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Modelling Cell Origami via a Tensegrity Model of the Cytoskeleton in Adherent Cells

Modelling Cell Origami via a Tensegrity Model of the Cytoskeleton in Adherent Cells Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 8541303, 9 pages https://doi.org/10.1155/2019/8541303 Research Article Modelling Cell Origami via a Tensegrity Model of the Cytoskeleton in Adherent Cells 1,2 1,2 Lili Wang and Weiyi Chen Shanxi Key Laboratory of Material Strength & Structural Impact, College of Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China National Demonstration Center for Experimental Mechanics Education, Taiyuan University of Technology, Taiyuan 030024, China Correspondence should be addressed to Weiyi Chen; chenweiyi211@163.com Received 13 May 2019; Revised 29 June 2019; Accepted 30 July 2019; Published 14 August 2019 Academic Editor: Mohammad Rahimi-Gorji Copyright © 2019 Lili Wang and Weiyi Chen. 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. Cell origami has been widely used in the field of three-dimensional (3D) cell-populated microstructures due to their multiple advantages, including high biocompatibility, the lack of special requirements for substrate materials, and the lack of damage to cells. A 3D finite element method (FEM) model of an adherent cell based on the tensegrity structure is constructed to describe cell origami by using the principle of the origami folding technique and cell traction forces. Adherent cell models contain a cytoskeleton (CSK), which is primarily composed of microtubules (MTs), microfilaments (MFs), intermediate filaments (IFs), and a nucleoskeleton (NSK), which is mainly made up of the nuclear lamina and chromatin. The microplate is assumed to be an isotropic linear-elastic solid material with a flexible joint that is connected to the cell tensegrity structure model by spring elements representing focal adhesion complexes (FACs). To investigate the effects of the degree of complexity of the tensegrity structure and NSK on the folding angle of the microplate, four models are established in the study. The results demonstrate that the inclusion of the NSK can increase the folding angle of the microplate, indicating that the cell is closer to its physiological environment, while increased complexity can reduce the folding angle of the microplate since the folding angle is depended on the cell types. The proposed adherent cell FEM models are validated by comparisons with reported results. These findings can provide theoretical guidance for the application of biotechnology and the analysis of 3D structures of cells and have profound implications for the self-assembly of cell-based microscale medical devices. 1. Introduction traction forces to fold many microstructures from two- dimensional (2D) to 3D. Recently, He et al. described an Cell origami is defined as a technique that harnesses the trac- origami-inspired self-folding method to form 3D microstruc- tion force of living cells as a biological driving force to fold a tures of cocultured cells and indicated that the origami-based variety of three-dimensional (3D) cell-populated microstruc- cell self-folding technique is useful in regenerative medicine tures [1]. In the field of microfabrication, the origami folding and the preclinical stage of drug development [6]. However, technique has received increasing attention due to its multi- none of these studies have investigated cell origami by using ple advantages, including simplicity, high biocompatibility, the finite element method (FEM). the lack of special requirements for substrate materials, and Cell traction forces, as the contractile forces pointing to the lack of damage to cells. For example, Davis et al. [2] the centre of the cell body, are generated by the cytoskeleton and Azam et al. [3] used surface tension to create microsized (CSK) [7]. The CSK is a complex biopolymer network com- posed of microtubules (MTs), microfilaments (MFs), and containers. Sirrine et al. [4] and Song et al. [5] used the same technique to produce artificial tissue scaffolds. In addition, intermediate filaments (IFs) [8]. The CSK is the major Kaori et al. [1] experimentally determined that cells applied mechanical component of cells and plays a key role in the principle of the origami folding technique and cell mechanotransduction and extracellular force transmission 2 Applied Bionics and Biomechanics view of the fact that cell origami is a large deformation pro- from/to attaching a substrate through focal adhesion com- plexes (FACs) [9]. The forces in the CSK are related to the cess and IFs can provide resilience against mechanical forces biological functions of cells, such as differentiation, growth, and ensure cellular integrity [33], it is necessary to consider metastasis, and apoptosis [10–13]. The nucleus is regarded the role of IFs. Computational models of adherent cells com- as an integral structure functionally enabled by nuclear ten- posed of the CSK which is made up of MTs, MFs, and IFs and segrity, with struts representing the nuclear lamina and the NSK which mainly consists of the nuclear lamina and cables representing chromatin [12], featuring a large volume chromatin have been developed based on the tensegrity occupancy and including genetic information [14]. For structure. Since IFs vary from cell type to cell type [33] and example, Bursa et al. simulated the nucleoskeleton (NSK) as some studies have shown that IFs form a dense filament net- a tensegrity structure to study the CSK to transfer the exter- work spanning from the nucleus to the cell membrane [34], IFs are modelled as radial cables from the centre of the nal mechanical load of the cell to NSK, thereby initiating the biochemical response of the cell [15]. In addition, the tensegrity structure to the outer nodes in the