Cellular processes

Our model can account for cell polarity and several cellular processes including cell growth and division, and cell migrations. We discuss the modelling of those processes below.

1. Cell Growth and Division: The cell grows by gradually increasing internal pres- sureP. The growth through internal pressure is inspired by the fact that animal cells modulate their internal hydrostatic pressure before mitosis [11]. The current imple- mentation of the model assumes that all cells have the same pressure which grows at the same constant rate, but the model can be easily extended to assume various growth mechanisms as well. The individual modelling of cells allows one to vary the internal pressure and the growth rate for each cell independently.

Once the cell area reaches a threshold valueAdiv_{, the cell divides into two daughter}

cells. The model is scale invariant, and without loss of generality we can choose the numerical value forAdiv _{to be unity. This sets the length scale in our model. The in-}

crease in the internal pressure for mitotic cells observed by Stewartet al.[11] serves as a basis for the growth control through pressure in our model. To ensure that daughter cells grow similarly to the mother cell, both cells are assigned a pressure identical to the parental cell. During the division, new mass points are added along the division line so that the resulting two cells form closed loops and have the same number of mass

CHAPTER4. MODEL AND METHODS 53

points as the initial cell. We consider three division mechanisms: random, divisions by the‘longest axis rule’orHertwig’s ruleandasymmetriccell division (Fig. 4.2). In

Figure 4.2: Three mechanisms of cell division. A random(AB) division line passes through the cell’s center of mass and has random orientation. Division based on Her- twig’s rule (CD) has a division line that is perpendicular perpendicular to the cell’s longest axis. An asymmetric division line (AE) splits the cell into two unequal sizes and has random orientation.

random division scheme, the division line passes through cell’s center of mass and has random orientation. This division line divides the cell into two roughly identical cells. Here we assume homogeneous cells, with all mass points having the same mass and all spring constants being the same. With that assumption, we consider a random division line to be one that connects a randomly chosen mass point to its diagonally opposite mass point (line AB in Fig. 4.2). Two other division mechanisms are chosen based on biological relevance. More than a century ago Hertwig proposed a division rule based on his observations on mitotic cells. For most cells the mitotic spindle aligns along the longest axis of the cell. Hence, the cell division plane is oriented perpen- dicular to the longest axis. This division scheme is known as the ‘longest axis rule’ or ‘Hertwig’s rule’. We implement the division according to the ‘longest axis rule’ by searching for the cell’s longest axis and then choosing the division line closest to the line that is perpendicular to the longest axis (line CD). Finally, in some cases, cells divide asymmetrically. During asymmetric division, two daughter cells end up with different sizes [69]. Numerically, we construct the asymmetric division line by consid- ering the line which connects a randomly selected mass point with another one that is not diagonally opposite to the first mass point (Fig. 4.2 (AE)).

2. Cell Polarity and Migration:The model can account for cell polarity and migra- tion (not included in this work). Polar cells are characterized by differences in shape,

structure or functionality of spatially different regions of the cell. An epithelial cell is an example of a polar cell, where an individual cell is divided into two distinct regions, apical and basolateral. The apical region faces the lumen or outer surface, while the basolateral region is in contact with the basal lamina or other cells. Exposure to dif- ferent environments requires having different structure and functionality of apical and basolateral membranes of epithelial cell. Since our model the cell is presented as a collection of sub-cellular mass points, we can model cell polarity by varying properties of mass points that belong to distinct parts of the cell.

Collective cell migrations during the an embryonic development are an essential part of morphogenesis [97, 98]. Cell polarity plays an important role in cell migrations. For a cell to move in a specific direction, it should have defined front and rear parts. As a cell migrates, the leading edge extends towards the direction of motion, while the opposite edge retracts [136]. We can model the migration through modulation of the pressure forces acting on mass points that belong to the leading and the retracting edges of cell. If the mass points that belong to the leading edge are assigned a higher internal pressureP than the mass points of the neighbor cells, the leading edge extends outwards. Similarly, if the internal pressure of the mass points that belong to the rear edge of the cell is set to be lower than that of the internal pressure of the neighboring cell mass points, the rear edge retracts. Consequently, the cell moves generally in the direction of the motion of the leading edge.