Jean-Marc Bassat
email: bassat@icmcb-bordeaux.cnrs.fr

Mońica Burriel
email: monicaburriel@gmail.com.

Olivia Wahyudi

Reḿi Castaing

Monica Ceretti

Philippe Veber

Isabelle Weill

Antoine Villesuzanne

Jean-Claude Grenier

Werner Paulus

John A Kilner

Anisotropic Oxygen Diffusion Properties in Pr 2 NiO 4+δ and Nd 2 NiO 4+δ Single Crystals

The anisotropy of the oxygen anionic conductivity was measured for two mixed ionic electronic conducting (MIEC) oxides with the 2D K 2 NiF 4 -type structure, i.e., Nd 2 NiO 4+δ and Pr 2 NiO 4+δ , using high quality single crystals. Measurements of the oxygen diffusivity and surface exchange performed parallel and perpendicularly to the [001] direction, from 450 to 700 °C, using the isotope exchange depth profile (IEDP) technique, combining 16 O/ 18 O exchange and secondary ion mass spectroscopy (SIMS) are reported. For both materials the diffusion is about 3 orders of magnitude higher along the (a,b)-plane compared to the perpendicular (caxis) direction. These values are among the highest when compared to several state-of-the-art MIEC materials. The diffusion along the (a,b)-plane for Pr 2 NiO 4+δ is higher than that of Nd 2 NiO 4+δ due to a much lower diffusion activation energy (0.5 and 1.4 eV for Pr 2 NiO 4+δ and Nd 2 NiO 4+δ , respectively). A large anisotropy is also observed in the surface exchange coefficient (k*) values for both materials, with (a,b)-plane coefficients being 1 to 1.5 orders of magnitude larger than those for the c-axis.

INTRODUCTION

The development of new oxide materials for application in high temperature electrochemical energy conversion devices continues to be an area of strong interest. Of the many materials under development, mixed ionic electronic conductors (MIEC) are the materials that show the most fascinating aspects of solid state chemistry due to their high degree of nonstoichiometry. In the past, attention was focused on perovskite structured materials such as La 1-x Sr x CoO 3-δ or La 1-x Sr x Co 1-y Fe y O 3-δ that show high degrees of oxygen deficiency. [START_REF] Sun | Cathode Materials for Solid Oxide Fuel Cells: a Review[END_REF][START_REF] Mai | Ferrite-Based Perovskites as Cathode Materials for Anode-Supported Solid Oxide Fuel Cells: Part I. Variation of Composition[END_REF] Although these materials display high performances, e.g., as fuel cell cathodes, due to the high mobility of the resulting oxygen vacancies, there has been a recognition that the materials suffer from a number of problems including high chemical expansion coefficients, susceptibility to Cr poisoning, and long-term stability problems due in part to the high degree of oxygen deficiency and aliovalent substitution, particularly by strontium. [START_REF] Tucker | A Fundamental Study of Chromium Deposition on Solid Oxide Fuel Cell Cathode Materials[END_REF] In the past decade this has sparked interest in alternative materials that display nonstoichiometry due to oxygen excess accommodated by oxygen interstitial species such as La 2 NiO 4+δ and Pr 2 NiO 4+δ , oxides with the K 2 NiF 4 -type structure (first member of the so-called Ruddlesden-Popper series). [START_REF] Tarancoń | Advances in Layered Oxide Cathodes for Intermediate Temperature Solid Oxide Fuel Cells[END_REF] These structures consist of offset ABO 3 perovskite layers interspersed by AO rocksalt layers. The oxygen interstitials are thought to have a highly anisotropic mobility migrating by an interstitialcy mechanism in the rocksalt layers. Oxygen intercalated K 2 NiF 4type oxides all show large displacements of the apical oxygen atoms (O ap ), the interstitial oxygen atoms (O int ) being located close to them, i.e., in the (1/4, 1/4, 1/4) site with respect to the Fmmm-type-cell. As a consequence of the resulting short O int -O ap oxygen distances (about 2.1 Å), the adjacent apical oxygen atoms are shifted away from their average positions in order to form a symmetrically enlarged O int (O ap ) 4 tetrahedron, thus leading to strong local distortions. [START_REF] Jorgensen | Structure of the Interstitial Oxygen Defect in La 2 NiO 4+δ[END_REF] It has been recently shown for La 2 CuO 4.07 that these O ap displacements are at least partially of dynamic origin, resulting in an apical oxygen position that is strongly delocalized covering a circle of about 1 Å in diameter and thus strongly favoring displacements toward the interstitial lattice sites. [START_REF] Villesuzanne | On the Role of Lattice Dynamics on Low Temperature Oxygen Mobility in Solid Oxides: a Neutron Diffraction and First-Principles Investigation of La 2 CuO 4+δ[END_REF] A similar scenario has been found for La 2 NiO 4+δ . [START_REF] Paulus | Neutron Diffraction Studies of Stoichiometric and Oxygen Intercalated La 2 NiO 4 Single Crystals[END_REF] The dynamic aspect of these displacements becomes important especially at low temperatures, as the related diffusion mechanism has been found to be phonon assisted involving low energy lattice modes, similar to what has been observed for oxides with brownmillerite-type structure. [START_REF] Paulus | Lattice Dynamics to Trigger Low Temperature Oxygen Mobility in Solid Oxide Ion Conductors[END_REF] For K 2 NiF 4 -type oxides the presence of interstitial oxygen atoms seems to be necessary for oxygen diffusion, as it has been modeled for La 2 NiO 4+δ , thus confirming the necessity of dynamic displacements of the apical oxygen atoms, as experimentally found by high-resolution neutron diffraction. The shallow diffusion potential of the apical oxygen atoms formed by this dynamic disorder scenario should be influenced especially by the c-lattice parameter, as significant variations are observed ranging from 13.15 Å for stoichiometric La 2 CuO 4.0 and 12.52 Å for La 2 NiO 4.0 to only 12.10 Å for Pr 2 NiO 4.0 .

In this way the aim of this study is to carefully measure the oxygen diffusion coefficients and associated activation energies for single crystals of both Pr 2 NiO 4+δ and Nd 2 NiO 4+δ along the two main crystallographic orientations in this highly anisotropic structure, in order to compare these values within this series of compounds.

EXPERIMENTAL SECTION

2.1. Crystal Growth. Single crystals of Pr 2 NiO 4+δ and Nd 2 NiO 4+δ were synthesized using the floating zone (FZ) method by means of an optical mirror furnace (NEC SC3-MDH11020). Specifically, the traveling solvent floating zone (TSFZ) technique in a vertical configuration was used. The apparatus consists of two gold-coated ellipsoidal mirrors that focus the light from halogen lamps onto a vertically held rodshaped sample; temperatures above 2200 °C were reached, high enough to melt the polycrystalline rods of the nickelates.

First, polycrystalline powders of both materials were prepared by the conventional solid state method. The starting materials used are neodymium oxide, Nd 2 O 3 (99.9%, Sigma-Aldrich), praseodymium oxide, Pr 6 O 11 (99.9%, Alfa-Aesar), and nickel oxide, NiO (99%, Alfa-Aesar). Because of the hygroscopic nature of Nd 2 O 3 and Pr 6 O 11 , the starting oxide powders were first heated in vacuum at 1173 K overnight to remove any volatile impurities. The NiO powder was also heated at 353 K in an oven overnight. The reactants were then mixed in a stoichiometric ratio and ground in acetone medium. The mixture was first heated at 1523 K for 24 h. After cooling down to room temperature, the mixture was ground and made into pellets (about 1 g each, Φ 10 mm), which were sintered at 1523 K for more than 24 h. The last step was repeated two times, which leads to a total of about 72 h of sintering. The purity of the Ln 2 NiO 4+δ powder samples was verified by conventional X-ray diffraction (XRD). Finally, seed and feed rods were prepared by pressing the powders under hydrostatic pressure (about 10 MPa) and then sintering again. The rods have to be uniform in length, diameter, and density.

The first step of crystal growth is to melt the starting material and then to connect, via the liquid phase called the molten zone, the seed and feed rods. The molten zone is thus located in between the two rods that rotate (about 30 rpm) in opposite direction, and it is then moved (about 2-3 mm/h) along the feed rod through the focus in order to crystallize the liquid phase on the seed. The stability of the molten zone is controlled by visual observation and manual adjustments, with a CDD camera. Since the crystallization was found to be more difficult specifically for Pr 2 NiO 4+δ , a seed crystal was used for the growth, instead of a polycrystalline sample, as was the case for Nd 2 NiO 4+δ . In contrast to what has been observed for La 2 NiO 4+δ , [START_REF] Dembinski | Crystal Growth of La 2 NiO 4 by the Floating Zone Method with a CW CO 2 Laser: Preliminary Characterizations[END_REF] the partial volatility of nickel creates instability during crystal growth. Therefore, the polycrystalline powders used to make the rods were synthesized adding excess of NiO; repeated attempts led to determine the optimum excess to be 2%.

Single crystals for both compositions were successfully grown under an O 2 atmosphere. Their length was usually about 80-95 mm with diameter of 6.5 mm, as shown in Figure 1.

X-ray diffraction data were taken on a small section of both crystals that was crushed into fine powders. It was found that the crystals consist of a mixture of two phases with similar lattice parameters, both belonging to the Fmmm space group. After heat treatment at about 700 K for four days, all samples were found to be single phase (space group Fmmm), as illustrated for Nd 2 NiO 4+δ in Figure 2.

