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DF and JE conceived and designed the experiments, performed the experiments, analyzed the data, contributed reagents/materials/analysis tools, and wrote the paper.

The authors have declared that no competing interests exist.

The spatiotemporal oscillations of the ^{ −})^{−})^{−}) phenotypes. For spherical cells, the mean-field model is bistable, and the system can get trapped in a non-oscillatory state. However, when the intrinsic noise is considered, only the experimentally observed oscillatory behavior remains. The stochastic model also reproduces the change in oscillation directions observed in the spherical phenotype and the occasional gliding of the MinD region along the inner membrane. For the PE^{−} mutant, the stochastic model explains the appearance of randomly localized and dense MinD clusters as a nucleation phenomenon, in which the stochastic kinetics at low copy number causes local discharges of the high MinD^{ATP} to MinD^{ADP} potential. We find that a simple five-reaction model of the Min system can explain all documented Min phenotypes, if stochastic kinetics and three-dimensional diffusion are accounted for. Our results emphasize that local copy number fluctuation may result in phenotypic differences although the total number of molecules of the relevant species is high.

Many molecules inside a living cell do not have time to diffuse through the whole cell in-between reactions. Furthermore, the chemical reactions are random and discrete events. In this study, the authors study an example in which these aspects of intracellular chemistry need to be considered when we try to understand how a biological system works.

The authors have investigated the spatial oscillation patterns that are displayed by the Min system of

Quantitative modeling of biological processes is becoming increasingly important as the processes we seek to understand become more and more complicated. The necessity for quantitative modeling is especially compelling when the process of interest displays spatiotemporal pattern formation, such as the oscillations of the Min proteins seen in

From top to bottom: time evolution. From left to right: wt, spherical ^{−}^{ −} (long), filamentous ^{ −} (short), PE^{−}

(A) Shows the Min system and (B) shows the corresponding reaction scheme and rate constants. The reaction scheme is essentially adapted from Huang et al. [^{−8} cm^{2}s^{−1} and in the membrane, 1 × 10^{−10} cm^{2}s^{−1}. In the PE^{−} mutant: _{d}^{−6} μms^{−1} and the membrane diffusion is 1 × 10^{−12} cm^{2}s^{−1}.

The Min system, which directs

However, from the recent stochastic analysis of the Min system in three dimensions [^{−} and phospathedylethanolamide-deficient (PE^{−})

The Min system consists of the MinC, MinD, and MinE proteins expressed from the

In order to understand what drives the oscillation, the components of the system have been biochemically characterized in some detail. MinD in association with ATP binds cooperatively to the membrane independently of the presence of MinE [

In addition to wt ^{ −}^{−} mutants with a spherical phenotype, the oscillation sometimes changes direction (^{−} mutants have a more diffuse localization of the MinE protein than the rod-like phenotypes [^{−} strain, MinD is localized in tight clusters (spots), which randomly appear and disappear at a minute timescale (

The first quantitative models of the Min system were developed at the same time by Meinhardt and de Boer [^{−})^{−}^{−} strains.

Chemical reactions are stochastic events, meaning that it is not possible to know when and where the next reaction will occur. The probabilities for the reaction events can, however, be modeled, and the time evolution of the system can therefore be described probabilistically. In this article, we model the stochastic reaction–diffusion kinetics of the Min systems using the framework provided by the reaction–diffusion master equation (RDME) [

In the RDME description, the total system volume is divided into a large number of small subvolumes (25^{3}–50^{3} nm^{3} in our model). The number of molecules of the different species in the different subvolumes describes the state of the system. The state changes when the molecules in any subvolume react or when a molecule diffuses between subvolumes. The RDME provides the probability distributions for these different events.

We compare the stochastic time evolution of the Min system with the corresponding mean-field approximation. The approximation is that the state, i.e., the number of molecules in different subvolumes, changes with the average rate at each point in time.

A more informative description of the difference between the stochastic and the mean-field approach is given in the

The elementary interactions between the different forms of the MinD and MinE proteins are described in _{mem}) or bound in complex with MinE (MinDE). MinE is either freely diffusing in the cytoplasm (MinE) or in complex with MinD on the membrane (MinDE). _{mem} (reaction 2). MinD_{mem} also recruits cytosolic MinE (reaction 3). MinD-associated MinE hydrolyzes the ATP on MinD_{mem}, which results in the release of ^{−8} cm^{2}s^{−1} [^{−10} cm^{2}s^{−1}. (See the discussion about membrane diffusion and polymerization in the

The system was simulated in three different geometries corresponding to wt, filamentous, and spherical _{mem}] = 0,

The reactions rates used are presented in ^{−}^{−} strain does, however, interfere with the Min systems' membrane interaction, and the parameters for MinD's interaction with the membrane were changed to accommodate this mutation (see section about PE^{−} below.)

