Biosensors based on peptide exposure show single molecule conformations in live cells.

We describe an approach to study the conformation of individual proteins during single particle tracking (SPT) in living cells. "Binder/tag" is based on incorporation of a 7-mer peptide (the tag) into a protein where its solvent exposure is controlled by protein conformation. Only upon exposure can the peptide specifically interact with a reporter protein (the binder). Thus, simple fluorescence localization reflects protein conformation. Through direct excitation of bright dyes, the trajectory and conformation of individual proteins can be followed. Simple protein engineering provides highly specific biosensors suitable for SPT and FRET. We describe tagSrc, tagFyn, tagSyk, tagFAK, and an orthogonal binder/tag pair. SPT showed slowly diffusing islands of activated Src within Src clusters and dynamics of activation in adhesions. Quantitative analysis and stochastic modeling revealed in vivo Src kinetics. The simplicity of binder/tag can provide access to diverse proteins.

Reciprocal encoding of signal intensity and duration in a glucose-sensing circuit.

Cells continuously adjust their behavior in response to changing environmental conditions. Both intensity and duration of external signals are critical factors in determining what response is initiated. To understand how intracellular signaling networks process such multidimensional information, we studied the AtRGS1-mediated glucose response system of Arabidopsis. By combining experiments with mathematical modeling, we discovered a reciprocal dose and duration response relying on the orchestrated action of three kinases (AtWNK1, AtWNK8, and AtWNK10) acting on distinct timescales and activation thresholds. Specifically, we find that high concentrations of D-glucose rapidly signal through AtWNK8 and AtWNK10, whereas low, sustained sugar concentration slowly activate the pathway through AtWNK1, allowing the cells to respond similarly to transient, high-intensity signals and sustained, low-intensity signals. This "dose-duration reciprocity" allows encoding of both the intensity and persistence of glucose as an important energy resource and signaling molecule.

Negative feedback enhances robustness in the yeast polarity establishment circuit.

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

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.

Molecular switch architecture determines response properties of signaling pathways.

Many intracellular signaling pathways are composed of molecular switches, proteins that transition between two states-on and off Typically, signaling is initiated when an external stimulus activates its cognate receptor that, in turn, causes downstream switches to transition from off to on using one of the following mechanisms: activation, in which the transition rate from the off state to the on state increases; derepression, in which the transition rate from the on state to the off state decreases; and concerted, in which activation and derepression operate simultaneously. We use mathematical modeling to compare these signaling mechanisms in terms of their dose-response curves, response times, and abilities to process upstream fluctuations. Our analysis elucidates several operating principles for molecular switches. First, activation increases the sensitivity of the pathway, whereas derepression decreases sensitivity. Second, activation generates response times that decrease with signal strength, whereas derepression causes response times to increase with signal strength. These opposing features allow the concerted mechanism to not only show dose-response alignment, but also to decouple the response time from stimulus strength. However, these potentially beneficial properties come at the expense of increased susceptibility to upstream fluctuations. We demonstrate that these operating principles also hold when the models are extended to include additional features, such as receptor removal, kinetic proofreading, and cascades of switches. In total, we show how the architecture of molecular switches govern their response properties. We also discuss the biological implications of our findings.

Towards resolution of a paradox in plant G-protein signaling.

G-proteins are molecular on-off switches that are involved in transmitting a variety of extracellular signals to their intracellular targets. In animal and yeast systems, the switch property is encoded through nucleotides: a GDP-bound state is the "off-state" and the GTP-bound state is the "on-state". The G-protein cycle consists of the switch turning on through nucleotide exchange facilitated by a G-protein coupled receptor and the switch turning off through hydrolysis of GTP back to GDP, facilitated by a protein designated REGULATOR OF G SIGNALING 1 (RGS). In plants, G-protein signaling dramatically differs from that in animals and yeast. Despite stringent conservation of the nucleotide binding and catalytic structures over the 1.6 billion years that separate the evolution of plants and animals, genetic and biochemical data indicate that nucleotide exchange is less critical for this switch to operate in plants. Also, the loss of the single RGS protein in Arabidopsis (Arabidopsis thaliana) confers unexpectedly weaker phenotypes consistent with a diminished role for the G cycle, at least under static conditions. However, under dynamic conditions, genetic ablation of RGS in Arabidopsis results in a strong phenotype. We explore explanations to this conundrum by formulating a mathematical model that takes into account the accruing evidence for the indispensable role of phosphorylation in G-protein signaling in plants and that the G-protein cycle is needed to process dynamic signal inputs. We speculate that the plant G-protein cycle and its attendant components evolved to process dynamic signals through signaling modulation rather than through on-off, switch-like regulation of signaling. This so-called change detection may impart greater fitness for plants due to their sessility in a dynamic light, temperature, and pest environment.

