Intra- and intercellular fluctuations in Min-protein dynamics decrease with cell length
Elisabeth Fischer-Friedrich, Giovanni Meacci, Joe Lutkenhaus, Hugues Chate, Karsten Kruse
aa r X i v : . [ q - b i o . S C ] S e p Intra- and intercellular fluctuations in Min protein dynamicsdecrease with cell length
Elisabeth Fischer-Friedrich , Giovanni Meacci , Joe Lutkenhaus , Hugues Chat´e , andKarsten Kruse Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Straße 38, 01187 Dresden,Germany. Present address: Department of Physical Chemistry, Weizmann Institute of Science,Rehovot 76100, Israel IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598. Present address:Department of Biological Sciences, Columbia University, New York, NY 10027 Department of Microbiology, Molecular Genetics, and Immunology, University of Kansas MedicalCenter, Kansas City, Kansas 66160 CEA-Saclay, Service de Physique de l’Etat Condens´e, 91191 Gif-sur-Yvette, France Theoretische Physik, Universit¨at des Saarlandes, Postfach 151150, 66041 Saarbr¨ucken, Germany
Abstract
Self-organization of proteins in space and time is of crucial importance for the functioningof cellular processes. Often, this organization takes place in the presence of strong randomfluctuations due to the small number of molecules involved. We report on stochastic switchingof the Min-protein distributions between the two cell-halves in short
Escherichia coli cells.A computational model provides strong evidence that the macroscopic switching is rooted inmicroscopic noise on the molecular scale. In longer bacteria, the switching turns into regularoscillations that are required for positioning of the division plane. As the pattern becomesmore regular, cell-to-cell variability also lessens, indicating cell length-dependent regulation ofMin-protein activity. keywords : cell division, spatiotemporal patterns, fluctuations, protein self-organization . INTRODUCTION Subcellular structures are often formed by a small number of proteins. This is notably the casein prokaryotes, as exemplified by the Soj proteins in
Bacillus subtilis that switch randomly betweenthe two distal parts of the nucleoid [1, 2], by the Min proteins in
Escherichia coli [3] that switchperiodically between the two halves of the bacterium, as well as by the rings and helices formed byFtsZ and MreB in many bacteria [4]. The concentrations of these proteins are in the micromolarrange, hence, the above structures, that may extend over several micrometers, are formed by afew hundred molecules. Such small numbers can imply large random fluctuations in space andtime [1, 2].However, there is currently a lack of quantitative experimental studies of random fluctuationsin spatially extended protein structures. In particular it remains largely unexplored how fluctua-tions on cellular scales originate from the interplay of molecular noise, i.e., the stochastic natureof the protein kinetics, and external noise, i.e., variations in the environment or in the numberof proteins present in the cell [4–7, 9]. In the context of gene expression, random fluctuationscontribute essentially to the dynamics [10]. In particular, they directly lead to differences in ge-netically identical cells [11], i.e., intercellular fluctuations, and can play an important role in cellfate decisions [11, 12]. In this context, cell size can act as a control parameter to regulate eitherthe amplitude of fluctuations [13] or the distribution of phenotypes they generate [13, 14].In this work, we report on intra- and intercellular fluctuations in the spatiotemporal patternsformed by the Min proteins in
E. coli . These proteins select the cell center as the site of division [3].Their distribution has been found to change periodically with time, such that most of the proteinsreside for about 40s in one cell half and subsequently for the same time in the opposite cell half [15].As one of the Min proteins, MinC, inhibits formation of the Z-ring, division is in this way suppressedat the cell poles. Computational analysis of the Min-protein dynamics indicate that the observedpattern results from self-organization of the ATPase MinD and MinE [16]. This idea is supportedby experiments in vitro in which planar and spiral waves of MinD and MinE emerged spontaneouslyin the presence of ATP on a supported lipid bilayer [3].We found, that in cells below a critical length of 2.7 µ m the Min distribution switches stochas-tically between the two cell halves. In this phase intercellular fluctuations are pronounced. Incells longer than the critical length, we observed regular oscillations and intercellular fluctuations ere significantly reduced. Computational analysis shows that stochastic switching can resultfrom molecular noise. Furthermore, it indicates that, in the stochastic state, the rate of ATPconsumption per MinD molecule is only half of that in the oscillatory state. II. RESULTS
