Model Formulation

Combining the mechanics for MinD dimerisation and MinE interactions, we have developed a partial differential equation (PDE) model. In the equations, the concentrations of the proteins MinD and MinE are represented by single letters: D and d, and E and e, respectively. Upper case letters (D, E) denote species in solution while lower case letters (d, e) are bound to the membrane. Dimers are indicated by a subscript ‘2’.

The binding of MinD to the membrane is cooperative, suggesting MinD self association [13]. Crystal structures of MinD demonstrate it is capable of forming a dimer in solution when ATP is bound, albeit under non-physiological conditions [14]. While MinD may partially dimerise in solution, FRET [15] and yeast two-hybrid [16] experiments show that MinD dimerisation is predominantly a two-step process. First, MinD monomers with ATP bound are able to bind to the membrane. Once bound, they are then able to dimerise which re-enforces the strength of the membrane binding, increasing its stability. This is depicted in the model with MinD monomers in solution (D) being able to bind to the membrane (d) before they can react to form a dimer (d₂)

The interactions of MinE have been characterised to a lesser extent. MinE is always found as a dimer. Crystal [18, 37] and NMR [17, 38] structures show a dramatic change in structure of MinE between its inactive solution state and its active state that is capable of binding MinD to form a MinD/E heterotetramer on the membrane.

There is evidence for the membrane binding of MinE being both a one and two step transition with the MinE dimer in solution (E₂) being able to bind directly to both the membrane (e₂), as well as membrane-bound MinD dimers to form a heterotetramer [19]. Either of these pathways on its own is sufficient to reproduce the experimental data (see S1 Text for a direct comparison). Thus, there is no gain in having both pathways. We did not include the two pathways in our model as it was not worth the increase in parameter space, and the probable over-fitting that would result. We have simplified our model by assuming that MinE binding to the membrane is the dominant pathway, and hence all MinE first binds to the membrane before reacting with MinD. The importance of MinE membrane binding has also been demonstrated theoretically with MinE processivity being a critical component in other modern models of the Min system [31, 39].

The configuration of the MinE dimer when it is bound to the membrane is not known. It could be active or inactive with respect to its ability to activate MinD and it is possible that the two MinE monomers are conformationally distinct. We assume that when membrane-bound, MinE is in the active conformation but only one of the two MinE monomers is able to interact with membrane-bound MinD at one time. We ignore the possibility of MinE and MinD to form a heteropolymer where both MinE monomers bind to and activate MinD simultaneously. Such polymers are seen in the MinD/E heterotetramer crystal structure [19], however, we note that there is a 90° rotation between successive MinD dimers in the crystal due the four-fold screw axis. This may preclude the formation of such heteropolymers when both MinD and MinE are tethered to the cell membrane.

Whether MinE is able to interact with MinD monomers is also unknown. As the ATP binding domain is part of the dimer interface, it is unlikely that any such interaction would stimulate ATP hydrolysis, however it may still instigate the release of MinD from the membrane. We assume that this is the case. It is possible to exclude this reaction and still have patterning, however, to date, all models without this interaction suffer the same problem as previous models, that is, they fail to reproduce patterning that is consistent with experimental kymographs.

MinE binding to a single site of MinD is sufficient to catalyze the hydrolysis of both ATP molecules in the dimer [40]. MinD hydrolysis causes the complex to dissociate. MinD in its ADP state returns to the cytosol as monomers (D) while MinE remains on the membrane as a homodimer (e₂). This entire ATP hydrolysis and MinD dissociation process is modelled by a single reaction characterised by the rate constant ω_hydr. The ability of two MinE dimers to bind simultaneously to a single MinD dimer is ignored as it is not required for ATPase activation.

We have made further simplifying assumptions to reduce the size of the parameter space of the model. In solution MinD exists in both an active MinD.ATP and an inactive MinD.ADP state. The inclusion of both of these states does not affect simulations greatly, and so, in our model we assumed that all MinD in solution is in the active state. The release of MinD monomers from the membrane was assumed to be essentially instantaneous upon interaction with MinE.

