Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 182918, 6 pages

http://dx.doi.org/10.1155/2015/182918

## Numerical Modeling of Force-Stiffness Response of Cross-Linked Actin Networks Using Tensegrity Systems

^{1}Department of Civil Engineering, Zhejiang University, A-823 Anzhong Building, 866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China^{2}Department of Civil Engineering, Zhejiang University, A-818 Anzhong Building, 866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China^{3}Department of Civil Engineering, Zhejiang University, A-821 Anzhong Building, 866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China

Received 5 August 2015; Revised 3 November 2015; Accepted 16 December 2015

Academic Editor: Jose J. Muñoz

Copyright © 2015 Xian Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A three-dimensional tensegrity structure is used as a computational model for cross-linked actin networks. The postbuckling behavior of the members under compression is considered and the constitutive relation of the postbuckling members is modeled as a second-order polynomial. A numerical scheme incorporating the equivalent constitution of the postbuckling members is used to predict the structural response of the tensegrity model under compression loads. The numerical simulation shows that the stiffness of the tensegrity structure nonlinearly increases before member buckling and abruptly decreases to a lower level as soon as members buckle. This result qualitatively mimics the experimentally observed stiffness to compression stress response of cross-linked actin networks. In order to take member length variety into account, a large number of simulations with the length of buckling members varying in the given range are also carried out. It is found that the mean response of the simulations using different buckling member length exhibits more resemblance to the experimental observation.

#### 1. Introduction

Tensegrity is a special class of pin-jointed assemblies, whose stability is provided by the self-stress state between tensioned elements and compressed elements. Sculptures containing the essential characteristics of tensegrity were first created by Snelson in 1948, while the term “tensegrity” was first used by Fuller in 1961 to name a class of cable-bar structures in a patent [1, 2]. Since it was invented a half century ago, tensegrity has attracted much attention from both researchers and practicers. Architectural application was believed as a promising area for tensegrity applications. But after many endeavors towards this direction, applications of this structural system in real projects are very few. The limited use of the system can be attributed to its inherently complex and flexible nature. However, it inspired some innovative structural systems such as cable domes [3] and suspended-domes [4], which have attained great success in architectural applications. Currently, new applications of tensegrity structures are taking place in the field of “smart” structures [5–10], locomotive robots [11], and cellular mechanics [12–15], due to their adjustability and resemblances to cellular structures. A comprehensive review and discussion on the development and applications of tensegrity can be found in Motro [16], Skelton and de Oliveira [17], and Sultan [18].

Tensegrity-based models have been used to successfully predict the mechanical behaviors of whole cells [12, 13, 19–22], the erythrocyte membrane [23], viruses [24], actin stress fibers [25], and actin networks [26]. Postbuckling behavior of compression elements especially has been considered to explain the linear stiffening behavior of living cells [27, 28]. Stretch force [27] and shear force [28] were considered and emphasis was put on the linear stiffening behavior exhibited after the buckling of compression elements. The behavior before and during the buckling of compression elements was short of analysis. The application of tensegrity structures as mechanical models of living cells is based on the assumption that the cytoskeleton enjoys tensegrity structures with the microtubules and microfilaments acting as compression elements and tension elements, respectively. Since living cells are initially tensed, prestress is always included in the tensegrity model of the cytoskeleton. Though tensegrity models with zero prestress were also examined in the previous studies, they were treated as comparison cases to the ones with prestress [27, 28]. Xu and Luo [26] used a planar 3-member tensegrity structure with a buckling member to explain the nonlinear stiffening and reversible softening behavior exhibited by dendritic actin networks. Compression load which has not been considered in the previous studies is applied to the tensegrity model to mimic the compression imposing on the actin networks [26]. Since the average stiffness of dendritic actin networks* in vitro* was found to be independent of prestressing by myosin II motors [29], no prestress was introduced into the tensegrity model of actin networks [26].

Based on the previous study [26], a more general tensegrity model in three dimensions is proposed in this paper. Instead of the analytical method used for the planar model, a numerical approach is used to predict the structural response of the 3D tensegrity model and the constitutive relation of the postbuckling members is modeled as a second-order polynomial. Numerical results show that the stiffness to compression force response of the proposed model qualitative corresponds to the behavior exhibited by the actin networks under compression. The work presented in this paper confirms the finding reported in the previous study [26] and extends the tensegrity model from a conceptual 2D system to a more general 3D system. The analytical method used in the previous study [26] is only fit for simple tensegrity system with a very small number of members. The numerical approach presented in this paper is a general scheme which is able to handle more complex tensegrity system with a large number of members and can be used in the future study of using more sophisticated tensegrity model to obtain quantitative correspondence to the experimental observation.

#### 2. Methods

##### 2.1. Derivation of Stiffness

Here, the basic stiffness formulation described in Guest [30] is used. The tangent stiffness matrix relates to the nodal displacement, written as a vector , and the applied load at each node, written as a vector , aswhich is well known as the equilibrium equation in displacement form. The equilibrium of the structural system also can be described in force form aswhere is the compatibility matrix of the structural system and is the internal force matrix [31]. The compatibility matrix can be determined from kinematic relationships by using the principle of virtual displacements [28]where is the elongation matrix of the members.

The tangent stiffness of the system can be obtained bySubstituting (2) and (3) into (4) yieldswhere

The tangent stiffness matrix also can be written as [32, 33]where ; is called the “material stiffness matrix”; and is called the “geometric stiffness matrix.” is a diagonal matrix of modified axial stiffness, with an entry for each member (, and is the number of members of the structural system),where is the axial stiffness of member and is the nominal strain of member . , , and are defined as the cross-sectional area, Young’s modulus, and length of member , respectively.

The geometric stiffness matrix can be written as the Kronecker product of stress matrix and a 3D identity matrix **;** that is,The components of matrix are given bywhere is the tension coefficient in the member connecting nodes and .

##### 2.2. Constitutive Relation

A constitutive relation as that used by Volokh et al. [28] for the strut of their tensegrity models for the cell cytoskeleton is adopted for all the members of the tensegrity system considered in this paper. We do not intentionally sort the members of a tensegrity system as cables or struts, because actin networks consist of one type of elements: actin filaments. Each member in the tensegrity system is modeled as a pinned elastic column (Figure 1). The rest length of the elastic column is notated as and the bending stiffness of it is notated as , where is Young’s modulus of actin filaments and is the moment of inertia of the sectional area of an actin filament. When it is tensioned, it behaves linearly in accordance with Hooke’s law. When it is compressed, it also behaves linearly in accordance with Hooke’s law as long as the compression in it is smaller than the Euler critical force . Once the compression in the member reaches , the member begins to buckle. The postbuckling behavior of a member is assumed to be in accordance with the solution of the “elastica” problem [34]where ; is the chord length after the buckling (Figure 1); and are complete elliptic integrals of the first and second kind, respectively; and is the end slope of the deformed bar (Figure 1).