The chemical composition and homogeneity of the crystals were checked by using EDS analysis and SEM observations. The cationic ratio Nd (or Pr) over Ni measured at each point of the analyzed cross-section was close to 2. The crystallinity of the single crystals was checked by Laue X-ray diffraction. The oxygen content was determined by two ways, i.e., (i) thermogravimetric analysis (TGA) carried out under a reducing atmosphere (flowing 5% H 2 in Ar); for this purpose about 100 mg of powder was heated up to 900 °C with heating rate of 1 °C/min, and (ii) iodometric titration of powder obtained by crushing a piece of the single crystal. Both methods give the amount of present Ni 3+ and thereby, by deduction, the oxygen content of the sample at room temperature. Using both methods, the oxygen excesses were estimated to be δ = 0.20 and δ = 0.22 for Nd 2 NiO 4+δ and Pr 2 NiO 4+δ , respectively. These high δ values, together with a mixed Ni 2+ /Ni 3+ valence for nickel, lead to the well-known MIEC properties for both materials. For both materials, oxygen will be progressively lost as the temperature is raised. Nevertheless relatively high oxygen nonstoichiometry values are maintained at high temperatures (e.g., δ = 0.12 for Pr 2 NiO 4+δ and δ = 0.15 for Nd 2 NiO 4+δ at 700 °C), as reported by Boehm et al. [START_REF] Boehm | Oxygen Diffusion and Transport Properties in Non-Stoichiometric Ln 2 NiO 4+δ Oxides[END_REF] Thus, it is expected that the mixed conducting properties will be maintained at the electrode operating temperatures (T > 500 °C).

2.2. Crystal Orientation. Because of the very high expected anisotropy ratio of the oxygen diffusion in these nickelates with regard to previous results, [START_REF] Burriel | Anisotropic Oxygen Diffusion Properties in Epitaxial Thin Films of La 2 NiO 4+δ[END_REF][START_REF] Bassat | Anisotropic Ionic Transport Properties in La 2 NiO 4+δ Single Crystals[END_REF] it has been shown that it is essential to have a misalignment angle, Θ, of the crystallographic faces lower than 1.5°in order to perform accurate measurements, especially for the short diffusion lengths along the c-axis. [START_REF] Gauquelin | Impact of the Structural Anisotropy of La 2 NiO 4+δ on High Temperature Surface Modifications and Diffusion of Oxygen[END_REF] In this way, single crystals fixed on a HUBER 1006 goniometer head were orientated along the main crystallographic axes using the Laue back scattering method. A 5203.10 HUBER 2-circle segment rotation device was used to correct the crystal misalignment during the orientation process. The diffraction patterns were collected either on KODAK photographic films or with a CCD-camera device using polychromatic X-rays Bremsstrahlung (white beam) supplied by a copper anticathode (40 mA; 40 kV). Experimental patterns were then indexed with the Orient-express software allowing us to orientate crystals with an accuracy of 0.01°.

The cutting of the crystals was performed with a diamond wire saw ESCIL equipped with a 2 circle Microcheck orientation system on which the goniometer head can be positioned.

Finally, after a slight polishing of the edges, the orientation of the cut faces was checked by the Laue diffraction method using a silicon single crystal as a reference. The misalignment between the (511) crystal face and its mechanical plane was considered to be less than 0.01°for the reference single crystal. This allowed to reproduce the positioning of the crystal cut faces parallel to the (511) silicon face and perpendicular to the X-rays beam with an absolute accuracy being in each case, thanks to our updated equipment, less than 1°along the 2circle-segment axis, which satisfies the initial requirement for performing the diffusion measurements.

2.3. Oxygen Isotopic Exchange and Secondary Ion Mass Spectrometry Measurements. The oxygen diffusion and surface exchange phenomena, characterized by the D* and k* coefficients, was determined by measuring the diffusion of an oxygen isotope ( 18 O) using the so-called isotope exchange depth profile (IEDP) technique. [START_REF] De Souza | Determining Oxygen Isotope Profiles in Oxides with Time-of-Flight SIMS[END_REF] The secondary ion mass spectrometry (SIMS) analyses were carried out in two different laboratories: Institut de Chimie de la Matiere Condenseé de Bordeaux (ICMCB) and Imperial College London (IC), while the oxygen exchanges were performed at ICMCB only. Prior to the oxygen exchange, polished crystal samples were first thermodynamically equilibrated in a quartz tube, under a partial pressure (0.21 atm) of natural oxygen ( 16 O, 99.75%) by annealing at given temperatures corresponding to the exchange temperatures, during several hours. To ensure that the material had reached thermodynamic equilibrium at the operating temperature and pressure, the duration of this anneal step was about 12 h, which is much longer than the isotope exchange length of time. At the end of the equilibration stage, the sample was quenched down to room temperature. The atmosphere in the tube was then replaced by [START_REF] Minervini | Oxygen Migration in La 2 NiO 4+δ[END_REF] O (97% from Euriso-top), and the samples were exchanged at the same temperature and same partial pressure (0.21 atm) for a given time. This exchange time was estimated on the basis of previous measurements carried out on La 2 NiO 4+δ single crystals and depends on the crystallographic direction and temperature. [START_REF] Bassat | Anisotropic Ionic Transport Properties in La 2 NiO 4+δ Single Crystals[END_REF] Isotopic exchange experiments were performed at several temperatures in the range 450-700 °C.

After the isotope exchange, the samples were first cut perpendicular to the exchanged surface, parallel to the direction of interest, and the cross-section was polished prior to the measurement. The samples were then embedded in Woods alloy (Goodfellow) and polished using abrasive paper of decreasing roughness (until 0.25 μm). The 18 O and [START_REF] Chroneos | Oxygen Diffusion in Solid Oxide Fuel Cell Cathode and Electrolyte Materials: Mechanistic Insights from Atomistic Simulations[END_REF] O concentration depth profiles were measured by SIMS. The SIMS analyses performed at ICMCB were done using a Cameca IMS 6F device with Cs + ions as incident ions. Two different modes of analysis were used: the linescan mode, when the penetration depth profiles are higher than 250 μm, and the surface scanning mode, when profiles are less than 250 μm. When surface scanning mode was used, an image over a zone of 50 × 50 μm 2 was digitized to obtain accurate profile determinations. All the exchanged samples studied at IC were measured on a ToF-SIMS 5 instrument (ION-TOF GmbH, Munster, Germany) equipped with bismuth pulsed liquid metal ion gun (25 keV) incident at 45°(analysis beam), using the burst alignment mode (7 or 8 peaks). The imaging mode, described in detail by De Souza et al., [START_REF] De Souza | Determining Oxygen Isotope Profiles in Oxides with Time-of-Flight SIMS[END_REF] was used for the measurement of the oxygen diffusion. This mode was selected as it permits the visualization of the distribution of the species in the analyzed area using a high lateral resolution. In both cases, the main experimental difficulty being to determine the perpendicular diffusion (along the c-axis) without overestimating it, it was of high importance to accurately orientate the crystals as previously stressed.

RESULTS

Since the sample thickness is sufficiently large in comparison to the [START_REF] Minervini | Oxygen Migration in La 2 NiO 4+δ[END_REF] O tracer penetration depth, the measured penetration profile can be fitted using a solution for Fick's second law proposed by Crank, [START_REF] Crank | The Mathematics of Diffusion[END_REF] applied for diffusion into a semi-infinite medium. It is assumed that the source concentration is constant with respect to time and that the oxygen flux is limited by the surface exchange reactions. Mathematically, the normalized concentration C(x) of [START_REF] Minervini | Oxygen Migration in La 2 NiO 4+δ[END_REF] O is

= * - * + * * + * * ⎜ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎧ ⎨ ⎩ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎞ ⎠ ⎫ ⎬ ⎭ C x x D t k D x k D t x D t k D D t ( ) erfc 2 exp erfc 2 2
where x is the profile depth, D* is the tracer oxygen diffusion coefficient, k* is the tracer surface exchange coefficient, and t is the exchange duration. Figure 3 shows, as an example, the color gradation in the concentration of [START_REF] Chroneos | Oxygen Diffusion in Solid Oxide Fuel Cell Cathode and Electrolyte Materials: Mechanistic Insights from Atomistic Simulations[END_REF] O and [START_REF] Minervini | Oxygen Migration in La 2 NiO 4+δ[END_REF] O obtained by the surface scanning method (here performed along the c-axis) on a Nd 2 NiO 4+δ crystal exchanged at 703 °C during 856 s. Normalized [START_REF] Minervini | Oxygen Migration in La 2 NiO 4+δ[END_REF] O concentration profiles recorded along the two main crystallographic directions, on the same crystal using the same exchange conditions, are also compared in Figure 3.

