Modelling Cell Origami via a Tensegrity Model of the Cytoskeleton in Adherent Cells
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.
Cell origami is defined as a technique that harnesses the traction force of living cells as a biological driving force to fold a variety of three-dimensional (3D) cell-populated microstructures . In the field of microfabrication, the origami folding technique has received increasing attention due to its multiple advantages, including simplicity, high biocompatibility, the lack of special requirements for substrate materials, and the lack of damage to cells. For example, Davis et al.  and Azam et al.  used surface tension to create microsized containers. Sirrine et al.  and Song et al.  used the same technique to produce artificial tissue scaffolds. In addition, Kaori et al.  experimentally determined that cells applied the principle of the origami folding technique and cell traction forces to fold many microstructures from two-dimensional (2D) to 3D. Recently, He et al. described an origami-inspired self-folding method to form 3D microstructures of cocultured cells and indicated that the origami-based cell self-folding technique is useful in regenerative medicine and the preclinical stage of drug development . However, none of these studies have investigated cell origami by using the finite element method (FEM).
Cell traction forces, as the contractile forces pointing to the centre of the cell body, are generated by the cytoskeleton (CSK) . The CSK is a complex biopolymer network composed of microtubules (MTs), microfilaments (MFs), and intermediate filaments (IFs) . The CSK is the major mechanical component of cells and plays a key role in mechanotransduction and extracellular force transmission from/to attaching a substrate through focal adhesion complexes (FACs) . The forces in the CSK are related to the biological functions of cells, such as differentiation, growth, metastasis, and apoptosis [10–13]. The nucleus is regarded as an integral structure functionally enabled by nuclear tensegrity, with struts representing the nuclear lamina and cables representing chromatin , featuring a large volume occupancy and including genetic information . For example, Bursa et al. simulated the nucleoskeleton (NSK) as a tensegrity structure to study the CSK to transfer the external mechanical load of the cell to NSK, thereby initiating the biochemical response of the cell . In addition, the important role of the NSK in cellular differentiation and development has been demonstrated . Since some researchers [9, 10, 17–28] have used both the spherical and flattened tensegrity structure models’ approach combined with computational and mathematical models to investigate the responses of cells to the substrate based on the assumption that individual cells can react by contraction and that the forces produced by cells can act on the extracellular matrix (ECM) by FACs. There are many cell models and models of cell-substrate interactions; however, FEM simulation of cell origami has never been performed. Therefore, it is necessary to establish a simple 3D FEM model of adherent cells composed of the CSK and NSK based on the tensegrity structures to simulate the cell origami.
In this study, a 3D FEM model of an adherent cell made up of the CSK and NSK is established and connected with a microplate to depict the cell origami. The CSK and NSK are represented by tensegrity structures of different levels of complexity, in which the CSK is composed of MTs, MFs, and IFs and the NSK consists of the nuclear lamina and chromatin. The cell model adheres to the microplate through the spring elements representing FACs [29, 30]. The effects of the level of complexity on the folding degree of the microplate are investigated by changing the degree of complexity of the cell tensegrity structure. Furthermore, the role of the NSK is studied by using a 12-node and a 24-node sphere-like tensegrity structure in comparison with the models without the NSK. The validity of the proposed models is validated by comparisons with the reported findings, demonstrating that the models can provide an attempt to measure the cell traction force in a 3D physiological environment and a new way of promoting a deeper understanding of cell origami.
2. Materials and Methods
2.1. Tensegrity Model
The process of cell origami is modelled by using the adherent cell to fold the microplate. Geometries of the models are created using UG NX 10.0 (Unigraphics NX 10.0) and then imported into the commercial finite element package ABAQUS (standard version 6.13, SIMULIA company, Germany) for simulations and analysis. Since some studies have shown that IFs, as one of the major structure components of the CSK, play important roles in biological functions such as cell contractility, migration, stiffness, and stiffening ; play key mechanical roles in providing structural stability of the cell; and can increase the cellular rigidity at high strains (>20% strain) [24, 32]. Moreover, in view of the fact that cell origami is a large deformation process and IFs can provide resilience against mechanical forces and ensure cellular integrity , it is necessary to consider the role of IFs. Computational models of adherent cells composed of the CSK which is made up of MTs, MFs, and IFs and the NSK which mainly consists of the nuclear lamina and chromatin have been developed based on the tensegrity structure. Since IFs vary from cell type to cell type  and some studies have shown that IFs form a dense filament network spanning from the nucleus to the cell membrane , IFs are modelled as radial cables from the centre of the tensegrity structure to the outer nodes in the models.
