Simple membrane-based model of the Min oscillator

Min proteins in E. coli bacteria organize into a dynamic pattern oscillating between the two cell poles. This process identifies the middle of the cell and enables symmetric cell division. In an experimental model system consisting of a flat membrane with effectively infinite supply of proteins and energy source, the Min proteins assemble into travelling waves. Here we propose a simple one-dimensional model of the Min dynamics that, unlike the existing models, reproduces the sharp decrease of Min concentration when the majority of protein detaches from the membrane, and even the narrow MinE maximum immediately preceding the detachment. The proposed model thus provides a possible mechanism for the formation of the MinE ring known from cells. The model is restricted to one dimension, with protein interactions described by chemical kinetics allowing at most bimolecular reactions, and explicitly considering only three, membrane-bound, species. The bulk solution above the membrane is approximated as being well-mixed, with constant concentrations of all species. Unlike other models, our proposal does not require autocatalytic binding of MinD to the membrane. Instead, it is assumed that two MinE molecules are necessary to induce the dissociation of the MinD dimer and its subsequent detachment from the membrane. We investigate which reaction schemes lead to unstable homogeneous steady states and limit cycle oscillations, and how diffusion affects their stability. The suggested model qualitatively describes the shape of the Min waves observed on flat membranes, and agrees with the experimental dependence of the wave period on the MinE concentration. These results highlight the importance of MinE presence on the membrane without being bound to MinD, and of the reactions of Min proteins on the membrane.


Introduction
The combination of chemical reactions with diffusion under non-equilibrium conditions can give rise to a range of complex phenomena, such as formation of steady patterns, travelling waves or periodic oscillations [1][2][3]. The resulting patterns and structures often have spatial dimensions exceeding the molecular size by many orders of magnitude, and temporal periods much longer than the time scales of the underlying chemical reactions. Still, their size and dynamics are precisely determined by the parameters describing the interactions and motion of individual molecules.
These self-organizing physico-chemical phenomena are important for the functioning of living matter, as they provide accurate 'rulers' and 'clocks' that then determine the size and dynamics of the whole organism or its part [4][5][6]. Most prominent examples include spatial gradients of molecular concentrations that control the growth and development of cells, embryos and tissues [7][8][9], and temporal concentration oscillations forming the basis of circadian rhythms that adjust the functioning of an organism to the day/night cycle [10,11].
The Min proteins in the rod-like E. coli bacteria bind to the inner membrane surface, where they organize into a dynamic pattern oscillating between the two cell poles. The resulting exclusion of one of the proteins, MinC, from the cell center is important for accurate positioning of the cell division ring and for symmetric cell division [12]. The Min system consists of three proteins: MinD, MinE and MinC, but only the first two are

Results
The model To capture the essential features of the Min dynamics and pattern formation we aim to keep the model as simple as possible. This means using small number of species and reactions, and a simple geometry of the reaction domain.
The two proteins, MinD and MinE, are necessary and sufficient to produce the dynamic patterns on lipid bilayers. The following experimental facts concerning their interactions are generally accepted: MinD in its ATP-form binds to the membrane [34]; once bound, it strongly enhances binding of MinE to the membrane [35,36]; and MinE on the membrane stimulates the ATP-ase activity of membrane-bound MinD, leading to the release of both MinD and MinE from the membrane [35,37].
These facts are usually incorporated into the kinetic models in form of the following three reactions: MinD binds to the membrane resulting in membrane-bound MinD species; MinE binds to MinD on the membrane, forming a MinD-MinE membrane-bound complex, and MinD-MinE complex dissociates (following ATP hydrolysis) and leaves the membrane. These reactions alone, if assumed to take place in their simplest form, are not sufficient to make the steady state of the system unstable, and no interesting behaviour, such as oscillations or creation of patterns, takes place. A certain increase in complexity is therefore necessary.
All models based on chemical kinetics (i.e. not using effective reaction terms) additionally assume enhancement of MinD binding by MinD already present on the membrane. This is typically represented as an autocatalytic term in the differential equation for the concentration c of the membrane-bound MinC of the form = + c kċ ..., where the rate constant k is a linear function of the concentration of MinD in the solution above the membrane.
