Membrane Binding of MinE Allows for a Comprehensive Description of Min-Protein Pattern Formation
[['Nature presents an overwhelming variety of forms and patterns .', 'While system specific conditions can play an important role for their formation , also a few general principles underlying biological pattern formation have been proposed in the past .', 'A particularly attractive concept is the spontaneous formation of patterns in reaction diffusion systems as proposed by Alan Turing [1] .', 'In this case , a ( small ) number of different constituents collectively form large-scale patterns .', 'So far , however , only a few biological examples of bona fide Turing patterns are known [2] .', 'An example of subcellular pattern formation due to reactions and diffusion of just two different constituents is provided by the Min system in the rod-shaped bacterium Escherichia coli [3] .', 'This protein system forms a spatiotemporal oscillation in the cell , that is , a standing wave with a node in the cell center [4] , [5] , see Figure 1A , which plays an essential role in division site selection in E . coli .', 'Whereas the oscillations emerge solely from the interactions between MinD , MinE and the membrane , the inhibitor of cell division MinC binds to MinD and is thus distributed similarly: it appears periodically at the cell poles , but is practically absent from the cell center .', 'In this way , division occurs in the cell center leading to two daughter cells of the same size .', 'While some models suggest that particular properties of the cell poles might play an essential role for Min-protein pattern formation [6] , a number of observations support the notion that the Min system can self-organize without any additional spatial cues .', 'First of all , depending on the cell geometry and the Min-protein expression level , the protein pattern can change: In longer cells , standing waves with several nodes form [4] , see Figure 1B , whereas in shorter cells and for slightly over-expressed Min proteins , oscillations are replaced by stochastic switching of the proteins between the two cell halves [7] , [8] .', "In Y-shaped cells , the proteins visit the different arms in a way that depends on the arms' lengths [9] .", 'Furthermore , in vitro studies of purified proteins found MinD and MinE to spontaneously organize into collective traveling waves [10] .', 'Together , these observations suggest that the Min-protein patterns emerge from the intrinsic dynamics of these proteins , in particular , the exchange of proteins between the membrane , driven by the high affinity of MinD for the membrane when ATP is bound and a low affinity with ADP bound [11] .', 'In addition , membrane-bound MinD recruits MinE , which in turn induces hydrolysis of the bound nucleotide by MinD and consequently MinD detachment from the membrane .', 'These well-established processes are at the core of a number of computational models reproducing the Min-protein oscillations observed in E . coli [12] .', 'The most popular mechanism studied through such models assumes that cooperative membrane-attachment of MinD is at the origin of pattern formation .', 'In the simplest version , the rate of MinD attachment to the membrane increases in presence of membrane-bound MinD [13] .', 'Several works on models implementing cooperative membrane attachment in various ways and complementing it by different side processes have shown that it can robustly generate the pole-to-pole oscillations observed in E . coli [14]–[16] even during septum closure [17] .', 'Other works rather emphasized cooperative effects between already membrane-bound MinD [18] , [19] .', 'However , in spite of more than a decade of theoretical analysis , there exists to date no comprehensive description of all Min-protein patterns observed in vivo and in vitro .', 'Some evidence suggested that an N-terminal helix allows MinE to also interact with the membrane [20] , [21] , however , it remained unclear if this property was important for the self-organization of the Min system .', 'Single molecule data obtained in vitro [22] as well as genetic , physiological , and structural analysis [23] finally provided evidence that the ability of MinE to interact with phospholipids allows it to remain bound to the membrane after MinD has detached , which could lead to the subsequent removal of several MinD dimers by one MinE dimer .', 'In analogy to molecular motors that can perform several subsequent steps on a cytoskeletal filament , we call this property “MinE processivity” .', 'This possibility had been proposed earlier on theoretical grounds as it offers a mechanism for the formation of MinE-rings [19] , [24] , [25] and was crucial for describing the guidance of Min-protein waves on patterned substrates [26] .', 'In the present work , we perform a computational study to explore the consequences of this molecular property for large-scale pattern formation .', 'To this end , we use deterministic and stochastic