8.8 Exploring the biochemical and mechanical effects of intestinal  (Page 6/6)

The SMC is modeled as a dynamic two-dimensional mass-spring system. The membrane and the various filaments of the cell are discretized into a series of springs with mass distributed amongst nodes located at the intersections of these springs. The spring stiffnesses and node masses vary depending on the material they represent.

The model includes the powerstroking process with myosin heads extending from nodes along the myosin filament and connecting to nodes along the actin filaments. The contractile force generated by the biochemical model is applied at these actin binding sites, in a direction parallel to the axis of the actin filament. After the duration of a typical powerstroke has passed, each myosin head detaches from its corresponding binding site on actin, and reattaches to the actin at the next node on the filament. The location at which the contractile force is applied changes according to which actin nodes are attached to the myosin heads at the current time in the simulation. Figure 4 depicts the powerstroking process as used in the model.

The filaments which participate in the powerstroking process are considered active filaments, as they are the components of the cell which produce the cellular contraction. The model also includes several types of passive elements, including the cell membrane, the cytoskeletal network, and the extracellular matrix of collagen fibers. Unlike the active filaments, these passive elements do not produce a contractile force based on a series of biochemical reactions, but rather experience forces only as a result of drag, spring deformation, and the cell's own hydrostatic pressure.

Thus the model gives rise to the following equations which describe the motion of the $i^{th}$ node $i$ n the system:

Equations 22 and 23 represent the velocity of the node in the x- and y-directions, respectively; $x_i$ is the x-coordinate of the position of the node, $y_i$ is the y-coordinate of the position of the node, $u_i$ is the velocity in the x-direction and $v_i$ is the velocity in the y-direction. Equations 24 and 25 are the equations of Newton's second law, again in the x- and y-directions. In these equations, $m_i$ is the mass of node $i$ , $N$ is the number of other nodes which are attached to node $i$ by a spring, $k_{ij}$ is the spring constant of the spring connecting nodes $i$ and $j$ , $L_{ij}$ is the current length of that spring, and $L{0}_{ij}$ is the equilibrium length of that spring. $P$ is the hydrostatic pressure of the cell, $A$ is the cell's surface area, and $n_x$ and $n_y$ are the x- and y-components of a unit vector normal to the cell membrane at node $i$ . $F_{myosin,x}$ and $F_{myosin,y}$ are the x- and y-components of the force on node $i$ generated by the biochemical reactions, and $\beta$ is the drag coefficient for node $i$ . The right-hand sides of these last two equations represent the sum of the forces on the node.

Taking equation 24, the equation for motion in the x-direction, as an example, the first term on the right side of the equation represents the sum of the x-components of the spring forces of any filaments attached to that node as determined by Hooke's Law. The second term represents the x-component of the force acting on the node due to the higher pressure inside the cell. This force is only nonzero in the case of membrane nodes, and it is directed along a line normal to the cell membrane in an outward direction. The magnitude of the pressure force is inversely proportional to the area enclosed by the membrane filaments at the given time. The third term is the x-component of the powerstroking force (as previously described), which applies only to nodes along actin filaments which are connected to myosin heads at the current time. The final term in the equation represents the x-component of the force resulting from drag as the node moves through the cytoplasm. The drag coefficient, $\beta$ , is approximated using slender-body theory [link] . Each of these terms has a y-directional analog which can be found in equation 25.

Results

The contractile behavior of the cell under normal conditions is depicted in figures 4 and 5. The figures show that contraction occurs after calcium levels have increased. The initial width of the relaxed cell under normal conditions was 200 microns, and the width of the cell at maximum contraction was 144.6 microns.

In testing hypothesis 1, shown in figure 6, increased synaptic cleft distance decreased ACH concentration at the cell membrane receptor site. In testing hypothesis 2, shown in figure 7, the cell's length at maximum contraction increased with an increase in interstitial area due to edema.

Discussion

The goal of this study was to explore a possible mechanical relationship between edema and ileus in intestinal SMCs. We have presented two hypotheses that may help explain the link between edema and ileus and developed a biochemical and mechanical model which analyzes the respective hypotheses. The model successfully replicates cellular contraction in non-edematous conditions. It also replicates the expected results of the two hypotheses, demonstrating decreased ACH levels at larger synaptic cleft widths and decreased contraction with stretched collagen fibers.

The next step in the project is to quantify the decrease in contraction from ACH decrease and collagen uncoiling. Further considerations include correcting ACH levels using ACHesterase or flow in three-dimensional space. These final results must then be compared to the decreased muscle activity in intestinal edema, to test if our model can satisfactorily explain the level of decreased activity.