Figure 4 summarizes the measured D* ab and D* c values for Nd 2 NiO 4+δ as a function of temperature. For the whole measured temperature range, the oxygen diffusion in the (a,b)plane of the single crystal is about 3 orders of magnitude higher than the corresponding value along the c-axis. The value of the diffusion coefficient for ceramic samples is close to the (a,b)plane one. [START_REF] Boehm | Oxygen Diffusion and Transport Properties in Non-Stoichiometric Ln 2 NiO 4+δ Oxides[END_REF] This suggests that (a,b)-plane diffusion process dominates for polycrystalline materials. The activation energy of the oxygen diffusion does not depend on the crystallographic orientation, being about E a = 1.4 (±0.2) and 1.3 (±0.2) eV for the (a,b)-plane and c-axis, respectively, which are unexpectedly high values, particularly for the (a,b)-plane. Finally, the diffusion in the (a,b)-plane, for example, at T = 600 °C, appears to be larger than the ones measured for most of the current MIEC materials studied as SOFC cathodes. [START_REF] Chroneos | Oxygen Diffusion in Solid Oxide Fuel Cell Cathode and Electrolyte Materials: Mechanistic Insights from Atomistic Simulations[END_REF] The diffusion in the (a,b)-plane is larger for Pr 2 NiO 4+δ compared to Nd 2 NiO 4+δ (Figure 5) and La 2 NiO 4+δ , [START_REF] Burriel | Anisotropic Oxygen Diffusion Properties in Epitaxial Thin Films of La 2 NiO 4+δ[END_REF][START_REF] Bassat | Anisotropic Ionic Transport Properties in La 2 NiO 4+δ Single Crystals[END_REF] especially in the low temperature range. This is due to the small activation energy of the diffusion in Pr 2 NiO 4+δ , which is about one-half the one for the neodymium compound (0.67 (±0.1) eV and 1.4 (±0.2) eV, respectively). Both sets of data (measurements performed at ICMCB and IC) give results that are in good agreement. Once again the value of the diffusivity for the ceramic D*(ceramic) is close to the diffusivity along the (a,b)-plane (D* ab ) for all temperatures, while the activation energy of the oxygen diffusion in the ceramics is intermediate between the values measured for the two main crystallographic orientations of the single crystal.

DISCUSSION

In this kind of oxygen overstoichiometric oxides, it seems to be well established that the key point for the oxygen mobility is the intrinsic disorder occurring between both the apical and interstitial sites. [START_REF] Chroneos | Anisotropic Oxygen Diffusion in Tetragonal La 2 NiO 4+δ : Molecular Dynamics Calculations[END_REF] Indeed the diffusion mechanism along the (a,b)-plane is not, as might be first expected, a direct displacement of the oxygen ions along interstitial site paths but is rather a so-called interstitialcy (or push-pull) mechanism in which the labile apical oxygen is also involved: first, the apical oxygen moves into the interstitial site in the rock-salt-type layer; then, in a second step, the resulting oxygen vacancy at apical site is filled by a nearby interstitial oxygen. This mechanism was suggested by atomistic studies on La 2 NiO 4+δ 18 and has been evidenced latter by neutron diffraction on (Pr 0.9 La 0.1 ) 2 (Ni 0.74 Cu 0.21 Ga 0.05 )O 4+δ at high temperatures. [START_REF] Yashima | Structural Disorder and Diffusional Pathway of Oxide Ions in a Doped Pr 2 NiO 4 -based Mixed Conductor[END_REF] According to molecular dynamics (MD) calculations, [START_REF] Chroneos | Anisotropic Oxygen Diffusion in Tetragonal La 2 NiO 4+δ : Molecular Dynamics Calculations[END_REF][START_REF] Chroneos | Oxygen Transport in Perovskite and Related Oxides: A Brief Review[END_REF] the corresponding activation energy would range between 0.5 and 0.7 eV, in good agreement with our measurements for La 2 NiO 4+δ . [START_REF] Bassat | Anisotropic Ionic Transport Properties in La 2 NiO 4+δ Single Crystals[END_REF] For Pr 2 NiO 4+δ , MD simulations concluded that the activation energy of the oxygen diffusion varies between 0.49 and 0.64 eV, depending on the content of oxygen overstoichiometry. [START_REF] Parfitt | Molecular Dynamics Study of Oxygen Diffusion in Pr 2 NiO 4+δ[END_REF] Such values are also in good agreement with our measurements (Figure 5).

On the contrary, the value of E a (a,b) measured for Nd 2 NiO 4+δ is surprisingly high, especially regarding the oxygen diffusion values themselves that are very close to those measured for Pr and La at the highest temperatures (T ≥ 600 °C). Indeed, the oxygen diffusion coefficients at T = 700 °C are quite similar in both materials (D* ≈ 1-2.10 -7 cm 2 •s -1 ), while they differ by 1 order of magnitude at T = 500 °C (D* ≈ 3.10 -8 cm 2 •s -1 for Pr 2 NiO 4+δ vs D* ≈ 3.10 -9 cm 2 •s -1 for Nd 2 NiO 4+δ ). This means that the oxygen mobility is less favored at low temperatures in Nd 2 NiO 4+δ . As a difference between both materials, the possible mixed valence state between 3+ and 4+ only for Pr cation could be considered. However, to the best of our knowledge, Pr 4+ has never been experimentally observed in Pr 2 NiO 4+δ . Our recent measurements performed by XANES confirmed that only Pr 3+ can be evidenced. This is in agreement with general steric considerations: because of its too small cationic size, Pr 4+ cannot be stabilized in this K 2 NiF 4 structure. However, an explanation can be attempted considering the results obtained by Bhavaraju et al.: [START_REF] Bhavaraju | Electrochemical Intercalation of Oxygen in La 2 NiO 4+x (0 ≤ x ≤ 0.18) at[END_REF] using electrochemical intercalation experiments, the authors evidenced that stoichiometric Nd 2 NiO 4.00 is obtained by the initial electrochemical reduction of the starting material Nd 2 NiO 4.18 , but further reoxidation leads to a composition of only Nd 2 NiO 4.10 (oxygen overstoichiometric). On the contrary, La 2 NiO 4+δ exhibits a full reversibility over a range extending up to 0.26. [START_REF] Grenier | Electrochemical Oxidation: a New Way for Preparing High Oxidation States of Transition Metals[END_REF][START_REF] Grenier | Electrochemical Synthesis: Oxygen Intercalation[END_REF] Full reversibility has also been reported for La 2 CuO 4+δ in the total range of δ max = 0.07. [START_REF] Paulus | In Situ Neutron Diffraction Studies on the Electrochemical Oxidation of Polycrystalline La 2 CuO 4[END_REF] Such a difference in behavior between both materials is assigned to the effect of structural differences. Comparing the structures of Nd 2 NiO 4+δ to those of La 2 CuO 4+δ and La 2 NiO 4+δ indicates that there might be a critical value for the c lattice parameter, below which oxygen intercalation is no longer reversible. From the refined structure, it is clearly seen that the short c parameter results in both a short interoctahedral space (compared to the corresponding La-nickelates) and in a significantly distorted tetrahedral site for interstitial oxygen. Both effects could lead to the hindering of the oxygen diffusion at low temperatures.

Furthermore, it is assumed that an interplay between lattice dynamics and structural instabilities in K 2 NiF 4 -type structures exists, which is intimately related to the different oxygen intercalation behaviors of Nd 2 NiO 4+δ , compared to La 2 NiO 4+δ and La 2 CuO 4+δ . Reduced apical oxygen mobility at moderate temperature would be due to the absence of low energy specific phonon modes. As already seen for La 2 CuO 4+δ , [START_REF] Villesuzanne | On the Role of Lattice Dynamics on Low Temperature Oxygen Mobility in Solid Oxides: a Neutron Diffraction and First-Principles Investigation of La 2 CuO 4+δ[END_REF] low energy phonon modes become a general prerequisite for oxygen mobility at moderate (i.e., room) temperature, when thermal vibrations are less significant. Accordingly, some of us demonstrated both experimentally and theoretically that the oxygen mobility occurring at moderate temperature in brownmillerite-type structures (such as SrCoO 2.5 and SrFeO 2.5 ) is related unambiguously to low energy phonon modes. [START_REF] Paulus | Lattice Dynamics to Trigger Low Temperature Oxygen Mobility in Solid Oxide Ion Conductors[END_REF] Although brownmillerite-and K 2 NiF 4 -type structures are different and have low energy phonons of different nature, as shown by previous calculations, [START_REF] Villesuzanne | On the Role of Lattice Dynamics on Low Temperature Oxygen Mobility in Solid Oxides: a Neutron Diffraction and First-Principles Investigation of La 2 CuO 4+δ[END_REF][START_REF] Paulus | Lattice Dynamics to Trigger Low Temperature Oxygen Mobility in Solid Oxide Ion Conductors[END_REF] it is believed that such modes play a significant role for oxygen mobility in Pr 2 NiO 4+δ ; however, their effect might be less pronounced in the case of Nd 2 NiO 4+δ related to the slightly smaller c-axis parameter compared to the Pr homologue. This assumption has to be checked by performing electronic structure and lattice dynamics calculations, in situ electrochemical oxygen intercalation coupled with neutron diffraction measurements, as well as careful inelastic neutron diffusion experiments on single crystals. Such differences in terms of interlayer distances, lattice dynamics, and oxygen disorder, as well as the inter-relation between these three factors, could explain the lower oxygen mobility along the (a,b)-plane in Nd 2 NiO 4+δ at low temperatures.

On the contrary, the oxygen diffusion coefficients of Pr 2 NiO 4+δ and Nd 2 NiO 4+δ measured along the c-axis have very similar values over the whole temperature range, with exactly the same activation energy. The migration along the caxis, occurring via a direct vacancy mechanism, requires significantly higher activation energy. Thus, to give a better insight of the exact diffusion mechanism and pathways along the c-axis, lattice dynamics studies are needed, for instance, by in situ measurements during redox reactions in Pr 2 NiO 4+δ and Nd 2 NiO 4+δ single crystals. Moreover, the delocalization of the apical oxygen preferentially in the (a,b)-plane of the structure should be evidenced from neutron single crystal diffraction data using a maximum entropy analysis. This work is in progress.

The thermal evolution of the surface exchange coefficients k* is shown in Figure 6 for both materials. A large anisotropy is observed for both materials over the whole temperature range, with (a,b)-plane coefficients being 1 to 1.5 orders of magnitude larger than the c-axis values. When compared with the data obtained for ceramic samples, 10 the polycrystalline surface exchange coefficients are closer to the k* values for the (a,b)plane, which would mean that the crystallographic orientations with the highest surface exchange coefficient are the ones that dominate the process in polycrystalline materials. The activation energies of the surface exchange are high in both directions (>1.3 eV, except along the c-direction for Pr 2 NiO 4+δ in which it is quite low, 0.8 eV), without clear trends as a function of the rare-earth or the crystallographic direction.