To summarise, the model is governed by the following reactions: MinE dimer in solution (E₂) is able to bind to the membrane (e₂) and, in turn, can be released back into the cytosol. MinD monomer in solution (D) can bind to the membrane (d). Membrane-bound MinD (d) can either form a dimer (d₂) or bind to membrane-bound MinE (e₂), which stimulates the MinD monomer’s release from the membrane. The MinD dimer (d₂) can form a heterotetramer complex (d₂e₂) by binding the membrane-bound MinE homodimer (e₂). In the heterotetramer (d₂e₂) MinD hydrolyses ATP causing the complex to dissociate with MinD being released into the cytosol as monomers (D) while MinE returns to being a membrane-bound homodimer (e₂). These simplified reactions are shown schematically in Fig 1.

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Fig 1. Overview of the Simplified Reactions Underlying the Model.

A schematic showing the different states of the system and the reaction pathways between them. ω denote the respective rate parameter for each reaction. The value used for each of these parameters and a description of each reaction is shown in Table 1.

https://doi.org/10.1371/journal.pone.0128148.g001

Where possible, parameters have been constrained to fit experimental measurements. Free parameters were found using a genetic algorithm that rated the fitness of parameter sets by the existence of critical transitions within the relevant cell length ranges. The transitions were: that from the stationary to oscillating patterning at 2.7 μm; the transition from the first to the second order mode at 5.5 μm; and the transition to midcell antinodes in the partially-labelled system at 3 μm. Determining the free parameters was achieved by solving for a reduced one-dimensional version of the equations at fixed lengths 0.25 μm either side of each transition. All parameters are unaffected, however, the cytosolic states are approximated as homogeneous radially and a coordinate transform is used to project the membrane diffusion onto a one-dimensional line. The resulting parameters and the reactions they control are summarised in Table 1.

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Table 1. Rate Parameters and Reactions Summary.

https://doi.org/10.1371/journal.pone.0128148.t001

MinE strongly favours its inactive state in solution, with membrane-bound MinE homodimers being highly unstable and hence only existing transiently [17]. This is reflected in our model by having the rate of release of bound MinE homodimers (ω_er) exceeding their rate of binding (ω_eb). As seen in Table 1, the ratio of these rates is 430 μm^-1. So at equilibrium in the absence of MinD, 99.75% of MinE will remain in the cytoplasm.

To date, no experiment has been performed to differentiate between the rate at which MinE interacts with membrane-bound MinD monomers versus dimers. With no other information, we assume the same association rate, ω_edf, for both species. This approximation also helps to minimize the size of the parameter space of the model.

All reactions in this model will have associated back rates, which have been set to zero so as to simplify the model and reduce the parameter space. There are also other reactions that are may occur (including MinD dimerisation in solution) which are also not included as experimental evidence suggests that they are not the prevalent interactions of the physical system [15, 16].

The diffusion coefficients of the fluorescent cytoplasmic MinD-GFP and MinE-GFP fusion proteins have been measured in vivo as 16 and 10 μm² s^-1, respectively [41]. This value for the MinD-GFP diffusion constant, D_D, was used in our simulations of the Min system where all of MinD is labelled.

For cytoplasmic MinE and for unlabelled MinD, the difference in the diffusion coefficient between the GFP labelled and unlabelled protein can be approximated. MinE-GFP is approximately four times the size of the wild type MinE (88 versus 326 amino acids). If both of these proteins were spherical this difference in size would result in wild type MinE having twice the diffusion coefficient measured for MinE-GFP [42], that is, D_E approximately 20 μm² s^-1. By a similar argument, the unlabelled diffusion constant for MinD, D_D, is approximately 24 μm² s^-1.

Membrane-bound diffusion coefficients are modelled by a single parameter D_m. In reality each membrane-bound species would have a different diffusion coefficient depending on the type of complex within which it resides. In our model, we assume that the diffusion coefficient of each membrane-bound species varies according to the inverse of the number of membrane targeting sequences present in the complex (i.e. MinD monomer, dimer, MinE dimer and the MinDE heterotetramer will have diffusion coefficients of 1, ½, ½, and ¼ D_m, respectively). This relationship was determined experimentally using single molecule measurements of diffusion coefficients for engineered proteins containing one to three pleckstrin homology domains coupled by flexible linkers [43]. The reduced diffusion rate of larger complexes (dimers and tetramers) effectively causes a slight agglomeration of the Min system. Although this is not required for this model to function, it has been used as the basis for other aggregation current models which utilize anomalous diffusion to give rise to Min patterning [22].

