Self-organization of the MinE ring in subcellular Min oscillations
Julien Derr, Jason T. Hopper, Anirban Sain, Andrew D. Rutenberg
aa r X i v : . [ q - b i o . S C ] J un Self-organization of the MinE ring in subcellular Min oscillations
Julien Derr , , ∗ Jason T. Hopper , Anirban Sain , and Andrew D. Rutenberg † Department of Physics and Atmospheric Science,Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada FAS Center for Systems Biology, Harvard University,Northwest Labs, 52 Oxford Street, Cambridge, MA 02138, USA Physics Department, Indian Institute of Technology-Bombay, Powai 400076, India (Dated: October 26, 2018)We model the self-organization of the MinE ring that is observed during subcellular oscillations ofthe proteins MinD and MinE within the rod-shaped bacterium
Escherichia coli . With a steady-stateapproximation, we can study the MinE-ring generically – apart from the other details of the Minoscillation. Rebinding of MinE to depolymerizing MinD filament tips controls MinE ring formationthrough a scaled cell shape parameter ˜ r . We find two types of E-ring profiles near the filamenttip: a strong plateau-like E-ring controlled by 1D diffusion of MinE along the bacterial length, ora weak cusp-like E-ring controlled by 3D diffusion near the filament tip. While the width of astrong E-ring depends on ˜ r , the occupation fraction of MinE at the MinD filament tip is saturatedand hence the depolymerization speed do not depend strongly on ˜ r . Conversely, for weak E-ringsboth ˜ r and the MinE to MinD stoichiometry strongly control the tip occupation and hence thedepolymerization 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, andthe MinE-rebinding rate. We also find that the transient to MinE-ring formation is quite long inthe appropriate open geometry for assays of ATPase activity in vitro , explaining the long delaysof ATPase activity observed for smaller MinE concentrations in those assays without the need toinvoke cooperative MinE activity. PACS numbers: 87.17.Ee, 87.16.A-, 87.16.dr
I. INTRODUCTION
The oscillation of the proteins MinD and MinE frompole to pole of individual cells of the bacterium
Es-cherichia coli is used to localize cellular division to mid-cell [1]. One cycle of the oscillation, lasting approxi-mately one minute, starts with ATP-associated MinDbinding to the bacterial inner membrane and polymer-izing into helical filaments [2, 3, 4] (see also [5]). Thisoccurs at alternating poles of the bacterium, with theMinD forming a polar “cap”. MinE is recruited to themembrane-bound MinD, where it forms a distinctive “E-ring” [6, 7, 8] at the edge of the MinD cap by accu-mulating near the MinD filament tips [2]. Because therate of hydrolysis and subsequent release of ATP-MinDis stimulated by MinE [3, 4, 9], the E-ring drives depoly-merization of the MinD filament which allows the oscil-lation to proceed. The depolymerization occurs with anapproximately fixed E-ring width and speed along thecell axis [7, 8], indicating an approximate steady-stateduring this part of the Min oscillation. However, littleis known about the mechanism of E-ring formation, itsdetailed structure, or how important it is for Min os-cillations. Indeed, Min oscillations have been observedwithout prominent E-rings [10].Most models proposed for Min oscillation do not have ∗ Electronic address: [email protected] † Electronic address: [email protected] explicit MinD filaments [11, 12, 13, 14, 15], thoughthey do have E-rings. Recently, several models of Minoscillations that include explicit MinD polymerizationhave been proposed [16, 17, 18, 19], two of which dis-play strong E-rings that track the tips of depolymeriz-ing MinD filament caps with constant speed and width[18, 19]. In these models, E-rings are the result of MinEpolymerization either orthogonal to [18], or along [19],MinD filaments. While MinD polymerization has beenobserved in vitro [3, 4], there have been no reports ofMinE polymerization in the experimental literature. In-deed, the faint MinE “zebra-stripes” associated with theMinD zones adjacent to the MinE ring [7, 8, 10] seemto imply sparse lateral binding of MinE to the body ofMinD filaments – not MinE polymerization.In this paper, with both stochastic 3D simulations,and a deterministic 1D model, we show that local (non-polymeric) rebinding of MinE released from depolymeriz-ing MinD filament tips is sufficient for E-ring formation.We impose and characterize a dynamical steady-state ofan E-ring on a depolymerizing semi-infinite MinD fila-ment in order to address the approximate steady-statespeed and width of the E-ring in vivo [7, 8]. We in-vestigate the roles of spatial dimension, cell length, andradius, and of multiple MinD filaments and their heli-cal pitch. We estimate the timescale of E-ring formationand obtain results consistent with the significant delaybefore ATPase activity seen with small MinE concentra-tions and large MinD membrane coverage in vitro [3, 4].Finally, we discuss how competition between the intrin-sic and the MinE-stimulated ATPase activity of MinD ρ z ( µ m) BA W zL − L pR zL − L pR zL − L pR
FIG. 1: Fractional occupancy of MinE on the MinD filament ρ vs. distance along the bacterial axis z for the 3D stochasticmodel (continuous lines) and the 1D model (dashed lines) forparameters typical of E. coli : L = 2 µ m, R = 0 . µ m, and ρ = 0 .