Finally, Pr 2 NiO 4+δ presents the highest values of both k* and D*, which leads to a material with very good electrochemical performances that can be used as an oxygen electrode either in a SOFC device operating at 600 °C, [START_REF] Ferchaud | High Performance Praseodymium Nickelate Oxide Cathode for Low Temperature Solid Oxide Fuel Cell[END_REF] or in a high temperature steam electrolysis (HTSE) one at 700-800 °C, [START_REF] Ogier | Enhanced Performances of Structured Oxygen Electrodes for High Temperature Steam Electrolysis[END_REF] as has been previously reported in the literature.

CONCLUSIONS

Ln 2 NiO 4+δ (Ln = La, Pr, Nd) rare-earth nickelates are considered as promising oxygen electrode materials. Among them, Nd 2 NiO 4+δ and especially Pr 2 NiO 4+δ exhibit the most interesting electrochemical properties. Knowledge of the relevant oxygen transport parameters (i.e., oxygen transport coefficients D* and surface exchange constants k*) in such oxides with MIEC properties is of high importance, especially for understanding the oxygen transport mechanisms in these materials with highly anisotropic properties.

By experimentally tracing the isotopic oxygen ion concentration as a function of depth using the IEDP/SIMS technique, it was possible to determine for the first time the oxygen transport coefficients D* and surface exchange constants k*, along the two main crystallographic directions (the (a,b)-plane and the c-axis direction) for both materials. Reliable data were obtained due to the very high quality of the single crystals and to the crystallographic orientations, which were beforehand performed with an accuracy of less than 1°along the two considered directions. Over the whole temperature range (450-700 °C) the oxygen diffusion along the (a,b)-plane is about 3 orders of magnitude larger than along the c-axis, for both materials. The value of the diffusion coefficient for ceramic samples is close to the value measured for the (a,b)-plane, suggesting that (a,b)-plane diffusion process dominates for ceramic materials. As a general trend, the diffusion along the (a,b)-plane is larger compared to most of the state-of-the-art MIEC materials studied as SOFC cathodes. Some differences were found between both materials when comparing the oxygen diffusion coefficients D*. The diffusion along the (a,b)-plane is larger for Pr 2 NiO 4+δ compared to Nd 2 NiO 4+δ . Moreover, the activation energy of the diffusion in Pr 2 NiO 4+δ is about half the value for the neodymium, which is surprisingly high (∼0.7 and 1.4 eV, respectively). On the contrary, the oxygen diffusion coefficients of Nd 2 NiO 4+δ and Pr 2 NiO 4+δ measured along the c-axis have very similar values for the whole temperature range, with similar activation energy (E a ≈ 1.1-1.3 eV) for the oxygen diffusion process.

A large anisotropy is also observed for the k* values for both materials, with (a,b)-plane coefficients 1 to 1.5 orders of magnitude larger than c-axis coefficients. Once again the polycrystalline surface exchange coefficients are closer to the k* values for the (a,b)-plane, indicating that the oxygen exchange process is dominating in this direction.

Finally, the material having the highest k* and D* values, i.e., Pr 2 NiO 4+δ , exhibits very good electrochemical performances as oxygen electrode in SOFC or HTSE devices.

In order to get a better insight in the diffusion mechanism, we are now investigating structural studies of Pr 2 NiO 4+δ and Nd 2 NiO 4+δ on single crystals by neutron diffraction, with the aim to directly access the delocalization of the apical oxygen atoms for comparison with previous studies on La 2 CuO 4.07 .

Figure 1 .

 1 Figure 1. Single crystals of Nd 2 NiO 4+δ (top) and Pr 2 NiO 4+δ (bottom) grown by the floating zone method.

Figure 2 .

 2 Figure 2. XRD of a crushed Nd 2 NiO 4+δ (δ = 0.22) single crystal after heat treatment at 700 K.

Figure 3 .

 3 Figure 3. (a)[START_REF] Chroneos | Oxygen Diffusion in Solid Oxide Fuel Cell Cathode and Electrolyte Materials: Mechanistic Insights from Atomistic Simulations[END_REF] O and[START_REF] Minervini | Oxygen Migration in La 2 NiO 4+δ[END_REF] O SIMS images (along the c-axis) showing the color gradation in the concentrations obtained by the surface scanning method, (b) normalized[START_REF] Minervini | Oxygen Migration in La 2 NiO 4+δ[END_REF] O concentration profile extract along the (a,b)-plane direction (recorded using the line-scanning method), and (c) normalized[START_REF] Minervini | Oxygen Migration in La 2 NiO 4+δ[END_REF] O concentration profile extract along the c-axis (obtained from previous images). Both profiles correspond to the Nd 2 NiO 4+δ crystal exchanged at 703 °C during 856 s.

Figure 4 .

 4 Figure 4. D* vs 1000/T for Nd 2 NiO 4+δ single crystal (solid lines) and comparison with polycrystalline ceramics from ref 10 (dotted line).

Figure 5 .

 5 Figure 5. D* vs 1000/T for Pr 2 NiO 4+δ single crystal (solid lines) and comparison with polycrystalline ceramics from ref 10 (dotted line).

Figure 6 .

 6 Figure 6. k* vs 1000/T for Nd 2 NiO 4+δ and Pr 2 NiO 4+δ single crystals.

This research was partly supported by a Marie Curie Intra European Fellowship within the seventh European Community Framework Program (PIEF-GA-2009-252711) (for M.B.). J.-M.B., O.W., M.C., I.W., A.V., and W.P. are grateful for financial support obtained through the ANR project FUSTOM (ANR-08-BLAN-0069).


Supratim Sengupta
email: supratim.sen@iiserkol.ac.in

Julien Derr
email: julien.derr@univ-paris-diderot.fr

Anirban Sain
email: asain@phy.iitb.ac.in

Andrew D Rutenberg

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

Keywords: numbers: 87.16.A-, 87.17.Ee, 87.18.Tt 26 spatio-temporal oscillation, stochastic modelling, polymerization, stuttering, Escherichia coli, Min oscillation

We have developed a 3D off-lattice stochastic polymerization model to study subcellular oscillation of Min proteins in the bacteria Escherichia coli, and used it to investigate the experimental phenomenon of Min oscillation stuttering. Stuttering was affected by the rate of immediate rebinding of MinE released from depolymerizing filament tips (processivity), protection of depolymerizing filament tips from MinD binding, and fragmentation of MinD filaments due to MinE. Each of processivity, protection, and fragmentation reduces stuttering, speeds oscillations, and reduces MinD filament lengths. Neither processivity or tip-protection were, on their own, sufficient to produce fast stutter-free oscillations. While filament fragmentation could, on its own, lead to fast oscillations with infrequent stuttering; high levels of fragmentation degraded oscillations. The infrequent stuttering observed in standard Min oscillations are consistent with short filaments of MinD, while we expect that mutants that exhibit higher stuttering frequencies will exhibit longer MinD filaments. Increased stuttering rate may be a useful diagnostic to find observable MinD polymerization in experimental conditions.

Introduction

Subcellular oscillations of the proteins MinD and MinE within the rod-shaped bacterium E. coli help restrict division to midcell [START_REF] Raskin | The MinE ring: an FtsZ-independent cell structure required for selection of the correct division site in E. coli[END_REF][START_REF] Raskin | MinDE-dependent pole-to-pole oscillation of division inhibitor MinC in Escherichia coli[END_REF][START_REF] Hu | Topological regulation of cell division in Escherichia coli involves rapid pole to pole oscillation of the division inhibitor MinC under the control of MinD and MinE[END_REF][START_REF] Rothfield | Spatial control of bacterial division-site placement[END_REF][START_REF] Howard | Cellular organization by self-organization: mechanisms and models for Min protein dynamics[END_REF][START_REF] Lutkenhaus | Assembly and dynamics of the bacterial MinCDE system and spatial regulation of the Z ring[END_REF]. Pole-to-pole Min oscillations arise from an interplay between diffusion and membrane binding/unbinding of MinD and MinE proteins in the confined bacterial geometry. MinD-ATP binds to the membrane in a cooperative manner, and MinE binds to the membrane-bound MinD-ATP. Subsequent MinE-stimulated hydrolysis of membrane-bound MinD-ATP leads to the release of the MinD-ADP into the cytoplasm, together with MinE. Released MinE can immediately rebind to nearby membrane bound MinD-ATP, but MinD-ADP must undergo nucleotide exchange before it can rebind. In rod-shaped E. coli, membrane associated MinD is observed to form polar caps at alternating poles, and MinE associates with the medial edge of these caps in ring-like structure (the "E-ring").

