Modeling cellular deformations using the level set formalism

Background Many cellular processes involve substantial shape changes. Traditional simulations of these cell shape changes require that grids and boundaries be moved as the cell's shape evolves. Here we demonstrate that accurate cell shape changes can be recreated using level set methods (LSM), in which the cellular shape is defined implicitly, thereby eschewing the need for updating boundaries. Results We obtain a viscoelastic model of Dictyostelium cells using micropipette aspiration and show how this viscoelastic model can be incorporated into LSM simulations to recreate the observed protrusion of cells into the micropipette faithfully. We also demonstrate the use of our techniques by simulating the cell shape changes elicited by the chemotactic response to an external chemoattractant gradient. Conclusion Our results provide a simple but effective means of incorporating cellular deformations into mathematical simulations of cell signaling. Such methods will be useful for simulating important cellular events such as chemotaxis and cytokinesis.

When the cell is aspirated into the micropipette, we assume that the curve formed inside the micropipette is an arc of a sphere of radius R cap . In this section, we derive the expression for R cap given the cell radius R c , pipette radius R p , and length of protrusion L p shown in Figure A1.
We start by noting that ACF is a right-angle triangle. Therefore, the length of segment AC is: Next we note that BCF is also a right-angle triangle. Let Y represent the length of segment BC. Then: Finally, This formula applies while the center of the arc is outside the pipette (Y ≥ 0). Thereafter, the radius of the cap equals the radius of the pipette (R cap = L p ).

Derivation of effective velocity v
Given that the potential function φ has evolved from φ(t) to φ(t + ∆t) in time ∆t, we can find the speed of change in φ with respect to time: Assuming that a velocity vector v that is outward normal to the cell membrane is responsible for this evolution in φ, v must satisfy the level set equation: Let |v| represent the magnitude of vector v. Its direction is the normal vector: This leads to: Solving for |v|, we have: Thus, incorporating the direction of the vector (n) lead to the equation: Fitting the pressure profiles generated from cell shape In the main text we computed the local pressure profiles needed to generate steady-state elongated or fan-like cell shapes while chemotaxing. This leads to the discrete points of Figure 7B,E. However, to generate the profile during the simulations requires a continuous function of local chemoattractant concentration. This requires that the computed pressure profiles be smoothed and fitted by a curve.
First, we normalize the local chemoattractant (cAMP) concentration: Thus, the normalized chemoattractant concentration ranges from −0.5 (at the rear) to +0.5 (at the front).
The computed profiles (red dots in Figure 7B,E) are now fitted to specific formulae. For the elongated cell, we found that the computed pressure profile can be accurately approximated by the ninth order polynomial: y = a 1 x 9 + a 2 x 8 + a 3 x 7 + a 4 x 6 + a 5 x 5 + a 6 x 4 + a 7 x 3 + a 8 x 2 + a 9 x + a 10 .
Using a fitting program (Matlab's Curve Fitting Toolbox, The Mathworks, Natick, MA) to minimize the least squares error, we found the following values for the polynomial's coefficients: For the fan shaped amiB − null cell, we found that the computed pressure profile could be approximated by a fifth order rational function: Least square fit provided the following values for coefficients: The approximations are shown by the blue lines in Figure 7B,E. Figure 1. Geometry of micropipette aspiration experiment. R c is the initial radius of the spherical cell, R p is the radius of the pipette and L p is the measured protrusion into the pipette.