International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 M C M S ODELING ELL OVEMENT ON A UBSTRATE W V R ITH ARIABLE IGIDITY Arkady Voloshin Department of Mechanical Engineering and Mechanics and Bioengineering Program Lehigh University ABSTRACT Live cells respond to the changes of their physiological environment as well as to the mechanical stimuli occurring in and out of the cell body. It is known that cell directional motion is influenced by the substrate stiffness. A finite element modelling based on the tensegrity approach is used here to describe the biomechanical behavior of cells. The effects of substrate stiffness and prestress on strain energy of a cell are investigated by defining several substrate stiffness values and prestress values. Numerical simulations reveal that the internal elastic strain energy of the cell decreases as the substrate stiffness increases. As prestress of cell increases, the strain energy increases as well. The change of prestress value does not change behavior pattern of the strain energy: strain energy of a cell will decrease when substrate stiffness increases. These findings indicate that both cell prestress and substrate stiffness influence the cell directional movement. KEYWORDS cell motion, strain energy, tensegrity, modelling, substrate stiffness. 1. INTRODUCTION Cell migration may be encouraged by various external to the cell factors, like chemical [1] (chemotaxis), mechanical [2] (mechanotaxis), thermal [3] (thermotaxis), electrical [4] (galvanotaxis) or topological [5] (topotaxis) to name a few. Migration requires interaction between the cell and substrate it is located on the exact mechanisms by which different environmental forces are transduced into cell biological responses are still unknown. However, in any physical environment changes of cell’s geometry and motion are influenced by its physical and internal balance [6] since a cell needs to maintain its morphological stability and molecular self-assembly. Cell attached to a substrate can sense mechanical stimuli [ 7,8 ,9 ], respond the stimuli in order to keep cell’s intracellular and extracellular forces in balance [10,11] and regulate many important physiological and pathological processes [12,13,14]. Current experimental works focus on developing and identifying the mechanism called “mechanotransduction”, on the processes by which cells sense mechanical force and transduce it into a biochemical signal. Some of these studies have shown that cell movement have been influenced by substrate’s rigidity [15,16]. Based on the hypothesis that a single cell can probe substrate stiffness and respond by exerting contractile forces, Lo and colleagues [15] referred to the process as “durotaxis”. On the other hand, both computational [17,18,19] and mathematical DOI : 10.5121/ijbes.2016.3102 19 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 models [20] have been developed for further understanding of the biomechanical cellular responses. Computational mechanical models, especially tensegrity structures [21,22], have been widely used to model the cellular responses to environment changes. Cell alterations in shape and structure caused by mechanical loads are critical to cell functions, such as growth, motility, differentiation, and proliferation [23,24]. It is also well known that the directional movement of a cell is an important component of developmental patterning, wound healing and tumor metastasis [25,26]. It is a critical question to understand the mechanisms by which the cells resist and react to the deformation under various physical conditions. The forces generated within the actin cytoskeleton and applied to the extracellular matrix through focal adhesions can influence cell interaction with extracellular environment, such as cell migration and formation of focal adhesion [27,28].Cells can sense physicochemical and biochemical signals of surrounding environments that can be transmitted between cell and substrate via focal adhesions. By probing environment parameters or connecting with environment directly, cells can guide their activities, such as changing morphologies and migratory directions, due to the environmental changes. The in-vitro studies[29, 30, 31] have shown that cells cultured on substrate are influenced by substrate mechanics. Georgeset al.