On the spontaneous emergence of cell polarity

Chemo-mechanical model of cell polarization initiated by structural polarity

Cell polarization is crucial in most physiological functions. Living cells at the extracellular matrix (ECM) actively coordinate a polarized morphology to target the preferred signals. In particular, the initial heterogeneity of subcellular components, termed as structural polarity, has been discovered to mediate the early attachment and transmigration of cells in tumour metastasis. However, how heterogeneous cells initiate the early polarization remains incompletely discovered. Here, we establish a multiscale model of a cell to explore the chemo-mechanical mechanisms of cell polarization initiated by structural polarity. The two-dimensional vertex model of the cell is built with the main mechanical components of eukaryotic cells. The initial structural polarity of the modeled cell is introduced by seeding heterogeneous actin filaments at the cell cortex and quantified by the ratio of the filamentous forces at the vertices. Then, the structural polarity is integrated in the reaction-diffusion system of Rho GTPase (Cdc42) at the cell cortex to obtain the traction forces at the leading vertices. Finally, the modeled cell is actuated to spread under the traction forces and discovered to develop into a characteristic polarized morphology. The results indicate that the cell polarization is initiated and dynamically developed by structural polarity through the reaction-diffusion system of Cdc42. In addition, the bistability of Cdc42 activation at the cell cortex is defined and discovered to dominate the polarization status of the cell. Furthermore, biphasic (i.e., positive and negative) durotaxis of the cell is successfully modeled at an ECM with a stiffness gradient, and concluded to be codetermined by the chemo-mechanical coupling of the initial structural polarity and ECM stiffness gradient. The proposed multiscale model provides a quantitative way to probe cell polarization coupled with mechanical stimuli, biochemical reaction and cytoskeletal reorganization, and holds the potential to guide studies of cell polarization under multiple stimuli.

Bimodality in Ras signaling originates from processivity of the Ras activator SOS without deterministic bistability

Ras is a small GTPase that is central to important functional decisions in diverse cell types. An important aspect of Ras signaling is its ability to exhibit bimodal or switch-like activity. We describe the total reconstitution of a receptor-mediated Ras activation-deactivation reaction catalyzed by SOS and p120-RasGAP on supported lipid membrane microarrays. The results reveal a bimodal Ras activation response, which is not a result of deterministic bistability but is rather driven by the distinct processivity of the Ras activator, SOS. Furthermore, the bimodal response is controlled by the condensation state of the scaffold protein, LAT, to which SOS is recruited. Processivity-driven bimodality leads to stochastic bursts of Ras activation even under strongly deactivating conditions. This behavior contrasts deterministic bistability and may be more resistant to pharmacological inhibition.

Understanding the Role of Connexins in Hepatocellular Carcinoma: Molecular and Prognostic Implications

Connexins, a family of tetraspan membrane proteins forming intercellular channels localized in gap junctions, play a pivotal role at the different stages of tumor progression presenting both pro- and anti-tumorigenic effects. Considering the potential role of connexins as tumor suppressors through multiple channel-independent mechanisms, their loss of expression may be associated with tumorigenic activity, while it is hypothesized that connexins favor the clonal expansion of tumor cells and promote cell migration, invasion, and proliferation, affecting metastasis and chemoresistance in some cases. Hepatocellular carcinoma (HCC), characterized by unfavorable prognosis and limited responsiveness to current therapeutic strategies, has been linked to gap junction proteins as tumorigenic factors with prognostic value. Notably, several members of connexins have emerged as promising markers for assessing the progression and aggressiveness of HCC, as well as the chemosensitivity and radiosensitivity of hepatocellular tumor cells. Our review sheds light on the multifaceted role of connexins in HCC pathogenesis, offering valuable insights on recent advances in determining their prognostic and therapeutic potential.

Patterning of the cell cortex by Rho GTPases

The Rho GTPases - RHOA, RAC1 and CDC42 - are small GTP binding proteins that regulate basic biological processes such as cell locomotion, cell division and morphogenesis by promoting cytoskeleton-based changes in the cell cortex. This regulation results from active (GTP-bound) Rho GTPases stimulating target proteins that, in turn, promote actin assembly and myosin 2-based contraction to organize the cortex. This basic regulatory scheme, well supported by in vitro studies, led to the natural assumption that Rho GTPases function in vivo in an essentially linear matter, with a given process being initiated by GTPase activation and terminated by GTPase inactivation. However, a growing body of evidence based on live cell imaging, modelling and experimental manipulation indicates that Rho GTPase activation and inactivation are often tightly coupled in space and time via signalling circuits and networks based on positive and negative feedback. In this Review, we present and discuss this evidence, and we address one of the fundamental consequences of coupled activation and inactivation: the ability of the Rho GTPases to self-organize, that is, direct their own transition from states of low order to states of high order. We discuss how Rho GTPase self-organization results in the formation of diverse spatiotemporal cortical patterns such as static clusters, oscillatory pulses, travelling wave trains and ring-like waves. Finally, we discuss the advantages of Rho GTPase self-organization and pattern formation for cell function.

From Homogeneity to Turing Pattern: Kinetically Controlled Self-Organization of Transmembrane Protein

Understanding the spatial organization of membrane proteins is crucial for unraveling key principles in cell biology. The reaction-diffusion model is commonly used to understand biochemical patterning; however, applying reaction-diffusion models to subcellular phenomena is challenging because of the difficulty in measuring protein diffusivity and interaction kinetics in the living cell. In this work, we investigated the self-organization of the plasmalemma vesicle-associated protein (PLVAP), which creates regular arrangements of fenestrated ultrastructures, using single-molecule tracking. We demonstrated that the spatial organization of the ultrastructures is associated with a decrease in the association rate by actin destabilization. We also constructed a reaction-diffusion model that accurately generates a hexagonal array with the same 130 nm spacing as the actual scale and informs the stoichiometry of the ultrastructure, which can be discerned only through electron microscopy. Through this study, we integrated single-molecule experiments and reaction-diffusion modeling to surpass the limitations of static imaging tools and proposed emergent properties of the PLVAP ultrastructure.

