Fig 1.
Pseudo code for diffusion within moving boundaries.
An algorithm for executing reaction-diffusion systems within moving boundaries. Please refer to Fig 2a for an accompanying visual representation.
Fig 2.
Schematics of proposed algorithms.
a) A visual representation of the proposed method in Fig 1, mapping escaped particles back into the domain for diffusion within moving boundaries. b) A visual representation of the proposed method in Fig 3, mapping diffusing particles on a moving domain.
Fig 3.
Pseudo code for diffusion on moving boundaries.
An algorithm for executing reaction-diffusion systems on moving boundaries. Please refer to Fig 2b for an accompanying visual representation.
Fig 4.
Cumulative density functions obtained from numerical and analytical methods.
The CDF of particles diffusing within a shrinking circle, where the circle radius shrinks from an initial value of 5 length units to a final radius of 3 length units uniformly over a 5 second interval. The CDF is plotted at times 0 (purple), 1.25 (light blue), 2.5 (red), 3.75 (green) and 5 (blue) seconds. Lines represent the solutions as calculated from PDEs. Black dots represent results from the algorithm of Fig 1 with k = 1, hollow dots represent the same except with k = 2. Yellow stars dots represent the predictions on the basis of a uniform distribution. Subplots a to c represent point particles with diffusion constants 5, 0.5 and 0.05 unit squared per second, respectively, and subplots d to f represent particles with a radius of 0.5 units, with diffusion constants 5, 0.5 and 0.05 unit squared per second, respectively. Time steps for simulations in plots are 0.001s (for D = 0.05), 0.0001s (for D = 0.5) and 0.00001 (for D = 5). Simulations were repeated 10000 times. Subplot g shows a log-log plot of the percentage difference in the CDFs of two different boundary correction schemes against simulation time step. Parameters are as per the other subplots, with D = 0.5. The CDFs were evaluated at a radius of 2.75 length units for subplot g. Error bars were not included, since they are not visible beyond the size of scatter points. The maximum absolute error for each of subplots a-f CDF is 0.013, with a 99% confidence interval. This corresponds to the point where each CDF evaluates to 0.5. See main text for details on error calculation.
Fig 5.
Mean-squared displacement for diffusion inside a shrinking circle.
Log-Log plots of the MSD against time for particles diffusing within a shrinking sphere, where the circle radius is shrinking from an initial value of 5 length units to a final radius of 3 length units uniformly over a 5 second interval. Subplots a-c show results for diffusion constants of 0.005, 0.05 and 1, respectively. Blue lines show simulation results of single particles on the basis of 10000 repeats. Red dots show a linear regression on the basis of data from the first two seconds, while black dots show linear regression on the basis of the last second. Green dots show the predictions of Eq 5. Error bars were not included, since they are hardly visible beyond the line-width of plots. The maximum absolute error per subplot, with 99% confidence intervals, are 0.037, 0..026 and 0.026 for subplots a-c, respectively. See main text for details on error calculation.
Fig 6.
Diffusion on the surface of a shrinking circle.
Subplots a-c: CDFs of the angular component of particles diffusing on the surface of shrinking sphere, where the circle radius shrinks from an initial value of 5 length units to a final radius of 3 length units uniformly over a 5 second interval at times 0 (blue), 1.25 (green), 2.5 (red), 3.75 (light blue) and 5 (purple) seconds. Lines show numerical results from PDEs, dots show results from simulations of single particles with 10000 repeats. Diffusion constants are 0.1 (subplot a), 1 (subplot b) and 10 (subplot c) units squared per second, respectively. Time steps for simulations in plots are 0.0005s (for D = 0.1), 0.00005s (for D = 1) and 0.000005 (for D = 10). Subplot d shows MSD curves for the same simulations from subplots a-c in green, red and light blue, respectively. In addition, the blue line at the bottom represents results with a diffusion constant of 0.01 length units squared per second. Also shown are predictions of Eqs 9 and 10 (dotted black and dotted purple line respectively), and a dotted yellow line with gradient 1. The latter represents a non-anomalous regime to aid judgement of where diffusion is super or sub-diffusive. Error bars were not included, since they are hardly visible beyond the line-width of plots. The maximum absolute error per subplot of any CDF is 0.013, with a 99% confidence interval. This corresponds to the point where each CDF evaluates to 0.5. Maximum error bar sizes for MSD curves are 0.025 (D = 0.01), 0.038 (D = 0.1), 0.036 (D = 1) and 0.035 (D = 10), all at 99% confidence intervals. See main text for details on error calculation.
Fig 7.
Bimodal reaction times in a growing domain.
A histogram showing bimodal reaction times in a growing volume of two spheres connected by an elongating bridge as per section 2.2.3, with R = 3, d = 0.5, L = 0.01 per second, T1 = 200000 seconds, T2 = 201000 seconds, λ = 0.2. Each particle has a diffusion rate of 0.0025 units squared per second. A projection of the geometry at T2 onto the x-y plane is shown, superimposed onto the location of all reaction events occurring during bridge elongation. The location of a reaction event is taken as the mean position of the two reactants. Simulation time steps are 1/3 seconds for the blue bars, and 1/6 seconds for red dots.
Fig 8.
A schematic of the idealisation of a dividing yeast nucleus.
Parameters rym,t and rxm,t represent the maximal radius of the mother nuclear lobe in the direction of the y and x axis at time t, respectively. We define ryd,t and rxd,t analogously for the bud nuclear lobe. The length and width of the bridge is given by lt and wt, respectively, and the radius of the spherical photobleaching spot is given by rphoto. Note that the geometry is in 3D, and only a 2D projection is depicted here.
Fig 9.
Photobleaching in a dividing yeast nucleus.
Plots showing qualitatively different behaviour between static and moving boundaries for diffusion on (subplot a) and within (subplot b) a dividing yeast nucleus. The population of particles in the mother and daughter lobes are in blue and green respectively, with dark hues and light hues representing the static and moving boundary cases, respectively. Diffusion coefficients for all plots are 0.25 μm2 / s, with rphoto = 0.17 μm and kdeg = 2. The change of geometries is depicted in the diagram above the plots, with accompanying images from confocal microscopy. Time steps for all simulations are 0.00005 seconds.