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

Simulation-based Reconstructed Diffusion unveils the effect of aging on protein diffusion in Escherichia coli

We have developed Simulation-based Reconstructed Diffusion (SbRD) to determine diffusion coefficients corrected for confinement effects and for the bias introduced by two-dimensional models describing a three-dimensional motion. We validate the method on simulated diffusion data in three-dimensional cell-shaped compartments. We use SbRD, combined with a new cell detection method, to determine the diffusion coefficients of a set of native proteins in Escherichia coli. We observe slower diffusion at the cell poles than in the nucleoid region of exponentially growing cells, which is independent of the presence of polysomes. Furthermore, we show that the newly formed pole of dividing cells exhibits a faster diffusion than the old one. We hypothesize that the observed slowdown at the cell poles is caused by the accumulation of aggregated or damaged proteins, and that the effect is asymmetric due to cell aging.

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'.

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

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

Minimal in vitro systems shed light on cell polarity

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

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.

Oscillations of Min-proteins in micropatterned environments: a three-dimensional particle-based stochastic simulation approach

The Min-proteins from E. coli and other bacteria are the best characterized pattern forming system in cells and their spatiotemporal oscillations have been successfully reconstituted in vitro. Different mathematical and computational models have been used to better understand these oscillations. Here we use particle-based stochastic simulations to study Min-oscillations in patterned environments. We simulate a rectangular box of length 10 μm and width 5 μm that is filled with grid or checkerboard patterns of different patch sizes and distances. For this geometry, we find different stable oscillation patterns, typically pole-to-pole oscillations along the minor axis and striped oscillations along the major axis. The Min-oscillations can switch from one pattern to the other, either effected by changes in pattern geometry or stochastically. By automatic analysis of large-scale computer simulations, we show quantitatively how the perturbing effect of increased patch distance can be rescued by increased patch size. We also show that striped oscillations occur robustly in arbitrarily shaped filamentous E. coli cells. Our results highlight the robustness and variability of Min-oscillations, put limits on the effect of putative division sites, and provide a powerful computational framework for future studies of protein self-organization in patterned environments.

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.

Stuttering Min oscillations within E. coli bacteria: a stochastic polymerization model

We have developed a 3D off-lattice stochastic polymerization model to study the subcellular oscillation of Min proteins in the bacteria Escherichia coli, and used it to investigate the experimental phenomenon of Min oscillation stuttering. Stuttering was affected by the rate of immediate rebinding of MinE released from depolymerizing filament tips (processivity), protection of depolymerizing filament tips from MinD binding and fragmentation of MinD filaments due to MinE. Processivity, protection and fragmentation each reduce stuttering, speed oscillations and MinD filament lengths. Neither processivity nor tip protection were, on their own, sufficient to produce fast stutter-free oscillations. While filament fragmentation could, on its own, lead to fast oscillations with infrequent stuttering; high levels of fragmentation degraded oscillations. The infrequent stuttering observed in standard Min oscillations is consistent with short filaments of MinD, while we expect that mutants that exhibit higher stuttering frequencies will exhibit longer MinD filaments. Increased stuttering rate may be a useful diagnostic to find observable MinD polymerization under experimental conditions.

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.

Self-organized partitioning of dynamically localized proteins in bacterial cell division

The Min proteins are equally partitioned between daughter cells at division.

The mechanism allowing this accurate distribution is intrinsic to the Min system.

Individual oscillations appear in each daughter cell before cytokinesis is completed.

Diffusion through the gradually constricting septum is key to this process.

One of the central problems of cell division is the proper distribution of all components to the progeny, which is essential to avoid the adverse effects that an unequal distribution—when not actively sought for differentiation purposes—would have on cell growth and regulation. Fast-growing bacterial cells are particularly exposed to this problem, as corrections of inequalities in protein distribution by biosynthesis could be too slow compared with the generation time. Moreover, bacterial proteins are usually stable and, therefore, their levels are not easily adjustable in one generation. Although for homogeneously distributed proteins an equal partitioning at division is readily achieved, dedicated mechanisms must exist to segregate proteins or cellular structures that possess a specific cellular location, but these mechanisms are largely unknown.

An extremely challenging case is represented by the Min proteins—MinC, MinD and MinE—that in Escherichia coli oscillate from pole to pole to inhibit the assembly of the cytokinetic ring anywhere except at mid-cell. The oscillations stem solely from local interactions among the proteins at the cytoplasmic membrane. In this work, we show that self-organization is also responsible for the distribution of Min proteins between daughter cells at division. Our combined experimental and computational results demonstrate that the equal protein partitioning stems from interplay between the self-organized oscillations and changes in the cell geometry during division, with no need for any additional regulatory network.