35. The MinE binding parameter was σ = 0 . µm /s ,while for the 1D model f = 0 .
06 was used. One MinD fila-ment supports either (A) a strong plateau-like E-ring for pitch p = 0 . µ m or (B) a weak cusp-like E-ring for pitch p = ∞ (straight filament). The width W of the strong E-ring, givenby ρ ( W ) = (1 + ρ ) / z = 0.The helical pitch p is indicated. controls the instability that leads to the initial formationof the E-ring from a uniformly decorated MinD filament.Qualitatively, we predict that the width of MinE-rings will increase as the MinD-filament depolymeriza-tion speed is increased through manipulation of cellshape, MinD to MinE stoichiometry, or mutations thataffect the MinE binding rate to MinD. Eventually, the de-polymerization speed will saturate but the E-ring widthcan still grow. Conversely, as the depolymerization speedis decreased, MinE-rings will undergo a transition froma plateau-like “strong” E-ring to a cusp-like “weak” E-ring. To our knowledge, systematic experimental studiesof the E-ring width have not yet been done. II. E-RING MODEL
As illustrated in the inset of Fig. 1, we represent thebacterial geometry as a cylinder of radius R and length2 L . In the right half (0 < z < L ), n filaments of MinD areplaced on the cylinder, each with the same helical pitch p but with random (unbundled) helical phases. MinDfilaments are composed of monomers of length a , eachof which can bind one MinE. We depolymerize MinDfrom filament tips at z = 0, and any released MinE dif-fuses in the cylinder interior (cytoplasm) with a diffu-sion constant D . Released MinE can bind to unoccupied MinD monomers; if not it is removed from the system at z = ± L . This open boundary condition represents thesinks for MinE provided by other MinD in the system.Depolymerized MinD is removed from the system with-out further interaction, reflecting the nucleotide exchangeneeded before MinD rebinding is possible. This dramat-ically simplifies our model, since we may then explicitlyconsider only MinE dynamics on an implicit MinD fil-ament. Both the boundary conditions and the neglectof depolymerized MinD will be addressed again in thediscussion.In order to study a steady-state E-ring, we keep thefilament tips centered at z = 0 – the “tip-frame”. In thetip-frame, bound MinE move along MinD filaments at aconstant depolymerization speed v while new monomersof MinD are introduced at z = L decorated with MinEwith a constant probability ρ (determined by the rela-tive cellular amounts of MinE and MinD particles). [Ef-fectively we are studying semi-infinite MinD filamentsunder the approximation of uniform MinE binding for z > L .] In the steady-state, the fraction of MinE releasedby the depolymerizing MinD filament tip that reach theabsorbing boundaries will then be ρ /ρ tip , where ρ tip is the fractional MinE occupation of the filament tip.The depolymerization speed v can be determined self-consistently by ρ tip , though we will see below that v issmall and can be practically ignored in terms of the E-ring structure. A. Stochastic 3D implementation
The dimensionless parameter α ℓ ≡ vℓ/D , the frac-tional axial distance one MinE advects at speed v whileit diffuses a distance ℓ , characterizes the importanceof the depolymerization speed. Even with ℓ = 4 µm , v = 0 . µm/s [7, 8], and D = 10 µm /s [20], α ℓ = 0 .
01 issmall and depolymerization is slow compared to diffusion.Accordingly, our stochastic 3D model quasiadiabaticallyfollows each released MinE until it either rebinds or isremoved from the system before allowing further depoly-merization. Each MinE diffuses by taking a randomlyoriented step of fixed length δ every timestep ∆ t , where D = δ / (6∆ t ). Diffusing MinE binds to a free MinD withprobability P stick when it hits the bacterial membranewithin a distance r bind of the MinD. We take r bind = a .This leads to an effective binding rate of σ ρ ,local , where ρ ,local is the local bulk concentration of MinE and thebulk reaction rate σ = 3 πDr bind P stick / (2 δ ). We take σ = 0 . µm /s (this is approximately the threshold be-tween strong and weak E-rings given the cell geometry,see below). The steady state reached after successive de-polymerization steps is independent of small δ if we vary P stick with δ to keep σ constant. B. Analytic 1D treatment
We also study a deterministic 1D model that exactlycorresponds to the 3D stochastic model in the limit R ≪ a . This enables us to explore the role of spa-tial dimension and stochastic effects in the E-ring, andalso helps us to identify the combinations of parametersthat control the E-ring structure. Our 1D model tracksboth the linear density of bound MinE (B) and of freelydiffusing MinE ( F ):˙ B − vB ′ = σ F ( B max − B ) , for z > F − vF ′ = DF ′′ − σ F ( B max − B ) + vB (0) δ ( z ) , (2)where the dots and primes indicate time and spatialderivatives, respectively. For z < B = 0. For z >
0, the linear density of potentialbinding sites (i.e. of MinD) is B max = 1 /a , and the 1Drebinding rate is σ . The v dependent terms on the leftside of the equations represent advection of bound MinEin the tip frame, while on the right of Eqn. 2 is a sourceterm due to MinE release at the depolymerizing filamenttip. If we rescale all lengths by L (so ˜ z ≡ z/L ) and definedimensionless fields ˜ B ≡ B/B max and ˜ F ≡ DaF/ ( vL ),then we can consider the scaled steady-state equations:˜ B ′ = − ˜ σ ˜ F (1 − ˜ B ) , for ˜ z > F ′′ = − α L ˜ F ′ − ˜ B ′ − ˜ B (0) δ (˜ z ) . (4)The boundary conditions are ˜ F ( ±
1) = 0 and ˜ B (1) = ρ .The behavior is controlled by the dimensionless param-eters ˜ σ ≡ σ L / ( Da ) and α L = vL/D , as well as by ρ . We integrate Eqn. 4 for ˜ z < B = 0, andimpose flux conservation of MinE at the boundaries with˜ F ′ ( − − ˜ F ′ (1) = ρ . For ˜ z > α L to be small, and anticipate thatit is irrelevant for the E-ring structure – leaving only ρ and ˜ σ as relevant control parameters. Nevertheless, the1D treatment allows us to explore this assumption. Wefind that α . .