Dynamic filamentous structures of MinD have also been reported in vivo [START_REF] Shih | Division site selection in Escherichia coli involves dynamic redistribution of Min proteins within coiled structures that extend between the two cell poles[END_REF][START_REF] Szeto | A conserved polar region in the cell division site determinant MinD is required for responding to MinE-induced oscillation but not for localization within coiled arrays[END_REF], and are consistent with the observation of MinD polymerization in vitro [START_REF] Hu | Dynamic assembly of MinD on phospholipid vesicles regulated by ATP and MinE[END_REF][START_REF] Suefuji | Dynamic assembly of MinD into filament bundles modulated by ATP, phospholipids, and MinE[END_REF] and the delays of MinE-stimulated MinD-ATPase activity seen in vitro [START_REF] Derr | Self-organization of the MinE protein ring in subcellular Min oscillations[END_REF]. The Min oscillation in vivo may thus involve the periodic polar nucleation, polymerization, and depolymerization of MinD filaments. Nevertheless, long polymeric structures are not seen in electron cryotomograms [START_REF] Swulius | Long helical filaments are not seen encircling cells in electron cryotomograms of rod-shaped bacteria Bba[END_REF], and while static filamentous MinD structures were reported in a reconstituted planar Min system [START_REF] Loose | Spatial regulators for bacterial cell division self-organize into surface waves in vitro[END_REF] (see Fig. S11) dynamic filaments were not reported. It remains unclear how ubiquitous membrane associated MinD polymers are in normal E.coli Min oscillations [START_REF] Cabeen | The bacterial cytoskeleton[END_REF][START_REF] Loose | Protein self-organization: lessons from the Min system[END_REF], and what length any polymeric filaments have. While short MinD polymers would not be easily observable, could they still significantly affect the observable phenotype of the Min oscillation?

The stuttering of the disassembly of the polar MinD caps that has been observed in wild-type (WT) Min oscillations [START_REF] Hale | Dynamic localization cycle of the cell division regulator MinE in Escherichia coli[END_REF] and that is common in certain mutant systems [START_REF] Shih | Division site placement in E. coli: mutations that prevent formation of the MinE ring lead to loss of the normal midcell arrest of growth of polar MinD membrane domains[END_REF][START_REF] Hsieh | Direct MinE-membrane interaction contributes to the proper localization of MinDE in E. coli[END_REF] has not been recovered in existing models of Min oscillation [START_REF] Fange | Noise-Induced Min Phenotypes in E. coli[END_REF]. Organization of a small number of Min molecules (roughly 2000 MinD monomers and 700 MinE dimers [START_REF] Shih | Division site placement in E. coli: mutations that prevent formation of the MinE ring lead to loss of the normal midcell arrest of growth of polar MinD membrane domains[END_REF]) into an even smaller number of filaments should enhance stochastic effects. Indeed, Min stuttering has a natural explanation in tip-directed depolymerization models, where the bistability of individual filaments selected by MinE tip-decoration allows switching of individual filaments between disassembly and growth [START_REF] Derr | Self-organization of the MinE protein ring in subcellular Min oscillations[END_REF]. If many of the filament tips are denuded of MinE at the same time, which switches them from depolymerization to polymerization, then the collective oscillation should stutter.

Surprisingly, existing stochastic polymerization models [START_REF] Drew | A polymerization-depolymerization model that accurately generates the self-sustained oscillatory system involved in bacterial division site placement[END_REF][START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF][START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF][START_REF] Tostevin | A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division[END_REF][START_REF] Borowski | Predictions from a stochastic polymer model for the MinDE protein dynamics in Escherichia coli[END_REF] have not reproduced the stuttering phenomenon. Instead, we believe that stuttering has been strongly suppressed in all existing models. The stochastically-switched 1d model of Borowski and Cytrynbaum [START_REF] Borowski | Predictions from a stochastic polymer model for the MinDE protein dynamics in Escherichia coli[END_REF] does not allow switching while the filament tip is decorated with any MinE and does not have any spatial distribution of free monomers, and so cannot capture temporary reversal i.e. stuttering. Similarly, the 1d model of Drew et al. [START_REF] Drew | A polymerization-depolymerization model that accurately generates the self-sustained oscillatory system involved in bacterial division site placement[END_REF] precludes temporary reversals by keeping all of the bound MinE at the MinD-tip. Alternatively, the 1d model of Tostevin and Howard [START_REF] Tostevin | A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division[END_REF] allows MinE-induced filament fragmentation or cutting by allowing slow MinD unbinding away from filament tips. The resulting proliferation of rapidly-depolymerizing tips is similar to what is seen in non-polymerizing reaction-diffusion models. This proliferation of tips avoids collective poisoning. A similar mechanism of filament fragmentation was used by Pavin et al. [START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF] and Krstić et al. [START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF] in 3D.

In this paper we explore three tuneable mechanisms to inhibit poisoning of individual filaments in a polymerizing model for Min oscillations. The first is to allow for processivity of MinE (with the parameter P passE , described below), such that tipbound MinE is not always released from the filament upon depolymerization but has a chance to immediately rebind [START_REF] Derr | Self-organization of the MinE protein ring in subcellular Min oscillations[END_REF][START_REF] Klein | Filament depolymerization by motor molecules[END_REF]. This is similar to the "Tarzan of the Jungle" mechanism proposed by Park et al [START_REF] Park | The Min oscillator uses MinD-dependent conformational changes in MinE to spatially regulate cytokinesis[END_REF], but applied to polymeric MinD. However, we find that this mechanism is not enough on its own to suppress stuttering. So in addition to processivity, our second mechanism is to allow for tip-bound MinE to protect the filament from further MinD-ATP binding (with the parameter P protect , described below). This protection mechanism was also included in the polymeric model of Tostevin and Howard [START_REF] Tostevin | A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division[END_REF]. The third mechanism is to allow MinE that are bound to MinD filaments to cut filaments away from the tip (with rate k f , described below) [START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF][START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF][START_REF] Tostevin | A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division[END_REF].

We investigate stuttering, or transient reversals of the disassembly of polar MinD caps, within the context of a stochastic polymerization model of the Min oscillation. This is a 3D, molecular-dynamics style, "all-molecules" model, where the stochastic effects due to shot-noise and stochastic binding and unbinding are all implicitly included.

Our Stochastic Polymerization Model

We model the 3D E. coli bacterium by a cylindrical membrane of length 2L and radius R, capped with two polar hemispheres of radius R -all enclosing the fluid bacterial cytoplasm where diffusion occurs. We work with experimental number densities scaled to an E. coli cell of length 4µm and diameter 1µm and so containing approximately 3500 MinD monomers and 1200 MinE dimers [START_REF] Shih | Division site placement in E. coli: mutations that prevent formation of the MinE ring lead to loss of the normal midcell arrest of growth of polar MinD membrane domains[END_REF]. Each cytoplasmic MinD or MinE diffuses by taking a randomly-oriented (isotropic) step of fixed length δ in every timestep ∆t, leading to a diffusion constant D = δ 2 /(6∆t). We use D D = 16µm 2 /s and D E = 10µm 2 /s for MinD and MinE [START_REF] Meacci | Mobility of Minproteins in Escherichia coli measured by fluorescence correlation spectroscopy[END_REF], respectively, and choose ∆t = 10 -2 s. Both MinD monomers and MinE dimers are treated as non-interacting particles while diffusing.

MinD-ATP monomers can bind as an isolated monomer (with rate σ D , see Fig. 1(a)) or can bind to an isolated bound MinD monomer (with rate σ nuc , see Fig. 1(b)). These rates naturally translate into probabilities (see next subsection) for diffusing particles that encounter the membrane. Cooperative binding to a bound MinD monomer initializes a polymer of length 2a 0 , where a 0 = 5nm is the bound subunit spacing along MinD protofilaments [START_REF] Suefuji | Dynamic assembly of MinD into filament bundles modulated by ATP, phospholipids, and MinE[END_REF]. The orientation of the growing polymer is determined at nucleation. In this paper, we have taken all polymers to be straight along the geodesic line from one cell pole to the other: i.e. on great circles on end-caps and axial along the cylindrical portion of the membrane. MinD-ATP monomers can extend an existing polymer by binding to its tip monomer with a rate σ dD (see Fig. 1(c)). Every monomer added extends the polymer by a 0 . MinD-ATP binds to the closest membrane-bound MinD, either monomer with rate σ nuc or tip with rate σ dD , that lies within the radial distance r nuc or r D , respectively, of the point where MinD strikes the membrane. We require this finite interaction range in order to speed our computational algorithm: it allows us to take larger random steps during diffusion, and it allows us to turn off membrane diffusivity of MinD. Since most MinD is membrane associated, this represents an enormous computational efficiency.

To include the effect of PL heterogeneity [START_REF] Mileykovskaya | Visualization of phospholipid domains in Escherichia coli by using the cardiolipin-specific fluorescent dye 10-N-Nonyl Acridine Orange[END_REF][START_REF] Mileykovskaya | Cardiolipin membrane domains in prokaryotes and eukaryotes[END_REF] and account for the observation that MinD associates with anionic CL-rich PL more than non-polar PL [START_REF] Mileykovskaya | Effects of phospholipid composition on MinD membrane interactions[END_REF], we have allowed the rate (σ Dcl ) of association of MinD to the CL-rich end-caps to be greater than its rate (σ D ) of association elsewhere on the membrane. We furthermore only allow MinD filament nucleation (via σ nuc ) only at the CL-rich polar caps. We also briefly consider a fully homogeneous model with σ Dcl = σ D and homogeneous nucleation.

MinE can bind (with rate σ E , see Fig. 1(d)) to the closest membrane bound MinD-ATP that lies within distance r E of the point where MinE strikes the membrane. We report results for r E = r D = 5a 0 , and r nuc = a 0 . However, we have explored a wide range of r D and r E values and observed Min oscillations for r D and r E values as low as a 0 and 2a 0 respectively. MinE can bind to any membrane associated MinD, whether they are in filaments or not. When MinE binds, it forms a MinDE complex. Unbinding of MinDE primarily proceeds through MinE stimulated MinD ATPase activity of the MinDE complex. If the MinE is bound to an isolated membrane-bound MinD monomer the release rate is k SM , the release rate at the tips of a filament is k S (see Fig. 1(e)). We investigate the effects of MinE-stimulated MinD release away from filament tips (through the rate k f , where k f ≤ k S ), which cuts or fragments the filament (see Fig. 1(h)). We also allow a small intrinsic (non-hydrolysed) release rate of bound MinD monomers with rate k I . MinE can also spontaneously release from a MinDE complex without hydrolysis of the associated MinD with rate k E . These small spontaneous release rates of MinD and MinE without hydrolysis represent the reversibility of binding interactions. Spontaneously released MinD, like MinE, need no recovery time before rebinding. In contrast, after stimulated release MinD-ADP spends time τ c in the cytoplasm before it is converted to MinD-ATP by nucleotide exchange and become capable of binding to the membrane again [START_REF] Huang | Dynamic structures in Escherichia coli: spontaneous formation of MinE rings and MinD polar zones[END_REF].