We observed the distribution of MinD for 40 minutes in each of 209 cells, see Materials andMethods, during which individual bacteria grew roughly 0.5 µ m. In cells shorter than 2.5 µ m, insteadof oscillating regularly, MinD typically shifted stochastically from one cell half to the other, seeFig. 1a. The residence times of MinD in one cell half varied widely in these cells, whereas completeswitching from one cell half to the other occurred in an interval of less than 15s. The extent of theregion covered by MinD in one cell half did not change notably between two exchange events, seeFig. 1g.For cell lengths between 2 . µ m and 3 µ m, the Min pattern typically changed from stochasticswitching to regular oscillations with a period of about 80s, see Fig. 1b and Fig. S1 in the Supp. Mat.The precise lengths at which this transition occurred differed between cells. In the oscillatoryregime, between two switching events, the region covered by MinD first grew monotonically fromthe cell end and then shrank monotonically towards the same end, see Fig. 1g. Analogously to thestochastic exchange regime, the transition of MinD from one half to the other was fast comparedto the oscillation period. In a small number of cells (N=5), we observed a transition back fromregular oscillations to stochastic switching.Cells longer than 3 . µ m invariably displayed regular oscillations, see Fig. 1c and Fig. S1 in theSupp. Mat. The oscillation period typically decreased slightly with increasing cell length. For thecell shown in Fig. 1c, the initial period is approximately 87s, while it is approximately 70s at theend. Similar behavior can be observed for MinE, see Supplementary Material.About 5% of the cells divided during the observation time, see Fig. 1d, e. In all these cases,MinD oscillated regularly prior to division. Consistent with our findings in non-dividing cells, thepattern displayed by the Min proteins in the daughter cells immediately after division correlatedwith their length: daughter cells shorter than 2.5 µ m typically displayed stochastic switching, whilethe Min proteins mostly oscillated in daughter cells longer than 3 µ m. In some cases, however,the two daughter cells showed different MinD patterns, in spite of having equal lengths: while in ne daughter we observed oscillations, the other displayed stochastic shifts. In these cases, judgedfrom the fluorescence intensities, the distribution of MinD between the two daughter cells wassignificantly uneven [31]. A. Distribution of residence times
The transition from stochastic switching to regular oscillations is accompanied by a qualitativechange in the distribution of residence times τ cumulated over cells of a given length, see Fig. 2.For cells longer than 3 µ m, this distribution shows a pronounced peak at ≈ s , which essentiallycoincides with the mean value, and a small tail including rare events of residence times of up to200 s . These rare events correspond to instances when one or more oscillations were missed andthus the residence time extends to values significantly larger than the mean, see Fig. 9 in theSupplementary Material. The left, main part of the distribution ( τ < .
9s and a geometric standard deviation of 1 . τ > α = 3 . ± . µ m, the distribution of residence times is qualitatively different: it isessentially algebraic, ∝ τ − α , with a decay exponent α = 2 . ± .
2. Consequently, the mean-valueis just at the edge of being well-defined, while the variance diverges, implying enormous variations.This is in contrast to a usual random telegraph process in which a system switches stochasticallybetween two states at given constant rates. In this case, the distribution of residence times inone state decays exponentially. We, therefore, also fitted an exponential decay to the tail of thedistribution of residence times. However, the logarithm of the likelihood ratio of an algebraic toan exponential decay was 35 and thus strongly indicative of an algebraic decay [18].In Fig. 3a, we show the functional dependence on the cell length of the mean residence time h τ i cumulated over all events and all cells in a given length interval. Cell lengths were binned intointervals of 0 . µ m length and residence times were assigned to the length of the cell at the beginningof the respective residence time interval. Note, that since within 500s, which is exceptionallylarge for a residence time, the bacteria grew at most 0.2 µ m in length, our results do not dependsignificantly on the exact assignment rule. Strikingly, the variation of h τ i is well-described bytwo exponential decreases: For cell lengths smaller than 2.7 µ m that are typically in the stochasticregime, h τ i ∝ exp( − L/λ ) with λ = 0 . µ m, while for longer, typically oscillating cells λ = 2 µ m. o characterize fluctuations around h τ i , we computed the effective standard deviation σ corre-sponding to our finite number of samples [32]. The effective standard deviation σ , too, decreasesexponentially with increasing cell length. Note that, for cells smaller than approximately 2.7 µ mit is larger than h τ i , while for longer cells it is smaller. For cells larger than 3.5 µ m, the standarddeviation is below 10s indicating that the oscillation period varies remarkably little within a singlecell and within an ensemble of cells. B. Cell-cell variability
To assess the contribution of cell-to-cell variability to the fluctuations, we compared the meanresidence time ¯ τ ic of individual cells with that of other cells sharing the same cell length. InFig. 3b, we represent the normalized standard deviation of the distribution of mean residencetimes, σ ic = q h ¯ τ i − h ¯ τ ic i / h ¯ τ ic i . It decreases with increasing cell length, first mildly then sharplyfor cell lengths larger than 3 µ m, indicating a clear reduction of cell-to-cell variability in longercells. This suggests a control mechanism that adjusts Min parameters towards optimal referencevalues when the cells grow. C. Mathematical modeling
