A Multi-stage Representation of Cell Proliferation as a Markov Process

Abstract

The stochastic simulation algorithm commonly known as Gillespie’s algorithm (originally derived for modelling well-mixed systems of chemical reactions) 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 (e.g. embryonic development, cancer formation) 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 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 non-spatial) 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.

Synchronised oscillations in growing cell populations are explained by demographic noise

“…In cell biology, this approach has been supported by classic experimental studies for large populations under favourable growth conditions [Monod, 1949, Laird, 1965. However, when smaller populations are considered -for example clones of a single progenitor cell -the classical model of exponential growth fails to capture the variable per capita growth rates caused by non-exponentially distributed cell cycle times and more sophisticated models are necessary [Baker and Simpson, 2010, Yates et al, 2017, Jafarpour, 2019, Pirjol et al, 2017, Lang et al, 2009, Kuritz et al, 2018.…”

Mathematical models incorporating a multi-stage cell cycle replicate normally-hidden inherent synchronisation in cell proliferation

“…we arrive at a system of differential equations describing the mean 4 population M i (t) in each stage [19]:…”

Isolating the sources of heterogeneity in nanoparticle-cell interactions

“…To describe the cell cycle, we implement a multi-stage model of cell cycle progression, as presented previously (33). At each point in time a cell belongs to one of m possible phases of the cell cycle.…”

“…To investigate whether the progression through the cell cycle during the experiment affects the apparent heterogeneity, we now incorporate the cell cycle explicitly in our modeling framework. We implement a multistage model of cell cycle progression, where cells exist in one of m states, representing different phases of the cell cycle (33). Cells transition between states at rates corresponding to the average time spent in a particular phase.…”