Z-ring force and cell shape during division in rod-like bacteria

Abstract

The life cycle of bacterial cells consists of repeated elongation, septum formation, and division. Before septum formation, a division ring called the Z-ring, which is made of a filamentous tubulin analog, FtsZ, is seen at the mid cell. Together with several other proteins, FtsZ is essential for cell division. Visualization of strains with GFP-labeled FtsZ shows that the Z-ring contracts before septum formation and pinches the cell into two equal halves. Thus, the Z-ring has been postulated to act as a force generator, although the magnitude of the contraction force is unknown. In this article, we develop a mathematical model to describe the process of growth and Z-ring contraction in rod-like bacteria. The elasticity and growth of the cell wall is incorporated in the model to predict the contraction speed, the cell shape, and the contraction force. With reasonable parameters, the model shows that a small force from the Z-ring (8 pN in Escherichia coli) is sufficient to accomplish division.

Fig. 1.

A mechanical model of bacterial cell division. The FtsZ-ring is positioned at the mid cell and generates a contractile force, f, in the inward radial direction (red arrow). The observed PG cell wall is inflated from the undeformed cell shape by internal turgor pressure, much like a balloon. The undeformed cell shape is defined by parametric profiles (R(s), Z(s)), which are the radial and longitudinal components of the material point coordinates; the deformed shape is similarly defined by profiles (r(s), z(s)); s is an arc-length measured from the mid cell. During division, deformed and undeformed shapes are changed by cell wall growth and remodeling (green regions indicate where most of the growth and remodeling occurs). Our model combines the mechanical properties of the cell wall with the growth and remodeling kinetics to compute the cell shape during division.

Fig. 2.

Detailed view of growth and turnover of the PG cell wall, which occur by adding new PG strands and removing old ones. The newly added strands conform with the current shape (r(s), z(s)). For example, in Upper, the new cell wall at Lower Right should conform to a smaller radius than that shown at Lower Left. This implies that the undeformed shape (R(s), Z(s)) is changing. In particular, R should approach r over time (Lower). The cell probably adds what may be regarded as a “prestretched” strand during growth to maintain the wall radius. The prestretched amount is described by the parameter d₀ (see Cell Wall Growth and Remodeling).

Fig. 3.

Cell shapes obtained from our model with different combinations of parameters. (Left) The blue line is the instantaneous cell shape (r(s), z(s)), and the red line is the undeformed cell shape (R(s), Z(s)). (Right) Rendered 3D instantaneous shapes. (A) The shape of E. coli after 8 min of contraction under 8 pN of Z-ring force. Growth and turnover are confined within ±50 nm around the Z-ring. (B) An E. coli cell shape with 80 pN of Z-ring force. The cell shape appears to be relatively independent of the force. The contraction time is 2 min. (C) An E. coli cell shape with broader growth and turnover regions (λ_1,2 ≈ 25 nm). The cell shape appears unrealistic. (D) An E. coli minicell produced by contracting near the spherical cap. This phenotype occurs in mutants. (E) A B. subtilis cell shape with 58 pN of Z-ring force. Further contraction in this case is difficult because of the thick cell wall.

Fig. 4.

Quantitative results from the model. (A) Average contraction velocity for E. coli as a function of the Z-ring force when the growth rate, τ₂⁻¹ is changed. Tripling the growth rate does not affect the contraction velocity appreciably. (Inset) Typical division furrow radius versus time. (B) The turnover rate has a pronounced effect on the contraction velocity. The cell actively remodels the furrow region to achieve division.

Fig. 5.

The model predicts that the cell will continually increase in radius when d₀ is set to zero (see The Model and SI Text). (Upper) Shown are the instantaneous (blue) and the undeformed (red) shapes of the cell before (Left) and after (Right) d₀ is set to zero. (Lower) 3D renderings showing the change in the cell aspect ratio.