Effective two-dimensional model does not account for geometry sensing by self-organized proteins patterns - Supplementary document
EEffective two-dimensional model does not account for geometry sens-ing by self-organized proteins patterns - Supplementary document
Authors:
Jacob Halatek and Erwin Frey
Affiliation:
Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Depart-ment of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München,Germany
Subject article:
Schweizer, J., Loose, M., Bonny, M., Kruse, K., Mönch, I., and Schwille, P., Ge-ometry sensing by self-organized protein patterns,
Proc. Natl. Acad. Sci. USA , , 15283-15288(2012) Introduction
Here we provide a thorough discussion of the model for Min protein dynamics proposed by Schweizeret al. [11]. The manuscript serves as supplementary document for our letter to the editor toappear in PNAS. Our analysis is based on the original COMSOL simulation files that were usedfor the publication. We show that all computational data in Schweizer et al. rely on exploitationof simulation artifacts and various unmentioned modifications of model parameters that strikinglycontradict the experimental setup and experimental data. We find that the model neither accountsfor MinE membrane interactions nor for any observed MinDE protein patterns. All conclusionsdrawn from the computational model are void. There is no evidence at all that persistent MinEmembrane binding has any role in geometry sensing.
Summary of results and conclusions • The authors do not use the same parameters that are given in the article but a lower MinE/MinDratio that also deviates from the experimental value. • With the experimental MinE/MinD ratio no patterns emerge. (cf. Figure 3) • With the lower MinE/MinD value patterns emerge but do not resemble the experiments northe computational data presented in the article. (cf. Figure 4/5) • The quantitative data presented in Figure 5B/C/D in the paper cannot be reproduced by theproposed model (cf. Figure 7B and section 1.3.2). • Alignment to the aspect ratio (Figure 5D in the paper) solely relies on self-coupling via theperiodic boundaries in horizontal and vertical directions. (cf. Fig. 2/4/5) • Alignment to curved membranes (Figures 5A/S7 in the paper) fails for gold layer sizes as usedin the experiment. (cf. Figure 7D) • Transient MinE membrane binding is an order of magnitude stronger than in the experiments.(cf. section 2.1.1) • Adjusting MinE membrane binding to meet the experiment leads to loss of any patterns. (cf.section 2.1.2) 1 a r X i v : . [ q - b i o . S C ] M a r The model assumes a very small bulk volume and is highly sensitive to volume effects incontrast to the experimental evidence and the claim in the paper. (cf. section 2.2)We investigated the proposed model by means of linear stability analysis and numerical sim-ulations. First, we note that the actual simulations provided to us by the authors use reducedMinE/MinD ratios ( C D /C E = 2 . and C D /C E = 4 . ) that deviate from the experiments([MinD]/[MinE]=1.6) and the published parameter values ( C D /C E = 1 . ). For the experimen-tal/published values we find pattern formation to be restricted to very small ratios of gold layer tomembrane, or equivalently bulk volume to membrane. The model does not account for cytosolicvolume explicitly, but the choice of the total protein densities indicates an effective bulk height below µm . We find that rescaling the effective bulk volume by a small factor O (1) or explicitly increasinggold layer size yields loss of instability. Hence, the model behavior described in the published simu-lations is limited to system sizes that deviate from the experiment by several orders of magnitude.Moreover, in striking contradiction to the accompanying experiments and to the claim in the article,bulk size does have severe effects on protein patterns. The authors have compensated for the effectsof reservoir size by adjusting intrinsic system parameters (total protein densities). This was notmentioned in the published article. It should go without saying that the need to adjust genuinelyintrinsic system properties to keep certain desired phenomena invariant to variations of system sizeclearly proves that those phenomena are not intrinsic to the system. This directly contradicts themain experimental findings the model claims to account for.However, even with these adjustments in place the model relies on employing simulation arti-facts to reproduce the published data. We find that alignment to the aspect ratio (Fig. 5D in thepaper) strictly requires periodic boundary conditions at the outer boundary of the gold layer. Thesecross-boundary couplings in horizontal and vertical directions controls the alignment angle, whilethe aspect ratio of the patch has a negligible effect on alignment. Wave alignment ceases and wavesbecome disordered propagating blobs if gold layer sizes are inclreased to match the experimentalsetup or if cross-boundary coupling is disabled by replacing periodic boundary conditions with no-flux conditions. Plainly put, for simulations to resemble the presented set of computational dataseveral distinct and independent artifacts and unphysical parameter adjustments have to be em-ployed for each dataset individually. We were unable to determine the specific combinations of goldlayer size, cross-boundary coupling, and total [MinE]/[MinD] density ratio, that yield the publisheddata. Altogether, this invalidates the model on a conceptual level.The model is claimed to extend and supersede all previous models by incorporating recent ex-perimental evidence [10, 6] regarding MinE membrane interactions. We note that MinE membranebinding was already considered in the computational model by Arjunan and Tomita [1]. Further-more, the model contradicts the cited experimental references [10, 6] in several implications regardingtransient MinE membrane binding.Park et al. [10] have shown that unmasking the anti-MinCD domains in MinE F E/I N restoresthe wild type phenotype without membrane binding. We find that the model loses instability if theMinE membrane affinity is reduced. In contrast to the experiment the instability cannot be restoredby any adjustment of the MinE recruitment rate (representing the unmasking of anti-MinCD do-mains). Hence, without any experimental support the model actually implies that MinE membranebinding is required for pattern formation in the first place. The claim that MinE membrane bindingis supposed to be responsible for geometry sensing in particular is thereby unsubstantiated.2y means of the ratio of MinE/MinD residence times the relative strength of MinE membranebinding can be quantified. The individual residence times have been determined experimentally byLoose et al.