Coupling FtsZ filaments and morphodynamics during bacterial cell division

Abstract

FtsZ filaments play a central role in bacterial cell division. A theoretical framework is developed by combining a phase field model for rod-shaped cells with a kinetic description for FtsZ ring maintenance, in order to investigate cell morphodynamics during bacterial cell division. The cell division time and cell shape are collectively determined by the curvature elastic energy of the cell membrane/wall and the constriction force generated by FtsZ rings. The dependence of cell morphodynamics during cell division on different initial states of rod-shaped cells and FtsZ rings, such as the aspect ratio and the FtsZ concentration in cells, Z_T, is extensively studied. The obtained results with the measured experimental parameters are found to be well comparable to the observed results physiologically. Likewise, it is found that the quasi-steady state of FtsZ rings accords with the theoretical result derived from the kinetic description of FtsZ rings. In addition, the morphological phase diagram is presented as a function of the FtsZ concentration in cells, Z_T, and the aspect ratio of rod-shaped cells, given that rod-shaped cells with high Z_Ts and/or aspect ratios tend to divide. It will be straightforward to extend the theoretical framework to other complicated systems relevant to filamentous proteins and cells, for example, maintaining structural integrity, serving as a template for cell growth, cell adhesion and motility, and mechanical signal transduction.

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I. Introduction

Filamenting temperature sensitive mutant Z (FtsZ) protein is a GTPase,¹ i.e. it binds guanosine triphosphate (GTP) and hydrolyzes it to guanosine diphosphate (GDP).^2–4 Under different in vitro experimental conditions, FtsZ proteins in the presence of GTP can assemble into a large variety of dynamic structures, such as short filaments, bundles, ribbons, minirings, helices and tubes.^5–9 A typical length of short FtsZ filaments, as observed experimentally at the physiological concentration, is about 125 nm, which generally consists of 30 FtsZ monomers.¹⁰ FtsZ filaments play an important role in providing mechanical stability and defining the bacterial shapes, for example, cell division in E. coli and other rod-like bacteria.¹¹ FtsZ filaments assemble into a thin ring in the midcell region (the bacterial cell-division site). The thin ring, called the FtsZ ring, recruits other proteins to form the septal ring that exerts the force to divide the cell. However, the mechanism remains unclear as to how the dynamic FtsZ ring forms underlying the cell membrane and plays a central role in the force generation during cell division.¹²

The FtsZ ring has a dynamic structure, where filaments assemble into the ring and disassemble from it.¹³ The mechanism of its dynamic behavior can not be completely explained by the dynamic instability of microtubules or treadmilling of other filamentous proteins.^14,15 Of course, the GDP-bound FtsZ monomers that have been depolymerized from the FtsZ ring can be rapidly exchanged with GTP-bound ones in the cytoplasm for subsequent polymerizations, so that the cytosolic filaments consist entirely of GTP-bound monomers. Likewise, FtsZ monomers are observed to be rapidly exchanged between FtsZ rings and the cytoplasmic pool, and a half time of turnover is measured experimentally as about 10 s.¹⁶

One mechanism for force generation is proposed in several theoretical models, arising from curvature transitions induced by GTP hydrolysis.^{5,7–9,17–20} However, the conformation switch is observed experimentally to be independent of the nucleotide binding states of the FstZ monomers. At the same time, crystal structures of single FtsZ monomers reveal the structure of the GTP-bound monomer to be similar to that of the GDP-bound one.¹⁷ Subsequently, the effects of nucleotide binding on the structure of a FtsZ dimer are extensively studied by molecular dynamic simulations, supposing that the FtsZ dimer structure depends on its nucleotide binding state.¹⁸ As a whole, in the presence of GTP hydrolysis, FtsZ filaments are expected to possess three different curved conformations.^{5,7–9,19,20} The first is a straight conformation with a curvature radius larger than 0.5 μm when the FtsZ protein binds GTP molecules. The intermediate curved conformation corresponds to a 2.5 degree bend between monomers, producing a curvature radius of about 200 nm. Finally, the highly-curved conformation (miniring) contains 16 monomers bound to GDP molecules with a 23 degree bend at each interface, and its intrinsic curvature radius ranges from 12.5 to 100 nm. Therefore, the coupling of GTP hydrolysis at each of the FtsZ monomers within the FtsZ filament could cause a transition from a straight conformation to a curved one, by which the FtsZ filament would provide the force to constrict the cell wall/membrane.

A completely different mechanism for force generation has been suggested in other theoretical models.^21–24 These models propose that the FtsZ ring consists of several short filaments, and these short filaments can interact with each other via lateral and longitudinal bonds. These lateral and longitudinal bonds provide a negative (favorable) free energy. Meanwhile, a constriction force would be generated as the cooperativity of the polymerization, the condensation and bundling of the FtsZ filaments proceed, together with the increased number of lateral and longitudinal bonds.