Coarsening and wavelength selection far from equilibrium: A unifying framework based on singular perturbation theory

Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the well-understood Turing instability, no general theory exists for the dynamics of fully nonlinear patterns. We develop a unifying theory for nonlinear wavelength-selection dynamics in (nearly) mass-conserving two-component reaction-diffusion systems independent of the specific mathematical model chosen. Previous work has shown that these systems support an extremely broad band of stable wavelengths, but the mechanism by which a specific wavelength is selected has remained unclear. We show that an interrupted coarsening process selects the wavelength at the threshold to stability. Based on the physical intuition that coarsening is driven by competition for mass and interrupted by weak source terms that break strict mass conservation, we develop a singular perturbation theory for the stability of stationary patterns. The resulting closed-form analytical expressions enable us to quantitatively predict the coarsening dynamics and the final pattern wavelength. We find excellent agreement with numerical results throughout the diffusion- and reaction-limited regimes of the dynamics, including the crossover region. Further, we show how, in these limits, the two-component reaction-diffusion systems map to generalized Cahn-Hilliard and conserved Allen-Cahn dynamics, therefore providing a link to these two fundamental scalar field theories. The systematic understanding of the length-scale dynamics of fully nonlinear patterns in two-component systems provided here builds the basis to reveal the mechanisms underlying wavelength selection in multicomponent systems with potentially several conservation laws.

Redundancy and the role of protein copy numbers in the cell polarization machinery of budding yeast

How can a self-organized cellular function evolve, adapt to perturbations, and acquire new sub-functions? To make progress in answering these basic questions of evolutionary cell biology, we analyze, as a concrete example, the cell polarity machinery of Saccharomyces cerevisiae. This cellular module exhibits an intriguing resilience: it remains operational under genetic perturbations and recovers quickly and reproducibly from the deletion of one of its key components. Using a combination of modeling, conceptual theory, and experiments, we propose that multiple, redundant self-organization mechanisms coexist within the protein network underlying cell polarization and are responsible for the module's resilience and adaptability. Based on our mechanistic understanding of polarity establishment, we hypothesize that scaffold proteins, by introducing new connections in the existing network, can increase the redundancy of mechanisms and thus increase the evolvability of other network components. Moreover, our work gives a perspective on how a complex, redundant cellular module might have evolved from a more rudimental ancestral form.

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.

The membrane surface as a platform that organizes cellular and biochemical processes

Membranes are essential for life. They act as semi-permeable boundaries that define cells and organelles. In addition, their surfaces actively participate in biochemical reaction networks, where they confine proteins, align reaction partners, and directly control enzymatic activities. Membrane-localized reactions shape cellular membranes, define the identity of organelles, compartmentalize biochemical processes, and can even be the source of signaling gradients that originate at the plasma membrane and reach into the cytoplasm and nucleus. The membrane surface is, therefore, an essential platform upon which myriad cellular processes are scaffolded. In this review, we summarize our current understanding of the biophysics and biochemistry of membrane-localized reactions with particular focus on insights derived from reconstituted and cellular systems. We discuss how the interplay of cellular factors results in their self-organization, condensation, assembly, and activity, and the emergent properties derived from them.

Bimodality in Ras signaling originates from processivity of the Ras activator SOS without classic kinetic bistability

Ras is a small GTPase that is central to important functional decisions in diverse cell types. An important aspect of Ras signaling is its ability to exhibit bimodal, or switch-like activity. We describe the total reconstitution of a receptor-mediated Ras activation-deactivation reaction catalyzed by SOS and p120-RasGAP on supported lipid membrane microarrays. The results reveal a bimodal Ras activation response, which is not a result of classic kinetic bistability, but is rather driven by the distinct processivity of the Ras activator, SOS. Furthermore, the bimodal response is controlled by the condensation state of the scaffold protein, LAT, to which SOS is recruited. Processivity-driven bimodality leads to stochastic bursts of Ras activation even under strongly deactivating conditions. This behavior contrasts classic kinetic bistability and is distinctly more resistant to pharmacological inhibition.

Polarization of brown algal zygotes

Brown algae are a group of multicellular, heterokont algae that have convergently evolved developmental complexity that rivals that of embryophytes, animals or fungi. Early in development, brown algal zygotes establish a basal and an apical pole, which will become respectively the basal system (holdfast) and the apical system (thallus) of the adult alga. Brown algae are interesting models for understanding the establishment of cell polarity in a broad evolutionary context, because they exhibit a large diversity of life cycles, reproductive strategies and, importantly, their zygotes are produced in large quantities free of parental tissue, with symmetry breaking and asymmetric division taking place in a highly synchronous manner. This review describes the current knowledge about the establishment of the apical-basal axis in the model brown seaweeds Ectocarpus, Dictyota, Fucus and Saccharina, highlighting the advantages and specific interests of each system. Ectocarpus is a genetic model system that allows access to the molecular basis of early development and life-cycle control over apical-basal polarity. The oogamous brown alga Fucus, together with emerging comparative models Dictyota and Saccharina, emphasize the diversity of strategies of symmetry breaking in determining a cell polarity vector in brown algae. A comparison with symmetry-breaking mechanisms in land plants, animals and fungi, reveals that the one-step zygote polarisation of Fucus compares well to Saccharomyces budding and Arabidopsis stomata development, while the two-phased symmetry breaking in the Dictyota zygote compares to Schizosaccharomyces fission, the Caenorhabditis anterior-posterior zygote polarisation and Arabidopsis prolate pollen polarisation. The apical-basal patterning in Saccharina zygotes on the other hand, may be seen as analogous to that of land plants. Overall, brown algae have the potential to bring exciting new information on how a single cell gives rise to an entire complex body plan.

Complexity and self-organization in the evolution of cell polarization

Cellular life exhibits order and complexity, which typically increase over the course of evolution. Cell polarization is a well-studied example of an ordering process that breaks the internal symmetry of a cell by establishing a preferential axis. Like many cellular processes, polarization is driven by self-organization, meaning that the macroscopic pattern emerges as a consequence of microscopic molecular interactions at the biophysical level. However, the role of self-organization in the evolution of complex protein networks remains obscure. In this Review, we provide an overview of the evolution of polarization as a self-organizing process, focusing on the model species Saccharomyces cerevisiae and its fungal relatives. Moreover, we use this model system to discuss how self-organization might relate to evolutionary change, offering a shift in perspective on evolution at the microscopic scale.