The published models additionally add at least one more species, in addition to the membrane-bound MinD and MinD-MinE complex. One possibility is based on the assumption that MinE from the dissociated MinD-MinE complex stays on the membrane for a certain time during which it can bind to another MinD and form a new MinD-MinE complex [16,27]. This idea results from experiments showing a longer residence time of individual MinE on the membrane in comparison to MinD [39], and on biochemical evidence [40].
A model combining autocatalytic MinD attachment with MinE as a third membrane-bound species (model M dD ) is depicted in figure 1(E). It is closely related to the models recently used to describe the Min dynamics [16,27].
The model proposed here includes MinE as a third species, but does not require autocatalytic MinD membrane attachment. It is inspired by the fact that MinD in its ATP-form is a dimer (figure 1(B)), with the two ATP molecules at the interface between the two monomers [38]. Upon ATP hydrolysis the dimer dissociates; one can imagine that the ATP molecules act as a 'glue' holding the dimer together as long as they are not hydrolysed. The dimer has two symmetrically positioned binding sites for MinE at the interface of the two monomers. We assume that MinE binding to one binding site stimulates the ATP-ase activity of MinD, leading to hydrolysis of one of the ATP molecules. Binding of another MinE to the other symmetric site stimulates hydrolysis of the other ATP. As long as at least one ATP is not hydrolyzed, the whole complex holds together. Only after both ATP are converted to ADP, the MinD dimer dissociates, releasing also both MinE molecules since their binding sites at the MinD monomer interfaces cease to exist.
While MinD dimer is stably attached to the membrane by two membrane-targeting sequences (mts), the MinD monomer is anchored by only one mts, providing only a weak membrane affinity, and therefore dissociates from the membrane quickly.
The susequent binding of two MinE molecules to MinD 2 and their simultaneous release from the complex makes the reaction system sufficiently complex to exhibit instability and consequently rich dynamics. As we show below, the requirement of two MinE molecules for the complex dissociation is not stringent, and a certain degree of the MinD 2 -MinE complex dissociation stimulated by only one MinE can be tolerated.

Well-mixed system
First, we consider a well-mixed system: the situation where the membrane diffusion is infinitely fast and the effects of diffusion-limited mixing are neglected. The concentrations of all species y i are then spatially homogeneous and depend only on time. Later on we will investigate the effects of finite diffusion.
The model proposed here assumes three membrane-bound species: MinD 2 (concentration y 1 ), MinD 2 -MinE complex (y 2 ), and MinE (y 3 ). Although MinD is thought of as a dimer, the only implication of this fact is the existence of two binding sites for MinE. No MinD dimerization kinetics is included in the model.
The following reactions constitute the model (figure 1(C)): MinD 2 binds to the membrane (rate constant k D ); MinE binds to one of the binding sites on MinD 2 , this being either MinE from the solution (rate constant k dE ) or MinE already present on the membrane (rate constant k de ); MinE binds to the second binding site on the MinD 2 (again, either MinE from the solution or from the membrane); this is followed immediately by the dissociation of the complex, and detachment of MinD from the membrane. Finally, MinE leaves the membrane with the rate constant k e (∅ in the following reactions stands for no membrane-bound reactants or products).
k MinE e The following system of differential equations describes the dynamical system: It is convenient to change to non-dimensional quantities t′ and ′ y i by expressing time in the units of k 1 e and concentrations in the units of k k D e : Then we obtain (after dropping the primes of t′ and ′ y i ) a system of equations that we will refer to as model M e2 (the two-step MinD dissociation model): where the non-dimensional rate constants κ de and κ dE are related to k i as: dE dE e The whole system depends now on only two parameters, κ de and κ dE , making the analysis of its behaviour in the whole parameter space tractable. The system of equations (12)- (14) has two steady states ( = y˙0 i ), with only one of them being positive: . Linear stability analysis shows that the positive steady state can be either stable or unstable, depending on the values of κ de and κ dE (figure 2( A)). The steady state is unstable in the region of relatively large κ de and relatively small κ dE . In other words, a stable state can be made unstable by increasing κ de or decreasing κ dE , within the parameter range in figure 2(A). This points to the importance of the presence of MinE on the membrane for the instability: the reactions (3) and (5) in which the membrane-bound MinE participates must proceed with sufficiently high rate compared to the reactions (2) and (4) (MinE binding from the bulk) for the system to become unstable.