[6]. We find that the value in the computational model exceeds the experimental valueby one order of magnitude. While experiments show that the MinE membrane desity is alwayslower than the MinD density [6, 7], the waves in the computational model contain up to ten timesmore MinE than MinD. In particular, we note that the computational data in Figure 5C cannotbe reproduced. The simulations yield a MinE/MinD density ratio which is increased up to 16-foldcompared to the published computational data in Figure 5C. This represents a 23-fold deviationfrom the experiments cited alongside [6]. The fact that the model assumes wave propagation basedon very high concentrations of membrane bound MinE not co-localized with MinD invalidates themodel on a qualitative level in addition to the various aforementioned quantitative discrepancies.We conclude that the model neither accounts for MinE membrane interactions nor for any observedMinDE protein patterns. Therefore, all conclusions drawn from the computational model are void.3 ontents C E = 1 . · /µm ( [MinD] / [MinE] = 1 . . . . . . . . . . . . . 51.2.1 Alignment to rectangular membrane patches . . . . . . . . . . . . . . . . . . . 51.2.2 Alignment to curved membrane patches . . . . . . . . . . . . . . . . . . . . . 61.3 Simulations with C E = 1 . · /µm ( [MinD] / [MinE] = 2 . . . . . . . . . . . . . 71.3.1 Alignment to rectangular membrane patches . . . . . . . . . . . . . . . . . . . 71.3.2 Alignment to curved membrane patches . . . . . . . . . . . . . . . . . . . . . 8 Since we were unable to reproduce the published results we asked the authors, following PNASjournal policies, to provide us with the full set of model files that were used to produce the datain the paper and accompanying supplement. The files we received are listed in Table 1 with thecorresponding parameter configurations. In contrast to the value given in the paper ( C E = 1 . · /µm ) the MinE density is set to C E = 1 . · /µm in all simulations except the one for theL-shaped membrane patch (c.f. Fig 5 in the paper), where it is set to C E = 0 . · /µm . We didnot receive an explanation or justification for the use of two different parameter values. Because themodification creates a conflict with the [MinD]/[MinE] ratio in the experiment , we will considerboth cases in the investigation of the computational model. Table 1 lists the protein densities, andthe ratios of gold layer and membrane area used in the simulations. One notices that the goldlayer area is always smaller or of comparable size as the membrane area but never very large as theexperiments suggest. It is largest for the L-shape simulation where the MinE density is smallest.We further note that the boundary conditions are periodic. In combination with the small goldlayer size a coupling of patch dynamics across the periodic boundaries appears likely. This will beconsidered below in the context of wave alignment to rectangular membrane patches. The aspectratio in the corresponding model file is preset to a = 0 . . The geometry is constructed by applyingscaling operations on two squares with sides d and p . The scaling operations are defined in Figure1. All simulations of rectangular patches with altered aspect ratio or increased gold layer size arederived from this model file. For all simulations we have computed the solutions for . · s fromonset and compared the results with the solution at . · s to ensure that the system reached asteady state. In simulations with increased gold layer sizes we have coarsened the gold layer meshwith increasing distance from the membrane patch. In these cases the mesh size in the gold layer In the experiment a [MinD]/[MinE] ratio of 1.6 is maintained (0 . µM MinD , . µM MinE ) . The original param-eters agree with that (2 . / . ≈ . , but the modified parameter do not ( . / . ≈ . and . / . ≈ . ). s ) for original and coarse gold layer mesh. Inaddition we have verified that the final pattern after . · s does not change if the simulation iscontinued for s with a constant mesh in the gold layer that equals the membrane mesh.filename C D [ µm − ] C E [ µm − ] g AspectRatio_Paper.mph 2.9 1.3 0.56circle_S7.mph 2.9 1.3 4.13CouplingD15_S11.mph 2.9 1.3 0.38CouplingD50_S11.mph 2.9 1.3 0.38L_shape_Paper.mph 2.9 0.72 6.68largeGaps_S10.mph 2.9 1.3 1.67LargePatch_S09.mph 2.9 1.3 0.35note_S7.mph 2.9 1.3 4.14serpentine_S7.mph 2.9 1.3 4.16smallGaps_S10.mph 2.9 1.3 0.61smallPatch_S9.mph 2.9 1.3 0.86Table 1: Comsol Multiphysics simulation files provided by the authors. We list the preset values forthe total MinD and MinE densities, C D and C E , as well as the gold/membrane ratio g (Fig. 1). C E = 1 . · /µm ( [MinD] / [MinE] = 1 . In all following simulations the total MinE density is set to C E = 1 . · /µm unless notedotherwise. The simulation file corresponding to Figure 5D in the original paper (AspectRatio_Paper.mph)produces the reported alignment to the diagonal for the preset aspect ratio a = 0 . (c.f. Figure 2A).We started out increasing the size of the surrounding gold layer ( d , g = 5 . ) and found thatthe system settles in an homogeneous stable state (Figure 