calculations in three dimensions .', 'We show that MinE processivity provides a key to obtain a unified description of all previously described Min-protein patterns in vivo and in vitro .', 'In addition , our analysis predicts hitherto unknown patterns , namely traveling waves in long and moving patches in aberrantly large cells .', 'We confirm the existence of these states by fluorescence microscopy of living E . coli cells .', 'Beyond the Min system , our findings highlight the importance of membrane-binding for subcellular pattern formation .'], ['We first studied the behavior of Min protein patterns in cellular geometries .', 'To this end , we solved the stochastic and deterministic dynamic equations in a cylindrical domain with hemispherical caps .', 'The parameters used in this section are given in Table 1 .', 'The values of the cytosolic diffusion constants have been measured in Ref .', '[45] .', 'While there is no direct measurement of the diffusion constants for membrane-bound MinD , MinE , and MinDE , diffusion on membranes is usually two to three orders of magnitude smaller than in the bulk [46] .', 'For larger values of these constants , the resulting patterns are broader and less well defined .', 'Decreasing their values does not affect the patterns significantly .', 'To determine the value of the maximal density of membrane-bound proteins , we use that close packing of MinD on the membrane would yield a density of about 1/ ( lateral extension of a MinD dimer ) , with the latter being approximately .', 'To account for crowding of the membrane by other molecules we use a value roughly 10 times smaller , .', 'The values of the various attachment and detachment rates have been chosen to match the experimentally observed patterns .', 'Note , that for the parameter values given in Table 1 , the dominant path for MinE-induced MinD detachment involves MinE staying on the membrane .', 'This corresponds to a high MinE processivity .', 'Finally , we mostly considered the Min patterns in geometries of fixed size .', 'Even under optimal growth conditions , E . coli gains only about 100 nm per oscillation period .', 'As we show below , the patterns are robust against such changes .', 'A major breakthrough in the understanding of Min-protein pattern formation has been achieved by studying the Min-dynamics in open geometries [10] , [22] , [26] , [54] .', 'Experimentally , in vitro studies using supported lipid bilayers have allowed us to clearly establish the propensity of the Min proteins to self-organize [10] .', 'Structural analysis suggested that binding to the membrane can also occur for MinE not associated with MinD [23] , providing a natural explanation for guiding Min-protein waves on structured surfaces [26] .', 'In Figure 8 we present the result of a numeric solution of the dynamic equations ( 8 ) – ( 12 ) , where we have employed periodic boundary conditions in the x- and y-directions .', 'Parameter values are given in Table 1 .', 'The differences between these values and those used for the in vivo geometries reflect differences in the environmental conditions , notably the presence respectively absence of other macromolecules .', 'Similar to the experimental observations , the Min proteins self-organize into traveling waves .', 'The calculated wave profile presents the same features as in the experiment: the MinD profile increases at the wave front and then saturates until it sharply drops .', 'The density of MinE increases more slowly than that of MinD .', "Towards the wave's trailing edge it exhibits a sharp increase and then drops rapidly .", 'The parameter is increased in comparison to the value determined in the section ‘Min-protein patterns in cellular geometries’ .', "The presentation of the distribution's z-dependence in Figure 8C shows that the pattern is confined to a layer of about above the membrane .", 'This result justifies a posteriori the use of effective 2d descriptions for the Min-protein dynamics [10] , [26] even though it is not obvious how to formally obtain the 2d equations from the 3d system .', 'The propagation of the wave fronts can be understood by interpreting the space coordinate in Figure 8B as time: First cytosolic MinD binds to the empty membrane .', 'The nonlinearity in the MinD binding term then leads to an increased binding rate and thus to an accelerated increase of the MinD density on the membrane .', 'As soon as membrane-bound MinD is present , MinE starts to attach .', 'As the MinE binding sites are abundant , the increase of the MinE density is roughly linear .', 'With increasing MinE density , the net rate of MinD attachment decreases .', 'Eventually , the MinE-induced detachment rate exceeds the attachment rate and the density of membrane-bound MinD decreases .', 'This decrease is sharp at the waves trailing edge , because MinE processivity leads to an accumulation of MinE in this region .', 'The sequence of Min protein patterns in vivo upon changing the system length can be