Min-oscillations in Escherichia coli induced by interactions of membrane-bound proteins

The mechanism of MinD stability modulation by MinE in Min protein dynamics

The patterns formed both in vivo and in vitro by the Min protein system have attracted much interest because of the complexity of their dynamic interactions given the apparent simplicity of the component parts. Despite both the experimental and theoretical attention paid to this system, the details of the biochemical interactions of MinD and MinE, the proteins responsible for the patterning, are still unclear. For example, no model consistent with the known biochemistry has yet accounted for the observed dual role of MinE in the membrane stability of MinD. Until now, a statistical comparison of models to the time course of Min protein concentrations on the membrane has not been carried out. Such an approach is a powerful way to test existing and novel models that are difficult to test using a purely experimental approach. Here, we extract time series from previously published fluorescence microscopy time lapse images of in vitro experiments and fit two previously described and one novel mathematical model to the data. We find that the novel model, which we call the Asymmetric Activation with Bridged Stability Model, fits the time-course data best. It is also consistent with known biochemistry and explains the dual MinE role via MinE-dependent membrane stability that transitions under the influence of rising MinE to membrane instability with positive feedback. Our results reveal a more complex network of interactions between MinD and MinE underlying Min-system dynamics than previously considered.

Influence of physical interactions on spatiotemporal patterns

Spatiotemporal patterns are often modeled using reaction-diffusion equations, which combine complex reactions between constituents with ideal diffusive motion. Such descriptions neglect physical interactions between constituents, which might affect resulting patterns. To overcome this, we study how physical interactions affect cyclic dominant reactions, like the seminal rock-paper-scissors game, which exhibits spiral waves for ideal diffusion. Generalizing diffusion to incorporate physical interactions, we find that weak interactions change the length- and time scales of spiral waves, consistent with a mapping to the complex Ginzburg-Landau equation. In contrast, strong repulsive interactions typically generate oscillating lattices, and strong attraction leads to an interplay of phase separation and chemical oscillations, like droplets co-locating with cores of spiral waves. Our work suggests that physical interactions are relevant for forming spatiotemporal patterns in nature, and it might shed light on how biodiversity is maintained in ecological settings.

Preferential localization of a single spot in reaction-diffusion systems on non-spherical surfaces

The present work systematically examines the effect of breaking the rotational symmetry of a surface on the spot positioning in reaction-diffusion (RD) systems. In particular, we study analytically and numerically the steady-state positioning of a single spot in RD systems on a prolate and an oblate ellipsoid. We adapt perturbative techniques to perform a linear stability analysis of the RD system on both ellipsoids. Furthermore, the spot positionings in the steady states of non-linear RD equations are obtained numerically on both ellipsoids. Our analysis suggests that preferential spot positioning can be observed on non-spherical surfaces. The present work may provide useful insights into the role of cell geometry on various symmetry-breaking mechanisms in cellular processes.

The Min-protein oscillations in Escherichia coli: an example of self-organized cellular protein waves

In the rod-shaped bacterium Escherichia coli, selection of the cell centre as the division site involves pole-to-pole oscillations of the proteins MinC, MinD and MinE. This spatio-temporal pattern emerges from interactions among the Min proteins and with the cytoplasmic membrane. Combining experimental studies in vivo and in vitro together with theoretical analysis has led to a fairly good understanding of Min-protein self-organization. In different geometries, the system can, in addition to standing waves, also produce travelling planar and spiral waves as well as coexisting stable stationary distributions. Today it stands as one of the best-studied examples of cellular self-organization of proteins.This article is part of the theme issue 'Self-organization in cell biology'.

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.

Mechanistic insights of the Min oscillator via cell-free reconstitution and imaging

The MinD and MinE proteins of Escherichia coli self-organize into a standing-wave oscillator on the membrane to help align division at mid-cell. When unleashed from cellular confines, MinD and MinE form a spectrum of patterns on artificial bilayers-static amoebas, traveling waves, traveling mushrooms, and bursts with standing-wave dynamics. We recently focused our cell-free studies on bursts because their dynamics recapitulate many features of Min oscillation observed in vivo. The data unveiled a patterning mechanism largely governed by MinE regulation of MinD interaction with membrane. We proposed that the MinD to MinE ratio on the membrane acts as a toggle switch between MinE-stimulated recruitment and release of MinD from the membrane. In this review, we summarize cell-free data on the Min system and expand upon a molecular mechanism that provides a biochemical explanation as to how these two 'simple' proteins can form the remarkable spectrum of patterns.

