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Multistability and dynamic transitions of intracellular Min protein patterns

2016; Springer Nature; Volume: 12; Issue: 6 Linguagem: Inglês

10.15252/msb.20156724

1744-4292

Fabai Wu, Jacob Halatek, Matthias Reiter, Enzo Kingma, Erwin Frey, Cees Dekker,

Photosynthetic Processes and Mechanisms

Article8 June 2016Open Access Transparent process Multistability and dynamic transitions of intracellular Min protein patterns Fabai Wu Fabai Wu Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands Search for more papers by this author Jacob Halatek Jacob Halatek Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, München, Germany Search for more papers by this author Matthias Reiter Matthias Reiter Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, München, Germany Search for more papers by this author Enzo Kingma Enzo Kingma Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands Search for more papers by this author Erwin Frey Corresponding Author Erwin Frey Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, München, Germany Search for more papers by this author Cees Dekker Corresponding Author Cees Dekker Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands Search for more papers by this author Fabai Wu Fabai Wu Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands Search for more papers by this author Jacob Halatek Jacob Halatek Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, München, Germany Search for more papers by this author Matthias Reiter Matthias Reiter Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, München, Germany Search for more papers by this author Enzo Kingma Enzo Kingma Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands Search for more papers by this author Erwin Frey Corresponding Author Erwin Frey Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, München, Germany Search for more papers by this author Cees Dekker Corresponding Author Cees Dekker Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands Search for more papers by this author Author Information Fabai Wu1,‡, Jacob Halatek2,‡, Matthias Reiter2, Enzo Kingma1, Erwin Frey 2 and Cees Dekker 1 1Department of Bionanoscience, Kavli Institute of Nanoscience, Delft University of Technology, Delft, The Netherlands 2Arnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, München, Germany ‡These authors contributed equally to this work *Corresponding author. Tel: +49 8921804537; E-mail: [email protected] *Corresponding author. Tel: +31 152786094; E-mail: [email protected] Molecular Systems Biology (2016)12:873https://doi.org/10.15252/msb.20156724 PDFDownload PDF of article text and main figures. Peer ReviewDownload a summary of the editorial decision process including editorial decision letters, reviewer comments and author responses to feedback. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InMendeleyWechatReddit Figures & Info Abstract Cells owe their internal organization to self-organized protein patterns, which originate and adapt to growth and external stimuli via a process that is as complex as it is little understood. Here, we study the emergence, stability, and state transitions of multistable Min protein oscillation patterns in live Escherichia coli bacteria during growth up to defined large dimensions. De novo formation of patterns from homogenous starting conditions is observed and studied both experimentally and in simulations. A new theoretical approach is developed for probing pattern stability under perturbations. Quantitative experiments and simulations show that, once established, Min oscillations tolerate a large degree of intracellular heterogeneity, allowing distinctly different patterns to persist in different cells with the same geometry. Min patterns maintain their axes for hours in experiments, despite imperfections, expansion, and changes in cell shape during continuous cell growth. Transitions between multistable Min patterns are found to be rare events induced by strong intracellular perturbations. The instances of multistability studied here are the combined outcome of boundary growth and strongly nonlinear kinetics, which are characteristic of the reaction–diffusion patterns that pervade biology at many scales. Synopsis Persistence and transitions of Min protein oscillations in diverse cell shapes reveal how reaction–diffusion patterns respond to cellular heterogeneity and boundary growth. Min protein oscillation patterns are analyzed in live Escherichia coli cells. Experiments and simulations show that multiple distinct Min patterns can be stable in different cells of the same shape. Pattern selection largely depends on the growth history of the cells. Theoretical analyses show that the observed multistability is not generic but relies on strong cooperative membrane binding. Introduction Many cells have characteristic