When the steady state is unstable, the system exhibits periodic oscillations of the concentrations y i of all three species. The oscillations are close to sinusoidal near the bifurcation boundary (point 1 in figure 2(A), figure 2(B)), and increasingly deviate from this shape as one moves deeper into the unstable region of the parameter space (points 2-4). Although the concentration profiles vary with κ de and κ dE , they share common features: starting at the minimum of MinD 2 , the concentration y 1 of MinD 2 increases, followed by a slower increase of the concentration y 2 of the MinD 2 -MinE complex. The concentration y 3 of MinE increases only slowly, until it is at some point sufficiently high to overturn the trend of increasing y 1 through the reaction (3) and shortly thereafter to make y 2 decrease through the reaction (5). This reaction generates even more MinE on the membrane, leading to a sharp increase in y 3 and depletion of both y 1 and y 2 . After the concentration of MinE decreases again due to spontaneous MinE detachment, the whole cycle can start again.
The depletion of the MinD 2 -MinE complex and of the total MinE can occur extremely fast relative to the time window where all three concentrations increase (figures 2(A), (B), points 2 and 3). This behaviour persists when diffusion is added to the model, as shown below, and is also experimentally observed as a sharp edge of travelling Min waves [14]. In contrast, we could not find a set of parameters that would lead to similarly abrupt concentration changes in a model including autocatalytic attachment of MinD (model M dD in figure 1(E)). Figure 3 shows the concentration oscillations in the model M dD with a relatively steep decrease of MinE concentration. However, the difference between the overall rates of MinE concentration increase and decrease is much less pronounced than in the model M e2 or in the experimentally observed Min waves (figure 6(E)). This failure to reproduce the sharp MinE decrease appears to be a typical feature of the models relying on the autocatalytic MinD attachment [14,27]. Figure 2(A) also shows the dependence of the oscillation period T on the rates κ de and κ dE . The rate κ dE can be assumed to depend linearly on the concentration of bulk MinE. We can therefore compare the dependence κ T ( ) dE of the period T on the rate κ dE with the experimental dependence of the wave period on the bulk MinE concentration, published previously [14]. Fitting the experimental data to the calculated κ T ( ) dE shows that the trend of κ T ( ) dE in the model M e2 agrees very well with the data (figure 2(C)). The parameters of the fit are: The rate k e = 0.3 s −1 implies a MinE residence time of 3.3 s, if detachment were the only process of removing MinE species. This is compatible with the reported membrane residence times of MinE [39] between 6 and 12 s, as these experiments cannot distinguish between MinE and MinD 2 -MinE complexes, and include also the effect of MinE binding to MinD 2 (reaction (3)) and therefore a longer residence time on the membrane.

Effect of direct MinE membrane binding
In the model M e2 it is assumed that MinE binds to MinD 2 present on the membrane (reactions (2) and (4)) but is not able to bind to the membrane directly. It is interesting to consider the opposite situation, when MinE binds directly to the membrane, and only this membrane-bound MinE, not MinE from the solution, can bind to MinD 2 present on the membrane. Thus, the reactions (2) and (4) are replaced by direct MinE attachment: k MinE E Assuming rate constant k E for the MinE binding to the membrane, we obtain the model M E (the direct MinE membrane attachment model) as a mathematically simpler alternative to M e2 (figure 1(D)): where the non-dimensional form of k E , κ = k k E E D is used. The system of equations (18)(19)(20) has only one steady state, which is positive: Figure 4( A) shows the results of linear stability analysis of the steady state in the two-dimensional parameter space (κ , de κ E ). For κ E values larger than 0.5 the steady state is stable, regardless of κ . de At κ < 0.5 E , a region of unstable steady states and oscillatory solutions exists. The temporal profiles of the concentrations y i (t) of these oscillating solutions are qualitatively similar to those of the model M e2 (figure 4(C)).
