STOCHASTIC MODELLING OF STRUCTURED POPULATIONS

Quantifying cellular growth is crucial to understand the dynamics of cell populations such as microbes and cancer cells. The standard behaviour of batch cultures is well known and it is usually characterised by a delay before the start of exponential growth, an exponential phase, and a steady phase; however, at the single-cell level, growth varies drastically from cell to cell due to the fluctuations in the cell cycle duration, variability caused by changing environments, and cells interactions.
At the present time, understanding how the cell-to-cell variability affects the evolution of the entire popu-lation is still a open challenge; de facto, there are still lacking solid theoretical and simulation methods to forecast the effects of cell heterogeneity on the population dynamics [1].
We propose a novel stochastic model where the cells are represented by agents who divide, die, convert to other species, rejuvenate in response to an internal continuous state which increases with time. While such models are usually only amenable to simulations, we show that the population structure can be characterized by a functional master equation which can be manipulated to obtain a novel integral renewal equation. Compared to the classic results about renewal theory, as the Bellman Harris branching process [2] and the Galton-Watson theory [3], the latter equation takes a step further. In fact, it provides a solid and compact stochastic description of the role played by cell heterogeneity on the population dynamics.
The analytical framework allowed us to fully describe the population size distribution, population growth rate, ancestor and division times distributions; it also enables to understand the role played by heterogeneity in the initial conditions. Moreover, we provide an analytical and numerical characterization of the extinction probability and first extinction times distribution for any cell-to-cell heterogeneity range. We also propose a novel way to simulate the evolution of cell populations affected by the variability of the individuals. Such computational tool allowed us to substantiate the analytical and numerical results obtained during this in-vestigation.
Our last results also provide novel methods to address the role of cell-to-cell variability in time depen-dent environments. We showed that the stochastic description of agent-based populations dynamics can be obtained in scenario where the reaction network rates depend explicitly on time in addition to the internal traits of the cells.
In conclusion, the following research project proposes a novel methodology to describe the stochastic be-haviour of cell structured population with numerical, computational and analytical methods. Our results open a new theoretical path to understanding stochastic mechanisms underlying fluctuations in various bi-ological and medical applications as: extinction of cancer cell populations under treatment, cell population growth in adverse environments, dormancy-awakening transition in breast cancer and microbial quiescence.
References
[1] Thomas Philipp (2017).Making sense of snapshot data: ergodic principle for clonal cell populationsJ. R. Soc. Interface.142017046720170467 http://doi.org/10.1098/rsif.2017.0467
[2] Harris, T. E. (1963). The theory of branching processes (Vol. 6). Berlin: Springer.
[3] Kesten, H.Ney, P., Spitzer, F. (1966). The Galton-Watson Process with Mean One and Finite Variance. Theory of Probability Its Applications. Society for Industrial and Applied Mathematics. 10.1137/1111059 https: //doi.org/10.1137/1111059