forms. To guide proper assembly of their subcellular structures, cells employ machineries that garner and transmit information of cell shape (Kholodenko & Kolch, 2008; Shapiro et al, 2009; Moseley & Nurse, 2010; Minc & Piel, 2012). But cells are not static objects: They grow, divide, and react to stimuli, and these processes are often accompanied by a change of cell shape. Hence, the means by which a cell gathers spatial information need to be adaptive. One versatile mechanism that is capable of such spatial adaptation is self-organized pattern formation (Cross & Hohenberg, 1993; Epstein & Pojman, 1998; Murray, 2003). Spontaneous emergence of spatial structures from initially homogeneous conditions is a major paradigm in biology, and Alan Turing's reaction–diffusion theory was the first to show how local chemical interactions could be coupled through diffusion to yield sustained, non-uniform patterns (Turing, 1952). In this way, the symmetry of the starting system can be broken. Reaction–diffusion mechanisms have been shown to account for the generation of many biological patterns (Kondo & Miura, 2010). However, how patterns change in response to noise and perturbations, be they chemical or geometrical, is poorly understood. Resolution of such issues is critical for an understanding of the role of reaction–diffusion systems in the context of the spatial confines and physiology of a cell (or an organism). To include the effects of geometry, the mathematical framework for reaction–diffusion theory has been extended to circular (Levine & Rappel, 2005), spherical (Klünder et al, 2013), and elliptical geometries (Halatek & Frey, 2012). However, focusing on pattern formation from homogeneity is not enough, as was noted by Turing himself at the end of his seminal article in 1952 (Turing, 1952): "Most of an organism, most of the time, is developing from one pattern into another, rather than from homogeneity into a pattern". Min proteins form dynamic spatial patterns that regulate the placement of division sites in prokaryotic cells and eukaryotic plastids (de Boer et al, 1989; Hu & Lutkenhaus, 1999; Raskin & de Boer, 1999; Colletti et al, 2000; Maple et al, 2002; Ramirez-Arcos et al, 2002; Szeto et al, 2002; Leisch et al, 2012; Leger et al, 2015; Makroczyová et al, 2016). In rod-shaped Escherichia coli cells, MinD and MinE form a reaction–diffusion network that drives pole-to-pole oscillations in their local concentrations (Hu & Lutkenhaus, 1999; Raskin & de Boer, 1999; Huang et al, 2003). Membrane-bound MinD binds MinC, which inhibits FtsZ polymerization (Dajkovic et al, 2008). The dynamic Min oscillation patterns thus result in maximal inhibition of FtsZ accumulation at the cell poles and minimal inhibition at the cell center, which, together with a nucleoid occlusion mechanism, restricts formation of the division apparatus to midcell (Adams & Errington, 2009). Because it exhibits a multitude of complex phenomena, which can be explored by experimental and theoretical means, the Min oscillator provides an informative reference system for the quantitative study of geometry-responsive pattern formation. The dynamic Min oscillations have been explained by reaction–diffusion models based on a minimal set of interactions between MinD, MinE, ATP, and the cell membrane (Howard et al, 2001; Meinhardt & de Boer, 2001; Kruse, 2002; Huang et al, 2003; Fange & Elf, 2006; Touhami et al, 2006; Loose et al, 2008; Halatek & Frey, 2012). MinD, in its ATP-bound form, cooperatively binds to the cytoplasmic membrane (Hu et al, 2002; Mileykovskaya et al, 2003). MinE interacts with membrane-bound MinD, triggering the hydrolysis of its bound ATP and releasing MinD from the membrane (Hu et al, 2002; Shih et al, 2002; Hsieh et al, 2010; Loose et al, 2011; Park et al, 2011). MinD then undergoes a nucleotide exchange cycle in the cytosol, which was initially incorporated into the modeling framework by Huang et al (Huang et al, 2003). Further theoretical analysis of the minimal reaction scheme suggested that the interplay between the rate of cytosolic nucleotide exchange and strong preference for membrane recruitment of MinD relative to MinE facilitates transitions from pole-to-pole oscillations in cells of normal size to multinode oscillations (striped mode) in filamentous cells (Halatek & Frey, 2012). Such transitions occur if proteins that have detached from one polar zone have a greater tendency to re-attach to the membrane in the other half of the cell rather than to the old polar zone—a process which has been termed canalized transfer. This leads to synchronized growth and depletion of MinD from spatially separated polar zones, enabling the simultaneous maintenance of multiple polar zones. Numerical simulations of a reaction–diffusion model based on this canalized transfer of Min proteins successfully explain a plethora of