Non-linear Min protein interactions generate harmonics that signal mid-cell division in Escherichia coli

The Min protein system creates a dynamic spatial pattern in Escherichia coli cells where the proteins MinD and MinE oscillate from pole to pole. MinD positions MinC, an inhibitor of FtsZ ring formation, contributing to the mid-cell localization of cell division. In this paper, Fourier analysis is used to decompose experimental and model MinD spatial distributions into time-dependent harmonic components. In both experiment and model, the second harmonic component is responsible for producing a mid-cell minimum in MinD concentration. The features of this harmonic are robust in both experiment and model. Fourier analysis reveals a close correspondence between the time-dependent behaviour of the harmonic components in the experimental data and model. Given this, each molecular species in the model was analysed individually. This analysis revealed that membrane-bound MinD dimer shows the mid-cell minimum with the highest contrast when averaged over time, carrying the strongest signal for positioning the cell division ring. This concurs with previous data showing that the MinD dimer binds to MinC inhibiting FtsZ ring formation. These results show that non-linear interactions of Min proteins are essential for producing the mid-cell positioning signal via the generation of second-order harmonic components in the time-dependent spatial protein distribution.

Mapping out Min protein patterns in fully confined fluidic chambers

The bacterial Min protein system provides a major model system for studying reaction-diffusion processes in biology. Here we present the first in vitro study of the Min system in fully confined three-dimensional chambers that are lithography-defined, lipid-bilayer coated and isolated through pressure valves. We identify three typical dynamical behaviors that occur dependent on the geometrical chamber parameters: pole-to-pole oscillations, spiral rotations, and traveling waves. We establish the geometrical selection rules and show that, surprisingly, Min-protein spiral rotations govern the larger part of the geometrical phase diagram. Confinement as well as an elevated temperature reduce the characteristic wavelength of the Min patterns, although even for confined chambers with a bacterial-level viscosity, the patterns retain a ~5 times larger wavelength than in vivo. Our results provide an essential experimental base for modeling of intracellular Min gradients in bacterial cell division as well as, more generally, for understanding pattern formation in reaction-diffusion systems.

DOI:http://dx.doi.org/10.7554/eLife.19271.001

Some proteins can spontaneously organize themselves into ordered patterns within living cells. One widely studied pattern is made in a rod-shaped bacterium called Escherichia coli by a group of proteins called the Min proteins. The pattern formed by the Min proteins allows an E. coli cell to produce two equally sized daughter cells when it divides by ensuring that the division machinery correctly assembles at the center of the parent cell. These proteins move back and forth between the two ends of the parent cell so that the levels of Min proteins are highest at the ends and lowest in the middle. Since the Min proteins act to inhibit the assembly of the cell division machinery, this machinery only assembles in locations where the level of Min proteins is at its lowest, that is, at the middle of the cell.

When Min proteins are purified and placed within an artificial compartment that contains a source of chemical energy and is covered by a membrane similar to the membranes that surround cells, they spontaneously form traveling waves on top of the membrane in many directions along to surface. It is not clear how these waves relate to the oscillations seen in E. coli. Caspi and Dekker now analyze the behavior of purified Min proteins inside chambers of various sizes that are fully enclosed by a membrane.

The results show that in narrow chambers, Min proteins move back and forth (i.e. oscillate) from one side to the other. However, in wider containers the wave motion is more common. In containers of medium width the Min proteins rotate in a spiral fashion. Caspi and Dekker propose that the spiral rotations are the underlying pattern formed by Min proteins and that the back and forth motion is caused by spirals being cut short. In other words, if a spiral cannot form because the compartment is too small then the back and forth motion emerges. Similarly, Caspi and Dekker propose that the waves emerge in larger containers when multiple spirals come together.

These findings suggest that the different patterns that Min proteins form in bacterial cells and artificial compartments arise from different underlying mechanisms. The next step will be to investigate other differences in how the patterns of Min proteins form in E. coli and in artificial compartments.

DOI:http://dx.doi.org/10.7554/eLife.19271.002

Physics of Intracellular Organization in Bacteria

With the realization that bacteria achieve exquisite levels of spatiotemporal organization has come the challenge of discovering the underlying mechanisms. In this review, we describe three classes of such mechanisms, each of which has physical origins: the use of landmarks, the creation of higher-order structures that enable geometric sensing, and the emergence of length scales from systems of chemical reactions coupled to diffusion. We then examine the diversity of geometric cues that exist even in cells with relatively simple geometries, and end by discussing both new technologies that could drive further discovery and the implications of our current knowledge for the behavior, fitness, and evolution of bacteria. The organizational strategies described here are employed in a wide variety of systems and in species across all kingdoms of life; in many ways they provide a general blueprint for organizing the building blocks of life.

Geometry-induced protein pattern formation

Protein patterns are known to adapt to cell shape and serve as spatial templates that choreograph downstream processes like cell polarity or cell division. However, how can pattern-forming proteins sense and respond to the geometry of a cell, and what mechanistic principles underlie pattern formation? Current models invoke mechanisms based on dynamic instabilities arising from nonlinear interactions between proteins but neglect the influence of the spatial geometry itself. Here, we show that patterns can emerge as a direct result of adaptation to cell geometry, in the absence of dynamical instability. We present a generic reaction module that allows protein densities robustly to adapt to the symmetry of the spatial geometry. The key component is an NTPase protein that cycles between nucleotide-dependent membrane-bound and cytosolic states. For elongated cells, we find that the protein dynamics generically leads to a bipolar pattern, which vanishes as the geometry becomes spherically symmetrical. We show that such a reaction module facilitates universal adaptation to cell geometry by sensing the local ratio of membrane area to cytosolic volume. This sensing mechanism is controlled by the membrane affinities of the different states. We apply the theory to explain AtMinD bipolar patterns in [Formula: see text] EcMinDE Escherichia coli. Due to its generic nature, the mechanism could also serve as a hitherto-unrecognized spatial template in many other bacterial systems. Moreover, the robustness of the mechanism enables self-organized optimization of protein patterns by evolutionary processes. Finally, the proposed module can be used to establish geometry-sensitive protein gradients in synthetic biological systems.