models. important role of the NSK in cellular differentiation and Although the sphere-like tensegrity structure model development has been demonstrated [16]. Since some derived from the polyhedron (cuboctahedron or octahedron) researchers [9, 10, 17–28] have used both the spherical and [9] is symmetrical, the flat tensegrity structure derived from flattened tensegrity structure models’ approach combined the truncated polyhedron is not completely symmetric. with computational and mathematical models to investigate Two asymmetrical tensegrity structures derived from the the responses of cells to the substrate based on the assump- truncated polyhedron (flat cuboctahedron or octahedron), tion that individual cells can react by contraction and that 12-node tensegrity and 24-node tensegrity, are established the forces produced by cells can act on the extracellular to represent the different levels of complexity of the adherent matrix (ECM) by FACs. There are many cell models and cell models. The 12-node tensegrity structure is composed of models of cell-substrate interactions; however, FEM simula- 6 struts representing MTs and 36 cables, 24 of which repre- tion of cell origami has never been performed. Therefore, it sent MFs and 12 of which represent IFs, as shown in is necessary to establish a simple 3D FEM model of adherent Figure 1(a). The number of cables in the 24-node tensegrity cells composed of the CSK and NSK based on the tensegrity structure is 60, and the number of struts is twice that of the structures to simulate the cell origami. 12-node tensegrity structure. Among the 60 cables, 36 cables In this study, a 3D FEM model of an adherent cell made represent MFs, whereas the other 24 cables represent IFs, as up of the CSK and NSK is established and connected with a demonstrated in Figure 1(b). In both tensegrity structures, microplate to depict the cell origami. The CSK and NSK are the cables (red elements) representing MFs are connected represented by tensegrity structures of different levels of by the nodes at both ends of the struts (blue elements), while complexity, in which the CSK is composed of MTs, MFs, the cables (yellow elements) representing IFs are connected and IFs and the NSK consists of the nuclear lamina and chro- by the nodes at one end of the struts and the particle at the matin. The cell model adheres to the microplate through the centre of the structures which is simplified by the nucleus. spring elements representing FACs [29, 30]. The effects of the The microplate on which the cell is adhered is treated as level of complexity on the folding degree of the microplate an isotropic linear-elastic solid material, and its dimensions are investigated by changing the degree of complexity of the are adapted to the cell model. The microplate has a length cell tensegrity structure. Furthermore, the role of the NSK (b) and a width (b)of 30 μm and a height (h) of 2.7 μm; fur- is studied by using a 12-node and a 24-node sphere-like ten- thermore, the dimensions of the joint are 6 μm in width (w), 30 μm in length (b), and 0.3 μm in thickness (t), which shows segrity structure in comparison with the models without the NSK. The validity of the proposed models is validated by that the flexible joint is 3~8 μm in width and 70~390 nm in comparisons with the reported findings, demonstrating that thickness [1]. The spring elements are selected to link the cell the models can provide an attempt to measure the cell trac- model with the microplate. The folding angle (θ)isdefined as tion force in a 3D physiological environment and a new the angle between the folded microplate and its initial posi- tion, which is an important parameter for producing desired way of promoting a deeper understanding of cell origami. 3D cell-populated microstructures, as shown in Figure 2. From a geometric point of view, the folding angle can be 2. Materials and Methods expressed by the thickness (t) and the width (w) of the joint 2.1. Tensegrity Model. The process of cell origami is modelled and the thickness of the microplate (h) as follows [1]: by using the adherent cell to fold the microplate. Geometries of the models are created using UG NX 10.0 (Unigraphics = , 1 max NX 10.0) and then imported into the commercial finite ele- h + t/2 ment package ABAQUS (standard version 6.13, SIMULIA company, Germany) for simulations and analysis. Since When w is 6 μm, h is 2.7 μm, and t is 0.3 μm, the maxi- some studies have shown that IFs, as one of the major struc- mum folding angle is 120 according to equation (1). The fol- ture components of the CSK, play important roles in biolog- lowing folding angle must be less than the aforementioned ical functions such as cell contractility, migration, stiffness, value (<120 ) since the cell traction force of the measurement and stiffening [31]; play key mechanical roles in providing is inaccurate when the microplate contacts. structural stability of the cell; and can increase the cellular Furthermore, in order to simulate the influence of the rigidity at high strains (>20% strain) [24, 32]. Moreover, in NSK on the folding angle, the 12-node (Figure 3(a)) and w Applied Bionics and Biomechanics 3 (a) 12-node CSK (b) 24-node CSK Figure 1: Schematic diagram of the CSK with two different levels of complexity: (a) the 12-node CSK is composed of 6 struts (blue), 24 cables (red), and 12 cables (yellow); (b) the 24-node CSK is made up of 12 struts (blue), 36 cables (red), and 24 cables (yellow). max t (a) Arbitrary folding angle (b) Maximum folding angle Figure 2: Schematic diagram of the folding angle of the microplate: the horizontal blue dashed line represents the initial position of the microplate; the oblique red dashed line represents the position of the folded microplate. 