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Master Thesis Title of Thesis 3D Numerical Modeling of Cell Migration and Cell-Cell Interaction Presented by SEYED JAMALEDDIN MOUSAVI Supervised by MOHAMED HAMDY DOWEIDAR Dr. Industrial Engineering Presented at University of Zaragoza Department of Mechanical Engineering September 2012 Acknowledgements I would like to express my greatest gratitude to the persons who have helped and supported me throughout this work. I am grateful to my supervisor, Dr. Mohamed Hamdy Doweidar, for his continuous support during my study, from initial advice and contacts in the early stages of conceptual inception and his encouragement to this day. Special thanks of mine goes to Professor Manule Doblaré who accepted me for PhD in GEMM group, without his support this dissertation could not have been done. I would like to thank Dr. Pilar Martín Duque for her kind collaboration in experimental part of this dissertation in "Instituto Aragonés de Ciencias de la Salud (I+CS)". I wish to appreciate Iñaki, Raquel, Clara, and Alan, without their kind support and contribution, the experimental part of this dissertation would not have implemented. Many friends have helped me to have a great time through these two years. Their support and care helped me overcome setbacks and stay focused on my graduate study. I greatly value their friendship and I deeply appreciate their belief in me. I am also grateful to Olfa for her moral support, to Sara who helped me a lot specially with administrative works, and to Siamak who shared his experience with me. Also great thanks to my family, specially my mother, who tried her best to support me by giving me a lot of encouragement during schooling. Words fail me to express my sincere appreciation to my wife Solmaz whose love, dedication and persistent condence in me, have taken the load o my shoulder. I owe for her great patience, that made this dissertation possible. Abstract Cell migration is one of the signicant aspects to be taken into account in many physiological processes, such as wound healing, cancer development, morphogenesis, and the immune response. It is well known that in these processes and many others, cell migration is partially guided by mechanical properties of its substrate as well as specic chemoattractants. Although many experimental works have been developed to understand the eect of the mechanical properties of the substrate onto cell migration, accurate 3D cell locomotion models have not presented yet. In this work I present a 3D model for migration of a single cell as well as population of cells. In the presented model I assume that the cell follows two main processes, rstly sensing its interface with the substrate to determine the migration direction, secondly exerting subsequent forces to move. The cell traction forces are considered to depend on the internal cell deformation during the sensing step. A random protrusion force is also considered that aect on the cell migration and speed. This model was applied for one single cell getting results in agreement with available experimental and numerical data. The model has been also applied to a substrate with stiness gradient and to a substrate with changing depth. The results corroborate previous experimental results in the sense that the cell tendency is always to migrate from softer to stier regions. In very special cases, however, cells may change their tendency and migrate towards softer parts of the substrate till an imaginary equilibrium plane whose location depends on the mechanical properties of the substrate. Furthermore, cells tend to migrate toward xed boundaries. In case of surface migration, cell tends to migrate toward less thicker substrates. In case of interaction between two cells, the results demonstrate that their interaction decreases the mean migration speed while increase local migration speed. This process also takes place for high cell population as cells tend to aggregate in small slugs and then these slugs join together in middle of the substrate creating bigger aggregations. Contents Introduction i 1 Finite Element Model 1 1.1 Modelformulation .................................. 1 1.1.1 Single cell orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 Cell-cellinteraction.............................. 7 2 Experimental Monitoring of The Cell 9 2.1 CellCulture...................................... 10 2.2 GelManipulation................................... 10 2.3 CellVisualization................................... 11 3 Numerical Experiments 13 3.1 Two dierent stiness substrates with dierent boundary conditions . . . . . . . 13 3.1.1 Case1..................................... 13 3.1.2 Case2..................................... 15 3.1.3 Case3..................................... 17 3.2 Eect of a stiness gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Eectofsubstratedepth............................... 19 3.4 Interaction between two cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5 Cellpopulation .................................... 23 Conclusions and Future Works 27 7 List of Figures 1.1 Schematic diagram of the relevant mechanical constituents of a cell . . . . . . . 