Theoretical Prediction of Disrupted Min Oscillation in Flattened Escherichia coli

The dynamics of the Min-protein system help Escherichia coli regulate the process of cell division by identifying the center of the cell. While this system exhibits robust bipolar oscillations in wild-type cell shapes, recent experiments have shown that when the cells are mechanically deformed into wide, flattened out, irregular shapes, the spatial regularity of these oscillations breaks down. We employ widely used stochastic and deterministic models of the Min system to simulate cells with flattened shapes. The deterministic model predicts strong bipolar oscillations, in contradiction with the experimentally observed behavior, while the stochastic model, which is based on the same reaction-diffusion equations, predicts more spatially irregular oscillations. We further report simulations of flattened but more symmetric shapes, which suggest that the flattening and lateral expansion may contribute as much to the irregular oscillation behavior as the asymmetry of the cell shapes.

An Optimal Free Energy Dissipation Strategy of the MinCDE Oscillator in Regulating Symmetric Bacterial Cell Division

Sustained molecular oscillations are ubiquitous in biology. The obtained oscillatory patterns provide vital functions as timekeepers, pacemakers and spacemarkers. Models based on control theory have been introduced to explain how specific oscillatory behaviors stem from protein interaction feedbacks, whereas the energy dissipation through the oscillating processes and its role in the regulatory function remain unexplored. Here we developed a general framework to assess an oscillator’s regulation performance at different dissipation levels. Using the Escherichia coli MinCDE oscillator as a model system, we showed that a sufficient amount of energy dissipation is needed to switch on the oscillation, which is tightly coupled to the system’s regulatory performance. Once the dissipation level is beyond this threshold, unlike stationary regulators’ monotonic performance-to-cost relation, excess dissipation at certain steps in the oscillating process damages the oscillator’s regulatory performance. We further discovered that the chemical free energy from ATP hydrolysis has to be strategically assigned to the MinE-aided MinD release and the MinD immobilization steps for optimal performance, and a higher energy budget improves the robustness of the oscillator. These results unfold a novel mode by which living systems trade energy for regulatory function.

This paper presents a unique dissipation mode of converting biochemical free energy in ATP to regulatory function through the MinCDE bio-oscillator that marks the mid-cell position for symmetric bacterial cell division. Through assessing the oscillator’s performance-to-cost relation, we demonstrate that some dissipation threshold needs to be satisfied to switch on the oscillation, but the oscillator’s performance can be damaged by excess free energy dissipation, which is distinct from the known monotonic tradeoff relation of stationary regulators. An optimal dissipation strategy has been unveiled: the ATP free energy must be precisely allocated to specific reaction steps for accurate mid-cell recognition, which also coincides with the dynamic requirements for robust oscillation to occur. These discoveries identify an optimizable operation scheme of free energy consumption in biological systems and provide deep insights into the evolution of dynamic regulatory networks.

Molecular Interactions of the Min Protein System Reproduce Spatiotemporal Patterning in Growing and Dividing Escherichia coli Cells

Oscillations of the Min protein system are involved in the correct midcell placement of the divisome during Escherichia coli cell division. Based on molecular interactions of the Min system, we formulated a mathematical model that reproduces Min patterning during cell growth and division. Specifically, the increase in the residence time of MinD attached to the membrane as its own concentration increases, is accounted for by dimerisation of membrane-bound MinD and its interaction with MinE. Simulation of this system generates unparalleled correlation between the waveshape of experimental and theoretical MinD distributions, suggesting that the dominant interactions of the physical system have been successfully incorporated into the model. For cells where MinD is fully-labelled with GFP, the model reproduces the stationary localization of MinD-GFP for short cells, followed by oscillations from pole to pole in larger cells, and the transition to the symmetric distribution during cell filamentation. Cells containing a secondary, GFP-labelled MinD display a contrasting pattern. The model is able to account for these differences, including temporary midcell localization just prior to division, by increasing the rate constant controlling MinD ATPase and heterotetramer dissociation. For both experimental conditions, the model can explain how cell division results in an equal distribution of MinD and MinE in the two daughter cells, and accounts for the temperature dependence of the period of Min oscillations. Thus, we show that while other interactions may be present, they are not needed to reproduce the main characteristics of the Min system in vivo.