experimentally observed Min oscillations in various geometries (Halatek & Frey, 2012). Essential for the robust function of Min proteins in ensuring symmetric cell division is their ability to respond to, and thus encode, information relating to cell shape. Upon cell-shape manipulation, Min proteins have been found to exhibit a range of phenotypes under different boundary conditions (Corbin et al, 2002; Touhami et al, 2006; Varma et al, 2008; Männik et al, 2012; Wu et al, 2015b). Recent development of a cell-sculpting technique allows accurate control of cell shape over a size range from 2 × 1 × 1 μm3 to 11 × 6 × 1 μm3, in which Min proteins show diverse oscillation patterns, including longitudinal, diagonal, rotational, striped, and even transverse modes (Wu et al, 2015b). These patterns were found to autonomously sense the symmetry and size of shaped cells. The longitudinal pole-to-pole mode was most stable in cells with widths of < 3 μm, and lengths of 3–6 μm. In cells of this size range, Min proteins form concentration gradients that scale with cell length, leading to central minima and polar maxima of the average Min concentration. Increasing cell length to 7 μm and above led to the emergence of striped oscillations. In cells wider than 3.5 μm, Min oscillations can align with the short axis of the lateral rectangular shape, yielding a transverse mode (Wu et al, 2015b). The existence of various oscillation modes has also been reconstituted in vitro with MinD, MinE, ATP, and lipid bilayers confined to microchambers (Zieske & Schwille, 2014). Numerical simulations based on an established reaction–diffusion model (Halatek & Frey, 2012) successfully recaptured the various oscillation modes in the experimentally sampled cell dimensions (Wu et al, 2015b). This further emphasizes the role of the two above-mentioned factors generic to reaction–diffusion processes in cells: cytosolic nucleotide exchange and membrane recruitment (Huang et al, 2003; Halatek & Frey, 2012). These data provided the first evidence that sensing of geometry is enabled by establishing an adaptive length scale through self-organized pattern formation. Given that Min proteins in all cells initially adopt the same regime of pole-to-pole oscillations, it is as yet unclear how diverse oscillation modes emerge during cell growth to large dimensions, and whether transitions occur between these patterns. Furthermore, more than one mode of oscillation was often observed in different cells with the same shape, presenting an intriguing example of the multistability of different complex patterns (Wu et al, 2015b). These unexplained phenomena provide us with the rare opportunity to quantitatively explore the basic principles of the dynamics of pattern formation in the context of geometric perturbations and cellular heterogeneities. In this study, we combine experiments and theory to systematically examine the emergence and dynamic switching of the distinct oscillatory Min protein patterns (longitudinal, transverse, and striped oscillations, cf. Fig 1A) observed in E. coli bacteria that are physically constrained to adopt defined cell shapes. Our primary aim was to investigate the origin of multistability (coexistence of stable patterns), and to further understand its relevance in the context of cell growth (i.e. changing cell shape). Furthermore, we hoped to identify the kinetic regimes and mechanisms that promote transitions between patterns and to probe their robustness against spatial variations in kinetic parameters. One striking discovery is the high degree of robustness of individual modes of oscillation even in the face of significant changes in geometry. Figure 1. Symmetry breaking of Min protein patterns in vivo A. Schematic showing Min protein patterns in a defined geometry originating from 1) a dynamic instability arising from an equilibrium state or 2) dynamic transitions from a pre-existing pattern associated with cell growth. Green and red particles represent MinD and MinE proteins, respectively. The green gradient depicts the MinD concentration gradient. B–D. Examples of Min protein patterns emerging from nearly homogeneous initial conditions in E. coli cells of different sizes. Lateral dimensions (in μm) from top to bottom: 2 × 6.5, 2 × 8.8, and 5.2 × 8.8, respectively. The gray-scale images show cytosolic near-infrared fluorescence emitted by the protein eqFP670 at the first (left) and last (right) time points. The color montages show the sfGFP-MinD intensity (indicated by the color scale at the bottom right) over time. The scale bar in panel (B) corresponds to 5 μm. Red arrows show the oscillation mode at the respective time point. E. Two early and two late frames depicting sfGFP-MinD patterns in a cell exhibiting stable transverse oscillations. The images share the scale bar in (B). F. Difference in sfGFP-MinD intensity between the top