As protein synthesis and degradation are not important for patterning during E. coli cell division [5], the average concentrations of both proteins in the cell were set to constant values, for MinD 1389 μm^-3 (2.3 μM) and for MinE homodimers 486 μm^-3 (0.8 μM) (that is, 972 μm^-3 of MinE monomers), consistent with reported values [44]. In growing domain simulations we maintained a constant average Min concentration in the cell through spatially homogenous protein production added to cytoplasmic D and E₂ states for MinD and MinE, respectively. These approximations are commensurate with previous models [24].

We note that the absolute concentrations of MinD and MinE used in our simulations may not correspond to the actual concentrations in the fully-labelled MinD experiments [6], where both MinD and MinE were overexpressed in E. coli strain JS964 containing the plasmid pAM238 encoding for MinE and GFP-MinD under the control of the lac promoter [45]. Due to overexpression, the actual concentrations may be higher [9], however, they were not measured. Higher concentrations of MinD and MinE can be compensated in our partial differential equation model by rescaling only two parameters: ω_edf and ω_dim which control the only non-linear terms. Dividing each of these rate constants by the ratio of the actual Min protein concentrations and our assumed values will leave the form of the solutions (as presented in the kymographs) unchanged.

We assume an idealized geometry for an E. coli cell to be a cylinder with spherical end caps. The radius of the cylinder and end caps was fixed at 0.5 μm with the length of the cylinder varying depending on the simulation.

The membrane was taken as the two dimensional manifold given by the boundary of the cytoplasm. This manifold is denoted by M in the equations where x denotes the position within the cytoplasm. As a result, the Dirac delta function in the equations, δ(||x—M||) defines a thin region of cytoplasm close to the membrane within which aqueous species can interact with the membrane.

The resulting model can be written as a set of partial differential reaction-diffusion equations: where we note that d₂e₂ represents the concentration of the MinDE heterotetramer (and not the product of the concentrations of MinD and MinD dimers, which is represented by e₂⋅d₂). The robustness of the parameter set (summarised in Table 1) to variation was investigated. Whilst holding the remainder of the parameters constant, each parameter was varied until the system was no longer maintained a definite node at the middle of 3.5 μm cell. Minimum and maximum values attained for each parameter are shown in Table 1.

Solving the PDEs

Solving the partial differential equations utilized the numerical software FlexPDE (version 6.32, PDE Solutions Inc.). Within this software, the cell geometry was reduced to two dimensions by assuming that solutions were cylindrically symmetric about the major axis of the cell. The Dirac delta function on the membrane was approximated by a rectangular function with the step occurring 0.05 μm from the boundary. FlexPDE cannot handle the coupling between a three-dimensional volume and a two-dimensional manifold, so the membrane was approximated by a thin three-dimensional shell. A similar approximation has been used in other models [29]. The model parameters were scaled to make the resulting reactions independent of the width of this shell. To do this, for each reaction occurring on the membrane, the concentration of each membrane-bound protein was multiplied by the width of the shell. Each complete reaction was then divided by the width of the shell. Finally each resulting constant was then incorporated into the parameter associated with that reaction for the simulation.

Growing Cell Simulations

In growing domain experiments, the cells are grown linearly in time by stretching the cylindrical section of the cell. If the equations were solved under these conditions, particles would effectively be created in the growing cylindrical section in proportion to their concentrations. As the system is not homogeneous, this would result in an alteration of the overall protein concentrations.

Two steps are taken to remedy the above problem. To avoid creating particles as a result of the physical growth process, a decay term was added to each state in the cylinder section of the domain, taking the form of -growth rate · species/length of cylinder. As a second step, the overall concentrations of MinD and MinE are maintained through homogenous production of cytoplamic MinD and MinE throughout the entire cell (see Model Formulation above).

