Self-organization of mortal filaments and its role in bacterial division ring formation

Cytoskeletal filaments are active polymers that continuously grow and shrink through nucleotide hydrolysis, producing treadmilling dynamics in which growth at one end and shrinkage at the other give the appearance of motion without individual monomer motility. Many essential filaments, including actin, microtubules, and bacterial FtsZ, exhibit treadmilling. Prior physical models often approximate treadmilling as self-propulsion of filament centers of mass, which can capture single-filament motion but neglect continuous turnover and can overestimate polymerization forces on obstacles. Thus, collective self-organization mechanisms of treadmilling polymers may differ from those of self-propelled filaments and remain insufficiently understood. The study investigates how treadmilling contributes to emergent order and cellular-scale structures, focusing on the bacterial tubulin homolog FtsZ that treadmills and organizes into the Z-ring at midcell, essential for division in multiple bacteria. Specifically, the research asks whether and how treadmilling-driven turnover can produce nematic alignment and robust Z-ring formation and dynamics in vivo, and how chemical/geometric biases in cells are integrated by such mortal filaments.

The authors situate their work within several strands of literature: (i) decades of biochemical and biophysical work establishing treadmilling in cytoskeletal systems (actin, microtubules, FtsZ) and its nucleotide-hydrolysis dependence; (ii) modeling approaches treating active filaments as self-propelled rods to explain collective order and banding, which may not capture mortality/turnover intrinsic to treadmilling or the small polymerization forces measured experimentally; (iii) kinetic models of FtsZ nucleation/polymerization/hydrolysis and experimental measurements of in vitro and in vivo filament lengths, velocities, and monomer lifetimes; (iv) cell biological studies showing treadmilling is essential for Z-ring condensation and septal constriction initiation in Bacillus subtilis and influences septal cell wall synthesis; (v) observations of high-speed AFM/TIRF dynamics of FtsZ networks and the role of GTPase activity, as well as the influence of cross-linkers (e.g., ZapA) on ring condensation; and (vi) spatial patterning systems (Min, MipZ, nucleoid occlusion) that modulate FtsZ polymerization to position division at midcell. These works motivate a model explicitly incorporating filament mortality and turnover to re-examine self-organization mechanisms and their cellular consequences.

The study develops a coarse-grained molecular dynamics model of treadmilling filaments on a 2D surface with periodic boundaries. Filaments are bead-spring polymers of spherical monomers with diameter σ=5 nm (FtsZ monomer size), joined by harmonic bonds and angle potentials to capture stiffness. Time evolution includes explicit kinetic rules applied at reaction intervals Δt_react=0.1 s: (1) head growth with probability P_on=r_on Δt_react, where r_on (monomers s⁻¹) reflects polymerization kinetics and free monomer concentration; (2) tail shrinkage with probability P_off=1−exp(−t_tail/τ_det), where τ_det accounts for hydrolysis and fast tail dissociation and t_tail is the age of the current tail monomer; depolymerized monomers detach into solution, so P_off is environment-independent; (3) nucleation of new dimers with probability p_nuc=r_nuc Δt_react. At steady state P_on≈P_off, defining an intrinsic filament size N_tr and typical monomer lifetime τ_c in terms of τ_det and P_on. Growth steps only occur if space is available at the head. Simulations explore single-filament dynamics across {r_on, τ_det} and multi-filament collective behavior at fixed nucleation rates r_nuc until steady surface density ρ and mean size ⟨l⟩ are reached. Quantitative comparisons are made to in vitro reconstituted FtsZ on supported lipid bilayers (TIRF microscopy) and to high-speed AFM imaging of FtsZ networks, using previously published image analysis to extract lengths, velocities, and monomer lifetimes. For cellular-scale organization, the model adds: (a) a weak curvature-sensing aligning force f_curv=5 k_B T/σ applied at filament ends to bias orientation along cell circumference, capturing intrinsic filament/membrane anchor curvature and cylindrical cell geometry; (b) spatial modulation of growth and nucleation rates to mimic in vivo chemical patterning (increasing rates at midcell and decreasing at poles via a Gaussian profile; τ_det held constant), switched from uniform to patterned at t=0; and (c) short-range attractive interfilament interactions (cross-linking) to promote condensation and tight packing, consistent with ZapA-like effects. Cell-scale simulations constrain total monomer number to N_max≈2000 (estimated for B. subtilis Z-rings) and use cylindrical geometry (e.g., L≈3 μm, R≈1 μm) and parameter values reported in figure captions (e.g., typical sets r_on≈8 s⁻¹, τ_det≈15 s; alternative calibration with τ_det≈5 s to match specific in vitro datasets). Analyses quantify nematic and polar order parameters, density–order coupling, filament lifetimes versus individual alignment S, velocity distributions, and ring condensation/maturation dynamics. Arrested-treadmilling conditions are simulated by halting hydrolysis-dependent shrinkage (e.g., mimicking reduced GTPase mutants such as FtsZ L169R).