05 does not change the observed E-ringsteady-state structure by eye, while we expect α L ≈ . in vivo — and even lower valuesfor weak E-rings. The four-fold speedup observed for theMin oscillation at body temperature [21] puts the de-polymerization speed (i.e. α ) closer to, but still under,relevance with respect to the structure of the steady-stateMinE ring. III. RESULTS
We can compare results of our 1D deterministic modelwith our 3D stochastic model using F ≡ πR ρ ,av , where ρ ,av is the bulk density averaged over the bacterial cross-section. Then the 1D and 3D binding rates of MinE arerelated by σ = σ f / ( πR ), where f ≡ ρ ,local /ρ ,av . We expect that f will vary with distance from the filamenttip due to local release at the tip followed by diffusionand capture. We find f . f & n ) andfilament pitch ( p ) can be included in the 1D model byusing the MinD monomer spacing projected along thebacterial axis a , where a = a / ( n p π R /p ) . (5)Differences between the two approaches are either due tothe 1D vs. 3D geometry or due to the deterministic vs.stochastic nature of the models. A. Strong and weak E-rings
Fig. 1 illustrates the fractional occupation ρ (equiva-lent to ˜ B in the 1D model) of MinE binding sites on theMinD filament vs. distance z along the bacterial axis.Occupation monotonically decreases from the tip value, ρ tip ≡ ρ (0), due to local rebinding of MinE followingdepolymerization from the tip. Following the quantifica-tion of Shih et al [10], there are a few thousand MinDmonomers within a typical bacteria. With L = 2 µ mand a = 5nm [3, 4], they can be arranged either in onesingle helical filament (with p ≈ . µ m [2]) or about7 straight filaments (with p = ∞ ). In either case, wefind (A) a “strong” E-ring ( n = 1 shown) with ρ tip ≈ ρ tip <
1) and no plateau. Strong or weak E-rings have,respectively, negative ( ρ ′′ (0) <
0) or positive ( ρ ′′ (0) > f = 0 .
06. Thisbest value of f depends on r bind . Using the same f near the tips, the 1D model systematically underesti-mates the fractional occupation. This implies that alarger f ≡ ρ ,local /ρ ,av is appropriate there, in agree-ment with the increased likelihood that MinE will befound near the tip shortly after it is released at the tip. B. Scaling collapse of E-ring width
For both strong and weak E-rings, we can define thewidth W of the E-ring such that ρ ( W ) = (1 + ρ ) /
2. Mo-tivated by the importance of the scaled MinE rebindingrate ˜ σ in the 1D deterministic equations and by the cor-respondence of σ and σ , we investigated the influence ofthe scaled aspect ratio ˜ r ≡ p f / ˜ σ = R/L p πDa/σ onthe profile shape, as characterized by W/L and by ρ tip ,in Fig. 2 for both the 3D stochastic model (symbols) andthe 1D deterministic model (lines). Two regimes are de-marcated by a vertical dashed line: for small ˜ r we have a WL (a)strongring weakring 00.10.20.30.40.50.6 WL (c)0.50.60.70.80.91 0 0.05 0.1 0.15 0.2 ρ tip ˜ r (b) 0.50.60.70.80.91 0 0.05 0.1 0.15 0.2 ρ tip ˜ r (d) FIG. 2: MinE profile, characterized by
W/L and ρ tip , as afunction of the scaled aspect ratio ˜ r . (a, b): Straight filaments( p = ∞ , ρ = 0 . f = 0 . n = 1, green data) for L = 1 µ m( (cid:3) ), L = 2 µ m ( ◦ ) and L = 3 µ m ( △ ); multiple filaments( n =2, or 5, blue data) for L = 1 µ m ( ▽ ). (c, d): Helicalfilaments ( n = 1, ρ = 0 .