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.

A Gaussian jump process formulation of the reaction–diffusion master equation enables faster exact stochastic simulations

We propose a Gaussian jump process model on a regular Cartesian lattice for the diffusion part of the Reaction–Diffusion Master Equation (RDME). We derive the resulting Gaussian RDME (GRDME) formulation from analogy with a kernel-based discretization scheme for continuous diffusion processes and quantify the limits of its validity relative to the classic RDME. We then present an exact stochastic simulation algorithm for the GRDME, showing that the accuracies of GRDME and RDME are comparable, but exact simulations of the GRDME require only a fraction of the computational cost of exact RDME simulations. We analyze the origin of this speedup and its scaling with problem dimension. The benchmarks suggest that the GRDME is a particularly beneficial model for diffusion-dominated systems in three dimensional spaces, often occurring in systems biology and cell biology.

Picking winners in cell-cell collisions: Wetting, speed, and contact

Groups of eukaryotic cells can coordinate their crawling motion to follow cues more effectively, stay together, or invade new areas. This collective cell migration depends on cell-cell interactions, which are often studied by colliding pairs of cells together. Can the outcome of these collisions be predicted? Recent experiments on trains of colliding epithelial cells suggest that cells with a smaller contact angle to the surface or larger speeds are more likely to maintain their direction ("win") upon collision. When should we expect shape or speed to correlate with the outcome of a collision? To investigate this question, we build a model for two-cell collisions within the phase field framework, which allows for cell shape changes. We can reproduce the observation that cells with high speed and small contact angles are more likely to win with two different assumptions for how cells interact: (1) velocity aligning, in which we hypothesize that cells sense their own velocity and align to it over a finite timescale, and (2) front-front contact repolarization, where cells polarize away from cell-cell contact, akin to contact inhibition of locomotion. Surprisingly, though we simulate collisions between cells with widely varying properties, in each case, the probability of a cell winning is completely captured by a single summary variable: its relative speed (in the velocity-aligning model) or its relative contact angle (in the contact repolarization model). Both models are currently consistent with reported experimental results, but they can be distinguished by varying cell contact angle and speed through orthogonal perturbations.

Size regulation of multiple organelles competing for a limiting subunit pool

How cells regulate the size of intracellular structures and organelles is a longstanding question. Recent experiments suggest that size control of intracellular structures is achieved through the depletion of a limiting subunit pool in the cytoplasm. While the limiting pool model ensures organelle-to-cell size scaling, it does not provide a mechanism for robust size control of multiple co-existing structures. Here we develop a generalized theory for size-dependent growth of intracellular structures to demonstrate that robust size control of multiple intracellular structures, competing for a limiting subunit pool, is achieved via a negative feedback between the growth rate and the size of the individual structure. This design principle captures size maintenance of a wide variety of subcellular structures, from cytoskeletal filaments to three-dimensional organelles. We identify the feedback motifs for structure size regulation based on known molecular processes, and compare our theory to existing models of size regulation in biological assemblies. Furthermore, we show that positive feedback between structure size and growth rate can lead to bistable size distribution and spontaneous size selection.

Organelle size control is essential for the proper physiological functioning of eukaryotic cells, but the underlying mechanisms of size regulation remain poorly understood. By developing a general theory for organelle size control, we show that robust size control of intracellular structures and organelles is achieved via a negative feedback between individual organelle size and their net growth rates. This design principle not only describes size maintenance of single organelles, but also ensures size stability of multiple co-existing organelles that are built from a limiting pool of subunits. Our results delineate the role of limiting pool as a size scaling mechanism rather than a size control mechanism, supporting the idea that negative feedback control of organelle size via depletion of a limiting subunit pool is not sufficient to maintain the size of multiple competing organelles. In the case of positive feedback between organelle size and growth rate, our model reproduces phenomena such as bistability in organelle size distribution and spontaneous emergence of cell polarity.

Constrained Nonlinear and Mixed Effects Integral Differential Equation Models for Dynamic Cell Polarity Signaling

Polar cell growth is a process that couples the establishment of cell polarity with growth and is extremely important in the growth, development, and reproduction of eukaryotic organisms, such as pollen tube growth during plant fertilization and neuronal axon growth in animals. Pollen tube growth requires dynamic but polarized distribution and activation of a signaling protein named ROP1 to the plasma membrane via three processes: positive feedback and negative feedback regulation of ROP1 activation and its lateral diffusion along the plasma membrane. In this paper, we introduce a mechanistic integro-differential equation (IDE) along with constrained semiparametric regression to quantitatively describe the interplay among these three processes that lead to the polar distribution of active ROP1 at a steady state. Moreover, we introduce a population variability by a constrained nonlinear mixed model. Our analysis of ROP1 activity distributions from multiple pollen tubes revealed that the equilibrium between the positive and negative feedbacks for pollen tubes with similar shapes are remarkably stable, permitting us to infer an inherent quantitative relationship between the positive and negative feedback loops that defines the tip growth of pollen tubes and the polarity of tip growth.

Subdiffusive Activity Spreading in the Diffusive Epidemic Process

The diffusive epidemic process is a paradigmatic example of an absorbing state phase transition in which healthy and infected individuals spread with different diffusion constants. Using stochastic activity spreading simulations in combination with finite-size scaling analyses we reveal two qualitatively different processes that characterize the critical dynamics: subdiffusive propagation of infection clusters and diffusive fluctuations in the healthy population. This suggests the presence of a strong-coupling regime and sheds new light on a long-standing debate about the theoretical classification of the system.

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.