So far, research focused on the force generation of FtsZ rings has mainly disregarded the shape transformations of cells. The coupling of FtsZ rings and cell shape is expressed in such a way that the constriction force is a free parameter, independent of the dynamic turnover of FtsZ rings.²⁵ The phase field method has a reputation to be very general and applicable to complex microstructural phenomena, and it has been extended to study vesicle shape dynamics recently,^26–29 in particular, cell morphodynamics due to phase transitions between cross-linked actin filaments and bundles.^30,31 In the present study, one theoretical framework is firstly developed to study the coupling of cells and FtsZ ring systems, and combines a kinetic description of FtsZ ring maintenance with phase field dynamics for cell morphodynamics. The greatest advantage of the theoretical framework is to capture the collective behaviors of shape transformations of cells and the constriction forces generated by dynamic FtsZ rings. Furthermore, the theoretical framework presented here is easily extended to investigate other complicated systems relevant to filamentous proteins and cells.

This paper is organized as follows. Section II contains a detailed description of the theoretical framework by combination of the phase field model for cells and one kinetic model for FtsZ ring turnover, together with the numerical calculation method for the combined theoretical model. In Section III, a typical dynamic process during bacterial cell division is presented, and the steady states of cells are discussed as functions of different initial states of cells and FtsZ rings, such as the total concentration of FtsZ in a cell, the total number of FtsZ monomers within the ring, the total number of FtsZ–GDP monomers within the ring, the initial mean length of FtsZ filaments within the ring, λ, and the aspect ratio of cells. Additionally, a morphological phase diagram is presented as a function of the total concentration of FtsZ in the cells and the aspect ratio of cells. Finally, a brief summary and outlook are given in Section IV.

II. Theoretical method

Our model is motivated by experiments on E. coli and FtsZ rings, where cell division for E. coli is observed in the presence of FtsZ rings in the midcell.¹¹ Therefore, one new theoretical framework that couples cell division of bacteria with force generation of FtsZ rings, is developed to study dynamic processes of cell division and changes in constriction forces. A phase field model that can avoid tracking the explicit boundary and has been successfully applied to solve free-boundary problems,^26–30 has been widely used to study cell morphodynamics. At the same time, a kinetic description of FtsZ ring maintenance has been developed,¹³ including the incorporation of new short filaments into the ring, GTP hydrolysis in the ring and rapid nucleotide exchange in the cytoplasmic pool, disassembly of GDP-bound monomers within the ring, and the mechanical characterization of GDP-bound monomers in the FtsZ ring. However, the disassembly of GTP-bound monomers and the contribution of GTP-bound monomers within the ring to the constriction force are neglected in the previous work.¹³ In this work, the phase field model and the kinetic description of FtsZ rings, including the effect of GTP-bound monomers within the ring, are combined for the first time.

A typical bacterium, such as E. coli, is approximately assumed as a two-dimensional rod-shaped cell with a fixed surface area A₀, which is equivalent to the volume in three dimensions, and the dynamic turnover of short FtsZ filaments is considered to couple with GTP hydrolysis, see Fig. 1. The FtsZ ring that consists of several short FtsZ filaments, as expected, is always distributed in the midcell. In order to distinguish the interior and exterior of the rod-shaped cell, an auxiliary phase field, ϕ, is introduced, where ϕ takes on the value of 1 in the interior of the cell wall but ϕ = 0 in the cell exterior. This field varies abruptly in the diffusive interface between two limited values, ϕ = 1 and ϕ = 0. The width of this interface, ε, is used to describe the thickness of the cell wall.

Fig. 1 (a) Schematic illustration of a rod-shaped cell with FtsZ rings located in the middle of the cell along the horizontal direction. The yellow color indicates the diffusion interface, with a width of ε, the grey color is the interior of the cell, where ϕ = 1, while in the white area ϕ = 0. In green is the FtsZ ring located in the middle of the rod-shaped cell. (b) Enlarged view of the FtsZ ring in the midcell that consists of short FtsZ filaments.

Therefore, the dynamic evolution equation for the phase field is given by^26,27,30

where the advection term couples the phase field to the local flow velocity, u, the parameter ε controls the width of the cell wall, Γ is Langrange multiplier, the local curvature is denoted by , and G = 18ϕ²(1 − ϕ²) is a double well potential with minima at ϕ = 1 and ϕ = 0.

In the present study, the shape of the cell membrane or cell wall can be determined by the cooperation of various forces, including the surface tension, the bending force, and the pressure that constrains the cell area of vesicles, as can the volume in three dimensions. In addition, the perimeter of the cell wall is observed experimentally not to be fixed,³³ allowing proteins or lipid molecules to enter or escape during either cell wall expansion or cell division. We also consider the radial and constriction forces generated from FtsZ rings in the midcell, and the effective friction caused by cell division and cell motility. At first, the surface energy that is proportional to the cell’s perimeter L, is able to be implemented in the phase field formulations as follows^28,29

where γ is the surface tension. Then, the area density of the surface tension force is derived as follows

Here, this area density can be converted into a line density with F′dr = F_te ε|∇ϕ|² dr.³⁰ Therefore, the surface tension force with a line density is defined as

The bending energy H_be of the cell wall³² is written in the phase field formulation as

where κ is the bending rigidity of the cell wall. Note that the spontaneous curvature is taken to be zero. The area density of the bending force can be derived and converted into a line density as above. The bending force with a line density is given as follows³⁰

The cell area in two dimensions, A = ∫ϕdr, which is equivalent to the volume in three dimensions, is observed experimentally³³ to be conserved during cell division, indicating that the perimeter is not highly conserved. Therefore, a constraint term is required to guarantee the cell area and can be expressed in the form of phase field.

