Emergence of Escherichia coli critically buckled motile helices under stress

Significance

We have discovered an emergent mechanism by which Escherichia coli can escape high-stress regions, such as near-lethal concentrations of antibiotics, by forming long motile helical filaments that are poized at the critical shear buckling point: 2$\pi$ twist rotations independent of the length of the filament. All filaments, independent of length, have the same twist, indicating that this is a highly evolved response. The helices do not tumble as nonstressed bacteria do but rather, over a period of tens of seconds, reverse direction. The result is that the persistence length of the filaments’ translational motion is much larger than in unstressed normal-size bacteria, giving filaments an extremely large effective diffusion coefficient, allowing the bacteria to escape high existential stress.

Abstract

Bacteria under external stress can reveal unexpected emergent phenotypes. We show that the intensely studied bacterium Escherichia coli can transform into long, highly motile helical filaments poized at a torsional buckling criticality when exposed to minimum inhibitory concentrations of several antibiotics. While the highly motile helices are physically either right- or left-handed, the motile helices always rotate with a right-handed angular velocity $\overrightarrow{\omega }$, which points in the same direction as the translational velocity ${\overrightarrow{v}}_T$ of the helix. Furthermore, these helical cells do not swim by a “run and tumble” but rather synchronously flip their spin $\overrightarrow{\omega }$ and thus translational velocity—backing up rather than tumbling. By increasing the translational persistence length, these dynamics give rise to an effective diffusion coefficient up to 20 times that of a normal E. coli cell. Finally, we propose an evolutionary mechanism for this phenotype’s emergence whereby the increased effective diffusivity provides a fitness advantage in allowing filamentous cells to more readily escape regions of high external stress.

Studies of bacterial morphology are typically conducted under nonphysiological conditions that partly or completely immobilize the cell on an agar surface or in a confined channel (1), which puts constraints on cellular morphology (2, 3). Likewise, the swimming behavior of Escherichia coli is usually studied with prototypically healthy (short), unstressed cells. Such studies of unstressed cells have given rise to the description of E. coli’s swimming behavior as “run and tumble” dynamics wherein unidirectional “runs” are intermittently disrupted by random reorientations (“tumbles”) (4).

But even well-studied bacteria such as E. coli can exhibit remarkably emergent and unexpected phenotypes when exposed to external stressors such as antibiotics. The physics and biological fitness of these emergent phenotypes can be surprising. Previous work has shown that both genotoxic (ciprofloxacin) (5, 6) and cell wall-targeting (cephalexin) antibiotics (7) give rise to filamentous E. coli. Here we use the nongenotoxic antibiotic cephalexin because cephalexin impairs division septum formation but does not disrupt normal DNA replication as does ciprofloxacin. In our experiments, we used the common laboratory E. coli strains MG1655 and W3110. Except where noted we used for detailed analysis a strain of W3110 E. coli that constitutively expresses GFP throughout the cytoplasm, allowing for accurate tracking of the bacteria under the microscope. Filamentous E. coli were produced by incubation in lysogeny broth (LB) with 10 $\mathrm{\mu }$g/mL of the antibiotic cephalexin at ${\mathrm{30}}^o$C. We note that while antibiotics can lead to near-complete emergence of filamented E. coli throughout the population, even in the complete absence of antibiotics, a small percentage of E. coli are filamentous, indicating the emergence of the phenomena even at zero stress.

Data were recorded on swimming filaments using a high-speed resonant scanning confocal microscope. After 24 h of incubation at a dose of 10 $\mathrm{\mu }$g/mL of the antibiotic cephalexin (a final OD of 1.0), we transferred 5 $\mathrm{\mu }$L of the culture to a concave glass slide at either room temperature T $\sim \hspace{0.17em}{\mathrm{25}}^o$C (highly motile filaments) or near ${\mathrm{0}}^{o}C$ (greatly slowed motile filaments) and imaged them using a Nikon A1R inverted confocal epifluorescence microscope (Nikon Instruments). The objective used was a Nikon CFI Plan Apo Lambda 60× Oil. Mechanical helices (springs) were imaged in the Nikon A1R to correct for any mirror reflection or camera readout inversion of actual helicity handedness.