Upregulation of the Cdc42 GTPase limits the replicative lifespan of budding yeast

Cdc42, a conserved Rho GTPase, plays a central role in polarity establishment in yeast and animals. Cell polarity is critical for asymmetric cell division, and asymmetric cell division underlies replicative aging of budding yeast. Yet how Cdc42 and other polarity factors impact life span is largely unknown. Here we show by live-cell imaging that the active Cdc42 level is sporadically elevated in wild type during repeated cell divisions but rarely in the long-lived bud8 deletion cells. We find a novel Bud8 localization with cytokinesis remnants, which also recruit Rga1, a Cdc42 GTPase activating protein. Genetic analyses and live-cell imaging suggest that Rga1 and Bud8 oppositely impact life span likely by modulating active Cdc42 levels. An rga1 mutant, which has a shorter life span, dies at the unbudded state with a defect in polarity establishment. Remarkably, Cdc42 accumulates in old cells, and its mild overexpression accelerates aging with frequent symmetric cell divisions, despite no harmful effects on young cells. Our findings implicate that the interplay among these positive and negative polarity factors limits the life span of budding yeast.

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

A picket fence function for adherens junctions in epithelial cell polarity

Adherens junctions are a defining feature of all epithelial cells, providing cell-cell adhesion and contractile ring formation that is essential for cell and tissue morphology. In Drosophila, adherens junctions are concentrated between the apical and basolateral plasma membrane domains, defined by aPKC-Par6-Baz and Lgl/Dlg/Scrib, respectively. Whether adherens junctions contribute to apical-basal polarization itself has been unclear because neuroblasts exhibit apical-basal polarization of aPKC-Par6-Baz and Lgl in the absence of adherens junctions. Here we show that, upon disruption of adherens junctions in epithelial cells, apical polarity determinants such as aPKC can still segregate from basolateral Lgl, but lose their sharp boundaries and also overlap with Dlg and Scrib - similar to neuroblasts. In addition, control of apical versus basolateral domain size is lost, along with control of cell shape, in the absence of adherens junctions. Manipulating the levels of apical Par3/Baz or basolateral Lgl polarity determinants in experiments and in computer simulations confirms that adherens junctions provide a 'picket fence' diffusion barrier that restricts the spread of polarity determinants along the membrane to enable precise domain size control. Movement of adherens junctions in response to mechanical forces during morphogenetic change thus enables spontaneous adjustment of apical versus basolateral domain size as an emergent property of the polarising system.

Wavelength Selection by Interrupted Coarsening in Reaction-Diffusion Systems

Wavelength selection in reaction-diffusion systems can be understood as a coarsening process that is interrupted by counteracting processes at certain wavelengths. We first show that coarsening in mass-conserving systems is driven by self-amplifying mass transport between neighboring high-density domains. We derive a general coarsening criterion and show that coarsening is generically uninterrupted in two-component systems that conserve mass. The theory is then generalized to study interrupted coarsening and anticoarsening due to weakly broken mass conservation, providing a general path to analyze wavelength selection in pattern formation far from equilibrium.

The Arabidopsis STE20/Hippo kinase SIK1 regulates polarity independently of PIN proteins

Polarity is a feature of life. In higher plants, non-autonomous polarity is largely directed by auxin, the morphogen that drives its own polarized flow, Polar Auxin Transport (PAT), to guide patterning events such as phyllotaxis and tropism. The plasma membrane-localized PIN-FORMED (PIN) auxin efflux carriers are rate-limiting factors in PAT. In yeasts and metazoans, the STE20 kinases are key players in cell polarity. We had previously characterized SIK1 as a STE20/Hippo orthologue in Arabidopsis and confirmed its function in mitotic exit and organ growth. Here we explore the possible link between SIK1, auxin, PIN, and polarity. Abnormal phyllotaxis and gravitropism were observed in sik1. sik1 was more sensitive to exogenous auxin in primary root elongation and lateral root emergence. RNA-Seq revealed reduced expression in auxin biosynthesis genes and induced expression of auxin flux carriers in sik1. However, normal tissue- and sub-cellular localization patterns of PIN1 and PIN2 were observed in sik1. The dark-induced vacuolar degradation of PIN2 also appeared normal in sik1. An additive phenotype was observed in the sik1 pin1 double mutant, indicating that SIK1 does not directly regulate PIN1. The polarity defects of sik1 are hence unlikely mediated by PINs and await future exploration.

Construction of intracellular asymmetry and asymmetric division in Escherichia coli

The design principle of establishing an intracellular protein gradient for asymmetric cell division is a long-standing fundamental question. While the major molecular players and their interactions have been elucidated via genetic approaches, the diversity and redundancy of natural systems complicate the extraction of critical underlying features. Here, we take a synthetic cell biology approach to construct intracellular asymmetry and asymmetric division in Escherichia coli, in which division is normally symmetric. We demonstrate that the oligomeric PopZ from Caulobacter crescentus can serve as a robust polarized scaffold to functionalize RNA polymerase. Furthermore, by using another oligomeric pole-targeting DivIVA from Bacillus subtilis, the newly synthesized protein can be constrained to further establish intracellular asymmetry, leading to asymmetric division and differentiation. Our findings suggest that the coupled oligomerization and restriction in diffusion may be a strategy for generating a spatial gradient for asymmetric cell division.

Developmental clock and mechanism of de novo polarization of the mouse embryo

Embryo polarization is critical for mouse development; however, neither the regulatory clock nor the molecular trigger that it activates is known. Here, we show that the embryo polarization clock reflects the onset of zygotic genome activation, and we identify three factors required to trigger polarization. Advancing the timing of transcription factor AP-2 gamma (Tfap2c) and TEA domain transcription factor 4 (Tead4) expression in the presence of activated Ras homolog family member A (RhoA) induces precocious polarization as well as subsequent cell fate specification and morphogenesis. Tfap2c and Tead4 induce expression of actin regulators that control the recruitment of apical proteins on the membrane, whereas RhoA regulates their lateral mobility, allowing the emergence of the apical domain. Thus, Tfap2c, Tead4, and RhoA are regulators for the onset of polarization and cell fate segregation in the mouse.

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.