Using high-resolution time-lapse microscopy, we detected changes in the Min oscillatory regime that correlate with the amount of septal constriction (Figure 3A, B, E and F). When the cell is unconstricted, oscillations run from pole to pole (Figure 3A). When the constriction reaches a certain degree, typically corresponding to a septum of 600–500 nm, the oscillations change into a ‘half-cell to half-cell' mode during which the fluorescence covers, alternatively, the entire membrane of one daughter cell (Figure 3A, B and E). This mode persists for several minutes and, just before cell division when the septum is smaller than 200 nm, gives way to yet another oscillatory pattern wherein oscillations split and run independently in each daughter cell (Figure 3A, B and F).

Our 3D stochastic computer simulations revealed that these different regimes are an outcome of impaired diffusion through the closing septum and that oscillations finally split because protein exchange between the two future daughter cells becomes critically slow, so that independent oscillations on both sides of the septum become the stable solution (Figure 6A and E). FRAP experiments confirmed that the presence of the septum is enough to slow down the passage of molecules from one side of the cell to the other (Figure 6F). As oscillations become independent in each daughter cell before completion of cytokinesis, diffusion through the septum can still occur, which further equilibrates the levels of the Min proteins in the daughter cells (Figure 3C and D and Figure 6B, C and D).

In summary, our results suggest that E. coli cells have evolved a very simple and elegant way to ensure equal concentrations of the Min proteins in the progeny, based entirely on the intrinsic self-organizing properties of the Min system. This provides a clear example of self-organizing partitioning, which we expect to be a widely used strategy given its simplicity and low evolutionary cost.

How cells manage to get equal distribution of their structures and molecules at cell division is a crucial issue in biology. In principle, a feedback mechanism could always ensure equality by measuring and correcting the distribution in the progeny. However, an elegant alternative could be a mechanism relying on self-organization, with the interplay between system properties and cell geometry leading to the emergence of equal partitioning. The problem is exemplified by the bacterial Min system that defines the division site by oscillating from pole to pole. Unequal partitioning of Min proteins at division could negatively impact system performance and cell growth because of loss of Min oscillations and imprecise mid-cell determination. In this study, we combine live cell and computational analyses to show that known properties of the Min system together with the gradual reduction of protein exchange through the constricting septum are sufficient to explain the observed highly precise spontaneous protein partitioning. Our findings reveal a novel and effective mechanism of protein partitioning in dividing cells and emphasize the importance of self-organization in basic cellular processes.

Reaction-diffusion systems in intracellular molecular transport and control

Chemical reactions make cells work only if the participating chemicals are delivered to desired locations in a timely and precise fashion. Most research to date has focused on active-transport mechanisms, although passive diffusion is often equally rapid and energetically less costly. Capitalizing on these advantages, cells have developed sophisticated reaction-diffusion (RD) systems that control a wide range of cellular functions-from chemotaxis and cell division, through signaling cascades and oscillations, to cell motility. These apparently diverse systems share many common features and are "wired" according to "generic" motifs such as nonlinear kinetics, autocatalysis, and feedback loops. Understanding the operation of these complex (bio)chemical systems requires the analysis of pertinent transport-kinetic equations or, at least on a qualitative level, of the characteristic times of the constituent subprocesses. Therefore, in reviewing the manifestations of cellular RD, we also describe basic theory of reaction-diffusion phenomena.

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.

Reaction-diffusion master equation, diffusion-limited reactions, and singular potentials

To model biochemical systems in which both noise in the chemical reaction process and spatial movement of molecules is important, both the reaction-diffusion master equation (RDME) and Smoluchowski diffusion-limited reaction (SDLR) partial differential equation (PDE) models have been used. In previous work we showed that the solution to the RDME may be interpreted as an asymptotic approximation in the reaction radius to the solution of the SDLR PDE [S. A. Isaacson, SIAM J. Appl. Math. 70, 77 (2009)]. The approximation was shown to be divergent in the limit that the lattice spacing in the RDME approached zero. In this work we expand upon these results for the special case of the two-molecule annihilation reaction, A+B-->Ø. We first introduce a third stochastic reaction-diffusion PDE model that incorporates a pseudopotential based bimolecular reaction mechanism. The solution to the pseudopotential model is then shown to be an asymptotic approximation to the solution of the SDLR PDE for small reaction radii. We next illustrate how the RDME may be obtained by a formal discretization of the pseudopotential model, motivating why the RDME is itself an asymptotic approximation of the SDLR PDE. Finally, we give a more detailed numerical analysis of the difference between solutions to the RDME and SDLR PDE models as a function of both the reaction-radius and the lattice spacing (in the RDME).