The filaments have a high motility and rotate as they swim, although in the 2D projection of a conventional microscope they appear to wiggle. Because of the highest rotational rate $f_R=10\hspace{0.17em}\text{Hz}$ of the filamentous bacteria at $25^{o}C$, direct 3D confocal visualization of the filament topology was difficult. Instead, we continuously imaged (30 Hz single z plane frame rate) in the z plane nearest the coverslip and measured the 2D projected length, the translational speed $v_F$, and the rotational frequency $f_F$ in that z plane. Although we imaged in a single confocal z plane for maximum speed, distance from the plane could be determined by the relative brightness of the object. We measured the intensity recorded by the confocal detector of 42 nm-diameter Flash Red fluorescent beads (Bangs Laboratories) stuck to the coverslip as a function of distance from the coverslip plane by moving the stage known distances in the vertical direction; this brightness vs. distance calibration scale was then used to determine the physical displacement of sections of the filament from the coverslip plane as they rotated in the fluid.

Fig. 1A shows a single frame snapshot of filamentous bacteria. Fig. 1B shows a kymograph of the translational velocity ${\overrightarrow{v}}_L$ and the right-handed rotational spin $\overrightarrow{\omega }$ of a filament, stepped at the 1/30 s framing period of the confocal microscope. The right-handed rotation of the filament and the left-handedness of the helix are quite evident. Note that as observed in a nonconfocal microscope, the rotation of the helices appears as a wiggle when projected into the 2D image plane. Two confocal videos in SI Appendix (Movies S1 and S2) show a single z plane video of the bacterial body helical dynamics from Fig. 1 A and B.

Fig. 1.

(A) Confocal image in a z plane set to the contact plane of the coverslip with the medium. (B) Confocal kymograph of a rotating and translating bacterial filament. The translational speed $v$ is approximately 15 $\mathrm{\mu }$m/s, and the rotational rate $f_r$ is approximately 3 Hz. The bacterium is a left-handed helix but is rotating with a right-handed spin.

There is a rather simple way to characterize the differential geometry of a helix that we use to quantify the helix topology. We assume a local cartesian coordinate system with the x–y plane orthogonal to the local tangent of the curve. The local curvature $\kappa$ of the centroid of a deformed cylinder determines the local bending in the x–y plane of the helix and has units of 1/length, while the differential torsion ${\tau }_d$ (not to be confused with physical torque ${\tau }_p$) determines the local rotational twist/length of the cylinder and has dimensions of 1/length (technically twist angle/length). The sign of the torsion ${\tau }_d$ tells whether the helix is left-handed (−) or right-handed ($+$). Let $z$ be the distance along the centroid of the helix and $x\prime =dx/dz$ etc. represent the differential rotation of a body-fixed cartesian coordinate as a function of movement along the centroid from one end. The local curvature $\kappa$ of the centroid is then

$$\kappa =\sqrt{\frac{{x″}^2+{y″}^2+{\left(x\prime y″-x″y\prime \right)}^2}{{\left(x\prime 2+y\prime 2+1\right)}^3}}$$ [1]

and the local torsion ${\tau }_d$ (twist/length) of the body is

$${\tau }_d=\frac{x^{″}y‴-x‴y^{″}}{{x^{″}}^2+{y^{″}}^2+{\left(x\prime y″-x″y\prime \right)}^2}.$$ [2]

These equations allow us to parameterize the topology of the helices. We measured the three-dimensional helical parameters $\kappa$ and ${\tau }_d$ of the filament by cooling the bacteria down to just above $0^{o}C$. This slowed down the rotation rate of the bacteria to under 1 Hz so that we were able to make full 3D scans of single bacteria without blurring due to motion. We fit these 3D conformations to that of a helix using Eqs. 1 and 2. Fig. 2A shows the handedness of the coordinate system in three dimensions for determining the velocity $\overrightarrow{v}$, the rotational angular velocity $\omega$ of filaments, and the handedness of the filaments.

Fig. 2.