[29] showed that cells are able to sense the substrate rigidity since certain types of cells have less rounded shape on stiffer substrate and are more likely to exhibit as rounded shape on softer substrate. However, the study by Saezet al. [30], got the opposite results by using epithelial cells - cells are likely to extend into branched morphologies on softer substrate than the same cells on stiffer substrate. Cell morphology was changing, but they migrated towards the area of larger stiffness. Recently it was shown that the chromatin plasticity in epithelial cells is dependent of the substrate stiffness [31].Previous experiments also have shown that cells tend to move towards stiffer region on certain patterned substrate, which is known as durotaxis [32].Substrate stiffness varies across cell types, from softer brain tissue to stiffer bone tissue. The differences in substrate stiffness are caused by the various substrates’ components and their concentrations. To get better understanding of these complex structures and processes, various mechanical computational models have been developed in recent decades [32,33,34,35,36,37,38]. Mechanical computational models could provide the insights into how the physical properties of the substrate or adherent cells influence cells’ morphologies and movements. There are two main categories of computational models to investigate cells’ responses to environmental changes: continuum approach [32,33,34,35] and micro/ nanostructral approach [33,36,37,38]. The first category of computational models can be defined as continuum mechanical model that includes four major different types: liquid drop model, power-law structural damping model, solid model, and biphasic model. The cells are treated as certain continuum material properties in this method. The continuum mechanical model has been used to model blood cell with cytoplasm as a viscous liquid and cortex as cortical membrane [34], to study small strain deformation characteristics of leukocytesas a linear visco elastic solid model [35] and to model single chondrocytes and their interaction with the extracellular cartilage matrix [36]. The second category of computational model is based on microstructural and nanostructural approach. This approach includes the tensegrity model, tensed cable network model, open-cell foam model and spectrin-network model for erythrocytes. The cytoskeleton is used as the main structural component in microstructural and nanostructural approach, especially for developing cytoskeletal mechanics in adherent cell [32]. Prestressed cable network model is used to model 20 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 the deformability of the adherent cell act in cytoskeleton based on the values observed from living cells and mechanical measurements on isolated act in filaments [32]. Another type of model – open-cell foam model– can be categorized into two types. The first type of open-cell foam model is used to evaluate the mechanical properties and homogenized behaviors of the foam [37,38,39]. However, on the basis of this model it is found that the bulk modulus and hydrostatic yield strength of real foams usually are over predicted. To circumvent this drawback, various morphological defects (e.g., non-uniform and wavy cell edges) have to be included [40]. The other one called “super-cell model” has been developed in order to give a better representation of the morphological structure of real foams which usually contain a number of irregular cells, especially Voronoi model, which has been developed by Gibson and co-workers [41,42]. Among microstructural and nanostructural approaches, tensegrity model in conjunction with the finite element method is the widely used. The tensegrity architecture was first described by Buckminster Fuller in 1961 [43]. The basic idea of tensegrity model has explained the stress- hardening taking into account internal tensions [44,45]. In some studies researchers concluded that tissues may act continuously at the macroscopic scale, but in fact, they are discrete structural entities when viewed at the scale of cell [46]. The cells also prefer to form attachments heterogeneously that distribute over the cell surface discretely [47]. Tensegrity structure can appropriately model such characteristics. Tensegrity structure has been used here as a computational model of cell-substrate system in order to explain dependency of cell motility on the substrate rigidity. When a cell attaches to a particular surface, geometry of a cell changes as if external forces are applied to the cell membrane. Cellular responses are variable and give rise to a variety of changes of morphology and movement. Cell’s morphology and movement that adheres to a substrate is determined by the internal strain energy and interfacial energy. The assumption is that cell can probe its external environment and guide its movement based on the stiffness of the underlying substrate. The hypothesis in this work is that the cell probes the substrate stiffness in different directions and when it finds a direction that leads to the higher stiffness of the substrate this results in the decrease of the internal elastic strain energy. Since the cell, as any structure, prefers to stay in a lower energy state [48,49] this will be its preferred direction of migration. In other words, giving the choice of stiffer or less