complex, three-dimensional, and time-dependent domains using the reaction-diffusion master equation formalism. In particular, we look to address the fully coupled problems that arise in systems biology where the shape and mechanical properties of a cell are determined by the state of the biochemistry and vice versa. To validate our method and characterize the error involved, we compare our results for a carefully constructed test problem to those of a microscale implementation. We demonstrate the effectiveness of our method by simulating a model of polarization and shmoo formation during the mating of yeast. The method is generally applicable to problems in systems biology where biochemistry and mechanics are coupled, and spatial stochastic effects are critical.

A Predictive Model for Yeast Cell Polarization in Pheromone Gradients

Budding yeast cells exist in two mating types, a and α, which use peptide pheromones to communicate with each other during mating. Mating depends on the ability of cells to polarize up pheromone gradients, but cells also respond to spatially uniform fields of pheromone by polarizing along a single axis. We used quantitative measurements of the response of a cells to α-factor to produce a predictive model of yeast polarization towards a pheromone gradient. We found that cells make a sharp transition between budding cycles and mating induced polarization and that they detect pheromone gradients accurately only over a narrow range of pheromone concentrations corresponding to this transition. We fit all the parameters of the mathematical model by using quantitative data on spontaneous polarization in uniform pheromone concentration. Once these parameters have been computed, and without any further fit, our model quantitatively predicts the yeast cell response to pheromone gradient providing an important step toward understanding how cells communicate with each other.

MOLNs: A CLOUD PLATFORM FOR INTERACTIVE, REPRODUCIBLE, AND SCALABLE SPATIAL STOCHASTIC COMPUTATIONAL EXPERIMENTS IN SYSTEMS BIOLOGY USING PyURDME

Computational experiments using spatial stochastic simulations have led to important new biological insights, but they require specialized tools and a complex software stack, as well as large and scalable compute and data analysis resources due to the large computational cost associated with Monte Carlo computational workflows. The complexity of setting up and managing a large-scale distributed computation environment to support productive and reproducible modeling can be prohibitive for practitioners in systems biology. This results in a barrier to the adoption of spatial stochastic simulation tools, effectively limiting the type of biological questions addressed by quantitative modeling. In this paper, we present PyURDME, a new, user-friendly spatial modeling and simulation package, and MOLNs, a cloud computing appliance for distributed simulation of stochastic reaction-diffusion models. MOLNs is based on IPython and provides an interactive programming platform for development of sharable and reproducible distributed parallel computational experiments.

A Cellular System for Spatial Signal Decoding in Chemical Gradients

Directional cell growth requires that cells read and interpret shallow chemical gradients, but how the gradient directional information is identified remains elusive. We use single-cell analysis and mathematical modeling to define the cellular gradient decoding network in yeast. Our results demonstrate that the spatial information of the gradient signal is read locally within the polarity site complex using double-positive feedback between the GTPase Cdc42 and trafficking of the receptor Ste2. Spatial decoding critically depends on low Cdc42 activity, which is maintained by the MAPK Fus3 through sequestration of the Cdc42 activator Cdc24. Deregulated Cdc42 or Ste2 trafficking prevents gradient decoding and leads to mis-oriented growth. Our work discovers how a conserved set of components assembles a network integrating signal intensity and directionality to decode the spatial information contained in chemical gradients.

Positive feedback can lead to dynamic nanometer-scale clustering on cell membranes

Clustering of molecules on biological membranes is a widely observed phenomenon. A key example is the clustering of the oncoprotein Ras, which is known to be important for signal transduction in mammalian cells. Yet, the mechanism by which Ras clusters form and are maintained remains unclear. Recently, it has been discovered that activated Ras promotes further Ras activation. Here we show using particle-based simulation that this positive feedback is sufficient to produce persistent clusters of active Ras molecules at the nanometer scale via a dynamic nucleation mechanism. Furthermore, we find that our cluster statistics are consistent with experimental observations of the Ras system. Interestingly, we show that our model does not support a Turing regime of macroscopic reaction-diffusion patterning, and therefore that the clustering we observe is a purely stochastic effect, arising from the coupling of positive feedback with the discrete nature of individual molecules. These results underscore the importance of stochastic and dynamic properties of reaction diffusion systems for biological behavior.

A hardwired HIV latency program

Biological circuits can be controlled by two general schemes: environmental sensing or autonomous programs. For viruses such as HIV, the prevailing hypothesis is that latent infection is controlled by cellular state (i.e., environment), with latency simply an epiphenomenon of infected cells transitioning from an activated to resting state. However, we find that HIV expression persists despite the activated-to-resting cellular transition. Mathematical modeling indicates that HIV's Tat positive-feedback circuitry enables this persistence and strongly controls latency. To overcome the inherent crosstalk between viral circuitry and cellular activation and to directly test this hypothesis, we synthetically decouple viral dependence on cellular environment from viral transcription. These circuits enable control of viral transcription without cellular activation and show that Tat feedback is sufficient to regulate latency independent of cellular activation. Overall, synthetic reconstruction demonstrates that a largely autonomous, viral-encoded program underlies HIV latency—potentially explaining why cell-targeted latency-reversing agents exhibit incomplete penetrance.