35) with
L/p =20 ( (cid:3) , red), 10( ◦ , green), 4 ( △ , blue) and 0 ( ⋄ , pink). For all these data, σ = 0 . µm /s , and R is varied to explore ˜ r . Similar resultsare obtained when σ is varied. strong E-ring with ρ ′′ (0) <
0, a saturated tip ( ρ tip ≈ r we have a weak E-ring with ρ ′′ (0) > ρ tip no longer saturated, and a smaller width W .The agreement between the 3D and 1D results for ρ tip and W at small ˜ r shows that the essential physics ofstrong E-rings is one-dimensional. For small enough R the bacterial cross-section is well explored by MinE bythe time it has diffused to free binding sites a distance W from the filament tip. However, by effectively averagingthe radial profile the 1D model systematically underesti-mates the occupation fraction near the tip, as seen with ρ tip in Fig. 2 and also in the profiles shown in Fig. 1. Thedisagreement becomes stronger as ˜ r increases, reflectingthe increasingly 3D character of the stochastic system atlarger aspect ratios. However, the system still exhibit aremarkable collapse for all values of ˜ r . This shows that al-though the 1D model misses important details about thetip enhancement, the scaling behaviour of the 3D systemwith straight filaments is similar to the 1D model.As shown in Fig. 2(c, d), ˜ r also captures the effects ofhelical MinD filaments. Smaller pitches lead to strongerE-rings. However, the 3D stochastic results do not showscaling collapse with respect to ˜ r as the monomer spacingalong the filament a is a relevant length-scale in additionto the projected axial monomer spacing a . Since the 1Dmodel only uses the effective a , it incorrectly exhibitsperfect scaling collapse.As shown in Fig. 3(a) and (b), ρ (the ratio of the WL ˜ r (a) 0.20.40.60.810 0.05 0.1 0.15 0.2 ρ tip ˜ r (b) FIG. 3: (a, b): MinE profile, as characterized by
W/L and ρ tip obtained by the 3D model (points) and the 1D model(lines, using f = 0 . r for different values ofstoichiometry; ρ = 0 . (cid:3) , continuous lines), 0.35 ( ◦ , dashedlines), 0.50 ( △ , dotted lines). For each stoichiometry, thesame collapse as Fig. 2(a,b) is obtained: data are compiled for n =1, 2, 3, 4 and 5, L =1, 2 and 3 µ m, p = ∞ , σ = 0 . µm /s ,and R varies to explore ˜ r . Similar results are obtained when σ is varied. number of MinE and MinD particles) also controls thescaling curves of W/L or ρ tip vs. ˜ r . Agreement between1D and 3D models for small ˜ r and scaling collapse arepreserved for each ρ . C. Correspondence with in vivo
Min oscillations
Experimentally,
W/L ≈ . L as half the bacterial length.Using ρ ≈ .
35, which is consistent with the ratio ofMinE to MinD if we assume MinE are always dimerized[10], then from Fig. 3(a) we see that
W/L ≈ . r ≈ .
07 — which corresponds to σ ≈ . µm /s .(This σ is of the same order of magnitude as used ina number of previous models in 3D [13, 16] and in 1D[15, 17, 18] if we assume R = 0 . µm .) Interestingly,this indicates that the E-ring of the normal wild-type(WT) Min oscillations is a strong E-ring (with a plateauof MinE occupation near the MinD filament tip) but nearthe margin between weak (with ρ tip <
1) and strong.This implies (see Eqn. 6 below) that the tip occupation ρ tip , and hence the depolymerization speed and the oscil-lation period, will strongly depend on the stoichiometryof MinE to MinD. Since k S /k I ≫
1, changes to ρ tip evenat the percent level should be significant. Indeed, MinDoverexpression leads to a 2.5-fold increase in the period[22]. This also implies from Fig. 3(a) that the width ofthe E-ring will strongly depend on the stoichiometry —though this has not (yet) been explored experimentally.At a fixed stoichiometry of MinE to MinD ( ρ ), we ex-pect that overexpression of Min will increase the numberof filaments and/or decrease the pitch. As a result, weexpect a slightly stronger E-ring, and a slightly fasterperiod – as seen [22].Optically reconstructed E-rings [2] show a plateau-likedecoration along the MinD filament, consistent with astrong E-ring. E-rings have also been seen in long fila-mentous cells [6, 7, 8] and exhibit approximately the samewidth W , though both the spacing between MinD capsand the cell length are considerably longer in filamen-tous cells than in rod-shaped cells. This indicates thatthe effective L may not be determined by cell shape, butrather by other processes preserved between rod-shapedand filamentous bacteria such as the length of the MinDfilaments or spontaneous lateral release (without MinDhydrolysis) of MinE away from the tip of the MinD fila-ment.Shih et al. [10] identified MinE point-mutants(MinE D45 A and MinE V49 A ) that led to fainter E-rings,and double mutants (MinE D45
A/V A ) that resulted inmost of the MinE being cytoplasmic with no strong E-rings. Assembly and disassembly of MinD polar zonescontinued with no more than doubled periods [10] —too rapid to be explained by intrinsic depolymerizationalone (in contrast, see [19]). From Eqn. 6 (below) theobserved disassembly rates would only require a moder-ately enhanced ρ tip ≈ .