24-node (Figure 3(b)) sphere-like tensegrity structures are manufacturing and biocompatibility and is commonly used for the NSK, with cables (cyan elements) representing used in the microfabrication [35]. The material and geo- chromatin and with the nuclear lamina modelled as struts metrical properties for all the components are based on (purple elements). Both tensegrity structures describing the the values published in the literature and summarized in NSK are symmetrical since they are derived from the cuboc- Table 1. tahedron and octahedron. The centre of the NSK coincides The cables and struts are depicted as truss elements that with the centre of the corresponding CSK, and each node of support only axial force and deformation, neglecting subcel- the NSK is connected to the node of the CSK pointing the lular bending. The prestress carried in a cable (F) with a same direction by IFs treated as the linker, as demonstrated current length (l) is [10] in Figures 3(c) and 3(d). l − l F + E A ,if l > l , 0 a a r 2.2. Material Properties and Boundary Conditions. Although F = 2 most components of the cell exhibit more or less nonlin- 0, if l ≤ l , ear constitutive behaviour, all materials are assumed to be linear elastic for simplicity. Moreover, the material parameters of parylene C are used for the microplates where l and l denote the resting and initial cable lengths, r 0 because this material has the advantages of ease of respectively, and E and A are Young’s modulus and a a 4 Applied Bionics and Biomechanics (a) 12-node NSK (b) 24-node NSK Y X Y X (c) 12-node CSK-NSK with microplate (d) 24-node CSK-NSK with microplate Figure 3: Schematic diagram of tensegrity structures with two different levels of complexity: (a) the 12-node NSK is composed of 6 struts (purple) and 24 cables (cyan); (b) the 24-node NSK is made up of 12 struts (purple) and 36 cables (cyan); (c) the 12-node CSK-NSK with a microplate; (d) the 24-node CSK-NSK with a microplate. where L and L denote the resting and initial lengths of r 0 Table 1: The material parameters and geometric dimensions. struts, respectively, and E , B , and A are Young’s modulus, s s s bending stiffness, and cross-section area of struts, respec- Elastic modulus (Pa) ν Dimensions tively. P and P are the initial strut tension and axial thrust, 9 2 0 c 1 2× 10 MTs [36] 0.3 190 nm respectively, described as 9 2 MFs [36] 2 6× 10 0.3 19 nm 2 E A s s IFs [37] 2×10 0.3 100 nm P = L − L , 0 r 0 6 2 Lamina [38] 1 4× 10 0.3 78.5 nm 6 2 π B Chromatin [38] 244 × 10 0.3 1.13 nm c P = 9 3 Microplate [1] 4×10 0.3 30 × 30 × 2 7 μm 9 3 Flexible joint [1] 4×10 0.3 30 × 6 × 0 3 μm The initial boundary conditions for the cell tensegrity Bond stiffness [39] k =0 025 nN/μm structure models are that the 12-node tensegrity structure has three nodes pinned to the microplate and three cables located on the microplate and coupled with the microplate, cross-section area of cables, respectively. F is the initial cable while the 24-node tensegrity structure has four nodes and tension, described as follows: four cables. The nodes closest to the microplate are anchored to the corresponding nodes of the microplate via spring ele- E A a a ments, and the other nodes are pinned as free moveable F = l − l 3 0 0 r joints. The number of nodes on the microplate is determined according to the z coordinate of the node. If z is equal to 0, it Meanwhile, the prestress carried in a strut (P) with a cur- is on the microplate, and if z is greater than 0, the node is not rent length (L)is on the microplate. The smaller the z is, the closer it is to the microplate. The centre of the microplate is constrained in all degrees of freedom. A concentrated force of 10 pN is applied L − L r c P + E A ,if L < L , P < P , 0 s s r at the farthest nodes parallel to the microplate [40]. Only one P = 4 truss element is used for all subcellular components, and the 0, if L ≥ L , microplate is meshed with 8-node hexahedral elements. Applied Bionics and Biomechanics 5 U, U3 +5.439e+00 +4.983e+00 +4.526e+00 +4.069e+00 +3.612e+00 +3.155e+00 +2.699e+00 +2.242e+00 +1.785e+00 +1.328e+00 +8.715e−01 +4.147e−01 −4.209e−02 (a) 12-node CSK tensegrity structure U, U3 +5.912e+00 +5.420e+00 +4.927e+00 +4.434e+00 +3.942e+00 +3.449e+00 +2.956e+00 +2.464e+00 +1.971e+00 +1.478e+00 +9.854e−01 +4.927e−01 +0.000e+00 (b) 24-node CSK tensegrity structure Figure 4: Schematic diagram of cell origami without the NSK. 3. Results 3.1. Influence of the Level of Complexity on the Folding Angle. 17.9 Since Kaori et al. have shown that the folding angle of the microplate is related to the cell types [1], the relationship 12.4 between the folding angle and levels of the complexity ten- 11.7 segrity structure in the cell origami process is studied using two different levels of complexity for the CSK and NSK. A 6.76 deformation diagram of cell origami without the NSK is shown in Figure 4. The maximum folding angles of the microplates of the ° ° 12-node and the 24-node CSK models are 12.4 and 6.76 , respectively, and the maximum values of the 12-node and 12-node tensegrity 24-node tensegrity ° ° the 24-node CSK-NSK models are 17.9 and 11.7 , respec- CSK tively, as demonstrated in Figure 5. CSK-NSK The result shows that the stiffness of the 12-node tensegr- ity structure model with/without the NSK is larger than that Figure 5: The maximum folding angle of the microplate. of the 24-node model with/without the NSK, indicating Folding angle (°) 6 Applied Bionics and Biomechanics U, U3 +8.381e+00 +7.681e+00 +6.981e+00 +6.281e+00 +5.582e+00 +4.882e+00 +4.182e+00 +3.482e+00 +2.782e+00 +2.082e+00 +1.382e+00 +6.826e−01 −1.728e−02 (a) 12-node CSK-NSK tensegrity structure U, U3 +7.903e+00 +7.245e+00 +6.586e+00 +5.928e+00 +5.269e+00 +4.610e+00 +3.952e+00 +3.293e+00 +2.634e+00 +1.976e+00 +1.317e+00 +6.586e−01 +0.000e+00 (b) 24-node CSK-NSK tensegrity structure Figure 6: Schematic diagram of cell origami with the NSK. indirectly that the