1. Buckling Instabilities and Complex Trajectories in a Simple Model of Uniflagellar Bacteria 
Frank T.M. Nguyen, Michael D. Graham 
Biophysical Journal 
Volume 112, Issue 5, Pages (March 2017)
DOI: /j.bpj
Copyright © 2017 Biophysical Society Terms and Conditions

2. Figure 1
Schematic of the main swimmer model consisting of a body connected to a flagellum by a hook. In this illustration, the body is a prolate spheroid and the flagellum is a helix with a taper (≪lf) at the hook. Vectors pi denote orientations and ωi rotation directions of the body (i=b) and flagellum (i=f). To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

3. Figure 2
(a) Static toy model showing the body and flagellum as two connected rods. Propulsion is either aligned with the x-axis (Case 1) or the flagellum (Case 2). (b) Dynamic toy model shows a spheroidal body and cylindrical flagellum. The propulsion is aligned with the latter. The cylinder is exaggerated for clarity, and the associated helix is the same tapered one as Fig. 1. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

4. Figure 3
(a) Close-up of cell at the body-flagellum connection showing stator, rotor, and hook. The green arrow shows the applied motor torque, and the black arrows show Tp. The narrow tapering of the flagellum aligns hook axis with flagellar helical axis pf at connection point. (b) Nonhydrodynamic torques in the bending plane resulting from motor torque propagation include components (left to right): stator/body, rotor, hook, flagellum. Green arrows are the motor torque propagation, dotted red arrows are the required offset for a torque-free hook, and black arrows are the torques explicitly used in equations of motion. Constraint torques and hydrodynamic torques are omitted for clarity. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

5. Figure 4
(a) Snapshot of straight trajectory with FlT=1. (b) Snapshot of helical trajectory with FlT=2 is provided. Solid lines trace the path of the cell body center of mass. (c) Time evolution of the instantaneous bending angle θ is provided. The zoom inset shows oscillations in θ above FlT,crit (d) Bifurcation of average bending angle θ¯ in Fl is shown. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

6. Figure 5
(a) Phase space of bending-related Euler angles ϕy and ϕz. Thick lines are limiting trajectories for FlT=1 and FlT=2, thin gray line shows full transient trajectory for FlT=2, and colored dotted contours show level curves of θ. (b) Close-up is provided of center region showing the stable limit cycle of straight-swimming at FlT=1 in black, with the dotted gray line showing the trajectory evolution from (0,0). (c) Close-up is provided of limiting trajectory showing quasi-periodicity at FlT=2 helical swimming. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions

7. Figure 6
Average bending angle θ¯ as a function of (a) (f,L), (b) (f,ψ), (c) (f,λ∗), and (d) (f,eb), obtained by running simulations for an initially straight swimmer. In each case, the constant parameters are taken from the base geometry. The stability boundary is marked by the white line for the full model and dashed red line for the dynamic toy model, denoting the critical flexibility FlT,crit. The oscillations in the plots are purely numerical, as for certain parameter regimes, the bending timescale is far greater than the timescale of flagellar rotation. To see this figure in color, go online.
Biophysical Journal  , DOI: ( /j.bpj )
Copyright © 2017 Biophysical Society Terms and Conditions