where M_A is large and A₀ is the prescribed area. The force derived by the constraint term is

The coupling of FtsZ rings with cells provides a retraction force to the cell wall/membrane, thereby leading to cell division. In order to simply describe the mechanical properties of the FtsZ ring, a stable and smooth FtsZ ring is assumed to form in the midcell. This assumption of a smooth ring is justified by the mechanical averaging across the thickness of the ring due to the uniform distribution of hydrolyzed and unhydrolyzed monomers. The mechanical energy H_z stored in the FtsZ ring as a function of its radius and the size of FtsZ monomers is assumed to obey linear elasticity, given by³⁴

where B is the bending modulus of a FtsZ filament. κ(s) is the preferred curvature as a function of the position in the ring (s) and is defined as either κ_T or κ_D dependent on the hydrolysis state of that position (GTP- or GDP-bound, respectively) in the ring. δ is the size of a FtsZ monomer and R is the radius of the ring or the radius of the midcell. S and S_D are the total number of FtsZ monomers and GDP-bound FtsZ monomers within the ring, respectively. Calculating the derivative of the mechanical energy with respect to the radius, R, the radial force generated by the ring is given bywhere n denotes the generated force along the normal direction to the cell wall.

In order to achieve the mechanical energy, H_z, of the FtsZ-ring given by eqn (9), the total number of FtsZ monomers and GDP-bound FtsZ monomers within the ring have to be obtained in advance. This involves a kinetic description of FtsZ ring turnover, including the incorporation of free short filaments and FtsZ monomers into the ring, hydrolysis of GTP-bound monomers within the ring, disassembly of both GTP-bound and GDP-bound monomers at the tip of the filament within the ring, as well as the mechanical characterization of the FtsZ ring in terms of force generation. Compared to the previous kinetic model proposed by Cytrynbaum,¹³ the contributions of GTP-bound monomers within the ring to disassembly and to force generation are considered. Two main physical quantities are introduced, and these are the length distribution of filaments in the ring, p(l, t), and the number of hydrolyzed monomers, S_D(t) in the ring, where l is the number of monomers within the filament at time t. Subsequently, the total number of FtsZ monomers, S, and filament tips, F, in the ring can be derived by and , respectively. The cytosolic concentration of FtsZ monomers can be obtained from Z_T − S/(N_cA), where Z_T is the total concentration of FtsZ monomers in the cell, and the factor N_cA is used to transform the number of molecules to concentration in micromoles.

The cytosolic filaments are experimentally observed to satisfy an exponential distribution when the cytosolic concentration is above the critical concentration.¹⁰ As concluded from polymer dynamics, independent of cooperative versus isodesmic assembly or fragmentation and annealing, filament lengths are also found to obey a quasi-steady exponential distribution with a mean length, λ.¹³ Therefore, the lengths of cytosolic filaments are reasonably assumed to possess an exponential distribution with a mean length of λ = 30 monomers (typical length), and the amplitude of the distribution can be determined by the cytosolic concentration. The incorporation rate of filaments with a length, l, is proportional to the cytosolic concentration of the same filament length. In addition, the GDP-bound monomers depolymerized from the ring can be rapidly exchanged with GTP-bound monomers, thereby giving cytosolic filaments that consist entirely of GTP-bound monomers. As previously presented,^14,15 the depolymerization of one monomer is governed by not only its position within a filament but also its nucleotide state. κ^T_off and κ^D_off are defined as the depolymerization rates of GTP-bound and GDP-bound FtsZ monomers at the filament tips, respectively. However, the GDP-bound monomers at the filament tips are experimentally found to disassemble more rapidly than GTP-bound ones.⁴ The fraction of GDP-bound monomers in the ring is f_D = S_D/S.

Filaments are assumed to be randomly distributed throughout the ring. Changes in the length distribution p(l, t) arise from the incorporation of both FtsZ filaments and free monomers in the cytosol, and the disassociation of monomers from filament tips dependent on their nucleotide states. Therefore, the filament length distribution p(l, t) in the ring obeys

where, 4πR is the number of available binding sites in which new filaments or free monomers can only associate, along the outer edges of the FtsZ ring, and κ_in is the membrane association rate.

The time evolution of the number of GDP-bound FtsZ monomers in the ring, S_D, is given by

The first term on the right side in eqn (12) accounts for the hydrolysis, in that any GTP-bound monomers in the FtsZ ring can hydrolyze in a stochastic manner, independent of the state of adjacent monomers. The second term on the right side in eqn (12) accounts for the disassembly of GDP-bound monomers at all filament tips within the ring F. The total number of filament tips F is obtained by integrating the length distribution of filaments within the ring, as presented in the preceding paragraph. κ_hy is the rate of GTP-bound monomers hydrolyzed into GDP-bound ones. By combining eqn (11) and (12), the quasi-steady state for this subsystem (FtsZ ring) is given by the following

where the subscript (qss) indicates the parameters at the quasi-steady state. It suggests from the quasi-steady state, eqn (13)–(16), that the fraction of hydrolyzed monomers in the ring at the steady state, f_Dqss, that is, the ratio of S_D to S, is independent of the radius, R, and attains a theoretical value of f_Dqss = 0.02, and the filament length in the ring is deduced to obey an exponential distribution relevant to the total number of FtsZ monomers in the ring, S_qss.