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.

The dynamic nature of the bacterial cytoskeleton

Three of the four well-established bacterial cytoskeletal systems-the MreB, MinCDE, and FtsZ systems-undergo a variety of short-range and long-range dynamic behaviors. These include the cellular reorganization of the cytoskeletal elements, in which the proteins redistribute from a predominantly helical pole-to-pole pattern into annular structures near midcell. Despite their apparent similarity, these dramatic redistributional events in the three systems are in large part independent of each other. In addition, some of the cytoskeletal structures undergo oscillatory behavior in which the helical elements move repetitively back-and-forth between the two ends of the cell. The details and mechanisms underlying these dynamic cellular events are just now being revealed by fluorescence microscopy of intact cells, fluorescence photobleaching recovery studies, single molecule tracking techniques, and in vitro studies of the purified proteins.

Self-organization of the MinE protein ring in subcellular Min oscillations

We model the self-organization of the MinE ring that is observed during subcellular oscillations of the proteins MinD and MinE within the rod-shaped bacterium Escherichia coli. With a steady-state approximation, we can study the MinE ring generically--apart from the other details of the Min oscillation. Rebinding of MinE to depolymerizing MinD-filament tips controls MinE-ring formation through a scaled cell shape parameter r. We find two types of E-ring profiles near the filament tip: either a strong plateaulike E ring controlled by one-dimensional diffusion of MinE along the bacterial length or a weak cusplike E ring controlled by three-dimensional diffusion near the filament tip. While the width of a strong E ring depends on r, the occupation fraction of MinE at the MinD-filament tip is saturated and hence the depolymerization speed does not depend strongly on r. Conversely, for weak E rings both r and the MinE to MinD stoichiometry strongly control the tip occupation and hence the depolymerization speed. MinE rings in vivo are close to the threshold between weak and strong, and so MinD-filament depolymerization speed should be sensitive to cell shape, stoichiometry, and MinE-rebinding rate. We also find that the transient to MinE-ring formation is quite long in the appropriate open geometry for assays of ATPase activity in vitro, explaining the long delays of ATPase activity observed for smaller MinE concentrations in those assays without the need to invoke cooperative MinE activity.

Mutual effects of MinD-membrane interaction: I. Changes in the membrane properties induced by MinD binding

In Escherichia coli and other bacteria, MinD, along with MinE and MinC, rapidly oscillates from one pole of the cell to the other controlling the correct placement of the division septum. MinD binds to the membrane through its amphipathic C-terminal alpha-helix. This binding, promoted by ATP-induced dimerization, may be further enhanced by a consequent attraction of acidic phospholipids and formation of a stable proteolipid domain. In the context of this hypothesis we studied changes in dynamics of a model membrane caused by MinD binding using membrane-embedded fluorescent probes as reporters. A remarkable increase in membrane viscosity and order upon MinD binding to acidic phospholipids was evident from the pyrene and DPH fluorescence changes. This viscosity increase is cooperative with regards to the concentration of MinD-ATP, but not of the ADP form, indicative of dimerization. Moreover, similar changes in the membrane dynamics were demonstrated in the native inverted cytoplasmic membranes of E. coli, with a different depth effect. The mobility of pyrene-labeled phosphatidylglycerol indicated formation of acidic phospholipid-enriched domains in a mixed acidic-zwitterionic membrane at specific MinD/phospholipid ratios. A comparison between MinD from E. coli and Neisseria gonorrhea is also presented.

Mutual effects of MinD-membrane interaction: II. Domain structure of the membrane enhances MinD binding

MinD, a well-conserved bacterial amphitropic protein involved in spatial regulation of cell division, has a typical feature of reversible binding to the membrane. MinD shows a clear preference for acidic phospholipids organized into lipid domains in bacterial membrane. We have shown that binding of MinD may change the dynamics of model and native membranes (see accompanying paper [1]). On the other hand, MinD dimerization and anchoring could be enhanced on pre-existing anionic phospholipid domains. We have tested MinD binding to model membranes in which acidic and zwitterionic phospholipids are either well-mixed or segregated to phase domains. The phase separation was achieved in binary mixtures of 1-Stearoyl-2-Oleoyl-sn-Glycero-3-[Phospho-rac-(1-glycerol] (SOPG) with 1,2-Distearoyl-sn-Glycero-3-Phosphocholine (DSPC) or 1,2-Distearoyl-sn-Glycero-3-[Phospho-rac-(1-glycerol)] (DSPG) and binding to these membranes was compared with that to a fluid mixture of SOPG with 1-Stearoyl-2-Oleoyl-sn-Glycero-3-Phosphocholine (SOPC). The results demonstrate that MinD binding to the membrane is enhanced by segregation of anionic phospholipids to fluid domains in a gel-phase environment and, moreover, the protein stabilizes such domains. This suggests that an uneven binding of MinD to the heterogeneous native membrane is possible, leading to formation of a lipid-specific distribution pattern of MinD and/or modulation of its temporal behavior.