(A) Coordinate system in three dimensions for determining the velocity $\overrightarrow{v}$, the rotational angular velocity $\omega$ of filaments, and the handedness of the filament. In the cartoon, the helix is left-handed with $\kappa$ of 0.14 per unit and $\tau$ of −0.23 1 per unit. (B) A left-handed helical bacteria of motile strain RP437 of length $L$ about $26\hspace{0.17em}\mathrm{\mu }m$. The helix fit shown in red solid line has radius 1.7 $\mathrm{\mu }m$, $\kappa =$ 0.08, and $\tau =$−0.20 $\mathrm{\mu }{m}^{-1}$. (C) A right-handed helical bacteria of length $L$ 20 $\mathrm{\mu }m$. The helix fit shown in red solid line has a radius 2.0 $\mathrm{\mu }m$, $\kappa =$ 0.10 $\mathrm{\mu }{m}^{-1}$, and $\tau =$ +0.20 $\mathrm{\mu }{m}^{-1}$. (D) A left-handed helix from nonmotile strain RP3098 of length 57 $\mathrm{\mu }m$. The helix fit in red solid line has a radius 1.3 $\mathrm{\mu }m$ and $\kappa =$ 0.032 $\mathrm{\mu }{m}^{-1}$.

Fig. 2B shows a fit of Eqs. 1 and 2 to the 3D confocal image of a left-handed filament helix and Fig. 2C to the image of a right-handed helix bacterial filament. As a further control, we also induced lamentation in a strain containing a deletion of the flagellar regulatory gene flhDC, which does not assemble flagella. This nonmotile strain still develops long helical filaments as shown in Fig. 2D. Inspection of the helices in Fig. 2 indicates that all cells have a net total twist angle ${\int }_0^L{\tau }_{d}dl\sim \pm 2\pi$. We measured the torsion of both motile and nonmotile filaments. Fig. 3 shows the torsion ${\tau }_d$ scales as $1/L$.

Fig. 3.

Torsion $\tau$ vs. 1/L of filamented bacteria. Red circles indicate a right-handed $\tau$, and blue triangles indicate left-handed $\tau$. The dashed line has a slope of 2$\pi /L$. Inset shows handedness.

Note that $\kappa$ and ${\tau }_d$ are a measure of local longitudinal and shear strain, respectively, which results in stored elastic energy in the helix via the Young’s modulus and shear modulus, respectively. Although it is admittedly a gross simplification for a complex, highly elongated bacterial body, we use a simple model of buckling from linear Kirchhoff–Love theory of elastic rods (8) to model the helical filamented bacterial bodies. A long straight 3D rod of length $L$ above a critical torsion ${\tau }_d^{*}$ will buckle into a helical rod with lower stored strain energy (9). Kirchhoff–Love theory finds that the critical torsion ${\tau }_d^{*}$ (in the limit of zero tensile force) is

$${\tau }_d^{*}=2\pi \left[\frac{E{I}_a}{G{I}_p}\right]\frac{1}{L},$$ [3]

where $E$ is the Young’s modulus of the rod, $I_A$ is the surface moment of inertia $\int xydxdy$, $G$ is the shear modulus, and $I_p$ is the polar moment of inertia $\int \left(x^2+y^2\right)dxdy$. Since $I_a$ and $I_p$ for a cylinder have the same value and $E=2G\left(1+\nu \right)$ (where $\nu$ is Poisson’s ratio) is approximately the same number, we can roughly say that at torsional buckling criticality, the total twist of the helix should be about $2\pi$ radians, independent of the length $L$ of the helix, and that the critical torsion ${\tau }_d^{*}\sim \frac{2\pi }{L}$. The instability occurs as the torque (internal or external), a physical quantity, increases the torsion of the straight rod until a critical value of the net twist is reached, upon which $\kappa$ jumps to a finite value, as the local torsion jumps to a lower value. Note that this critical torsion ${\tau }_d^{*}$ is a very general number approximately of the same value for most materials, as can be verified by playing with rubber hoses. The physics reason for the instability is that the strain energy of a bent rod scales as the fourth power of the bending radius, while the strain energy of a twisted rod goes as the square of the twist (in linear elastic theory), so transferring energy from pure twist to twist + bend can lower the net stored energy. Filaments of other bacteria such as Bacillus subtilis can achieve very high torsion values far beyond $2\pi /L$, resulting in immobile plectonemic supercoils (10).