Although the sphere-like tensegrity structure model derived from the polyhedron (cuboctahedron or octahedron)  is symmetrical, the flat tensegrity structure derived from the truncated polyhedron is not completely symmetric. Two asymmetrical tensegrity structures derived from the truncated polyhedron (flat cuboctahedron or octahedron), 12-node tensegrity and 24-node tensegrity, are established to represent the different levels of complexity of the adherent cell models. The 12-node tensegrity structure is composed of 6 struts representing MTs and 36 cables, 24 of which represent MFs and 12 of which represent IFs, as shown in Figure 1(a). The number of cables in the 24-node tensegrity structure is 60, and the number of struts is twice that of the 12-node tensegrity structure. Among the 60 cables, 36 cables represent MFs, whereas the other 24 cables represent IFs, as demonstrated in Figure 1(b). In both tensegrity structures, the cables (red elements) representing MFs are connected by the nodes at both ends of the struts (blue elements), while the cables (yellow elements) representing IFs are connected by the nodes at one end of the struts and the particle at the centre of the structures which is simplified by the nucleus.
(a) 12-node CSK
(b) 24-node CSK
The microplate on which the cell is adhered is treated as an isotropic linear-elastic solid material, and its dimensions are adapted to the cell model. The microplate has a length () and a width () of 30 μm and a height () of 2.7 μm; furthermore, the dimensions of the joint are 6 μm in width (), 30 μm in length (), and 0.3 μm in thickness (), which shows that the flexible joint is 3~8 μm in width and 70~390 nm in thickness . The spring elements are selected to link the cell model with the microplate. The folding angle () is defined as the angle between the folded microplate and its initial position, which is an important parameter for producing desired 3D cell-populated microstructures, as shown in Figure 2.
(a) Arbitrary folding angle
(b) Maximum folding angle
From a geometric point of view, the folding angle can be expressed by the thickness () and the width () of the joint and the thickness of the microplate () as follows :
When is 6 μm, is 2.7 μm, and is 0.3 μm, the maximum folding angle is 120° according to equation (1). The following folding angle must be less than the aforementioned value (<120°) since the cell traction force of the measurement is inaccurate when the microplate contacts.
Furthermore, in order to simulate the influence of the NSK on the folding angle, the 12-node (Figure 3(a)) and 24-node (Figure 3(b)) sphere-like tensegrity structures are used for the NSK, with cables (cyan elements) representing chromatin and with the nuclear lamina modelled as struts (purple elements). Both tensegrity structures describing the NSK are symmetrical since they are derived from the cuboctahedron and octahedron. The centre of the NSK coincides with the centre of the corresponding CSK, and each node of the NSK is connected to the node of the CSK pointing the same direction by IFs treated as the linker, as demonstrated in Figures 3(c) and 3(d).
(a) 12-node NSK
(b) 24-node NSK
(c) 12-node CSK-NSK with microplate
(d) 24-node CSK-NSK with microplate
2.2. Material Properties and Boundary Conditions
Although most components of the cell exhibit more or less nonlinear constitutive behaviour, all materials are assumed to be linear elastic for simplicity. Moreover, the material parameters of parylene C are used for the microplates because this material has the advantages of ease of manufacturing and biocompatibility and is commonly used in the microfabrication . The material and geometrical properties for all the components are based on the values published in the literature and summarized in Table 1.
The cables and struts are depicted as truss elements that support only axial force and deformation, neglecting subcellular bending. The prestress carried in a cable () with a current length () is  where and denote the resting and initial cable lengths, respectively, and and are Young’s modulus and cross-section area of cables, respectively. is the initial cable tension, described as follows:
Meanwhile, the prestress carried in a strut () with a current length () is where and denote the resting and initial lengths of struts, respectively, and , , and are Young’s modulus, bending stiffness, and cross-section area of struts, respectively. and are the initial strut tension and axial thrust, respectively, described as
The initial boundary conditions for the cell tensegrity 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, while the 24-node tensegrity structure has four nodes and four cables. The nodes closest to the microplate are anchored to the corresponding nodes of the microplate via spring elements, and the other nodes are pinned as free moveable joints. The number of nodes on the microplate is determined according to the coordinate of the node. If is equal to 0, it is on the microplate, and if is greater than 0, the node is not on the microplate. The smaller the 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 at the farthest nodes parallel to the microplate . Only one truss element is used for all subcellular components, and the microplate is meshed with 8-node hexahedral elements.
3.1. Influence of the Level of Complexity on the Folding Angle
Since Kaori et al. have shown that the folding angle of the microplate is related to the cell types , the relationship between the folding angle and levels of the complexity tensegrity structure in the cell origami process is studied using two different levels of complexity for the CSK and NSK. A deformation diagram of cell origami without the NSK is shown in Figure 4.
(a) 12-node CSK tensegrity structure
(b) 24-node CSK tensegrity structure
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 the 24-node CSK-NSK models are 17.9° and 11.7°, respectively, as demonstrated in Figure 5.