2 1.2 Mechano-sensing model of an adherent cell . . . . . . . . . . . . . . . . . . . . . 3 1.3 3D spherical shape conguration of the cell . . . . . . . . . . . . . . . . . . . . . 4 1.4 Deformed cell subjected to sensing forces . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Computational algorithm of cell migration . . . . . . . . . . . . . . . . . . . . . 7 1.6 Distance between the centroids of two cells . . . . . . . . . . . . . . . . . . . . . 8 2.1 Elasticities of cellular environments . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Samples of manipulated gels in the lab for cell culture . . . . . . . . . . . . . . . 10 2.3 Visualization of a single cell in a 3D collagen based substrate . . . . . . . . . . . 11 3.1 Cell migration in two dierent stiness substrates, case 1 . . . . . . . . . . . . . 15 3.2 Cell migration in two dierent stiness substrates, case 2 . . . . . . . . . . . . . 16 3.3 Cell migration in two dierent stiness substrates, case 3 . . . . . . . . . . . . . 18 3.4 Cell migration in a substrate with a stiness gradient . . . . . . . . . . . . . . . 19 3.5 The eect of the substrate stiness on the cell net traction force during migration 20 3.6 The eect of the substrate stiness on the cell net traction force during migration 20 3.7 The eect of the substrate stiness on the cell migration velocity . . . . . . . . . 20 3.8 Migration of the cell due to the variance of substrate depth . . . . . . . . . . . . 21 3.9 Stress and deformation in x direction during sensing process . . . . . . . . . . . 21 3.10 Interaction between two cells inside a substrate with constant elastic modulus . 22 3.11 Von Mises stresses due to mechanosensing of two cells during migration . . . . . 22 3.12 Interaction between two cells inside a substrate with stiness gradient . . . . . . 23 3.13 Comparison of cell migration velocities for two cells and single cell migration . . 23 3.14 Interaction between 40 cells inside a substrate . . . . . . . . . . . . . . . . . . . 25 9 Chapter 1 Finite Element Model Cells have a special internal structure which is able to sense the stiness of the matrix in which they reside. For instance, broblasts preferentially move toward stier substrates [20, 28]. This phenomenon is known as mechanotaxis in which a cell moves directionally following a mechanosensing process [35]. In the mechano-sensing step, the cell senses its substrate by exerting a sensing force to diagnose its surroundings, and consequently, gets some information about its substrate rigidity. Once the cell has determined its surrounding mechanical conditions, it starts to pull itself towards the stier and/or more xed region. The cellular elements with a relevant function in the cell mechano-sensing mechanics are the actin bundles, the actomyosin contractile apparatus and the passive mechanical strength of the rest of the cell body, whose main contribution is related to the action of the cytoskeleton microtubules and the membrane (Fig. 1.1) [9, 28, 32, 34, 35]. The cytoplasmic CSK is linked with the external ECM through focal adhesions and trans-membrane integrins that are assumed totally rigid for the present model. This scheme agrees with the tensegrity hypothesis [20], since deformation of external substrate is balanced by tensile forces generated in the actin CSK. External forces are also considered to be another possible cause of the deformation of the substrate and cell. The presented model can be used to simulate adherent cells cultured on 2D substrates, cultured in 3D hydro-gels, on the surface of a scaold or attached to the ECM of a connective tissue, regardless of their real environment. ECM and substrate will be used therefore without distinction in this work. In this chapter I will present a numerical model to describe eective stresses and forces on a single cell migration and then I will discuss how to extend it to cell population. 1.1 Model formulation In the same line of Borau et al. [5, 6], the one-dimensional model represented in Fig. 1.2-a can be particularized to octahedral or hydrostatic stresses. Assuming that contractile forces exerted by cells are isotropic [28], the change of length of each element is then interpreted as its corresponding volumetric strain [48]. In such a case, the characteristic spring constant can be identied with the volumetric stiness modulus of the representative element. The physical interpretation of the variables in each branch of the model is then: Kpas , Kact and Ksubs denote the stinesses of microtubules, myosin II and substrate respectively, ϵ1 represents the minimum volumetric strain, ϵ2 is the maximum volumetric strain of the cell [28], σact stands for the mean contractile stress generated internally by the myosin II machinery and transmitted through the actin bundles; σpas denotes the contractile stress supported by the passive resistance of the cell, 1 1.1. Model formulation Figure 1.1: Schematic diagram of the relevant mechanical constituents of a cell [28]. essentially corresponding to the CSK microtubules and to the membrane; σs is the stress of the ECM, and fext denotes external forces. The eective stress transmitted by the cell to the ECM, σcell , is therefore given by σcell =σpas +σact (1.1) It may be