2B). This implies that pattern formationrequires a very small gold layer to membrane ratio. Next, we went to investigate wave couplingacross periodic boundaries and the impact on wave alignment in the original model simulations. Toeliminate any possible coupling across periodic boundaries we replaced the periodic condition by ano-flux condition. As a result waves ceased to align to the diagonal in contrast to the data in theoriginal paper. Instead we observed a disordered pattern of oscillatory blots without any apparentalignment but with preference for the edges and center of the patch (Figure 2C). This leads us tothe conclusion that the alignment observed in the simulation must result from the coupling acrossperiodic boundaries. To shed light on this matter we changed the aspect ratio of the surroundinggold layer, leaving the aspect ratio and size of the membrane patch unaltered. Increasing either thehorizontal ( x ) or vertical ( y ) width of the gold layer by a factor 1.5 or more resulted in a stablehomogeneous state without any pattern on the membrane patches. Increasing the width merely bya factor 1.25 led to a standing wave pattern aligned to the major axis of the patch (Figure 2D).This complex change in dynamics due to a small perturbation of the surrounding geometry strongly5ndicates that the ratio of gold layer area to membrane area as well as the anisotropic coupling acrossboundaries mainly regulate the formation and selection of patterns on the rectangular membranepatches. Since the alignment angle is regulated by the aspect ratio of the patch in the experimentone might expect a similar result in the simulation. The paper clearly states that the model accountsfor these experimental findings. In our attempts to reproduce this claim we rescaled the membranepatch along with the surrounding gold layer in order to obtain a membrane patch and gold layerwith an aspect ratio of 0.2. For this case the paper reports alignment of waves to the long axis inthe experiment as well as in the simulation. In contrast, we find that waves start at the edges of thepatch and align to the diagonals in the initial phase. After . · s the system consists of a mixtureof disordered waves and imperfect drifting spirals with no alignment to a specific axis (Figure 2E).Again, slightly increasing the size of surrounding gold layer leads to a complete loss of any patternsand a spatially uniform density. Increasing the vertical width by a factor 1.25 leads to a pattern ofstanding nodes attached to the edges of the long axis with odd symmetry with respect to the shortaxis (Figure 2F). Increasing the horizontal width instead leads to domains forming consecutively inthe upper and lower half of the patch (Figure 2G).To conclude, we find that the published data (Figure 5D in the paper) cannot be recovered.Waves do not align to the aspect ratio of the membrane patch. On the contrary, alignment arises aspure result of the cross-boundary coupling, and only if the surrounding gold layer area is very smallcompared to the patch size. Hence, alignment is not intrinsic to the membrane geometry as theauthors claim but, in fact, the exact opposite: there is no intrinsic alignment (i.e. independent ofcross-boundary coupling) to the membrane geometry at all. This point is further emphasized by thesimulation that the authors used to investigate decoupling of patters due to increased patch distance(large_gaps_S10.mph). In contrast to the published data (Figure S10B in the paper), we find thatthe system settles in a homogeneous stationary state (Figure 3A) for the published parameter set. The second quantitative dataset provided by the computational analysis in the paper concerns thealignment of Min protein waves to curved membrane patches. In the simulation files the membranepatches are embedded in rectangular domains representing surrounding gold layer. The preset ratioof gold layer area to membrane area g can be found in Table 1. Again, we ran all simulations withthe total MinE and MinD concentrations as published in the paper and found that the system settledin a stable homogeneous state (Figure 3A-C). It is obvious that a rectangular domain surroundingcurved patches cannot be made arbitrarily small to yield dynamical instabilities. Therefore, otherparameters need to be changed to trigger any pattern formation process and enable comparisonwith the published results. To reproduce the published results the initial MinE density has beenreduced by the authors to C E = 1 . · /µm ([MinD]/[MinE] = 2 . ) in all simulations exceptthe one for the L-shape simulation, in which case the initial MinE concentration had been set to C E = 0 . · /µm , i.e. [MinD]/[MinE] = 4 . . This does not come as a surprise, given thatthe sensitivity of Min protein patterns to [MinD]/[MinE] ratios is well documented in the literature[3, 5, 4, 9]. Nonetheless, the necessity to change this parameter in this particular case directlycontradicts the accompanying experiments the model strives to account for. To gather insight intothe reported model dynamics we repeated the simulations with the reduced total MinE density thatwas preset in the simulations files. The results are discussed in the next section.6 .3 Simulations with C E = 1 . · /µm ( [MinD] / [MinE] = 2 . In all following simulations the total MinE density is set to C E = 1 . · /µm unless notedotherwise. Running the simulation of rectangular patches with the preset MinE concentration C E = 1 . · /µm yields target patterns forming approximately at the center of one of the long edges. Thetarget waves propagate along the short axis (Figure 4A) and the pattern remains disordered. In-creasing the size of surrounding gold layer first stabilizes the target pattern and yields clean targetwaves centered at the edge of the long axis for d (Figure 4B). However, further increase of the goldlayer size leads to a destabilization of the target pattern and we observe disordered