9, i.e. a weak E-ring. Indeed,in all of these constructs there appears to be enhancedco-localization of MinE with the MinD polar zones [10].We believe that the lack of visible E-rings in these mu-tants can be explained with decreased σ (as suggestedpreviously by [13]) and/or enhanced spontaneous MinEunbinding away from filament tips. Local rebinding ofMinE near filament tips would still lead to an enhanced ρ tip . We predict that the oscillation period in these mu-tants should be strongly susceptible to the MinE to MinDstoichiometry. IV. TRANSIENTS
We may use our models to check that the transientsbefore steady-state are fast enough in the context of thenormal Min oscillation. If we initially decorate the MinDfilament with MinE monomers released from z = − L consistent with MinE released from a different depoly-merizing MinD cap, we find (data not shown) an ini-tial decoration pattern that has a plateau-like strong E-ring from the beginning (as previously noted [23]), sothat we expect rapid E-ring formation without appre-ciable delay during Min oscillations (as also observedexperimentally[7, 8]). A. Transients before the steady-state in vitro
While delays are not observed for E-ring formationduring Min oscillations in vivo , significant delays are ob-served in vitro . MinD binds to phospholipid vesicles inthe presence of ATP and undergoes self-assembly, con-stricting the vesicles into tubes with diameters on theorder of 100 nm [3]. Electron-microscopy revealed thatMinD assembles into a tightly wound helix on the sur-face of these tubulated vesicles with a pitch (helical re-peat distance) of only 5 nm. Hu et al. [3] report a sig- nificant delay (several minutes) for stimulated ATPaseactivity when small concentrations of MinE were added,while this delay vanished for larger MinE concentrations.Similar delays were seen in vitro by Suefuji et al. [4].Furthermore, the eventual steady-state ATPase activitywas smaller for smaller concentrations of MinE [3, 4].This has led to the hypothesis of explicit cooperativ-ity of MinE binding, which has then been explicitly in-cluded in reaction-diffusion models[14, 24] and in MinEpolymerization in models with MinD polymers [18, 19].Here we show that our stochastic model for the MinEring, with no explicit MinE cooperativity, can recoverthe MinE concentration dependent ATPase delays andactivities observed in vivo . We conclude that MinE co-operativity is not needed to explain the in vitro results,apart from cooperative effects that arise implicitly fromthe self-organization of the MinE ring.We use an “inside-out” open geometry correspondingto what is reported in vitro [3], with a narrow phospho-lipid cylinder that is tightly wound by MinD filaments.MinE, when released by a depolymerizing filament tip,will diffuse outside the cylinder. We consider a heli-cal MinD filament of radius R = 50 nm and pitch 5nm (equal to a ). Upon MinD depolymerization, we al-low any released MinE to diffuse until either it binds toan available MinD binding site or it is absorbed by theboundaries at z = ± L . We impose reflecting boundaryconditions at r = R , but otherwise allow MinE to diffusefreely for r > R . Our stochastic 3D model is otherwisethe same as before though with an emphasis on the tran-sients approaching steady-state.The transient to steady-state is shown in Fig. 4(a),with the fractional occupation of MinE at the MinD fil-ament tip ( ρ tip ) shown as a function of the number ofdepolymerized monomers from the filament tip, N . TheMinE occupation fraction at the MinD filament tip is ex-perimentally observable through the ATPase activity (i.e.the MinD depolymerization rate). The initial conditionis a uniform occupation ρ , corresponding to an initiallyrandom binding of MinE on the MinD filament. Thelarger ρ is, the shorter the transient and the strongerthe eventual steady-state ρ tip . Significant enhancementof ρ tip is obtained even for small fractions of MinE. For ρ & . ρ tip > .
