The macroscale movement behaviour of a wide range of isolated migrating cells has been well characterised experimentally. Recently, attention has turned to understanding the behaviour of cells in crowded environments. In such scenarios it is possible for cells to interact mechanistically, inducing neighbouring cells to move in order to make room for their own movements or progeny.Although the behaviour of interacting cells has been modelled extensively through volume-exclusion processes, no models, thus far, have explicitly accounted for the ability of cells to actively displace each other.In this work we consider both on and off-lattice volume-exclusion position-jump processes in which cells are explicitly allowed to induce movements in their near neighbours in order to create space for themselves (which we refer to as pushing). From these simple individual-level representations we derive continuum partial differential equations for the average occupancy of the domain.We find that, for limited amounts of pushing, the comparison between the averaged individual-level simulations and the population-level model is nearly as good as in the scenario without pushing but, that for larger and more complicated pushing events the assumptions used to derive the population-level model begin to break down. Interestingly, we find that, in the on-lattice case, the diffusion coefficient of the population-level model is increased by pushing, whereas, for the particular off-lattice model that we investigate, the diffusion coefficient is reduced. We conclude therefore, that it is important to consider carefully the appropriate individual-level model to use when representing complex cell-cell interactions such as pushing.
Understanding synchrony in growing populations is important for applications as diverse as epidemiology and cancer treatment. Recent experiments employing fluorescent reporters in melanoma cell lines have uncovered growing subpopulations exhibiting sustained oscillations, with nearby cells appearing to synchronise their cycles. In this study we demonstrate that the behaviour observed is consistent with long-lasting transient phenomenon initiated, and amplified by the finite-sample effects and demographic noise. We present a novel mathematical analysis of a multi-stage model of cell growth which accurately reproduces the synchronised oscillations. As part of the analysis, we elucidate the transient and asymptotic phases of the dynamics and derive an analytical formula to quantify the effect of demographic noise in the appearance of the oscillations. The implications of these findings are broad, such as providing insight into experimental protocols that are used to study the growth of asynchronous populations and, in particular, those investigations relating to anti-cancer drug discovery.
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