During cell division, an effective friction between cell wall/membrane and fluid environment is proportional to the local speed, u, obeying F_fr = −τu. At the quasi-steady state, the total force approaches zero, that is F_te + F_be + F_ar + F_z + F_fr = 0. Therefore, the evolution equation of the phase field model in the presence of FtsZ rings according to eqn (1)–(10) can be derived as follows

Due to the presence of MinCDE proteins in cells, the spatial distribution of the FtsZ rings is regulated to the middle of cells.^1,35 An FtsZ ring, which generally consists laterally of 6–10 filaments, localizes in the midcell along the horizontal axis at a wide range of 25 nm.¹¹ Therefore, the force generated by FtsZ rings in other positions is very small, and can be approximately neglected. The evolution equation of the phase field model in the absence of FtsZ rings is given by

The fourth-order nonlinear partial differential equations (17) and (18) are solved using an alternating direction-implicit scheme and a second order backward differentiation formula, while the partial differential equations (11) and (12) are explicitly solved using a forward Euler differential scheme. These position-dependent equations are solved on a 400 × 200 rectangle with the box size of 40r₀ × 20r₀, where the unit length r₀ is about 0.1 μm. Therefore, the average length of the sides of the box is about 3 μm, which is available to in vivo bacteria.^1,11 The simulation time step of these evolution equations is Δt = 4.0 × 10⁻⁵ s. A typical simulation starts with a stationary rod-shaped cell with surface area A₀ = 2.78 μm² and an aspect ratio equal to 3.0, which are also comparable with in vivo bacteria.^1,11 The diffusive interface that describes the thickness of the cell wall is chosen as one unit length, r₀ = 100 nm, which is comparable to the experimentally observed thickness of the cell wall in the range of 20 to 50 nm.¹¹ Compared with the experimental result for the lateral size of the FtsZ rings,¹¹ the dynamic FtsZ ring is assumed to be distributed in the middle of a rod-shaped cell, and has a width of 0.3r₀ (about 30 nm). Therefore, the evolution equation of the phase field in the presence of the force generated by FtsZ rings, eqn (17), is only solved on three grid sizes (about 30 nm) in the middle of a rod-shaped cell. The lengths of the FtsZ filaments are presented to satisfy a quasi-steady exponential distribution: 1/λe^−l/λ, see eqn (16), where λ is the mean length of the FtsZ filaments. Therefore, the total number of the monomers within the ring at the initial state is obtained by S_initial = ∫lp(l, 0)dl. Likewise, the filament length distribution within the ring is determined by p(l, 0) = S_initial/λ²e^−l/λ in the initial state. The length distribution and the total number of GDP-bound FtsZ monomers within the ring for the next time step are able to be obtained according to the evolution of eqn (11) and (12). The resulting values of S and S_D are inserted into the evolution eqn (17) and (18) of the phase field model to produce a new shape of a cell. These equations are solved iteratively to produce the new cell shape, as well as the updated distribution of FtsZ filaments. The steps are finished when a steady state is eventually reached where the cell has a stationary shape and the FtsZ filaments possess a stable distribution. The parameters used in the numerical calculations, which are all comparable to the experimental values,^13,31 are provided in Table 1.

Table 1 Model parameters

	Description	Value
γ	Surface tension	50 pN μm^−1
κ	Bending rigidity	0.5 pN μm
τ	Friction coefficient	5 × 10^3 pN s μm^−3
B	Bending modulus of FtsZ filament	1.2 × 10^−2 pN μm^2
ε	Boundary width	0.1 μm
δ	Size of FtsZ monomer	4 nm
Γ	Langrange multiplier	0.1
κT	Intrinsic curvature of GTP-bound FtsZ	0.8 μm^−1
κD	Intrinsic curvature of GDP-bound FtsZ	80 μm^−1
Nc	Conversion factor	600 μM^−1 μm^−3
κin	Membrane association rate	0.0021 μM^−1 μm^−1 s^−1
κ^Toff	Depolymerization rate of GTP-bound FtsZ	0.1 s^−1
κ^Doff	Depolymerization rate of GDP-bound FtsZ	3.5 s^−1
κhy	GTP hydrolysis rate	0.03 s^−1
MA	Area constraint	1.0 pN μm^−2
λ	Average number of monomers in a filament	30 monomers
ZT	Total concentration of FtsZ monomers in cell	13 μM

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III. Results and discussions