Several attempts at mathematically modeling of the Min system have been successful in re-producing essential aspects of the Min dynamics [16]. Based on different underlying microscopicpictures, these models uniformly explain the Min oscillations as an emergent property of inter-acting MinD and MinE in presence of a membrane and ATP. In particular, self-enhanced bindingof MinD and/or MinE to the membrane, or aggregation of membrane-bound proteins can triggeran instability towards oscillations. Biochemically, MinD and MinE are not yet characterized wellenough to exclude one possibility or the other. The validity of a model can alternatively be testedby verifying predictions of the macroscopic behavior. For example, the aggregation models inRefs. [19, 20] predict the existence of stationary heterogeneous protein distributions.We tested several stochastic versions of models presented in the past [3, 6, 20] and checked ifthey showed stochastic switching of Min proteins in certain parameter regimes, see Supp. Mat.Out of these, only the model suggested in [20] produced stochastic switching similar to what is bserved in the cell [33]. Since stochastic simulations in three dimensions yield essentially the sameresults as in one dimension, cf. Refs. [7] and [9], we restricted our attention to simulations in onedimension.What is at the origin of the transition between stochastic switching and oscillations of MinDand the corresponding change in the residence-time distribution? The experimental data presentedabove show that the spatiotemporal structure of the MinD pattern correlates with cell length. Asa naive guess one might think that the cell length directly controls the pattern. However, in ourexperiments, we occasionally observed that two daughter cells of about the same length showeddifferent MinD patterns right after division of the mother cell. In addition, in some instances, cellsshorter than 2.5 µ m also showed oscillations, while in some cells longer than 3 µ m MinD switchedstochastically between the two cell halves. This suggests that, in addition to the cell length, otherfactors control the transition from stochastic switching to regular oscillations.In order to better understand which parameters of the system are able to trigger stochasticity inMin switching, we used a one-dimensional particle-based stochastic version of the one model whichwe identified as being able to produce stochastic switching, see Fig. 4 and Material and Methods.We find that, increasing the cell length is not sufficient to generate a transition from stochasticswitching to oscillatory Min dynamics if, simultaneously, protein numbers increase proportionallyto the cell length. Changing other parameters of the Min dynamics, however, can trigger sucha transition. As an example we now discuss alterations in the rate ω E of MinE binding to themembrane, see Fig. 5. Let us stress that in the following we keep the total protein concentrationsconstant, that is, protein numbers vary proportionally to the cell length. For ω E = 0 . s − and acell length of 2 µ m, our simulations show stochastic switching, while for ω E = 0 . s − and a celllength of 3 µ m, we find oscillations [34], see Fig. 5a. The model is thus capable of reproducing theobserved transition by changing MinE activity. It is important to note, that this transition is nota consequence of approaching the deterministic limit, but is inherent to the Min-protein dynamics.Indeed, for the respective binding rates, the deterministic model is either bistable and settling intoone of two mirror-symmetric stationary states or it generates oscillations [19, 20]. We concludethat in short cells the macroscopic switching behavior is rooted in intrinsic fluctuations of themolecular processes due to the relatively small number of proteins involved. The transition fromstochastic switching to oscillations, however, is not due to the discreteness of molecule numbers,which in general is possible [23]. uantitative analysis of simulations corresponding to a single cell in the stochastic switchingstate, presented in Fig 5a, yield an exponential decay of the residence times as expected for a ran-dom telegraph process. This is however not in contradiction with the experimental data, for which ensemble averaging over different cells yields algebraic decay, Fig. 2a. Indeed, experiments revealedcell-to-cell variability, particularly strong amid short cells exhibiting stochastic switching, Fig. 3b.As we show below, introducing such variability in the model resolves the apparent contradiction.It is conceivable that in the experiments the cell-to-cell variations in the Min dynamics for a givencell length result mainly from differences in the numbers of MinD and MinE. Indeed, we found in ≈
65% of the observed divisions that the numbers in the daughter cells, as judged from fluorescenceintensities, differed by more than 10%, see Supplementary Material.In order to test the effects of varying protein numbers in the model, we performed severalsimulation runs with MinD protein numbers drawn at random from a Gaussian distribution witha standard deviation of 10% of the mean [11]. The ratio of MinD to MinE was fixed to 8/3 [2] andall other parameters kept constant. Lumping together the observed residence times of 70 runs withdifferent protein numbers, we now observe an algebraic decay with exponent 2 . ± .
2, see Fig. 5b,which is very close to the experimental value of 2.1, see Fig. 2a. As in the experiments, the meanis barely defined and the standard deviation diverges. This result constitutes a second instanceof the general finding that power law distributions can result from Gaussian variations of systemparameters as discussed by Tu and Grinstein in the context of the bacterial flagellar motor [25].Analogously, we calculated the distribution of residence times in the oscillatory regime, seeFig. 5c. It is remarkably similar to that found experimentally, see Fig. 2b: a large hump, wellfitted to a log normal distribution with geometric mean 31s and geometric standard deviation 1.2s,plus a small tail at large residence times which can be fitted by an algebraic decay with exponent α = 4 . ± .