When MinE is released from the filament tip it is passed on to the next MinD site on the filament with a "passing", or processivity, probability P passE , provided the site is not already occupied by a MinE -see Fig. 1(f). Processivity arises naturally from the efficient exploration of the tip-environment by a continuously diffusing tip-released MinE [START_REF] Derr | Self-organization of the MinE protein ring in subcellular Min oscillations[END_REF]. Processivity may also be enhanced by MinE-membrane interactions [START_REF] Park | The Min oscillator uses MinD-dependent conformational changes in MinE to spatially regulate cytokinesis[END_REF]. In addition to processivity, we inhibit binding of MinD to the tips of filaments that are already decorated by MinE with probability P protect (see Fig. 1(g)). Both P passE and P protect range from 0 to 1, and serve as tuneable parameters that inhibit poisoning of individual filaments.

Implementation details

As mentioned above, we implement diffusive motion through a randomly oriented step of fixed length δ 0 , where D = δ 2 0 /(6∆t 0 ). We keep the timestep ∆t 0 fixed to allow synchronous motion, and so adjust the step size δ 0 to give the desired cytoplasmic diffusivities of MinD and MinE. If the step would cross the cytoplasmic membrane then binding is checked (see below). If the particle does not bind, it is reflected specularly from the membrane. This ensures a uniform volume density in the non-interacting limit. For computational efficiency we have chosen a relatively large maximal timestep ∆t 0 = 10 -2 s that is still much less than the Min oscillation period. This timestep is used for unbinding, for nucleotide exchange, and for diffusing proteins far from the membrane. For bulk MinD-ATP or MinE close to the membrane, we use a random spatial step length δ equal to half-the separation of the protein from the closest membrane -but no less then a 0 = 5nm. We adjust the timestep ∆t = δ 2 /(6D) accordingly, and take these smaller steps until ∆t 0 is reached. As a result the simulation is efficient and synchronous, but retains a relatively fine spatial scale close to the membrane.

Reaction-diffusion membrane association rates are mapped to "sticking" probabilities of MinD-ATP and MinE upon collision with the cytoplasmic membrane [START_REF] Derr | Self-organization of the MinE protein ring in subcellular Min oscillations[END_REF]. The mapping agrees dimensionally with the one given by Pavin et al. [START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF][START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF] but our dimensionless prefactors differ. For a particle a distance z < δ from a membrane, steps within a polar angle θ z = cos -1 (z/δ) will hit the membrane. The corresponding solid angle gives a fraction f (z) = (1 -z/δ)/2 of particles hitting the surface with one randomly oriented step of length δ. For a bulk density ρ, integrating over z gives a sticking rate per unit area of P ρδ/4 -where P is the desired sticking probability. Equating this to the expected change in one timestep using reaction-diffusion rates, σρ∆t, gives P = 4σ∆t/δ = 2σδ/(3D). This is used for MinD-ATP binding to the membrane. For binding to a small patch of area πr 2 , e.g. at the filament tip, we can use the previous rate per unit area to obtain the binding rate πr 2 P tip ρδ/4 and equate that to the reactiondiffusion rate σ tip ρ∆t to obtain P tip = 2σ tip δ/(3πr 2 D). We use this for MinD-ATP binding to the filament tip, filament nucleation, and for MinE binding to membrane associated MinD. We return particles released from the membrane to the position that they originally bound from in order to recover uniform bulk densities in the absence of other interactions.

We have investigated a variety of initial conditions, such as randomly distributed MinD and MinE or an inhomogeneous condition with all MinD randomly placed near one pole and all MinE randomly placed near the other. Our results are unaffected by the initial conditions, though we typically used inhomogeneous initial conditions to minimize the duration of initial transients before steady-state data could be taken.

Data analysis

To simply characterize oscillations in a manner amenable to experimental measurement, we have recorded the amount of membrane-associated MinD, n D (t), in each polar hemispherical cap at 1s intervals. We have also recorded the number of filaments, and their length. The plots of MinD in one pole vs time, as illustrated in Figs. 2 and3, showed an initial transient and then periodic oscillations. We ignored the earliest 10% of the data (similar results were obtained with 20%) to avoid initial transients. We then characterized the time-average n D , the variance

σ 2 D = n 2 D -n D 2
, and the corresponding standard deviation σ D ≡ σ 2 D . From the n D (t) plot shown in Fig. 2, we identified local minima and maxima that were below n D -σ D or above n D + σ D , respectively in order to accurately identify the peak and trough of the oscillations. These local minima or maxima sometimes clustered, but the clusters alternated between minima and maxima. Within each cluster we took the smallest minimum or largest maximum as the corresponding extremum of one oscillation. The region n D ∈ [ n D -σ D , n D + σ D ], from one maximum to the next minimum, was identified as the disassembly interval. Several disassembly intervals are indicated in Fig. 2 with thicker red lines. Between 130 and 1300 oscillations were analyzed for every parameter set, depending upon the period and the computational efficiency.

Statistics of stuttering, polymer number, and length were extracted only from disassembly intervals. Polymers of MinD were counted if they were of length two or more, i.e. monomers were excluded. Stuttering was defined by a transient increase of polar MinD during the disassembly interval, and the duration of the transient increase was the stutter duration. While excluding short stutters reduced the number of stutters, and degraded their statistics, it did not appear to change their overall functional dependence on various model parameters. Accordingly, we typically counted any stutter that lasted for 1s or more -accessible to experimental timescales and avoiding fluctuations due to individual polymerization events. The stutter rate was defined to be the average number of stutters observed per disassembly interval.

Results

We recovered pole-to-pole Min oscillations for a wide range of parameters. The oscillation period was strongly dependent (data not shown) on the ratio of MinD to MinE copy numbers, with a critical minimum ratio required to sustain oscillations [START_REF] Raskin | MinDE-dependent pole-to-pole oscillation of division inhibitor MinC in Escherichia coli[END_REF][START_REF] Huang | Dynamic structures in Escherichia coli: spontaneous formation of MinE rings and MinD polar zones[END_REF]. We found oscillations with both helically pitched (data not shown) and with straight MinD filaments, with both filament fragmentation and with tip-directed depolymerization, and with a variety of bacterial lengths and widths. In all cases the oscillations were end-to-end, and were observed in both length L = 2µm and L = 4µm cells. Typically the shorter cells stuttered more. With some parameter sets we observed oscillations only with heterogeneous binding. For simplicity, and in lieu of direct biophysical measurements of most of the interaction parameters, we have restricted ourselves to one core parameter set (see caption of Fig. 2) that exhibits oscillations with both homogeneous and inhomogeneous PL patches, one bacterial size, and with axiallyoriented MinD filaments.

Fig. 2 shows spatio-temporal Min oscillations in a cell of length 4 micron cell through the process of periodic growth and decay of several polymer filaments at alternate ends of the cell. Filament nucleation was restricted to the hemispherical poles and MinD monomer binding was enhanced there as well-this is motivated by the inhomogeneous CL distribution seen in vivo. For the sake of clarity, only membranebound Min molecules are shown in the figure with blue representing MinD-ATP and red representing MinE bound to MinD-ATP. MinD filament scan be observed to start forming in the left end of the cell in Fig. 2(a). As time progresses, the filaments gradually grow longer and become more numerous (panels (b) and (c)) and many of them are also decorated by MinE, though the MinE are still sparsely distributed in panels (b) and (c). The gradual shrinking of the filaments that are predominantly capped by MinE (the E-ring) can also be observed in the opposite (right) end of the cell, in synchrony with the growth of MinD-ATP filaments in the left end of the cell. The growth of the filaments on the left end is eventually halted and shrinking of filaments, primarily by the MinE stimulated hydrolysis of MinD-ATP, is observed in panels (d), (e) and (f). This E-ring driven shrinking is associated with growth of the MinD filaments at the opposite pole. The period of oscillations was approximately 42 seconds, which is consistent with observations in WT cells at room temperature.

In Fig. 3 we consider the same set of parameters but with homogeneous nucleation and binding of MinD along the membrane (i.e. σ Dcl = σ D ) -corresponding to the absence of anionic PL patches at the cell poles. Panels (a-f) show snapshots of the resulting oscillation; the filaments are more uniformly distributed along the length of the cell and are both shorter and more numerous. The oscillation period is approximately double the inhomogeneous case.

Effects of P protect and P passE on the Min oscillation

In Fig. 4 we consider the effects of relaxing either P passE or P protect from P passE = P protect = 1, where we always have processivity of released MinE from filament tips and always protect MinE-bound tips from growth. As either P protect (red squares) or P passE (blue stars) decrease, oscillation periods grow both longer and more variable. The effect is more pronounced as P protect is decreased. In the inset, we see that the amplitude of the oscillation, as measured by twice the standard deviation of the polar MinD content (2σ D ), remains appreciable over the parameter ranges shown. We note that regular oscillations were not observed with no processivity (P passE = 0), and that (data not shown) shorter L = 2µm cells had appreciably longer and more variable periods for P passE < 1 -which was not seen with any other parameter variation in this figure or subsequent.