Initial conditions were generated by setting all MinD and MinE homogeneously distributed in the D and E₂ states, respectively, throughout the cell cytoplasm with cell length set equal to the starting length of the corresponding experimental kymograph. All other states were set to zero. This system was then run for 1,000 seconds without any growth to allow it to converge to a stable solution (either oscillating or stationary). The resulting distributions for each species were then taken as the initial conditions for the growing domain simulations. This procedure was justified by solving the model for a complete cell cycle (see Results) and comparing this complete simulation to individual segments used for comparison to experimental kymographs.

Dividing Cell Simulations

To approximate the division process, we implemented a moving mesh to pinch in at the cell centre. We approximate the pinch as two semicircles with radius 0.5 μm (concave with respect to the cytoplasm) that are joined by a third small semicircle of 0.1 μm (convex with respect to the cytoplasm). At its thinnest, the radius of the cytoplasm at the pinch was 0.077 μm. The ratio of the expanded to constricted septum radii is 15.4% which corresponds to a ratio in the cross-sectional areas of 2.4%. This was the smallest value possible before numerical errors escalated uncontrollably. The time taken for constriction to occur was taken to be 512 s, as used by Sengupta [46]. As we cannot completely divide the cell in our simulations, we linearly decreased the pinching radius from the initial radius of 0.5 μm, to our maximum constriction at the same rate as wild type i.e. over 432 s. Once the minimum radius was reached, this shape was maintained for the remainder of the simulation. By not growing the cell during binary fission, it was possible to obtain a greater constriction of the septum, as the mesh was only being distorted in one direction rather than two.

Growing Domain with Fully-Labelled MinD

To examine how the model develops throughout a cell cycle, a simulation was run for the lifetime of a cell, from the length of a short newborn cell to a filamentous cell using the parameters for fully-labelled MinD. This was started at 2 μm and grown all the way through to 6.5 μm at a growth rate of 0.625 nm s^-1 over 120 minutes. The resulting kymograph showing the distribution of MinD as a function of time, and subsequently cell length, is presented in Fig 2, where yellow is high MinD concentration while dark blue is low (see S1 Movie).

This simulation has several defining characteristics. There is a transition from stationary to oscillating MinD distribution at approximately 2.7 μm. As the simulation continues past this critical point, there is a gradual transition in waveform from a rectangular shape (2.75–3.25 μm) to a triangular shape (4.5–5.25 μm). This continues until the cell length reaches approximately 5.4 μm, where a distinct pattern appears with MinD alternately occupying the poles and the midcell position, forming a second order breather mode (Fig 2).

Transition From Stationary to Oscillating Pattern Arises Naturally From Model Dynamics

To evaluate the accuracy of this model for the fully-labelled MinD system, representative experimental kymographs from the work of Fischer-Friedrich [6] were selected. These kymographs do not span a complete cell division cycle due to experimental limitations. Theoretical simulations were then run between the same starting and finishing lengths as the experimental kymographs and over the same time period to provide a direct comparison. These are shown in Fig 3A (experimental kymographs in 3Ai and 3Aiii and corresponding simulations in 3Aii and 3Aiv, respectively).

Many of the physical characteristics of the Min system as seen in experimental kymographs are present in the theoretical model. Fig 3Ai illustrates how the MinD distribution in the experimental kymograph is essentially stationary at lengths less than 2.7μm, with MinD residing at one pole, with the phase of this first order mode switching stochastically, abruptly moving MinD to the opposite pole (Fig 3Ai at 250 s and twice around 1200 s). The equivalent theoretical simulation (Fig 3Aii) contains no stochastic terms (other than noise due to discrete computation). Hence, there is no mechanism to give rise to phase switching while the pattern is in a stationary mode i.e. MinD remains localized at the top pole from the start of the simulation continuously through to 1440 s (Fig 3Aii).

Once the cell grew past a critical length (2.7 μm in Fig 3Ai and 3Aii), both experimental (Fig 3Ai) and theoretical (Fig 3Aii) kymographs started to oscillate. As the cell elongated further, the experimental oscillations become more regular, and the agreement in period between experiment and theory improves (Fig 3A).

