The interaction parameters in the fine-grained lattice model can be tuned to reproduce the experimental AFM results (see 25 and Additional file 1). Comparison of the shape and length of the filaments at low protein coverage with the structures obtained from the model allows calibrating the different energy values. The set of values that provide a good description of the observed structures are: a bond energy U_b = 12(± 2)kT ≈ 7.2 kcal/mol, a small left-bent penalization U_- = 0.5(± 0.5)kT, U₊ = U_b to cancel the right-bent and a lateral attraction U_a = 0.5(± 0.1)kT ≈ 0.3 kcal/mol. This longitudinal bond energy is comparable to available experimental estimates 26. The lateral interaction is 20 times weaker than the polymeric bond, explaining the hierarchical structure observed: monomers forming filaments that bundle to form aggregates. A more complete description of the energy calibration procedure is presented in 25 and in Additional file 1. Fig. 2 shows that the same set of parameters used at different protein coverage gives a good match between theory and experiments over the whole range of observed structures, from isolated spiral clusters at low protein coverage to complex percolating (interconnecting) networks at high protein coverage.

The quantitative representation of curvature, longitudinal bonding and lateral interactions of the filaments considered in this integrative model correspond to the minimal set of parameters needed to reproduce the rich polymorphism observed in the high resolution AFM images. All these effects had been considered separately in previous FtsZ models 891012131416, but they had never been combined to explore their joint effect. Fig. 3 illustrates the aggregates formed when some of the monomer-monomer interactions considered here are turned off, as implicitly done in previous theoretical models. Each panel corresponds to a typical MC snapshot for the same protein coverage as in Fig. 3(a). Ideal polymer treatments, described in terms of the reaction rates for the growth, splitting and cyclation of the filaments, e.g. 14, would describe a version of our model in which repulsive excluded area and lateral interactions are eliminated. The MC result for this ideal polymer situation is presented in Fig. 3(b) with overlapping filaments and no tendency to form condensed aggregates. The inclusion of excluded area effects but not lateral interactions between monomers produces the structures in panel (c), still unable to reproduce the experimental observations. The formation of the observed dense aggregates only occurs in the presence of a lateral attraction between FtsZ filaments (panel (d)).

Fig. 3(b-d), with the same longitudinal bond energy and preferential angle between monomers, shows that the excluded area effects and lateral attractions act cooperatively to stabilize longer filaments. Some reaction rate models represent the cooperative polymerization in terms of effective "bundling coefficients" 13. However those coefficients would depend on protein coverage since the packing constraints shift the energy/entropy balance between the different structures. Our model includes all the "bundling effects" through the value of a unique parameter U_a = 0.5 kT, the lateral interaction, that accurately accounts for the bundling and observed cooperative lengthening effect independently of protein density, as shown in Fig. 3

A polymerization curve presenting the amount of unpolymerized FtsZ, ρ₁ as a function of the total protein concentration can be estimated from the model (see Fig.4). The curve agrees with previous experimental results in bulk solution that show a cooperative polymerization behavior. The insets are snapshots of the structures obtained along MC simulations at equilibrium for low surface coverage (between 0.26 and 1.06 picomol/cm²) of the FtsZ protein.

The effective bundling coefficients is included here through the unique parameter U_a = 0.5 kT, the lateral interaction, and accurately accounts for the bundling and observed cooperative lengthening effect at any protein density. Below ρ_T ≈ 0.4 picomol/cm² the model follows an ideal (isodesmic) behavior. The extrapolation (broken line) leads to the upper bound of unpolymerized protein concentration of ρ_o ≈ 0.0106 picomol/cm², at which the polymerized fraction would fill the system. The cooperativity induced by the lateral attractions, balanced by packing constraints and filament curvature, produces the saturation of ρ_o at 0.0091 picomol/cm² (86% of its ideal limit). The snapshots in the insets are typical configurations of our model in the regions I (ideal polymers), II (non-ideal polymers), and III (condensed aggregates). Notice that from the rough information given by ρ_o(ρ_T) we could not see any difference between the regimes II and III, although they clearly differ in the degree of condensation. In region II the interaction between filaments leads to the formation of longer filaments stabilized by lateral interactions, whereas in region III higher order roll-shaped aggregates with a well defined typical size are present. The AFM experimental information, equivalent to that given in the insets, provides more structural detail than any global measurement of the FtsZ polymerized fraction. It is able to clearly distinguish the condensation of clusters from the mere cooperative polymerization, thus providing experimental support to the predictions of this simple but accurate model of FtsZ polymerization. To illustrate the contribution of this model to previously existing ones, the square-lattice described in a very recent work 9 is used to simulate filaments formed on a planar surface of similar size and protein coverage than the ones observed experimentally (Fig. 3(e)). The square-lattice model includes a very strong longitudinal bond (17 kT) and a weaker lateral attraction (0.2 kT) between parallel filaments and neglects any flexibility or curvature of the filaments. Furthermore, the lateral attraction is only considered between filaments with the same orientation. These assumptions lead to the formation of isolated elongated drop condensates of different orientation interacting only through excluded area effects very different from the percolating and open polymorphic structures observed experimentally.

The inclusion of curvature and flexibility, together with the lateral interactions, is therefore crucial for the assembly of aggregates with the global structures and filament sharing observed experimentally.