folding angle is related to the cell type from the NSK. The deformation diagram of cell origami with the NSK is shown in Figure 6. the perspective of simulation. This result may be due to the larger number of nodes, the greater complexity of the struc- The maximum folding angle of the model with the ture, the additional degree of freedom, the additional energy NSK is larger than that without the NSK, as shown in required for deformation, and the smaller folding angle of the Figure 5. The results show that the augmentation of the microplate. Furthermore, the folding angle is asymmetrical, NSK can increase the stiffness of the model, independent which may be caused by the asymmetry of the model, as of the complexity. An increase in stiffness means an shown in Figure 4. increase of the folding angle, indicating that the cell is closer to its physiological environment. The above results 3.2. Effect of the NSK on the Folding Angle. In view of the fact imply that the 24-node tensegrity structure is sufficiently that the nucleus represented by the NSK has a large volume complex to describe the flat tensegrity structure representing the adherent cell morphology, and the results are consistent occupancy and includes genetic information, to investigate the effect of the NSK structure on the folding angle, two with the conclusions of Pugh [41], who demonstrated tensegrity structure models with different levels of NSK com- that when the levels of complexity of the structure increase plexity are established and compared with the model without further, the tensegrity structure becomes more analogous Applied Bionics and Biomechanics 7 to a cylinder and does not represent the geometry of 5. Conclusions suspended cells. In the present study, a 3D FEM model of adherent cells with different levels of tensegrity structure complexity is devel- oped. The cell origami model is constructed on the basis of 4. Discussions the principle of the origami folding technique and cell trac- tion forces. The process of cell origami is first performed by The proposed tensegrity structure models of adherent cells is using the spring elements to connect the model with the aimed at better understanding the cell origami. In compari- microplate in order to fold a microplate from a 2D configura- son with cells, the cables of the cellular tensegrity structure tion to form a 3D cell-populated microstructure. The simula- may be viewed as analogous to the CSK tension elements tion results are as follows: (e.g., MFs), the struts as the CSK compression elements (e.g. MTs), the microplate as the ECM, and spring elements (a) The inclusion of the NSK can enhance the folding as the FACs. Since the tensegrity structure is a simplification angle of the microplate. The larger the folding angle of the CSK morphology, the FEM models help us to under- is, the closer it is to the real situation of the cells in stand the cell origami in a simple way, which provide an the 3D environment, which cannot be described in attempt to measure the cell traction force in a 3D physiolog- the 2D environment ical environment and a new method for further study on cell origami. Unfortunately, the simulation of cell origami has (b) Increasing the level of complexity of the model can many limitations compared with the existing cell models reduce the folding angle, indirectly demonstrating and the folding of living cells. The main limitations can be that the folding angle depends on the cell type from succinctly summarized as follows. the perspective of simulation First, some studies have shown that a more complex computational model, such as bendo-tensegrity models, can In other words, both the level of complexity of the ten- better understand the mechanotransduction mechanism segrity structures and the NSK have an important influence and can be used to determine the mechanical contribution on the behaviour of cell origami. The proposed FEM models of individual cytoskeletal components to the cellular overall can provide theoretical guidance for the application of bio- structural responses [42]. Therefore, although the tensegrity technology and the analysis of the 3D structures of cells structure contains many features consistent with living cells and have a great potential to be implemented for the self- assembly of cell-based microscale medical devices. and could be used to represent a reasonable starting point for describing CSK mechanics, it is an oversimplification of the CSK morphology. The following work is to establish a Data Availability more sophisticated and accurate adherent cell model to All data used and analyzed during the current study are avail- describe the cell origami. Second, although some studies have able from the corresponding author on reasonable request. shown that the mechanical properties of IFs are far from lin- ear elastic and that IFs and FACs show obvious strain stiffen- ing behaviour [33, 43], for simplification, the strain stiffening Conflicts of Interest behaviours of IFs and FACs are not considered and the The authors declare that they have no competing interests. mechanical properties of IFs and FACs are still assumed to be linear elastic, which is consistent with the actual situation. Acknowledgments Third, some studies have shown that MTs of unequal lengths originate from centrosomes near the nucleus and spread out- This work has been funded by the National Natural Science ward through the cytoplasm to the cell cortex where they Foundation of China (Grant No. 11572213). 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Modelling Cell Origami via a Tensegrity Model of the Cytoskeleton in Adherent Cells

Applied Bionics and Biomechanics , Volume 2019 – Aug 14, 2019

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Copyright © 2019 Lili Wang and Weiyi Chen. 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.