In cells just long enough to oscillate (2.75–3.25 μm) both experimental (Fig 3Ai) and theoretical (Fig 3Aii) kymographs displayed rectangular waveforms. As the cells grew, the waveform gradually changed from square wave to a more triangular shape (see the experimental and theoretical kymographs in Fig 3Aiii and 3Aiv, respectively). In both theoretical and experimental kymographs, this transition occurred by initial lengthening of the leading edge of the rectangle for cell lengths between 3.2 and 3.8 μm before the peak moves to the centre of the wave.

While these features are unlikely to exist in the wild type system [8, 9], they do represent a set of critical phenomena that any accurate model should be able to recreate after accounting for the fully-labelled conditions.

Variations in Transition to the Second Order Mode in Filamentous Cells

Typically, shortly after the end of the kymograph in Fig 3Aiii cell division occurs for the fully-labelled MinD system. This creates two daughter cells in which the Min patterning returns back to the regime seen in Fig 3Ai [6]. However, occasionally a cell did not divide when it reached ~5 μm and continued to grow past lengths normally seen, known as filamentous growth [2].

As filamentous cells continued to grow, eventually the first order breather mode gave way to a second order breather mode. In this regime, MinD oscillated between localizing simultaneously at both poles before localizing to the centre of the cell. Examples of second order breather modes are seen in every kymograph in Fig 3B after 2000 s and in Fig 3C.

Visual inspection of the kymographs of filamentous cells provided by Fischer-Friedrich for the fully-labelled MinD system suggested that there are two broad classes of transition from the first to the second order breather modes. The first type of transition contained an intermediary travelling wave pattern (diagonal stripes in Fig 3Bi & 3Bii between 600 and 1400 s). This transitional pattern appeared to be a superposition of a first and second order breather mode. The travelling wave was moving from the bottom of the cell to the top, giving rise to the striped kymograph. After further cell growth, the first order mode decayed, leaving a simple second order breather mode (Fig 3Bi & 3Bii at times > 1800 s).

In contrast, the second type of transition skipped this travelling wave phase and went straight from a first to a second order breather mode. An example of this type of transition is shown in Fig 3Biii where the first order breather mode was maintained all the way to 1900 seconds before a sudden transition to a stable second order breather mode occurs.

By varying the starting length and growth rate of our simulations, we were able to replicate both of these types of transition types. These are shown in Fig 3Bii and 3Biv respectively. We note that the transition length of the model, approximately 5.5 μm, differs from that seen in the experimental kymographs (Fig 3Bi and 3Biii) however, it was within the range seen experimentally (approximately 4.8–7.2 μm).

Min Patterning is Dependent on the MinE:MinD Ratio

To test the robustness of the model to variations in the relative concentrations of MinD and MinE, a simulation was run with 90% of the wild type MinE concentration (Fig 3Cii). Although it shows similar features to wild type simulations, distinct differences are evident. The reduction of MinE concentration resulted in an increase in the period of oscillation compared to the wild type kymographs (approximately 100 s versus 80 s) which was accompanied by a proportional increase in width of the waveforms as a function of time. This is in qualitative agreement with previous stochastic [26] and deterministic [24, 30] models that have also shown an increase in period with a decrease in MinD:MinE ratio. Unfortunately, to date, there is no quantitative data relating protein ratios to the period of oscillation [30].

Transitions to the second order breather mode occur at a cell length of 6 μm compared to 5.5 μm for wild type MinE:MinD ratios. The triangular waveforms of high MinD concentration interdigitate (Fig 3Cii, t < 1200s), indicating that high MinD concentrations are seen all the way from the cell pole to beyond the cell midline. High MinD concentration crossing the cell midline was also seen in some simulations with wild type MinE:MinD ratios (Fig 3Biv near 1500 s), however, the extent to which MinD crossed the midline was smaller. The extent to which MinD crosses the cell midline increases as the MinE:MinD ratio was decreased until eventually the waveforms originating at each pole begin to merge (see S2 Text for details).

Within the set of experimental data with fully-labelled MinD [6], one distinct experimental kymograph displayed the same characteristics as the simulation with reduced MinE concentration (Fig 3Ci). Comparing this to the kymographs in Fig 3Bi and 3Biii, one sees that the period has increased. Also, in the first part of the kymograph where the first order breather mode dominates the pattern, the regions of high MinD concentration extend past cell midline and partially merge as per the simulation. The transition to the second order breather mode in the experimental kymograph occurs at a cell length of approximately 6.4 μm which was longer than the median length for transitions seen in the experimental set.