Qualitatively, both P passE and P protect improve the quality of oscillation by strengthening the decoration of filament tips with MinE. With a fixed microscopic stimulated disassembly rate k S , oscillations are slowed with decreased P passE or P protect by interrupting the rapid disassembly of MinD filaments. Interestingly, oscillations are much more sensitive to P protect than to P passE because of the approximately uniform concentration of MinD -AT P monomers ready to assemble unprotected filaments. The interruptions are stochastic, and lead to an increased variability of the cycle-tocycle duration. We expect that this will also be associated with an increased stutter rate.

In Fig. 5 we consider the stutter rate, as measured by the number of transient reversals of polar disassembly per oscillation, for the same parameter ranges as in Fig. 4. We show stutters of longer than 1s, 2s, and 3s with solid, dashed, and dotted lines respectively. We see that the functional form of the different stutter durations are similar, and we subsequently show all stutters of 1s or longer duration. We also see that very few stutters are observed until P passE 0.8, while significant stutters are seen for any P protect < 1. We note that no more than about 1000 oscillation periods were recorded for any parameter set, so stutter rates below 10 -3 were not observable. In the inset we show the average filament length during the disassembly phase, measured in monomers, at corresponding values of P protect (red squares) or P passE (blue stars). Reduce stuttering corresponds to shorter filaments, on average. Tip protection, through P protect , and processivity, through P passE , both reduce tippoisoning and hence stuttering. This echoes what was seen in the previous figure with the degradation of the oscillation period. Better MinE coverage of filament tips during disassembly leads to quicker disassembly, shorter periods, shorter filaments, less stochastic pausing during disassembly, and less stuttering. Both P protect and P passE must be close to unity for reliable oscillations with infrequent stuttering.

Effects of filament cutting (k f ) on the Min oscillation

In the previous section we explored models with unbroken MinD filaments by only allowing tip-directed disassembly. In this section we allow MinE-stimulated ATPase activity to break filaments away from from MinD filament tips through the cutting or fragmentation rate k f . As shown in Fig. 6, small levels of filament cutting significantly decreases stuttering even in systems (green circles) with no processivity or tip-protection. As shown in the inset, the corresponding oscillations are regular with large amplitudes. Small levels of cutting also reduces stuttering in conjunction with processivity (purple squares) and with partial protection and processivity (orange triangles). We see a broad minimum of the stutter rate near k f ≈ 0.1/sec, and the subsequent increase of apparent stuttering is associated with (see inset) a significant reduction of the oscillation amplitude. Qualitatively, when k f 0.5/sec the traces for polar MinD appear (not shown) to be noisy and irregular. Nevertheless, for k f 0.2/sec good oscillations are observed for all of the systems investigated.

As shown in Fig. 7, increasing k f leads to shorter MinD filaments (solid lines, same colour and point type as Fig. 6) and more MinD filaments (dashed lines). As shown in the inset, this also leads to shorter oscillation periods -following from faster disassembly of the shorter filaments. Further increasing k f continues the trend. For our model parameters, regular oscillations are not observed with filaments that are shorter than approximately 10 monomers.

Qualitatively, excessively high fragmentation (k f 0.1/sec) degrades the oscillation amplitude. Microscopic stochastic effects, such as transient polymerization of individual filaments or nucleation of new filaments, are then more likely to lead to transient increase of the total MinD (i.e. stuttering, as we have measured it). This results in the increase of measured stuttering at larger k f 0.1/sec, and a broad minimum of the stuttering rate. Interestingly, the minimal stuttering rate without any tip-protection or processivity (green circles) is significantly increased if processivity (P passE = 1) is added without tipprotection (purple squares). How could processivity increase stuttering? We speculate that in this case processivity leads to some filaments retaining more bound MinE as they rapidly disassemble -allowing other filaments to experience increased tip poisoning which then leads to increased collective stuttering. This increase is avoided when tipprotection is also added (orange triangles). Clearly, there is a rich interplay between multiple filament tips mediated by the association dynamics of MinE and cytoplasmic MinD.

Discussion and conclusions

We have demonstrated a polymeric model of Min oscillations with MinD filaments, consistent with structures seen both in vivo [START_REF] Shih | Division site selection in Escherichia coli involves dynamic redistribution of Min proteins within coiled structures that extend between the two cell poles[END_REF] and in vitro [START_REF] Hu | Dynamic assembly of MinD on phospholipid vesicles regulated by ATP and MinE[END_REF][START_REF] Suefuji | Dynamic assembly of MinD into filament bundles modulated by ATP, phospholipids, and MinE[END_REF]. Like the polymeric Min model of Pavin et al [START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF][START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF], our model is 3D and fully stochastic, including random motion of individual cytoplasmic proteins within the bacterial volume. However, we also systematically explore stuttering -the transient reversal of polar MinD disassembly that has been observed in vivo [START_REF] Shih | Division site placement in E. coli: mutations that prevent formation of the MinE ring lead to loss of the normal midcell arrest of growth of polar MinD membrane domains[END_REF][START_REF] Hale | Dynamic localization cycle of the cell division regulator MinE in Escherichia coli[END_REF][START_REF] Hsieh | Direct MinE-membrane interaction contributes to the proper localization of MinDE in E. coli[END_REF]. Stuttering has not been recovered in any non-polymeric model of Min oscillations, and we believe it to be intrinsically a polymeric phenotype [START_REF] Fange | Noise-Induced Min Phenotypes in E. coli[END_REF].

Frequent poisoning (i.e. binding of MinD to a DE-complex at the tip of a polymer filament) of a rapidly depolymerizing MinD filament by denuding the filament tip of MinE is an issue facing normal Min oscillations in quantitative polymeric models. The existence within the cell of filament tips that are polymerizing at the same time that others are depolymerizing means that a depolymerizing MinD tip that is poisoned will rapidly switch to polymerization, slowing the oscillation and causing a stutter.

To recover fast regular oscillations, we needed to suppress stuttering. Three mechanisms in our model reduced stuttering. The first was MinE processivity, the immediate rebinding of MinE associated with MinD filament tips upon MinD depolymerization -through P passE . The second was to protect MinD from binding to the tip of an already depolymerizing (MinE decorated) MinD filament -through P protect . We found that both of these mechanisms together were sufficient to recover fast oscillations with long filaments, as illustrated in Fig. 2 with P passE = 0.9 and P protect = 1. However, oscillations were more sensitive to changes in P protect than P passE with oscillations degenerating even for a value of P protect as large as ≈ 0.80. Independently of processivity and tip-protection, we were also able to suppress stuttering by allowing MinE bound away from filament tips to cut MinD filaments. This mechanism has been included in previous polymeric Min models [START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF][START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF][START_REF] Tostevin | A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division[END_REF], but was not systematically explored.

In general we found that more stuttering is associated with longer oscillation periods and longer filaments. For parameter values that recovered typically observed oscillation periods in the range of 10-100s a stutter rate of approximately one stutter in every 100 oscillations was observed. For some parameter values no stutters were observed in hundreds of oscillations. Combining several stutter suppression mechanisms, such as filament cutting and protecting filament tips from poisoning, led to lower stuttering rates.

How does this compare with stuttering observed in vivo? WT oscillations "occasionally" stutter [START_REF] Hale | Dynamic localization cycle of the cell division regulator MinE in Escherichia coli[END_REF], which we take as no more than 1% stutter per cycle. The "C1" MinE mutant (R10G/K11E/K12E) [START_REF] Hsieh | Direct MinE-membrane interaction contributes to the proper localization of MinDE in E. coli[END_REF] exhibited frequent stuttering, extended MinD polar zones, slower oscillations, and weaker E-rings. The D45A/V49A MinE mutant was qualitatively similar [START_REF] Shih | Division site placement in E. coli: mutations that prevent formation of the MinE ring lead to loss of the normal midcell arrest of growth of polar MinD membrane domains[END_REF][START_REF] Hsieh | Direct MinE-membrane interaction contributes to the proper localization of MinDE in E. coli[END_REF]. The quantification is crude, but oscillation periods of 2-3x slower and stutter rates of approx 50% of cells over 1 hour (corresponding to approximately 10% per cycle) are consistent with all of our mechanisms. However, the observation of longer MinD polar zones in the stutter mutants [START_REF] Shih | Division site placement in E. coli: mutations that prevent formation of the MinE ring lead to loss of the normal midcell arrest of growth of polar MinD membrane domains[END_REF][START_REF] Hsieh | Direct MinE-membrane interaction contributes to the proper localization of MinDE in E. coli[END_REF] together with the infrequent reports of long filaments in WT Min oscillations (only [START_REF] Shih | Division site selection in Escherichia coli involves dynamic redistribution of Min proteins within coiled structures that extend between the two cell poles[END_REF][START_REF] Szeto | A conserved polar region in the cell division site determinant MinD is required for responding to MinE-induced oscillation but not for localization within coiled arrays[END_REF]), could both be explained by significant filament fragmentation operating in WT oscillations, and reduced fragmentation and longer MinD filaments in the stutter mutants.