• Single-filament dynamics: Simulations reproduce steady-state treadmilling with realistic lengths (100–500 nm), velocities (15–50 nm s⁻¹), and monomer lifetimes (~8 s) spanning reported experimental ranges. Parameterization with r_on≈8 monomers s⁻¹ and τ_det≈5 s yields velocity and lifetime distributions closely matching in vitro TIRF measurements; directional autocorrelation of individual monomers decays rapidly to zero in both simulations and experiments, confirming monomer immobility despite apparent filament motion.
• Collective behavior: As intrinsic size N_tr increases, systems transition from quasi-empty (filaments rapidly die) to an unstable low-density treadmilling regime, and then to a stable, persistent treadmilling regime with higher densities. Over a broad parameter space, treadmilling filaments develop strong nematic order and polar bands. Unlike passive or self-propelled systems where density drives ordering, here turnover kinetics (N_tr) set both order and density. Arresting treadmilling (e.g., reduced depolymerization) yields highly populated but disordered states with persistent nematic defects, mirroring HS-AFM observations with FtsZ L169R.
• Mechanism of ordering: Collisions halt head growth but do not halt shrinkage; misaligned filaments thus preferentially shrink and dissolve because they more often encounter neighbors, while aligned filaments persist. Filament lifetime correlates strongly with individual nematic order S: highly aligned filaments (S≈1) survive for minutes, whereas misaligned ones die within seconds; aligned filaments maintain velocities near their single-filament values, misaligned ones show near-zero velocities.
• Experimental agreement: HS-AFM of FtsZ on SLBs and simulations show the same coupling between surface density and nematic order, and similar evolution toward ordered states; mutants with arrested treadmilling fail to heal defects and remain less ordered.
• Z-ring formation in vivo: Incorporating curvature bias, spatially patterned growth/nucleation (midcell enrichment, polar depletion), and cross-linking reproduces key live-cell dynamics in Bacillus subtilis: (i) rapid ring condensation to midcell within ~1 minute post-switch arises from fast dissolution of filaments in growth-inhibited regions (set by monomer lifetime/offs rates); (ii) slow maturation over several minutes results from turnover-driven alignment and bundling into tightly packed bands, increasing local midcell density; (iii) mature rings exhibit higher average treadmilling velocities than nascent rings due to a decreased fraction of trapped/low-velocity misaligned filaments, not because individual filaments speed up; (iv) arresting treadmilling before patterning prevents condensation (diffuse filaments), whereas arresting after patterning yields frozen, aberrant rings. Cross-linking stabilizes rings and increases packing, consistent with experimental effects of ZapA and interfilament interactions.

The work addresses how treadmilling polymers self-organize without self-propelled motion. The key mechanism is ordering via mortality: misaligned filaments disproportionately shrink and dissolve when their head growth is blocked by neighbors, while aligned filaments persist and replenish the population. This turnover-driven selection heals nematic defects and produces large-scale nematic order and polar banding, distinct from force-based alignment in self-propelled rod models. In cellular context, the same mechanism integrates geometric and chemical biases—curvature and midcell-enriched polymerization—enabling rapid, minutes-scale reorganization: misaligned filaments in disfavored regions quickly die out, yielding fast midcell condensation, while slow replacement by aligned bundles drives maturation and increased ring density. The model thereby explains observed in vivo phenomena, including faster apparent treadmilling in mature rings due to selection against trapped filaments, and the failure of ordering when treadmilling is arrested. More broadly, the results position treadmilling filaments as a distinct class of active matter that expends energy to maintain turnover and achieve order through local dissolution, with implications for responses to obstacles, perturbation healing, and information propagation in active materials.

This study introduces and validates a coarse-grained, turnover-explicit model for treadmilling filaments that quantitatively reproduces single-filament FtsZ dynamics, collective nematic ordering via defect healing, and the essential features of bacterial Z-ring formation in vivo. It identifies a general self-organization mechanism—order through birth–death dynamics that selectively removes misaligned filaments—distinct from self-propulsion-based alignment. In cells, coupling of treadmilling with curvature sensing, spatial modulation of polymerization, and interfilament attraction yields rapid ring condensation and slow maturation consistent with experiments. The framework suggests new routes for engineering programmable active matter and synthetic cells using mortal filaments capable of self-healing through local dissolution. Future work can leverage the model to study coupling between Z-ring dynamics and septal cell wall synthesis machinery, responses to obstacles and perturbations, and parameter regimes across diverse cytoskeletal polymers and cellular geometries.

• The model is coarse-grained and two-dimensional, with monomers in solution treated implicitly and with assumed constant free monomer concentration (except for imposed total-monomer limits at cell scale). This simplification may neglect detailed biochemical feedbacks and crowding effects.
• Filament curvature sensing is implemented via a simplified aligning force, and interfilament attraction is modeled as an effective short-range interaction; specific molecular determinants and heterogeneity of cross-linkers are not explicitly represented.
• Polymerization forces on obstacles and mechanical coupling to membranes/cell walls are not explicitly modeled; the model focuses on turnover-driven dynamics rather than force generation.
• The coupling between the Z-ring and downstream peptidoglycan synthesis enzymes is not included; thus, implications for septal growth are inferential.
• Many kinetic parameters are calibrated to FtsZ under specific in vitro/in vivo conditions; generalization to other systems may require reparameterization.