Spatial models of pattern formation during phagocytosis.

Phagocytosis, the biological process in which cells ingest large particles such as bacteria, is a key component of the innate immune response. Fcγ receptor (FcγR)-mediated phagocytosis is initiated when these receptors are activated after binding immunoglobulin G (IgG). Receptor activation initiates a signaling cascade that leads to the formation of the phagocytic cup and culminates with ingestion of the foreign particle. In the experimental system termed "frustrated phagocytosis", cells attempt to internalize micropatterned disks of IgG. Cells that engage in frustrated phagocytosis form "rosettes" of actin-enriched structures called podosomes around the IgG disk. The mechanism that generates the rosette pattern is unknown. We present data that supports the involvement of Cdc42, a member of the Rho family of GTPases, in pattern formation. Cdc42 acts downstream of receptor activation, upstream of actin polymerization, and is known to play a role in polarity establishment. Reaction-diffusion models for GTPase spatiotemporal dynamics exist. We demonstrate how the addition of negative feedback and minor changes to these models can generate the experimentally observed rosette pattern of podosomes. We show that this pattern formation can occur through two general mechanisms. In the first mechanism, an intermediate species forms a ring of high activity around the IgG disk, which then promotes rosette organization. The second mechanism does not require initial ring formation but relies on spatial gradients of intermediate chemical species that are selectively activated over the IgG patch. Finally, we analyze the models to suggest experiments to test their validity.

Recurrent mismatch binding by MutS mobile clamps on DNA localizes repair complexes nearby.

DNA mismatch repair (MMR), the guardian of the genome, commences when MutS identifies a mismatch and recruits MutL to nick the error-containing strand, allowing excision and DNA resynthesis. Dominant MMR models posit that after mismatch recognition, ATP converts MutS to a hydrolysis-independent, diffusive mobile clamp that no longer recognizes the mismatch. Little is known about the postrecognition MutS mobile clamp and its interactions with MutL. Two disparate frameworks have been proposed: One in which MutS-MutL complexes remain mobile on the DNA, and one in which MutL stops MutS movement. Here we use single-molecule FRET to follow the postrecognition states of MutS and the impact of MutL on its properties. In contrast to current thinking, we find that after the initial mobile clamp formation event, MutS undergoes frequent cycles of mismatch rebinding and mobile clamp reformation without releasing DNA. Notably, ATP hydrolysis is required to alter the conformation of MutS such that it can recognize the mismatch again instead of bypassing it; thus, ATP hydrolysis licenses the MutS mobile clamp to rebind the mismatch. Moreover, interaction with MutL can both trap MutS at the mismatch en route to mobile clamp formation and stop movement of the mobile clamp on DNA. MutS's frequent rebinding of the mismatch, which increases its residence time in the vicinity of the mismatch, coupled with MutL's ability to trap MutS, should increase the probability that MutS-MutL MMR initiation complexes localize near the mismatch.

Ratiometric GPCR signaling enables directional sensing in yeast.

Accurate detection of extracellular chemical gradients is essential for many cellular behaviors. Gradient sensing is challenging for small cells, which can experience little difference in ligand concentrations on the up-gradient and down-gradient sides of the cell. Nevertheless, the tiny cells of the yeast Saccharomyces cerevisiae reliably decode gradients of extracellular pheromones to find their mates. By imaging the behavior of polarity factors and pheromone receptors, we quantified the accuracy of initial polarization during mating encounters. We found that cells bias the orientation of initial polarity up-gradient, even though they have unevenly distributed receptors. Uneven receptor density means that the gradient of ligand-bound receptors does not accurately reflect the external pheromone gradient. Nevertheless, yeast cells appear to avoid being misled by responding to the fraction of occupied receptors rather than simply the concentration of ligand-bound receptors. Such ratiometric sensing also serves to amplify the gradient of active G protein. However, this process is quite error-prone, and initial errors are corrected during a subsequent indecisive phase in which polarity clusters exhibit erratic mobile behavior.