While more structural evidence is accumulating on how MinE binds to MinD [START_REF] Park | The Min oscillator uses MinD-dependent conformational changes in MinE to spatially regulate cytokinesis[END_REF][START_REF] Wu | Determination of the structure of the MinD-ATP complex reveals the orientation of MinD on the membrane and the relative location of the binding sites for MinE and MinC[END_REF], there are essentially no measurements of biochemical rates or of MinD polymer lengths that might distinguish our mechanisms. We note that our processivity mechanism is qualitatively similar to the "Tarzan of the jungle" model of Park et al [START_REF] Park | The Min oscillator uses MinD-dependent conformational changes in MinE to spatially regulate cytokinesis[END_REF], in which active MinE is passed from one membrane associated MinD to another. We do note that such processivity is likely to be local, and so may be limited to polymeric models: even very slow protein conformational timescales of 1ms would only allow several nm of diffusion -much less than the expected separation of isolated MinD on the membrane but not so different from monomer spacing within MinD polymers. Hence, we believe processivity is inherently polymeric.

The observed minimal stutter rate of less than 1% per cycle at intermediate filament cutting rates (k f ∈ [0.01, 0.1]/sec, see Fig. 6), corresponding to short MinD polymers with length between 10 and 20 monomers (see Fig. 7) leads us to believe that a polymeric model with filament cutting is appropriate for describing WT Min oscillations in vivo [START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF][START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF][START_REF] Tostevin | A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division[END_REF]. The expected filament cutting rate, k f , depends on other mechanisms that can suppress stuttering, such as processivity P passE [START_REF] Park | The Min oscillator uses MinD-dependent conformational changes in MinE to spatially regulate cytokinesis[END_REF] and tip-protection P protect .

It would be interesting to compare our results with stutter rates in other stochastic polymeric models of Min oscillations [START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF][START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF][START_REF] Tostevin | A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division[END_REF] as fragmentation rates are varied. We anticipate that 1D models [START_REF] Tostevin | A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division[END_REF] may effectively enhance the true processivity P passE due to recurrence of random walks in one-dimension -so studies in 3D are called for. Unfortunately 3D molecular-dynamics simulations such as our study, and those of Pavin et al [START_REF] Pavin | Min oscillation in Escherichia coli with spontaneous formation of two-stranded filaments in 3D stochastic reaction-diffusion model[END_REF][START_REF] Krstić | Min-proteins oscillations in E. coli: three-dimensional off-lattice stochastic reaction-diffusion model[END_REF] are slow -and extensive parameter searches to achieve quantitative agreement with experimental phenomenology are not practical. Given the variety of mechanisms that are needed to restrict stuttering, and the number of other parameters in Min oscillation models, what is needed is careful experimental characterization of microscopic rates and structures akin to what is available for actin polymerization [START_REF] Pollard | Rate constants for the reactions of ATP-and ADP-actin with the ends of actin filaments[END_REF]. We expect that this is possible in in vitro systems [START_REF] Loose | Spatial regulators for bacterial cell division self-organize into surface waves in vitro[END_REF]. will processively associate with an adjacent MinD-ATP rather than being released into the cytoplasm; (g) With probability P protect , MinD-ATP binding will be blocked from filament tips that have MinDE; (h) With rate k f , MinDE away from filament tips will disassociate and fragment the filament into two. We find that the last three processes (f-h) control stochastic stuttering of Min oscillations. We see that with maximal processivity and tip protection, oscillations are fast and precise, but that decreasing either P protect or P passE both slows and degrades the oscillation -this is particularly pronounced with smaller P protect . In the inset, we show the corresponding oscillation amplitude (2σ D ) vs. P . While the oscillation amplitude does degrade somewhat as P protect decreases, the amplitudes remain large for periods less than 1000s. Parameters for this and subsequent figures are as in Fig. 2, unless otherwise mentioned. or longer stutters per oscillation (the stutter rate) vs. the filament fragmentation rate k f . Statistical errors are as indicated. The various curves are for no protection but full processivity (purple squares), partial protection with P protect = 0.9 and full processivity (orange triangles), and neither protection nor processivity (green circles). For all three curves, stuttering decreases as k f increases from small values, reaches a broad minimum, then increases for further increases of k f . The inset shows the oscillation amplitude (2σ D ), which sharply decreases at k f 0.1/sec -corresponding to when the stutter rate begins to increase again. Interestingly, processivity significantly increases the minimal stuttering rate (purple vs green curves) unless tip-protection is also present (orange curves). We see that increased fragmentation leads to shorter filaments, more filaments, and shorter oscillation periods.

Figure 1 .

 1 Figure 1. Schematic of kinetic parameters in our Min model (a) Cytoplasmic MinD-ATP (yellow circles) binds to the membrane with rate proportional to σ D ; (b) Filaments nucleate by binding of second MinD-ATP to a membrane-associated MinD-ATP, with rate proportional to σ nuc ; (c) Filaments elongate by tip-binding of MinD-ATP, with rate proportional to σ dD ; (d) MinE (green square) binds to any of the filamentous MinD (binding to the tip is illustrated) with rate proportional to σ E , leading to MinDE); (e) MinE stimulated MinD-ATPase activity of MinDE leads to disassociation of MinD-ADP (darker, rose coloured, circle) and MinE from the filament tip, with rate k S ; (f) With probability P passE the tip-released MinE from (e)will processively associate with an adjacent MinD-ATP rather than being released into the cytoplasm; (g) With probability P protect , MinD-ATP binding will be blocked from filament tips that have MinDE; (h) With rate k f , MinDE away from filament tips will disassociate and fragment the filament into two. We find that the last three processes (f-h) control stochastic stuttering of Min oscillations.

Figure 2 .

 2 Figure 2. Min oscillations -inhomogeneous nucleation. Consecutive snapshots (a-f) from our simulation video shows periodic polymerization and depolymerization of MinD filaments at the two ends of a rod-shaped cell. The cylindrical portion of the cell has length L = 4µm with radius 0.5µm; the hemispherical end-caps have the same radius. Only membrane-associated molecules are shown: MinD-ATP in blue and MinD-MinE in red. We apply inhomogeneous nucleation where MinD filaments can only nucleate at the hemispherical poles, and σ Dcl = 10 σ D . The oscillation of the number of membrane-bound MinD in the left third of the cell is shown in (g). The oscillation period is approximately 43 seconds, after a short initial transient from inhomogeneous initial conditions. Also shown are the mean n D with a dashed blue line, and n D ± σ D with the two dotted green lines. Four disassembly regions, which go between the dotted green lines, are illustrated with thicker red lines. The parameters are: ρ D = 1200µm -3 , ρ E = 400µm -3 , D D = 16µm 2 /sec, D E = 10µm 2 /sec, σ D = 100µm/sec, σ dD = 5.5×10 8 µm 3 /sec, σ E = 8.0×10 8 µm 3 /sec, σ nuc = 1.6 × 10 7 µm 3 /sec, k I = 0.1/sec, k SM = 10/sec, k S = 6/sec, k E = 0.01/sec, τ c = 1.0sec, r D = r E = 25nm, r nuc = 5nm, δt = 0.01 sec, P passE = 0.9, P protect = 1.0.

Figure 3 .

 3 Figure 3. Min oscillations -homogeneous nucleation. Consecutive snapshots (a-f) from our simulation video shows periodic polymerization and depolymerization of MinD filaments at the two ends of a rod-shaped cell. Only membrane-associated molecules are shown: MinD-ATP in blue and MinD-MinE in red. We apply homogeneous nucleation where MinD filaments can nucleate everywhere and MinD monomers bind homogeneously σ Dcl = σ D . Other parameters are the same as in Fig. 2. The oscillation of the number of membrane-bound MinD in the left third of the cell is shown in (g). The oscillation period is approximately 85 seconds, and is reached after a short initial transient from inhomogeneous initial conditions.

Figure 4 .

 4 Figure 4. Effect of P protect and P passE on the oscillation period. We show the mean oscillation period as either tip protection or processivity, P protect (red squares) or P passE (blue stars) respectively, is varied from P protect = P passE = 1. The vertical bars indicate the standard deviation of the period, σ T -statistical errors are much smaller. We see that with maximal processivity and tip protection, oscillations are fast and precise, but that decreasing either P protect or P passE both slows and degrades the oscillation -this is particularly pronounced with smaller P protect . In the inset, we show the corresponding oscillation amplitude (2σ D ) vs. P . While the oscillation amplitude does degrade somewhat as P protect decreases, the amplitudes remain large for periods less than 1000s. Parameters for this and subsequent figures are as in Fig.2, unless otherwise mentioned.

Figure 5 .Figure 6 .

 56 Figure 5. Effect of P protect and P passE on the stutter rate. The points show the stutter rate per oscillation period as either tip protection or processivity, P protect (red squares) or P passE (blue stars) respectively, is varied from P protect = P passE = 1. Statistical error bars are shown. The stutter rate is the average number of transient reversals of the indicated duration (1s, 2s, or 3s indicated by solid, dashed, and dotted lines, respectively) or longer during polar disassembly per oscillation. We see that moving away from full protection or processivity leads to significant rates of stuttering. We also see that stutters of different duration are similarly distributed. The inset shows the corresponding average filament length, measured in number of monomers.

Figure 7 .

 7 Figure 7. Number of polymers and polymer length (in numbers of monomers) vs fragmentation k f . The solid lines show the average length of MinD filaments during the disassembly phase of oscillations at either pole, while the dashed lines show the corresponding average number of filaments. The points and colours are the same as the previous figure. The inset shows the corresponding oscillation periods.We see that increased fragmentation leads to shorter filaments, more filaments, and shorter oscillation periods.

Acknowledgments

This work was supported financially by the Natural Sciences and Engineering Research Council (NSERC), and with fellowship support for JD from the Atlantic Computational Excellence Network (ACEnet). Computational resources were provided by ACEnet. SS would also like to acknowledge use of the high performance computing facility, at Jawaharlal Nehru University, that was funded by the University Grants Commission (UGC), India through an UPOE grant.