The exact model descriptions, i.e., the SBML files that were used to make the simulations, are supplied in Datasets (^{8} events takes 25 min on a Intel Xeon 3.06 GHz and requires 20 Mb of RAM.

The wt geometry (A) and (B), filamentous geometry ^{ −})^{−})^{−} cells with filamentous geometry (G) and (H) are shown. (A), (C), (E), and (G) show the stochastic simulations and (B), (D), (F), and (H) show the mean-field simulations. Membrane-bound MinD is shown in blue, and MinE in complex with MinD on the membrane is shown in red. The cells in (B), (D), (F), and (H) have been divided into two halves (upper and lower) to show both the MinD and the MinE concentration fields in the same plot. The discretized solution of the mean-field simulations has been mapped onto a smooth surface to facilitate visualization. In (G), the cell surface is transparent to allow visualization of the clusters on the back.

The first thing to notice for the wt cells in

The wt geometry shows MinD oscillating between the poles, followed by a narrow MinE ring. In agreement with experiments, both the stochastic and the mean-field models display clear growth of membrane-associated MinD zones from the poles to mid-cell and shrinkage back again. The growth phase is fast and due to rapid recruitment of the abundant ^{ATP}. Thus, if the concentration of MinE is increased, the oscillation frequency also increases. In the wt simulations, the average period times are 53, 40, 31, 26, and 22 s for 0.8, 0.9, 1.0, 1.1, and 1.2 times variation in the concentration of MinE, respectively. The corresponding numbers for variations in MinD concentration are 20, 25, 31, 38, and 48 s for 0.8, 0.9, 1.0, 1.1, and 1.2 times variation in the concentration of MinD. These responses to variations in MinD or MinE correspond to what has been observed experimentally [

Although the stochastic properties are not prominent in the wt system, some predictions can only be made using the stochastic description. For instance we estimate that the standard deviation in period time divided by the mean is 2.75%. This phase drift is more pronounced for a smaller number of molecules. When the concentrations of MinD and MinE are simultaneously reduced by 50% or 67%, the phase drift is increased by 4.5% and 6.9%, respectively. In

The number of membrane-bound MinD molecules in one half of a cell for wt (A) and 25% of wt concentration (B). Time-averaged localization of MinD for stochastic and deterministic simulation for wt concentration (C) and for 25% of wt concentration (D). Stochastic simulations are show in solid gray, and deterministic simulations in dashed black.

Since one physiological role of the MinD protein is to keep the co-localized protein MinC away from mid-cell, one may also be interested in how the MinD protein is distributed over the length of the cell and how this distribution depends on the copy numbers. In

The filamentous 10-μm

Percentage of deviation from equal partitioning (i.e., 50% of the total molecule number goes to each pole) of membrane-bound MinD molecules. The percentage of deviation for one half of the cell is plotted as solid black and the percentage of deviation in for other half is plotted as dashed black.

For wt and filamentous ^{−}^{−} mutants, the stochastic models explain experimental observations that cannot be accounted for by mean-field models. Starting with the spherical ^{−} mutant in

Membrane-bound MinD is shown in red, and cytosolic MinD^{ATP} is shown in blue. The color intensity is proportional to the integrated concentration of molecules along the axis perpendicular to the projection plane.

In (A) (wt) and (B) (spherical), the series are initialized with 3/4 of the MinD in the cytoplasm in one half of the cell and 1/4 in the other half.

In (C) (spherical), the series is initialized with most of the MinD membrane bound at one side of the cell.

The requirements for bistability are further characterized in ^{−9} cm^{2}s^{−1} whereas the rod-shaped 4.5-μm wt cell is bistable for diffusion rate constants over 5 × 10^{−8} cm^{2}s^{−1}. It seems like a fast redistribution of molecules in the cytoplasm prevents formation of sufficiently large local membrane occupancy to initiate cooperative recruitment of MinD and thereby start the oscillations. When the fluctuations are included in the model, the potential non-oscillatory state is not a problem since a few spontaneously associated MinD molecules are sufficient to initiate the oscillations (see the PE^{−} phenotype below).