propagating blotsfor d (Figure 4C) and disordered standing waves/blots for d (Figure 4D). These waves and blotsare typically attached to the inner boundaries of the patch and tend to form and propagate ratherchaotically within the patch domain. As none of these patterns resembles the reported data from thepaper we investigated the effect of cross-boundary coupling. Increasing the width of the gold layer ineither direction yields waves roughly aligned to the diagonal (Figure 4E/F). Leaving the gold layerunaltered but replacing the periodic boundary conditions with no-flux boundary conditions yieldsdisordered waves and blots that are aligned to the long axis (Figure 4G). Once more, the resultsindicate that the reported alignment to the diagonal requires careful tuning of the cross-boundarycoupling and gold layer size.To investigate the effect of the patch geometry we repeated the simulations for different aspectratios of the membrane patch. Rescaling the whole system to obtain a membrane patch with aspectratio 0.2 yields counter-propagating waves aligned to the diagonal and originating at two diagonallyopposed corners (Figure 5A). Note that the paper reports alignment to the long axis for this aspectratio in the experiment and corresponding simulations. Increasing the horizontal width of the goldlayer yields diagonally aligned waves originating from horizontally opposed corners (Figure 5B). Wefind that the alignment to the long axis requires increasing the vertical width of the rescaled goldlayer (Figure 5C). In this case wave originate from all four corners of the patch, and waves originatingfrom vertically opposed corners merge to align to the long axis. However, the pattern is symmetricas counter-propagating waves annihilate each other. As in the previously discussed simulations forpatches with aspect ratio a = 0 . we find disordered waves and blots attached to the membraneedges if the area of the gold layer is increased uniformly (Figure 5D). Similar results are obtainedfor membrane patches with aspect ratio 1 (Figure 5E/F). Hence, any realistically sized gold layerwhere cross-boundary coupling can be excluded results in a disordered state that does not resemblethe experimental or computational data from the paper.As the experimental data for large membrane patches indicates a much broader distribution ofalignment angles one might suspect that the failure of the model originates from the choice of patchsize. In particular, the experiments imply that misalignment occurs if the patch is much larger thanthe wavelength of the patterns. Therefore, we investigated the model dynamics for large gold layersand smaller patches with constant aspect ratio. We find that reducing the patch size to . p and . p results in a pure standing wave pattern aligned to the long axis for aspect ratios a = 0 . (Figure 6A/B) and a = 0 . (Figure 6C/D). Hence, the patch size is not the cause for the disorderedwaves and failed alignment.We conclude that a commensurable alignment can only be achieved by exploiting cross-boundarycoupling effects and tuning the total MinE density. For instance, to recover alignment to the longaxis one needs to employ an anisotropic rescaling of the cross-boundary coupling in y-direction and7educe the MinE density. This demonstrates that the [MinE]/[MinD] ratio sensitively regulates thealignment angle in presence of active cross-boundary coupling. Moreover, reducing the total MinEdensity entails that the dynamical instability is not lost when the gold layer is increased. Thisenables us to study the alignment to curved membrane patches in the following section. As mentioned above the MinE concentration in the L-shape model file provided by the authors ispreset to C E = 0 . · /µm while it is set to C E = 1 . · /µm in all other model files (cf.Table 1). For the sake of a systematic study we adjusted the MinE concentration to match all othersimulations. With an otherwise unaltered model file we find that wave trains align to the patch asreported in the paper after about s (Figure 7A). While the phenomenology is quite similar, i.e.waves are aligned to the long rectangular sections then turn into the kink and realign afterwards,neither wave velocities nor [MinE]/[MinD] ratios along the patch match the data from the paper.The paper reports velocities in the range . µm/s − . µm/s (on the rectangular section and at theouter edge of the curve) while we find waves with velocities in the range . µm/s − . µm/s . Thisshows that the data in the paper is not recovered with the current parameters. The ability of wavesto realign is ascribed to the dynamics that locally increase the [MinE]/[MinD] ratio (and therebythe wave velocity) at the outer part of the curve. This mechanism for geometry sensing is ascribedto MinE membrane binding, quoting the paper:“When we plotted the ratio between the activator MinE and the membrane-bound AT-Pase MinD along the travel path of the membrane, we found that this ratio is significantlyhigher at the outer part compared to the inner part of the wave, thereby acceleratingthe detachment of the proteins from the membrane (23) (Fig. 5C). [...] Importantly, wecould only reproduce this behavior when we considered transient binding of MinE to themembrane in our model.”In this context the authors cite their previous work [6] where the local [MinE]/[MinD] ratio within awave had been quantified. These previous experiments revealed that the [MinE]/[MinD] membranedensity ratio peaks at the rear of the wave. The maximal [MinE]/[MinD] ratio is about 0.9 and itmarks the point where the protein flux off the membrane is maximized and drives wave propagation.For the computational model the authors report a [MinE]/[MinD] ratio about 1.33 at the outer partvs. 