A. Dynamic process for cell division

A typical FtsZ ring has been reported, where approximately 30% of the total FtsZ monomers are found within the FtsZ ring in an in vivo cell.¹⁶ According to the concentration presented in Table 1, we find that the total number of FtsZ monomers within the ring, about 30% of Z_T, is S = 5504 monomers, which fits the value of 5000–15 000 FtsZ monomers estimated to be present in bacterial cells.¹⁶ Fig. 2 presents an example for the particular set of parameter values given in Table 1. The cell’s area, equivalent to the volume in three dimensions, is found to have changed less than 0.15% throughout the dynamic process. It is seen from these snapshots that a cell starting with a rod shape at a radius, R, of 0.5 μm retracts in the midcell down to a radius about 0.095 μm at the simulation time t = 160 s, and then maintains this shape for a long time, eventually reaching the steady state. The time for achieving this deformation has the same magnitude as the contraction time of minutes, as experimentally observed in live cells.¹¹ Several physical quantities, such as the constriction force generated by FtsZ rings, the total number of FtsZ monomers, S, and GDP-bound FtsZ monomers, S_D, within the ring, the fraction of hydrolyzed monomers in the ring, f_D, and length distributions p(l, t), are thoroughly studied during the dynamic process of cell division. We find that the constriction force at first shifts from 28 pN to 48 pN along with a decrease in the radius at the midcell, R, down to 25 nm, and then drops to 32 pN with a further decrease of R to 10 nm. The obtained constriction force has the same magnitude as the previous result, that cell division can succeed for a wide range of FtsZ rings with a force between 8 pN and 80 pN.²⁵ Along with the process of cell division, the total number of FtsZ monomers, S, GDP-bound FtsZ monomers, S_D, and the fraction of hydrolyzed monomers in the ring, f_D, are found to achieve stable values when the cell reaches the steady sate, where S is equal to 2643 ± 20 monomers, S_D is 421 ± 5 monomers and f_D is 0.1610. These are close to the theoretical values, S_qss = 3235 ± 150 monomers, S_Dqss = 661 ± 30 monomers and f_Dqss = 0.204 derived from eqn (13)–(15). Likewise, the fraction of hydrolyzed monomers in the ring is consistent with the value observed experimentally¹⁰, about 20%. In addition, we find that the length distributions, p(l, t), for FtsZ rings, whether in the initial state or steady state, satisfy the exponential distributions, with the characteristic length about 30 monomers.

Fig. 2 Snapshots of the numerical evolution of a cell shape during cell division.

B. The effect of initial states

The distribution of cytosolic FtsZ filaments, as observed in FRAP experiments, is found to be consistent with the theoretical prediction for an exponential length distribution.¹⁰ Likewise, both the mean length and its amplitude are found to be determined by the current cytosolic concentration,¹⁰ where the cytosolic concentration is well above the critical concentration. Fig. 3 gives the effect of the mean length for FtsZ filaments, λ, on the dynamic process during cell division. The particular set of parameter values is given in Table 1, and several physical quantities at the steady state, such as S, S_D, f_D and R, are listed in Fig. 3. Fig. 3(a) presents the dependences of the constriction forces generated by FtsZ rings at the initial state and steady state, F_z,initial and F_z,final, on λ. It is found that the constriction forces generated by FtsZ rings at the initial state remain at approximately 28.53 pN, and are independent of the mean length for FtsZ filaments, λ. This result is attributed to the number of GTP-bound and GDP-bound FtsZ monomers within the rings being kept constant at the initial state. Whereas, the constriction forces at the steady state at first increase slowly, and then increase abruptly as a function of the mean length, λ. At λ = 30 monomers, the constriction force increases up to 32.81 ± 0.92 pN, close to the force required for cell division. Likewise, we find in the inset of Fig. 3(a) that the contraction rates, dR/dt, approach zero when the mean length, λ, is less than 24 monomers. Further, it can be concluded from eqn (11) that the mean length, the value of p(l, t), and the corresponding total number of FtsZ and GDP-bound FtsZ monomers within the ring, S and S_D, are relatively small at the steady state, thereby leading to the reduction of constriction forces. In addition, it is seen from Fig. 3(b) that the total number of FtsZ and GDP-bound FtsZ monomers within the ring at the steady state, S and S_D, would be enhanced along with the increase of the mean length. However, these start to slightly decrease when the radius at the midcell rapidly retracts. At the same time, it is found in the inset of Fig. 3(b) that the fraction of hydrolyzed monomers in the ring, f_D, ranges from 0.0752 to 0.01626 when the mean length of FtsZ filaments shifts from 1 to 30 monomers. Additionally, the total number of FtsZ and GDP-bound FtsZ monomers, S and S_D, and the fraction of hydrolyzed monomers in the ring, f_D, obtained in the numerical calculation (solid symbols) are found to be comparable to the theoretical curves derived from eqn (13)–(15) (red line). Likewise, the length distributions of FtsZ filaments at the steady state are presented in Fig. 3(d), where the mean lengths of FtsZ filaments are initially chosen as 10 and 30 monomers. The distributions of FstZ filaments are found to satisfy exponential distributions with their characteristic lengths close to the preset mean lengths. As a result, these results reveal that cell division depends on the mean length of FtsZ filaments, because the total number of FtsZ and GDP-bound FtsZ monomers, S and S_D, and the fraction of hydrolyzed monomers in the ring, f_D, would be changed as a function of the mean length of FtsZ filaments, thereby leading to the alteration of the constriction forces generated by FtsZ filaments within the ring.

Fig. 3 The effect of the mean length for FtsZ filaments, λ: (a) constriction forces generated by FtsZ rings at the initial and steady states, and contraction rates in the inset; (b) S, S_D and f_D at the steady state, the red lines are obtained from the theoretical values, eqn (13)–(15); (c) the radius of the constriction ring at the steady state, cell shapes at the steady state given in the insets; (d) length distribution with λ = 10 and λ = 30 at the steady state.