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Hindawi Applied Bionics and Biomechanics Volume 2019, Article ID 8541303, 9 pages https://doi.org/10.1155/2019/8541303 Research Article Modelling Cell Origami via a Tensegrity Model of the Cytoskeleton in Adherent Cells 1,2 1,2 Lili Wang and Weiyi Chen Shanxi Key Laboratory of Material Strength & Structural Impact, College of Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024, China National Demonstration Center for Experimental Mechanics Education, Taiyuan University of Technology, Taiyuan 030024, China Correspondence should be addressed to Weiyi Chen; chenweiyi211@163.com Received 13 May 2019; Revised 29 June 2019; Accepted 30 July 2019; Published 14 August 2019 Academic Editor: Mohammad Rahimi-Gorji Copyright © 2019 Lili Wang and Weiyi Chen. 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. Cell origami has been widely used in the field of three-dimensional (3D) cell-populated microstructures due to their multiple advantages, including high biocompatibility, the lack of special requirements for substrate materials, and the lack of damage to cells. A 3D finite element method (FEM) model of an adherent cell based on the tensegrity structure is constructed to describe cell origami by using the principle of the origami folding technique and cell traction forces. Adherent cell models contain a cytoskeleton (CSK), which is primarily composed of microtubules (MTs), microfilaments (MFs), intermediate filaments (IFs), and a nucleoskeleton (NSK), which is mainly made up of the nuclear lamina and chromatin. The microplate is assumed to be an isotropic linear-elastic solid material with a flexible joint that is connected to the cell tensegrity structure model by spring elements representing focal adhesion complexes (FACs). To investigate the effects of the degree of complexity of the tensegrity structure and NSK on the folding angle of the microplate, four models are established in the study. The results demonstrate that the inclusion of the NSK can increase the folding angle of the microplate, indicating that the cell is closer to its physiological environment, while increased complexity can reduce the folding angle of the microplate since the folding angle is depended on the cell types. The proposed adherent cell FEM models are validated by comparisons with reported results. These findings can provide theoretical guidance for the application of biotechnology and the analysis of 3D structures of cells and have profound implications for the self-assembly of cell-based microscale medical devices. 1. Introduction traction forces to fold many microstructures from two- dimensional (2D) to 3D. Recently, He et al. described an Cell origami is defined as a technique that harnesses the trac- origami-inspired self-folding method to form 3D microstruc- tion force of living cells as a biological driving force to fold a tures of cocultured cells and indicated that the origami-based variety of three-dimensional (3D) cell-populated microstruc- cell self-folding technique is useful in regenerative medicine tures [1]. In the field of microfabrication, the origami folding and the preclinical stage of drug development [6]. However, technique has received increasing attention due to its multi- none of these studies have investigated cell origami by using ple advantages, including simplicity, high biocompatibility, the finite element method (FEM). the lack of special requirements for substrate materials, and Cell traction forces, as the contractile forces pointing to the lack of damage to cells. For example, Davis et al. [2] the centre of the cell body, are generated by the cytoskeleton and Azam et al. [3] used surface tension to create microsized (CSK) [7]. The CSK is a complex biopolymer network com- posed of microtubules (MTs), microfilaments (MFs), and containers. Sirrine et al. [4] and Song et al. [5] used the same technique to produce artificial tissue scaffolds. In addition, intermediate filaments (IFs) [8]. The CSK is the major Kaori et al. [1] experimentally determined that cells applied mechanical component of cells and plays a key role in the principle of the origami folding technique and cell mechanotransduction and extracellular force transmission 2 Applied Bionics and Biomechanics view of the fact that cell origami is a large deformation pro- from/to attaching a substrate through focal adhesion com- plexes (FACs) [9]. The forces in the CSK are related to the cess and IFs can provide resilience against mechanical forces biological functions of cells, such as differentiation, growth, and ensure cellular integrity [33], it is necessary to consider metastasis, and apoptosis [10–13]. The nucleus is regarded the role of IFs. Computational models of adherent cells com- as an integral structure functionally enabled by nuclear ten- posed of the CSK which is made up of MTs, MFs, and IFs and segrity, with struts representing the nuclear lamina and the NSK which mainly consists of the nuclear lamina and cables representing chromatin [12], featuring a large volume chromatin have been developed based on the tensegrity occupancy and including genetic information [14]. For structure. Since IFs vary from cell type to cell type [33] and example, Bursa et al. simulated the nucleoskeleton (NSK) as some studies have shown that IFs form a dense filament net- a tensegrity structure to study the CSK to transfer the exter- work spanning from the nucleus to the cell membrane [34], IFs are modelled as radial cables from the centre of the nal mechanical load of the cell to NSK, thereby initiating the biochemical response of the cell [15]. In addition, the tensegrity structure to the outer nodes in the models. important role of the NSK in cellular differentiation and Although the sphere-like tensegrity structure model development has been demonstrated [16]. Since some derived from the polyhedron (cuboctahedron or octahedron) researchers [9, 10, 17–28] have used both the