Mitosis and involved forces.

<p>a) Several bio-mechanical aspects of cell proliferation are included in our model. First, the cell area (or volume in 3D) is doubled. The axis of cell division is selected (dash-line), in a dividing unpolarized cell, this axis usually is perpendicular to the cell elongation direction in such a way as to split the cell into two approximately equal parts. In a partially polarized cell, however, the axis of cell division is orthogonal to the part of the cell membrane that is in contact with the ECM. Two new daughter nuclei are then placed orthogonal to the axis of cell division. After the selection of division axis, the model finds the nearest membrane point to this axis, i.e. point In mitosis there are two major mechanical forces, first in the Anaphase stage of mitosis, shortening the spindle fibers caused by the kinetochores separation, and the chromatids (daughter chromosomes) are pulled apart and begin moving to the cell poles <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0012097#pone.0012097-Lodish1" target="_blank">[65]</a>. Second, a contractile ring is formed by contractile forces acting on the opposite sites of the cell boundary in the cytokinesis process <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0012097#pone.0012097-Lodish1" target="_blank">[65]</a>. This results in the formation of a contractile furrow and causes division of the cell into two daughter cells. The cell points can therefore be divided into two groups, A and B, where group A consists of membrane points from to and nucleus points from to and the remaining points belong to group B. To model the first mechanical force, the points of the nucleus in A and B are pulled apart, in the orthogonal direction to the division axis with force (Eq.(8) ). During the nucleus separation, the contractile force, , acts on boundary points of A and B groups to model the second mechanical force(Eq. (9)). b) The main phases of cell growth and division. (I) Cell growth. To implement cell growth in the proposed model, the number of membrane points, i.e. the number of viscoelastic compartments, is allowed to increase. When we add two points on each the cell and nuclear membranes, four subunits are added to the system, with the parameters of these new subunits calculated from the average of the first neighbor's homogeneous subunit parameters. With the additional ‘growth’ point, the circumferential length of the membrane increases in proportion to . Hence, the rest volume i. e. the volume of the cell when it grows freely without any inner or outer constraint, must increase proportional to . Therefore, the rest length of radial springs is increased in proportion to . When the area (or volume) of cell doubles, the number of defining membrane points increases to , where is the number of membrane points on the initial cell. (II,III) Mitotic process: two types of forces act on points to divide the cell. Due to these forces the cell elongates and prepares for division. (IV) Two new daughter nuclei are then placed orthogonal to the axis of cell division. After the nucleus separates, i.e. the distance between the center of the mass of nuclear points exceeds a certain value, , the cell will divide into two daughter cells, i.e. the subunits which join the boundary points will be eliminated and will bind to a new first neighbor point in the same group with a new subunit. V) After division takes place, each daughter cell will only have points, and as a result it is possible to simultaneously add points to each cell. To add membrane points, two consecutive points in the membrane are found that have the longest distance and a new point is inserted between them, and this process is repeated until the number of cell points becomes . f) Adhesion of the two daughter cells.