Mathematically Modeling Chondrocyte Orientation and division in Relation to Primary Cilium

Confocal microscopy images show that growth plate chondrocytes will tend to form columns after cellular division[2], and the primary cilium may be responsible for the migration of the cells[4][5]. These cells have the ability to move within the growth plate, possibly influenced by the environment, and the primary cilium may direct the orientation and positioning of the cell in response to the environment or extracellular matrix[13][10][15][19][8][22][24]. The primary cilium may also point in the direction of movement of cells or cell formation[20], and that cell movement may be related to hardness or softness of the surface[23]. It has been demonstrated that cells divide perpendicular to the growth plate before moving into columns[6]. The cells may react differently to stimuli based upon their level of development[12], and the development of the cells can be influenced by a mutation. In particular, the disruption of IFT can cause chondrocytes to no longer form in a columnar structure[21] and knockouts of Smad 1, 5, and 8 cause increased apoptosis[18]. The primary cilium has been shown to be related to the centrioles, and that the centrioles are related to cell division and positioning[1][7]. Previous mathematical models are broken into three categories; ones that look at longitudinal bone length as a differential equation with respect to time, ones that view the bone as broken into various regions and models the growth of each of them, and ones that model individual cellular division without concern to the large-scale full bone growth. Classic examples include modeling the length of gull wings as they age [11], as well as fluid and mass flow through the zones of the growth plate using coupled partial differential equations [9]. Individual cell division was modeled using ovals of Cassini to represent one mother cell dividing into two daughter cells [14]. The value of the φci for the wild type clusters at around both 10 degrees and 80 degrees, as opposed to the value of the φci of the mutant whose φci clusters only at approximately 80 degrees. Because the growth plate of the wild type mouse is more organized than the mutant mouse, we utilized the idea that the φci clustering at around 10 degrees, which does not occur in the mutant, controls the organization of the growth plate of the wild type mouse. Therefore, we made the assumption that the closer the value of φci is to 0 degrees, the stronger the signal emitted.