Abstract
The stochastic simulation algorithm commonly known as Gillespie's algorithm (originally derived for modelling wellmixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In wellmixed scenarios at the subcellular 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 (e.g. embryonic development, cancer formation) should not be simulated naively using the Gillespie algorithm since the historydependent 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 as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one nonspatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation  vital to the accurate modelling of many biological processes  whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm.
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 as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one nonspatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation  vital to the accurate modelling of many biological processes  whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm.
Original language  English 

Pages (fromto)  2905–2928 
Number of pages  24 
Journal  Bulletin of Mathematical Biology 
Volume  79 
Issue number  12 
Early online date  13 Oct 2017 
DOIs  
Publication status  Published  1 Dec 2017 
Keywords
 Algorithms
 Animals
 Cell Cycle/physiology
 Cell Proliferation/physiology
 Computer Simulation
 Humans
 Markov Chains
 Mathematical Concepts
 Mice
 Models, Biological
 NIH 3T3 Cells
 Neoplastic Stem Cells/pathology
 Stochastic Processes
 Time Factors
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Kit Yates
 Department of Mathematical Sciences  Senior Lecturer
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Centre for Mathematical Biology  CoDirector
 Institute for Mathematical Innovation (IMI)
Person: Research & Teaching