spherical and [9] is symmetrical, the flat tensegrity structure derived from flattened tensegrity structure models’ approach combined the truncated polyhedron is not completely symmetric. with computational and mathematical models to investigate Two asymmetrical tensegrity structures derived from the the responses of cells to the substrate based on the assump- truncated polyhedron (flat cuboctahedron or octahedron), tion that individual cells can react by contraction and that 12-node tensegrity and 24-node tensegrity, are established the forces produced by cells can act on the extracellular to represent the different levels of complexity of the adherent matrix (ECM) by FACs. There are many cell models and cell models. The 12-node tensegrity structure is composed of models of cell-substrate interactions; however, FEM simula- 6 struts representing MTs and 36 cables, 24 of which repre- tion of cell origami has never been performed. Therefore, it sent MFs and 12 of which represent IFs, as shown in is necessary to establish a simple 3D FEM model of adherent Figure 1(a). The number of cables in the 24-node tensegrity cells composed of the CSK and NSK based on the tensegrity structure is 60, and the number of struts is twice that of the structures to simulate the cell origami. 12-node tensegrity structure. Among the 60 cables, 36 cables In this study, a 3D FEM model of an adherent cell made represent MFs, whereas the other 24 cables represent IFs, as up of the CSK and NSK is established and connected with a demonstrated in Figure 1(b). In both tensegrity structures, microplate to depict the cell origami. The CSK and NSK are the cables (red elements) representing MFs are connected represented by tensegrity structures of different levels of by the nodes at both ends of the struts (blue elements), while complexity, in which the CSK is composed of MTs, MFs, the cables (yellow elements) representing IFs are connected and IFs and the NSK consists of the nuclear lamina and chro- by the nodes at one end of the struts and the particle at the matin. The cell model adheres to the microplate through the centre of the structures which is simplified by the nucleus. spring elements representing FACs [29, 30]. The effects of the The microplate on which the cell is adhered is treated as level of complexity on the folding degree of the microplate an isotropic linear-elastic solid material, and its dimensions are investigated by changing the degree of complexity of the are adapted to the cell model. The microplate has a length cell tensegrity structure. Furthermore, the role of the NSK (b) and a width (b)of 30 μm and a height (h) of 2.7 μm; fur- is studied by using a 12-node and a 24-node sphere-like ten- thermore, the dimensions of the joint are 6 μm in width (w), 30 μm in length (b), and 0.3 μm in thickness (t), which shows segrity structure in comparison with the models without the NSK. The validity of the proposed models is validated by that the flexible joint is 3~8 μm in width and 70~390 nm in comparisons with the reported findings, demonstrating that thickness [1]. The spring elements are selected to link the cell the models can provide an attempt to measure the cell trac- model with the microplate. The folding angle (θ)isdefined as tion force in a 3D physiological environment and a new the angle between the folded microplate and its initial posi- tion, which is an important parameter for producing desired way of promoting a deeper understanding of cell origami. 3D cell-populated microstructures, as shown in Figure 2. From a geometric point of view, the folding angle can be 2. Materials and Methods expressed by the thickness (t) and the width (w) of the joint 2.1. Tensegrity Model. The process of cell origami is modelled and the thickness of the microplate (h) as follows [1]: by using the adherent cell to fold the microplate. Geometries of the models are created using UG NX 10.0 (Unigraphics = , 1 max NX 10.0) and then imported into the commercial finite ele- h + t/2 ment package ABAQUS (standard version 6.13, SIMULIA company, Germany) for simulations and analysis. Since When w is 6 μm, h is 2.7 μm, and t is 0.3 μm, the maxi- some studies have shown that IFs, as one of the major struc- mum folding angle is 120 according to equation (1). The fol- ture components of the CSK, play important roles in biolog- lowing folding angle must be less than the aforementioned ical functions such as cell contractility, migration, stiffness, value (<120 ) since the cell traction force of the measurement and stiffening [31]; play key mechanical roles in providing is inaccurate when the microplate contacts. structural stability of the cell; and can increase the cellular Furthermore, in order to simulate the influence of the rigidity at high strains (>20% strain) [24, 32]. Moreover, in NSK on the folding angle, the 12-node (Figure 3(a)) and w Applied Bionics and Biomechanics 3 (a) 12-node CSK (b) 24-node CSK Figure 1: Schematic diagram of the CSK with two different levels of complexity: (a) the 12-node CSK is composed of 6 struts (blue), 24 cables (red), and 12 cables (yellow); (b) the 24-node CSK is made up of 12 struts (blue), 36 cables (red), and 24 cables (yellow). max t (a) Arbitrary folding angle (b) Maximum folding angle Figure 2: Schematic diagram of the folding angle of the microplate: the horizontal blue dashed line represents the initial position of the microplate; the oblique red dashed line represents the position of the folded microplate. 