The result shows that the stiffness of the 12-node tensegrity structure model with/without the NSK is larger than that of the 24-node model with/without the NSK, indicating indirectly that the folding angle is related to the cell type from the perspective of simulation. This result may be due to the larger number of nodes, the greater complexity of the structure, the additional degree of freedom, the additional energy required for deformation, and the smaller folding angle of the microplate. Furthermore, the folding angle is asymmetrical, which may be caused by the asymmetry of the model, as shown in Figure 4.
3.2. Effect of the NSK on the Folding Angle
In view of the fact that the nucleus represented by the NSK has a large volume occupancy and includes genetic information, to investigate the effect of the NSK structure on the folding angle, two tensegrity structure models with different levels of NSK complexity are established and compared with the model without the NSK. The deformation diagram of cell origami with the NSK is shown in Figure 6.
(a) 12-node CSK-NSK tensegrity structure
(b) 24-node CSK-NSK tensegrity structure
The maximum folding angle of the model with the NSK is larger than that without the NSK, as shown in Figure 5. The results show that the augmentation of the NSK can increase the stiffness of the model, independent of the complexity. An increase in stiffness means an increase of the folding angle, indicating that the cell is closer to its physiological environment. The above results imply that the 24-node tensegrity structure is sufficiently complex to describe the flat tensegrity structure representing the adherent cell morphology, and the results are consistent with the conclusions of Pugh , who demonstrated that when the levels of complexity of the structure increase further, the tensegrity structure becomes more analogous to a cylinder and does not represent the geometry of suspended cells.
The proposed tensegrity structure models of adherent cells is aimed at better understanding the cell origami. In comparison with cells, the cables of the cellular tensegrity structure may be viewed as analogous to the CSK tension elements (e.g., MFs), the struts as the CSK compression elements (e.g. MTs), the microplate as the ECM, and spring elements as the FACs. Since the tensegrity structure is a simplification of the CSK morphology, the FEM models help us to understand the cell origami in a simple way, which provide an attempt to measure the cell traction force in a 3D physiological environment and a new method for further study on cell origami. Unfortunately, the simulation of cell origami has many limitations compared with the existing cell models and the folding of living cells. The main limitations can be succinctly summarized as follows.
First, some studies have shown that a more complex computational model, such as bendo-tensegrity models, can better understand the mechanotransduction mechanism and can be used to determine the mechanical contribution of individual cytoskeletal components to the cellular overall structural responses . Therefore, although the tensegrity structure contains many features consistent with living cells 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 more sophisticated and accurate adherent cell model to describe the cell origami. Second, although some studies have shown that the mechanical properties of IFs are far from linear elastic and that IFs and FACs show obvious strain stiffening behaviour [33, 43], for simplification, the strain stiffening behaviours of IFs and FACs are not considered and the mechanical properties of IFs and FACs are still assumed to be linear elastic, which is consistent with the actual situation. Third, some studies have shown that MTs of unequal lengths originate from centrosomes near the nucleus and spread outward through the cytoplasm to the cell cortex where they interact with other cytoskeletal filaments at FACs , while others have shown that cytoplasmic IFs are radially distributed from the nuclear membrane towards the cell surface . In view of the fact that the distribution of CSK varies with the cell type, the following work is to create a different CSK model to describe cell origami. Finally, although most cellular components behave more or less in a nonlinear constitutive behaviour, linear elastic properties are assigned to all components of the model, which is far from the real behaviour of cells.
Therefore, the present model can be further improved by considering the more complex and different distribution of FEM modelling of the CSK and NSK with viscoelasticity or hyperelasticity properties, so that we can better understand the mechanotransduction mechanism and accurately measure the traction force of living cells through the folding angle of the microplate.
In the present study, a 3D FEM model of adherent cells with different levels of tensegrity structure complexity is developed. The cell origami model is constructed on the basis of the principle of the origami folding technique and cell traction forces. The process of cell origami is first performed by using the spring elements to connect the model with the microplate in order to fold a microplate from a 2D configuration to form a 3D cell-populated microstructure. The simulation results are as follows: (a)The inclusion of the NSK can enhance the folding angle of the microplate. The larger the folding angle is, the closer it is to the real situation of the cells in the 3D environment, which cannot be described in the 2D environment(b)Increasing the level of complexity of the model can reduce the folding angle, indirectly demonstrating that the folding angle depends on the cell type from the perspective of simulation
In other words, both the level of complexity of the tensegrity structures and the NSK have an important influence on the behaviour of cell origami. The proposed FEM models can provide theoretical guidance for the application of biotechnology and the analysis of the 3D structures of cells and have a great potential to be implemented for the self-assembly of cell-based microscale medical devices.
All data used and analyzed during the current study are available from the corresponding author on reasonable request.
Conflicts of Interest
The authors declare that they have no competing interests.
This work has been funded by the National Natural Science Foundation of China (Grant No. 11572213). I would like to thank my supervisor, Professor Weiyi Chen, for his continued patience and valuable guidance.
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