As κ E becomes lower, approximately coinciding with the white region in figure 4(A), the concentrations y 1 and y 2 reach unrealistically high maximum values, and the period T becomes very long. In reality, this behaviour would be prevented by saturation of the membrane surface with proteins. The unbounded growth in simplified models, such as M E , can be eliminated by taking into account the finite number of binding sites at the membrane (that is, by including surface saturation), or by adding reverse reactions, which have been neglected so far [41]. Such modifications can in some cases turn the unstable steady state into stable. If we include spontaneous detachment of MinD 2 from the membrane with the rate constant k d , the steady state stays however unstable, and the concentrations of all species remain finite and within reasonable limits. This back-reaction of reaction (1) results in an additional term κ − y d 1 in equation (18). The stability diagram with instability region extended in this way for a relatively small value of κ = 0.01 d is shown in figure 4(B). The comparison of models M e2 and M E shows that the way MinE attaches to the membrane is not crucial for the existence of instabilities, and also for the basic characteristics of the shapes of the concentration profiles of the reacting species. It is therefore reasonable to speculate that the combination of the models M e2 and M E , where both ways of MinE attachment to the membrane are allowed, would also display instability and similar concentration oscillations. This is relevant, because even though it is usually assumed that MinE does not bind directly to the membrane, there is experimental evidence that MinE can bind to the membrane weakly [42].

Effect of hydrolysis stimulation by a single MinE
The M e2 model assumes that the dissociation of the membrane MinD 2 -MinE complex requires subsequent binding of two MinE to the MinD dimer. This requirement can be relaxed without destroying the instability of the steady state and losing the oscillatory behaviour.
Let us add the possibility that binding of the first MinE to the MinD dimer can be followed by hydrolysis of both ATPs, complex dissociation and detachment of MinD from the membrane. We combine all these reactions into a single step, described by the rate constant κ d1 : This reaction enters the model equations by adding the term κ − y d1 2 to equation (13) and the term κ + y d1 2 to equation (14). The system has now three steady states, only one of which is positive.
The MinD 2 -MinE membrane complex can now disappear in essentially two ways: the newly added reaction (κ d1 ) competes with the two reactions (4) and (5)

Including diffusion on infinite domain
So far we have considered a well-mixed chemical system without spatial coordinates. Finite diffusion prevents efficient mixing of the reacting species and can in general lead to inhomogeneous concentrations and formation of spatial patterns. The obvious questions one might ask are: can the system of chemical reactions in the model M , e2 together with finite diffusion, exhibit travelling wave patterns as observed in the experiments on supported membranes [14], and how similar are the shapes of these waves in the simple model M e2 to the experimental waves?
To account for diffusion, the diffusion terms with generally different diffusion coefficients D i are added to the reaction equations (18)-(20): In the following, it is assumed for simplicity that the diffusion coefficients of MinD 2 and MinD 2 -MinE complex on the membrane are equal: = p 1.
The parameters b i can be understood as a relative measure of diffusion speed compared to the wave propagation velocity. Low b i means small dispersion of molecules due to diffusion compared to the propagation of the phase of the wave over the intrinsic time scale of the reaction-diffusion system (determined by the reaction rates), and therefore a weak effect of mixing due to diffusion.
Solving equations (25)- (27) numerically gives a periodic wave solution for every value of b i . This means that for a fixed values of the reaction rates and the diffusion coefficients D i there is a range of solutions with different propagation velocities v and wavelengths λ = vT . The parameters of the propagating wave are therefore not determined uniquely.
Some solutions can be excluded on basis of their stability with respect to small perturbations: in unstable solutions even the smallest perturbations will grow, eventually causing the system to converge to a stable solution. In two dimensions the waves are usually circular or spiral, and only approximately planar; the stability conditions are therefore likely to differ from the 1D case considered here. Nevertheless, investigating the stability of solutions in 1D still provides important insights. Linear stability analysis of equations (22)-(24) identified a region in the space of the parameters b 1 and p 3 where the solutions are stable ( figure 6(A)). In general, the waves are stable for small values of b i , that is, for large propagation velocity and wavelength. The size of the stability region depends on the rate constants of the chemical reactions; for a fixed value of κ de it becomes larger with decreasing κ dE . The Min dynamics are spatially synchronized into coherent waves due to diffusional coupling. In the stable region, this coupling is not strong enough to destabilize the waves through strong mixing, which would otherwise lead to a homogeneous oscillating state.