Toward Spatially Regulated Division of Protocells: Insights into the E. coli Min System from in Vitro Studies

For reconstruction of controlled cell division in a minimal cell model, or protocell, a positioning mechanism that spatially regulates division is indispensable. In Escherichia coli, the Min proteins oscillate from pole to pole to determine the division site by inhibition of the primary divisome protein FtsZ anywhere but in the cell middle. Remarkably, when reconstituted under defined conditions in vitro, the Min proteins self-organize into spatiotemporal patterns in the presence of a lipid membrane and ATP. We review recent progress made in studying the Min system in vitro, particularly focusing on the effects of various physicochemical parameters and boundary conditions on pattern formation. Furthermore, we discuss implications and challenges for utilizing the Min system for division site placement in protocells.

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.

Traveling waves and global oscillations triggered by attractive molecular interactions in an excitable system

During pattern formation in spatially extended systems, different mechanisms with different characteristic length scales, e.g., reaction-diffusion processes or molecular interactions, can be active. Such multiscale effects may generate new phenomena, which are not observed in systems where pattern formation occurs on a single scale. Here, we derive and analyze a reaction-diffusion model of the FitzHugh-Nagumo type with short-range attractive molecular interactions of the activator species. The model exhibits a wave instability. Simulations in one and two dimensions show traveling waves with a wavelength set by the parameters of the molecular interaction in the model. In two dimensions, simulations reveal a labyrinthine arrangement of the waves in systems with isotropic diffusion, whereas parallel bands of counterpropagating waves are formed in simulations of a model with anisotropic diffusion. The latter findings are in good qualitative agreement with experimental observation in the catalytic NO+H_{2} reaction on an anisotropic Rh(110) surface. In addition we have identified a transition regime in the simulations, where a short scale instability triggers global oscillations in an excitable regime.

Membrane binding of MinE allows for a comprehensive description of Min-protein pattern formation

The rod-shaped bacterium Escherichia coli selects the cell center as site of division with the help of the proteins MinC, MinD, and MinE. This protein system collectively oscillates between the two cell poles by alternately binding to the membrane in one of the two cell halves. This dynamic behavior, which emerges from the interaction of the ATPase MinD and its activator MinE on the cell membrane, has become a paradigm for protein self-organization. Recently, it has been found that not only the binding of MinD to the membrane, but also interactions of MinE with the membrane contribute to Min-protein self-organization. Here, we show that by accounting for this finding in a computational model, we can comprehensively describe all observed Min-protein patterns in vivo and in vitro. Furthermore, by varying the system's geometry, our computations predict patterns that have not yet been reported. We confirm these predictions experimentally.

Cellular protein structures have long been suggested to form by protein self-organization. A particularly clear example is provided by the proteins MinC, MinD, and MinE selecting the center as site of cell division in the rod-shaped bacterium Escherichia coli. Based on binding of MinD to the cytoplasmic membrane and an antagonistic action of MinE, which induces the release of MinD into the cytoplasm, these proteins oscillate from pole to pole, where they inhibit cell division. Supporting the idea of self-organization being the cause of the Min oscillations, purified Min proteins were found to spontaneously form traveling waves on supported lipid bilayers. A comprehensive understanding of the Min patterns formed under various conditions remains elusive. We have performed a computational analysis of Min-protein dynamics taking into account the recently discovered persistent action of MinE. We show that this property allows to reproduce all observed Min-protein patterns in a unified framework. Furthermore, our analysis predicts new structures, which we observed experimentally. Our study highlights that mechanisms underlying the spontaneous formation of protein patterns under purified in vitro conditions can also generate patterns inside complex intracellular environments.

Propagation of MinCDE waves on free-standing membranes

As a spatial modulator of cytokinesis in Escherichia coli, the Min system cooperates with the nucleoid occlusion mechanism to target the divisome assembly towards mid-cell. Based on a reaction-diffusion mechanism powered by ATP (adenosine triphosphate) hydrolysis, the Min proteins propagate in waves on the cell membrane, resulting in oscillations between the cell poles, thus preventing the formation of the division ring everywhere but in the cell centre. The dynamic behaviour of Min proteins has been successfully reconstructed in vitro on supported lipid bilayers (SLBs), reproducing many of the features observed in the cell. However, there has been a marked discrepancy between the speed of propagation of Min protein waves in vitro, compared with the cellular system. A very plausible explanation is the different mobility of proteins on model membranes, compared with the inner membrane of bacteria. To quantitatively demonstrate how membrane diffusion influences Min wave propagation, we compared Min waves on SLBs with free-standing giant unilamellar vesicles (GUV) membranes which display higher fluidity. Intriguingly, the propagation velocity and wavelength on GUVs are three times higher than those reported on supported bilayers, but the wave period is conserved. This suggests that the shorter spatial period of the patterns in vivo might indeed be primarily explained by lower diffusion coefficients of proteins on the bacterial inner membrane.