interpreted as this part of the active stress that is not absorbed by the microtubules. Forces from both microtubules, σpas , and actin bundles, σact , are exerted to the proteins plaque (Fig. 1.1) [28]. Thereby, σcell can also be interpreted as the average cell stress that bears the submembrane plaque in agreement with the integrin-mediated mechano-sensing hypothesis [4]. In addition, ϵ denotes local volumetric strain. In the model herein presented, local strain is computed from the deformation of the cell external nodes along the direction of the traction force exerted at the corresponding node. ϵact stands for deformation of the active contractile element. This deformation relates to the fact that the real physical change of the overlap between myosin and actin laments occurs when active forces are applied. Eventually, ϵa represents the deformation of the actin bundles that promote the active forces transmitted. We approximated the cell unidimensional constitutive behavior by a simple linear-elastic spring [28], which is a reasonable simplication of cell-substrate structure under moderate cell and substrate strains. Besides, interaction between the actin and myosin is considered as an active force arisen from a relative sloping between actin and myosin laments. It is motivated by myosin cross-bridges on hydrolysis of adenosine triphosphate (ATP). The reaction by which chemical energy stored transmits into high energy of phosphoanhydric bonds in ATP [45] is maximal for an optimal lament overlap and decreases proportionally with decreasing overlapping [39]. Therefore, for calculation of the contractile stress, σact , as a function of deformation of the contractile elements, a simple piecewise linear constitutive model has been used. As seen in Fig. 1.2-b, if deformation of cell is in the ϵ1 - ϵ2 range, the active stress has eect on net stress transmitted by the cell, else it will be zero. Thus in the ϵ1 - ϵ2 range, it can be calculated as σact =Kactσmax(ϵi−ϵ) Kactϵi−σmax (1.2) 2 1. Finite Element Model (a) (b) Figure 1.2: Mechano-sensing model of an adherent cell. (a) Cell mechanical model. Kact , Kpas , and Ksubs , denote the stiness modulus of actin laments, of the passive components of the cell and of the substrate, respectively. fext stands for external forces applied to the cell or the substrate. (b) Dependence of the contractile stress, σact , on the deformation of the contractile element, ϵact . σmax , stands for the maximum contractile stress exerted by the actin-myosin machinery, ϵ1 and ϵ2 are the corresponding shortening and lengthening strains of the contractile elements with respect to the unloaded length at which active stress becomes zero [28]. where i =1 for ϵ1≤ϵ≤˜ϵ and i =2 for ˜ϵ≤ϵ≤ϵ2 , while ˜ϵ=σmax/Kact . The passive resistance of the cell, σpas , is given by σpas =Kpasϵ (1.3) Therefore by substitution of equations (1.2) and (1.3) in (1.1) the net stress transmitted to the ECM by a single cell as a function of the ECM volumetric strain, ϵ , can be calculated as σcell =     Kpasϵ ϵ < ϵ1 or ϵ > ϵ2 Kpasϵ+Kact σmax (ϵ1−ϵ) Kactϵ1−σmax ϵ1≤ϵ≤˜ϵ Kpasϵ+Kact σmax (ϵ2−ϵ) Kactϵ2−σmax ˜ϵ≤ϵ≤ϵ2 (1.4) In the physical process of cell locomotion, the contraction of the actin-myosin apparatus drives forward the translocation of the cell body and causes traction forces on the substrate [26, 44]. Actually, the movement of cells within a complex embryo or organism is guided by a complex interplay between chemical and physical signals like substrate stiness [17, 26], boundary conditions and generated forces due to cell-cell and cell-substrate interactions [26, 44]. Anyway, the simplied model described above is here used to predict cell migration as a function of cell internal deformation. The prominent aspect of the presented approach for cell modeling is that the cell can have any shape and can be represented by any number of nite elements. For this purpose, an algorithm has been used to track the key parameters required for migration at each time step considering important processes for cell migration, such as asymmetry of the cell, traction force, and traction force generation. Moreover, several aspects associated to the substrate such as stiness, boundary conditions and their eects on the direction of cell locomotion have been taken into account. To calculate the velocity and the new position of the cell within the substrate at each time step, the total net force during cell movement is balanced. In this case, the time step needed for model discretization is equal to the time of one cycle of cell migration that is the time taken by the cell to become suciently attached to its substrate in its advancing front and simultaneously break the adhesion with the substrate in the trailing back. It can change regarding to the 3 1.1. Model formulation (a) (b) Figure 1.3: (a) 3D spherical shape conguration of the cell. (b) Exerted sensing forces at every external node of the cell toward its centroid. deformation of the cell, traction and protrusion forces and the cell speed at every step. Time of one complete cycle of cell migration for broblast, epithelial or endothelial cells is around 600 seconds [44]. The considered eective forces which act on cell and