1.15 at the inner part, (cf. Figure 5C in the paper).We find that the simulations yield a local [MinE]/[MinD] ratio about 21 at the outer part ofthe curve and 15 at the inner part (Figure 7B). This represents a striking 16-fold deviation fromthe simulation data in the paper and a 23-fold deviation from the experimental data the authorscite in this particular context to support the model. We have also plotted the mean profile of the[MinE]/[MinD] density ratio on the patch in Figure 7C. Time-integration was performed over 950swhich is the period of the envelope modulating the amplitude of the protein waves [2]. As expectedthe mean [MinE]/[MinD] profile shows a maximum at the outer part of the curve and a minimum atthe inside. The mean [MinE]/[MinD] density ratio takes values between 3.6 and 9.5 throughout thepatch. Comparison of the wave profile for MinE and MinD densities and [MinE]/[MinD] ratios fromthe rectangular section of the L-shaped patches with experimental data published by the authors(see supplementary figure 1 in [6]) reveals a similar quantitative inconsistency, cf. Figure 8. We notethat these results are equally recovered if the simulations are performed with a further reduced totalMinE concentration ( C E = 0 . · /µm ) that was preset in the model file we received from theauthors. The most notable difference was that alignment was already established after about s C E = 0 . · /µm , hence, one order of magnitude earlier than with C E = 1 . · /µm .The origin of the quantitative discrepancies between [MinE]/[MinD] membrane density ratios in thepublished dataset and the simulations remains elusive. It appears to be intrinsic to the model as itrelies on very strong MinE membrane binding. This will be discussed in the next section.Before we go into that, we address the question if waves are sustained for realistically large goldlayers as the experiments dictate. We note that the variation of the cytosolic protein densities isalready very small at the system boundary (well below 1%). Still, when the surrounding gold layeris increased ( g = 72 ) waves become disordered and cease to align (Figure 7D). This observation isconsistent with the previous simulations of rectangular patches (cf. Figure 4C/D and 5D/F). Thisobservation emphasizes that coupling across the periodic boundary and the gold/membrane ratio g are two distinct aspects of the geometry affecting the model dynamics. The irregular waves foundfor the increased gold layer do not span the patch width but are attached to the interior edges.Also, waves do not align and realign after passing through the curve. Apparently, transient MinEmembrane binding does not facilitate the local increase of the [MinE]/[MinD] ratio at the outer partof the curve (Figure 7E/F) any more. To conclude: For simulations to resemble the data depictedin the paper, both, the total MinE density as well as the size of the surrounding gold layer need tobe fine tuned. While this procedure already conflicts with the experiments in the first place, thesimulations one recovers deviate from the published experimental and computational data by oneorder of magnitude in the very quantity that is argued to be key for the phenomenon of geometrysensing that the computational analysis tries to explain. So far we have established by means of numerical simulations that the model does not account forany experimental observation in the paper. Clearly, this conflicts with the final conclusion reachedby the authors in the paper’s abstract:“Using a computational model we quantitatively analyzed our experimental findings andidentified persistent binding of MinE to the membrane as requirement for the Min systemto sense geometry. Our results give insight into the interplay between geometrical con-finement and biochemical patterns emerging from a nonlinear reaction-diffusion system.”The molecular basis of the model is set by the authors’ previous experimental work [6] and the workby Park et al. [10]. The main intention is to address the role of MinE membrane binding by meansof the computational analysis, quoting from the paper:“Recently, two reports have shown that MinE persists at the MinD-membrane surfaceafter activation of the MinD ATPase (23, 30). Although persistent binding of MinEappears to be important for its ability to completely remove MinD from the membrane(23), its possible role for the ability of the Min system to organize the interior of thecell has so far not been addressed. Our model extends previous models by incorporatingthat MinE transiently interacts with the membrane during the activation of MinD. Thisdescription gives a unified account of all currently known stable Min-protein patterns invivo and in vitro as will be discussed in detail elsewhere.”The final statement is proven false by the results from the previous sections. In this context thequestion arises whether the model can be used for an assessment of MinE membrane binding at all.The research by Park et al. [10] strongly suggests that MinE membrane interactions take place. We9ant to emphasize that we do not question these experimental findings in any way. On the contrarywe find these results very helpful to understand the molecular basis of MinE protein dynamics asreflected in our own research [3].In this section we show that the model conflicts with the available experimental evidence in it’smolecular basis. Our analysis will show that the model operates in a kinetic regime that contradictsthe experimental data. As such, the model is invalid as a theory for MinE membrane interactions. Noconclusions about MinE membrane interaction can be drawn from it. To establish a mutual basis interminology we start by describing how the proposed model introduces MinE membrane interactions.We discuss the distinct molecular processes the model is based on and compare the qualitative andquantitative implications with experimental data. We limit the comparison to experimental researchpapers that are explicitly cited in the paper to motivate the model. In the last part we will addressthe role of particle numbers, bulk-membrane ratio, and the validity of an effective 2D modeling.