Cell wall expansion is experimentally observed to be mainly contributed to by cell-wall biosynthesis, such as peptidoglycan synthesis proteins.^36,37 Meanwhile, the turgor pressure is found not to be required for cell-wall biosynthesis, but does play a simple role, whereby it stretches recently assembled, unextended peptidoglycan.^36,37 Once the FtsZ ring is formed and is anchored to cell wall/membrane, other proteins, such as the septum synthesis proteins, are recruited to the ring site, and cell wall growth and remodeling can proceed. In addition, many bacteria, for example, E. coli, generally elongate with both an exponential growth and a constant diameter. Therefore, the dynamic processes of cell division and steady states of cells are extensively investigated in response to the preset aspect ratio for rod-shaped cells. The results are given in Fig. 4. The cell area of rod-shaped cells, as expected, is altered upon the change in the aspect ratio. However, the initial concentration of FtsZ monomers in cells is chosen as a constant, 13 μM, the fraction of FtsZ monomers within the ring contains 30% of the total number of FtsZ monomers, and the fraction of hydrolyzed monomers in the ring, f_D, is 0.4. As shown in Fig. 4(c), when the aspect ratio for rod-shaped cells shifts from 1.6 to 3.8, the radius at the midcell in the steady state, R, at first increases from 0.5 μm to close to 1.0 μm, where the shape is almost spherical, and then decreases gradually until cell division finishes. Typical shapes of cells at the initial and steady states are given in the inset of Fig. 4(c) and their aspect ratios are 2.4 and 3.8. It is noted that one rod-shaped cell would be inclined to divide at the ring site when it has a high aspect ratio in the initial state. That is, a rod-shaped cell that has a higher aspect ratio, would tend to undergo cell division more easily. The result is in good agreement with experiments that show that cell division of prokaryotes happens as a result of cell wall expansion with a constant radius.³⁷ In the present study, the horizontal length of rod-shaped cells is kept constant, while the longitudinal length has to adjust to satisfy the preset aspect ratio. Therefore, the radius of rod-shaped cells, R, would reduce along with the increase in the aspect ratio, as well as the fact that the areas of rod-shaped cells would respond to the transformation of the aspect ratios. If the area of rod-shaped cells with different aspect ratios is fixed, the radius of a rod-shaped cell would decrease along with the increase in the aspect ratio. The initial constriction force generated by FtsZ rings would be augmented with respect to the contractile radius, when S and S_D are kept unchanged in the initial state, see eqn (10). As shown in Fig. 4(a), the initial constriction force is confirmed to increase as a function of the aspect ratio. Additionally, it can be inferred from eqn (13) and (14) that the total number of FtsZ and GDP-bound FtsZ monomers within the ring would be decreased along with the reduction of R in the steady state, which is confirmed in Fig. 4(b). However, the constriction force at the steady state would depend on the competition between its radius and the number of FtsZ monomers within the ring. Therefore, the forces at the steady state shown in Fig. 4(a) exhibit evident fluctuations in response to the aspect ratio. As a whole, cell division depends on the aspect ratio for rod-shaped cells, as well as cell wall expansion in vivo.

Fig. 4 The effect of aspect ratio for rod-shaped cells: (a) constriction forces generated by FtsZ rings at the initial and steady states, and contraction rates in the inset; (b) S, S_D and f_D for FtsZ rings at the steady state; (c) the radius of the midcell at the steady state, a cell shape at the initial and steady states given in the insets.

Fig. 5 illustrates the dependence of the dynamic process in cell division on the total concentration of FtsZ monomers in cells, Z_T, where the number of FtsZ monomers in the ring, S, is 30% of the total number of FtsZ monomers in the cell, and the fraction of hydrolyzed monomers in the ring, f_D, is 40%. It is demonstrated experimentally^10,11 that at saturated FtsZ concentrations, overlapping FtsZ filaments remain, allowing additional rings to form, whereas at low FtsZ concentrations of about 3–5 μM, a broad distribution of membrane-bound FtsZ monomers is observed in the midcell, but no FtsZ rings form. A critical concentration for ring formation is required, given that only a single ring forms when the concentration of FtsZ monomers is about 8–10 μM, and multiple rings form when the concentration is above 10 μM. Therefore, not only the formation of FtsZ rings but also the constriction force are expected to depend on the concentration of FtsZ monomers in cells. Fig. 5(a) presents the dependences of the constriction forces at the initial state and steady state on Z_T. It can be seen that the constriction forces, regardless of the initial and final states, F_z,initial and F_z,final, increase along with the increase of Z_T. When Z_T is less than 12 μM, the constriction force generated by the FtsZ rings is found not to be large enough to overcome the curvature elastic energy for cells, therefore, the rod-shaped cell would eventually become spherical in shape (energy benefit). As Z_T further increases, the cell tends to retract at the midcell, and the radius at the midcell becomes less. When Z_T increases up to 13 μM, the constriction force continuously maintains a high level, and then cell division proceeds. The radius at the midcell is presented in Fig. 5(c) as a function of Z_T, and the variation trend is found to be in accordance with the previous discussions. Fig. 5(b) gives the dependences of the total number of FtsZ and GDP-bound monomers, S and S_D, on Z_T. It can be seen that at Z_T < 13 μM, S and S_D increase as a function of Z_T, but at Z_T > 13 μM, both S and S_D gradually decline. Likewise, it is found that at Z_T > 13 μM, cell division starts to happen, and the radius at the midcell rapidly reduces, thereby leading to the decrease in both S and S_D. In addition, it is found in the inset of Fig. 5(b) that the fraction of hydrolyzed monomers in the ring, f_D, keeps fluctuating as a function of Z_T, consistent with the theoretical result given in eqn (15). At the same time, as shown in Fig. 5(b), the numerical results of S and S_D (symbols) are found to obey the relations of eqn (13) and (14) (red lines).