Aster swarming by symmetry breaking of cortical dynein transport and coupling kinesins

Microtubule (MT) radial arrays or asters establish the internal topology of a cell by interacting with organelles and molecular motors. We proceed to understand the general pattern forming potential of aster-motor systems using a computational model of multiple MT asters interacting with motors in cellular confinement. In this model dynein motors are attached to the cell cortex and plus-ended motors resembling kinesin-5 diffuse in the cell interior. The introduction of 'noise' in the form of MT length fluctuations spontaneously results in the emergence of coordinated, achiral vortex-like rotation of asters. The coherence and persistence of rotation require a threshold density of both cortical dyneins and coupling kinesins, while the onset is diffusion-limited with relation to the cortical dynein mobility. The coordinated rotational motion emerges due to the resolution of a 'tug-of-war' of multiple cortical dynein motors bound to MTs of the same aster by 'noise' in the form of MT dynamic instability. This transient symmetry breaking is amplified by local coupling by kinesin-5 complexes. The lack of widespread aster rotation across cell types suggests that biophysical mechanisms that suppress such intrinsic dynamics may have evolved. This model is analogous to more general models of locally coupled self-propelled particles (SPP) that spontaneously undergo collective transport in the presence of 'noise' that have been invoked to explain swarming in birds and fish. However, the aster-motor system is distinct from SPP models with regard to the particle density and 'noise' dependence, providing a set of experimentally testable predictions for a novel sub-cellular pattern forming system.

Noise-induced effects in collective dynamics and inferring local interactions from data

In animal groups, individual decisions are best characterized by probabilistic rules. Furthermore, animals of many species live in small groups. Probabilistic interactions among small numbers of individuals lead to a so-called intrinsic noise at the group level. Theory predicts that the strength of intrinsic noise is not a constant but often depends on the collective state of the group; hence, it is also called a state-dependent noise or a multiplicative noise. Surprisingly, such noise may produce collective order. However, only a few empirical studies on collective behaviour have paid attention to such effects owing to the lack of methods that enable us to connect data with theory. Here, we demonstrate a method to characterize the role of stochasticity directly from high-resolution time-series data of collective dynamics. We do this by employing two well-studied individual-based toy models of collective behaviour. We argue that the group-level noise may encode important information about the underlying processes at the individual scale. In summary, we describe a method that enables us to establish connections between empirical data of animal (or cellular) collectives and the phenomenon of noise-induced states, a field that is otherwise largely limited to the theoretical literature. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

A hybrid stochastic-deterministic mechanochemical model of cell polarization

Polarization is a crucial component in cell differentiation, development, and motility, but its details are not yet well understood. At the onset of cell locomotion, cells break symmetry to form well-defined cell fronts and rears. This polarity establishment varies across cell types: in Dictyostelium discoideum cells, it is mediated by biochemical signaling pathways and can function in the absence of a cytoskeleton, while in keratocytes, it is tightly connected to cytoskeletal dynamics and mechanics. Theoretical models that have been developed to understand the onset of polarization have explored either signaling or mechanical pathways, yet few have explored mechanochemical mechanisms. However, many motile cells rely on both signaling modules and actin cytoskeleton to break symmetry and achieve a stable polarized state. We propose a general mechanochemical polarization model based on coupling between a stochastic model for the segregation of signaling molecules and a simplified mechanical model for actin cytoskeleton network competition. We find that local linear coupling between minimally nonlinear signaling and cytoskeletal systems, separately not supporting stable polarization, yields a robustly polarized cell state. The model captures the essence of spontaneous polarization of neutrophils, which has been proposed to emerge due to the competition between frontness and backness pathways.

Size-Regulated Symmetry Breaking in Reaction-Diffusion Models of Developmental Transitions

The development of multicellular organisms proceeds through a series of morphogenetic and cell-state transitions, transforming homogeneous zygotes into complex adults by a process of self-organisation. Many of these transitions are achieved by spontaneous symmetry breaking mechanisms, allowing cells and tissues to acquire pattern and polarity by virtue of local interactions without an upstream supply of information. The combined work of theory and experiment has elucidated how these systems break symmetry during developmental transitions. Given that such transitions are multiple and their temporal ordering is crucial, an equally important question is how these developmental transitions are coordinated in time. Using a minimal mass-conserved substrate-depletion model for symmetry breaking as our case study, we elucidate mechanisms by which cells and tissues can couple reaction-diffusion-driven symmetry breaking to the timing of developmental transitions, arguing that the dependence of patterning mode on system size may be a generic principle by which developing organisms measure time. By analysing different regimes of our model, simulated on growing domains, we elaborate three distinct behaviours, allowing for clock-, timer- or switch-like dynamics. Relating these behaviours to experimentally documented case studies of developmental timing, we provide a minimal conceptual framework to interrogate how developing organisms coordinate developmental transitions.

Shaping Organs: Shared Structural Principles Across Kingdoms

Development encapsulates the morphogenesis of an organism from a single fertilized cell to a functional adult. A critical part of development is the specification of organ forms. Beyond the molecular control of morphogenesis, shape in essence entails structural constraints and thus mechanics. Revisiting recent results in biophysics and development, and comparing animal and plant model systems, we derive key overarching principles behind the formation of organs across kingdoms. In particular, we highlight how growing organs are active rather than passive systems and how such behavior plays a role in shaping the organ. We discuss the importance of considering different scales in understanding how organs form. Such an integrative view of organ development generates new questions while calling for more cross-fertilization between scientific fields and model system communities.

The Roles of Signaling in Cytoskeletal Changes, Random Movement, Direction-Sensing and Polarization of Eukaryotic Cells

Movement of cells and tissues is essential at various stages during the lifetime of an organism, including morphogenesis in early development, in the immune response to pathogens, and during wound-healing and tissue regeneration. Individual cells are able to move in a variety of microenvironments (MEs) (A glossary of the acronyms used herein is given at the end) by suitably adapting both their shape and how they transmit force to the ME, but how cells translate environmental signals into the forces that shape them and enable them to move is poorly understood. While many of the networks involved in signal detection, transduction and movement have been characterized, how intracellular signals control re-building of the cyctoskeleton to enable movement is not understood. In this review we discuss recent advances in our understanding of signal transduction networks related to direction-sensing and movement, and some of the problems that remain to be solved.