4, similar to the experimental value α = 3 .
6. Moreover, an inspection of the rare eventsat the origin of this algebraic tail showed that they are instances of “missed” oscillations, just likein the experimental kymographs, see Fig. 9 in the Supplementary Material. We conclude that theoscillation period is a robust property of the Min System with respect to fluctuations in proteinnumbers.We then investigated the transition from the switching to the oscillatory state in more detail.Fig. 5d presents the mean value and the standard deviation as a function of system length. Therate ω E was varied with the system length according to a sigmoidal dependence, see Fig. 5e inset nd Supplementary Material. Similar to the experimental results, we find a decreasing meanand standard deviation with increasing length and the transition from stochastic switching tooscillations is accompanied by a drop of the standard deviation below the mean value. The simpleparticle model of the Min dynamics thus semi-quantitatively captures the effects of length changesand of fluctuations in the system.Finally, we used the computational model to infer the ATP consumption rate as a function of celllength. Interestingly, the average ATP consumption rate per unit length, which is proportional tothe ATP consumption rate per MinD, increases with cell length until the transition from stochasticswitching to regular oscillations and then remains roughly constant, see Fig. 5e. Our calculationsthus let us hypothesize that is energetically advantageous for a short cell to keep the Min systemin the stochastic switching regime. III. DISCUSSION
In summary, we have shown that in short
E. coli cells the distribution of MinD proteins stochas-tically switches between two mirror-symmetric states, while it oscillates regularly in longer cells.The Min system provides thus an intriguing example of a spatiotemporal pattern under physiolog-ical conditions that combines regularity with stochastic elements.Stochastic switching between two states is also known for the Soj/Spo0J proteins which in
B. subtilis relocates irregularly between the two sides of the bacterial nucleoid [1, 2]. A computa-tional model suggests that this phenomenon, too, is rooted in molecular fluctuations [6]. Similaritiesbetween MinD and Soj had already been pointed out due to resemblance in structure, polymeriza-tion and ATPase activity [26]. However, notable differences exist as, for example, Spo0J seems tobe always bound to the nucleoid unlike MinE that detaches from the membrane. Furthermore, nooscillations have been found for Soj/Spo0J.Based on our theoretical analysis, we can speculate about a possible regulatory mechanism con-trolling the dynamics of the Min proteins in
E. coli . We interpret the findings of our computationsas indications of changes in the activity of the Min proteins as the cells grow. Specifically, theability of MinE to bind to membrane bound MinD might be reduced in early phases of the cellcycle. According to this view, the Min system is kept in a stand-by mode with reduced ATPconsumption in short cells and gets fully activated only as a cell approaches division, when the in system needs to be functional. Why does stochastic switching of the Min proteins not lead tominicelling? The central Z-ring needs about 5 to 10 minutes to mature and polar Z-rings are lessstable than the central one. Consequently, polar rings formed in an early phase of the cell cyclemight be disassembled by MinC after oscillations have set in, which occurs significantly beforeseptation starts.Obviously, the transition might be also due to other evolutionary constraints or be just a side-effect of the Min-protein dynamics. Independently of the transition’s physiological reason, wepropose that changing system parameters such that the dynamic behavior is qualitatively modifiedthrough a bifurcation - in the present case from bistable to oscillatory - presents an interestinggeneric mechanism to tame detrimental fluctuations as subcellular processes become vital. MaterialsA. Data acquisition
We used cells of the
E. coli strain JS964 containing the plasmid pAM238 encoding for MinEand GFP-MinD under the control of the lac-Promoter [27]. Bacteria were grown overnight in a3ml LB medium at 37 ◦ C. Cells were induced with Isopropyl- β -D-thiogalactopyranosid (IPTG) ata concentration of 200 µ M and incubated for 4 hours. During measurements, cells were in theexponential growth phase. The samples were kept at a temperature of 30 ◦ C using a Bachhofferchamber. 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 back-ground fluorescence). The fluorescence recordings were taken with an Olympus FV 1000 confocalmicroscope, at an excitation wavelength of 488nm from a helium laser at low power. We usedan Olympus UPLSAPO 60x, NA 1.35 oil immersion objective and recorded a frame every 3s. Ameasurement lasted 40min. During this period, the focus was manually readjusted at irregularintervals. We were not able to determine the protein numbers in individual cells.In total, we extracted data from 209 MinD-fluorescent cells obtained from 5 different measure-ments. We extracted data only for cells which, at the beginning of the measurement, were smallerthan 3 µ m. Cell lengths were determined from differential interference contrast (DIC) images withan accuracy of ± as determined by linear interpolation. Some of the cells in the field of view divided during themeasurement time. If division occurred after more than 20min of measurement, they were in-cluded in the data analysis. In these cases, fluorescence recordings were used until cell constrictionterminated.To observe MinE distribution in cells, we used the strain WM1079 expressing MinD and MinE-GFP on a plasmid under the control of the pBAD-promoter [1]. Bacteria were induced withArabinose but otherwise grown in the same way as described above. B. Data analysis