Fig. 5 The effect of total concentration of FtsZ monomers in cells, Z_T: (a) constriction forces generated by FtsZ rings at the initial and steady states, and contraction rates in the inset; (b) S, S_D and f_D for FtsZ rings at the steady state, the red lines are obtained from the theoretical values, eqn (13)–(15); (c) the radius of FtsZ rings at the steady state, a cell shape at the steady state given in the inset.

Fig. 6 presents the dependence of cell morphodynamics on the initial number of FtsZ monomers within the FtsZ rings, S_initial, where the initial fraction of hydrolyzed monomers in the ring, f_D, is still kept at 40%. It can be seen from Fig. 6(a) that the initial constriction force generated by FtsZ rings increases linearly as a function of S_initial, as concluded from eqn (10). However, we find that at S_initial < 6500 monomers, the constriction force, S, S_D and R in the steady state are independent of S_initial, where the constriction force of about 8.4 pN at the steady state is far less than the force required for cell division. At S_initial > 6500 monomers, the initial force (about 28 pN) is larger than the critical force for cell division, the rod-shaped cell gradually shrinks at the midcell, and then cell division begins. In addition, the fraction of hydrolyzed monomers in the ring, f_D, is found to be close to 0.162 ± 0.002 and independent of S_initial, see the inset of Fig. 6(b).

Fig. 6 The effect of the initial number of monomers in the FtsZ rings, S_initial: (a) constriction forces generated by FtsZ rings at the initial and steady states, and contraction rates in the inset; (b) S, S_D and f_D for FtsZ rings at the steady state; (c) the radius of constriction rings at the steady state, a cell shape at the steady state is given in the inset.

Fig. 7 gives the effect of the initial number of GDP-bound FtsZ monomers within the ring, S_D,initial, on cell morphodynamics, where Z_T and S_initial are chosen as 13 μM and 6500 monomers, respectively. In Fig. 7(a), the initial constriction force generated by the FtsZ ring, F_z,initial, is found to increase linearly along with the enhancement of S_D,initial. This trend obeys the relation of eqn (10). Likewise, it is also found that at S_D,initial < 2600 monomer, the constriction force, S, S_D and R in the steady state remain almost unchanged, and are independent of S_D,initial, where the constriction force is about 8.4 pN and far less than the critical force (∼28 pN) for cell division. At S_D,initial > 2600 monomers, the initial force is about 28 pN, close to the critical force for cell division. The radius at the midcell becomes much less, and then cell division is under way. In addition, the fraction of hydrolyzed monomers in the ring, f_D, is found to be approximately 0.162 ± 0.002 regardless of the alteration in S_D,initial, see the inset of Fig. 7(b).

Fig. 7 The effect of the initial number of GDP-bound FtsZ monomers in the FtsZ rings, S_D,initial: (a) constriction forces generated by FtsZ rings at the initial and steady states, and contraction rates in the inset; (b) S, S_D and f_D for FtsZ rings at the steady state; (c) the radius of constriction rings at the steady state, a cell shape at the steady state given in the inset.

As a result, the constriction force generated by the FtsZ rings, whether in the initial or in the steady state, is required to be above the critical force of 28 pN in order to arrive at cell division, regardless of the changes in the total concentration of FtsZ monomers in cells, the mean length of FtsZ filaments, and the aspect ratio for rod-shaped cells. This critical constriction force has the same magnitude as in the previous result, indicating that cell division can succeed for a wide range of FtsZ rings with a force between 8 pN and 80 pN.²⁵ Additionally, it is found that when the initial constriction force generated is less than the critical force, the constriction forces, S, S_D, f_D and the radius at the midcell in the steady state are independent of the initial number of FtsZ monomers and GDP-bound FtsZ monomers in the FtsZ rings, S_initial and S_D,initial. However, except for the mean length of FtsZ filaments, the initial constriction forces, F_z,initial, are found to increase along with the enhancement of Z_T, the aspect ratio for rod-shaped cells, S_initial and S_D,initial. In addition, it is noted that S and S_D to some extent start to decrease at the beginning of such a reduction in the radius at the midcell. At the same time, the fraction of hydrolyzed monomers in the steady state, f_D is observed to be dependent on the mean length of FtsZ filaments, λ, but independent of Z_T, the aspect ratio for rod-shaped cells, S_initial and S_D,initial. Whereas, S and S_D in the steady state are observed to depend on λ, Z_T and radius at the midcell, R. As a whole, these changes in S, S_D and f_D obtained in the numerical calculations are found to agree well with the theoretical results derived from eqn (13)–(15), which describe the quasi-steady state of the FtsZ ring subsystem.