8, though, as shown inFig. 4(b), strong E-rings are predicted only for very largestoichiometry ( ρ & . in vitro geome-try includes some small radius features (the helical wind-ing of the MinD filament) and some large radius features(no closed boundary at large r ). The tight helical wind-ing of the MinD filament contributes to long transients,while the semi-infinite radial geometry contributes to theweak E-ring for small and moderate ρ .To convert the number of depolymerization steps N toa time t ( N ) we need to sum the average time for eachstep, which will depend on ρ tip : t ( N ) = P Nn =1 ∆ t ( n )where, ∆ t ( n ) = ρ tip ( n ) /k S + (1 − ρ tip ( n )) /k I . (6) ρ z (nm) (b)0.40.50.60.70.80.91 0 10 20 30 40 50 60 ρ tip N ( × ) (a) ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . FIG. 4: Transients and E-ring structure for an “inside-out”open geometry appropriate for in vitro experiments, wherea MinD filament is tightly wound on the outside of a cylin-der of small radius ( R = 50 nm) with open boundaries at R = ∞ . (a) Evolution of ρ tip as a function of the number ofdepolymerization steps N (measured in thousands) after theuniform intial conditions for ρ equal to 0 . . . . ρ ( z ) as a function of axial distance z along the helical axis for the same ρ . N ( × ) t (min) ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . FIG. 5: For the same inside-out in vitro geometry describedin the previous figure. Cumulative ATP-ase activity N ( t )(measured in thousands of depolymerization steps ) versustime t for various ρ . Asymptotic behavior are plotted asthin dotted lines. The timesteps are determined by k I when the tip of theMinD polymer is unoccupied by MinE and k S when it isoccupied. Using k S /k I = 20 [4] and k S = 1 / (20 ms ) givenby the maximal depolymerization speed in vivo (assum-ing ρ tip ≈
1, with a strong E-ring) [25], we plot the cumu-lative total ATPase activity N ( t ) (equal to the numberof depolymerization steps) vs. elapsed time t in Fig. 5. The stoichiometric ratio of MinE to MinD correspondsto ρ if the MinE mostly binds to available MinD be-fore depolymerization proceeds significantly. For smallamount of MinE (typically ρ . .
3) we obtain a sig-nificant delay of about 5 minutes, corresponding to theATPase delay seen in vitro [3, 4] ; and for larger MinEamounts ( ρ going to 1) the delays decrease towards zeroalso in agreement with in vitro studies. When the steady-state ρ tip is reached, the ATPase rate will also be ina steady-state as indicated by the linear asymptotes inFig. 5. Since ρ tip can be large even for smaller ρ , weexpect the ATPase rates to be comparable for moderateor larger ρ , as seen in vitro [3, 4]. For smaller ρ thesteady-state ATPase activity is reduced, as also observed.We conclude that the delay of ATPase activity seen in vitro is determined by the time needed to reach thesteady-state ρ tip . We see that it is considerably longerin an open than in a closed geometry. Our local re-binding model recovers the delays seen in vitro with-out any explicit MinE cooperativity (see, conversely,[3, 4, 14, 18, 19, 24]). B. E-ring instability
In the tip-frame, the MinD filament tip is bistable dur-ing Min oscillations [19] and the formation of the E-ringswitches the filament tip between polymerization and de-polymerization. While long transients for this switch-ing are not expected during Min oscillations in vivo be-cause of initially non-uniform tip decoration [23], we mayask about the transient to form the E-ring from a non-oscillating state — such as seen experimentally after ex-posure to high levels of extracellular cations [26]. Weconsider a MinD filament that is initially uniformly dec-orated with MinE. To tractably include the MinD poly-merization dynamics, we use a uniform (mean-field) bulkMinD density ρ D . Because we are interested in the ini-tial slow stages of E-ring formation, we consider MinEbinding only near the tip with occupation fraction ρ tip (initially equal to ρ )The net polymerization rate of a MinD filament is R ≡ k + ρ D − ( ρ tip k S +(1 − ρ tip ) k I ), where ρ D is the bulk MinDmonomer concentration and k + controls MinD monomeraddition. Depolymerization of n monomers from a singletip will enhance ρ tip due to local rebinding of MinE, sothat dR/dn = k + /V − ( k S − k I ) dρ tip /dn for cell volume V . The depolymerization time per monomer is ∆ t ≈ /k I for an initially weak E-ring (with ρ tip small), andthe change in tip occupation in one depolymerization stepwill be proportional to both the number of MinE released( ρ tip ) and the locally available binding sites (1 − ρ tip ), sothat dRdt = k + k I /V − A ( k S − k I ) k I ρ tip (1 − ρ tip ) , (7)where the constant A is the fraction of MinE that re-bind to available sites at the filament tip. For k S suf-ficiently greater than k I this represents an instability( dR/dt growing more negative with time) that will leadto E-ring formation. We therefore expect that both a sig-nificant difference between intrinsic and stimulated AT-Pase activity of MinD and significant intrinsic ATPaseactivity are needed for E-ring formation, and hence forthe initiation of Min oscillations.We have neglected any lateral unbinding of MinE fromthe MinD filament, which will kill the instability if dR/dt is small enough. We also neglect the presence of otherMinD filament tips, which will buffer the bulk MinD den-sity and reduce the effect of the k + term in Eqn. 7. Theseeffects will shift the threshold, but will not change thepresence of the E-ring instability.Since ρ tip ≃ ρ initially, we also predict from Eqn. 7that both low and high proportions of MinE to MinD willalso preclude Min oscillations by making the MinD fila-ment tip initially stable against depolymerization. How-ever, using k + = 100 / ( µM s ) [19], A ≈
1, and V = 1 µm we estimate a tiny stoichiometry threshold of 0 .
