Figure 48.3 Computational time as a function of the number of mtDNA molecules per cell. (A) mtDNA Molecule Level simulation. One cell was simulated for 100 years. The computational time requirements for this method are approximately 48 seconds per cell per century per 1,000 mtDNA molecules. (B) Cell Level simulation. One thousand cells were simulated for 100 years. In the plateau region for more than 10,000 mtDNA molecules per cell, the simulation requirements are approximately nine seconds per simulated cell per century.

In Figure 48.3 we show the dependence of the computation time on the number of mtDNA molecules per compartment.

The mtDNA Molecule Level simulation shows a linear increase in computational time with increasing numbers of mtDNA molecules per cell. This is expected since each mtDNA molecule is modeled separately in this simulation. For the Cell Level simulation (see Figure 48.3B) the behavior is more complicated. The change in behavior from a linear increase up to 10,000 mtDNA per compartment to a relatively constant computation time for a higher number of mtDNA is due to a change in the numerical methods used to calculate the binomial and Poisson distributions (shifting from the direct method to the rejection method). This shift in methods is done automatically by the standard numerical methods described in detail in Numerical Recipes in C (Press et al., 1988). The direct method is used when the number returned by the binomial or Poisson distribution subroutines is small, so the location of this switch in numerical methods depends on the time step used. The values in Figure 48.3 are for a time step of one hour. By coincidence, with this time step the computation time flattens out at about 10,000 mtDNA molecules per cell, so the computational requirements of this simulation method are insensitive to the number of mtDNA molecules per cell over an important physiological range (10,000 to 100,000 mtDNA molecules per cell).

Generally, the simulated cells do not interact with each other, and we are interested only in the statistics taken over a large number of independent cells. In this case I recommend that simulations of multiple cells be run sequentially, with each simulated cell run one at a time. Data may be saved at set time points along the simulation, and then statistics over the set of cells can be calculated at these time points after all simulated cells are run. By running the cells sequentially, instead of all at once, the RAM memory requirements of both simulation methods are negligible. However, in simulations where the cells interact in some way, the full set of simulated cells must be calculated simultaneously. For example, we have had to do this in simulations of cell cultures where the cell division rate depends on the number of cells in the simulated culture. In these cases, the Cell Level simulation is the only practical choice since less data must be stored per simulated cell in that method compared to the mtDNA Molecule Level models. On a PC with 512 MB RAM the memory requirements of the Cell Level simulation are not a limitation until the number of simulated cells reaches 8 million.

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