Modelling post-bleaching diffusion inside a sphere: Monte-Carlo simulations and direct calculations.

<p><i>A</i>, Graphical representation of the bleaching experiment, showing an impermeable sphere of unit radius, in which the horizontal polar axis distance <i>µ</i> = cos(<i>θ</i>) and the normalised radius <i>r</i> are spherical coordinates describing fluorophore concentration distribution. A bleached region covering the horizontal axis interval <i>α</i>≤<i>µ</i>≤1 is shown in white, the rest of the sphere contains fluorophore (grey). <i>B–C</i> Monte-Carlo FRAP recovery simulations. <i>B,</i> Concentration profiles for simulated redistribution of 20000 particles, initially positioned at time <i>τ</i> = 0 s in the left hemisphere (black trace). Recovery times are coded by the trace colours. Simulation parameters were: sphere radius <i>R</i>₀ = 1 µm, prescribed diffusion coefficient <i>D</i> = 3 µm²/s, bleaching parameter <i>α</i> = 0, simulation step 5·10⁻⁴s. Profiles were constructed from images representing the 2D projections of the sphere, each simulated particle was represented by a pixel of value 1. The plot of normalised moment kinetics is shown in <i>C</i>. Red line- monoexponential fit, the fitted time constant <i>τ</i> = 83 ms corresponds to the <i>D</i> value of 2.79 µm²/s (Eq.8). <i>E–F,</i> Analysis of the same problem from images representing projections of concentration distributions calculated using Eq.10. <i>E</i>, Concentration profiles for simulated distributions, time-dependent colour coding as in <i>B</i>. Oscillating patterns (Gibb’s phenomena) at short times (black trace) are due to approximation of discontinuity by finite Legendre sums. <i>F,</i> The normalised moment kinetics fitted either by a single exponent (red trace, <i>τ</i> = 66 ms, evaluated <i>D</i> = 3.5 µm²/s) or by a sum of two exponents (blue trace, <i>τ</i>₁ = 3.5 ms, <i>τ</i>₂ = 75 ms, <i>D</i> evaluated from <i>τ</i>₂ was 3.1 µm²/s; the amplitude of fast component was 0.105 of that for slow component). <i>D,</i> The relationship between the values of <i>D</i> prescribed in Monte-Carlo simulations and results obtained from fits as in <i>C</i> and <i>F.</i> The results were compared with identity line (red). The slopes of the best fits were: for Monte-Carlo simulations (as in <i>B–C</i>) 0.93±0.009 (circles), for the best monoexponential fit to calculated values 1.18±0.0006 (open triangles), for the slower mode of bi-exponential fit (as in <i>E–F</i>) 1.03±0.0004 (filled blue triangles).