stiff substrate the cell will adhere to the stiffer substrate and its internal energy will be smaller than if it would adhere to the softer substrate. On the basis of the above referenced experimental observation a tensegrity based model of cell- substrate interaction was developed that allows it to mimic how the cell probes the stiffness of a substrate and how different substrate stiffness modified the relative elastic strain energy of the cell. The cytoskeleton of a cell is modeled as a tensegrity structure under prestress and its elastic strain energy is calculated by using finite element method as a function of the stiffness of substrate which it is attached to. 2. MATERIALS AND METHODS 2.1 TENSEGRITY MODEL Tensegrity structure consists of a set of interconnected members carrying compression or tension to provide a mechanical force balance environment, stable volume and shape in the space. The tensegrity structure is used to explain cell motility and shape changes since it provides a comprehensive approach based on a fact that the mechanical integrity is maintained and a self- 21 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 equilibrium is obtained through the contribution of actin filaments that are under tension and microtubules that are under compression [50,51]. A simple tensegrity structure placed on a flat surface is shown in Figure 1. The principle of tensegrity structure is based on the set of isolated components in compression embedded inside of a net of connected tension components in order to separate the compressed members from each other. The role of tension elements carrying “prestress” (i.e., initial stress) is to confer load-supporting capability to the entire structure. Several models based on tensegrity architecture have been used to successfully predict the mechanical responses of whole cells, such as the erythrocytes and viruses [44]. The tensegrity structure representing the cell consists of 30 elements. It is used here to represent a 3T3 cell. The 3T3 cells come from a cell line established in 1962 by George Todaro and Howard Green [52]. There are 6 pre-compressed struts that represent microtubule members under compressional loads. The rest of the 24 elements are pre-tensed cables that are homologous with microfilament members that carry tensional load. A spring element between nodes 2 and 13 (Figure 1) is used to model an elastic substrate that cytoskeleton is anchored to via focal adhesion. Polydimethylsiloxane (PDMS) is the basic material for cell substrate fabrication in many in-vitro studies and its properties were used to model the substrate. Node 3 is the origin of the coordinate system and node 1 is fixed to the substrate. Figure 1. Tensegrity model with spring element (between nodes 2 and 13) representing the substrate stiffness. Node 3 represents focal adhesion that is linking the substrate and cell. The nodes are allowed to move in three dimensional space without rotation, representing the active movement of a living cell. The length of cables and struts allowed to increase or decrease in this model as a function of the applied prestress and external deformation. The initial length of microtubule and microfilament were selected to be 10 µm and 6.12 µm, respectively. The distance between the so- called “superior plane” (the farthest plane with respect to the X-Y plane) and the “inferior plane” (the nearest plane with respect to the X-Y plane) for un-deformed structure represent the initial height of 8.7 µm and these two planes were set to be parallel to each other in the initial state. 2.2 MATERIAL PROPERTIES OF THE ELEMENTS To model the cell, we choose Link180, Beam188 and Combine14 as element components from the ANSYS elements library. Link180 and Beam188 are three dimensional truss elements. Link180 element was used to model the cable system, Beam188 element was used to model the 22 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 strut system, the Combine14 element was used to model the substrate stiffness. All components of basic model are treated as elastic. Material properties for microtubules and microfilaments are not known precisely. In Gittes et al. [53] study of the thrice-cycled phosphocellulise-purified tubulin, the Young’s modulus was found to be 1.2 GPa for microtubules and 2.6 GPa for microfilaments, the Poisson’s ratio for microtubules and microfilaments is 0.3. Tubulin was thought to be specific to eukaryotic cell, 3T3 cells that were obtained from mouse embryo tissue were also a kind of eukaryotic cell. Thus, we use these values as our elements’ material properties (Table 1). Table 1.Modeling and Mechanical Properties of Microtubules and Microfilaments Substrate Microtubules Microfilaments ANSYS Element Type Link 180 Beam 188 Combine 14 Cross-section Area (µm2) 1.9×10-4 1.9×10-5 Length (µm) 10 6.12 6.12 E (GPa) 1.2 2.6 ν 