intuitively understood from the mechanism underlying traveling waves in vitro .', 'To this end , we introduce the diffusion length , which is the length a molecule typically diffuses before attaching to the membrane .', 'For a diffusion constant D and an attachment rate ω it is given by .', 'Now , consider a wave in a cell propagating in the direction of the long axis .', "The wave is sustained by molecules binding to the wave's leading edge after they have been released from the trailing edge .", 'When the wave reaches a pole , the MinD dimers released from the membrane at the trailing edge can no longer bind at its leading edge .', 'Instead , they diffuse away form the cell pole .', 'If the cell length is on the order of , the proteins will preferentially bind at the opposite pole [55] , see Figure 9A .', 'Similarly , with some delay , MinE released from the original wave , will bind at this pole , too , and a new wave traveling in the same direction as the original one is generated .', 'If the system size is shorter , MinD binding will occur in a zone extending further from the new pole to the cell center because the ratio of diffusion length to the cell length has increased .', 'As the affinity for MinE binding to MinD on the membrane is large , MinE will preferentially bind to the part of the MinD zone proximal to the cell center and the wave will move into the opposite direction compared to the original wave , see Figure 9B , thus giving rise to pole-to-pole oscillations .', 'For even shorter cells , the distribution of cytosolic MinD and MinE is essentially homogenous as the diffusion lengths significantly exceeds the cell length .', 'MinD and also MinE thus bind preferentially to zones of the highest MinD concentrations on the membrane and a stationary profile emerges , see Figure 9C .', 'The picture presented here is thus somewhat different from the mechanism underlying the pole-to-pole oscillations proposed in Ref .', '[15] as we discuss below .', 'Let us finally note , that it is harder to get an intuitive picture of the dependence of the Min-protein patterns on the total protein concentration and we refrain here from discussing this topic further .'], ['In this work , we presented a computational study of self-organized pattern formation by MinD and MinE from E . coli .', 'The equations , which notably account for membrane-binding of MinE , generate the patterns previously observed in living cells as well as Min protein waves on flat surfaces observed in reconstitution experiments .', 'In addition , our analysis yielded two patterns that had not been reported before: In sufficiently long cells and for elevated protein levels , traveling waves emanating from one cell pole and propagating to the opposite pole should emerge .', 'Secondly , in aberrantly large cells , the rotational symmetry of the pattern should be lost and a moving spot should form instead .', 'Both predictions were confirmed experimentally .', 'We conclude that the membrane-binding of MinE is an essential molecular feature to comprehensively describe large-scale pattern formation of the Min proteins .', 'In vitro experiments on micropatterned membranes suggest an important role of MinE processivity for Min-protein pattern formation [26] , but it remains to be seen whether this is the case in vivo .', 'In fact , comparing our system to the one proposed by Huang et al . [15] shows that MinE processivity can at least in part be replaced by a high rate of MinE binding to membrane-bound MinD ( they chose a rate orders of magnitude higher than we did ) .', 'This leads to a different mechanism underlying the pole-to-pole oscillations and requires a finite MinD-ADP to MinD-ATP exchange rate for stabilizing standing waves with several nodes .', 'It will be interesting to test experimentally which of the two possibilities is realized in living E . coli .', 'Our description neglects many molecular details .', 'For example , we did not consider explicitly a MinD dimerization step or the finite exchange rate of ADP for ATP for cytosolic MinD .', 'Also , different expressions accounting for the binding of cytosolic MinE to membrane-bound MinD might be used .', 'We analyzed several different expressions describing the effect that a single MinE dimer can induce detachment of several MinD dimers from the membrane .', 'While these modifications led to quantitative differences , their analysis also revealed that details of the corresponding expressions are rather unimportant for the overall behavior of the system .', 'As a consequence of the relatively simple reaction terms employed in our description , our model reveals some quantitative discrepancies compared to experimental observations .', 'For example , the fluctuations present in the kymographs in Figure 3A and B are apparently larger than in the experimental kymographs in Figure 1 .', 'In addition , the wave profile shown in Figure 8 differs from the experimentally determined [22] .', 'However , complete