Calculating the Shape Constraint of a Cell – the Elongation Term

Learning Objectives:

Learn how CC3D leverages inertia tensors to elongate cells

The shape of a single cell immersed in medium and not subject to too drastic surface or surface constraints will be spherical (circular in 2D). However, in certain situations, we may want to use cells that are elongated along one of their body axes. To facilitate, this we can place constraints on the principal lengths of the cell. In 2D, it is sufficient to constrain one of the principal lengths of the cell. However, in 3D, we need to constrain 2 out of 3 principal lengths.

Our first task is to diagonalize the inertia tensor (i.e. find a coordinate frame transformation which brings the inertia tensor to a diagonal form)

Diagonalizing inertia tensor

We will consider here a more difficult 3D case. The 2D case is described in detail in M. Zajac, G.L. Jones, and J.A. Glazier in “Simulating convergent extension by way of anisotropic differential adhesion” Journal of Theoretical Biology 222 (2003) 247–259.

In order to diagonalize the inertia tensor, we need to solve the following eigenvalue equation:

\begin{eqnarray} det(I-\lambda) = 0 \end{eqnarray}

or in full form

\begin{eqnarray} 0 = \begin{vmatrix} \sum_i y_i^2+z_i^2 - \lambda & -\sum_i x_i y_i & -\sum_i x_i z_i \\ -\sum_i x_i y_i & \sum_i x_i^2+z_i^2 - \lambda & -\sum_i y_i z_i \\ -\sum_i x_i z_i & -\sum_i y_i z_i & \sum_i x_i^2+y_i^2 - \lambda \\ \end{vmatrix} = \begin{vmatrix} I_{xx} - \lambda & I_{xy} & I_xz \\ I_{xy} & I_{yy} - \lambda & I_yz \\ I_{xz} & I_{yz} & I_zz - \lambda \end{vmatrix} \end{eqnarray}

The eigenvalue equation will be in the form of 3^rd order polynomial. The roots of it are guaranteed to be real. The polynomial itself can be found either by explicit derivation, using symbolic calculation, or simply in Wikipedia ( http://en.wikipedia.org/wiki/Eigenvalue_algorithm )

so in our case, the eigenvalue equation takes the form:

\begin{eqnarray} -L^2+L^2(I_{xx}+I_{yy}+I_{zz})+L(I_{xy}^2+I_{yz}^2+I_{xz}^2-I_{xx}I_{yy}-I_{yy}I_{zz}-I_{xx}I_{zz}) \\ +I_{xx}I_{yy}I_{zz} - I_{xx}I_{yz}^2 -I_{yy}I_{zz}^2 I_{zz}I_{xy}^2 + 2I_{xy}I_{yz}I_{zx} \end{eqnarray}

This equation can be solved analytically, again we may use Wikipedia ( http://en.wikipedia.org/wiki/Cubic_function )

Now, the eigenvalues found that way are the principal moments of inertia of a cell. That is they are components of the inertia tensor in a coordinate frame rotated in such a way that off-diagonal elements of the inertia tensor are 0:

I = \begin{eqnarray} \begin{vmatrix} I_{xx} & 0 & 0 \\ 0 & I_{yy} & 0 \\ 0 & 0 & I_zz \end{vmatrix} \end{eqnarray}

In our cell shape constraint, we will want to obtain ellipsoidal cells. Therefore the target tensor of inertia for the cell should be the tensor if inertia for the ellipsoid:

I = \begin{eqnarray} \begin{vmatrix} \frac{1}{5}(b^2+c^2) & 0 & 0 \\ 0 & \frac{1}{5}(a^2+c^2) & 0 \\ 0 & 0 & \frac{1}{5}(a^2+b^2) \end{vmatrix} \end{eqnarray}

where a, b, c are parameters describing the surface of an ellipsoid:

\begin{eqnarray} \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \end{eqnarray}

In other words a, b, c are half lengths of principal axes (they are analogues of the circle’s radius).

Now we can determine semi axes lengths in terms of principal moments of inertia by inverting the following set of equations:

\begin{eqnarray} I_{xx} = \frac{1}{5}(b^2+c^2) \\ I_{yy} = \frac{1}{5}(a^2+c^2) \\ I_{zz} = \frac{1}{5}(a^2+b^2) \end{eqnarray}

Once we have calculated semiaxes lengths in terms of moments of inertia, we can plug in actual numbers for moment of inertia (the ones for actual cell) and obtain lengths of semiaxes.

Next, we apply a quadratic constraint on largest (semimajor) and smallest (semiminor axes). This is what the elongation plugin does.