substrate are traction forces, drag forces, and protrusion forces [26, 35, 44]. The traction forces are the result of traction at the front and the rear of the cell and depend on the force per ligand-receptor complex due to their different adhesiveness. These two forces at the back and front of the cell can be mathematically represented as [52] Ftrac |f,b=σcellSζ |f,b (1.5) where σcell , cell stress, can be calculated from Eq. 1.4. S stands for a proportionality model parameter with units of area and ζ is the adhesiveness that takes into account dierent numbers of receptors at the front and the rear of cell and binding strength of these receptors to the ligand in the ECM [52] and is given by ζ=knψ (1.6) The value of ζ is dierent at the front and the rear of the cell because there are dierent number of receptors and the binding stress of each of those receptors may be also dierent. k is the binding constant for the integrins at the front and end of the cell to the ligands in the ECM (in mol −1 ). For present simulations, I assume that the binding constant is equal for both front and rear of the cell. n is the total number of available receptors at the front or back of the cell and it is assumed that nf> nb (in mol). This means that as cell polarizes, the distribution of integrins will be asymmetrically distributed on the cell surface [52]. Finally, ψ represents the concentration of the ligands at the leading edge of the cell in the ECM. Consequently, Ftrac may be written as Ftrac =σcellS(knψ |f−knψ |b) (1.7) The second force which aects the cell is the resistive force (drag force) which comes from the viscous resistance to motility. In a Maxwell solid, the needed force for deforming the ECM depends on the deformation rate and accordingly the velocity. As the main objective here is to imply a velocity dependent opposing force associated to the viscoelastic character of the cell surrounding ECM, so, and for simplication, I assume the ECM as a viscoelastic medium [52]. 4 1. Finite Element Model In this case, the drag force can be dened as Fdrag =βµv (1.8) where µ is the eective viscosity of the viscoelastic matrix, it is considered as a constant throughout the substrate, v denotes the velocity of cell relative to the substrate. The constant β depends on the cell shape [52]. In the ideal case of a spherical cell moving through Newtonian innitely viscose medium, β can be approximated as [52] β= 6πr (1.9) where r is the radius of the cell. It is necessary to note that if the cell migrates through a purely elastic substrate, the force required for deforming the matrix will not depend on the velocity. Therefore a more realistic representation of the opposing force would be the summation of two contributions, one depending on cell velocity and the other independent [18, 21]. This is why the protrusion force has been introduced to be independent from cell velocity. Therefore, an eective guidance system appears in which cells send out local protrusions to probe the mechanical properties of the environment [26]. This force is generated by actin polymerization and cell or substrate attachments at the new location of lamellipodia protrusion, distinct from the cytoskeletal contractile force transmitted to the ECM [52]. It is a random force whose order of magnitude is the same to that of the traction force and less than it at every time step [21, 38]. Therefore, force equilibrium should be satised during cell locomotion, hence Ftrac +Fprot +Fdrag = 0 (1.10) Through this presented work, neither degradation nor remodeling of the ECM during cell locomotion are considered. 1.1.1 Single cell orientation At every step, the cell exerts a sensing force to diagnose its environment and hence it can determine the direction of migration within the substrate. I suppose that this sensing force is exerted at each external nite element node of the elements that represent the cell toward the cell centroid (Fig. 1.3-b). The deformed cell subjected to those sensing forces is represented by dotted lines in Fig. 1.4. So, the cell internal strain for each external node of the cell can be written as ϵcell =AB OA (1.11) Once the displacements of all nodes are calculated, information needed to compute ϵcell is available. Using Eq. 1.4, σcell at every external node of the cell can be calculated. Therefore, at each external node of the cell, the traction force vector can be represented by Ftrac i=Ftrac iei (1.12) where Ftrac i is the magnitude of the traction force corresponding to the i th external node of the cell obtained from Eq. 1.7. ei is a unit vector standing along the i th external node and the cell centroid which can be obtained by ei=xo−xi ∥xo−xi∥ (1.13) 5 1.1. Model formulation Figure 1.4: Deformed cell (dashed line) subjected to sensing forces. epol stands for the unit vector of the polarization direction calculated via resultant traction force, Ftrac R , and random protrusion force, Fprot . It is of interest to note that the eect of the resultant traction force on polarization direction is more important than the one of the random protrusion force. xo is the cell centroid position vector and xi is the position vector of the i th external node of the cell. Consequently, the