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Theoretical Analysis of Twist/Bend Ratio and Mechanical Moduli of Bacterial Flagellar Hook and Filament

Terence C. Flynn ^* and Jianpeng Ma ^*  

^* Department of Bioengineering, Rice University, Houston, Texas 77005; and Graduate Program of Structural and Computational Biology and Molecular Biophysics and Verna and Marrs McLean Department of Biochemistry and Molecular Biology, Baylor College of Medicine, Houston, Texas 77030

Correspondence: Address reprint requests to Jianpeng Ma, One Baylor Plaza, BCM-125, Baylor College of Medicine, Houston, TX 77030. Tel.: 713-798-8187; Fax: 713-796-9438; E-mail: jpma{at}bcm.tmc.edu.

Certain motile bacteria employ rotating flagella for propulsion. The relative flexibility of two key components of the flagellum, filament and hook, is partially responsible for the mechanistic workings of this motor. A new computational method, the quantized elastic deformational model, was employed in this article to calculate the dimensionless twist/bend ratio (EI/GJ) of the filament and hook, providing a quantitative means to compare their relative stiffness. Both ratios were much <1.0, an average of 0.0440 for the filament and 0.0512 for the hook, indicating that within each structure bending is favored over twisting. These two ratios, along with previous experimental measurements, allowed us to propose a theoretical Young's modulus (E) between 10⁶ and 10⁷ dyn/cm² for the hook. This value is orders of magnitude smaller than experimentally determined Young's moduli of the filament, hence in agreement with empirical evidence linking compliance in the flagellum mainly to the hook.

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