0.3 0.3 10-3 to 1000 Stiffness (N/m) N/m In this study we model attachments of the cell to surfaces with various stiffness values via focal adhesion. Number of different substrates were used for various cell types. For example, Trichet et.al. [54] chose flat PDMS substrate to investigate REF52 fibroblast cell migration response to the stiffness gradients of the substrate. The stiffness of substrate ranges from 0.003 N/m to 1.4 N/m. In another study, Gray et al. [55] have evaluated two different substrates: acrylamide and PDMS. The values of substrate Young’s moduli in this study were 2.5±0.2 MPa and 12±1 kPa for PDMS material. The cross section area of each material substrate was approximately 10 mm×50 mm, while the height was 1 mm. Stiffness can be calculated by using Young’s modulus and its structure: K = (AE) / L, where A = cross sectional area, L = length, and E is Young’s modulus. The stiffness value of PDMS substrate in this study was between 2.6 N/m to 540 N/m. Even though the main material of substrate is PDMS, various concentrations of PDMS lead to substrates with different rigidities. The flexural rigidity of substrate in the present study was assigned in the wide range of 10-3 to 1000 N/m and the discrete values selected for analysis are presented in Table 2. 23 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 Table 2 .Values of the Substrate Stiffness used in the Analysis Model Stiffness (N/m) Model Stiffness (N/m) 1 0.001 14 0.09 2 0.002 15 0.1 3 0.004 16 0.2 4 0.006 17 0.3 5 0.008 18 0.4 6 0.01 19 0.5 7 0.02 20 1 8 0.03 21 5 9 0.04 22 10 10 0.05 23 50 11 0.06 24 100 12 0.07 25 500 13 0.08 26 1000 2.3 PRESTRESS Cellular prestress has a structural importance in resisting extracellular forces and maintaining cell morphology. If there is an external load acting on the structure, the components of structure move relative to one another until attaining a new equilibrium position between cell and external environment. The cytoskeleton prestress plays a key role in mechanotransduction [52]. Gardel et al. [56] assumed that the cells’ mechanical properties, especially cell prestress, can influence cells’ deformation. The critical importance of prestress makes the model even more similar to the behavior of living cells as the degree of prestress determines the cells stiffness. Therefore, it is necessary to apply prestress forces into cytoskeleton modeling. In this study, prestress values assigned to microfilament and microtubule elements of model are varied to study their effects on the cell strain energy (Table 3). The values represent different cases of the prestress values for the 3T3 cell. Table 3. Range of the Prestress Values used in the Analysis Prestress Case Microfilament(pN) Microtubules(pN) Number 1 1.6 3.92 2 1.0 3.92 3 2.2 3.92 4 1.6 3.42 5 1.6 4.42 24 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 2.4 SIMULATION PROCEDURE The gravity and magnetic fields are neglected in the simulation. The model includes possibility of focal adhesion. By extending one of the nodes, the cell can probe and sense the stiffness rigidity of substrate. The strain energy of the cell changes when it extends one of the nodes in order to probe the stiffness of the substrate. This strain energy is computed for number of stiffness values. We model the substrate stiffness by a spring element. In our model, the displacement of the node attached to the substrate and total strain energy of the whole cell will be used to explain the cell preference (from the point of view of minimizing the cell’s internal elastic energy) to move to a stiffer substrate. The simulation process is performed as follows. The first step is to apply prestress forces to the structure. The length of struts or cables will change according to the applied prestress forces. The location of each node in the tensegrity structure is redefined due to the changes of node location in every step. The deformed location of each node in the first step has been stored and used as the new initial location for the second step of the cell simulation. The second step is to give to the node 2 and the other side of the spring element (node 13) an equal displacement in X direction. We apply the same displacement to assure that there is no force in the spring element. Therefore the spring element will not influence the cell energy until the third step. The displacement was selected to be 1 µm. Since the length of each element in the model is greater than the applied displacement, the shape of the cell will not change significantly due to this displacement. The location of each node has been stored as the initial location for the third step. In the second step, we also get the reaction forces at node 2 which were generated due to the displacement in X direction. In the third step we fix one side of the spring element (node 13)and apply the