The mean-field model was solved for three different geometries: Spherical mutant (length/width = 1), wt (length/width = 4.5), and an intermediate geometry (length/width = 1.75; cylinder of radius 1 μm and length 1.5 μm padded with half spheres of radius 1 μm). The combinations indicated with black squares display oscillations when started from the initial condition in which 3/4 of the MinD molecules are started in one half of the cell and 1/4 started in the other. The gray circles represent combinations that display bistability: When the simulation is started from the initial condition described above, the system will go to a stationary homogeneous state. When the simulation is started from an existing MinD accumulation at one of the poles, it will start oscillating.

The stochastic model also suggests a simple explanation for the change in oscillation direction that is seen experimentally in spherical cells (

In the stochastic model, we also initially observe that the MinD molecules are gliding around the cell membrane in a rotational motion (see

The most interesting differences between the stochastic and mean-field descriptions are seen in the “spotty” PE^{−} phenotype. Here, there are large qualitative differences between the models, and only the stochastic model can explain the experimentally observed behavior.

To account for the abnormal MinD membrane interactions in the spotty (PE^{−}) phenotype, we reduced the rate of spontaneous MinD association to the membrane and the membrane diffusion constant (see

With the modified MinD membrane interaction, the stochastic model nicely reproduces the appearance and disappearance of dense MinD clusters in the PE^{−} phenotype (

To characterize the phenomenon further, we initialize the stochastic simulation with 1–12 MinD_{mem} bound to a well-defined membrane location and determine the probability that a certain number of initiator molecules will lead to the formation of a MinD spot (more than 100 membrane-bound MinD). The result is presented in _{mem} has bound to the membrane is about 10%, whereas the probability of spot formation is about 90% if ten MinD_{mem} are initially bound. The critical nuclei size for which there is a 50% chance to get a spot is between five and six initiator molecules. There is no simple way to express the nucleation probabilities explicitly in terms of the rate constants since there are many paths through state space to spot formation. From

The stochastic model with PE^{−} parameters was solved for a box (5 μm × 1 μm × 1 μm), with membrane on one of the 1-μm × 1-μm sides. The simulations were initialized with a membrane occupancy of 1–12 MinD molecules. A total of 100 trajectories were gathered for each number (1–12) of initiator molecules. The probability of nucleation is defined as the fraction of trajectories reaching more than 100 membrane-bound MinD molecules. More detail is given in

The reason for the activation threshold and the excitability of the system is most clearly seen in a deterministic model, which can be initialized above or below the threshold. In _{mem} in the membrane. After initialization, some of the cytosolic MinE and MinD^{ATP} are rapidly recruited to the membrane. There is, however, always a sufficiently high local concentration of MinE close to the membrane to get a strictly decreasing concentration of membrane-bound MinD (red curve in the leftmost column of _{mem}, which is above the activation threshold. In this case, a sufficiently large amount of MinD is recruited to the membrane to sequester the local MinE pool into MinDE complexes, corresponding to a saturation of the MinE activity. When the local intracellular MinE supply is depleted, there is still plenty of _{mem} accumulates unhindered by the saturated MinE system. After some time, the supply of _{mem} proteins are hydrolyzed in this stage, which results in a burst of cytosolic _{mem} cluster as the

The mean-field equations with PE^{−} parameters were solved for a box (5 μm × 1 μm × 1 μm), with membrane on one of the 1-μm × 1-μm sides. The simulations were initialized with a membrane occupancy of MinD corresponding to 5, 7.5, and 10 molecules in the center of the membrane for (A), (B), and (C), respectively. The right-most column displays how the number of membrane-bound MinD (red) and MinDE (blue) complexes change in time. In the other plots, we see the concentrations of the cytosolic proteins at different distances from the membrane for different points in time.

In ^{ATP} on the membrane. The result is indistinguishable from the case with 7.5 initiator molecules, which demonstrates that essentially the same response is exited as soon as the activation threshold is passed.

We have developed computational methods (see ^{−}^{−} phenotype and the PE^{−} phenotype (

For wt

In agreement with previous observations by Howard and Rutenberg [

Our simulations further predict a noise-induced phase drift of 2.75% per oscillation in wt cells (

Both the mean-field and the stochastic models accurately reproduce the striped patterning in filamentous cells without adding any “topological markers.” Such markers have, in some studies, been introduced to position the MinD zones at the desired places [

For the parameters that we found optimal for reproducing the experimental oscillations in wt and filamentous cells, the mean-field model of the spherical cell is bistable (

The formation of dense MinD clusters in the PE-lipid–deficient strain is explained here as a nucleation phenomenon. The low spontaneous ^{ATP} solution supersaturated until the number of membrane-associated MinD reaches a threshold in which MinE-mediated hydrolysis of