resultant traction force, Ftrac R , can be calculated as Ftrac R= n ∑ i=1 Ftrac i (1.14) where n is the number of the external node of the cell. To calculate the magnitude of the cell velocity, the vector of the protrusion force should be estimated at every step. It can be dened as Fprot =λerand (1.15) erand is a random unit vector. λ is the magnitude of the protrusion force which is estimated as λ=κ∥Ftrac R∥ (1.16) where κ is a random number, 0<κ<1 . It is important to note that the magnitude of the protrusion force, λ , should be less than the traction force [26, 52]. From Eq. 1.8 and Eq. 1.10, the cell velocity can be dened as v=∥Ftrac R+Fprot ∥ βµ (1.17) Thereby, at every time step, the distance through which the cell migrates to locate in its new position can be calculated. Hence, the displacement vector of the cell can be dened as d=vτepol (1.18) where τ is time step of simulation and epol is a unit vector which represents direction of cell migration. It is important to note that, at every time step, the internal deformation at every cell external node, caused by the sensing force, is negative (cell exerts contraction forces toward 6 1. Finite Element Model Figure 1.5: Computational algorithm of cell migration. First, the cell senses its environment. By information evaluated at that sensing step, cell can polarize and migrate after calculating traction force and estimating protrusion force. its centroid and always tries to compress itself). Nodes with less internal deformation will have a higher traction force (note that the cell internal deformation has to be within the hatched area of Fig. 1.2-b). As all the traction forces are acting toward the cell centroid, the resultant of these traction forces will have the direction of minimum internal deformation. So that, the resultant of the traction force opposite direction and random protrusion force presents the polarization direction of the cell. Therefore, the unit vector of cell polarization, epol , can be dened as epol =Fprot −Ftrac R ∥Fprot −Ftrac R∥ (1.19) It is remarkable that according to the eect of the random protrusion force, Fprot , the cell will move toward stier and/or more xed region of the substrate (minimum deformation) in a random directed motion. 1.1.2 Cell-cell interaction The same previous formulation is used to dene traction force, protrusion force, velocity and reorientation of each individual cell. A model that denes the interaction of cells will be presented along this section. Let us dene rij as a vector passing through centroid of two cells i and j (Fig. 1.6-a). rij =rj−ri (1.20) A useful simplication to avoid interference of two cells is that the magnitude of the rij should be greater than or equal to the cell diameter. In reality the cells inside a multicellular system do not preserve a spherical shape but deform to be tangent to each other and cover all the matrix [32]. In our case and for discretization, when two cells touch each other they can have a maximum of four common nodes (Fig. 1.6-b). In vivo, the cell drives out a pseudopod to sense its environment better. Once the cell nds the stier region of the substrate it pulls up whole body in direction of the pseudopod [47]. 7 1.1. Model formulation (a) (b) Figure 1.6: (a) Distance between the centroids of two cells. (b) Interaction of two cells when they contact each other. The distance between their centroid is equal to the proposed cell diameter. Here, for assumed shape of the cell, two cells can have maximum four common nodes in this situation. Therefore, when two or several cells touch each other, the common points of both cells (for instance nodes n1:n4 in Fig. 1.6-b) are not able to drive out the pseudopod to substrate [8, 47]. Therefore, for two or more cells, I assume that the cells do not exert any sensing force at those nodes unless they get separated again due to the protrusion force (see Fig. 1.6-b). It is worth to note that in such situation these common nodes do not have any role to sense their environment but traction forces in those nodes are not zero. 8 Chapter 2 Experimental Monitoring of The Cell Cellular microenvironments through dierent tissues are characterized in terms of protein composition, protein-protein interactions and the collective properties. These characteristics create variety of local elasticity and structure which make specic each tissue. The elasticity of microenvironments within brain, fat, muscle, cartilage and pre-calcied bone is ranged at Fig. 2.1 [10]. Cells within tissues constantly probe the mechanical properties of their surroundings by adhering, actively pulling, and sensing of their substrate to induced deformations. The introduction of novel techniques have not only opened up completely new perspectives regarding biological function, but also presented a new quantitative element into this eld. For example, the availability of soft elastic substrates with controlled stiness and variety of optical and orescence microscopies allows me to culture cells in dierent stiness substrates to track the cell behavior. Besides numerical model approaches, these progresses enable us to work in close contact with experimental data. To experimentally visualize cell migration in 3D