The model assumes that cytosolic MinE is able to sense membrane bound MinD-ATP and form aMinDE complex upon recruitment. These two steps are introduced and quantified by the MinErecruitment rate ω E = 5 · − µm /s . Upon MinDE complex formation, MinE stimulates MinDATPase activity which leads to the detachment of MinD-ATP (nucleotide exchange is neglectedin the model) from the membrane. The stimulation of MinD ATPase with MinD detachment isquantified by the sum of detachment rates ω de ≡ ω de,c + ω de,m = 0 . s − . Persistent MinE bindingis introduced by enabling the MinE dimer bound in a MinDE complex to directly interact withthe membrane. Upon stimulation of MinD ATPase activity MinE remains bound to the membranewith a certain probability p m = ω de,m /ω de = 0 . or detaches instantly along with MinD withprobability p c = 1 − p m = ω de,c /ω de = 0 . . Of course, within the notion of a deterministicmodel p m and p c can be interpreted as fraction of MinE concentration that remains membranebound or becomes cytosolic upon stimulation of MinD ATPase activity, respectively. The strengthof the MinE-membrane bond is characterized by the MinE detachment rate ω e = 0 . s − . Theparameter ω ed = 2 . · − µm /s quantifies MinE-MinD reassociation at the membrane. All theseparameters can be tuned individually to study the model dynamics. However, no experimental dataexists to support any particular parameter choice. Moreover, mutations studies (e.g. regarding theMTS of MinE) are unlikely to affect only one corresponding model parameter alone. To constrainMinE membrane binding and MinE-MinD interactions in the model we will take the experimentallydetermined Min protein residence times into account. This enables us to establish a direct relationbetween quantitative experimental data and the corresponding model parameters. The residence < τ D > and < τ E > times of MinD and MinE along the protein wave have beenquantified by the authors in previous experiments [6]. The experiments revealed that MinE remainslonger in a membrane bound state than MinD which indicates transient MinE membrane binding.While the individual residence times increase from front to rear of the wave, the ratio of residencetimes ∆ τ = < τ E > / < τ D > appears to be constant throughout the wave (∆ τ exp ≈ . .Therefore, we can use the ratio of residence times as characteristic parameter to quantify transientMinE membrane interaction in context of any specific model.The mean residence time of MinD < τ D > is given by the time it takes cytosolic or membrane-bound MinE to sense and attach to a membrane-bound MinD, and the time MinE needs to drive10inD off the membrane. Hence, < τ D > = ( ω E c E + ω ed c e ) − + ω − de . (1)For the mean MinE residence time < τ E > one has to consider the conditional branches whetherMinE detaches alongside MinD (with probability p c ) or remains on the membrane (with probability p m = 1 − p c ), and whether membrane-bound MinE detaches from the membrane (with probability q c = ω e / ( ω e + ω ed c d ) ) or reassociates with MinD (with probability q m = 1 − q c ): < τ E > = p c ω − de,c + p m ( ω − de,m + q c ω − e + q m (( ω ed c d ) − + < τ E > )) . (2)After a few algebraic manipulations one obtains an expression for the mean MinE residence time < τ E > = 2( ω de,m + ω ed c d + ω e ) ω de,c ω ed c d + ω de ω e , (3)and for the ratio of the residence times ∆ τ ∆ τ = 2 ω de ( ω de,m + ω ed c d + ω e )( ω ed c e + ω E c E )( ω de,c ω ed c d + ω de ω e )( ω de + ω ed c e + ω E c E ) . (4)By comparison with the experiment (∆ τ exp ≈ . we obtain an expression relating all processesthat characterize MinE interaction with the membrane to interactions with MinD. Due to the ex-plicit dependency on the protein concentration c d , c e , and c E numerical simulations or additionalapproximations are required for further progression. We note that for the specific parameter choicein the present model ∆ τ becomes independent of c d . The reason is that both exit processes forMinE occur on the same time scale, i.e. ω de,c = ω e , such that < τ E > → /ω e and ∆ τ → p c · ω ed c e + ω E c E ) ω de + ω ed c e + ω E c E . (5)To obtain a first estimate we approximate the missing values for the protein concentrations by thestationary solution that represents the mean concentrations in linear approximation. For the pa-rameters provided in the paper we find ∆ τ ≈ . , hence an 11-fold deviation from the experimentalvalue ∆ τ exp ≈ . . We conclude that the model’s assumption regarding MinE dynamics (in termsof parameter choice) clearly conflicts with experimental data. It represent the underlying molecularprocesses. In the following section we will investigate the model’s linear stability upon varying theMinE dynamics. The research by Park et al. [10] indicates that membrane associated MinE has an increased MinE-MinD interaction rate due to an exposed contact helix compared to the stable conformation insolution where the contact helix is buried in the dimeric interface. It is our understanding thatthis is the main idea which defines persistent binding [6] and tarzan of the jungle [10] models. Tokeep these mechanisms unaltered while trying to reduce the relative residence times we focus on theisolated effect of MinE membrane interactions. The membrane affinity of MinE can be expressedby p m = 1 − p c alone. Tuning this parameter enables a smooth reduction of the relative residencetime ∆ τ without altering the reassociation process with MinD. By decreasing the MinE membraneaffinity we find that the dynamical instability is lost for p m < . at ∆ τ = 3 . , c.f. Figure 9A.11eleasing the constraint ω de,c = ω e and increasing ω e to weaken