half and bottom half of the cell plotted against time. Download figure Download PowerPoint To present our results, we first show experimentally that different patterns can emerge out of near-homogeneous initial states in living cells with different dimensions, thus providing further support for an underlying Turing instability. We then use computational approaches to capture the dependence of pattern selection on geometry. Using stability analysis, we establish kinetic and geometric parameter regimes that allow both longitudinal and transverse patterns to coexist. Furthermore, we evaluate the emergence and stability of these patterns in computer simulations and compare the results with experimental data. Remarkably, we find that the experimentally observed multistability is reproduced by the theoretical model in its original parameter regime characterized by canalized transfer. In experiments, we trace pattern development during the cell-shape changes that accompany cell growth, and we quantitatively assess the persistence and transition of patterns in relation to cell shape. These analyses reveal that Min patterns are remarkably robust against shape imperfections, size expansion, and even changes in cell axes induced by cell growth. Transitions between multistable patterns occur (albeit infrequently), driving the system from one stable oscillatory pattern to another. Altogether, this study provides a comprehensive framework for understanding pattern formation in the context of spatial perturbations induced by intracellular fluctuations and cellular growth. Results Symmetry breaking of Min patterns from homogeneity in live E. coli cells One of the most striking examples of the accessibility of multiple stable states observed in shaped E. coli cells is the emergence of different—transverse and longitudinal—Min oscillation modes in rectangular cells with identical dimensions (Wu et al, 2015b). The existence of a transverse mode has also been noted in reconstituted in vitro systems (Zieske & Schwille, 2014). In live cells, this phenomenon is most prominent in cells with widths of about 5 μm and lengths of between 7 and 11 μm (Wu et al, 2015b). To probe the emergence and stability of these different stable states, we began this study by monitoring the temporal evolution of Min protein patterns in deformable cells growing in rectangular microchambers. Improving upon our previous shaping and imaging method (see Materials and Methods), we recorded cytosolic eqFP670 (a near-infrared fluorescent protein) and sfGFP-MinD fluorescence signals over the entire course of cell growth (~6–8 h). Owing to the superior brightness and photostability of these two fluorescent probes (Wu et al, 2015a), we were able to image the cells at 2-min intervals without affecting cell growth. Given that an oscillation cycle (or period) takes 68 ± 13 s (mean ± SD) at our experimental temperature (26°C), shorter intervals were subsequently used to capture the detailed dynamics within one oscillation cycle. We first grew cells with the above-mentioned lateral dimensions (7–11 × 5 × 1 μm3) in microchambers of the appropriate form. Of the 126 cells examined, almost all (n = 121) showed clear MinD polar zones in all times prior to cell death or growth beyond the confines of the chambers, demonstrating the striking persistence of the oscillation cycles. In some cells, transition states between different patterns were also captured, which are described below (see Sections Persistent directionality traps Min oscillations in a stable state during cell growth and Experimental observations of pattern transitions between multistable states). Interestingly, imaging of the remaining five cells captured 1–2 frames in which the sfGFP-MinD fluorescence was distributed homogeneously (Fig EV1 and Video EV1). Such a homogeneous state phenomenologically resembles the initial conditions chosen in the majority of chemical and theoretical studies on pattern formation. However, in the present case, Min proteins re-established oscillations exclusively in the transverse mode, irrespective of their preceding oscillation mode (Fig EV1). Why the system should "revert" to such a homogeneous state in the first place is unknown, although the rapid recovery of patterns leads us to speculate that it most probably results from a transient effect, such as a change in membrane potential or a rearrangement of chromosomes, rather than from a drastic depletion of ATP. Nonetheless, such an intermittent state provides a unique opportunity to study the emergence of patterns from a spatially uniform background. Click here to expand this figure. Figure EV1. Disruption and re-emergence of Min patterns in cells of 5 μm in widthThe red boxes show the near-homogeneous state. The color scale indicates MinD concentration. Scale bar = 5 μm. Download figure Download PowerPoint We therefore explored