As is predicted by Eq. 3, all filaments we have studied, both right-handed and left-handed, motile and immotile, obey Eq. 3, indicating they all are slightly past the critical torsional buckling point with net integrated twist angle of $2\pi$ radians. The fact that E. coli RP3098 ($\mathrm{\Delta }$flhA–flhD) also is poized at criticality suggests that the buckling instability we see here is not due to a hydrodynamic shear loading of the swimming filaments and thus is probably not an externally imposed dynamical bucking transition (11) but rather due to internally generated strain registration. However, unlike the strain registration topology observed by Mendelson (12) in stationary B. subtilis, in our case the buckling always stops exactly at the critical value independent of the length of the bacterium.

The rotational handedness of the helices is also unusual: The filaments always rotate with angular velocity $\overrightarrow{\omega }$ parallel to the translational velocity $\overrightarrow{v}$. However, as seen in Fig. 2, not all filaments are physically right-handed as would be expected if the filaments are “screwing” through the liquid at low $R_e$ (13). That is, a left-handed helix should rotate with $\overrightarrow{{\omega }_L}$ in the opposite direction from $\overrightarrow{v_T}$ if the handedness of the helix determines the direction of $\overrightarrow{v_L}$. But this is not observed: All helices have a right-handed $\overrightarrow{\omega }$ when moving independent of their physical handedness.

Further, we found that the radius $r_h$ of the helices increases in a nonlinear way with the length of the helix $L$ as shown in Fig. 4A. However, another constant of the motion, not as surprising as the criticality of the buckling of the filament, can be found from the relation between the rotation rate of the helical bodies ${\omega }_R$ and the radius of the helix $r_h$. Measurement of the rotational rate $f_R=\omega \left(R\right)/2\pi$ shows that $f_R$ varies inversely with the radius $r_H$ of the helix as seen in Fig. 4B, which reveals that $r_H\sim f_R^{-1.0}$. This implies that the tangential velocity ${\overrightarrow{v}}_T=\overrightarrow{\omega }R=20\hspace{0.17em}\mathrm{\mu }m/\sec$ is both constant and independent of not only the length $L$ of the filament but also the radius $r_H$ of the helix and equal to the translation speed $v_T$ of the motile helices.

Fig. 4.

(A) Helix radius vs. length $L$. (B) Rotational rate $f_R$ vs. radius $R_H$ of the helix.

The ultimate propulsion mechanism of the helical E. coli filaments is believed to be left-handed rotating helical flagella bundles (14, 15). The motion of a normal 2 $\mathrm{\mu }m$ × 0.5 $\mathrm{\mu }m$ cylindrical E. coli cell in the absence of a chemical gradient is described by a simple, elegant random-walk model based on alternating sequences of linear runs and random-direction tumbles (4). The helices of the flagella are left-handed and during linear runs form a single left-handed bundle that pushes the bacterium by rotating with angular velocity $\overrightarrow{{\omega }_F}$, which is directed opposite to the translational velocity $\overrightarrow{v_L}$ of the normal bacteria, because of the left-handed torsion $\tau$ of the flagella helix bundle. Normal bacteria always swim with angular velocity $\overrightarrow{{\mathit{\omega }}_{\mathbf{N}}}$ of the flagella that is oppositely directed from $\overrightarrow{v_L}$ because the bundle is left-handed. The reaction torque on the bacterium body gives it a right-handed angular velocity $\omega$, which is always directed in the direction of $\overrightarrow{v_L}$ for normal bacteria. Since the helical bacterial body filaments also always rotate with angular velocity $\overrightarrow{\omega }$ in the same direction as $v_L$, presumably the same reaction torque drives their rotation, independent of the body torsion. This finding is in agreement with other work on flagella-driven helical swimming bacteria that indicates that the helical shape has very little influence on speed of the bacteria (16).