Even though the solutions in the unstable region can be excluded, there is still a broad range of stable solutions that could in principle be realized. The experimentally observed Min waves on supported bilayers sometimes exhibit variations of wavelengths within one sample [43], although in many cases this range is rather narrow. Other chemical reaction-diffusion systems commonly show a broad range of wavelengths in different parts of the same sample and at the same time [44]. The experimental waves are likely to be to some extent influenced by the boundary conditions imposed by the sample container, and by the initial condition defined by the way the wave was initiated. In experiments, one strives to minimize the effects of these factors by using a large membrane area and by introducing a sufficient delay between the wave initiation and observation.
The period T of the waves depends rather weakly on b 1 and p 3 ( figure 6(B)), in comparison with its dependence on the rates κ de and κ dE (figure 2(A)). One can therefore state that the wave period T is determined predominantly by the reaction part of the system. ) approach the concentration profiles in the absence of diffusion. With increasing b 1 , that is, with decreasing velocity and also wavelength, the wave maxima and the wave modulation depth (the relative difference between the concentration minima and maxima) decrease. On the boundary between the stable and unstable regions, the wave modulation is still relatively high (more than 85% for the concentration profiles shown in thick lines in figure 6(C)). This is in agreement with the observed fact that the experimental Min waves exhibit deep modulation, that is, relatively high contrast between the concentration minima and maxima [14,15,39,43]. The MinE molecule is smaller than the MinD dimer or the MinD 2 -MinE complex; it is therefore interesting to consider the situation when MinE diffuses faster on the membrane than the other two species ( > p 1 3 ). ). When MinE diffuses faster, the concentration profile y 2 of the MinD 2 -MinE complex and, more dramatically, the total MinE concentration ( + y y 2 3 ) exhibit a sharp narrow maximum before a steep decay. Such a sharp peak near the falling edge of the MinE wave on a supported bilayer is indeed observed experimentally [14,39] (figure 6(E)), and is reminiscent of the MinE ring observed in cells [45].
Observing the MinE reactions around the narrow peak of MinE (y 3 , red curve in figure 6(D)) provides an insight into the mechanism of the Min wave propagation: at the falling edge of the MinE peak the dominant reaction of MinE is its detachment from the membrane (reaction (6)). Contrary to this, at the rising edge of the MinE peak the concentration of MinD 2 -MinE is high, leading to net production of new MinE mainly via the reaction (5). The balance between these processes at the rising and falling edges of the MinE peak results in a peak of stable shape moving in the direction of high MinD 2 -MinE concentration. The localized MinE peak (or a line, in 2D) resembles a propagating dissipative soliton [46]. The reaction-diffusion system of equations (22)-(24) then becomes:   The lengths x are expressed in the units of the domain length L; the integrals in equation (31) therefore cover the whole domain. Consequently, the units of the diffusion coefficients are L k e 2 . Since the parameter space became rather large through the addition of two more parameters (d t and e t ) we did not perform a detailed analysis of equations (28) A closer look at the concentration profiles of all three species ( figure 7(B)) shows that the wave progressing from the center towards the end of the domain is similar to the travelling waves on infinite domain ( figure 6(D)). Interestingly, the wave always propagates from the center towards the alternating ends, not from one end towards the other. This behaviour is known from living cells [13] and synthetic enclosed compartments [19], but is not observed on finite membrane patches with unlimited supply of Min molecules [16].
In long filamentous cells [27,47] and in long artificial enclosed microcompartments [19], a standing-wavelike pattern with the number of nodes depending on the cell length has been observed experimentally. This pattern can be thought of as composed of several oscillating short-cell patterns aligned along each other. The periodic oscillation between the two cell poles in the shortest cells corresponds to one half of the wavelength of the patterns in the long cells.