For a quantitative analysis, we mapped the time-lapse data of MinD-fluorescence from
E. coli cells onto the states of a two-state process. We transformed the fluorescence data into a real-valuedtime-dependent function by subtracting the integrated fluorescence intensities from the left andthe right cell half, see Fig. 6. Then, the moving average over four time points was taken in orderto reduce noise. The resultant function f is positive when the fluorescence maximum is in one cellhalf and negative in the opposite case. The residence time τ of MinD in one cell half is definedas the interval between two consecutive zeros of f . To each residence time, we assigned the celllength at the start of the respective residence period. C. Computational model of the Min dynamics
For the computational analysis of the Min-protein dynamics, we use a particle-based versionof the model suggested in [20]. It describes the formation of Min oscillations on the basis ofan aggregation current of bound MinD that is generated by mutual attraction of the proteins.Furthermore, it accounts for the exchange of MinD and MinE between the membrane and thecytoplasm, where MinE only binds to membrane-bound MinD and where MinD detaches fromthe membrane only in the presence of MinE. Measurements of the cytoplasmic diffusion constantsof MinD and MinE yielded values larger than 10 µ m /s [29]. We thus consider the limit of largecytoplasmic diffusion which effectively leads to homogeneous cytoplasmic concentrations. Thediffusion constant of membrane-bound proteins is about two orders of magnitude smaller than forcytosolic diffusion [29]. This mobility suffices to generate a sufficiently strong aggregation current enerating an instability of the homogenous protein distribution [20].A corresponding stochastic particle-based description is defined on a one-dimensional latticewith N sitesrepresenting the long axis of the cell. The lattice spacing l b is chosen such that itis much larger than the protein size and much smaller than the characteristic length of the Minpattern. Each site can contain at most n max proteins, a number which can be understood as thecircumference of the cell. We assume diffusional mixing such that proteins are indistinguishable ona site. For site j , the probability of attachment of cytoplasmic MinD and MinE during a sufficientlysmall time step ∆ t is given by P D → d = ∆ t ω D (cid:18) N D N (cid:19) (1 − n d,j + n de,j n max ) (1) P E → de = ∆ t ω E (cid:18) N E N (cid:19) n d,j n max , (2)respectively. Here, ω D , ω E are the corresponding attachment rates and N D and N E are, respec-tively, the numbers of cytoplasmic MinD and MinE. The numbers of membrane-bound MinD andMinDE complexes on site j are n d,j and n de,j . The detachment probability is P de → E + D = ∆ t ω de n de,j , (3)where ω de is the detachment rate. The exchange of particles between sites is governed by P j → j ± = D d ∆ tl b n d,j (1 − n d,j ± + n de,j ± n max ) I j → j ± , (4)where I j → j ± = E j < − ∆ E j k B T ) if ∆ E j > E j = V j ± − V j . The potential V describes the interaction strength between Min-proteinson the membrane. We assume a square hole potential V j = − n max g d (2 R d + 1) R d X j = − R d n d,j + g de (2 R de + 1) R de X j = − R de n de,j . (6)Here, the integers R d and R de relate to the ranges r d of the MinD-MinD interaction and r de of theMinD-MinDE interaction through R d ≃ r d /l b and R de ≃ r de /l b . The parameters g d and g de tunethe interaction strength. The diffusion constant of membrane-bound MinD is D d .In Fig. 5d, the attachment rate ω E is increased jointly with the cell length in the simulations. e chose a sigmoidal increase of ω E in dependence of the cell length ℓ c according to the Goldbeter-Koshland function, which gives the mole fraction of modified proteins that are under control of amodifying enzyme [30] G ( ℓ c ) = ω sat E × ℓ c KvJ + ℓ c ( K −
1) + p ( vJ + ℓ c ( K − − v − ℓ c ) ℓ c K . (7)We chose parameters v = 2 . , J = 1 .
116 and K = 0 . ω sat E determines the saturationvalue and was chosen as 0 . − . The precise functional form of ω E ( ℓ c ), however, is not of importanceas long as it is sigmoidal. Acknowledgments
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20 50 100 200 500110100 residence times (s) residence times (s) a) b)N
30 100 200 N FIG. 2: Ensemble distributions of residence times. a) Residence times for initial cell lengthsbetween 2 and 2.2 µ m (total: 358) and b) for initial lengths greater than 3 µ m (total: 2157). Timesare respectively binned into 20s (a) and into 3s intervals (b). Solid lines result from fits of thedistribution tail ( τ < s ). (a) Fitting of an algebraic tail yields an exponent − . ± . .