Molecular mechanisms of cell polarity in a range of model systems and in migrating neurons

Cell polarity is defined as the asymmetric distribution of cellular components along an axis. Most cells, from the simplest single-cell organisms to highly specialized mammalian cells, are polarized and use similar mechanisms to generate and maintain polarity. Cell polarity is important for cells to migrate, form tissues, and coordinate activities. During development of the mammalian cerebral cortex, cell polarity is essential for neurogenesis and for the migration of newborn but as-yet undifferentiated neurons. These oriented migrations include both the radial migration of excitatory projection neurons and the tangential migration of inhibitory interneurons. In this review, I will first describe the development of the cerebral cortex, as revealed at the cellular level. I will then define the core molecular mechanisms - the Par/Crb/Scrib polarity complexes, small GTPases, the actin and microtubule cytoskeletons, and phosphoinositides/PI3K signaling - that are required for asymmetric cell division, apico-basal and front-rear polarity in model systems, including C elegans zygote, Drosophila embryos and cultured mammalian cells. As I go through each core mechanism I will explain what is known about its importance in radial and tangential migration in the developing mammalian cerebral cortex.

Identification of TUBB2A by quantitative proteomic analysis as a novel biomarker for the prediction of distant metastatic breast cancer

Background

Metastasis of breast cancer to distal organs is fatal. However, few studies have identified biomarkers that are associated with distant metastatic breast cancer. Furthermore, the inability of current biomarkers, such as HER2, ER, and PR, to differentiate between distant and nondistant metastatic breast cancers accurately has necessitated the development of novel biomarker candidates.

Methods

An integrated proteomics approach that combined filter-aided sample preparation, tandem mass tag labeling (TMT), high pH fractionation, and high-resolution MS was applied to acquire in-depth proteomic data from FFPE distant metastatic breast cancer tissues. A bioinformatics analysis was performed with regard to gene ontology and signaling pathways using differentially expressed proteins (DEPs) to examine the molecular characteristics of distant metastatic breast cancer. In addition, real-time polymerase chain reaction (RT-PCR) and invasion/migration assays were performed to validate the differential regulation and function of our protein targets.

Results

A total of 9441 and 8746 proteins were identified from the pooled and individual sample sets, respectively. Based on our criteria, TUBB2A was selected as a novel biomarker candidate. The metastatic activities of TUBB2A were subsequently validated. In our bioinformatics analysis using DEPs, we characterized the overall molecular features of distant metastasis and measured differences in the molecular functions of distant metastatic breast cancer between breast cancer subtypes.

Conclusions

Our report is the first study to examine the distant metastatic breast cancer proteome using FFPE tissues. The depth of our dataset allowed us to discover a novel biomarker candidate and a proteomic characteristics of distant metastatic breast cancer. Distinct molecular features of various breast cancer subtypes were also established. Our proteomic data constitute a valuable resource for research on distant metastatic breast cancer.

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.

The cytoskeletal motor proteins Dynein and MyoV direct apical transport of Crumbs

Crumbs (Crb in Drosophila; CRB1-3 in mammals) is a transmembrane determinant of epithelial cell polarity and a regulator of Hippo signalling. Crb is normally localized to apical cell-cell contacts, just above adherens junctions, but how apical trafficking of Crb is regulated in epithelial cells remains unclear. We use the Drosophila follicular epithelium to demonstrate that polarized trafficking of Crb is mediated by transport along microtubules by the motor protein Dynein and along actin filaments by the motor protein Myosin-V (MyoV). Blocking transport of Crb-containing vesicles by Dynein or MyoV leads to accumulation of Crb within Rab11 endosomes, rather than apical delivery. The final steps of Crb delivery and stabilisation at the plasma membrane requires the exocyst complex and three apical FERM domain proteins - Merlin, Moesin and Expanded - whose simultaneous loss disrupts apical localization of Crb. Accordingly, a knock-in deletion of the Crb FERM-binding motif (FBM) also impairs apical localization. Finally, overexpression of Crb challenges this system, creating a sensitized background to identify components involved in cytoskeletal polarization, apical membrane trafficking and stabilisation of Crb at the apical domain.

Exocytosis and Endocytosis: Yin-Yang Crosstalk for Sculpting a Dynamic Growing Pollen Tube Tip

The growing pollen tube has become one of the most fascinating model cell systems for investigations into cell polarity and polar cell growth in plants. Rapidly growing pollen tubes achieve tip-focused cell expansion by vigorous anterograde exocytosis, through which various newly synthesized macromolecules are directionally transported and deposited at the cell apex. Meanwhile, active retrograde endocytosis counter balances the exocytosis at the tip which is believed to recycle the excessive exocytic components for multiple rounds of secretion. Therefore, apical exocytosis and endocytosis are the frontline cellular processes which drive the polar growth of pollen tubes, although they represent opposite vesicular trafficking events with distinct underpinning mechanisms. Nevertheless, the molecular basis governing the spatiotemporal crosstalk and counterbalance of exocytosis and endocytosis during pollen tube polarization and growth remains elusive. Here we discuss recent insight into exocytosis and endocytosis in sculpturing high rates of polarized pollen tube growth. In addition, we especially introduce the novel integration of mathematical modeling in uncovering the mysteries of cell polarity and polar cell growth.

Phosphorylation of nephrin induces phase separated domains that move through actomyosin contraction

The plasma membrane of eukaryotic cells is organized into lipid and protein microdomains, whose assembly mechanisms and functions are incompletely understood. We demonstrate that proteins in the nephrin/Nck/N-WASP actin-regulatory pathway cluster into micron-scale domains at the basal plasma membrane upon triggered phosphorylation of transmembrane protein nephrin. The domains are persistent but readily exchange components with their surroundings, and their formation is dependent on the number of Nck SH3 domains, suggesting they are phase separated polymers assembled through multivalent interactions among the three proteins. The domains form independent of the actin cytoskeleton, but acto-myosin contractility induces their rapid lateral movement. Nephrin phosphorylation induces larger clusters at the cell periphery, which are associated with extensive actin assembly and dense filopodia. Our studies illustrate how multivalent interactions between proteins at the plasma membrane can produce micron-scale organization of signaling molecules, and how the resulting clusters can both respond to and control the actin cytoskeleton.

