TY - JOUR
T1 - The shape and dynamics of the Leptospiraceae
AU - Kan, Wanxi
AU - Wolgemuth, Charles W.
N1 - Funding Information:
Here we have presented a model for the end morphology and dynamics of the Leptospiraceae . This model assumes that competition between the preferred shapes of the cell cylinder and the periplasmic flagellum determines the morphology of the ends of the bacterium. Since the PF is constrained to reside at the radius of the cell cylinder, both the PF and the CC must deform elastically. The minimum energy configuration in the absence of applied forces or torques is a hook shape. The model predicts that the bending modulus of the PF is ∼7 times smaller than that of the CC. The Young’s modulus of the cell wall of the MPa MPa MPa for MPa for = MPa and a thickness of 10 nm, we estimate × 10 × 10 × 10 Leptospiraceae has not been measured; however, the modulus of other bacteria has been. For example, optical trapping experiments estimated the Young’s modulus of Bacillus subtilis to be 5.0 (24) and atomic force microscopy has found the modulus for the cell wall of Myxococcus xanthus to be 1.3 (25) , 25.0 Escherichia coli (26) , and 0.085–0.15 Magnetospirillum gryphiswaldense (27) . The bending modulus of the cell can be estimated using the radius of the cell, a , and the thickness, t , of the cell wall as A cc πEa 3 t . If we use a moderate value of the Young’s modulus for the cell wall of 1.0 A cc to be ∼2 −23 N m 2 . Therefore, our model predicts the bending modulus of the PF to be ∼3 −24 N m 2 . Estimates based on experiments using Salmonella flagellar filaments reported values ranging from 10 −24 N m 2 (28) to 10 −22 N m 2 (29) . Kim and Powers reanalyzed the data from (29) using slenderbody theory and estimated a value of 3.2 −24 N m 2 for the flagellar bending modulus (30) . We showed that clockwise rotation of the PF driven by a torque applied by a rotary motor located at the end of the cell can maintain a hook shape. As well, rotation of the flagellum in a counter-clockwise direction can produce left-handed, spiral-shaped end morphology. These morphologies are a result of linear elastic deformation induced by the applied torque from the flagellar motor and the internalization of the PF inside the periplasmic space. Our model produces realistic cell shapes when the torque of the bacterial flagellar motor of the Leptospiraceae is between 0.1 and 0.3 A cc μ m −1 . Using the estimate for the bending modulus of the CC given above, we calculate a torque of 2000–6000 pN nm. Berry and Berg measured the stall torque of the flagellar motor of Escherichia coli to be ∼4500 pN nm (31) . Therefore, our model predicts realistic cell shapes for reasonable values of the applied torque. To test this model, the bending moduli of the PF and CC should be measured. One possible method would be to use an optical trap to apply forces to these structures. By measuring the end-to-end displacement as a function of forcing, the bending moduli can be estimated. The authors thank N. W. Charon, R. E. Goldstein, and S. F. Goldstein for comments and useful discussions and T. R. Powers for discussions pertaining to the relationship between functional derivatives and the moment. This work was supported by the National Institutes of Health (grant No. R01 GM072004). Appendix A
PY - 2007/7
Y1 - 2007/7
N2 - Most swimming bacteria produce thrust by rotating helical filaments called flagella. Typically, the flagella stick out into the external fluid environment; however, in the spirochetes, a unique group that includes some highly pathogenic species of bacteria, the flagella are internalized, being incased in the periplasmic space; i.e., between the outer membrane and the cell wall. This coupling between the periplasmic flagella and the cell wall allows the flagella to serve a skeletal, as well as a motile, function. In this article, we propose a mathematical model for spirochete morphology based on the elastic interaction between the cell body and the periplasmic flagella. This model describes the mechanics of the composite structure of the cell cylinder and periplasmic flagella and accounts for the morphology of Leptospiraceae. This model predicts that the cell cylinder should be roughly seven times stiffer than the flagellum. In addition, we explore how rotation of the periplasmic flagellum deforms the cell cylinder during motility. We show that the transition between hook-shaped and spiral-shaped ends is purely a consequence of the change in direction of the flagellar motor and does not require flagellar polymorphism.
AB - Most swimming bacteria produce thrust by rotating helical filaments called flagella. Typically, the flagella stick out into the external fluid environment; however, in the spirochetes, a unique group that includes some highly pathogenic species of bacteria, the flagella are internalized, being incased in the periplasmic space; i.e., between the outer membrane and the cell wall. This coupling between the periplasmic flagella and the cell wall allows the flagella to serve a skeletal, as well as a motile, function. In this article, we propose a mathematical model for spirochete morphology based on the elastic interaction between the cell body and the periplasmic flagella. This model describes the mechanics of the composite structure of the cell cylinder and periplasmic flagella and accounts for the morphology of Leptospiraceae. This model predicts that the cell cylinder should be roughly seven times stiffer than the flagellum. In addition, we explore how rotation of the periplasmic flagellum deforms the cell cylinder during motility. We show that the transition between hook-shaped and spiral-shaped ends is purely a consequence of the change in direction of the flagellar motor and does not require flagellar polymorphism.
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U2 - 10.1529/biophysj.106.103143
DO - 10.1529/biophysj.106.103143
M3 - Article
C2 - 17434949
AN - SCOPUS:34447271221
SN - 0006-3495
VL - 93
SP - 54
EP - 61
JO - Biophysical Journal
JF - Biophysical Journal
IS - 1
ER -