Because of the reflective boundary condition, the same effect can be obtained with the system of equations (28)- (30): doubling the domain length L, or equivalently, reducing the diffusion coefficient by a factor of four, allows solutions with one period per domain length (figure 7(C)). Depending on the exact value of D, this pattern may become unstable, converting into a travelling-wave pattern (figure 7(D)), which appears to be stable. A similar travelling-wave pattern has been observed in longer cells [27].
Transitions from one stable oscillating pattern to another as the domain length slowly increases are observed in growing cells before the division. The likely explanation is that the initially stable pattern becomes unstable with longer domain length and converts to a new stable state. The initial state in figure 7(D) is however unstable for the used parameters, because no change of domain length or perturbation was necessary to induce the transition to a different pattern. Transitions between two stable patterns without any changes of parameters have been observed in particle-based simulations of the Min dynamics in two dimensions [48]. This result points to the importance of stochastic fluctuations for the conversion between stable patterns separated by an unstable region, a phenomenon not directly revealed by deterministic models.
The Min dynamics shown in figure 7(A) and (C) can be viewed as waves propagating in alternating directions. The wave propagates until it collides with the boundary or annihilates with another wave travelling towards it; in its wake a new wave emerges, moving in the opposite direction. It appears that in closed-volume geometries where the supply of bulk Min proteins is limited this dynamics is preffered to waves travelling in the same direction. Indeed, this is the predominant behaviour observed in short cells [13] and in enclosed compartments [19], in long filamentous cells [27,47], and even in larger 2D compartments where the total volume is limited by their small height [49].

Discussion
In this work we explored simple one-dimensional models of Min dynamics that consider only the species and reactions on the membrane, and do not contain direct autocatalytic reactions. In living cells, the supply of Min molecules in the bulk is limited, therefore it is conceivable that bulk reactions, such as conversion of MinD-ADP to MinD-ATP, could be a vital part of the reaction system underlying the Min oscillations. However, Min proteins exhibit complex dynamics also when the supply of Min molecules is virtually infinite, as the experiments with wave patterns on supported lipid bilayers show [15,39]. The fast protein diffusion in the solution above the membrane in relation to the speed of the propagating waves in these experiments guarantees substantial mixing of the bulk solution on the relevant time scales. Although reactions and diffusion in bulk may still influence the observed patterns, as suggested by experiments [16] and theoretical work [27], they are unlikely to be the decisive factor behind the Min dynamics. It is further reasonable to assume that the molecular mechanism of the Min dynamics in cells and on the planar bilayers is the same. For these reasons we considered only reactions and diffusion on the membrane.
Experiments show that the concentration of membrane-bound MinD increases more than linearly with the concentration of the MinD in solution [36,50]. There is, however, no evidence that this happens as a consequence of an autocatalytic binding process as it is commonly implemented in the models [16,26,27]. Similar nonlinear increase could be caused, for example, by MinD aggregation on the membrane following simple membrane binding. As noted before [24], this type of aggregation is, however, not sufficient to generate instabilities of the steady state leading to the observed Min behaviour. Furthermore, the autocatalytic binding would have to be a complex process with at least two steps: transient interaction (binding) of the membranebound and bulk MinD (since one promotes membrane binding of the other), and subsequent dissociation of this complex into two membrane-bound MinD molecules. Although this process can in principle take place, it is certainly worth inspecting alternative Min models that do not rely on its existence.
The models presented here are relatively simple in comparison with other suggested models of Min dynamics. All of them have three reacting species. This is the minimum number: it has been shown that systems with only two reactants and at most bimolecular reactions cannot show limit-cycle oscillations [51]. Wilhelm [33] identified the smallest chemical reaction system that exhibits instability via Hopf bifurcation. The models M e2 and especially the direct MinE membrane attachment model M E are close to this minimal system in terms of their complexity, defined in terms of the number of reactive species, the number of total and bimolecular reactions and the number of quadratic terms in the reaction equations.
The Wilhelm's model contains an autocatalytic term; its instability can however be traced to a negative feedback loop. Contrary to the autocatalytic MinD attachment model M dD , the models M e2 and M E have no direct autocatalytic term, it is therefore interesting to see if it is possible to pinpoint the cause of their instability.