MinE membrane binding similarlyleads to loss of dynamical instability for ω e > . at ∆ τ = 4 . , c.f. Figure 9B. We draw twoconclusions from these results. First, tweaking the MinE membrane affinity alone proves to beinsufficient to recover consistency with the experimental data. It appears that the problems are atleast inherent to the parameter configuration as a whole (in particular, the MinE-MinD interactions)such that the model requires a broad reinvestigation in it’s vast parameter space. Second, thefact that the model requires at least 72% of MinE remaining membrane bound after stimulatingMinD ATPase activity implies that strong membrane interactions are a key requirement for patternformation in the proposed model. Translated into an experimental test a hypothetical MinE mutantwith slightly reduced membrane binding affinity but otherwise unaltered protein-protein interactionsshould lack any Min oscillations. In that case one would expect to observe the correspondingphenotypes (filamentous cells and minicells). However, we couldn’t find any experimental evidencethat supports this major model prediction. In Park et al. [10] four mutations of the MinE MTS( minE L E , minE F E , minE L E , minE F E ) are investigated, quoting [10]:“Surprisingly, the strains containing minE L E and minE F E were extremely filamentousand could not form colonies on plates with arabinose (Figure 4B and data not shown),indicating that MinE function was absent. In contrast, strains containing minE L E and minE F E formed colonies normally on plates with arabinose, but the morphologies ofthe cells were heterogeneous in length with some minicells. The average cell lengthof an exponential culture of the strain with minE W T was . ± . µm compared to . ± . µm for the strain lacking Min function. The strains containing minE F E and minE L E had average cell lengths of . ± . µm and . ± . µm , respectively ( N ∼ for each). In summary, each of the four charge substitution mutations eliminatedmembrane binding of the MinE I R mutant. However, two of the mutations, minE L E and minE F E , completely eliminated the ability of MinE to counteract MinC/MinD,whereas the other two, minE L E and minE F E , did not, although they did reduce theability of MinE to spatially regulate division as evidenced by the increases in the averagecell length and the standard deviation.”Hence, removing MinE membrane binding affects the performance of the Min system rather thandisabling it’s function altogether. As discussed in our previous work [3], altering Min protein re-cruitment rates (which represent the sensing step discussed before) has significant impact on midcelllocalization by pole-to-pole oscillations. Therefore, the observed filamentous phenotypes in Park etal. [10] might be explained by an altered Min oscillation with reduced midcell localization precisiondue to weakened MinE-MinD interactions. The explanation offered by Park et al. [10] appears toagree with this line of thought, quoting [10]:“ [...] two of the MinE mutants we described above, MinE F E and MinE L E , were unableto rescue cells from expression of MinC/ MinD (Figure 4B). This was surprising becausethese residues lie beyond the putative interacting helix. We reasoned that these residuescould play a role in sensing MinD and therefore might have a defect in MinD-MinE inter-action similar to that of the MinD M L mutant. If so, the minE I N mutation shouldsuppress these mutations. As shown in Figure 2D, the double mutant MinE F E/I N rescued cells from expression of MinC/MinD, demonstrating that the minE I N muta-tion is an intragenic suppressor of minE F E . It also suppressed minE LE [ sic ] (data notshown).” We note that reducing the MinE-MinD interaction parameters ω E and ω ed leads to loss of instability as well. I R lacks the ability to sense MinD. In turn, this leads to the loss of Min-oscillations andresults in the filamentous phenotype. The fact that minE L E , minE F E , and MinE F E/I N seemsto retain the function of the Min system, while lacking the ability to bind to the membrane clearlyimplies that not membrane binding itself but the effect it has on MinD sensing enables proper Minoscillations. For minE L E and minE F E the specific effect on MinD sensing cannot be deduced fromthe experimental data. This impedes an unequivocal comparison with the model. However, this isnot the case for MinE F E/I N . MinE I N shows significantly increased MinD sensing ability throughunmasking of the anti-MinCD domains ( β strands) which are buried in the dimeric interface in WTMinE. So, on one hand MinE F E/I N lacks the ability to interact with the membrane, such thatmembrane binding effects can be excluded. On the other hand the unmasked anti-MinCD domainsenable it to interact with MinD as if it were in it’s membrane bound conformation. As the experimentindicates that this mutation restores proper function of the Min system this scenario can be directlytranslated into a test of the model dynamics: We incorporate the corresponding modification bypreventing MinE to bind to the membrane ω de,m = 0 /s , ω de,c = 0 . /s and setting the recruitmentrate ω E for cytosolic MinE equal to the reassociation rate ω ed of former membrane bound MinEspecies, i.e. ω E = ω ed = 2 . · − µm /s . The experimental observation implies that the dynamicalinstability should be restored. In contrast, the linear stability analysis reveals no instabilities inthis kinetic configuration, cf. Figure 9C. Moreover, we find that without MinE membrane bindingthe instability cannot be restored by changing the MinE recruitment rate. Therefore, not themodified MinE-MinD interaction but MinE membrane binding