Crumbs protein homolog 3 (CRB3) expression is associated with oestrogen and progesterone receptor positivity in breast cancer

The crumbs protein homolog 3 (CRB3) regulates the tight junction to help maintain epithelial polarity. Altered CRB3 expression was associated with carcinogenesis of epithelial cells. This study detected CRB3 expression in 192 cases of breast cancer tissues and in the Molecular Taxonomy of Breast Cancer International Consortium (Metabric) and The Cancer Genome Atlas (TCGA) datasets for association with triple negative breast cancer (TNBC) phenotypes. The in vitro experiments confirm the ex vivo data. The data showed that levels of both CRB3 mRNA and protein were associated with TNBC phenotypes, ie, 41.1% (39/95) of ER+ breast cancer was CRB3-positive, whereas 26.9% (25/93) ER- tumour was CRB3-positive (P = 0.046). Moreover, 47.6% (30/63) of PR+ breast cancer was CRB3-positive vs 28.4% (33/116) PR- tumours positive for CRB3 (P = 0.013). In addition, 40.1% (27/66) of ER+/PR+ tumour was CRB3-positive, but only 22.4% (19/85) of TNBC showed CRB3 expression (P = 0.048). Indeed, levels of CRB3 mRNA were higher in non-TNBC than TNBC in both Metabric (P = 3.682e-10) and TCGA datasets (P = 2.501e-07). The in vitro data showed that CRB3 expression was higher in luminal (MCF7 and T47D) than in HER2 (MDA-MB-453 and SK-BR-3) and basal (MDA-MB-231 and BT-549) breast cancer cell lines. More interestingly, ERα regulated expression of CRB3 protein in MCF7 and BT-549 cells and ERα expression was associated with CRB3 expression in breast cancer tissues specimens. This study demonstrated that ERα could be a novel regulator for CRB3 expression in breast cancer.

Temporal regulation of cell polarity via the interaction of the Ras GTPase Rsr1 and the scaffold protein Bem1

The Cdc42 guanosine triphosphatase (GTPase) plays a central role in polarity development in species ranging from yeast to humans. In budding yeast, a specific growth site is selected in the G1 phase. Rsr1, a Ras GTPase, interacts with Cdc42 and its associated proteins to promote polarized growth at the proper bud site. Yet how Rsr1 regulates cell polarization is not fully understood. Here, we show that Rsr1-GDP interacts with the scaffold protein Bem1 in early G1, likely hindering the role of Bem1 in Cdc42 polarization and polarized secretion. Consistent with these in vivo observations, mathematical modeling predicts that Bem1 is unable to promote Cdc42 polarization in early G1 in the presence of Rsr1-GDP. We find that a part of the Bem1 Phox homology domain, which overlaps with a region interacting with the exocyst component Exo70, is necessary for the association of Bem1 with Rsr1-GDP. Overexpression of the GDP-locked Rsr1 interferes with Bem1-dependent Exo70 polarization. We thus propose that Rsr1 functions in spatial and temporal regulation of polarity establishment by associating with distinct polarity factors in its GTP- and GDP-bound states.

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.

Enterotoxigenic Escherichia coli Interferes FATP4-Dependent Long-Chain Fatty Acid Uptake of Intestinal Epithelial Enterocytes via Phosphorylation of ERK1/2-PPARγ Pathway

Sufficient fatty acid (FA) uptake from jejunal lumen is closely associated with pediatric growth. Enterotoxigenic Escherichia coli (ETEC), which poses a big threat to young mammals’ health, is also targeted on the jejunum, however, the effects on FA uptake is not understood yet. To explore the impacts of ETEC on the FA uptake ability of jejunum epithelial enterocytes during early life, we orally gavaged weaning piglets with ETEC K88 and found intestinal inflammation combined with compromised uptake of LCFA (C16:0, C18:0, C20:3, C20:4) except for C14:0 whose chain length is similar to medium chain fatty acid (MCFA). Furthermore, we observed reduced protein expression of TJs, fatty acid transport protein 4 (FATP4), peroxisome proliferator-activated receptor γ (PPARγ), phosphorylated extracellular signal-regulated kinase 1/2 (p-ERK1/2), and upregulated expression of p-PPARγ. In the in vitro study, we challenged polarized porcine intestine jejunum cell line IPEC-J2 with ETEC K88 and discovered similar results on intestinal barrier and expression of associated genes combined with morphological changes. Based on the constructed cellular model, we then determined lower uptake of BODIPY-labeled C16:0 without any difference in the uptake of BODIPY-labeled C12:0. The content of intracellular triglyceride which was mainly synthesized by LCFA concomitantly lowered down. Using gene knock down and overexpression, FATP4 was confirmed to be responsible for LCFA uptake. Moreover, ERK1/2 inhibitor U0126 and PPARγ antagonist T0070907 revealed ETEC could initiate cascaded phosphorylation of ERK1/2 and PPARγ resulting in hindered expression of FATP4. These results indicate ETEC challenge will cause dysfunction in FATP4-dependent LCFA uptake by phosphorylation of ERK1/2 and PPARγ. Furthermore, intestinal uptake of MCFA is in a FATP4-independent manner which is not easily disturbed by ETEC.

Leukocyte Cytoskeleton Polarization Is Initiated by Plasma Membrane Curvature from Cell Attachment

Cell polarization is important for various biological processes. However, its regulation, particularly initiation, is incompletely understood. Here, we investigated mechanisms by which neutrophils break their symmetry and initiate their cytoskeleton polarization from an apolar state in circulation for their extravasation during inflammation. We show here that a local increase in plasma membrane (PM) curvature resulting from cell contact to a surface triggers the initial breakage of the symmetry of an apolar neutrophil and is required for subsequent polarization events induced by chemical stimulation. This local increase in PM curvature recruits SRGAP2 via its F-BAR domain, which in turn activates PI4KA and results in PM PtdIns4P polarization. Polarized PM PtdIns4P is targeted by RPH3A, which directs PIP5K1C90 and subsequent phosphorylated myosin light chain polarization, and this polarization signaling axis regulates neutrophil firm attachment to endothelium. Thus, this study reveals a mechanism for the initiation of cell cytoskeleton polarization.