Aggregation-fragmentation model of robust concentration gradient formation

Concentration gradients of signaling molecules are essential for patterning during development and they have been observed in both unicellular and multicellular systems. In subcellular systems, clustering of the signaling molecule has been observed. We develop a theoretical model of cluster-mediated concentration gradient formation based on the Becker-Döring equations of aggregation-fragmentation processes. We show that such a mechanism produces robust concentration gradients on realistic time and spatial scales so long as the process of clustering does not significantly stabilize the signaling molecule. Finally, we demonstrate that such a model is applicable to the pom1p subcellular gradient in fission yeast.

Spatial segregation of polarity factors into distinct cortical clusters is required for cell polarity control

Cell polarity is regulated by evolutionarily conserved polarity factors whose precise higher-order organization at the cell cortex is largely unknown. Here we image frontally the cortex of live fission yeast cells using time-lapse and super-resolution microscopy. Interestingly, we find that polarity factors are organized in discrete cortical clusters resolvable to ~50–100 nm in size, which can form and become cortically enriched by oligomerization. We show that forced co-localization of the polarity factors Tea1 and Tea3 results in polarity defects, suggesting that the maintenance of both factors in distinct clusters is required for polarity. However, during mitosis, their co-localization increases, and Tea3 helps to retain the cortical localization of the Tea1 growth landmark in preparation for growth reactivation following mitosis. Thus, regulated spatial segregation of polarity factor clusters provides a means to spatio-temporally control cell polarity at the cell cortex. We observe similar clusters in Saccharomyces cerevisiae and Caenorhabditis elegans cells, indicating this could be a universal regulatory feature.

Cell polarity is generated and maintained by the spatial accumulation of polarity factors. By imaging fission yeast cells ‘end-on’, the authors show that the polarity factors Tea1 and Tea3 segregate into distinct clusters, and that surprisingly, their segregation is critical for cell polarization.

Establishment of a robust single axis of cell polarity by coupling multiple positive feedback loops

Establishment of cell polarity—or symmetry breaking—relies on local accumulation of polarity regulators. Although simple positive feedback is sufficient to drive symmetry breaking, it is highly sensitive to stochastic fluctuations typical for living cells. Here, by integrating mathematical modelling with quantitative experimental validations, we show that in the yeast Saccharomyces cerevisiae a combination of actin- and guanine nucleotide dissociation inhibitor-dependent recycling of the central polarity regulator Cdc42 is needed to establish robust cell polarity at a single site during yeast budding. The guanine nucleotide dissociation inhibitor pathway consistently generates a single-polarization site, but requires Cdc42 to cycle rapidly between its active and inactive form, and is therefore sensitive to perturbations of the GTPase cycle. Conversely, actin-mediated recycling of Cdc42 induces robust symmetry breaking but cannot restrict polarization to a single site. Our results demonstrate how cells optimize symmetry breaking through coupling between multiple feedback loops.

A positive feedback loop which results in localized accumulation of the small GTPase Cdc42 generates cell polarity in budding yeast; however, such loops are inherently susceptible to noise. Here the authors demonstrate how two pathways that mediate Cdc42 recycling work together to ensure the robustness of symmetry breaking.

Stochastic transitions in a bistable reaction system on the membrane

Transitions between steady states of a multi-stable stochastic system in the perfectly mixed chemical reactor are possible only because of stochastic switching. In realistic cellular conditions, where diffusion is limited, transitions between steady states can also follow from the propagation of travelling waves. Here, we study the interplay between the two modes of transition for a prototype bistable system of kinase-phosphatase interactions on the plasma membrane. Within microscopic kinetic Monte Carlo simulations on the hexagonal lattice, we observed that for finite diffusion the behaviour of the spatially extended system differs qualitatively from the behaviour of the same system in the well-mixed regime. Even when a small isolated subcompartment remains mostly inactive, the chemical travelling wave may propagate, leading to the activation of a larger compartment. The activating wave can be induced after a small subdomain is activated as a result of a stochastic fluctuation. Such a spontaneous onset of activity is radically more probable in subdomains characterized by slower diffusion. Our results show that a local immobilization of substrates can lead to the global activation of membrane proteins by the mechanism that involves stochastic fluctuations followed by the propagation of a semi-deterministic travelling wave.

Spatial simulations in systems biology: from molecules to cells

Cells are highly organized objects containing millions of molecules. Each biomolecule has a specific shape in order to interact with others in the complex machinery. Spatial dynamics emerge in this system on length and time scales which can not yet be modeled with full atomic detail. This review gives an overview of methods which can be used to simulate the complete cell at least with molecular detail, especially Brownian dynamics simulations. Such simulations require correct implementation of the diffusion-controlled reaction scheme occurring on this level. Implementations and applications of spatial simulations are presented, and finally it is discussed how the atomic level can be included for instance in multi-scale simulation methods.

Membrane tension maintains cell polarity by confining signals to the leading edge during neutrophil migration

Little is known about how neutrophils and other cells establish a single zone of actin assembly during migration. A widespread assumption is that the leading edge prevents formation of additional fronts by generating long-range diffusible inhibitors or by sequestering essential polarity components. We use morphological perturbations, cell-severing experiments, and computational simulations to show that diffusion-based mechanisms are not sufficient for long-range inhibition by the pseudopod. Instead, plasma membrane tension could serve as a long-range inhibitor in neutrophils. We find that membrane tension doubles during leading-edge protrusion, and increasing tension is sufficient for long-range inhibition of actin assembly and Rac activation. Furthermore, reducing membrane tension causes uniform actin assembly. We suggest that tension, rather than diffusible molecules generated or sequestered at the leading edge, is the dominant source of long-range inhibition that constrains the spread of the existing front and prevents the formation of secondary fronts.