Finally, these helical filaments do not tumble in random directions every second or so but rather simply reverse direction, as shown in Fig. 5; have the same swimming speed $v_L$ as normal cells independent of the filament length $L$, as shown in Fig. 5A; and reverse direction much less frequently than normal E. coli. Tracking individual filaments allowed us to determine the swimming persistence length $P_f$ of the filaments using

$$<\cos \theta >=e^{-s/P_F},$$ [4]

where $s$ is the arc length of a path of a filament and $<\cos \theta >$ is the average value of the angle $\theta$ between a vector that is tangent to the path at a distance $s$ away from the initial position. Fig. 5C shows the distribution histogram of persistence lengths recorded using this tracking method. There is a large range of persistence lengths, with basically an exponential distribution. Using this distribution of the persistence lengths and knowing the constant swimming speed of the filaments, we can compute the effective diffusion constant $D_F$ of the filament. For example, in our dataset, for $P_F\sim 500\hspace{0.17em}\mathrm{\mu }m$, $K_F=2P_F$ (the Kuhn length of the filamentous E. coli motion) and d = 3 dimensional space:

$$t_F=\frac{K_F}{v_F},D_F=\frac{K_F^2}{2d{t}_F}=\frac{v_F{P}_F}{d}\sim 3000\hspace{0.17em}\mathrm{\mu }{m}^2/s.$$ [5]

We do not yet know the underlying biological and physical mechanisms of how the helicity is generated, but the evidence is that internal strain buckles the filaments. The programmatic nature of filamentous response to heat stress was originally studied in the classic work of Mendelson and Cole (17), and we assume a related response occurs in E. coli under antibiotic stress. Cell shape in both gram-negative and -positive bacteria is determined by the rigid peptidoglycan (PG) cell wall. The PG cell wall is constructed of glycan strands covalently cross-linked by peptide units that generate a material that can withstand the internal turgor pressure of the cell and maintain its shape (18). It was demonstrated that cells grown in confined geometries retained their shape after release but relaxed after periods of growth (19). How the PG layer is modified during growth and division has been a long-studied problem. MreB is a cytoskeletal protein that is a homolog of actin and has been shown to form helical constructs within the cell that direct and recruit the synthesis of PG (20–22). Several papers have shown that during PG synthesis, MreB induces a chiral ordering and physical twisting of the PG within the cell wall. The handedness of the chiral ordering/twist has been observed to be both left- and right-handed, with often some bias toward one or the other (23–25). Computational work has shown that defects or spatial variation in the cross-linking of the PG can result in different cellular morphologies (26, 27). Indeed, the shape of the helical bacteria Helicobacter pylori is a result of directed relaxation of cross-links in the PG cell wall (28).

Fig. 5.

(A) Translational speed $v_T$ of filaments within a fixed z plane 0.5 $\mathrm{\mu }$m from the coverslip surface as a function of filament length. The dashed (blue) line is the average speed of a normal E. coli between tumbles. (B) Reversal event for a filament. The filament simply reverses spin and velocity. (C) Semi-log histogram of measured persistence length $L_p$ of motile filaments.

It is very possible that the helices we observe in this work are a result of both the intrinsic chiral twist that is generated during PG synthesis and possible defects that result from the stress imposed by the environment. Interestingly, highly spiraled/helical filaments have been observed in certain mutant E. coli that lack machinery for PG synthesis and septation (29). These higher spiraled structures are reminiscent of different, noncritical modes of a torsional instability of a thin rod and may provide another route in understanding how the molecular machinery of PG deposition couples to the resultant physical morphology of the cell.

Ultimately, emergent phenotypes in biology must have a fitness advantage. The extremely large filament diffusion coefficients, up to 20 times that of a normal bacterium, mean that the filaments cover much larger regions of space for a fixed time than a normal bacterium despite their similar translational speeds. We suggest that motile filamentous E. coli can explore environments much faster and more widely than normal-length bacteria. Helical motile filamentation is an emergent property (30, 31): Genetically, the helical bacterial filaments are identical to their parents in the absence of mutations, yet their phenotype is greatly different from the parent’s with a possibly substantially changed fitness mission.