Vibration and length-dependent flexural rigidity of protein microtubules using higher order shear deformation theory Abdelouahed Tounsi a,b,n , Houari Heireche a,c , Hachemi Benhassaini d , Miloud Missouri e a Laboratoire des Mate´riaux et Hydrologie, Universite´ de Sidi Bel Abbe ´s, BP 89 Cite ´ Ben M’hidi, 22000 Sidi Bel Abbe ´s, Algeria b De´partement de Ge´nie Civil, Faculte´ des Sciences de l’inge´nieur, Universite´ de Sidi Bel Abbe ´s,Alge´rie c De´partement de Physique, Faculte´ des Sciences, Universite´ de Sidi Bel Abbe ´s,Alge´rie d Laboratoire de Biodiversite´ Ve ´ge´tale, Conservation et valorisation, Universite´ de Sidi Bel Abbe ´s, BP 89 Cite ´ Ben M’hidi, 22000 Sidi Bel Abbe ´s, Algeria e De´partement de Biologie, Faculte´ des Sciences, Universite´ de Sidi Bel Abbe´s, Alge´rie article info Article history: Received 7 March 2010 Received in revised form 27 June 2010 Accepted 29 June 2010 Available online 11 July 2010 Keywords: Cell mechanics Microtubules Flexural rigidity Vibration Parabolic shear deformation theory abstract Microtubules are hollow cylindrical filaments of the eukaryotic cytoskeleton characterized by extremely low shear modulus. A remarkable controversy has occurred in the literature, regarding the length dependence of flexural rigidity of microtubules predicted by the classical elastic beam model. In this study, a higher order shear deformable beam model for microtubules is employed to study unexplained length-dependent flexural rigidity and Young’s modulus of microtubules reported in the literature. The formulation allows for warping of the cross-section of the microtubule and eliminates the need for using arbitrary shear correction coefficients as in other theories. It is showed that vibration frequencies predicted by the present parabolic shear deformation theory (PSDT) are much lower than that given by the approximate isotropic beam model for shorter microtubules, although the two models give almost identical results for sufficiently long microtubules. It is confirmed that transverse shearing and the warping of the cross-section of microtubules are mainly responsible for the length-dependent flexural rigidity of an isolated microtubule reported in the literature, which cannot be explained by the widely used Euler–Bernoulli beam model. Indeed, the length-dependent flexural rigidity predicted by the present model is found to be in qualitative agreement with the existing experimental data (Kurachi et al., 1995; Pampaloni et al., 2006). These results recommend that the parabolic shear deformation- beam theory offers a unified simple 1D model, which can capture the length dependence of flexural rigidity and be applied to various static and dynamic problems of microtubule mechanics. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction The ability of eukaryotic cells to adopt a variety of shapes and to carry out coordinated and directed movements depends on the cytoskeleton. Microtubules (MTs) belong to one of the three major classes of filaments in such a complex polymer network and play essential roles in many cellular functions (Alberts et al., 2005), such as separating chromosomes during mitosis, facilitating long- distance intracellular transport of both proteins and organelles, providing mechanical strength to maintain the shape of a cell and support the motor proteins to generate the force required for cell movement and changes in shapes, and acting as important targets for anticancer drugs (Jordan and Wilson, 2004). Biological functions of MTs largely depend on their mechanical properties. Recently, Pampaloni and Florin (2008) described key structural aspects of microtubules and presented recent results on their mechanics. Furthermore, Keten et al. (2010) showed the importance of studying biopolymer structure and their impor- tance to advance in material sciences. In their study, silk which is a biological protein fiber is investigated. In order to probe mechanical properties of MTs, continuum isotropic (Euler–Bernoulli) elastic beam model has been widely used to analyze 1D rod-like deformation of MTs, such as column- like buckling (Kurachi et al., 1995; Dogterom and Yurke, 1997; Wang et al., 2001; Takasone et al., 2002; Kikumoto et al., 2006) and beam-like vibration (Gittes et al., 1995; Venier et al., 1994; Vinckier, 1996; Janson and Dogterom, 2004; Pampaloni et al., 2006; Kasas et al., 2004), from which the flexural rigidity of MTs and the Young’s modulus were estimated. However, a remarkable controversy has occurred in the literature, regarding the flexural rigidity and the associated Young’s modulus of MTs. As com- mented recently by Kasas et al. (2004), ‘‘two decades of measurements involving different techniquesy resulted in values of Young’s modulus between 1 MPa (see Takasone et al., 2002; Venier et al., 1994)) and 7 GPa (see Kurachi et al., 1995; Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi Journal of Theoretical Biology 0022-5193/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2010.06.037 n Corresponding author at: Laboratoire des Mate ´ riaux et Hydrologie, Universite ´ de Sidi Bel Abbe ´s, BP 89 Cite ´ Ben M’hidi, 22000 Sidi Bel Abbe ´ s, Algeria. E-mail address: tou_abdel@yahoo.com (A. Tounsi). Journal of Theoretical Biology 266 (2010) 250–255