symmetry breaking by Min proteins over a larger range of cell sizes and found that different cell dimensions gave rise to different patterns from an intermittent homogeneous state. Because homogeneous distributions of MinD are observed at low frequency, we manually searched for cells in such a state. Once targeted, such cells were subsequently imaged at short time intervals of between 5 and 20 s until an oscillation pattern stabilized. As shown in Fig 1B–D, the uniform distribution of sfGFP-MinD seen in cells of different sizes and shapes became inhomogeneous, and always re-established stable oscillations within a few minutes. In the 6.5 × 2 × 1 μm3 cell shown in Fig 1B, the homogeneous sfGFP-MinD signal first became concentrated at the periphery of the cell, indicating a transition from the cytosolic state to the membrane-bound form. At t = 20 s, a minor degree of asymmetry was observed. Within the next 30 s, a clear sfGFP-MinD binding zone developed on the left-hand side of the top cell half. This zone persisted for 40 s, until a new binding zone was established at the top cell pole, which then recruited the majority of the sfGFP-MinD molecules. This pattern rapidly evolved into longitudinal pole-to-pole oscillations which lasted for the rest of the time course of our time-lapse imaging (10 min). In an 8.8 × 2 × 1 μm3 cell (Fig 1C), the initial membrane binding of sfGFP-MinD was accompanied by the formation of several local patches of enhanced density (see, e.g. t = 30 s), which went on to form one large patch that was asymmetrically positioned in relation to the cell axes (t = 110 s). This MinD binding zone further evolved into a few cycles of asymmetric oscillations before converging into striped oscillations, with sfGFP-MinD oscillating between two polar caps and a central striped. In the 8.8 × 5.2 × 1 μm3 cell (Fig 1D), persistent transverse oscillations emerged within ~2.5 min after clusters of sfGFP-MinD had begun to emerge as randomly localized, membrane-bound patches from the preceding homogeneous state. To further examine the stability of the transverse mode, we tracked transverse oscillations in 5-μm-wide cells with a time resolution of 20 s. We found that these indeed persisted, with a very robust oscillation frequency, for at least 17 cycles (i.e. the maximum duration of our experiment) under our imaging conditions (Fig 1E and F, and Video EV2). This indicates that, once established, the transverse mode in these large cells is just as robust as the longitudinal pole-to-pole mode in a regular rod-shaped E. coli cell. In order to probe the effect of MinE in the process of symmetry breaking, we engineered a strain that co-expresses sfGFP-MinD and MinE-mKate2 from the endogenous minDE genomic locus (see Materials and Methods). In shaped bacteria, MinE-mKate2 proteins oscillate in concert with MinD (Video EV3). After the loss of oscillatory activities of both sfGFP-MinD and MinE-mKate2, no heterogeneous MinE pattern was observed prior to the emergence of MinD patches that dictate the axis of symmetry breaking (Video EV2). This is in agreement with the previous finding that MinE relies on MinD for its recruitment to the membrane (Hu et al, 2002). The observed emergence of Min protein patterns from homogeneous states shows several striking features. First of all, after the early stage of MinD membrane binding, which appears to be rather uniform across the cell, the first patch with enhanced MinD density that forms is neither aligned with the symmetry axes nor does it show a preference for the highly curved polar regions. Secondly, Min patterns converge into a stable pattern within a few oscillation cycles. Emerging patterns align with symmetry axes, and exhibit a preference for the characteristic length range discovered previously (Wu et al, 2015b), confirming that the geometry-sensing ability of Min proteins is intrinsic and self-organized. The fast emergence and stabilization of Min protein patterns indicates an intrinsic robustness of Min oscillations and an ability to adjust oscillatory patterns dynamically to changes in cell geometry. Analytical and computational approach to probe the geometry-dependent symmetry breaking and pattern selection The experimental observations described above showed that symmetry breaking in spatially almost-homogeneous states can result in stable oscillation patterns of Min proteins. These spatiotemporal configurations are longitudinal and transverse oscillation patterns whose detailed features are dependent on the geometry of the system, in accordance with our previous study (Wu et al, 2015b). We therefore set out to gain a deeper understanding of the mechanisms underlying the phenomenon of multistability and the role of cell geometry in determining, regulating, and guiding the pattern formation process and the ensuing stable spatiotemporal patterns. To this