The origins of instability in chemical reaction systems have been classified by Tyson [52]. This classification is based on the Jacobian matrix = ∂ ∂ a F y { } ij i j of the chemical reaction system = y F ẏ ( ) i i j . For the steady state to be qualitatively stable (meaning stable regardless of the parameter values) the sign pattern of the Jacobian matrix in the steady state has to fulfil several conditions. These conditions guarantee that the eigenvalues of the Jacobian matrix have negative real parts, that is, that any perturbation of the steady state dies out. The unstable steady states are classified depending on the stability conditions that are violated.
The steady-state sign patterns of the Jacobian matrices for the three models considered here (figure 1) are: The models M e2 and M E have two destabilizing elements: a negative feedback loop: MinD 2 → MinD 2 -MinE → MinE ⊣MinD 2 ( < a a a 0 21 13 32 ) and an indirect autocatalysis (competition): MinD 2 ⊣ MinE ⊣MinD 2 ( > a a 0 13 31  ). This third destabilizing factor can also appear in the model M e2 if the approximation of equal rate constants of the reaction pairs (2)-(4) and (3)-(5) is not made.
Since there are two or even three violated stability conditions, it is not possible to identify a single reaction or an element of these models as a sole reason for the instability. It is, however, interesting to note that the direct autocatalysis of MinD in the model M dD is not one of the destabilizing elements, since < a 0 11 . In other words, in the steady state of the model M dD , an infinitesimal increase of MinD concentration will induce its decrease back towards the steady state value and not further away from it, as the autocatalytic attachment term might seem to suggest. This happens because the combined rates of the two MinE-attachment reactions (reactions (2) and (4), both decreasing the MinD concentration) will increase more strongly than the autocatalytic MinD attachment (increasing the MinD concentration). The presence of the MinE species on the membrane and its bimolecular interaction with MinD appears therefore to be more relevant for the Min dynamics than the mode of MinD attachment (plain or autocatalytic) to the membrane.
The model M e2 inintroduced here leads to concentration profiles of Min proteins qualitatively similar to those observed on supported lipid bilayers. Most importantly, it reproduces the sharp concentration decrease at the end of the wave, and even the narrow sharp maximum of MinE preceding this decrease. It also agrees with the experimental dependence of the wave period T on the bulk MinE concentration. The rate constant of MinE detachment from the membrane k e obtained from the fit to these data is consistent with other experimental data.
An important feature of the model is its robustness to modifications. It appears not to be particularly important how MinE reaches the membrane-either by binding to membrane-bound MinD or by direct binding-the existence of instability and the general shapes of the profiles are not affected. Also the assumption that two MinE molecules are required to induce MinD detachment from the membrane is not absolute; the instability is preserved even if this is only one of the possible pathways.
The model also exhibits qualitatively similar dynamics to those in living cells when formulated for finite interval and constant total number of Min molecules. The difference between the model solutions and the oscillation patterns observed in cells [53] can possibly be linked to the rather crude approximation of the full 3D cell geometry by the one-dimensional interval. For a typical cell shape, the hemispherical end cups closing the cylindrical cell volume, completely ignored in the 1D approximation, can amount to more than 1/3 of the total membrane surface.
The reduction of dimensionality to 1D is also relevant for the dynamics on infinite domain: while the results shown in figure 6 may provide a good description of the spiral waves far from the spiral center, where they can be locally well approximated by a planar wave, the 1D model cannot accurately describe the two-dimensional center of the spiral. When using 1D approximation, particularly when dealing with finite domains, solutions that cannot be simply reduced to one dimension may exist in the original dimension [24], and the patterns stable in 1D approximation may become unstable in higher dimensions due to the existence of additional perturbation modes.
An obvious extension of this work is the solution of the presented model in various 3D geometries: normal and filamentous cell shapes, and synthetic compartments of different aspect ratios. Exploring the theoretical dependence of the wave parameters (period, wavelength, the shape of the concentration profile) on the Min protein concentrations, membrane diffusion coefficients and membrane binding affinities will provide predictions that can be tested experimentally. These experiments will ultimately show whether the proposed interaction of two MinE molecules with MinD before the detachment of MinD from the membrane is indeed the key factor behind the Min dynamics.