quantitative agreement likely requires knowledge of more molecular details of the reactions involved .', 'Note , however , that a quantitative comparison on the single cell level also requires precise measurements of the corresponding amount of MinD and MinE , which are currently not available .', 'On a coarser level , though , our description seems to match the topology of the phase space .', 'That is , we present one set of parameters , that correctly reproduces the sequence of patterns as cells grow and also correctly describes the appearance of stochastic switching and traveling waves in living cells with increasing protein levels .', 'In contrast , the exact transition points differ in general from those observed in experiment and any coincidence would be fortuitous .', 'Let us also emphasize that , experiments are now very much needed to constrain possible parameter values .', 'Only with such data we can expect to make further significant progress in understanding Min protein patterns .', 'In agreement with previous work , our analysis also showed that molecular noise has only a minor effect on the Min-protein patterns .', 'Macroscopic signatures of molecular noise were only found under special conditions , namely , in short cells presenting stochastic switching and in large cells , where the Min proteins formed a rotating patch with a stochastically switching sense of rotation .', 'Our description of the Min-protein dynamics can now be used to to design new experiments , for example , to test the interplay between the Min oscillations and Z-ring assembly in vivo or to determine conditions to generate Min-protein patterns inside vesicles in vitro .', 'Such experiments could present important steps on the way to synthesize a system that is able to divide autonomously , that is , a minimal synthetic cell .'], ['We used cells of the E . coli strain JS964 containing the plasmid pAM238 encoding for MinE and GFP-MinD under the control of the lac-Promoter [5] .', 'Bacteria were grown overnight in a 3ml LB medium at 37°C .', 'Cells were induced with Isopropyl – β – D-thiogalactopyranosid ( IPTG ) at a concentration of and incubated for 3–4 hours prior to measurements .', 'During 1–2 hours prior to measurement , cells were kept at 30°C for better fluorescence .', 'The optical density was less than 0 . 6 .', 'During measurements , cells were in the exponential growth phase .', 'The samples were kept at a temperature of 30°C using a Bachhoffer chamber .', 'To keep bacteria from moving under the cover slip , we put them on an agar pad ( 1% agar solution in LB medium with a reduced yeast extract fraction , 10% , in order to lower background fluorescence ) .', 'The fluorescence recordings were taken with an Olympus FV 1000 confocal microscope , at an excitation wavelength of 488 nm from a helium laser at low power .', 'We used an Olympus UPLSAPO 60× , NA 1 . 35 oil immersion objective and recorded a frame every 3s .', 'A measurement lasted 40min .', 'During this period , the focus was manually readjusted at irregular intervals .', 'A22 ( S- ( 3 , 4-Dichlorobenzyl ) isothiourea , HCl ) was purchased from Merck Millipore .', 'Cells were imaged 2–3 hours after adding of A22 .', 'We solve the dynamic equations ( 8 ) – ( 14 ) in the in vitro as well as in the in vivo geometry by using Comsol Multiphysics 4 . 1 which is a solver for partial differential equations based on the finite element method ( FEM ) .', 'All computations with exception of those for Figure 4C were performed in 3d and no assumption was made about the symmetries of the solutions .', 'For the calculations for the patterns in a bacterial geometry the maximal grid size was .', 'For the calculations in the in vitro geometry , we used a maximal grid size of in the surface domain and of in the buffer domain .', 'As initial condition we used homogenous distributions of cytosolic proteins with a random perturbation of 5–10% .', 'The initial surface densities were chosen to be zero for the in vivo geometries .', 'For the in vitro geometry the surface densities were different from zero in a semi-annulus to rapidly induce a spiral .', 'The calculations for the growing cell presented in Figure 4C , the system length was increased by adding discrete pieces at one end of the interval .', 'For the pattern shown in Figure 4 , the rate of growth was .', 'On the added pieces , the protein densities of cytosolic MinD and MinE were initialized with the values and , respectively , whereas the densities of membrane-bound proteins were initially set to zero .', 'To simulate the stochastic reaction diffusion kinetics ( 1 ) – ( 7 ) in three dimensions , we used MesoRD [56] , a tool to solve the stochastic Master Equation using a reaction diffusion Master Equation .', 'It is based on the Next Subvolume Method [57] .']]
['Introduction', 'Results', 'Discussion', 'Materials and Methods']
['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 .']
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