Min Oscillations with Partially-Labelled MinD

Touhami demonstrated that there is a large discrepancy in the period of oscillation between the fully and partially-GFP-labelled Min systems [10]. The difference in period reported is much larger than the reported changes in the period of oscillation of the Min system throughout the cell cycle [6]. As different base strains were used for the different types of labelling experiments, there are many potential sources for the discrepancy in oscillation period.

It has been proposed that the appearance of the stationary phase in short cells as observed by Fischer-Friedrich is a result of over expression of the Min system [9]. If Min protein concentration is the sole source of differences between the experimental results for fully-labelled versus partially-labelled cells, we should be able to model partial-labelling by reducing Min protein concentration. While we find that reducing the concentration of the Min system is sufficient to abolish this stationary phase of patterning and decrease the period of oscillation, it is not capable recreating the partially-labelled data of Juarez [7]. A kymograph of the reduced concentration system is shown in S1 Fig. In particular, the transition from first to second order mode remains longer than observed experimentally in partially-labelled cells. In the fully-labelled Fischer-Friedrich experiment this transition occurs between 4.7 and 5.5 μm (Fig 2), while in the Juarez experiment the transition to mid cell antinodes occurs at 3 μm (Fig 4). This transition to mid cell antinode patterning has been independently observed by fluorescently labelling MinC [45]. Using our model that fits the fully-labelled kymographs and reducing the Min protein concentration the transition occurs at 4.5 μm (S1 Fig) which is not consistent with the partially-labelled experimental kymographs. Furthermore, there is only a minimal region where mid cell antinodes are observed in this model (S1 Fig from 4000 to 5000 s), again differing from experiment. Thus, a reduction in Min protein concentration alone is not sufficient to allow our model to fit both fully- and partially-labelled experimental data.

Plotting the change in period due to perturbations in each parameter in our model showed that the ATPase/heterotetramer dissociation parameter, ω_hydr, had the greatest influence on the period of oscillation (see S3 Text for details). It has recently been shown experimentally that GFP tagging can cause artefactual aggregation of native homomultimeric proteins [47]. If GFP were having a stabilizing effect on the membrane-bound MinD-GFP dimer then we would expect that GFP labelling would decrease the rate of monomerisation, which, in our model, is a component of the ATPase and MinDE heterotetramer dissociation reaction modelled by the rate constant ω_hydr. Two approaches to avoid GFP artefacts have been the use of GFP variants that prevent GFP self-association [47], and minimally labelling proteins with GFP in vivo [48].

Combined, these results indicate that, as a first approximation, altering this parameter (ω_hydr) could counteract the effects of GFP-labelling on the function of MinD. Rescaling this rate from 0.12 s^-1 for fully-labelled MinD to 0.5 s^-1 for the partially-labelled system resulted in a decrease in the oscillation period so that simulations matched experimental data (Fig 4). In addition, GFP labelling alters the diffusion coefficient of MinD in solution; hence it was increased from 16 to 24 μm² s^-1. We note that changing the diffusion constant had a minimal effect on simulations (see S4 Text for comparison).

Results of a simulation with this modified set of parameters are shown in Fig 4A (see S2 Movie). In this simulation, the cell was grown at a rate of 1.05 nm s^-1 which is faster than the growth rates seen in the fully-labelled system but correlates to a biomass doubling time of approximately 30 minutes. This was chosen to be consistent with E. coli in rich media. In any case, changes in growth rate have a negligible effect on transition lengths and patterning within stable patterning regions. This can be seen by comparing the theoretical kymographs in Fig 4B where there was no cell growth, with the corresponding lengths in Fig 4A. Many of the characteristics typically associated with the partially-labelled Min system are seen in this simulation (Fig 4A).