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.

Cellular noise suppression by the regulator of G protein signaling Sst2.

G proteins and their associated receptors process information from a variety of environmental stimuli to induce appropriate cellular responses. Generally speaking, each cell in a population responds within defined limits, despite large variation in the expression of protein signaling components. Therefore, we postulated that noise suppression is encoded within the signaling system. Using the yeast mating pathway as a model, we evaluated the ability of a regulator of G protein signaling (RGS) protein to suppress noise. We found that the RGS protein Sst2 limits variability in transcription and morphogenesis in response to pheromone stimulation. While signal suppression is a result of both the GAP (GTPase accelerating) and receptor binding functions of Sst2, noise suppression requires only the GAP activity. Taken together, our findings reveal a hitherto overlooked role of RGS proteins as noise suppressors and demonstrate an ability to uncouple signal and noise in a prototypical stimulus-response pathway.

Mechanistic models of PLC/PKC signaling implicate phosphatidic acid as a key amplifier of chemotactic gradient sensing.

Chemotaxis of fibroblasts and other mesenchymal cells is critical for embryonic development and wound healing. Fibroblast chemotaxis directed by a gradient of platelet-derived growth factor (PDGF) requires signaling through the phospholipase C (PLC)/protein kinase C (PKC) pathway. Diacylglycerol (DAG), the lipid product of PLC that activates conventional PKCs, is focally enriched at the up-gradient leading edge of fibroblasts responding to a shallow gradient of PDGF, signifying polarization. To explain the underlying mechanisms, we formulated reaction-diffusion models including as many as three putative feedback loops based on known biochemistry. These include the previously analyzed mechanism of substrate-buffering by myristoylated alanine-rich C kinase substrate (MARCKS) and two newly considered feedback loops involving the lipid, phosphatidic acid (PA). DAG kinases and phospholipase D, the enzymes that produce PA, are identified as key regulators in the models. Paradoxically, increasing DAG kinase activity can enhance the robustness of DAG/active PKC polarization with respect to chemoattractant concentration while decreasing their whole-cell levels. Finally, in simulations of wound invasion, efficient collective migration is achieved with thresholds for chemotaxis matching those of polarization in the reaction-diffusion models. This multi-scale modeling framework offers testable predictions to guide further study of signal transduction and cell behavior that affect mesenchymal chemotaxis.

Systematic analysis of F-box proteins reveals a new branch of the yeast mating pathway.

The mating pathway in yeast Saccharomyces cerevisiae has long been used to reveal new mechanisms of signal transduction. The pathway comprises a pheromone receptor, a heterotrimeric G protein, and intracellular effectors of morphogenesis and transcription. Polarized cell growth, in the direction of a potential mating partner, is accomplished by the G-protein ßγ subunits and the small G-protein Cdc42. Transcription induction, needed for cell-cell fusion, is mediated by Gßγ and the mitogen-activated protein kinase (MAPK) scaffold protein Ste5. A potential third pathway is initiated by the G-protein α subunit Gpa1. Gpa1 signaling was shown previously to involve the F-box adaptor protein Dia2 and an endosomal effector protein, the phosphatidylinositol 3-kinase Vps34. Vps34 is also required for proper vacuolar sorting and autophagy. Here, using a panel of reporter assays, we demonstrate that mating pheromone stimulates vacuolar targeting of a cytoplasmic reporter protein and that this process depends on Vps34. Through a systematic analysis of F-box deletion mutants, we show that Dia2 is required to sustain pheromone-induced vacuolar targeting. We also found that other F-box proteins selectively regulate morphogenesis (Ydr306, renamed Pfu1) and transcription (Ucc1). These findings point to the existence of a new and distinct branch of the pheromone-signaling pathway, one that likely leads to vacuolar engulfment of cytoplasmic proteins and recycling of cellular contents in preparation for mating.

