  The stochastic simulation algorithm commonly known as Gillespie's algorithm
is now used ubiquitously in the modelling of biological processes in which
stochastic effects play an important role. In well-mixed scenarios at the
sub-cellular level it is often reasonable to assume that times between
successive reaction/interaction events are exponentially distributed and can be
appropriately modelled as a Markov process and hence simulated by the Gillespie
algorithm. However, Gillespie's algorithm is routinely applied to model
biological systems for which it was never intended. In particular, processes in
which cell proliferation is important should not be simulated naively using the
Gillespie algorithm since the history-dependent nature of the cell cycle breaks
the Markov process. The variance in experimentally measured cell cycle times is
far less than in an exponential cell cycle time distribution with the same
mean.
  Here we suggest a method of modelling the cell cycle that restores the
memoryless property to the system and is therefore consistent with simulation
via the Gillespie algorithm. By breaking the cell cycle into a number of
independent exponentially distributed stages we can restore the Markov property
at the same time as more accurately approximating the appropriate cell cycle
time distributions. The consequences of our revised mathematical model are
explored analytically. We demonstrate the importance of employing the correct
cell cycle time distribution by considering two models incorporating cellular
proliferation (one spatial and one non-spatial) and demonstrating that changing
the cell cycle time distribution makes quantitative and qualitative differences
to their outcomes. Our adaptation will allow modellers and experimentalists
alike to appropriately represent cellular proliferation, whilst still being
able to take advantage of the Gillespie algorithm.