Self-organization principles of intracellular pattern formation

Dynamic patterning of specific proteins is essential for the spatio-temporal regulation of many important intracellular processes in prokaryotes, eukaryotes and multicellular organisms. The emergence of patterns generated by interactions of diffusing proteins is a paradigmatic example for self-organization. In this article, we review quantitative models for intracellular Min protein patterns in Escherichia coli, Cdc42 polarization in Saccharomyces cerevisiae and the bipolar PAR protein patterns found in Caenorhabditis elegans. By analysing the molecular processes driving these systems we derive a theoretical perspective on general principles underlying self-organized pattern formation. We argue that intracellular pattern formation is not captured by concepts such as ‘activators’, ‘inhibitors’ or ‘substrate depletion’. Instead, intracellular pattern formation is based on the redistribution of proteins by cytosolic diffusion, and the cycling of proteins between distinct conformational states. Therefore, mass-conserving reaction–diffusion equations provide the most appropriate framework to study intracellular pattern formation. We conclude that directed transport, e.g. cytosolic diffusion along an actively maintained cytosolic gradient, is the key process underlying pattern formation. Thus the basic principle of self-organization is the establishment and maintenance of directed transport by intracellular protein dynamics.

This article is part of the theme issue ‘Self-organization in cell biology’.

eGFRD in all dimensions

Biochemical reactions often occur at low copy numbers but at once in crowded and diverse environments. Space and stochasticity therefore play an essential role in biochemical networks. Spatial-stochastic simulations have become a prominent tool for understanding how stochasticity at the microscopic level influences the macroscopic behavior of such systems. While particle-based models guarantee the level of detail necessary to accurately describe the microscopic dynamics at very low copy numbers, the algorithms used to simulate them typically imply trade-offs between computational efficiency and biochemical accuracy. eGFRD (enhanced Green's Function Reaction Dynamics) is an exact algorithm that evades such trade-offs by partitioning the N-particle system into M ≤ N analytically tractable one- and two-particle systems; the analytical solutions (Green's functions) then are used to implement an event-driven particle-based scheme that allows particles to make large jumps in time and space while retaining access to their state variables at arbitrary simulation times. Here we present "eGFRD2," a new eGFRD version that implements the principle of eGFRD in all dimensions, thus enabling efficient particle-based simulation of biochemical reaction-diffusion processes in the 3D cytoplasm, on 2D planes representing membranes, and on 1D elongated cylinders representative of, e.g., cytoskeletal tracks or DNA; in 1D, it also incorporates convective motion used to model active transport. We find that, for low particle densities, eGFRD2 is up to 6 orders of magnitude faster than conventional Brownian dynamics. We exemplify the capabilities of eGFRD2 by simulating an idealized model of Pom1 gradient formation, which involves 3D diffusion, active transport on microtubules, and autophosphorylation on the membrane, confirming recent experimental and theoretical results on this system to hold under genuinely stochastic conditions.

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.

Minimal in vitro systems shed light on cell polarity

Cell polarity - the morphological and functional differentiation of cellular compartments in a directional manner - is required for processes such as orientation of cell division, directed cellular growth and motility. How the interplay of components within the complexity of a cell leads to cell polarity is still heavily debated. In this Review, we focus on one specific aspect of cell polarity: the non-uniform accumulation of proteins on the cell membrane. In cells, this is achieved through reaction-diffusion and/or cytoskeleton-based mechanisms. In reaction-diffusion systems, components are transformed into each other by chemical reactions and are moving through space by diffusion. In cytoskeleton-based processes, cellular components (i.e. proteins) are actively transported by microtubules (MTs) and actin filaments to specific locations in the cell. We examine how minimal systems - in vitro reconstitutions of a particular cellular function with a minimal number of components - are designed, how they contribute to our understanding of cell polarity (i.e. protein accumulation), and how they complement in vivo investigations. We start by discussing the Min protein system from Escherichia coli, which represents a reaction-diffusion system with a well-established minimal system. This is followed by a discussion of MT-based directed transport for cell polarity markers as an example of a cytoskeleton-based mechanism. To conclude, we discuss, as an example, the interplay of reaction-diffusion and cytoskeleton-based mechanisms during polarity establishment in budding yeast.

A Tuba/Cdc42/Par6A complex is required to ensure singularity in apical domain formation during enterocyte polarization

Apico-basal polarity establishment is a seminal process in tissue morphogenesis. To function properly it is often imperative that epithelial cells limit apical membrane formation to a single domain. We previously demonstrated that signaling by the small GTPase Cdc42, together with its guanine nucleotide exchange factor (GEF) Tuba, is required to prevent the formation of multiple apical domains in polarized Ls174T:W4 cells, a single cell model for enterocyte polarization. To further chart the molecular signaling mechanisms that safeguard singularity during enterocyte polarization we generated knockout cells for the Cdc42 effector protein Par6A. Par6A loss results in the formation of multiple apical domains, similar to loss of Cdc42. In Par6A knockout cells, we find that active Cdc42 is more mobile at the apical membrane compared to control cells and that wild type Cdc42 is more diffusely localized throughout the cell, indicating that Par6A is required to restrict Cdc42 signaling. Par6A, Cdc42 and its GEF Tuba bind in a co-immunoprecipitation experiment and they partially colocalize at the apical membrane in polarized Ls174T:W4 cells, suggesting the formation of a trimeric complex. Indeed, in a rescue experiment using Par6A mutants, we show that the ability to establish this trimeric complex correlates with the ability to restore singularity in Par6A knockout cells. Together, these experiments therefore indicate that a Tuba/Cdc42/Par6A complex is required to ensure the formation of a single apical domain during enterocyte polarization.