Highly canalized MinD transfer and MinE sequestration explain the origin of robust MinCDE-protein dynamics

Min-protein oscillations in Escherichia coli are characterized by the remarkable robustness with which spatial patterns dynamically adapt to variations of cell geometry. Moreover, adaption, and therefore proper cell division, is independent of temperature. These observations raise fundamental questions about the mechanisms establishing robust Min oscillations, and about the role of spatial cues, as they are at odds with present models. Here, we introduce a robust model based on experimental data, consistently explaining the mechanisms underlying pole-to-pole, striped, and circular patterns, as well as the observed temperature dependence of the oscillation period. Contrary to prior conjectures, the model predicts that MinD and cardiolipin domains are not colocalized. The transient sequestration of MinE and highly canalized transfer of MinD between polar zones are the key mechanisms underlying oscillations. MinD channeling enhances midcell localization and facilitates stripe formation, revealing the potential optimization process from which robust Min-oscillations originally arose.

Protein self-organization: lessons from the min system

One of the most fundamental features of biological systems is probably their ability to self-organize in space and time on different scales. Despite many elaborate theoretical models of how molecular self-organization can come about, only a few experimental systems of biological origin have so far been rigorously described, due mostly to their inherent complexity. The most promising strategy of modern biophysics is thus to identify minimal biological systems showing self-organized emergent behavior. One of the best-understood examples of protein self-organization, which has recently been successfully reconstituted in vitro, is represented by the oscillations of the Min proteins in Escherichia coli. In this review, we summarize the current understanding of the mechanism of Min protein self-organization in vivo and in vitro. We discuss the potential of the Min oscillations to sense the geometry of the cell and suggest that spontaneous protein waves could be a general means of intracellular organization. We hypothesize that cooperative membrane binding and unbinding, e.g., as an energy-dependent switch, may act as an important regulatory mechanism for protein oscillations and pattern formation in the cell.

Min protein patterns emerge from rapid rebinding and membrane interaction of MinE

In Escherichia coli, the pole-to-pole oscillation of the Min proteins directs septum formation to midcell, which is required for symmetric cell division. In vitro, protein waves emerge from the self-organization of MinD, a membrane-binding ATPase, and its activator MinE. For wave propagation, the proteins need to cycle through states of collective membrane binding and unbinding. Although MinD presumably undergoes cooperative membrane attachment, it is unclear how synchronous detachment is coordinated. We used confocal and single-molecule microscopy to elucidate the order of events during Min wave propagation. We propose that protein detachment at the rear of the wave, and the formation of the E-ring, are accomplished by two complementary processes: first, local accumulation of MinE due to rapid rebinding, leading to dynamic instability; and second, a structural change induced by membrane-interaction of MinE in an equimolar MinD-MinE (MinDE) complex, which supports the robustness of pattern formation.

High-throughput, subpixel precision analysis of bacterial morphogenesis and intracellular spatio-temporal dynamics

Bacteria display various shapes and rely on complex spatial organization of their intracellular components for many cellular processes. This organization changes in response to internal and external cues. Quantitative, unbiased study of these spatio-temporal dynamics requires automated image analysis of large microscopy datasets. We have therefore developed MicrobeTracker, a versatile and high-throughput image analysis program that outlines and segments cells with subpixel precision, even in crowded images and mini-colonies, enabling cell lineage tracking. MicrobeTracker comes with an integrated accessory tool, SpotFinder, which precisely tracks foci of fluorescently labelled molecules inside cells. Using MicrobeTracker, we discover that the dynamics of the extensively studied Escherichia coli Min oscillator depends on Min protein concentration, unveiling critical limitations in robustness within the oscillator. We also find that the fraction of MinD proteins oscillating increases with cell length, indicating that the oscillator has evolved to be most effective when cells attain an appropriate length. MicrobeTracker was also used to uncover novel aspects of morphogenesis and cell cycle regulation in Caulobacter crescentus. By tracking filamentous cells, we show that the chromosomal origin at the old-pole is responsible for most replication/separation events while the others remain largely silent despite contiguous cytoplasm. This surprising position-dependent silencing is regulated by division.

Intra- and intercellular fluctuations in Min-protein dynamics decrease with cell length

Self-organization of proteins in space and time is of crucial importance for the functioning of cellular processes. Often, this organization takes place in the presence of strong random fluctuations due to the small number of molecules involved. We report on stochastic switching of the Min-protein distributions between the two cell halves in short Escherichia coli cells. A computational model provides strong evidence that the macroscopic switching is rooted in microscopic noise on the molecular scale. In longer bacteria, the switching turns into regular oscillations that are required for positioning of the division plane. As the pattern becomes more regular, cell-to-cell variability also lessens, indicating cell length-dependent regulation of Min-protein activity.

Multiple modes of interconverting dynamic pattern formation by bacterial cell division proteins

Min proteins of the Escherichia coli cell division system oscillate between the cell poles in vivo. In vitro on a solid-surface supported lipid bilayer, these proteins exhibit a number of interconverting modes of collective ATP-driven dynamic pattern formation including not only the previously described propagating waves, but also near uniformity in space surface concentration oscillation, propagating filament like structures with a leading head and decaying tail and moving and dividing amoeba-like structures with sharp edges. We demonstrate that the last behavior most closely resembles in vivo system behavior. The simple reaction-diffusion models previously proposed for the Min system fail to explain the results of the in vitro self-organization experiments. We propose the hypotheses that initiation of MinD binding to the surface is controlled by counteraction of initiation and dissociation complexes; the binding of MinD/E is stimulated by MinE and involves polymerization-depolymerization dynamics; polymerization of MinE over MinD oligomers triggers dynamic instability leading to detachment from the membrane. The physical properties of the lipid bilayer are likely to be one of the critical determinants of certain aspects of the dynamic patterns observed.