Nucleation occurs infrequently in relation to all other events in the system. It is therefore computationally demanding to accurately determine the rate at with MinD clusters are formed using brute force simulations. Inspired by the methods for Forward Flux Sampling [^{ATP} to the membrane in the PE^{−} model with the probability of cluster formation given that one MinD^{ATP} has bound to the membrane. The association rate is estimated by the average ^{3} × 32.99 μm^{2} × 5 10^{5} μm s^{−1} = 1.39 s^{−1}. The probability of cluster formation after that one MinD has bound is approximated by the probability of cluster formation if one MinD has bound to the membrane and the other molecules are randomly distributed (≈10%), as given by ^{−1}.

With some notable exceptions ([

The first lesson from the Min system is that the fluctuations can destabilize one of several stable attractors of the mean-field model and make that attractor practically unimportant for the real system. This was observed in the Min system for the round cell, in which the mean-field model has one stable fixed point and a limit cycle attractor, whereas the stochastic system only uses the limit cycle.

The second lesson is that a stochastic system can explore different parts of neutrally stable attractors, when the mean-field model will be confined to a stationary point or a limit cycle. In the stochastic model of the Min system, this was exemplified by the phase drift in the wt cell and the change in oscillation directions seen in the round cell.

The final lesson is that the discrete and probabilistic aspect of stochastic kinetics sometimes causes a sufficiently high local concentration to initiate a process with an activation threshold. This was seen in the Min systems in the formation of dense clusters after spontaneous membrane association of a few MinD molecules in the PE^{−} mutant.

As with all other stochastic phenomena in intracellular kinetics, the possibility of localized stochastic nucleation is a constraint for the wiring of intracellular reaction networks, but also a potentially useful process. It is, for instance, a constraint for signaling systems that depend on local activation of sensory systems at the membrane followed by signal amplification for efficient propagation to intracellular targets [^{−} mutant.

On the other hand, spatially localized, low copy number fluctuations could be used to generate variability in cell shape, for instance by nucleating the formation of morphogen clusters at random localization in the cell. Such a mechanism would be a spatial analog to the stochastically activated excitable systems that are used to generate variability in a cell population, e.g., the sporulation process in

Chemical reactions are stochastic events, meaning that it is not possible to know when and where the next reaction will occur. The probabilities for the reaction events can, however, be modeled, and the time evolution of the system can therefore be described probabilistically. Stochastic reaction–diffusion kinetics is commonly modeled by the RDME [

The number of molecules in the different subvolumes, i.e., the state of the system, changes when the molecules in any subvolume react or when a molecule diffuses between subvolumes. In the stochastic framework, the reaction and diffusion events are probabilistic, and the state changes in discrete steps when an event occurs. The probability that a certain reaction occurs in a subvolume of volume Ω during the next time interval _{i}_{i}_{a}a_{i}b_{i},_{i}_{A,i}^{−1} and _{i}_{B,i}^{−1} are the concentrations of A and B in subvolume _{a}_{A,i},_{B,i},

The event that a molecule diffuses to a neighboring subvolume is treated as a first-order reaction with a rate constant of _{diff}^{2}_{diff}a_{i}_{diff}n_{A,i},_{i}

The rates, i.e., the probabilities per time unit, of all different diffusion and reaction events define a stochastic process. When one event occurs, some of the probabilities for the next event will change. The RDME describes how the probability that the system is in a certain state changes in time. Unfortunately, the RDME can not be solved analytically except for very simple systems [

In this study, we compare the stochastic time evolutions with the corresponding mean-field approximation. The approximation is that the state, i.e., number of molecules, can change continuously and that the state changes with the average rate at each point in time. To see what this means, consider the events that change the number of A molecules in subvolume _{a}a_{i}b_{i}_{diff} a_{i}_{diff} a_{i−1}_{diff} a_{i+1}_{A,i}_{A,i}_{a}a_{i}b_{i}_{diff} a_{i}_{diff} a_{i−1}_{diff} a_{i+1}_{diff}^{2}_{i}_{A,i}_{a}a_{i}b_{i}_{i}_{−1} − 2_{i}_{i}_{+1})/^{2}). Similar expressions can be straightforwardly derived for all species and subvolumes in the system. If _{a}^{2}^{2}

We have developed the MesoRD software [

For stochastic simulations, MesoRD uses the Next Subvolume Method (NSM) essentially as described in [

In NSM, the rates of all elementary events are summed for each subvolume, and the time of the next event in each subvolume is sampled from their respective exponential distributions. Based on these event times, the subvolumes are ordered in a priority queue (stored as a binary tree). The next reaction or diffusion event occurs in the subvolume that is first in the queue. This event will only change the states, rates, and next event time for maximally two different subvolumes. Therefore, in each iteration, only the one or two queue elements that correspond to the subvolumes with a state change in the last event need to be updated and sorted.

