Numerical approach to solving the diffusion equations

3.4.1 Numerical approach

There are three ways the model can be numerically implemented:

1. Assume the outer ULe and inner ULi boundaries are fixed relative to the cell axis, and the thickness of the ULs (δUL) and numerical grids change over time as the cell expands or contracts;

2. Assume the outer ULe and inner ULi boundaries move in space with the membrane, and the thickness of the ULs and numerical grids remain constant over time;

3. Assume that the outer ULe and inner ULi boundaries are fixed, and all the

numerical grids in the ULs are also fixed, except for the two immediately adjacent to the membrane which change width over time.

The first two are more mathematically rigorous than the third, since the numerical scheme should place no conditions on the relative width of the grids, for which

in the limit. However, as the third approach has been used elsewhere (Tyree et al. 2005) this model is included here to check its results with a more rigorous approach. All three schemes should give the same results, since the membrane moves only slightly during the

0 r

time step is small relative to the total cell volume. However, a limitation of the second approach is that the ULi thickness is only able to come close to but not equal the entire cell volume, for the center of the cell is fixed and cannot move. A limitation of the third approach is that it may become numerically unstable when Δr is made sufficiently small such that the membrane moves further than the width of a grid.

All three approaches were implemented, and found to give the same quantitative results. Runtimes for a simple simulation of CPP dynamics were the same to within 3%.

Runtimes for parameter estimations, involving many simulation runs, varied between the methods by differing amounts depending on the model conditions.

Implementation of the first 2 methods is described in §3.5. For ease of reference the first method is named the “Varying δUL method”, and the second method named the “Fixed

δUL method”. Characteristics of each are that:

1. Varying δUL method: Non-membrane UL boundaries are fixed, δUL changes over time. dr_j/dt differs for each radial point rj in the UL.The shell spacing Δr

is dependent on t. Radial points rj, the surface area Aj, and volume Vjof the shells

change with time.

2. Fixed δUL method: Non-membrane UL boundaries move with the membrane, and δUL remains constant over time. drj/dt=dR dt/ where R is the cell radius. In

the frame-of-reference of the membrane the shells are fixed so that the shell spacing Δr remains constant over time. Radial points rj, the surface area Aj, and

volume Vjof the shells change with time.

3.4.2 Indexing

Let R be the radial distance to the membrane, Ra be the radial distance to the inner ULi

boundary, and Rb the radial distance to the outer ULe boundary. The ULs are segmented

cylindrical geometry as the cell. Let the number of shells in the ULe be m-1 and the number of shells in the ULi be n-1.

The indexing was chosen to step inwards from the bulk solution towards the cell axis, in both the ULe and ULi, since in an OPP the perturbation occurs outside the cell and propagates inwards. The midpoint of each ULe shell is at a distance rjfrom the center of

the cell, where j = 1:m. The index 1 corresponds to a point outside the ULe (in the bulk solution), and the index m corresponds to the midpoint of the shell externally adjacent to the membrane. This is illustrated in Fig. 3.2. Similarly, the midpoint of each ULi shell is at a distance rjfrom the center of the cell, where j = 1:n. The index 1 corresponds to the

midpoint of the shell internally adjacent to the membrane, and the index n corresponds to a point outside the ULi in the central region of the cell.

The jth shell has volume Vj, and inner and outer surface areas Aj+1 and Aj respectively.

The indexing of Vj corresponds to that of rj. The volume of the bulk solution is assumed

to be infinite. The total number of Aj values is m-1 for the ULe and n-1 for the ULi, 1 less

than that for rjand Vj in each of the ULe and ULi regions.

r_m r₁ r₁ r_n a _{R b} bulk solution cell core r_j A₁ A_n-1 A_m-1 A₁ C_j V_j J_j Δr A_j

Fig. 3.2 Indexing for the numerical UL model

The distance between the midpoint of 2 adjacent shells is equal to the width of a shell. Cj

is the concentration at the midpoint of the shells (at each rj), and Jj is the flux across the

shell boundaries (at each Aj).

The numerical code was made compatible with the no-ULs case, by making m=1 when no ULe is present, and n=1 when no ULi is present.