Using these parameters, the length at which the cell transitioned from a stationary to an oscillating MinD pattern occurred at less than 1.5 μm, which is below the minimum length of typical E. coli cells. From 1.5 μm through to 3 μm the model predicts that partially-labelled MinD rapidly oscillates from pole to pole (Fig 4A). MinD resides close to one of the cell poles giving rise to a definite bare zone at the midcell. A direct comparison between a short cell of 1.8 μm from Juarez’ experiments [7] and the model is made in Fig 4Bi and 4Bii, respectively. As partial labelling provides a much weaker signal compared to the fully-labelled system, it is difficult to characterize the experimental waveform. However, the antinodes of both experimental and theoretical kymographs extend a similar distance from the poles of the cell and have a similar period. Such characteristics continue as the cell grows, as demonstrated for a 2.5 μm cell (Fig 4Biii and 4Biv).

Experiments on cells with partially-labelled MinD show a transition to a different type of patterning at 3 μm [7]. This length is marked on the second line of the model kymograph in Fig 4A by a red line. This regime is dominated by patterning where the high concentration of MinD at one end of the cell moves away from the pole to a region adjacent to the cell midline. This type of patterning was previously coined “midcell pausing” [7], however, we feel that this is misleading as the word “pausing” suggests a change in the time dependent behaviour of the Min proteins. To avoid this, we refer to this phenomenon as “midcell antinodes” as this accurately describes an increase in MinD concentration in the midcell region that arises from the dynamics of the Min system.

Midcell antinodes are seen in both experiments and simulations (Fig 4Bv and 4Bvi). Our simulation for a full cell growth demonstrates a strong tendency for anitinode formation at midcell to favour one end of the cell with only occasional switching to the opposite pole (Fig 4A. cell length > 3 μm). This is consistent with qualitative experimental reports [7]. Such switching indicates that there is an instability underlying this behaviour with the switching in the model likely being triggered by numerical imprecision in the computation.

Temperature Dependence of Period of Oscillation

It has been suggested that ATP hydrolysis is the rate-limiting step in the Min cycle [24]. The ability for changes to the rate of ATP hydrolysis to account for changes in the period of oscillation have also been explored in similar models [30]. If this is correct for our model, we expect that the temperature dependence of the system could be modelled by changing the rate of ATP hydrolysis. The model implicitly contains the ATP hydrolysis step within the parameter ω_hydr which also includes heterotetramer dissociation (see Model Formulation for details). A Boltzmann factor was subsequently applied to this term. The Boltzmann factor takes the form where ε is the activation energy for that the reaction that includes ATP hydrolysis and tetramer dissociation, T₀ is the temperature under which the experiments of Juarez were conducted (32°C) [7]) and k is Boltzmann’s constant.

The resulting relationship between temperature and MinD oscillation period of the model is shown in red in Fig 5. Close examination reveals that the model results fit the experimental data for the dependence of oscillation period on temperature [49] with a standard deviation of 1.03 s between model (Fig 5, red) and experimental (Fig 5, blue) data points. The activation energy for the best fit was equal to 11.5 kcal/mol, which coincidentally is approximately equal to the free energy of ATP hydrolysis.

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Fig 5. Temperature Dependence of the Period of Oscillation for the Min Protein System.

Experimental data with error bars is shown in blue [49]. The periods extracted from individual simulations at each temperature are shown in red with straight lines joining simulation points.

https://doi.org/10.1371/journal.pone.0128148.g005

Simulating Min Patterning During Cell Division

Juarez measured the changes in Min dynamics during cell division (using partially-labelled MinD) to investigate the mechanism behind the cell’s ability to equally partition the Min proteins between daughter cells [7]. It was shown that, in constricting cells, patterning dominated by antinodes forming at midcell seen in Fig 4A soon gives way to regular, second order oscillation around the time of septum closure. This second order breather mode is symmetric, so on average both MinD and MinE would be divided equally between each cell half throughout the oscillation cycle and hence partition equally to each daughter cell. A similar process has also been reported to occur in cells where MinD is fully labelled, with Min patterning continues unimpeded for both daughter cells following cell division [6].