24-node (Figure 3(b)) sphere-like tensegrity structures are manufacturing and biocompatibility and is commonly used for the NSK, with cables (cyan elements) representing used in the microfabrication [35]. The material and geo- chromatin and with the nuclear lamina modelled as struts metrical properties for all the components are based on (purple elements). Both tensegrity structures describing the the values published in the literature and summarized in NSK are symmetrical since they are derived from the cuboc- Table 1. tahedron and octahedron. The centre of the NSK coincides The cables and struts are depicted as truss elements that with the centre of the corresponding CSK, and each node of support only axial force and deformation, neglecting subcel- the NSK is connected to the node of the CSK pointing the lular bending. The prestress carried in a cable (F) with a same direction by IFs treated as the linker, as demonstrated current length (l) is [10] in Figures 3(c) and 3(d). l − l F + E A ,if l > l , 0 a a r 2.2. Material Properties and Boundary Conditions. Although F = 2 most components of the cell exhibit more or less nonlin- 0, if l ≤ l , ear constitutive behaviour, all materials are assumed to be linear elastic for simplicity. Moreover, the material parameters of parylene C are used for the microplates where l and l denote the resting and initial cable lengths, r 0 because this material has the advantages of ease of respectively, and E and A are Young’s modulus and a a 4 Applied Bionics and Biomechanics (a) 12-node NSK (b) 24-node NSK Y X Y X (c) 12-node CSK-NSK with microplate (d) 24-node CSK-NSK with microplate Figure 3: Schematic diagram of tensegrity structures with two different levels of complexity: (a) the 12-node NSK is composed of 6 struts (purple) and 24 cables (cyan); (b) the 24-node NSK is made up of 12 struts (purple) and 36 cables (cyan); (c) the 12-node CSK-NSK with a microplate; (d) the 24-node CSK-NSK with a microplate. where L and L denote the resting and initial lengths of r 0 Table 1: The material parameters and geometric dimensions. struts, respectively, and E , B , and A are Young’s modulus, s s s bending stiffness, and cross-section area of struts, respec- Elastic modulus (Pa) ν Dimensions tively. P and P are the initial strut tension and axial thrust, 9 2 0 c 1 2× 10 MTs [36] 0.3 190 nm respectively, described as 9 2 MFs [36] 2 6× 10 0.3 19 nm 2 E A s s IFs [37] 2×10 0.3 100 nm P = L − L , 0 r 0 6 2 Lamina [38] 1 4× 10 0.3 78.5 nm 6 2 π B Chromatin [38] 244 × 10 0.3 1.13 nm c P = 9 3 Microplate [1] 4×10 0.3 30 × 30 × 2 7 μm 9 3 Flexible joint [1] 4×10 0.3 30 × 6 × 0 3 μm The initial boundary conditions for the cell tensegrity Bond stiffness [39] k =0 025 nN/μm structure models are that the 12-node tensegrity structure has three nodes pinned to the microplate and three cables located on the microplate and coupled with the microplate, cross-section area of cables, respectively. F is the initial cable while the 24-node tensegrity structure has four nodes and tension, described as follows: four cables. The nodes closest to the microplate are anchored to the corresponding nodes of the microplate via spring ele- E A a a ments, and the other nodes are pinned as free moveable F = l − l 3 0 0 r joints. The number of nodes on the microplate is determined according to the z coordinate of the node. If z is equal to 0, it Meanwhile, the prestress carried in a strut (P) with a cur- is on the microplate, and if z is greater than 0, the node is not rent length (L)is on the microplate. The smaller the z is, the closer it is to the microplate. The centre of the microplate is constrained in all degrees of freedom. A concentrated force of 10 pN is applied L − L r c P + E A ,if L < L , P < P , 0 s s r at the farthest nodes parallel to the microplate [40]. Only one P = 4 truss element is used for all subcellular components, and the 0, if L ≥ L , microplate is meshed with 8-node hexahedral elements. Applied Bionics and Biomechanics 5 U, U3 +5.439e+00 +4.983e+00 +4.526e+00 +4.069e+00 +3.612e+00 +3.155e+00 +2.699e+00 +2.242e+00 +1.785e+00 +1.328e+00 +8.715e−01 +4.147e−01 −4.209e−02 (a) 12-node CSK tensegrity structure U, U3 +5.912e+00 +5.420e+00 +4.927e+00 +4.434e+00 +3.942e+00 +3.449e+00 +2.956e+00 +2.464e+00 +1.971e+00 +1.478e+00 +9.854e−01 +4.927e−01 +0.000e+00 (b) 24-node CSK tensegrity structure Figure 4: Schematic diagram of cell origami without the NSK. 3. Results 3.1. Influence of the Level of Complexity on the Folding Angle. 17.9 Since Kaori et al. have shown that the folding angle of the microplate is related to the cell types [1], the relationship 12.4 between the folding angle and levels of the complexity ten- 11.7 segrity structure in the cell origami process is studied using two different levels of complexity for the CSK and NSK. A 6.76 deformation diagram of cell origami without the NSK is shown in Figure 4. The maximum folding angles of the microplates of the ° ° 12-node and the 24-node CSK models are 12.4 and 6.76 , respectively, and the maximum values of the 12-node and 12-node tensegrity 24-node tensegrity ° ° the 24-node CSK-NSK models are 17.9 and 11.7 , respec- CSK tively, as demonstrated in Figure 5. CSK-NSK The result shows that the stiffness of the 12-node tensegr- ity structure model with/without the NSK is larger than that Figure 5: The maximum folding angle of the microplate. of the 24-node model with/without the NSK, indicating Folding angle (°) 6 Applied Bionics and Biomechanics U, U3 +8.381e+00 +7.681e+00 +6.981e+00 +6.281e+00 +5.582e+00 +4.882e+00 +4.182e+00 +3.482e+00 +2.782e+00 +2.082e+00 +1.382e+00 +6.826e−01 −1.728e−02 (a) 12-node CSK-NSK tensegrity structure U, U3 +7.903e+00 +7.245e+00 +6.586e+00 +5.928e+00 +5.269e+00 +4.610e+00 +3.952e+00 +3.293e+00 +2.634e+00 +1.976e+00 +1.317e+00 +6.586e−01 +0.000e+00 (b) 24-node CSK-NSK tensegrity structure Figure 6: Schematic diagram of cell origami with the NSK. indirectly that the folding angle is related to the cell type from the NSK. The deformation diagram of cell origami with the NSK is shown in Figure 6. the perspective of simulation. This result may be due to the larger number of nodes, the greater complexity of the struc- The maximum folding angle of the model with the ture, the additional degree of freedom, the additional energy NSK is larger than that without the NSK, as shown in required for deformation, and the smaller folding angle of the Figure 5. The results show that the augmentation