5s and geometric standard deviation 1 .
2s (blue line); the tail of rare events of duration longerthan 60s is reasonably well fitted to an algebraic law with exponent − . ± . ell length ( µ m) a) b)< τ > in (s) σ in (s) cell length ( µ m) σ ic FIG. 3: a) Mean h τ i and effective standard deviation σ of ensemble distribution of residence timesas a function of cell length L . Cell lengths have been binned into 0 . µ m intervals. Straight linesare exponential fits to the h τ i data, τ = τ exp( − L/λ ). For cell lengths below 2 . µ m we find λ = 0 . µ m, for larger lengths λ = 2 µ m. b) Relative standard deviation σ ic = q h ¯ τ i − h ¯ τ ic i / h ¯ τ ic i of the distribution of individual cell mean residence times τ ic as a function of cell length. ω de ω D ω E j-1 j j+1 j+2 MinD-ATPMinEMinD
FIG. 4: Stochastic model of Min-protein dynamics. The cell membrane is represented by a one-dimensional array of bins that can take up MinD-ATP (green) and MinE (red). Cytosolic MinD-ATP binds at rate ω D , cytosolic MinE binds to membrane-bound MinD-ATP at rate ω E , MinD-ATP-MinE complexes detach at rate ω de . The distribution of proteins in the cytosol is assumedto be homogenous [29] and the exchange of MinD-ADP to MinD-ATP infinitely fast. On themembrane MinD-ATP diffuses and is attracted towards other MinD-ATP. For details see Materialsand Methods. IG. 5: Theoretical analysis of the Min-protein dynamics. a) Kymographs obtained from sim-ulations with ω E = 0 . − and length 2 µ m (top) and ω E = 0 . − and length 3 µ m (bottom),other parameters as below. Scale bar indicates 5min. b,c) Ensemble distribution of residencetimes for the case of stochastic switching ( ω E = 0 . − , cell length: 2 µ m) and oscillatory dynamics( ω E = 0 . − , cell length:3 µ m). Each histogram was obtained from 70 runs where the numberof MinD was drawn from Gaussian distributions with mean 1440 (b) and 2160 (c), respectivelyand 10% standard deviation. The ratio of MinD to MinE proteins was fixed at 8/3. The red linesrepresent algebraic fits τ − α for residence times τ > s . For (b) α = 2 . ± .
2, for (c) α = 4 . ± . . h τ i (green circles) and effectivestandard deviation σ (black squares) on the cell length obtained from ensemble simulations as in(b) and (c). e) The rate of ATP hydrolysis per unit length as a function of cell length. Inset:dependence of the MinE attachment rate ω E on cell length. Other parameters are ω D = 0 . s − , ω de = 0 . s − , D d = 0 . µ m /s , r d = 1 . µ m, r de = 0 . µ m, g d = 35 k B T , g de = − k B T , n max = 43and bin length l b = 33nm. IG. 6: Kymograph of GFP-MinD fluorescence and corresponding intensity curve f obtained bysubtracting the fluorescence intensity of one cell half from the intensity of the other cell half. V. SUPPLEMENTARY MATERIALA. Stochasticity of Min Switching Decreases in Individual Cells During Cell Elon-gation/Aging
We analyzed the stochasticity of MinD switching in individual cells for time intervals in whichthe cell grew less than ∆ l . For the associated Min dynamics in such an interval, we calculated theratio r ic of the standard deviation σ ic of the residence times and the corresponding mean ¯ τ ic , r ic = σ ic / ¯ τ ic . This quantity measures the stochasticity of the Min dynamics in an individual cell – the higher itis, the more stochastic is the switching behavior. In Fig. 7, we show a histogram of the numberof cells which switch stochastically/regularly in a length interval of ∆ l = 0 . µ m according to thestochasticity measure r ic .We averaged the quantity r ic over ensembles of cells in a cell length interval ∆ l = 0 . µ m.In Fig. 8, we present the average ratio h r ic i as a function of the cell length, (yellow triangles),where h . . . i denotes the average over all measured cells in the respective length interval. To get ameaningful estimate of r ic , only cells showing at least 5 switching events within a length intervalof 0 . µ m were considered. Very long residence times are thus neglected, which, together with thesmall number of events entering r ic , tends to systematically underestimate the standard deviationof the ”real” distribution. The value of r initially drops monotonically and stays constant for celllengths larger than 3.5 µ m. The ratio of the mean residence time and the corresponding standarddeviation decreases thus in individual cells.For comparison, we took the residence times of the same cells and pooled them directly accordingto the appropriate cell length interval. This means that the averaging over cell populations isdone first, before any statistical analysis. From the obtained sets of residence time distributions,we calculated again standard deviation σ and mean h τ i and their ratio r = σ/ h τ i . Again, thisquantity r drops for increasing cell length, Fig. 8 (green triangles), but takes on average highervalues than h r ic i (yellow triangles). This is because the standard deviation results in this case alsofrom cell-cell variability. . Regularly Oscillating Cells, Protein Translocation Events Are OccasionallyMissed Rarely, in regularly oscillating cells, a switching event is missed and the fluorescence maximumstays longer in one cell half. An example of such an event is shown in the top panel of Fig. 9. Thisphenomenon can also be observed in the simulated Min oscillations as shown in the bottom panelof Fig. 9. These events give rise to a small tail in the residence time distribution of oscillating cells.