The Min system as a general cell geometry detection mechanism: branch lengths in Y-shaped Escherichia coli cells affect Min oscillation patterns and division dynamics

In Escherichia coli, division site placement is regulated by the dynamic behavior of the MinCDE proteins, which oscillate from pole to pole and confine septation to the centers of normal rod-shaped cells. Some current mathematical models explain these oscillations by considering interactions among the Min proteins without recourse to additional localization signals. So far, such models have been applied only to regularly shaped bacteria, but here we test these models further by employing aberrantly shaped E. coli cells as miniature reactors. The locations of MinCDE proteins fused to derivatives of green fluorescent protein were monitored in branched cells with at least three conspicuous poles. MinCDE most often moved from one branch to another in an invariant order, following a nonreversing clockwise or counterclockwise direction over the time periods observed. In cells with two short branches or nubs, the proteins oscillated symmetrically from one end to the other. The locations of FtsZ rings were consistent with a broad MinC-free zone near the branch junctions, and Min rings exhibited the surprising behavior of moving quickly from one possible position to another. Using a reaction-diffusion model that reproduces the observed MinCD oscillations in rod-shaped and round E. coli, we predict that the oscillation patterns in branched cells are a natural response of Min behavior in cellular geometries having different relative branch lengths. The results provide further evidence that Min protein oscillations act as a general cell geometry detection mechanism that can locate poles even in branched cells.

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.

Assembly dynamics of the bacterial MinCDE system and spatial regulation of the Z ring

The positioning of a cytoskeletal element that dictates the division plane is a fundamental problem in biology. The assembly and positioning of this cytoskeletal element has to be coordinated with DNA segregation and cell growth to ensure that equal-sized progeny cells are produced, each with a copy of the chromosome. In most prokaryotes, cytokinesis involves positioning a Z ring assembled from FtsZ, the ancestral homologue of tubulin. The position of the Z ring is determined by a gradient of negative regulators of Z-ring assembly. In Escherichia coli, the Min system consists of three proteins that cooperate to position the Z ring through a fascinating oscillation, which inhibits the formation of the Z ring away from midcell. Additional gradients of negative regulators of FtsZ assembly are used by E. coli and other bacteria to achieve spatial control of Z-ring assembly.

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.

Chromosome segregation in Escherichia coli division: a free energy-driven string model

Although the mechanisms of eukaryotic chromosome segregation and cell division have been elucidated to a certain extent, those for bacteria remain largely unknown. Here we present a computational string model for simulating the dynamics of Escherichia coli chromosome segregation. A novel thermal-average force field accounting for stretching, bending, volume exclusion, friction and random fluctuation is introduced. A Langevin equation is used to simulate the chromosome structural changes. The mechanism of chromosome segregation is thereby postulated as a result of free energy-driven structural optimization with replication introduced chromosomal mass increase. Predictions of the model agree well with observations of fluorescence labeled chromosome loci movement in living cells. The results demonstrate the possibility of a mechanism of chromosome segregation that does not involve cytoskeletal guidance or advanced apparatus in an E. coli cell. The model also shows that DNA condensation of locally compacted domains is a requirement for successful chromosome segregation. Simulations also imply that the shape-determining protein MreB may play a role in the segregation via modification of the membrane pressure.

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.

Temperature dependence of MinD oscillation in Escherichia coli: running hot and fast

We observed that the oscillation period of MinD within rod-like and filamentous cells of Escherichia coli varied by a factor of 4 in the temperature range from 20 degrees C to 40 degrees C. The detailed dependence was Arrhenius, with a slope similar to the overall temperature-dependent growth curve of E. coli. The detailed pattern of oscillation, including the characteristic wavelength in filamentous cells, remained independent of temperature. A quantitative model of MinDE oscillation exhibited similar behavior, with an activated temperature dependence of the MinE-stimulated MinD-ATPase rate.

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.