An evolutionary-game model of tumour–cell interactions: possible relevance to gene therapy
Introduction
Throughout the development of a tumour, selection of specific mutations is traditionally believed to reflect the ability of these mutations to promote growth or inhibit death by cell-autonomous modes of action 1, 2, 3, 4. An improved understanding of the genetic and biochemical mechanisms, by which mutations confer a growth advantage to cancer cells, has led to the formulation of mathematical models for tumour initiation, growth and metastatic spread 5, 6, 7, 8, 9. These models typically incorporate the effect of putative paracrine and/or autocrine growth factors, but rely nevertheless on the cell-autonomous consequences of mutations in the malignant cells. Recently, Tomlinson and Bodmer 10, 11 presented evolutionary-game mathematical models for the time evolution of a tumour cell population which incorporated non-autonomous effects of tumour cell mutations in terms of interactions among cells that have adopted individual genetic strategies. Following the work by Axelrod and Hamilton [12], evolutionary games have been studied as simple mathematical models of the evolution of cooperation in a population. The prototypical example is the Prisoner's Dilemma, where the payoff from a specific strategy depends on whether the other prisoner collaborates or defects. A popular introduction to the Prisoner's Dilemma and other evolutionary games may be found in Nowak and colleagues [13].
Tomlinson and Bodmer discussed several models, and one of these considered the production of a growth factor, exemplified by an angiogenic promoter, that was associated with a cost of production and a replication advantage. The benefit was obtained by the cell itself, as well as other cells in the population. As long as the benefit was greater than the cost of production, a stable polymorphism between producers and non-producers were predicted.
Section snippets
Materials and methods
Here, an extension of the evolutionary model by Tomlinson and Bodmer 10, 11 is proposed and its behaviour investigated. Specifically, this model incorporates a local threshold condition for deriving a benefit from collaboration between tumour cells. One concrete example could be the production of an angiogenic promoter where it is conceivable that producer cells have to collaborate in order to provide a sufficiently high local concentration of the stimulating factor. We describe the simplest
Evolutionary games and tumour population dynamics
Tomlinson and Bodmer [10] developed a mathematical model of the dynamics of a population consisting of two types of cells with strategies A+ and A−. An A+ cell produces a certain factor which conveys a proliferative advantage both to the cell itself and its neighbours. In contrast, the A− cell does not produce the factor and a population of A− cells will only have the baseline reproductive rate. The detailed mechanism of this interaction needs not to be specified, but the resulting net cost or
Discussion
The most interesting feature of the current model is the existence of an unstable internal equilibrium which forms a barrier between two stable states of the population, one with a stable polymorphism between A+ and A− cells and one where only A− cells are present. During tumorigenesis, it must be assumed that local collaboration is possible which may allow this critical threshold to be crossed locally. At a later stage, the selection pressure against the tumour cells may have increased and
Conclusion
In the light of recent advances in the description of cell–cell interactions and their biochemical signalling pathways, this kind of model is, in our view, a potentially useful tool for understanding the behaviour of populations of tumour cells. Eventually, these models could lead to more elaborate applications of spatial simulations, where local synergism and clustering effects determine the final composition of the cell population. We propose that some gene therapy approaches may benefit from
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