In the NSM, the number of computations scales logarithmically with the number of subvolumes instead of linearly as in the

The most important algorithmic improvement in MesoRD, as compared to our original formulation of the NSM, is that we now use a hash table to look up reaction rates corresponding to the commonly occurring combinations of reactants per subvolume. Since the number of reactants of each species per subvolume usually is very low (zero, one, or two), the rates for all commonly occurring combinations can be precalculated. Furthermore, in the case that two subvolumes needs to be sorted, the one with the earliest next event time is sorted from the top of the queue.

A set of different partial differential equation (PDE) solvers was implemented in MesoRD. All the solvers are based on the method of lines [

As an alternative to the RDME [

Although the GFRD algorithm is a computational breakthrough, it is likely to be too computationally demanding for a direct application to the Min system. However, if one does not need exact interaction information in time and space, there are at least three simulation tools that can be used: Mcell [

As a point of reference to the RDME treatment, one can consider the case in which the time step for the Brownian diffusion is chosen as the mean time between diffusion events in the RDME description, i.e. ^{2}/2

The MinD protein forms polymers in vitro if both ATP and phospholipids are present [^{ATP} almost equally efficiently in monomer and polymer form and that MinE can hydrolyze MinD^{ATP} anywhere in the polymers. If this is the case, there is no practical difference between our model and a model with polymerized MinD except that the polymerization makes the MinD spread along the membrane, which we account for by slow membrane diffusion. The exact value of the membrane diffusion rate is, however, unimportant as long as it is significantly lower than the cytosolic diffusion rate but not zero, which would lead to unphysiologically high local concentrations.

To justify the membrane diffusion model we have made a control simulation using a more detailed membrane translocation model in which the membrane-associated MinD is immobile unless the local membrane occupancy is higher than ten MinD molecule per 2,500 nm^{2} (one molecule per 15 nm × 15 nm) in which case one MinD molecule is moved to a neighboring, less crowded, location. This translocation model, which approximates the effects of MinD polymerization on the membrane, gives very similar results to the more simple model with a slow diffusion of membrane-bound MinD.

One experimental observation that we believe may require polymerization for quantitative modeling is the stuttering in the shrinkage phase for the MinD zone. This stuttering is observed especially in certain MinE mutants [

This file is also deposited in the BioModels Database ID: MODEL5974712823.

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Membrane-bound MinD molecules are shown in blue and membrane-bound MinDE complexes are shown in red.

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The stochastic model with PE^{−} parameters was solved for a box (5 μm × 1 μm × 1 μm), with membrane on the 1-μm × 1-μm side. The simulations were initialized with a membrane occupancy of 1–12 MinD molecules. A total of 100 trajectories were gathered for each number (1–12) of initiator molecules. The probability of nucleation is defined as the fraction of trajectories reaching more than 100 membrane-bound MinD molecules.

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Initial conditions as described in the text. Membrane-bound MinD is shown in blue, and MinD in complex with MinE on the membrane is shown in red.

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Initial conditions as described in the text. Membrane-bound MinD is shown in blue, and MinD in complex with MinE on the membrane is shown in red.

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Initial conditions as described in the text. Membrane-bound MinD is shown in red, and MinD in complex with MinE on the membrane is shown in blue.

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Initial conditions as described in the text. Membrane-bound MinD is shown in blue, and MinD in complex with MinE on the membrane is shown in red.

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The simulation is started with most of the MinD molecules bound to the membrane on one side of the cell. Membrane-bound MinD is shown in blue, and MinD in complex with MinE on the membrane is shown in red.

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The Ecogene Database of

We are grateful to Johan Hattne and Prof. Per Lötstedt for their contribution in developing MesoRD, to Prof. Måns Ehrenberg and Drs. Johan Paulsson, Martin Lovmar, Arvi Jõers, and Peter Sims, and an anonymous reviewer for many helpful suggestions and to Profs. Piet de Boer, William Margolin, and Eugenia Mileykovskaya for generously sharing images for

three dimensional

Green's Function Reaction Diffusion algorithm

Next Subvolume Method

phospathedylethanolamide

^{−}

phospathedylethanolamide deficient

reaction–diffusion master equation

Systems Biology Markup Language

wild type