substrate, I manipulated a 3D substrate with constant stiness (Fig. 2.2). The substrate is composed of collagen I which forms a rm gel at a neutral pH and 37 ◦ C when diluted. It is a brous protein that is composed of three α chains which form a rope-like triple helix, providing tensile strength of ECM. The α chains contain GXY. G refers to glycine which is a small amino acid and ts well to the triple helix. X and Y are typically proline and hydroxyproline which are critical for collagen stability. Collagen type I is the most common brillar collagen and is mostly found in skin, bone, tendons, and other connective tissues. After preparation of the gel, visualize of the cell in 3D collage based substrate has been performed by collaboration of Dr. Pilar Martín Duque in "Instituto Aragonés de Ciencias de la Salud (I+CS), departamento de Hospital Miguel Servet, Unidad de investigación Traslacional". I have used the orescence microscopy of this center to record the cell motility during about 24 hours. Figure 2.1: Elasticities of cellular environments [10]. 9 3.1. Two dierent stiness substrates with dierent boundary conditions (a) (b) (c) (d) Figure 3.2: Cell migration in two dierent stiness substrates, case 2. The two surfaces perpendicular to the x -axis at the both sides are constrained. In this case, an IEP appears through where the cell changes the direction of polarization. If the cell is placed in the left region of IEP, it moves toward the constrained surface in the softer part (a and b). By contrast, if the cell is initially located in the right side of the IEP the behavior of the cell changes and migrates toward the stier region and toward the constrained surface in the stier part (c and d). surfaces or dierent stinesses regions. The location of this IEP slightly changes between dierent simulations due to random protrusion force. In this case, when the cell is located close enough to the constrained surface in the softer side (left hand region of the IEP), the signal coming from that constrained wall is higher, so it moves toward this constrained boundary of the softer side (Fig. 3.2-a and 3.2-b). Alternatively, when the cell is placed in the other side of the IEP, it rstly sense the stiness of the stier part of the substrate (right hand one) so it migrates toward the stier part and then toward the constrained surface in this side (Fig. 3.2-c and 3.2-d). In all cases, when the cell arrives to constrained surface in the stier or softer part it keeps moving near to this surface randomly. If the cell is initially positioned in the stier part it never goes to the softer part of the substrate. It is remarkable that if the cell is placed close to the IEP, its behavior depends on the relative amplitude of the protrusion force in the initial steps of its movement. 16 3. Numerical Experiments 3.1.3 Case 3 Here for the same substrate I change the boundary conditions again. In this case, I restrain (zero displacements) the surface perpendicular again to the x -axis but now the one closer to the softer region leaving the rest of the boundary surfaces free (zero tractions). Two IEPs do appear (one in the softer part and another in the stier part) (Fig. 3.3). Therefore, there are three zones separated by these IEPs where the cell receives dierent feedbacks [26]. In Fig. 3.3-a and 3.3-b, when the cell is located close enough to the free surfaces in the stier side of the substrate, it moves away toward the interior of the substrate. Once the cell nds the rst IEP, in the stier region, it keeps moving near to it randomly. If the cell is initially positioned between the two IEPs (Fig. 3.3-c and 3.3-d), no matter if in the stier or softer parts of the substrate, it again moves toward the IEP in the stier part. By contrast, if the cell is placed in the left hand side of the IEP placed in soft part (left hand IEP) the polarization of the cell changes and the cell moves toward the constrained surface in the softer side as seen in Fig. 3.3-e and 3.3-f. 3.2 Eect of a stiness gradient To fully understand the eect of the stiness on cell migration I applied the proposed model to a substrate with linear stiness gradient which changes from 100 kPa at x= 0 to 200 kPa at x= 400 µ m through which all the substrate surfaces are considered free. Hadjipanayi et al. [17] analyzed the eect of a 3D substrate with linear stiness gradient on cell migration. They divided this substrate into soft, middle, and sti regions. First, they inserted about the same number of cells inside these zones. Since the number of cells was few, they assumed that there was no interaction among cells. After 6 days, they observed a signicant dierence in cell concentration in these three regions. They observed that the higher cell concentration resulted in the stier part while the least concentration was identied in the softer. In our simulation, and as expected when a cell is placed in this substrate, the cell tendency is to migrate toward the direction of the higher stiness (Fig. 3.4). When all boundary surfaces are considered free, the cell randomly moves around an IEP far enough from those free surfaces located in the stier side. In this case, the results neither depend on the initial location of the