A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of E. coli MinE to E-ring formation

Many important cellular processes are regulated by reaction-diffusion (RD) of molecules that takes place both in the cytoplasm and on the membrane. To model and analyze such multicompartmental processes, we developed a lattice-based Monte Carlo method, Spatiocyte that supports RD in volume and surface compartments at single molecule resolution. Stochasticity in RD and the excluded volume effect brought by intracellular molecular crowding, both of which can significantly affect RD and thus, cellular processes, are also supported. We verified the method by comparing simulation results of diffusion, irreversible and reversible reactions with the predicted analytical and best available numerical solutions. Moreover, to directly compare the localization patterns of molecules in fluorescence microscopy images with simulation, we devised a visualization method that mimics the microphotography process by showing the trajectory of simulated molecules averaged according to the camera exposure time. In the rod-shaped bacterium Escherichia coli, the division site is suppressed at the cell poles by periodic pole-to-pole oscillations of the Min proteins (MinC, MinD and MinE) arising from carefully orchestrated RD in both cytoplasm and membrane compartments. Using Spatiocyte we could model and reproduce the in vivo MinDE localization dynamics by accounting for the previously reported properties of MinE. Our results suggest that the MinE ring, which is essential in preventing polar septation, is largely composed of MinE that is transiently attached to the membrane independently after recruited by MinD. Overall, Spatiocyte allows simulation and visualization of complex spatial and reaction-diffusion mediated cellular processes in volumes and surfaces. As we showed, it can potentially provide mechanistic insights otherwise difficult to obtain experimentally.

Cytokinesis and the contractile ring in fission yeast: towards a systems-level understanding

Cytokinesis, the final stage of the cell division cycle, requires the proper placement, assembly and contraction of an actomyosin-based contractile ring. Conserved sets of cytokinesis proteins and pathways have now been identified and characterized functionally. Additionally, fluorescent protein fusion technology enables quantitative high-resolution imaging of protein dynamics in living cells. For these reasons, the study of cytokinesis is now ripe for quantitative, systems-level approaches. Here, we review our current understanding of the molecular mechanisms of contractile ring dynamics in the model organism Schizosaccharomyces pombe (fission yeast), focusing on recent examples that illustrate a synergistic integration of quantitative experimental data with computational modeling. A picture of a highly dynamic and integrated system consisting of overlapping networks is beginning to emerge, the detailed nature of which remains to be elucidated.

Predictions from a stochastic polymer model for the MinDE protein dynamics in Escherichia coli

The spatiotemporal oscillations of the Min proteins in the bacterium Escherichia coli play an important role in cell division. A number of different models have been proposed to explain the dynamics from the underlying biochemistry. Here, we extend a previously described discrete polymer model from a deterministic to a stochastic formulation. We express the stochastic evolution of the oscillatory system as a map from the probability distribution of maximum polymer length in one period of the oscillation to the probability distribution of maximum polymer length half a period later and solve for the fixed point of the map with a combined analytical and numerical technique. This solution gives a theoretical prediction of the distributions of both lengths of the polar MinD zones and periods of oscillations--both of which are experimentally measurable. The model provides an interesting example of a stochastic hybrid system that is, in some limits, analytically tractable.

Modelling of processes governing subcellular localisation of MinD in Escherichia coli

Recent research has highlighted several examples wherein bacterial cell fate is determined by precise subcellular localisation of proteins. A prominent example is the polar localisation and oscillation of the Min proteins which is necessary for accurate cell division in Escherichia coli. Several computational models have been proposed which reproduce the oscillatory behaviour and observed phenotypes. However, these models use varying assumptions to do so leading to different mechanisms for precise polar localisation of MinD zones. To gain further insight, the authors extend a simplified model which focused on some key processes to explain the observed length scale for MinD zone formation. Using analytical approaches and numerical simulations, the authors explore cellular MinD distributions produced by these processes and propose a mechanism for precise polar localisation of MinD.

Spatial regulators for bacterial cell division self-organize into surface waves in vitro

In the bacterium Escherichia coli, the Min proteins oscillate between the cell poles to select the cell center as division site. This dynamic pattern has been proposed to arise by self-organization of these proteins, and several models have suggested a reaction-diffusion type mechanism. Here, we found that the Min proteins spontaneously formed planar surface waves on a flat membrane in vitro. The formation and maintenance of these patterns, which extended for hundreds of micrometers, required adenosine 5'-triphosphate (ATP), and they persisted for hours. We present a reaction-diffusion model of the MinD and MinE dynamics that accounts for our experimental observations and also captures the in vivo oscillations.