reaction force that was calculated in the previous step at the site of node 2 in X direction. In this step, the total strain energy of the cell is calculated and stored in the database. The flow chart of the analysis is presented in Figure 2. Figure 2. Flow Chart of the Simulation Process 25 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 3. RESULTS To understand the relationship between preferential direction of cell movement and substrate stiffness during spreading process, the cell model was placed on a particular substrate with a certain stiffness. To evaluate the influence of the prestress on cell preferential direction while cell is anchored on a particular substrate, the model was solved for different values of prestress forces. Simulations were performed on flat substrate which has no topological preference thus there are no other factors that guide the cell motion in a preferential direction, except the substrate stiffness. Hence, the direction of cell motion is defined by the substrate rigidity and cell prestress only. 3.1 EFFECT OF THE SUBSTRATE RIGIDITY ON THE CELL’S STRAIN ENERGY By running the model with ANSYS Mechanical APDL solver, the tendency of cell movement in the substrates with different rigidities was analyzed. To explore the relationship between substrate stiffness and cell directional movement with the certain prestress values for the cable (representing microtubules) and strut (representing microfilaments) elements, number of different values of substrate stiffness had been used in the study. For the first case the prestress values of the model structure were selected to be 1.6 pN for microfilaments and 3.92 pN for microtubules, while the substrate rigidity was selected to be relatively small, 0.001 N/m (Case 1). The displacements of each node were around 0.1 µm after applying the prestress. Since lengths of the cables and the struts are around 1-5 µm, the location of each node changed very little after the first step (application of prestress). The deformation of cell structure after the second step (nodes 2 and 13 were given a specific, 1 µm displacement in X direction) is shown in Figure 3. It is obvious that the node 2 and node 13 moved the same distance. The spring element (between node 2 and node 13) has no internal force due to this movement. Figure 3.Comparison of node displacements due to the 2ndstep. It is obvious that the segment 2-13 just translated but did not change its length. Dotted lines represent original location, solid – after the 2nd step. The third step simulating attachment of the cell’s filopodia to the substrate results in the new configuration. The comparison of cell’s geometry before and after the third step has been shown in Figure 4. The dash lines show the geometry before the third step, while the solid lines shows 26 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 the geometry after the third step. For the third step, the node 13 is located at the same location since the node 13 (one side of the spring element) was fixed to model the cell’s adhesion to the substrate. The node 2 displaced 0.878 µm in X direction since the substrate rigidity is relatively very small, just 0.001 N/m (Figure4). Softer spring element represents softer substrate. Figure 4. Comparison of node displacements due to the 3rd Step The displacement of node 2 and strain energy of cell has been influenced by substrate stiffness. The simulation results (cell’s strain energy and node 2 displacement) of models with various values of the substrate stiffness for the first case of prestress values are shown in Appendix (Table A1). The displacement of the node 2 in X direction as a function of substrate stiffness is shown in Figure 5. It is obvious that the displacement of node 2 is a function of substrate stiffness. With the substrate stiffness increases, the displacement of node 2 decreases significantly. Figure 5. Displacement of Node 2 after the3rd Step in X direction for Substrate Stiffness in the range of 10-3 N/m to 103 N/m (Prestress Case 1) 27 International Journal of Biomedical Engineering and Science (IJBES), Vol. 3, No. 1, January 2016 Figure 6. Cell Strain Energy for Substrate Stiffness in the range of 10-3 N/m to 103 N/m (Prestress Case 1) The total cell strain energy that includes elastic energies of all cell elements is presented in Figure 6. For the substrate stiffness values over 0.5 N/m, the cell strain energy changes is hard to visualize thus they are shown in the following Figures. Figure 7 shows the results for the stiffness range of 0.01 N/m to 0.1 N/m, while Figure 8 shows the results for the stiffness range of 0.1 N/m to 1 N/m. Figure 7. Cell Strain Energy for Substrate Stiffness in the range of 0.01 N/m to 0.1 N/m (Prestress Case 1) Figure 8. Cell Strain Energy for Substrate Stiffness in the range of 0.1 N/m to 1 N/m (Prestress Case 1). 28