cell (Fig. 3.4) nor the stiness gradient. Here, only the results of the simulation corresponding to two initial dierent locations of the cell are presented (Fig. 3.4). This simulation was repeated for several values of the stiness gradient and initial position of the cell and all results were consistent and in agreement with the experimental results [17]. While the cell migrates from a soft part to stier one, the nodal traction forces at external nodes increase. This happens because in stier regions, cell internal strain decreases while cell stress increases (hatched area in Fig. 1.2-b). In Fig. 3.5, the nodal traction force corresponding to one of the external nodes of the cell has been plotted. On the contrary, the net traction force aecting on cell migration decreases (Fig. 3.6). This happens because while the nodal traction forces increase, dierences between them decrease and consequently the net traction force reduces. This explains why the cell in stier parts generally remains round and symmetric whereas it exerts higher nodal traction forces. This result is corroborated by ndings of Ehrbar et al. [13]. It is worth to remark that curve uctuation in Figs. 3.5 and 3.6 is due to the eect 17 3.2. Eect of a stiness gradient (a) (b) (c) (d) (e) (f) Figure 3.3: Cell migration in two dierent stiness substrates, case 3. The surface in the softer part has been restrained leaving the rest free. In this case, two IEPs appear, one in the stier part and another in the softer part. If the cell is initially placed in the right side of the IEP located in the stier part (a and b) or between these IEPs (c and d), it migrates toward the IEP placed inside the stier region. If the cell is rstly seeded in the left side of the IEP located in the softer part it migrates toward the constrained surface in the softer region (e and f). 18 3. Numerical Experiments (a) (b) (c) (d) Figure 3.4: Cell migration in a substrate with a stiness gradient and free boundary surfaces. In the rst case (a) and (b), cell has been embedded in one of the corners of the substrate (softer side), it migrates toward the IEP (dotted line) far enough from free surfaces and keeps moving randomly around it. In the second case (c) and (d), the cell starts to migrates from the stier side toward the softer one until the IEP and keeps moving around it. of the cell protrusion force. Fig. 3.7 shows how the overall cell velocity decreases during cell migration toward stier regions. This means that the low net traction force in stier zones causes low cell speed which can even stop cell migration in very dense substrates [13, 52]. On the other hand, the oscillations of the cell velocity in Fig. 3.7 seem to have higher amplitude than that of nodal traction forces and net traction force in Figs. 3.5 and 3.6. This is because the protrusion force aects not only on the direction of cell migration but also on the magnitude of the cell velocity. It is clear that when the cell approaches the IEP, after about 80 time steps, the studded parameters tend to be more stable since the local strains of the cell do not change too much. 3.3 Eect of substrate depth The proposed model was applied for a substrate with a slope as seen Fig. 3.8 to study the eect of substrate depth on cell migration. Here the sloped surface has been constrained and 19 3.3. Eect of substrate depth Figure 3.5: The eect of the substrate stiness on the cell nodal traction force during migration. Figure 3.6: The eect of the substrate stiness on the cell net traction force during migration. Figure 3.7: The eect of the substrate stiness on the cell migration velocity. 20 3. Numerical Experiments Figure 3.8: Migration of the cell due to the variance of substrate depth. The sloped surface is constrained. (a) (b) Figure 3.9: Stress and deformation in x direction during sensing process. the rest of surfaces are free. Elastic modulus of the substrate is assumed to be 100 kPa. In this experiment the cell moves over the substrate surface. Because the sloped surface is constrained, the cell will sense less internal deformation for lower depth. The cell started the migration process in a point on the substrate surface that has maximum depth. When the cell exerts sensing forces, it recognizes the direction of the slope, therefore it migrates toward minimum depth in a random path. Fig. 3.9 demonstrates an intense dierence of sensing domain between the substrate surface and substrate depth. Here, since the cell exerts sensing force at the substrate surface, the z component of the sensing force is close to zero, so that the cell can not sense too deep. Consequently, due to this limitation, the speed of propagation of the cell on the surface is less than that of the cell inside three dimensional domains. This explains why the cell moves more randomly in surface migration [26]. 