003 (for ρ or 1 − ρ ). While our predicted stoichiometry thresh-olds are unlikely to be relevant in vivo , they may beapproachable in vitro . We also note that initially slowE-ring formation dynamics near the instability thresholdshould be observable when Min oscillations are restartedafter being halted [26].Previous models of the full Min oscillation have foundlimiting MinE:MinD stoichiometries, either both low andhigh [12, 15, 18] or just high [13, 17]. Sufficiently lowstoichiometries may not have been explored in the latermodels. Conversely, Min oscillations have always beenseen in vivo with moderate stoichiometry changes [22].It would be desirable for a more systematic explorationof the role of stoichiometry on Min oscillations, giventhe predicted stoichiometry limits for the existence ofoscillations predicted in this and other models. V. DISCUSSION
We have presented a model of the self-assembly of theMinE-ring within single
E. coli bacteria, without invok-ing either MinE cooperativity or MinE polymerization.We highlight the difference between strong E-rings, with ρ tip ≈
1, essentially 1D physics and a maximal depoly-merization speed, and weak E-rings with ρ tip < a enters. Since theexperimentally measured E-ring width indicates that E- rings in vivo are close to the threshold between weakand strong, the detailed response of the E-ring structure(i.e. the width W , or the depolymerization speed via thetip occupation ρ tip ) to experimental manipulations thatchange the oscillation period (stoichiometry through ρ or, e.g., [26]) is unlikely to be correctly captured by 1Dor non-filamentous models.We have explained the anomalous delays of MinE stim-ulated MinD ATPase activity seen in vitro [3, 4], andhave also identified an instability of MinE ring forma-tion that is required to develop from a disordered initialstate to the full Min oscillation. We have shown thatMinE-ring structure and dynamics can be treated inde-pendently of a full Min oscillation model. The instabil-ity to E-ring formation, and subsequent MinD filamentdepolymerization, that we identify neither depends onnor determines the spatial pattern of Min oscillation –which could be selected by either diffusion and rebindingof MinD [27] or by phospholipid heterogeneities [28].We have constructed our E-ring model to obtain asteady-state. The steady-state is formed by balancingthe MinE entering the system as a bound fraction ρ onthe MinD filament with the MinD lost by diffusing acrossthe open boundaries at z ± L . Other geometries, such asan open boundary at z = − L and closed at z = L , or a fil-ament tip placed asymmetrically (away from z = 0), willalso lead to a steady-state E-ring that should be qualita-tively similar to the one we have described. An extremeexample of this is the inside-out geometry we used todescribe in vitro ATPase experiments. What we have ac-complished is to characterize the steady-state, and use itto explore the effects of cell-shape, helical pitch, MinErebinding rate, and stoichiometry on the E-ring struc-ture. Our model is expected to be a generic part of fulloscillation models that exhibit E-rings.It is worth speculating on how our simplified E-ringmodel would be modified by possible additional ingredi-ents within a full model of the Min oscillation. (1) We donot expect that filament cutting (see e.g. [16, 17]) willqualitatively affect our results, though it would lead tomany more free ends and faster depolymerization. TheMinE ring would still only be expected to form near thevery end of the MinD filament, and significant depoly-merization would only occur within its width W fromthe end. Similarly, our results should apply to modelswithout filaments (see e.g. [12, 13, 14, 15, 16]). In thatcase, we expect that our analytic 1D treatment to bea better approximation due to the absence of an intrin-sic monomer spacing a that is relevant near the fila-ment tip. (2) We expect that lateral release of boundMinE away from filament tips, without associated cut-ting, would affect the E-ring profile a distance ℓ = vτ away from the tip (where v is the depolymerization rate,and τ − is the lateral release rate). This can be crudelyincluded in our model by placing our boundary condi-tions at L ≈ ℓ . (3) We have neglected the rebinding ofMinD to the filament tip. We would expect rebinding to“poison” the E-ring by significantly reducing the depoly-merization rate – which would allow further rebinding.This appears to be observed in the occasional E-ring re-versal in vivo [7, 10]. While interesting, poisoning ap-pears to be typically avoided during Min oscillations —perhaps by filament cutting or