Signaling cascades as cellular devices for spatial computations

Signaling networks usually include protein-modification cycles. Cascades of such cycles are the backbones of multiple signaling pathways. Protein gradients emerge from the spatial separation of opposing enzymes, such as kinases and phosphatases, or guanine nucleotide exchange factors (GEFs) and GTPase activating proteins (GAPs) for GTPase cycles. We show that different diffusivities of an active protein form and an inactive form leads to spatial gradients of protein abundance in the cytoplasm. For a cascade of cycles, using a discrete approximation of the space, we derive an analytical expression for the spatial gradients and show that it converges to an exact solution with decreasing the size of the quantization. Our results facilitate quantitative analysis of the dependence of spatial gradients on the network topology and reaction kinetics. We demonstrate how different cascade designs filter and process the input information to generate precise, complex spatial guidance for multiple GTPase effector processes. Thus, protein-modification cascades may serve as devices to compute complex spatial distributions of target proteins within intracellular space.

An experimentalist's guide to computational modelling of the Min system

Spatio-temporal oscillations of the Min proteins are essential for selecting the cell division site in Escherichia coli. These oscillations are a key example of a biological phenomenon that can only be understood on a systems level rather than on the level of its individual components. Here, we review the key concepts that mathematical modelling has added to our understanding of the Min system. While several different mechanisms have been proposed, in all cases the oscillations emerge from a dynamic instability of a uniform protein distribution. To generate this instability, however, the various mechanisms rely on different features of Min protein interactions and transport. We critically evaluate these mechanisms in light of recent experimental evidence. We also review the effects of fluctuations caused by low cellular concentration of Min proteins, and describe how stochastic effects may potentially influence Min protein dynamics.

Modeling partitioning of Min proteins between daughter cells after septation in Escherichia coli

Ongoing sub-cellular oscillation of Min proteins is required to block minicelling in Escherichia coli. Experimentally, Min oscillations are seen in newly divided cells and no minicells are produced. In model Min systems many daughter cells do not oscillate following septation because of unequal partitioning of Min proteins between the daughter cells. Using the 3D model of Huang et al, we investigate the septation process in detail to determine the cause of the asymmetric partitioning of Min proteins between daughter cells. We find that this partitioning problem arises at certain phases of the MinD and MinE oscillations with respect to septal closure and it persists independently of parameter variation. At most 85% of the daughter cells exhibit Min oscillation following septation. Enhanced MinD binding at the static polar and dynamic septal regions, consistent with cardiolipin domains, does not substantially increase this fraction of oscillating daughters. We believe that this problem will be shared among all existing Min models and discuss possible biological mechanisms that may minimize partitioning errors of Min proteins following septation.

A multistranded polymer model explains MinDE dynamics in E. coli cell division

In Escherichia coli, the location of the site for cell division is regulated by the action of the Min proteins. These proteins undergo a periodic pole-to-pole oscillation that involves polymerization and ATPase activity of MinD under the controlling influence of MinE. This oscillation suppresses division near the poles while permitting division at midcell. Here, we propose a multistranded polymer model for MinD and MinE dynamics that quantitatively agrees with the experimentally observed dynamics in wild-type cells and in several well-studied mutant phenotypes. The model also provides new explanations for several phenotypes that have never been addressed by previous modeling attempts. In doing so, the model bridges a theoretical gap between protein structure, biochemistry, and mutant phenotypes. Finally, the model emphasizes the importance of nonequilibrium polymer dynamics in cell function by demonstrating how behavior analogous to the dynamic instability of microtubules is used by E. coli to achieve a sufficiently rapid timescale in controlling division site selection.

Surface waves of Min-proteins

In the bacterium Escherichia coli, the Min-proteins show pronounced pole-to-pole oscillations. They are functional for suppressing cell division at the cell ends, leaving the center as the only possible site for division. Analyzing different models of Min-protein dynamics in a bacterial geometry, we find waves on the cytoplasmic membrane. Interestingly, the surface wave solutions of different models belong to different symmetry classes. We suggest that experiments on Min-protein surface waves in vitro are helpful in distinguishing between different classes of models of Min-protein dynamics.

Mobility of Min-proteins in Escherichia coli measured by fluorescence correlation spectroscopy

In the bacterium Escherichia coli, selection of the division site involves pole-to-pole oscillations of the proteins MinD and MinE. Different oscillation mechanisms based on cooperative effects between Min-proteins and on the exchange of Min-proteins between the cytoplasm and the cytoplasmic membrane have been proposed. The parameters characterizing the dynamics of the Min-proteins in vivo are not known. It has therefore been difficult to compare the models quantitatively with experiments. Here, we present in vivo measurements of the mobility of MinD and MinE using fluorescence correlation spectroscopy. Two distinct timescales are clearly visible in the correlation curves. While the faster timescale can be attributed to cytoplasmic diffusion, the slower timescale could result from diffusion of membrane-bound proteins or from protein exchange between the cytoplasm and the membrane. We determine the diffusion constant of cytoplasmic MinD to be approximately 16 microm(2) s(-1), while for MinE we find about 10 microm(2) s(-1), independently of the processes responsible for the slower time-scale. The implications of the measured values for the oscillation mechanism are discussed.