3.4 Interaction between two cells As cited in previous sections, the developed model can simulate cell migration with any number of cells in the populations. Hence, to fully understand the interaction between two cells, a substrate with the same dimensions of the one with stiness gradient is employed to simulate the behavior of two cells in the same substrate (Fig. 3.10). All boundary surfaces of the substrate are considered free. The substrate elastic modulus is constant and equal to 100 kPa. 21 3.4. Interaction between two cells (a) (b) Figure 3.10: Interaction between two cells inside a substrate with constant elastic modulus. All boundary surfaces of the substrate are considered free. Cells start to move from two corners of the substrate. They migrate toward the center of the substrate and keep move around each other randomly. Figure 3.11: Von Mises stresses due to mechanosensing of two cells during migration. The stretched zone between two cells is clear because of their interaction. Near to the free surfaces, one cell is located in the top corner of the substrate and another in the bottom corner of the substrate. It should be again mentioned that a red and yellow sphere represent the centroid of the each cell. As we observe in Fig. 3.10, both cells migrate in a random path toward each other. Once they sense each other in the middle of the substrate, they keep moving around each other randomly. As mentioned earlier, the cell exerts contraction sensing forces at its external nodes toward its centroid to feel its environment. So, when cells are sucient near to each other, the region between them will be under tension (Fig. 3.11). Consequently, cells will feel less internal deformation in this direction. They will detect this zone as stier region of the substrate and migrate toward each other. Once they contact each other as long as their polarization direction is either toward each other or in the same direction they remain in contact. When, their polarization directions become dierent they separate until they regain contact and so on. To better understand the eect of this phenomenon of cell migration, simulations were repeated for the previous substrate with stiness gradient. The results are in agreement with migration of one cell inside the substrate with constant stiness as seen in Fig. 3.12. Firstly, the cells try to reach each other and after crossing the rst quarter of the x -axis, they continue to move together toward the stier region of their substrate. Again, and as in case of individual cell, there exists an IEP where cells maintain moving together randomly around it. Fig. 3.13 shows the curve of cell velocity for one and two cells migration. To compare the cell velocities in both 22 3. Numerical Experiments (a) (b) Figure 3.12: Interaction between two cells inside a substrate with stiness gradient. All the boundary surfaces of the substrate are considered free. Two cells start to move from two corners of the substrate. Figure 3.13: Comparison of cell migration velocities for two cells and single cell migration. cases, I tted an average curve for each cell velocity to eliminate the velocity uctuation due to the protrusion force. The simulations were repeated several times and the result was similar. The gure shows that the overall cell velocity in case of single cell migration is less than two-cell migration. This is because in the case of two cells, the nodes near to the stretched area between the two cells experience less deformation (Fig. 3.11). On the contrary, the nodes that are away from this area in the other side of the cell will experience more deformation and less traction force. The local cell velocity for two-cell migration is higher than the one for single cell. In this case, the number of steps needed by the cell to catch the IEP was about 130 time steps which is higher than in case of one cell. 3.5 Cell population In this experiment, the proposed model is employed to simulate a cell population with 40 cells simultaneously embedded inside a substrate with the same dimensions as that of the previous experiment. All boundary surfaces of the substrate are considered free. Elastic modulus is considered to be constant throughout the substrate (100 kPa). At the beginning, the cells are randomly distributed near the boundary surfaces (Fig. 3.14-a). When cells exert the sensing 23 3.5. Cell population force to check their environment, they recognize the middle of the substrate as a more stable region. Fig. 3.14 represents cells locomotion for dierent time steps. Here, for better representation of cell-cell interaction and their contacts, I represented the proposed sphere shape conguration of the cells in each step. During migration, several stretched zones exist between cells that aect cell locomotion. So that, in primary steps of migration, they aggregate in small groups (Fig. 3.14-b). Joining these slugs to each other, several big slugs of the cells are formed (Fig. 3.14-c). Afterward, all this slugs contact each other in the middle of the substrate (Fig. 3.14-d). Internal cells which are enveloped by several cells stop moving, but boundary cells may move around enveloped cells. Upon a cell inside the slug reach circumference of the cell aggregation, it will have possibility to move around the cell aggregation. In this simulation the cells respond similarly and consistently as previous experimental [30] and numerical works [32]. 24 3. Numerical Experiments (a) (b) (c) (d) Figure 3.14: Interaction between 40 cells inside a substrate with free boundary surfaces. 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