To simulate the cell division process in both the fully-labelled and partially-labelled systems, the model equations were solved for a fixed cell length while the midcell region was continuously constricted (see Methods; Fig 6). This builds on the work of Di Ventura who solved a model of the Min system for various static constriction radii and showed that it was sufficient to induce a transition to a second order mode [8]. To be consistent with Juarez’s observation of high MinD concentration near the cell midline prior to cell division, we performed the partially-labelled study at a cell length 3.5 μm, where the system is firmly in the regime supporting midcell antinodes (Fig 4A). The fully-labelled system was modelled with a 5 μm cell, which is within the range where cell division is observed. However, from our simulations, a fully-labelled 5 μm cell shows a stable first order breather mode (Figs 2, 3B and 3C) which will continue indefinitely in the absence of septum constriction.

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Fig 6. Simulation of Cell Division.

Septum constriction begins after 300 s and is halted after 732 s. If constriction continued, cytokinesis would occur at 812 s. (A) Fully-labelled system run at 5 μm. (B) Partially-labelled system run at 3.5 μm. (i) The kymograph of the MinD distribution over the division process. Lengths on top of the kymograph denote the minimum radius of the septum at each point in time. (ii) The proportion of MinD (blue) and MinE (purple) in the future top daughter cell. The green and orange vertical lines mark the beginning and end of constriction respectively. The red vertical line marks the theoretical binary fission point. (iii) The cell geometry before (left) and after (right) constriction.

https://doi.org/10.1371/journal.pone.0128148.g006

Results of the simulations of the fully-labelled and partially-labelled systems are shown in Fig 6A and 6B, respectively (see S3 and S4 Movies, respectively). For the fully-labelled cell prior to the initiation of division (t < 300 s, green line in Fig 6A), the proportion of Min proteins in the top cell half varies from approximately 13% to 81% for MinD and 17% to 78% for MinE across a single period (Fig 6Aii). With the onset of septum constriction, oscillations are initially unaffected as the septum radius is reduced (Fig 6Ai and 6Aii). Once the radius of the septum has reduced to approximately 0.25 μm, half its initial value, we see a critical point, following which an erratic transition period ensues. Eventually this stabilises into a stationary second order mode where MinD is symmetrically localized to the centre of the cell. After 432 seconds of constriction (732 seconds of simulation), when the septum has reached the maximum constriction possible, indicated in the figure with an orange line, the proportion of MinD and MinE in the top cell is within 4% of parity, following which it then continues to converge towards equal distribution. At this stage, oscillations cease and the MinD distributions adopts a stationary second order mode (Fig 6Ai with t > 750 s).

Both MinD and MinE asymptotically approach symmetric distribution in the two halves of the cell from the same side, such that even if their individual partitioning has not completely converged to 50%, as is the case in this simulation, their stoichiometric ratio is approximately constant. Consequently this increases the stability of the patterning in the daughter cells, as the stability of future patterning is more susceptible to differences in the MinE to MinD ratio than their absolute concentrations [11]. The root mean square of the difference between the division ratios of MinD and MinE is 0.35%,

A model calculation for cell division in the partially-labelled system is shown in Fig 6B. As seen in Fig 6Bii partially-labelled cell leads to a much greater asymmetry in the distribution of Min proteins in each half of the cell when compared to the fully-labelled system (Fig 6Aii), with MinD varying from 30% to 90% and MinE from 37% to 76% in the top half of the cell. As per the fully-labelled system, the oscillations are initially unaffected by the divisome constriction. However, as constriction continues, the distribution of Min proteins in the two halves of the cell becomes more symmetric over an oscillation period. Fig 6Bii shows this clearly around 600 seconds where the percentages oscillate from 20% and 90% for MinD and 28% and 76% for MinE.

At approximately 700 s, a transition to a second order breather mode occurs. As seen in the kymograph in Fig 6Bi, this second order mode is not completely symmetric which leads to the small scale fluctuations in Min protein distributions (seen after 700 seconds in Fig 6Bii). Between the time that the constriction is halted due to numerical constraints (shown by the orange line) and when binary fission is likely to occur (denoted by the red line), the root mean square of the difference between parity and the percentage of Min protein in both halves was 7.7% and 6.1% for MinD and MinE respectively. These results are consistent with the experimental measurements of the distribution of MinD between daughter cells [7]. Similar to the fully-labelled model, MinE closely follows the MinD oscillations, so the root mean square of the difference between the MinD and MinE percentages is 4.75%, which is less than the absolute fluctuation.