C. Phase diagram

In in vitro experiments, the concentration of FtsZ monomers in cells and the aspect ratio for rod-shaped cells are observed to be different in a variety of biological processes due to cell wall expansion and protein synthesis.³⁶ In order to explicitly elucidate the relevance of cell division to the concentration of FtsZ monomers in cells, Z_T, and the aspect ratio for rod-shaped cells, a phase diagram is presented in Fig. 8 that describes the steady state of rod-shaped cells, where divided and undivided are employed to describe the final states of cells. It is found that when the total concentration is less than 10 μM, rod-shaped cells, even with an aspect ratio as high as 3.8, can not realize cell division. It is previously reported^10,11 that only a single ring is experimentally observed to form when FtsZ concentration is in the range from 8 to 10 μM. Therefore, the constriction force generated by a single ring is not large enough to attain the critical force for cell division. In addition, we find that cell division would be easily realized along with an increase in the concentration, Z_T, even with a much lower aspect ratio. Therefore, we can conclude that one rod-shaped cell with a high aspect ratio and/or Z_T, is prone to divide.

Fig. 8 The phase diagram as a function of aspect ratio for rod-shaped cells and total concentrations of FtsZ in cells, Z_T.

IV. Summary and outlook

We have developed a new theoretical framework, by a combination of a phase field model for cells and a kinetics description for FtsZ ring maintenance, to understand shape transformations and the mechanics of a dividing rod-shaped cell. The kinetic description of the dynamic FtsZ ring takes GTP-bound monomers in the ring into account. The dynamic turnover of FtsZ rings therefore includes the association of short filaments or free monomers in the cytoplasimc pool to the ring, GTP-bound FtsZ monomers hydrolyzed to GDP-bound FtsZ ones in the ring, rapid nucleotide exchange in the cytoplasmic pool, and detachments of both GTP-bound and GDP-bound FtsZ monomer at filament tips in the ring. The difference in the mechanical characterization of GTP-bound and GDP-bound monomers within the FtsZ ring reflects the transformation changes of FtsZ monomers due to GTP hydrolysis. Cell divisions are determined by cooperation of the constriction force generated by dynamic FtsZ rings, the bending elastic energy and cell wall expansion.

The constriction force generated by FtsZ rings determines the contractile direction, velocity, as well as cell shape. The constriction force presented here are altered along with the dynamic process during cell division, as well as the changes in cell shapes and dynamic turnover of FtsZ rings. We find that with reasonable parameters given in Table 1, a rod-shaped cell retracts at the midcell when the constriction forces are above the critical force of about 28 pN that has the same magnitude as in the previous report.²⁵ Cell shapes and cell division times obtained in our calculations are all comparable to physiologically observed results.¹¹ FtsZ filaments within the ring are also found to obey an exponential distribution that has its characteristic length close to the preset mean length of FtsZ filaments, λ. Likewise, cell morphodynamics during cell division are extensively investigated as functions of different initial states of rod-shaped cells and FtsZ rings, such as the aspect ratio, the mean length of FtsZ filaments, the concentration of FtsZ monomers in cells, Z_T, and the total number of FtsZ and GDP-bound FtsZ monomers within the ring. It is found that with the increase in the aspect ratio, the mean length of FtsZ filaments, and Z_T in the initial state, rod-shaped cells contract at the midcell and eventually divide when the constriction force is above the force required for cell division. However, when the constriction force is below the critical force, the cell shape eventually becomes close to spherical, and is independent of the total number of FtsZ and GDP-bound FtsZ monomers within the ring in the initial state. It is noted that the total number of FtsZ and GDP-bound FtsZ monomers within the ring, S and S_D, start to reduce to some extent at the beginning of the reduction of the radius at the midcell. In addition, the quasi-steady state for the dynamic turnover of FtsZ rings is derived, thereby giving the theoretical expressions for S_qss, S_Dqss and f_Dqss, see eqn (13)–(15). The obtained results for S, S_D and f_D in the numerical calculations are found to be consistent with the theoretical results, and f_D approaches the experimental value of about 20%.¹¹

In order to explicitly elucidate the relation of cell division and some parameters, the morphological phase diagram is presented as a function of the FtsZ concentration in cells, Z_T, and the aspect ratio of rod-shaped cells, given that cell division tends to be observed in rod-shaped cells with high Z_T and/or high aspect ratios. As experimentally observed,^10,11 only a single ring forms at Z_T < 10 μM, but at this concentration, the constriction force generated by a single ring is not high enough to initiate cell division. Likewise, one rod-shaped cell that possesses a higher aspect ratio would easily undergo cell division, in good agreement with the experimental result that cell division of prokaryotes happens due to cell wall expansion with a constant radius.³⁷ Therefore, the theoretical framework presented here can effectively predict cell morphodynamics during division with the coupling of dynamic FtsZ rings and cell morphology.

In the present work, the dependence of cell morphodynamics during division on different initial states of cells and FtsZ rings is the focus. A more detailed discussion relevant to various kinetic rates within the FtsZ ring and division times will be presented in future work. The basic theoretical framework developed here, which has the evident advantage to integrate cell shape with dynamic FtsZ rings, can be extended to study division in eukaryotic cells, such as budding and fission in yeast and plant cells, which can provide valuable insights into critical cellular functions. In future research, it is also of great interest to develop other space-dependent theoretical frameworks aimed to study the spatial structure and dynamic turnover of FtsZ rings.

Acknowledgements

The financial support for this work was provided by the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China (Grant no. 21274038).

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