Figure 4.

Comparison of calculations based on numerical and analytical solutions of the diffusion equation. In both panels, all curves of a single color correspond to a specific choice of a realistic geometry and associated idealized geometries (see text), combined with a specific choice of diffusional parameters. For each color the solutions are identified by line type: numerical solution on a realistic geometry (solid), analytical solution on the corresponding equivalent cylinder (short dashed) and analytical solution on the corresponding minimal cylinder (long dashed). Red curves refer to a realistic region with volume 10 μl and a permeability parameter c of the solvent/atmosphere interface of −5 (cf. Eq 4), green curves refer to a realistic region with volume 10 μl and c = −2, and blue curves refer to a realistic region with volume 14 μl and c = −5. Realistic regions with volume 10 μl have a minimum well depth (depth at the center of the meniscus, and also the height of the minimal cylinder) of 0.140 cm, and a height of the equivalent cylinder of 0.155 cm. Realistic regions with volume 14 μl have a minimum well depth of 0.192 cm, and a height of the equivalent cylinder of 0.214 cm. a) Oxygen partial pressure at the bottom of the well vs. time. b) Fraction sickled vs. time, computed as described in Methods. As discussed in the text, the numerical solutions for realistic geometries are incorporated into the calculations by interpolating on pre-computed values saved at discrete time intervals; the relative lack of smoothness of the corresponding (solid) curves arises from potential instabilities in the solution of eq. (2) arising from this interpolation process.