Noise-induced Min phenotypes in E. coli

The spatiotemporal oscillations of the Escherichia coli proteins MinD and MinE direct cell division to the region between the chromosomes. Several quantitative models of the Min system have been suggested before, but no one of them accounts for the behavior of all documented mutant phenotypes. We analyzed the stochastic reaction-diffusion kinetics of the Min proteins for several E. coli mutants and compared the results to the corresponding deterministic mean-field description. We found that wild-type (wt) and filamentous (ftsZ⁻) cells are well characterized by the mean-field model, but that a stochastic model is necessary to account for several of the characteristics of the spherical (rodA⁻) and phospathedylethanolamide-deficient (PE⁻) phenotypes. For spherical cells, the mean-field model is bistable, and the system can get trapped in a non-oscillatory state. However, when the intrinsic noise is considered, only the experimentally observed oscillatory behavior remains. The stochastic model also reproduces the change in oscillation directions observed in the spherical phenotype and the occasional gliding of the MinD region along the inner membrane. For the PE⁻ mutant, the stochastic model explains the appearance of randomly localized and dense MinD clusters as a nucleation phenomenon, in which the stochastic kinetics at low copy number causes local discharges of the high MinD^ATP to MinD^ADP potential. We find that a simple five-reaction model of the Min system can explain all documented Min phenotypes, if stochastic kinetics and three-dimensional diffusion are accounted for. Our results emphasize that local copy number fluctuation may result in phenotypic differences although the total number of molecules of the relevant species is high.

Many molecules inside a living cell do not have time to diffuse through the whole cell in-between reactions. Furthermore, the chemical reactions are random and discrete events. In this study, the authors study an example in which these aspects of intracellular chemistry need to be considered when we try to understand how a biological system works.

The authors have investigated the spatial oscillation patterns that are displayed by the Min system of Escherichia coli. In wild-type E. coli, the Min proteins oscillate back and forth between the cell poles to help the bacterium find its middle before cell division. The authors used computer simulations to explain why the oscillation patterns change the way they do in different mutants of E. coli. They find that two of the mutant phenotypes can only be explained if one considers the randomness and discreteness of chemical reactions in addition to the spatial characteristics of the process. Particularly interesting is the phospathedylethanolamide-deficient phenotype, in which large dense clusters of MinD protein appear for some time at random locations on the membrane. The authors believe that this phenotype is due to a nucleation phenomenon, in which the stochastic kinetics at low copy number is amplified to macroscopic proportions.

Min-protein oscillations in Escherichia coli with spontaneous formation of two-stranded filaments in a three-dimensional stochastic reaction-diffusion model

We introduce a three-dimensional stochastic reaction-diffusion model to describe MinD/MinE dynamical structures in Escherichia coli. This model spontaneously generates pole-to-pole oscillations of the membrane-associated MinD proteins, MinE ring, as well as filaments of the membrane-associated MinD proteins. Experimental data suggest MinD filaments are two-stranded. In order to model them we assume that each membrane-associated MinD protein can form up to three bonds with adjacent membrane-associated MinD molecules and that MinE induced hydrolysis strongly depends on the number of bonds MinD has established.

Division accuracy in a stochastic model of Min oscillations in Escherichia coli

Accurate cell division in Escherichia coli requires the Min proteins MinC, MinD, and MinE as well as the presence of nucleoids. MinD and MinE exhibit spatial oscillations, moving from pole to pole of the bacterium, resulting in an average MinD concentration that is low at the center of the cell and high at the poles. This concentration minimum is thought to signal the site of cell division. Deterministic models of the Min oscillations reproduce many observed features of the system, including the concentration minimum of MinD. However, there are only a few thousand Min proteins in a bacterium, so stochastic effects are likely to play an important role. Here, we show that Monte Carlo simulations with a large number of proteins agree well with the results from a deterministic treatment of the equations. The location of minimum local MinD concentration is too variable to account for cell division accuracy in wild-type, but is consistent with the accuracy of cell division in cells without nucleoids. This finding confirms the need to include additional mechanisms, such as reciprocal interactions with the cell division ring or positioning of the nucleoids, to explain wild-type accuracy.

A stochastic model of Min oscillations inEscherichia coliand Min protein segregation during cell division

The Min system in Escherichia coli directs division to the centre of the cell through pole-to-pole oscillations of the MinCDE proteins. We present a one-dimensional stochastic model of these oscillations which incorporates membrane polymerization of MinD into linear chains. This model reproduces much of the observed phenomenology of the Min system, including pole-to-pole oscillations of the Min proteins. We then apply this model to investigate the Min system during cell division. Oscillations continue initially unaffected by the closing septum, before cutting off rapidly. The fractions of Min proteins in the daughter cells vary widely, from 50%-50% up to 85%-15% of the total from the parent cell, suggesting that there may be another mechanism for regulating these levels in vivo.