The main biophysical interest in modeling filament interactions on a surface is to extrapolate this understanding to explore their behaviour on a cylindrical surface comparable to bacterial geometry. This extrapolation requires extra assumptions for which we have no experimental information. We have explored different possibilities to see how generic or specific is the formation of cylindrical rings. The direct transfer of the planar lattice model parameters to a cylindrical geometry, without any off-plane spontaneous curvature, does not produce the formation of ring structures. In this case we assume that the polymer does not feel the curvature of the cylinder and that the energy of the bond would not change if a piece of the cylindrical surface containing that bond were cut along a line to flatten on a plane. As shown in Fig. 5(a-b) the filaments form rolls or open interconnecting structures similar to those observed on a planar substrate. The formation of ring structures requires, in addition to the condensation effects of lateral attractions 910, either the presence of a spontaneous off-plane curvature 1516 or minimization of the on-plane curvature. The incomplete representations of the filaments in previous models prevented estimating the balance between both contributions but our fine-grained description now shows that tuning both, curvature and lateral interactions, can easily give rise to the formation of stable rings.

We first deal with modulations affecting filament curvature while the longitudinal and lateral interactions are directly extrapolated from the planar to the cylindrical geometry. Again, in this case we assume that the polymer does not feel the curvature of the cylinder. Eliminating the on-plane preferential curvature of the filaments by setting equal values for both bent bond penalizations allows the spontaneous formation of rings, as shown in Fig. 5(c). The results are quite robust with respect to the particular value of U_±, which may be tuned to produce very flexible filaments (like in this figure with U _± = kT) or much rigid ones (setting larger bending penalizations).

Alternatively, in Fig. 5(d) ring formation is also induced when adding an off-plane preferential curvature while keeping the on-plane curvature. Although we have no experimental data to quantify this off-plane energy, we can give an orientational dependence to the bond energy, and explore its effect using our lattice model. The result shown in the figure corresponds to a small bond modulation of ΔU_b = 0.2 kT, over the planar bond estimate U_b = 12 kT. This weak energetic bias in the alignment of the bonds is enough to overcome the tendency of the filaments to form rolls, and drives them to form ring condensates (see 11 and Additional file 1).

Affecting longitudinal bond energies can also condense the filaments into rings. Lowering filament longitudinal interactions outside a narrow cylindrical segment was used recently to simulate localized ring formation 9. The effect was associated to the role played in vivo by the presence of MinC, known to induce filament shortening 27, in the cell regions where the ring is not formed. Longer FtsZ filaments would migrate to the MinC-free segment, where the FtsZ coverage would become very large. In ref. 9 the formation of the ring demands a drastic reduction of the energetic value of the longitudinal interaction from 17 kT = 10.2 kcal/mol to 3.6 kcal/mol under the assumed influence of the MinC system. The same concept applied to our fine-grained lattice model is much more efficient inducing the formation of rings and a small reduction of the bonding energy produces an equivalent effect. The ring in Fig. 5(e) is formed with the energy parameters obtained for FtsZ on mica, without any added off-plane spontaneous curvature and only reducing by 2.5% the bond energy outside the central part of the cylinder (Fig.S3 in Additional file 1 refers to the movies presented as additional files 2, 3, 4, 5, 6 that illustrate the formation of rings on a cylindrical surface for each of the conditions described in Fig. 5(a-e)).

The condensed aggregates formed under the effect of lateral attractions are driven towards a ring shape by any of the weak effects presented above. The rings obtained through the three alternative minor modifications (Fig. 5(c-e)) are formed as collections of parallel short filaments that actively exchange monomers and span only a fraction of the circumference. The parameters governing the FtsZ polymers on mica directly transferred to a cylindrical surface show that fine tuning the energetic balance confers the filament aggregates the high plasticity needed to form dynamic rings.

Extrapolating the model to explain the formation of filament aggregates on a cylindrical surface also allows exploring possible force generating mechanism and quantitatively estimating the generated forces. The two interactions identified above as important in determining the ring forming capacity, lateral attraction between filaments and filament flexibility supporting a spontaneous curvature, have also been proposed as alternative force generating mechanism 916. Our model indicates that in fact both could be relevant and there is no need to consider them as being mutually exclusive (see Additional file 1).

The radial force produced by single FtsZ filament ring kept around a cylinder by lateral attractions was previously evaluated within Langevin dynamics simulations with an off-lattice model 10. We are now interested in exploring differences between the force generated by a long stabilized filament, and that of the Z-rings described here formed under GTP hydrolyzing conditions. These living polymers now retain their monomer exchanging capacity and form rings constituted by a collections of condensed disordered short filaments. We have used a mixed-approach in which the typical filament length distribution obtained with the triangular lattice model is transferred to the off-lattice model in ref. 10, for Langevin simulations of the equilibrated rings (see Additional file 1). The global structure of the ring is maintained by the attractive lateral interactions, and the distribution of the instantaneous radial force gives F_R = 90(± 45)pN, i.e. slightly larger and more spread than in the single filament case (see Figure S4 and text in Additional file 1)). The force mechanism based on the lateral attractions, proposed and analyzed in ref. 10 for a single filament, is therefore not essentially modified by the multi-filamentary composition of the Z-ring. Within the calibration of that lateral interaction obtained from the experimental AFM data, the estimated radial forces fall within the correct order of magnitude to explain the observed deformation of the cylindrical liposomes under the action of FtsZ. Fig. 5(f) shows a central deformation of a cylinder similar to the one described in 7. The figure was obtained from estimating the deformation of a multilamellar tubular vesicle subjected to the radial force of a ring formed when the bonding energy U_b decreases by 2.5% away from the central cylindrical segment (see Additional file 1 for details).