of the microplate. Furthermore, the folding angle is asymmetrical, NSK can increase the stiffness of the model, independent which may be caused by the asymmetry of the model, as of the complexity. An increase in stiffness means an shown in Figure 4. increase of the folding angle, indicating that the cell is closer to its physiological environment. The above results 3.2. Effect of the NSK on the Folding Angle. In view of the fact imply that the 24-node tensegrity structure is sufficiently that the nucleus represented by the NSK has a large volume complex to describe the flat tensegrity structure representing the adherent cell morphology, and the results are consistent occupancy and includes genetic information, to investigate the effect of the NSK structure on the folding angle, two with the conclusions of Pugh [41], who demonstrated tensegrity structure models with different levels of NSK com- that when the levels of complexity of the structure increase plexity are established and compared with the model without further, the tensegrity structure becomes more analogous Applied Bionics and Biomechanics 7 to a cylinder and does not represent the geometry of 5. Conclusions suspended cells. In the present study, a 3D FEM model of adherent cells with different levels of tensegrity structure complexity is devel- oped. The cell origami model is constructed on the basis of 4. Discussions the principle of the origami folding technique and cell trac- tion forces. The process of cell origami is first performed by The proposed tensegrity structure models of adherent cells is using the spring elements to connect the model with the aimed at better understanding the cell origami. In compari- microplate in order to fold a microplate from a 2D configura- son with cells, the cables of the cellular tensegrity structure tion to form a 3D cell-populated microstructure. The simula- may be viewed as analogous to the CSK tension elements tion results are as follows: (e.g., MFs), the struts as the CSK compression elements (e.g. MTs), the microplate as the ECM, and spring elements (a) The inclusion of the NSK can enhance the folding as the FACs. Since the tensegrity structure is a simplification angle of the microplate. The larger the folding angle of the CSK morphology, the FEM models help us to under- is, the closer it is to the real situation of the cells in stand the cell origami in a simple way, which provide an the 3D environment, which cannot be described in attempt to measure the cell traction force in a 3D physiolog- the 2D environment ical environment and a new method for further study on cell origami. Unfortunately, the simulation of cell origami has (b) Increasing the level of complexity of the model can many limitations compared with the existing cell models reduce the folding angle, indirectly demonstrating and the folding of living cells. The main limitations can be that the folding angle depends on the cell type from succinctly summarized as follows. the perspective of simulation First, some studies have shown that a more complex computational model, such as bendo-tensegrity models, can In other words, both the level of complexity of the ten- better understand the mechanotransduction mechanism segrity structures and the NSK have an important influence and can be used to determine the mechanical contribution on the behaviour of cell origami. The proposed FEM models of individual cytoskeletal components to the cellular overall can provide theoretical guidance for the application of bio- structural responses [42]. Therefore, although the tensegrity technology and the analysis of the 3D structures of cells structure contains many features consistent with living cells and have a great potential to be implemented for the self- assembly of cell-based microscale medical devices. and could be used to represent a reasonable starting point for describing CSK mechanics, it is an oversimplification of the CSK morphology. The following work is to establish a Data Availability more sophisticated and accurate adherent cell model to All data used and analyzed during the current study are avail- describe the cell origami. Second, although some studies have able from the corresponding author on reasonable request. shown that the mechanical properties of IFs are far from lin- ear elastic and that IFs and FACs show obvious strain stiffen- ing behaviour [33, 43], for simplification, the strain stiffening Conflicts of Interest behaviours of IFs and FACs are not considered and the The authors declare that they have no competing interests. mechanical properties of IFs and FACs are still assumed to be linear elastic, which is consistent with the actual situation. Acknowledgments Third, some studies have shown that MTs of unequal lengths originate from centrosomes near the nucleus and spread out- This work has been funded by the National Natural Science ward through the cytoplasm to the cell cortex where they Foundation of China (Grant No. 11572213). I would like to interact with other cytoskeletal filaments at FACs [42], while thank my supervisor, Professor Weiyi Chen, for his contin- others have shown that cytoplasmic IFs are radially distrib- ued patience and valuable guidance. uted from the nuclear membrane towards the cell surface [34]. In view of the fact that the distribution of CSK varies References with the cell type, the following work is to create a different CSK model to describe cell origami. Finally, although most [1] K. Kuribayashi-Shigetomi, H. Onoe, and S. Takeuchi, “Cell cellular components behave more or less in a nonlinear con- origami: self-folding of three-dimensional cell-laden micro- stitutive behaviour, linear elastic properties are assigned to structures driven by cell traction force,” PLoS One, vol. 7, all components of the model, which is far from the real no. 12, article e51085, 2012. behaviour of cells. [2] D. Davis, B. Chen, M. D. Dickey, and J. 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