Stochastic modeling of human papillomavirusearly promoter gene regulation.

High risk forms of human papillomaviruses (HPVs) promote cancerous lesions and are implicated in almost all cervical cancer. Of particular relevance to cancer progression is regulation of the early promoter that controls gene expression in the initial phases of infection and can eventually lead to pre-cancer progression. Our goal was to develop a stochastic model to investigate the control mechanisms that regulate gene expression from the HPV early promoter. Our model integrates modules that account for transcriptional, post-transcriptional, translational and post-translational regulation of E1 and E2 early genes to form a functioning gene regulatory network. Each module consists of a set of biochemical steps whose stochastic evolution is governed by a chemical Master Equation and can be simulated using the Gillespie algorithm. To investigate the role of noise in gene expression, we compared our stochastic simulations with solutions to ordinary differential equations for the mean behavior of the system that are valid under the conditions of large molecular abundances and quasi-equilibrium for fast reactions. The model produced results consistent with known HPV biology. Our simulation results suggest that stochasticity plays a pivotal role in determining the dynamics of HPV gene expression. In particular, the combination of positive and negative feedback regulation generates stochastic bursts of gene expression. Analysis of the model reveals that regulation at the promoter affects burst amplitude and frequency, whereas splicing is more specialized to regulate burst frequency. Our results also suggest that splicing enhancers are a significant source of stochasticity in pre-mRNA abundance and that the number of viruses infecting the host cell represents a third important source of stochasticity in gene expression.

Mathematical model reveals role of nucleotide signaling in airway surface liquid homeostasis and its dysregulation in cystic fibrosis.

Mucociliary clearance is composed of three components (i.e., mucin secretion, airway surface hydration, and ciliary-activity) which function coordinately to clear inhaled microbes and other foreign particles from airway surfaces. Airway surface hydration is maintained by water fluxes driven predominantly by active chloride and sodium ion transport. The ion channels that mediate electrogenic ion transport are regulated by extracellular purinergic signals that signal through G protein-coupled receptors. These purinoreceptors and the signaling pathways they activate have been identified as possible therapeutic targets for treating lung disease. A systems-level description of airway surface liquid (ASL) homeostasis could accelerate development of such therapies. Accordingly, we developed a mathematical model to describe the dynamic coupling of ion and water transport to extracellular purinergic signaling. We trained our model from steady-state and time-dependent experimental measurements made using normal and cystic fibrosis (CF) cultured human airway epithelium. To reproduce CF conditions, reduced chloride secretion, increased potassium secretion, and increased sodium absorption were required. The model accurately predicted ASL height under basal normal and CF conditions and the collapse of surface hydration due to the accelerated nucleotide metabolism associated with CF exacerbations. Finally, the model predicted a therapeutic strategy to deliver nucleotide receptor agonists to effectively rehydrate the ASL of CF airways.

Quantitative analysis of the yeast pheromone pathway.

The pheromone response pathway of the yeast Saccharomyces cerevisiae is a well-established model for the study of G proteins and mitogen-activated protein kinase (MAPK) cascades. Our longstanding ability to combine sophisticated genetic approaches with established functional assays has provided a thorough understanding of signalling mechanisms and regulation. In this report, we compare new and established methods used to quantify pheromone-dependent MAPK phosphorylation, transcriptional induction, mating morphogenesis, and gradient tracking. These include both single-cell and population-based assays of activity. We describe several technical advances, provide example data for benchmark mutants, highlight important differences between newer and established methodologies, and compare the advantages and disadvantages of each as applied to the yeast model. Quantitative measurements of pathway activity have been used to develop mathematical models and reveal new regulatory mechanisms in yeast. It is our expectation that experimental and computational approaches developed in yeast may eventually be adapted to human systems biology and pharmacology.

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.

Principles that govern competition or co-existence in Rho-GTPase driven polarization.