C. MinE Ring Stochastically Switches in Short Cells
During our measurements of MinD fluorescence in cells, we could not assess the distributionof MinE simultaneously. Instead, we observed the distribution of fluorescently labeled MinE ina different Escherichia coli strain, WM1079 (S1). This strain produces on average larger cellsand does not show a clear transition from stochastic Min switching to Min oscillations during cellgrowth. We did however observe stochastic switching of MinE in individual cells, Fig. 10. This facttogether with known results of simultaneously recorded distributions of MinD and MinE in vivo(S2) and in vitro (S3) suggests strongly that MinE is in the same way distributed in stochasticallyswitching cells as in normally oscillating cells: A MinE ring at the rim of the MinD tube and ashallow MinE layer on the remainder of the MinD tube.
D. MinD is Partitioned Unevenly Between the Daughter Cells During Cell Divi-sion
We measured the total fluorescence intensity from daughter cells after cell division (the estimatederror is about 10%). We then calculated the associated deviation of the daughter fluorescence fromthe equipartition value, i.e., the total fluorescence intensity which would be expected if MinD waspartitioned equally to both daughter cells. Data from 37 cell divisions give a distribution of relativedeviations from the equipartition value with a standard deviation of 21%, Fig. 11. The distributionpeaks in the center meaning that a zero deviation is most likely. This is in contrast to the Mindistribution suggested by the theoretical results presented by Tostevin and Howard (S4). There,the authors probed the partitioning of Min proteins using a one-dimensional stochastic modelincluding MinD polymerization and the formation of MinD/MinE complexes on the membrane. hey simulated the division process assuming a gradually decreased diffusion constant in the cellmiddle due to cell constriction. As a result, they predicted a distribution of MinD fractions in thedaughter cells which peaked at 20% deviation from the equipartition value. E. Increasing MinE Attachment Rate can Generate a Transition from StochasticSwitching to Regular Oscillations in Cells of Fixed Length
We tested how the Min-protein dynamics changes within our stochastic one-dimensional modelwith an increasing MinE attachment rate ω E in cells of a fixed length of 2 . µ m. We chose amean MinD protein number of 1800 and a fixed MinD/MinE ratio of 8 /
3. For each value of ω E we performed 70 simulations with simulation time 160min each. To mimic the variation ofactual protein numbers in real cells, we drew the protein numbers of MinD and MinE from aGaussian distribution with 10% standard deviation. The remaining parameters of the simulationwere ω D = 0 . s − , ω de = 0 . s − , D d = 0 . µ m /s , r d = 1 . µ m, r de = 0 . µ m, g d = 35 k B T , g de = − k B T , n max = 43 and bin length l b = 33nm. As a result, we find that the stochasticityof the switching is reduced for increasing ω E coupled with a drop in mean residence time, Fig. 12.For ω E > . s − we observe regular oscillations. F. Stochastic Model Switching Can be Triggered by Other Parameters Than ω E . As mentioned in the main text and in the previous paragraph, our model predicts a transitionfrom stochastic Min switching to regular oscillations by an increase of the binding rate of MinEto the cytoplasmic membrane. In addition, other model parameters can trigger such a transitionin the dynamics (S5). It can be induced by i) an increase of the MinE/MinD ratio in the cell,by ii) a decrease in the density of binding sites on the membrane n max , by iii) a joint increase ofMinE and MinD concentrations, and by iv) a change in MinD binding or MinD/MinE detachmentfrom the membrane (data not shown). A pure increase of the cell length keeping the remainingsystem parameters constant (including protein concentrations, not protein numbers) results in atrend towards more stochastic switching for longer cells. This is the contrary effect to what weobserve experimentally. This effect can be (over)compensated though by an increase of e.g. ω E ,as has been shown in Fig. 5 (main text). Only for very short cells smaller than 1 . µ m, the model redictions deviate from the experimental data for the parameter choice presented in Fig. 5. Inthis length regime, it predicts a decrease of the mean residence time towards smaller cell lengths. G. Analysis of Alternative Models with Regard to Stochastic Switching