is the crucial component in themodel. Assuming that the MinE F E/I N mutant most likely restores pole-to-pole Min oscillations(hence, geometry sensing) without requiring membrane binding further questions the claim thatMinE membrane binding is responsible for geometry sensing, we conclude that the experiments byPark et al. [10] disprove the proposed model. We stress that a loss of dynamical instability withoutMinE membrane binding does not imply that MinE membrane bindings is required or responsiblefor geometry sensing in any way. The demonstration is given by the model simulations above: Thereone observes pattern formation based on dynamical instabilities but no geometry sensing. In this last section we focus on questions about volume effects, spacial dimensions, and effectivesystem size. A main claim of the theoretical investigation in the paper is that“ [...] Min protein waves sense the geometry of the flat, two-dimensional membrane,rather than the three-dimensional space of the cell or the curvature of the membrane. ”Certainly, there is no doubt that the available experimental data offers such a conjecture as wavesare found for various system/bulk heights and the gold layer size does not seem to have any impactbeyond enabling and disabling patch-to-patch coupling. However, in light of the numerical inves-tigation above this statement raises the question why the size of the space above the membraneshould not matter while additional space around the membrane (in form of the surrounding goldlayer) has significant impact on the model dynamics. Increasing gold layer size leads to loss of pat-terns for the experimental [MinE]/[MinD] ratio and impedes wave alignment (i.e. geometry sensingphenomena) for fine tuned [MinE]/[MinD] ratios. This indicates that bulk size (via bulk/membrane Figure 9D shows the same data as Figure 9C but with the hydrolysis rate ω de reduced by 50%. This correspondsto the measurement of reduced MinD ATPase activity stimulation by MinE I N -h ∗ in the experiment by Park et al.[10]. Min proteins waves sense the geometry of the flat, two-dimensionalmembrane ”. The foundation of this conclusion appear inscrutable to us. The authors state in thesupplementary document to the paper that they “ have checked on specific examples that the samephenomena [...] can also be observed in the full three-dimensional description ”. Obviously this re-quires an explicit mapping between the full 3D dynamics and the effective 2D reduction. However,no such mapping is provided in the model definition. Inspection of the model parameters revealsthat the cytosolic protein concentrations are treated as surface densities: C D = 2 . · µm − and C E = 1 . · µm − . This indicates an underlying bulk integration. The protein concentrationsin the experiment are c D = 0 . µM ≈ . µm − MinD and c E = 0 . µM ≈ . µm − MinE.Comparison with the model parameters implies integration of µm bulk, i.e. C D /E ≈ c D/E . Thepaper does not provide the experimental bulk height explicitly, however, it is stated that the bulkheight is very large, quoting [11]:“Although the space above the membrane was not limited in our experiments, the proteinswere located only in a small layer above the membrane during pattern formation.”The corresponding figure (Figure S1 in the supplement of the paper) clearly shows that the experi-mental bulk height is much larger than µm , and previous in vitro experiments were performed witha total bulk height about h = 5 · µm [7]. Regarding particle numbers the model accounts for asystem that is three orders of magnitude smaller than the typical experimental setup.Without any notion of bulk volume in the model definition we are left with ad hoc approximationsthat maintain the mathematical structure of the model. The bulk dynamics in normal direction tothe membrane can be eliminated via integration or averaging to yield a 2D reaction-diffusion system.In the first case we increase the total densities of MinD ( C D ) and MinE ( C E ) keeping the ratioconstant. In the second case we keep the total densities of MinD and MinE fixed and introduce ascaling factor h between between membrane and bulk dynamics, i.e. ∂ t c B = D B ∇ c B + 1 h f B (6)where c B denotes any bulk species with membrane reactions given by f B . Note that in this case bulkdensities are (mean) volume densities and not surface densities as the reported parameters suggest.Using the effective system size as parameter we find that dynamical instabilities are rapidly lostin both approximation (Figure 9E/F). This proves that the model’s dynamics are actually highlysensitive to volume effects in contrast to the authors’ claim in the paper. We further note that thesefindings are supported by the fact that patterns vanish if the gold layer size is increased.We conclude that the model cannot account for experimental system sizes. Moreover there is noexplicit notion of bulk volume and no relation to the full three-dimensional dynamics. Increasingthe effective system size leads to loss of any pattern forming instabilities which directly contradictsthe claim that three dimensional dynamics do not affect pattern formation.14 Figures and ReferencesReferences [1] A. Arjunan and M. Tomita. A new multicompartmental reaction-diffusion modeling methodlinks transient membrane attachment of E. coli MinE to E-ring formation.
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Proc. Natl. Acad. Sci. USA , 109(38):15283–15288, 2012.15 d d x d y p y p x Figure 1: Model geometry and scaling operations.16 : increased gold layer (g=5.25)F: increased gold layer width (vertical) D: increased gold layer width (vertical) G: increased gold layer width (horizontal)