Rho-GTPases are master regulators of polarity establishment and cell morphology. Positive feedback enables concentration of Rho-GTPases into clusters at the cell cortex, from where they regulate the cytoskeleton. Different cell types reproducibly generate either one (e.g. the front of a migrating cell) or several clusters (e.g. the multiple dendrites of a neuron), but the mechanistic basis for unipolar or multipolar outcomes is unclear. The design principles of Rho-GTPase circuits are captured by two-component reaction-diffusion models based on conserved aspects of Rho-GTPase biochemistry. Some such models display rapid winner-takes-all competition between clusters, yielding a unipolar outcome. Other models allow prolonged co-existence of clusters. We investigate the behavior of a simple class of models and show that while the timescale of competition varies enormously depending on model parameters, a single factor explains a large majority of this variation. The dominant factor concerns the degree to which the maximal active GTPase concentration in a cluster approaches a "saturation point" determined by model parameters. We suggest that both saturation and the effect of saturation on competition reflect fundamental properties of the Rho-GTPase polarity machinery, regardless of the specific feedback mechanism, which predict whether the system will generate unipolar or multipolar outcomes.

An RNAi screen of Rho signalling networks identifies RhoH as a regulator of Rac1 in prostate cancer cell migration.

BACKGROUND: Cell migration is essential for development and tissue repair, but it also contributes to disease. Rho GTPases regulate cell migration, but a comprehensive analysis of how each Rho signalling component affects migration has not been carried out. RESULTS: Through an RNA interference screen, and using a prostate cancer cell line, we find that approximately 25% of Rho network components alter migration. Some genes enhance migration while others decrease basal and/or hepatocyte growth factor-stimulated migration. Surprisingly, we identify RhoH as a screen hit. RhoH expression is normally restricted to haematopoietic cells, but we find it is expressed in multiple epithelial cancer cell lines. High RhoH expression in samples from prostate cancer patients correlates with earlier relapse. RhoH depletion reduces cell speed and persistence and decreases migratory polarity. Rac1 activity normally localizes to the front of migrating cells at areas of dynamic membrane movement, but in RhoH-depleted cells active Rac1 is localised around the whole cell periphery and associated with membrane regions that are not extending or retracting. RhoH interacts with Rac1 and with several p21-activated kinases (PAKs), which are Rac effectors. Similar to RhoH depletion, PAK2 depletion increases cell spread area and reduces cell migration. In addition, RhoH depletion reduces lamellipodium extension induced by PAK2 overexpression. CONCLUSIONS: We describe a novel role for RhoH in prostate cancer cell migration. We propose that RhoH promotes cell migration by coupling Rac1 activity and PAK2 to membrane protrusion. Our results also suggest that RhoH expression levels correlate with prostate cancer progression.

Expression of an S phase-stabilized version of the CDK inhibitor Dacapo can alter endoreplication.

In developing organisms, divergence from the canonical cell division cycle is often necessary to ensure the proper growth, differentiation, and physiological function of a variety of tissues. An important example is endoreplication, in which endocycling cells alternate between G and S phase without intervening mitosis or cytokinesis, resulting in polyploidy. Although significantly different from the canonical cell cycle, endocycles use regulatory pathways that also function in diploid cells, particularly those involved in S phase entry and progression. A key S phase regulator is the Cyclin E-Cdk2 kinase, which must alternate between periods of high (S phase) and low (G phase) activity in order for endocycling cells to achieve repeated rounds of S phase and polyploidy. The mechanisms that drive these oscillations of Cyclin E-Cdk2 activity are not fully understood. Here, we show that the Drosophila Cyclin E-Cdk2 inhibitor Dacapo (Dap) is targeted for destruction during S phase via a PIP degron, contributing to oscillations of Dap protein accumulation during both mitotic cycles and endocycles. Expression of a PIP degron mutant Dap attenuates endocycle progression but does not obviously affect proliferating diploid cells. A mathematical model of the endocycle predicts that the rate of destruction of Dap during S phase modulates the endocycle by regulating the length of G phase. We propose from this model and our in vivo data that endo S phase-coupled destruction of Dap reduces the threshold of Cyclin E-Cdk2 activity necessary to trigger the subsequent G-S transition, thereby influencing endocycle oscillation frequency and the extent of polyploidy.