by lateral MinE releaseand re-binding, neither of which have been experimen-tally characterized — and so we are justified in neglect-ing it for steady-state E-rings. Poisoning may howeverweaken the E-ring instability described by Eqn. 7, andthis deserves further study. The next step is to developa full 3D Min oscillation model with MinD filaments butwithout MinE polymerization.Previous work has considered the steady-states of semi-infinite filaments with tip-directed depolymerization en-hanced by bound motors (in this paper, bound MinE)[29]. That work used a uniform (mean-field) cytoplas-mic motor distribution, and obtained tip-enhanced motordensity by a combination of diffusion and directed motionalong the filament together with a “processivity” reten-tion probability ¯ p for motors at the depolymerizing tip.In contrast, in our model MinE remains immobile on thefilament. [Note that advection ( v ) represents the drag-ging of MinE along with the MinD filament, not motionwith respect to the filament.] Furthermore, we explicitly consider the cytoplasmic MinE random-walk or diffusionupon release from the filament tip. While this does leadto implicit processivity (local retention of MinE), it alsocorrectly allows for rebinding of MinE away from the fil-ament tip. This physical modeling of the cytoplasmicMinE allows us to consider, e.g., the 3D vs. 1D cross-over, realistic transients for the inside-out in vitro geom-etry, and the E-ring width. Note that the enhanced localrebinding of MinE to the MinD filament upon releaseis related to ligand rebinding (see, e.g., [30]), and sim-ilar dimension and geometry dependent effects are seenthere. Acknowledgments
This work was supported financially by Natural Sci-ences and Engineering Research Council (NSERC),Canadian Institutes for Health Research (CIHR), andAtlantic Computational Excellence Network (ACENET);computational resources came from ACENET and the In-stitute for Research in Materials (IRM). We acknowledgeuseful discussions with Manfred Jericho. [1] J. Lutkenhaus, Annu. Rev. Biochem. , 539 (2007); K.Kruse, M. Howard, and W. Margolin, Mol. Micro. ,1279 (2007); M. Howard and K. Kruse, J. Cell. Biol. ,533 (2005).[2] Y-L. Shih, T. Le, and L. Rothfield, Proc. Natl. Acad. Sci.USA , 7865 (2003).[3] Z. Hu, E. P. Gogol, and J. Lutkenhaus, Proc. Natl. Acad.Sci. USA , 6761 (2002).[4] K. Suefuji, R. Valluzzi, and D. RayChaudhuri, Proc.Natl. Acad. Sci. USA , 16776 (2002).[5] J. Szeto, N. F. Eng, S. Acharya, M. D. Rigden, and J.-A.R. Dillon, Res. Microbiol. ; 17 (2005).[6] D. M. Raskin and P. A. J. de Boer, Cell , 685 (1997).[7] C. A. Hale, H. Meinhardt and P. A. J. de Boer, EMBOJ. , 1563 (2001).[8] X. Fu, Y.-L. Shih, Y. Zhang, and L. Rothfield, Proc.Natl. Acad. Sci. USA, , 980 (2001).[9] Z. Hu and J. Lutkenhaus, Mol. Cell , 1337 (2001).[10] Y-L. Shih, X. Fu, G. F. King, T. Le, and L. Rothfield,EMBO J. , 3347 (2002).[11] M. Howard, A. D. Rutenberg, and S. de Vet, Phys. Rev.Lett. , 278102 (2001)[12] M. Howard and A. D. Rutenberg, Phys. Rev. Lett. ,128102 (2003).[13] K. C. Huang, Y. Meir, and N. S. Wingreen, Proc. Natl.Acad. Sci. USA , 12724 (2003).[14] H. Meinhardt and P. A. J. de Boer, Proc. Natl. Acad.Sci. USA , 14202 (2001).[15] K. Kruse, Biophys. J. , 021904 (2006).[17] F. Tostevin and M. Howard, Phys. Biol. , 1 (2006). [18] D. A. Drew, M. J. Osborn and L. I. Rothfield, Proc. Natl.Acad. Sci. USA , 6114-6118 (2005).[19] E. N. Cytrynbaum and B. D. L. Marshall, Biophys. J. , 1134 (2007).[20] G. Meacci, J. Ries, E. Fischer-Friedrich, N. Kahya, P.Schwille, and K. Kruse, Phys. Biol. , 255 (2006).[21] A. Touhami, M. H. Jericho, and A. D. Rutenberg, J.Bacteriol. , 7661 (2006)[22] D. M. Raskin and P. A. J. de Boer, Proc. Natl. Acad.Sci. USA , 4971 (1999).[23] K. C. Huang and N. S. Wingreen, Phys. Biol. , 229(2004).[24] M. Loose, E. Fischer-Friedrich, J. Ries, K. Kruse, and P.Schwille, Science , 789 (2008).[25] The depolymerization speed along the bacterial axis isapproximately 30nm/s [2, 7, 8]. By taking into accountthe pitch, it gives an approximate speed along the fila-ment of 250nm/s which gives, with a = 5nm, an averagetime per depolymerization on the order of 20ms.[26] B. P. B. Downing, A. D. Rutenberg, A. Touhami, and M.Jericho , submitted for publication (2009).[27] R. V. Kulkarni, K. C. Huang, M. Kloster, and N. S.Wingreen, Phys. Rev. Lett. , 228103 (2004).[28] E. Mileykovskaya and W. Dowhan, Curr. Opin. Micro-biol. ,135 (2005).[29] G. A. Klein, K. Kruse, G. Cuniberti, and F. J¨ulicher,Phys. Rev. Lett. , 108102 (2005).[30] M. Gopalakrishnan, K. Forsten-Williams, T. R. Cassino,L. Padro, T. E. Ryan, U. C. T¨auber, Eur. Biophys. J.34