end, we performed a theoretical analysis, building on previous investigations of symmetry breaking induced by the oscillatory Turing instability in bounded geometries (Halatek & Frey, 2012). The results presented in this Section are based on the observation that the selection of the initial pattern (which does not necessarily coincide with the final pattern) depends on both the Turing instability and the system's geometry. While we focus on the latter aspect in the main text, we review in Box 1 how, more generally, a Turing instability facilitates symmetry breaking in a planar geometry, which may help the reader to understand why the interconnection between geometry and the classical Turing mechanism is crucial. Box 1: Symmetry breaking by the Turing instability in cellular geometries The initial phase of a "symmetry-breaking" process in a nonlinear, spatially extended system is determined by a mode-selection mechanism. Consider an initial steady state of the corresponding well-mixed system that is weakly perturbed spatially, by some spatially white noise, for instance. For the planar geometry considered in textbooks and review articles, the initial state is typically a spatially uniform state (Cross & Hohenberg, 1993; Epstein & Pojman, 1998; Murray, 2003). The spectral decomposition of this state gives equal weight to all Fourier modes and, therefore, sets no bias for a particular mode. A system is referred to as being "Turing unstable" if any spatially non-uniform perturbation of a uniform equilibrium fails to decay (as expected due to diffusion) but instead grows into a patterned state. The collection of growth rates plotted as a function of the wave number of the corresponding Fourier modes is called the dispersion relation, and can be computed by a linear stability analysis. The mode with the fastest growth rate is called the critical mode. It sets the length scale of the initial pattern if there is no other bias for a different mode. Such a bias could, for instance, be provided by a specific initial condition that is non-uniform. It has been shown recently that, in the context of realistic biological systems, a well-mixed state is generically non-uniform for reaction–diffusion systems based on membrane–cytosol cycling and an NTPase activity (Thalmeier et al, 2016). Hence, in this generic case, the symmetry of the stationary state is already broken—in the sense that it is adapted to the geometry of the cell. Consequently, any downstream instabilities—such as the Turing instability—will inherit the symmetry of this spatially non-uniform steady state. In this paper, we discuss how the analysis of the instability of such a non-uniform steady state differs from that of the traditional Turing instabilities of uniform states. The non-uniformity of the well-mixed state in cell geometries (as noted in Box 1) is not the only salient difference relative to the classical case of a planar geometry. To perform linear stability analysis on a particular system, a set of Fourier modes must be derived that is specific for the boundary geometry of the system. Hence, both the well-mixed state and the spectrum of Fourier modes are generically geometry-dependent. Only a few geometries are amenable to an analytical treatment. A recent advance was the derivation of eigenfunctions for reaction–diffusion systems with reactive boundaries (the cell membrane) and diffusive bulks (the cytosol) in an elliptical geometry (Halatek & Frey, 2012). This geometry, being analytically accessible, permits broad, systematic parameter studies. At the same time, it shares the symmetries of interest with rod-shaped, circular, and rectangular cells. The eigenfunctions or modes of the ellipse are classified into even and odd functions by their symmetry with respect to reflections through a plane along the long axis; the lowest-order modes are shown in Fig 2A. Even functions are symmetric, and odd functions are anti-symmetric with respect to long-axis reflection. As such, even functions correspond to longitudinal modes, and odd functions to transverse modes. More subtle than the separation into two symmetry classes, but no less significant, is the strict absence of any homogeneous steady states in elliptical systems undergoing cytosolic nucleotide exchange (Thalmeier et al, 2016). This can be understood intuitively from a source–degradation picture: Proteins detach from the membrane and undergo cytosolic ADP-ATP exchange. The concentration of ADP-bound MinD drops with increasing distance from the membrane as the diphosphate is replaced by ATP. This yields cytosolic concentration gradients at the membrane that determine the densities of membrane-bound proteins. In an equilibrium state confined to an elliptical geometry, the cytosolic gradients at the membrane cannot be constant, but will vary along the cell's circumference. Hence, a uniform density a