Figure 5. Simulations of morphogen diffusion.
(A) Free diffusion with a diffusion coefficient D = 60 μm2/s. (Dashed circles indicate positions of cells in later simulations but not taken account of in this case). (B) Diffusion in the presence of cells. (C) Diffusion in the presence of cells and binding with an average number of free particles of 0.003, i.e. 99.7% of all particles are bound on average. Simulations were done in a 3D space as described in the text and the diffusion coefficient was D = 60 μm2/s. (D) Comparison of the spread of the particles as a function of the distance from the source (the left border in panels A–C). The concentration curves were fit with a bell curve that describes the diffusion of particles from the source. For free diffusion (A) we recover a diffusion coefficient of D = 63.4 μm2/s close to the input value, and in the presence of cells (B) this reduces to an effective diffusion coefficient of Deff = 33.8 μm2/s, demonstrating the effect of tortuosity. (E) Simulations of diffusion in the presence of cells, and with different amounts of binding. The simulated diffusion coefficient was D = 60 μm2/s. The concentration curves were fit with Equation 8. The recovered effective diffusion coefficients for a fraction of free particles of 0.1, 0.003, and 0.0015 were Deff = 2.99 μm2/s, Deff = 0.09 μm2/s, and Deff = 0.042 μm2/s, respectively, demonstrating the effect of binding on the effective diffusion coefficient. (F) Gradient formation using the effective diffusion coefficients determined from graph E and degradation rates of 0.0001/s and 0.0005/s, respectively. The blue curve represents Cyc, the red curve Sqt. Although Sqt has higher binding affinity and consequently a lower free mobile fraction, its lower degradation rate ensures that Sqt has a less steep gradient. The data was fit with an exponential function yielding gradients of 19 μm for Cyc and and 30 μm for Sqt, respectively.