We also analyzed stochastic versions of models presented in (S6) and in (S3) with regard tostationary states and stochastic switching in small E. coli cells. The stochastic simulations that weperformed were one-dimensional and relied on a particle based description together with spatialbinning (∆ x =0.13 µ m). The reaction dynamics and particle exchange between the spatial binsdue to diffusion was implemented with a Gillespie algorithm (S8). The rates for the attachment,detachment and diffusion processes were chosen as indicated in the refs. S3, S6, S7.In these models, we did not find switching behavior similar to what we observed in cells. Wesystematically scanned the neighborhood of the parameter space of the models given in (S6, S7)and (S3), respectively, including independent increase of the parameters up to a factor of 2 anddecline up to a factor of 0.5. Particle numbers were varied in the simulations up to 20% from theoriginally given values.We did not chose a broader range, because the particle densities are chosenaccording to experimental findings (S2).We always tested cell lengths between 2 and 6 microns.The solutions we found for the models presented in (S3, S6, S7) were either oscillations or noisyhomogeneous states.[1] B. D. Corbin, X. C. Yu, and W. Margolin, EMBO J. , 1998 (2002).[2] Y. Shih, X. Fu, G. King, T. Le, and L. Rothfield, EMBO J. , 3347 (2002).[3] M. Loose, E. Fischer-Friedrich, J. Ries, K. Kruse, and P. Schwille, Science , 789 (2008).[4] F. Tostevin and M. Howard, Phys. Biol. , 1 (2006).[5] G. Meacci, Min oscillations in Escherichia coli. Physical Aspects: Experiments and TheoreticalDescriptions (VDM, Saarbr¨ucken, Germany), pp. 67-68, (2009).[6] K. C. Huang, Y. Meir, and N. S. Wingreen, Proc. Natl. Acad. Sci. USA , 12724 (2003).[7] K. C. Huang and N. S. Wingreen, Phys. Biol. , 229 (2004).[8] D. Gillespie, J. Phys. Chem. , 23402361 (1977). µ m) FIG. 7: Histogram of Min switching behavior of cells in a given cell length interval. The red columncounts stochastically switching cells whereas the blue column represents the regularly switchingcells. The decision whether a cell switches stochastically or not was taken as follows: residencetimes from a cell where sorted according to the cell length interval which the cell was in at thebeginning of the residence time. If there were more than 3 residence times associated to a givencell length interval, with mean smaller than 80s, than we calculated r ic = σ ic / ¯ τ ic , where ¯ τ ic is themean and σ ic the standard deviation. If this quantity was greater than one, the cell was decidedto switch stochastically in this cell length interval. If there were less than 3 residence times inthe corresponding length interval or if the mean ¯ τ ic was greater than 80s, then we looked at themaximal residence time occurring. If it was greater than 100s, we labeled the cell as stochasticallyswitching in the respective length interval. .0 2.5 3.0 3.5 4.0 cell length ( µ m) σ/<τ> , for pooled data < σ ic / τ ic > , for individual cellsStochasticity measure of MinD switching FIG. 8: The evolution of stochasticity of Min dynamics with increasing cell length. Yellow triangles:population average of r ic , where r ic denotes the ratio of the standard deviation to the mean ofresidence times for an individual cell in a time intervall in which the cell grew less than 0 . µ m,see text for details. Green triangles: The same data of residence times were first pooled for cellsin the same length interval. Then, ratio r = σ/ h τ i of the standard deviation to the mean of theseresidence times were calculated.FIG. 9: Examples of missed switching events in otherwise regularly oscillating cells. The top panelshows a kymograph from a real cell. The bottom panel was obtained from a simulation. IG. 10: Kymograph of stochastic switching of fluorescent labeled MinE in two cells of the
E.coli strain WM1079. The kymographs cover a time span of 40 minutes. -40 -20 0 20 4051015 Relative deviation of MinD concentrationfrom equipartition value in daughter cells % FIG. 11: Histogram of the deviation from the equipartition value I eq = ( I D int + I D int ) / I Dint in a daughter cell after cell division. Given is the relative deviation I Dint /I eq in percent. The red part of the columns indicates the number of daughter cells whichswitched stochastically after division and the blue part of the columns those which oscillated regu-larly. Note that, since we assume that no GFP-MinD are lost during division, the overall histogramhas to be symmetric around zero. ean residence time (s)Standard deviation (s) FIG. 12: Mean residence time (green dots) and standard deviation (black squares) in a simulatedcell for increasing attachment rate ω E of MinE but at constant cell length. The MinD/MinE ratio iskept constant as 8/3. Standard deviation and residence time both fall for increasing concentrations.Their ratio also decreases, implicating reduced stochasticity of Min dynamics.of MinE but at constant cell length. The MinD/MinE ratio iskept constant as 8/3. Standard deviation and residence time both fall for increasing concentrations.Their ratio also decreases, implicating reduced stochasticity of Min dynamics.